Finite Group Databasebeta

← #44409137

Subgroups — #44409137

Order 29056.

772 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 772 conjugacy classes of subgroups — too many to draw. Only its 145 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
1#28409868C211yesmaximal, characteristic, complemented
2#28409601C212yesmaximal, complemented
3#28409596C211yesmaximal, characteristic, complemented
4#28409220C218yesmaximal, complemented
5#28409005C212yesmaximal, complemented
6#410×C227C211yesmaximal, characteristic, complemented
7#20300336—21no—
8#20300335—22no—
9#20300334—22no—
10#20300333—21no—
11#20300333C2214yes—
12C2×#16813884—22no—
13C2×#16813884C2212yes—
14#41×D227—24no—
15C22×D908C2211yescharacteristic, complemented
16#20300285—22no—
17#20300284—21no—
18#20300277—21no—
19#20300276—22no—
20#20300263—24no—
21#20300262—22no—
22#20300258C2218yescomplemented
23#20300256C2214yescomplemented
24#20300247C2211yescharacteristic, complemented
25#20300202C2218yescomplemented
26C2×#41×C227C2211yescharacteristic
27#117×C227C2212yescomplemented
28#99×C227C2214yescomplemented
29C23×D227C2312yes—
30#16813885—24no—
31#16813884—24no—
32#16813884—24no—
33#16813884—22no—
34#16813884—22no—
35#16813884—24no—
36#16813884—22no—
37#16813884—22no—
38#16813883—24no—
39#16813883—28no—
40C2×Dic454—22no—
41C2×Dic454D412yescomplemented
42Q8×D227—24no—
43C2×D908—22no—
44C2×D908—28no—
45C2×D908—21no—
46C2×D908—22no—
47C2×D908—22no—
48C2×D908—21no—
49C2×D908D412yescomplemented
50C2×C22⋊D227—24no—
51C2×C22⋊D227—24no—
52C2×C22⋊D227C2314yes—
53D4×D227—24no—
54D4×D227—28no—
55C2×C4×D227—24no—
56C2×C4×D227D414yescomplemented
57C4⋊Dic227—21no—
58C22⋊Dic227C2312yes—
59(C2×D227)⋊C4—28no—
60(C2×D227)⋊C4—24no—
61C4×Dic227—21no—
62#41×C227—22no—
63#41×C227—22no—
64C2×Q8×C227D411yescharacteristic, complemented
65C2×D4×C227D411yescharacteristic, complemented
66C2×D4×C227C2312yes—
67C22×C4×C227D412yescomplemented
68C22×C4×C227C2311yescharacteristic
69C4⋊C4×C227—22no—
70C22⋊C4×C227C2314yes—
71C42×C227—22no—
72C22×D227—24no—
73C22×D227—22no—
74C22×D227—24no—
75C22×D227—82no—
76C22×D227C2×D414yes—
77Dic454—24no—
78Dic454—22no—
79Dic454—22no—
80D908—24no—
81D908—22no—
82D908—22no—
83D908—41no—
84D908—41no—
85D908—42no—
86D908—88no—
87C22⋊D227—28no—
88C22⋊D227—24no—
89C22⋊D227—24no—
90C22⋊D227—84no—
91C2×Dic227—24no—
92C2×Dic227—22no—
93C2×Dic227C2×D414yes—
94C4×D227—28no—
95C4×D227—24no—
96C4×D227—24no—
97C23×C227C2×D412yes—
98C23×C227C2411yescharacteristic
99Q8×C227—22no—
100D4×C227—24no—
101D4×C227—22no—
102C2×C4×C227—22no—
103C2×C4×C227—22no—
104C2×C4×C227—24no—
105C2×C4×C227C2×D412yes—
106C2×C4×C227C2×D412yes—
107C2×C4×C227C2×D412yes—
108C2×C4×C227C2×D411yescharacteristic
109C2×C4×C227C2×D411yescharacteristic
110C22×C227—22no—
111C22×C227—41no—
112C22×C227—42no—
113C22×C227C22×D412yes—
114C22×C227C22×D411yescharacteristic
115C2×D227—24no—
116C2×D227—44no—
117C2×D227—44no—
118C2×D227—48no—
119C2×D227—42no—
120C2×D227—84no—
121C4×C227—22no—
122C4×C227—44no—
123C4×C227#12011yescharacteristic
124C4×C227#12011yescharacteristic
125C4×C227#12012yes—
126Dic227—24no—
127Dic227—42no—
128C2×C227—21no—
129C2×C227—22no—
130C2×C227—42no—
131C2×C227C2×#12011yescharacteristic
132D227—44no—
133D227—82no—
134C227#302711yescharacteristic, complemented, Sylow
135#3027—2271nomaximal, Sylow
136#585—2271no—
137#535—2271no—
138#535—2272no—
139B(4,𝔽2)—2278no—
140#410—2272no—
141#410D22711yescharacteristic, complemented
1422−1+4—4542no—
1432−1+4—4541no—
144H(5,𝔽2)—4542no—
145H(5,𝔽2)—4541no—
146H(5,𝔽2)—2274no—
147C2×#41—4544no—
148C2×#41—4542no—
149C2×#41—2272no—
150C2×#41C2×D22711yescharacteristic
151C22×D4—2271no—
152#127—4542no—
153#127—4541no—
154#124—4542no—
155#124—4541no—
156#121—4544no—
157#121—4542no—
158#120—2274no—
159#120—2278no—
160#117—2271no—
161#117C2×D22712yescomplemented
162#99—2278no—
163#99C2×D22714yescomplemented
164C24—2272no—
165#41—22no—
166#41—22no—
167#41—4544no—
168#41—4544no—
169#41—4544no—
170#41—4544no—
171#41—4548no—
172#41—4542no—
173#41—4542no—
174#41—4542no—
175#41—4544no—
176#41—4542no—
177C2×Q8—4542no—
178C2×Q8—4544no—
179C2×Q8—2272no—
180C2×Q8D90811yescharacteristic, complemented
181C2×D4—4548no—
182C2×D4—4542no—
183C2×D4—4544no—
184C2×D4—4548no—
185C2×D4—4544no—
186C2×D4—4544no—
187C2×D4—4542no—
188C2×D4—4542no—
189C2×D4—4541no—
190C2×D4—4541no—
191C2×D4—2274no—
192C2×D4—2272no—
193C2×D4D90811yescharacteristic, complemented
194C2×D4C22×D22712yes—
195C22×C4—4544no—
196C22×C4—2274no—
197C22×C4D90812yescomplemented
198C22×C4C22×D22711yescharacteristic
199C4⋊C4—22no—
200C4⋊C4—4541no—
201C22⋊C4—4548no—
202C22⋊C4—4544no—
203C22⋊C4—2272no—
204C22⋊C4C22×D22714yes—
205C42—22no—
206C42—4541no—
207C23—18162no—
208C23—4544no—
209C23—4544no—
210C23—4542no—
211C23—2274no—
212C23C2×D90812yes—
213C23C23×D22711yescharacteristic
214Q8—22no—
215Q8—4544no—
216Q8—4542no—
217Q8—4542no—
218D4—24no—
219D4—22no—
220D4—18168no—
221D4—18164no—
222D4—9081no—
223D4—9081no—
224D4—9082no—
225D4—4544no—
226D4—4548no—
227D4—4544no—
228D4—4544no—
229D4—4542no—
230D4—4542no—
231C2×C4—22no—
232C2×C4—22no—
233C2×C4—24no—
234C2×C4—4548no—
235C2×C4—4542no—
236C2×C4—4544no—
237C2×C4—4544no—
238C2×C4—4544no—
239C2×C4—2274no—
240C2×C4D4×D22712yes—
241C2×C4D4×D22712yes—
242C2×C4C2×D90812yes—
243C2×C4C2×D90811yescharacteristic
244C2×C4C2×D90811yescharacteristic
245C22—22no—
246C22—41no—
247C22—42no—
248C22—18164no—
249C22—9088no—
250C22—9084no—
251C22—9084no—
252C22—9082no—
253C22—4544no—
254C22C2×D4×D22712yes—
255C22C22×D90811yescharacteristic
256C4—22no—
257C4—44no—
258C4—9082no—
259C4—4544no—
260C4#2030025811yescharacteristic
261C4#2030025811yescharacteristic
262C4#2030025812yes—
263C2—22no—
264C2—21no—
265C2—42no—
266C2—18162no—
267C2—9084no—
268C2C2×#2030025811yescharacteristic
subgroupG/H
centerC2C2×#20300258
commutator, cosocleC23×C227C24
FrattiniC23C23×D227
Fitting#410×C227C2
socleC2×C227C2×#120
hypercenter#410D227
hypocenterC227#3027
LCSG ▷ C23×C227 ▷ C2×C227 ▷ C227
UCSC1 ◁ C2 ◁ C22×C4 ◁ #410

Complements

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

NH ≅ G/Ntype
1. #28409868C2semidirect
2. #28409601C2semidirect
3. #28409596C2semidirect
4. #28409220C2semidirect
5. #28409005C2semidirect
6. #410×C227C2semidirect
15. C22×D908C22semidirect
22. #20300258C22semidirect
23. #20300256C22semidirect
24. #20300247C22semidirect
25. #20300202C22semidirect
27. #117×C227C22semidirect
28. #99×C227C22semidirect
41. C2×Dic454D4semidirect
49. C2×D908D4semidirect
56. C2×C4×D227D4semidirect
64. C2×Q8×C227D4semidirect
65. C2×D4×C227D4semidirect
67. C22×C4×C227D4semidirect
134. C227#3027semidirect
141. #410D227semidirect
161. #117C2×D227semidirect
163. #99C2×D227semidirect
180. C2×Q8D908semidirect
193. C2×D4D908semidirect
197. C22×C4D908semidirect

Maximal subgroups

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

subgroupindex
#284098682
#284096012
#284095962
#284092202
#284090052
#410×C2272
#3027227