Finite Group Databasebeta

← #44408959

Subgroups — #44408959

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×#20300313C211yesmaximal, characteristic, complemented
3#28409539C211yesmaximal, characteristic, complemented
4#28409181C214yesmaximal, complemented
5#28409164C214yesmaximal, complemented
6#28409038C211yesmaximal, characteristic, complemented
7#28409037C211yesmaximal, characteristic, complemented
8#28408995C211yesmaximal, characteristic, complemented
9C2×AGL(1,ℤ/8)×C227C211yesmaximal, characteristic, complemented
10C24×D227—21no—
11C22×D908C2211yescharacteristic
12C22×C22⋊D227—21no—
13C2×D4×D227—24no—
14C2×D4×D227—22no—
15C2×D4×D227—24no—
16C2×D4×D227C2214yes—
17C22×C4×D227C2211yescharacteristic
18#20300313—24no—
19C2×#16813871C2212yescomplemented
20C2×#16813869C2212yescomplemented
21C2×#16813868C2211yescharacteristic, complemented
22#20300269—21no—
23#20300264C2212yescomplemented
24#20300259C2212yescomplemented
25#20300256—21no—
26C4×C22⋊D227—22no—
27#20300247C2211yescharacteristic, complemented
28#20300210C2214yescomplemented
29#20300209C2214yescomplemented
30(C2×D227)⋊C8C2214yescomplemented
31C2×#41×C227C2211yescharacteristic, complemented
32C22×D4×C227C2211yescharacteristic
33AGL(1,ℤ/8)×C227—24no—
34C2×SD16×C227C2212yescomplemented
35C2×D8×C227C2212yescomplemented
36C2×M16×C227C2211yescharacteristic, complemented
37C23×D227—28no—
38C23×D227—24no—
39C23×D227—21no—
40C23×D227—48no—
41C23×D227D411yescharacteristic
42C2×D908—21no—
43C2×D908—22no—
44C2×D908D414yescomplemented
45C2×D908C2312yes—
46C2×C22⋊D227—28no—
47C2×C22⋊D227—22no—
48C2×C22⋊D227—24no—
49D4×D227—216no—
50D4×D227—28no—
51D4×D227—28no—
52C22×Dic227D411yescharacteristic
53C2×C4×D227—22no—
54C2×C4×D227—22no—
55C2×C4×D227D412yes—
56C2×C4×D227C2312yes—
57#16813871—24no—
58#16813869—24no—
59#16813868—22no—
60C2×C227⋊C8C2312yescomplemented
61C4⋊Dic227C2312yescomplemented
62Dic227⋊C4—22no—
63C22⋊Dic227—22no—
64C22⋊Dic227—21no—
65(C2×D227)⋊C4—22no—
66C4×Dic227—21no—
67C24×C227—21no—
68#41×C227—22no—
69#41×C227—22no—
70C2×Q8×C227C2311yescharacteristic, complemented
71C2×D4×C227—21no—
72C2×D4×C227—21no—
73C2×D4×C227—22no—
74C2×D4×C227D414yescomplemented
75C2×D4×C227C2312yes—
76C2×D4×C227C2311yescharacteristic, complemented
77C22×C4×C227—21no—
78C22×C4×C227C2311yescharacteristic
79SD16×C227—24no—
80D8×C227—24no—
81M16×C227—22no—
82C2×C8×C227C2312yescomplemented
83C22×D227—22no—
84C22×D227—24no—
85C22×D227—216no—
86C22×D227—24no—
87C22×D227—22no—
88C22×D227—432no—
89C22×D227—48no—
90C22×D227—44no—
91C22×D227—48no—
92C22×D227C2×D412yes—
93D908—24no—
94D908—22no—
95D908C2×D414yes—
96C22⋊D227—216no—
97C22⋊D227—44no—
98C2×Dic227—22no—
99C2×Dic227—22no—
100C2×Dic227C2×D412yes—
101C4×D227—24no—
102C4×D227—22no—
103C4×D227—22no—
104C227⋊C8—22no—
105C23×C227—24no—
106C23×C227—22no—
107C23×C227—21no—
108C23×C227—44no—
109C23×C227C2×D411yescharacteristic
110Q8×C227—21no—
111Q8×C227—21no—
112D4×C227—24no—
113D4×C227—22no—
114D4×C227—21no—
115D4×C227—21no—
116D4×C227—42no—
117D4×C227C2×D414yes—
118C2×C4×C227—21no—
119C2×C4×C227—42no—
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—41no—
130C22×C227—42no—
131C22×C227#12012yes—
132C22×C227C22×D411yescharacteristic
133C2×D227—28no—
134C2×D227—22no—
135C2×D227—22no—
136C2×D227—24no—
137C2×D227—432no—
138C2×D227—48no—
139C2×D227—44no—
140C2×D227AGL(1,ℤ/8)14yescomplemented
141C4×C227—41no—
142C4×C227#12012yes—
143C4×C227C22×D411yescharacteristic
144C4×C227C22×D411yescharacteristic
145Dic227—22no—
146Dic227—42no—
147C2×C227—22no—
148C2×C227—44no—
149C2×C227—81no—
150C2×C227C2×#12011yescharacteristic
151C2×C227C2×AGL(1,ℤ/8)12yes—
152D227—24no—
153D227—44no—
154C227#300311yescharacteristic, complemented, Sylow
155#3003—2271nomaximal, Sylow
156C23×D4—2271no—
157C2×AGL(1,ℤ/8)—2271no—
158C2×AGL(1,ℤ/8)D22711yescharacteristic, complemented
159#525—2271no—
160#450—2274no—
161#447—2274no—
162#418—2271no—
163#418—2271no—
164#407—2271no—
165C25—4541no—
166C2×#41C2×D22711yescharacteristic, complemented
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)—24no—
176AGL(1,ℤ/8)—4544no—
177C2×SD16—2272no—
178C2×SD16C2×D22712yescomplemented
179C2×D8—2272no—
180C2×D8C2×D22712yescomplemented
181C2×M16—2271no—
182C2×M16C2×D22711yescharacteristic, complemented
183#123—4541no—
184C22⋊Q8—2272no—
185#121—2272no—
186#120—4541no—
187C4×D4—4542no—
188#117—2271no—
189#102—2274no—
190#102—2274no—
191C22⋊C8—2274no—
192C24—21no—
193C24—9088no—
194C24—4548no—
195C24—4544no—
196C24—4541no—
197C24—2271no—
198#41—22no—
199#41—22no—
200C2×Q8C22×D22711yescharacteristic, complemented
201C2×D4—21no—
202C2×D4—22no—
203C2×D4—21no—
204C2×D4—45416no—
205C2×D4—4548no—
206C2×D4—4542no—
207C2×D4—4548no—
208C2×D4—4548no—
209C2×D4—4544no—
210C2×D4—4541no—
211C2×D4—4542no—
212C2×D4—2274no—
213C2×D4—2272no—
214C2×D4C22⋊D22714yescomplemented
215C2×D4C22×D22712yes—
216C2×D4C22×D22711yescharacteristic, complemented
217C22×C4—21no—
218C22×C4—4542no—
219C22×C4—4542no—
220C22×C4—2271no—
221C22×C4—2272no—
222C22×C4—2272no—
223C22×C4C22×D22711yescharacteristic
224SD16—24no—
225SD16—4544no—
226D8—24no—
227D8—4544no—
228M16—22no—
229M16—4542no—
230C2×C8—2272no—
231C2×C8C22×D22712yescomplemented
232C4⋊C4—4542no—
233C4⋊C4—2272no—
234C22⋊C4—4542no—
235C22⋊C4—4541no—
236C22⋊C4—4542no—
237C42—4541no—
238C23—22no—
239C23—24no—
240C23—21no—
241C23—44no—
242C23—9084no—
243C23—90832no—
244C23—9088no—
245C23—9088no—
246C23—45416no—
247C23—4544no—
248C23—4542no—
249C23—4544no—
250C23—4542no—
251C23—2272no—
252C23D4×D22711yescharacteristic
253Q8—21no—
254Q8—21no—
255D4—22no—
256D4—21no—
257D4—21no—
258D4—24no—
259D4—42no—
260D4—9084no—
261D4—45416no—
262D4—4542no—
263D4—4544no—
264D4—2274no—
265D4C2×C22⋊D22714yes—
266C2×C4—21no—
267C2×C4—42no—
268C2×C4—4542no—
269C2×C4—4542no—
270C2×C4—4542no—
271C2×C4—4542no—
272C2×C4—4544no—
273C2×C4—2272no—
274C2×C4D4×D22712yes—
275C2×C4D4×D22711yescharacteristic
276C2×C4C2×C22⋊D22712yes—
277C2×C4C23×D22711yescharacteristic
278C8—22no—
279C8—4542no—
280C22—22no—
281C22—24no—
282C22—48no—
283C22—44no—
284C22—41no—
285C22—42no—
286C22—9088no—
287C22—90832no—
288C22—9084no—
289C22—4548no—
290C22—4544no—
291C22—4542no—
292C22—4542no—
293C22—2274no—
294C22#2030025612yes—
295C22C2×D4×D22711yescharacteristic
296C4—41no—
297C4—9082no—
298C4—4542no—
299C4#2030025612yes—
300C4C2×D4×D22711yescharacteristic
301C4C22×C22⋊D22711yescharacteristic
302C2—22no—
303C2—44no—
304C2—81no—
305C2—9084no—
306C2—4544no—
307C2C2×#2030025611yescharacteristic
308C2AGL(1,ℤ/8)×D22712yes—
subgroupG/H
centerC22C2×D4×D227
commutator, cosocleC2×C4×C227C24
FrattiniC2×C4C23×D227
FittingC2×AGL(1,ℤ/8)×C227C2
socleC22×C227C22×D4
hypercenterC2×AGL(1,ℤ/8)D227
hypocenterC227#3003
LCSG ▷ C2×C4×C227 ▷ C2×C227 ▷ C227
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. C22×D4×D227C2semidirect
2. C2×#20300313C2semidirect
3. #28409539C2semidirect
4. #28409181C2semidirect
5. #28409164C2semidirect
6. #28409038C2semidirect
7. #28409037C2semidirect
8. #28408995C2semidirect
9. C2×AGL(1,ℤ/8)×C227C2semidirect
19. C2×#16813871C22semidirect
20. C2×#16813869C22semidirect
21. C2×#16813868C22semidirect
23. #20300264C22semidirect
24. #20300259C22semidirect
27. #20300247C22semidirect
28. #20300210C22semidirect
29. #20300209C22semidirect
30. (C2×D227)⋊C8C22semidirect
31. C2×#41×C227C22semidirect
34. C2×SD16×C227C22semidirect
35. C2×D8×C227C22semidirect
36. C2×M16×C227C22semidirect
44. C2×D908D4semidirect
60. C2×C227⋊C8C23semidirect
61. C4⋊Dic227C23semidirect
70. C2×Q8×C227C23semidirect
74. C2×D4×C227D4semidirect
76. C2×D4×C227C23semidirect
82. C2×C8×C227C23semidirect
140. C2×D227AGL(1,ℤ/8)semidirect
154. C227#3003semidirect
158. C2×AGL(1,ℤ/8)D227semidirect
166. C2×#41C2×D227semidirect
178. C2×SD16C2×D227semidirect
180. C2×D8C2×D227semidirect
182. C2×M16C2×D227semidirect
200. C2×Q8C22×D227semidirect
214. C2×D4C22⋊D227semidirect
216. C2×D4C22×D227semidirect
231. C2×C8C22×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×#203003132
#284095392
#284091812
#284091642
#284090382
#284090372
#284089952
C2×AGL(1,ℤ/8)×C2272
#3003227