Finite Group Databasebeta

← #29735

Subgroups — #29735

Order 256.

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

normal subgroups only

This group has 600 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×#465C211yesmaximal, characteristic, complemented
2C22×C22⋊C8C211yesmaximal, characteristic, complemented
3#2848C211yesmaximal, characteristic, complemented
4#2847C211yesmaximal, characteristic
5#2077C212yesmaximal, complemented
6#2036C212yesmaximal, complemented
7#2033C212yesmaximal, complemented
8#2000C212yesmaximal, complemented
9#1893C211yesmaximal, characteristic, complemented
10#1892C211yesmaximal, characteristic, complemented
11#1808C211yesmaximal, characteristic, complemented
12C24×C4C2211yescharacteristic
13C22×SD16C2211yescharacteristic, complemented
14C23×C8C2211yescharacteristic
15C23×C8C2211yescharacteristic
16C2×C22⋊Q8C2211yescharacteristic
17C2×C22⋊Q8C2212yes—
18C2×#121C2211yescharacteristic
19C2×#121C2212yes—
20C2×#120—21no—
21#465—24no—
22C2×#107C2212yescomplemented
23C2×#106C2211yescharacteristic, complemented
24C2×#103C2211yescharacteristic, complemented
25C2×#103C2212yescomplemented
26C2×#102C2211yescharacteristic, complemented
27C2×#102C2212yescomplemented
28C2×C22⋊C8—24no—
29C2×C22⋊C8C2212yes—
30C2×C22⋊C8C2212yes—
31#386—21no—
32#386—22no—
33#385—21no—
34#385—22no—
35#380C2211yescharacteristic
36#380C2211yescharacteristic
37#380C2212yes—
38#380C2212yes—
39#379—21no—
40#340C2212yescomplemented
41#340C2212yescomplemented
42#340C2212yescomplemented
43C25D411yescharacteristic
44C22×Q8C2311yescharacteristic
45C22×D4—22no—
46C22×D4—21no—
47C22×D4C2311yescharacteristic
48C23×C4—22no—
49C23×C4—22no—
50C23×C4—42no—
51C23×C4—42no—
52C23×C4D411yescharacteristic
53C23×C4D412yes—
54C23×C4C2312yes—
55C23×C4C2311yescharacteristic
56C2×SD16—22no—
57C2×SD16—42no—
58C22×C8—22no—
59C22×C8—42no—
60C22×C8—42no—
61C22×C8D414yescomplemented
62C22×C8C2312yes—
63C22×C8C2311yescharacteristic
64C22×C8C2311yescharacteristic
65C22⋊Q8—22no—
66C22⋊Q8—22no—
67C22⋊Q8—42no—
68C22⋊Q8—42no—
69#121—22no—
70#121—22no—
71#121—42no—
72#121—42no—
73#120—44no—
74C2×C4⋊C4—22no—
75C2×C4⋊C4—21no—
76C2×C4⋊C4C2311yescharacteristic
77C2×C4⋊C4C2312yes—
78C2×C4⋊C4C2311yescharacteristic
79C2×C4⋊C4C2312yes—
80C2×C22⋊C4—22no—
81C2×C22⋊C4—21no—
82C2×C22⋊C4—22no—
83C2×C22⋊C4—21no—
84C2×C22⋊C4—22no—
85C2×C22⋊C4—21no—
86C2×C22⋊C4—22no—
87C2×C22⋊C4—21no—
88#107—42no—
89#106—22no—
90#103—22no—
91#103—42no—
92#102—22no—
93#102—42no—
94C22⋊C8—24no—
95C22⋊C8—24no—
96#95—22no—
97#95—21no—
98#95—22no—
99#95—21no—
100C24—22no—
101C24—24no—
102C24—22no—
103C24—22no—
104C24—21no—
105C24—42no—
106C24C2×D412yes—
107C24C2×D411yescharacteristic
108C2×Q8—21no—
109C2×Q8—21no—
110C2×Q8—22no—
111C2×Q8—22no—
112C2×D4—21no—
113C2×D4—21no—
114C2×D4—22no—
115C2×D4—22no—
116C2×D4—44no—
117C2×D4—44no—
118C2×D4—42no—
119C2×D4—44no—
120C2×D4—42no—
121C2×D4—42no—
122C22×C4—22no—
123C22×C4—22no—
124C22×C4—22no—
125C22×C4—24no—
126C22×C4—22no—
127C22×C4—22no—
128C22×C4—22no—
129C22×C4—22no—
130C22×C4—22no—
131C22×C4—22no—
132C22×C4—21no—
133C22×C4—22no—
134C22×C4—21no—
135C22×C4—22no—
136C22×C4—21no—
137C22×C4—42no—
138C22×C4—42no—
139C22×C4—42no—
140C22×C4—44no—
141C22×C4—44no—
142C22×C4—48no—
143C22×C4C2×D412yes—
144C22×C4C2×D411yescharacteristic
145C22×C4C2×D412yes—
146C22×C4#4112yes—
147C22×C4C2411yescharacteristic
148SD16—44no—
149C2×C8—22no—
150C2×C8—22no—
151C2×C8—22no—
152C2×C8—22no—
153C2×C8—22no—
154C2×C8—44no—
155C2×C8—44no—
156C2×C8C2×D412yes—
157C2×C8C2×D412yes—
158C4⋊C4—21no—
159C4⋊C4—21no—
160C4⋊C4—21no—
161C4⋊C4—21no—
162C4⋊C4—22no—
163C4⋊C4—22no—
164C4⋊C4—22no—
165C4⋊C4—22no—
166C4⋊C4—44no—
167C4⋊C4—42no—
168C22⋊C4—44no—
169C22⋊C4—44no—
170C22⋊C4—44no—
171C22⋊C4—42no—
172C22⋊C4—44no—
173C22⋊C4—42no—
174C22⋊C4—42no—
175C22⋊C4—42no—
176C23—22no—
177C23—22no—
178C23—28no—
179C23—22no—
180C23—22no—
181C23—22no—
182C23—22no—
183C23—22no—
184C23—22no—
185C23—48no—
186C23—48no—
187C23—44no—
188C23—44no—
189C23—41no—
190C23—42no—
191C23—42no—
192C23—41no—
193C23—41no—
194C23#12112yes—
195C23#13514yes—
196C23AGL(1,ℤ/8)12yes—
197C23#13712yes—
198C23C22×D411yescharacteristic
199Q8—41no—
200Q8—41no—
201Q8—42no—
202D4—41no—
203D4—41no—
204D4—42no—
205D4—88no—
206D4—84no—
207C2×C4—22no—
208C2×C4—22no—
209C2×C4—22no—
210C2×C4—24no—
211C2×C4—22no—
212C2×C4—22no—
213C2×C4—22no—
214C2×C4—48no—
215C2×C4—44no—
216C2×C4—42no—
217C2×C4—42no—
218C2×C4—42no—
219C2×C4—42no—
220C2×C4—41no—
221C2×C4—41no—
222C2×C4—41no—
223C2×C4—42no—
224C2×C4—42no—
225C2×C4—42no—
226C2×C4—42no—
227C2×C4—42no—
228C2×C4—42no—
229C2×C4—42no—
230C2×C4—41no—
231C2×C4—42no—
232C2×C4—41no—
233C2×C4#12112yes—
234C2×C4C22×D411yescharacteristic
235C2×C4C2×#4111yescharacteristic
236C2×C4H(5,𝔽2)12yes—
237C2×C4H(5,𝔽2)12yes—
238C8—22no—
239C8—22no—
240C8—42no—
241C22—22no—
242C22—24no—
243C22—24no—
244C22—22no—
245C22—24no—
246C22—24no—
247C22—24no—
248C22—416no—
249C22—42no—
250C22—42no—
251C22—42no—
252C22—42no—
253C22—81no—
254C22—82no—
255C22—82no—
256C22—81no—
257C22—81no—
258C22#46814yes—
259C22C2×#12111yescharacteristic
260C22#53412yes—
261C22C2×#13512yes—
262C22C2×AGL(1,ℤ/8)11yescharacteristic
263C22C2×#13711yescharacteristic
264C4—22no—
265C4—42no—
266C4—81no—
267C4—82no—
268C4—82no—
269C4—81no—
270C4—81no—
271C4#53411yescharacteristic
272C4#53511yescharacteristic
273C4#53812yes—
274C2—24no—
275C2—44no—
276C2—161no—
277C2#261711yescharacteristic
278C2#305011yescharacteristic
279C2C2×#46811yescharacteristic
280C2#318812yes—
281C2#319012yes—
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radical#2847C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#465C2semidirect
2. C22×C22⋊C8C2semidirect
3. #2848C2semidirect
5. #2077C2semidirect
6. #2036C2semidirect
7. #2033C2semidirect
8. #2000C2semidirect
9. #1893C2semidirect
10. #1892C2semidirect
11. #1808C2semidirect
13. C22×SD16C22semidirect
22. C2×#107C22semidirect
23. C2×#106C22semidirect
24. C2×#103C22semidirect
25. C2×#103C22semidirect
26. C2×#102C22semidirect
27. C2×#102C22semidirect
40. #340C22semidirect
41. #340C22semidirect
42. #340C22semidirect
61. C22×C8D4semidirect

Maximal subgroups

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

subgroupindex
C2×#4652
C22×C22⋊C82
#28482
#28472
#20772
#20362
#20332
#20002
#18932
#18922
#18082