Subgroups — #63808

Order 288.

135 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#3799	C₂	1	2	yes	maximal, complemented
2	#3745	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×Dic₁₂	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×D₃×C₈	C₂	1	1	yes	maximal, characteristic, complemented
5	#3717	C₂	1	2	yes	maximal, complemented
6	#919	—	3	1	no	maximal
7	#904	—	3	1	no	maximal
8	C₄⋊(C₃⋊D₃)	C₂²	1	2	yes	—
9	C₃×C₄×D₃	C₂²	1	1	yes	characteristic
10	C₃×Dic₆	C₂²	1	2	yes	—
11	#626	—	2	2	no	—
12	#624	—	2	2	no	—
13	C₃²×C₈	C₂²	1	1	yes	characteristic
14	C₃×C₃⋊C₈	C₂²	1	1	yes	characteristic
15	#239	—	3	2	no	—
16	#235	—	3	2	no	—
17	C₃×Q₁₆	—	3	1	no	—
18	C₂×C₃×C₈	—	3	1	no	—
19	#215	—	3	2	no	—
20	Dic₁₂	D₃	1	1	yes	characteristic, complemented
21	D₂₄	—	3	1	no	—
22	D₂₄	—	3	1	no	—
23	D₂₄	—	6	1	no	—
24	#204	—	3	2	no	—
25	D₃×C₈	D₃	1	1	yes	characteristic, complemented
26	C₂×C₃⋊D₃	—	2	2	no	—
27	C₂×C₃×D₃	D₄	1	1	yes	characteristic
28	C₃²×C₄	C₂³	1	1	yes	characteristic
29	C₃×Dic₃	—	2	2	no	—
30	C₃×Dic₃	D₄	1	1	yes	characteristic
31	#135	—	9	1	no	Sylow
32	C₃×Q₈	—	3	2	no	—
33	C₂×C₃×C₄	—	3	1	no	—
34	#67	—	6	2	no	—
35	D₁₂	—	3	2	no	—
36	D₁₂	—	3	2	no	—
37	D₁₂	—	6	2	no	—
38	D₁₂	—	6	2	no	—
39	C₄×D₃	—	6	2	no	—
40	C₄×D₃	—	6	2	no	—
41	C₄×D₃	C₂×D₃	1	1	yes	characteristic
42	Dic₆	C₂×D₃	1	2	yes	—
43	C₃×C₈	—	2	1	no	—
44	C₃×C₈	—	3	1	no	—
45	C₃×C₈	C₂×D₃	1	1	yes	characteristic
46	C₃×C₈	C₂×D₃	1	1	yes	characteristic
47	C₃⋊C₈	C₂×D₃	1	1	yes	characteristic
48	C₂×C₃²	C₂×D₄	1	1	yes	characteristic
49	C₃⋊D₃	—	4	2	no	—
50	C₃×D₃	—	2	1	no	—
51	#41	—	9	2	no	—
52	Q₁₆	—	3	1	no	—
53	SD₁₆	—	9	2	no	—
54	D₈	—	9	1	no	—
55	C₂×C₈	—	3	1	no	—
56	C₂²×C₃	—	3	1	no	—
57	C₂×D₃	—	6	2	no	—
58	C₂×D₃	—	6	2	no	—
59	C₂×D₃	—	12	2	no	—
60	C₂×D₃	D₁₂	1	1	yes	characteristic
61	C₃×C₄	—	2	1	no	—
62	C₃×C₄	—	3	1	no	—
63	C₃×C₄	—	6	2	no	—
64	C₃×C₄	C₂²×D₃	1	1	yes	characteristic
65	C₃×C₄	C₂²×D₃	1	1	yes	characteristic
66	Dic₃	—	2	2	no	—
67	Dic₃	D₁₂	1	1	yes	characteristic
68	C₃²	#135	1	1	yes	characteristic, complemented, Sylow
69	Q₈	—	3	2	no	—
70	D₄	—	9	2	no	—
71	D₄	—	18	2	no	—
72	C₂×C₄	—	3	1	no	—
73	C₂×C₄	—	18	2	no	—
74	C₈	—	3	1	no	—
75	C₈	D₃²	1	1	yes	characteristic
76	C₂×C₃	—	2	1	no	—
77	C₂×C₃	—	6	1	no	—
78	C₂×C₃	C₂×D₁₂	1	1	yes	characteristic
79	C₂×C₃	D₃×D₄	1	1	yes	characteristic
80	D₃	—	2	1	no	—
81	D₃	—	12	2	no	—
82	D₃	—	12	2	no	—
83	D₃	—	24	2	no	—
84	C₂²	—	3	1	no	—
85	C₂²	—	18	2	no	—
86	C₄	—	3	1	no	—
87	C₄	—	6	2	no	—
88	C₄	C₂×D₃²	1	1	yes	characteristic
89	C₃	—	2	1	no	—
90	C₃	#904	1	1	yes	characteristic, complemented
91	C₃	#919	1	1	yes	characteristic, complemented
92	C₂	—	6	1	no	—
93	C₂	—	36	2	no	—
94	C₂	D₃×D₁₂	1	1	yes	characteristic

	subgroup	G/H
center	C₂	D₃×D₁₂
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. #3799	C₂	semidirect
2. #3745	C₂	semidirect
3. C₃×Dic₁₂	C₂	semidirect
4. C₃×D₃×C₈	C₂	semidirect
5. #3717	C₂	semidirect
20. Dic₁₂	D₃	semidirect
25. D₃×C₈	D₃	semidirect
68. C₃²	#135	semidirect
90. C₃	#904	semidirect
91. C₃	#919	semidirect

Maximal subgroups

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

subgroup	index
#3799	2
#3745	2
C₃×Dic₁₂	2
C₃×D₃×C₈	2
#3717	2
#919	3
#904	3