Subgroups — #11786234

Order 1152.

641 conjugacy classes of subgroups.

The subgroup lattice has 226 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₈×C₃⋊Dic₃	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×#818	C₂	1	2	yes	maximal
3	#10591359	C₂	1	2	yes	maximal
4	C₂×#63576	C₂	1	2	yes	maximal
5	#70606	—	3	2	no	maximal
6	C₂²×C₃²×C₈	C₂²	1	1	yes	characteristic
7	C₂×C₄×C₃⋊Dic₃	C₂²	1	1	yes	characteristic
8	C₂²×C₃⋊(C₃⋊C₈)	C₂²	1	1	yes	characteristic
9	C₂×C₃×C₄⋊Dic₃	C₂²	1	4	yes	—
10	C₂×#3724	—	2	4	no	—
11	C₈×C₃⋊Dic₃	C₄	1	4	yes	complemented
12	C₃×#818	—	2	4	no	—
13	#63576	—	2	4	no	—
14	C₂×C₃×#107	—	3	2	no	—
15	C₂×#818	D₃	1	2	yes	—
16	C₂×C₈×Dic₃	—	3	2	no	—
17	C₂×C₈×Dic₃	—	6	1	no	—
18	#4975	—	3	4	no	—
19	C₂×#807	—	3	4	no	—
20	C₂²×C₃²×C₄	C₂³	1	1	yes	characteristic
21	C₂²×C₃⋊Dic₃	D₄	1	2	yes	—
22	C₂²×C₃×Dic₃	—	2	4	no	—
23	C₂×C₃²×C₈	C₂×C₄	1	2	yes	—
24	C₂×C₃²×C₈	D₄	1	2	yes	—
25	C₂×C₃²×C₈	Q₈	1	2	yes	—
26	C₄×C₃⋊Dic₃	C₂×C₄	1	2	yes	—
27	C₄×C₃⋊Dic₃	Q₈	1	2	yes	—
28	C₂×C₃⋊(C₃⋊C₈)	C₂×C₄	1	2	yes	—
29	C₂×C₃⋊(C₃⋊C₈)	D₄	1	2	yes	—
30	C₂×C₃⋊(C₃⋊C₈)	Q₈	1	2	yes	—
31	C₃×C₄⋊Dic₃	—	2	4	no	—
32	C₃×C₄⋊Dic₃	—	2	4	no	—
33	#3724	—	4	8	no	—
34	#1845	—	9	1	no	Sylow
35	C₂²×C₃×C₈	—	2	1	no	—
36	C₂²×C₃×C₈	C₂×D₃	1	2	yes	—
37	C₂×C₃×C₄⋊C₄	—	3	4	no	—
38	C₂×C₄⋊Dic₃	C₂×D₃	1	4	yes	—
39	C₂×Dic₃⋊C₄	—	6	8	no	—
40	C₂×C₄×Dic₃	—	3	2	no	—
41	C₂×C₄×Dic₃	—	6	1	no	—
42	C₂²×C₃⋊C₈	—	3	2	no	—
43	C₂²×C₃⋊C₈	—	6	1	no	—
44	C₃×#107	—	6	4	no	—
45	#818	—	2	4	no	—
46	C₈×Dic₃	—	3	8	no	—
47	C₈×Dic₃	—	6	4	no	—
48	#807	—	6	8	no	—
49	C₂³×C₃²	C₂×D₄	1	1	yes	characteristic
50	C₂×C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
51	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
52	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
53	C₂×C₃²×C₄	C₂×Q₈	1	1	yes	characteristic
54	C₂×C₃²×C₄	C₂×Q₈	1	1	yes	characteristic
55	C₂×C₃²×C₄	C₂×Q₈	1	1	yes	characteristic
56	C₂×C₃⋊Dic₃	C₄⋊C₄	1	4	yes	—
57	C₂×C₃⋊Dic₃	D₈	1	4	yes	—
58	C₂×C₃⋊Dic₃	Q₁₆	1	4	yes	—
59	C₂×C₃×Dic₃	—	4	4	no	—
60	C₂×C₃×Dic₃	—	4	8	no	—
61	C₃²×C₈	C₄⋊C₄	1	4	yes	complemented
62	C₃⋊(C₃⋊C₈)	C₄⋊C₄	1	4	yes	complemented
63	C₂×#107	—	3	2	no	—
64	C₂×#107	—	9	2	no	—
65	C₂×C₄×C₈	—	9	1	no	—
66	#384	—	9	2	no	—
67	C₂²×C₃×C₄	—	2	1	no	—
68	C₂²×C₃×C₄	—	6	4	no	—
69	C₂²×C₃×C₄	C₂²×D₃	1	2	yes	—
70	C₂²×Dic₃	—	2	4	no	—
71	C₂²×Dic₃	—	3	4	no	—
72	C₂²×Dic₃	—	6	2	no	—
73	C₂×C₃×C₈	—	2	2	no	—
74	C₂×C₃×C₈	—	2	2	no	—
75	C₂×C₃×C₈	—	2	2	no	—
76	C₂×C₃×C₈	Dic₆	1	4	yes	—
77	C₂×C₃×C₈	C₄×D₃	1	4	yes	—
78	C₂×C₃×C₈	#67	1	4	yes	—
79	C₃×C₄⋊C₄	—	6	4	no	—
80	C₃×C₄⋊C₄	—	6	4	no	—
81	C₄⋊Dic₃	—	2	4	no	—
82	C₄⋊Dic₃	—	2	4	no	—
83	Dic₃⋊C₄	—	12	16	no	—
84	C₄×Dic₃	—	3	4	no	—
85	C₄×Dic₃	—	3	4	no	—
86	C₄×Dic₃	—	6	2	no	—
87	C₄×Dic₃	—	6	2	no	—
88	C₂×C₃⋊C₈	—	3	4	no	—
89	C₂×C₃⋊C₈	—	3	4	no	—
90	C₂×C₃⋊C₈	—	3	4	no	—
91	C₂×C₃⋊C₈	—	6	2	no	—
92	C₂×C₃⋊C₈	—	6	2	no	—
93	C₂×C₃⋊C₈	—	6	2	no	—
94	C₂²×C₃²	C₂×C₄⋊C₄	1	1	yes	characteristic
95	C₂²×C₃²	#127	1	1	yes	characteristic
96	C₂²×C₃²	C₄⋊Q₈	1	1	yes	characteristic
97	C₂²×C₃²	C₂×D₈	1	2	yes	—
98	C₂²×C₃²	C₂×Q₁₆	1	2	yes	—
99	C₃²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
100	C₃²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
101	C₃²×C₄	C₄⋊Q₈	1	1	yes	characteristic
102	C₃²×C₄	C₄⋊Q₈	1	1	yes	characteristic
103	C₃⋊Dic₃	#107	1	8	yes	complemented
104	C₃×Dic₃	—	8	4	no	—
105	C₂²×C₈	—	9	1	no	—
106	C₂²×C₈	D₃²	1	1	yes	characteristic
107	C₂×C₄⋊C₄	—	3	4	no	—
108	C₂×C₄⋊C₄	—	18	4	no	—
109	C₂×C₄²	—	9	1	no	—
110	#107	—	6	4	no	—
111	#107	—	18	4	no	—
112	C₄×C₈	—	9	4	no	—
113	C₂³×C₃	—	2	1	no	—
114	C₂³×C₃	D₃×D₄	1	2	yes	—
115	C₂×C₃×C₄	—	2	1	no	—
116	C₂×C₃×C₄	—	2	1	no	—
117	C₂×C₃×C₄	—	2	1	no	—
118	C₂×C₃×C₄	—	2	1	no	—
119	C₂×C₃×C₄	—	2	1	no	—
120	C₂×C₃×C₄	—	2	1	no	—
121	C₂×C₃×C₄	—	12	8	no	—
122	C₂×C₃×C₄	—	12	4	no	—
123	C₂×C₃×C₄	C₂×Dic₆	1	2	yes	—
124	C₂×C₃×C₄	C₂×C₄×D₃	1	2	yes	—
125	C₂×C₃×C₄	D₃×D₄	1	2	yes	—
126	C₂×C₃×C₄	D₃×Q₈	1	2	yes	—
127	C₂×C₃×C₄	D₃×Q₈	1	2	yes	—
128	C₂×C₃×C₄	C₂×#67	1	2	yes	—
129	C₂×Dic₃	—	3	8	no	—
130	C₂×Dic₃	—	3	8	no	—
131	C₂×Dic₃	—	3	8	no	—
132	C₂×Dic₃	—	4	8	no	—
133	C₂×Dic₃	—	4	4	no	—
134	C₂×Dic₃	—	6	4	no	—
135	C₂×Dic₃	—	6	4	no	—
136	C₂×Dic₃	—	6	4	no	—
137	C₃×C₈	—	2	4	no	—
138	C₃×C₈	Dic₃⋊C₄	1	8	yes	complemented
139	C₃⋊C₈	—	3	8	no	—
140	C₃⋊C₈	—	6	4	no	—
141	C₂×C₃²	#384	1	1	yes	characteristic
142	C₂×C₃²	C₂×#107	1	2	yes	—
143	C₂×C₃²	#493	1	1	yes	characteristic
144	C₂×C₃²	#494	1	1	yes	characteristic
145	C₂×C₃²	#500	1	2	yes	—
146	C₂²×C₄	—	6	4	no	—
147	C₂²×C₄	—	9	2	no	—
148	C₂²×C₄	C₂×D₃²	1	1	yes	characteristic
149	C₂×C₈	—	9	2	no	—
150	C₂×C₈	—	9	2	no	—
151	C₂×C₈	—	9	2	no	—
152	C₂×C₈	#624	1	2	yes	—
153	C₂×C₈	#625	1	2	yes	—
154	C₂×C₈	#627	1	2	yes	—
155	C₄⋊C₄	—	6	4	no	—
156	C₄⋊C₄	—	6	4	no	—
157	C₄⋊C₄	—	36	8	no	—
158	C₄²	—	9	2	no	—
159	C₄²	—	9	2	no	—
160	C₂²×C₃	—	2	2	no	—
161	C₂²×C₃	—	2	2	no	—
162	C₂²×C₃	—	2	1	no	—
163	C₂²×C₃	—	2	1	no	—
164	C₂²×C₃	—	2	1	no	—
165	C₂²×C₃	#888	1	2	yes	—
166	C₂²×C₃	D₃×C₄⋊C₄	1	2	yes	—
167	C₂²×C₃	D₃×D₈	1	4	yes	—
168	C₂²×C₃	D₃×Q₁₆	1	4	yes	—
169	C₂²×C₃	#940	1	2	yes	—
170	C₃×C₄	—	2	1	no	—
171	C₃×C₄	—	2	1	no	—
172	C₃×C₄	—	2	1	no	—
173	C₃×C₄	—	2	1	no	—
174	C₃×C₄	—	24	4	no	—
175	C₃×C₄	#888	1	2	yes	—
176	C₃×C₄	D₃×C₄⋊C₄	1	2	yes	—
177	C₃×C₄	C₂×Dic₃⋊C₄	1	2	yes	—
178	C₃×C₄	#944	1	2	yes	—
179	Dic₃	—	3	16	no	—
180	Dic₃	—	6	8	no	—
181	Dic₃	—	8	4	no	—
182	C₃²	#1845	1	1	yes	characteristic, complemented, Sylow
183	C₂³	#3802	1	1	yes	characteristic
184	C₂×C₄	—	9	4	no	—
185	C₂×C₄	—	9	4	no	—
186	C₂×C₄	—	9	4	no	—
187	C₂×C₄	—	12	8	no	—
188	C₂×C₄	—	12	4	no	—
189	C₂×C₄	#3797	1	1	yes	characteristic
190	C₂×C₄	#3797	1	1	yes	characteristic
191	C₂×C₄	#3802	1	1	yes	characteristic
192	C₂×C₄	C₂×#624	1	1	yes	characteristic
193	C₂×C₄	C₂×#625	1	1	yes	characteristic
194	C₂×C₄	C₂×#627	1	1	yes	characteristic
195	C₈	—	9	4	no	—
196	C₈	#3724	1	4	yes	complemented
197	C₂×C₃	—	2	1	no	—
198	C₂×C₃	—	2	1	no	—
199	C₂×C₃	—	2	2	no	—
200	C₂×C₃	—	2	1	no	—
201	C₂×C₃	—	2	2	no	—
202	C₂×C₃	#4874	1	4	yes	—
203	C₂×C₃	D₃×#107	1	4	yes	—
204	C₂×C₃	#4975	1	2	yes	—
205	C₂×C₃	#5151	1	2	yes	—
206	C₂×C₃	#5183	1	2	yes	—
207	C₂²	#63797	1	2	yes	—
208	C₂²	#63802	1	2	yes	—
209	C₂²	#63900	1	1	yes	characteristic
210	C₂²	#63914	1	1	yes	characteristic
211	C₂²	#63919	1	1	yes	characteristic
212	C₄	—	9	8	no	—
213	C₄	—	24	4	no	—
214	C₄	#63873	1	1	yes	characteristic
215	C₄	#63900	1	1	yes	characteristic
216	C₄	#63919	1	1	yes	characteristic
217	C₄	C₂×#3724	1	1	yes	characteristic
218	C₃	—	2	1	no	—
219	C₃	#70606	1	2	yes	complemented
220	C₂	#10590687	1	2	yes	—
221	C₂	#10590886	1	1	yes	characteristic
222	C₂	#10590902	1	2	yes	—
223	C₂	#10590905	1	1	yes	characteristic
224	C₂	#10591359	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#3802
commutator	C₂×C₃²×C₄	C₂²×C₄
Frattini	C₂²×C₄	C₂×D₃²
Fitting	C₂²×C₃²×C₈	C₂²
socle, prime radical	C₂³×C₃²	C₂×D₄
cosocle	C₂²×C₃²×C₄	C₂³
hypercenter	C₂²×C₈	D₃²
hypocenter	C₃²	#1845
LCS	G ▷ C₂×C₃²×C₄ ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×C₈

Complements

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

N	H ≅ G/N	type
11. C₈×C₃⋊Dic₃	C₄	semidirect
61. C₃²×C₈	C₄⋊C₄	semidirect
62. C₃⋊(C₃⋊C₈)	C₄⋊C₄	semidirect
103. C₃⋊Dic₃	#107	semidirect
138. C₃×C₈	Dic₃⋊C₄	semidirect
182. C₃²	#1845	semidirect
196. C₈	#3724	semidirect
219. C₃	#70606	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₈×C₃⋊Dic₃	2
C₂×C₃×#818	2
#10591359	2
C₂×#63576	2
#70606	3