Finite Group Databasebeta

← #24478

Subgroups — #24478

Order 256.

616 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 616 conjugacy classes of subgroups — too many to draw. Only its 139 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×#481C211yesmaximal, characteristic, complemented
2C22×#102C211yesmaximal, characteristic, complemented
3#2433C211yesmaximal, characteristic
4#2345C211yesmaximal, characteristic, complemented
5#2065C212yesmaximal, complemented
6#2041C212yesmaximal, complemented
7#1982C212yesmaximal, complemented
8#1926C212yesmaximal, complemented
9#1893C211yesmaximal, characteristic, complemented
10#1850C211yesmaximal, characteristic, complemented
11#1786C211yesmaximal, characteristic, complemented
12C23×D4C2211yescharacteristic
13C23×C8C2211yescharacteristic
14C2×#127C2212yes—
15C2×#121C2211yescharacteristic
16C2×#120—21no—
17C2×#120—21no—
18C22×C4⋊C4C2211yescharacteristic
19#481—24no—
20C2×#106C2211yescharacteristic, complemented
21C2×#106C2211yescharacteristic, complemented
22C2×C4⋊C8C2212yescomplemented
23C2×#102—24no—
24C2×#102C2212yes—
25C2×#102C2212yes—
26C2×#102C2212yes—
27C2×#102C2212yes—
28C2×#102C2211yescharacteristic, complemented
29C2×#102C2211yescharacteristic, complemented
30C2×#102C2212yescomplemented
31C2×C22⋊C8C2211yescharacteristic
32#392—22no—
33#390C2212yes—
34#389C2212yescomplemented
35#387—22no—
36#380—21no—
37#380C2211yescharacteristic
38C4×C4⋊C4C2212yescomplemented
39C4×C22⋊C4—21no—
40C4×C22⋊C4C2211yescharacteristic
41#340C2212yescomplemented
42#336C2212yescomplemented
43C25—21no—
44C22×D4—24no—
45C22×D4—21no—
46C22×D4—22no—
47C22×D4—21no—
48C22×D4—22no—
49C22×D4—22no—
50C22×D4—22no—
51C22×D4—42no—
52C22×D4C2311yescharacteristic
53C22×D4C2312yes—
54C23×C4—21no—
55C23×C4C2311yescharacteristic
56C22×C8—22no—
57C22×C8—42no—
58C22×C8C2312yes—
59C22×C8C2311yescharacteristic
60C22×C8C2311yescharacteristic
61#127—24no—
62#127—42no—
63#121—22no—
64#121—22no—
65#120—44no—
66#120—44no—
67C2×C4⋊C4—21no—
68C2×C4⋊C4—22no—
69C2×C4⋊C4—22no—
70C2×C4⋊C4D414yescomplemented
71C2×C4⋊C4C2311yescharacteristic
72C2×C4⋊C4C2312yes—
73C2×C4⋊C4C2311yescharacteristic
74C2×C4⋊C4C2311yescharacteristic
75C2×C22⋊C4—21no—
76C2×C22⋊C4—22no—
77C2×C22⋊C4—22no—
78C2×C22⋊C4—21no—
79C2×C22⋊C4D411yescharacteristic
80C2×C22⋊C4D411yescharacteristic
81C2×C42—22no—
82C2×C42C2312yes—
83#106—22no—
84#106—22no—
85C4⋊C8—42no—
86#102—24no—
87#102—24no—
88#102—24no—
89#102—22no—
90#102—22no—
91#102—42no—
92#102—42no—
93C22⋊C8—22no—
94#95—22no—
95#95—22no—
96#95—21no—
97#95D412yes—
98C24—22no—
99C24—21no—
100C24—44no—
101C24—42no—
102C24—42no—
103C24—42no—
104C24—82no—
105C24C2×D411yescharacteristic
106C2×D4—28no—
107C2×D4—22no—
108C2×D4—21no—
109C2×D4—22no—
110C2×D4—22no—
111C2×D4—21no—
112C2×D4—22no—
113C2×D4—22no—
114C2×D4—22no—
115C2×D4—22no—
116C2×D4—48no—
117C2×D4—44no—
118C2×D4—44no—
119C2×D4—44no—
120C2×D4—44no—
121C2×D4—44no—
122C2×D4—42no—
123C2×D4—42no—
124C2×D4—42no—
125C2×D4—42no—
126C2×D4—44no—
127C2×D4—82no—
128C2×D4—82no—
129C2×D4#4114yes—
130C22×C4—22no—
131C22×C4—22no—
132C22×C4—21no—
133C22×C4—21no—
134C22×C4—24no—
135C22×C4—22no—
136C22×C4—21no—
137C22×C4—82no—
138C22×C4C2×D412yes—
139C22×C4C2×D411yescharacteristic
140C22×C4C2×D412yes—
141C22×C4C2×D412yes—
142C22×C4#4112yes—
143C22×C4C2411yescharacteristic
144C2×C8—21no—
145C2×C8—21no—
146C2×C8—22no—
147C2×C8—22no—
148C2×C8—22no—
149C2×C8—22no—
150C2×C8—22no—
151C2×C8—44no—
152C4⋊C4—24no—
153C4⋊C4—21no—
154C4⋊C4—21no—
155C4⋊C4—21no—
156C4⋊C4—22no—
157C4⋊C4—21no—
158C4⋊C4—22no—
159C4⋊C4—82no—
160C4⋊C4C2×D412yes—
161C4⋊C4C2×D412yes—
162C22⋊C4—22no—
163C22⋊C4—44no—
164C22⋊C4—42no—
165C22⋊C4—42no—
166C22⋊C4—44no—
167C22⋊C4SD1614yes—
168C42—22no—
169C42—22no—
170C42—82no—
171C23—22no—
172C23—22no—
173C23—24no—
174C23—48no—
175C23—42no—
176C23—42no—
177C23—42no—
178C23—42no—
179C23—42no—
180C23—42no—
181C23—41no—
182C23—42no—
183C23—41no—
184C23—41no—
185C23—42no—
186C23—88no—
187C23—88no—
188C23—84no—
189C23—84no—
190C23#12112yes—
191C23C2×SD1612yes—
192C23AGL(1,ℤ/8)12yes—
193C23C22×D411yescharacteristic
194D4—24no—
195D4—24no—
196D4—48no—
197D4—41no—
198D4—42no—
199D4—41no—
200D4—42no—
201D4—42no—
202D4—88no—
203D4—88no—
204D4—84no—
205C2×C4—22no—
206C2×C4—22no—
207C2×C4—22no—
208C2×C4—22no—
209C2×C4—22no—
210C2×C4—24no—
211C2×C4—22no—
212C2×C4—22no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—22no—
216C2×C4—42no—
217C2×C4—42no—
218C2×C4—41no—
219C2×C4—42no—
220C2×C4—41no—
221C2×C4—41no—
222C2×C4—42no—
223C2×C4—42no—
224C2×C4—84no—
225C2×C4#12112yes—
226C2×C4C2×SD1614yes—
227C2×C4C22×D411yescharacteristic
228C2×C4C2×#4112yes—
229C2×C4C2×#4111yescharacteristic
230C2×C4H(5,𝔽2)12yes—
231C8—41no—
232C8—41no—
233C8—42no—
234C22—22no—
235C22—24no—
236C22—22no—
237C22—24no—
238C22—44no—
239C22—44no—
240C22—44no—
241C22—816no—
242C22—82no—
243C22—82no—
244C22—82no—
245C22—82no—
246C22—81no—
247C22—81no—
248C22—82no—
249C22—81no—
250C22—82no—
251C22#46014yes—
252C22C2×#12111yescharacteristic
253C22#54612yes—
254C22C22×SD1611yescharacteristic
255C22C2×AGL(1,ℤ/8)11yescharacteristic
256C22#57612yes—
257C4—22no—
258C4—24no—
259C4—42no—
260C4—42no—
261C4—42no—
262C4—81no—
263C4—82no—
264C4#52511yescharacteristic
265C4#53511yescharacteristic
266C4#55112yes—
267C2—24no—
268C2—84no—
269C2—161no—
270C2#241111yescharacteristic
271C2C2×#46011yescharacteristic
272C2#304111yescharacteristic
273C2#329512yes—
274C2#331812yes—
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radical#2433C2
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×#481C2semidirect
2. C22×#102C2semidirect
4. #2345C2semidirect
5. #2065C2semidirect
6. #2041C2semidirect
7. #1982C2semidirect
8. #1926C2semidirect
9. #1893C2semidirect
10. #1850C2semidirect
11. #1786C2semidirect
20. C2×#106C22semidirect
21. C2×#106C22semidirect
22. C2×C4⋊C8C22semidirect
28. C2×#102C22semidirect
29. C2×#102C22semidirect
30. C2×#102C22semidirect
34. #389C22semidirect
38. C4×C4⋊C4C22semidirect
41. #340C22semidirect
42. #336C22semidirect
70. C2×C4⋊C4D4semidirect

Maximal subgroups

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

subgroupindex
C2×#4812
C22×#1022
#24332
#23452
#20652
#20412
#19822
#19262
#18932
#18502
#17862