Subgroups — #11896024

Order 1152.

507 conjugacy classes of subgroups.

The subgroup lattice has 351 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	#10589686	C₂	1	1	yes	maximal, characteristic
3	#10588774	C₂	1	2	yes	maximal
4	#10588769	C₂	1	1	yes	maximal, characteristic
5	#10588767	C₂	1	2	yes	maximal, complemented
6	#10588766	C₂	1	1	yes	maximal, characteristic, complemented
7	#10588711	C₂	1	2	yes	maximal
8	C₄×#3675	C₂	1	1	yes	maximal, characteristic
9	#10588707	C₂	1	1	yes	maximal, characteristic, complemented
10	#10588706	C₂	1	2	yes	maximal, complemented
11	#10588701	C₂	1	1	yes	maximal, characteristic, complemented
12	#83704	—	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	2	yes	—
19	C₄×Q₈×C₉	C₂²	1	1	yes	characteristic
20	C₄×Q₈×C₉	C₂²	1	1	yes	characteristic
21	#63508	—	2	1	no	—
22	C₂×#3675	C₂²	1	1	yes	characteristic
23	C₂×#3674	C₂²	1	2	yes	—
24	#63460	—	2	2	no	—
25	#63454	—	2	1	no	—
26	#63437	C₂²	1	2	yes	—
27	C₄×D₃₆	C₂²	1	1	yes	characteristic
28	#63431	C₂²	1	1	yes	characteristic
29	C₄×Dic₁₈	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	#63368	C₂²	1	2	yes	—
34	#63368	C₂²	1	2	yes	—
35	#63368	C₂²	1	2	yes	—
36	#63367	C₂²	1	2	yes	—
37	#63367	C₂²	1	1	yes	characteristic
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	#5584	—	3	1	no	—
44	#5094	—	3	2	no	—
45	#5093	—	3	1	no	—
46	#5089	—	3	2	no	—
47	#5084	—	3	1	no	—
48	#5031	—	3	1	no	—
49	#5030	—	3	2	no	—
50	#5028	—	3	2	no	—
51	C₄×#215	—	3	1	no	—
52	#5024	—	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×Dic₁₈	C₂³	1	2	yes	—
60	C₉×C₄⋊C₄	—	2	2	no	—
61	C₉×C₄⋊C₄	—	2	1	no	—
62	C₉×C₄⋊C₄	—	2	2	no	—
63	C₉×C₄⋊C₄	—	2	1	no	—
64	C₉×C₄⋊C₄	D₄	1	1	yes	characteristic
65	C₉×C₄⋊C₄	D₄	1	2	yes	—
66	C₉×C₄⋊C₄	C₂³	1	2	yes	—
67	C₉×C₄⋊C₄	C₂³	1	1	yes	characteristic
68	C₄²×C₉	—	2	2	no	—
69	C₄²×C₉	—	2	1	no	—
70	C₄²×C₉	C₂³	1	1	yes	characteristic
71	C₄²×C₉	C₂³	1	2	yes	—
72	#3675	—	2	2	no	—
73	#3674	—	2	4	no	—
74	(C₂×D₉)⋊C₄	—	2	2	no	—
75	(C₂×D₉)⋊C₄	—	2	1	no	—
76	(C₂×D₉)⋊C₄	—	2	2	no	—
77	C₄⋊Dic₉	—	2	1	no	—
78	C₄⋊Dic₉	—	2	1	no	—
79	C₄⋊Dic₉	C₂³	1	1	yes	characteristic
80	Dic₉⋊C₄	—	2	2	no	—
81	C₄×Dic₉	—	2	2	no	—
82	C₂×C₉⋊C₈	C₂³	1	2	yes	—
83	C₂×C₉⋊C₈	C₂³	1	1	yes	characteristic
84	C₂×C₉⋊C₈	C₂³	1	1	yes	characteristic
85	#3369	—	9	1	no	Sylow
86	C₃×C₄⋊Q₈	—	2	1	no	—
87	C₃×C₄⋊Q₈	—	2	2	no	—
88	C₃×C₄⋊Q₈	C₂×D₃	1	1	yes	characteristic
89	C₃×C₄⋊Q₈	C₂×D₃	1	2	yes	—
90	C₃×C₄×Q₈	—	2	1	no	—
91	C₃×C₄×Q₈	C₂×D₃	1	1	yes	characteristic
92	C₃×C₄×Q₈	C₂×D₃	1	2	yes	—
93	C₃×C₄×Q₈	C₂×D₃	1	1	yes	characteristic
94	#946	—	6	1	no	—
95	C₂×#216	—	3	2	no	—
96	C₂×#215	—	3	1	no	—
97	#898	—	6	2	no	—
98	#892	—	6	1	no	—
99	#875	—	3	2	no	—
100	C₄×D₁₂	—	3	1	no	—
101	#869	—	3	1	no	—
102	C₄×Dic₆	—	3	2	no	—
103	#835	—	3	1	no	—
104	#810	—	3	2	no	—
105	#810	—	3	2	no	—
106	#810	—	3	2	no	—
107	#809	—	3	1	no	—
108	#809	—	3	2	no	—
109	#808	—	3	1	no	—
110	#808	—	3	2	no	—
111	C₄⋊(C₃⋊C₈)	—	3	2	no	—
112	C₄⋊(C₃⋊C₈)	—	3	1	no	—
113	#803	—	3	2	no	—
114	C₄×C₃⋊C₈	—	3	1	no	—
115	C₂²×D₉	—	2	1	no	—
116	Q₈×C₉	—	2	2	no	—
117	Q₈×C₉	—	2	1	no	—
118	Q₈×C₉	—	8	2	no	—
119	Q₈×C₉	—	8	1	no	—
120	Q₈×C₉	#41	1	2	yes	—
121	C₂×C₄×C₉	—	2	1	no	—
122	C₂×C₄×C₉	—	2	2	no	—
123	C₂×C₄×C₉	—	2	1	no	—
124	C₂×C₄×C₉	C₂×D₄	1	2	yes	—
125	C₂×C₄×C₉	C₂×D₄	1	2	yes	—
126	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
127	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
128	C₂×C₄×C₉	C₂⁴	1	1	yes	characteristic
129	C₂×Dic₉	—	2	1	no	—
130	C₂×Dic₉	—	2	2	no	—
131	D₃₆	—	2	1	no	—
132	D₃₆	—	2	1	no	—
133	C₄×D₉	—	4	2	no	—
134	Dic₁₈	—	2	2	no	—
135	Dic₁₈	—	2	2	no	—
136	C₉⋊C₈	—	2	2	no	—
137	C₉⋊C₈	—	2	1	no	—
138	C₉⋊C₈	—	2	1	no	—
139	Q₈²	D₉	1	1	yes	characteristic, complemented
140	#556	—	9	1	no	—
141	#489	—	9	2	no	—
142	#487	—	9	1	no	—
143	#476	—	9	1	no	—
144	#475	—	9	2	no	—
145	#464	—	9	2	no	—
146	#462	—	9	1	no	—
147	#446	—	9	1	no	—
148	#441	—	9	2	no	—
149	C₄×SD₁₆	—	9	1	no	—
150	C₂×C₃×Q₈	—	2	2	no	—
151	C₂×C₃×Q₈	—	2	1	no	—
152	C₂×C₃×Q₈	#67	1	1	yes	characteristic
153	C₂×C₃×Q₈	C₂²×D₃	1	1	yes	characteristic
154	C₂×D₁₂	—	3	1	no	—
155	C₂×C₄×D₃	—	6	1	no	—
156	C₂×Dic₆	—	3	2	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	#216	—	6	4	no	—
170	#215	—	6	2	no	—
171	(C₂×D₃)⋊C₄	—	6	1	no	—
172	(C₂×D₃)⋊C₄	—	6	2	no	—
173	(C₂×D₃)⋊C₄	—	6	2	no	—
174	C₄⋊Dic₃	—	3	1	no	—
175	C₄⋊Dic₃	—	6	1	no	—
176	C₄⋊Dic₃	—	6	1	no	—
177	Dic₃⋊C₄	—	6	2	no	—
178	C₄×Dic₃	—	6	2	no	—
179	C₂×C₃⋊C₈	—	3	1	no	—
180	C₂×C₃⋊C₈	—	3	1	no	—
181	C₂×C₃⋊C₈	—	3	2	no	—
182	C₂²×C₉	C₂²×D₄	1	1	yes	characteristic
183	C₂×D₉	—	4	1	no	—
184	C₂×D₉	—	4	1	no	—
185	C₂×D₉	—	4	1	no	—
186	C₄×C₉	—	2	2	no	—
187	C₄×C₉	—	2	2	no	—
188	C₄×C₉	—	2	2	no	—
189	C₄×C₉	—	2	1	no	—
190	C₄×C₉	—	4	2	no	—
191	C₄×C₉	—	4	1	no	—
192	C₄×C₉	#137	1	2	yes	—
193	C₄×C₉	C₂×#41	1	1	yes	characteristic
194	C₄×C₉	2₋¹⁺⁴	1	1	yes	characteristic
195	Dic₉	—	4	2	no	—
196	Dic₉	—	4	1	no	—
197	C₂×Q₁₆	—	9	2	no	—
198	C₂×SD₁₆	—	9	1	no	—
199	C₄⋊Q₈	—	2	1	no	—
200	C₄⋊Q₈	—	2	2	no	—
201	C₄⋊Q₈	—	9	1	no	—
202	C₄⋊Q₈	C₂×D₉	1	2	yes	—
203	C₄⋊Q₈	C₂×D₉	1	1	yes	characteristic
204	#126	—	18	2	no	—
205	#124	—	9	2	no	—
206	C₂²⋊Q₈	—	18	1	no	—
207	C₄×Q₈	—	2	1	no	—
208	C₄×Q₈	—	9	2	no	—
209	C₄×Q₈	C₂×D₉	1	1	yes	characteristic
210	C₄×Q₈	C₂×D₉	1	2	yes	—
211	C₄×Q₈	C₂×D₉	1	1	yes	characteristic
212	C₄×D₄	—	9	1	no	—
213	#117	—	18	1	no	—
214	#106	—	9	1	no	—
215	#106	—	9	2	no	—
216	C₄⋊C₈	—	9	1	no	—
217	C₄⋊C₈	—	9	2	no	—
218	#103	—	9	1	no	—
219	#103	—	9	2	no	—
220	#103	—	9	2	no	—
221	#103	—	9	2	no	—
222	#102	—	9	1	no	—
223	#102	—	9	2	no	—
224	#97	—	9	2	no	—
225	C₄×C₈	—	9	1	no	—
226	C₂²×D₃	—	6	1	no	—
227	C₃×Q₈	—	2	2	no	—
228	C₃×Q₈	—	2	1	no	—
229	C₃×Q₈	—	8	2	no	—
230	C₃×Q₈	—	8	1	no	—
231	C₃×Q₈	#239	1	2	yes	—
232	C₂×C₃×C₄	—	2	1	no	—
233	C₂×C₃×C₄	—	2	2	no	—
234	C₂×C₃×C₄	—	2	1	no	—
235	C₂×C₃×C₄	C₂×#67	1	2	yes	—
236	C₂×C₃×C₄	C₂×#67	1	2	yes	—
237	C₂×C₃×C₄	C₂×#67	1	1	yes	characteristic
238	C₂×C₃×C₄	C₂×#67	1	1	yes	characteristic
239	C₂×C₃×C₄	C₂³×D₃	1	1	yes	characteristic
240	C₂×Dic₃	—	6	1	no	—
241	C₂×Dic₃	—	6	2	no	—
242	D₁₂	—	6	1	no	—
243	D₁₂	—	6	1	no	—
244	C₄×D₃	—	12	2	no	—
245	Dic₆	—	6	2	no	—
246	Dic₆	—	6	2	no	—
247	C₃⋊C₈	—	6	1	no	—
248	C₃⋊C₈	—	6	2	no	—
249	C₃⋊C₈	—	6	1	no	—
250	C₂×C₉	#548	1	1	yes	characteristic
251	C₂×C₉	C₂×#137	1	1	yes	characteristic
252	C₂×C₉	#577	1	1	yes	characteristic
253	D₉	—	8	1	no	—
254	C₂×Q₈	—	2	2	no	—
255	C₂×Q₈	—	2	1	no	—
256	C₂×Q₈	—	9	2	no	—
257	C₂×Q₈	#611	1	1	yes	characteristic
258	C₂×Q₈	C₂²×D₉	1	1	yes	characteristic
259	C₂×D₄	—	9	1	no	—
260	C₂²×C₄	—	18	1	no	—
261	Q₁₆	—	18	4	no	—
262	SD₁₆	—	18	2	no	—
263	C₂×C₈	—	9	1	no	—
264	C₂×C₈	—	9	1	no	—
265	C₂×C₈	—	9	2	no	—
266	C₄⋊C₄	—	2	2	no	—
267	C₄⋊C₄	—	2	1	no	—
268	C₄⋊C₄	—	2	2	no	—
269	C₄⋊C₄	—	2	1	no	—
270	C₄⋊C₄	—	9	1	no	—
271	C₄⋊C₄	—	18	1	no	—
272	C₄⋊C₄	—	18	2	no	—
273	C₄⋊C₄	—	18	1	no	—
274	C₄⋊C₄	#611	1	1	yes	characteristic
275	C₄⋊C₄	#611	1	2	yes	—
276	C₄⋊C₄	C₂²×D₉	1	1	yes	characteristic
277	C₄⋊C₄	C₂²×D₉	1	2	yes	—
278	C₂²⋊C₄	—	18	2	no	—
279	C₂²⋊C₄	—	18	1	no	—
280	C₂²⋊C₄	—	18	2	no	—
281	C₄²	—	2	2	no	—
282	C₄²	—	2	1	no	—
283	C₄²	—	18	2	no	—
284	C₄²	C₂²×D₉	1	1	yes	characteristic
285	C₄²	C₂²×D₉	1	2	yes	—
286	C₂²×C₃	C₂²×#67	1	1	yes	characteristic
287	C₂×D₃	—	12	1	no	—
288	C₂×D₃	—	12	1	no	—
289	C₂×D₃	—	12	1	no	—
290	C₃×C₄	—	2	2	no	—
291	C₃×C₄	—	2	2	no	—
292	C₃×C₄	—	2	2	no	—
293	C₃×C₄	—	2	1	no	—
294	C₃×C₄	—	4	2	no	—
295	C₃×C₄	—	4	1	no	—
296	C₃×C₄	#942	1	2	yes	—
297	C₃×C₄	C₂×#239	1	1	yes	characteristic
298	C₃×C₄	#1007	1	1	yes	characteristic
299	Dic₃	—	12	1	no	—
300	Dic₃	—	12	2	no	—
301	C₉	#3369	1	1	yes	characteristic, complemented, Sylow
302	C₂³	—	18	1	no	—
303	Q₈	—	2	2	no	—
304	Q₈	—	2	1	no	—
305	Q₈	—	8	2	no	—
306	Q₈	—	8	1	no	—
307	Q₈	—	18	2	no	—
308	Q₈	—	18	2	no	—
309	Q₈	#3701	1	2	yes	—
310	D₄	—	18	1	no	—
311	D₄	—	18	1	no	—
312	C₂×C₄	—	2	1	no	—
313	C₂×C₄	—	2	2	no	—
314	C₂×C₄	—	2	1	no	—
315	C₂×C₄	—	18	1	no	—
316	C₂×C₄	—	18	2	no	—
317	C₂×C₄	—	36	2	no	—
318	C₂×C₄	C₂×#611	1	2	yes	—
319	C₂×C₄	C₂×#611	1	2	yes	—
320	C₂×C₄	C₂×#611	1	1	yes	characteristic
321	C₂×C₄	C₂×#611	1	1	yes	characteristic
322	C₂×C₄	C₂³×D₉	1	1	yes	characteristic
323	C₈	—	18	1	no	—
324	C₈	—	18	2	no	—
325	C₈	—	18	1	no	—
326	C₂×C₃	C₂×#942	1	1	yes	characteristic
327	C₂×C₃	#5817	1	1	yes	characteristic
328	C₂×C₃	#5836	1	1	yes	characteristic
329	D₃	—	24	1	no	—
330	C₂²	—	36	1	no	—
331	C₂²	—	36	1	no	—
332	C₂²	—	36	1	no	—
333	C₂²	C₂²×#611	1	1	yes	characteristic
334	C₄	—	2	2	no	—
335	C₄	—	2	2	no	—
336	C₄	—	2	2	no	—
337	C₄	—	2	1	no	—
338	C₄	—	4	2	no	—
339	C₄	—	4	1	no	—
340	C₄	—	36	1	no	—
341	C₄	—	36	2	no	—
342	C₄	#63505	1	2	yes	—
343	C₄	C₂×#3701	1	1	yes	characteristic
344	C₄	#63713	1	1	yes	characteristic
345	C₃	#83704	1	1	yes	characteristic
346	C₂	—	72	1	no	—
347	C₂	C₂×#63505	1	1	yes	characteristic
348	C₂	#10589919	1	1	yes	characteristic
349	C₂	#10589938	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	#3369
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. #10588767	C₂	semidirect
6. #10588766	C₂	semidirect
9. #10588707	C₂	semidirect
10. #10588706	C₂	semidirect
11. #10588701	C₂	semidirect
139. Q₈²	D₉	semidirect
301. C₉	#3369	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
#10589686	2
#10588774	2
#10588769	2
#10588767	2
#10588766	2
#10588711	2
C₄×#3675	2
#10588707	2
#10588706	2
#10588701	2
#83704	3