Subgroups — #11868720

Order 1152.

564 conjugacy classes of subgroups.

The subgroup lattice has 321 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×Q₈×C₉	C₂	1	1	yes	maximal, characteristic
2	#10589572	C₂	1	1	yes	maximal, characteristic
3	#10588906	C₂	1	2	yes	maximal
4	#10588704	C₂	1	4	yes	maximal, complemented
5	#10588703	C₂	1	2	yes	maximal, complemented
6	#10588702	C₂	1	2	yes	maximal, complemented
7	#10588644	C₂	1	2	yes	maximal
8	#10588604	C₂	1	1	yes	maximal, characteristic
9	#82431	—	3	1	no	maximal
10	C₂²×Q₈×C₉	C₂²	1	1	yes	characteristic
11	C₄×Q₈×C₉	—	2	1	no	—
12	C₄×Q₈×C₉	—	2	1	no	—
13	C₄×Q₈×C₉	C₂²	1	2	yes	—
14	C₄×Q₈×C₉	C₂²	1	2	yes	—
15	C₂×C₉×C₄⋊C₄	—	2	1	no	—
16	C₂×C₉×C₄⋊C₄	C₂²	1	1	yes	characteristic
17	C₂×C₄²×C₉	—	2	1	no	—
18	C₂×C₄²×C₉	C₂²	1	1	yes	characteristic
19	#63488	C₂²	1	2	yes	—
20	#63486	—	2	1	no	—
21	#63486	—	2	1	no	—
22	C₂×#3667	C₂²	1	2	yes	—
23	#63432	C₂²	1	2	yes	—
24	#63431	C₂²	1	1	yes	characteristic
25	#63431	C₂²	1	1	yes	characteristic
26	C₄×Dic₁₈	—	2	1	no	—
27	C₄×Dic₁₈	—	2	1	no	—
28	#63395	C₂²	1	4	yes	—
29	#63395	C₂²	1	4	yes	—
30	#63367	C₂²	1	4	yes	—
31	#63366	C₂²	1	4	yes	—
32	C₄⋊(C₉⋊C₈)	C₂²	1	2	yes	—
33	C₄⋊(C₉⋊C₈)	C₂²	1	2	yes	—
34	C₂×C₃×C₄×Q₈	D₃	1	1	yes	characteristic
35	#5470	—	3	1	no	—
36	#5225	—	3	2	no	—
37	#5022	—	3	4	no	—
38	#5021	—	3	2	no	—
39	#5020	—	3	2	no	—
40	#4964	—	3	2	no	—
41	#4924	—	3	1	no	—
42	C₂×Q₈×C₉	—	2	1	no	—
43	C₂×Q₈×C₉	—	2	2	no	—
44	C₂×Q₈×C₉	Q₈	1	4	yes	complemented
45	C₂×Q₈×C₉	C₂³	1	2	yes	—
46	C₂²×C₄×C₉	—	2	1	no	—
47	C₂²×C₄×C₉	—	2	1	no	—
48	C₂²×C₄×C₉	D₄	1	1	yes	characteristic
49	C₂²×C₄×C₉	D₄	1	1	yes	characteristic
50	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
51	C₂×Dic₁₈	—	2	1	no	—
52	C₂×Dic₁₈	—	2	1	no	—
53	C₉×C₄⋊C₄	—	2	2	no	—
54	C₉×C₄⋊C₄	—	2	2	no	—
55	C₉×C₄⋊C₄	—	2	1	no	—
56	C₉×C₄⋊C₄	C₂³	1	2	yes	—
57	C₄²×C₉	—	2	2	no	—
58	C₄²×C₉	—	2	2	no	—
59	C₄²×C₉	D₄	1	1	yes	characteristic
60	C₄²×C₉	D₄	1	1	yes	characteristic
61	C₄²×C₉	C₂³	1	1	yes	characteristic
62	C₄²×C₉	C₂³	1	1	yes	characteristic
63	#3676	—	2	2	no	—
64	C₄⋊Dic₉	—	2	1	no	—
65	C₄⋊Dic₉	—	2	1	no	—
66	C₄⋊Dic₉	C₂³	1	2	yes	—
67	C₄⋊Dic₉	C₂³	1	2	yes	—
68	Dic₉⋊C₄	—	2	2	no	—
69	Dic₉⋊C₄	—	2	2	no	—
70	C₄×Dic₉	—	2	2	no	—
71	#3667	—	2	4	no	—
72	C₂×C₉⋊C₈	C₂³	1	4	yes	—
73	#3078	—	9	1	no	Sylow
74	C₂²×C₃×Q₈	C₂×D₃	1	1	yes	characteristic
75	C₃×C₄×Q₈	—	2	1	no	—
76	C₃×C₄×Q₈	—	2	1	no	—
77	C₃×C₄×Q₈	C₂×D₃	1	2	yes	—
78	C₃×C₄×Q₈	C₂×D₃	1	2	yes	—
79	C₂×C₃×C₄⋊C₄	—	2	1	no	—
80	C₂×C₃×C₄⋊C₄	C₂×D₃	1	1	yes	characteristic
81	C₂×C₃×C₄²	—	2	1	no	—
82	C₂×C₃×C₄²	C₂×D₃	1	1	yes	characteristic
83	#926	—	3	2	no	—
84	#924	—	6	1	no	—
85	#924	—	6	1	no	—
86	C₂×#208	—	3	2	no	—
87	#870	—	3	2	no	—
88	#869	—	3	1	no	—
89	#869	—	3	1	no	—
90	C₄×Dic₆	—	6	1	no	—
91	C₄×Dic₆	—	6	1	no	—
92	#835	—	3	4	no	—
93	#835	—	3	4	no	—
94	#808	—	3	4	no	—
95	#807	—	3	4	no	—
96	C₄⋊(C₃⋊C₈)	—	3	2	no	—
97	C₄⋊(C₃⋊C₈)	—	3	2	no	—
98	C₂³×C₉	C₂×D₄	1	1	yes	characteristic
99	Q₈×C₉	—	2	4	no	—
100	Q₈×C₉	—	2	2	no	—
101	Q₈×C₉	C₂×Q₈	1	4	yes	—
102	C₂×C₄×C₉	—	2	4	no	—
103	C₂×C₄×C₉	—	2	2	no	—
104	C₂×C₄×C₉	—	2	2	no	—
105	C₂×C₄×C₉	—	2	2	no	—
106	C₂×C₄×C₉	—	4	1	no	—
107	C₂×C₄×C₉	—	4	2	no	—
108	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
109	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
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×Q₈	1	2	yes	—
114	C₂×C₄×C₉	#41	1	2	yes	—
115	C₂×C₄×C₉	C₂⁴	1	1	yes	characteristic
116	C₂×Dic₉	—	2	2	no	—
117	C₂×Dic₉	—	2	2	no	—
118	Dic₁₈	—	4	2	no	—
119	Dic₁₈	—	4	2	no	—
120	C₉⋊C₈	—	2	4	no	—
121	#533	—	9	1	no	—
122	C₂×C₄×Q₈	D₉	1	1	yes	characteristic
123	#479	—	9	4	no	—
124	#477	—	9	2	no	—
125	#475	—	9	2	no	—
126	#428	—	9	2	no	—
127	#423	—	9	1	no	—
128	#419	—	9	2	no	—
129	C₂×C₃×Q₈	—	2	2	no	—
130	C₂×C₃×Q₈	—	2	1	no	—
131	C₂×C₃×Q₈	Dic₆	1	4	yes	—
132	C₂×C₃×Q₈	C₂²×D₃	1	2	yes	—
133	C₂²×C₃×C₄	—	2	1	no	—
134	C₂²×C₃×C₄	—	2	1	no	—
135	C₂²×C₃×C₄	#67	1	1	yes	characteristic
136	C₂²×C₃×C₄	#67	1	1	yes	characteristic
137	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
138	C₂×Dic₆	—	6	1	no	—
139	C₂×Dic₆	—	6	1	no	—
140	C₃×C₄⋊C₄	—	2	2	no	—
141	C₃×C₄⋊C₄	—	2	2	no	—
142	C₃×C₄⋊C₄	—	2	1	no	—
143	C₃×C₄⋊C₄	C₂²×D₃	1	2	yes	—
144	C₃×C₄²	—	2	2	no	—
145	C₃×C₄²	—	2	2	no	—
146	C₃×C₄²	#67	1	1	yes	characteristic
147	C₃×C₄²	#67	1	1	yes	characteristic
148	C₃×C₄²	C₂²×D₃	1	1	yes	characteristic
149	C₃×C₄²	C₂²×D₃	1	1	yes	characteristic
150	#217	—	6	2	no	—
151	C₄⋊Dic₃	—	3	2	no	—
152	C₄⋊Dic₃	—	3	2	no	—
153	C₄⋊Dic₃	—	6	1	no	—
154	C₄⋊Dic₃	—	6	1	no	—
155	Dic₃⋊C₄	—	6	2	no	—
156	Dic₃⋊C₄	—	6	2	no	—
157	C₄×Dic₃	—	6	2	no	—
158	#208	—	6	4	no	—
159	C₂×C₃⋊C₈	—	3	4	no	—
160	C₂²×C₉	—	2	1	no	—
161	C₂²×C₉	—	2	1	no	—
162	C₂²×C₉	C₂²⋊Q₈	1	2	yes	—
163	C₂²×C₉	C₂²×D₄	1	1	yes	characteristic
164	C₄×C₉	—	2	4	no	—
165	C₄×C₉	—	2	1	no	—
166	C₄×C₉	—	2	1	no	—
167	C₄×C₉	—	2	1	no	—
168	C₄×C₉	—	4	2	no	—
169	C₄×C₉	C₂²⋊Q₈	1	2	yes	—
170	C₄×C₉	#137	1	2	yes	—
171	C₄×C₉	C₂²×Q₈	1	1	yes	characteristic
172	C₄×C₉	C₂×#41	1	1	yes	characteristic
173	Dic₉	—	4	2	no	—
174	Dic₉	—	4	2	no	—
175	C₂²×Q₈	C₂×D₉	1	1	yes	characteristic
176	C₂×M₁₆	—	9	2	no	—
177	C₄⋊Q₈	—	9	1	no	—
178	C₄⋊Q₈	—	9	1	no	—
179	#125	—	9	2	no	—
180	C₂²⋊Q₈	—	18	1	no	—
181	C₂²⋊Q₈	—	18	1	no	—
182	C₄×Q₈	—	2	1	no	—
183	C₄×Q₈	—	2	1	no	—
184	C₄×Q₈	—	18	1	no	—
185	C₄×Q₈	—	18	1	no	—
186	C₄×Q₈	C₂×D₉	1	2	yes	—
187	C₄×Q₈	C₂×D₉	1	2	yes	—
188	#117	—	9	2	no	—
189	C₂×C₄⋊C₄	—	2	1	no	—
190	C₂×C₄⋊C₄	C₂×D₉	1	1	yes	characteristic
191	C₂×C₄²	—	2	1	no	—
192	C₂×C₄²	C₂×D₉	1	1	yes	characteristic
193	#107	—	9	4	no	—
194	#106	—	9	4	no	—
195	C₄⋊C₈	—	9	2	no	—
196	C₄⋊C₈	—	9	2	no	—
197	#103	—	9	4	no	—
198	#103	—	9	4	no	—
199	C₂³×C₃	C₂×#67	1	1	yes	characteristic
200	C₃×Q₈	—	2	4	no	—
201	C₃×Q₈	—	2	2	no	—
202	C₃×Q₈	C₂×Dic₆	1	4	yes	—
203	C₂×C₃×C₄	—	2	4	no	—
204	C₂×C₃×C₄	—	2	2	no	—
205	C₂×C₃×C₄	—	2	2	no	—
206	C₂×C₃×C₄	—	2	2	no	—
207	C₂×C₃×C₄	—	4	1	no	—
208	C₂×C₃×C₄	—	4	2	no	—
209	C₂×C₃×C₄	C₂×Dic₆	1	2	yes	—
210	C₂×C₃×C₄	#235	1	2	yes	—
211	C₂×C₃×C₄	C₂×#67	1	1	yes	characteristic
212	C₂×C₃×C₄	C₂×#67	1	1	yes	characteristic
213	C₂×C₃×C₄	C₂×#67	1	1	yes	characteristic
214	C₂×C₃×C₄	C₂×#67	1	1	yes	characteristic
215	C₂×C₃×C₄	C₂×#67	1	1	yes	characteristic
216	C₂×C₃×C₄	C₂³×D₃	1	1	yes	characteristic
217	C₂×Dic₃	—	6	2	no	—
218	C₂×Dic₃	—	6	2	no	—
219	Dic₆	—	12	2	no	—
220	Dic₆	—	12	2	no	—
221	C₃⋊C₈	—	6	4	no	—
222	C₂×C₉	—	2	2	no	—
223	C₂×C₉	C₂×C₂²⋊Q₈	1	1	yes	characteristic
224	C₂×C₉	C₂×#137	1	1	yes	characteristic
225	C₂×C₉	#575	1	1	yes	characteristic
226	C₂×Q₈	—	2	2	no	—
227	C₂×Q₈	—	2	1	no	—
228	C₂×Q₈	—	18	1	no	—
229	C₂×Q₈	—	18	1	no	—
230	C₂×Q₈	Dic₁₈	1	4	yes	complemented
231	C₂×Q₈	C₂²×D₉	1	2	yes	—
232	C₂²×C₄	—	2	1	no	—
233	C₂²×C₄	—	2	1	no	—
234	C₂²×C₄	#611	1	1	yes	characteristic
235	C₂²×C₄	#611	1	1	yes	characteristic
236	C₂²×C₄	C₂²×D₉	1	1	yes	characteristic
237	M₁₆	—	18	4	no	—
238	C₂×C₈	—	9	4	no	—
239	C₄⋊C₄	—	2	2	no	—
240	C₄⋊C₄	—	2	2	no	—
241	C₄⋊C₄	—	2	1	no	—
242	C₄⋊C₄	—	9	2	no	—
243	C₄⋊C₄	—	9	2	no	—
244	C₄⋊C₄	—	18	2	no	—
245	C₄⋊C₄	—	18	2	no	—
246	C₄⋊C₄	—	18	1	no	—
247	C₄⋊C₄	—	18	1	no	—
248	C₄⋊C₄	C₂²×D₉	1	2	yes	—
249	C₂²⋊C₄	—	18	2	no	—
250	C₄²	—	2	2	no	—
251	C₄²	—	2	2	no	—
252	C₄²	—	18	2	no	—
253	C₄²	#611	1	1	yes	characteristic
254	C₄²	#611	1	1	yes	characteristic
255	C₄²	C₂²×D₉	1	1	yes	characteristic
256	C₄²	C₂²×D₉	1	1	yes	characteristic
257	C₂²×C₃	—	2	1	no	—
258	C₂²×C₃	—	2	1	no	—
259	C₂²×C₃	#924	1	2	yes	—
260	C₂²×C₃	C₂²×#67	1	1	yes	characteristic
261	C₃×C₄	—	2	4	no	—
262	C₃×C₄	—	2	1	no	—
263	C₃×C₄	—	2	1	no	—
264	C₃×C₄	—	2	1	no	—
265	C₃×C₄	—	4	2	no	—
266	C₃×C₄	#924	1	2	yes	—
267	C₃×C₄	#942	1	2	yes	—
268	C₃×C₄	C₂²×Dic₆	1	1	yes	characteristic
269	C₃×C₄	C₂×#235	1	1	yes	characteristic
270	Dic₃	—	12	2	no	—
271	Dic₃	—	12	2	no	—
272	C₉	#3078	1	1	yes	characteristic, complemented, Sylow
273	C₂³	C₂×#611	1	1	yes	characteristic
274	Q₈	—	2	2	no	—
275	Q₈	—	2	4	no	—
276	Q₈	—	36	2	no	—
277	Q₈	—	36	2	no	—
278	Q₈	C₂×Dic₁₈	1	4	yes	—
279	C₂×C₄	—	2	2	no	—
280	C₂×C₄	—	2	4	no	—
281	C₂×C₄	—	2	2	no	—
282	C₂×C₄	—	2	2	no	—
283	C₂×C₄	—	4	1	no	—
284	C₂×C₄	—	4	2	no	—
285	C₂×C₄	—	18	2	no	—
286	C₂×C₄	—	18	2	no	—
287	C₂×C₄	C₂×Dic₁₈	1	2	yes	—
288	C₂×C₄	#3697	1	2	yes	—
289	C₂×C₄	C₂×#611	1	1	yes	characteristic
290	C₂×C₄	C₂×#611	1	1	yes	characteristic
291	C₂×C₄	C₂×#611	1	1	yes	characteristic
292	C₂×C₄	C₂×#611	1	1	yes	characteristic
293	C₂×C₄	C₂×#611	1	1	yes	characteristic
294	C₂×C₄	C₂³×D₉	1	1	yes	characteristic
295	C₈	—	18	4	no	—
296	C₂×C₃	—	2	2	no	—
297	C₂×C₃	C₂×#924	1	1	yes	characteristic
298	C₂×C₃	C₂×#942	1	1	yes	characteristic
299	C₂×C₃	#5822	1	1	yes	characteristic
300	C₂²	—	2	1	no	—
301	C₂²	—	2	1	no	—
302	C₂²	#63486	1	2	yes	—
303	C₂²	C₂²×#611	1	1	yes	characteristic
304	C₄	—	2	4	no	—
305	C₄	—	2	1	no	—
306	C₄	—	2	1	no	—
307	C₄	—	2	1	no	—
308	C₄	—	4	2	no	—
309	C₄	—	36	2	no	—
310	C₄	—	36	2	no	—
311	C₄	#63486	1	2	yes	—
312	C₄	#63505	1	2	yes	—
313	C₄	C₂²×Dic₁₈	1	1	yes	characteristic
314	C₄	C₂×#3697	1	1	yes	characteristic
315	C₃	#82431	1	1	yes	characteristic
316	C₂	—	2	2	no	—
317	C₂	C₂×#63486	1	1	yes	characteristic
318	C₂	C₂×#63505	1	1	yes	characteristic
319	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	C₂×C₄×Q₈×C₉	C₂
socle	C₂²×C₃	C₂²×#67
prime radical	C₂³×C₃	C₂×#67
hypercenter	C₂×C₄×Q₈	D₉
hypocenter	C₉	#3078
LCS	G ▷ C₂×C₄×C₉ ▷ C₂×C₉ ▷ C₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×Q₈

Complements

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

N	H ≅ G/N	type
4. #10588704	C₂	semidirect
5. #10588703	C₂	semidirect
6. #10588702	C₂	semidirect
44. C₂×Q₈×C₉	Q₈	semidirect
230. C₂×Q₈	Dic₁₈	semidirect
272. C₉	#3078	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×Q₈×C₉	2
#10589572	2
#10588906	2
#10588704	2
#10588703	2
#10588702	2
#10588644	2
#10588604	2
#82431	3