Subgroups — #70516

Order 384.

188 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	#4924	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×C₃⋊C₈	C₂	1	1	yes	maximal, characteristic
4	#4558	C₂	1	4	yes	maximal
5	#1837	—	3	1	no	maximal, Sylow
6	C₂×C₃×M₁₆	C₂²	1	2	yes	—
7	C₂×C₃×C₄²	C₂²	1	1	yes	characteristic
8	C₂×#208	C₂²	1	2	yes	—
9	C₂²×C₃⋊C₈	C₂²	1	2	yes	—
10	C₃×C₄⋊C₈	—	2	2	no	—
11	C₄⋊(C₃⋊C₈)	—	2	2	no	—
12	C₄×C₃⋊C₈	C₄	1	2	yes	—
13	C₄×C₃⋊C₈	C₄	1	2	yes	—
14	#423	—	3	1	no	—
15	#423	D₃	1	1	yes	characteristic
16	C₂×C₄×C₈	—	3	1	no	—
17	#341	—	3	4	no	—
18	C₂²×C₃×C₄	Q₈	1	2	yes	—
19	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
20	C₃×M₁₆	—	2	4	no	—
21	C₂×C₃×C₈	—	2	2	no	—
22	C₃×C₄²	C₂×C₄	1	1	yes	characteristic
23	C₃×C₄²	C₂×C₄	1	1	yes	characteristic
24	C₃×C₄²	D₄	1	2	yes	—
25	#208	—	2	4	no	—
26	C₂×C₃⋊C₈	—	2	2	no	—
27	C₂×C₃⋊C₈	—	2	2	no	—
28	C₂×C₃⋊C₈	—	2	2	no	—
29	C₂×C₃⋊C₈	C₂×C₄	1	4	yes	—
30	C₂×M₁₆	—	3	2	no	—
31	C₂×M₁₆	C₂×D₃	1	2	yes	—
32	C₂²×C₈	—	3	2	no	—
33	C₂×C₄²	C₂×D₃	1	1	yes	characteristic
34	C₄⋊C₈	—	2	2	no	—
35	C₄⋊C₈	—	6	2	no	—
36	C₄×C₈	—	3	2	no	—
37	C₄×C₈	—	3	2	no	—
38	C₂³×C₃	C₂×Q₈	1	1	yes	characteristic
39	C₂×C₃×C₄	—	2	4	no	—
40	C₂×C₃×C₄	C₄⋊C₄	1	2	yes	—
41	C₂×C₃×C₄	C₄⋊C₄	1	2	yes	—
42	C₂×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
43	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
44	C₂×C₃×C₄	#41	1	2	yes	—
45	C₂×C₃×C₄	#41	1	2	yes	—
46	C₃×C₈	—	4	2	no	—
47	C₃⋊C₈	—	2	4	no	—
48	C₃⋊C₈	—	4	2	no	—
49	C₂²×C₄	Dic₆	1	2	yes	—
50	C₂²×C₄	C₂²×D₃	1	1	yes	characteristic
51	M₁₆	—	2	4	no	—
52	M₁₆	—	6	4	no	—
53	C₂×C₈	—	2	2	no	—
54	C₂×C₈	—	3	4	no	—
55	C₂×C₈	—	6	2	no	—
56	C₂×C₈	—	6	2	no	—
57	C₂×C₈	—	6	2	no	—
58	C₄²	C₄×D₃	1	1	yes	characteristic
59	C₄²	C₄×D₃	1	1	yes	characteristic
60	C₄²	#67	1	2	yes	—
61	C₂²×C₃	—	2	2	no	—
62	C₂²×C₃	C₂×C₄⋊C₄	1	1	yes	characteristic
63	C₂²×C₃	#125	1	2	yes	—
64	C₃×C₄	—	2	2	no	—
65	C₃×C₄	#108	1	4	yes	—
66	C₃×C₄	#117	1	2	yes	—
67	C₃×C₄	#124	1	1	yes	characteristic
68	C₃×C₄	#124	1	1	yes	characteristic
69	C₂³	C₂×Dic₆	1	1	yes	characteristic
70	C₂×C₄	—	2	4	no	—
71	C₂×C₄	Dic₃⋊C₄	1	2	yes	—
72	C₂×C₄	Dic₃⋊C₄	1	2	yes	—
73	C₂×C₄	C₂×C₄×D₃	1	1	yes	characteristic
74	C₂×C₄	#237	1	2	yes	—
75	C₂×C₄	#239	1	2	yes	—
76	C₂×C₄	C₂×#67	1	1	yes	characteristic
77	C₈	—	4	2	no	—
78	C₈	—	6	4	no	—
79	C₈	—	12	2	no	—
80	C₂×C₃	—	2	2	no	—
81	C₂×C₃	#382	1	1	yes	characteristic
82	C₂×C₃	C₂×#108	1	2	yes	—
83	C₂²	—	2	2	no	—
84	C₂²	#890	1	2	yes	—
85	C₂²	C₂×Dic₃⋊C₄	1	1	yes	characteristic
86	C₄	—	2	2	no	—
87	C₄	#822	1	4	yes	—
88	C₄	#892	1	2	yes	—
89	C₄	#936	1	1	yes	characteristic
90	C₄	#947	1	1	yes	characteristic
91	C₃	#1837	1	1	yes	characteristic, complemented, Sylow
92	C₂	—	2	2	no	—
93	C₂	#4977	1	1	yes	characteristic
94	C₂	C₂×#822	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×#67
commutator	C₂×C₃×C₄	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₃
Fitting	C₃×#423	C₂
socle	C₂²×C₃	C₂×C₄⋊C₄
prime radical	C₂³×C₃	C₂×Q₈
cosocle	C₂²×C₃×C₄	C₂³
hypercenter	#423	D₃
hypocenter	C₃	#1837
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #423

Complements

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

N	H ≅ G/N	type
91. C₃	#1837	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
#4924	2
C₂×C₄×C₃⋊C₈	2
#4558	2
#1837	3