Finite Group Databasebeta

← #44409010

Subgroups — #44409010

Order 29056.

752 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 752 conjugacy classes of subgroups — too many to draw. Only its 149 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×#20300334C211yesmaximal, characteristic
2C22×#16813871C211yesmaximal, characteristic, complemented
3#28409606C211yesmaximal, characteristic, complemented
4#28409193C214yesmaximal, complemented
5#28409181C214yesmaximal, complemented
6#28409038C211yesmaximal, characteristic, complemented
7#28409028C211yesmaximal, characteristic, complemented
8#28409000C211yesmaximal, characteristic, complemented
9C2×#137×C227C211yesmaximal, characteristic, complemented
10#20300334—24no—
11#20300334—24no—
12C2×#16813885C2212yes—
13C2×#16813884—21no—
14C2×#16813884C2211yescharacteristic
15C22×D908—21no—
16C22×D908C2211yescharacteristic
17C2×D4×D227—22no—
18C2×D4×D227C2212yes—
19C2×#16813871—24no—
20C2×#16813871C2214yescomplemented
21C22×C227⋊C8C2211yescharacteristic, complemented
22#20300286—21no—
23#20300277—21no—
24#20300271—22no—
25#20300264C2214yescomplemented
26#20300247C2211yescharacteristic, complemented
27#20300212C2212yescomplemented
28#20300210C2214yescomplemented
29#20300209C2212yescomplemented
30(C2×D227)⋊C8C2214yescomplemented
31C2×#41×C227C2211yescharacteristic
32C22×Q8×C227C2211yescharacteristic, complemented
33#137×C227—24no—
34C2×Q16×C227C2212yescomplemented
35C2×SD16×C227C2212yescomplemented
36C2×M16×C227C2211yescharacteristic, complemented
37C23×D227—22no—
38C23×D227—21no—
39#16813885—24no—
40#16813885—24no—
41#16813884—28no—
42#16813884—22no—
43#16813884—22no—
44C2×Dic454D411yescharacteristic
45C2×D908—28no—
46C2×D908—22no—
47C2×D908—21no—
48C2×D908—22no—
49C2×D908D411yescharacteristic
50C2×D908D414yes—
51C2×D908C2312yes—
52C2×C22⋊D227—22no—
53C2×C22⋊D227D412yes—
54D4×D227—216no—
55D4×D227—24no—
56D4×D227—24no—
57C2×C4×D227—22no—
58C2×C4×D227C2312yes—
59#16813871—28no—
60C2×C227⋊C8—22no—
61C2×C227⋊C8C2312yes—
62C4⋊Dic227C2312yes—
63Dic227⋊C4—24no—
64C22⋊Dic227—21no—
65(C2×D227)⋊C4—24no—
66C4×Dic227—21no—
67#41×C227—22no—
68#41×C227D414yes—
69C2×Q8×C227—21no—
70C2×Q8×C227—22no—
71C2×Q8×C227—22no—
72C2×Q8×C227C2312yes—
73C2×Q8×C227C2311yescharacteristic
74C2×D4×C227—21no—
75C2×D4×C227C2311yescharacteristic
76C22×C4×C227—21no—
77C22×C4×C227—21no—
78C22×C4×C227C2311yescharacteristic
79Q16×C227—24no—
80SD16×C227—24no—
81M16×C227—22no—
82C2×C8×C227C2312yes—
83C22×D227—22no—
84C22×D227—28no—
85C22×D227—24no—
86C22×D227—84no—
87C22×D227—82no—
88C22×D227C2×D412yes—
89Dic454—22no—
90D908—28no—
91D908—22no—
92D908—24no—
93D908—22no—
94D908C2×D414yes—
95C22⋊D227—28no—
96C22⋊D227—24no—
97C2×Dic227—22no—
98C2×Dic227C2×D412yes—
99C4×D227—28no—
100C4×D227—22no—
101C4×D227—22no—
102C227⋊C8—22no—
103C23×C227—21no—
104C23×C227C2×D411yescharacteristic
105Q8×C227—21no—
106Q8×C227—22no—
107Q8×C227—22no—
108Q8×C227—42no—
109Q8×C227C2×D412yes—
110D4×C227—24no—
111D4×C227—21no—
112D4×C227C2×D412yes—
113C2×C4×C227—21no—
114C2×C4×C227—24no—
115C2×C4×C227—22no—
116C2×C4×C227—42no—
117C2×C4×C227C2×D411yescharacteristic
118C2×C4×C227C2×D412yes—
119C2×C4×C227C2×D412yes—
120C2×C4×C227C2411yescharacteristic
121C8×C227—22no—
122C22×C227—22no—
123C22×C227—22no—
124C22×C227—42no—
125C22×C227#12012yes—
126C22×C227C22×D411yescharacteristic
127C2×D227—22no—
128C2×D227—24no—
129C2×D227—24no—
130C2×D227—44no—
131C2×D227—84no—
132C2×D227—88no—
133C4×C227—22no—
134C4×C227—42no—
135C4×C227#12012yes—
136C4×C227C22×D411yescharacteristic
137C4×C227C22×D411yescharacteristic
138Dic227—22no—
139Dic227—42no—
140C2×C227—22no—
141C2×C227—42no—
142C2×C227C2×#12011yescharacteristic
143C2×C227#57712yes—
144D227—42no—
145D227—44no—
146C227#300911yescharacteristic, complemented, Sylow
147#3009—2271nomaximal, Sylow
148C2×H(5,𝔽2)—2271no—
149C2×#137D22711yescharacteristic, complemented
150C22×SD16—2271no—
151#536—2271no—
152#452—2274no—
153#450—2274no—
154#418—2271no—
155#416—2271no—
156#408—2271no—
157H(5,𝔽2)—4544no—
158H(5,𝔽2)—4544no—
159C2×#41—4541no—
160C2×#41—2272no—
161C2×#41—2271no—
162C2×#41C2×D22711yescharacteristic
163C22×Q8C2×D22711yescharacteristic, complemented
164C22×D4—4541no—
165C22×D4—4542no—
166C22×D4—2272no—
167C22×D4—2271no—
168#137—24no—
169C2×Q16C2×D22712yescomplemented
170C2×SD16—4544no—
171C2×SD16—2274no—
172C2×SD16C2×D22712yescomplemented
173C2×M16C2×D22711yescharacteristic, complemented
174C22×C8—2271no—
175C4⋊Q8—4541no—
176#124—4541no—
177#123—4542no—
178C22⋊Q8—2274no—
179#117—2271no—
180#103—2272no—
181#102—2272no—
182#102—2274no—
183C22⋊C8—2274no—
184C24—4542no—
185C24—4541no—
186#41—22no—
187#41—4548no—
188#41—4544no—
189#41—4544no—
190#41—4542no—
191#41—4542no—
192#41C22⋊D22714yes—
193C2×Q8—21no—
194C2×Q8—22no—
195C2×Q8—22no—
196C2×Q8—2271no—
197C2×Q8C22×D22712yes—
198C2×Q8C22×D22711yescharacteristic
199C2×D4—21no—
200C2×D4—4548no—
201C2×D4—4542no—
202C2×D4—45416no—
203C2×D4—4544no—
204C2×D4—4544no—
205C2×D4—4542no—
206C2×D4—4541no—
207C2×D4—4542no—
208C2×D4—2272no—
209C2×D4—2271no—
210C2×D4—2274no—
211C2×D4—2272no—
212C2×D4C22×D22711yescharacteristic
213C22×C4—21no—
214C22×C4—21no—
215C22×C4—4542no—
216C22×C4—2272no—
217C22×C4C22×D22711yescharacteristic
218Q16—24no—
219SD16—24no—
220SD16—4548no—
221M16—22no—
222C2×C8—4542no—
223C2×C8—2272no—
224C2×C8C22×D22712yes—
225C4⋊C4—4544no—
226C4⋊C4—2272no—
227C22⋊C4—4544no—
228C22⋊C4—4541no—
229C42—4541no—
230C23—21no—
231C23—18162no—
232C23—18164no—
233C23—4548no—
234C23—4544no—
235C23—4542no—
236C23—2272no—
237C23D4×D22711yescharacteristic
238Q8—21no—
239Q8—22no—
240Q8—22no—
241Q8—42no—
242Q8—4542no—
243Q8C2×C22⋊D22712yes—
244D4—24no—
245D4—21no—
246D4—4548no—
247D4—4548no—
248D4—4544no—
249D4—4542no—
250D4—4542no—
251D4—4544no—
252D4—2274no—
253D4C2×C22⋊D22712yes—
254C2×C4—21no—
255C2×C4—24no—
256C2×C4—22no—
257C2×C4—42no—
258C2×C4—4548no—
259C2×C4—4542no—
260C2×C4—4542no—
261C2×C4—4542no—
262C2×C4—2272no—
263C2×C4D4×D22712yes—
264C2×C4D4×D22711yescharacteristic
265C2×C4C2×C22⋊D22712yes—
266C2×C4C23×D22711yescharacteristic
267C8—22no—
268C8—4542no—
269C22—22no—
270C22—22no—
271C22—42no—
272C22—18164no—
273C22—18168no—
274C22—9084no—
275C22—4542no—
276C22—4544no—
277C22—4544no—
278C22#2030025612yes—
279C22C2×D4×D22711yescharacteristic
280C4—22no—
281C4—42no—
282C4—9082no—
283C4—4542no—
284C4#2030025612yes—
285C4C2×D4×D22711yescharacteristic
286C4C22×C22⋊D22711yescharacteristic
287C2—22no—
288C2—42no—
289C2—9084no—
290C2—9082no—
291C2C2×#2030025611yescharacteristic
292C2#2840984112yes—
subgroupG/H
centerC22C2×D4×D227
commutator, cosocleC2×C4×C227C24
FrattiniC2×C4C23×D227
FittingC2×#137×C227C2
socleC22×C227C22×D4
prime radicalC2×#20300334C2
hypercenterC2×#137D227
hypocenterC227#3009
LCSG ▷ C2×C4×C227 ▷ C2×C227 ▷ C227
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#137

Complements

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

NH ≅ G/Ntype
2. C22×#16813871C2semidirect
3. #28409606C2semidirect
4. #28409193C2semidirect
5. #28409181C2semidirect
6. #28409038C2semidirect
7. #28409028C2semidirect
8. #28409000C2semidirect
9. C2×#137×C227C2semidirect
20. C2×#16813871C22semidirect
21. C22×C227⋊C8C22semidirect
25. #20300264C22semidirect
26. #20300247C22semidirect
27. #20300212C22semidirect
28. #20300210C22semidirect
29. #20300209C22semidirect
30. (C2×D227)⋊C8C22semidirect
32. C22×Q8×C227C22semidirect
34. C2×Q16×C227C22semidirect
35. C2×SD16×C227C22semidirect
36. C2×M16×C227C22semidirect
146. C227#3009semidirect
149. C2×#137D227semidirect
163. C22×Q8C2×D227semidirect
169. C2×Q16C2×D227semidirect
172. C2×SD16C2×D227semidirect
173. C2×M16C2×D227semidirect

Maximal subgroups

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

subgroupindex
C2×#203003342
C22×#168138712
#284096062
#284091932
#284091812
#284090382
#284090282
#284090002
C2×#137×C2272
#3009227