Subgroups — #67768

Order 384.

94 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₈×Q₈	C₂	1	1	yes	maximal, characteristic
2	C₄⋊(C₃⋊C₁₆)	C₂	1	1	yes	maximal, characteristic
3	C₄×C₃⋊C₁₆	C₂	1	1	yes	maximal, characteristic
4	Q₈⋊C₁₆	—	3	1	no	maximal, Sylow
5	C₃×C₄×Q₈	C₄	1	1	yes	characteristic
6	C₃×C₄⋊C₈	—	2	1	no	—
7	C₃×C₄⋊C₈	C₄	1	1	yes	characteristic
8	C₃×C₄×C₈	—	2	1	no	—
9	C₃×C₄×C₈	C₂²	1	1	yes	characteristic
10	C₂×C₃⋊C₁₆	—	2	1	no	—
11	C₂×C₃⋊C₁₆	—	2	1	no	—
12	C₈×Q₈	D₃	1	1	yes	characteristic
13	C₄⋊C₁₆	—	3	1	no	—
14	C₄×C₁₆	—	3	1	no	—
15	C₂×C₃×Q₈	C₈	1	1	yes	characteristic
16	C₂×C₃×C₈	—	2	1	no	—
17	C₂×C₃×C₈	D₄	1	1	yes	characteristic
18	C₂×C₃×C₈	D₄	1	1	yes	characteristic
19	C₃×C₄⋊C₄	—	2	1	no	—
20	C₃×C₄⋊C₄	C₈	1	1	yes	characteristic
21	C₃×C₄²	—	2	1	no	—
22	C₃×C₄²	C₂×C₄	1	1	yes	characteristic
23	C₃⋊C₁₆	—	2	2	no	—
24	C₃⋊C₁₆	—	4	1	no	—
25	C₄×Q₈	Dic₃	1	1	yes	characteristic
26	C₂×C₁₆	—	6	1	no	—
27	C₂×C₁₆	—	6	1	no	—
28	C₄⋊C₈	—	2	1	no	—
29	C₄⋊C₈	Dic₃	1	1	yes	characteristic
30	C₄×C₈	—	2	1	no	—
31	C₄×C₈	C₂×D₃	1	1	yes	characteristic
32	C₃×Q₈	—	2	1	no	—
33	C₃×Q₈	C₁₆	1	2	yes	complemented
34	C₂×C₃×C₄	—	2	1	no	—
35	C₂×C₃×C₄	—	2	1	no	—
36	C₂×C₃×C₄	C₂²⋊C₄	1	1	yes	characteristic
37	C₂×C₃×C₄	C₂×C₈	1	1	yes	characteristic
38	C₂×C₃×C₄	M₁₆	1	1	yes	characteristic
39	C₃×C₈	—	2	1	no	—
40	C₃×C₈	—	4	1	no	—
41	C₃×C₈	SD₁₆	1	1	yes	characteristic
42	C₃×C₈	Q₁₆	1	1	yes	characteristic
43	C₂×Q₈	C₃⋊C₈	1	1	yes	characteristic
44	C₂×C₈	—	2	1	no	—
45	C₂×C₈	#67	1	1	yes	characteristic
46	C₂×C₈	#67	1	1	yes	characteristic
47	C₄⋊C₄	—	2	1	no	—
48	C₄⋊C₄	C₃⋊C₈	1	1	yes	characteristic
49	C₄²	—	2	1	no	—
50	C₄²	C₂×Dic₃	1	1	yes	characteristic
51	C₁₆	—	6	2	no	—
52	C₁₆	—	12	1	no	—
53	C₂²×C₃	C₂²⋊C₈	1	1	yes	characteristic
54	C₃×C₄	—	2	2	no	—
55	C₃×C₄	—	2	1	no	—
56	C₃×C₄	—	4	1	no	—
57	C₃×C₄	#103	1	1	yes	characteristic
58	C₃×C₄	#104	1	1	yes	characteristic
59	C₃×C₄	C₂×C₁₆	1	1	yes	characteristic
60	C₃×C₄	M₃₂	1	1	yes	characteristic
61	Q₈	—	2	1	no	—
62	Q₈	C₃⋊C₁₆	1	2	yes	complemented
63	C₂×C₄	—	2	1	no	—
64	C₂×C₄	—	2	1	no	—
65	C₂×C₄	C₂×C₃⋊C₈	1	1	yes	characteristic
66	C₂×C₄	#208	1	1	yes	characteristic
67	C₂×C₄	#217	1	1	yes	characteristic
68	C₈	—	2	1	no	—
69	C₈	—	4	1	no	—
70	C₈	#215	1	1	yes	characteristic
71	C₈	#216	1	1	yes	characteristic
72	C₂×C₃	Q₈⋊C₈	1	1	yes	characteristic
73	C₂×C₃	C₂²⋊C₁₆	1	1	yes	characteristic
74	C₂×C₃	#350	1	1	yes	characteristic
75	C₂²	#830	1	1	yes	characteristic
76	C₄	—	2	2	no	—
77	C₄	—	2	1	no	—
78	C₄	—	4	1	no	—
79	C₄	C₂×C₃⋊C₁₆	1	1	yes	characteristic
80	C₄	#812	1	1	yes	characteristic
81	C₄	#835	1	1	yes	characteristic
82	C₄	#837	1	1	yes	characteristic
83	C₃	Q₈⋊C₁₆	1	1	yes	characteristic, complemented, Sylow
84	C₂	#4535	1	1	yes	characteristic
85	C₂	#4549	1	1	yes	characteristic
86	C₂	#4561	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	#67
commutator	C₃×C₄	C₂×C₁₆
Frattini	C₄×C₈	C₂×D₃
Fitting	C₃×C₈×Q₈	C₂
socle, prime radical	C₂²×C₃	C₂²⋊C₈
cosocle	C₃×C₄×C₈	C₂²
hypercenter	C₈×Q₈	D₃
hypocenter	C₃	Q₈⋊C₁₆
LCS	G ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₈ ◁ C₄×C₈ ◁ C₈×Q₈

Complements

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

N	H ≅ G/N	type
33. C₃×Q₈	C₁₆	semidirect
62. Q₈	C₃⋊C₁₆	semidirect
83. C₃	Q₈⋊C₁₆	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₈×Q₈	2
C₄⋊(C₃⋊C₁₆)	2
C₄×C₃⋊C₁₆	2
Q₈⋊C₁₆	3