Subgroups — #71254

Order 384.

214 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#492	C₂	1	1	yes	maximal, characteristic
2	#5080	C₂	1	2	yes	maximal, complemented
3	#5076	C₂	1	2	yes	maximal
4	#4680	C₂	1	1	yes	maximal, characteristic, complemented
5	C₈×C₃⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
6	#1705	—	3	1	no	maximal, Sylow
7	C₂×C₃×SD₁₆	—	2	2	no	—
8	C₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
9	C₃×#127	C₂²	1	1	yes	characteristic
10	C₂×#214	—	2	4	no	—
11	#869	C₂²	1	2	yes	—
12	C₃×C₄×C₈	C₂²	1	1	yes	characteristic
13	#817	—	2	2	no	—
14	#808	—	2	4	no	—
15	C₄×C₃⋊C₈	C₂²	1	2	yes	—
16	#498	—	3	2	no	—
17	#498	—	3	1	no	—
18	#492	—	3	2	no	—
19	#492	D₃	1	1	yes	characteristic
20	C₈²	—	3	1	no	—
21	C₂×C₃×Q₈	—	2	1	no	—
22	C₂×C₃×D₄	—	2	2	no	—
23	C₂×C₃×D₄	—	2	1	no	—
24	C₂×Dic₆	—	2	2	no	—
25	C₃×SD₁₆	—	4	4	no	—
26	C₂×C₃×C₈	D₄	1	2	yes	—
27	C₃×C₄⋊C₄	—	2	2	no	—
28	C₃×C₄²	C₂³	1	1	yes	characteristic
29	#214	—	4	8	no	—
30	C₄⋊Dic₃	—	2	2	no	—
31	C₄⋊Dic₃	—	2	2	no	—
32	C₂×C₃⋊C₈	D₄	1	4	yes	—
33	C₂×SD₁₆	—	2	2	no	—
34	C₂×SD₁₆	—	6	4	no	—
35	C₄⋊Q₈	—	3	2	no	—
36	C₄⋊Q₈	C₂×D₃	1	1	yes	characteristic
37	#127	C₂×D₃	1	1	yes	characteristic
38	#106	—	6	4	no	—
39	#106	—	6	2	no	—
40	C₄×C₈	—	3	2	no	—
41	C₄×C₈	C₂×D₃	1	1	yes	characteristic
42	C₂³×C₃	—	4	1	no	—
43	C₃×Q₈	—	4	2	no	—
44	C₃×D₄	—	4	4	no	—
45	C₃×D₄	—	4	2	no	—
46	C₂×C₃×C₄	—	4	1	no	—
47	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
48	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
49	C₂×Dic₃	—	4	2	no	—
50	Dic₆	—	4	4	no	—
51	C₃×C₈	SD₁₆	1	4	yes	complemented
52	C₃⋊C₈	SD₁₆	1	8	yes	complemented
53	C₂×Q₈	—	2	1	no	—
54	C₂×Q₈	—	6	2	no	—
55	C₂×D₄	—	2	2	no	—
56	C₂×D₄	—	2	1	no	—
57	SD₁₆	—	4	4	no	—
58	SD₁₆	—	12	8	no	—
59	C₂×C₈	—	3	4	no	—
60	C₂×C₈	#67	1	2	yes	—
61	C₄⋊C₄	—	2	2	no	—
62	C₄⋊C₄	—	6	2	no	—
63	C₄⋊C₄	—	6	2	no	—
64	C₄²	C₂²×D₃	1	1	yes	characteristic
65	C₂²×C₃	—	8	2	no	—
66	C₂²×C₃	—	8	1	no	—
67	C₂²×C₃	#127	1	1	yes	characteristic
68	C₃×C₄	—	8	1	no	—
69	C₃×C₄	C₂×SD₁₆	1	4	yes	—
70	C₃×C₄	C₂×SD₁₆	1	2	yes	—
71	Dic₃	—	8	2	no	—
72	C₂³	—	4	1	no	—
73	Q₈	—	4	2	no	—
74	Q₈	—	12	4	no	—
75	D₄	—	4	4	no	—
76	D₄	—	4	2	no	—
77	C₂×C₄	—	4	1	no	—
78	C₂×C₄	—	12	2	no	—
79	C₂×C₄	D₃×D₄	1	2	yes	—
80	C₂×C₄	C₂×#67	1	1	yes	characteristic
81	C₈	—	3	8	no	—
82	C₈	#214	1	4	yes	complemented
83	C₂×C₃	—	16	1	no	—
84	C₂×C₃	#492	1	2	yes	—
85	C₂×C₃	#492	1	1	yes	characteristic
86	C₂²	—	8	2	no	—
87	C₂²	—	8	1	no	—
88	C₂²	#940	1	1	yes	characteristic
89	C₄	—	8	1	no	—
90	C₄	—	24	2	no	—
91	C₄	D₃×SD₁₆	1	4	yes	—
92	C₄	C₂×#214	1	2	yes	—
93	C₃	#1705	1	1	yes	characteristic, complemented, Sylow
94	C₂	—	16	1	no	—
95	C₂	#5076	1	1	yes	characteristic
96	C₂	#5175	1	2	yes	—

	subgroup	G/H
center	C₂²	#940
commutator, cosocle	C₃×C₄²	C₂³
Frattini	C₄²	C₂²×D₃
Fitting	C₃×#492	C₂
socle	C₂²×C₃	#127
prime radical	C₃×#127	C₂²
hypercenter	#492	D₃
hypocenter	C₃	#1705
LCS	G ▷ C₃×C₄² ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #492

Complements

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

N	H ≅ G/N	type
2. #5080	C₂	semidirect
4. #4680	C₂	semidirect
5. C₈×C₃⋊C₈	C₂	semidirect
51. C₃×C₈	SD₁₆	semidirect
52. C₃⋊C₈	SD₁₆	semidirect
82. C₈	#214	semidirect
93. C₃	#1705	semidirect

Maximal subgroups

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

subgroup	index
C₃×#492	2
#5080	2
#5076	2
#4680	2
C₈×C₃⋊C₈	2
#1705	3