Subgroups — #11868638

Order 1152.

660 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	C₉×#533	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₄×D₃₆	C₂	1	1	yes	maximal, characteristic
3	#10588772	C₂	1	2	yes	maximal, complemented
4	#10588769	C₂	1	2	yes	maximal, complemented
5	#10588747	C₂	1	4	yes	maximal, complemented
6	#10588682	C₂	1	2	yes	maximal
7	#10588679	C₂	1	2	yes	maximal, complemented
8	#10588604	C₂	1	1	yes	maximal, characteristic, complemented
9	#83350	—	3	1	no	maximal
10	C₂²×D₃₆	C₂²	1	1	yes	characteristic
11	C₂²×C₄×D₉	—	2	1	no	—
12	C₉×C₄⋊Q₈	C₂²	1	1	yes	characteristic, complemented
13	C₉×C₄⋊Q₈	C₂²	1	1	yes	characteristic, complemented
14	C₉×#125	C₂²	1	2	yes	complemented
15	C₉×C₂²⋊Q₈	—	2	1	no	—
16	C₉×C₂²⋊Q₈	—	2	1	no	—
17	C₄×Q₈×C₉	—	2	1	no	—
18	C₄×Q₈×C₉	—	2	1	no	—
19	C₉×#117	C₂²	1	2	yes	—
20	C₂×C₄²×C₉	C₂²	1	1	yes	characteristic
21	C₂×(C₂×D₉)⋊C₄	—	2	1	no	—
22	C₂×C₄⋊Dic₉	C₂²	1	1	yes	characteristic
23	C₂×#3667	C₂²	1	2	yes	—
24	C₄×D₃₆	—	2	1	no	—
25	C₄×D₃₆	—	2	1	no	—
26	C₄×D₃₆	C₂²	1	2	yes	—
27	C₄×D₃₆	C₂²	1	2	yes	—
28	#63369	C₂²	1	4	yes	—
29	#63369	C₂²	1	4	yes	—
30	#63367	C₂²	1	4	yes	complemented
31	#63366	C₂²	1	4	yes	complemented
32	C₄⋊(C₉⋊C₈)	C₂²	1	2	yes	complemented
33	C₄⋊(C₉⋊C₈)	C₂²	1	2	yes	complemented
34	C₃×#533	D₃	1	1	yes	characteristic
35	C₂×C₄×D₁₂	—	3	1	no	—
36	#5087	—	3	2	no	—
37	#5084	—	3	2	no	—
38	#5066	—	3	4	no	—
39	#5001	—	3	2	no	—
40	#4995	—	3	2	no	—
41	#4924	—	3	1	no	—
42	C₂³×D₉	—	2	1	no	—
43	C₂×Q₈×C₉	—	2	1	no	—
44	C₂×Q₈×C₉	—	2	1	no	—
45	C₂²×C₄×C₉	D₄	1	1	yes	characteristic
46	C₂²×C₄×C₉	D₄	1	1	yes	characteristic
47	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
48	C₂²×Dic₉	—	2	1	no	—
49	C₂×D₃₆	—	2	1	no	—
50	C₂×D₃₆	—	2	2	no	—
51	C₂×D₃₆	Q₈	1	4	yes	complemented
52	C₂×D₃₆	C₂³	1	2	yes	—
53	C₂×C₄×D₉	—	2	2	no	—
54	C₂×C₄×D₉	—	2	2	no	—
55	C₂×C₄×D₉	—	4	4	no	—
56	C₉×C₄⋊C₄	—	2	2	no	—
57	C₉×C₄⋊C₄	—	2	2	no	—
58	C₉×C₄⋊C₄	—	2	1	no	—
59	C₉×C₄⋊C₄	—	2	1	no	—
60	C₉×C₄⋊C₄	C₂³	1	2	yes	—
61	C₉×C₄⋊C₄	C₂³	1	2	yes	—
62	C₉×C₂²⋊C₄	—	2	2	no	—
63	C₄²×C₉	—	2	2	no	—
64	C₄²×C₉	D₄	1	1	yes	characteristic
65	C₄²×C₉	D₄	1	1	yes	characteristic
66	C₄²×C₉	C₂³	1	1	yes	characteristic
67	C₄²×C₉	C₂³	1	1	yes	characteristic
68	(C₂×D₉)⋊C₄	—	2	2	no	—
69	(C₂×D₉)⋊C₄	—	2	2	no	—
70	C₄⋊Dic₉	—	2	1	no	—
71	C₄⋊Dic₉	C₂³	1	2	yes	—
72	#3667	—	2	4	no	—
73	C₂×C₉⋊C₈	C₂³	1	4	yes	—
74	#3077	—	9	1	no	Sylow
75	C₂²×D₁₂	—	3	1	no	—
76	C₂²×C₄×D₃	—	6	1	no	—
77	C₃×C₄⋊Q₈	C₂×D₃	1	1	yes	characteristic
78	C₃×C₄⋊Q₈	C₂×D₃	1	1	yes	characteristic
79	C₃×#125	C₂×D₃	1	2	yes	—
80	C₃×C₂²⋊Q₈	—	2	1	no	—
81	C₃×C₂²⋊Q₈	—	2	1	no	—
82	C₃×C₄×Q₈	—	2	1	no	—
83	C₃×C₄×Q₈	—	2	1	no	—
84	C₃×#117	C₂×D₃	1	2	yes	—
85	C₂×C₃×C₄²	C₂×D₃	1	1	yes	characteristic
86	C₂×(C₂×D₃)⋊C₄	—	6	1	no	—
87	C₂×C₄⋊Dic₃	—	3	1	no	—
88	C₂×#208	—	3	2	no	—
89	C₄×D₁₂	—	3	2	no	—
90	C₄×D₁₂	—	3	2	no	—
91	C₄×D₁₂	—	6	1	no	—
92	C₄×D₁₂	—	6	1	no	—
93	#809	—	3	4	no	—
94	#809	—	3	4	no	—
95	#808	—	3	4	no	—
96	#807	—	3	4	no	—
97	C₄⋊(C₃⋊C₈)	—	3	2	no	—
98	C₄⋊(C₃⋊C₈)	—	3	2	no	—
99	C₂³×C₉	C₂×D₄	1	1	yes	characteristic
100	C₂²×D₉	—	2	4	no	—
101	C₂²×D₉	—	2	2	no	—
102	C₂²×D₉	—	4	2	no	—
103	C₂²×D₉	—	4	2	no	—
104	Q₈×C₉	—	4	2	no	—
105	Q₈×C₉	—	4	2	no	—
106	C₂×C₄×C₉	—	2	2	no	—
107	C₂×C₄×C₉	—	2	2	no	—
108	C₂×C₄×C₉	—	2	2	no	—
109	C₂×C₄×C₉	—	4	1	no	—
110	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
111	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
112	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
113	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
114	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
115	C₂×C₄×C₉	C₂×Q₈	1	2	yes	—
116	C₂×C₄×C₉	#41	1	2	yes	—
117	C₂×C₄×C₉	C₂⁴	1	1	yes	characteristic
118	C₂×Dic₉	—	2	2	no	—
119	C₂×Dic₉	—	4	2	no	—
120	D₃₆	—	2	4	no	—
121	D₃₆	—	2	2	no	—
122	D₃₆	C₂×Q₈	1	4	yes	—
123	C₄×D₉	—	4	4	no	—
124	C₄×D₉	—	4	4	no	—
125	C₉⋊C₈	—	2	4	no	—
126	#533	D₉	1	1	yes	characteristic, complemented
127	C₂×C₄×D₄	—	9	1	no	—
128	#478	—	9	4	no	—
129	#476	—	9	2	no	—
130	#474	—	9	2	no	—
131	#428	—	9	2	no	—
132	#423	—	9	1	no	—
133	#418	—	9	2	no	—
134	C₂³×D₃	—	6	1	no	—
135	C₂×C₃×Q₈	—	2	1	no	—
136	C₂×C₃×Q₈	—	2	1	no	—
137	C₂²×C₃×C₄	#67	1	1	yes	characteristic
138	C₂²×C₃×C₄	#67	1	1	yes	characteristic
139	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
140	C₂²×Dic₃	—	6	1	no	—
141	C₂×D₁₂	—	3	4	no	—
142	C₂×D₁₂	—	3	2	no	—
143	C₂×D₁₂	—	6	2	no	—
144	C₂×D₁₂	—	6	1	no	—
145	C₂×C₄×D₃	—	6	2	no	—
146	C₂×C₄×D₃	—	6	2	no	—
147	C₂×C₄×D₃	—	12	4	no	—
148	C₃×C₄⋊C₄	—	2	2	no	—
149	C₃×C₄⋊C₄	—	2	2	no	—
150	C₃×C₄⋊C₄	—	2	1	no	—
151	C₃×C₄⋊C₄	—	2	1	no	—
152	C₃×C₄⋊C₄	C₂²×D₃	1	2	yes	—
153	C₃×C₄⋊C₄	C₂²×D₃	1	2	yes	—
154	C₃×C₂²⋊C₄	—	2	2	no	—
155	C₃×C₄²	—	2	2	no	—
156	C₃×C₄²	#67	1	1	yes	characteristic
157	C₃×C₄²	#67	1	1	yes	characteristic
158	C₃×C₄²	C₂²×D₃	1	1	yes	characteristic
159	C₃×C₄²	C₂²×D₃	1	1	yes	characteristic
160	(C₂×D₃)⋊C₄	—	6	2	no	—
161	(C₂×D₃)⋊C₄	—	6	2	no	—
162	C₄⋊Dic₃	—	3	2	no	—
163	C₄⋊Dic₃	—	6	1	no	—
164	#208	—	6	4	no	—
165	C₂×C₃⋊C₈	—	3	4	no	—
166	C₂²×C₉	—	2	1	no	—
167	C₂²×C₉	—	2	1	no	—
168	C₂²×C₉	C₂²⋊Q₈	1	2	yes	—
169	C₂²×C₉	C₂²×D₄	1	1	yes	characteristic
170	C₂×D₉	—	2	4	no	—
171	C₂×D₉	—	4	8	no	—
172	C₂×D₉	—	4	2	no	—
173	C₂×D₉	—	4	2	no	—
174	C₄×C₉	—	2	1	no	—
175	C₄×C₉	—	2	1	no	—
176	C₄×C₉	—	2	1	no	—
177	C₄×C₉	—	4	2	no	—
178	C₄×C₉	—	4	2	no	—
179	C₄×C₉	C₂²⋊Q₈	1	2	yes	—
180	C₄×C₉	AGL(1,ℤ/8)	1	2	yes	—
181	C₄×C₉	C₂²×Q₈	1	1	yes	characteristic
182	C₄×C₉	C₂×#41	1	1	yes	characteristic
183	Dic₉	—	4	2	no	—
184	C₂²×D₄	—	9	1	no	—
185	C₂³×C₄	—	18	1	no	—
186	C₂×M₁₆	—	9	2	no	—
187	C₄⋊Q₈	C₂×D₉	1	1	yes	characteristic, complemented
188	C₄⋊Q₈	C₂×D₉	1	1	yes	characteristic, complemented
189	#125	C₂×D₉	1	2	yes	complemented
190	C₂²⋊Q₈	—	2	1	no	—
191	C₂²⋊Q₈	—	2	1	no	—
192	C₄×Q₈	—	2	1	no	—
193	C₄×Q₈	—	2	1	no	—
194	C₄×D₄	—	9	2	no	—
195	C₄×D₄	—	9	2	no	—
196	C₄×D₄	—	18	1	no	—
197	C₄×D₄	—	18	1	no	—
198	#117	C₂×D₉	1	2	yes	—
199	C₂×C₄⋊C₄	—	9	1	no	—
200	C₂×C₂²⋊C₄	—	18	1	no	—
201	C₂×C₄²	C₂×D₉	1	1	yes	characteristic
202	#107	—	9	4	no	—
203	#106	—	9	4	no	—
204	C₄⋊C₈	—	9	2	no	—
205	C₄⋊C₈	—	9	2	no	—
206	#102	—	9	4	no	—
207	#102	—	9	4	no	—
208	C₂³×C₃	C₂×#67	1	1	yes	characteristic
209	C₂²×D₃	—	6	4	no	—
210	C₂²×D₃	—	6	2	no	—
211	C₂²×D₃	—	12	2	no	—
212	C₂²×D₃	—	12	2	no	—
213	C₃×Q₈	—	4	2	no	—
214	C₃×Q₈	—	4	2	no	—
215	C₂×C₃×C₄	—	2	2	no	—
216	C₂×C₃×C₄	—	2	2	no	—
217	C₂×C₃×C₄	—	2	2	no	—
218	C₂×C₃×C₄	—	4	1	no	—
219	C₂×C₃×C₄	D₃×Q₈	1	2	yes	—
220	C₂×C₃×C₄	#239	1	2	yes	—
221	C₂×C₃×C₄	C₂×#67	1	1	yes	characteristic
222	C₂×C₃×C₄	C₂×#67	1	1	yes	characteristic
223	C₂×C₃×C₄	C₂×#67	1	1	yes	characteristic
224	C₂×C₃×C₄	C₂×#67	1	1	yes	characteristic
225	C₂×C₃×C₄	C₂×#67	1	1	yes	characteristic
226	C₂×C₃×C₄	C₂³×D₃	1	1	yes	characteristic
227	C₂×Dic₃	—	6	2	no	—
228	C₂×Dic₃	—	12	2	no	—
229	D₁₂	—	3	4	no	—
230	D₁₂	—	6	4	no	—
231	D₁₂	—	6	2	no	—
232	C₄×D₃	—	12	4	no	—
233	C₄×D₃	—	12	4	no	—
234	C₃⋊C₈	—	6	4	no	—
235	C₂×C₉	—	2	2	no	—
236	C₂×C₉	C₂×C₂²⋊Q₈	1	1	yes	characteristic
237	C₂×C₉	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
238	C₂×C₉	#575	1	1	yes	characteristic
239	D₉	—	4	4	no	—
240	C₂⁴	—	18	1	no	—
241	C₂×Q₈	—	2	1	no	—
242	C₂×Q₈	—	2	1	no	—
243	C₂×D₄	—	9	4	no	—
244	C₂×D₄	—	9	2	no	—
245	C₂×D₄	—	18	2	no	—
246	C₂×D₄	—	18	1	no	—
247	C₂²×C₄	—	18	1	no	—
248	C₂²×C₄	—	18	2	no	—
249	C₂²×C₄	—	18	2	no	—
250	C₂²×C₄	—	36	4	no	—
251	C₂²×C₄	#611	1	1	yes	characteristic
252	C₂²×C₄	#611	1	1	yes	characteristic
253	C₂²×C₄	C₂²×D₉	1	1	yes	characteristic
254	M₁₆	—	18	4	no	—
255	C₂×C₈	—	9	4	no	—
256	C₄⋊C₄	—	2	2	no	—
257	C₄⋊C₄	—	2	2	no	—
258	C₄⋊C₄	—	2	1	no	—
259	C₄⋊C₄	—	2	1	no	—
260	C₄⋊C₄	—	9	2	no	—
261	C₄⋊C₄	—	18	1	no	—
262	C₄⋊C₄	C₂²×D₉	1	2	yes	—
263	C₄⋊C₄	C₂²×D₉	1	2	yes	—
264	C₂²⋊C₄	—	2	2	no	—
265	C₂²⋊C₄	—	18	2	no	—
266	C₂²⋊C₄	—	18	2	no	—
267	C₄²	—	2	2	no	—
268	C₄²	#611	1	1	yes	characteristic
269	C₄²	#611	1	1	yes	characteristic
270	C₄²	C₂²×D₉	1	1	yes	characteristic
271	C₄²	C₂²×D₉	1	1	yes	characteristic
272	C₂²×C₃	—	2	1	no	—
273	C₂²×C₃	—	2	1	no	—
274	C₂²×C₃	#946	1	2	yes	—
275	C₂²×C₃	C₂²×#67	1	1	yes	characteristic
276	C₂×D₃	—	6	4	no	—
277	C₂×D₃	—	12	8	no	—
278	C₂×D₃	—	12	2	no	—
279	C₂×D₃	—	12	2	no	—
280	C₃×C₄	—	2	1	no	—
281	C₃×C₄	—	2	1	no	—
282	C₃×C₄	—	2	1	no	—
283	C₃×C₄	—	4	2	no	—
284	C₃×C₄	—	4	2	no	—
285	C₃×C₄	#946	1	2	yes	—
286	C₃×C₄	#949	1	2	yes	—
287	C₃×C₄	C₂×D₃×Q₈	1	1	yes	characteristic
288	C₃×C₄	C₂×#239	1	1	yes	characteristic
289	Dic₃	—	12	2	no	—
290	C₉	#3077	1	1	yes	characteristic, complemented, Sylow
291	C₂³	—	18	4	no	—
292	C₂³	—	18	2	no	—
293	C₂³	—	36	2	no	—
294	C₂³	—	36	2	no	—
295	C₂³	C₂×#611	1	1	yes	characteristic
296	Q₈	—	4	2	no	—
297	Q₈	—	4	2	no	—
298	D₄	—	9	4	no	—
299	D₄	—	18	4	no	—
300	D₄	—	18	2	no	—
301	C₂×C₄	—	2	2	no	—
302	C₂×C₄	—	2	2	no	—
303	C₂×C₄	—	2	2	no	—
304	C₂×C₄	—	4	1	no	—
305	C₂×C₄	—	18	2	no	—
306	C₂×C₄	—	36	4	no	—
307	C₂×C₄	—	36	2	no	—
308	C₂×C₄	—	36	4	no	—
309	C₂×C₄	Q₈×D₉	1	2	yes	—
310	C₂×C₄	#3701	1	2	yes	—
311	C₂×C₄	C₂×#611	1	1	yes	characteristic
312	C₂×C₄	C₂×#611	1	1	yes	characteristic
313	C₂×C₄	C₂×#611	1	1	yes	characteristic
314	C₂×C₄	C₂×#611	1	1	yes	characteristic
315	C₂×C₄	C₂×#611	1	1	yes	characteristic
316	C₂×C₄	C₂³×D₉	1	1	yes	characteristic
317	C₈	—	18	4	no	—
318	C₂×C₃	—	2	2	no	—
319	C₂×C₃	C₂×#946	1	1	yes	characteristic
320	C₂×C₃	C₂×#949	1	1	yes	characteristic
321	C₂×C₃	#5822	1	1	yes	characteristic
322	D₃	—	12	4	no	—
323	C₂²	—	2	1	no	—
324	C₂²	—	2	1	no	—
325	C₂²	—	18	4	no	—
326	C₂²	—	36	8	no	—
327	C₂²	—	36	2	no	—
328	C₂²	—	36	2	no	—
329	C₂²	#63508	1	2	yes	—
330	C₂²	C₂²×#611	1	1	yes	characteristic
331	C₄	—	2	1	no	—
332	C₄	—	2	1	no	—
333	C₄	—	2	1	no	—
334	C₄	—	4	2	no	—
335	C₄	—	4	2	no	—
336	C₄	—	36	2	no	—
337	C₄	#63508	1	2	yes	—
338	C₄	#63512	1	2	yes	—
339	C₄	C₂×Q₈×D₉	1	1	yes	characteristic
340	C₄	C₂×#3701	1	1	yes	characteristic
341	C₃	#83350	1	1	yes	characteristic
342	C₂	—	2	2	no	—
343	C₂	—	36	4	no	—
344	C₂	C₂×#63508	1	1	yes	characteristic
345	C₂	C₂×#63512	1	1	yes	characteristic
346	C₂	#10589925	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×#611
commutator, cosocle	C₂×C₄×C₉	C₂⁴
Frattini	C₂×C₃×C₄	C₂³×D₃
Fitting	C₉×#533	C₂
socle	C₂²×C₃	C₂²×#67
prime radical	C₂²×D₃₆	C₂²
hypercenter	#533	D₉
hypocenter	C₉	#3077
LCS	G ▷ C₂×C₄×C₉ ▷ C₂×C₉ ▷ C₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #533

Complements

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

N	H ≅ G/N	type
1. C₉×#533	C₂	semidirect
3. #10588772	C₂	semidirect
4. #10588769	C₂	semidirect
5. #10588747	C₂	semidirect
7. #10588679	C₂	semidirect
8. #10588604	C₂	semidirect
12. C₉×C₄⋊Q₈	C₂²	semidirect
13. C₉×C₄⋊Q₈	C₂²	semidirect
14. C₉×#125	C₂²	semidirect
30. #63367	C₂²	semidirect
31. #63366	C₂²	semidirect
32. C₄⋊(C₉⋊C₈)	C₂²	semidirect
33. C₄⋊(C₉⋊C₈)	C₂²	semidirect
51. C₂×D₃₆	Q₈	semidirect
126. #533	D₉	semidirect
187. C₄⋊Q₈	C₂×D₉	semidirect
188. C₄⋊Q₈	C₂×D₉	semidirect
189. #125	C₂×D₉	semidirect
290. C₉	#3077	semidirect

Maximal subgroups

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

subgroup	index
C₉×#533	2
C₂×C₄×D₃₆	2
#10588772	2
#10588769	2
#10588747	2
#10588682	2
#10588679	2
#10588604	2
#83350	3