Subgroups — #67709

Order 384.

108 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
2	#4549	C₂	1	2	yes	maximal
3	#1322	—	3	1	no	maximal, Sylow
4	C₂²×C₃×C₈	C₄	1	1	yes	characteristic
5	C₂²×C₃×C₈	C₂²	1	1	yes	characteristic
6	C₂×C₃×C₄²	C₄	1	1	yes	characteristic
7	C₃×C₄⋊C₈	—	2	2	no	—
8	C₂×C₃⋊C₁₆	—	2	2	no	—
9	C₂×C₄⋊C₈	D₃	1	1	yes	characteristic
10	C₂²⋊C₁₆	—	3	2	no	—
11	C₂²×C₃×C₄	C₈	1	1	yes	characteristic
12	C₂²×C₃×C₄	C₈	1	1	yes	characteristic
13	C₂²×C₃×C₄	C₂×C₄	1	1	yes	characteristic
14	C₂×C₃×C₈	—	2	1	no	—
15	C₂×C₃×C₈	—	4	1	no	—
16	C₂×C₃×C₈	—	4	1	no	—
17	C₂×C₃×C₈	D₄	1	2	yes	—
18	C₃×C₄²	—	2	1	no	—
19	C₃×C₄²	—	2	1	no	—
20	C₃⋊C₁₆	—	4	2	no	—
21	C₂²×C₈	Dic₃	1	1	yes	characteristic
22	C₂²×C₈	C₂×D₃	1	1	yes	characteristic
23	C₂×C₄²	Dic₃	1	1	yes	characteristic
24	C₂×C₁₆	—	6	2	no	—
25	C₄⋊C₈	—	2	2	no	—
26	C₂³×C₃	C₂×C₈	1	1	yes	characteristic
27	C₂×C₃×C₄	—	2	1	no	—
28	C₂×C₃×C₄	—	2	1	no	—
29	C₂×C₃×C₄	—	2	1	no	—
30	C₂×C₃×C₄	—	2	1	no	—
31	C₂×C₃×C₄	—	2	1	no	—
32	C₂×C₃×C₄	—	4	1	no	—
33	C₂×C₃×C₄	C₁₆	1	2	yes	complemented
34	C₂×C₃×C₄	C₂²⋊C₄	1	1	yes	characteristic
35	C₂×C₃×C₄	M₁₆	1	1	yes	characteristic
36	C₃×C₈	—	2	2	no	—
37	C₃×C₈	—	4	1	no	—
38	C₂²×C₄	C₃⋊C₈	1	1	yes	characteristic
39	C₂²×C₄	C₃⋊C₈	1	1	yes	characteristic
40	C₂²×C₄	C₂×Dic₃	1	1	yes	characteristic
41	C₂×C₈	—	2	1	no	—
42	C₂×C₈	—	4	1	no	—
43	C₂×C₈	—	4	1	no	—
44	C₂×C₈	#67	1	2	yes	—
45	C₄²	—	2	1	no	—
46	C₄²	—	2	1	no	—
47	C₁₆	—	12	2	no	—
48	C₂²×C₃	—	2	1	no	—
49	C₂²×C₃	—	2	1	no	—
50	C₂²×C₃	C₂²⋊C₈	1	1	yes	characteristic
51	C₂²×C₃	C₂×C₁₆	1	1	yes	characteristic
52	C₂²×C₃	M₃₂	1	1	yes	characteristic
53	C₃×C₄	—	2	1	no	—
54	C₃×C₄	—	2	2	no	—
55	C₃×C₄	—	4	1	no	—
56	C₃×C₄	#99	1	1	yes	characteristic
57	C₃×C₄	#101	1	1	yes	characteristic
58	C₂³	C₂×C₃⋊C₈	1	1	yes	characteristic
59	C₂×C₄	—	2	1	no	—
60	C₂×C₄	—	2	1	no	—
61	C₂×C₄	—	2	1	no	—
62	C₂×C₄	—	2	1	no	—
63	C₂×C₄	—	2	1	no	—
64	C₂×C₄	—	4	1	no	—
65	C₂×C₄	C₃⋊C₁₆	1	2	yes	complemented
66	C₂×C₄	#208	1	1	yes	characteristic
67	C₂×C₄	#217	1	1	yes	characteristic
68	C₈	—	2	2	no	—
69	C₈	—	4	1	no	—
70	C₂×C₃	—	2	1	no	—
71	C₂×C₃	—	2	1	no	—
72	C₂×C₃	#324	1	1	yes	characteristic
73	C₂×C₃	C₂²⋊C₁₆	1	1	yes	characteristic
74	C₂×C₃	#349	1	1	yes	characteristic
75	C₂²	—	2	1	no	—
76	C₂²	—	2	1	no	—
77	C₂²	C₂×C₃⋊C₁₆	1	1	yes	characteristic
78	C₂²	#812	1	1	yes	characteristic
79	C₂²	#830	1	1	yes	characteristic
80	C₄	—	2	1	no	—
81	C₄	—	2	2	no	—
82	C₄	—	4	1	no	—
83	C₄	#834	1	1	yes	characteristic
84	C₄	#836	1	1	yes	characteristic
85	C₃	#1322	1	1	yes	characteristic, complemented, Sylow
86	C₂	—	2	1	no	—
87	C₂	—	2	1	no	—
88	C₂	#4528	1	1	yes	characteristic
89	C₂	#4549	1	1	yes	characteristic
90	C₂	#4553	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#217
commutator	C₂²×C₃	C₂×C₁₆
Frattini	C₂²×C₈	C₂×D₃
Fitting	C₂×C₃×C₄⋊C₈	C₂
socle	C₂²×C₃	C₂²⋊C₈
prime radical	C₂³×C₃	C₂×C₈
cosocle	C₂²×C₃×C₈	C₂²
hypercenter	C₂×C₄⋊C₈	D₃
hypocenter	C₃	#1322
LCS	G ▷ C₂²×C₃ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ C₂×C₄⋊C₈

Complements

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

N	H ≅ G/N	type
33. C₂×C₃×C₄	C₁₆	semidirect
65. C₂×C₄	C₃⋊C₁₆	semidirect
85. C₃	#1322	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃×C₄⋊C₈	2
#4549	2
#1322	3