Subgroups — #30236

Order 256.

223 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#2082	C₂	1	3	yes	maximal, complemented
2	#2081	C₂	1	3	yes	maximal
3	#2070	C₂	1	1	yes	maximal, characteristic, complemented
4	#2069	C₂	1	1	yes	maximal, characteristic
5	#1977	C₂	1	3	yes	maximal
6	#1953	C₂	1	3	yes	maximal, complemented
7	#1747	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×SD₁₆	C₂²	1	1	yes	characteristic
9	#430	C₂²	1	3	yes	—
10	#430	C₂²	1	3	yes	—
11	C₂×#108	C₂²	1	3	yes	—
12	#419	C₂²	1	3	yes	—
13	#418	C₂²	1	3	yes	—
14	C₂×#101	C₂²	1	3	yes	—
15	C₂×#100	C₂²	1	3	yes	—
16	#405	C₂²	1	3	yes	—
17	#344	C₂²	1	6	yes	—
18	#338	C₂²	1	1	yes	characteristic
19	H(3,ℤ/4)	C₂²	1	3	yes	—
20	C₂²×Q₈	C₂³	1	1	yes	characteristic
21	C₂²×D₄	C₂³	1	1	yes	characteristic
22	C₂×SD₁₆	—	2	3	no	—
23	C₂×SD₁₆	—	2	3	no	—
24	C₂×M₁₆	C₂³	1	3	yes	—
25	C₂×M₁₆	C₂³	1	3	yes	—
26	C₂×M₁₆	C₂³	1	3	yes	—
27	C₂²×C₈	C₂³	1	1	yes	characteristic
28	#117	C₂³	1	3	yes	—
29	#108	—	2	3	no	—
30	#108	—	2	3	no	—
31	#108	—	4	3	no	—
32	#108	—	4	3	no	—
33	#103	—	4	3	no	—
34	#102	—	4	3	no	—
35	#101	—	4	3	no	—
36	#100	—	4	3	no	—
37	#97	—	2	3	no	—
38	C₄×C₈	—	2	3	no	—
39	C₂⁴	—	2	1	no	—
40	C₂×Q₈	—	2	3	no	—
41	C₂×Q₈	—	2	3	no	—
42	C₂×D₄	—	2	3	no	—
43	C₂×D₄	—	2	3	no	—
44	C₂²×C₄	—	2	1	no	—
45	C₂²×C₄	C₂⁴	1	1	yes	characteristic
46	SD₁₆	—	4	1	no	—
47	SD₁₆	—	4	3	no	—
48	M₁₆	—	2	3	no	—
49	M₁₆	—	2	3	no	—
50	M₁₆	—	2	3	no	—
51	M₁₆	—	2	3	no	—
52	M₁₆	—	2	3	no	—
53	M₁₆	—	2	3	no	—
54	C₂×C₈	—	2	3	no	—
55	C₂×C₈	—	2	3	no	—
56	C₂×C₈	—	2	3	no	—
57	C₂×C₈	#41	1	3	yes	—
58	C₂×C₈	#41	1	3	yes	—
59	C₄⋊C₄	—	2	3	no	—
60	C₂²⋊C₄	—	2	3	no	—
61	C₄²	—	2	3	no	—
62	C₂³	—	4	3	no	—
63	C₂³	—	16	1	no	—
64	C₂³	2₋¹⁺⁴	1	1	yes	characteristic
65	Q₈	—	4	1	no	—
66	Q₈	—	4	3	no	—
67	D₄	—	4	1	no	—
68	D₄	—	4	3	no	—
69	C₂×C₄	—	4	3	no	—
70	C₂×C₄	—	4	3	no	—
71	C₂×C₄	C₂×#41	1	3	yes	—
72	C₂×C₄	H(5,𝔽₂)	1	3	yes	—
73	C₈	—	4	3	no	—
74	C₈	—	4	3	no	—
75	C₈	—	4	3	no	—
76	C₈	#126	1	1	yes	characteristic
77	C₈	#126	1	3	yes	—
78	C₂²	—	4	1	no	—
79	C₂²	—	8	1	no	—
80	C₂²	—	16	3	no	—
81	C₂²	#542	1	3	yes	—
82	C₄	—	8	3	no	—
83	C₄	—	8	1	no	—
84	C₄	C₂×#126	1	1	yes	characteristic
85	C₄	#540	1	3	yes	—
86	C₂	—	2	3	no	—
87	C₂	—	16	1	no	—
88	C₂	#2644	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂	#2644
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴
prime radical	C₂²×D₄	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
1. #2082	C₂	semidirect
3. #2070	C₂	semidirect
6. #1953	C₂	semidirect
7. #1747	C₂	semidirect

Maximal subgroups

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

subgroup	index
#2082	2
#2081	2
#2070	2
#2069	2
#1977	2
#1953	2
#1747	2