Subgroups — #67705

Order 384.

152 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	#4924	C₂	1	2	yes	maximal
3	#1292	—	3	1	no	maximal, Sylow
4	C₂²×C₃×C₈	C₄	1	2	yes	—
5	C₂×C₃×C₄²	C₂²	1	1	yes	characteristic
6	C₂×#208	—	2	2	no	—
7	C₃×C₄⋊C₈	—	2	2	no	—
8	C₄⋊(C₃⋊C₈)	—	2	2	no	—
9	C₄⋊(C₃⋊C₈)	C₄	1	4	yes	—
10	#423	—	3	2	no	—
11	C₂×C₄⋊C₈	D₃	1	1	yes	characteristic
12	C₂²×C₃×C₄	C₂×C₄	1	1	yes	characteristic
13	C₂²×C₃×C₄	D₄	1	1	yes	characteristic
14	C₂²×C₃×C₄	Q₈	1	1	yes	characteristic
15	C₂×C₃×C₈	—	2	2	no	—
16	C₂×C₃×C₈	—	4	2	no	—
17	C₃×C₄²	C₂×C₄	1	2	yes	—
18	C₃×C₄²	D₄	1	2	yes	—
19	#208	—	4	2	no	—
20	#208	—	4	2	no	—
21	C₂×C₃⋊C₈	—	2	4	no	—
22	C₂×M₁₆	—	6	2	no	—
23	C₂²×C₈	Dic₃	1	2	yes	—
24	C₂×C₄²	C₂×D₃	1	1	yes	characteristic
25	C₄⋊C₈	—	2	2	no	—
26	C₄⋊C₈	—	3	4	no	—
27	C₄⋊C₈	—	6	2	no	—
28	C₂³×C₃	C₄⋊C₄	1	1	yes	characteristic
29	C₂×C₃×C₄	—	2	1	no	—
30	C₂×C₃×C₄	—	2	1	no	—
31	C₂×C₃×C₄	—	4	1	no	—
32	C₂×C₃×C₄	C₄²	1	1	yes	characteristic
33	C₂×C₃×C₄	C₂²⋊C₄	1	2	yes	—
34	C₂×C₃×C₄	C₂²⋊C₄	1	1	yes	characteristic
35	C₂×C₃×C₄	C₄⋊C₄	1	2	yes	—
36	C₂×C₃×C₄	D₈	1	1	yes	characteristic
37	C₂×C₃×C₄	SD₁₆	1	2	yes	—
38	C₂×C₃×C₄	Q₁₆	1	1	yes	characteristic
39	C₃×C₈	—	4	2	no	—
40	C₃⋊C₈	—	4	4	no	—
41	C₂²×C₄	Dic₆	1	1	yes	characteristic
42	C₂²×C₄	D₁₂	1	1	yes	characteristic
43	C₂²×C₄	C₂×Dic₃	1	1	yes	characteristic
44	M₁₆	—	12	2	no	—
45	M₁₆	—	12	2	no	—
46	C₂×C₈	—	2	2	no	—
47	C₂×C₈	—	4	2	no	—
48	C₂×C₈	—	6	4	no	—
49	C₄²	C₄×D₃	1	2	yes	—
50	C₄²	#67	1	2	yes	—
51	C₂²×C₃	—	2	1	no	—
52	C₂²×C₃	—	2	1	no	—
53	C₂²×C₃	#95	1	1	yes	characteristic
54	C₂²×C₃	#106	1	1	yes	characteristic
55	C₂²×C₃	#107	1	1	yes	characteristic
56	C₃×C₄	—	2	2	no	—
57	C₃×C₄	—	2	1	no	—
58	C₃×C₄	#100	1	1	yes	characteristic
59	C₃×C₄	#101	1	1	yes	characteristic
60	C₃×C₄	#102	1	2	yes	—
61	C₃×C₄	#103	1	2	yes	—
62	C₂³	C₄⋊Dic₃	1	1	yes	characteristic
63	C₂×C₄	—	2	1	no	—
64	C₂×C₄	—	2	1	no	—
65	C₂×C₄	—	4	1	no	—
66	C₂×C₄	#204	1	2	yes	—
67	C₂×C₄	D₂₄	1	1	yes	characteristic
68	C₂×C₄	Dic₁₂	1	1	yes	characteristic
69	C₂×C₄	C₄×Dic₃	1	1	yes	characteristic
70	C₂×C₄	Dic₃⋊C₄	1	2	yes	—
71	C₂×C₄	(C₂×D₃)⋊C₄	1	2	yes	—
72	C₂×C₄	#217	1	1	yes	characteristic
73	C₈	—	4	2	no	—
74	C₈	—	12	4	no	—
75	C₂×C₃	—	2	1	no	—
76	C₂×C₃	—	2	1	no	—
77	C₂×C₃	#338	1	1	yes	characteristic
78	C₂×C₃	#340	1	1	yes	characteristic
79	C₂×C₃	#343	1	1	yes	characteristic
80	C₂²	—	2	1	no	—
81	C₂²	—	2	1	no	—
82	C₂²	#817	1	1	yes	characteristic
83	C₂²	#818	1	1	yes	characteristic
84	C₂²	C₃⋊#95	1	1	yes	characteristic
85	C₄	—	2	1	no	—
86	C₄	—	2	2	no	—
87	C₄	#816	1	2	yes	—
88	C₄	#821	1	2	yes	—
89	C₄	#833	1	1	yes	characteristic
90	C₄	#836	1	1	yes	characteristic
91	C₃	#1292	1	1	yes	characteristic, complemented, Sylow
92	C₂	—	2	1	no	—
93	C₂	—	2	1	no	—
94	C₂	#4530	1	1	yes	characteristic
95	C₂	#4551	1	1	yes	characteristic
96	C₂	C₃⋊#338	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₃⋊#95
commutator	C₂×C₃×C₄	C₄²
Frattini	C₂×C₄²	C₂×D₃
Fitting	C₂×C₃×C₄⋊C₈	C₂
socle	C₂²×C₃	#95
prime radical	C₂³×C₃	C₄⋊C₄
cosocle	C₂×C₃×C₄²	C₂²
hypercenter	C₂×C₄⋊C₈	D₃
hypocenter	C₃	#1292
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄⋊C₈

Complements

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

N	H ≅ G/N	type
91. C₃	#1292	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
#4924	2
#1292	3