Subgroups — #69065

Order 384.

140 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#4699	C₂	1	1	yes	maximal, characteristic
2	#4682	C₂	1	1	yes	maximal, characteristic
3	C₃×C₄×C₁₆	C₂	1	1	yes	maximal, characteristic
4	#4503	C₂	1	4	yes	maximal
5	#2241	—	3	1	no	maximal, Sylow
6	C₂×Dic₁₂	—	2	1	no	—
7	C₂×Dic₁₂	C₂²	1	2	yes	—
8	#869	—	2	1	no	—
9	#869	—	2	1	no	—
10	C₂×C₃×C₁₆	C₂²	1	2	yes	—
11	C₃×C₄×C₈	C₂²	1	1	yes	characteristic
12	#818	—	2	1	no	—
13	#818	C₂²	1	2	yes	—
14	#500	—	3	1	no	—
15	#494	—	3	1	no	—
16	#358	—	3	4	no	—
17	C₄×C₁₆	D₃	1	1	yes	characteristic
18	C₂×Dic₆	—	2	2	no	—
19	C₂×Dic₆	—	4	1	no	—
20	C₂×C₃×C₈	D₄	1	1	yes	characteristic
21	C₂×C₃×C₈	C₂³	1	1	yes	characteristic
22	C₃×C₄²	D₄	1	1	yes	characteristic
23	C₄⋊Dic₃	—	2	2	no	—
24	C₄⋊Dic₃	—	4	1	no	—
25	C₄⋊Dic₃	—	4	1	no	—
26	C₄⋊Dic₃	—	4	1	no	—
27	Dic₁₂	—	2	4	no	—
28	Dic₁₂	—	4	2	no	—
29	C₃×C₁₆	—	2	2	no	—
30	C₂×Q₁₆	—	3	2	no	—
31	C₂×Q₁₆	—	6	1	no	—
32	C₄⋊Q₈	—	6	1	no	—
33	C₄⋊Q₈	—	6	1	no	—
34	C₂×C₁₆	C₂×D₃	1	2	yes	—
35	#107	—	3	2	no	—
36	#107	—	6	1	no	—
37	C₄×C₈	C₂×D₃	1	1	yes	characteristic
38	C₂×C₃×C₄	D₈	1	1	yes	characteristic
39	C₂×C₃×C₄	D₈	1	1	yes	characteristic
40	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
41	C₂×Dic₃	—	4	2	no	—
42	C₂×Dic₃	—	4	2	no	—
43	Dic₆	—	4	2	no	—
44	Dic₆	—	4	2	no	—
45	Dic₆	—	8	2	no	—
46	C₃×C₈	SD₁₆	1	2	yes	—
47	C₃×C₈	#41	1	2	yes	—
48	C₂×Q₈	—	6	2	no	—
49	C₂×Q₈	—	12	1	no	—
50	Q₁₆	—	6	4	no	—
51	Q₁₆	—	12	2	no	—
52	C₂×C₈	D₁₂	1	1	yes	characteristic
53	C₂×C₈	C₂²×D₃	1	1	yes	characteristic
54	C₄⋊C₄	—	6	2	no	—
55	C₄⋊C₄	—	12	1	no	—
56	C₄⋊C₄	—	12	1	no	—
57	C₄⋊C₄	—	12	1	no	—
58	C₄²	D₁₂	1	1	yes	characteristic
59	C₁₆	—	2	2	no	—
60	C₂²×C₃	C₂×D₈	1	1	yes	characteristic
61	C₃×C₄	SD₃₂	1	2	yes	—
62	C₃×C₄	Q₃₂	1	2	yes	—
63	C₃×C₄	#124	1	1	yes	characteristic
64	C₃×C₄	C₂×SD₁₆	1	1	yes	characteristic
65	Dic₃	—	8	2	no	—
66	Dic₃	—	8	2	no	—
67	Q₈	—	12	2	no	—
68	Q₈	—	12	2	no	—
69	Q₈	—	24	2	no	—
70	C₂×C₄	—	12	2	no	—
71	C₂×C₄	—	12	2	no	—
72	C₂×C₄	D₂₄	1	1	yes	characteristic
73	C₂×C₄	D₂₄	1	1	yes	characteristic
74	C₂×C₄	C₂×D₁₂	1	1	yes	characteristic
75	C₈	#204	1	2	yes	—
76	C₈	#235	1	2	yes	—
77	C₂×C₃	#486	1	1	yes	characteristic
78	C₂×C₃	C₂×SD₃₂	1	1	yes	characteristic
79	C₂×C₃	C₂×Q₃₂	1	1	yes	characteristic
80	C₂²	C₂×D₂₄	1	1	yes	characteristic
81	C₄	—	24	2	no	—
82	C₄	—	24	2	no	—
83	C₄	#800	1	2	yes	—
84	C₄	Dic₂₄	1	2	yes	—
85	C₄	#875	1	1	yes	characteristic
86	C₄	C₂×#204	1	1	yes	characteristic
87	C₃	#2241	1	1	yes	characteristic, complemented, Sylow
88	C₂	#4694	1	1	yes	characteristic
89	C₂	C₂×#800	1	1	yes	characteristic
90	C₂	C₂×Dic₂₄	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₂₄
commutator, cosocle	C₂×C₃×C₈	C₂³
Frattini	C₂×C₈	C₂²×D₃
Fitting	C₃×C₄×C₁₆	C₂
socle, prime radical	C₂²×C₃	C₂×D₈
hypercenter	C₄×C₁₆	D₃
hypocenter	C₃	#2241
LCS	G ▷ C₂×C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×C₁₆

Complements

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

N	H ≅ G/N	type
87. C₃	#2241	semidirect

Maximal subgroups

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

subgroup	index
#4699	2
#4682	2
C₃×C₄×C₁₆	2
#4503	2
#2241	3