Finite Group Databasebeta

← #44408963

Subgroups — #44408963

Order 29056.

994 conjugacy classes of subgroups; only the 153 normal subgroups are shown.

normal subgroups only

This group has 994 conjugacy classes of subgroups — too many to draw. Only its 153 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×D4×D227C211yesmaximal, characteristic, complemented
2C2×#20300316C211yesmaximal, characteristic, complemented
3C2×#20300314C211yesmaximal, characteristic, complemented
4#28409544C211yesmaximal, characteristic, complemented
5#28409184C214yesmaximal, complemented
6#28409167C214yesmaximal, complemented
7#28409036C211yesmaximal, characteristic, complemented
8#28408995C211yesmaximal, characteristic, complemented
9#418×C227C211yesmaximal, characteristic, complemented
10C24×D227—21no—
11C2×#16813884C2211yescharacteristic, complemented
12C22×D908C2211yescharacteristic
13C22×C22⋊D227—21no—
14C2×D4×D227—24no—
15C2×D4×D227—22no—
16C2×D4×D227—24no—
17C2×D4×D227C2214yes—
18C22×C4×D227C2211yescharacteristic
19#20300316—24no—
20#20300314—24no—
21C2×#16813873C2212yescomplemented
22C2×#16813872C2212yescomplemented
23C2×D1816C2212yescomplemented
24C2×#16813869C2212yescomplemented
25C2×#16813868C2211yescharacteristic, complemented
26#20300272—21no—
27#20300268C2212yescomplemented
28#20300263C2212yescomplemented
29#20300258—21no—
30C4×D908—22no—
31#20300208C2214yescomplemented
32(C2×D227)⋊C8C2214yescomplemented
33C22×D4×C227C2211yescharacteristic
34C2×M16×C227C2211yescharacteristic, complemented
35#117×C227C2211yescharacteristic, complemented
36#102×C227C2214yescomplemented
37C23×D227—28no—
38C23×D227—24no—
39C23×D227—21no—
40C23×D227—48no—
41C23×D227D411yescharacteristic
42#16813884—22no—
43#16813884—22no—
44C2×Dic454C2311yescharacteristic, complemented
45C2×D908—22no—
46C2×D908—21no—
47C2×D908—22no—
48C2×D908D414yescomplemented
49C2×D908C2312yes—
50C2×D908C2311yescharacteristic, complemented
51C2×C22⋊D227—28no—
52C2×C22⋊D227—24no—
53C2×C22⋊D227—21no—
54D4×D227—216no—
55D4×D227—28no—
56D4×D227—28no—
57C22×Dic227D411yescharacteristic
58C2×C4×D227—22no—
59C2×C4×D227—22no—
60C2×C4×D227—21no—
61C2×C4×D227D412yes—
62C2×C4×D227C2312yes—
63#16813873—24no—
64#16813872—24no—
65D1816—24no—
66#16813869—24no—
67#16813868—22no—
68C2×C227⋊C8C2312yescomplemented
69C4⋊Dic227—22no—
70(C2×D227)⋊C4—22no—
71(C2×D227)⋊C4—22no—
72C24×C227—21no—
73C2×D4×C227—21no—
74C2×D4×C227—22no—
75C2×D4×C227D414yescomplemented
76C2×D4×C227C2312yes—
77C22×C4×C227C2311yescharacteristic
78M16×C227—22no—
79C2×C8×C227C2312yescomplemented
80C4⋊C4×C227C2312yescomplemented
81C22⋊C4×C227—21no—
82C42×C227—21no—
83C22×D227—22no—
84C22×D227—24no—
85C22×D227—216no—
86C22×D227—24no—
87C22×D227—22no—
88C22×D227—21no—
89C22×D227—432no—
90C22×D227—48no—
91C22×D227—44no—
92C22×D227—48no—
93C22×D227C2×D412yes—
94Dic454—21no—
95Dic454—21no—
96D908—24no—
97D908—22no—
98D908—21no—
99D908—21no—
100D908—44no—
101D908C2×D414yes—
102C22⋊D227—216no—
103C22⋊D227—42no—
104C2×Dic227—22no—
105C2×Dic227—21no—
106C2×Dic227C2×D412yes—
107C4×D227—24no—
108C4×D227—22no—
109C4×D227—22no—
110C4×D227—42no—
111C227⋊C8—22no—
112C23×C227—24no—
113C23×C227—22no—
114C23×C227—44no—
115C23×C227C2×D411yescharacteristic
116D4×C227—24no—
117D4×C227—22no—
118D4×C227C2×D414yes—
119C2×C4×C227—22no—
120C2×C4×C227C2×D411yescharacteristic
121C2×C4×C227C2×D412yes—
122C2×C4×C227C2×D412yes—
123C2×C4×C227C2411yescharacteristic
124C8×C227—22no—
125C22×C227—22no—
126C22×C227—24no—
127C22×C227—48no—
128C22×C227—44no—
129C22×C227#12012yes—
130C22×C227C22×D411yescharacteristic
131C2×D227—28no—
132C2×D227—22no—
133C2×D227—22no—
134C2×D227—24no—
135C2×D227—432no—
136C2×D227—48no—
137C2×D227—44no—
138C2×D227—41no—
139C2×D227—42no—
140C2×D227AGL(1,ℤ/8)14yescomplemented
141C4×C227—42no—
142C4×C227#12012yes—
143C4×C227C22×D411yescharacteristic
144C4×C227C22×D411yescharacteristic
145Dic227—22no—
146Dic227—41no—
147C2×C227—22no—
148C2×C227—44no—
149C2×C227C2×#12011yescharacteristic
150C2×C227C2×AGL(1,ℤ/8)12yes—
151D227—24no—
152D227—44no—
153D227—81no—
154C227#300311yescharacteristic, complemented, Sylow
155#3003—2271nomaximal, Sylow
156C23×D4—2271no—
157C2×AGL(1,ℤ/8)—2271no—
158C2×AGL(1,ℤ/8)—2271no—
159#525—2271no—
160#450—2274no—
161#447—2274no—
162#418—2271no—
163#418D22711yescharacteristic, complemented
164#407—2271no—
165C25—4541no—
166C2×#41—2271no—
167C22×D4—4544no—
168C22×D4—4544no—
169C22×D4—4542no—
170C22×D4—4541no—
171C22×D4—2274no—
172C22×D4—2271no—
173C22×D4C2×D22711yescharacteristic
174C23×C4—2271no—
175AGL(1,ℤ/8)—4544no—
176AGL(1,ℤ/8)—4544no—
177C2×SD16—2272no—
178C2×SD16—2272no—
179C2×D8—2272no—
180C2×D8—2272no—
181C2×M16—2271no—
182C2×M16C2×D22711yescharacteristic, complemented
183#123—4541no—
184C22⋊Q8—2272no—
185#121—2272no—
186#120—4541no—
187C4×D4—4542no—
188#117C2×D22711yescharacteristic, complemented
189#102—2274no—
190#102C2×D22714yescomplemented
191C22⋊C8—2274no—
192C24—21no—
193C24—9088no—
194C24—4548no—
195C24—4544no—
196C24—4541no—
197C24—2271no—
198#41—4542no—
199#41—4542no—
200C2×Q8—2271no—
201C2×D4—22no—
202C2×D4—21no—
203C2×D4—4542no—
204C2×D4—45416no—
205C2×D4—4548no—
206C2×D4—4541no—
207C2×D4—4548no—
208C2×D4—4548no—
209C2×D4—4544no—
210C2×D4—4541no—
211C2×D4—4542no—
212C2×D4—2274no—
213C2×D4—2271no—
214C2×D4—2272no—
215C2×D4D90814yescomplemented
216C2×D4C22×D22712yes—
217C22×C4—4541no—
218C22×C4—4542no—
219C22×C4—4542no—
220C22×C4—2271no—
221C22×C4—2272no—
222C22×C4—2272no—
223C22×C4C22×D22711yescharacteristic
224SD16—4544no—
225SD16—4544no—
226D8—4544no—
227D8—4544no—
228M16—22no—
229M16—4542no—
230C2×C8—2272no—
231C2×C8C22×D22712yescomplemented
232C4⋊C4—4542no—
233C4⋊C4C22×D22712yescomplemented
234C22⋊C4—21no—
235C22⋊C4—4542no—
236C22⋊C4—4542no—
237C42—21no—
238C23—22no—
239C23—24no—
240C23—44no—
241C23—9084no—
242C23—90832no—
243C23—9088no—
244C23—9088no—
245C23—45416no—
246C23—4544no—
247C23—4541no—
248C23—4542no—
249C23—4544no—
250C23—4542no—
251C23—2272no—
252C23D4×D22711yescharacteristic
253Q8—4541no—
254Q8—4541no—
255D4—22no—
256D4—24no—
257D4—9084no—
258D4—9082no—
259D4—45416no—
260D4—4541no—
261D4—4541no—
262D4—4542no—
263D4—4544no—
264D4—2274no—
265D4C2×D90814yes—
266C2×C4—22no—
267C2×C4—9082no—
268C2×C4—4541no—
269C2×C4—4542no—
270C2×C4—4542no—
271C2×C4—4542no—
272C2×C4—4544no—
273C2×C4—2272no—
274C2×C4D4×D22712yes—
275C2×C4D4×D22711yescharacteristic
276C2×C4C2×D90812yes—
277C2×C4C23×D22711yescharacteristic
278C8—22no—
279C8—4542no—
280C22—22no—
281C22—24no—
282C22—48no—
283C22—44no—
284C22—9088no—
285C22—90832no—
286C22—9081no—
287C22—9082no—
288C22—9084no—
289C22—4548no—
290C22—4544no—
291C22—4542no—
292C22—4542no—
293C22—2274no—
294C22#2030025812yes—
295C22C2×D4×D22711yescharacteristic
296C4—42no—
297C4—9081no—
298C4—4542no—
299C4#2030025812yes—
300C4C2×D4×D22711yescharacteristic
301C4C22×D90811yescharacteristic
302C2—22no—
303C2—44no—
304C2—18161no—
305C2—9084no—
306C2—4544no—
307C2C2×#2030025811yescharacteristic
308C2AGL(1,ℤ/8)×D22712yes—
subgroupG/H
centerC22C2×D4×D227
commutator, cosocleC2×C4×C227C24
FrattiniC2×C4C23×D227
Fitting#418×C227C2
socleC22×C227C22×D4
hypercenter#418D227
hypocenterC227#3003
LCSG ▷ C2×C4×C227 ▷ C2×C227 ▷ C227
UCSC1 ◁ C22 ◁ C22×C4 ◁ #418

Complements

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

NH ≅ G/Ntype
1. C22×D4×D227C2semidirect
2. C2×#20300316C2semidirect
3. C2×#20300314C2semidirect
4. #28409544C2semidirect
5. #28409184C2semidirect
6. #28409167C2semidirect
7. #28409036C2semidirect
8. #28408995C2semidirect
9. #418×C227C2semidirect
11. C2×#16813884C22semidirect
21. C2×#16813873C22semidirect
22. C2×#16813872C22semidirect
23. C2×D1816C22semidirect
24. C2×#16813869C22semidirect
25. C2×#16813868C22semidirect
27. #20300268C22semidirect
28. #20300263C22semidirect
31. #20300208C22semidirect
32. (C2×D227)⋊C8C22semidirect
34. C2×M16×C227C22semidirect
35. #117×C227C22semidirect
36. #102×C227C22semidirect
44. C2×Dic454C23semidirect
48. C2×D908D4semidirect
50. C2×D908C23semidirect
68. C2×C227⋊C8C23semidirect
75. C2×D4×C227D4semidirect
79. C2×C8×C227C23semidirect
80. C4⋊C4×C227C23semidirect
140. C2×D227AGL(1,ℤ/8)semidirect
154. C227#3003semidirect
163. #418D227semidirect
182. C2×M16C2×D227semidirect
188. #117C2×D227semidirect
190. #102C2×D227semidirect
215. C2×D4D908semidirect
231. C2×C8C22×D227semidirect
233. C4⋊C4C22×D227semidirect

Maximal subgroups

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

subgroupindex
C22×D4×D2272
C2×#203003162
C2×#203003142
#284095442
#284091842
#284091672
#284090362
#284089952
#418×C2272
#3003227