Finite Group Databasebeta

← #24074

Subgroups — #24074

Order 256.

533 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 533 conjugacy classes of subgroups — too many to draw. Only its 131 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×#542C211yesmaximal, characteristic
2C2×#481C211yesmaximal, characteristic
3C2×#483C211yesmaximal, characteristic, complemented
4#2317C211yesmaximal, characteristic, complemented
5#2058C212yesmaximal, complemented
6#2042C212yesmaximal, complemented
7#1875C211yesmaximal, characteristic
8#1873C211yesmaximal, characteristic, complemented
9#1865C212yesmaximal, complemented
10#1864C212yesmaximal, complemented
11#1818C211yesmaximal, characteristic, complemented
12#542—24no—
13#542—24no—
14C2×C4⋊Q8C2211yescharacteristic
15C2×#126—21no—
16C2×#124—21no—
17C2×#124C2211yescharacteristic
18C2×#123—21no—
19C2×C22⋊Q8—21no—
20C2×C22⋊Q8C2211yescharacteristic
21C2×#121C2211yescharacteristic
22C2×C4×Q8C2211yescharacteristic
23C2×C4×D4C2211yescharacteristic
24C2×#117C2211yescharacteristic
25C2×#117C2211yescharacteristic
26C22×C4⋊C4C2211yescharacteristic
27#483—24no—
28#481—24no—
29C2×#106C2212yes—
30C2×#106C2212yes—
31C2×#103C2212yes—
32C2×#103C2212yes—
33C2×#102C2212yes—
34C2×#102C2212yes—
35C2×C22⋊C8C2211yescharacteristic
36C2×C22⋊C8C2211yescharacteristic
37#389C2212yes—
38#389C2212yes—
39#385—21no—
40#385—21no—
41#384—21no—
42#382—21no—
43#340C2212yes—
44#340C2212yes—
45#336C2212yes—
46#336C2212yes—
47C22×Q8—21no—
48C22×Q8C2311yescharacteristic
49C22×D4—21no—
50C22×D4C2311yescharacteristic
51C23×C4—21no—
52C23×C4—21no—
53C23×C4C2311yescharacteristic
54C22×C8C2312yes—
55C22×C8C2312yes—
56C4⋊Q8—22no—
57C4⋊Q8—22no—
58#126—28no—
59#124—28no—
60#124—22no—
61#124—22no—
62#123—28no—
63C22⋊Q8—28no—
64C22⋊Q8—22no—
65C22⋊Q8D414yes—
66#121—22no—
67#121D414yes—
68C4×Q8—22no—
69C4×Q8—22no—
70C4×D4—22no—
71C4×D4—22no—
72#117—22no—
73#117—22no—
74#117—24no—
75C2×C4⋊C4—21no—
76C2×C4⋊C4—21no—
77C2×C4⋊C4—21no—
78C2×C4⋊C4—22no—
79C2×C4⋊C4—22no—
80C2×C4⋊C4—21no—
81C2×C4⋊C4—22no—
82C2×C4⋊C4—22no—
83C2×C4⋊C4D411yescharacteristic
84C2×C4⋊C4D411yescharacteristic
85C2×C4⋊C4C2312yes—
86C2×C4⋊C4C2311yescharacteristic
87C2×C4⋊C4C2311yescharacteristic
88C2×C4⋊C4C2312yes—
89C2×C22⋊C4—21no—
90C2×C22⋊C4—21no—
91C2×C22⋊C4—21no—
92C2×C22⋊C4—21no—
93C2×C22⋊C4—21no—
94C2×C22⋊C4—21no—
95C2×C22⋊C4D411yescharacteristic
96C2×C22⋊C4D411yescharacteristic
97C2×C42—21no—
98C2×C42—21no—
99C2×C42C2311yescharacteristic
100C2×C42C2311yescharacteristic
101#106—24no—
102#106—24no—
103#103—24no—
104#103—24no—
105#102—24no—
106#102—24no—
107C22⋊C8—22no—
108C22⋊C8—22no—
109#95—22no—
110#95—22no—
111C24—21no—
112C24C2×D411yescharacteristic
113C2×Q8—24no—
114C2×Q8—21no—
115C2×Q8—22no—
116C2×Q8—22no—
117C2×Q8—82no—
118C2×Q8C2×D412yes—
119C2×D4—24no—
120C2×D4—21no—
121C2×D4—22no—
122C2×D4—22no—
123C2×D4—82no—
124C2×D4C2×D412yes—
125C22×C4—22no—
126C22×C4—22no—
127C22×C4—22no—
128C22×C4—21no—
129C22×C4—21no—
130C22×C4—21no—
131C22×C4—24no—
132C22×C4—22no—
133C22×C4—42no—
134C22×C4—82no—
135C22×C4—82no—
136C22×C4C2×D411yescharacteristic
137C22×C4C2×D411yescharacteristic
138C22×C4C2×D411yescharacteristic
139C22×C4C2×D411yescharacteristic
140C22×C4C2×D411yescharacteristic
141C22×C4C2×D412yes—
142C22×C4C2×D412yes—
143C22×C4C2411yescharacteristic
144C2×C8—22no—
145C2×C8—22no—
146C2×C8—22no—
147C2×C8—22no—
148C2×C8—22no—
149C2×C8—22no—
150C4⋊C4—24no—
151C4⋊C4—24no—
152C4⋊C4—24no—
153C4⋊C4—22no—
154C4⋊C4—22no—
155C4⋊C4—21no—
156C4⋊C4—21no—
157C4⋊C4—22no—
158C4⋊C4—22no—
159C4⋊C4—22no—
160C4⋊C4—42no—
161C4⋊C4—82no—
162C4⋊C4—82no—
163C4⋊C4C2×D412yes—
164C4⋊C4C2×D412yes—
165C4⋊C4#4114yes—
166C22⋊C4—24no—
167C22⋊C4—24no—
168C22⋊C4—24no—
169C22⋊C4—24no—
170C22⋊C4—24no—
171C22⋊C4—22no—
172C22⋊C4—22no—
173C22⋊C4—24no—
174C42—24no—
175C42—24no—
176C42—21no—
177C42—21no—
178C42—21no—
179C42—21no—
180C23—21no—
181C23—21no—
182C23—22no—
183C23—22no—
184C23—22no—
185C23—41no—
186C23—41no—
187C23—42no—
188C23—42no—
189C23#12012yes—
190C23C22×D411yescharacteristic
191Q8—24no—
192Q8—42no—
193Q8—84no—
194D4—24no—
195D4—42no—
196D4—84no—
197C2×C4—21no—
198C2×C4—21no—
199C2×C4—24no—
200C2×C4—24no—
201C2×C4—21no—
202C2×C4—21no—
203C2×C4—22no—
204C2×C4—21no—
205C2×C4—21no—
206C2×C4—22no—
207C2×C4—22no—
208C2×C4—24no—
209C2×C4—22no—
210C2×C4—22no—
211C2×C4—22no—
212C2×C4—22no—
213C2×C4—41no—
214C2×C4—41no—
215C2×C4—42no—
216C2×C4—42no—
217C2×C4—42no—
218C2×C4—42no—
219C2×C4—42no—
220C2×C4—42no—
221C2×C4—42no—
222C2×C4—84no—
223C2×C4—84no—
224C2×C4#12012yes—
225C2×C4C22×D411yescharacteristic
226C2×C4C22×D411yescharacteristic
227C2×C4C2×#4112yes—
228C2×C42−1+412yes—
229C8—42no—
230C8—42no—
231C22—22no—
232C22—22no—
233C22—22no—
234C22—42no—
235C22—42no—
236C22—42no—
237C22—42no—
238C22—42no—
239C22—82no—
240C22—82no—
241C22C2×#12011yescharacteristic
242C22#52511yescharacteristic
243C22#53611yescharacteristic
244C22#57712yes—
245C22#57712yes—
246C4—22no—
247C4—44no—
248C4—42no—
249C4—41no—
250C4—41no—
251C4—42no—
252C4—42no—
253C4—41no—
254C4—41no—
255C4—82no—
256C4#54712yes—
257C4#54712yes—
258C2—42no—
259C2—82no—
260C2#240611yescharacteristic
261C2#300911yescharacteristic
262C2#301011yescharacteristic
263C2#312512yes—
264C2#324412yes—
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC2×#121C22
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
3. C2×#483C2semidirect
4. #2317C2semidirect
5. #2058C2semidirect
6. #2042C2semidirect
8. #1873C2semidirect
9. #1865C2semidirect
10. #1864C2semidirect
11. #1818C2semidirect

Maximal subgroups

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

subgroupindex
C2×#5422
C2×#4812
C2×#4832
#23172
#20582
#20422
#18752
#18732
#18652
#18642
#18182