Subgroups — #63004

Order 256.

3369 conjugacy classes of subgroups.

The subgroup lattice has 140 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#3504	C₂	1	20	yes	maximal, complemented
2	#3503	C₂	1	10	yes	maximal, complemented
3	#3502	C₂	1	5	yes	maximal, complemented
4	#3502	C₂	1	10	yes	maximal, complemented
5	Q₈×#41	C₂	1	10	yes	maximal, complemented
6	C₂×#557	C₂	1	5	yes	maximal, complemented
7	C₄×2₋¹⁺⁴	C₂	1	1	yes	maximal, characteristic
8	C₄×2₋¹⁺⁴	C₂	1	2	yes	maximal
9	C₂×2₋¹⁺⁴	C₂²	1	1	yes	characteristic
10	Q₈²	C₂²	1	10	yes	—
11	Q₈²	C₂²	1	20	yes	—
12	#557	C₂²	1	10	yes	—
13	#557	C₂²	1	20	yes	—
14	#557	C₂²	1	40	yes	—
15	#557	C₂²	1	20	yes	—
16	#557	C₂²	1	40	yes	—
17	#554	C₂²	1	10	yes	—
18	#554	C₂²	1	60	yes	—
19	#554	C₂²	1	60	yes	—
20	D₄×Q₈	C₂²	1	30	yes	—
21	#544	C₂²	1	60	yes	—
22	#544	C₂²	1	30	yes	—
23	#533	C₂²	1	30	yes	—
24	#533	C₂²	1	60	yes	—
25	C₂×C₄⋊Q₈	C₂²	1	15	yes	—
26	C₂×#125	C₂²	1	30	yes	—
27	#520	C₂²	1	20	yes	—
28	#520	C₂²	1	10	yes	—
29	#519	C₂²	1	5	yes	—
30	#519	C₂²	1	10	yes	—
31	C₄×#41	C₂²	1	10	yes	—
32	C₄×#41	C₂²	1	20	yes	—
33	C₂×C₄×Q₈	C₂²	1	10	yes	—
34	C₂×C₄×Q₈	C₂²	1	5	yes	—
35	C₂×C₄×Q₈	C₂²	1	15	yes	—
36	2₋¹⁺⁴	Q₈	1	16	yes	complemented
37	C₂×#41	C₂³	1	10	yes	—
38	C₂²×Q₈	C₂³	1	15	yes	—
39	C₂²×Q₈	C₂³	1	5	yes	—
40	C₄⋊Q₈	C₂³	1	30	yes	—
41	C₄⋊Q₈	C₂³	1	60	yes	—
42	C₄⋊Q₈	C₂³	1	10	yes	—
43	C₄⋊Q₈	C₂³	1	60	yes	—
44	C₄⋊Q₈	C₂³	1	60	yes	—
45	#125	C₂³	1	60	yes	—
46	#125	C₂³	1	120	yes	—
47	#125	C₂³	1	20	yes	—
48	#125	C₂³	1	60	yes	—
49	C₂²⋊Q₈	C₂³	1	20	yes	—
50	C₂²⋊Q₈	C₂³	1	40	yes	—
51	C₂²⋊Q₈	C₂³	1	120	yes	—
52	C₂²⋊Q₈	C₂³	1	60	yes	—
53	C₄×Q₈	C₂³	1	10	yes	—
54	C₄×Q₈	C₂³	1	20	yes	—
55	C₄×Q₈	C₂³	1	10	yes	—
56	C₄×Q₈	C₂³	1	20	yes	—
57	C₄×Q₈	C₂³	1	40	yes	—
58	C₄×Q₈	C₂³	1	30	yes	—
59	C₄×Q₈	C₂³	1	60	yes	—
60	C₄×Q₈	C₂³	1	20	yes	—
61	C₄×Q₈	C₂³	1	10	yes	—
62	C₄×Q₈	C₂³	1	60	yes	—
63	C₄×D₄	C₂³	1	10	yes	—
64	C₄×D₄	C₂³	1	20	yes	—
65	C₄×D₄	C₂³	1	60	yes	—
66	C₄×D₄	C₂³	1	30	yes	—
67	#117	C₂³	1	30	yes	—
68	#117	C₂³	1	60	yes	—
69	C₂×C₄⋊C₄	C₂³	1	5	yes	—
70	C₂×C₄⋊C₄	C₂³	1	15	yes	—
71	C₂×C₄⋊C₄	C₂³	1	30	yes	—
72	C₂×C₄⋊C₄	C₂³	1	30	yes	—
73	C₂×C₄⋊C₄	C₂³	1	30	yes	—
74	C₂×C₄²	C₂³	1	30	yes	—
75	C₂×C₄²	C₂³	1	15	yes	—
76	#41	C₂×Q₈	1	80	yes	complemented
77	C₂×Q₈	—	4	30	no	—
78	C₂×Q₈	C₂×Q₈	1	40	yes	complemented
79	C₂×Q₈	C₂⁴	1	10	yes	—
80	C₂×Q₈	C₂⁴	1	20	yes	—
81	C₂×Q₈	C₂⁴	1	30	yes	—
82	C₂×Q₈	C₂⁴	1	10	yes	—
83	C₂×D₄	C₂⁴	1	10	yes	—
84	C₂²×C₄	C₂⁴	1	5	yes	—
85	C₂²×C₄	C₂⁴	1	10	yes	—
86	C₂²×C₄	C₂⁴	1	15	yes	—
87	C₂²×C₄	C₂⁴	1	30	yes	—
88	C₂²×C₄	C₂⁴	1	15	yes	—
89	C₄⋊C₄	C₂⁴	1	1	yes	characteristic
90	C₄⋊C₄	C₂⁴	1	20	yes	—
91	C₄⋊C₄	C₂⁴	1	30	yes	—
92	C₄⋊C₄	C₂⁴	1	60	yes	—
93	C₄⋊C₄	C₂⁴	1	60	yes	—
94	C₄⋊C₄	C₂⁴	1	10	yes	—
95	C₄⋊C₄	C₂⁴	1	5	yes	—
96	C₄⋊C₄	C₂⁴	1	60	yes	—
97	C₄⋊C₄	C₂⁴	1	20	yes	—
98	C₄⋊C₄	C₂⁴	1	10	yes	—
99	C₄⋊C₄	C₂⁴	1	40	yes	—
100	C₂²⋊C₄	C₂⁴	1	40	yes	—
101	C₂²⋊C₄	C₂⁴	1	20	yes	—
102	C₄²	C₂⁴	1	20	yes	—
103	C₄²	C₂⁴	1	10	yes	—
104	C₄²	C₂⁴	1	60	yes	—
105	C₄²	C₂⁴	1	30	yes	—
106	C₂³	C₂⁵	1	5	yes	—
107	Q₈	—	2	20	no	—
108	Q₈	—	4	30	no	—
109	Q₈	—	4	20	no	—
110	Q₈	C₂²×Q₈	1	40	yes	—
111	D₄	C₂²×Q₈	1	40	yes	—
112	C₂×C₄	—	2	10	no	—
113	C₂×C₄	—	4	10	no	—
114	C₂×C₄	—	4	15	no	—
115	C₂×C₄	—	4	30	no	—
116	C₂×C₄	C₂²×Q₈	1	60	yes	—
117	C₂×C₄	C₂⁵	1	2	yes	—
118	C₂×C₄	C₂⁵	1	10	yes	—
119	C₂×C₄	C₂⁵	1	5	yes	—
120	C₂×C₄	C₂⁵	1	10	yes	—
121	C₂×C₄	C₂⁵	1	20	yes	—
122	C₂×C₄	C₂⁵	1	10	yes	—
123	C₂×C₄	C₂⁵	1	1	yes	characteristic
124	C₂²	—	2	5	no	—
125	C₂²	—	2	5	no	—
126	C₂²	C₂³×Q₈	1	10	yes	—
127	C₂²	C₂⁶	1	1	yes	characteristic
128	C₄	—	2	5	no	—
129	C₄	—	2	20	no	—
130	C₄	—	2	2	no	—
131	C₄	—	2	10	no	—
132	C₄	—	2	10	no	—
133	C₄	C₂³×Q₈	1	20	yes	—
134	C₄	#585	1	2	yes	—
135	C₂	—	2	10	no	—
136	C₂	C₂⁴×Q₈	1	1	yes	characteristic
137	C₂	C₂×#585	1	1	yes	characteristic
138	C₂	2₋¹⁺⁶	1	1	yes	characteristic

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂²	C₂⁶
prime radical	C₂×2₋¹⁺⁴	C₂²
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ G
LECS	G ▷ C₂² ▷ C₁

Complements

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

N	H ≅ G/N	type
1. #3504	C₂	semidirect
2. #3503	C₂	semidirect
3. #3502	C₂	semidirect
4. #3502	C₂	semidirect
5. Q₈×#41	C₂	semidirect
6. C₂×#557	C₂	semidirect
36. 2₋¹⁺⁴	Q₈	semidirect
76. #41	C₂×Q₈	semidirect
78. C₂×Q₈	C₂×Q₈	semidirect

Maximal subgroups

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

subgroup	index
#3504	2
#3503	2
#3502	2
Q₈×#41	2
C₂×#557	2
C₄×2₋¹⁺⁴	2