Subgroups — #12019

Order 256.

113 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#346	C₂	1	1	yes	maximal, characteristic
2	#1568	C₂	1	1	yes	maximal, characteristic
3	#1566	C₂	1	1	yes	maximal, characteristic
4	#1368	C₂	1	4	yes	maximal, complemented
5	C₂²×C₁₆	C₄	1	2	yes	—
6	#432	—	2	1	no	—
7	#423	—	2	1	no	—
8	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
9	#346	C₄	1	1	yes	characteristic
10	#346	C₄	1	1	yes	characteristic
11	#346	C₂²	1	2	yes	—
12	#335	—	2	1	no	—
13	#334	—	2	1	no	—
14	#334	C₂²	1	2	yes	—
15	#334	C₂²	1	2	yes	—
16	C₂×M₁₆	—	4	1	no	—
17	C₂²×C₈	C₂×C₄	1	1	yes	characteristic
18	C₂²×C₈	Q₈	1	1	yes	characteristic
19	C₂×C₄²	D₄	1	1	yes	characteristic
20	C₂×C₁₆	—	2	4	no	—
21	C₂×C₁₆	C₂×C₄	1	2	yes	—
22	C₂×C₁₆	C₂×C₄	1	2	yes	—
23	C₄⋊C₈	—	2	2	no	—
24	C₄⋊C₈	—	2	2	no	—
25	C₄⋊C₈	—	4	1	no	—
26	C₂²⋊C₈	—	4	1	no	—
27	#97	—	4	1	no	—
28	C₄×C₈	C₂×C₄	1	1	yes	characteristic
29	C₄×C₈	D₄	1	1	yes	characteristic
30	C₄×C₈	Q₈	1	1	yes	characteristic
31	C₄×C₈	C₂³	1	1	yes	characteristic
32	C₂²×C₄	C₄⋊C₄	1	1	yes	characteristic
33	C₂²×C₄	SD₁₆	1	1	yes	characteristic
34	C₂²×C₄	SD₁₆	1	1	yes	characteristic
35	M₁₆	—	8	2	no	—
36	C₂×C₈	—	2	2	no	—
37	C₂×C₈	—	2	2	no	—
38	C₂×C₈	—	4	2	no	—
39	C₂×C₈	—	4	2	no	—
40	C₂×C₈	C₄⋊C₄	1	1	yes	characteristic
41	C₂×C₈	C₄⋊C₄	1	1	yes	characteristic
42	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
43	C₂×C₈	C₂×Q₈	1	1	yes	characteristic
44	C₄²	C₄⋊C₄	1	1	yes	characteristic
45	C₄²	SD₁₆	1	1	yes	characteristic
46	C₄²	SD₁₆	1	1	yes	characteristic
47	C₄²	C₂×D₄	1	1	yes	characteristic
48	C₁₆	—	2	2	no	—
49	C₁₆	M₁₆	1	4	yes	complemented
50	C₂³	#106	1	1	yes	characteristic
51	C₂×C₄	—	2	2	no	—
52	C₂×C₄	—	2	1	no	—
53	C₂×C₄	—	2	1	no	—
54	C₂×C₄	—	4	1	no	—
55	C₂×C₄	#106	1	1	yes	characteristic
56	C₂×C₄	#106	1	1	yes	characteristic
57	C₂×C₄	#106	1	1	yes	characteristic
58	C₂×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
59	C₂×C₄	C₂×SD₁₆	1	1	yes	characteristic
60	C₂×C₄	C₂×SD₁₆	1	1	yes	characteristic
61	C₈	—	2	1	no	—
62	C₈	—	2	1	no	—
63	C₈	—	8	2	no	—
64	C₈	—	8	2	no	—
65	C₈	#108	1	2	yes	—
66	C₈	C₂×M₁₆	1	2	yes	—
67	C₂²	—	2	1	no	—
68	C₂²	—	2	1	no	—
69	C₂²	#365	1	2	yes	—
70	C₂²	C₂×#106	1	1	yes	characteristic
71	C₄	—	2	1	no	—
72	C₄	—	2	1	no	—
73	C₄	—	2	1	no	—
74	C₄	—	2	1	no	—
75	C₄	—	2	1	no	—
76	C₄	#423	1	1	yes	characteristic
77	C₄	C₂×#108	1	1	yes	characteristic
78	C₂	—	2	2	no	—
79	C₂	#1566	1	1	yes	characteristic
80	C₂	C₂×#365	1	1	yes	characteristic
81	C₂	#2155	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	C₂×#106
commutator	C₂×C₈	C₂²×C₄
Frattini, cosocle	C₄×C₈	C₂³
prime radical	C₂³	#106
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
4. #1368	C₂	semidirect
49. C₁₆	M₁₆	semidirect

Maximal subgroups

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

subgroup	index
C₂×#346	2
#1568	2
#1566	2
#1368	2