Subgroups — C₆₇×#25884 #30705135

Order 17152.

1302 conjugacy classes of subgroups.

The subgroup lattice has 440 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₄×Q₈×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₂×#487×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#475×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₆₇×#2547	C₂	1	1	yes	maximal, characteristic
5	C₆₇×#2035	C₂	1	4	yes	maximal, complemented
6	C₆₇×#1981	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₇×#1954	C₂	1	1	yes	maximal, characteristic
8	C₆₇×#1787	C₂	1	4	yes	maximal, complemented
9	C₄×#103×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂³×Q₈×C₆₇	C₂²	1	2	yes	—
11	D₄×Q₈×C₆₇	—	2	4	no	—
12	D₄×Q₈×C₆₇	—	2	4	no	—
13	C₂×C₄⋊Q₈×C₆₇	—	2	1	no	—
14	C₂×C₄⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
16	C₂×#124×C₆₇	—	2	1	no	—
17	C₂×C₂²⋊Q₈×C₆₇	—	2	2	no	—
18	C₂×C₂²⋊Q₈×C₆₇	C₂²	1	2	yes	—
19	C₂×C₄×Q₈×C₆₇	C₂²	1	1	yes	characteristic
20	C₂×C₄×D₄×C₆₇	—	2	1	no	—
21	C₂×C₄×D₄×C₆₇	C₂²	1	1	yes	characteristic
22	#487×C₆₇	—	2	4	no	—
23	#475×C₆₇	—	2	4	no	—
24	C₂×#106×C₆₇	C₂²	1	2	yes	—
25	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
26	C₂×#103×C₆₇	C₂²	1	2	yes	—
27	C₂×#103×C₆₇	C₂²	1	4	yes	—
28	C₂×#103×C₆₇	C₂²	1	2	yes	—
29	C₂×#103×C₆₇	C₂²	1	4	yes	—
30	C₂×C₂²⋊C₈×C₆₇	C₂²	1	4	yes	—
31	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
32	#397×C₆₇	—	2	2	no	—
33	#391×C₆₇	—	2	1	no	—
34	#380×C₆₇	C₂²	1	4	yes	—
35	C₄×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
36	#340×C₆₇	C₂²	1	2	yes	—
37	C₂²×Q₈×C₆₇	—	2	4	no	—
38	C₂²×Q₈×C₆₇	—	2	4	no	—
39	C₂²×Q₈×C₆₇	—	2	2	no	—
40	C₂²×Q₈×C₆₇	—	2	1	no	—
41	C₂²×Q₈×C₆₇	—	2	8	no	—
42	C₂²×Q₈×C₆₇	—	4	4	no	—
43	C₂²×Q₈×C₆₇	C₂³	1	2	yes	—
44	C₂²×Q₈×C₆₇	C₂³	1	1	yes	characteristic
45	C₂²×D₄×C₆₇	D₄	1	1	yes	characteristic
46	C₂³×C₄×C₆₇	—	2	2	no	—
47	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
48	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
49	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
50	C₄⋊Q₈×C₆₇	—	2	8	no	—
51	C₄⋊Q₈×C₆₇	—	2	2	no	—
52	C₄⋊Q₈×C₆₇	—	2	2	no	—
53	C₄⋊Q₈×C₆₇	—	2	2	no	—
54	C₄⋊Q₈×C₆₇	D₄	1	4	yes	—
55	#124×C₆₇	—	4	4	no	—
56	C₂²⋊Q₈×C₆₇	—	2	16	no	—
57	C₂²⋊Q₈×C₆₇	—	2	4	no	—
58	C₂²⋊Q₈×C₆₇	—	2	4	no	—
59	C₄×Q₈×C₆₇	—	2	2	no	—
60	C₄×Q₈×C₆₇	D₄	1	4	yes	—
61	C₄×D₄×C₆₇	—	2	8	no	—
62	C₄×D₄×C₆₇	—	2	2	no	—
63	C₄×D₄×C₆₇	—	2	2	no	—
64	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
65	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
66	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
67	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
68	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
69	C₂×C₄⋊C₄×C₆₇	D₄	1	1	yes	characteristic
70	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
71	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
72	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
73	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
74	C₂×C₂²⋊C₄×C₆₇	—	2	4	no	—
75	C₂×C₂²⋊C₄×C₆₇	D₄	1	2	yes	—
76	C₂×C₄²×C₆₇	—	2	1	no	—
77	C₂×C₄²×C₆₇	—	2	1	no	—
78	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
79	#106×C₆₇	—	2	4	no	—
80	C₄⋊C₈×C₆₇	—	2	2	no	—
81	#103×C₆₇	—	2	4	no	—
82	#103×C₆₇	—	2	8	no	—
83	#103×C₆₇	—	2	4	no	—
84	#103×C₆₇	—	4	4	no	—
85	C₂²⋊C₈×C₆₇	—	4	4	no	—
86	C₄×C₈×C₆₇	—	2	2	no	—
87	#95×C₆₇	—	2	1	no	—
88	#95×C₆₇	—	2	4	no	—
89	C₂⁴×C₆₇	C₂×D₄	1	2	yes	—
90	C₂×Q₈×C₆₇	—	2	8	no	—
91	C₂×Q₈×C₆₇	—	2	1	no	—
92	C₂×Q₈×C₆₇	—	2	2	no	—
93	C₂×Q₈×C₆₇	—	2	2	no	—
94	C₂×Q₈×C₆₇	—	2	4	no	—
95	C₂×Q₈×C₆₇	—	2	4	no	—
96	C₂×Q₈×C₆₇	—	4	8	no	—
97	C₂×Q₈×C₆₇	—	4	16	no	—
98	C₂×Q₈×C₆₇	—	4	8	no	—
99	C₂×Q₈×C₆₇	—	4	8	no	—
100	C₂×Q₈×C₆₇	—	4	2	no	—
101	C₂×Q₈×C₆₇	—	4	2	no	—
102	C₂×Q₈×C₆₇	—	4	2	no	—
103	C₂×Q₈×C₆₇	C₂×D₄	1	4	yes	—
104	C₂×Q₈×C₆₇	C₂×D₄	1	2	yes	—
105	C₂×Q₈×C₆₇	#41	1	4	yes	—
106	C₂×D₄×C₆₇	—	2	2	no	—
107	C₂×D₄×C₆₇	—	2	2	no	—
108	C₂×D₄×C₆₇	SD₁₆	1	4	yes	—
109	C₂²×C₄×C₆₇	—	2	1	no	—
110	C₂²×C₄×C₆₇	—	2	8	no	—
111	C₂²×C₄×C₆₇	—	2	1	no	—
112	C₂²×C₄×C₆₇	—	2	2	no	—
113	C₂²×C₄×C₆₇	—	2	2	no	—
114	C₂²×C₄×C₆₇	—	2	2	no	—
115	C₂²×C₄×C₆₇	—	2	4	no	—
116	C₂²×C₄×C₆₇	—	2	4	no	—
117	C₂²×C₄×C₆₇	—	2	2	no	—
118	C₂²×C₄×C₆₇	—	2	2	no	—
119	C₂²×C₄×C₆₇	—	4	8	no	—
120	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
121	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
122	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
123	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
124	C₂×C₈×C₆₇	—	2	2	no	—
125	C₂×C₈×C₆₇	—	2	2	no	—
126	C₂×C₈×C₆₇	—	2	2	no	—
127	C₂×C₈×C₆₇	—	2	4	no	—
128	C₂×C₈×C₆₇	—	2	2	no	—
129	C₄⋊C₄×C₆₇	—	2	8	no	—
130	C₄⋊C₄×C₆₇	—	2	4	no	—
131	C₄⋊C₄×C₆₇	—	2	4	no	—
132	C₄⋊C₄×C₆₇	—	2	4	no	—
133	C₄⋊C₄×C₆₇	—	2	2	no	—
134	C₄⋊C₄×C₆₇	—	2	1	no	—
135	C₄⋊C₄×C₆₇	—	2	2	no	—
136	C₄⋊C₄×C₆₇	—	2	2	no	—
137	C₄⋊C₄×C₆₇	—	2	2	no	—
138	C₄⋊C₄×C₆₇	—	2	2	no	—
139	C₄⋊C₄×C₆₇	—	4	2	no	—
140	C₄⋊C₄×C₆₇	C₂×D₄	1	2	yes	—
141	C₂²⋊C₄×C₆₇	—	2	8	no	—
142	C₂²⋊C₄×C₆₇	—	2	4	no	—
143	C₂²⋊C₄×C₆₇	—	4	8	no	—
144	C₄²×C₆₇	—	2	4	no	—
145	C₄²×C₆₇	—	4	2	no	—
146	C₄²×C₆₇	C₂×D₄	1	2	yes	—
147	C₄²×C₆₇	C₂×D₄	1	2	yes	—
148	C₂³×C₆₇	—	2	2	no	—
149	C₂³×C₆₇	—	2	4	no	—
150	C₂³×C₆₇	—	2	2	no	—
151	C₂³×C₆₇	—	2	4	no	—
152	C₂³×C₆₇	C₂×SD₁₆	1	4	yes	—
153	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
154	Q₈×C₆₇	—	2	2	no	—
155	Q₈×C₆₇	—	2	4	no	—
156	Q₈×C₆₇	—	2	8	no	—
157	Q₈×C₆₇	—	4	16	no	—
158	Q₈×C₆₇	—	4	2	no	—
159	Q₈×C₆₇	—	4	2	no	—
160	Q₈×C₆₇	—	8	4	no	—
161	Q₈×C₆₇	#124	1	4	yes	complemented
162	D₄×C₆₇	—	2	8	no	—
163	C₂×C₄×C₆₇	—	2	2	no	—
164	C₂×C₄×C₆₇	—	2	4	no	—
165	C₂×C₄×C₆₇	—	2	2	no	—
166	C₂×C₄×C₆₇	—	2	2	no	—
167	C₂×C₄×C₆₇	—	2	4	no	—
168	C₂×C₄×C₆₇	—	2	2	no	—
169	C₂×C₄×C₆₇	—	2	4	no	—
170	C₂×C₄×C₆₇	—	4	16	no	—
171	C₂×C₄×C₆₇	—	4	4	no	—
172	C₂×C₄×C₆₇	—	4	2	no	—
173	C₂×C₄×C₆₇	—	4	2	no	—
174	C₂×C₄×C₆₇	—	4	2	no	—
175	C₂×C₄×C₆₇	—	4	4	no	—
176	C₂×C₄×C₆₇	—	4	2	no	—
177	C₂×C₄×C₆₇	—	4	2	no	—
178	C₂×C₄×C₆₇	—	4	2	no	—
179	C₂×C₄×C₆₇	—	4	2	no	—
180	C₂×C₄×C₆₇	—	4	2	no	—
181	C₂×C₄×C₆₇	—	4	2	no	—
182	C₂×C₄×C₆₇	#120	1	2	yes	—
183	C₂×C₄×C₆₇	#120	1	2	yes	—
184	C₂×C₄×C₆₇	C₂×SD₁₆	1	2	yes	—
185	C₂×C₄×C₆₇	#137	1	2	yes	—
186	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
187	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
188	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
189	C₂×C₄×C₆₇	H(5,𝔽₂)	1	2	yes	—
190	C₈×C₆₇	—	4	2	no	—
191	C₈×C₆₇	—	4	2	no	—
192	C₂²×C₆₇	—	2	4	no	—
193	C₂²×C₆₇	—	2	8	no	—
194	C₂²×C₆₇	—	4	4	no	—
195	C₂²×C₆₇	—	4	2	no	—
196	C₂²×C₆₇	—	4	2	no	—
197	C₂²×C₆₇	C₂×#120	1	1	yes	characteristic
198	C₂²×C₆₇	#546	1	2	yes	—
199	C₂²×C₆₇	C₂²×SD₁₆	1	1	yes	characteristic
200	C₂²×C₆₇	C₂×#137	1	1	yes	characteristic
201	C₂²×C₆₇	#578	1	2	yes	—
202	C₄×C₆₇	—	2	4	no	—
203	C₄×C₆₇	—	2	2	no	—
204	C₄×C₆₇	—	4	2	no	—
205	C₄×C₆₇	—	4	4	no	—
206	C₄×C₆₇	—	4	2	no	—
207	C₄×C₆₇	—	8	2	no	—
208	C₄×C₆₇	—	8	2	no	—
209	C₄×C₆₇	#448	1	4	yes	—
210	C₄×C₆₇	C₂×#124	1	1	yes	characteristic
211	C₄×C₆₇	#535	1	1	yes	characteristic
212	C₄×C₆₇	#546	1	2	yes	—
213	#25884	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
214	C₂×C₆₇	—	4	4	no	—
215	C₂×C₆₇	#2435	1	1	yes	characteristic
216	C₂×C₆₇	C₂×#448	1	1	yes	characteristic
217	C₂×C₆₇	#3011	1	1	yes	characteristic
218	C₂×C₆₇	#3298	1	2	yes	—
219	C₂×C₆₇	#3328	1	2	yes	—
220	C₂×D₄×Q₈	C₂×C₆₇	1	1	yes	characteristic
221	C₂×#487	C₂×C₆₇	1	1	yes	characteristic, complemented
222	C₂×#475	C₂×C₆₇	1	1	yes	characteristic, complemented
223	#2547	C₂×C₆₇	1	1	yes	characteristic
224	#2035	C₂×C₆₇	1	4	yes	complemented
225	#1981	C₂×C₆₇	1	1	yes	characteristic, complemented
226	#1954	C₂×C₆₇	1	1	yes	characteristic
227	#1787	C₂×C₆₇	1	4	yes	complemented
228	C₄×#103	C₂×C₆₇	1	1	yes	characteristic, complemented
229	C₆₇	#25884	1	1	yes	characteristic, direct product factor, Sylow
230	C₂³×Q₈	C₂²×C₆₇	1	2	yes	—
231	D₄×Q₈	—	2	4	no	—
232	D₄×Q₈	—	2	4	no	—
233	C₂×C₄⋊Q₈	—	2	1	no	—
234	C₂×C₄⋊Q₈	C₂²×C₆₇	1	1	yes	characteristic
235	C₂×C₄⋊Q₈	C₂²×C₆₇	1	1	yes	characteristic
236	C₂×#124	—	2	1	no	—
237	C₂×C₂²⋊Q₈	—	2	2	no	—
238	C₂×C₂²⋊Q₈	C₂²×C₆₇	1	2	yes	—
239	C₂×C₄×Q₈	C₂²×C₆₇	1	1	yes	characteristic
240	C₂×C₄×D₄	—	2	1	no	—
241	C₂×C₄×D₄	C₂²×C₆₇	1	1	yes	characteristic
242	#487	—	2	4	no	—
243	#475	—	2	4	no	—
244	C₂×#106	C₂²×C₆₇	1	2	yes	—
245	C₂×C₄⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
246	C₂×#103	C₂²×C₆₇	1	2	yes	—
247	C₂×#103	C₂²×C₆₇	1	4	yes	—
248	C₂×#103	C₂²×C₆₇	1	2	yes	—
249	C₂×#103	C₂²×C₆₇	1	4	yes	—
250	C₂×C₂²⋊C₈	C₂²×C₆₇	1	4	yes	—
251	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
252	#397	—	2	2	no	—
253	#391	—	2	1	no	—
254	#380	C₂²×C₆₇	1	4	yes	—
255	C₄×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
256	#340	C₂²×C₆₇	1	2	yes	—
257	C₂²×Q₈	—	2	4	no	—
258	C₂²×Q₈	—	2	4	no	—
259	C₂²×Q₈	—	2	2	no	—
260	C₂²×Q₈	—	2	1	no	—
261	C₂²×Q₈	—	2	8	no	—
262	C₂²×Q₈	—	4	4	no	—
263	C₂²×Q₈	C₂³×C₆₇	1	2	yes	—
264	C₂²×Q₈	C₂³×C₆₇	1	1	yes	characteristic
265	C₂²×D₄	D₄×C₆₇	1	1	yes	characteristic
266	C₂³×C₄	—	2	2	no	—
267	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
268	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
269	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
270	C₄⋊Q₈	—	2	8	no	—
271	C₄⋊Q₈	—	2	2	no	—
272	C₄⋊Q₈	—	2	2	no	—
273	C₄⋊Q₈	—	2	2	no	—
274	C₄⋊Q₈	D₄×C₆₇	1	4	yes	—
275	#124	—	4	4	no	—
276	C₂²⋊Q₈	—	2	16	no	—
277	C₂²⋊Q₈	—	2	4	no	—
278	C₂²⋊Q₈	—	2	4	no	—
279	C₄×Q₈	—	2	2	no	—
280	C₄×Q₈	D₄×C₆₇	1	4	yes	—
281	C₄×D₄	—	2	8	no	—
282	C₄×D₄	—	2	2	no	—
283	C₄×D₄	—	2	2	no	—
284	C₂×C₄⋊C₄	—	2	1	no	—
285	C₂×C₄⋊C₄	—	2	1	no	—
286	C₂×C₄⋊C₄	—	2	1	no	—
287	C₂×C₄⋊C₄	—	2	2	no	—
288	C₂×C₄⋊C₄	—	2	1	no	—
289	C₂×C₄⋊C₄	D₄×C₆₇	1	1	yes	characteristic
290	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
291	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
292	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
293	C₂×C₂²⋊C₄	—	2	2	no	—
294	C₂×C₂²⋊C₄	—	2	4	no	—
295	C₂×C₂²⋊C₄	D₄×C₆₇	1	2	yes	—
296	C₂×C₄²	—	2	1	no	—
297	C₂×C₄²	—	2	1	no	—
298	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
299	#106	—	2	4	no	—
300	C₄⋊C₈	—	2	2	no	—
301	#103	—	2	4	no	—
302	#103	—	2	8	no	—
303	#103	—	2	4	no	—
304	#103	—	4	4	no	—
305	C₂²⋊C₈	—	4	4	no	—
306	C₄×C₈	—	2	2	no	—
307	#95	—	2	1	no	—
308	#95	—	2	4	no	—
309	C₂⁴	C₂×D₄×C₆₇	1	2	yes	—
310	C₂×Q₈	—	2	8	no	—
311	C₂×Q₈	—	2	1	no	—
312	C₂×Q₈	—	2	2	no	—
313	C₂×Q₈	—	2	2	no	—
314	C₂×Q₈	—	2	4	no	—
315	C₂×Q₈	—	2	4	no	—
316	C₂×Q₈	—	4	8	no	—
317	C₂×Q₈	—	4	16	no	—
318	C₂×Q₈	—	4	8	no	—
319	C₂×Q₈	—	4	8	no	—
320	C₂×Q₈	—	4	2	no	—
321	C₂×Q₈	—	4	2	no	—
322	C₂×Q₈	—	4	2	no	—
323	C₂×Q₈	C₂×D₄×C₆₇	1	4	yes	—
324	C₂×Q₈	C₂×D₄×C₆₇	1	2	yes	—
325	C₂×Q₈	#41×C₆₇	1	4	yes	—
326	C₂×D₄	—	2	2	no	—
327	C₂×D₄	—	2	2	no	—
328	C₂×D₄	SD₁₆×C₆₇	1	4	yes	—
329	C₂²×C₄	—	2	1	no	—
330	C₂²×C₄	—	2	8	no	—
331	C₂²×C₄	—	2	1	no	—
332	C₂²×C₄	—	2	2	no	—
333	C₂²×C₄	—	2	2	no	—
334	C₂²×C₄	—	2	2	no	—
335	C₂²×C₄	—	2	4	no	—
336	C₂²×C₄	—	2	4	no	—
337	C₂²×C₄	—	2	2	no	—
338	C₂²×C₄	—	2	2	no	—
339	C₂²×C₄	—	4	8	no	—
340	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
341	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
342	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
343	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
344	C₂×C₈	—	2	2	no	—
345	C₂×C₈	—	2	2	no	—
346	C₂×C₈	—	2	2	no	—
347	C₂×C₈	—	2	2	no	—
348	C₂×C₈	—	2	4	no	—
349	C₄⋊C₄	—	2	8	no	—
350	C₄⋊C₄	—	2	4	no	—
351	C₄⋊C₄	—	2	4	no	—
352	C₄⋊C₄	—	2	4	no	—
353	C₄⋊C₄	—	2	2	no	—
354	C₄⋊C₄	—	2	1	no	—
355	C₄⋊C₄	—	2	2	no	—
356	C₄⋊C₄	—	2	2	no	—
357	C₄⋊C₄	—	2	2	no	—
358	C₄⋊C₄	—	2	2	no	—
359	C₄⋊C₄	—	4	2	no	—
360	C₄⋊C₄	C₂×D₄×C₆₇	1	2	yes	—
361	C₂²⋊C₄	—	2	8	no	—
362	C₂²⋊C₄	—	2	4	no	—
363	C₂²⋊C₄	—	4	8	no	—
364	C₄²	—	2	4	no	—
365	C₄²	—	4	2	no	—
366	C₄²	C₂×D₄×C₆₇	1	2	yes	—
367	C₄²	C₂×D₄×C₆₇	1	2	yes	—
368	C₂³	—	2	2	no	—
369	C₂³	—	2	4	no	—
370	C₂³	—	2	2	no	—
371	C₂³	—	2	4	no	—
372	C₂³	C₂×SD₁₆×C₆₇	1	4	yes	—
373	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
374	Q₈	—	2	2	no	—
375	Q₈	—	2	4	no	—
376	Q₈	—	2	8	no	—
377	Q₈	—	4	16	no	—
378	Q₈	—	4	2	no	—
379	Q₈	—	4	2	no	—
380	Q₈	—	8	4	no	—
381	Q₈	#124×C₆₇	1	4	yes	complemented
382	D₄	—	2	8	no	—
383	C₂×C₄	—	2	2	no	—
384	C₂×C₄	—	2	4	no	—
385	C₂×C₄	—	2	2	no	—
386	C₂×C₄	—	2	2	no	—
387	C₂×C₄	—	2	4	no	—
388	C₂×C₄	—	2	2	no	—
389	C₂×C₄	—	2	4	no	—
390	C₂×C₄	—	4	16	no	—
391	C₂×C₄	—	4	4	no	—
392	C₂×C₄	—	4	2	no	—
393	C₂×C₄	—	4	2	no	—
394	C₂×C₄	—	4	2	no	—
395	C₂×C₄	—	4	4	no	—
396	C₂×C₄	—	4	2	no	—
397	C₂×C₄	—	4	2	no	—
398	C₂×C₄	—	4	2	no	—
399	C₂×C₄	—	4	2	no	—
400	C₂×C₄	—	4	2	no	—
401	C₂×C₄	—	4	2	no	—
402	C₂×C₄	#120×C₆₇	1	2	yes	—
403	C₂×C₄	#120×C₆₇	1	2	yes	—
404	C₂×C₄	C₂×SD₁₆×C₆₇	1	2	yes	—
405	C₂×C₄	#137×C₆₇	1	2	yes	—
406	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
407	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
408	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
409	C₂×C₄	H(5,𝔽₂)×C₆₇	1	2	yes	—
410	C₈	—	4	2	no	—
411	C₈	—	4	2	no	—
412	C₂²	—	2	8	no	—
413	C₂²	—	2	4	no	—
414	C₂²	—	4	2	no	—
415	C₂²	—	4	4	no	—
416	C₂²	—	4	2	no	—
417	C₂²	C₂×#120×C₆₇	1	1	yes	characteristic
418	C₂²	#546×C₆₇	1	2	yes	—
419	C₂²	C₂²×SD₁₆×C₆₇	1	1	yes	characteristic
420	C₂²	C₂×#137×C₆₇	1	1	yes	characteristic
421	C₂²	#578×C₆₇	1	2	yes	—
422	C₄	—	2	2	no	—
423	C₄	—	2	4	no	—
424	C₄	—	4	2	no	—
425	C₄	—	4	2	no	—
426	C₄	—	4	4	no	—
427	C₄	—	8	2	no	—
428	C₄	—	8	2	no	—
429	C₄	#448×C₆₇	1	4	yes	—
430	C₄	C₂×#124×C₆₇	1	1	yes	characteristic
431	C₄	#535×C₆₇	1	1	yes	characteristic
432	C₄	#546×C₆₇	1	2	yes	—
433	C₂	—	4	4	no	—
434	C₂	C₆₇×#2435	1	1	yes	characteristic
435	C₂	C₂×#448×C₆₇	1	1	yes	characteristic
436	C₂	C₆₇×#3011	1	1	yes	characteristic
437	C₂	C₆₇×#3298	1	2	yes	—
438	C₂	C₆₇×#3328	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₆₇	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
prime radical	C₂²×D₄×C₆₇	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₆₇ ◁ C₂×C₄×D₄×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×#487×C₆₇	C₂	semidirect
3. C₂×#475×C₆₇	C₂	semidirect
5. C₆₇×#2035	C₂	semidirect
6. C₆₇×#1981	C₂	semidirect
8. C₆₇×#1787	C₂	semidirect
9. C₄×#103×C₆₇	C₂	semidirect
161. Q₈×C₆₇	#124	semidirect
213. #25884	C₆₇	direct (N × H)
221. C₂×#487	C₂×C₆₇	semidirect
222. C₂×#475	C₂×C₆₇	semidirect
224. #2035	C₂×C₆₇	semidirect
225. #1981	C₂×C₆₇	semidirect
227. #1787	C₂×C₆₇	semidirect
228. C₄×#103	C₂×C₆₇	semidirect
229. C₆₇	#25884	direct (N × H)
381. Q₈	#124×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×D₄×Q₈×C₆₇	2
C₂×#487×C₆₇	2
C₂×#475×C₆₇	2
C₆₇×#2547	2
C₆₇×#2035	2
C₆₇×#1981	2
C₆₇×#1954	2
C₆₇×#1787	2
C₄×#103×C₆₇	2
#25884	67

Subgroups — C67×#25884 #30705135

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#25884 #30705135