Finite Group Databasebeta

← #24495

Subgroups — #24495

Order 256.

534 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 534 conjugacy classes of subgroups — too many to draw. Only its 123 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×#485C211yesmaximal, characteristic, complemented
2C2×#418C211yesmaximal, characteristic
3#2435C211yesmaximal, characteristic
4#2346C211yesmaximal, characteristic, complemented
5#2067C212yesmaximal, complemented
6#2042C212yesmaximal, complemented
7#1983C212yesmaximal, complemented
8#1926C212yesmaximal, complemented
9#1894C211yesmaximal, characteristic, complemented
10#1851C211yesmaximal, characteristic, complemented
11#1786C211yesmaximal, characteristic
12C23×D4C2211yescharacteristic
13C22×M16C2211yescharacteristic
14C2×#124C2212yes—
15C2×C22⋊Q8C2211yescharacteristic
16C2×#117C2211yescharacteristic
17#485—24no—
18C2×#107C2211yescharacteristic, complemented
19C2×#106C2211yescharacteristic, complemented
20C2×C4⋊C8C2212yescomplemented
21#418—24no—
22C2×#103C2211yescharacteristic, complemented
23C2×#103C2211yescharacteristic, complemented
24C2×#103C2212yescomplemented
25C2×#102C2212yes—
26C2×#102C2212yes—
27C2×#102C2212yes—
28C2×#102C2212yes—
29C2×C22⋊C8C2211yescharacteristic
30#394—21no—
31#394—21no—
32#390C2212yes—
33#389C2212yescomplemented
34#388—22no—
35#382—21no—
36#380C2211yescharacteristic
37#379—22no—
38C4×C4⋊C4C2212yescomplemented
39C4×C22⋊C4C2211yescharacteristic
40#376—21no—
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×C4C2311yescharacteristic
52C2×M16—22no—
53C2×M16—42no—
54C22×C8C2312yes—
55C22×C8C2311yescharacteristic
56C22×C8C2311yescharacteristic
57#124—24no—
58#124—42no—
59C22⋊Q8—22no—
60C22⋊Q8—22no—
61#117—22no—
62#117D414yes—
63C2×C4⋊C4—21no—
64C2×C4⋊C4—22no—
65C2×C4⋊C4C2311yescharacteristic
66C2×C4⋊C4C2312yes—
67C2×C4⋊C4C2311yescharacteristic
68C2×C4⋊C4C2311yescharacteristic
69C2×C22⋊C4—21no—
70C2×C22⋊C4—21no—
71C2×C22⋊C4—22no—
72C2×C22⋊C4—22no—
73C2×C22⋊C4—21no—
74C2×C22⋊C4—22no—
75C2×C22⋊C4—22no—
76C2×C22⋊C4D412yes—
77C2×C42—21no—
78C2×C42—22no—
79C2×C42C2312yes—
80#107—22no—
81#106—22no—
82C4⋊C8—42no—
83#103—22no—
84#103—22no—
85#103—42no—
86#102—24no—
87#102—24no—
88#102—24no—
89#102—42no—
90C22⋊C8—22no—
91#95—21no—
92#95—22no—
93#95—22no—
94#95D411yescharacteristic
95#95D411yescharacteristic
96C24—22no—
97C24—44no—
98C24—44no—
99C24—42no—
100C24—81no—
101C24—81no—
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—22no—
123C22×C4—21no—
124C22×C4—21no—
125C22×C4C2×D412yes—
126C22×C4C2×D412yes—
127C22×C4C2×D411yescharacteristic
128C22×C4C2×D412yes—
129C22×C4#4112yes—
130C22×C4C2411yescharacteristic
131M16—44no—
132C2×C8—21no—
133C2×C8—21no—
134C2×C8—22no—
135C2×C8—22no—
136C2×C8—22no—
137C2×C8—22no—
138C2×C8—22no—
139C4⋊C4—21no—
140C4⋊C4—21no—
141C4⋊C4—21no—
142C4⋊C4—22no—
143C4⋊C4—21no—
144C4⋊C4—22no—
145C4⋊C4—42no—
146C4⋊C4—82no—
147C4⋊C4C2×D412yes—
148C4⋊C4C2×D412yes—
149C22⋊C4—24no—
150C22⋊C4—42no—
151C22⋊C4—44no—
152C22⋊C4—44no—
153C22⋊C4—42no—
154C22⋊C4—82no—
155C22⋊C4—82no—
156C22⋊C4—81no—
157C42—24no—
158C42—22no—
159C42—22no—
160C42—44no—
161C23—22no—
162C23—22no—
163C23—21no—
164C23—21no—
165C23—24no—
166C23—48no—
167C23—42no—
168C23—42no—
169C23—42no—
170C23—42no—
171C23—42no—
172C23—42no—
173C23—84no—
174C23—84no—
175C23—88no—
176C23—88no—
177C23#12112yes—
178C23C22×D411yescharacteristic
179Q8—41no—
180Q8—41no—
181Q8—42no—
182D4—24no—
183D4—24no—
184D4—48no—
185D4—42no—
186D4—42no—
187C2×C4—22no—
188C2×C4—22no—
189C2×C4—22no—
190C2×C4—24no—
191C2×C4—22no—
192C2×C4—22no—
193C2×C4—22no—
194C2×C4—22no—
195C2×C4—22no—
196C2×C4—22no—
197C2×C4—22no—
198C2×C4—22no—
199C2×C4—42no—
200C2×C4—41no—
201C2×C4—41no—
202C2×C4—41no—
203C2×C4—41no—
204C2×C4—42no—
205C2×C4—42no—
206C2×C4—42no—
207C2×C4—41no—
208C2×C4—42no—
209C2×C4—41no—
210C2×C4#12112yes—
211C2×C4C22×D411yescharacteristic
212C2×C4C2×#4112yes—
213C2×C4C2×#4111yescharacteristic
214C2×C4H(5,𝔽2)12yes—
215C8—41no—
216C8—41no—
217C8—42no—
218C22—22no—
219C22—22no—
220C22—22no—
221C22—42no—
222C22—42no—
223C22—44no—
224C22—44no—
225C22—44no—
226C22—816no—
227C22—82no—
228C22—82no—
229C22—82no—
230C22—82no—
231C22C2×#12111yescharacteristic
232C22#54612yes—
233C22#57512yes—
234C22#57611yescharacteristic
235C22#57711yescharacteristic
236C4—22no—
237C4—42no—
238C4—42no—
239C4—42no—
240C4—42no—
241C4—81no—
242C4—81no—
243C4—82no—
244C4#52511yescharacteristic
245C4#53511yescharacteristic
246C4#55112yes—
247C2—42no—
248C2—84no—
249C2#241111yescharacteristic
250C2#304011yescharacteristic
251C2#304511yescharacteristic
252C2#331912yes—
253C2#332312yes—
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×#485C2semidirect
4. #2346C2semidirect
5. #2067C2semidirect
6. #2042C2semidirect
7. #1983C2semidirect
8. #1926C2semidirect
9. #1894C2semidirect
10. #1851C2semidirect
18. C2×#107C22semidirect
19. C2×#106C22semidirect
20. C2×C4⋊C8C22semidirect
22. C2×#103C22semidirect
23. C2×#103C22semidirect
24. C2×#103C22semidirect
33. #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×#4852
C2×#4182
#24352
#23462
#20672
#20422
#19832
#19262
#18942
#18512
#17862