Subgroups — #67688

Order 384.

190 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#407	C₂	1	1	yes	maximal, characteristic
2	C₂×#830	C₂	1	2	yes	maximal
3	#1298	—	3	1	no	maximal, Sylow
4	C₂³×C₃×C₄	C₂²	1	1	yes	characteristic
5	C₂×C₃×M₁₆	C₄	1	2	yes	—
6	C₂²×C₃⋊C₈	—	2	2	no	—
7	C₃×C₂²⋊C₈	—	2	2	no	—
8	#830	—	2	2	no	—
9	#830	C₄	1	4	yes	—
10	#407	D₃	1	1	yes	characteristic
11	C₂×C₂²⋊C₈	—	3	2	no	—
12	C₂⁴×C₃	Q₈	1	1	yes	characteristic
13	C₂²×C₃×C₄	—	4	2	no	—
14	C₂²×C₃×C₄	C₂×C₄	1	2	yes	—
15	C₂²×C₃×C₄	C₂×C₄	1	1	yes	characteristic
16	C₂²×C₃×C₄	D₄	1	1	yes	characteristic
17	C₂²×C₃×C₄	D₄	1	2	yes	—
18	C₃×M₁₆	—	4	2	no	—
19	C₂×C₃×C₈	—	2	2	no	—
20	C₂×C₃⋊C₈	—	2	4	no	—
21	C₂×C₃⋊C₈	—	4	4	no	—
22	C₂³×C₄	C₂×D₃	1	1	yes	characteristic
23	C₂×M₁₆	Dic₃	1	2	yes	—
24	C₂²×C₈	—	6	2	no	—
25	C₂²⋊C₈	—	2	2	no	—
26	C₂²⋊C₈	—	3	4	no	—
27	C₂²⋊C₈	—	6	2	no	—
28	C₂³×C₃	—	2	4	no	—
29	C₂³×C₃	—	4	1	no	—
30	C₂³×C₃	C₄⋊C₄	1	2	yes	—
31	C₂³×C₃	C₄⋊C₄	1	1	yes	characteristic
32	C₂×C₃×C₄	—	2	4	no	—
33	C₂×C₃×C₄	—	2	2	no	—
34	C₂×C₃×C₄	—	4	2	no	—
35	C₂×C₃×C₄	—	4	1	no	—
36	C₂×C₃×C₄	C₄²	1	1	yes	characteristic
37	C₂×C₃×C₄	C₂²⋊C₄	1	2	yes	—
38	C₂×C₃×C₄	C₂²⋊C₄	1	1	yes	characteristic
39	C₃×C₈	—	4	2	no	—
40	C₃⋊C₈	—	4	4	no	—
41	C₂⁴	Dic₆	1	1	yes	characteristic
42	C₂²×C₄	—	4	2	no	—
43	C₂²×C₄	C₄×D₃	1	2	yes	—
44	C₂²×C₄	D₁₂	1	1	yes	characteristic
45	C₂²×C₄	C₂×Dic₃	1	1	yes	characteristic
46	C₂²×C₄	#67	1	2	yes	—
47	M₁₆	—	4	2	no	—
48	C₂×C₈	—	2	2	no	—
49	C₂×C₈	—	6	4	no	—
50	C₂×C₈	—	12	4	no	—
51	C₂²×C₃	—	2	2	no	—
52	C₂²×C₃	—	2	2	no	—
53	C₂²×C₃	—	2	2	no	—
54	C₂²×C₃	—	2	1	no	—
55	C₂²×C₃	—	4	4	no	—
56	C₂²×C₃	#95	1	1	yes	characteristic
57	C₂²×C₃	#108	1	4	yes	—
58	C₃×C₄	—	2	2	no	—
59	C₃×C₄	—	2	1	no	—
60	C₃×C₄	#99	1	2	yes	—
61	C₂³	—	2	4	no	—
62	C₂³	—	4	1	no	—
63	C₂³	Dic₃⋊C₄	1	2	yes	—
64	C₂³	C₄⋊Dic₃	1	1	yes	characteristic
65	C₂×C₄	—	2	4	no	—
66	C₂×C₄	—	2	2	no	—
67	C₂×C₄	—	4	2	no	—
68	C₂×C₄	—	4	1	no	—
69	C₂×C₄	C₄×Dic₃	1	1	yes	characteristic
70	C₂×C₄	(C₂×D₃)⋊C₄	1	2	yes	—
71	C₂×C₄	#217	1	1	yes	characteristic
72	C₈	—	4	2	no	—
73	C₈	—	12	4	no	—
74	C₂×C₃	—	2	4	no	—
75	C₂×C₃	—	4	1	no	—
76	C₂×C₃	#341	1	2	yes	—
77	C₂×C₃	#342	1	1	yes	characteristic
78	C₂²	—	2	2	no	—
79	C₂²	—	2	2	no	—
80	C₂²	—	2	1	no	—
81	C₂²	—	2	2	no	—
82	C₂²	—	4	4	no	—
83	C₂²	#822	1	4	yes	—
84	C₂²	C₃⋊#95	1	1	yes	characteristic
85	C₄	—	2	2	no	—
86	C₄	—	2	1	no	—
87	C₄	#834	1	2	yes	—
88	C₃	#1298	1	1	yes	characteristic, complemented, Sylow
89	C₂	—	2	4	no	—
90	C₂	—	4	1	no	—
91	C₂	#4526	1	1	yes	characteristic
92	C₂	#4558	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	#217
commutator	C₂×C₃×C₄	C₄²
Frattini	C₂³×C₄	C₂×D₃
Fitting	C₃×#407	C₂
socle	C₂²×C₃	#95
prime radical	C₂⁴×C₃	Q₈
cosocle	C₂³×C₃×C₄	C₂²
hypercenter	#407	D₃
hypocenter	C₃	#1298
LCS	G ▷ C₂×C₃×C₄ ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ #407

Complements

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

N	H ≅ G/N	type
88. C₃	#1298	semidirect

Maximal subgroups

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

subgroup	index
C₃×#407	2
C₂×#830	2
#1298	3