Finite Group Databasebeta

← #23457

Subgroups — #23457

Order 256.

601 conjugacy classes of subgroups; only the 175 normal subgroups are shown.

normal subgroups only

This group has 601 conjugacy classes of subgroups — too many to draw. Only its 175 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#544C211yesmaximal, characteristic
2C2×#419C211yesmaximal, characteristic
3C22×#103C211yesmaximal, characteristic
4#2310C211yesmaximal, characteristic
5#2026C214yesmaximal, complemented
6#2024C214yesmaximal, complemented
7#1736C211yesmaximal, characteristic
8#1735C211yesmaximal, characteristic
9#1729C211yesmaximal, characteristic
10C23×Q8C2211yescharacteristic
11C22×M16C2211yescharacteristic
12C23×C8C2211yescharacteristic
13#544—24no—
14#544—24no—
15C2×C4⋊Q8—21no—
16C2×C4⋊Q8C2212yes—
17C2×#125—21no—
18C2×#125C2212yes—
19C2×C22⋊Q8—22no—
20C2×#117C2211yescharacteristic
21C2×#117C2211yescharacteristic
22C2×#117C2211yescharacteristic
23C22×C4⋊C4C2211yescharacteristic
24#419—24no—
25C2×#103—24no—
26C2×#103C2214yes—
27C2×#103C2212yes—
28C2×#103C2212yes—
29#391—21no—
30#391—21no—
31#391C2214yes—
32#381—22no—
33#340C2212yes—
34#340C2212yes—
35#340C2214yes—
36#336C2214yes—
37C22×Q8—21no—
38C22×Q8—22no—
39C22×Q8—22no—
40C22×Q8—22no—
41C22×Q8—22no—
42C22×Q8—22no—
43C22×Q8—22no—
44C22×Q8Q814yescomplemented
45C22×Q8C2312yes—
46C23×C4—21no—
47C23×C4—21no—
48C23×C4C2311yescharacteristic
49C2×M16—22no—
50C2×M16—22no—
51C2×M16—22no—
52C22×C8—22no—
53C22×C8—22no—
54C22×C8—22no—
55C22×C8C2312yes—
56C22×C8C2312yes—
57C4⋊Q8—28no—
58C4⋊Q8—24no—
59C4⋊Q8—24no—
60#125—28no—
61#125—24no—
62#125—24no—
63C22⋊Q8—216no—
64#117—22no—
65#117—22no—
66#117—22no—
67#117—22no—
68#117—22no—
69#117D414yes—
70C2×C4⋊C4—21no—
71C2×C4⋊C4—22no—
72C2×C4⋊C4—22no—
73C2×C4⋊C4—22no—
74C2×C4⋊C4—22no—
75C2×C4⋊C4D411yescharacteristic
76C2×C4⋊C4D411yescharacteristic
77C2×C4⋊C4D414yes—
78C2×C4⋊C4C2312yes—
79C2×C4⋊C4C2312yes—
80C2×C4⋊C4C2311yescharacteristic
81C2×C4⋊C4C2311yescharacteristic
82C2×C22⋊C4—21no—
83C2×C22⋊C4—21no—
84C2×C22⋊C4D412yes—
85C2×C42—21no—
86C2×C42—21no—
87C2×C42C2312yes—
88#103—28no—
89#103—24no—
90#103—24no—
91#95—24no—
92#95—24no—
93C24C2×D411yescharacteristic
94C2×Q8—28no—
95C2×Q8—24no—
96C2×Q8—22no—
97C2×Q8—28no—
98C2×Q8—28no—
99C2×Q8—22no—
100C2×Q8—22no—
101C2×Q8—22no—
102C2×Q8—22no—
103C2×Q8—42no—
104C2×Q8—42no—
105C2×Q8—44no—
106C2×Q8—44no—
107C2×Q8—84no—
108C2×Q8C2×Q814yes—
109C22×C4—24no—
110C22×C4—22no—
111C22×C4—21no—
112C22×C4—21no—
113C22×C4—22no—
114C22×C4—24no—
115C22×C4—22no—
116C22×C4—42no—
117C22×C4—42no—
118C22×C4—82no—
119C22×C4C2×D412yes—
120C22×C4C2×D412yes—
121C22×C4C2×D411yescharacteristic
122C22×C4C2×D412yes—
123C22×C4C2×D412yes—
124C22×C4C2×Q812yes—
125C22×C4#4112yes—
126C22×C4#4112yes—
127C22×C4#4112yes—
128C22×C4C2411yescharacteristic
129M16—44no—
130C2×C8—22no—
131C2×C8—22no—
132C2×C8—22no—
133C2×C8—22no—
134C2×C8—22no—
135C2×C8—22no—
136C2×C8—44no—
137C4⋊C4—28no—
138C4⋊C4—28no—
139C4⋊C4—28no—
140C4⋊C4—24no—
141C4⋊C4—22no—
142C4⋊C4—22no—
143C4⋊C4—22no—
144C4⋊C4—21no—
145C4⋊C4—21no—
146C4⋊C4—22no—
147C4⋊C4—22no—
148C4⋊C4C2×D414yes—
149C4⋊C4C2×D412yes—
150C4⋊C4C2×D412yes—
151C22⋊C4—24no—
152C22⋊C4—24no—
153C22⋊C4—42no—
154C42—24no—
155C42—22no—
156C42—22no—
157C42—42no—
158C23—22no—
159C23—22no—
160C23#12012yes—
161C23C22⋊Q812yes—
162C23#12412yes—
163C23C22×D411yescharacteristic
164Q8—24no—
165Q8—24no—
166Q8—42no—
167Q8—42no—
168Q8—48no—
169Q8—88no—
170C2×C4—22no—
171C2×C4—22no—
172C2×C4—22no—
173C2×C4—22no—
174C2×C4—24no—
175C2×C4—24no—
176C2×C4—24no—
177C2×C4—24no—
178C2×C4—42no—
179C2×C4—42no—
180C2×C4—44no—
181C2×C4—42no—
182C2×C4—42no—
183C2×C4—48no—
184C2×C4—42no—
185C2×C4—42no—
186C2×C4—42no—
187C2×C4—84no—
188C2×C4#12012yes—
189C2×C4C22⋊Q814yes—
190C2×C4C22⋊Q814yes—
191C2×C4C22⋊Q812yes—
192C2×C4#12414yes—
193C2×C4#12414yes—
194C2×C4#12412yes—
195C2×C4C22×D411yescharacteristic
196C2×C4C22×D411yescharacteristic
197C2×C4C22×Q811yescharacteristic
198C2×C4C2×#4111yescharacteristic
199C2×C4C2×#4111yescharacteristic
200C2×C4C2×#4111yescharacteristic
201C8—42no—
202C8—42no—
203C22—24no—
204C22—24no—
205C22—22no—
206C22—22no—
207C22#39314yes—
208C22C2×#12011yescharacteristic
209C22C2×C22⋊Q811yescharacteristic
210C22C2×#12411yescharacteristic
211C22#57712yes—
212C22#57812yes—
213C4—44no—
214C4—42no—
215C4—44no—
216C4—42no—
217C4—42no—
218C4—42no—
219C4—82no—
220C4#39314yes—
221C4C2×C22⋊Q811yescharacteristic
222C4C2×C22⋊Q811yescharacteristic
223C4C2×#12411yescharacteristic
224C4C2×#12411yescharacteristic
225C2—24no—
226C2C2×#39311yescharacteristic
227C2#300911yescharacteristic
228C2#301111yescharacteristic
229C2#308212yes—
230C2#313712yes—
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC24C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
5. #2026C2semidirect
6. #2024C2semidirect
44. C22×Q8Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#5442
C2×#4192
C22×#1032
#23102
#20262
#20242
#17362
#17352
#17292