Finite Group Databasebeta

← #29884

Subgroups — #29884

Order 256.

377 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 377 conjugacy classes of subgroups — too many to draw. Only its 123 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×#501C211yesmaximal, characteristic, complemented
2C2×#432C211yesmaximal, characteristic
3#2857C211yesmaximal, characteristic
4#2851C211yesmaximal, characteristic
5#2088C212yesmaximal, complemented
6#2086C212yesmaximal, complemented
7#2077C212yesmaximal
8#2076C212yesmaximal
9#1993C211yesmaximal, characteristic, complemented
10#1992C211yesmaximal, characteristic, complemented
11#1836C211yesmaximal, characteristic, complemented
12C2×C4⋊Q8C2211yescharacteristic
13C2×#125C2211yescharacteristic
14C2×C22⋊Q8—21no—
15C22×C42C2211yescharacteristic
16#501—24no—
17#432—24no—
18C2×#107C2212yes—
19C2×#106C2212yes—
20C2×C4⋊C8C2212yes—
21C2×C22⋊C8C2212yes—
22C2×#97C2211yescharacteristic
23C2×#97C2211yescharacteristic
24#391—21no—
25#389—21no—
26#389C2211yescharacteristic
27#389C2211yescharacteristic
28#389C2212yes—
29#387—21no—
30#387—21no—
31#385—21no—
32#385—21no—
33#384C2211yescharacteristic
34#382C2211yescharacteristic
35#380—21no—
36#380C2211yescharacteristic
37#380C2212yes—
38#380C2211yescharacteristic
39#340C2212yes—
40#340C2212yes—
41#340C2212yes—
42#340C2212yes—
43#340C2212yes—
44#340C2212yes—
45C22×Q8—21no—
46C23×C4D411yescharacteristic
47C23×C4D411yescharacteristic
48C23×C4C2311yescharacteristic
49C22×C8C2312yes—
50C22×C8C2312yes—
51C4⋊Q8—22no—
52C4⋊Q8—22no—
53#125—22no—
54#125—22no—
55C22⋊Q8—44no—
56C2×C4⋊C4—21no—
57C2×C4⋊C4—22no—
58C2×C4⋊C4—21no—
59C2×C4⋊C4—21no—
60C2×C4⋊C4—21no—
61C2×C4⋊C4—21no—
62C2×C4⋊C4—21no—
63C2×C4⋊C4—21no—
64C2×C4⋊C4C2311yescharacteristic
65C2×C4⋊C4C2311yescharacteristic
66C2×C4⋊C4C2312yes—
67C2×C4⋊C4C2311yescharacteristic
68C2×C4⋊C4C2311yescharacteristic
69C2×C4⋊C4C2312yes—
70C2×C22⋊C4—22no—
71C2×C22⋊C4—21no—
72C2×C22⋊C4—21no—
73C2×C42—22no—
74C2×C42—22no—
75C2×C42D411yescharacteristic
76C2×C42D411yescharacteristic
77C2×C42C2311yescharacteristic
78C2×C42C2311yescharacteristic
79#107—24no—
80#106—24no—
81C4⋊C8—24no—
82C22⋊C8—24no—
83#97—22no—
84#97D414yes—
85#95—22no—
86#95—21no—
87#95—21no—
88#95—22no—
89#95—21no—
90#95—21no—
91#95—21no—
92#95—21no—
93C24C2×D411yescharacteristic
94C2×Q8—42no—
95C2×Q8—42no—
96C2×Q8—42no—
97C22×C4—22no—
98C22×C4—22no—
99C22×C4—22no—
100C22×C4—22no—
101C22×C4—21no—
102C22×C4—21no—
103C22×C4—22no—
104C22×C4—22no—
105C22×C4—21no—
106C22×C4—21no—
107C22×C4—21no—
108C22×C4—21no—
109C22×C4—22no—
110C22×C4—22no—
111C22×C4—41no—
112C22×C4—41no—
113C22×C4C2×D411yescharacteristic
114C22×C4C2×D411yescharacteristic
115C22×C4C2×D411yescharacteristic
116C22×C4C2×D411yescharacteristic
117C22×C4C2×D411yescharacteristic
118C22×C4C2411yescharacteristic
119C2×C8—22no—
120C2×C8—22no—
121C2×C8—22no—
122C2×C8—22no—
123C2×C8—22no—
124C2×C8C2×D414yes—
125C4⋊C4—21no—
126C4⋊C4—22no—
127C4⋊C4—21no—
128C4⋊C4—21no—
129C4⋊C4—21no—
130C4⋊C4—22no—
131C4⋊C4—22no—
132C4⋊C4—22no—
133C4⋊C4—24no—
134C4⋊C4—42no—
135C4⋊C4—42no—
136C4⋊C4—42no—
137C4⋊C4—42no—
138C4⋊C4—42no—
139C4⋊C4—42no—
140C4⋊C4—42no—
141C4⋊C4—44no—
142C22⋊C4—42no—
143C22⋊C4—44no—
144C22⋊C4—42no—
145C42—22no—
146C42—22no—
147C42—21no—
148C42—21no—
149C42C2×D412yes—
150C42#4112yes—
151C23—21no—
152C23—21no—
153C23—21no—
154C23—21no—
155C23—21no—
156C23—21no—
157C23#13712yes—
158C23C22×D411yescharacteristic
159Q8—84no—
160C2×C4—21no—
161C2×C4—21no—
162C2×C4—21no—
163C2×C4—21no—
164C2×C4—21no—
165C2×C4—21no—
166C2×C4—21no—
167C2×C4—21no—
168C2×C4—21no—
169C2×C4—21no—
170C2×C4—21no—
171C2×C4—21no—
172C2×C4—44no—
173C2×C4—44no—
174C2×C4—42no—
175C2×C4—42no—
176C2×C4—41no—
177C2×C4—41no—
178C2×C4—42no—
179C2×C4—42no—
180C2×C4—41no—
181C2×C4—41no—
182C2×C4—42no—
183C2×C4—42no—
184C2×C4—41no—
185C2×C4—41no—
186C2×C4—41no—
187C2×C4—41no—
188C2×C4—41no—
189C2×C4—41no—
190C2×C4—42no—
191C2×C4—41no—
192C2×C4—41no—
193C2×C4—42no—
194C2×C4#12112yes—
195C2×C4#12112yes—
196C2×C4AGL(1,ℤ/8)12yes—
197C2×C4C22×D411yescharacteristic
198C2×C4C2×#4111yescharacteristic
199C2×C42−1+412yes—
200C2×C42−1+412yes—
201C8—24no—
202C8—42no—
203C22—22no—
204C22—22no—
205C22—22no—
206C22—41no—
207C22—41no—
208C22—41no—
209C22—41no—
210C22C2×#12111yescharacteristic
211C22#53611yescharacteristic
212C22#53611yescharacteristic
213C22C2×AGL(1,ℤ/8)11yescharacteristic
214C22C2×#13711yescharacteristic
215C22#57511yescharacteristic
216C22#57511yescharacteristic
217C4—22no—
218C4—22no—
219C4—22no—
220C4—41no—
221C4—41no—
222C4—82no—
223C4—81no—
224C4—81no—
225C4—81no—
226C4—82no—
227C4—81no—
228C4#53612yes—
229C4#54112yes—
230C2—42no—
231C2#262111yescharacteristic
232C2#305511yescharacteristic
233C2#305611yescharacteristic
234C2#323511yescharacteristic
235C2#323611yescharacteristic
236C2#323911yescharacteristic
237C2#324011yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC24C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#501C2semidirect
5. #2088C2semidirect
6. #2086C2semidirect
9. #1993C2semidirect
10. #1992C2semidirect
11. #1836C2semidirect

Maximal subgroups

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

subgroupindex
C2×#5012
C2×#4322
#28572
#28512
#20882
#20862
#20772
#20762
#19932
#19922
#18362