Subgroups — #30864508

Order 17152.

802 conjugacy classes of subgroups.

The subgroup lattice has 432 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16863527	C₂	1	2	yes	maximal
2	#22561150	C₂	1	2	yes	maximal
3	#22560933	C₂	1	2	yes	maximal
4	#22560928	C₂	1	1	yes	maximal, characteristic
5	#22560928	C₂	1	1	yes	maximal, characteristic
6	#22560361	C₂	1	1	yes	maximal, characteristic
7	#22560084	C₂	1	2	yes	maximal
8	#22560078	C₂	1	1	yes	maximal, characteristic
9	C₄⋊C₄×C₆₇⋊C₈	C₂	1	1	yes	maximal, characteristic
10	C₄×#14523954	C₂	1	1	yes	maximal, characteristic
11	C₄×C₄⋊Q₈×C₆₇	C₂	1	1	yes	maximal, characteristic
12	#16863527	—	2	8	no	—
13	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	2	yes	—
16	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	2	yes	—
17	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	2	yes	—
18	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	2	yes	—
19	C₂×#14523954	C₂²	1	2	yes	—
20	C₂×#14523954	C₂²	1	1	yes	characteristic
21	C₂×#14523954	C₂²	1	1	yes	characteristic
22	C₂×#14523954	C₂²	1	1	yes	characteristic
23	C₂×#14523954	C₂²	1	1	yes	characteristic
24	C₂×C₄×C₆₇⋊C₈	C₂²	1	2	yes	—
25	C₂×C₄×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
26	C₂×C₄×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
27	#16863151	C₂²	1	2	yes	—
28	#16863151	C₂²	1	2	yes	—
29	#16863151	C₂²	1	2	yes	—
30	#16863151	C₂²	1	2	yes	—
31	C₂×C₄⋊Q₈×C₆₇	C₄	1	1	yes	characteristic
32	C₂×C₄×Q₈×C₆₇	C₂²	1	2	yes	—
33	#391×C₆₇	C₄	1	2	yes	—
34	#389×C₆₇	C₄	1	1	yes	characteristic
35	#389×C₆₇	C₄	1	1	yes	characteristic
36	#389×C₆₇	C₄	1	2	yes	—
37	#384×C₆₇	C₄	1	1	yes	characteristic
38	C₄×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
39	C₄×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
40	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
41	C₄³×C₆₇	C₂²	1	1	yes	characteristic
42	C₂²×C₆₇⋊C₈	C₂³	1	2	yes	—
43	C₂²×C₆₇⋊C₈	C₂³	1	2	yes	—
44	C₂²×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
45	C₂²×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
46	C₂²×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
47	C₂²×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
48	C₄⋊(C₆₇⋊C₈)	—	2	4	no	—
49	C₄⋊(C₆₇⋊C₈)	—	2	4	no	—
50	C₄⋊(C₆₇⋊C₈)	—	2	4	no	—
51	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
52	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
53	C₄⋊(C₆₇⋊C₈)	—	2	4	no	—
54	#14523954	—	2	4	no	—
55	#14523954	—	2	2	no	—
56	#14523954	—	2	2	no	—
57	#14523954	D₄	1	4	yes	—
58	#14523954	Q₈	1	4	yes	complemented
59	C₄×C₆₇⋊C₈	—	2	4	no	—
60	C₄×C₆₇⋊C₈	—	2	2	no	—
61	C₄×C₆₇⋊C₈	Q₈	1	4	yes	complemented
62	C₂²×Q₈×C₆₇	C₂×C₄	1	2	yes	—
63	C₄⋊Q₈×C₆₇	—	2	4	no	—
64	C₄×Q₈×C₆₇	—	2	4	no	—
65	C₄×Q₈×C₆₇	—	2	4	no	—
66	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
67	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
68	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
69	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
70	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
71	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
72	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
73	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
74	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
75	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
76	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
77	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
78	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
79	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
80	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
81	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
82	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
83	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
84	#95×C₆₇	C₂×C₄	1	2	yes	—
85	#95×C₆₇	C₂×C₄	1	2	yes	—
86	#95×C₆₇	C₂×C₄	1	2	yes	—
87	#95×C₆₇	C₂×C₄	1	2	yes	—
88	C₂×C₆₇⋊C₈	—	2	1	no	—
89	C₂×C₆₇⋊C₈	—	2	1	no	—
90	C₂×C₆₇⋊C₈	—	2	1	no	—
91	C₂×C₆₇⋊C₈	—	2	1	no	—
92	C₂×C₆₇⋊C₈	—	2	1	no	—
93	C₂×C₆₇⋊C₈	—	2	1	no	—
94	C₂×C₆₇⋊C₈	—	2	2	no	—
95	C₂×C₆₇⋊C₈	—	2	2	no	—
96	C₂×C₆₇⋊C₈	—	2	2	no	—
97	C₂×C₆₇⋊C₈	—	2	2	no	—
98	C₂×C₆₇⋊C₈	—	2	2	no	—
99	C₂×C₆₇⋊C₈	—	2	2	no	—
100	C₂×C₆₇⋊C₈	C₂×D₄	1	2	yes	—
101	C₂×C₆₇⋊C₈	C₂×D₄	1	2	yes	—
102	C₂×C₆₇⋊C₈	C₂×Q₈	1	2	yes	—
103	C₂×C₆₇⋊C₈	C₂×Q₈	1	2	yes	—
104	C₂×C₆₇⋊C₈	C₂×Q₈	1	2	yes	—
105	C₂×C₆₇⋊C₈	C₂×Q₈	1	2	yes	—
106	C₂×Q₈×C₆₇	—	2	4	no	—
107	C₂×Q₈×C₆₇	—	2	4	no	—
108	C₂×Q₈×C₆₇	—	4	2	no	—
109	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
110	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
111	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
112	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
113	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
114	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
115	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
116	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
117	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
118	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
119	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
120	C₄⋊C₄×C₆₇	—	2	4	no	—
121	C₄⋊C₄×C₆₇	—	2	4	no	—
122	C₄⋊C₄×C₆₇	—	2	4	no	—
123	C₄⋊C₄×C₆₇	—	2	4	no	—
124	C₄⋊C₄×C₆₇	—	2	2	no	—
125	C₄⋊C₄×C₆₇	—	2	2	no	—
126	C₄⋊C₄×C₆₇	—	2	2	no	—
127	C₄⋊C₄×C₆₇	—	4	2	no	—
128	C₄⋊C₄×C₆₇	M₁₆	1	4	yes	—
129	C₄²×C₆₇	—	2	2	no	—
130	C₄²×C₆₇	—	2	2	no	—
131	C₄²×C₆₇	—	2	2	no	—
132	C₄²×C₆₇	—	2	2	no	—
133	C₄²×C₆₇	—	2	2	no	—
134	C₄²×C₆₇	—	2	2	no	—
135	C₄²×C₆₇	—	4	1	no	—
136	C₄²×C₆₇	—	4	1	no	—
137	C₄²×C₆₇	C₂×D₄	1	2	yes	—
138	C₄²×C₆₇	C₂×Q₈	1	2	yes	—
139	C₄²×C₆₇	C₂×Q₈	1	2	yes	—
140	C₄²×C₆₇	#41	1	2	yes	—
141	C₄²×C₆₇	#41	1	2	yes	—
142	C₄²×C₆₇	#41	1	2	yes	—
143	C₂³×C₆₇	C₂³×C₄	1	1	yes	characteristic
144	Q₈×C₆₇	—	4	8	no	—
145	C₂×C₄×C₆₇	—	2	2	no	—
146	C₂×C₄×C₆₇	—	2	1	no	—
147	C₂×C₄×C₆₇	—	2	1	no	—
148	C₂×C₄×C₆₇	—	2	1	no	—
149	C₂×C₄×C₆₇	—	2	1	no	—
150	C₂×C₄×C₆₇	—	2	1	no	—
151	C₂×C₄×C₆₇	—	2	1	no	—
152	C₂×C₄×C₆₇	—	2	1	no	—
153	C₂×C₄×C₆₇	—	2	1	no	—
154	C₂×C₄×C₆₇	—	2	1	no	—
155	C₂×C₄×C₆₇	—	2	1	no	—
156	C₂×C₄×C₆₇	—	2	2	no	—
157	C₂×C₄×C₆₇	—	2	2	no	—
158	C₂×C₄×C₆₇	—	2	2	no	—
159	C₂×C₄×C₆₇	—	2	2	no	—
160	C₂×C₄×C₆₇	—	2	2	no	—
161	C₂×C₄×C₆₇	—	2	2	no	—
162	C₂×C₄×C₆₇	—	2	2	no	—
163	C₂×C₄×C₆₇	—	2	2	no	—
164	C₂×C₄×C₆₇	—	2	2	no	—
165	C₂×C₄×C₆₇	—	2	2	no	—
166	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
167	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
168	C₂×C₄×C₆₇	C₄×Q₈	1	2	yes	—
169	C₂×C₄×C₆₇	C₄×Q₈	1	2	yes	—
170	C₂×C₄×C₆₇	C₄×Q₈	1	2	yes	—
171	C₂×C₄×C₆₇	C₄×Q₈	1	2	yes	—
172	C₂×C₄×C₆₇	C₂×M₁₆	1	4	yes	—
173	C₂×C₄×C₆₇	C₂×M₁₆	1	2	yes	—
174	C₂×C₄×C₆₇	#131	1	2	yes	—
175	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
176	C₂×C₄×C₆₇	C₂²×Q₈	1	1	yes	characteristic
177	C₂×C₄×C₆₇	C₂²×Q₈	1	1	yes	characteristic
178	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
179	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
180	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
181	C₆₇⋊C₈	—	2	2	no	—
182	C₆₇⋊C₈	—	2	2	no	—
183	C₆₇⋊C₈	—	2	2	no	—
184	C₆₇⋊C₈	—	4	2	no	—
185	C₆₇⋊C₈	—	4	2	no	—
186	C₆₇⋊C₈	C₄⋊Q₈	1	4	yes	complemented
187	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
188	C₂²×C₆₇	C₂×C₄×Q₈	1	1	yes	characteristic
189	C₂²×C₆₇	C₂×C₄×Q₈	1	1	yes	characteristic
190	C₂²×C₆₇	C₂²×M₁₆	1	1	yes	characteristic
191	C₂²×C₆₇	C₂×#131	1	1	yes	characteristic
192	C₂²×C₆₇	#568	1	1	yes	characteristic
193	C₂²×C₆₇	#568	1	1	yes	characteristic
194	C₄×C₆₇	—	2	2	no	—
195	C₄×C₆₇	—	2	4	no	—
196	C₄×C₆₇	—	2	2	no	—
197	C₄×C₆₇	—	2	2	no	—
198	C₄×C₆₇	—	2	2	no	—
199	C₄×C₆₇	—	2	2	no	—
200	C₄×C₆₇	—	4	2	no	—
201	C₄×C₆₇	—	4	2	no	—
202	C₄×C₆₇	—	4	2	no	—
203	C₄×C₆₇	#446	1	4	yes	complemented
204	C₄×C₆₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
205	C₄×C₆₇	#532	1	1	yes	characteristic
206	C₄×C₆₇	#533	1	1	yes	characteristic
207	C₄×C₆₇	#533	1	1	yes	characteristic
208	#21713	—	67	1	no	maximal, Sylow
209	C₂×C₆₇	C₄×C₄⋊Q₈	1	1	yes	characteristic
210	C₂×C₆₇	D₄×M₁₆	1	1	yes	characteristic
211	C₂×C₆₇	#2935	1	1	yes	characteristic
212	C₂×C₆₇	C₂×#446	1	1	yes	characteristic
213	C₂×C₆₇	#2961	1	1	yes	characteristic
214	C₂×C₆₇	#2962	1	1	yes	characteristic
215	C₂×C₆₇	Q₈×M₁₆	1	1	yes	characteristic
216	C₂×#446	—	67	2	no	—
217	C₄×C₄⋊Q₈	D₆₇	1	1	yes	characteristic
218	#1985	—	67	2	no	—
219	#1940	—	67	2	no	—
220	#1939	—	67	1	no	—
221	#1939	—	67	1	no	—
222	#1842	—	67	1	no	—
223	#1771	—	67	2	no	—
224	#1770	—	67	1	no	—
225	C₈×C₄⋊C₄	—	67	1	no	—
226	C₄×#97	—	67	1	no	—
227	C₆₇	#21713	1	1	yes	characteristic, complemented, Sylow
228	C₂×C₄⋊Q₈	Dic₆₇	1	1	yes	characteristic
229	C₂×C₄×Q₈	C₂×D₆₇	1	2	yes	—
230	#446	—	134	8	no	—
231	C₂×C₄⋊C₈	—	67	2	no	—
232	C₂×C₄⋊C₈	—	67	1	no	—
233	C₂×C₄⋊C₈	—	67	1	no	—
234	C₂×C₄⋊C₈	—	67	2	no	—
235	C₂×C₄⋊C₈	—	67	2	no	—
236	C₂×C₄⋊C₈	—	67	2	no	—
237	C₂×#97	—	67	1	no	—
238	C₂×#97	—	67	1	no	—
239	C₂×#97	—	67	2	no	—
240	C₂×#97	—	67	1	no	—
241	C₂×#97	—	67	1	no	—
242	C₂×C₄×C₈	—	67	1	no	—
243	C₂×C₄×C₈	—	67	2	no	—
244	C₂×C₄×C₈	—	67	1	no	—
245	#391	Dic₆₇	1	2	yes	—
246	#389	Dic₆₇	1	1	yes	characteristic
247	#389	Dic₆₇	1	1	yes	characteristic
248	#389	Dic₆₇	1	2	yes	—
249	#384	Dic₆₇	1	1	yes	characteristic
250	C₄×C₄⋊C₄	C₂×D₆₇	1	1	yes	characteristic
251	C₄×C₄⋊C₄	C₂×D₆₇	1	1	yes	characteristic
252	C₄×C₄⋊C₄	C₂×D₆₇	1	2	yes	—
253	C₄³	C₂×D₆₇	1	1	yes	characteristic
254	#336	—	67	2	no	—
255	#336	—	67	2	no	—
256	#336	—	67	2	no	—
257	#336	—	67	2	no	—
258	C₂²×Q₈	C₂×Dic₆₇	1	2	yes	—
259	C₂²×C₈	—	67	1	no	—
260	C₂²×C₈	—	67	1	no	—
261	C₂²×C₈	—	67	1	no	—
262	C₂²×C₈	—	67	1	no	—
263	C₂²×C₈	—	67	2	no	—
264	C₂²×C₈	—	67	2	no	—
265	C₄⋊Q₈	—	2	4	no	—
266	C₄×Q₈	—	2	4	no	—
267	C₄×Q₈	—	2	4	no	—
268	C₂×C₄⋊C₄	C₂×Dic₆₇	1	2	yes	—
269	C₂×C₄⋊C₄	C₂×Dic₆₇	1	2	yes	—
270	C₂×C₄⋊C₄	C₂×Dic₆₇	1	2	yes	—
271	C₂×C₄⋊C₄	C₂×Dic₆₇	1	1	yes	characteristic
272	C₂×C₄⋊C₄	C₂×Dic₆₇	1	2	yes	—
273	C₂×C₄⋊C₄	C₂×Dic₆₇	1	1	yes	characteristic
274	C₂×C₄⋊C₄	C₂×Dic₆₇	1	1	yes	characteristic
275	C₂×C₄⋊C₄	C₂×Dic₆₇	1	2	yes	—
276	C₂×C₄⋊C₄	C₂×Dic₆₇	1	1	yes	characteristic
277	C₂×C₄²	C₂×Dic₆₇	1	1	yes	characteristic
278	C₂×C₄²	C₂×Dic₆₇	1	1	yes	characteristic
279	C₂×C₄²	C₂×Dic₆₇	1	1	yes	characteristic
280	C₂×C₄²	C₂×Dic₆₇	1	1	yes	characteristic
281	C₂×C₄²	C₂²×D₆₇	1	2	yes	—
282	C₂×C₄²	C₂²×D₆₇	1	2	yes	—
283	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
284	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
285	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
286	C₄⋊C₈	—	134	4	no	—
287	C₄⋊C₈	—	134	4	no	—
288	C₄⋊C₈	—	134	4	no	—
289	C₄⋊C₈	—	134	2	no	—
290	C₄⋊C₈	—	134	2	no	—
291	C₄⋊C₈	—	134	4	no	—
292	#97	—	67	4	no	—
293	#97	—	67	4	no	—
294	#97	—	134	4	no	—
295	#97	—	134	2	no	—
296	#97	—	134	2	no	—
297	C₄×C₈	—	67	4	no	—
298	C₄×C₈	—	134	4	no	—
299	C₄×C₈	—	134	2	no	—
300	#95	C₂×Dic₆₇	1	2	yes	—
301	#95	C₂×Dic₆₇	1	2	yes	—
302	#95	C₂×Dic₆₇	1	2	yes	—
303	#95	C₂×Dic₆₇	1	2	yes	—
304	C₂×Q₈	—	2	4	no	—
305	C₂×Q₈	—	2	4	no	—
306	C₂×Q₈	—	4	2	no	—
307	C₂²×C₄	C₂²×Dic₆₇	1	2	yes	—
308	C₂²×C₄	C₂²×Dic₆₇	1	2	yes	—
309	C₂²×C₄	C₂²×Dic₆₇	1	2	yes	—
310	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
311	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
312	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
313	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
314	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
315	C₂²×C₄	C₂²×Dic₆₇	1	2	yes	—
316	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
317	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
318	C₂×C₈	—	67	2	no	—
319	C₂×C₈	—	67	2	no	—
320	C₂×C₈	—	67	2	no	—
321	C₂×C₈	—	67	2	no	—
322	C₂×C₈	—	67	2	no	—
323	C₂×C₈	—	67	2	no	—
324	C₂×C₈	—	134	2	no	—
325	C₂×C₈	—	134	2	no	—
326	C₂×C₈	—	134	2	no	—
327	C₂×C₈	—	134	2	no	—
328	C₂×C₈	—	134	2	no	—
329	C₂×C₈	—	134	2	no	—
330	C₂×C₈	—	134	1	no	—
331	C₂×C₈	—	134	1	no	—
332	C₂×C₈	—	134	1	no	—
333	C₂×C₈	—	134	1	no	—
334	C₂×C₈	—	134	1	no	—
335	C₂×C₈	—	134	1	no	—
336	C₄⋊C₄	—	2	4	no	—
337	C₄⋊C₄	—	2	4	no	—
338	C₄⋊C₄	—	2	4	no	—
339	C₄⋊C₄	—	2	4	no	—
340	C₄⋊C₄	—	2	2	no	—
341	C₄⋊C₄	—	2	2	no	—
342	C₄⋊C₄	—	2	2	no	—
343	C₄⋊C₄	—	4	2	no	—
344	C₄⋊C₄	#11761612	1	4	yes	—
345	C₄²	—	2	2	no	—
346	C₄²	—	2	2	no	—
347	C₄²	—	2	2	no	—
348	C₄²	—	2	2	no	—
349	C₄²	—	2	2	no	—
350	C₄²	—	2	2	no	—
351	C₄²	—	4	1	no	—
352	C₄²	—	4	1	no	—
353	C₄²	D₄×D₆₇	1	2	yes	—
354	C₄²	Q₈×D₆₇	1	2	yes	—
355	C₄²	Q₈×D₆₇	1	2	yes	—
356	C₄²	#11761627	1	2	yes	—
357	C₄²	#11761629	1	2	yes	—
358	C₄²	#11761629	1	2	yes	—
359	C₂³	C₂³×Dic₆₇	1	1	yes	characteristic
360	Q₈	—	4	8	no	—
361	C₂×C₄	—	2	2	no	—
362	C₂×C₄	—	2	1	no	—
363	C₂×C₄	—	2	1	no	—
364	C₂×C₄	—	2	1	no	—
365	C₂×C₄	—	2	1	no	—
366	C₂×C₄	—	2	1	no	—
367	C₂×C₄	—	2	1	no	—
368	C₂×C₄	—	2	1	no	—
369	C₂×C₄	—	2	1	no	—
370	C₂×C₄	—	2	1	no	—
371	C₂×C₄	—	2	1	no	—
372	C₂×C₄	—	2	2	no	—
373	C₂×C₄	—	2	2	no	—
374	C₂×C₄	—	2	2	no	—
375	C₂×C₄	—	2	2	no	—
376	C₂×C₄	—	2	2	no	—
377	C₂×C₄	—	2	2	no	—
378	C₂×C₄	—	2	2	no	—
379	C₂×C₄	—	2	2	no	—
380	C₂×C₄	—	2	2	no	—
381	C₂×C₄	—	2	2	no	—
382	C₂×C₄	D₄×Dic₆₇	1	2	yes	—
383	C₂×C₄	D₄×Dic₆₇	1	2	yes	—
384	C₂×C₄	Q₈×Dic₆₇	1	2	yes	—
385	C₂×C₄	Q₈×Dic₆₇	1	2	yes	—
386	C₂×C₄	Q₈×Dic₆₇	1	2	yes	—
387	C₂×C₄	Q₈×Dic₆₇	1	2	yes	—
388	C₂×C₄	C₂×#11761612	1	4	yes	—
389	C₂×C₄	C₂×#11761612	1	2	yes	—
390	C₂×C₄	#14524050	1	2	yes	—
391	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
392	C₂×C₄	C₂×Q₈×D₆₇	1	1	yes	characteristic
393	C₂×C₄	C₂×Q₈×D₆₇	1	1	yes	characteristic
394	C₂×C₄	C₂×#11761627	1	1	yes	characteristic
395	C₂×C₄	C₂×#11761629	1	1	yes	characteristic
396	C₂×C₄	C₂×#11761629	1	1	yes	characteristic
397	C₈	—	268	2	no	—
398	C₈	—	268	2	no	—
399	C₈	—	67	4	no	—
400	C₈	—	134	2	no	—
401	C₈	—	134	2	no	—
402	C₈	—	134	2	no	—
403	C₂²	C₂×D₄×Dic₆₇	1	1	yes	characteristic
404	C₂²	C₂×Q₈×Dic₆₇	1	1	yes	characteristic
405	C₂²	C₂×Q₈×Dic₆₇	1	1	yes	characteristic
406	C₂²	C₂²×#11761612	1	1	yes	characteristic
407	C₂²	C₂×#14524050	1	1	yes	characteristic
408	C₂²	#16864163	1	1	yes	characteristic
409	C₂²	#16864163	1	1	yes	characteristic
410	C₄	—	2	2	no	—
411	C₄	—	2	2	no	—
412	C₄	—	2	2	no	—
413	C₄	—	2	2	no	—
414	C₄	—	2	4	no	—
415	C₄	—	2	2	no	—
416	C₄	—	4	2	no	—
417	C₄	—	4	2	no	—
418	C₄	—	4	2	no	—
419	C₄	#16863527	1	4	yes	complemented
420	C₄	C₄⋊Q₈×D₆₇	1	1	yes	characteristic
421	C₄	#16863947	1	1	yes	characteristic
422	C₄	#16863951	1	1	yes	characteristic
423	C₄	#16863951	1	1	yes	characteristic
424	C₂	C₄⋊Q₈×Dic₆₇	1	1	yes	characteristic
425	C₂	D₄×#11761612	1	1	yes	characteristic
426	C₂	#22568953	1	1	yes	characteristic
427	C₂	C₂×#16863527	1	1	yes	characteristic
428	C₂	#22569150	1	1	yes	characteristic
429	C₂	#22569158	1	1	yes	characteristic
430	C₂	Q₈×#11761612	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₆₇
commutator, socle, prime radical	C₂³×C₆₇	C₂³×C₄
Fitting	C₄×C₄⋊Q₈×C₆₇	C₂
cosocle	C₂²×C₄×C₆₇	C₂⁴
hypercenter	C₄×C₄⋊Q₈	D₆₇
hypocenter	C₆₇	#21713
LCS	G ▷ C₂³×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₄×C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
58. #14523954	Q₈	semidirect
61. C₄×C₆₇⋊C₈	Q₈	semidirect
186. C₆₇⋊C₈	C₄⋊Q₈	semidirect
203. C₄×C₆₇	#446	semidirect
227. C₆₇	#21713	semidirect
419. C₄	#16863527	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16863527	2
#22561150	2
#22560933	2
#22560928	2
#22560361	2
#22560084	2
#22560078	2
C₄⋊C₄×C₆₇⋊C₈	2
C₄×#14523954	2
C₄×C₄⋊Q₈×C₆₇	2
#21713	67