Subgroups — #12554

Order 256.

293 conjugacy classes of subgroups.

The subgroup lattice has 161 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×M₃₂	C₂	1	1	yes	maximal, characteristic
2	C₂²×M₃₂	C₂	1	1	yes	maximal, characteristic
3	C₂×#405	C₂	1	1	yes	maximal, characteristic
4	#1379	C₂	1	4	yes	maximal, complemented
5	C₂²×M₁₆	C₄	1	1	yes	characteristic
6	C₂³×C₈	C₂²	1	1	yes	characteristic
7	C₂×#117	C₄	1	1	yes	characteristic
8	C₂×M₃₂	—	2	2	no	—
9	C₂×M₃₂	—	2	2	no	—
10	C₂×M₃₂	—	2	1	no	—
11	C₂×M₃₂	—	2	1	no	—
12	C₂×M₃₂	—	2	1	no	—
13	C₂×M₃₂	C₄	1	4	yes	complemented
14	C₂×M₃₂	C₄	1	4	yes	complemented
15	C₂×M₃₂	C₂²	1	2	yes	—
16	C₂²×C₁₆	—	2	1	no	—
17	C₂²×C₁₆	C₂²	1	2	yes	—
18	#405	—	2	1	no	—
19	#405	—	2	1	no	—
20	#405	C₄	1	2	yes	—
21	#405	C₂²	1	2	yes	—
22	C₂×#97	—	2	1	no	—
23	C₂×C₄×C₈	—	2	1	no	—
24	C₂³×C₄	C₂×C₄	1	1	yes	characteristic
25	C₂×M₁₆	—	2	1	no	—
26	C₂×M₁₆	—	2	2	no	—
27	C₂×M₁₆	C₈	1	4	yes	—
28	C₂×M₁₆	C₂×C₄	1	2	yes	—
29	C₂²×C₈	—	2	1	no	—
30	C₂²×C₈	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₈	C₂×C₄	1	2	yes	—
32	C₂²×C₈	C₂×C₄	1	2	yes	—
33	C₂²×C₈	D₄	1	2	yes	—
34	C₂²×C₈	D₄	1	1	yes	characteristic
35	C₂²×C₈	D₄	1	1	yes	characteristic
36	C₂²×C₈	D₄	1	1	yes	characteristic
37	C₂²×C₈	D₄	1	1	yes	characteristic
38	C₂²×C₈	Q₈	1	2	yes	—
39	C₂²×C₈	C₂³	1	1	yes	characteristic
40	#117	—	2	1	no	—
41	#117	—	2	1	no	—
42	#117	C₈	1	2	yes	—
43	#117	C₂×C₄	1	2	yes	—
44	C₂×C₄⋊C₄	C₈	1	2	yes	—
45	C₂×C₂²⋊C₄	—	2	1	no	—
46	C₂×C₄²	—	2	1	no	—
47	M₃₂	—	2	4	no	—
48	M₃₂	—	2	4	no	—
49	M₃₂	—	2	4	no	—
50	M₃₂	—	2	2	no	—
51	M₃₂	C₂×C₄	1	4	yes	complemented
52	C₂×C₁₆	—	2	4	no	—
53	C₂×C₁₆	—	2	2	no	—
54	C₂×C₁₆	—	2	2	no	—
55	C₂×C₁₆	—	2	2	no	—
56	C₂×C₁₆	C₂×C₄	1	4	yes	complemented
57	#97	—	2	2	no	—
58	#97	—	2	2	no	—
59	C₄×C₈	—	2	2	no	—
60	C₄×C₈	—	2	2	no	—
61	C₂⁴	M₁₆	1	1	yes	characteristic
62	C₂²×C₄	—	2	2	no	—
63	C₂²×C₄	C₄²	1	2	yes	—
64	C₂²×C₄	C₂²⋊C₄	1	1	yes	characteristic
65	C₂²×C₄	C₂²⋊C₄	1	1	yes	characteristic
66	C₂²×C₄	C₄⋊C₄	1	2	yes	—
67	C₂²×C₄	C₂×C₈	1	2	yes	—
68	C₂²×C₄	C₂×C₈	1	1	yes	characteristic
69	C₂²×C₄	C₂×C₈	1	1	yes	characteristic
70	C₂²×C₄	M₁₆	1	1	yes	characteristic
71	C₂²×C₄	M₁₆	1	2	yes	—
72	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
73	M₁₆	—	2	4	no	—
74	M₁₆	—	2	2	no	—
75	M₁₆	C₂×C₈	1	4	yes	—
76	C₂×C₈	—	2	4	no	—
77	C₂×C₈	—	2	2	no	—
78	C₂×C₈	C₄²	1	2	yes	—
79	C₂×C₈	C₂²⋊C₄	1	1	yes	characteristic
80	C₂×C₈	C₂²⋊C₄	1	1	yes	characteristic
81	C₂×C₈	C₂²⋊C₄	1	2	yes	—
82	C₂×C₈	C₂²⋊C₄	1	2	yes	—
83	C₂×C₈	C₂²⋊C₄	1	2	yes	—
84	C₂×C₈	C₂²⋊C₄	1	2	yes	—
85	C₂×C₈	C₄⋊C₄	1	2	yes	—
86	C₂×C₈	C₄⋊C₄	1	2	yes	—
87	C₂×C₈	C₄⋊C₄	1	2	yes	—
88	C₂×C₈	C₄⋊C₄	1	2	yes	—
89	C₂×C₈	C₄⋊C₄	1	2	yes	—
90	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
91	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
92	C₂×C₈	C₂×D₄	1	1	yes	characteristic
93	C₂×C₈	C₂×D₄	1	1	yes	characteristic
94	C₂×C₈	C₂×D₄	1	1	yes	characteristic
95	C₂×C₈	C₂×Q₈	1	1	yes	characteristic
96	C₄⋊C₄	—	2	2	no	—
97	C₄⋊C₄	C₂×C₈	1	4	yes	—
98	C₂²⋊C₄	—	2	2	no	—
99	C₂²⋊C₄	—	2	2	no	—
100	C₄²	—	2	2	no	—
101	C₄²	—	2	2	no	—
102	C₁₆	—	2	4	no	—
103	C₁₆	—	2	4	no	—
104	C₂³	—	4	2	no	—
105	C₂³	#97	1	2	yes	—
106	C₂³	C₂²⋊C₈	1	1	yes	characteristic
107	C₂³	C₂²⋊C₈	1	1	yes	characteristic
108	C₂³	C₄⋊C₈	1	2	yes	—
109	C₂³	C₂×M₁₆	1	1	yes	characteristic
110	C₂×C₄	—	2	4	no	—
111	C₂×C₄	—	4	4	no	—
112	C₂×C₄	#95	1	2	yes	—
113	C₂×C₄	#95	1	2	yes	—
114	C₂×C₄	C₄×C₈	1	2	yes	—
115	C₂×C₄	C₄×C₈	1	2	yes	—
116	C₂×C₄	#97	1	2	yes	—
117	C₂×C₄	C₂²⋊C₈	1	1	yes	characteristic
118	C₂×C₄	C₂²⋊C₈	1	1	yes	characteristic
119	C₂×C₄	C₂²⋊C₈	1	2	yes	—
120	C₂×C₄	C₂²⋊C₈	1	2	yes	—
121	C₂×C₄	C₄⋊C₈	1	2	yes	—
122	C₂×C₄	C₄⋊C₈	1	2	yes	—
123	C₂×C₄	C₄⋊C₈	1	2	yes	—
124	C₂×C₄	C₂×C₄²	1	1	yes	characteristic
125	C₂×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
126	C₂×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
127	C₂×C₄	C₂²×C₈	1	1	yes	characteristic
128	C₂×C₄	C₂²×C₈	1	1	yes	characteristic
129	C₂×C₄	C₂×M₁₆	1	1	yes	characteristic
130	C₈	—	2	4	no	—
131	C₈	#95	1	2	yes	—
132	C₈	#95	1	2	yes	—
133	C₈	C₂×C₂²⋊C₄	1	1	yes	characteristic
134	C₈	C₂×C₂²⋊C₄	1	1	yes	characteristic
135	C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
136	C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
137	C₂²	—	2	2	no	—
138	C₂²	—	2	2	no	—
139	C₂²	—	2	2	no	—
140	C₂²	—	4	2	no	—
141	C₂²	—	4	2	no	—
142	C₂²	#336	1	2	yes	—
143	C₂²	#336	1	2	yes	—
144	C₂²	C₂×#97	1	1	yes	characteristic
145	C₂²	C₂×C₂²⋊C₈	1	1	yes	characteristic
146	C₂²	C₂×C₄⋊C₈	1	1	yes	characteristic
147	C₄	—	4	4	no	—
148	C₄	#336	1	2	yes	—
149	C₄	#336	1	2	yes	—
150	C₄	C₂×#95	1	1	yes	characteristic
151	C₄	C₂×C₄×C₈	1	1	yes	characteristic
152	C₄	C₂×C₂²⋊C₈	1	1	yes	characteristic
153	C₄	C₂×C₄⋊C₈	1	1	yes	characteristic
154	C₂	—	2	2	no	—
155	C₂	—	2	2	no	—
156	C₂	—	2	1	no	—
157	C₂	—	2	1	no	—
158	C₂	C₂×#336	1	1	yes	characteristic
159	C₂	#2117	1	2	yes	—

	subgroup	G/H
center	C₂×C₈	C₂×D₄
commutator	C₄	C₂×C₄×C₈
Frattini, cosocle	C₂²×C₈	C₂³
socle	C₂²	C₂×C₂²⋊C₈
prime radical	C₂⁴	M₁₆
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
4. #1379	C₂	semidirect
13. C₂×M₃₂	C₄	semidirect
14. C₂×M₃₂	C₄	semidirect
51. M₃₂	C₂×C₄	semidirect
56. C₂×C₁₆	C₂×C₄	semidirect

Maximal subgroups

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

subgroup	index
C₂²×M₃₂	2
C₂×#405	2
#1379	2