Subgroups — #7924

Order 256.

357 conjugacy classes of subgroups.

The subgroup lattice has 193 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#431	C₂	1	1	yes	maximal, characteristic
2	C₂×#431	C₂	1	1	yes	maximal, characteristic
3	C₄×#117	C₂	1	1	yes	maximal, characteristic
4	#1282	C₂	1	4	yes	maximal
5	C₂×#117	C₄	1	2	yes	—
6	C₂²×C₄²	C₂²	1	1	yes	characteristic
7	#431	—	2	1	no	—
8	#431	—	2	1	no	—
9	#431	—	2	1	no	—
10	#431	—	2	1	no	—
11	#431	—	2	1	no	—
12	#431	—	2	1	no	—
13	#431	C₄	1	2	yes	complemented
14	#431	C₄	1	2	yes	complemented
15	C₂×C₄⋊C₈	—	2	1	no	—
16	C₂×C₄⋊C₈	C₄	1	2	yes	complemented
17	C₂×C₂²⋊C₈	C₂²	1	2	yes	—
18	C₂×C₂²⋊C₈	C₂²	1	2	yes	—
19	C₂×C₄×C₈	—	2	1	no	—
20	C₂×C₄×C₈	C₄	1	2	yes	complemented
21	C₄×C₄⋊C₄	—	2	1	no	—
22	C₄×C₄⋊C₄	—	2	1	no	—
23	C₄×C₂²⋊C₄	C₄	1	2	yes	—
24	C₄×C₂²⋊C₄	C₂²	1	2	yes	—
25	#376	—	2	1	no	—
26	C₄³	—	2	1	no	—
27	C₂³×C₄	C₂×C₄	1	1	yes	characteristic
28	C₂³×C₄	C₂×C₄	1	1	yes	characteristic
29	C₂³×C₄	C₂³	1	1	yes	characteristic
30	C₂²×C₈	—	2	2	no	—
31	C₂²×C₈	C₂×C₄	1	4	yes	—
32	#117	—	2	4	no	—
33	#117	C₈	1	8	yes	complemented
34	C₂×C₄⋊C₄	—	2	2	no	—
35	C₂×C₂²⋊C₄	C₂×C₄	1	4	yes	—
36	C₂×C₄²	—	2	2	no	—
37	C₂×C₄²	—	2	2	no	—
38	C₂×C₄²	—	2	2	no	—
39	C₂×C₄²	C₂×C₄	1	1	yes	characteristic
40	C₂×C₄²	C₂×C₄	1	1	yes	characteristic
41	C₂×C₄²	C₂×C₄	1	1	yes	characteristic
42	C₂×C₄²	C₂×C₄	1	1	yes	characteristic
43	C₂×C₄²	D₄	1	1	yes	characteristic
44	C₂×C₄²	D₄	1	1	yes	characteristic
45	C₂×C₄²	D₄	1	1	yes	characteristic
46	C₂×C₄²	D₄	1	1	yes	characteristic
47	C₂×C₄²	D₄	1	1	yes	characteristic
48	C₂×C₄²	D₄	1	1	yes	characteristic
49	C₂×C₄²	Q₈	1	1	yes	characteristic
50	C₂×C₄²	Q₈	1	1	yes	characteristic
51	C₄⋊C₈	—	2	2	no	—
52	C₄⋊C₈	—	2	2	no	—
53	C₄⋊C₈	—	2	2	no	—
54	C₄⋊C₈	—	2	2	no	—
55	C₂²⋊C₈	—	2	2	no	—
56	C₂²⋊C₈	—	2	2	no	—
57	C₂²⋊C₈	—	2	2	no	—
58	C₂²⋊C₈	C₂×C₄	1	4	yes	—
59	C₄×C₈	—	2	2	no	—
60	C₄×C₈	—	2	2	no	—
61	C₄×C₈	—	2	2	no	—
62	C₄×C₈	—	2	2	no	—
63	#95	—	2	2	no	—
64	#95	—	2	2	no	—
65	C₂⁴	C₂²×C₄	1	1	yes	characteristic
66	C₂²×C₄	—	2	4	no	—
67	C₂²×C₄	—	4	2	no	—
68	C₂²×C₄	—	4	2	no	—
69	C₂²×C₄	C₄²	1	1	yes	characteristic
70	C₂²×C₄	C₄²	1	1	yes	characteristic
71	C₂²×C₄	C₄²	1	1	yes	characteristic
72	C₂²×C₄	C₄²	1	1	yes	characteristic
73	C₂²×C₄	C₂²⋊C₄	1	1	yes	characteristic
74	C₂²×C₄	C₂²⋊C₄	1	1	yes	characteristic
75	C₂²×C₄	C₂²⋊C₄	1	1	yes	characteristic
76	C₂²×C₄	C₂²⋊C₄	1	1	yes	characteristic
77	C₂²×C₄	C₄⋊C₄	1	1	yes	characteristic
78	C₂²×C₄	C₄⋊C₄	1	1	yes	characteristic
79	C₂²×C₄	C₄⋊C₄	1	1	yes	characteristic
80	C₂²×C₄	C₄⋊C₄	1	1	yes	characteristic
81	C₂²×C₄	C₂×C₈	1	4	yes	—
82	C₂²×C₄	M₁₆	1	4	yes	—
83	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
84	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
85	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
86	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
87	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
88	C₂²×C₄	C₂×Q₈	1	1	yes	characteristic
89	C₂×C₈	—	2	4	no	—
90	C₂×C₈	—	2	4	no	—
91	C₂×C₈	—	4	4	no	—
92	C₂×C₈	—	4	4	no	—
93	C₄⋊C₄	—	2	8	no	—
94	C₂²⋊C₄	—	2	4	no	—
95	C₂²⋊C₄	C₂×C₈	1	8	yes	—
96	C₄²	—	2	8	no	—
97	C₄²	—	4	4	no	—
98	C₄²	—	4	4	no	—
99	C₄²	C₂²⋊C₄	1	1	yes	characteristic
100	C₄²	C₂²⋊C₄	1	1	yes	characteristic
101	C₄²	C₂²⋊C₄	1	1	yes	characteristic
102	C₄²	C₂²⋊C₄	1	1	yes	characteristic
103	C₄²	C₂²⋊C₄	1	1	yes	characteristic
104	C₄²	C₂²⋊C₄	1	1	yes	characteristic
105	C₄²	C₂²⋊C₄	1	1	yes	characteristic
106	C₄²	C₂²⋊C₄	1	1	yes	characteristic
107	C₄²	C₄⋊C₄	1	1	yes	characteristic
108	C₄²	C₄⋊C₄	1	1	yes	characteristic
109	C₄²	C₄⋊C₄	1	1	yes	characteristic
110	C₄²	C₄⋊C₄	1	1	yes	characteristic
111	C₄²	C₄⋊C₄	1	1	yes	characteristic
112	C₄²	C₄⋊C₄	1	1	yes	characteristic
113	C₄²	C₄⋊C₄	1	1	yes	characteristic
114	C₄²	C₄⋊C₄	1	1	yes	characteristic
115	C₂³	—	4	1	no	—
116	C₂³	—	4	1	no	—
117	C₂³	C₂×C₄²	1	1	yes	characteristic
118	C₂³	C₂×C₂²⋊C₄	1	1	yes	characteristic
119	C₂³	C₂×C₄⋊C₄	1	1	yes	characteristic
120	C₂³	C₂²×C₈	1	2	yes	—
121	C₂³	C₂×M₁₆	1	2	yes	—
122	C₂×C₄	—	2	2	no	—
123	C₂×C₄	—	2	2	no	—
124	C₂×C₄	—	2	2	no	—
125	C₂×C₄	—	2	2	no	—
126	C₂×C₄	—	2	2	no	—
127	C₂×C₄	—	2	2	no	—
128	C₂×C₄	—	2	8	no	—
129	C₂×C₄	—	4	1	no	—
130	C₂×C₄	—	4	1	no	—
131	C₂×C₄	—	4	1	no	—
132	C₂×C₄	—	4	1	no	—
133	C₂×C₄	—	4	1	no	—
134	C₂×C₄	—	4	1	no	—
135	C₂×C₄	—	4	4	no	—
136	C₂×C₄	—	4	4	no	—
137	C₂×C₄	#95	1	1	yes	characteristic
138	C₂×C₄	#95	1	1	yes	characteristic
139	C₂×C₄	#95	1	1	yes	characteristic
140	C₂×C₄	#95	1	1	yes	characteristic
141	C₂×C₄	#95	1	1	yes	characteristic
142	C₂×C₄	#95	1	1	yes	characteristic
143	C₂×C₄	#95	1	1	yes	characteristic
144	C₂×C₄	#95	1	1	yes	characteristic
145	C₂×C₄	C₄×C₈	1	2	yes	—
146	C₂×C₄	C₄×C₈	1	2	yes	—
147	C₂×C₄	#97	1	2	yes	—
148	C₂×C₄	#97	1	2	yes	—
149	C₂×C₄	C₂²⋊C₈	1	4	yes	—
150	C₂×C₄	C₂²⋊C₈	1	4	yes	—
151	C₂×C₄	C₄⋊C₈	1	4	yes	—
152	C₂×C₄	C₄⋊C₈	1	4	yes	—
153	C₂×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
154	C₂×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
155	C₂×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
156	C₂×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
157	C₈	—	4	4	no	—
158	C₈	—	4	4	no	—
159	C₂²	—	2	1	no	—
160	C₂²	—	2	1	no	—
161	C₂²	—	2	1	no	—
162	C₂²	—	2	1	no	—
163	C₂²	—	2	1	no	—
164	C₂²	—	2	1	no	—
165	C₂²	—	4	2	no	—
166	C₂²	—	4	2	no	—
167	C₂²	C₂×#95	1	1	yes	characteristic
168	C₂²	C₂×C₄×C₈	1	1	yes	characteristic
169	C₂²	C₂×#97	1	1	yes	characteristic
170	C₂²	C₂×C₂²⋊C₈	1	2	yes	—
171	C₂²	C₂×C₄⋊C₈	1	2	yes	—
172	C₄	—	2	1	no	—
173	C₄	—	2	1	no	—
174	C₄	—	2	1	no	—
175	C₄	—	2	1	no	—
176	C₄	—	2	1	no	—
177	C₄	—	2	1	no	—
178	C₄	—	4	8	no	—
179	C₄	#336	1	2	yes	—
180	C₄	#336	1	2	yes	—
181	C₄	#336	1	2	yes	—
182	C₄	#336	1	2	yes	—
183	C₄	#344	1	2	yes	—
184	C₄	#410	1	2	yes	—
185	C₂	—	2	2	no	—
186	C₂	—	2	1	no	—
187	C₂	—	2	1	no	—
188	C₂	—	2	2	no	—
189	C₂	C₂×#336	1	1	yes	characteristic
190	C₂	#1740	1	1	yes	characteristic
191	C₂	C₂×#344	1	1	yes	characteristic

	subgroup	G/H
center	C₄²	C₂²⋊C₄
commutator	C₂²	C₂×C₄×C₈
Frattini, cosocle	C₂³×C₄	C₂³
socle	C₂²	C₂×#95
prime radical	C₂⁴	C₂²×C₄
LCS	G ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

Complements

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

N	H ≅ G/N	type
13. #431	C₄	semidirect
14. #431	C₄	semidirect
16. C₂×C₄⋊C₈	C₄	semidirect
20. C₂×C₄×C₈	C₄	semidirect
33. #117	C₈	semidirect

Maximal subgroups

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

subgroup	index
C₂×#431	2
C₄×#117	2
#1282	2