Subgroups — #60058

Order 256.

1589 conjugacy classes of subgroups.

The subgroup lattice has 134 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#568	C₂	1	1	yes	maximal, characteristic
2	C₂²×#131	C₂	1	1	yes	maximal, characteristic
3	#3428	C₂	1	1	yes	maximal, characteristic
4	#2875	C₂	1	8	yes	maximal, complemented
5	C₄×#131	C₂	1	8	yes	maximal, complemented
6	#2873	C₂	1	6	yes	maximal, complemented
7	C₂×#405	C₂	1	6	yes	maximal, complemented
8	C₂²×#41	C₂²	1	1	yes	characteristic
9	#568	C₄	1	16	yes	complemented
10	C₂×#131	C₄	1	16	yes	complemented
11	C₂×#131	C₂²	1	4	yes	—
12	C₂×#131	C₂²	1	8	yes	—
13	C₂×#131	C₂²	1	4	yes	—
14	C₂²×M₁₆	C₂²	1	6	yes	—
15	C₂²×M₁₆	C₂²	1	3	yes	—
16	C₂³×C₈	C₂²	1	3	yes	—
17	#520	C₄	1	8	yes	—
18	#519	C₄	1	2	yes	—
19	#518	C₄	1	6	yes	—
20	C₄×#41	C₂²	1	8	yes	—
21	C₂×#117	C₂²	1	6	yes	—
22	#405	C₂²	1	4	yes	—
23	#405	C₂²	1	12	yes	—
24	#405	C₂²	1	12	yes	complemented
25	#405	C₂²	1	12	yes	complemented
26	#405	C₂²	1	24	yes	complemented
27	C₄×M₁₆	C₂²	1	24	yes	complemented
28	C₂×#97	C₂²	1	12	yes	complemented
29	C₂×C₄×C₈	C₂²	1	12	yes	complemented
30	C₂×#41	C₂×C₄	1	8	yes	—
31	C₂×#41	C₂×C₄	1	8	yes	—
32	C₂×#41	C₂×C₄	1	4	yes	—
33	C₂×#41	C₂³	1	4	yes	—
34	C₂²×Q₈	C₂×C₄	1	1	yes	characteristic
35	C₂²×D₄	C₂×C₄	1	3	yes	—
36	C₂³×C₄	C₂³	1	3	yes	—
37	#131	C₂×C₄	1	32	yes	complemented
38	#131	C₂×C₄	1	32	yes	complemented
39	#131	C₂×C₄	1	64	yes	complemented
40	C₂×M₁₆	C₂×C₄	1	24	yes	complemented
41	C₂×M₁₆	C₂×C₄	1	48	yes	complemented
42	C₂×M₁₆	C₂³	1	4	yes	—
43	C₂×M₁₆	C₂³	1	12	yes	—
44	C₂×M₁₆	C₂³	1	12	yes	—
45	C₂²×C₈	C₂×C₄	1	24	yes	complemented
46	C₂²×C₈	C₂³	1	3	yes	—
47	C₂²×C₈	C₂³	1	6	yes	—
48	C₂²×C₈	C₂³	1	12	yes	—
49	C₂²×C₈	C₂³	1	1	yes	characteristic
50	C₂²×C₈	C₂³	1	6	yes	—
51	C₄×Q₈	C₂×C₄	1	16	yes	—
52	C₄×D₄	C₂×C₄	1	48	yes	—
53	#117	C₂×C₄	1	12	yes	—
54	#117	C₂×C₄	1	12	yes	—
55	#117	C₂³	1	4	yes	—
56	#117	C₂³	1	12	yes	—
57	C₂×C₄⋊C₄	C₂×C₄	1	12	yes	—
58	C₂×C₂²⋊C₄	C₂×C₄	1	12	yes	—
59	C₂×C₄²	C₂³	1	12	yes	—
60	#97	C₂³	1	8	yes	—
61	#97	C₂³	1	24	yes	—
62	C₄×C₈	C₂³	1	24	yes	—
63	C₄×C₈	C₂³	1	8	yes	—
64	C₂⁴	C₂²×C₄	1	3	yes	—
65	#41	C₄²	1	32	yes	—
66	#41	C₂²×C₄	1	16	yes	—
67	#41	C₂²×C₄	1	16	yes	—
68	C₂×Q₈	C₄²	1	8	yes	—
69	C₂×Q₈	C₂²×C₄	1	4	yes	—
70	C₂×D₄	C₄²	1	24	yes	—
71	C₂×D₄	C₂²×C₄	1	12	yes	—
72	C₂²×C₄	C₂²×C₄	1	24	yes	—
73	C₂²×C₄	C₂²×C₄	1	6	yes	—
74	C₂²×C₄	C₂²×C₄	1	3	yes	—
75	C₂²×C₄	C₂²×C₄	1	12	yes	—
76	C₂²×C₄	C₂²×C₄	1	12	yes	—
77	C₂²×C₄	C₂⁴	1	6	yes	—
78	C₂²×C₄	C₂⁴	1	1	yes	characteristic
79	M₁₆	C₂²×C₄	1	16	yes	—
80	M₁₆	C₂²×C₄	1	48	yes	—
81	M₁₆	C₂²×C₄	1	48	yes	—
82	C₂×C₈	C₂²×C₄	1	4	yes	—
83	C₂×C₈	C₂²×C₄	1	12	yes	—
84	C₂×C₈	C₂²×C₄	1	24	yes	—
85	C₂×C₈	C₂²×C₄	1	24	yes	—
86	C₂×C₈	C₂²×C₄	1	48	yes	—
87	C₂×C₈	C₂⁴	1	6	yes	—
88	C₂×C₈	C₂⁴	1	8	yes	—
89	C₂×C₈	C₂⁴	1	1	yes	characteristic
90	C₂×C₈	C₂⁴	1	1	yes	characteristic
91	C₄⋊C₄	C₂²×C₄	1	8	yes	—
92	C₄⋊C₄	C₂²×C₄	1	24	yes	—
93	C₂²⋊C₄	C₂²×C₄	1	8	yes	—
94	C₂²⋊C₄	C₂²×C₄	1	24	yes	—
95	C₄²	C₂²×C₄	1	24	yes	—
96	C₄²	C₂⁴	1	8	yes	—
97	C₂³	—	4	6	no	—
98	C₂³	C₂×C₄²	1	12	yes	—
99	C₂³	C₂³×C₄	1	1	yes	characteristic
100	C₂³	C₂³×C₄	1	6	yes	—
101	Q₈	C₂×C₄²	1	16	yes	—
102	D₄	C₂×C₄²	1	48	yes	—
103	C₂×C₄	—	4	24	no	—
104	C₂×C₄	C₂×C₄²	1	12	yes	—
105	C₂×C₄	C₂×C₄²	1	24	yes	—
106	C₂×C₄	C₂³×C₄	1	16	yes	—
107	C₂×C₄	C₂³×C₄	1	2	yes	—
108	C₂×C₄	C₂³×C₄	1	12	yes	—
109	C₂×C₄	C₂³×C₄	1	12	yes	—
110	C₂×C₄	C₂³×C₄	1	2	yes	—
111	C₂×C₄	C₂³×C₄	1	6	yes	—
112	C₂×C₄	C₂³×C₄	1	1	yes	characteristic
113	C₂×C₄	C₂⁵	1	1	yes	characteristic
114	C₈	C₂³×C₄	1	2	yes	—
115	C₈	C₂³×C₄	1	2	yes	—
116	C₈	C₂³×C₄	1	12	yes	—
117	C₈	C₂³×C₄	1	16	yes	—
118	C₂²	—	2	12	no	—
119	C₂²	—	2	2	no	—
120	C₂²	—	4	12	no	—
121	C₂²	C₂²×C₄²	1	2	yes	—
122	C₂²	C₂²×C₄²	1	12	yes	—
123	C₂²	C₂⁴×C₄	1	1	yes	characteristic
124	C₄	—	2	16	no	—
125	C₄	C₂²×C₄²	1	2	yes	—
126	C₄	C₂²×C₄²	1	12	yes	—
127	C₄	C₂⁴×C₄	1	1	yes	characteristic
128	C₄	C₂⁴×C₄	1	1	yes	characteristic
129	C₂	—	2	2	no	—
130	C₂	—	2	12	no	—
131	C₂	C₂³×C₄²	1	1	yes	characteristic
132	C₂	#3573	1	2	yes	—

	subgroup	G/H
center	C₂×C₈	C₂⁴
commutator	C₂	C₂³×C₄²
Frattini, cosocle	C₂×C₄	C₂⁵
socle	C₂²	C₂⁴×C₄
prime radical	C₂²×#41	C₂²
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ G
LECS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
4. #2875	C₂	semidirect
5. C₄×#131	C₂	semidirect
6. #2873	C₂	semidirect
7. C₂×#405	C₂	semidirect
9. #568	C₄	semidirect
10. C₂×#131	C₄	semidirect
24. #405	C₂²	semidirect
25. #405	C₂²	semidirect
26. #405	C₂²	semidirect
27. C₄×M₁₆	C₂²	semidirect
28. C₂×#97	C₂²	semidirect
29. C₂×C₄×C₈	C₂²	semidirect
37. #131	C₂×C₄	semidirect
38. #131	C₂×C₄	semidirect
39. #131	C₂×C₄	semidirect
40. C₂×M₁₆	C₂×C₄	semidirect
41. C₂×M₁₆	C₂×C₄	semidirect
45. C₂²×C₈	C₂×C₄	semidirect

Maximal subgroups

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

subgroup	index
C₂×#568	2
C₂²×#131	2
#3428	2
#2875	2
C₄×#131	2
#2873	2
C₂×#405	2