Subgroups — #67717

Order 384.

96 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#423	C₂	1	1	yes	maximal, characteristic
2	#4553	C₂	1	2	yes	maximal
3	#1327	—	3	1	no	maximal, Sylow
4	C₂×C₃×M₁₆	C₄	1	1	yes	characteristic
5	C₂×C₃×M₁₆	C₂²	1	1	yes	characteristic
6	C₂×C₃×C₄²	C₄	1	1	yes	characteristic
7	C₃×C₄⋊C₈	—	2	2	no	—
8	#812	—	2	2	no	—
9	#423	D₃	1	1	yes	characteristic
10	#349	—	3	2	no	—
11	C₂²×C₃×C₄	—	2	1	no	—
12	C₂²×C₃×C₄	C₂×C₄	1	1	yes	characteristic
13	C₃×M₁₆	—	4	1	no	—
14	C₃×M₁₆	—	4	1	no	—
15	C₂×C₃×C₈	—	2	1	no	—
16	C₂×C₃×C₈	D₄	1	2	yes	—
17	C₃×C₄²	—	2	1	no	—
18	C₃×C₄²	C₈	1	1	yes	characteristic
19	C₃×C₄²	C₈	1	1	yes	characteristic
20	C₃⋊C₁₆	—	4	2	no	—
21	C₂×M₁₆	Dic₃	1	1	yes	characteristic
22	C₂×M₁₆	C₂×D₃	1	1	yes	characteristic
23	C₂×C₄²	Dic₃	1	1	yes	characteristic
24	M₃₂	—	6	2	no	—
25	C₄⋊C₈	—	2	2	no	—
26	C₂³×C₃	M₁₆	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	—	4	2	no	—
32	C₂×C₃×C₄	C₂²⋊C₄	1	1	yes	characteristic
33	C₂×C₃×C₄	C₂×C₈	1	1	yes	characteristic
34	C₃×C₈	—	2	2	no	—
35	C₃×C₈	—	4	1	no	—
36	C₂²×C₄	—	2	1	no	—
37	C₂²×C₄	C₂×Dic₃	1	1	yes	characteristic
38	M₁₆	—	4	1	no	—
39	M₁₆	—	4	1	no	—
40	C₂×C₈	—	2	1	no	—
41	C₂×C₈	#67	1	2	yes	—
42	C₄²	—	2	1	no	—
43	C₄²	C₃⋊C₈	1	1	yes	characteristic
44	C₄²	C₃⋊C₈	1	1	yes	characteristic
45	C₁₆	—	12	2	no	—
46	C₂²×C₃	—	2	1	no	—
47	C₂²×C₃	—	4	1	no	—
48	C₂²×C₃	C₂²⋊C₈	1	1	yes	characteristic
49	C₃×C₄	—	2	2	no	—
50	C₃×C₄	—	2	1	no	—
51	C₃×C₄	—	4	1	no	—
52	C₃×C₄	#99	1	1	yes	characteristic
53	C₃×C₄	#101	1	1	yes	characteristic
54	C₂³	#208	1	1	yes	characteristic
55	C₂×C₄	—	2	1	no	—
56	C₂×C₄	—	2	1	no	—
57	C₂×C₄	—	2	1	no	—
58	C₂×C₄	—	2	1	no	—
59	C₂×C₄	—	4	2	no	—
60	C₂×C₄	C₂×C₃⋊C₈	1	1	yes	characteristic
61	C₂×C₄	#217	1	1	yes	characteristic
62	C₈	—	2	2	no	—
63	C₈	—	4	1	no	—
64	C₂×C₃	—	2	1	no	—
65	C₂×C₃	—	4	1	no	—
66	C₂×C₃	#324	1	1	yes	characteristic
67	C₂²	—	2	1	no	—
68	C₂²	—	4	1	no	—
69	C₂²	#830	1	1	yes	characteristic
70	C₄	—	2	2	no	—
71	C₄	—	2	1	no	—
72	C₄	—	4	1	no	—
73	C₄	#834	1	1	yes	characteristic
74	C₄	#836	1	1	yes	characteristic
75	C₃	#1327	1	1	yes	characteristic, complemented, Sylow
76	C₂	—	2	1	no	—
77	C₂	—	4	1	no	—
78	C₂	#4528	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#834
commutator	C₂×C₃×C₄	C₂×C₈
Frattini	C₂×M₁₆	C₂×D₃
Fitting	C₃×#423	C₂
socle	C₂×C₃	#324
prime radical	C₂³×C₃	M₁₆
cosocle	C₂×C₃×M₁₆	C₂²
hypercenter	#423	D₃
hypocenter	C₃	#1327
LCS	G ▷ C₂×C₃×C₄ ▷ C₂²×C₃ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×M₁₆ ◁ #423

Complements

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

N	H ≅ G/N	type
75. C₃	#1327	semidirect

Maximal subgroups

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

subgroup	index
C₃×#423	2
#4553	2
#1327	3