Subgroups — #29230

Order 256.

204 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#2102	C₂	1	4	yes	maximal
2	#1953	C₂	1	4	yes	maximal
3	#1848	C₂	1	1	yes	maximal, characteristic
4	#1833	C₂	1	4	yes	maximal
5	#1772	C₂	1	2	yes	maximal
6	#430	C₂²	1	4	yes	—
7	C₂×#108	C₂²	1	4	yes	—
8	C₂×#108	C₂²	1	2	yes	—
9	#427	C₂²	1	2	yes	—
10	#423	C₂²	1	2	yes	—
11	#405	C₂²	1	4	yes	—
12	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
13	#344	C₂²	1	8	yes	—
14	#339	C₂²	1	4	yes	—
15	H(3,ℤ/4)	C₂²	1	4	yes	—
16	C₂×M₁₆	C₂³	1	4	yes	—
17	C₂×M₁₆	C₂³	1	4	yes	—
18	C₂²×C₈	C₂³	1	2	yes	—
19	#117	C₂³	1	4	yes	—
20	C₂×C₄²	C₂³	1	1	yes	characteristic
21	#108	—	2	2	no	—
22	#108	—	2	2	no	—
23	#108	—	2	4	no	—
24	#108	—	2	4	no	—
25	#108	—	4	4	no	—
26	#107	—	2	2	no	—
27	#106	—	2	2	no	—
28	C₄⋊C₈	—	4	2	no	—
29	#97	—	2	4	no	—
30	C₄×C₈	—	2	4	no	—
31	C₄×C₈	Q₈	1	1	yes	characteristic
32	C₄×C₈	Q₈	1	1	yes	characteristic
33	C₄×C₈	Q₈	1	2	yes	—
34	C₂²×C₄	—	2	1	no	—
35	C₂²×C₄	C₂⁴	1	1	yes	characteristic
36	M₁₆	—	2	4	no	—
37	M₁₆	—	2	8	no	—
38	M₁₆	—	2	4	no	—
39	C₂×C₈	—	2	4	no	—
40	C₂×C₈	—	2	4	no	—
41	C₂×C₈	C₂×Q₈	1	2	yes	—
42	C₂×C₈	C₂×Q₈	1	2	yes	—
43	C₂×C₈	#41	1	2	yes	—
44	C₂×C₈	#41	1	2	yes	—
45	C₂×C₈	#41	1	2	yes	—
46	C₂×C₈	#41	1	2	yes	—
47	C₄⋊C₄	—	2	4	no	—
48	C₂²⋊C₄	—	2	4	no	—
49	C₄²	—	2	1	no	—
50	C₄²	—	2	4	no	—
51	C₄²	C₂×Q₈	1	1	yes	characteristic
52	C₄²	C₂×Q₈	1	1	yes	characteristic
53	C₂³	2₋¹⁺⁴	1	1	yes	characteristic
54	C₂×C₄	—	2	1	no	—
55	C₂×C₄	—	2	1	no	—
56	C₂×C₄	—	4	2	no	—
57	C₂×C₄	—	4	4	no	—
58	C₂×C₄	C₂²×Q₈	1	1	yes	characteristic
59	C₂×C₄	C₂×#41	1	2	yes	—
60	C₂×C₄	C₂×#41	1	2	yes	—
61	C₂×C₄	H(5,𝔽₂)	1	1	yes	characteristic
62	C₈	—	4	4	no	—
63	C₈	—	4	4	no	—
64	C₈	#125	1	2	yes	—
65	C₈	#125	1	2	yes	—
66	C₈	#125	1	2	yes	—
67	C₈	#125	1	2	yes	—
68	C₂²	—	4	1	no	—
69	C₂²	#544	1	1	yes	characteristic
70	C₂²	#556	1	2	yes	—
71	C₄	—	2	2	no	—
72	C₄	—	4	1	no	—
73	C₄	—	8	4	no	—
74	C₄	C₂×#125	1	2	yes	—
75	C₄	#555	1	1	yes	characteristic
76	C₄	#559	1	1	yes	characteristic
77	C₂	—	2	1	no	—
78	C₂	—	2	2	no	—
79	C₂	#2571	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#555
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴
socle	C₂	#2571
prime radical	C₂³	2₋¹⁺⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

Maximal subgroups

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

subgroup	index
#2102	2
#1953	2
#1848	2
#1833	2
#1772	2