Subgroups — #12893

Order 256.

117 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#2108	C₂	1	1	yes	maximal, characteristic
2	#1833	C₂	1	2	yes	maximal
3	#1385	C₂	1	1	yes	maximal, characteristic
4	#1384	C₂	1	1	yes	maximal, characteristic
5	#1382	C₂	1	2	yes	maximal
6	C₂×M₃₂	C₂²	1	1	yes	characteristic
7	C₂²×C₁₆	C₂²	1	1	yes	characteristic
8	C₂×#108	C₂²	1	2	yes	—
9	#427	C₂²	1	2	yes	—
10	#405	C₂²	1	1	yes	characteristic
11	#346	C₄	1	2	yes	complemented
12	C₄×C₁₆	C₄	1	2	yes	complemented
13	#344	—	2	2	no	—
14	H(3,ℤ/4)	—	2	2	no	—
15	C₂×M₁₆	—	2	2	no	—
16	C₂×M₁₆	D₄	1	1	yes	characteristic
17	C₂²×C₈	C₂³	1	1	yes	characteristic
18	#117	—	2	2	no	—
19	#117	D₄	1	1	yes	characteristic
20	M₃₂	—	2	2	no	—
21	C₂×C₁₆	—	2	2	no	—
22	C₂×C₁₆	C₂×C₄	1	1	yes	characteristic
23	C₂×C₁₆	C₂×C₄	1	2	yes	—
24	C₂×C₁₆	C₂×C₄	1	1	yes	characteristic
25	#108	—	2	2	no	—
26	#108	—	2	2	no	—
27	#107	—	2	2	no	—
28	#106	—	2	2	no	—
29	#97	Q₈	1	2	yes	—
30	C₄×C₈	C₂×C₄	1	2	yes	—
31	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
32	M₁₆	—	2	2	no	—
33	M₁₆	—	4	2	no	—
34	M₁₆	—	4	2	no	—
35	C₂×C₈	—	4	2	no	—
36	C₂×C₈	C₄⋊C₄	1	2	yes	—
37	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
38	C₂×C₈	C₂×Q₈	1	1	yes	characteristic
39	C₂×C₈	#41	1	2	yes	—
40	C₂×C₈	#41	1	2	yes	—
41	C₄⋊C₄	—	2	1	no	—
42	C₄⋊C₄	—	4	2	no	—
43	C₂²⋊C₄	—	2	1	no	—
44	C₂²⋊C₄	—	4	2	no	—
45	C₄²	—	4	2	no	—
46	C₄²	C₄⋊C₄	1	2	yes	—
47	C₁₆	—	2	1	no	—
48	C₁₆	—	2	1	no	—
49	C₁₆	—	2	2	no	—
50	C₂³	#135	1	1	yes	characteristic
51	C₂×C₄	—	2	2	no	—
52	C₂×C₄	—	8	2	no	—
53	C₂×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
54	C₂×C₄	#124	1	2	yes	—
55	C₂×C₄	#135	1	2	yes	—
56	C₂×C₄	#135	1	1	yes	characteristic
57	C₈	—	2	2	no	—
58	C₈	—	8	2	no	—
59	C₈	#117	1	1	yes	characteristic
60	C₈	#117	1	1	yes	characteristic
61	C₈	#125	1	2	yes	—
62	C₂²	—	4	1	no	—
63	C₂²	#427	1	1	yes	characteristic
64	C₂²	#488	1	2	yes	—
65	C₄	—	4	2	no	—
66	C₄	—	16	2	no	—
67	C₄	#382	1	1	yes	characteristic
68	C₄	#427	1	1	yes	characteristic
69	C₄	#488	1	2	yes	—
70	C₂	—	2	1	no	—
71	C₂	—	2	2	no	—
72	C₂	#1835	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#427
commutator	C₂×C₈	C₂²×C₄
Frattini, cosocle	C₂²×C₈	C₂³
socle	C₂	#1835
prime radical	C₂³	#135
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ #117 ◁ #405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
11. #346	C₄	semidirect
12. C₄×C₁₆	C₄	semidirect

Maximal subgroups

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

subgroup	index
#2108	2
#1833	2
#1385	2
#1384	2
#1382	2