Subgroups — #67790

Order 384.

118 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#493	C₂	1	1	yes	maximal, characteristic
2	C₄⋊(C₃⋊C₁₆)	C₂	1	1	yes	maximal, characteristic, complemented
3	#4458	C₂	1	1	yes	maximal, characteristic, complemented
4	#1361	—	3	1	no	maximal, Sylow
5	C₂×C₃×D₈	—	2	1	no	—
6	C₂×C₃×D₈	C₄	1	2	yes	—
7	C₃×#127	—	2	1	no	—
8	C₃×C₄×C₈	C₂²	1	1	yes	characteristic
9	C₂×C₃⋊C₁₆	—	2	1	no	—
10	C₄⋊(C₃⋊C₈)	—	2	1	no	—
11	#493	D₃	1	1	yes	characteristic
12	C₄⋊C₁₆	—	3	1	no	—
13	#335	—	3	1	no	—
14	C₂×C₃×D₄	—	2	2	no	—
15	C₂×C₃×D₄	—	4	1	no	—
16	C₂×C₃×D₄	—	4	1	no	—
17	C₃×D₈	—	4	2	no	—
18	C₃×D₈	—	4	1	no	—
19	C₃×D₈	—	4	1	no	—
20	C₂×C₃×C₈	C₂×C₄	1	1	yes	characteristic
21	C₂×C₃×C₈	D₄	1	1	yes	characteristic
22	C₃×C₄²	D₄	1	1	yes	characteristic
23	C₂×C₃⋊C₈	—	4	1	no	—
24	C₃⋊C₁₆	—	4	1	no	—
25	C₂×D₈	—	2	1	no	—
26	C₂×D₈	Dic₃	1	2	yes	—
27	#127	—	2	1	no	—
28	C₂×C₁₆	—	6	1	no	—
29	C₄⋊C₈	—	6	1	no	—
30	C₄×C₈	C₂×D₃	1	1	yes	characteristic
31	C₂³×C₃	—	4	2	no	—
32	C₃×D₄	—	4	2	no	—
33	C₃×D₄	—	4	2	no	—
34	C₃×D₄	—	8	2	no	—
35	C₃×D₄	—	8	1	no	—
36	C₃×D₄	—	8	1	no	—
37	C₂×C₃×C₄	C₂²⋊C₄	1	1	yes	characteristic
38	C₂×C₃×C₄	D₈	1	1	yes	characteristic
39	C₂×C₃×C₄	SD₁₆	1	1	yes	characteristic
40	C₃×C₈	—	2	1	no	—
41	C₃×C₈	D₈	1	1	yes	characteristic
42	C₃×C₈	SD₁₆	1	1	yes	characteristic
43	C₃⋊C₈	—	8	1	no	—
44	C₂×D₄	—	2	2	no	—
45	C₂×D₄	—	4	1	no	—
46	C₂×D₄	—	4	1	no	—
47	D₈	—	4	2	no	—
48	D₈	—	4	1	no	—
49	D₈	—	4	1	no	—
50	C₂×C₈	—	12	1	no	—
51	C₂×C₈	C₂×Dic₃	1	1	yes	characteristic
52	C₂×C₈	#67	1	1	yes	characteristic
53	C₄²	#67	1	1	yes	characteristic
54	C₁₆	—	12	1	no	—
55	C₂²×C₃	—	8	2	no	—
56	C₂²×C₃	—	8	2	no	—
57	C₂²×C₃	—	8	2	no	—
58	C₂²×C₃	#102	1	1	yes	characteristic
59	C₃×C₄	—	2	1	no	—
60	C₃×C₄	#100	1	1	yes	characteristic
61	C₃×C₄	#102	1	1	yes	characteristic
62	C₃×C₄	D₁₆	1	1	yes	characteristic
63	C₃×C₄	SD₃₂	1	1	yes	characteristic
64	C₂³	—	4	2	no	—
65	D₄	—	4	2	no	—
66	D₄	—	4	2	no	—
67	D₄	—	8	2	no	—
68	D₄	—	8	1	no	—
69	D₄	—	8	1	no	—
70	C₂×C₄	#213	1	1	yes	characteristic
71	C₂×C₄	#214	1	1	yes	characteristic
72	C₂×C₄	#217	1	1	yes	characteristic
73	C₈	—	2	1	no	—
74	C₈	—	24	1	no	—
75	C₈	#213	1	1	yes	characteristic
76	C₈	#214	1	1	yes	characteristic
77	C₂×C₃	—	16	2	no	—
78	C₂×C₃	#331	1	1	yes	characteristic
79	C₂×C₃	#357	1	1	yes	characteristic
80	C₂×C₃	#361	1	1	yes	characteristic
81	C₂²	—	8	2	no	—
82	C₂²	—	8	2	no	—
83	C₂²	—	8	2	no	—
84	C₂²	#832	1	1	yes	characteristic
85	C₄	—	2	1	no	—
86	C₄	#826	1	1	yes	characteristic
87	C₄	#827	1	1	yes	characteristic
88	C₄	#832	1	1	yes	characteristic
89	C₄	#833	1	1	yes	characteristic
90	C₃	#1361	1	1	yes	characteristic, complemented, Sylow
91	C₂	—	16	2	no	—
92	C₂	#4544	1	1	yes	characteristic
93	C₂	#4562	1	1	yes	characteristic
94	C₂	#4563	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#832
commutator	C₂×C₃×C₈	C₂×C₄
Frattini	C₄×C₈	C₂×D₃
Fitting, prime radical	C₃×#493	C₂
socle	C₂²×C₃	#102
cosocle	C₃×C₄×C₈	C₂²
hypercenter	#493	D₃
hypocenter	C₃	#1361
LCS	G ▷ C₂×C₃×C₈ ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #493

Complements

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

N	H ≅ G/N	type
2. C₄⋊(C₃⋊C₁₆)	C₂	semidirect
3. #4458	C₂	semidirect
90. C₃	#1361	semidirect

Maximal subgroups

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

subgroup	index
C₃×#493	2
C₄⋊(C₃⋊C₁₆)	2
#4458	2
#1361	3