Subgroups — #12027

Order 256.

90 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#2166	C₂	1	2	yes	maximal
2	#2108	C₂	1	1	yes	maximal, characteristic
3	#1373	C₂	1	2	yes	maximal, complemented
4	#1369	C₂	1	2	yes	maximal, complemented
5	C₂×M₃₂	C₄	1	1	yes	characteristic
6	C₂²×C₁₆	C₄	1	1	yes	characteristic
7	#405	C₂²	1	1	yes	characteristic
8	#364	C₂²	1	4	yes	—
9	#349	—	2	2	no	—
10	#347	—	2	2	no	—
11	#346	C₄	1	1	yes	characteristic
12	#346	C₂²	1	1	yes	characteristic
13	C₄×C₁₆	C₄	1	1	yes	characteristic
14	C₄×C₁₆	C₂²	1	1	yes	characteristic
15	C₂×M₁₆	D₄	1	1	yes	characteristic
16	C₂²×C₈	C₂×C₄	1	1	yes	characteristic
17	#117	Q₈	1	1	yes	characteristic
18	M₃₂	—	2	2	no	—
19	M₃₂	—	2	4	no	—
20	C₂×C₁₆	—	2	2	no	—
21	C₂×C₁₆	C₂×C₄	1	1	yes	characteristic
22	C₂×C₁₆	C₂×C₄	1	1	yes	characteristic
23	C₂×C₁₆	C₂×C₄	1	1	yes	characteristic
24	C₂×C₁₆	C₂×C₄	1	1	yes	characteristic
25	#97	D₄	1	1	yes	characteristic
26	#97	Q₈	1	1	yes	characteristic
27	C₄×C₈	C₂×C₄	1	1	yes	characteristic
28	C₄×C₈	C₂³	1	1	yes	characteristic
29	C₂²×C₄	C₄⋊C₄	1	1	yes	characteristic
30	M₁₆	—	2	2	no	—
31	C₂×C₈	—	2	1	no	—
32	C₂×C₈	—	2	1	no	—
33	C₂×C₈	C₄⋊C₄	1	1	yes	characteristic
34	C₂×C₈	C₄⋊C₄	1	1	yes	characteristic
35	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
36	C₂×C₈	C₂×D₄	1	1	yes	characteristic
37	C₄⋊C₄	—	2	1	no	—
38	C₂²⋊C₄	—	2	1	no	—
39	C₄²	C₄⋊C₄	1	1	yes	characteristic
40	C₄²	C₂×Q₈	1	1	yes	characteristic
41	C₁₆	—	2	1	no	—
42	C₁₆	—	2	1	no	—
43	C₁₆	—	4	4	no	—
44	C₁₆	M₁₆	1	2	yes	—
45	C₁₆	M₁₆	1	2	yes	—
46	C₂³	#108	1	1	yes	characteristic
47	C₂×C₄	—	2	1	no	—
48	C₂×C₄	—	2	1	no	—
49	C₂×C₄	—	2	1	no	—
50	C₂×C₄	—	2	1	no	—
51	C₂×C₄	#108	1	1	yes	characteristic
52	C₂×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
53	C₈	—	2	1	no	—
54	C₈	—	2	1	no	—
55	C₈	C₂×M₁₆	1	1	yes	characteristic
56	C₈	C₂×M₁₆	1	1	yes	characteristic
57	C₈	#135	1	2	yes	—
58	C₂²	—	2	1	no	—
59	C₂²	—	4	1	no	—
60	C₂²	C₂×#108	1	1	yes	characteristic
61	C₄	—	2	1	no	—
62	C₄	—	4	1	no	—
63	C₄	—	4	1	no	—
64	C₄	#423	1	1	yes	characteristic
65	C₄	#427	1	1	yes	characteristic
66	C₂	—	2	1	no	—
67	C₂	—	4	1	no	—
68	C₂	#1568	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#427
commutator	C₂×C₈	C₂²×C₄
Frattini, cosocle	C₄×C₈	C₂³
socle	C₂	#1568
prime radical	C₂³	#108
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ #97 ◁ #405 ◁ 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
3. #1373	C₂	semidirect
4. #1369	C₂	semidirect

Maximal subgroups

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

subgroup	index
#2166	2
#2108	2
#1373	2
#1369	2