Subgroups — #69280

Order 384.

134 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₈×Dic₆	C₂	1	1	yes	maximal, characteristic
2	C₃×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic
3	Dic₃⋊C₁₆	C₂	1	2	yes	maximal
4	Dic₃×C₁₆	C₂	1	2	yes	maximal
5	C₄×C₃⋊C₁₆	C₂	1	1	yes	maximal, characteristic
6	Q₈×C₁₆	—	3	1	no	maximal, Sylow
7	C₄×Dic₆	C₄	1	1	yes	characteristic
8	C₂×C₃×C₁₆	C₂²	1	2	yes	—
9	C₃×C₄×C₈	C₂²	1	1	yes	characteristic
10	Dic₃⋊C₈	C₄	1	2	yes	—
11	C₈×Dic₃	C₂²	1	2	yes	—
12	C₂×C₃⋊C₁₆	C₂²	1	1	yes	characteristic
13	C₂×C₃⋊C₁₆	C₂²	1	1	yes	characteristic
14	C₄⋊(C₃⋊C₈)	C₄	1	1	yes	characteristic
15	C₈×Q₈	—	3	1	no	—
16	C₄⋊C₁₆	—	3	2	no	—
17	C₄⋊C₁₆	D₃	1	1	yes	characteristic
18	C₄×C₁₆	—	3	2	no	—
19	C₄×C₁₆	—	3	1	no	—
20	C₂×Dic₆	C₈	1	1	yes	characteristic
21	C₂×C₃×C₈	C₂×C₄	1	1	yes	characteristic
22	C₂×C₃×C₈	C₂³	1	1	yes	characteristic
23	C₃×C₄²	C₂×C₄	1	1	yes	characteristic
24	C₄⋊Dic₃	C₈	1	1	yes	characteristic
25	Dic₃⋊C₄	C₈	1	2	yes	—
26	C₄×Dic₃	C₂×C₄	1	2	yes	—
27	C₂×C₃⋊C₈	C₂×C₄	1	2	yes	—
28	C₃×C₁₆	—	2	2	no	—
29	C₃⋊C₁₆	—	2	1	no	—
30	C₃⋊C₁₆	Q₈	1	2	yes	complemented
31	C₄×Q₈	—	3	1	no	—
32	C₂×C₁₆	—	3	1	no	—
33	C₂×C₁₆	—	3	1	no	—
34	C₂×C₁₆	C₂×D₃	1	2	yes	—
35	C₄⋊C₈	—	3	2	no	—
36	C₄⋊C₈	—	3	1	no	—
37	C₄×C₈	—	3	2	no	—
38	C₄×C₈	C₂×D₃	1	1	yes	characteristic
39	C₂×C₃×C₄	C₂×C₈	1	1	yes	characteristic
40	C₂×C₃×C₄	C₂×C₈	1	1	yes	characteristic
41	C₂×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
42	C₂×Dic₃	C₂×C₈	1	2	yes	—
43	C₂×Dic₃	C₂×C₈	1	2	yes	—
44	Dic₆	C₁₆	1	4	yes	complemented
45	C₃×C₈	—	2	1	no	—
46	C₃×C₈	C₂×Q₈	1	1	yes	characteristic
47	C₃×C₈	#41	1	1	yes	characteristic
48	C₃⋊C₈	—	2	2	no	—
49	C₂×Q₈	—	3	1	no	—
50	C₂×C₈	—	3	2	no	—
51	C₂×C₈	C₄×D₃	1	1	yes	characteristic
52	C₂×C₈	C₂²×D₃	1	1	yes	characteristic
53	C₄⋊C₄	—	3	2	no	—
54	C₄⋊C₄	—	3	1	no	—
55	C₄²	—	3	2	no	—
56	C₄²	C₄×D₃	1	1	yes	characteristic
57	C₁₆	—	2	2	no	—
58	C₁₆	—	3	2	no	—
59	C₁₆	—	6	1	no	—
60	C₂²×C₃	C₂²×C₈	1	1	yes	characteristic
61	C₃×C₄	—	2	1	no	—
62	C₃×C₄	C₂×C₁₆	1	2	yes	—
63	C₃×C₄	C₄×Q₈	1	1	yes	characteristic
64	C₃×C₄	#131	1	1	yes	characteristic
65	Dic₃	—	2	2	no	—
66	Dic₃	C₂×C₁₆	1	4	yes	—
67	Q₈	—	3	4	no	—
68	C₂×C₄	—	3	2	no	—
69	C₂×C₄	—	3	2	no	—
70	C₂×C₄	D₃×C₈	1	1	yes	characteristic
71	C₂×C₄	D₃×C₈	1	1	yes	characteristic
72	C₂×C₄	C₂×C₄×D₃	1	1	yes	characteristic
73	C₈	—	2	1	no	—
74	C₈	—	6	2	no	—
75	C₈	#237	1	1	yes	characteristic
76	C₈	D₃×Q₈	1	1	yes	characteristic
77	C₂×C₃	C₈×Q₈	1	1	yes	characteristic
78	C₂×C₃	C₂²×C₁₆	1	1	yes	characteristic
79	C₂×C₃	#504	1	1	yes	characteristic
80	C₂²	C₂×D₃×C₈	1	1	yes	characteristic
81	C₄	—	2	1	no	—
82	C₄	—	3	4	no	—
83	C₄	—	6	2	no	—
84	C₄	D₃×C₁₆	1	2	yes	—
85	C₄	#887	1	1	yes	characteristic
86	C₄	#907	1	1	yes	characteristic
87	C₃	Q₈×C₁₆	1	1	yes	characteristic, complemented, Sylow
88	C₂	#4830	1	1	yes	characteristic
89	C₂	C₂×D₃×C₁₆	1	1	yes	characteristic
90	C₂	#4907	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂²×D₃
commutator	C₂×C₃	C₂²×C₁₆
Fitting	C₃×C₄⋊C₁₆	C₂
socle, prime radical	C₂²×C₃	C₂²×C₈
cosocle	C₂×C₃×C₈	C₂³
hypercenter	C₄⋊C₁₆	D₃
hypocenter	C₃	Q₈×C₁₆
LCS	G ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₈ ◁ C₄⋊C₁₆

Complements

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

N	H ≅ G/N	type
30. C₃⋊C₁₆	Q₈	semidirect
44. Dic₆	C₁₆	semidirect
87. 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
C₈×Dic₆	2
C₃×C₄⋊C₁₆	2
Dic₃⋊C₁₆	2
Dic₃×C₁₆	2
C₄×C₃⋊C₁₆	2
Q₈×C₁₆	3