Subgroups — #11772884

Order 1152.

234 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×#491	C₂	1	1	yes	maximal, characteristic
2	#10589414	C₂	1	2	yes	maximal
3	#67786	—	3	4	no	maximal
4	C₃²×C₄⋊Q₈	C₂²	1	1	yes	characteristic
5	C₃²×#125	C₄	1	1	yes	characteristic
6	C₃²×#103	—	2	2	no	—
7	C₃²×#97	C₄	1	1	yes	characteristic
8	#63663	—	2	2	no	—
9	C₃×#491	D₃	1	4	yes	—
10	#4548	—	3	8	no	—
11	C₂×C₃²×Q₈	D₄	1	2	yes	—
12	C₂×C₃²×C₈	—	2	1	no	—
13	C₃²×C₄⋊C₄	—	2	1	no	—
14	C₃²×C₄⋊C₄	—	4	1	no	—
15	C₃²×C₄⋊C₄	—	4	1	no	—
16	C₃²×C₄²	C₂×C₄	1	1	yes	characteristic
17	#3748	—	4	2	no	—
18	#1413	—	9	1	no	Sylow
19	C₃×C₄⋊Q₈	C₂×D₃	1	4	yes	—
20	C₃×#125	Dic₃	1	4	yes	—
21	C₃×#103	—	2	8	no	—
22	C₃×#97	Dic₃	1	4	yes	—
23	#836	—	6	8	no	—
24	C₃²×Q₈	—	4	2	no	—
25	C₂×C₃²×C₄	—	2	2	no	—
26	C₂×C₃²×C₄	—	2	1	no	—
27	C₂×C₃²×C₄	—	4	1	no	—
28	C₂×C₃²×C₄	C₂²⋊C₄	1	1	yes	characteristic
29	C₃²×C₈	—	4	1	no	—
30	C₃⋊(C₃⋊C₈)	—	8	2	no	—
31	#491	C₃⋊D₃	1	1	yes	characteristic
32	#356	—	9	2	no	—
33	C₂×C₃×Q₈	#67	1	8	yes	—
34	C₂×C₃×C₈	—	2	4	no	—
35	C₃×C₄⋊C₄	—	2	4	no	—
36	C₃×C₄⋊C₄	—	4	4	no	—
37	C₃×C₄⋊C₄	—	4	4	no	—
38	C₃×C₄²	C₂×Dic₃	1	4	yes	—
39	#208	—	12	8	no	—
40	C₂²×C₃²	#99	1	1	yes	characteristic
41	C₃²×C₄	—	2	1	no	—
42	C₃²×C₄	—	4	1	no	—
43	C₃²×C₄	—	4	2	no	—
44	C₃²×C₄	—	8	1	no	—
45	C₄⋊Q₈	C₂×C₃⋊D₃	1	1	yes	characteristic
46	#125	C₃⋊Dic₃	1	1	yes	characteristic
47	#103	—	2	2	no	—
48	#101	—	18	2	no	—
49	#97	C₃⋊Dic₃	1	1	yes	characteristic
50	C₃×Q₈	—	4	8	no	—
51	C₂×C₃×C₄	—	2	8	no	—
52	C₂×C₃×C₄	—	2	4	no	—
53	C₂×C₃×C₄	—	4	4	no	—
54	C₂×C₃×C₄	#217	1	4	yes	—
55	C₃×C₈	—	4	4	no	—
56	C₃⋊C₈	—	24	8	no	—
57	C₂×C₃²	—	2	1	no	—
58	C₂×C₃²	#354	1	1	yes	characteristic
59	C₂×Q₈	#638	1	2	yes	—
60	M₁₆	—	36	2	no	—
61	C₂×C₈	—	2	1	no	—
62	C₄⋊C₄	—	2	1	no	—
63	C₄⋊C₄	—	4	1	no	—
64	C₄⋊C₄	—	4	1	no	—
65	C₄²	C₂×C₃⋊Dic₃	1	1	yes	characteristic
66	C₂²×C₃	#834	1	4	yes	—
67	C₃×C₄	—	2	4	no	—
68	C₃×C₄	—	4	4	no	—
69	C₃×C₄	—	4	8	no	—
70	C₃×C₄	—	8	4	no	—
71	C₃²	#1413	1	1	yes	characteristic, complemented, Sylow
72	Q₈	—	4	2	no	—
73	C₂×C₄	—	2	1	no	—
74	C₂×C₄	—	2	2	no	—
75	C₂×C₄	—	4	1	no	—
76	C₂×C₄	#3757	1	1	yes	characteristic
77	C₈	—	4	1	no	—
78	C₈	—	72	2	no	—
79	C₂×C₃	—	2	4	no	—
80	C₂×C₃	#4541	1	4	yes	—
81	C₂²	#63661	1	1	yes	characteristic
82	C₄	—	2	1	no	—
83	C₄	—	4	1	no	—
84	C₄	—	4	2	no	—
85	C₄	—	8	1	no	—
86	C₃	#67786	1	4	yes	complemented
87	C₂	—	2	1	no	—
88	C₂	#10589407	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#10589407
commutator	C₃²×C₄²	C₂×C₄
Frattini	C₄⋊Q₈	C₂×C₃⋊D₃
Fitting	C₃²×#491	C₂
socle	C₂×C₃²	#354
prime radical	C₂²×C₃²	#99
cosocle	C₃²×C₄⋊Q₈	C₂²
hypercenter	#491	C₃⋊D₃
hypocenter	C₃²	#1413
LCS	G ▷ C₃²×C₄² ▷ C₂×C₃²×C₄ ▷ C₂²×C₃² ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂×C₄ ◁ C₄⋊Q₈ ◁ #491

Complements

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

N	H ≅ G/N	type
71. C₃²	#1413	semidirect
86. C₃	#67786	semidirect

Maximal subgroups

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

subgroup	index
C₃²×#491	2
#10589414	2
#67786	3