Subgroups — #83780

Order 384.

360 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×#131	C₂	1	1	yes	maximal, characteristic
2	#5140	C₂	1	8	yes	maximal, complemented
3	C₂×#812	C₂	1	6	yes	maximal, complemented
4	#3407	—	3	1	no	maximal, Sylow
5	C₂×C₃×#41	C₄	1	1	yes	characteristic
6	C₃×#131	C₄	1	4	yes	—
7	C₃×#131	C₂²	1	4	yes	—
8	C₂×C₃×M₁₆	C₄	1	3	yes	—
9	C₂²×C₃×C₈	C₂²	1	3	yes	—
10	#812	C₂²	1	4	yes	—
11	#812	C₂²	1	12	yes	complemented
12	C₂×C₃⋊C₁₆	C₂²	1	12	yes	complemented
13	C₂×#131	D₃	1	1	yes	characteristic
14	#504	—	3	8	no	—
15	C₂×M₃₂	—	3	6	no	—
16	C₃×#41	C₈	1	4	yes	—
17	C₃×#41	C₂×C₄	1	4	yes	—
18	C₂×C₃×Q₈	C₈	1	1	yes	characteristic
19	C₂×C₃×D₄	C₈	1	3	yes	—
20	C₂²×C₃×C₄	C₂×C₄	1	3	yes	—
21	C₃×M₁₆	C₂×C₄	1	12	yes	—
22	C₂×C₃×C₈	C₂×C₄	1	3	yes	—
23	C₂×C₃×C₈	C₂×C₄	1	6	yes	—
24	C₂×C₃×C₈	C₂³	1	6	yes	—
25	C₂×C₃×C₈	C₂³	1	1	yes	characteristic
26	C₃⋊C₁₆	C₂³	1	8	yes	—
27	C₂×#41	Dic₃	1	1	yes	characteristic
28	#131	Dic₃	1	4	yes	—
29	#131	C₂×D₃	1	4	yes	—
30	C₂×M₁₆	Dic₃	1	3	yes	—
31	C₂²×C₈	C₂×D₃	1	3	yes	—
32	M₃₂	—	3	4	no	—
33	M₃₂	—	3	12	no	—
34	C₂×C₁₆	—	3	12	no	—
35	C₂³×C₃	C₂×C₈	1	3	yes	—
36	C₃×Q₈	C₂×C₈	1	4	yes	—
37	C₃×D₄	C₂×C₈	1	12	yes	—
38	C₂×C₃×C₄	C₂×C₈	1	3	yes	—
39	C₂×C₃×C₄	C₂×C₈	1	6	yes	—
40	C₂×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
41	C₂×C₃×C₄	C₂²×C₄	1	6	yes	—
42	C₃×C₈	C₂²×C₄	1	1	yes	characteristic
43	C₃×C₈	C₂²×C₄	1	6	yes	—
44	C₃×C₈	C₂⁴	1	1	yes	characteristic
45	#41	C₃⋊C₈	1	4	yes	—
46	#41	C₂×Dic₃	1	4	yes	—
47	C₂×Q₈	C₃⋊C₈	1	1	yes	characteristic
48	C₂×D₄	C₃⋊C₈	1	3	yes	—
49	C₂²×C₄	C₂×Dic₃	1	3	yes	—
50	M₁₆	C₂×Dic₃	1	12	yes	—
51	C₂×C₈	C₂×Dic₃	1	3	yes	—
52	C₂×C₈	C₂×Dic₃	1	6	yes	—
53	C₂×C₈	C₂²×D₃	1	6	yes	—
54	C₂×C₈	C₂²×D₃	1	1	yes	characteristic
55	C₁₆	—	3	8	no	—
56	C₂²×C₃	—	4	3	no	—
57	C₂²×C₃	C₂²×C₈	1	1	yes	characteristic
58	C₂²×C₃	C₂²×C₈	1	6	yes	—
59	C₃×C₄	C₂²×C₈	1	1	yes	characteristic
60	C₃×C₄	C₂²×C₈	1	6	yes	—
61	C₃×C₄	C₂³×C₄	1	1	yes	characteristic
62	C₂³	C₂×C₃⋊C₈	1	3	yes	—
63	Q₈	C₂×C₃⋊C₈	1	4	yes	—
64	D₄	C₂×C₃⋊C₈	1	12	yes	—
65	C₂×C₄	C₂×C₃⋊C₈	1	3	yes	—
66	C₂×C₄	C₂×C₃⋊C₈	1	6	yes	—
67	C₂×C₄	C₂²×Dic₃	1	1	yes	characteristic
68	C₂×C₄	C₂²×Dic₃	1	6	yes	—
69	C₈	C₂²×Dic₃	1	1	yes	characteristic
70	C₈	C₂²×Dic₃	1	6	yes	—
71	C₈	C₂³×D₃	1	1	yes	characteristic
72	C₂×C₃	—	2	1	no	—
73	C₂×C₃	—	2	6	no	—
74	C₂×C₃	C₂³×C₈	1	1	yes	characteristic
75	C₂²	—	4	3	no	—
76	C₂²	C₂²×C₃⋊C₈	1	1	yes	characteristic
77	C₂²	C₂²×C₃⋊C₈	1	6	yes	—
78	C₄	C₂²×C₃⋊C₈	1	1	yes	characteristic
79	C₄	C₂²×C₃⋊C₈	1	6	yes	—
80	C₄	C₂³×Dic₃	1	1	yes	characteristic
81	C₃	#3407	1	1	yes	characteristic, complemented, Sylow
82	C₂	—	2	1	no	—
83	C₂	—	2	6	no	—
84	C₂	C₂³×C₃⋊C₈	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₈	C₂³×D₃
commutator, socle	C₂×C₃	C₂³×C₈
Fitting	C₂×C₃×#131	C₂
prime radical	C₂×C₃×#41	C₄
cosocle	C₃×C₈	C₂⁴
hypercenter	C₂×#131	D₃
hypocenter	C₃	#3407
LCS	G ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₈ ◁ C₂×#131

Complements

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

N	H ≅ G/N	type
2. #5140	C₂	semidirect
3. C₂×#812	C₂	semidirect
11. #812	C₂²	semidirect
12. C₂×C₃⋊C₁₆	C₂²	semidirect
81. C₃	#3407	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃×#131	2
#5140	2
C₂×#812	2
#3407	3