Finite Group Databasebeta

← #44409013

Subgroups — #44409013

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
2C2×#20300321C211yesmaximal, characteristic, complemented
3C22×#16813873C211yesmaximal, characteristic, complemented
4#28409603C211yesmaximal, characteristic, complemented
5#28409196C214yesmaximal, complemented
6#28409185C214yesmaximal, complemented
7#28409038C211yesmaximal, characteristic, complemented
8#28409029C211yesmaximal, characteristic, complemented
9#408×C227C211yesmaximal, characteristic, complemented
10#20300334—24no—
11#20300334—24no—
12C2×#16813885—21no—
13C2×#16813884C2212yes—
14C2×#16813884C2211yescharacteristic
15C22×Dic454C2211yescharacteristic, complemented
16C22×D908C2212yes—
17C22×D908C2211yescharacteristic
18C2×D4×D227—22no—
19C2×D4×D227—21no—
20#20300321—24no—
21C2×Dic908C2212yescomplemented
22C2×#16813873—24no—
23C2×#16813873C2212yescomplemented
24C2×#16813873C2214yescomplemented
25#20300286—21no—
26#20300275—21no—
27#20300269—22no—
28#20300267C2214yescomplemented
29#20300247C2211yescharacteristic, complemented
30#20300213C2212yescomplemented
31#20300210C2212yescomplemented
32#20300210C2214yescomplemented
33C2×#41×C227C2211yescharacteristic
34C2×M16×C227C2211yescharacteristic, complemented
35C22×C8×C227C2211yescharacteristic, complemented
36C22⋊C8×C227C2214yescomplemented
37C23×D227—22no—
38C23×D227—21no—
39#16813885—28no—
40#16813884—24no—
41#16813884—24no—
42#16813884—22no—
43#16813884D414yes—
44C2×Dic454—21no—
45C2×Dic454—22no—
46C2×Dic454—22no—
47C2×Dic454C2312yes—
48C2×Dic454C2311yescharacteristic
49C2×D908—24no—
50C2×D908—24no—
51C2×D908—22no—
52C2×D908—21no—
53C2×D908—22no—
54C2×D908D414yes—
55C2×D908C2311yescharacteristic
56C2×D908C2312yes—
57C2×C22⋊D227—22no—
58C2×C22⋊D227—21no—
59D4×D227—216no—
60D4×D227—28no—
61C22×Dic227—21no—
62C2×C4×D227—22no—
63C2×C4×D227—21no—
64Dic908—24no—
65#16813873—28no—
66#16813873—24no—
67C4⋊Dic227C2312yes—
68Dic227⋊C4—24no—
69C22⋊Dic227—21no—
70C22⋊Dic227—24no—
71C4×Dic227—21no—
72#41×C227—22no—
73#41×C227—22no—
74C2×Q8×C227D411yescharacteristic
75C2×D4×C227D412yes—
76C2×D4×C227D411yescharacteristic
77C22×C4×C227C2312yes—
78C22×C4×C227C2311yescharacteristic
79M16×C227—22no—
80C2×C8×C227—22no—
81C2×C8×C227C2312yes—
82C2×C8×C227C2312yes—
83C22×D227—22no—
84C22×D227—21no—
85C22×D227—28no—
86C22×D227—24no—
87C22×D227—82no—
88C22×D227—84no—
89Dic454—21no—
90Dic454—22no—
91Dic454—22no—
92Dic454—42no—
93Dic454C2×D412yes—
94D908—28no—
95D908—24no—
96D908—22no—
97D908—21no—
98D908C2×D412yes—
99D908C2×D414yes—
100C22⋊D227—28no—
101C22⋊D227—24no—
102C2×Dic227—21no—
103C2×Dic227—22no—
104C2×Dic227—22no—
105C2×Dic227—42no—
106C4×D227—28no—
107C4×D227—24no—
108C23×C227C2×D412yes—
109C23×C227C2×D411yescharacteristic
110Q8×C227—22no—
111D4×C227—24no—
112D4×C227—22no—
113C2×C4×C227—22no—
114C2×C4×C227—22no—
115C2×C4×C227C2×D412yes—
116C2×C4×C227C2×D411yescharacteristic
117C2×C4×C227C2×D412yes—
118C2×C4×C227C2×D412yes—
119C2×C4×C227C2411yescharacteristic
120C8×C227—22no—
121C8×C227—22no—
122C22×C227—22no—
123C22×C227—22no—
124C22×C227—24no—
125C22×C227#12012yes—
126C22×C227C22×D411yescharacteristic
127C2×D227—24no—
128C2×D227—22no—
129C2×D227—44no—
130C2×D227—42no—
131C2×D227—84no—
132C2×D227—88no—
133C4×C227—22no—
134C4×C227#12012yes—
135C4×C227C22×D411yescharacteristic
136C4×C227C22×D411yescharacteristic
137Dic227—22no—
138Dic227—42no—
139Dic227—42no—
140C2×C227—22no—
141C2×C227—42no—
142C2×C227C2×#12011yescharacteristic
143C2×C227#57712yes—
144D227—44no—
145D227—42no—
146C227#300911yescharacteristic, complemented, Sylow
147#3009—2271nomaximal, Sylow
148C2×H(5,𝔽2)—2271no—
149C2×#137—2271no—
150C22×SD16—2271no—
151#536—2271no—
152#452—2274no—
153#450—2274no—
154#418—2271no—
155#416—2271no—
156#408D22711yescharacteristic, complemented
157H(5,𝔽2)—4544no—
158H(5,𝔽2)—4544no—
159C2×#41—4541no—
160C2×#41—2272no—
161C2×#41—2271no—
162C2×#41C2×D22711yescharacteristic
163C22×Q8—2271no—
164C22×D4—4542no—
165C22×D4—4541no—
166C22×D4—2272no—
167C22×D4—2271no—
168#137—4544no—
169C2×Q16—2272no—
170C2×SD16—4544no—
171C2×SD16—2274no—
172C2×SD16—2272no—
173C2×M16C2×D22711yescharacteristic, complemented
174C22×C8C2×D22711yescharacteristic, complemented
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⋊C8C2×D22714yescomplemented
184C24—4542no—
185C24—4541no—
186#41—22no—
187#41—22no—
188#41—4544no—
189#41—4544no—
190#41—4548no—
191#41—4542no—
192#41—2274no—
193C2×Q8—4542no—
194C2×Q8—4542no—
195C2×Q8—4541no—
196C2×Q8—2272no—
197C2×Q8—2271no—
198C2×Q8D90811yescharacteristic
199C2×D4—4544no—
200C2×D4—4544no—
201C2×D4—45416no—
202C2×D4—4548no—
203C2×D4—4542no—
204C2×D4—4541no—
205C2×D4—4542no—
206C2×D4—4542no—
207C2×D4—4541no—
208C2×D4—2274no—
209C2×D4—2271no—
210C2×D4—2272no—
211C2×D4D90812yes—
212C2×D4D90811yescharacteristic
213C22×C4—4541no—
214C22×C4—4542no—
215C22×C4—4541no—
216C22×C4C22×D22712yes—
217C22×C4C22×D22711yescharacteristic
218Q16—4544no—
219SD16—4548no—
220SD16—4544no—
221M16—22no—
222C2×C8—22no—
223C2×C8C22×D22712yes—
224C2×C8C22×D22712yes—
225C4⋊C4—4544no—
226C4⋊C4—2272no—
227C22⋊C4—4541no—
228C22⋊C4—4544no—
229C42—4541no—
230C23—18162no—
231C23—18164no—
232C23—4548no—
233C23—4544no—
234C23—4542no—
235C23—4541no—
236C23C2×D90812yes—
237C23C2×D90811yescharacteristic
238Q8—22no—
239Q8—9082no—
240Q8—4542no—
241Q8—4542no—
242Q8—4541no—
243Q8—2272no—
244D4—24no—
245D4—22no—
246D4—4548no—
247D4—4548no—
248D4—4544no—
249D4—4542no—
250D4—4541no—
251D4—4544no—
252D4—2272no—
253D4—2274no—
254C2×C4—22no—
255C2×C4—22no—
256C2×C4—9082no—
257C2×C4—4548no—
258C2×C4—4542no—
259C2×C4—4542no—
260C2×C4—4541no—
261C2×C4—4544no—
262C2×C4D4×D22712yes—
263C2×C4D4×D22712yes—
264C2×C4C2×D90812yes—
265C2×C4C2×D90811yescharacteristic
266C2×C4C23×D22711yescharacteristic
267C8—22no—
268C8—22no—
269C22—22no—
270C22—22no—
271C22—24no—
272C22—18164no—
273C22—18168no—
274C22—9084no—
275C22—9082no—
276C22—4544no—
277C22—4542no—
278C22#2030025812yes—
279C22C22×D90811yescharacteristic
280C4—22no—
281C4—9082no—
282C4—9082no—
283C4—4542no—
284C4#2030025812yes—
285C4C2×D4×D22711yescharacteristic
286C4C2×D4×D22711yescharacteristic
287C2—22no—
288C2—42no—
289C2—9084no—
290C2—9082no—
291C2C2×#2030025811yescharacteristic
292C2#2840984412yes—
subgroupG/H
centerC22C22×D908
commutator, cosocleC2×C4×C227C24
FrattiniC2×C4C23×D227
Fitting#408×C227C2
socleC22×C227C22×D4
prime radicalC2×#20300334C2
hypercenter#408D227
hypocenterC227#3009
LCSG ▷ C2×C4×C227 ▷ C2×C227 ▷ C227
UCSC1 ◁ C22 ◁ C2×#41 ◁ #408

Complements

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

NH ≅ G/Ntype
2. C2×#20300321C2semidirect
3. C22×#16813873C2semidirect
4. #28409603C2semidirect
5. #28409196C2semidirect
6. #28409185C2semidirect
7. #28409038C2semidirect
8. #28409029C2semidirect
9. #408×C227C2semidirect
15. C22×Dic454C22semidirect
21. C2×Dic908C22semidirect
23. C2×#16813873C22semidirect
24. C2×#16813873C22semidirect
28. #20300267C22semidirect
29. #20300247C22semidirect
30. #20300213C22semidirect
31. #20300210C22semidirect
32. #20300210C22semidirect
34. C2×M16×C227C22semidirect
35. C22×C8×C227C22semidirect
36. C22⋊C8×C227C22semidirect
146. C227#3009semidirect
156. #408D227semidirect
173. C2×M16C2×D227semidirect
174. C22×C8C2×D227semidirect
183. C22⋊C8C2×D227semidirect

Maximal subgroups

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

subgroupindex
C2×#203003342
C2×#203003212
C22×#168138732
#284096032
#284091962
#284091852
#284090382
#284090292
#408×C2272
#3009227