Finite Group Databasebeta

← #24110

Subgroups — #24110

Order 256.

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

normal subgroups only

This group has 544 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×#484C211yesmaximal, characteristic, complemented
3C2×#428C211yesmaximal, characteristic, complemented
4#2486C211yesmaximal, characteristic
5#2059C212yesmaximal, complemented
6#2035C212yesmaximal, complemented
7#1894C211yesmaximal, characteristic, complemented
8#1870C211yesmaximal, characteristic
9#1863C212yesmaximal, complemented
10#1852C212yesmaximal, 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×#123—21no—
20C2×C22⋊Q8C2211yescharacteristic
21C2×C22⋊Q8C2211yescharacteristic
22C2×C4×Q8C2211yescharacteristic
23C2×C4×D4C2211yescharacteristic
24C2×#117—21no—
25C2×#117C2211yescharacteristic
26C22×C4⋊C4C2211yescharacteristic
27C22×C4⋊C4C2211yescharacteristic
28#484—24no—
29#428—24no—
30C2×#107C2212yes—
31C2×#107C2212yes—
32C2×#106C2212yes—
33C2×#103C2212yes—
34C2×#103C2212yes—
35C2×C22⋊C8C2211yescharacteristic
36C2×C22⋊C8C2212yes—
37#398—21no—
38#397—21no—
39#389C2212yes—
40#381—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—
62#123—44no—
63C22⋊Q8—22no—
64C22⋊Q8—22no—
65C22⋊Q8—22no—
66C22⋊Q8D414yes—
67C4×Q8—22no—
68C4×Q8—22no—
69C4×D4—22no—
70C4×D4—22no—
71#117—28no—
72#117—22no—
73#117D414yes—
74C2×C4⋊C4—21no—
75C2×C4⋊C4—21no—
76C2×C4⋊C4—21no—
77C2×C4⋊C4—21no—
78C2×C4⋊C4—21no—
79C2×C4⋊C4—21no—
80C2×C4⋊C4—22no—
81C2×C4⋊C4—22no—
82C2×C4⋊C4—22no—
83C2×C4⋊C4—22no—
84C2×C4⋊C4—21no—
85C2×C4⋊C4—22no—
86C2×C4⋊C4D411yescharacteristic
87C2×C4⋊C4C2312yes—
88C2×C4⋊C4C2312yes—
89C2×C4⋊C4C2311yescharacteristic
90C2×C4⋊C4C2311yescharacteristic
91C2×C4⋊C4C2311yescharacteristic
92C2×C22⋊C4—21no—
93C2×C22⋊C4—21no—
94C2×C22⋊C4—21no—
95C2×C22⋊C4—21no—
96C2×C22⋊C4—22no—
97C2×C22⋊C4D411yescharacteristic
98C2×C22⋊C4D411yescharacteristic
99C2×C42—21no—
100C2×C42—21no—
101C2×C42C2311yescharacteristic
102#107—24no—
103#107—24no—
104#106—24no—
105#103—24no—
106#103—24no—
107C22⋊C8—22no—
108C22⋊C8—24no—
109#95—22no—
110#95—21no—
111#95—22no—
112C24C2×D411yescharacteristic
113C24C2×D411yescharacteristic
114C2×Q8—21no—
115C2×Q8—22no—
116C2×Q8—22no—
117C2×Q8C2×D412yes—
118C2×D4—22no—
119C2×D4—22no—
120C2×D4—41no—
121C22×C4—22no—
122C22×C4—22no—
123C22×C4—24no—
124C22×C4—21no—
125C22×C4—21no—
126C22×C4—21no—
127C22×C4—21no—
128C22×C4—21no—
129C22×C4—21no—
130C22×C4—22no—
131C22×C4—22no—
132C22×C4—22no—
133C22×C4—22no—
134C22×C4—42no—
135C22×C4—42no—
136C22×C4—42no—
137C22×C4—42no—
138C22×C4—42no—
139C22×C4C2×D411yescharacteristic
140C22×C4C2×D411yescharacteristic
141C22×C4C2×D411yescharacteristic
142C22×C4C2×D411yescharacteristic
143C22×C4C2×D412yes—
144C22×C4C2×D412yes—
145C22×C4C2411yescharacteristic
146M16—28no—
147C2×C8—22no—
148C2×C8—22no—
149C2×C8—22no—
150C2×C8—22no—
151C2×C8—22no—
152C2×C8#4114yes—
153C4⋊C4—24no—
154C4⋊C4—24no—
155C4⋊C4—24no—
156C4⋊C4—24no—
157C4⋊C4—24no—
158C4⋊C4—24no—
159C4⋊C4—22no—
160C4⋊C4—21no—
161C4⋊C4—21no—
162C4⋊C4—21no—
163C4⋊C4—22no—
164C4⋊C4—22no—
165C4⋊C4—22no—
166C4⋊C4—22no—
167C4⋊C4—44no—
168C4⋊C4—42no—
169C4⋊C4C2×D412yes—
170C4⋊C4C2×D412yes—
171C4⋊C4C2×D412yes—
172C22⋊C4—24no—
173C22⋊C4—24no—
174C22⋊C4—24no—
175C22⋊C4—22no—
176C22⋊C4—22no—
177C22⋊C4—42no—
178C22⋊C4—44no—
179C42—24no—
180C42—24no—
181C42—21no—
182C42—21no—
183C23—21no—
184C23—22no—
185C23—22no—
186C23—21no—
187C23—21no—
188C23—21no—
189C23—22no—
190C23—22no—
191C23#12012yes—
192C23AGL(1,ℤ/8)12yes—
193C23C22×D411yescharacteristic
194Q8—24no—
195Q8—41no—
196Q8—41no—
197D4—44no—
198C2×C4—22no—
199C2×C4—22no—
200C2×C4—22no—
201C2×C4—24no—
202C2×C4—21no—
203C2×C4—22no—
204C2×C4—22no—
205C2×C4—22no—
206C2×C4—21no—
207C2×C4—21no—
208C2×C4—22no—
209C2×C4—22no—
210C2×C4—21no—
211C2×C4—21no—
212C2×C4—22no—
213C2×C4—48no—
214C2×C4—42no—
215C2×C4—41no—
216C2×C4—41no—
217C2×C4—41no—
218C2×C4—41no—
219C2×C4—42no—
220C2×C4—41no—
221C2×C4—41no—
222C2×C4—42no—
223C2×C4—42no—
224C2×C4—42no—
225C2×C4—42no—
226C2×C4—42no—
227C2×C4—42no—
228C2×C4—42no—
229C2×C4—84no—
230C2×C4#12012yes—
231C2×C4C22×D411yescharacteristic
232C2×C4C22×D411yescharacteristic
233C2×C4C2×#4112yes—
234C2×C42−1+412yes—
235C8—24no—
236C8—42no—
237C22—24no—
238C22—22no—
239C22—22no—
240C22—22no—
241C22—22no—
242C22—42no—
243C22—42no—
244C22—42no—
245C22—41no—
246C22—41no—
247C22C2×#12011yescharacteristic
248C22#54712yes—
249C22C2×AGL(1,ℤ/8)11yescharacteristic
250C22#57511yescharacteristic
251C22#57812yes—
252C4—22no—
253C4—41no—
254C4—42no—
255C4—41no—
256C4—42no—
257C4—42no—
258C4—41no—
259C4—42no—
260C4—82no—
261C4—82no—
262C4#52511yescharacteristic
263C4#53611yescharacteristic
264C4#54712yes—
265C2—42no—
266C2—42no—
267C2#240611yescharacteristic
268C2#300611yescharacteristic
269C2#301111yescharacteristic
270C2#334312yes—
271C2#335212yes—
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×#484C2semidirect
3. C2×#428C2semidirect
5. #2059C2semidirect
6. #2035C2semidirect
7. #1894C2semidirect
9. #1863C2semidirect
10. #1852C2semidirect
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×#4842
C2×#4282
#24862
#20592
#20352
#18942
#18702
#18632
#18522
#18252