Subgroups — C₂×C₆₇×#3384 #30740609

Order 17152.

2082 conjugacy classes of subgroups.

The subgroup lattice has 534 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#554×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×D₄×Q₈×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₆₇×#3384	C₂	1	16	yes	maximal, direct product factor
4	C₂×#501×C₆₇	C₂	1	2	yes	maximal, complemented
5	C₂×#498×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#478×C₆₇	C₂	1	2	yes	maximal, complemented
7	C₂×#476×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#476×C₆₇	C₂	1	2	yes	maximal, complemented
9	C₂×#474×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#446×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×#442×C₆₇	C₂	1	2	yes	maximal
12	C₂×C₄×D₈×C₆₇	C₂	1	1	yes	maximal, characteristic
13	C₂³×Q₈×C₆₇	—	2	1	no	—
14	C₂²×D₈×C₆₇	C₂²	1	1	yes	characteristic
15	#554×C₆₇	—	2	4	no	—
16	#554×C₆₇	C₂²	1	8	yes	complemented
17	D₄×Q₈×C₆₇	—	2	4	no	—
18	D₄×Q₈×C₆₇	C₂²	1	8	yes	complemented
19	C₂×C₄⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
20	C₂×C₄⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
21	C₂×C₄⋊Q₈×C₆₇	C₂²	1	2	yes	—
22	C₂×#125×C₆₇	C₂²	1	2	yes	—
23	C₂×C₂²⋊Q₈×C₆₇	—	2	1	no	—
24	C₂×C₂²⋊Q₈×C₆₇	—	2	1	no	—
25	C₂×C₂²⋊Q₈×C₆₇	—	2	2	no	—
26	C₂×C₂²⋊Q₈×C₆₇	—	2	2	no	—
27	C₂×C₄×Q₈×C₆₇	C₂²	1	1	yes	characteristic
28	C₂×C₄×D₄×C₆₇	C₂²	1	1	yes	characteristic
29	C₂×C₄×D₄×C₆₇	C₂²	1	1	yes	characteristic
30	C₂×C₄×D₄×C₆₇	C₂²	1	2	yes	—
31	C₂×C₄×D₄×C₆₇	C₂²	1	2	yes	—
32	C₂²×C₄⋊C₄×C₆₇	—	2	1	no	—
33	#501×C₆₇	C₂²	1	16	yes	complemented
34	#498×C₆₇	C₂²	1	8	yes	complemented
35	#478×C₆₇	C₂²	1	16	yes	complemented
36	#476×C₆₇	C₂²	1	16	yes	complemented
37	#476×C₆₇	C₂²	1	8	yes	complemented
38	#474×C₆₇	C₂²	1	8	yes	complemented
39	#446×C₆₇	C₂²	1	8	yes	complemented
40	#442×C₆₇	C₂²	1	16	yes	—
41	C₄×D₈×C₆₇	C₂²	1	8	yes	—
42	C₂×#107×C₆₇	C₂²	1	2	yes	—
43	C₂×#107×C₆₇	C₂²	1	1	yes	characteristic
44	C₂×#106×C₆₇	C₂²	1	2	yes	—
45	C₂×#106×C₆₇	C₂²	1	2	yes	—
46	C₂×#106×C₆₇	C₂²	1	2	yes	—
47	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
48	C₂×C₄⋊C₈×C₆₇	C₂²	1	2	yes	—
49	C₂×#102×C₆₇	C₂²	1	2	yes	—
50	C₂×#102×C₆₇	C₂²	1	1	yes	characteristic
51	C₂×#102×C₆₇	C₂²	1	2	yes	—
52	C₂×#102×C₆₇	C₂²	1	1	yes	characteristic
53	C₂×#97×C₆₇	C₂²	1	2	yes	—
54	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
55	C₂²×Q₈×C₆₇	—	2	8	no	—
56	C₂²×Q₈×C₆₇	—	2	2	no	—
57	C₂²×Q₈×C₆₇	—	2	1	no	—
58	C₂²×Q₈×C₆₇	—	2	1	no	—
59	C₂²×Q₈×C₆₇	—	4	8	no	—
60	C₂²×Q₈×C₆₇	D₄	1	1	yes	characteristic
61	C₂²×D₄×C₆₇	C₂³	1	1	yes	characteristic
62	C₂²×D₄×C₆₇	C₂³	1	1	yes	characteristic
63	C₂³×C₄×C₆₇	—	2	1	no	—
64	C₂³×C₄×C₆₇	—	2	2	no	—
65	C₂³×C₄×C₆₇	—	2	1	no	—
66	C₂³×C₄×C₆₇	—	2	2	no	—
67	C₂×D₈×C₆₇	Q₈	1	8	yes	complemented
68	C₂×D₈×C₆₇	C₂³	1	4	yes	—
69	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
70	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
71	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
72	C₄⋊Q₈×C₆₇	—	2	4	no	—
73	C₄⋊Q₈×C₆₇	—	2	2	no	—
74	C₄⋊Q₈×C₆₇	—	2	2	no	—
75	C₄⋊Q₈×C₆₇	C₂³	1	4	yes	—
76	C₄⋊Q₈×C₆₇	C₂³	1	4	yes	—
77	C₄⋊Q₈×C₆₇	C₂³	1	8	yes	—
78	#125×C₆₇	—	2	4	no	—
79	#125×C₆₇	C₂³	1	8	yes	—
80	C₂²⋊Q₈×C₆₇	—	2	8	no	—
81	C₂²⋊Q₈×C₆₇	—	2	8	no	—
82	C₂²⋊Q₈×C₆₇	—	2	16	no	—
83	C₂²⋊Q₈×C₆₇	—	2	16	no	—
84	C₄×Q₈×C₆₇	D₄	1	4	yes	—
85	C₄×Q₈×C₆₇	C₂³	1	4	yes	—
86	C₄×D₄×C₆₇	—	2	4	no	—
87	C₄×D₄×C₆₇	—	2	2	no	—
88	C₄×D₄×C₆₇	—	2	4	no	—
89	C₄×D₄×C₆₇	—	2	2	no	—
90	C₄×D₄×C₆₇	C₂³	1	4	yes	—
91	C₄×D₄×C₆₇	C₂³	1	4	yes	—
92	C₄×D₄×C₆₇	C₂³	1	8	yes	—
93	C₄×D₄×C₆₇	C₂³	1	8	yes	—
94	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
95	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
96	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
97	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
98	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
99	C₂×C₄⋊C₄×C₆₇	—	2	8	no	—
100	C₂×C₄⋊C₄×C₆₇	D₄	1	1	yes	characteristic
101	C₂×C₄⋊C₄×C₆₇	D₄	1	2	yes	—
102	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
103	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
104	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
105	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
106	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
107	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
108	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
109	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
110	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
111	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
112	#107×C₆₇	C₂³	1	8	yes	—
113	#107×C₆₇	C₂³	1	4	yes	—
114	#106×C₆₇	C₂³	1	8	yes	—
115	#106×C₆₇	C₂³	1	8	yes	—
116	#106×C₆₇	C₂³	1	8	yes	—
117	C₄⋊C₈×C₆₇	C₂³	1	4	yes	—
118	C₄⋊C₈×C₆₇	C₂³	1	8	yes	—
119	#102×C₆₇	C₂³	1	8	yes	—
120	#102×C₆₇	C₂³	1	4	yes	—
121	#102×C₆₇	C₂³	1	8	yes	—
122	#102×C₆₇	C₂³	1	4	yes	—
123	#97×C₆₇	C₂³	1	8	yes	—
124	C₄×C₈×C₆₇	C₂³	1	4	yes	—
125	C₂⁴×C₆₇	—	2	1	no	—
126	C₂⁴×C₆₇	—	2	1	no	—
127	C₂×Q₈×C₆₇	—	2	4	no	—
128	C₂×Q₈×C₆₇	—	2	8	no	—
129	C₂×Q₈×C₆₇	—	2	4	no	—
130	C₂×Q₈×C₆₇	—	2	4	no	—
131	C₂×Q₈×C₆₇	—	4	8	no	—
132	C₂×Q₈×C₆₇	—	4	32	no	—
133	C₂×Q₈×C₆₇	—	4	4	no	—
134	C₂×Q₈×C₆₇	—	8	2	no	—
135	C₂×Q₈×C₆₇	C₂×D₄	1	4	yes	—
136	C₂×D₄×C₆₇	—	2	1	no	—
137	C₂×D₄×C₆₇	—	2	1	no	—
138	C₂×D₄×C₆₇	—	2	2	no	—
139	C₂×D₄×C₆₇	—	2	2	no	—
140	C₂×D₄×C₆₇	C₂×Q₈	1	4	yes	—
141	C₂×D₄×C₆₇	C₂×Q₈	1	4	yes	complemented
142	C₂×D₄×C₆₇	C₂⁴	1	2	yes	—
143	C₂×D₄×C₆₇	C₂⁴	1	2	yes	—
144	C₂²×C₄×C₆₇	—	2	1	no	—
145	C₂²×C₄×C₆₇	—	2	2	no	—
146	C₂²×C₄×C₆₇	—	2	2	no	—
147	C₂²×C₄×C₆₇	—	2	1	no	—
148	C₂²×C₄×C₆₇	—	2	4	no	—
149	C₂²×C₄×C₆₇	—	2	8	no	—
150	C₂²×C₄×C₆₇	—	2	4	no	—
151	C₂²×C₄×C₆₇	—	2	8	no	—
152	C₂²×C₄×C₆₇	—	4	8	no	—
153	C₂²×C₄×C₆₇	—	4	4	no	—
154	C₂²×C₄×C₆₇	—	4	4	no	—
155	C₂²×C₄×C₆₇	—	8	4	no	—
156	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
157	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
158	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
159	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
160	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
161	D₈×C₆₇	C₂×Q₈	1	16	yes	complemented
162	C₂×C₈×C₆₇	—	2	2	no	—
163	C₂×C₈×C₆₇	—	2	4	no	—
164	C₂×C₈×C₆₇	C₂×Q₈	1	4	yes	complemented
165	C₂×C₈×C₆₇	C₂⁴	1	2	yes	—
166	C₂×C₈×C₆₇	C₂⁴	1	4	yes	—
167	C₂×C₈×C₆₇	C₂⁴	1	2	yes	—
168	C₄⋊C₄×C₆₇	—	2	8	no	—
169	C₄⋊C₄×C₆₇	—	2	4	no	—
170	C₄⋊C₄×C₆₇	—	2	8	no	—
171	C₄⋊C₄×C₆₇	—	2	4	no	—
172	C₄⋊C₄×C₆₇	—	2	8	no	—
173	C₄⋊C₄×C₆₇	—	2	1	no	—
174	C₄⋊C₄×C₆₇	—	2	2	no	—
175	C₄⋊C₄×C₆₇	—	2	1	no	—
176	C₄⋊C₄×C₆₇	—	2	2	no	—
177	C₄⋊C₄×C₆₇	C₂×D₄	1	4	yes	—
178	C₄⋊C₄×C₆₇	C₂×D₄	1	8	yes	—
179	C₄⋊C₄×C₆₇	C₂⁴	1	4	yes	—
180	C₄⋊C₄×C₆₇	C₂⁴	1	4	yes	—
181	C₄⋊C₄×C₆₇	C₂⁴	1	2	yes	—
182	C₄⋊C₄×C₆₇	C₂⁴	1	2	yes	—
183	C₂²⋊C₄×C₆₇	—	2	4	no	—
184	C₂²⋊C₄×C₆₇	—	2	8	no	—
185	C₂²⋊C₄×C₆₇	—	2	4	no	—
186	C₂²⋊C₄×C₆₇	—	2	8	no	—
187	C₄²×C₆₇	C₂×D₄	1	4	yes	—
188	C₄²×C₆₇	C₂×D₄	1	2	yes	—
189	C₄²×C₆₇	C₂⁴	1	4	yes	—
190	C₄²×C₆₇	C₂⁴	1	2	yes	—
191	C₂³×C₆₇	—	2	4	no	—
192	C₂³×C₆₇	—	2	4	no	—
193	C₂³×C₆₇	—	2	2	no	—
194	C₂³×C₆₇	—	2	2	no	—
195	C₂³×C₆₇	—	4	2	no	—
196	C₂³×C₆₇	—	4	2	no	—
197	C₂³×C₆₇	—	4	2	no	—
198	C₂³×C₆₇	—	4	2	no	—
199	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
200	Q₈×C₆₇	—	2	8	no	—
201	Q₈×C₆₇	—	4	16	no	—
202	Q₈×C₆₇	—	4	8	no	—
203	Q₈×C₆₇	—	8	4	no	—
204	D₄×C₆₇	—	2	4	no	—
205	D₄×C₆₇	—	2	4	no	—
206	D₄×C₆₇	—	2	2	no	—
207	D₄×C₆₇	—	2	2	no	—
208	D₄×C₆₇	C₂²×Q₈	1	4	yes	—
209	D₄×C₆₇	C₂²×Q₈	1	4	yes	complemented
210	C₂×C₄×C₆₇	—	2	4	no	—
211	C₂×C₄×C₆₇	—	2	4	no	—
212	C₂×C₄×C₆₇	—	2	2	no	—
213	C₂×C₄×C₆₇	—	2	2	no	—
214	C₂×C₄×C₆₇	—	2	4	no	—
215	C₂×C₄×C₆₇	—	2	4	no	—
216	C₂×C₄×C₆₇	—	2	2	no	—
217	C₂×C₄×C₆₇	—	4	16	no	—
218	C₂×C₄×C₆₇	—	4	2	no	—
219	C₂×C₄×C₆₇	—	4	4	no	—
220	C₂×C₄×C₆₇	—	4	2	no	—
221	C₂×C₄×C₆₇	—	4	4	no	—
222	C₂×C₄×C₆₇	—	4	8	no	—
223	C₂×C₄×C₆₇	—	4	8	no	—
224	C₂×C₄×C₆₇	—	8	8	no	—
225	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	4	yes	—
226	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
227	C₂×C₄×C₆₇	C₂²×D₄	1	4	yes	—
228	C₂×C₄×C₆₇	C₂²×D₄	1	4	yes	—
229	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
230	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
231	C₂×C₄×C₆₇	C₂²×Q₈	1	2	yes	—
232	C₂×C₄×C₆₇	2₋¹⁺⁴	1	2	yes	—
233	C₂×C₄×C₆₇	C₂⁵	1	1	yes	characteristic
234	C₈×C₆₇	—	2	2	no	—
235	C₈×C₆₇	—	2	4	no	—
236	C₈×C₆₇	C₂²×Q₈	1	4	yes	complemented
237	C₂²×C₆₇	—	2	4	no	—
238	C₂²×C₆₇	—	2	4	no	—
239	C₂²×C₆₇	—	4	8	no	—
240	C₂²×C₆₇	—	4	8	no	—
241	C₂²×C₆₇	—	4	2	no	—
242	C₂²×C₆₇	—	4	2	no	—
243	C₂²×C₆₇	—	4	2	no	—
244	C₂²×C₆₇	—	4	2	no	—
245	C₂²×C₆₇	D₄×Q₈	1	2	yes	—
246	C₂²×C₆₇	C₂×AGL(1,ℤ/8)	1	2	yes	—
247	C₂²×C₆₇	#577	1	2	yes	—
248	C₂²×C₆₇	C₂³×D₄	1	1	yes	characteristic
249	C₄×C₆₇	—	2	4	no	—
250	C₄×C₆₇	—	2	4	no	—
251	C₄×C₆₇	—	2	2	no	—
252	C₄×C₆₇	—	4	2	no	—
253	C₄×C₆₇	—	4	4	no	—
254	C₄×C₆₇	—	4	2	no	—
255	C₄×C₆₇	—	4	4	no	—
256	C₄×C₆₇	D₄×Q₈	1	2	yes	—
257	C₄×C₆₇	C₂×AGL(1,ℤ/8)	1	4	yes	—
258	C₄×C₆₇	C₂³×Q₈	1	1	yes	characteristic
259	C₄×C₆₇	C₂×2₋¹⁺⁴	1	1	yes	characteristic
260	C₂×#3384	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
261	C₂×C₆₇	—	4	4	no	—
262	C₂×C₆₇	—	4	4	no	—
263	C₂×C₆₇	#3384	1	4	yes	direct product factor
264	C₂×C₆₇	C₂×D₄×Q₈	1	1	yes	characteristic
265	C₂×C₆₇	C₂²×AGL(1,ℤ/8)	1	1	yes	characteristic
266	C₂×C₆₇	C₂×#577	1	1	yes	characteristic
267	C₂×#554	C₂×C₆₇	1	1	yes	characteristic, complemented
268	C₂×D₄×Q₈	C₂×C₆₇	1	1	yes	characteristic, complemented
269	#3384	C₂×C₆₇	1	16	yes	direct product factor
270	C₂×#501	C₂×C₆₇	1	2	yes	complemented
271	C₂×#498	C₂×C₆₇	1	1	yes	characteristic, complemented
272	C₂×#478	C₂×C₆₇	1	2	yes	complemented
273	C₂×#476	C₂×C₆₇	1	1	yes	characteristic, complemented
274	C₂×#476	C₂×C₆₇	1	2	yes	complemented
275	C₂×#474	C₂×C₆₇	1	1	yes	characteristic, complemented
276	C₂×#446	C₂×C₆₇	1	1	yes	characteristic, complemented
277	C₂×#442	C₂×C₆₇	1	2	yes	—
278	C₂×C₄×D₈	C₂×C₆₇	1	1	yes	characteristic
279	C₆₇	C₂×#3384	1	1	yes	characteristic, direct product factor, Sylow
280	C₂³×Q₈	—	2	1	no	—
281	C₂²×D₈	C₂²×C₆₇	1	1	yes	characteristic
282	#554	—	2	4	no	—
283	#554	C₂²×C₆₇	1	8	yes	complemented
284	D₄×Q₈	—	2	4	no	—
285	D₄×Q₈	C₂²×C₆₇	1	8	yes	complemented
286	C₂×C₄⋊Q₈	C₂²×C₆₇	1	1	yes	characteristic
287	C₂×C₄⋊Q₈	C₂²×C₆₇	1	1	yes	characteristic
288	C₂×C₄⋊Q₈	C₂²×C₆₇	1	2	yes	—
289	C₂×#125	C₂²×C₆₇	1	2	yes	—
290	C₂×C₂²⋊Q₈	—	2	1	no	—
291	C₂×C₂²⋊Q₈	—	2	1	no	—
292	C₂×C₂²⋊Q₈	—	2	2	no	—
293	C₂×C₂²⋊Q₈	—	2	2	no	—
294	C₂×C₄×Q₈	C₂²×C₆₇	1	1	yes	characteristic
295	C₂×C₄×D₄	C₂²×C₆₇	1	2	yes	—
296	C₂×C₄×D₄	C₂²×C₆₇	1	1	yes	characteristic
297	C₂×C₄×D₄	C₂²×C₆₇	1	1	yes	characteristic
298	C₂×C₄×D₄	C₂²×C₆₇	1	2	yes	—
299	C₂²×C₄⋊C₄	—	2	1	no	—
300	#501	C₂²×C₆₇	1	16	yes	complemented
301	#498	C₂²×C₆₇	1	8	yes	complemented
302	#478	C₂²×C₆₇	1	16	yes	complemented
303	#476	C₂²×C₆₇	1	16	yes	complemented
304	#476	C₂²×C₆₇	1	8	yes	complemented
305	#474	C₂²×C₆₇	1	8	yes	complemented
306	#446	C₂²×C₆₇	1	8	yes	complemented
307	#442	C₂²×C₆₇	1	16	yes	—
308	C₄×D₈	C₂²×C₆₇	1	8	yes	—
309	C₂×#107	C₂²×C₆₇	1	2	yes	—
310	C₂×#107	C₂²×C₆₇	1	1	yes	characteristic
311	C₂×#106	C₂²×C₆₇	1	2	yes	—
312	C₂×#106	C₂²×C₆₇	1	2	yes	—
313	C₂×#106	C₂²×C₆₇	1	2	yes	—
314	C₂×C₄⋊C₈	C₂²×C₆₇	1	2	yes	—
315	C₂×C₄⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
316	C₂×#102	C₂²×C₆₇	1	1	yes	characteristic
317	C₂×#102	C₂²×C₆₇	1	2	yes	—
318	C₂×#102	C₂²×C₆₇	1	1	yes	characteristic
319	C₂×#102	C₂²×C₆₇	1	2	yes	—
320	C₂×#97	C₂²×C₆₇	1	2	yes	—
321	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
322	C₂²×Q₈	—	2	8	no	—
323	C₂²×Q₈	—	2	2	no	—
324	C₂²×Q₈	—	2	1	no	—
325	C₂²×Q₈	—	2	1	no	—
326	C₂²×Q₈	—	4	8	no	—
327	C₂²×Q₈	D₄×C₆₇	1	1	yes	characteristic
328	C₂²×D₄	C₂³×C₆₇	1	1	yes	characteristic
329	C₂²×D₄	C₂³×C₆₇	1	1	yes	characteristic
330	C₂³×C₄	—	2	1	no	—
331	C₂³×C₄	—	2	2	no	—
332	C₂³×C₄	—	2	1	no	—
333	C₂³×C₄	—	2	2	no	—
334	C₂×D₈	Q₈×C₆₇	1	8	yes	complemented
335	C₂×D₈	C₂³×C₆₇	1	4	yes	—
336	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
337	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
338	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
339	C₄⋊Q₈	—	2	4	no	—
340	C₄⋊Q₈	—	2	2	no	—
341	C₄⋊Q₈	—	2	2	no	—
342	C₄⋊Q₈	C₂³×C₆₇	1	4	yes	—
343	C₄⋊Q₈	C₂³×C₆₇	1	8	yes	—
344	C₄⋊Q₈	C₂³×C₆₇	1	4	yes	—
345	#125	—	2	4	no	—
346	#125	C₂³×C₆₇	1	8	yes	—
347	C₂²⋊Q₈	—	2	8	no	—
348	C₂²⋊Q₈	—	2	8	no	—
349	C₂²⋊Q₈	—	2	16	no	—
350	C₂²⋊Q₈	—	2	16	no	—
351	C₄×Q₈	D₄×C₆₇	1	4	yes	—
352	C₄×Q₈	C₂³×C₆₇	1	4	yes	—
353	C₄×D₄	—	2	4	no	—
354	C₄×D₄	—	2	2	no	—
355	C₄×D₄	—	2	4	no	—
356	C₄×D₄	—	2	2	no	—
357	C₄×D₄	C₂³×C₆₇	1	4	yes	—
358	C₄×D₄	C₂³×C₆₇	1	8	yes	—
359	C₄×D₄	C₂³×C₆₇	1	4	yes	—
360	C₄×D₄	C₂³×C₆₇	1	8	yes	—
361	C₂×C₄⋊C₄	—	2	2	no	—
362	C₂×C₄⋊C₄	—	2	1	no	—
363	C₂×C₄⋊C₄	—	2	2	no	—
364	C₂×C₄⋊C₄	—	2	1	no	—
365	C₂×C₄⋊C₄	—	2	2	no	—
366	C₂×C₄⋊C₄	—	2	8	no	—
367	C₂×C₄⋊C₄	D₄×C₆₇	1	1	yes	characteristic
368	C₂×C₄⋊C₄	D₄×C₆₇	1	2	yes	—
369	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
370	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
371	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
372	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
373	C₂×C₂²⋊C₄	—	2	1	no	—
374	C₂×C₂²⋊C₄	—	2	2	no	—
375	C₂×C₂²⋊C₄	—	2	1	no	—
376	C₂×C₂²⋊C₄	—	2	2	no	—
377	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
378	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
379	#107	C₂³×C₆₇	1	4	yes	—
380	#107	C₂³×C₆₇	1	8	yes	—
381	#106	C₂³×C₆₇	1	8	yes	—
382	#106	C₂³×C₆₇	1	8	yes	—
383	#106	C₂³×C₆₇	1	8	yes	—
384	C₄⋊C₈	C₂³×C₆₇	1	4	yes	—
385	C₄⋊C₈	C₂³×C₆₇	1	8	yes	—
386	#102	C₂³×C₆₇	1	4	yes	—
387	#102	C₂³×C₆₇	1	4	yes	—
388	#102	C₂³×C₆₇	1	8	yes	—
389	#102	C₂³×C₆₇	1	8	yes	—
390	#97	C₂³×C₆₇	1	8	yes	—
391	C₄×C₈	C₂³×C₆₇	1	4	yes	—
392	C₂⁴	—	2	1	no	—
393	C₂⁴	—	2	1	no	—
394	C₂×Q₈	—	2	4	no	—
395	C₂×Q₈	—	2	8	no	—
396	C₂×Q₈	—	2	4	no	—
397	C₂×Q₈	—	2	4	no	—
398	C₂×Q₈	—	4	8	no	—
399	C₂×Q₈	—	4	32	no	—
400	C₂×Q₈	—	4	4	no	—
401	C₂×Q₈	—	8	2	no	—
402	C₂×Q₈	C₂×D₄×C₆₇	1	4	yes	—
403	C₂×D₄	—	2	1	no	—
404	C₂×D₄	—	2	1	no	—
405	C₂×D₄	—	2	2	no	—
406	C₂×D₄	—	2	2	no	—
407	C₂×D₄	C₂×Q₈×C₆₇	1	4	yes	—
408	C₂×D₄	C₂×Q₈×C₆₇	1	4	yes	complemented
409	C₂×D₄	C₂⁴×C₆₇	1	2	yes	—
410	C₂×D₄	C₂⁴×C₆₇	1	2	yes	—
411	C₂²×C₄	—	2	1	no	—
412	C₂²×C₄	—	2	2	no	—
413	C₂²×C₄	—	2	2	no	—
414	C₂²×C₄	—	2	1	no	—
415	C₂²×C₄	—	2	4	no	—
416	C₂²×C₄	—	2	8	no	—
417	C₂²×C₄	—	2	4	no	—
418	C₂²×C₄	—	2	8	no	—
419	C₂²×C₄	—	4	8	no	—
420	C₂²×C₄	—	4	4	no	—
421	C₂²×C₄	—	4	4	no	—
422	C₂²×C₄	—	8	4	no	—
423	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
424	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
425	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
426	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
427	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
428	D₈	C₂×Q₈×C₆₇	1	16	yes	complemented
429	C₂×C₈	—	2	2	no	—
430	C₂×C₈	—	2	4	no	—
431	C₂×C₈	C₂×Q₈×C₆₇	1	4	yes	complemented
432	C₂×C₈	C₂⁴×C₆₇	1	2	yes	—
433	C₂×C₈	C₂⁴×C₆₇	1	4	yes	—
434	C₂×C₈	C₂⁴×C₆₇	1	2	yes	—
435	C₄⋊C₄	—	2	8	no	—
436	C₄⋊C₄	—	2	4	no	—
437	C₄⋊C₄	—	2	8	no	—
438	C₄⋊C₄	—	2	4	no	—
439	C₄⋊C₄	—	2	8	no	—
440	C₄⋊C₄	—	2	1	no	—
441	C₄⋊C₄	—	2	2	no	—
442	C₄⋊C₄	—	2	2	no	—
443	C₄⋊C₄	—	2	1	no	—
444	C₄⋊C₄	C₂×D₄×C₆₇	1	4	yes	—
445	C₄⋊C₄	C₂×D₄×C₆₇	1	8	yes	—
446	C₄⋊C₄	C₂⁴×C₆₇	1	4	yes	—
447	C₄⋊C₄	C₂⁴×C₆₇	1	2	yes	—
448	C₄⋊C₄	C₂⁴×C₆₇	1	4	yes	—
449	C₄⋊C₄	C₂⁴×C₆₇	1	2	yes	—
450	C₂²⋊C₄	—	2	4	no	—
451	C₂²⋊C₄	—	2	8	no	—
452	C₂²⋊C₄	—	2	4	no	—
453	C₂²⋊C₄	—	2	8	no	—
454	C₄²	C₂×D₄×C₆₇	1	4	yes	—
455	C₄²	C₂×D₄×C₆₇	1	2	yes	—
456	C₄²	C₂⁴×C₆₇	1	2	yes	—
457	C₄²	C₂⁴×C₆₇	1	4	yes	—
458	C₂³	—	2	4	no	—
459	C₂³	—	2	4	no	—
460	C₂³	—	2	2	no	—
461	C₂³	—	2	2	no	—
462	C₂³	—	4	2	no	—
463	C₂³	—	4	2	no	—
464	C₂³	—	4	2	no	—
465	C₂³	—	4	2	no	—
466	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
467	Q₈	—	2	8	no	—
468	Q₈	—	4	16	no	—
469	Q₈	—	4	8	no	—
470	Q₈	—	8	4	no	—
471	D₄	—	2	4	no	—
472	D₄	—	2	4	no	—
473	D₄	—	2	2	no	—
474	D₄	—	2	2	no	—
475	D₄	C₂²×Q₈×C₆₇	1	4	yes	—
476	D₄	C₂²×Q₈×C₆₇	1	4	yes	complemented
477	C₂×C₄	—	2	4	no	—
478	C₂×C₄	—	2	4	no	—
479	C₂×C₄	—	2	2	no	—
480	C₂×C₄	—	2	2	no	—
481	C₂×C₄	—	2	4	no	—
482	C₂×C₄	—	2	4	no	—
483	C₂×C₄	—	2	2	no	—
484	C₂×C₄	—	4	16	no	—
485	C₂×C₄	—	4	4	no	—
486	C₂×C₄	—	4	2	no	—
487	C₂×C₄	—	4	2	no	—
488	C₂×C₄	—	4	4	no	—
489	C₂×C₄	—	4	8	no	—
490	C₂×C₄	—	4	8	no	—
491	C₂×C₄	—	8	8	no	—
492	C₂×C₄	AGL(1,ℤ/8)×C₆₇	1	4	yes	—
493	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
494	C₂×C₄	C₂²×D₄×C₆₇	1	4	yes	—
495	C₂×C₄	C₂²×D₄×C₆₇	1	4	yes	—
496	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
497	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
498	C₂×C₄	C₂²×Q₈×C₆₇	1	2	yes	—
499	C₂×C₄	2₋¹⁺⁴×C₆₇	1	2	yes	—
500	C₂×C₄	C₂⁵×C₆₇	1	1	yes	characteristic
501	C₈	—	2	2	no	—
502	C₈	—	2	4	no	—
503	C₈	C₂²×Q₈×C₆₇	1	4	yes	complemented
504	C₂²	—	2	4	no	—
505	C₂²	—	2	4	no	—
506	C₂²	—	4	2	no	—
507	C₂²	—	4	2	no	—
508	C₂²	—	4	2	no	—
509	C₂²	—	4	2	no	—
510	C₂²	—	4	8	no	—
511	C₂²	—	4	8	no	—
512	C₂²	D₄×Q₈×C₆₇	1	2	yes	—
513	C₂²	C₂×AGL(1,ℤ/8)×C₆₇	1	2	yes	—
514	C₂²	#577×C₆₇	1	2	yes	—
515	C₂²	C₂³×D₄×C₆₇	1	1	yes	characteristic
516	C₄	—	2	4	no	—
517	C₄	—	2	4	no	—
518	C₄	—	2	2	no	—
519	C₄	—	4	4	no	—
520	C₄	—	4	2	no	—
521	C₄	—	4	2	no	—
522	C₄	—	4	4	no	—
523	C₄	D₄×Q₈×C₆₇	1	2	yes	—
524	C₄	C₂×AGL(1,ℤ/8)×C₆₇	1	4	yes	—
525	C₄	C₂³×Q₈×C₆₇	1	1	yes	characteristic
526	C₄	C₂×2₋¹⁺⁴×C₆₇	1	1	yes	characteristic
527	C₂	—	4	4	no	—
528	C₂	—	4	4	no	—
529	C₂	C₆₇×#3384	1	4	yes	direct product factor
530	C₂	C₂×D₄×Q₈×C₆₇	1	1	yes	characteristic
531	C₂	C₂²×AGL(1,ℤ/8)×C₆₇	1	1	yes	characteristic
532	C₂	C₂×#577×C₆₇	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₆₇	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₆₇
prime radical	C₂²×D₈×C₆₇	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₆₇ ◁ C₂×C₄×Q₈×C₆₇ ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. C₂×#554×C₆₇	C₂	semidirect
2. C₂×D₄×Q₈×C₆₇	C₂	semidirect
3. C₆₇×#3384	C₂	direct (N × H)
4. C₂×#501×C₆₇	C₂	semidirect
5. C₂×#498×C₆₇	C₂	semidirect
6. C₂×#478×C₆₇	C₂	semidirect
7. C₂×#476×C₆₇	C₂	semidirect
8. C₂×#476×C₆₇	C₂	semidirect
9. C₂×#474×C₆₇	C₂	semidirect
10. C₂×#446×C₆₇	C₂	semidirect
16. #554×C₆₇	C₂²	semidirect
18. D₄×Q₈×C₆₇	C₂²	semidirect
33. #501×C₆₇	C₂²	semidirect
34. #498×C₆₇	C₂²	semidirect
35. #478×C₆₇	C₂²	semidirect
36. #476×C₆₇	C₂²	semidirect
37. #476×C₆₇	C₂²	semidirect
38. #474×C₆₇	C₂²	semidirect
39. #446×C₆₇	C₂²	semidirect
67. C₂×D₈×C₆₇	Q₈	semidirect
141. C₂×D₄×C₆₇	C₂×Q₈	semidirect
161. D₈×C₆₇	C₂×Q₈	semidirect
164. C₂×C₈×C₆₇	C₂×Q₈	semidirect
209. D₄×C₆₇	C₂²×Q₈	semidirect
236. C₈×C₆₇	C₂²×Q₈	semidirect
260. C₂×#3384	C₆₇	direct (N × H)
263. C₂×C₆₇	#3384	direct (N × H)
267. C₂×#554	C₂×C₆₇	semidirect
268. C₂×D₄×Q₈	C₂×C₆₇	semidirect
269. #3384	C₂×C₆₇	direct (N × H)
270. C₂×#501	C₂×C₆₇	semidirect
271. C₂×#498	C₂×C₆₇	semidirect
272. C₂×#478	C₂×C₆₇	semidirect
273. C₂×#476	C₂×C₆₇	semidirect
274. C₂×#476	C₂×C₆₇	semidirect
275. C₂×#474	C₂×C₆₇	semidirect
276. C₂×#446	C₂×C₆₇	semidirect
279. C₆₇	C₂×#3384	direct (N × H)
283. #554	C₂²×C₆₇	semidirect
285. D₄×Q₈	C₂²×C₆₇	semidirect
300. #501	C₂²×C₆₇	semidirect
301. #498	C₂²×C₆₇	semidirect
302. #478	C₂²×C₆₇	semidirect
303. #476	C₂²×C₆₇	semidirect
304. #476	C₂²×C₆₇	semidirect
305. #474	C₂²×C₆₇	semidirect
306. #446	C₂²×C₆₇	semidirect
334. C₂×D₈	Q₈×C₆₇	semidirect
408. C₂×D₄	C₂×Q₈×C₆₇	semidirect
428. D₈	C₂×Q₈×C₆₇	semidirect
431. C₂×C₈	C₂×Q₈×C₆₇	semidirect
476. D₄	C₂²×Q₈×C₆₇	semidirect
503. C₈	C₂²×Q₈×C₆₇	semidirect
529. C₂	C₆₇×#3384	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×#554×C₆₇	2
C₂×D₄×Q₈×C₆₇	2
C₆₇×#3384	2
C₂×#501×C₆₇	2
C₂×#498×C₆₇	2
C₂×#478×C₆₇	2
C₂×#476×C₆₇	2
C₂×#474×C₆₇	2
C₂×#446×C₆₇	2
C₂×#442×C₆₇	2
C₂×C₄×D₈×C₆₇	2
C₂×#3384	67

Subgroups — C2×C67×#3384 #30740609

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₆₇×#3384 #30740609