Finite Group Databasebeta

← #25821

Subgroups — #25821

Order 256.

495 conjugacy classes of subgroups; only the 151 normal subgroups are shown.

normal subgroups only

This group has 495 conjugacy classes of subgroups — too many to draw. Only its 151 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×#501C211yesmaximal, characteristic, complemented
3C2×#500C211yesmaximal, characteristic, complemented
4#2628C211yesmaximal, characteristic
5#2059C212yesmaximal, complemented
6#2057C212yesmaximal, complemented
7#1993C211yesmaximal, characteristic, complemented
8#1991C211yesmaximal, characteristic, complemented
9#1854C212yesmaximal
10#1852C212yesmaximal
11#1750C211yesmaximal, characteristic
12#544—24no—
13#544—24no—
14C2×C4⋊Q8—21no—
15C2×C4⋊Q8C2211yescharacteristic
16C2×C4⋊Q8C2211yescharacteristic
17C2×C4⋊Q8C2211yescharacteristic
18C2×#125—21no—
19C2×#125C2211yescharacteristic
20C2×#125C2211yescharacteristic
21C2×C22⋊Q8—21no—
22C2×C22⋊Q8—21no—
23C2×#117C2211yescharacteristic
24C2×#117C2211yescharacteristic
25C22×C4⋊C4C2211yescharacteristic
26#501—24no—
27#500—24no—
28C2×#107C2212yes—
29C2×#107C2212yes—
30C2×#107C2212yes—
31C2×#106C2212yes—
32C2×C22⋊C8C2212yes—
33C2×#97C2211yescharacteristic
34C2×C4×C8C2211yescharacteristic
35#400—21no—
36#399—21no—
37#395—21no—
38#393—21no—
39#389C2212yes—
40#382C2211yescharacteristic
41#380C2212yes—
42#340C2212yes—
43#340C2212yes—
44#340C2212yes—
45#340C2212yes—
46#336C2212yes—
47C22×Q8—21no—
48C22×Q8—21no—
49C22×Q8—21no—
50C23×C4—21no—
51C23×C4C2311yescharacteristic
52C22×C8C2312yes—
53C22×C8C2312yes—
54C4⋊Q8—28no—
55C4⋊Q8—22no—
56C4⋊Q8—22no—
57C4⋊Q8—22no—
58C4⋊Q8—22no—
59C4⋊Q8—22no—
60C4⋊Q8D414yes—
61#125—28no—
62#125—22no—
63#125—22no—
64#125—22no—
65#125D414yes—
66C22⋊Q8—28no—
67C22⋊Q8—28no—
68#117—22no—
69#117—22no—
70#117—22no—
71#117—22no—
72C2×C4⋊C4—21no—
73C2×C4⋊C4—21no—
74C2×C4⋊C4—21no—
75C2×C4⋊C4—21no—
76C2×C4⋊C4—21no—
77C2×C4⋊C4—21no—
78C2×C4⋊C4—21no—
79C2×C4⋊C4—22no—
80C2×C4⋊C4—22no—
81C2×C4⋊C4—21no—
82C2×C4⋊C4—22no—
83C2×C4⋊C4D411yescharacteristic
84C2×C4⋊C4D411yescharacteristic
85C2×C4⋊C4C2312yes—
86C2×C4⋊C4C2312yes—
87C2×C4⋊C4C2311yescharacteristic
88C2×C4⋊C4C2311yescharacteristic
89C2×C4⋊C4C2311yescharacteristic
90C2×C4⋊C4C2311yescharacteristic
91C2×C22⋊C4—21no—
92C2×C22⋊C4—22no—
93C2×C22⋊C4D411yescharacteristic
94C2×C22⋊C4D411yescharacteristic
95C2×C42—21no—
96C2×C42C2311yescharacteristic
97C2×C42C2311yescharacteristic
98#107—24no—
99#107—24no—
100#107—24no—
101#106—24no—
102C22⋊C8Q818yescomplemented
103#97—22no—
104C4×C8—22no—
105#95—22no—
106#95—21no—
107#95—21no—
108#95—22no—
109C24C2×D411yescharacteristic
110C2×Q8—24no—
111C2×Q8—24no—
112C2×Q8—42no—
113C2×Q8—82no—
114C2×Q8—82no—
115C2×Q8—82no—
116C22×C4—21no—
117C22×C4—21no—
118C22×C4—22no—
119C22×C4—21no—
120C22×C4—21no—
121C22×C4—21no—
122C22×C4—21no—
123C22×C4—24no—
124C22×C4—22no—
125C22×C4—22no—
126C22×C4—82no—
127C22×C4C2×D411yescharacteristic
128C22×C4C2×D411yescharacteristic
129C22×C4C2×D411yescharacteristic
130C22×C4C2×D411yescharacteristic
131C22×C4C2×D411yescharacteristic
132C22×C4C2×Q814yes—
133C22×C4C2411yescharacteristic
134C2×C8—22no—
135C2×C8—22no—
136C2×C8—22no—
137C2×C8—22no—
138C2×C8C2×Q814yes—
139C2×C8C2×Q814yes—
140C4⋊C4—24no—
141C4⋊C4—24no—
142C4⋊C4—24no—
143C4⋊C4—24no—
144C4⋊C4—24no—
145C4⋊C4—24no—
146C4⋊C4—24no—
147C4⋊C4—22no—
148C4⋊C4—22no—
149C4⋊C4—21no—
150C4⋊C4—21no—
151C4⋊C4—21no—
152C4⋊C4—21no—
153C4⋊C4—22no—
154C4⋊C4—22no—
155C4⋊C4—24no—
156C4⋊C4—42no—
157C4⋊C4—82no—
158C4⋊C4C2×D412yes—
159C4⋊C4C2×D412yes—
160C4⋊C4C2×D412yes—
161C4⋊C4C2×D412yes—
162C22⋊C4—24no—
163C22⋊C4—22no—
164C22⋊C4—22no—
165C22⋊C4—82no—
166C42—24no—
167C42—21no—
168C42—21no—
169C42C2×D412yes—
170C42C2×D412yes—
171C23—21no—
172C23—21no—
173C23—22no—
174C23—22no—
175C23C4⋊Q812yes—
176C23C22×D411yescharacteristic
177Q8—84no—
178Q8—84no—
179Q8—84no—
180C2×C4—21no—
181C2×C4—21no—
182C2×C4—22no—
183C2×C4—22no—
184C2×C4—24no—
185C2×C4—24no—
186C2×C4—21no—
187C2×C4—22no—
188C2×C4—21no—
189C2×C4—22no—
190C2×C4—22no—
191C2×C4—22no—
192C2×C4—22no—
193C2×C4—22no—
194C2×C4—42no—
195C2×C4—42no—
196C2×C4—42no—
197C2×C4—42no—
198C2×C4—42no—
199C2×C4—42no—
200C2×C4—42no—
201C2×C4—42no—
202C2×C4—42no—
203C2×C4—42no—
204C2×C4—84no—
205C2×C4#12012yes—
206C2×C4#12012yes—
207C2×C4C4⋊Q812yes—
208C2×C4C22×D411yescharacteristic
209C2×C4C22×D411yescharacteristic
210C2×C4C22×Q812yes—
211C2×C42−1+412yes—
212C8—24no—
213C8—24no—
214C22—24no—
215C22—22no—
216C22—42no—
217C22—41no—
218C22—41no—
219C22C2×#12011yescharacteristic
220C22C2×C4⋊Q811yescharacteristic
221C22#53611yescharacteristic
222C22#57612yes—
223C22#57812yes—
224C4—22no—
225C4—22no—
226C4—22no—
227C4—41no—
228C4—42no—
229C4—42no—
230C4—42no—
231C4—41no—
232C4—42no—
233C4—82no—
234C4—82no—
235C4D4×Q812yes—
236C4D4×Q812yes—
237C2—42no—
238C2#243411yescharacteristic
239C2#300811yescharacteristic
240C2#301111yescharacteristic
241C2#316412yes—
242C2#324312yes—
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
2. C2×#501C2semidirect
3. C2×#500C2semidirect
5. #2059C2semidirect
6. #2057C2semidirect
7. #1993C2semidirect
8. #1991C2semidirect
102. C22⋊C8Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#5442
C2×#5012
C2×#5002
#26282
#20592
#20572
#19932
#19912
#18542
#18522
#17502