Subgroups — #67472

Order 384.

98 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×Dic₁₂	C₂	1	1	yes	maximal, characteristic
2	C₃×#335	C₂	1	1	yes	maximal, characteristic
3	C₄×C₃⋊C₁₆	C₂	1	1	yes	maximal, characteristic
4	#1332	—	3	1	no	maximal, Sylow
5	C₂×Dic₁₂	C₄	1	1	yes	characteristic
6	C₄×Dic₆	—	2	1	no	—
7	C₃×C₄⋊C₈	—	2	1	no	—
8	C₃×C₄×C₈	C₂²	1	1	yes	characteristic
9	#818	C₄	1	1	yes	characteristic
10	#816	—	2	1	no	—
11	C₂×C₃⋊C₁₆	—	2	1	no	—
12	C₄×Q₁₆	—	3	1	no	—
13	C₄×C₁₆	—	3	1	no	—
14	#335	D₃	1	1	yes	characteristic
15	C₂×Dic₆	—	2	1	no	—
16	C₂×C₃×C₈	—	4	1	no	—
17	C₂×C₃×C₈	C₂×C₄	1	1	yes	characteristic
18	C₂×C₃×C₈	D₄	1	1	yes	characteristic
19	C₃×C₄²	D₄	1	1	yes	characteristic
20	C₄⋊Dic₃	—	2	1	no	—
21	Dic₃⋊C₄	—	4	1	no	—
22	C₄×Dic₃	—	4	1	no	—
23	Dic₁₂	—	2	1	no	—
24	Dic₁₂	C₈	1	2	yes	complemented
25	C₃⋊C₁₆	—	2	2	no	—
26	C₂×Q₁₆	—	3	1	no	—
27	C₄×Q₈	—	6	1	no	—
28	C₂×C₁₆	—	6	1	no	—
29	#107	—	3	1	no	—
30	C₄⋊C₈	—	2	1	no	—
31	#103	—	6	1	no	—
32	C₄×C₈	C₂×D₃	1	1	yes	characteristic
33	C₂×C₃×C₄	C₂²⋊C₄	1	1	yes	characteristic
34	C₂×C₃×C₄	D₈	1	1	yes	characteristic
35	C₂×C₃×C₄	SD₁₆	1	1	yes	characteristic
36	C₂×Dic₃	—	4	1	no	—
37	C₂×Dic₃	—	4	1	no	—
38	Dic₆	—	2	2	no	—
39	Dic₆	—	4	1	no	—
40	C₃×C₈	—	2	1	no	—
41	C₃×C₈	—	8	1	no	—
42	C₃×C₈	C₂×C₈	1	1	yes	characteristic
43	C₃×C₈	M₁₆	1	1	yes	characteristic
44	C₂×Q₈	—	6	1	no	—
45	Q₁₆	—	3	2	no	—
46	Q₁₆	—	6	1	no	—
47	C₂×C₈	—	4	1	no	—
48	C₂×C₈	C₄×D₃	1	1	yes	characteristic
49	C₂×C₈	D₁₂	1	1	yes	characteristic
50	C₄⋊C₄	—	6	1	no	—
51	C₄⋊C₄	—	12	1	no	—
52	C₄²	—	12	1	no	—
53	C₄²	#67	1	1	yes	characteristic
54	C₁₆	—	6	2	no	—
55	C₂²×C₃	#102	1	1	yes	characteristic
56	C₃×C₄	—	2	1	no	—
57	C₃×C₄	C₂²⋊C₈	1	1	yes	characteristic
58	C₃×C₄	#104	1	1	yes	characteristic
59	C₃×C₄	SD₃₂	1	1	yes	characteristic
60	C₃×C₄	Q₃₂	1	1	yes	characteristic
61	Dic₃	—	4	2	no	—
62	Dic₃	—	8	1	no	—
63	Q₈	—	6	2	no	—
64	Q₈	—	12	1	no	—
65	C₂×C₄	—	12	1	no	—
66	C₂×C₄	—	12	1	no	—
67	C₂×C₄	(C₂×D₃)⋊C₄	1	1	yes	characteristic
68	C₂×C₄	#213	1	1	yes	characteristic
69	C₂×C₄	#215	1	1	yes	characteristic
70	C₈	—	2	1	no	—
71	C₈	—	8	1	no	—
72	C₈	D₃×C₈	1	1	yes	characteristic
73	C₈	ΓL(1,𝔽₂₅)	1	1	yes	characteristic
74	C₂×C₃	D₄⋊C₈	1	1	yes	characteristic
75	C₂×C₃	#358	1	1	yes	characteristic
76	C₂×C₃	#359	1	1	yes	characteristic
77	C₂²	#809	1	1	yes	characteristic
78	C₄	—	2	1	no	—
79	C₄	—	12	2	no	—
80	C₄	—	24	1	no	—
81	C₄	(C₂×D₃)⋊C₈	1	1	yes	characteristic
82	C₄	#825	1	1	yes	characteristic
83	C₄	#827	1	1	yes	characteristic
84	C₄	#829	1	1	yes	characteristic
85	C₃	#1332	1	1	yes	characteristic, complemented, Sylow
86	C₂	#4483	1	1	yes	characteristic
87	C₂	#4492	1	1	yes	characteristic
88	C₂	#4497	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#213
commutator	C₃×C₈	C₂×C₈
Frattini	C₄×C₈	C₂×D₃
Fitting	C₃×#335	C₂
socle, prime radical	C₂²×C₃	#102
cosocle	C₃×C₄×C₈	C₂²
hypercenter	#335	D₃
hypocenter	C₃	#1332
LCS	G ▷ C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×C₈ ◁ #335

Complements

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

N	H ≅ G/N	type
24. Dic₁₂	C₈	semidirect
85. C₃	#1332	semidirect

Maximal subgroups

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

subgroup	index
C₄×Dic₁₂	2
C₃×#335	2
C₄×C₃⋊C₁₆	2
#1332	3