Subgroups — #33418

Order 256.

341 conjugacy classes of subgroups.

The subgroup lattice has 142 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄×C₈	C₂	1	1	yes	maximal, characteristic
2	#2163	C₂	1	4	yes	maximal, complemented
3	#2162	C₂	1	4	yes	maximal, complemented
4	#2150	C₂	1	2	yes	maximal, complemented
5	#2112	C₂	1	2	yes	maximal
6	C₄×M₃₂	C₂	1	2	yes	maximal, complemented
7	C₂³×C₈	C₄	1	1	yes	characteristic
8	C₂³×C₈	C₂²	1	1	yes	characteristic
9	C₂²×C₄²	C₄	1	1	yes	characteristic
10	C₂×M₃₂	C₂²	1	4	yes	—
11	C₂×C₄×C₈	C₄	1	1	yes	characteristic
12	C₂×C₄×C₈	C₄	1	2	yes	—
13	C₂×C₄×C₈	C₄	1	2	yes	—
14	C₂×C₄×C₈	C₄	1	1	yes	characteristic
15	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
16	C₂×C₄×C₈	C₂²	1	2	yes	—
17	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
18	C₂×C₄×C₈	C₂²	1	2	yes	—
19	C₄⋊C₁₆	C₂²	1	8	yes	complemented
20	C₂²⋊C₁₆	C₂²	1	8	yes	—
21	#346	C₂²	1	4	yes	complemented
22	C₄×C₁₆	C₂²	1	4	yes	complemented
23	C₂³×C₄	C₈	1	2	yes	—
24	C₂³×C₄	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₈	—	2	4	no	—
26	C₂²×C₈	—	2	4	no	—
27	C₂²×C₈	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₈	C₂×C₄	1	2	yes	—
29	C₂²×C₈	C₂×C₄	1	2	yes	—
30	C₂²×C₈	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₈	C₂×C₄	1	2	yes	—
32	C₂²×C₈	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₈	C₂³	1	2	yes	—
34	C₂²×C₈	C₂³	1	1	yes	characteristic
35	C₂×C₄²	C₈	1	2	yes	—
36	C₂×C₄²	C₈	1	2	yes	—
37	C₂×C₄²	C₈	1	2	yes	—
38	C₂×C₄²	C₂×C₄	1	2	yes	—
39	C₂×C₄²	C₂×C₄	1	1	yes	characteristic
40	C₂×C₄²	C₂×C₄	1	1	yes	characteristic
41	C₂×C₄²	C₂×C₄	1	2	yes	—
42	M₃₂	—	2	8	no	—
43	C₂×C₁₆	C₂³	1	8	yes	—
44	C₄×C₈	C₂×C₄	1	2	yes	—
45	C₄×C₈	C₂×C₄	1	2	yes	—
46	C₄×C₈	C₂×C₄	1	2	yes	—
47	C₄×C₈	C₂×C₄	1	2	yes	—
48	C₄×C₈	C₂×C₄	1	1	yes	characteristic
49	C₄×C₈	C₂×C₄	1	2	yes	—
50	C₄×C₈	C₂×C₄	1	1	yes	characteristic
51	C₄×C₈	C₂³	1	1	yes	characteristic
52	C₄×C₈	C₂³	1	2	yes	—
53	C₄×C₈	C₂³	1	1	yes	characteristic
54	C₂⁴	C₂×C₈	1	1	yes	characteristic
55	C₂²×C₄	—	2	8	no	—
56	C₂²×C₄	—	2	2	no	—
57	C₂²×C₄	—	2	2	no	—
58	C₂²×C₄	C₂×C₈	1	1	yes	characteristic
59	C₂²×C₄	C₂×C₈	1	2	yes	—
60	C₂²×C₄	C₂×C₈	1	2	yes	—
61	C₂²×C₄	C₂×C₈	1	2	yes	—
62	C₂²×C₄	C₂×C₈	1	2	yes	—
63	C₂²×C₄	C₂×C₈	1	2	yes	—
64	C₂²×C₄	C₂×C₈	1	2	yes	—
65	C₂²×C₄	C₂×C₈	1	2	yes	—
66	C₂²×C₄	C₂²×C₄	1	2	yes	—
67	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
68	C₂×C₈	—	2	2	no	—
69	C₂×C₈	—	2	2	no	—
70	C₂×C₈	—	2	4	no	—
71	C₂×C₈	—	2	4	no	—
72	C₂×C₈	—	2	4	no	—
73	C₂×C₈	—	2	4	no	—
74	C₂×C₈	M₁₆	1	4	yes	—
75	C₂×C₈	C₂²×C₄	1	2	yes	—
76	C₂×C₈	C₂²×C₄	1	2	yes	—
77	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
78	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
79	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
80	C₂×C₈	#41	1	4	yes	—
81	C₂×C₈	C₂⁴	1	1	yes	characteristic
82	C₄²	C₂×C₈	1	2	yes	—
83	C₄²	C₂×C₈	1	2	yes	—
84	C₄²	C₂×C₈	1	2	yes	—
85	C₄²	C₂×C₈	1	2	yes	—
86	C₄²	C₂×C₈	1	1	yes	characteristic
87	C₄²	C₂×C₈	1	2	yes	—
88	C₄²	C₂×C₈	1	1	yes	characteristic
89	C₄²	C₂²×C₄	1	1	yes	characteristic
90	C₄²	C₂²×C₄	1	2	yes	—
91	C₄²	C₂²×C₄	1	1	yes	characteristic
92	C₁₆	—	2	8	no	—
93	C₂³	—	2	2	no	—
94	C₂³	—	2	4	no	—
95	C₂³	C₂²×C₈	1	1	yes	characteristic
96	C₂³	C₂²×C₈	1	2	yes	—
97	C₂×C₄	—	2	2	no	—
98	C₂×C₄	—	2	4	no	—
99	C₂×C₄	—	2	2	no	—
100	C₂×C₄	—	2	2	no	—
101	C₂×C₄	—	2	4	no	—
102	C₂×C₄	—	2	4	no	—
103	C₂×C₄	—	2	4	no	—
104	C₂×C₄	—	2	4	no	—
105	C₂×C₄	—	2	4	no	—
106	C₂×C₄	M₃₂	1	8	yes	complemented
107	C₂×C₄	#117	1	2	yes	—
108	C₂×C₄	C₂²×C₈	1	2	yes	—
109	C₂×C₄	C₂²×C₈	1	2	yes	—
110	C₂×C₄	C₂²×C₈	1	2	yes	—
111	C₂×C₄	C₂²×C₈	1	1	yes	characteristic
112	C₂×C₄	C₂²×C₈	1	2	yes	—
113	C₂×C₄	C₂²×C₈	1	2	yes	—
114	C₂×C₄	C₂×M₁₆	1	2	yes	—
115	C₂×C₄	C₂³×C₄	1	1	yes	characteristic
116	C₈	—	2	2	no	—
117	C₈	—	2	2	no	—
118	C₈	#117	1	2	yes	—
119	C₈	C₂×M₁₆	1	2	yes	—
120	C₈	C₂×M₁₆	1	2	yes	—
121	C₈	C₂×#41	1	2	yes	—
122	C₂²	—	2	8	no	—
123	C₂²	—	2	2	no	—
124	C₂²	—	2	2	no	—
125	C₂²	—	2	2	no	—
126	C₂²	#431	1	2	yes	—
127	C₂²	C₂×M₃₂	1	4	yes	—
128	C₂²	C₂³×C₈	1	1	yes	characteristic
129	C₄	—	2	2	no	—
130	C₄	—	2	2	no	—
131	C₄	—	2	2	no	—
132	C₄	#431	1	2	yes	—
133	C₄	C₂×M₃₂	1	4	yes	—
134	C₄	C₂×M₃₂	1	4	yes	—
135	C₄	C₂×#117	1	1	yes	characteristic
136	C₄	C₂²×M₁₆	1	1	yes	characteristic
137	C₂	—	2	2	no	—
138	C₂	—	2	4	no	—
139	C₂	C₂×#431	1	1	yes	characteristic
140	C₂	C₂²×M₃₂	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
2. #2163	C₂	semidirect
3. #2162	C₂	semidirect
4. #2150	C₂	semidirect
6. C₄×M₃₂	C₂	semidirect
19. C₄⋊C₁₆	C₂²	semidirect
21. #346	C₂²	semidirect
22. C₄×C₁₆	C₂²	semidirect
106. C₂×C₄	M₃₂	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₄×C₈	2
#2163	2
#2162	2
#2150	2
#2112	2
C₄×M₃₂	2