Subgroups — #63803

Order 288.

135 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#3798	C₂	1	1	yes	maximal, characteristic
2	C₈×C₃⋊D₃	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×D₃×C₈	C₂	1	2	yes	maximal, complemented
4	#3712	C₂	1	1	yes	maximal, characteristic, complemented
5	#3711	C₂	1	2	yes	maximal, complemented
6	#901	—	3	2	no	maximal
7	C₄×C₃⋊D₃	C₂²	1	1	yes	characteristic
8	C₃×C₄×D₃	C₂²	1	2	yes	—
9	#627	C₄	1	1	yes	characteristic
10	#626	C₄	1	2	yes	—
11	#625	C₄	1	1	yes	characteristic
12	C₃²×C₈	C₂²	1	1	yes	characteristic
13	C₃⋊(C₃⋊C₈)	C₂²	1	1	yes	characteristic
14	C₃×C₃⋊C₈	C₂²	1	2	yes	—
15	#235	—	3	2	no	—
16	C₂×C₃×C₈	—	3	2	no	—
17	#208	—	3	2	no	—
18	ΓL(1,𝔽₂₅)	—	3	2	no	—
19	ΓL(1,𝔽₂₅)	—	3	2	no	—
20	D₃×C₈	—	3	2	no	—
21	D₃×C₈	—	6	1	no	—
22	D₃×C₈	D₃	1	2	yes	complemented
23	C₂×C₃⋊D₃	C₂×C₄	1	1	yes	characteristic
24	C₂×C₃×D₃	C₂×C₄	1	2	yes	—
25	C₃²×C₄	C₂³	1	1	yes	characteristic
26	C₃⋊Dic₃	C₂×C₄	1	1	yes	characteristic
27	C₃×Dic₃	C₂×C₄	1	2	yes	—
28	#131	—	9	1	no	Sylow
29	C₂×C₃×C₄	—	3	2	no	—
30	#67	—	3	2	no	—
31	#67	—	3	2	no	—
32	D₁₂	—	3	2	no	—
33	C₄×D₃	—	3	2	no	—
34	C₄×D₃	—	6	1	no	—
35	C₄×D₃	C₂×D₃	1	2	yes	—
36	Dic₆	—	3	2	no	—
37	C₃×C₈	—	2	1	no	—
38	C₃×C₈	—	3	2	no	—
39	C₃×C₈	C₂×D₃	1	2	yes	—
40	C₃⋊C₈	—	3	2	no	—
41	C₃⋊C₈	—	6	1	no	—
42	C₃⋊C₈	C₂×D₃	1	2	yes	—
43	C₂×C₃²	C₂²×C₄	1	1	yes	characteristic
44	C₃⋊D₃	—	2	1	no	—
45	C₃×D₃	—	2	2	no	—
46	#41	—	9	1	no	—
47	M₁₆	—	9	1	no	—
48	M₁₆	—	9	2	no	—
49	C₂×C₈	—	3	2	no	—
50	C₂×C₈	—	9	1	no	—
51	C₂²×C₃	—	3	2	no	—
52	C₂×D₃	—	3	2	no	—
53	C₂×D₃	—	6	1	no	—
54	C₂×D₃	C₄×D₃	1	2	yes	—
55	C₃×C₄	—	2	1	no	—
56	C₃×C₄	—	3	2	no	—
57	C₃×C₄	C₂²×D₃	1	2	yes	—
58	Dic₃	—	3	2	no	—
59	Dic₃	—	6	1	no	—
60	Dic₃	C₄×D₃	1	2	yes	—
61	C₃²	#131	1	1	yes	characteristic, complemented, Sylow
62	Q₈	—	9	1	no	—
63	D₄	—	9	2	no	—
64	D₄	—	9	1	no	—
65	C₂×C₄	—	3	2	no	—
66	C₂×C₄	—	9	1	no	—
67	C₈	—	3	2	no	—
68	C₈	—	9	1	no	—
69	C₈	D₃²	1	1	yes	characteristic
70	C₂×C₃	—	2	1	no	—
71	C₂×C₃	—	6	2	no	—
72	C₂×C₃	C₂×C₄×D₃	1	2	yes	—
73	D₃	—	2	2	no	—
74	D₃	—	6	2	no	—
75	D₃	—	6	2	no	—
76	C₂²	—	3	2	no	—
77	C₂²	—	9	1	no	—
78	C₄	—	3	2	no	—
79	C₄	—	9	1	no	—
80	C₄	C₂×D₃²	1	1	yes	characteristic
81	C₃	—	2	1	no	—
82	C₃	#901	1	2	yes	complemented
83	C₂	—	6	2	no	—
84	C₂	—	18	1	no	—
85	C₂	C₄×D₃²	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₈	D₃²
commutator, socle	C₂×C₃²	C₂²×C₄
Frattini	C₄	C₂×D₃²
Fitting	C₃²×C₈	C₂²
prime radical	#3798	C₂
cosocle	C₃²×C₄	C₂³
hypocenter	C₃²	#131
LCS	G ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₈

Complements

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

N	H ≅ G/N	type
2. C₈×C₃⋊D₃	C₂	semidirect
3. C₃×D₃×C₈	C₂	semidirect
4. #3712	C₂	semidirect
5. #3711	C₂	semidirect
22. D₃×C₈	D₃	semidirect
61. C₃²	#131	semidirect
82. C₃	#901	semidirect

Maximal subgroups

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

subgroup	index
#3798	2
C₈×C₃⋊D₃	2
C₃×D₃×C₈	2
#3712	2
#3711	2
#901	3