Finite Group Databasebeta

← #24494

Subgroups — #24494

Order 256.

599 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 599 conjugacy classes of subgroups — too many to draw. Only its 127 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×#482C211yesmaximal, characteristic, complemented
2C2×#418C211yesmaximal, characteristic, complemented
3#2433C211yesmaximal, characteristic
4#2346C211yesmaximal, characteristic, complemented
5#2065C212yesmaximal, complemented
6#2041C212yesmaximal, complemented
7#1982C212yesmaximal, complemented
8#1925C212yesmaximal, complemented
9#1895C211yesmaximal, characteristic, complemented
10#1851C211yesmaximal, characteristic, complemented
11#1786C211yesmaximal, characteristic, complemented
12C23×D4C2211yescharacteristic
13C22×M16C2211yescharacteristic
14C2×#127C2212yes—
15C2×#121C2211yescharacteristic
16C2×#120—22no—
17C2×#117C2211yescharacteristic
18#482—24no—
19C2×#107C2211yescharacteristic, complemented
20C2×#106C2211yescharacteristic, complemented
21C2×C4⋊C8C2212yescomplemented
22#418—24no—
23C2×#102C2212yes—
24C2×#102C2212yes—
25C2×#102C2212yes—
26C2×#102C2212yes—
27C2×#102C2211yescharacteristic, complemented
28C2×#102C2211yescharacteristic, complemented
29C2×#102C2212yescomplemented
30C2×C22⋊C8C2211yescharacteristic
31#392—21no—
32#392—21no—
33#390C2212yes—
34#389C2212yescomplemented
35#388—22no—
36#382—21no—
37#380C2211yescharacteristic
38C4×C4⋊C4C2212yescomplemented
39C4×C22⋊C4C2211yescharacteristic
40#376—21no—
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×C4C2311yescharacteristic
55C2×M16—22no—
56C2×M16—42no—
57C22×C8C2312yes—
58C22×C8C2311yescharacteristic
59C22×C8C2311yescharacteristic
60#127—24no—
61#127—42no—
62#121—22no—
63#121—22no—
64#120—48no—
65#117—22no—
66#117D414yescomplemented
67C2×C4⋊C4—22no—
68C2×C4⋊C4C2311yescharacteristic
69C2×C4⋊C4C2312yes—
70C2×C4⋊C4C2311yescharacteristic
71C2×C4⋊C4C2311yescharacteristic
72C2×C22⋊C4—21no—
73C2×C22⋊C4—22no—
74C2×C22⋊C4—22no—
75C2×C22⋊C4—21no—
76C2×C22⋊C4—21no—
77C2×C22⋊C4D412yes—
78C2×C42—21no—
79C2×C42—22no—
80C2×C42C2312yes—
81#107—22no—
82#106—22no—
83C4⋊C8—42no—
84#102—24no—
85#102—24no—
86#102—24no—
87#102—22no—
88#102—22no—
89#102—42no—
90#102—42no—
91C22⋊C8—22no—
92#95—21no—
93#95—22no—
94#95—22no—
95#95D411yescharacteristic
96#95D411yescharacteristic
97C24—22no—
98C24—21no—
99C24—44no—
100C24—44no—
101C24—42no—
102C24—81no—
103C24—81no—
104C24C2×D411yescharacteristic
105C2×D4—28no—
106C2×D4—22no—
107C2×D4—21no—
108C2×D4—22no—
109C2×D4—22no—
110C2×D4—21no—
111C2×D4—22no—
112C2×D4—22no—
113C2×D4—22no—
114C2×D4—22no—
115C2×D4—48no—
116C2×D4—44no—
117C2×D4—44no—
118C2×D4—44no—
119C2×D4—44no—
120C2×D4—44no—
121C2×D4—44no—
122C2×D4—42no—
123C2×D4—42no—
124C2×D4—44no—
125C2×D4—82no—
126C2×D4—82no—
127C2×D4#4114yes—
128C22×C4—22no—
129C22×C4—22no—
130C22×C4—21no—
131C22×C4—21no—
132C22×C4—22no—
133C22×C4—21no—
134C22×C4C2×D412yes—
135C22×C4C2×D411yescharacteristic
136C22×C4C2×D412yes—
137C22×C4C2×D412yes—
138C22×C4#4112yes—
139C22×C4C2411yescharacteristic
140M16—44no—
141C2×C8—21no—
142C2×C8—21no—
143C2×C8—22no—
144C2×C8—22no—
145C2×C8—22no—
146C2×C8—22no—
147C2×C8—22no—
148C4⋊C4—21no—
149C4⋊C4—21no—
150C4⋊C4—21no—
151C4⋊C4—22no—
152C4⋊C4—21no—
153C4⋊C4—22no—
154C4⋊C4—82no—
155C4⋊C4C2×D412yes—
156C4⋊C4C2×D412yes—
157C22⋊C4—24no—
158C22⋊C4—24no—
159C22⋊C4—44no—
160C22⋊C4—42no—
161C22⋊C4—44no—
162C22⋊C4—81no—
163C42—24no—
164C42—22no—
165C42—22no—
166C42—44no—
167C23—22no—
168C23—22no—
169C23—21no—
170C23—21no—
171C23—24no—
172C23—48no—
173C23—42no—
174C23—42no—
175C23—42no—
176C23—42no—
177C23—42no—
178C23—42no—
179C23—41no—
180C23—42no—
181C23—42no—
182C23—41no—
183C23—41no—
184C23—84no—
185C23—84no—
186C23—88no—
187C23—88no—
188C23#12112yes—
189C23C22×D411yescharacteristic
190D4—24no—
191D4—24no—
192D4—48no—
193D4—41no—
194D4—42no—
195D4—41no—
196D4—42no—
197D4—42no—
198D4—88no—
199D4—88no—
200D4—84no—
201C2×C4—22no—
202C2×C4—22no—
203C2×C4—22no—
204C2×C4—22no—
205C2×C4—22no—
206C2×C4—24no—
207C2×C4—22no—
208C2×C4—22no—
209C2×C4—22no—
210C2×C4—22no—
211C2×C4—22no—
212C2×C4—42no—
213C2×C4—41no—
214C2×C4—41no—
215C2×C4—41no—
216C2×C4—42no—
217C2×C4—42no—
218C2×C4—42no—
219C2×C4—42no—
220C2×C4#12112yes—
221C2×C4AGL(1,ℤ/8)14yes—
222C2×C4C22×D411yescharacteristic
223C2×C4C2×#4112yes—
224C2×C4C2×#4111yescharacteristic
225C2×C4H(5,𝔽2)12yes—
226C8—41no—
227C8—41no—
228C8—42no—
229C22—22no—
230C22—22no—
231C22—22no—
232C22—42no—
233C22—42no—
234C22—44no—
235C22—44no—
236C22—44no—
237C22—816no—
238C22—82no—
239C22—82no—
240C22—82no—
241C22—82no—
242C22—81no—
243C22—81no—
244C22—81no—
245C22—82no—
246C22—82no—
247C22C2×#12111yescharacteristic
248C22#54612yes—
249C22C2×AGL(1,ℤ/8)12yes—
250C22#57611yescharacteristic
251C22#57711yescharacteristic
252C4—22no—
253C4—24no—
254C4—42no—
255C4—42no—
256C4—42no—
257C4—81no—
258C4—82no—
259C4#52511yescharacteristic
260C4#53511yescharacteristic
261C4#55112yes—
262C2—42no—
263C2—84no—
264C2—161no—
265C2#241111yescharacteristic
266C2#303811yescharacteristic
267C2#304511yescharacteristic
268C2#331012yes—
269C2#332512yes—
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×#482C2semidirect
2. C2×#418C2semidirect
4. #2346C2semidirect
5. #2065C2semidirect
6. #2041C2semidirect
7. #1982C2semidirect
8. #1925C2semidirect
9. #1895C2semidirect
10. #1851C2semidirect
11. #1786C2semidirect
19. C2×#107C22semidirect
20. C2×#106C22semidirect
21. C2×C4⋊C8C22semidirect
27. C2×#102C22semidirect
28. C2×#102C22semidirect
29. C2×#102C22semidirect
34. #389C22semidirect
38. C4×C4⋊C4C22semidirect
41. #340C22semidirect
42. #336C22semidirect
66. #117D4semidirect

Maximal subgroups

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

subgroupindex
C2×#4822
C2×#4182
#24332
#23462
#20652
#20412
#19822
#19252
#18952
#18512
#17862