Finite Group Databasebeta

← #44409133

Subgroups — #44409133

Order 29056.

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

normal subgroups only

This group has 800 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
1C2×#20300333C211yesmaximal, characteristic, complemented
2#28409594C212yesmaximal, complemented
3#28409591C211yesmaximal, characteristic, complemented
4#28409223C214yesmaximal, complemented
5#28409220C214yesmaximal, complemented
6C2×#20300202C212yesmaximal, complemented
7C2×#99×C227C211yesmaximal, characteristic, complemented
8#20300333—24no—
9#20300333—22no—
10#20300333C2214yes—
11C2×#16813884—21no—
12C2×#16813883—22no—
13C22×C22⋊D227—21no—
14C22×C22⋊D227C2212yes—
15C22×C22⋊D227C2211yescharacteristic, complemented
16C2×D4×D227—22no—
17#20300272—22no—
18#20300272C2214yescomplemented
19#20300271—21no—
20#20300269C2212yescomplemented
21#20300262—24no—
22#20300260—22no—
23#20300258—22no—
24#20300258C2214yescomplemented
25#20300257—21no—
26#20300256C2212yescomplemented
27C2×C22⋊Dic227C2211yescharacteristic, complemented
28#20300202C2218yescomplemented
29C22×D4×C227C2211yescharacteristic
30C2×C22⋊C4×C227C2212yescomplemented
31#99×C227C2214yescomplemented
32C23×D227—21no—
33C23×D227D412yescomplemented
34C23×D227C2311yescharacteristic, complemented
35#16813884—24no—
36#16813884—24no—
37#16813883—28no—
38#16813883—28no—
39C2×Dic454—21no—
40C2×D908—21no—
41C2×D908—24no—
42C2×C22⋊D227—22no—
43C2×C22⋊D227—24no—
44C2×C22⋊D227—22no—
45C2×C22⋊D227—22no—
46C2×C22⋊D227—24no—
47C2×C22⋊D227—22no—
48C2×C22⋊D227—22no—
49C2×C22⋊D227—21no—
50C2×C22⋊D227—22no—
51C2×C22⋊D227—21no—
52C2×C22⋊D227—22no—
53C2×C22⋊D227—22no—
54C2×C22⋊D227D414yescomplemented
55C2×C22⋊D227C2314yes—
56D4×D227—28no—
57D4×D227—28no—
58C22×Dic227—21no—
59C22×Dic227—21no—
60C22×Dic227D412yescomplemented
61C22×Dic227C2311yescharacteristic, complemented
62C2×C4×D227—22no—
63C4⋊Dic227—24no—
64Dic227⋊C4—22no—
65C22⋊Dic227—21no—
66C22⋊Dic227—22no—
67C22⋊Dic227C2312yescomplemented
68(C2×D227)⋊C4—24no—
69(C2×D227)⋊C4—24no—
70(C2×D227)⋊C4—22no—
71C24×C227D411yescharacteristic, complemented
72C24×C227C2311yescharacteristic
73C2×D4×C227—22no—
74C2×D4×C227—22no—
75C2×D4×C227D412yescomplemented
76C2×D4×C227C2312yes—
77C22×C4×C227—22no—
78C22×C4×C227D411yescharacteristic, complemented
79C22⋊C4×C227—22no—
80C22⋊C4×C227C2314yescomplemented
81C22×D227—22no—
82C22×D227—24no—
83C22×D227—22no—
84C22×D227—21no—
85C22×D227—22no—
86C22×D227—81no—
87C22×D227—82no—
88C22×D227—82no—
89C22×D227C2×D414yes—
90Dic454—24no—
91D908—24no—
92D908—84no—
93C22⋊D227—28no—
94C22⋊D227—24no—
95C22⋊D227—24no—
96C22⋊D227—24no—
97C22⋊D227—24no—
98C22⋊D227—41no—
99C22⋊D227—41no—
100C22⋊D227—42no—
101C22⋊D227—82no—
102C2×Dic227—24no—
103C2×Dic227—22no—
104C2×Dic227—22no—
105C2×Dic227—22no—
106C2×Dic227—22no—
107C2×Dic227—21no—
108C2×Dic227—81no—
109C2×Dic227C2×D414yes—
110C4×D227—28no—
111C23×C227—21no—
112C23×C227—21no—
113C23×C227—81no—
114C23×C227—81no—
115C23×C227C2×D412yes—
116C23×C227C2×D412yes—
117C23×C227C2×D412yes—
118C23×C227C2×D411yescharacteristic
119C23×C227C2×D411yescharacteristic
120C23×C227C2411yescharacteristic
121D4×C227—24no—
122D4×C227—24no—
123C2×C4×C227—21no—
124C2×C4×C227—21no—
125C2×C4×C227—24no—
126C2×C4×C227—82no—
127C2×C4×C227C2×D412yes—
128C22×C227—22no—
129C22×C227—42no—
130C22×C227—41no—
131C22×C227—41no—
132C22×C227—42no—
133C22×C227—41no—
134C22×C227—42no—
135C22×C227—42no—
136C22×C227—81no—
137C22×C227#12011yescharacteristic
138C22×C227#12011yescharacteristic
139C22×C227#12012yes—
140C22×C227C22×D412yes—
141C22×C227C22×D411yescharacteristic
142C2×D227—24no—
143C2×D227—44no—
144C2×D227—44no—
145C2×D227—42no—
146C2×D227—44no—
147C2×D227—41no—
148C2×D227—82no—
149C2×D227—84no—
150C4×C227—22no—
151C4×C227—44no—
152Dic227—24no—
153Dic227—42no—
154Dic227—41no—
155C2×C227—21no—
156C2×C227—22no—
157C2×C227—21no—
158C2×C227—22no—
159C2×C227—21no—
160C2×C227—42no—
161C2×C227C2×#12011yescharacteristic
162D227—44no—
163D227—81no—
164C227#302611yescharacteristic, complemented, Sylow
165#3026—2271nomaximal, Sylow
166C2×H(5,𝔽2)—2271no—
167#534—2271no—
168#534—2272no—
169#458—2274no—
170B(4,𝔽2)—2274no—
171C2×#99—2272no—
172C2×#99D22711yescharacteristic, complemented
173H(5,𝔽2)—4544no—
174H(5,𝔽2)—4542no—
175H(5,𝔽2)—2274no—
176C2×#41—4542no—
177C2×#41—4541no—
178C22×D4—4542no—
179C22×D4—4541no—
180C22×D4—2271no—
181C22×D4—2272no—
182C22×D4C2×D22711yescharacteristic
183#123—4542no—
184#123—4541no—
185#123—2272no—
186#123—2274no—
187#121—4544no—
188#121—4542no—
189#120—4542no—
190#120—4541no—
191#120—2272no—
192#120—2274no—
193C2×C22⋊C4—2271no—
194C2×C22⋊C4C2×D22712yescomplemented
195#99—2278no—
196#99C2×D22714yescomplemented
197C24—4541no—
198C24—2271no—
199C24—2272no—
200C24D90811yescharacteristic, complemented
201C24C22×D22711yescharacteristic
202#41—4544no—
203#41—4548no—
204#41—4548no—
205#41—4544no—
206C2×Q8—4541no—
207C2×D4—22no—
208C2×D4—22no—
209C2×D4—4544no—
210C2×D4—4541no—
211C2×D4—4548no—
212C2×D4—4548no—
213C2×D4—4542no—
214C2×D4—4542no—
215C2×D4—4542no—
216C2×D4—4541no—
217C2×D4—4541no—
218C2×D4—4542no—
219C2×D4—4544no—
220C2×D4—4542no—
221C2×D4—4542no—
222C2×D4—4542no—
223C2×D4—4544no—
224C2×D4—4542no—
225C2×D4—2274no—
226C2×D4—2274no—
227C2×D4D90812yescomplemented
228C2×D4C22×D22712yes—
229C22×C4—22no—
230C22×C4—4542no—
231C22×C4—4541no—
232C22×C4—4541no—
233C22×C4—2271no—
234C22×C4—2272no—
235C22×C4D90811yescharacteristic, complemented
236C4⋊C4—4544no—
237C4⋊C4—4542no—
238C22⋊C4—22no—
239C22⋊C4—4544no—
240C22⋊C4—4544no—
241C22⋊C4—4541no—
242C22⋊C4—4542no—
243C22⋊C4—4542no—
244C22⋊C4—2272no—
245C22⋊C4C22×D22714yescomplemented
246C23—21no—
247C23—21no—
248C23—81no—
249C23—81no—
250C23—18162no—
251C23—18162no—
252C23—18161no—
253C23—4544no—
254C23—4542no—
255C23—4542no—
256C23—4541no—
257C23—4542no—
258C23—2274no—
259C23D4×D22712yes—
260C23D4×D22712yes—
261C23C2×D90812yes—
262C23C2×D90811yescharacteristic
263C23C2×D90811yescharacteristic
264C23C23×D22711yescharacteristic
265Q8—4544no—
266D4—24no—
267D4—24no—
268D4—18164no—
269D4—18162no—
270D4—9081no—
271D4—9081no—
272D4—9082no—
273D4—4544no—
274D4—4548no—
275D4—4544no—
276D4—4544no—
277D4—4544no—
278D4—4544no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—24no—
282C2×C4—82no—
283C2×C4—18161no—
284C2×C4—4548no—
285C2×C4—4542no—
286C2×C4—4542no—
287C2×C4—4541no—
288C2×C4—4544no—
289C2×C4—4542no—
290C2×C4—4542no—
291C2×C4—2274no—
292C2×C4C2×D90812yes—
293C22—22no—
294C22—42no—
295C22—41no—
296C22—41no—
297C22—42no—
298C22—41no—
299C22—42no—
300C22—42no—
301C22—81no—
302C22—18162no—
303C22—18164no—
304C22—9084no—
305C22—9082no—
306C22—9084no—
307C22—9084no—
308C22—9081no—
309C22—4544no—
310C22#2030025811yescharacteristic
311C22#2030025811yescharacteristic
312C22#2030025812yes—
313C22C2×D4×D22712yes—
314C22C22×D90811yescharacteristic
315C4—22no—
316C4—44no—
317C4—9082no—
318C4—9081no—
319C4—4544no—
320C2—22no—
321C2—21no—
322C2—21no—
323C2—21no—
324C2—22no—
325C2—42no—
326C2—18161no—
327C2—9084no—
328C2C2×#2030025811yescharacteristic
subgroupG/H
centerC2C2×#20300258
commutator, cosocleC23×C227C24
FrattiniC23C23×D227
FittingC2×#99×C227C2
socleC2×C227C2×#120
hypercenterC2×#99D227
hypocenterC227#3026
LCSG ▷ C23×C227 ▷ C2×C227 ▷ C227
UCSC1 ◁ C2 ◁ C24 ◁ C2×#99

Complements

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

NH ≅ G/Ntype
1. C2×#20300333C2semidirect
2. #28409594C2semidirect
3. #28409591C2semidirect
4. #28409223C2semidirect
5. #28409220C2semidirect
6. C2×#20300202C2semidirect
7. C2×#99×C227C2semidirect
15. C22×C22⋊D227C22semidirect
18. #20300272C22semidirect
20. #20300269C22semidirect
24. #20300258C22semidirect
26. #20300256C22semidirect
27. C2×C22⋊Dic227C22semidirect
28. #20300202C22semidirect
30. C2×C22⋊C4×C227C22semidirect
31. #99×C227C22semidirect
33. C23×D227D4semidirect
34. C23×D227C23semidirect
54. C2×C22⋊D227D4semidirect
60. C22×Dic227D4semidirect
61. C22×Dic227C23semidirect
67. C22⋊Dic227C23semidirect
71. C24×C227D4semidirect
75. C2×D4×C227D4semidirect
78. C22×C4×C227D4semidirect
80. C22⋊C4×C227C23semidirect
164. C227#3026semidirect
172. C2×#99D227semidirect
194. C2×C22⋊C4C2×D227semidirect
196. #99C2×D227semidirect
200. C24D908semidirect
227. C2×D4D908semidirect
235. C22×C4D908semidirect
245. C22⋊C4C22×D227semidirect

Maximal subgroups

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

subgroupindex
C2×#203003332
#284095942
#284095912
#284092232
#284092202
C2×#203002022
C2×#99×C2272
#3026227