Subgroups — #23258

Order 256.

332 conjugacy classes of subgroups.

The subgroup lattice has 205 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#125	C₂	1	1	yes	maximal, characteristic
2	#1940	C₂	1	2	yes	maximal
3	#1939	C₂	1	1	yes	maximal, characteristic
4	#1939	C₂	1	1	yes	maximal, characteristic
5	#1917	C₂	1	2	yes	maximal
6	#1916	C₂	1	2	yes	maximal
7	#1770	C₂	1	1	yes	maximal, characteristic
8	#1770	C₂	1	2	yes	maximal
9	C₈×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic
10	#1767	C₂	1	1	yes	maximal, characteristic
11	C₄×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
12	C₂×#125	C₄	1	1	yes	characteristic
13	C₂×C₄⋊C₈	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊C₈	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊C₈	C₂²	1	2	yes	—
16	C₂×C₄⋊C₈	C₂²	1	1	yes	characteristic
17	C₂×C₄⋊C₈	C₂²	1	2	yes	—
18	C₂×C₄⋊C₈	C₂²	1	1	yes	characteristic
19	C₂×#97	C₂²	1	2	yes	—
20	C₂×#97	C₂²	1	2	yes	—
21	C₂×C₄×C₈	C₂²	1	2	yes	—
22	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
23	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
24	#389	C₄	1	1	yes	characteristic
25	#389	C₄	1	1	yes	characteristic
26	#387	C₄	1	2	yes	—
27	#387	C₄	1	2	yes	—
28	#382	C₄	1	1	yes	characteristic
29	C₄×C₄⋊C₄	C₂²	1	2	yes	—
30	C₄×C₄⋊C₄	C₂²	1	2	yes	—
31	C₄×C₄⋊C₄	C₂²	1	1	yes	characteristic
32	C₄×C₄⋊C₄	C₂²	1	1	yes	characteristic
33	C₄³	C₂²	1	1	yes	characteristic
34	#336	C₂²	1	2	yes	—
35	#336	C₂²	1	2	yes	—
36	#336	C₂²	1	2	yes	—
37	#336	C₂²	1	2	yes	—
38	#336	C₂²	1	1	yes	characteristic
39	#336	C₂²	1	2	yes	—
40	#336	C₂²	1	1	yes	characteristic
41	C₂²×C₈	C₂³	1	1	yes	characteristic
42	C₂²×C₈	C₂³	1	1	yes	characteristic
43	C₂²×C₈	C₂³	1	2	yes	—
44	C₂²×C₈	C₂³	1	2	yes	—
45	C₂²×C₈	C₂³	1	1	yes	characteristic
46	C₂²×C₈	C₂³	1	1	yes	characteristic
47	#125	—	2	4	no	—
48	C₂×C₄⋊C₄	C₂×C₄	1	2	yes	—
49	C₂×C₄⋊C₄	C₂×C₄	1	2	yes	—
50	C₂×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
51	C₂×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
52	C₂×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
53	C₂×C₄⋊C₄	C₂×C₄	1	2	yes	—
54	C₂×C₄⋊C₄	C₂×C₄	1	2	yes	—
55	C₂×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
56	C₂×C₄²	C₂×C₄	1	1	yes	characteristic
57	C₂×C₄²	C₂×C₄	1	1	yes	characteristic
58	C₂×C₄²	C₂×C₄	1	1	yes	characteristic
59	C₂×C₄²	C₂×C₄	1	1	yes	characteristic
60	C₂×C₄²	C₂³	1	1	yes	characteristic
61	C₂×C₄²	C₂³	1	1	yes	characteristic
62	C₂×C₄²	C₂³	1	2	yes	—
63	C₂×C₄²	C₂³	1	2	yes	—
64	C₂×C₄²	C₂³	1	1	yes	characteristic
65	C₄⋊C₈	—	2	2	no	—
66	C₄⋊C₈	—	2	4	no	—
67	C₄⋊C₈	—	2	2	no	—
68	C₄⋊C₈	—	2	2	no	—
69	C₄⋊C₈	—	4	2	no	—
70	C₄⋊C₈	Q₈	1	4	yes	—
71	#97	—	4	2	no	—
72	#97	—	4	2	no	—
73	C₄×C₈	—	2	4	no	—
74	C₄×C₈	—	2	2	no	—
75	C₄×C₈	—	2	2	no	—
76	#95	C₂×C₄	1	2	yes	—
77	#95	C₂×C₄	1	2	yes	—
78	#95	C₂×C₄	1	2	yes	—
79	#95	C₂×C₄	1	2	yes	—
80	#95	C₂×C₄	1	2	yes	—
81	#95	C₂×C₄	1	2	yes	—
82	C₂²×C₄	C₂²×C₄	1	2	yes	—
83	C₂²×C₄	C₂²×C₄	1	2	yes	—
84	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
85	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
86	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
87	C₂²×C₄	C₂²×C₄	1	2	yes	—
88	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
89	C₂²×C₄	C₂²×C₄	1	2	yes	—
90	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
91	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
92	C₂²×C₄	C₂⁴	1	1	yes	characteristic
93	C₂×C₈	—	2	1	no	—
94	C₂×C₈	—	2	1	no	—
95	C₂×C₈	—	2	2	no	—
96	C₂×C₈	—	2	1	no	—
97	C₂×C₈	—	2	2	no	—
98	C₂×C₈	—	2	1	no	—
99	C₂×C₈	—	2	1	no	—
100	C₂×C₈	—	2	1	no	—
101	C₂×C₈	—	2	1	no	—
102	C₂×C₈	—	2	2	no	—
103	C₂×C₈	—	2	1	no	—
104	C₂×C₈	—	2	2	no	—
105	C₂×C₈	—	2	2	no	—
106	C₂×C₈	—	2	1	no	—
107	C₂×C₈	—	2	1	no	—
108	C₂×C₈	C₂×Q₈	1	4	yes	—
109	C₂×C₈	#41	1	2	yes	—
110	C₂×C₈	#41	1	2	yes	—
111	C₄⋊C₄	—	2	4	no	—
112	C₄⋊C₄	—	2	4	no	—
113	C₄⋊C₄	—	2	2	no	—
114	C₄⋊C₄	—	2	4	no	—
115	C₄⋊C₄	—	2	2	no	—
116	C₄⋊C₄	—	2	2	no	—
117	C₄⋊C₄	—	4	2	no	—
118	C₄⋊C₄	M₁₆	1	4	yes	—
119	C₄²	—	2	2	no	—
120	C₄²	—	2	2	no	—
121	C₄²	—	2	2	no	—
122	C₄²	—	2	2	no	—
123	C₄²	—	2	1	no	—
124	C₄²	—	2	1	no	—
125	C₄²	—	2	2	no	—
126	C₄²	—	2	1	no	—
127	C₄²	—	2	2	no	—
128	C₄²	—	2	1	no	—
129	C₄²	—	2	2	no	—
130	C₄²	—	4	1	no	—
131	C₄²	C₂×Q₈	1	2	yes	—
132	C₄²	#41	1	2	yes	—
133	C₂³	C₂³×C₄	1	1	yes	characteristic
134	C₂×C₄	—	2	1	no	—
135	C₂×C₄	—	2	2	no	—
136	C₂×C₄	—	2	1	no	—
137	C₂×C₄	—	2	1	no	—
138	C₂×C₄	—	2	1	no	—
139	C₂×C₄	—	2	2	no	—
140	C₂×C₄	—	2	2	no	—
141	C₂×C₄	—	2	1	no	—
142	C₂×C₄	—	2	1	no	—
143	C₂×C₄	—	2	2	no	—
144	C₂×C₄	—	2	1	no	—
145	C₂×C₄	—	2	2	no	—
146	C₂×C₄	—	2	1	no	—
147	C₂×C₄	—	2	1	no	—
148	C₂×C₄	—	2	2	no	—
149	C₂×C₄	—	2	1	no	—
150	C₂×C₄	—	2	1	no	—
151	C₂×C₄	—	2	2	no	—
152	C₂×C₄	—	2	2	no	—
153	C₂×C₄	—	2	2	no	—
154	C₂×C₄	—	2	2	no	—
155	C₂×C₄	—	2	1	no	—
156	C₂×C₄	—	2	1	no	—
157	C₂×C₄	—	2	2	no	—
158	C₂×C₄	C₄×Q₈	1	2	yes	—
159	C₂×C₄	C₄×Q₈	1	2	yes	—
160	C₂×C₄	C₂×M₁₆	1	4	yes	—
161	C₂×C₄	C₂×M₁₆	1	2	yes	—
162	C₂×C₄	#131	1	2	yes	—
163	C₂×C₄	#131	1	2	yes	—
164	C₂×C₄	C₂²×Q₈	1	1	yes	characteristic
165	C₂×C₄	C₂×#41	1	1	yes	characteristic
166	C₂×C₄	C₂×#41	1	1	yes	characteristic
167	C₂×C₄	C₂×#41	1	1	yes	characteristic
168	C₂×C₄	H(5,𝔽₂)	1	1	yes	characteristic
169	C₂×C₄	2₋¹⁺⁴	1	1	yes	characteristic
170	C₈	—	2	2	no	—
171	C₈	—	2	2	no	—
172	C₈	—	2	4	no	—
173	C₈	—	4	1	no	—
174	C₈	—	4	1	no	—
175	C₈	—	4	2	no	—
176	C₂²	C₂×C₄×Q₈	1	1	yes	characteristic
177	C₂²	C₄×#41	1	1	yes	characteristic
178	C₂²	#520	1	1	yes	characteristic
179	C₂²	C₂²×M₁₆	1	1	yes	characteristic
180	C₂²	C₂×#131	1	1	yes	characteristic
181	C₂²	C₂×#131	1	1	yes	characteristic
182	C₂²	#568	1	1	yes	characteristic
183	C₄	—	2	2	no	—
184	C₄	—	2	2	no	—
185	C₄	—	2	4	no	—
186	C₄	—	4	2	no	—
187	C₄	—	4	1	no	—
188	C₄	—	4	1	no	—
189	C₄	—	4	2	no	—
190	C₄	—	4	1	no	—
191	C₄	—	4	2	no	—
192	C₄	#446	1	4	yes	—
193	C₄	#552	1	1	yes	characteristic
194	C₄	#553	1	1	yes	characteristic
195	C₄	#554	1	1	yes	characteristic
196	C₄	#557	1	1	yes	characteristic
197	C₂	#2370	1	1	yes	characteristic
198	C₂	C₂×#446	1	1	yes	characteristic
199	C₂	#2962	1	1	yes	characteristic
200	C₂	#2966	1	1	yes	characteristic
201	C₂	#2970	1	1	yes	characteristic
202	C₂	#2991	1	1	yes	characteristic
203	C₂	#2993	1	1	yes	characteristic

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

Maximal subgroups

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

subgroup	index
C₄×#125	2
#1940	2
#1939	2
#1917	2
#1916	2
#1770	2
C₈×C₄⋊C₄	2
#1767	2
C₄×C₄⋊C₈	2