Subgroups — #62813

Order 256.

3465 conjugacy classes of subgroups.

The subgroup lattice has 176 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#3504	C₂	1	8	yes	maximal, complemented
2	#3503	C₂	1	8	yes	maximal, complemented
3	#3502	C₂	1	4	yes	maximal, complemented
4	#3501	C₂	1	12	yes	maximal, complemented
5	Q₈×#41	C₂	1	16	yes	maximal, complemented
6	#3458	C₂	1	6	yes	maximal, complemented
7	C₂×#533	C₂	1	6	yes	maximal, complemented
8	#3428	C₂	1	2	yes	maximal
9	C₂×C₄×#41	C₂	1	1	yes	maximal, characteristic
10	C₂²×#41	C₂²	1	1	yes	characteristic
11	Q₈²	C₂²	1	16	yes	complemented
12	#557	C₂²	1	16	yes	complemented
13	#557	C₂²	1	32	yes	complemented
14	#554	C₂²	1	48	yes	complemented
15	#554	C₂²	1	96	yes	complemented
16	D₄×Q₈	C₂²	1	48	yes	complemented
17	#544	C₂²	1	24	yes	—
18	#544	C₂²	1	48	yes	complemented
19	#543	C₂²	1	24	yes	—
20	#533	C₂²	1	4	yes	—
21	#533	C₂²	1	12	yes	—
22	#533	C₂²	1	24	yes	—
23	#533	C₂²	1	24	yes	complemented
24	#533	C₂²	1	12	yes	complemented
25	#533	C₂²	1	12	yes	complemented
26	#533	C₂²	1	48	yes	complemented
27	C₂×C₄⋊Q₈	C₂²	1	12	yes	complemented
28	C₂×#125	C₂²	1	12	yes	complemented
29	C₂×C₂²⋊Q₈	C₂²	1	24	yes	—
30	#520	C₂²	1	16	yes	—
31	#519	C₂²	1	4	yes	—
32	#518	C₂²	1	12	yes	—
33	C₄×#41	C₂²	1	8	yes	—
34	C₄×#41	C₂²	1	4	yes	—
35	C₄×#41	C₂²	1	4	yes	—
36	C₄×#41	C₂²	1	16	yes	—
37	C₂×C₄×Q₈	C₂²	1	2	yes	—
38	C₂×C₄×Q₈	C₂²	1	24	yes	complemented
39	C₂×C₄×D₄	C₂²	1	6	yes	—
40	C₂×#117	C₂²	1	3	yes	—
41	C₂×#117	C₂²	1	12	yes	—
42	C₂²×C₄²	C₂²	1	3	yes	—
43	C₂×#41	Q₈	1	16	yes	complemented
44	C₂×#41	C₂³	1	4	yes	—
45	C₂×#41	C₂³	1	4	yes	—
46	C₂²×Q₈	C₂³	1	24	yes	—
47	C₂²×Q₈	C₂³	1	1	yes	characteristic
48	C₂²×D₄	C₂³	1	3	yes	—
49	C₂³×C₄	C₂³	1	6	yes	—
50	C₂³×C₄	C₂³	1	3	yes	—
51	C₄⋊Q₈	C₂³	1	8	yes	—
52	C₄⋊Q₈	C₂³	1	24	yes	—
53	C₄⋊Q₈	C₂³	1	24	yes	—
54	C₄⋊Q₈	C₂³	1	24	yes	—
55	C₄⋊Q₈	C₂³	1	96	yes	—
56	#125	C₂³	1	8	yes	—
57	#125	C₂³	1	24	yes	—
58	#125	C₂³	1	48	yes	—
59	#125	C₂³	1	96	yes	—
60	C₂²⋊Q₈	C₂³	1	16	yes	—
61	C₂²⋊Q₈	C₂³	1	48	yes	—
62	C₂²⋊Q₈	C₂³	1	48	yes	—
63	C₂²⋊Q₈	C₂³	1	48	yes	—
64	C₂²⋊Q₈	C₂³	1	192	yes	—
65	C₄×Q₈	C₂³	1	8	yes	—
66	C₄×Q₈	C₂³	1	8	yes	—
67	C₄×Q₈	C₂³	1	16	yes	—
68	C₄×Q₈	C₂³	1	48	yes	—
69	C₄×Q₈	C₂³	1	48	yes	—
70	C₄×Q₈	C₂³	1	48	yes	—
71	C₄×Q₈	C₂³	1	32	yes	—
72	C₄×D₄	C₂³	1	24	yes	—
73	C₄×D₄	C₂³	1	24	yes	—
74	C₄×D₄	C₂³	1	96	yes	—
75	#117	C₂³	1	8	yes	—
76	#117	C₂³	1	12	yes	—
77	#117	C₂³	1	24	yes	—
78	#117	C₂³	1	12	yes	—
79	#117	C₂³	1	24	yes	—
80	#117	C₂³	1	24	yes	—
81	C₂×C₄⋊C₄	C₂³	1	6	yes	—
82	C₂×C₄⋊C₄	C₂³	1	24	yes	—
83	C₂×C₄⋊C₄	C₂³	1	48	yes	—
84	C₂×C₄⋊C₄	C₂³	1	24	yes	—
85	C₂×C₂²⋊C₄	C₂³	1	6	yes	—
86	C₂×C₂²⋊C₄	C₂³	1	24	yes	—
87	C₂×C₄²	C₂³	1	3	yes	—
88	C₂×C₄²	C₂³	1	24	yes	—
89	C₂×C₄²	C₂³	1	6	yes	—
90	C₂×C₄²	C₂³	1	12	yes	—
91	C₂×C₄²	C₂³	1	6	yes	—
92	C₂×C₄²	C₂³	1	6	yes	—
93	C₂×C₄²	C₂³	1	1	yes	characteristic
94	C₂⁴	C₂⁴	1	3	yes	—
95	#41	C₂×Q₈	1	32	yes	complemented
96	#41	C₂×Q₈	1	32	yes	complemented
97	C₂×Q₈	—	4	48	no	—
98	C₂×Q₈	C₂×Q₈	1	8	yes	complemented
99	C₂×Q₈	C₂⁴	1	16	yes	—
100	C₂×Q₈	C₂⁴	1	48	yes	—
101	C₂×Q₈	C₂⁴	1	4	yes	—
102	C₂×D₄	C₂×Q₈	1	24	yes	complemented
103	C₂×D₄	C₂⁴	1	12	yes	—
104	C₂²×C₄	—	4	12	no	—
105	C₂²×C₄	C₂×Q₈	1	24	yes	complemented
106	C₂²×C₄	C₂⁴	1	2	yes	—
107	C₂²×C₄	C₂⁴	1	12	yes	—
108	C₂²×C₄	C₂⁴	1	6	yes	—
109	C₂²×C₄	C₂⁴	1	12	yes	—
110	C₂²×C₄	C₂⁴	1	48	yes	—
111	C₂²×C₄	C₂⁴	1	6	yes	—
112	C₂²×C₄	C₂⁴	1	3	yes	—
113	C₂²×C₄	C₂⁴	1	6	yes	—
114	C₂²×C₄	C₂⁴	1	1	yes	characteristic
115	C₄⋊C₄	C₂⁴	1	16	yes	—
116	C₄⋊C₄	C₂⁴	1	32	yes	—
117	C₄⋊C₄	C₂⁴	1	16	yes	—
118	C₄⋊C₄	C₂⁴	1	24	yes	—
119	C₄⋊C₄	C₂⁴	1	48	yes	—
120	C₄⋊C₄	C₂⁴	1	96	yes	—
121	C₄⋊C₄	C₂⁴	1	48	yes	—
122	C₂²⋊C₄	C₂⁴	1	16	yes	—
123	C₂²⋊C₄	C₂⁴	1	24	yes	—
124	C₂²⋊C₄	C₂⁴	1	48	yes	—
125	C₄²	C₂⁴	1	1	yes	characteristic
126	C₄²	C₂⁴	1	1	yes	characteristic
127	C₄²	C₂⁴	1	6	yes	—
128	C₄²	C₂⁴	1	12	yes	—
129	C₄²	C₂⁴	1	12	yes	—
130	C₄²	C₂⁴	1	48	yes	—
131	C₄²	C₂⁴	1	2	yes	—
132	C₄²	C₂⁴	1	6	yes	—
133	C₄²	C₂⁴	1	16	yes	—
134	C₂³	—	4	6	no	—
135	C₂³	C₂²×Q₈	1	12	yes	—
136	C₂³	C₂⁵	1	1	yes	characteristic
137	C₂³	C₂⁵	1	6	yes	—
138	Q₈	—	2	32	no	—
139	Q₈	—	4	48	no	—
140	Q₈	C₂²×Q₈	1	16	yes	—
141	D₄	C₂²×Q₈	1	48	yes	—
142	C₂×C₄	—	2	4	no	—
143	C₂×C₄	—	2	24	no	—
144	C₂×C₄	—	4	12	no	—
145	C₂×C₄	—	4	6	no	—
146	C₂×C₄	—	4	48	no	—
147	C₂×C₄	C₂²×Q₈	1	4	yes	—
148	C₂×C₄	C₂²×Q₈	1	12	yes	—
149	C₂×C₄	C₂²×Q₈	1	24	yes	—
150	C₂×C₄	C₂²×Q₈	1	24	yes	—
151	C₂×C₄	C₂⁵	1	2	yes	—
152	C₂×C₄	C₂⁵	1	12	yes	—
153	C₂×C₄	C₂⁵	1	32	yes	—
154	C₂×C₄	C₂⁵	1	6	yes	—
155	C₂×C₄	C₂⁵	1	1	yes	characteristic
156	C₂×C₄	C₂⁵	1	2	yes	—
157	C₂×C₄	C₂⁵	1	1	yes	characteristic
158	C₂²	—	2	12	no	—
159	C₂²	—	2	2	no	—
160	C₂²	—	4	12	no	—
161	C₂²	C₂³×Q₈	1	2	yes	—
162	C₂²	C₂³×Q₈	1	12	yes	—
163	C₂²	C₂⁶	1	1	yes	characteristic
164	C₄	—	2	2	no	—
165	C₄	—	2	12	no	—
166	C₄	—	2	32	no	—
167	C₄	C₂³×Q₈	1	2	yes	—
168	C₄	C₂³×Q₈	1	2	yes	—
169	C₄	C₂³×Q₈	1	12	yes	—
170	C₄	#585	1	4	yes	—
171	C₂	—	2	2	no	—
172	C₂	—	2	12	no	—
173	C₂	C₂⁴×Q₈	1	1	yes	characteristic
174	C₂	C₂×#585	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂⁵
commutator, Frattini, socle, cosocle	C₂²	C₂⁶
prime radical	C₂²×#41	C₂²
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ 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. #3501	C₂	semidirect
5. Q₈×#41	C₂	semidirect
6. #3458	C₂	semidirect
7. C₂×#533	C₂	semidirect
11. Q₈²	C₂²	semidirect
12. #557	C₂²	semidirect
13. #557	C₂²	semidirect
14. #554	C₂²	semidirect
15. #554	C₂²	semidirect
16. D₄×Q₈	C₂²	semidirect
18. #544	C₂²	semidirect
23. #533	C₂²	semidirect
24. #533	C₂²	semidirect
25. #533	C₂²	semidirect
26. #533	C₂²	semidirect
27. C₂×C₄⋊Q₈	C₂²	semidirect
28. C₂×#125	C₂²	semidirect
38. C₂×C₄×Q₈	C₂²	semidirect
43. C₂×#41	Q₈	semidirect
95. #41	C₂×Q₈	semidirect
96. #41	C₂×Q₈	semidirect
98. C₂×Q₈	C₂×Q₈	semidirect
102. C₂×D₄	C₂×Q₈	semidirect
105. C₂²×C₄	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
#3501	2
Q₈×#41	2
#3458	2
C₂×#533	2
#3428	2
C₂×C₄×#41	2