Subgroups — #67632

Order 384.

118 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#382	C₂	1	1	yes	maximal, characteristic
2	#4528	C₂	1	2	yes	maximal
3	#1324	—	3	1	no	maximal, Sylow
4	C₂×C₃×C₄⋊C₄	C₄	1	1	yes	characteristic
5	C₂×C₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
6	C₂×C₃×C₄²	C₄	1	1	yes	characteristic
7	C₃×#95	—	2	2	no	—
8	#830	—	2	2	no	—
9	#382	D₃	1	1	yes	characteristic
10	#324	—	3	2	no	—
11	C₂²×C₃×C₄	—	2	1	no	—
12	C₂²×C₃×C₄	—	2	1	no	—
13	C₂²×C₃×C₄	C₂×C₄	1	1	yes	characteristic
14	C₂²×C₃×C₄	D₄	1	2	yes	—
15	C₃×C₄⋊C₄	—	4	1	no	—
16	C₃×C₄⋊C₄	—	4	1	no	—
17	C₃×C₄²	—	2	1	no	—
18	C₃×C₄²	C₈	1	2	yes	complemented
19	C₂×C₃⋊C₈	—	4	2	no	—
20	C₂×C₄⋊C₄	Dic₃	1	1	yes	characteristic
21	C₂×C₄⋊C₄	C₂×D₃	1	1	yes	characteristic
22	C₂×C₄²	Dic₃	1	1	yes	characteristic
23	C₂²⋊C₈	—	6	2	no	—
24	#95	—	2	2	no	—
25	C₂³×C₃	C₂²⋊C₄	1	1	yes	characteristic
26	C₂×C₃×C₄	—	2	1	no	—
27	C₂×C₃×C₄	—	2	1	no	—
28	C₂×C₃×C₄	—	2	2	no	—
29	C₂×C₃×C₄	—	2	2	no	—
30	C₂×C₃×C₄	—	4	1	no	—
31	C₂×C₃×C₄	—	4	1	no	—
32	C₂×C₃×C₄	—	4	2	no	—
33	C₂×C₃×C₄	—	4	2	no	—
34	C₂×C₃×C₄	—	4	1	no	—
35	C₂×C₃×C₄	C₂×C₈	1	1	yes	characteristic
36	C₂×C₃×C₄	M₁₆	1	1	yes	characteristic
37	C₃⋊C₈	—	8	2	no	—
38	C₂²×C₄	—	2	1	no	—
39	C₂²×C₄	—	2	1	no	—
40	C₂²×C₄	C₂×Dic₃	1	1	yes	characteristic
41	C₂²×C₄	#67	1	2	yes	—
42	C₂×C₈	—	12	2	no	—
43	C₄⋊C₄	—	4	1	no	—
44	C₄⋊C₄	—	4	1	no	—
45	C₄²	—	2	1	no	—
46	C₄²	C₃⋊C₈	1	2	yes	complemented
47	C₂²×C₃	—	2	1	no	—
48	C₂²×C₃	—	2	1	no	—
49	C₂²×C₃	C₂²⋊C₈	1	1	yes	characteristic
50	C₂²×C₃	#99	1	1	yes	characteristic
51	C₂²×C₃	#101	1	1	yes	characteristic
52	C₃×C₄	—	2	1	no	—
53	C₃×C₄	—	2	1	no	—
54	C₃×C₄	—	4	2	no	—
55	C₃×C₄	—	4	2	no	—
56	C₃×C₄	—	8	1	no	—
57	C₂³	#217	1	1	yes	characteristic
58	C₂×C₄	—	2	1	no	—
59	C₂×C₄	—	2	1	no	—
60	C₂×C₄	—	2	2	no	—
61	C₂×C₄	—	2	2	no	—
62	C₂×C₄	—	4	1	no	—
63	C₂×C₄	—	4	1	no	—
64	C₂×C₄	—	4	2	no	—
65	C₂×C₄	—	4	2	no	—
66	C₂×C₄	—	4	1	no	—
67	C₂×C₄	C₂×C₃⋊C₈	1	1	yes	characteristic
68	C₂×C₄	#208	1	1	yes	characteristic
69	C₈	—	24	2	no	—
70	C₂×C₃	—	2	1	no	—
71	C₂×C₃	—	2	1	no	—
72	C₂×C₃	#324	1	1	yes	characteristic
73	C₂×C₃	#354	1	1	yes	characteristic
74	C₂×C₃	#355	1	1	yes	characteristic
75	C₂²	—	2	1	no	—
76	C₂²	—	2	1	no	—
77	C₂²	#830	1	1	yes	characteristic
78	C₂²	#834	1	1	yes	characteristic
79	C₂²	#836	1	1	yes	characteristic
80	C₄	—	2	1	no	—
81	C₄	—	2	1	no	—
82	C₄	—	4	2	no	—
83	C₄	—	4	2	no	—
84	C₄	—	8	1	no	—
85	C₃	#1324	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	#4541	1	1	yes	characteristic
90	C₂	#4542	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#834
commutator	C₂×C₃×C₄	C₂×C₈
Frattini	C₂×C₄⋊C₄	C₂×D₃
Fitting	C₃×#382	C₂
socle	C₂²×C₃	#99
prime radical	C₂³×C₃	C₂²⋊C₄
cosocle	C₂×C₃×C₄⋊C₄	C₂²
hypercenter	#382	D₃
hypocenter	C₃	#1324
LCS	G ▷ C₂×C₃×C₄ ▷ C₂²×C₃ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₂³ ◁ C₂×C₄⋊C₄ ◁ #382

Complements

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

N	H ≅ G/N	type
18. C₃×C₄²	C₈	semidirect
46. C₄²	C₃⋊C₈	semidirect
85. C₃	#1324	semidirect

Maximal subgroups

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

subgroup	index
C₃×#382	2
#4528	2
#1324	3