Finite Group Databasebeta

← #32935

Subgroups — #32935

Order 256.

790 conjugacy classes of subgroups; only the 163 normal subgroups are shown.

normal subgroups only

This group has 790 conjugacy classes of subgroups — too many to draw. Only its 163 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
1C23×M16C211yesmaximal, characteristic, complemented
2C2×#468C212yesmaximal, complemented
3C2×#465C211yesmaximal, characteristic, complemented
4#2848C211yesmaximal, characteristic, complemented
5#2847C211yesmaximal, characteristic
6#1895C212yesmaximal, complemented
7#1894C212yesmaximal, complemented
8#1893C211yesmaximal, characteristic, complemented
9#1892C211yesmaximal, characteristic, complemented
10#1809C212yesmaximal, complemented
11#1808C211yesmaximal, characteristic, complemented
12C24×C4C2211yescharacteristic
13C22×SD16C2211yescharacteristic, complemented
14C22×M16—28no—
15C22×M16—24no—
16C22×M16C2212yes—
17C22×M16C2212yes—
18C23×C8C2211yescharacteristic
19C23×C8C2211yescharacteristic
20C2×C22⋊Q8C2211yescharacteristic
21C2×C22⋊Q8C2212yes—
22C2×#121C2211yescharacteristic
23C2×#121C2212yes—
24C2×#120—21no—
25#468—28no—
26#465—24no—
27C2×#107C2212yescomplemented
28C2×#106C2211yescharacteristic, complemented
29C2×#103C2212yescomplemented
30C2×#103C2211yescharacteristic, complemented
31C2×#102C2212yescomplemented
32C2×#102C2211yescharacteristic, complemented
33#386—22no—
34#386—21no—
35#385—22no—
36#385—21no—
37#380C2211yescharacteristic
38#380C2211yescharacteristic
39#380C2212yes—
40#380C2212yes—
41#379—21no—
42#340C2212yescomplemented
43#340C2212yescomplemented
44#340C2212yescomplemented
45C25D411yescharacteristic
46C22×Q8C2311yescharacteristic
47C22×D4—22no—
48C22×D4—21no—
49C22×D4C2311yescharacteristic
50C23×C4—24no—
51C23×C4—22no—
52C23×C4—24no—
53C23×C4—22no—
54C23×C4D412yes—
55C23×C4D411yescharacteristic
56C23×C4C2311yescharacteristic
57C23×C4C2312yes—
58C2×SD16—22no—
59C2×SD16—24no—
60C2×M16—232no—
61C2×M16—24no—
62C2×M16—24no—
63C2×M16—24no—
64C2×M16—24no—
65C2×M16—24no—
66C2×M16D418yescomplemented
67C22×C8—22no—
68C22×C8—24no—
69C22×C8—24no—
70C22×C8D414yescomplemented
71C22×C8C2312yes—
72C22×C8C2311yescharacteristic
73C22×C8C2311yescharacteristic
74C22⋊Q8—22no—
75C22⋊Q8—24no—
76C22⋊Q8—22no—
77C22⋊Q8—24no—
78#121—22no—
79#121—24no—
80#121—22no—
81#121—24no—
82#120—44no—
83C2×C4⋊C4—21no—
84C2×C4⋊C4—22no—
85C2×C4⋊C4C2311yescharacteristic
86C2×C4⋊C4C2312yes—
87C2×C4⋊C4C2311yescharacteristic
88C2×C4⋊C4C2312yes—
89C2×C22⋊C4—22no—
90C2×C22⋊C4—21no—
91C2×C22⋊C4—22no—
92C2×C22⋊C4—21no—
93C2×C22⋊C4—22no—
94C2×C22⋊C4—21no—
95C2×C22⋊C4—21no—
96C2×C22⋊C4—22no—
97#107—24no—
98#106—22no—
99#103—22no—
100#103—24no—
101#102—22no—
102#102—24no—
103#95—21no—
104#95—22no—
105#95—21no—
106#95—22no—
107C24—24no—
108C24—22no—
109C24—22no—
110C24—22no—
111C24—21no—
112C24—42no—
113C24C2×D412yes—
114C24C2×D411yescharacteristic
115C2×Q8—21no—
116C2×Q8—22no—
117C2×Q8—21no—
118C2×Q8—22no—
119C2×D4—21no—
120C2×D4—22no—
121C2×D4—21no—
122C2×D4—22no—
123C2×D4—44no—
124C2×D4—44no—
125C2×D4—44no—
126C2×D4—42no—
127C2×D4—42no—
128C2×D4—42no—
129C22×C4—28no—
130C22×C4—28no—
131C22×C4—216no—
132C22×C4—24no—
133C22×C4—22no—
134C22×C4—24no—
135C22×C4—24no—
136C22×C4—24no—
137C22×C4—22no—
138C22×C4—22no—
139C22×C4—22no—
140C22×C4—22no—
141C22×C4—22no—
142C22×C4—22no—
143C22×C4—22no—
144C22×C4—21no—
145C22×C4—21no—
146C22×C4—22no—
147C22×C4—21no—
148C22×C4C2×D411yescharacteristic
149C22×C4C2×D412yes—
150C22×C4C2×D412yes—
151C22×C4C2×D414yes—
152C22×C4#4114yes—
153C22×C4#4112yes—
154C22×C4C2411yescharacteristic
155SD16—44no—
156M16—216no—
157M16—216no—
158C2×C8—28no—
159C2×C8—22no—
160C2×C8—22no—
161C2×C8—22no—
162C2×C8—28no—
163C2×C8C2×D414yes—
164C2×C8C2×D412yes—
165C2×C8C2×D412yes—
166C2×C8C2×D414yes—
167C4⋊C4—21no—
168C4⋊C4—22no—
169C4⋊C4—21no—
170C4⋊C4—22no—
171C4⋊C4—21no—
172C4⋊C4—22no—
173C4⋊C4—21no—
174C4⋊C4—22no—
175C4⋊C4—42no—
176C4⋊C4—44no—
177C22⋊C4—44no—
178C22⋊C4—44no—
179C22⋊C4—44no—
180C22⋊C4—42no—
181C22⋊C4—42no—
182C22⋊C4—42no—
183C22⋊C4—42no—
184C22⋊C4—44no—
185C23—22no—
186C23—22no—
187C23—28no—
188C23—22no—
189C23—22no—
190C23—22no—
191C23—22no—
192C23—22no—
193C23—22no—
194C23—48no—
195C23—48no—
196C23—44no—
197C23—44no—
198C23—41no—
199C23—42no—
200C23—41no—
201C23—41no—
202C23—42no—
203C23#12114yes—
204C23#12112yes—
205C23AGL(1,ℤ/8)12yes—
206C23#13712yes—
207C23C22×D411yescharacteristic
208Q8—41no—
209Q8—41no—
210Q8—42no—
211D4—41no—
212D4—41no—
213D4—42no—
214D4—88no—
215D4—84no—
216C2×C4—28no—
217C2×C4—216no—
218C2×C4—22no—
219C2×C4—22no—
220C2×C4—22no—
221C2×C4—22no—
222C2×C4—24no—
223C2×C4—22no—
224C2×C4—24no—
225C2×C4—24no—
226C2×C4—24no—
227C2×C4—24no—
228C2×C4—41no—
229C2×C4—42no—
230C2×C4—41no—
231C2×C4—42no—
232C2×C4—41no—
233C2×C4—42no—
234C2×C4—41no—
235C2×C4—42no—
236C2×C4—42no—
237C2×C4—41no—
238C2×C4—42no—
239C2×C4—42no—
240C2×C4—42no—
241C2×C4#12112yes—
242C2×C4#12114yes—
243C2×C4C22×D411yescharacteristic
244C2×C4C22×D412yes—
245C2×C4C2×#4111yescharacteristic
246C2×C4C2×#4112yes—
247C8—24no—
248C8—22no—
249C8#12014yescomplemented
250C22—22no—
251C22—22no—
252C22—24no—
253C22—24no—
254C22—24no—
255C22—24no—
256C22—24no—
257C22—416no—
258C22—42no—
259C22—42no—
260C22—42no—
261C22—42no—
262C22—81no—
263C22—81no—
264C22—81no—
265C22—82no—
266C22—82no—
267C22#46814yescomplemented
268C22C2×#12111yescharacteristic
269C22C2×#12112yes—
270C22C2×AGL(1,ℤ/8)11yescharacteristic
271C22C2×#13711yescharacteristic
272C22#57512yes—
273C4—22no—
274C4—24no—
275C4—81no—
276C4—81no—
277C4—81no—
278C4—82no—
279C4—82no—
280C4C2×#12011yescharacteristic
281C4#52511yescharacteristic
282C4#52512yes—
283C2—24no—
284C2—44no—
285C2—161no—
286C2#284711yescharacteristic
287C2C2×#46811yescharacteristic
288C2#305411yescharacteristic
289C2#305512yes—
290C2#305612yes—
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. C23×M16C2semidirect
2. C2×#468C2semidirect
3. C2×#465C2semidirect
4. #2848C2semidirect
6. #1895C2semidirect
7. #1894C2semidirect
8. #1893C2semidirect
9. #1892C2semidirect
10. #1809C2semidirect
11. #1808C2semidirect
13. C22×SD16C22semidirect
27. C2×#107C22semidirect
28. C2×#106C22semidirect
29. C2×#103C22semidirect
30. C2×#103C22semidirect
31. C2×#102C22semidirect
32. C2×#102C22semidirect
42. #340C22semidirect
43. #340C22semidirect
44. #340C22semidirect
66. C2×M16D4semidirect
70. C22×C8D4semidirect
249. C8#120semidirect
267. C22#468semidirect

Maximal subgroups

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

subgroupindex
C23×M162
C2×#4682
C2×#4652
#28482
#28472
#18952
#18942
#18932
#18922
#18092
#18082