Finite Group Databasebeta

← #33217

Subgroups — #33217

Order 256.

441 conjugacy classes of subgroups; only the 163 normal subgroups are shown.

normal subgroups only

This group has 441 conjugacy classes of subgroups — too many to draw. Only its 163 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
2C22×#97C211yesmaximal, characteristic
3#2857C211yesmaximal, characteristic
4#2851C211yesmaximal, characteristic
5#1993C211yesmaximal, characteristic, complemented
6#1992C211yesmaximal, characteristic, complemented
7#1944C212yesmaximal, complemented
8#1943C212yesmaximal, complemented
9#1847C211yesmaximal, characteristic, complemented
10#1808C212yesmaximal
11#1807C212yesmaximal
12C23×C8C2212yes—
13C2×C4⋊Q8C2211yescharacteristic
14C2×#125C2211yescharacteristic
15C2×C22⋊Q8—21no—
16C22×C42C2211yescharacteristic
17#501—24no—
18C2×#107C2212yes—
19C2×#107C2212yes—
20C2×#106C2212yes—
21C2×#106C2212yes—
22C2×#97—24no—
23C2×#97C2212yes—
24C2×#97C2211yescharacteristic
25C2×#97C2211yescharacteristic
26#391—21no—
27#389—21no—
28#389C2211yescharacteristic
29#389C2211yescharacteristic
30#389C2212yes—
31#387—21no—
32#387—21no—
33#385—21no—
34#385—21no—
35#384C2211yescharacteristic
36#382C2211yescharacteristic
37#380—21no—
38#380C2211yescharacteristic
39#380C2212yes—
40#380C2211yescharacteristic
41#340C2212yes—
42#340C2212yes—
43#340C2212yes—
44#340C2212yes—
45C22×Q8—21no—
46C23×C4D411yescharacteristic
47C23×C4D411yescharacteristic
48C23×C4C2311yescharacteristic
49C22×C8—24no—
50C22×C8—24no—
51C22×C8Q818yescomplemented
52C22×C8C2312yes—
53C22×C8C2312yes—
54C4⋊Q8—22no—
55C4⋊Q8—22no—
56#125—22no—
57#125—22no—
58C22⋊Q8—44no—
59C2×C4⋊C4—21no—
60C2×C4⋊C4—22no—
61C2×C4⋊C4—21no—
62C2×C4⋊C4—21no—
63C2×C4⋊C4—21no—
64C2×C4⋊C4—21no—
65C2×C4⋊C4—21no—
66C2×C4⋊C4—21no—
67C2×C4⋊C4C2311yescharacteristic
68C2×C4⋊C4C2311yescharacteristic
69C2×C4⋊C4C2312yes—
70C2×C4⋊C4C2311yescharacteristic
71C2×C4⋊C4C2311yescharacteristic
72C2×C4⋊C4C2312yes—
73C2×C22⋊C4—22no—
74C2×C22⋊C4—21no—
75C2×C22⋊C4—21no—
76C2×C42—22no—
77C2×C42—22no—
78C2×C42D411yescharacteristic
79C2×C42D411yescharacteristic
80C2×C42C2311yescharacteristic
81C2×C42C2311yescharacteristic
82#107—24no—
83#107—24no—
84#106—24no—
85#106—24no—
86#97—24no—
87#97—22no—
88#97D414yes—
89#95—22no—
90#95—21no—
91#95—21no—
92#95—22no—
93#95—21no—
94#95—21no—
95#95—21no—
96#95—21no—
97C24C2×D411yescharacteristic
98C2×Q8—42no—
99C2×Q8—42no—
100C2×Q8—42no—
101C22×C4—21no—
102C22×C4—21no—
103C22×C4—22no—
104C22×C4—22no—
105C22×C4—22no—
106C22×C4—22no—
107C22×C4—22no—
108C22×C4—21no—
109C22×C4—21no—
110C22×C4—21no—
111C22×C4—21no—
112C22×C4—22no—
113C22×C4—22no—
114C22×C4—41no—
115C22×C4—41no—
116C22×C4C2×D411yescharacteristic
117C22×C4C2×D411yescharacteristic
118C22×C4C2×D411yescharacteristic
119C22×C4C2×D411yescharacteristic
120C22×C4C2×D411yescharacteristic
121C22×C4C2×Q814yes—
122C22×C4C2411yescharacteristic
123C2×C8—22no—
124C2×C8—22no—
125C2×C8—216no—
126C2×C8C2×D414yes—
127C2×C8C2×Q814yes—
128C2×C8C2×Q814yes—
129C2×C8#4114yes—
130C4⋊C4—21no—
131C4⋊C4—22no—
132C4⋊C4—21no—
133C4⋊C4—22no—
134C4⋊C4—22no—
135C4⋊C4—22no—
136C4⋊C4—22no—
137C4⋊C4—21no—
138C4⋊C4—21no—
139C4⋊C4—22no—
140C4⋊C4—42no—
141C4⋊C4—42no—
142C4⋊C4—42no—
143C4⋊C4—42no—
144C4⋊C4—44no—
145C4⋊C4—42no—
146C4⋊C4—42no—
147C4⋊C4—42no—
148C22⋊C4—42no—
149C22⋊C4—44no—
150C22⋊C4—42no—
151C42—22no—
152C42—22no—
153C42—21no—
154C42—21no—
155C42C2×D412yes—
156C42#4112yes—
157C23—21no—
158C23—21no—
159C23—21no—
160C23—21no—
161C23C4⋊Q812yes—
162C23AGL(1,ℤ/8)12yes—
163C23#13712yes—
164C23C22×D411yescharacteristic
165Q8—84no—
166C2×C4—21no—
167C2×C4—21no—
168C2×C4—21no—
169C2×C4—21no—
170C2×C4—21no—
171C2×C4—21no—
172C2×C4—24no—
173C2×C4—24no—
174C2×C4—21no—
175C2×C4—21no—
176C2×C4—21no—
177C2×C4—21no—
178C2×C4—21no—
179C2×C4—21no—
180C2×C4—44no—
181C2×C4—44no—
182C2×C4—41no—
183C2×C4—41no—
184C2×C4—42no—
185C2×C4—42no—
186C2×C4—41no—
187C2×C4—41no—
188C2×C4—42no—
189C2×C4—42no—
190C2×C4—41no—
191C2×C4—41no—
192C2×C4—41no—
193C2×C4—41no—
194C2×C4—41no—
195C2×C4—42no—
196C2×C4—41no—
197C2×C4—41no—
198C2×C4—41no—
199C2×C4—42no—
200C2×C4#12112yes—
201C2×C4#12112yes—
202C2×C4C4⋊Q812yes—
203C2×C4C22×D411yescharacteristic
204C2×C4C22×Q812yes—
205C2×C4C2×#4112yes—
206C2×C4C2×#4111yescharacteristic
207C8—24no—
208C8C22⋊Q818yescomplemented
209C22—22no—
210C22—22no—
211C22—22no—
212C22—22no—
213C22—22no—
214C22—22no—
215C22#50114yescomplemented
216C22C2×#12111yescharacteristic
217C22C2×C4⋊Q811yescharacteristic
218C22#53211yescharacteristic
219C22C2×AGL(1,ℤ/8)11yescharacteristic
220C22C2×#13711yescharacteristic
221C22#57511yescharacteristic
222C22#57511yescharacteristic
223C4—22no—
224C4—22no—
225C4—22no—
226C4—41no—
227C4—41no—
228C4—82no—
229C4—81no—
230C4—81no—
231C4—81no—
232C4—82no—
233C4—81no—
234C4C2×C22⋊Q812yes—
235C4#53312yes—
236C2—24no—
237C2#285711yescharacteristic
238C2#305511yescharacteristic
239C2#305611yescharacteristic
240C2C2×#50111yescharacteristic
241C2#316211yescharacteristic
242C2#318011yescharacteristic
243C2#318111yescharacteristic
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. #1993C2semidirect
6. #1992C2semidirect
7. #1944C2semidirect
8. #1943C2semidirect
9. #1847C2semidirect
51. C22×C8Q8semidirect
208. C8C22⋊Q8semidirect
215. C22#501semidirect

Maximal subgroups

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

subgroupindex
C2×#5012
C22×#972
#28572
#28512
#19932
#19922
#19442
#19432
#18472
#18082
#18072