Finite Group Databasebeta

← #24111

Subgroups — #24111

Order 256.

534 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 534 conjugacy classes of subgroups — too many to draw. Only its 133 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×#552C211yesmaximal, characteristic
2C2×#483C211yesmaximal, characteristic, complemented
3C2×#428C211yesmaximal, characteristic, complemented
4#2487C211yesmaximal, characteristic
5#2059C212yesmaximal, complemented
6#2036C212yesmaximal, complemented
7#1894C211yesmaximal, characteristic, complemented
8#1870C211yesmaximal, characteristic
9#1865C212yesmaximal, complemented
10#1853C212yesmaximal, complemented
11#1825C211yesmaximal, characteristic, complemented
12C22×M16C2211yescharacteristic
13#552—24no—
14#552—24no—
15C2×#126—21no—
16C2×#125—21no—
17C2×#125C2211yescharacteristic
18C2×#123—21no—
19C2×C22⋊Q8C2211yescharacteristic
20C2×C22⋊Q8C2211yescharacteristic
21C2×C4×Q8C2211yescharacteristic
22C2×C4×D4C2211yescharacteristic
23C2×#117—21no—
24C2×#117C2211yescharacteristic
25C22×C4⋊C4C2211yescharacteristic
26C22×C4⋊C4C2211yescharacteristic
27#483—24no—
28#428—24no—
29C2×#107C2212yes—
30C2×#106C2212yes—
31C2×#106C2212yes—
32C2×#103C2212yes—
33C2×#103C2212yes—
34C2×C22⋊C8C2211yescharacteristic
35C2×C22⋊C8C2212yes—
36#400—21no—
37#399—21no—
38#389C2212yes—
39#386—21no—
40#385—21no—
41#380C2211yescharacteristic
42#380C2212yes—
43#340C2212yes—
44#340C2212yes—
45#340C2212yes—
46#336C2212yes—
47C22×Q8C2311yescharacteristic
48C22×D4D411yescharacteristic
49C23×C4—21no—
50C23×C4—21no—
51C23×C4C2311yescharacteristic
52C23×C4C2311yescharacteristic
53C2×M16—22no—
54C2×M16—24no—
55C22×C8C2312yes—
56C22×C8C2312yes—
57#126—28no—
58#125—28no—
59#125—22no—
60#125—22no—
61#123—28no—
62C22⋊Q8—22no—
63C22⋊Q8—22no—
64C22⋊Q8—22no—
65C22⋊Q8D414yes—
66C4×Q8—22no—
67C4×Q8—22no—
68C4×D4—22no—
69C4×D4—22no—
70#117—28no—
71#117—22no—
72#117D414yes—
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—22no—
82C2×C4⋊C4—22no—
83C2×C4⋊C4—21no—
84C2×C4⋊C4—22no—
85C2×C4⋊C4D411yescharacteristic
86C2×C4⋊C4C2312yes—
87C2×C4⋊C4C2312yes—
88C2×C4⋊C4C2311yescharacteristic
89C2×C4⋊C4C2311yescharacteristic
90C2×C4⋊C4C2311yescharacteristic
91C2×C22⋊C4—21no—
92C2×C22⋊C4—21no—
93C2×C22⋊C4—21no—
94C2×C22⋊C4—21no—
95C2×C22⋊C4—22no—
96C2×C22⋊C4D411yescharacteristic
97C2×C22⋊C4D411yescharacteristic
98C2×C42—21no—
99C2×C42—21no—
100C2×C42C2311yescharacteristic
101#107—24no—
102#106—24no—
103#106—24no—
104#103—24no—
105#103—24no—
106C22⋊C8—22no—
107C22⋊C8—24no—
108#95—22no—
109#95—21no—
110#95—22no—
111C24C2×D411yescharacteristic
112C24C2×D411yescharacteristic
113C2×Q8—21no—
114C2×Q8—22no—
115C2×Q8—22no—
116C2×Q8C2×D412yes—
117C2×D4—22no—
118C2×D4—22no—
119C2×D4—41no—
120C22×C4—22no—
121C22×C4—22no—
122C22×C4—24no—
123C22×C4—21no—
124C22×C4—21no—
125C22×C4—21no—
126C22×C4—21no—
127C22×C4—21no—
128C22×C4—21no—
129C22×C4—22no—
130C22×C4—22no—
131C22×C4—22no—
132C22×C4—22no—
133C22×C4—42no—
134C22×C4—42no—
135C22×C4—42no—
136C22×C4—82no—
137C22×C4C2×D411yescharacteristic
138C22×C4C2×D411yescharacteristic
139C22×C4C2×D411yescharacteristic
140C22×C4C2×D411yescharacteristic
141C22×C4C2×D412yes—
142C22×C4C2×D412yes—
143C22×C4C2411yescharacteristic
144M16—28no—
145C2×C8—22no—
146C2×C8—22no—
147C2×C8—22no—
148C2×C8—22no—
149C2×C8—22no—
150C2×C8#4114yes—
151C4⋊C4—24no—
152C4⋊C4—24no—
153C4⋊C4—24no—
154C4⋊C4—24no—
155C4⋊C4—24no—
156C4⋊C4—24no—
157C4⋊C4—22no—
158C4⋊C4—21no—
159C4⋊C4—21no—
160C4⋊C4—21no—
161C4⋊C4—22no—
162C4⋊C4—22no—
163C4⋊C4—22no—
164C4⋊C4—22no—
165C4⋊C4—42no—
166C4⋊C4—82no—
167C4⋊C4C2×D412yes—
168C4⋊C4C2×D412yes—
169C4⋊C4C2×D412yes—
170C22⋊C4—24no—
171C22⋊C4—24no—
172C22⋊C4—24no—
173C22⋊C4—22no—
174C22⋊C4—22no—
175C22⋊C4—42no—
176C22⋊C4—82no—
177C42—24no—
178C42—24no—
179C42—21no—
180C42—21no—
181C23—21no—
182C23—22no—
183C23—22no—
184C23—21no—
185C23—21no—
186C23—21no—
187C23—22no—
188C23—22no—
189C23#12012yes—
190C23#13712yes—
191C23C22×D411yescharacteristic
192Q8—24no—
193Q8—41no—
194Q8—41no—
195D4—44no—
196C2×C4—22no—
197C2×C4—22no—
198C2×C4—22no—
199C2×C4—24no—
200C2×C4—21no—
201C2×C4—22no—
202C2×C4—22no—
203C2×C4—22no—
204C2×C4—21no—
205C2×C4—21no—
206C2×C4—22no—
207C2×C4—22no—
208C2×C4—21no—
209C2×C4—21no—
210C2×C4—22no—
211C2×C4—48no—
212C2×C4—42no—
213C2×C4—41no—
214C2×C4—41no—
215C2×C4—41no—
216C2×C4—41no—
217C2×C4—42no—
218C2×C4—41no—
219C2×C4—41no—
220C2×C4—42no—
221C2×C4—42no—
222C2×C4—42no—
223C2×C4—42no—
224C2×C4—42no—
225C2×C4—42no—
226C2×C4—42no—
227C2×C4—84no—
228C2×C4#12012yes—
229C2×C4C22×D411yescharacteristic
230C2×C4C22×D411yescharacteristic
231C2×C4C2×#4112yes—
232C2×C42−1+412yes—
233C8—24no—
234C8—42no—
235C22—24no—
236C22—22no—
237C22—22no—
238C22—22no—
239C22—22no—
240C22—42no—
241C22—42no—
242C22—42no—
243C22—41no—
244C22—41no—
245C22C2×#12011yescharacteristic
246C22#54712yes—
247C22C2×#13711yescharacteristic
248C22#57511yescharacteristic
249C22#57712yes—
250C4—22no—
251C4—41no—
252C4—42no—
253C4—41no—
254C4—42no—
255C4—42no—
256C4—41no—
257C4—42no—
258C4—82no—
259C4—82no—
260C4#52511yescharacteristic
261C4#53611yescharacteristic
262C4#54712yes—
263C2—42no—
264C2—42no—
265C2#240611yescharacteristic
266C2#300711yescharacteristic
267C2#301011yescharacteristic
268C2#334412yes—
269C2#335112yes—
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×#483C2semidirect
3. C2×#428C2semidirect
5. #2059C2semidirect
6. #2036C2semidirect
7. #1894C2semidirect
9. #1865C2semidirect
10. #1853C2semidirect
11. #1825C2semidirect

Maximal subgroups

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

subgroupindex
C2×#5522
C2×#4832
C2×#4282
#24872
#20592
#20362
#18942
#18702
#18652
#18532
#18252