Finite Group Databasebeta

← #24477

Subgroups — #24477

Order 256.

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

normal subgroups only

This group has 565 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×#483C211yesmaximal, characteristic, complemented
2C22×#102C211yesmaximal, characteristic
3#2435C211yesmaximal, characteristic
4#2344C211yesmaximal, characteristic, complemented
5#2065C212yesmaximal, complemented
6#2042C212yesmaximal, complemented
7#1983C212yesmaximal, complemented
8#1926C212yesmaximal, complemented
9#1892C211yesmaximal, characteristic, complemented
10#1850C211yesmaximal, characteristic, complemented
11#1786C211yesmaximal, characteristic
12C23×D4C2211yescharacteristic
13C23×C8C2211yescharacteristic
14C2×#124C2212yes—
15C2×C22⋊Q8C2211yescharacteristic
16C2×#117—21no—
17C22×C4⋊C4C2211yescharacteristic
18#483—24no—
19C2×#106C2211yescharacteristic, complemented
20C2×#106C2211yescharacteristic, complemented
21C2×C4⋊C8C2212yescomplemented
22C2×#103C2211yescharacteristic, complemented
23C2×#103C2211yescharacteristic, complemented
24C2×#103C2212yescomplemented
25C2×#102—24no—
26C2×#102C2212yes—
27C2×#102C2212yes—
28C2×#102C2212yes—
29C2×#102C2212yes—
30C2×C22⋊C8C2211yescharacteristic
31#394—22no—
32#390C2212yes—
33#389C2212yescomplemented
34#387—22no—
35#381—21no—
36#380C2211yescharacteristic
37#379—21no—
38#379—21no—
39C4×C4⋊C4C2212yescomplemented
40C4×C22⋊C4C2211yescharacteristic
41#340C2212yescomplemented
42#336C2212yescomplemented
43C25—21no—
44C22×Q8C2311yescharacteristic
45C22×D4—24no—
46C22×D4—22no—
47C22×D4—21no—
48C22×D4—22no—
49C22×D4—42no—
50C22×D4C2312yes—
51C23×C4—21no—
52C23×C4C2311yescharacteristic
53C22×C8—22no—
54C22×C8—42no—
55C22×C8C2312yes—
56C22×C8C2311yescharacteristic
57C22×C8C2311yescharacteristic
58#124—24no—
59#124—42no—
60C22⋊Q8—22no—
61C22⋊Q8—22no—
62#117—44no—
63C2×C4⋊C4—21no—
64C2×C4⋊C4—22no—
65C2×C4⋊C4—22no—
66C2×C4⋊C4—21no—
67C2×C4⋊C4D414yes—
68C2×C4⋊C4C2311yescharacteristic
69C2×C4⋊C4C2312yes—
70C2×C4⋊C4C2311yescharacteristic
71C2×C4⋊C4C2311yescharacteristic
72C2×C22⋊C4—21no—
73C2×C22⋊C4—21no—
74C2×C22⋊C4—22no—
75C2×C22⋊C4—22no—
76C2×C22⋊C4—22no—
77C2×C22⋊C4—22no—
78C2×C22⋊C4D411yescharacteristic
79C2×C22⋊C4D411yescharacteristic
80C2×C42—22no—
81C2×C42C2312yes—
82#106—22no—
83#106—22no—
84C4⋊C8—42no—
85#103—22no—
86#103—22no—
87#103—42no—
88#102—24no—
89#102—24no—
90#102—24no—
91#102—42no—
92C22⋊C8—22no—
93#95—22no—
94#95—21no—
95#95—22no—
96#95D412yes—
97C24—22no—
98C24—44no—
99C24—42no—
100C24—42no—
101C24—42no—
102C24—82no—
103C24C2×D411yescharacteristic
104C2×Q8—21no—
105C2×Q8—21no—
106C2×Q8—22no—
107C2×Q8—22no—
108C2×D4—28no—
109C2×D4—22no—
110C2×D4—22no—
111C2×D4—22no—
112C2×D4—22no—
113C2×D4—22no—
114C2×D4—48no—
115C2×D4—44no—
116C2×D4—44no—
117C2×D4—42no—
118C2×D4#4114yes—
119C22×C4—22no—
120C22×C4—22no—
121C22×C4—24no—
122C22×C4—21no—
123C22×C4—21no—
124C22×C4—22no—
125C22×C4—21no—
126C22×C4—21no—
127C22×C4—42no—
128C22×C4—42no—
129C22×C4C2×D412yes—
130C22×C4C2×D412yes—
131C22×C4C2×D411yescharacteristic
132C22×C4C2×D412yes—
133C22×C4#4112yes—
134C22×C4C2411yescharacteristic
135C2×C8—21no—
136C2×C8—21no—
137C2×C8—22no—
138C2×C8—22no—
139C2×C8—22no—
140C2×C8—22no—
141C2×C8—22no—
142C2×C8—44no—
143C4⋊C4—24no—
144C4⋊C4—21no—
145C4⋊C4—21no—
146C4⋊C4—21no—
147C4⋊C4—22no—
148C4⋊C4—21no—
149C4⋊C4—22no—
150C4⋊C4—44no—
151C4⋊C4—42no—
152C4⋊C4C2×D412yes—
153C4⋊C4C2×D412yes—
154C22⋊C4—22no—
155C22⋊C4—22no—
156C22⋊C4—44no—
157C22⋊C4—44no—
158C22⋊C4—42no—
159C22⋊C4—42no—
160C22⋊C4—82no—
161C22⋊C4—82no—
162C42—22no—
163C42—22no—
164C42—44no—
165C23—22no—
166C23—22no—
167C23—24no—
168C23—48no—
169C23—42no—
170C23—42no—
171C23—42no—
172C23—42no—
173C23—42no—
174C23—42no—
175C23—88no—
176C23—88no—
177C23—84no—
178C23—84no—
179C23#12112yes—
180C23#13512yes—
181C23AGL(1,ℤ/8)12yes—
182C23C22×D411yescharacteristic
183Q8—41no—
184Q8—41no—
185Q8—42no—
186D4—24no—
187D4—24no—
188D4—48no—
189D4—42no—
190D4—42no—
191C2×C4—22no—
192C2×C4—22no—
193C2×C4—22no—
194C2×C4—24no—
195C2×C4—22no—
196C2×C4—22no—
197C2×C4—22no—
198C2×C4—22no—
199C2×C4—22no—
200C2×C4—22no—
201C2×C4—22no—
202C2×C4—22no—
203C2×C4—41no—
204C2×C4—41no—
205C2×C4—41no—
206C2×C4—41no—
207C2×C4—48no—
208C2×C4—41no—
209C2×C4—41no—
210C2×C4—42no—
211C2×C4—42no—
212C2×C4—42no—
213C2×C4—42no—
214C2×C4—42no—
215C2×C4#12112yes—
216C2×C4C22×D411yescharacteristic
217C2×C4C2×#4112yes—
218C2×C4C2×#4111yescharacteristic
219C2×C4H(5,𝔽2)12yes—
220C8—41no—
221C8—41no—
222C8—42no—
223C22—22no—
224C22—24no—
225C22—22no—
226C22—24no—
227C22—44no—
228C22—44no—
229C22—44no—
230C22—816no—
231C22—82no—
232C22—82no—
233C22—82no—
234C22—82no—
235C22#46314yes—
236C22C2×#12111yescharacteristic
237C22#54612yes—
238C22C2×#13511yescharacteristic
239C22C2×AGL(1,ℤ/8)11yescharacteristic
240C22#57712yes—
241C4—22no—
242C4—42no—
243C4—42no—
244C4—42no—
245C4—42no—
246C4—81no—
247C4—81no—
248C4—82no—
249C4#52511yescharacteristic
250C4#53511yescharacteristic
251C4#55112yes—
252C2—24no—
253C2—84no—
254C2#241111yescharacteristic
255C2C2×#46311yescharacteristic
256C2#304311yescharacteristic
257C2#329612yes—
258C2#330612yes—
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC23×D4C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C2×#483C2semidirect
4. #2344C2semidirect
5. #2065C2semidirect
6. #2042C2semidirect
7. #1983C2semidirect
8. #1926C2semidirect
9. #1892C2semidirect
10. #1850C2semidirect
19. C2×#106C22semidirect
20. C2×#106C22semidirect
21. C2×C4⋊C8C22semidirect
22. C2×#103C22semidirect
23. C2×#103C22semidirect
24. C2×#103C22semidirect
33. #389C22semidirect
39. C4×C4⋊C4C22semidirect
41. #340C22semidirect
42. #336C22semidirect

Maximal subgroups

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

subgroupindex
C2×#4832
C22×#1022
#24352
#23442
#20652
#20422
#19832
#19262
#18922
#18502
#17862