Subgroups — #11899701

Order 1152.

540 conjugacy classes of subgroups.

The subgroup lattice has 381 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#10589873	C₂	1	1	yes	maximal, characteristic, complemented
2	#10589871	C₂	1	1	yes	maximal, characteristic, complemented
3	C₉×#508	C₂	1	1	yes	maximal, characteristic, complemented
4	#10588875	C₂	1	1	yes	maximal, characteristic, complemented
5	#10588873	C₂	1	1	yes	maximal, characteristic, complemented
6	#10588598	C₂	1	2	yes	maximal, complemented
7	#10588596	C₂	1	2	yes	maximal, complemented
8	D₉×SD₃₂	C₂	1	2	yes	maximal, complemented
9	#10588591	C₂	1	2	yes	maximal, complemented
10	C₂×#63359	C₂	1	1	yes	maximal, characteristic, complemented
11	#10588581	C₂	1	1	yes	maximal, characteristic, complemented
12	#82032	—	3	1	no	maximal
13	#63716	—	2	1	no	—
14	#63715	—	2	1	no	—
15	C₉×#135	C₂²	1	1	yes	characteristic
16	C₉×#135	C₂²	1	1	yes	characteristic, complemented
17	#63512	—	2	1	no	—
18	#63511	—	2	1	no	—
19	#63481	C₂²	1	2	yes	—
20	Q₁₆×D₉	C₂²	1	2	yes	complemented
21	#63477	—	2	2	no	—
22	#63476	—	2	2	no	—
23	#63474	C₂²	1	2	yes	complemented
24	D₈×D₉	C₂²	1	2	yes	—
25	C₂×D₇₂	C₂²	1	1	yes	characteristic
26	#63464	C₂²	1	1	yes	characteristic
27	C₂×Dic₃₆	C₂²	1	1	yes	characteristic, complemented
28	C₉×Q₃₂	C₂²	1	1	yes	characteristic, complemented
29	C₉×SD₃₂	C₂²	1	1	yes	characteristic, complemented
30	C₉×SD₃₂	C₂²	1	1	yes	characteristic, complemented
31	C₉×D₁₆	C₂²	1	1	yes	characteristic, complemented
32	C₂×C₉×C₁₆	C₂²	1	1	yes	characteristic, complemented
33	#63388	C₂²	1	2	yes	complemented
34	#63387	C₂²	1	2	yes	complemented
35	#63386	C₂²	1	2	yes	complemented
36	#63385	C₂²	1	2	yes	complemented
37	#63371	C₂²	1	1	yes	characteristic, complemented
38	#63359	C₂²	1	2	yes	complemented
39	#63359	C₂²	1	2	yes	complemented
40	#63357	C₂²	1	2	yes	complemented
41	C₁₆×D₉	C₂²	1	2	yes	complemented
42	#5771	—	3	1	no	—
43	#5769	—	3	1	no	—
44	C₃×#508	D₃	1	1	yes	characteristic
45	#5195	—	3	1	no	—
46	#5193	—	3	1	no	—
47	#4918	—	3	2	no	—
48	#4916	—	3	2	no	—
49	D₃×SD₃₂	—	3	2	no	—
50	#4911	—	3	2	no	—
51	C₂×#800	—	3	1	no	—
52	#4901	—	3	1	no	—
53	C₉×#41	—	2	1	no	—
54	C₉×#41	—	2	1	no	—
55	#3701	—	2	2	no	—
56	Q₈×D₉	—	2	2	no	—
57	#3699	—	2	2	no	—
58	#3699	—	4	2	no	—
59	D₄×D₉	—	2	2	no	—
60	D₄×D₉	—	4	2	no	—
61	#3697	—	4	1	no	—
62	#3697	—	4	1	no	—
63	#3697	D₄	1	1	yes	characteristic
64	C₂×D₃₆	—	2	1	no	—
65	C₂×D₃₆	—	4	1	no	—
66	C₂×Dic₁₈	—	2	1	no	—
67	C₂×Dic₁₈	—	4	1	no	—
68	C₉×Q₁₆	C₂³	1	1	yes	characteristic
69	C₉×Q₁₆	C₂³	1	1	yes	characteristic
70	C₉×SD₁₆	—	2	1	no	—
71	C₉×SD₁₆	—	2	1	no	—
72	C₉×D₈	C₂³	1	1	yes	characteristic
73	C₉×D₈	C₂³	1	1	yes	characteristic
74	C₂×C₈×C₉	C₂³	1	1	yes	characteristic
75	#3675	—	2	2	no	—
76	#3674	—	2	2	no	—
77	#3673	—	2	2	no	—
78	#3672	—	2	2	no	—
79	#3667	D₄	1	1	yes	characteristic
80	D₇₂	—	2	1	no	—
81	D₇₂	C₂³	1	2	yes	—
82	#3663	D₄	1	2	yes	—
83	C₈×D₉	C₂³	1	2	yes	—
84	Dic₃₆	—	2	1	no	—
85	Dic₃₆	C₂³	1	2	yes	—
86	C₉×C₁₆	C₂³	1	1	yes	characteristic
87	C₉×C₁₆	C₂³	1	1	yes	characteristic
88	C₉⋊C₁₆	C₂³	1	2	yes	—
89	#3416	—	9	1	no	Sylow
90	#1010	—	6	1	no	—
91	#1009	—	6	1	no	—
92	C₃×#135	C₂×D₃	1	1	yes	characteristic
93	C₃×#135	C₂×D₃	1	1	yes	characteristic
94	#951	—	6	1	no	—
95	#949	—	6	1	no	—
96	#919	—	3	2	no	—
97	D₃×Q₁₆	—	3	2	no	—
98	#915	—	6	2	no	—
99	#914	—	6	2	no	—
100	#912	—	3	2	no	—
101	D₃×D₈	—	3	2	no	—
102	C₂×Dic₁₂	—	3	1	no	—
103	C₂×D₂₄	—	3	1	no	—
104	#901	—	3	1	no	—
105	C₃×Q₃₂	C₂×D₃	1	1	yes	characteristic
106	C₃×SD₃₂	C₂×D₃	1	1	yes	characteristic
107	C₃×SD₃₂	C₂×D₃	1	1	yes	characteristic
108	C₃×D₁₆	C₂×D₃	1	1	yes	characteristic
109	C₂×C₃×C₁₆	C₂×D₃	1	1	yes	characteristic
110	#829	—	3	2	no	—
111	#828	—	3	2	no	—
112	#827	—	3	2	no	—
113	#826	—	3	2	no	—
114	#812	—	3	1	no	—
115	#800	—	3	2	no	—
116	#800	—	3	2	no	—
117	#798	—	3	2	no	—
118	D₃×C₁₆	—	3	2	no	—
119	C₂²×D₉	—	4	1	no	—
120	C₂²×D₉	—	4	2	no	—
121	Q₈×C₉	—	2	1	no	—
122	Q₈×C₉	—	2	1	no	—
123	D₄×C₉	—	2	1	no	—
124	D₄×C₉	—	2	1	no	—
125	D₄×C₉	—	4	1	no	—
126	D₄×C₉	—	4	1	no	—
127	C₂×C₄×C₉	—	4	1	no	—
128	C₂×C₄×C₉	—	4	1	no	—
129	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
130	#611	—	4	2	no	—
131	#611	—	4	2	no	—
132	#611	D₈	1	2	yes	—
133	C₂×Dic₉	—	4	1	no	—
134	C₂×Dic₉	—	4	2	no	—
135	D₃₆	—	2	2	no	—
136	D₃₆	—	4	2	no	—
137	D₃₆	—	4	1	no	—
138	D₃₆	D₈	1	1	yes	characteristic
139	C₄×D₉	—	4	2	no	—
140	C₄×D₉	—	4	2	no	—
141	C₄×D₉	C₂×D₄	1	2	yes	—
142	Dic₁₈	—	2	2	no	—
143	Dic₁₈	—	4	1	no	—
144	Dic₁₈	—	4	2	no	—
145	Dic₁₈	D₈	1	1	yes	characteristic
146	C₈×C₉	C₂×D₄	1	1	yes	characteristic
147	C₈×C₉	C₂⁴	1	1	yes	characteristic
148	C₉⋊C₈	C₂×D₄	1	2	yes	—
149	#578	—	9	1	no	—
150	#576	—	9	1	no	—
151	#510	—	9	1	no	—
152	#510	—	9	2	no	—
153	#509	—	9	1	no	—
154	#509	—	9	2	no	—
155	#508	—	9	2	no	—
156	#508	D₉	1	1	yes	characteristic, complemented
157	C₂×SD₃₂	—	9	2	no	—
158	C₂×SD₃₂	—	9	1	no	—
159	#504	—	9	1	no	—
160	C₃×#41	—	2	1	no	—
161	C₃×#41	—	2	1	no	—
162	#239	—	6	2	no	—
163	D₃×Q₈	—	6	2	no	—
164	#237	—	6	2	no	—
165	#237	—	12	2	no	—
166	D₃×D₄	—	6	2	no	—
167	D₃×D₄	—	12	2	no	—
168	#235	—	3	1	no	—
169	#235	—	12	1	no	—
170	#235	—	12	1	no	—
171	C₂×D₁₂	—	6	1	no	—
172	C₂×D₁₂	—	12	1	no	—
173	C₂×Dic₆	—	6	1	no	—
174	C₂×Dic₆	—	12	1	no	—
175	C₃×Q₁₆	C₂²×D₃	1	1	yes	characteristic
176	C₃×Q₁₆	C₂²×D₃	1	1	yes	characteristic
177	C₃×SD₁₆	—	2	1	no	—
178	C₃×SD₁₆	—	2	1	no	—
179	C₃×D₈	C₂²×D₃	1	1	yes	characteristic
180	C₃×D₈	C₂²×D₃	1	1	yes	characteristic
181	C₂×C₃×C₈	C₂²×D₃	1	1	yes	characteristic
182	#216	—	6	2	no	—
183	#215	—	6	2	no	—
184	#214	—	6	2	no	—
185	#213	—	6	2	no	—
186	#208	—	3	1	no	—
187	Dic₁₂	—	3	2	no	—
188	Dic₁₂	—	6	1	no	—
189	D₂₄	—	3	2	no	—
190	D₂₄	—	6	1	no	—
191	ΓL(1,𝔽₂₅)	—	3	2	no	—
192	D₃×C₈	—	3	2	no	—
193	C₃×C₁₆	C₂²×D₃	1	1	yes	characteristic
194	C₃×C₁₆	C₂²×D₃	1	1	yes	characteristic
195	C₃⋊C₁₆	—	3	2	no	—
196	C₂²×C₉	—	4	1	no	—
197	C₂²×C₉	—	4	1	no	—
198	C₂²×C₉	C₂×D₈	1	1	yes	characteristic
199	C₂×D₉	—	4	2	no	—
200	C₂×D₉	—	16	2	no	—
201	C₂×D₉	—	16	1	no	—
202	C₂×D₉	C₂×D₈	1	2	yes	—
203	C₄×C₉	—	4	1	no	—
204	C₄×C₉	—	4	1	no	—
205	C₄×C₉	C₂×D₈	1	1	yes	characteristic
206	C₄×C₉	C₂²×D₄	1	1	yes	characteristic
207	Dic₉	—	4	2	no	—
208	Dic₉	C₂×D₈	1	2	yes	—
209	2₋¹⁺⁴	—	18	1	no	—
210	H(5,𝔽₂)	—	18	1	no	—
211	#137	—	18	2	no	—
212	#137	—	18	1	no	—
213	AGL(1,ℤ/8)	—	18	2	no	—
214	AGL(1,ℤ/8)	—	18	1	no	—
215	#135	—	9	2	no	—
216	#135	—	9	2	no	—
217	#135	C₂×D₉	1	1	yes	characteristic
218	#135	C₂×D₉	1	1	yes	characteristic, complemented
219	C₂×Q₁₆	—	9	2	no	—
220	C₂×Q₁₆	—	9	1	no	—
221	C₂×D₈	—	9	2	no	—
222	C₂×D₈	—	9	1	no	—
223	#131	—	9	1	no	—
224	Q₃₂	—	9	2	no	—
225	Q₃₂	C₂×D₉	1	1	yes	characteristic, complemented
226	SD₃₂	—	9	2	no	—
227	SD₃₂	—	9	2	no	—
228	SD₃₂	—	9	2	no	—
229	SD₃₂	—	9	2	no	—
230	SD₃₂	C₂×D₉	1	1	yes	characteristic, complemented
231	SD₃₂	C₂×D₉	1	1	yes	characteristic, complemented
232	D₁₆	—	9	2	no	—
233	D₁₆	C₂×D₉	1	1	yes	characteristic, complemented
234	M₃₂	—	9	1	no	—
235	M₃₂	—	9	2	no	—
236	C₂×C₁₆	—	9	2	no	—
237	C₂×C₁₆	C₂×D₉	1	1	yes	characteristic, complemented
238	C₂²×D₃	—	12	2	no	—
239	C₂²×D₃	—	12	1	no	—
240	C₃×Q₈	—	2	1	no	—
241	C₃×Q₈	—	2	1	no	—
242	C₃×D₄	—	2	1	no	—
243	C₃×D₄	—	2	1	no	—
244	C₃×D₄	—	4	1	no	—
245	C₃×D₄	—	4	1	no	—
246	C₂×C₃×C₄	—	4	1	no	—
247	C₂×C₃×C₄	—	4	1	no	—
248	C₂×C₃×C₄	D₃×D₄	1	1	yes	characteristic
249	#67	—	3	2	no	—
250	#67	—	12	2	no	—
251	#67	—	12	2	no	—
252	C₂×Dic₃	—	12	2	no	—
253	C₂×Dic₃	—	12	1	no	—
254	D₁₂	—	3	1	no	—
255	D₁₂	—	6	2	no	—
256	D₁₂	—	12	2	no	—
257	D₁₂	—	12	1	no	—
258	C₄×D₃	—	3	2	no	—
259	C₄×D₃	—	12	2	no	—
260	C₄×D₃	—	12	2	no	—
261	Dic₆	—	3	1	no	—
262	Dic₆	—	6	2	no	—
263	Dic₆	—	12	2	no	—
264	Dic₆	—	12	1	no	—
265	C₃×C₈	D₃×D₄	1	1	yes	characteristic
266	C₃×C₈	C₂³×D₃	1	1	yes	characteristic
267	C₃⋊C₈	—	3	2	no	—
268	C₂×C₉	—	2	1	no	—
269	C₂×C₉	—	8	1	no	—
270	C₂×C₉	—	8	1	no	—
271	C₂×C₉	C₂²×D₈	1	1	yes	characteristic
272	D₉	—	2	2	no	—
273	D₉	—	8	2	no	—
274	#41	—	2	1	no	—
275	#41	—	2	1	no	—
276	#41	—	9	1	no	—
277	#41	—	18	2	no	—
278	#41	—	18	2	no	—
279	#41	—	36	1	no	—
280	#41	—	36	1	no	—
281	#41	—	36	2	no	—
282	C₂×Q₈	—	18	2	no	—
283	C₂×Q₈	—	18	1	no	—
284	C₂×Q₈	—	36	1	no	—
285	C₂×D₄	—	18	2	no	—
286	C₂×D₄	—	18	1	no	—
287	C₂×D₄	—	36	1	no	—
288	C₂×D₄	—	36	2	no	—
289	Q₁₆	—	9	2	no	—
290	Q₁₆	—	18	2	no	—
291	Q₁₆	—	18	1	no	—
292	Q₁₆	C₂²×D₉	1	1	yes	characteristic
293	Q₁₆	C₂²×D₉	1	1	yes	characteristic
294	SD₁₆	—	2	1	no	—
295	SD₁₆	—	2	1	no	—
296	SD₁₆	—	18	2	no	—
297	SD₁₆	—	18	2	no	—
298	D₈	—	9	2	no	—
299	D₈	—	18	2	no	—
300	D₈	—	18	1	no	—
301	D₈	C₂²×D₉	1	1	yes	characteristic
302	D₈	C₂²×D₉	1	1	yes	characteristic
303	M₁₆	—	9	1	no	—
304	M₁₆	—	9	2	no	—
305	C₂×C₈	—	9	2	no	—
306	C₂×C₈	C₂²×D₉	1	1	yes	characteristic
307	C₁₆	—	9	2	no	—
308	C₁₆	C₂²×D₉	1	1	yes	characteristic
309	C₁₆	C₂²×D₉	1	1	yes	characteristic
310	C₂²×C₃	—	4	1	no	—
311	C₂²×C₃	—	4	1	no	—
312	C₂²×C₃	D₃×D₈	1	1	yes	characteristic
313	C₂×D₃	—	3	2	no	—
314	C₂×D₃	—	12	2	no	—
315	C₂×D₃	—	48	2	no	—
316	C₂×D₃	—	48	1	no	—
317	C₃×C₄	—	4	1	no	—
318	C₃×C₄	—	4	1	no	—
319	C₃×C₄	D₃×D₈	1	1	yes	characteristic
320	C₃×C₄	C₂×D₃×D₄	1	1	yes	characteristic
321	Dic₃	—	3	2	no	—
322	Dic₃	—	12	2	no	—
323	C₉	#3416	1	1	yes	characteristic, complemented, Sylow
324	C₂³	—	36	2	no	—
325	C₂³	—	36	1	no	—
326	Q₈	—	2	1	no	—
327	Q₈	—	2	1	no	—
328	Q₈	—	9	1	no	—
329	Q₈	—	18	2	no	—
330	Q₈	—	36	2	no	—
331	Q₈	—	36	1	no	—
332	D₄	—	2	1	no	—
333	D₄	—	2	1	no	—
334	D₄	—	4	1	no	—
335	D₄	—	4	1	no	—
336	D₄	—	9	2	no	—
337	D₄	—	9	1	no	—
338	D₄	—	18	2	no	—
339	D₄	—	36	2	no	—
340	D₄	—	36	2	no	—
341	D₄	—	36	2	no	—
342	D₄	—	36	1	no	—
343	C₂×C₄	—	4	1	no	—
344	C₂×C₄	—	4	1	no	—
345	C₂×C₄	—	9	2	no	—
346	C₂×C₄	—	36	2	no	—
347	C₂×C₄	—	36	2	no	—
348	C₂×C₄	—	36	2	no	—
349	C₂×C₄	—	36	1	no	—
350	C₂×C₄	D₄×D₉	1	1	yes	characteristic
351	C₈	—	9	2	no	—
352	C₈	D₄×D₉	1	1	yes	characteristic
353	C₈	C₂³×D₉	1	1	yes	characteristic
354	C₂×C₃	—	2	1	no	—
355	C₂×C₃	—	8	1	no	—
356	C₂×C₃	—	8	1	no	—
357	C₂×C₃	C₂×D₃×D₈	1	1	yes	characteristic
358	D₃	—	6	2	no	—
359	D₃	—	24	2	no	—
360	C₂²	—	4	1	no	—
361	C₂²	—	4	1	no	—
362	C₂²	—	9	2	no	—
363	C₂²	—	36	2	no	—
364	C₂²	—	144	1	no	—
365	C₂²	—	144	2	no	—
366	C₂²	D₈×D₉	1	1	yes	characteristic
367	C₄	—	4	1	no	—
368	C₄	—	4	1	no	—
369	C₄	—	9	2	no	—
370	C₄	—	36	2	no	—
371	C₄	D₈×D₉	1	1	yes	characteristic
372	C₄	C₂×D₄×D₉	1	1	yes	characteristic
373	C₃	#82032	1	1	yes	characteristic
374	C₂	—	2	1	no	—
375	C₂	—	8	1	no	—
376	C₂	—	8	1	no	—
377	C₂	—	18	2	no	—
378	C₂	—	72	2	no	—
379	C₂	C₂×D₈×D₉	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×D₈×D₉
commutator, cosocle	C₈×C₉	C₂⁴
Frattini	C₃×C₈	C₂³×D₃
Fitting	C₉×#508	C₂
socle	C₂×C₃	C₂×D₃×D₈
hypercenter	#508	D₉
hypocenter	C₉	#3416
LCS	G ▷ C₈×C₉ ▷ C₄×C₉ ▷ C₂×C₉ ▷ C₉
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ #508

Complements

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

N	H ≅ G/N	type
1. #10589873	C₂	semidirect
2. #10589871	C₂	semidirect
3. C₉×#508	C₂	semidirect
4. #10588875	C₂	semidirect
5. #10588873	C₂	semidirect
6. #10588598	C₂	semidirect
7. #10588596	C₂	semidirect
8. D₉×SD₃₂	C₂	semidirect
9. #10588591	C₂	semidirect
10. C₂×#63359	C₂	semidirect
11. #10588581	C₂	semidirect
16. C₉×#135	C₂²	semidirect
20. Q₁₆×D₉	C₂²	semidirect
23. #63474	C₂²	semidirect
27. C₂×Dic₃₆	C₂²	semidirect
28. C₉×Q₃₂	C₂²	semidirect
29. C₉×SD₃₂	C₂²	semidirect
30. C₉×SD₃₂	C₂²	semidirect
31. C₉×D₁₆	C₂²	semidirect
32. C₂×C₉×C₁₆	C₂²	semidirect
33. #63388	C₂²	semidirect
34. #63387	C₂²	semidirect
35. #63386	C₂²	semidirect
36. #63385	C₂²	semidirect
37. #63371	C₂²	semidirect
38. #63359	C₂²	semidirect
39. #63359	C₂²	semidirect
40. #63357	C₂²	semidirect
41. C₁₆×D₉	C₂²	semidirect
156. #508	D₉	semidirect
218. #135	C₂×D₉	semidirect
225. Q₃₂	C₂×D₉	semidirect
230. SD₃₂	C₂×D₉	semidirect
231. SD₃₂	C₂×D₉	semidirect
233. D₁₆	C₂×D₉	semidirect
237. C₂×C₁₆	C₂×D₉	semidirect
323. C₉	#3416	semidirect

Maximal subgroups

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

subgroup	index
#10589873	2
#10589871	2
C₉×#508	2
#10588875	2
#10588873	2
#10588598	2
#10588596	2
D₉×SD₃₂	2
#10588591	2
C₂×#63359	2
#10588581	2
#82032	3