Subgroups — #70949

Order 384.

126 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	2	yes	maximal
3	C₄⋊(C₃⋊C₁₆)	C₂	1	1	yes	maximal, characteristic
4	#4461	C₂	1	2	yes	maximal
5	C₄×C₃⋊C₁₆	C₂	1	1	yes	maximal, characteristic
6	#2183	—	3	1	no	maximal, Sylow
7	C₃×C₄×Q₈	C₄	1	1	yes	characteristic
8	C₃×C₄⋊C₈	C₄	1	2	yes	—
9	C₃×C₄⋊C₈	C₄	1	1	yes	characteristic
10	C₃×C₄×C₈	C₂²	1	1	yes	characteristic
11	C₃×C₄×C₈	C₂²	1	2	yes	—
12	C₂×C₃⋊C₁₆	C₂²	1	1	yes	characteristic
13	C₂×C₃⋊C₁₆	C₂²	1	2	yes	—
14	C₂×C₃⋊C₁₆	C₂²	1	1	yes	characteristic
15	C₈×Q₈	D₃	1	1	yes	characteristic
16	C₄⋊C₁₆	—	3	1	no	—
17	C₄⋊C₁₆	—	3	2	no	—
18	#346	—	3	2	no	—
19	C₄×C₁₆	—	3	1	no	—
20	C₂×C₃×Q₈	C₈	1	1	yes	characteristic
21	C₂×C₃×C₈	C₂×C₄	1	1	yes	characteristic
22	C₂×C₃×C₈	C₂×C₄	1	2	yes	—
23	C₂×C₃×C₈	C₂³	1	1	yes	characteristic
24	C₃×C₄⋊C₄	C₈	1	2	yes	—
25	C₃×C₄⋊C₄	C₈	1	1	yes	characteristic
26	C₃×C₄²	C₂×C₄	1	2	yes	—
27	C₃×C₄²	C₂×C₄	1	1	yes	characteristic
28	C₃⋊C₁₆	—	2	1	no	—
29	C₃⋊C₁₆	—	2	2	no	—
30	C₃⋊C₁₆	Q₈	1	2	yes	complemented
31	C₄×Q₈	Dic₃	1	1	yes	characteristic
32	C₂×C₁₆	—	3	1	no	—
33	C₂×C₁₆	—	3	1	no	—
34	C₂×C₁₆	—	3	2	no	—
35	C₄⋊C₈	Dic₃	1	2	yes	—
36	C₄⋊C₈	Dic₃	1	1	yes	characteristic
37	C₄×C₈	C₂×D₃	1	1	yes	characteristic
38	C₄×C₈	C₂×D₃	1	2	yes	—
39	C₃×Q₈	—	2	2	no	—
40	C₂×C₃×C₄	C₂×C₈	1	2	yes	—
41	C₂×C₃×C₄	C₂×C₈	1	1	yes	characteristic
42	C₂×C₃×C₄	C₂×C₈	1	2	yes	—
43	C₂×C₃×C₄	C₂×C₈	1	1	yes	characteristic
44	C₂×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
45	C₃×C₈	—	2	1	no	—
46	C₃×C₈	—	2	2	no	—
47	C₃×C₈	C₂×Q₈	1	1	yes	characteristic
48	C₃×C₈	#41	1	1	yes	characteristic
49	C₂×Q₈	C₃⋊C₈	1	1	yes	characteristic
50	C₂×C₈	C₂×Dic₃	1	1	yes	characteristic
51	C₂×C₈	C₂×Dic₃	1	2	yes	—
52	C₂×C₈	C₂²×D₃	1	1	yes	characteristic
53	C₄⋊C₄	C₃⋊C₈	1	2	yes	—
54	C₄⋊C₄	C₃⋊C₈	1	1	yes	characteristic
55	C₄²	C₂×Dic₃	1	2	yes	—
56	C₄²	C₂×Dic₃	1	1	yes	characteristic
57	C₁₆	—	3	2	no	—
58	C₁₆	—	6	1	no	—
59	C₁₆	—	6	2	no	—
60	C₂²×C₃	C₂²×C₈	1	1	yes	characteristic
61	C₃×C₄	—	2	2	no	—
62	C₃×C₄	—	2	2	no	—
63	C₃×C₄	—	2	1	no	—
64	C₃×C₄	M₃₂	1	2	yes	—
65	C₃×C₄	C₄×Q₈	1	1	yes	characteristic
66	C₃×C₄	#131	1	1	yes	characteristic
67	Q₈	—	2	2	no	—
68	C₂×C₄	C₂×C₃⋊C₈	1	2	yes	—
69	C₂×C₄	C₂×C₃⋊C₈	1	1	yes	characteristic
70	C₂×C₄	C₂×C₃⋊C₈	1	2	yes	—
71	C₂×C₄	C₂×C₃⋊C₈	1	1	yes	characteristic
72	C₂×C₄	C₂²×Dic₃	1	1	yes	characteristic
73	C₈	—	2	1	no	—
74	C₈	—	2	2	no	—
75	C₈	D₃×Q₈	1	1	yes	characteristic
76	C₈	#239	1	1	yes	characteristic
77	C₂×C₃	C₈×Q₈	1	1	yes	characteristic
78	C₂×C₃	C₂×M₃₂	1	1	yes	characteristic
79	C₂×C₃	#504	1	1	yes	characteristic
80	C₂²	C₂²×C₃⋊C₈	1	1	yes	characteristic
81	C₄	—	2	2	no	—
82	C₄	—	2	1	no	—
83	C₄	—	2	2	no	—
84	C₄	#812	1	2	yes	—
85	C₄	Q₈×Dic₃	1	1	yes	characteristic
86	C₄	#948	1	1	yes	characteristic
87	C₃	#2183	1	1	yes	characteristic, complemented, Sylow
88	C₂	Q₈×C₃⋊C₈	1	1	yes	characteristic
89	C₂	C₂×#812	1	1	yes	characteristic
90	C₂	#5140	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂²×D₃
commutator, socle, prime radical	C₂²×C₃	C₂²×C₈
Fitting	C₃×C₈×Q₈	C₂
cosocle	C₂×C₃×C₈	C₂³
hypercenter	C₈×Q₈	D₃
hypocenter	C₃	#2183
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₈ ◁ C₈×Q₈

Complements

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

N	H ≅ G/N	type
30. C₃⋊C₁₆	Q₈	semidirect
87. C₃	#2183	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
#4461	2
C₄×C₃⋊C₁₆	2
#2183	3