Finite Group Databasebeta

Subgroups — #11772968

Order 1152.

190 conjugacy classes of subgroups; only the 56 normal subgroups are shown.

normal subgroups only

This group has 190 conjugacy classes of subgroups — too many to draw. Only its 56 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#10592603	C₂	1	1	yes	maximal, characteristic
2	C₃×#4505	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×#4499	C₂	1	1	yes	maximal, characteristic, complemented
4	#67606	—	3	1	no	maximal
5	#67580	—	3	1	no	maximal
6	#64113	—	2	1	no	—
7	C₂×C₃²×C₁₆	C₂²	1	1	yes	characteristic
8	#63628	C₄	1	1	yes	characteristic, complemented
9	#63627	—	2	1	no	—
10	#63626	C₄	1	1	yes	characteristic, complemented
11	C₃×#818	—	2	1	no	—
12	C₃×C₃⋊C₃₂	—	2	1	no	—
13	#4904	—	3	1	no	—
14	#4904	—	3	1	no	—
15	#4904	—	6	1	no	—
16	C₃×M₆₄	—	3	1	no	—
17	C₃×#367	—	3	1	no	—
18	#4505	D₃	1	1	yes	characteristic, complemented
19	#4499	D₃	1	1	yes	characteristic, complemented
20	#3828	—	4	1	no	—
21	C₂×C₃²×C₈	D₄	1	1	yes	characteristic
22	#3746	—	2	1	no	—
23	#3745	—	2	1	no	—
24	#3744	—	4	1	no	—
25	C₃×C₄⋊Dic₃	—	4	1	no	—
26	C₃²×C₁₆	C₂×C₄	1	1	yes	characteristic
27	C₃²×C₁₆	D₄	1	1	yes	characteristic
28	#1418	—	9	1	no	Sylow
29	#904	—	6	2	no	—
30	#904	—	6	1	no	—
31	#904	—	6	1	no	—
32	C₂×C₃×C₁₆	—	2	1	no	—
33	C₂×C₃×C₁₆	C₂×D₃	1	1	yes	characteristic
34	C₂×C₃×C₁₆	C₂×D₃	1	1	yes	characteristic
35	C₃×#107	—	6	1	no	—
36	#818	—	2	1	no	—
37	Dic₂₄	—	3	1	no	—
38	Dic₂₄	—	3	1	no	—
39	Dic₂₄	—	6	1	no	—
40	#800	—	6	2	no	—
41	#800	—	6	1	no	—
42	#800	—	6	1	no	—
43	D₄₈	—	3	1	no	—
44	D₄₈	—	3	1	no	—
45	D₄₈	—	6	1	no	—
46	C₃×C₃₂	—	6	1	no	—
47	C₃⋊C₃₂	—	2	1	no	—
48	C₂×C₃²×C₄	D₈	1	1	yes	characteristic
49	#638	—	8	1	no	—
50	C₄⋊(C₃⋊D₃)	—	4	1	no	—
51	C₄×C₃⋊D₃	—	8	1	no	—
52	#634	—	4	1	no	—
53	C₂×C₃×Dic₃	—	8	1	no	—
54	C₃²×C₈	C₂²⋊C₄	1	1	yes	characteristic
55	C₃²×C₈	SD₁₆	1	1	yes	characteristic
56	#508	—	9	1	no	—
57	M₆₄	—	3	1	no	—
58	#367	—	3	1	no	—
59	#235	—	12	2	no	—
60	#235	—	12	1	no	—
61	#235	—	12	1	no	—
62	C₂×C₃×C₈	—	2	1	no	—
63	C₂×C₃×C₈	D₁₂	1	1	yes	characteristic
64	C₂×C₃×C₈	#67	1	1	yes	characteristic
65	C₃×C₄⋊C₄	—	12	1	no	—
66	C₄⋊Dic₃	—	4	1	no	—
67	Dic₁₂	—	6	2	no	—
68	Dic₁₂	—	6	1	no	—
69	Dic₁₂	—	6	1	no	—
70	D₂₄	—	6	2	no	—
71	D₂₄	—	6	1	no	—
72	D₂₄	—	6	1	no	—
73	#204	—	12	2	no	—
74	#204	—	12	1	no	—
75	#204	—	12	1	no	—
76	C₃×C₁₆	—	2	1	no	—
77	C₃×C₁₆	—	2	1	no	—
78	C₃×C₁₆	C₄×D₃	1	1	yes	characteristic
79	C₃×C₁₆	C₄×D₃	1	1	yes	characteristic
80	C₃×C₁₆	D₁₂	1	1	yes	characteristic
81	C₃×C₁₆	#67	1	1	yes	characteristic
82	C₂²×C₃²	D₁₆	1	1	yes	characteristic
83	C₂×C₃⋊D₃	—	8	1	no	—
84	C₃²×C₄	#102	1	1	yes	characteristic
85	C₃²×C₄	SD₃₂	1	1	yes	characteristic
86	C₃⋊Dic₃	—	8	1	no	—
87	C₃×Dic₃	—	16	1	no	—
88	#135	—	18	1	no	—
89	Q₃₂	—	9	1	no	—
90	SD₃₂	—	18	1	no	—
91	D₁₆	—	9	1	no	—
92	C₂×C₁₆	D₃²	1	1	yes	characteristic
93	#107	—	6	1	no	—
94	C₃₂	—	6	1	no	—
95	C₂×C₃×C₄	—	2	1	no	—
96	C₂×C₃×C₄	—	24	1	no	—
97	C₂×C₃×C₄	D₂₄	1	1	yes	characteristic
98	C₂×C₃×C₄	#213	1	1	yes	characteristic
99	#67	—	24	2	no	—
100	#67	—	24	1	no	—
101	#67	—	24	1	no	—
102	C₂×Dic₃	—	8	1	no	—
103	D₁₂	—	12	2	no	—
104	D₁₂	—	12	1	no	—
105	D₁₂	—	12	1	no	—
106	C₄×D₃	—	24	2	no	—
107	C₄×D₃	—	24	1	no	—
108	C₄×D₃	—	24	1	no	—
109	Dic₆	—	12	2	no	—
110	Dic₆	—	12	1	no	—
111	Dic₆	—	12	1	no	—
112	C₃×C₈	—	2	1	no	—
113	C₃×C₈	—	2	1	no	—
114	C₃×C₈	#204	1	1	yes	characteristic
115	C₃×C₈	(C₂×D₃)⋊C₄	1	1	yes	characteristic
116	C₃×C₈	(C₂×D₃)⋊C₄	1	1	yes	characteristic
117	C₃×C₈	#215	1	1	yes	characteristic
118	C₂×C₃²	—	2	1	no	—
119	C₂×C₃²	#357	1	1	yes	characteristic
120	C₃⋊D₃	—	16	1	no	—
121	#41	—	36	1	no	—
122	Q₁₆	—	18	1	no	—
123	SD₁₆	—	36	1	no	—
124	D₈	—	18	1	no	—
125	C₂×C₈	#626	1	1	yes	characteristic
126	C₄⋊C₄	—	12	1	no	—
127	C₁₆	#624	1	1	yes	characteristic
128	C₁₆	#626	1	1	yes	characteristic
129	C₂²×C₃	—	2	1	no	—
130	C₂²×C₃	D₄₈	1	1	yes	characteristic
131	C₂²×C₃	#826	1	1	yes	characteristic
132	C₂×D₃	—	24	2	no	—
133	C₂×D₃	—	24	1	no	—
134	C₂×D₃	—	24	1	no	—
135	C₃×C₄	—	2	1	no	—
136	C₃×C₄	—	2	1	no	—
137	C₃×C₄	—	48	1	no	—
138	C₃×C₄	#800	1	1	yes	characteristic
139	C₃×C₄	#809	1	1	yes	characteristic
140	C₃×C₄	#821	1	1	yes	characteristic
141	C₃×C₄	#828	1	1	yes	characteristic
142	Dic₃	—	16	1	no	—
143	Dic₃	—	24	2	no	—
144	Dic₃	—	24	1	no	—
145	Dic₃	—	24	1	no	—
146	C₃²	#1418	1	1	yes	characteristic, complemented, Sylow
147	Q₈	—	36	1	no	—
148	D₄	—	36	1	no	—
149	D₄	—	72	1	no	—
150	C₂×C₄	—	24	1	no	—
151	C₂×C₄	—	72	1	no	—
152	C₂×C₄	#3714	1	1	yes	characteristic
153	C₈	#3717	1	1	yes	characteristic
154	C₈	#3722	1	1	yes	characteristic
155	C₂×C₃	—	2	1	no	—
156	C₂×C₃	—	2	1	no	—
157	C₂×C₃	—	2	1	no	—
158	C₂×C₃	—	4	1	no	—
159	C₂×C₃	#4491	1	1	yes	characteristic
160	C₂×C₃	#4509	1	1	yes	characteristic
161	D₃	—	48	2	no	—
162	D₃	—	48	1	no	—
163	D₃	—	48	1	no	—
164	C₂²	—	72	1	no	—
165	C₂²	#63546	1	1	yes	characteristic
166	C₄	—	48	1	no	—
167	C₄	—	72	1	no	—
168	C₄	#63549	1	1	yes	characteristic
169	C₄	#63564	1	1	yes	characteristic
170	C₃	—	2	1	no	—
171	C₃	#67580	1	1	yes	characteristic, complemented
172	C₃	#67606	1	1	yes	characteristic, complemented
173	C₂	—	2	1	no	—
174	C₂	—	144	1	no	—
175	C₂	#10589116	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#10589116
commutator	C₃²×C₁₆	C₂×C₄
Frattini, hypercenter	C₂×C₁₆	D₃²
Fitting, cosocle	C₂×C₃²×C₁₆	C₂²
socle	C₂×C₃²	#357
prime radical	#10592603	C₂
hypocenter	C₃²	#1418
LCS	G ▷ C₃²×C₁₆ ▷ C₃²×C₈ ▷ C₃²×C₄ ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆

Complements

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

N	H ≅ G/N	type
2. C₃×#4505	C₂	semidirect
3. C₃×#4499	C₂	semidirect
8. #63628	C₄	semidirect
10. #63626	C₄	semidirect
18. #4505	D₃	semidirect
19. #4499	D₃	semidirect
146. C₃²	#1418	semidirect
171. C₃	#67580	semidirect
172. C₃	#67606	semidirect

Maximal subgroups

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

subgroup	index
#10592603	2
C₃×#4505	2
C₃×#4499	2
#67606	3
#67580	3