Subgroups — C₂×#10588796 #11866734

Order 1152.

924 conjugacy classes of subgroups.

The subgroup lattice has 288 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₈×C₉	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#63492	C₂	1	2	yes	maximal, complemented
3	C₂²×D₇₂	C₂	1	1	yes	maximal, characteristic, complemented
4	#10588796	C₂	1	8	yes	maximal, direct product factor
5	C₂×#63380	C₂	1	2	yes	maximal, complemented
6	C₂×#63378	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#5115	—	3	1	no	maximal
8	C₂³×C₄×C₉	C₂²	1	1	yes	characteristic
9	C₂²×#611	—	2	2	no	—
10	C₂²×D₃₆	C₂²	1	2	yes	—
11	C₂²×C₈×C₉	—	2	4	no	—
12	C₂²×C₈×C₉	C₂²	1	1	yes	characteristic
13	C₂²×C₈×C₉	C₂²	1	1	yes	characteristic
14	C₂²×C₈×C₉	C₂²	1	4	yes	complemented
15	#63492	—	2	4	no	—
16	#63492	C₂²	1	8	yes	complemented
17	C₂×(C₂×D₉)⋊C₄	—	2	2	no	—
18	C₂×C₄⋊Dic₉	C₂²	1	2	yes	—
19	C₂×D₇₂	—	2	4	no	—
20	C₂×D₇₂	C₂²	1	4	yes	complemented
21	#63380	C₂²	1	8	yes	complemented
22	#63378	C₂²	1	4	yes	complemented
23	C₂³×C₃×C₈	D₃	1	1	yes	characteristic
24	C₂×#930	—	3	2	no	—
25	C₂²×D₂₄	—	3	1	no	—
26	#5115	—	3	8	no	—
27	C₂×#821	—	3	2	no	—
28	C₂×#818	—	3	1	no	—
29	C₂⁴×C₉	D₄	1	1	yes	characteristic
30	C₂³×D₉	—	2	2	no	—
31	C₂²×C₄×C₉	—	2	2	no	—
32	C₂²×C₄×C₉	—	2	2	no	—
33	C₂²×C₄×C₉	D₄	1	1	yes	characteristic
34	C₂²×C₄×C₉	D₄	1	2	yes	—
35	C₂²×C₄×C₉	C₂³	1	2	yes	—
36	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
37	C₂×#611	—	2	8	no	—
38	C₂×#611	—	4	8	no	—
39	C₂²×Dic₉	—	2	2	no	—
40	C₂×D₃₆	—	2	4	no	—
41	C₂×D₃₆	—	2	4	no	—
42	C₂×D₃₆	—	2	2	no	—
43	C₂×D₃₆	C₂³	1	4	yes	—
44	C₂×C₈×C₉	—	2	2	no	—
45	C₂×C₈×C₉	—	2	8	no	—
46	C₂×C₈×C₉	D₄	1	4	yes	complemented
47	C₂×C₈×C₉	C₂³	1	2	yes	—
48	C₂×C₈×C₉	C₂³	1	2	yes	—
49	(C₂×D₉)⋊C₄	—	2	8	no	—
50	C₄⋊Dic₉	—	2	2	no	—
51	C₄⋊Dic₉	C₂³	1	4	yes	—
52	D₇₂	—	2	8	no	—
53	C₂×#466	—	9	1	no	Sylow
54	C₂³×C₃×C₄	C₂×D₃	1	1	yes	characteristic
55	C₂²×#67	—	6	2	no	—
56	C₂²×D₁₂	—	3	2	no	—
57	C₂²×C₃×C₈	—	2	4	no	—
58	C₂²×C₃×C₈	C₂×D₃	1	1	yes	characteristic
59	C₂²×C₃×C₈	C₂×D₃	1	1	yes	characteristic
60	C₂²×C₃×C₈	C₂×D₃	1	4	yes	—
61	#930	—	3	8	no	—
62	#930	—	6	4	no	—
63	C₂×(C₂×D₃)⋊C₄	—	6	2	no	—
64	C₂×C₄⋊Dic₃	—	3	2	no	—
65	C₂×D₂₄	—	3	4	no	—
66	C₂×D₂₄	—	6	4	no	—
67	#821	—	3	8	no	—
68	#818	—	3	4	no	—
69	C₂³×C₉	—	2	2	no	—
70	C₂³×C₉	—	2	2	no	—
71	C₂³×C₉	D₈	1	4	yes	complemented
72	C₂³×C₉	C₂×D₄	1	2	yes	—
73	C₂³×C₉	C₂×D₄	1	1	yes	characteristic
74	C₂²×D₉	—	2	4	no	—
75	C₂²×D₉	—	4	4	no	—
76	C₂²×D₉	—	4	4	no	—
77	C₂²×D₉	—	4	4	no	—
78	C₂×C₄×C₉	—	2	2	no	—
79	C₂×C₄×C₉	—	2	4	no	—
80	C₂×C₄×C₉	—	2	2	no	—
81	C₂×C₄×C₉	—	2	2	no	—
82	C₂×C₄×C₉	C₂×D₄	1	2	yes	—
83	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
84	C₂×C₄×C₉	C₂×D₄	1	2	yes	—
85	C₂×C₄×C₉	#41	1	2	yes	—
86	C₂×C₄×C₉	C₂⁴	1	1	yes	characteristic
87	#611	—	4	16	no	—
88	C₂×Dic₉	—	2	4	no	—
89	C₂×Dic₉	—	4	4	no	—
90	D₃₆	—	2	4	no	—
91	D₃₆	—	2	4	no	—
92	D₃₆	—	4	4	no	—
93	C₈×C₉	—	2	2	no	—
94	C₈×C₉	C₂×D₄	1	4	yes	complemented
95	C₂²×D₈	—	9	1	no	—
96	C₂³×C₈	D₉	1	1	yes	characteristic, complemented
97	C₂×#121	—	9	2	no	—
98	#466	—	9	8	no	—
99	C₂×#107	—	9	1	no	—
100	C₂×#102	—	9	2	no	—
101	C₂⁴×C₃	D₁₂	1	1	yes	characteristic
102	C₂³×D₃	—	6	2	no	—
103	C₂²×C₃×C₄	—	2	2	no	—
104	C₂²×C₃×C₄	—	2	2	no	—
105	C₂²×C₃×C₄	D₁₂	1	1	yes	characteristic
106	C₂²×C₃×C₄	D₁₂	1	2	yes	—
107	C₂²×C₃×C₄	C₂²×D₃	1	2	yes	—
108	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
109	C₂×#67	—	6	8	no	—
110	C₂×#67	—	12	8	no	—
111	C₂²×Dic₃	—	6	2	no	—
112	C₂×D₁₂	—	3	4	no	—
113	C₂×D₁₂	—	6	4	no	—
114	C₂×D₁₂	—	6	4	no	—
115	C₂×D₁₂	—	6	2	no	—
116	C₂×C₃×C₈	—	2	2	no	—
117	C₂×C₃×C₈	—	2	8	no	—
118	C₂×C₃×C₈	#67	1	4	yes	—
119	C₂×C₃×C₈	C₂²×D₃	1	2	yes	—
120	C₂×C₃×C₈	C₂²×D₃	1	2	yes	—
121	(C₂×D₃)⋊C₄	—	6	8	no	—
122	C₄⋊Dic₃	—	3	4	no	—
123	C₄⋊Dic₃	—	6	2	no	—
124	D₂₄	—	6	8	no	—
125	C₂²×C₉	—	2	2	no	—
126	C₂²×C₉	—	2	2	no	—
127	C₂²×C₉	—	2	8	no	—
128	C₂²×C₉	#121	1	2	yes	—
129	C₂²×C₉	C₂×D₈	1	2	yes	—
130	C₂²×C₉	C₂×D₈	1	4	yes	complemented
131	C₂²×C₉	#135	1	2	yes	—
132	C₂²×C₉	C₂²×D₄	1	1	yes	characteristic
133	C₂×D₉	—	4	4	no	—
134	C₂×D₉	—	4	4	no	—
135	C₂×D₉	—	4	4	no	—
136	C₂×D₉	—	8	8	no	—
137	C₄×C₉	—	2	2	no	—
138	C₄×C₉	#121	1	2	yes	—
139	C₄×C₉	C₂²×D₄	1	1	yes	characteristic
140	C₄×C₉	C₂×#41	1	1	yes	characteristic
141	Dic₉	—	4	4	no	—
142	C₂²×D₄	—	9	2	no	—
143	C₂²×D₄	—	18	2	no	—
144	C₂³×C₄	C₂×D₉	1	1	yes	characteristic
145	C₂×D₈	—	9	4	no	—
146	C₂×D₈	—	18	4	no	—
147	C₂²×C₈	—	2	4	no	—
148	C₂²×C₈	C₂×D₉	1	1	yes	characteristic
149	C₂²×C₈	C₂×D₉	1	1	yes	characteristic
150	C₂²×C₈	C₂×D₉	1	4	yes	complemented
151	#121	—	9	8	no	—
152	#121	—	18	4	no	—
153	C₂×C₄⋊C₄	—	9	2	no	—
154	C₂×C₂²⋊C₄	—	18	2	no	—
155	#107	—	9	4	no	—
156	#102	—	9	8	no	—
157	C₂³×C₃	—	2	2	no	—
158	C₂³×C₃	—	2	2	no	—
159	C₂³×C₃	D₂₄	1	4	yes	—
160	C₂³×C₃	C₂×D₁₂	1	2	yes	—
161	C₂³×C₃	C₂×D₁₂	1	1	yes	characteristic
162	C₂²×D₃	—	6	4	no	—
163	C₂²×D₃	—	12	4	no	—
164	C₂²×D₃	—	12	4	no	—
165	C₂²×D₃	—	12	4	no	—
166	C₂×C₃×C₄	—	2	2	no	—
167	C₂×C₃×C₄	—	2	4	no	—
168	C₂×C₃×C₄	—	2	2	no	—
169	C₂×C₃×C₄	—	2	2	no	—
170	C₂×C₃×C₄	C₂×D₁₂	1	2	yes	—
171	C₂×C₃×C₄	C₂×D₁₂	1	1	yes	characteristic
172	C₂×C₃×C₄	#235	1	2	yes	—
173	C₂×C₃×C₄	C₂×#67	1	2	yes	—
174	C₂×C₃×C₄	C₂³×D₃	1	1	yes	characteristic
175	#67	—	12	16	no	—
176	C₂×Dic₃	—	6	4	no	—
177	C₂×Dic₃	—	12	4	no	—
178	D₁₂	—	6	4	no	—
179	D₁₂	—	6	4	no	—
180	D₁₂	—	12	4	no	—
181	C₃×C₈	—	2	2	no	—
182	C₃×C₈	C₂×#67	1	4	yes	—
183	C₂×C₉	—	2	4	no	—
184	C₂×C₉	#466	1	4	yes	complemented
185	C₂×C₉	C₂×#121	1	1	yes	characteristic
186	C₂×C₉	C₂²×D₈	1	1	yes	characteristic
187	C₂×C₉	C₂×#135	1	1	yes	characteristic
188	D₉	—	8	4	no	—
189	C₂⁴	—	18	2	no	—
190	C₂⁴	D₃₆	1	1	yes	characteristic
191	C₂×D₄	—	9	4	no	—
192	C₂×D₄	—	18	8	no	—
193	C₂×D₄	—	18	2	no	—
194	C₂×D₄	—	18	4	no	—
195	C₂×D₄	—	18	4	no	—
196	C₂×D₄	—	36	8	no	—
197	C₂²×C₄	—	2	2	no	—
198	C₂²×C₄	—	2	2	no	—
199	C₂²×C₄	—	18	2	no	—
200	C₂²×C₄	D₃₆	1	2	yes	—
201	C₂²×C₄	D₃₆	1	1	yes	characteristic
202	C₂²×C₄	C₂²×D₉	1	2	yes	—
203	C₂²×C₄	C₂²×D₉	1	1	yes	characteristic
204	D₈	—	18	8	no	—
205	C₂×C₈	—	2	2	no	—
206	C₂×C₈	—	2	8	no	—
207	C₂×C₈	#611	1	4	yes	complemented
208	C₂×C₈	C₂²×D₉	1	2	yes	—
209	C₂×C₈	C₂²×D₉	1	2	yes	—
210	C₄⋊C₄	—	9	4	no	—
211	C₄⋊C₄	—	18	2	no	—
212	C₂²⋊C₄	—	18	8	no	—
213	C₂²×C₃	—	2	2	no	—
214	C₂²×C₃	—	2	2	no	—
215	C₂²×C₃	—	2	8	no	—
216	C₂²×C₃	C₂×D₂₄	1	2	yes	—
217	C₂²×C₃	C₂×D₂₄	1	4	yes	—
218	C₂²×C₃	#904	1	2	yes	—
219	C₂²×C₃	#930	1	2	yes	—
220	C₂²×C₃	C₂²×D₁₂	1	1	yes	characteristic
221	C₂×D₃	—	12	4	no	—
222	C₂×D₃	—	12	4	no	—
223	C₂×D₃	—	12	4	no	—
224	C₂×D₃	—	24	8	no	—
225	C₃×C₄	—	2	2	no	—
226	C₃×C₄	#930	1	2	yes	—
227	C₃×C₄	C₂×#235	1	1	yes	characteristic
228	C₃×C₄	C₂²×#67	1	1	yes	characteristic
229	Dic₃	—	12	4	no	—
230	C₉	C₂×#466	1	1	yes	characteristic, complemented, Sylow
231	C₂³	—	2	2	no	—
232	C₂³	—	2	2	no	—
233	C₂³	—	18	4	no	—
234	C₂³	—	36	4	no	—
235	C₂³	—	36	4	no	—
236	C₂³	—	36	4	no	—
237	C₂³	D₇₂	1	4	yes	complemented
238	C₂³	C₂×D₃₆	1	2	yes	—
239	C₂³	C₂×D₃₆	1	1	yes	characteristic
240	D₄	—	18	4	no	—
241	D₄	—	18	4	no	—
242	D₄	—	36	16	no	—
243	D₄	—	36	4	no	—
244	C₂×C₄	—	2	2	no	—
245	C₂×C₄	—	2	4	no	—
246	C₂×C₄	—	2	2	no	—
247	C₂×C₄	—	2	2	no	—
248	C₂×C₄	—	18	4	no	—
249	C₂×C₄	—	36	4	no	—
250	C₂×C₄	C₂×D₃₆	1	2	yes	—
251	C₂×C₄	C₂×D₃₆	1	1	yes	characteristic
252	C₂×C₄	#3697	1	2	yes	—
253	C₂×C₄	C₂×#611	1	2	yes	—
254	C₂×C₄	C₂³×D₉	1	1	yes	characteristic
255	C₈	—	2	2	no	—
256	C₈	C₂×#611	1	4	yes	complemented
257	C₂×C₃	—	2	4	no	—
258	C₂×C₃	#5115	1	4	yes	—
259	C₂×C₃	C₂²×D₂₄	1	1	yes	characteristic
260	C₂×C₃	C₂×#904	1	1	yes	characteristic
261	C₂×C₃	C₂×#930	1	1	yes	characteristic
262	D₃	—	24	4	no	—
263	C₂²	—	2	8	no	—
264	C₂²	—	2	2	no	—
265	C₂²	—	2	2	no	—
266	C₂²	—	36	4	no	—
267	C₂²	—	36	4	no	—
268	C₂²	—	36	4	no	—
269	C₂²	—	72	8	no	—
270	C₂²	C₂×D₇₂	1	2	yes	—
271	C₂²	C₂×D₇₂	1	4	yes	complemented
272	C₂²	#63467	1	2	yes	—
273	C₂²	#63492	1	2	yes	—
274	C₂²	C₂²×D₃₆	1	1	yes	characteristic
275	C₄	—	2	2	no	—
276	C₄	—	36	4	no	—
277	C₄	#63492	1	2	yes	—
278	C₄	C₂×#3697	1	1	yes	characteristic
279	C₄	C₂²×#611	1	1	yes	characteristic
280	C₃	C₂×#5115	1	1	yes	characteristic
281	C₂	—	2	4	no	—
282	C₂	—	72	4	no	—
283	C₂	#10588796	1	4	yes	direct product factor
284	C₂	C₂²×D₇₂	1	1	yes	characteristic
285	C₂	C₂×#63467	1	1	yes	characteristic
286	C₂	C₂×#63492	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×D₃₆
commutator, cosocle	C₂×C₄×C₉	C₂⁴
Frattini	C₂×C₃×C₄	C₂³×D₃
Fitting	C₂³×C₈×C₉	C₂
socle	C₂³×C₃	C₂×D₁₂
hypercenter	C₂³×C₈	D₉
hypocenter	C₉	C₂×#466
LCS	G ▷ C₂×C₄×C₉ ▷ C₂×C₉ ▷ C₉
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂³×C₈

Complements

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

N	H ≅ G/N	type
1. C₂³×C₈×C₉	C₂	semidirect
2. C₂×#63492	C₂	semidirect
3. C₂²×D₇₂	C₂	semidirect
4. #10588796	C₂	direct (N × H)
5. C₂×#63380	C₂	semidirect
6. C₂×#63378	C₂	semidirect
14. C₂²×C₈×C₉	C₂²	semidirect
16. #63492	C₂²	semidirect
20. C₂×D₇₂	C₂²	semidirect
21. #63380	C₂²	semidirect
22. #63378	C₂²	semidirect
46. C₂×C₈×C₉	D₄	semidirect
71. C₂³×C₉	D₈	semidirect
94. C₈×C₉	C₂×D₄	semidirect
96. C₂³×C₈	D₉	semidirect
130. C₂²×C₉	C₂×D₈	semidirect
150. C₂²×C₈	C₂×D₉	semidirect
184. C₂×C₉	#466	semidirect
207. C₂×C₈	#611	semidirect
230. C₉	C₂×#466	semidirect
237. C₂³	D₇₂	semidirect
256. C₈	C₂×#611	semidirect
271. C₂²	C₂×D₇₂	semidirect
283. C₂	#10588796	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×C₈×C₉	2
C₂×#63492	2
C₂²×D₇₂	2
#10588796	2
C₂×#63380	2
C₂×#63378	2
C₂×#5115	3

Subgroups — C2×#10588796 #11866734

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#10588796 #11866734