Subgroups — C₂×C₃×#5231 #11808109

Order 1152.

2497 conjugacy classes of subgroups.

The subgroup lattice has 278 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₃²×Q₈	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₃×(C₂×D₃)⋊C₄	C₂	1	3	yes	maximal
3	C₃×#5231	C₂	1	8	yes	maximal, direct product factor
4	C₂×C₃×C₃⋊#95	C₂	1	3	yes	maximal, complemented
5	C₂×C₃×#393	—	3	1	no	maximal
6	C₂×#5231	C₃	1	1	yes	maximal, characteristic, direct product factor
7	C₂⁴×C₃×D₃	C₂²	1	1	yes	characteristic
8	C₂²×C₃²×Q₈	—	2	12	no	—
9	C₂²×C₃²×Q₈	C₂²	1	4	yes	complemented
10	C₂³×C₃²×C₄	C₂²	1	3	yes	—
11	C₂³×C₃×Dic₃	C₂²	1	3	yes	—
12	C₂×C₃×(C₂×D₃)⋊C₄	—	2	12	no	—
13	C₂×C₃×(C₂×D₃)⋊C₄	C₂²	1	12	yes	—
14	C₃×C₃⋊#95	C₂²	1	12	yes	complemented
15	C₂³×C₃×Q₈	—	2	1	no	—
16	C₂³×C₃×Q₈	D₃	1	1	yes	characteristic, complemented
17	C₂³×C₃×Q₈	C₂×C₃	1	1	yes	characteristic, complemented
18	C₂²×C₃×C₂²⋊C₄	—	3	3	no	—
19	C₂²×(C₂×D₃)⋊C₄	C₂×C₃	1	3	yes	—
20	C₃×#393	—	3	8	no	—
21	C₂×C₃×#95	—	3	3	no	—
22	#5231	C₂×C₃	1	8	yes	direct product factor
23	C₂×C₃⋊#95	C₂×C₃	1	3	yes	complemented
24	C₂⁴×C₃²	C₂³	1	1	yes	characteristic
25	C₂³×C₃×D₃	—	2	6	no	—
26	C₂³×C₃×D₃	—	2	6	no	—
27	C₂³×C₃×D₃	Q₈	1	4	yes	complemented
28	C₂³×C₃×D₃	C₂³	1	2	yes	—
29	C₂×C₃²×Q₈	—	2	32	no	—
30	C₂×C₃²×Q₈	—	2	24	no	—
31	C₂²×C₃²×C₄	—	2	12	no	—
32	C₂²×C₃²×C₄	D₄	1	12	yes	—
33	C₂²×C₃²×C₄	C₂³	1	6	yes	—
34	C₂²×C₃×Dic₃	—	2	6	no	—
35	C₂²×C₃×Dic₃	—	2	12	no	—
36	C₂²×C₃×Dic₃	C₂³	1	6	yes	—
37	C₃×(C₂×D₃)⋊C₄	—	2	24	no	—
38	C₂×#393	—	3	1	no	Sylow
39	C₂⁵×C₃	—	3	1	no	—
40	C₂⁴×D₃	C₂²×C₃	1	1	yes	characteristic
41	C₂²×C₃×Q₈	—	2	24	no	—
42	C₂²×C₃×Q₈	—	2	12	no	—
43	C₂²×C₃×Q₈	—	2	12	no	—
44	C₂²×C₃×Q₈	—	2	4	no	—
45	C₂²×C₃×Q₈	C₂×D₃	1	4	yes	complemented
46	C₂²×C₃×Q₈	C₂²×C₃	1	4	yes	complemented
47	C₂³×C₃×C₄	—	2	3	no	—
48	C₂³×C₃×C₄	—	3	3	no	—
49	C₂³×C₃×C₄	C₂×D₃	1	3	yes	—
50	C₂³×C₃×C₄	C₂²×C₃	1	3	yes	—
51	C₂³×Dic₃	C₂²×C₃	1	3	yes	—
52	C₂×C₃×C₂²⋊C₄	—	3	12	no	—
53	C₂×C₃×C₂²⋊C₄	—	6	12	no	—
54	C₂×(C₂×D₃)⋊C₄	—	2	12	no	—
55	C₂×(C₂×D₃)⋊C₄	C₂²×C₃	1	12	yes	—
56	C₃×#95	—	3	12	no	—
57	C₃⋊#95	C₂²×C₃	1	12	yes	complemented
58	C₂³×C₃²	C₂×D₄	1	6	yes	—
59	C₂³×C₃²	C₂×Q₈	1	2	yes	—
60	C₂³×C₃²	#41	1	6	yes	—
61	C₂³×C₃²	C₂⁴	1	1	yes	characteristic
62	C₂²×C₃×D₃	—	2	24	no	—
63	C₂²×C₃×D₃	—	2	6	no	—
64	C₂²×C₃×D₃	—	2	6	no	—
65	C₂²×C₃×D₃	—	4	4	no	—
66	C₂²×C₃×D₃	—	4	12	no	—
67	C₂²×C₃×D₃	C₂×Q₈	1	4	yes	complemented
68	C₃²×Q₈	—	2	32	no	—
69	C₂×C₃²×C₄	—	2	24	no	—
70	C₂×C₃²×C₄	—	2	12	no	—
71	C₂×C₃²×C₄	C₂×D₄	1	12	yes	—
72	C₂×C₃×Dic₃	—	2	6	no	—
73	C₂×C₃×Dic₃	—	2	12	no	—
74	C₂×C₃×Dic₃	—	4	12	no	—
75	C₂³×Q₈	C₃×D₃	1	1	yes	characteristic, complemented
76	C₂²×C₂²⋊C₄	—	3	3	no	—
77	#393	—	3	8	no	—
78	C₂×#95	—	3	3	no	—
79	C₂⁴×C₃	—	2	1	no	—
80	C₂⁴×C₃	—	3	4	no	—
81	C₂⁴×C₃	—	3	2	no	—
82	C₂⁴×C₃	—	6	6	no	—
83	C₂⁴×C₃	—	6	6	no	—
84	C₂⁴×C₃	C₂²×D₃	1	1	yes	characteristic
85	C₂⁴×C₃	C₂³×C₃	1	1	yes	characteristic
86	C₂³×D₃	—	2	6	no	—
87	C₂³×D₃	—	2	6	no	—
88	C₂³×D₃	C₃×Q₈	1	4	yes	complemented
89	C₂³×D₃	C₂³×C₃	1	2	yes	—
90	C₂×C₃×Q₈	—	2	64	no	—
91	C₂×C₃×Q₈	—	2	32	no	—
92	C₂×C₃×Q₈	—	2	32	no	—
93	C₂×C₃×Q₈	—	2	48	no	—
94	C₂×C₃×Q₈	—	2	24	no	—
95	C₂×C₃×Q₈	—	2	24	no	—
96	C₂²×C₃×C₄	—	2	24	no	—
97	C₂²×C₃×C₄	—	2	12	no	—
98	C₂²×C₃×C₄	—	2	12	no	—
99	C₂²×C₃×C₄	—	2	12	no	—
100	C₂²×C₃×C₄	—	2	6	no	—
101	C₂²×C₃×C₄	—	3	6	no	—
102	C₂²×C₃×C₄	—	6	12	no	—
103	C₂²×C₃×C₄	—	6	6	no	—
104	C₂²×C₃×C₄	#67	1	12	yes	—
105	C₂²×C₃×C₄	C₃×D₄	1	12	yes	—
106	C₂²×C₃×C₄	C₂²×D₃	1	6	yes	—
107	C₂²×C₃×C₄	C₂³×C₃	1	6	yes	—
108	C₂²×Dic₃	—	2	6	no	—
109	C₂²×Dic₃	—	2	12	no	—
110	C₂²×Dic₃	C₂³×C₃	1	6	yes	—
111	C₃×C₂²⋊C₄	—	6	24	no	—
112	(C₂×D₃)⋊C₄	—	2	24	no	—
113	C₂²×C₃²	#120	1	4	yes	—
114	C₂²×C₃²	C₂²⋊Q₈	1	12	yes	—
115	C₂²×C₃²	#124	1	12	yes	—
116	C₂²×C₃²	C₂²×D₄	1	3	yes	—
117	C₂²×C₃²	C₂²×Q₈	1	1	yes	characteristic
118	C₂²×C₃²	C₂×#41	1	3	yes	—
119	C₂×C₃×D₃	—	2	12	no	—
120	C₂×C₃×D₃	—	4	2	no	—
121	C₂×C₃×D₃	—	4	6	no	—
122	C₂×C₃×D₃	—	4	16	no	—
123	C₃²×C₄	—	2	12	no	—
124	C₃×Dic₃	—	4	6	no	—
125	C₂⁵	—	3	1	no	—
126	C₂²×Q₈	—	2	12	no	—
127	C₂²×Q₈	C₂×C₃×D₃	1	4	yes	complemented
128	C₂³×C₄	—	3	3	no	—
129	C₂³×C₄	C₂×C₃×D₃	1	3	yes	—
130	C₂×C₂²⋊C₄	—	3	12	no	—
131	C₂×C₂²⋊C₄	—	6	12	no	—
132	#95	—	3	12	no	—
133	C₂³×C₃	—	2	1	no	—
134	C₂³×C₃	—	2	2	no	—
135	C₂³×C₃	—	2	6	no	—
136	C₂³×C₃	—	2	6	no	—
137	C₂³×C₃	—	3	4	no	—
138	C₂³×C₃	—	6	24	no	—
139	C₂³×C₃	—	6	6	no	—
140	C₂³×C₃	—	6	6	no	—
141	C₂³×C₃	—	12	12	no	—
142	C₂³×C₃	—	12	4	no	—
143	C₂³×C₃	D₃×Q₈	1	2	yes	—
144	C₂³×C₃	#239	1	6	yes	—
145	C₂³×C₃	C₂×#67	1	6	yes	—
146	C₂³×C₃	C₂×C₃×D₄	1	6	yes	—
147	C₂³×C₃	C₂×C₃×Q₈	1	2	yes	—
148	C₂³×C₃	C₃×#41	1	6	yes	—
149	C₂³×C₃	C₂³×D₃	1	1	yes	characteristic
150	C₂³×C₃	C₂⁴×C₃	1	1	yes	characteristic
151	C₂²×D₃	—	2	24	no	—
152	C₂²×D₃	—	2	6	no	—
153	C₂²×D₃	—	2	6	no	—
154	C₂²×D₃	—	4	4	no	—
155	C₂²×D₃	—	4	12	no	—
156	C₂²×D₃	C₂×C₃×Q₈	1	4	yes	complemented
157	C₃×Q₈	—	2	64	no	—
158	C₃×Q₈	—	2	32	no	—
159	C₃×Q₈	—	2	32	no	—
160	C₂×C₃×C₄	—	2	48	no	—
161	C₂×C₃×C₄	—	2	24	no	—
162	C₂×C₃×C₄	—	2	24	no	—
163	C₂×C₃×C₄	—	2	24	no	—
164	C₂×C₃×C₄	—	2	12	no	—
165	C₂×C₃×C₄	—	2	12	no	—
166	C₂×C₃×C₄	—	2	12	no	—
167	C₂×C₃×C₄	—	6	12	no	—
168	C₂×C₃×C₄	—	6	6	no	—
169	C₂×C₃×C₄	—	12	12	no	—
170	C₂×C₃×C₄	C₂×#67	1	12	yes	—
171	C₂×C₃×C₄	C₂×C₃×D₄	1	12	yes	—
172	C₂×Dic₃	—	2	6	no	—
173	C₂×Dic₃	—	2	12	no	—
174	C₂×Dic₃	—	4	12	no	—
175	C₂×C₃²	#393	1	8	yes	complemented
176	C₂×C₃²	C₂×#120	1	1	yes	characteristic
177	C₂×C₃²	C₂×C₂²⋊Q₈	1	3	yes	—
178	C₂×C₃²	C₂×#124	1	3	yes	—
179	C₃×D₃	—	4	4	no	—
180	C₂⁴	—	3	2	no	—
181	C₂⁴	—	3	4	no	—
182	C₂⁴	—	6	6	no	—
183	C₂⁴	—	6	6	no	—
184	C₂⁴	C₂²×C₃×D₃	1	1	yes	characteristic
185	C₂×Q₈	—	2	32	no	—
186	C₂×Q₈	—	2	24	no	—
187	C₂²×C₄	—	2	12	no	—
188	C₂²×C₄	—	3	6	no	—
189	C₂²×C₄	—	6	12	no	—
190	C₂²×C₄	—	6	6	no	—
191	C₂²×C₄	C₃×#67	1	12	yes	—
192	C₂²×C₄	C₂²×C₃×D₃	1	6	yes	—
193	C₂²⋊C₄	—	6	24	no	—
194	C₂²×C₃	—	2	12	no	—
195	C₂²×C₃	—	2	12	no	—
196	C₂²×C₃	—	2	4	no	—
197	C₂²×C₃	—	2	1	no	—
198	C₂²×C₃	—	2	3	no	—
199	C₂²×C₃	—	2	3	no	—
200	C₂²×C₃	—	6	12	no	—
201	C₂²×C₃	—	12	16	no	—
202	C₂²×C₃	—	12	6	no	—
203	C₂²×C₃	—	12	2	no	—
204	C₂²×C₃	#946	1	12	yes	—
205	C₂²×C₃	#947	1	12	yes	—
206	C₂²×C₃	#953	1	4	yes	—
207	C₂²×C₃	C₃×#120	1	4	yes	—
208	C₂²×C₃	C₃×C₂²⋊Q₈	1	12	yes	—
209	C₂²×C₃	C₃×#124	1	12	yes	—
210	C₂²×C₃	C₂×D₃×Q₈	1	1	yes	characteristic
211	C₂²×C₃	C₂×#239	1	3	yes	—
212	C₂²×C₃	C₂²×#67	1	3	yes	—
213	C₂²×C₃	C₂²×C₃×D₄	1	3	yes	—
214	C₂²×C₃	C₂²×C₃×Q₈	1	1	yes	characteristic
215	C₂²×C₃	C₂×C₃×#41	1	3	yes	—
216	C₂×D₃	—	2	12	no	—
217	C₂×D₃	—	4	16	no	—
218	C₂×D₃	—	4	2	no	—
219	C₂×D₃	—	4	6	no	—
220	C₃×C₄	—	2	24	no	—
221	C₃×C₄	—	2	12	no	—
222	C₃×C₄	—	2	12	no	—
223	C₃×C₄	—	12	6	no	—
224	Dic₃	—	4	6	no	—
225	C₃²	C₂×#393	1	1	yes	characteristic, complemented, Sylow
226	C₂³	—	3	4	no	—
227	C₂³	—	6	24	no	—
228	C₂³	—	6	6	no	—
229	C₂³	—	6	6	no	—
230	C₂³	—	12	12	no	—
231	C₂³	—	12	4	no	—
232	C₂³	C₃×D₃×Q₈	1	2	yes	—
233	C₂³	C₃×#239	1	6	yes	—
234	C₂³	C₂×C₃×#67	1	6	yes	—
235	C₂³	C₂³×C₃×D₃	1	1	yes	characteristic
236	Q₈	—	2	32	no	—
237	C₂×C₄	—	2	24	no	—
238	C₂×C₄	—	2	12	no	—
239	C₂×C₄	—	6	12	no	—
240	C₂×C₄	—	6	6	no	—
241	C₂×C₄	—	12	12	no	—
242	C₂×C₄	C₂×C₃×#67	1	12	yes	—
243	C₂×C₃	—	2	3	no	—
244	C₂×C₃	—	2	3	no	—
245	C₂×C₃	—	2	8	no	—
246	C₂×C₃	—	2	1	no	—
247	C₂×C₃	—	12	4	no	—
248	C₂×C₃	#5231	1	8	yes	direct product factor
249	C₂×C₃	C₃×#393	1	8	yes	complemented
250	C₂×C₃	C₂×#946	1	3	yes	—
251	C₂×C₃	C₂×#947	1	3	yes	—
252	C₂×C₃	C₂×#953	1	1	yes	characteristic
253	C₂×C₃	C₂×C₃×#120	1	1	yes	characteristic
254	C₂×C₃	C₂×C₃×C₂²⋊Q₈	1	3	yes	—
255	C₂×C₃	C₂×C₃×#124	1	3	yes	—
256	D₃	—	4	4	no	—
257	C₂²	—	6	12	no	—
258	C₂²	—	12	6	no	—
259	C₂²	—	12	16	no	—
260	C₂²	—	12	2	no	—
261	C₂²	C₃×#946	1	12	yes	—
262	C₂²	C₃×#947	1	12	yes	—
263	C₂²	C₃×#953	1	4	yes	—
264	C₂²	C₂×C₃×D₃×Q₈	1	1	yes	characteristic
265	C₂²	C₂×C₃×#239	1	3	yes	—
266	C₂²	C₂²×C₃×#67	1	3	yes	—
267	C₄	—	2	12	no	—
268	C₄	—	12	6	no	—
269	C₃	—	2	1	no	—
270	C₃	C₂×#5231	1	1	yes	characteristic, direct product factor
271	C₃	C₂×C₃×#393	1	1	yes	characteristic, complemented
272	C₂	—	12	4	no	—
273	C₂	C₃×#5231	1	8	yes	direct product factor
274	C₂	C₂×C₃×#946	1	3	yes	—
275	C₂	C₂×C₃×#947	1	3	yes	—
276	C₂	C₂×C₃×#953	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴×C₃	C₂²×D₃
commutator, cosocle	C₂³×C₃	C₂⁴×C₃
Frattini	C₂³	C₂³×C₃×D₃
Fitting	C₂³×C₃²×Q₈	C₂
socle	C₂⁴×C₃²	C₂³
prime radical	C₂⁴×C₃×D₃	C₂²
hypercenter	C₂³×C₃×Q₈	D₃
hypocenter	C₃	C₂×C₃×#393
LCS	G ▷ C₂³×C₃ ▷ C₃
UCS	C₁ ◁ C₂⁴×C₃ ◁ C₂³×C₃×Q₈

Complements

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

N	H ≅ G/N	type
1. C₂³×C₃²×Q₈	C₂	semidirect
3. C₃×#5231	C₂	direct (N × H)
4. C₂×C₃×C₃⋊#95	C₂	semidirect
6. C₂×#5231	C₃	direct (N × H)
9. C₂²×C₃²×Q₈	C₂²	semidirect
14. C₃×C₃⋊#95	C₂²	semidirect
16. C₂³×C₃×Q₈	D₃	semidirect
17. C₂³×C₃×Q₈	C₂×C₃	semidirect
22. #5231	C₂×C₃	direct (N × H)
23. C₂×C₃⋊#95	C₂×C₃	semidirect
27. C₂³×C₃×D₃	Q₈	semidirect
45. C₂²×C₃×Q₈	C₂×D₃	semidirect
46. C₂²×C₃×Q₈	C₂²×C₃	semidirect
57. C₃⋊#95	C₂²×C₃	semidirect
67. C₂²×C₃×D₃	C₂×Q₈	semidirect
75. C₂³×Q₈	C₃×D₃	semidirect
88. C₂³×D₃	C₃×Q₈	semidirect
127. C₂²×Q₈	C₂×C₃×D₃	semidirect
156. C₂²×D₃	C₂×C₃×Q₈	semidirect
175. C₂×C₃²	#393	semidirect
225. C₃²	C₂×#393	semidirect
248. C₂×C₃	#5231	direct (N × H)
249. C₂×C₃	C₃×#393	semidirect
270. C₃	C₂×#5231	direct (N × H)
271. C₃	C₂×C₃×#393	semidirect
273. C₂	C₃×#5231	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×C₃²×Q₈	2
C₂²×C₃×(C₂×D₃)⋊C₄	2
C₃×#5231	2
C₂×C₃×C₃⋊#95	2
C₂×C₃×#393	3
C₂×#5231	3

Subgroups — C2×C3×#5231 #11808109

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×#5231 #11808109