Finite Group Databasebeta

← #24134

Subgroups — #24134

Order 256.

560 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 560 conjugacy classes of subgroups — too many to draw. Only its 135 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×#554C211yesmaximal, characteristic
2C2×#475C211yesmaximal, characteristic, complemented
3C2×#441C211yesmaximal, characteristic, complemented
4#2536C211yesmaximal, characteristic
5#2035C212yesmaximal, complemented
6#2001C212yesmaximal, complemented
7#1993C211yesmaximal, characteristic, complemented
8#1955C211yesmaximal, characteristic
9#1872C212yesmaximal, complemented
10#1853C212yesmaximal, complemented
11#1805C211yesmaximal, characteristic, complemented
12C22×Q16C2211yescharacteristic
13#554—24no—
14#554—24no—
15C2×C4⋊Q8C2211yescharacteristic
16C2×#125—21no—
17C2×C22⋊Q8—22no—
18C2×C22⋊Q8C2212yes—
19C2×C22⋊Q8C2212yes—
20C2×C4×Q8C2211yescharacteristic
21C2×C4×Q8C2211yescharacteristic
22C2×C4×D4—21no—
23C2×C4×D4C2211yescharacteristic
24C22×C4⋊C4C2212yes—
25#475—24no—
26#441—24no—
27C2×#106C2212yes—
28C2×#106C2211yescharacteristic
29C2×C4⋊C8C2211yescharacteristic
30C2×#103C2212yes—
31C2×#103C2211yescharacteristic
32C2×#103C2211yescharacteristic
33C2×#103C2212yes—
34C2×C22⋊C8C2212yes—
35C2×C22⋊C8C2212yes—
36C2×#97C2211yescharacteristic
37#398—21no—
38#394—21no—
39#388—22no—
40#382C2211yescharacteristic
41#380C2212yes—
42#340C2212yes—
43#340C2212yes—
44#340C2212yes—
45C22×Q8—21no—
46C22×Q8C2311yescharacteristic
47C22×Q8C2311yescharacteristic
48C22×D4D411yescharacteristic
49C23×C4—22no—
50C23×C4C2312yes—
51C2×Q16—22no—
52C2×Q16—22no—
53C2×Q16—22no—
54C22×C8C2311yescharacteristic
55C22×C8C2312yes—
56C22×C8C2311yescharacteristic
57C4⋊Q8—22no—
58C4⋊Q8D414yes—
59#125—28no—
60C22⋊Q8—216no—
61C22⋊Q8—24no—
62C22⋊Q8—24no—
63C22⋊Q8—24no—
64C22⋊Q8—42no—
65C4×Q8—22no—
66C4×Q8—22no—
67C4×Q8—22no—
68C4×Q8D414yes—
69C4×D4—28no—
70C4×D4—22no—
71C4×D4—22no—
72C2×C4⋊C4—22no—
73C2×C4⋊C4—21no—
74C2×C4⋊C4—21no—
75C2×C4⋊C4—22no—
76C2×C4⋊C4—24no—
77C2×C4⋊C4—24no—
78C2×C4⋊C4—21no—
79C2×C4⋊C4—22no—
80C2×C4⋊C4D411yescharacteristic
81C2×C4⋊C4C2311yescharacteristic
82C2×C4⋊C4C2312yes—
83C2×C4⋊C4C2311yescharacteristic
84C2×C4⋊C4C2312yes—
85C2×C22⋊C4—22no—
86C2×C22⋊C4—22no—
87C2×C22⋊C4—22no—
88C2×C22⋊C4D412yes—
89C2×C42—21no—
90C2×C42—21no—
91C2×C42C2311yescharacteristic
92#106—22no—
93#106—24no—
94C4⋊C8—22no—
95#103—22no—
96#103—22no—
97#103—24no—
98#103—42no—
99C22⋊C8—24no—
100C22⋊C8—42no—
101#97—22no—
102#95—22no—
103#95—21no—
104#95—21no—
105C24C2×D412yes—
106C2×Q8—24no—
107C2×Q8—21no—
108C2×Q8—21no—
109C2×Q8—21no—
110C2×Q8—21no—
111C2×Q8—21no—
112C2×Q8—21no—
113C2×Q8—21no—
114C2×Q8—21no—
115C2×Q8—21no—
116C2×Q8—21no—
117C2×Q8—82no—
118C2×Q8C2×D412yes—
119C2×Q8#4112yes—
120C2×D4—24no—
121C2×D4—22no—
122C22×C4—21no—
123C22×C4—21no—
124C22×C4—21no—
125C22×C4—22no—
126C22×C4—28no—
127C22×C4—22no—
128C22×C4—22no—
129C22×C4—22no—
130C22×C4—22no—
131C22×C4—22no—
132C22×C4—22no—
133C22×C4—22no—
134C22×C4—21no—
135C22×C4—82no—
136C22×C4—82no—
137C22×C4C2×D412yes—
138C22×C4C2×D411yescharacteristic
139C22×C4C2×D411yescharacteristic
140C22×C4C2411yescharacteristic
141Q16—28no—
142C2×C8—21no—
143C2×C8—21no—
144C2×C8—21no—
145C2×C8—22no—
146C2×C8—22no—
147C2×C8—21no—
148C2×C8—22no—
149C2×C8—21no—
150C2×C8#4112yes—
151C4⋊C4—28no—
152C4⋊C4—28no—
153C4⋊C4—24no—
154C4⋊C4—24no—
155C4⋊C4—22no—
156C4⋊C4—21no—
157C4⋊C4—22no—
158C4⋊C4—21no—
159C4⋊C4—21no—
160C4⋊C4—22no—
161C4⋊C4—22no—
162C4⋊C4—44no—
163C4⋊C4—42no—
164C4⋊C4C2×D412yes—
165C4⋊C4C2×D414yes—
166C22⋊C4—28no—
167C22⋊C4—24no—
168C22⋊C4—44no—
169C22⋊C4—44no—
170C42—24no—
171C42—42no—
172C42C2×D412yes—
173C42C2×D412yes—
174C23—22no—
175C23—22no—
176C23—22no—
177C23—22no—
178C23—22no—
179C23—22no—
180C23#13714yes—
181C23C22×D411yescharacteristic
182Q8—22no—
183Q8—22no—
184Q8—22no—
185Q8—22no—
186Q8—41no—
187Q8—41no—
188Q8—41no—
189Q8—41no—
190Q8—84no—
191D4—28no—
192C2×C4—22no—
193C2×C4—22no—
194C2×C4—22no—
195C2×C4—22no—
196C2×C4—22no—
197C2×C4—24no—
198C2×C4—22no—
199C2×C4—22no—
200C2×C4—22no—
201C2×C4—41no—
202C2×C4—41no—
203C2×C4—41no—
204C2×C4—41no—
205C2×C4—42no—
206C2×C4—42no—
207C2×C4—42no—
208C2×C4—42no—
209C2×C4—42no—
210C2×C4—42no—
211C2×C4—42no—
212C2×C4—41no—
213C2×C4—42no—
214C2×C4—42no—
215C2×C4—42no—
216C2×C4—42no—
217C2×C4—88no—
218C2×C4#12012yes—
219C2×C4#12012yes—
220C2×C4C22×D411yescharacteristic
221C2×C4C22×D411yescharacteristic
222C2×C4C2×#4111yescharacteristic
223C2×C4C2×#4111yescharacteristic
224C2×C4H(5,𝔽2)11yescharacteristic
225C2×C42−1+411yescharacteristic
226C8—22no—
227C8—41no—
228C8—42no—
229C22—24no—
230C22—24no—
231C22—24no—
232C22—42no—
233C22—42no—
234C22—42no—
235C22—42no—
236C22C2×#12011yescharacteristic
237C22#54611yescharacteristic
238C22#54711yescharacteristic
239C22C2×#13712yes—
240C22#57712yes—
241C4—22no—
242C4—22no—
243C4—42no—
244C4—42no—
245C4—42no—
246C4—42no—
247C4—44no—
248C4—81no—
249C4—82no—
250C4#52511yescharacteristic
251C4#53711yescharacteristic
252C4#54811yescharacteristic
253C4#55011yescharacteristic
254C2—44no—
255C2#240711yescharacteristic
256C2#300411yescharacteristic
257C2#300911yescharacteristic
258C2#331612yes—
259C2#334412yes—
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC22×D4D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#475C2semidirect
3. C2×#441C2semidirect
5. #2035C2semidirect
6. #2001C2semidirect
7. #1993C2semidirect
9. #1872C2semidirect
10. #1853C2semidirect
11. #1805C2semidirect

Maximal subgroups

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

subgroupindex
C2×#5542
C2×#4752
C2×#4412
#25362
#20352
#20012
#19932
#19552
#18722
#18532
#18052