Subgroups — #63809

Order 288.

135 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#3796	C₂	1	2	yes	maximal, complemented
2	C₈×C₃⋊D₃	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×#204	C₂	1	2	yes	maximal, complemented
4	#3718	C₂	1	1	yes	maximal, characteristic, complemented
5	#3713	C₂	1	1	yes	maximal, characteristic, complemented
6	#916	—	3	2	no	maximal
7	C₄×C₃⋊D₃	C₂²	1	1	yes	characteristic
8	C₃×D₁₂	C₂²	1	2	yes	—
9	C₃×Dic₆	C₂²	1	2	yes	—
10	#626	—	2	2	no	—
11	D₃×Dic₃	—	2	2	no	—
12	C₃²×C₈	C₂²	1	1	yes	characteristic
13	C₃⋊(C₃⋊C₈)	C₂²	1	1	yes	characteristic
14	#239	—	3	2	no	—
15	#237	—	3	2	no	—
16	C₃×SD₁₆	—	3	2	no	—
17	#216	—	3	2	no	—
18	#213	—	3	2	no	—
19	#204	D₃	1	2	yes	complemented
20	D₃×C₈	—	3	2	no	—
21	D₃×C₈	—	6	1	no	—
22	C₂×C₃⋊D₃	D₄	1	1	yes	characteristic
23	C₂×C₃×D₃	—	2	2	no	—
24	C₃²×C₄	C₂³	1	1	yes	characteristic
25	C₃⋊Dic₃	D₄	1	1	yes	characteristic
26	C₃×Dic₃	—	2	2	no	—
27	#135	—	9	1	no	Sylow
28	C₃×Q₈	—	3	2	no	—
29	C₃×D₄	—	3	2	no	—
30	#67	—	6	2	no	—
31	C₂×Dic₃	—	6	2	no	—
32	D₁₂	—	6	2	no	—
33	D₁₂	C₂×D₃	1	2	yes	—
34	C₄×D₃	—	3	2	no	—
35	C₄×D₃	—	6	1	no	—
36	C₄×D₃	—	6	2	no	—
37	Dic₆	C₂×D₃	1	2	yes	—
38	C₃×C₈	—	2	1	no	—
39	C₃×C₈	C₂×D₃	1	2	yes	—
40	C₃⋊C₈	—	3	2	no	—
41	C₃⋊C₈	—	6	1	no	—
42	C₂×C₃²	C₂×D₄	1	1	yes	characteristic
43	C₃⋊D₃	—	2	1	no	—
44	C₃×D₃	—	4	2	no	—
45	#41	—	9	2	no	—
46	Q₁₆	—	9	1	no	—
47	SD₁₆	—	3	2	no	—
48	D₈	—	9	1	no	—
49	C₂×C₈	—	9	1	no	—
50	C₂²×C₃	—	6	2	no	—
51	C₂×D₃	—	2	2	no	—
52	C₂×D₃	—	3	2	no	—
53	C₂×D₃	—	6	1	no	—
54	C₃×C₄	—	2	1	no	—
55	C₃×C₄	—	6	2	no	—
56	C₃×C₄	C₂²×D₃	1	2	yes	—
57	Dic₃	—	2	2	no	—
58	Dic₃	—	3	2	no	—
59	Dic₃	—	6	1	no	—
60	C₃²	#135	1	1	yes	characteristic, complemented, Sylow
61	Q₈	—	3	2	no	—
62	D₄	—	3	2	no	—
63	D₄	—	18	2	no	—
64	C₂×C₄	—	9	1	no	—
65	C₂×C₄	—	18	2	no	—
66	C₈	—	9	1	no	—
67	C₈	D₃²	1	1	yes	characteristic
68	C₂×C₃	—	2	1	no	—
69	C₂×C₃	—	12	2	no	—
70	C₂×C₃	D₃×D₄	1	2	yes	—
71	D₃	—	4	2	no	—
72	D₃	—	6	2	no	—
73	D₃	—	6	2	no	—
74	C₂²	—	6	2	no	—
75	C₂²	—	9	1	no	—
76	C₄	—	6	2	no	—
77	C₄	—	9	1	no	—
78	C₄	C₂×D₃²	1	1	yes	characteristic
79	C₃	—	2	1	no	—
80	C₃	#916	1	2	yes	complemented
81	C₂	—	12	2	no	—
82	C₂	—	18	1	no	—
83	C₂	#3802	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#3802
commutator, cosocle	C₃²×C₄	C₂³
Frattini	C₄	C₂×D₃²
Fitting	C₃²×C₈	C₂²
socle	C₂×C₃²	C₂×D₄
hypercenter	C₈	D₃²
hypocenter	C₃²	#135
LCS	G ▷ C₃²×C₄ ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈

Complements

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

N	H ≅ G/N	type
1. #3796	C₂	semidirect
2. C₈×C₃⋊D₃	C₂	semidirect
3. C₃×#204	C₂	semidirect
4. #3718	C₂	semidirect
5. #3713	C₂	semidirect
19. #204	D₃	semidirect
60. C₃²	#135	semidirect
80. C₃	#916	semidirect

Maximal subgroups

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

subgroup	index
#3796	2
C₈×C₃⋊D₃	2
C₃×#204	2
#3718	2
#3713	2
#916	3