Finite Group Databasebeta

← #24479

Subgroups — #24479

Order 256.

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

normal subgroups only

This group has 551 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×#484C211yesmaximal, characteristic, complemented
2C22×#102C211yesmaximal, characteristic
3#2435C211yesmaximal, characteristic
4#2345C211yesmaximal, characteristic, complemented
5#2067C212yesmaximal, complemented
6#2042C212yesmaximal, complemented
7#1983C212yesmaximal, complemented
8#1925C212yesmaximal, complemented
9#1892C211yesmaximal, characteristic, complemented
10#1850C211yesmaximal, characteristic, complemented
11#1786C211yesmaximal, characteristic
12C23×D4C2211yescharacteristic
13C23×C8C2211yescharacteristic
14C2×#124C2212yes—
15C2×C22⋊Q8C2211yescharacteristic
16C22×C4⋊C4C2211yescharacteristic
17#484—24no—
18C2×#107C2211yescharacteristic, complemented
19C2×#107C2211yescharacteristic, complemented
20C2×C4⋊C8C2212yescomplemented
21C2×#103C2211yescharacteristic, complemented
22C2×#103C2211yescharacteristic, complemented
23C2×#103C2212yescomplemented
24C2×#102—24no—
25C2×#102C2212yes—
26C2×#102C2212yes—
27C2×#102C2212yes—
28C2×#102C2212yes—
29C2×C22⋊C8C2211yescharacteristic
30#394—22no—
31#390C2212yes—
32#389C2212yescomplemented
33#387—22no—
34#380—21no—
35#380C2211yescharacteristic
36#379—21no—
37#379—21no—
38C4×C4⋊C4C2212yescomplemented
39C4×C22⋊C4—21no—
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—
62C2×C4⋊C4—21no—
63C2×C4⋊C4—22no—
64C2×C4⋊C4—21no—
65C2×C4⋊C4—22no—
66C2×C4⋊C4D414yes—
67C2×C4⋊C4C2311yescharacteristic
68C2×C4⋊C4C2312yes—
69C2×C4⋊C4C2311yescharacteristic
70C2×C4⋊C4C2311yescharacteristic
71C2×C22⋊C4—21no—
72C2×C22⋊C4—21no—
73C2×C22⋊C4—22no—
74C2×C22⋊C4—22no—
75C2×C22⋊C4—22no—
76C2×C22⋊C4—22no—
77C2×C22⋊C4D411yescharacteristic
78C2×C22⋊C4D411yescharacteristic
79C2×C42—22no—
80C2×C42C2312yes—
81#107—22no—
82#107—22no—
83C4⋊C8—42no—
84#103—22no—
85#103—22no—
86#103—42no—
87#102—24no—
88#102—24no—
89#102—24no—
90#102—42no—
91C22⋊C8—22no—
92#95—22no—
93#95—22no—
94#95—21no—
95#95D412yes—
96C24—22no—
97C24—44no—
98C24—42no—
99C24—42no—
100C24—42no—
101C24—82no—
102C24C2×D411yescharacteristic
103C2×Q8—21no—
104C2×Q8—21no—
105C2×Q8—22no—
106C2×Q8—22no—
107C2×D4—28no—
108C2×D4—22no—
109C2×D4—22no—
110C2×D4—22no—
111C2×D4—22no—
112C2×D4—22no—
113C2×D4—48no—
114C2×D4—44no—
115C2×D4—44no—
116C2×D4—42no—
117C2×D4#4114yes—
118C22×C4—22no—
119C22×C4—22no—
120C22×C4—21no—
121C22×C4—21no—
122C22×C4—24no—
123C22×C4—22no—
124C22×C4—21no—
125C22×C4—21no—
126C22×C4—82no—
127C22×C4C2×D412yes—
128C22×C4C2×D412yes—
129C22×C4C2×D411yescharacteristic
130C22×C4C2×D412yes—
131C22×C4#4112yes—
132C22×C4C2411yescharacteristic
133C2×C8—21no—
134C2×C8—21no—
135C2×C8—22no—
136C2×C8—22no—
137C2×C8—22no—
138C2×C8—22no—
139C2×C8—22no—
140C2×C8—44no—
141C4⋊C4—24no—
142C4⋊C4—21no—
143C4⋊C4—21no—
144C4⋊C4—21no—
145C4⋊C4—22no—
146C4⋊C4—21no—
147C4⋊C4—22no—
148C4⋊C4—42no—
149C4⋊C4—82no—
150C4⋊C4C2×D412yes—
151C4⋊C4C2×D412yes—
152C22⋊C4—22no—
153C22⋊C4—22no—
154C22⋊C4—44no—
155C22⋊C4—44no—
156C22⋊C4—42no—
157C22⋊C4—42no—
158C22⋊C4—82no—
159C22⋊C4—82no—
160C42—22no—
161C42—22no—
162C42—82no—
163C23—22no—
164C23—22no—
165C23—24no—
166C23—48no—
167C23—42no—
168C23—42no—
169C23—42no—
170C23—42no—
171C23—42no—
172C23—42no—
173C23—88no—
174C23—88no—
175C23—84no—
176C23—84no—
177C23#12112yes—
178C23#13512yes—
179C23#13712yes—
180C23C22×D411yescharacteristic
181Q8—41no—
182Q8—41no—
183Q8—42no—
184D4—24no—
185D4—24no—
186D4—48no—
187D4—42no—
188D4—42no—
189C2×C4—22no—
190C2×C4—22no—
191C2×C4—22no—
192C2×C4—24no—
193C2×C4—22no—
194C2×C4—22no—
195C2×C4—22no—
196C2×C4—22no—
197C2×C4—22no—
198C2×C4—22no—
199C2×C4—22no—
200C2×C4—22no—
201C2×C4—42no—
202C2×C4—42no—
203C2×C4—41no—
204C2×C4—41no—
205C2×C4—42no—
206C2×C4—42no—
207C2×C4—42no—
208C2×C4—42no—
209C2×C4—42no—
210C2×C4—84no—
211C2×C4#12112yes—
212C2×C4C22×D411yescharacteristic
213C2×C4C2×#4112yes—
214C2×C4C2×#4111yescharacteristic
215C2×C4H(5,𝔽2)12yes—
216C8—41no—
217C8—41no—
218C8—42no—
219C22—22no—
220C22—22no—
221C22—24no—
222C22—24no—
223C22—44no—
224C22—44no—
225C22—44no—
226C22—816no—
227C22—82no—
228C22—82no—
229C22—82no—
230C22—82no—
231C22#46414yes—
232C22C2×#12111yescharacteristic
233C22#54612yes—
234C22C2×#13511yescharacteristic
235C22C2×#13711yescharacteristic
236C22#57612yes—
237C4—22no—
238C4—42no—
239C4—42no—
240C4—42no—
241C4—42no—
242C4—81no—
243C4—81no—
244C4—82no—
245C4#52511yescharacteristic
246C4#53511yescharacteristic
247C4#55112yes—
248C2—24no—
249C2—84no—
250C2#241111yescharacteristic
251C2C2×#46411yescharacteristic
252C2#304111yescharacteristic
253C2#329712yes—
254C2#331412yes—
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×#484C2semidirect
4. #2345C2semidirect
5. #2067C2semidirect
6. #2042C2semidirect
7. #1983C2semidirect
8. #1925C2semidirect
9. #1892C2semidirect
10. #1850C2semidirect
18. C2×#107C22semidirect
19. C2×#107C22semidirect
20. C2×C4⋊C8C22semidirect
21. C2×#103C22semidirect
22. C2×#103C22semidirect
23. C2×#103C22semidirect
32. #389C22semidirect
38. 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×#4842
C22×#1022
#24352
#23452
#20672
#20422
#19832
#19252
#18922
#18502
#17862