Subgroups — #8824

Order 256.

161 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#1837	C₂	1	2	yes	maximal
2	#1567	C₂	1	2	yes	maximal
3	#1566	C₂	1	2	yes	maximal
4	C₂×C₈²	C₂	1	1	yes	maximal, characteristic
5	#423	C₂²	1	4	yes	—
6	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
7	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
8	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
9	#341	—	2	4	no	—
10	#335	—	2	4	no	—
11	#334	—	2	4	no	—
12	C₈²	C₄	1	1	yes	characteristic
13	C₈²	C₄	1	1	yes	characteristic
14	C₈²	C₄	1	1	yes	characteristic
15	C₈²	C₄	1	1	yes	characteristic
16	C₂×M₁₆	—	2	4	no	—
17	C₂²×C₈	D₄	1	2	yes	—
18	C₂²×C₈	Q₈	1	2	yes	—
19	C₂²×C₈	Q₈	1	2	yes	—
20	C₂×C₄²	C₂³	1	1	yes	characteristic
21	C₄⋊C₈	—	2	4	no	—
22	C₄⋊C₈	—	2	4	no	—
23	C₄×C₈	C₂×C₄	1	1	yes	characteristic
24	C₄×C₈	C₂×C₄	1	1	yes	characteristic
25	C₄×C₈	C₂×C₄	1	1	yes	characteristic
26	C₄×C₈	C₂×C₄	1	1	yes	characteristic
27	C₄×C₈	C₂×C₄	1	1	yes	characteristic
28	C₄×C₈	C₂×C₄	1	1	yes	characteristic
29	C₄×C₈	D₄	1	2	yes	—
30	C₄×C₈	D₄	1	2	yes	—
31	C₄×C₈	Q₈	1	2	yes	—
32	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
33	C₂²×C₄	C₂×Q₈	1	1	yes	characteristic
34	C₂²×C₄	C₂×Q₈	1	1	yes	characteristic
35	M₁₆	—	4	8	no	—
36	C₂×C₈	—	2	4	no	—
37	C₂×C₈	—	2	4	no	—
38	C₂×C₈	—	2	4	no	—
39	C₂×C₈	—	4	4	no	—
40	C₂×C₈	C₄⋊C₄	1	2	yes	—
41	C₂×C₈	C₄⋊C₄	1	2	yes	—
42	C₂×C₈	C₄⋊C₄	1	2	yes	—
43	C₂×C₈	C₄⋊C₄	1	2	yes	—
44	C₂×C₈	C₄⋊C₄	1	2	yes	—
45	C₂×C₈	C₄⋊C₄	1	2	yes	—
46	C₄²	C₂²×C₄	1	1	yes	characteristic
47	C₄²	C₂×D₄	1	1	yes	characteristic
48	C₄²	C₂×D₄	1	1	yes	characteristic
49	C₄²	C₂×Q₈	1	1	yes	characteristic
50	C₂³	C₄⋊Q₈	1	1	yes	characteristic
51	C₂×C₄	—	2	2	no	—
52	C₂×C₄	—	2	2	no	—
53	C₂×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
54	C₂×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
55	C₂×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
56	C₂×C₄	#127	1	1	yes	characteristic
57	C₂×C₄	C₄⋊Q₈	1	1	yes	characteristic
58	C₂×C₄	C₄⋊Q₈	1	1	yes	characteristic
59	C₂×C₄	#135	1	4	yes	—
60	C₈	—	2	2	no	—
61	C₈	—	2	2	no	—
62	C₈	—	2	2	no	—
63	C₈	—	2	2	no	—
64	C₈	—	8	4	no	—
65	C₈	#108	1	4	yes	—
66	C₈	#108	1	4	yes	—
67	C₂²	—	2	1	no	—
68	C₂²	—	2	1	no	—
69	C₂²	#384	1	1	yes	characteristic
70	C₂²	#499	1	2	yes	—
71	C₄	—	2	1	no	—
72	C₄	—	2	1	no	—
73	C₄	#427	1	2	yes	—
74	C₄	C₂×#108	1	2	yes	—
75	C₄	C₂×#108	1	2	yes	—
76	C₄	#495	1	2	yes	—
77	C₂	—	2	2	no	—
78	C₂	#1846	1	1	yes	characteristic
79	C₂	#1848	1	1	yes	characteristic
80	C₂	#1848	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#127
commutator	C₄²	C₂²×C₄
Frattini, cosocle	C₂×C₄²	C₂³
socle	C₂²	#384
prime radical	C₂³	C₄⋊Q₈
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

Maximal subgroups

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

subgroup	index
#1837	2
#1567	2
#1566	2
C₂×C₈²	2