Subgroups — #78479

Order 384.

348 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#5470	C₂	1	2	yes	maximal
2	C₂×C₃×#97	C₂	1	1	yes	maximal, characteristic
3	#5106	C₂	1	4	yes	maximal
4	#4702	C₂	1	8	yes	maximal, complemented
5	#3162	—	3	1	no	maximal, Sylow
6	C₂²×C₃×C₈	C₂²	1	2	yes	—
7	C₂×C₃×C₄²	C₂²	1	1	yes	characteristic
8	#926	C₂²	1	4	yes	—
9	#924	—	2	4	no	—
10	#870	C₂²	1	4	yes	—
11	#869	C₂²	1	4	yes	—
12	C₄×Dic₆	—	2	4	no	—
13	C₃×#97	C₂²	1	4	yes	—
14	#818	C₂²	1	8	yes	—
15	#817	C₂²	1	8	yes	—
16	#533	—	3	2	no	—
17	#501	—	3	8	no	—
18	#427	—	3	4	no	—
19	C₂×#97	D₃	1	1	yes	characteristic
20	C₂²×C₃×C₄	D₄	1	2	yes	—
21	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
22	C₂×Dic₆	—	2	4	no	—
23	C₂×C₃×C₈	Q₈	1	8	yes	complemented
24	C₂×C₃×C₈	C₂³	1	4	yes	—
25	C₃×C₄²	D₄	1	2	yes	—
26	C₃×C₄²	C₂³	1	2	yes	—
27	#217	—	2	4	no	—
28	C₄⋊Dic₃	—	2	4	no	—
29	C₄⋊Dic₃	C₂³	1	8	yes	—
30	Dic₃⋊C₄	—	2	8	no	—
31	C₄×Dic₃	—	2	4	no	—
32	C₂²×C₈	C₂×D₃	1	2	yes	—
33	C₄⋊Q₈	—	3	4	no	—
34	#125	—	3	4	no	—
35	C₂²⋊Q₈	—	6	4	no	—
36	C₄×Q₈	—	6	4	no	—
37	#117	—	3	4	no	—
38	C₂×C₄²	C₂×D₃	1	1	yes	characteristic
39	#107	—	3	8	no	—
40	#106	—	3	8	no	—
41	#97	C₂×D₃	1	4	yes	—
42	C₂³×C₃	C₂×D₄	1	1	yes	characteristic
43	C₂×C₃×C₄	—	4	2	no	—
44	C₂×C₃×C₄	C₂×D₄	1	4	yes	—
45	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
46	C₂×C₃×C₄	C₂×Q₈	1	4	yes	—
47	C₂×C₃×C₄	C₂⁴	1	1	yes	characteristic
48	C₂×Dic₃	—	2	8	no	—
49	Dic₆	—	4	8	no	—
50	C₃×C₈	C₂×Q₈	1	8	yes	—
51	C₂×Q₈	—	6	4	no	—
52	C₂²×C₄	D₁₂	1	2	yes	—
53	C₂²×C₄	C₂²×D₃	1	1	yes	characteristic
54	C₂×C₈	Dic₆	1	8	yes	complemented
55	C₂×C₈	C₂²×D₃	1	4	yes	—
56	C₄⋊C₄	—	3	8	no	—
57	C₄⋊C₄	—	6	8	no	—
58	C₄⋊C₄	—	6	4	no	—
59	C₂²⋊C₄	—	6	4	no	—
60	C₄²	—	6	4	no	—
61	C₄²	D₁₂	1	2	yes	—
62	C₄²	C₂²×D₃	1	2	yes	—
63	C₂²×C₃	—	2	2	no	—
64	C₂²×C₃	C₄⋊Q₈	1	2	yes	—
65	C₂²×C₃	C₂²×D₄	1	1	yes	characteristic
66	C₃×C₄	—	2	4	no	—
67	C₃×C₄	C₄⋊Q₈	1	2	yes	—
68	C₃×C₄	C₂²×Q₈	1	2	yes	—
69	Dic₃	—	4	8	no	—
70	C₂³	C₂×D₁₂	1	1	yes	characteristic
71	Q₈	—	12	8	no	—
72	C₂×C₄	—	4	2	no	—
73	C₂×C₄	—	6	8	no	—
74	C₂×C₄	C₂×Dic₆	1	4	yes	—
75	C₂×C₄	C₂×D₁₂	1	4	yes	—
76	C₂×C₄	C₂×D₁₂	1	1	yes	characteristic
77	C₂×C₄	C₂³×D₃	1	1	yes	characteristic
78	C₈	C₂×Dic₆	1	8	yes	—
79	C₂×C₃	—	2	2	no	—
80	C₂×C₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
81	C₂×C₃	#575	1	2	yes	—
82	C₂²	—	2	2	no	—
83	C₂²	#869	1	2	yes	—
84	C₂²	C₂²×D₁₂	1	1	yes	characteristic
85	C₄	—	2	4	no	—
86	C₄	—	12	8	no	—
87	C₄	#869	1	2	yes	—
88	C₄	C₂²×Dic₆	1	2	yes	—
89	C₃	#3162	1	1	yes	characteristic, complemented, Sylow
90	C₂	—	2	2	no	—
91	C₂	C₂×#869	1	1	yes	characteristic
92	C₂	#5748	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×D₁₂
commutator, cosocle	C₂×C₃×C₄	C₂⁴
Frattini	C₂×C₄	C₂³×D₃
Fitting	C₂×C₃×#97	C₂
socle	C₂²×C₃	C₂²×D₄
prime radical	C₂³×C₃	C₂×D₄
hypercenter	C₂×#97	D₃
hypocenter	C₃	#3162
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×#97

Complements

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

N	H ≅ G/N	type
4. #4702	C₂	semidirect
23. C₂×C₃×C₈	Q₈	semidirect
54. C₂×C₈	Dic₆	semidirect
89. C₃	#3162	semidirect

Maximal subgroups

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

subgroup	index
#5470	2
C₂×C₃×#97	2
#5106	2
#4702	2
#3162	3