Subgroups — #64121

Order 288.

186 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#3830	C₂	1	2	yes	maximal, complemented
2	C₃²×D₈	C₂	1	1	yes	maximal, characteristic, complemented
3	#3754	C₂	1	2	yes	maximal, complemented
4	#3746	C₂	1	1	yes	maximal, characteristic, complemented
5	C₈×C₃⋊D₃	C₂	1	1	yes	maximal, characteristic, complemented
6	#912	—	3	4	no	maximal
7	C₃²×D₄	C₂²	1	2	yes	—
8	#638	—	2	2	no	—
9	C₂×C₃⋊Dic₃	—	2	2	no	—
10	C₄×C₃⋊D₃	C₂²	1	1	yes	characteristic
11	#634	C₂²	1	2	yes	—
12	C₃²×C₈	C₂²	1	1	yes	characteristic
13	C₃⋊(C₃⋊C₈)	C₂²	1	1	yes	characteristic
14	#237	—	3	8	no	—
15	C₃×D₈	D₃	1	4	yes	complemented
16	#214	—	3	8	no	—
17	Dic₁₂	—	3	4	no	—
18	D₃×C₈	—	3	4	no	—
19	C₂²×C₃²	—	2	2	no	—
20	C₂×C₃⋊D₃	D₄	1	1	yes	characteristic
21	C₃²×C₄	C₂³	1	1	yes	characteristic
22	C₃⋊Dic₃	—	2	2	no	—
23	C₃⋊Dic₃	D₄	1	1	yes	characteristic
24	#135	—	9	1	no	Sylow
25	C₃×D₄	C₂×D₃	1	8	yes	—
26	#67	—	6	8	no	—
27	C₂×Dic₃	—	6	8	no	—
28	C₄×D₃	—	3	4	no	—
29	Dic₆	—	3	8	no	—
30	C₃×C₈	C₂×D₃	1	4	yes	—
31	C₃⋊C₈	—	3	4	no	—
32	C₂×C₃²	—	4	2	no	—
33	C₂×C₃²	C₂×D₄	1	1	yes	characteristic
34	C₃⋊D₃	—	2	1	no	—
35	#41	—	9	2	no	—
36	Q₁₆	—	9	1	no	—
37	SD₁₆	—	9	2	no	—
38	D₈	C₃⋊D₃	1	1	yes	characteristic, complemented
39	C₂×C₈	—	9	1	no	—
40	C₂²×C₃	—	2	8	no	—
41	C₂×D₃	—	3	4	no	—
42	C₃×C₄	C₂²×D₃	1	4	yes	—
43	Dic₃	—	3	4	no	—
44	Dic₃	—	6	8	no	—
45	C₃²	#135	1	1	yes	characteristic, complemented, Sylow
46	Q₈	—	9	2	no	—
47	D₄	—	18	2	no	—
48	D₄	C₂×C₃⋊D₃	1	2	yes	—
49	C₂×C₄	—	9	1	no	—
50	C₂×C₄	—	18	2	no	—
51	C₈	—	9	1	no	—
52	C₈	C₂×C₃⋊D₃	1	1	yes	characteristic
53	C₂×C₃	—	4	8	no	—
54	C₂×C₃	D₃×D₄	1	4	yes	—
55	D₃	—	6	4	no	—
56	C₂²	—	2	2	no	—
57	C₂²	—	9	1	no	—
58	C₄	—	9	1	no	—
59	C₄	—	18	2	no	—
60	C₄	C₂²×C₃⋊D₃	1	1	yes	characteristic
61	C₃	#912	1	4	yes	complemented
62	C₂	—	4	2	no	—
63	C₂	—	18	1	no	—
64	C₂	D₄×C₃⋊D₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂	D₄×C₃⋊D₃
commutator, cosocle	C₃²×C₄	C₂³
Frattini	C₄	C₂²×C₃⋊D₃
Fitting	C₃²×D₈	C₂
socle	C₂×C₃²	C₂×D₄
hypercenter	D₈	C₃⋊D₃
hypocenter	C₃²	#135
LCS	G ▷ C₃²×C₄ ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂ ◁ C₄ ◁ D₈

Complements

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

N	H ≅ G/N	type
1. #3830	C₂	semidirect
2. C₃²×D₈	C₂	semidirect
3. #3754	C₂	semidirect
4. #3746	C₂	semidirect
5. C₈×C₃⋊D₃	C₂	semidirect
15. C₃×D₈	D₃	semidirect
38. D₈	C₃⋊D₃	semidirect
45. C₃²	#135	semidirect
61. C₃	#912	semidirect

Maximal subgroups

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

subgroup	index
#3830	2
C₃²×D₈	2
#3754	2
#3746	2
C₈×C₃⋊D₃	2
#912	3