Finite Group Databasebeta

← #23480

Subgroups — #23480

Order 256.

951 conjugacy classes of subgroups; only the 179 normal subgroups are shown.

normal subgroups only

This group has 951 conjugacy classes of subgroups — too many to draw. Only its 179 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
1C22×#121C211yesmaximal, characteristic
2C2×#428C211yesmaximal, characteristic, complemented
3C2×#418C211yesmaximal, characteristic, complemented
4C2×#418C211yesmaximal, characteristic, complemented
5C2×#407C211yesmaximal, characteristic, complemented
6#2280C211yesmaximal, characteristic, complemented
7#2034C214yesmaximal, complemented
8#2033C214yesmaximal, complemented
9#1736C211yesmaximal, characteristic, complemented
10C23×D4—21no—
11C23×D4C2211yescharacteristic
12C24×C4C2211yescharacteristic
13C22×M16C2211yescharacteristic
14C22×M16C2211yescharacteristic
15C2×#121—24no—
16C2×#121—22no—
17C2×#121—24no—
18C2×#121C2214yes—
19C2×#117C2211yescharacteristic
20C2×#117C2211yescharacteristic
21C22×C4⋊C4C2211yescharacteristic
22C22×C22⋊C4—21no—
23#428—24no—
24C2×#107C2212yescomplemented
25C2×#106C2212yescomplemented
26#418—24no—
27#418—24no—
28C2×#102C2212yescomplemented
29C2×#102C2214yescomplemented
30C2×#102C2212yescomplemented
31#407—24no—
32C2×C22⋊C8C2214yescomplemented
33#381—21no—
34#380C2212yescomplemented
35#380C2212yescomplemented
36#379—21no—
37C4×C22⋊C4—22no—
38#340C2214yescomplemented
39C25—21no—
40C25D411yescharacteristic
41C22×D4—28no—
42C22×D4—24no—
43C22×D4—22no—
44C22×D4—21no—
45C22×D4—22no—
46C22×D4—22no—
47C22×D4—22no—
48C22×D4—22no—
49C22×D4—48no—
50C22×D4D414yes—
51C22×D4C2312yes—
52C23×C4—21no—
53C23×C4—22no—
54C23×C4—22no—
55C23×C4—22no—
56C23×C4—22no—
57C23×C4—22no—
58C23×C4—22no—
59C23×C4D411yescharacteristic
60C23×C4D412yes—
61C23×C4C2312yes—
62C23×C4C2311yescharacteristic
63C2×M16—22no—
64C2×M16—22no—
65C2×M16—22no—
66C2×M16—22no—
67C2×M16—22no—
68C2×M16D414yescomplemented
69C22×C8C2312yes—
70C22×C8C2312yes—
71#121—216no—
72#121—28no—
73#121—28no—
74#117—22no—
75#117—22no—
76#117—22no—
77#117—22no—
78C2×C4⋊C4—21no—
79C2×C4⋊C4—22no—
80C2×C4⋊C4D414yes—
81C2×C4⋊C4C2312yes—
82C2×C4⋊C4C2311yescharacteristic
83C2×C4⋊C4C2311yescharacteristic
84C2×C4⋊C4C2312yes—
85C2×C22⋊C4—28no—
86C2×C22⋊C4—24no—
87C2×C22⋊C4—21no—
88C2×C22⋊C4—21no—
89C2×C22⋊C4—22no—
90C2×C22⋊C4—22no—
91C2×C42—21no—
92C2×C42—21no—
93#107—24no—
94#106—24no—
95#102—24no—
96#102—24no—
97#102—28no—
98C22⋊C8—28no—
99#95—22no—
100#95—22no—
101C24—22no—
102C24—24no—
103C24—24no—
104C24—22no—
105C24—22no—
106C24—22no—
107C24—42no—
108C24—44no—
109C24—44no—
110C24—42no—
111C24—42no—
112C24C2×D412yes—
113C24C2×D411yescharacteristic
114C2×D4—216no—
115C2×D4—24no—
116C2×D4—28no—
117C2×D4—28no—
118C2×D4—22no—
119C2×D4—22no—
120C2×D4—22no—
121C2×D4—22no—
122C2×D4—22no—
123C2×D4—432no—
124C2×D4—48no—
125C2×D4—48no—
126C2×D4—44no—
127C2×D4—42no—
128C2×D4—42no—
129C2×D4—44no—
130C2×D4C2×D414yes—
131C22×C4—22no—
132C22×C4—24no—
133C22×C4—22no—
134C22×C4—22no—
135C22×C4—22no—
136C22×C4—24no—
137C22×C4—24no—
138C22×C4—24no—
139C22×C4—22no—
140C22×C4—22no—
141C22×C4—22no—
142C22×C4—22no—
143C22×C4—22no—
144C22×C4—22no—
145C22×C4—21no—
146C22×C4—21no—
147C22×C4—22no—
148C22×C4—44no—
149C22×C4—44no—
150C22×C4—44no—
151C22×C4—44no—
152C22×C4—44no—
153C22×C4C2×D412yes—
154C22×C4C2×D411yescharacteristic
155C22×C4C2×D412yes—
156C22×C4C2×D412yes—
157C22×C4C2×D412yes—
158C22×C4#4112yes—
159C22×C4#4112yes—
160C22×C4#4112yes—
161C22×C4C2411yescharacteristic
162M16—28no—
163M16—44no—
164C2×C8—22no—
165C2×C8—22no—
166C2×C8—22no—
167C2×C8—22no—
168C2×C8—22no—
169C2×C8C2×D414yescomplemented
170C4⋊C4—24no—
171C4⋊C4—22no—
172C4⋊C4—21no—
173C4⋊C4—21no—
174C4⋊C4—22no—
175C4⋊C4—21no—
176C4⋊C4—21no—
177C4⋊C4—22no—
178C4⋊C4C2×D414yes—
179C22⋊C4—216no—
180C22⋊C4—44no—
181C22⋊C4—42no—
182C22⋊C4—42no—
183C22⋊C4—44no—
184C42—42no—
185C42—42no—
186C23—22no—
187C23—22no—
188C23—28no—
189C23—22no—
190C23—22no—
191C23—22no—
192C23—22no—
193C23—22no—
194C23—22no—
195C23—24no—
196C23—48no—
197C23—48no—
198C23—44no—
199C23—48no—
200C23—48no—
201C23—44no—
202C23—42no—
203C23—42no—
204C23—44no—
205C23—44no—
206C23—48no—
207C23—88no—
208C23—88no—
209C23#12012yes—
210C23#12112yes—
211C23#12312yes—
212C23AGL(1,ℤ/8)14yes—
213C23C22×D411yescharacteristic
214D4—24no—
215D4—24no—
216D4—432no—
217D4—42no—
218D4—42no—
219D4—48no—
220C2×C4—22no—
221C2×C4—24no—
222C2×C4—22no—
223C2×C4—22no—
224C2×C4—48no—
225C2×C4—44no—
226C2×C4—44no—
227C2×C4—42no—
228C2×C4—42no—
229C2×C4—42no—
230C2×C4—42no—
231C2×C4—48no—
232C2×C4—41no—
233C2×C4—41no—
234C2×C4—42no—
235C2×C4—42no—
236C2×C4—42no—
237C2×C4—42no—
238C2×C4—42no—
239C2×C4#12012yes—
240C2×C4#12114yes—
241C2×C4#12114yes—
242C2×C4#12112yes—
243C2×C4#12314yes—
244C2×C4#12312yes—
245C2×C4#12414yes—
246C2×C4C22×D411yescharacteristic
247C2×C4C22×D411yescharacteristic
248C2×C4C22×D411yescharacteristic
249C2×C4C2×#4111yescharacteristic
250C2×C4C2×#4111yescharacteristic
251C2×C4C2×#4111yescharacteristic
252C8—24no—
253C8—42no—
254C22—24no—
255C22—22no—
256C22—22no—
257C22—24no—
258C22—24no—
259C22—24no—
260C22—24no—
261C22—416no—
262C22—42no—
263C22—42no—
264C22—42no—
265C22—42no—
266C22—44no—
267C22—44no—
268C22—44no—
269C22—816no—
270C22—84no—
271C22—84no—
272C22#39414yes—
273C22C2×#12011yescharacteristic
274C22C2×#12111yescharacteristic
275C22C2×#12311yescharacteristic
276C22C2×AGL(1,ℤ/8)12yes—
277C22#57512yes—
278C4—42no—
279C4—44no—
280C4—81no—
281C4—81no—
282C4—82no—
283C4#39414yes—
284C4C2×#12111yescharacteristic
285C4C2×#12111yescharacteristic
286C4C2×#12311yescharacteristic
287C4C2×#12411yescharacteristic
288C2—24no—
289C2—44no—
290C2—84no—
291C2C2×#39411yescharacteristic
292C2#300311yescharacteristic
293C2#300511yescharacteristic
294C2#305512yes—
295C2#309412yes—
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC22×#121C2
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
2. C2×#428C2semidirect
3. C2×#418C2semidirect
4. C2×#418C2semidirect
5. C2×#407C2semidirect
6. #2280C2semidirect
7. #2034C2semidirect
8. #2033C2semidirect
9. #1736C2semidirect
24. C2×#107C22semidirect
25. C2×#106C22semidirect
28. C2×#102C22semidirect
29. C2×#102C22semidirect
30. C2×#102C22semidirect
32. C2×C22⋊C8C22semidirect
34. #380C22semidirect
35. #380C22semidirect
38. #340C22semidirect
68. C2×M16D4semidirect
169. C2×C8C2×D4semidirect

Maximal subgroups

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

subgroupindex
C22×#1212
C2×#4282
C2×#4182
C2×#4072
#22802
#20342
#20332
#17362