Subgroups — #64128

Order 288.

186 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#3832	C₂	1	2	yes	maximal, complemented
2	C₃²×Q₁₆	C₂	1	1	yes	maximal, characteristic, complemented
3	#3755	C₂	1	2	yes	maximal, complemented
4	#3745	C₂	1	1	yes	maximal, characteristic, complemented
5	C₈×C₃⋊D₃	C₂	1	1	yes	maximal, characteristic, complemented
6	#919	—	3	4	no	maximal
7	C₃²×Q₈	C₂²	1	2	yes	—
8	C₄⋊(C₃⋊D₃)	—	2	2	no	—
9	C₄⋊(C₃⋊D₃)	C₂²	1	2	yes	—
10	C₄×C₃⋊D₃	—	2	2	no	—
11	C₄×C₃⋊D₃	C₂²	1	1	yes	characteristic
12	C₃²×C₈	C₂²	1	1	yes	characteristic
13	C₃⋊(C₃⋊C₈)	C₂²	1	1	yes	characteristic
14	#239	—	3	8	no	—
15	C₃×Q₁₆	D₃	1	4	yes	complemented
16	#215	—	3	8	no	—
17	D₂₄	—	3	4	no	—
18	D₃×C₈	—	3	4	no	—
19	C₂×C₃⋊D₃	—	2	2	no	—
20	C₂×C₃⋊D₃	D₄	1	1	yes	characteristic
21	C₃²×C₄	—	2	2	no	—
22	C₃²×C₄	C₂³	1	1	yes	characteristic
23	C₃⋊Dic₃	D₄	1	1	yes	characteristic
24	#135	—	9	1	no	Sylow
25	C₃×Q₈	C₂×D₃	1	8	yes	—
26	D₁₂	—	3	8	no	—
27	D₁₂	—	6	8	no	—
28	C₄×D₃	—	3	4	no	—
29	C₄×D₃	—	6	8	no	—
30	C₃×C₈	C₂×D₃	1	4	yes	—
31	C₃⋊C₈	—	3	4	no	—
32	C₂×C₃²	C₂×D₄	1	1	yes	characteristic
33	C₃⋊D₃	—	2	1	no	—
34	C₃⋊D₃	—	4	2	no	—
35	#41	—	9	2	no	—
36	Q₁₆	C₃⋊D₃	1	1	yes	characteristic, complemented
37	SD₁₆	—	9	2	no	—
38	D₈	—	9	1	no	—
39	C₂×C₈	—	9	1	no	—
40	C₂×D₃	—	3	4	no	—
41	C₂×D₃	—	6	8	no	—
42	C₃×C₄	—	2	8	no	—
43	C₃×C₄	C₂²×D₃	1	4	yes	—
44	Dic₃	—	3	4	no	—
45	C₃²	#135	1	1	yes	characteristic, complemented, Sylow
46	Q₈	C₂×C₃⋊D₃	1	2	yes	—
47	D₄	—	9	2	no	—
48	D₄	—	18	2	no	—
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	D₃×D₄	1	4	yes	—
54	D₃	—	6	4	no	—
55	D₃	—	12	8	no	—
56	C₂²	—	9	1	no	—
57	C₂²	—	18	2	no	—
58	C₄	—	2	2	no	—
59	C₄	—	9	1	no	—
60	C₄	C₂²×C₃⋊D₃	1	1	yes	characteristic
61	C₃	#919	1	4	yes	complemented
62	C₂	—	18	1	no	—
63	C₂	—	36	2	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×Q₁₆	C₂
socle	C₂×C₃²	C₂×D₄
hypercenter	Q₁₆	C₃⋊D₃
hypocenter	C₃²	#135
LCS	G ▷ C₃²×C₄ ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂ ◁ C₄ ◁ Q₁₆

Complements

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

N	H ≅ G/N	type
1. #3832	C₂	semidirect
2. C₃²×Q₁₆	C₂	semidirect
3. #3755	C₂	semidirect
4. #3745	C₂	semidirect
5. C₈×C₃⋊D₃	C₂	semidirect
15. C₃×Q₁₆	D₃	semidirect
36. Q₁₆	C₃⋊D₃	semidirect
45. C₃²	#135	semidirect
61. C₃	#919	semidirect

Maximal subgroups

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

subgroup	index
#3832	2
C₃²×Q₁₆	2
#3755	2
#3745	2
C₈×C₃⋊D₃	2
#919	3