Subgroups — #64124

Order 288.

180 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₈×C₃⋊D₃	C₂	1	1	yes	maximal, characteristic, complemented
2	#3830	C₂	1	1	yes	maximal, characteristic
3	C₃²×SD₁₆	C₂	1	1	yes	maximal, characteristic, complemented
4	#3756	C₂	1	1	yes	maximal, characteristic, complemented
5	#3754	C₂	1	1	yes	maximal, characteristic, complemented
6	#3746	C₂	1	1	yes	maximal, characteristic, complemented
7	#3743	C₂	1	1	yes	maximal, characteristic, complemented
8	#915	—	3	4	no	maximal
9	C₃²×Q₈	C₂²	1	1	yes	characteristic
10	C₃²×D₄	C₂²	1	1	yes	characteristic
11	#638	—	2	1	no	—
12	C₂×C₃⋊Dic₃	—	2	1	no	—
13	C₄×C₃⋊D₃	—	2	1	no	—
14	C₄×C₃⋊D₃	C₂²	1	1	yes	characteristic
15	#634	—	2	1	no	—
16	#634	C₂²	1	1	yes	characteristic
17	#634	C₂²	1	1	yes	characteristic
18	C₃²×C₈	C₂²	1	1	yes	characteristic
19	C₃⋊(C₃⋊C₈)	C₂²	1	1	yes	characteristic
20	D₃×Q₈	—	3	4	no	—
21	#237	—	3	4	no	—
22	C₃×SD₁₆	D₃	1	4	yes	complemented
23	#216	—	3	4	no	—
24	#214	—	3	4	no	—
25	Dic₁₂	—	3	4	no	—
26	ΓL(1,𝔽₂₅)	—	3	4	no	—
27	C₂²×C₃²	—	2	1	no	—
28	C₂×C₃⋊D₃	D₄	1	1	yes	characteristic
29	C₃²×C₄	—	2	1	no	—
30	C₃²×C₄	C₂³	1	1	yes	characteristic
31	C₃⋊Dic₃	—	2	1	no	—
32	C₃⋊Dic₃	—	2	1	no	—
33	C₃⋊Dic₃	D₄	1	1	yes	characteristic
34	#137	—	9	1	no	Sylow
35	C₃×Q₈	C₂×D₃	1	4	yes	—
36	C₃×D₄	C₂×D₃	1	4	yes	—
37	#67	—	6	4	no	—
38	C₂×Dic₃	—	6	4	no	—
39	C₄×D₃	—	3	4	no	—
40	C₄×D₃	—	6	4	no	—
41	Dic₆	—	3	4	no	—
42	Dic₆	—	3	4	no	—
43	Dic₆	—	6	4	no	—
44	C₃×C₈	C₂×D₃	1	4	yes	—
45	C₃⋊C₈	—	3	4	no	—
46	C₂×C₃²	—	4	1	no	—
47	C₂×C₃²	C₂×D₄	1	1	yes	characteristic
48	C₃⋊D₃	—	2	1	no	—
49	#41	—	9	1	no	—
50	C₂×Q₈	—	9	1	no	—
51	Q₁₆	—	9	1	no	—
52	Q₁₆	—	9	1	no	—
53	SD₁₆	—	9	1	no	—
54	SD₁₆	C₃⋊D₃	1	1	yes	characteristic, complemented
55	M₁₆	—	9	1	no	—
56	C₂²×C₃	—	2	4	no	—
57	C₂×D₃	—	3	4	no	—
58	C₃×C₄	—	2	4	no	—
59	C₃×C₄	C₂²×D₃	1	4	yes	—
60	Dic₃	—	3	4	no	—
61	Dic₃	—	6	4	no	—
62	Dic₃	—	6	4	no	—
63	C₃²	#137	1	1	yes	characteristic, complemented, Sylow
64	Q₈	—	9	1	no	—
65	Q₈	—	9	1	no	—
66	Q₈	—	18	1	no	—
67	Q₈	C₂×C₃⋊D₃	1	1	yes	characteristic
68	D₄	—	18	1	no	—
69	D₄	C₂×C₃⋊D₃	1	1	yes	characteristic
70	C₂×C₄	—	9	1	no	—
71	C₂×C₄	—	18	1	no	—
72	C₂×C₄	—	18	1	no	—
73	C₈	—	9	1	no	—
74	C₈	C₂×C₃⋊D₃	1	1	yes	characteristic
75	C₂×C₃	—	4	4	no	—
76	C₂×C₃	D₃×D₄	1	4	yes	—
77	D₃	—	6	4	no	—
78	C₂²	—	2	1	no	—
79	C₂²	—	9	1	no	—
80	C₄	—	2	1	no	—
81	C₄	—	9	1	no	—
82	C₄	—	18	1	no	—
83	C₄	—	18	1	no	—
84	C₄	C₂²×C₃⋊D₃	1	1	yes	characteristic
85	C₃	#915	1	4	yes	complemented
86	C₂	—	4	1	no	—
87	C₂	—	18	1	no	—
88	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×SD₁₆	C₂
socle	C₂×C₃²	C₂×D₄
prime radical	#3830	C₂
hypercenter	SD₁₆	C₃⋊D₃
hypocenter	C₃²	#137
LCS	G ▷ C₃²×C₄ ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂ ◁ C₄ ◁ SD₁₆

Complements

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

N	H ≅ G/N	type
1. Q₈×C₃⋊D₃	C₂	semidirect
3. C₃²×SD₁₆	C₂	semidirect
4. #3756	C₂	semidirect
5. #3754	C₂	semidirect
6. #3746	C₂	semidirect
7. #3743	C₂	semidirect
22. C₃×SD₁₆	D₃	semidirect
54. SD₁₆	C₃⋊D₃	semidirect
63. C₃²	#137	semidirect
85. C₃	#915	semidirect

Maximal subgroups

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

subgroup	index
Q₈×C₃⋊D₃	2
#3830	2
C₃²×SD₁₆	2
#3756	2
#3754	2
#3746	2
#3743	2
#915	3