Subgroups — #73310

Order 384.

400 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₄³	C₂	1	1	yes	maximal, characteristic
2	#4936	C₂	1	2	yes	maximal
3	#4933	C₂	1	8	yes	maximal
4	#4928	C₂	1	4	yes	maximal
5	#2864	—	3	1	no	maximal, Sylow
6	C₂×C₃×C₄²	C₂²	1	2	yes	—
7	C₂×C₃×C₄²	C₂²	1	4	yes	—
8	C₂×C₃×C₄²	C₂²	1	1	yes	characteristic
9	C₂×C₄⋊Dic₃	C₂²	1	4	yes	—
10	C₂×Dic₃⋊C₄	C₂²	1	8	yes	—
11	C₃⋊#95	C₂²	1	8	yes	—
12	C₃⋊#95	C₂²	1	8	yes	—
13	#389	—	3	4	no	—
14	#387	—	3	8	no	—
15	#382	—	3	2	no	—
16	C₄³	D₃	1	1	yes	characteristic
17	C₂²×C₃×C₄	C₂³	1	2	yes	—
18	C₂²×C₃×C₄	C₂³	1	4	yes	—
19	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
20	C₂²×Dic₃	C₂³	1	8	yes	—
21	C₃×C₄²	—	2	8	no	—
22	C₃×C₄²	—	2	4	no	—
23	C₃×C₄²	Q₈	1	4	yes	—
24	C₄⋊Dic₃	—	2	8	no	—
25	Dic₃⋊C₄	—	2	16	no	—
26	C₂×C₄⋊C₄	—	3	4	no	—
27	C₂×C₄⋊C₄	—	3	8	no	—
28	C₂×C₄²	C₂×D₃	1	4	yes	—
29	C₂×C₄²	C₂×D₃	1	2	yes	—
30	C₂×C₄²	C₂×D₃	1	1	yes	characteristic
31	#95	—	3	8	no	—
32	#95	—	3	8	no	—
33	C₂³×C₃	C₂⁴	1	1	yes	characteristic
34	C₂×C₃×C₄	—	2	4	no	—
35	C₂×C₃×C₄	—	2	4	no	—
36	C₂×C₃×C₄	—	2	4	no	—
37	C₂×C₃×C₄	C₂×Q₈	1	4	yes	—
38	C₂×C₃×C₄	C₂×Q₈	1	2	yes	—
39	C₂×C₃×C₄	#41	1	8	yes	—
40	C₂×C₃×C₄	#41	1	4	yes	—
41	C₂×Dic₃	—	2	16	no	—
42	C₂×Dic₃	—	2	8	no	—
43	C₂²×C₄	—	3	8	no	—
44	C₂²×C₄	C₂²×D₃	1	2	yes	—
45	C₂²×C₄	C₂²×D₃	1	4	yes	—
46	C₂²×C₄	C₂²×D₃	1	1	yes	characteristic
47	C₄⋊C₄	—	6	16	no	—
48	C₄⋊C₄	—	6	8	no	—
49	C₄²	—	2	8	no	—
50	C₄²	—	2	4	no	—
51	C₄²	Dic₆	1	4	yes	—
52	C₂²×C₃	C₂²×Q₈	1	1	yes	characteristic
53	C₂²×C₃	C₂×#41	1	2	yes	—
54	C₂²×C₃	C₂×#41	1	4	yes	—
55	C₃×C₄	—	2	8	no	—
56	C₃×C₄	—	2	4	no	—
57	C₃×C₄	#125	1	4	yes	—
58	Dic₃	—	4	8	no	—
59	C₂³	C₂³×D₃	1	1	yes	characteristic
60	C₂×C₄	—	2	4	no	—
61	C₂×C₄	—	2	4	no	—
62	C₂×C₄	—	2	4	no	—
63	C₂×C₄	—	6	8	no	—
64	C₂×C₄	—	6	16	no	—
65	C₂×C₄	C₂×Dic₆	1	4	yes	—
66	C₂×C₄	C₂×Dic₆	1	2	yes	—
67	C₂×C₄	#235	1	8	yes	—
68	C₂×C₄	#235	1	4	yes	—
69	C₂×C₃	C₂×#125	1	1	yes	characteristic
70	C₂×C₃	#529	1	4	yes	—
71	C₂×C₃	#533	1	2	yes	—
72	C₂²	C₂²×Dic₆	1	1	yes	characteristic
73	C₂²	C₂×#235	1	2	yes	—
74	C₂²	C₂×#235	1	4	yes	—
75	C₄	—	2	4	no	—
76	C₄	—	2	8	no	—
77	C₄	—	12	8	no	—
78	C₄	#870	1	4	yes	—
79	C₃	#2864	1	1	yes	characteristic, complemented, Sylow
80	C₂	C₂×#870	1	1	yes	characteristic
81	C₂	#5470	1	2	yes	—
82	C₂	#5479	1	4	yes	—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₃
commutator, socle, prime radical, cosocle	C₂³×C₃	C₂⁴
Fitting	C₃×C₄³	C₂
hypercenter	C₄³	D₃
hypocenter	C₃	#2864
LCS	G ▷ C₂³×C₃ ▷ C₃
UCS	C₁ ◁ C₂³ ◁ C₄³

Complements

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

N	H ≅ G/N	type
79. C₃	#2864	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₄³	2
#4936	2
#4933	2
#4928	2
#2864	3