Subgroups — #11897607

Order 1152.

510 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₈²×C₉	C₂	1	1	yes	maximal, characteristic, complemented
2	#10589681	C₂	1	1	yes	maximal, characteristic
3	#10588772	C₂	1	2	yes	maximal
4	#10588769	C₂	1	1	yes	maximal, characteristic
5	#10588761	C₂	1	2	yes	maximal, complemented
6	#10588758	C₂	1	1	yes	maximal, characteristic, complemented
7	#10588709	C₂	1	2	yes	maximal
8	C₄×#3675	C₂	1	1	yes	maximal, characteristic
9	#10588704	C₂	1	2	yes	maximal, complemented
10	#10588703	C₂	1	1	yes	maximal, characteristic, complemented
11	#10588701	C₂	1	1	yes	maximal, characteristic, complemented
12	#83706	—	3	1	no	maximal
13	C₉×C₄⋊Q₈	—	2	1	no	—
14	C₉×C₄⋊Q₈	—	2	2	no	—
15	C₉×C₄⋊Q₈	C₂²	1	1	yes	characteristic
16	C₉×C₄⋊Q₈	C₂²	1	2	yes	—
17	C₄×Q₈×C₉	—	2	1	no	—
18	C₄×Q₈×C₉	C₂²	1	1	yes	characteristic
19	C₄×Q₈×C₉	C₂²	1	2	yes	—
20	C₄×Q₈×C₉	C₂²	1	1	yes	characteristic
21	#63508	—	2	1	no	—
22	C₂×#3675	C₂²	1	1	yes	characteristic
23	#63458	—	2	2	no	—
24	C₄⋊C₄×D₉	—	2	1	no	—
25	C₄×D₃₆	C₂²	1	1	yes	characteristic
26	C₄×D₃₆	C₂²	1	2	yes	—
27	#63432	C₂²	1	2	yes	—
28	#63431	C₂²	1	1	yes	characteristic
29	#63395	C₂²	1	2	yes	—
30	#63395	C₂²	1	1	yes	characteristic
31	#63369	C₂²	1	1	yes	characteristic
32	#63369	C₂²	1	2	yes	—
33	#63367	C₂²	1	2	yes	—
34	#63367	C₂²	1	1	yes	characteristic
35	#63366	C₂²	1	2	yes	—
36	#63366	C₂²	1	2	yes	—
37	#63366	C₂²	1	2	yes	—
38	C₄⋊(C₉⋊C₈)	C₂²	1	1	yes	characteristic
39	C₄⋊(C₉⋊C₈)	C₂²	1	2	yes	—
40	#63362	C₂²	1	2	yes	—
41	C₄×C₉⋊C₈	C₂²	1	1	yes	characteristic
42	C₃×Q₈²	D₃	1	1	yes	characteristic
43	#5579	—	3	1	no	—
44	#5087	—	3	2	no	—
45	#5084	—	3	1	no	—
46	#5081	—	3	2	no	—
47	#5078	—	3	1	no	—
48	#5026	—	3	2	no	—
49	C₄×#215	—	3	1	no	—
50	#5024	—	3	1	no	—
51	#5022	—	3	2	no	—
52	#5021	—	3	1	no	—
53	C₂×Q₈×C₉	—	2	2	no	—
54	C₂×Q₈×C₉	—	2	1	no	—
55	C₂×Q₈×C₉	D₄	1	1	yes	characteristic
56	C₂×Q₈×C₉	C₂³	1	1	yes	characteristic
57	C₂×D₃₆	C₂³	1	1	yes	characteristic
58	C₂×C₄×D₉	—	2	1	no	—
59	C₂×C₄×D₉	—	2	2	no	—
60	C₂×Dic₁₈	—	2	1	no	—
61	C₉×C₄⋊C₄	—	2	2	no	—
62	C₉×C₄⋊C₄	—	2	1	no	—
63	C₉×C₄⋊C₄	—	2	2	no	—
64	C₉×C₄⋊C₄	—	2	1	no	—
65	C₉×C₄⋊C₄	D₄	1	1	yes	characteristic
66	C₉×C₄⋊C₄	D₄	1	2	yes	—
67	C₉×C₄⋊C₄	C₂³	1	1	yes	characteristic
68	C₉×C₄⋊C₄	C₂³	1	2	yes	—
69	C₄²×C₉	—	2	2	no	—
70	C₄²×C₉	—	2	1	no	—
71	C₄²×C₉	C₂³	1	1	yes	characteristic
72	C₄²×C₉	C₂³	1	2	yes	—
73	#3675	Q₈	1	4	yes	complemented
74	(C₂×D₉)⋊C₄	—	2	2	no	—
75	(C₂×D₉)⋊C₄	—	2	1	no	—
76	C₄⋊Dic₉	—	2	1	no	—
77	C₄⋊Dic₉	C₂³	1	2	yes	—
78	C₄⋊Dic₉	C₂³	1	1	yes	characteristic
79	Dic₉⋊C₄	—	2	2	no	—
80	Dic₉⋊C₄	—	2	2	no	—
81	C₂×C₉⋊C₈	C₂³	1	1	yes	characteristic
82	C₂×C₉⋊C₈	C₂³	1	2	yes	—
83	C₂×C₉⋊C₈	C₂³	1	1	yes	characteristic
84	#3385	—	9	1	no	Sylow
85	C₃×C₄⋊Q₈	—	2	1	no	—
86	C₃×C₄⋊Q₈	—	2	2	no	—
87	C₃×C₄⋊Q₈	C₂×D₃	1	1	yes	characteristic
88	C₃×C₄⋊Q₈	C₂×D₃	1	2	yes	—
89	C₃×C₄×Q₈	—	2	1	no	—
90	C₃×C₄×Q₈	C₂×D₃	1	1	yes	characteristic
91	C₃×C₄×Q₈	C₂×D₃	1	2	yes	—
92	C₃×C₄×Q₈	C₂×D₃	1	1	yes	characteristic
93	#946	—	6	1	no	—
94	C₂×#215	—	3	1	no	—
95	#896	—	6	2	no	—
96	D₃×C₄⋊C₄	—	6	1	no	—
97	C₄×D₁₂	—	3	1	no	—
98	C₄×D₁₂	—	3	2	no	—
99	#870	—	3	2	no	—
100	#869	—	3	1	no	—
101	#835	—	3	2	no	—
102	#835	—	3	1	no	—
103	#809	—	3	1	no	—
104	#809	—	3	2	no	—
105	#808	—	3	2	no	—
106	#808	—	3	1	no	—
107	#807	—	3	2	no	—
108	#807	—	3	2	no	—
109	#807	—	3	2	no	—
110	C₄⋊(C₃⋊C₈)	—	3	1	no	—
111	C₄⋊(C₃⋊C₈)	—	3	2	no	—
112	#803	—	3	2	no	—
113	C₄×C₃⋊C₈	—	3	1	no	—
114	C₂²×D₉	—	2	1	no	—
115	Q₈×C₉	—	2	2	no	—
116	Q₈×C₉	—	2	1	no	—
117	Q₈×C₉	—	8	1	no	—
118	Q₈×C₉	—	8	2	no	—
119	Q₈×C₉	C₂×Q₈	1	2	yes	—
120	C₂×C₄×C₉	—	2	1	no	—
121	C₂×C₄×C₉	—	2	2	no	—
122	C₂×C₄×C₉	—	2	1	no	—
123	C₂×C₄×C₉	C₂×D₄	1	2	yes	—
124	C₂×C₄×C₉	C₂×D₄	1	2	yes	—
125	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
126	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
127	C₂×C₄×C₉	C₂⁴	1	1	yes	characteristic
128	C₂×Dic₉	—	2	1	no	—
129	C₂×Dic₉	—	2	2	no	—
130	D₃₆	—	2	1	no	—
131	D₃₆	C₂×Q₈	1	2	yes	—
132	C₄×D₉	—	4	2	no	—
133	C₄×D₉	—	8	2	no	—
134	Dic₁₈	—	8	1	no	—
135	C₉⋊C₈	—	2	1	no	—
136	C₉⋊C₈	—	2	2	no	—
137	C₉⋊C₈	C₂×Q₈	1	2	yes	—
138	Q₈²	D₉	1	1	yes	characteristic, complemented
139	#554	—	9	1	no	—
140	#501	—	9	2	no	—
141	#500	—	9	1	no	—
142	#479	—	9	2	no	—
143	#477	—	9	1	no	—
144	#476	—	9	1	no	—
145	#474	—	9	2	no	—
146	#446	—	9	1	no	—
147	#440	—	9	2	no	—
148	C₄×SD₁₆	—	9	1	no	—
149	C₂×C₃×Q₈	—	2	2	no	—
150	C₂×C₃×Q₈	—	2	1	no	—
151	C₂×C₃×Q₈	#67	1	1	yes	characteristic
152	C₂×C₃×Q₈	C₂²×D₃	1	1	yes	characteristic
153	C₂×D₁₂	—	3	1	no	—
154	C₂×C₄×D₃	—	6	1	no	—
155	C₂×C₄×D₃	—	6	2	no	—
156	C₂×Dic₆	—	6	1	no	—
157	C₃×C₄⋊C₄	—	2	2	no	—
158	C₃×C₄⋊C₄	—	2	1	no	—
159	C₃×C₄⋊C₄	—	2	2	no	—
160	C₃×C₄⋊C₄	—	2	1	no	—
161	C₃×C₄⋊C₄	#67	1	1	yes	characteristic
162	C₃×C₄⋊C₄	#67	1	2	yes	—
163	C₃×C₄⋊C₄	C₂²×D₃	1	1	yes	characteristic
164	C₃×C₄⋊C₄	C₂²×D₃	1	2	yes	—
165	C₃×C₄²	—	2	2	no	—
166	C₃×C₄²	—	2	1	no	—
167	C₃×C₄²	C₂²×D₃	1	1	yes	characteristic
168	C₃×C₄²	C₂²×D₃	1	2	yes	—
169	#215	—	3	4	no	—
170	(C₂×D₃)⋊C₄	—	6	1	no	—
171	(C₂×D₃)⋊C₄	—	6	2	no	—
172	C₄⋊Dic₃	—	3	2	no	—
173	C₄⋊Dic₃	—	3	1	no	—
174	C₄⋊Dic₃	—	6	1	no	—
175	Dic₃⋊C₄	—	6	2	no	—
176	Dic₃⋊C₄	—	6	2	no	—
177	C₂×C₃⋊C₈	—	3	1	no	—
178	C₂×C₃⋊C₈	—	3	1	no	—
179	C₂×C₃⋊C₈	—	3	2	no	—
180	C₂²×C₉	C₂²×D₄	1	1	yes	characteristic
181	C₂×D₉	—	2	2	no	—
182	C₂×D₉	—	4	1	no	—
183	C₂×D₉	—	4	1	no	—
184	C₄×C₉	—	2	2	no	—
185	C₄×C₉	—	2	2	no	—
186	C₄×C₉	—	2	1	no	—
187	C₄×C₉	—	2	2	no	—
188	C₄×C₉	—	4	2	no	—
189	C₄×C₉	—	4	1	no	—
190	C₄×C₉	#137	1	2	yes	—
191	C₄×C₉	C₂²×Q₈	1	1	yes	characteristic
192	C₄×C₉	2₋¹⁺⁴	1	1	yes	characteristic
193	Dic₉	—	4	1	no	—
194	Dic₉	—	4	2	no	—
195	C₂×SD₁₆	—	9	1	no	—
196	C₄⋊Q₈	—	2	1	no	—
197	C₄⋊Q₈	—	2	2	no	—
198	C₄⋊Q₈	—	9	1	no	—
199	C₄⋊Q₈	C₂×D₉	1	2	yes	—
200	C₄⋊Q₈	C₂×D₉	1	1	yes	characteristic
201	#125	—	9	2	no	—
202	C₂²⋊Q₈	—	18	1	no	—
203	C₂²⋊Q₈	—	18	2	no	—
204	C₄×Q₈	—	2	1	no	—
205	C₄×Q₈	C₂×D₉	1	1	yes	characteristic
206	C₄×Q₈	C₂×D₉	1	2	yes	—
207	C₄×Q₈	C₂×D₉	1	1	yes	characteristic
208	C₄×D₄	—	9	1	no	—
209	C₄×D₄	—	9	2	no	—
210	C₂×C₄⋊C₄	—	18	1	no	—
211	#107	—	9	2	no	—
212	#107	—	9	2	no	—
213	#107	—	9	2	no	—
214	#106	—	9	1	no	—
215	#106	—	9	2	no	—
216	C₄⋊C₈	—	9	2	no	—
217	C₄⋊C₈	—	9	1	no	—
218	#103	—	9	1	no	—
219	#103	—	9	2	no	—
220	#102	—	9	1	no	—
221	#102	—	9	2	no	—
222	#97	—	9	2	no	—
223	C₄×C₈	—	9	1	no	—
224	C₂²×D₃	—	6	1	no	—
225	C₃×Q₈	—	2	2	no	—
226	C₃×Q₈	—	2	1	no	—
227	C₃×Q₈	—	8	1	no	—
228	C₃×Q₈	—	8	2	no	—
229	C₃×Q₈	D₃×Q₈	1	2	yes	—
230	C₂×C₃×C₄	—	2	1	no	—
231	C₂×C₃×C₄	—	2	2	no	—
232	C₂×C₃×C₄	—	2	1	no	—
233	C₂×C₃×C₄	C₂×#67	1	2	yes	—
234	C₂×C₃×C₄	C₂×#67	1	2	yes	—
235	C₂×C₃×C₄	C₂×#67	1	1	yes	characteristic
236	C₂×C₃×C₄	C₂×#67	1	1	yes	characteristic
237	C₂×C₃×C₄	C₂³×D₃	1	1	yes	characteristic
238	C₂×Dic₃	—	6	1	no	—
239	C₂×Dic₃	—	6	2	no	—
240	D₁₂	—	3	2	no	—
241	D₁₂	—	6	1	no	—
242	C₄×D₃	—	12	2	no	—
243	C₄×D₃	—	24	2	no	—
244	Dic₆	—	24	1	no	—
245	C₃⋊C₈	—	3	2	no	—
246	C₃⋊C₈	—	6	1	no	—
247	C₃⋊C₈	—	6	2	no	—
248	C₂×C₉	D₄×Q₈	1	1	yes	characteristic
249	C₂×C₉	C₂×#137	1	1	yes	characteristic
250	C₂×C₉	#576	1	1	yes	characteristic
251	D₉	—	4	2	no	—
252	C₂×Q₈	—	2	1	no	—
253	C₂×Q₈	—	2	2	no	—
254	C₂×Q₈	—	18	1	no	—
255	C₂×Q₈	#611	1	1	yes	characteristic
256	C₂×Q₈	C₂²×D₉	1	1	yes	characteristic
257	C₂×D₄	—	9	1	no	—
258	C₂²×C₄	—	18	1	no	—
259	C₂²×C₄	—	18	2	no	—
260	SD₁₆	—	9	4	no	—
261	C₂×C₈	—	9	1	no	—
262	C₂×C₈	—	9	1	no	—
263	C₂×C₈	—	9	2	no	—
264	C₄⋊C₄	—	2	2	no	—
265	C₄⋊C₄	—	2	1	no	—
266	C₄⋊C₄	—	2	1	no	—
267	C₄⋊C₄	—	2	2	no	—
268	C₄⋊C₄	—	9	1	no	—
269	C₄⋊C₄	—	9	2	no	—
270	C₄⋊C₄	—	18	2	no	—
271	C₄⋊C₄	—	18	1	no	—
272	C₄⋊C₄	—	18	2	no	—
273	C₄⋊C₄	#611	1	1	yes	characteristic
274	C₄⋊C₄	#611	1	2	yes	—
275	C₄⋊C₄	C₂²×D₉	1	1	yes	characteristic
276	C₄⋊C₄	C₂²×D₉	1	2	yes	—
277	C₂²⋊C₄	—	18	1	no	—
278	C₂²⋊C₄	—	18	2	no	—
279	C₄²	—	2	2	no	—
280	C₄²	—	2	1	no	—
281	C₄²	C₂²×D₉	1	1	yes	characteristic
282	C₄²	C₂²×D₉	1	2	yes	—
283	C₂²×C₃	C₂²×#67	1	1	yes	characteristic
284	C₂×D₃	—	6	2	no	—
285	C₂×D₃	—	12	1	no	—
286	C₂×D₃	—	12	1	no	—
287	C₃×C₄	—	2	2	no	—
288	C₃×C₄	—	2	2	no	—
289	C₃×C₄	—	2	1	no	—
290	C₃×C₄	—	2	2	no	—
291	C₃×C₄	—	4	2	no	—
292	C₃×C₄	—	4	1	no	—
293	C₃×C₄	#942	1	2	yes	—
294	C₃×C₄	C₂×D₃×Q₈	1	1	yes	characteristic
295	C₃×C₄	#1007	1	1	yes	characteristic
296	Dic₃	—	12	1	no	—
297	Dic₃	—	12	2	no	—
298	C₉	#3385	1	1	yes	characteristic, complemented, Sylow
299	C₂³	—	18	1	no	—
300	Q₈	—	2	2	no	—
301	Q₈	—	2	1	no	—
302	Q₈	—	8	2	no	—
303	Q₈	—	8	1	no	—
304	Q₈	—	72	1	no	—
305	Q₈	Q₈×D₉	1	2	yes	—
306	D₄	—	9	2	no	—
307	D₄	—	18	1	no	—
308	C₂×C₄	—	2	1	no	—
309	C₂×C₄	—	2	2	no	—
310	C₂×C₄	—	2	1	no	—
311	C₂×C₄	—	18	1	no	—
312	C₂×C₄	—	18	2	no	—
313	C₂×C₄	—	36	2	no	—
314	C₂×C₄	—	72	2	no	—
315	C₂×C₄	C₂×#611	1	2	yes	—
316	C₂×C₄	C₂×#611	1	1	yes	characteristic
317	C₂×C₄	C₂×#611	1	2	yes	—
318	C₂×C₄	C₂×#611	1	1	yes	characteristic
319	C₂×C₄	C₂³×D₉	1	1	yes	characteristic
320	C₈	—	9	2	no	—
321	C₈	—	18	1	no	—
322	C₈	—	18	2	no	—
323	C₂×C₃	C₂×#942	1	1	yes	characteristic
324	C₂×C₃	Q₈×#67	1	1	yes	characteristic
325	C₂×C₃	#5835	1	1	yes	characteristic
326	D₃	—	12	2	no	—
327	C₂²	—	18	2	no	—
328	C₂²	—	36	1	no	—
329	C₂²	—	36	1	no	—
330	C₂²	C₂²×#611	1	1	yes	characteristic
331	C₄	—	2	2	no	—
332	C₄	—	2	2	no	—
333	C₄	—	2	2	no	—
334	C₄	—	2	1	no	—
335	C₄	—	4	1	no	—
336	C₄	—	4	2	no	—
337	C₄	—	36	1	no	—
338	C₄	—	36	2	no	—
339	C₄	#63505	1	2	yes	—
340	C₄	C₂×Q₈×D₉	1	1	yes	characteristic
341	C₄	#63713	1	1	yes	characteristic
342	C₃	#83706	1	1	yes	characteristic
343	C₂	—	36	2	no	—
344	C₂	C₂×#63505	1	1	yes	characteristic
345	C₂	Q₈×#611	1	1	yes	characteristic
346	C₂	#10589937	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×#611
commutator, cosocle	C₂×C₄×C₉	C₂⁴
Frattini	C₂×C₃×C₄	C₂³×D₃
Fitting	Q₈²×C₉	C₂
socle	C₂²×C₃	C₂²×#67
prime radical	C₂×D₃₆	C₂³
hypercenter	Q₈²	D₉
hypocenter	C₉	#3385
LCS	G ▷ C₂×C₄×C₉ ▷ C₂×C₉ ▷ C₉
UCS	C₁ ◁ C₂² ◁ C₄×Q₈ ◁ Q₈²

Complements

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

N	H ≅ G/N	type
1. Q₈²×C₉	C₂	semidirect
5. #10588761	C₂	semidirect
6. #10588758	C₂	semidirect
9. #10588704	C₂	semidirect
10. #10588703	C₂	semidirect
11. #10588701	C₂	semidirect
73. #3675	Q₈	semidirect
138. Q₈²	D₉	semidirect
298. C₉	#3385	semidirect

Maximal subgroups

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

subgroup	index
Q₈²×C₉	2
#10589681	2
#10588772	2
#10588769	2
#10588761	2
#10588758	2
#10588709	2
C₄×#3675	2
#10588704	2
#10588703	2
#10588701	2
#83706	3