Subgroups — #70519

Order 384.

212 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	C₂×#822	C₂	1	4	yes	maximal
3	#4924	C₂	1	1	yes	maximal, characteristic
4	C₂×#803	C₂	1	1	yes	maximal, characteristic
5	#1849	—	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	#822	—	2	8	no	—
12	C₄⋊(C₃⋊C₈)	—	2	2	no	—
13	#803	C₄	1	4	yes	—
14	C₂×#108	—	3	4	no	—
15	#423	—	3	1	no	—
16	#423	D₃	1	1	yes	characteristic
17	C₂×#97	—	3	1	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	2	yes	—
23	C₃×C₄²	D₄	1	2	yes	—
24	#208	—	2	4	no	—
25	C₂×C₃⋊C₈	—	2	2	no	—
26	C₂×C₃⋊C₈	C₂×C₄	1	4	yes	—
27	C₂×C₃⋊C₈	D₄	1	4	yes	—
28	C₂×C₃⋊C₈	Q₈	1	4	yes	—
29	C₂×M₁₆	—	3	2	no	—
30	C₂×M₁₆	C₂×D₃	1	2	yes	—
31	C₂²×C₈	—	3	2	no	—
32	C₂×C₄²	C₂×D₃	1	1	yes	characteristic
33	#108	—	6	8	no	—
34	C₄⋊C₈	—	2	2	no	—
35	C₄⋊C₈	—	6	2	no	—
36	#97	—	3	4	no	—
37	C₂³×C₃	C₂×Q₈	1	1	yes	characteristic
38	C₂×C₃×C₄	—	4	2	no	—
39	C₂×C₃×C₄	C₄⋊C₄	1	4	yes	—
40	C₂×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
41	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
42	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
43	C₂×C₃×C₄	C₂×Q₈	1	2	yes	—
44	C₃×C₈	—	4	2	no	—
45	C₃⋊C₈	—	4	2	no	—
46	C₃⋊C₈	C₄⋊C₄	1	8	yes	—
47	C₂²×C₄	Dic₆	1	2	yes	—
48	C₂²×C₄	C₂²×D₃	1	1	yes	characteristic
49	M₁₆	—	2	4	no	—
50	M₁₆	—	6	4	no	—
51	C₂×C₈	—	2	2	no	—
52	C₂×C₈	—	3	4	no	—
53	C₂×C₈	—	3	4	no	—
54	C₂×C₈	—	3	4	no	—
55	C₂×C₈	—	6	2	no	—
56	C₄²	C₄×D₃	1	2	yes	—
57	C₄²	#67	1	2	yes	—
58	C₂²×C₃	—	2	2	no	—
59	C₂²×C₃	C₂×C₄⋊C₄	1	1	yes	characteristic
60	C₂²×C₃	C₄⋊Q₈	1	2	yes	—
61	C₃×C₄	—	2	4	no	—
62	C₃×C₄	C₂×C₄⋊C₄	1	2	yes	—
63	C₃×C₄	#127	1	1	yes	characteristic
64	C₃×C₄	C₄⋊Q₈	1	1	yes	characteristic
65	C₂³	C₂×Dic₆	1	1	yes	characteristic
66	C₂×C₄	—	4	2	no	—
67	C₂×C₄	Dic₃⋊C₄	1	4	yes	—
68	C₂×C₄	C₂×C₄×D₃	1	1	yes	characteristic
69	C₂×C₄	D₃×D₄	1	2	yes	—
70	C₂×C₄	D₃×Q₈	1	2	yes	—
71	C₂×C₄	C₂×#67	1	1	yes	characteristic
72	C₈	—	3	8	no	—
73	C₈	—	4	2	no	—
74	C₈	—	12	2	no	—
75	C₂×C₃	—	2	2	no	—
76	C₂×C₃	#384	1	1	yes	characteristic
77	C₂×C₃	#430	1	2	yes	—
78	C₂²	—	2	2	no	—
79	C₂²	#888	1	2	yes	—
80	C₂²	C₂×Dic₃⋊C₄	1	1	yes	characteristic
81	C₄	—	2	4	no	—
82	C₄	D₃×C₄⋊C₄	1	2	yes	—
83	C₄	#940	1	1	yes	characteristic
84	C₄	#944	1	1	yes	characteristic
85	C₃	#1849	1	1	yes	characteristic, complemented, Sylow
86	C₂	—	2	2	no	—
87	C₂	#4975	1	1	yes	characteristic
88	C₂	#5124	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	#1849
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
85. C₃	#1849	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
C₂×#822	2
#4924	2
C₂×#803	2
#1849	3