Subgroups — #67630

Order 384.

180 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	C₂×#803	C₂	1	2	yes	maximal
3	#1295	—	3	1	no	maximal, Sylow
4	C₂×C₃×C₄⋊C₄	C₄	1	2	yes	—
5	C₂×C₃×C₄²	C₂²	1	1	yes	characteristic
6	C₂²×C₃⋊C₈	—	2	2	no	—
7	C₃×#95	—	2	2	no	—
8	#803	—	2	2	no	—
9	#803	C₄	1	4	yes	complemented
10	C₂×#97	—	3	2	no	—
11	#382	D₃	1	1	yes	characteristic
12	C₂²×C₃×C₄	—	2	2	no	—
13	C₂²×C₃×C₄	C₂×C₄	1	1	yes	characteristic
14	C₂²×C₃×C₄	D₄	1	2	yes	—
15	C₃×C₄⋊C₄	—	4	2	no	—
16	C₃×C₄²	C₂×C₄	1	2	yes	—
17	C₃×C₄²	D₄	1	1	yes	characteristic
18	C₃×C₄²	Q₈	1	1	yes	characteristic
19	C₂×C₃⋊C₈	—	2	8	no	—
20	C₂×C₃⋊C₈	—	2	4	no	—
21	C₂²×C₈	—	6	2	no	—
22	C₂×C₄⋊C₄	Dic₃	1	2	yes	—
23	C₂×C₄²	C₂×D₃	1	1	yes	characteristic
24	#97	—	3	4	no	—
25	#97	—	6	2	no	—
26	#95	—	2	2	no	—
27	C₂³×C₃	C₂²⋊C₄	1	1	yes	characteristic
28	C₂×C₃×C₄	—	2	1	no	—
29	C₂×C₃×C₄	—	2	1	no	—
30	C₂×C₃×C₄	—	2	4	no	—
31	C₂×C₃×C₄	—	4	2	no	—
32	C₂×C₃×C₄	—	4	4	no	—
33	C₂×C₃×C₄	C₄²	1	1	yes	characteristic
34	C₂×C₃×C₄	C₂²⋊C₄	1	2	yes	—
35	C₂×C₃×C₄	C₄⋊C₄	1	2	yes	—
36	C₂×C₃×C₄	C₄⋊C₄	1	1	yes	characteristic
37	C₃⋊C₈	—	2	8	no	—
38	C₂²×C₄	—	2	2	no	—
39	C₂²×C₄	C₂×Dic₃	1	1	yes	characteristic
40	C₂²×C₄	#67	1	2	yes	—
41	C₂×C₈	—	6	8	no	—
42	C₂×C₈	—	6	4	no	—
43	C₄⋊C₄	—	4	2	no	—
44	C₄²	Dic₆	1	1	yes	characteristic
45	C₄²	C₄×D₃	1	2	yes	—
46	C₄²	D₁₂	1	1	yes	characteristic
47	C₂²×C₃	#95	1	1	yes	characteristic
48	C₂²×C₃	#100	1	1	yes	characteristic
49	C₂²×C₃	#101	1	1	yes	characteristic
50	C₂²×C₃	#104	1	4	yes	—
51	C₃×C₄	—	2	2	no	—
52	C₃×C₄	—	2	2	no	—
53	C₃×C₄	—	2	1	no	—
54	C₃×C₄	—	2	1	no	—
55	C₃×C₄	—	8	2	no	—
56	C₂³	#217	1	1	yes	characteristic
57	C₂×C₄	—	2	1	no	—
58	C₂×C₄	—	2	1	no	—
59	C₂×C₄	—	2	4	no	—
60	C₂×C₄	—	4	2	no	—
61	C₂×C₄	—	4	4	no	—
62	C₂×C₄	C₄×Dic₃	1	1	yes	characteristic
63	C₂×C₄	Dic₃⋊C₄	1	2	yes	—
64	C₂×C₄	C₄⋊Dic₃	1	1	yes	characteristic
65	C₂×C₄	(C₂×D₃)⋊C₄	1	2	yes	—
66	C₈	—	6	8	no	—
67	C₂×C₃	#329	1	2	yes	—
68	C₂×C₃	#330	1	2	yes	—
69	C₂×C₃	#339	1	2	yes	—
70	C₂×C₃	#343	1	1	yes	characteristic
71	C₂²	C₃⋊#95	1	1	yes	characteristic
72	C₂²	#833	1	1	yes	characteristic
73	C₂²	#836	1	1	yes	characteristic
74	C₂²	#837	1	4	yes	—
75	C₄	—	2	1	no	—
76	C₄	—	2	1	no	—
77	C₄	—	2	2	no	—
78	C₄	—	2	2	no	—
79	C₄	—	8	2	no	—
80	C₃	#1295	1	1	yes	characteristic, complemented, Sylow
81	C₂	#4530	1	1	yes	characteristic
82	C₂	#4532	1	2	yes	—
83	C₂	#4540	1	2	yes	—
84	C₂	#4543	1	2	yes	—

	subgroup	G/H
center	C₂³	#217
commutator	C₂×C₃×C₄	C₄²
Frattini	C₂×C₄²	C₂×D₃
Fitting	C₃×#382	C₂
socle, prime radical	C₂³×C₃	C₂²⋊C₄
cosocle	C₂×C₃×C₄²	C₂²
hypercenter	#382	D₃
hypocenter	C₃	#1295
LCS	G ▷ C₂×C₃×C₄ ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #382

Complements

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

N	H ≅ G/N	type
9. #803	C₄	semidirect
80. C₃	#1295	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
C₂×#803	2
#1295	3