Subgroups — #69320

Order 384.

128 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#4874	C₂	1	2	yes	maximal
2	C₃×#366	C₂	1	1	yes	maximal, characteristic
3	#4504	C₂	1	1	yes	maximal, characteristic
4	Dic₃×C₁₆	C₂	1	1	yes	maximal, characteristic
5	#4489	C₂	1	2	yes	maximal
6	#2252	—	3	1	no	maximal, Sylow
7	#888	—	2	2	no	—
8	C₂×C₃×C₁₆	C₂²	1	1	yes	characteristic
9	C₃×#107	C₂²	1	2	yes	—
10	#818	C₂²	1	2	yes	—
11	C₈×Dic₃	C₂²	1	1	yes	characteristic
12	C₂×C₃⋊C₁₆	C₂²	1	1	yes	characteristic
13	#807	—	2	2	no	—
14	#500	—	3	2	no	—
15	#366	—	3	1	no	—
16	#366	—	3	2	no	—
17	#366	D₃	1	1	yes	characteristic
18	C₄×C₁₆	—	3	1	no	—
19	C₂×Dic₆	—	4	2	no	—
20	C₂×C₃×C₈	C₂³	1	1	yes	characteristic
21	C₃×C₄⋊C₄	—	2	2	no	—
22	C₄⋊Dic₃	—	2	2	no	—
23	Dic₃⋊C₄	—	4	2	no	—
24	C₄×Dic₃	D₄	1	1	yes	characteristic
25	C₂×C₃⋊C₈	D₄	1	1	yes	characteristic
26	C₃×C₁₆	Q₈	1	2	yes	complemented
27	C₃⋊C₁₆	Q₈	1	2	yes	complemented
28	C₄⋊Q₈	—	6	2	no	—
29	C₂×C₁₆	—	3	1	no	—
30	C₂×C₁₆	C₂×D₃	1	1	yes	characteristic
31	#107	—	3	2	no	—
32	#107	—	6	2	no	—
33	#107	C₂×D₃	1	2	yes	—
34	C₄×C₈	—	3	1	no	—
35	C₂×C₃×C₄	—	4	2	no	—
36	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
37	C₂×Dic₃	—	4	2	no	—
38	C₂×Dic₃	D₈	1	1	yes	characteristic
39	C₂×Dic₃	D₈	1	1	yes	characteristic
40	Dic₆	—	8	4	no	—
41	C₃×C₈	C₂×Q₈	1	1	yes	characteristic
42	C₃×C₈	C₂×Q₈	1	1	yes	characteristic
43	C₃⋊C₈	Q₁₆	1	2	yes	—
44	C₂×Q₈	—	12	2	no	—
45	C₂×C₈	—	3	1	no	—
46	C₂×C₈	C₂²×D₃	1	1	yes	characteristic
47	C₄⋊C₄	—	2	2	no	—
48	C₄⋊C₄	—	6	2	no	—
49	C₄⋊C₄	—	12	2	no	—
50	C₄²	—	3	1	no	—
51	C₁₆	—	3	2	no	—
52	C₁₆	Dic₆	1	2	yes	complemented
53	C₂²×C₃	C₂×D₈	1	1	yes	characteristic
54	C₃×C₄	—	8	2	no	—
55	C₃×C₄	C₄⋊Q₈	1	1	yes	characteristic
56	C₃×C₄	C₂×Q₁₆	1	1	yes	characteristic
57	Dic₃	—	8	2	no	—
58	Dic₃	D₁₆	1	2	yes	—
59	Dic₃	Q₃₂	1	2	yes	—
60	Q₈	—	24	4	no	—
61	C₂×C₄	—	3	1	no	—
62	C₂×C₄	—	3	1	no	—
63	C₂×C₄	—	4	2	no	—
64	C₂×C₄	—	12	2	no	—
65	C₂×C₄	D₃×D₄	1	1	yes	characteristic
66	C₈	—	3	2	no	—
67	C₈	C₂×Dic₆	1	1	yes	characteristic
68	C₈	D₃×Q₈	1	1	yes	characteristic
69	C₂×C₃	#500	1	1	yes	characteristic
70	C₂×C₃	C₂×D₁₆	1	1	yes	characteristic
71	C₂×C₃	C₂×Q₃₂	1	1	yes	characteristic
72	C₂²	D₃×D₈	1	1	yes	characteristic
73	C₄	—	3	2	no	—
74	C₄	—	3	2	no	—
75	C₄	—	8	2	no	—
76	C₄	—	24	2	no	—
77	C₄	#888	1	1	yes	characteristic
78	C₄	D₃×Q₁₆	1	1	yes	characteristic
79	C₃	#2252	1	1	yes	characteristic, complemented, Sylow
80	C₂	#4874	1	1	yes	characteristic
81	C₂	D₃×D₁₆	1	1	yes	characteristic
82	C₂	D₃×Q₃₂	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₃×D₈
commutator, cosocle	C₂×C₃×C₈	C₂³
Frattini	C₂×C₈	C₂²×D₃
Fitting	C₃×#366	C₂
socle, prime radical	C₂²×C₃	C₂×D₈
hypercenter	#366	D₃
hypocenter	C₃	#2252
LCS	G ▷ C₂×C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ #366

Complements

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

N	H ≅ G/N	type
26. C₃×C₁₆	Q₈	semidirect
27. C₃⋊C₁₆	Q₈	semidirect
52. C₁₆	Dic₆	semidirect
79. C₃	#2252	semidirect

Maximal subgroups

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

subgroup	index
#4874	2
C₃×#366	2
#4504	2
Dic₃×C₁₆	2
#4489	2
#2252	3