Subgroups — #60276

Order 256.

1153 conjugacy classes of subgroups.

The subgroup lattice has 152 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×#41	C₂	1	1	yes	maximal, characteristic
2	#2988	C₂	1	8	yes	maximal, complemented
3	#2982	C₂	1	2	yes	maximal, complemented
4	#2972	C₂	1	6	yes	maximal, complemented
5	C₈×#41	C₂	1	8	yes	maximal, complemented
6	C₂×#431	C₂	1	6	yes	maximal, complemented
7	C₂²×#41	C₄	1	1	yes	characteristic
8	C₄×#41	C₄	1	4	yes	—
9	C₄×#41	C₄	1	4	yes	—
10	C₄×#41	C₂²	1	4	yes	—
11	C₄×#41	C₂²	1	4	yes	—
12	C₂×C₄×Q₈	C₄	1	1	yes	characteristic
13	C₂×C₄×Q₈	C₂²	1	1	yes	characteristic
14	C₂×C₄×D₄	C₄	1	3	yes	—
15	C₂×C₄×D₄	C₂²	1	3	yes	—
16	C₂×#117	C₄	1	3	yes	—
17	C₂²×C₄²	C₂²	1	3	yes	—
18	C₈×Q₈	C₂²	1	16	yes	complemented
19	C₈×D₄	C₂²	1	48	yes	complemented
20	#431	C₂²	1	4	yes	—
21	#431	C₂²	1	12	yes	—
22	#431	C₂²	1	12	yes	complemented
23	#431	C₂²	1	12	yes	complemented
24	C₂×C₄⋊C₈	C₂²	1	12	yes	complemented
25	C₂×C₂²⋊C₈	C₂²	1	12	yes	—
26	C₂×C₄×C₈	C₂²	1	12	yes	complemented
27	C₂×#41	C₈	1	16	yes	complemented
28	C₂×#41	C₂×C₄	1	4	yes	—
29	C₂×#41	C₂×C₄	1	4	yes	—
30	C₂²×Q₈	C₂×C₄	1	1	yes	characteristic
31	C₂²×D₄	C₂×C₄	1	3	yes	—
32	C₂³×C₄	C₂×C₄	1	3	yes	—
33	C₂³×C₄	C₂×C₄	1	3	yes	—
34	C₂³×C₄	C₂³	1	3	yes	—
35	C₂²×C₈	C₂³	1	24	yes	—
36	C₄×Q₈	C₂×C₄	1	4	yes	—
37	C₄×Q₈	C₂×C₄	1	4	yes	—
38	C₄×Q₈	C₂×C₄	1	4	yes	—
39	C₄×Q₈	C₂³	1	4	yes	—
40	C₄×D₄	C₂×C₄	1	12	yes	—
41	C₄×D₄	C₂×C₄	1	12	yes	—
42	C₄×D₄	C₂×C₄	1	12	yes	—
43	C₄×D₄	C₂³	1	12	yes	—
44	#117	C₂×C₄	1	12	yes	—
45	#117	C₂×C₄	1	12	yes	—
46	C₂×C₄⋊C₄	C₂×C₄	1	3	yes	—
47	C₂×C₄⋊C₄	C₂×C₄	1	3	yes	—
48	C₂×C₂²⋊C₄	C₂×C₄	1	6	yes	—
49	C₂×C₄²	C₂×C₄	1	3	yes	—
50	C₂×C₄²	C₂×C₄	1	3	yes	—
51	C₂×C₄²	C₂×C₄	1	6	yes	—
52	C₂×C₄²	C₂×C₄	1	6	yes	—
53	C₂×C₄²	C₂³	1	3	yes	—
54	C₂×C₄²	C₂³	1	6	yes	—
55	C₂×C₄²	C₂³	1	6	yes	—
56	C₂×C₄²	C₂³	1	1	yes	characteristic
57	C₄⋊C₈	C₂³	1	8	yes	—
58	C₄⋊C₈	C₂³	1	24	yes	—
59	C₂²⋊C₈	C₂³	1	8	yes	—
60	C₂²⋊C₈	C₂³	1	24	yes	—
61	C₄×C₈	C₂³	1	24	yes	—
62	C₄×C₈	C₂³	1	8	yes	—
63	C₂⁴	C₂²×C₄	1	3	yes	—
64	#41	C₂×C₈	1	32	yes	complemented
65	#41	C₂×C₈	1	32	yes	complemented
66	C₂×Q₈	C₂×C₈	1	8	yes	complemented
67	C₂×Q₈	C₂²×C₄	1	4	yes	—
68	C₂×D₄	C₂×C₈	1	24	yes	complemented
69	C₂×D₄	C₂²×C₄	1	12	yes	—
70	C₂²×C₄	—	4	6	no	—
71	C₂²×C₄	—	4	6	no	—
72	C₂²×C₄	C₂×C₈	1	24	yes	complemented
73	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
74	C₂²×C₄	C₂²×C₄	1	6	yes	—
75	C₂²×C₄	C₂²×C₄	1	3	yes	—
76	C₂²×C₄	C₂²×C₄	1	6	yes	—
77	C₂²×C₄	C₂²×C₄	1	3	yes	—
78	C₂²×C₄	C₂²×C₄	1	6	yes	—
79	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
80	C₂²×C₄	C₂²×C₄	1	3	yes	—
81	C₂²×C₄	C₂²×C₄	1	6	yes	—
82	C₂²×C₄	C₂²×C₄	1	6	yes	—
83	C₂²×C₄	C₂⁴	1	1	yes	characteristic
84	C₂²×C₄	C₂⁴	1	6	yes	—
85	C₂×C₈	—	4	24	no	—
86	C₂×C₈	C₂⁴	1	16	yes	—
87	C₄⋊C₄	C₂²×C₄	1	12	yes	—
88	C₄⋊C₄	C₂²×C₄	1	12	yes	—
89	C₂²⋊C₄	C₂²×C₄	1	24	yes	—
90	C₄²	C₂²×C₄	1	1	yes	characteristic
91	C₄²	C₂²×C₄	1	1	yes	characteristic
92	C₄²	C₂²×C₄	1	6	yes	—
93	C₄²	C₂²×C₄	1	6	yes	—
94	C₄²	C₂²×C₄	1	6	yes	—
95	C₄²	C₂²×C₄	1	6	yes	—
96	C₄²	C₂²×C₄	1	6	yes	—
97	C₄²	C₂⁴	1	6	yes	—
98	C₄²	C₂⁴	1	1	yes	characteristic
99	C₄²	C₂⁴	1	1	yes	characteristic
100	C₂³	—	4	3	no	—
101	C₂³	—	4	3	no	—
102	C₂³	C₂²×C₈	1	12	yes	—
103	C₂³	C₂³×C₄	1	6	yes	—
104	C₂³	C₂³×C₄	1	1	yes	characteristic
105	Q₈	C₂²×C₈	1	16	yes	—
106	D₄	C₂²×C₈	1	48	yes	—
107	C₂×C₄	—	2	2	no	—
108	C₂×C₄	—	2	12	no	—
109	C₂×C₄	—	2	12	no	—
110	C₂×C₄	—	2	2	no	—
111	C₂×C₄	—	4	6	no	—
112	C₂×C₄	—	4	3	no	—
113	C₂×C₄	—	4	6	no	—
114	C₂×C₄	—	4	3	no	—
115	C₂×C₄	C₂²×C₈	1	4	yes	—
116	C₂×C₄	C₂²×C₈	1	12	yes	—
117	C₂×C₄	C₂²×C₈	1	24	yes	—
118	C₂×C₄	C₂²×C₈	1	24	yes	—
119	C₂×C₄	C₂³×C₄	1	1	yes	characteristic
120	C₂×C₄	C₂³×C₄	1	1	yes	characteristic
121	C₂×C₄	C₂³×C₄	1	6	yes	—
122	C₂×C₄	C₂³×C₄	1	6	yes	—
123	C₂×C₄	C₂³×C₄	1	1	yes	characteristic
124	C₂×C₄	C₂³×C₄	1	1	yes	characteristic
125	C₂×C₄	C₂³×C₄	1	6	yes	—
126	C₂×C₄	C₂³×C₄	1	1	yes	characteristic
127	C₂×C₄	C₂⁵	1	1	yes	characteristic
128	C₈	—	2	16	no	—
129	C₂²	—	2	1	no	—
130	C₂²	—	2	6	no	—
131	C₂²	—	2	1	no	—
132	C₂²	—	2	6	no	—
133	C₂²	—	4	12	no	—
134	C₂²	C₂³×C₈	1	2	yes	—
135	C₂²	C₂³×C₈	1	12	yes	—
136	C₂²	C₂⁴×C₄	1	1	yes	characteristic
137	C₄	—	2	1	no	—
138	C₄	—	2	6	no	—
139	C₄	—	2	1	no	—
140	C₄	—	2	6	no	—
141	C₄	C₂³×C₈	1	2	yes	—
142	C₄	C₂³×C₈	1	2	yes	—
143	C₄	C₂³×C₈	1	12	yes	—
144	C₄	#568	1	2	yes	—
145	C₄	#585	1	2	yes	—
146	C₂	—	2	2	no	—
147	C₂	—	2	12	no	—
148	C₂	#3428	1	1	yes	characteristic
149	C₂	C₂⁴×C₈	1	1	yes	characteristic
150	C₂	C₂×#568	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
2. #2988	C₂	semidirect
3. #2982	C₂	semidirect
4. #2972	C₂	semidirect
5. C₈×#41	C₂	semidirect
6. C₂×#431	C₂	semidirect
18. C₈×Q₈	C₂²	semidirect
19. C₈×D₄	C₂²	semidirect
22. #431	C₂²	semidirect
23. #431	C₂²	semidirect
24. C₂×C₄⋊C₈	C₂²	semidirect
26. C₂×C₄×C₈	C₂²	semidirect
27. C₂×#41	C₈	semidirect
64. #41	C₂×C₈	semidirect
65. #41	C₂×C₈	semidirect
66. C₂×Q₈	C₂×C₈	semidirect
68. C₂×D₄	C₂×C₈	semidirect
72. C₂²×C₄	C₂×C₈	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×#41	2
#2988	2
#2982	2
#2972	2
C₈×#41	2
C₂×#431	2