Finite Group Databasebeta

← #44409211

Subgroups — #44409211

Order 29056.

584 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 584 conjugacy classes of subgroups — too many to draw. Only its 133 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×#20300315C212yesmaximal, complemented
2#127×D227C211yesmaximal, characteristic, complemented
3C4×#16813885C211yesmaximal, characteristic, complemented
4#28409232C211yesmaximal, characteristic, complemented
5#28409230C212yesmaximal, complemented
6#28409228C211yesmaximal, characteristic, complemented
7#28409226C212yesmaximal, complemented
8#28409224C211yesmaximal, characteristic, complemented
9#28409054C211yesmaximal, characteristic, complemented
10#28409035C212yesmaximal, complemented
11#460×C227C211yesmaximal, characteristic, complemented
12C2×#16813885C2211yescharacteristic
13C2×D4×D227—21no—
14C2×D4×D227—21no—
15C2×D4×D227C2212yes—
16#20300315—28no—
17C2×#16813871C2212yescomplemented
18C2×D1816C2212yescomplemented
19C2×#16813869C2212yescomplemented
20C2×#16813867C2212yes—
21#20300285C2211yescharacteristic
22#20300284—21no—
23#20300284—21no—
24#20300284C2212yes—
25Q8×Dic227C2211yescharacteristic, complemented
26C4×D908—21no—
27C4×D908C2211yescharacteristic, complemented
28#20300250—21no—
29#20300250C2211yescharacteristic, complemented
30#20300246—21no—
31#20300246C2211yescharacteristic
32C42×D227—21no—
33C42×D227C2211yescharacteristic
34C4⋊(C227⋊C8)C2211yescharacteristic, complemented
35Dic227⋊C8C2212yescomplemented
36#20300210C2212yescomplemented
37#20300209C2212yescomplemented
38#20300208C2212yescomplemented
39C2×SD16×C227C2212yescomplemented
40#127×C227C2211yescharacteristic
41C4×Q8×C227C2211yescharacteristic, complemented
42C4⋊C8×C227C2211yescharacteristic, complemented
43#102×C227C2212yescomplemented
44C23×D227—22no—
45#16813885—22no—
46#16813885D414yescomplemented
47C2×D908—21no—
48C2×D908—21no—
49C2×D908—21no—
50C2×D908C2312yes—
51C2×D908C2311yescharacteristic
52C2×C22⋊D227—22no—
53C2×C22⋊D227—22no—
54C2×C22⋊D227—22no—
55D4×D227—24no—
56D4×D227—24no—
57D4×D227—44no—
58D4×D227—44no—
59C2×C4×D227—21no—
60C2×C4×D227—21no—
61C2×C4×D227—21no—
62C2×C4×D227D411yescharacteristic
63C2×C4×D227D411yescharacteristic
64C2×C4×D227C2311yescharacteristic
65#16813871—24no—
66D1816—24no—
67#16813869—24no—
68#16813867—24no—
69C2×C227⋊C8C2312yes—
70C4⋊Dic227—21no—
71C4⋊Dic227C2311yescharacteristic
72(C2×D227)⋊C4—21no—
73(C2×D227)⋊C4—21no—
74(C2×D227)⋊C4—21no—
75C4×Dic227—21no—
76C4×Dic227—21no—
77C4×Dic227D411yescharacteristic
78C4×Dic227D411yescharacteristic
79C4×Dic227C2311yescharacteristic
80C2×Q8×C227C2311yescharacteristic
81C2×D4×C227—21no—
82C2×D4×C227—21no—
83C2×D4×C227C2312yes—
84SD16×C227—24no—
85C2×C8×C227C2312yes—
86C4⋊C4×C227—21no—
87C4⋊C4×C227C2311yescharacteristic
88C42×C227—21no—
89C42×C227C2311yescharacteristic
90C22×D227—21no—
91C22×D227—22no—
92C22×D227—44no—
93C22×D227—44no—
94C22×D227—44no—
95C22×D227C2×D411yescharacteristic
96D908—24no—
97D908—22no—
98D908—21no—
99D908—22no—
100D908—42no—
101D908—42no—
102D908C2×D412yescomplemented
103C22⋊D227—44no—
104C22⋊D227—44no—
105C22⋊D227—44no—
106C2×Dic227—21no—
107C2×Dic227C2×D411yescharacteristic
108C2×Dic227C2×D411yescharacteristic
109C2×Dic227C2×D411yescharacteristic
110C4×D227—22no—
111C4×D227—24no—
112C4×D227—22no—
113C4×D227—42no—
114C4×D227—42no—
115C4×D227C2×D412yes—
116C4×D227#4112yes—
117C227⋊C8—22no—
118C23×C227—22no—
119Q8×C227—21no—
120Q8×C227C2×D412yescomplemented
121D4×C227—22no—
122D4×C227—22no—
123D4×C227—42no—
124D4×C227—42no—
125C2×C4×C227—21no—
126C2×C4×C227—21no—
127C2×C4×C227C2×D411yescharacteristic
128C2×C4×C227C2×D411yescharacteristic
129C2×C4×C227C2411yescharacteristic
130C8×C227—22no—
131C22×C227—42no—
132C22×C227—42no—
133C22×C227—42no—
134C22×C227C22×D411yescharacteristic
135C2×D227—21no—
136C2×D227—21no—
137C2×D227—22no—
138C2×D227—41no—
139C2×D227—41no—
140C2×D227—42no—
141C2×D227—42no—
142C2×D227—42no—
143C2×D227—88no—
144C2×D227#12112yes—
145C4×C227—22no—
146C4×C227—21no—
147C4×C227—41no—
148C4×C227AGL(1,ℤ/8)12yescomplemented
149C4×C227C22×D411yescharacteristic
150C4×C227C2×#4111yescharacteristic
151Dic227—21no—
152Dic227—41no—
153Dic227#12112yes—
154Dic227AGL(1,ℤ/8)12yescomplemented
155C2×C227—82no—
156C2×C227C2×#12111yescharacteristic
157C2×C227C2×AGL(1,ℤ/8)11yescharacteristic
158C2×C227C2×AGL(1,ℤ/8)11yescharacteristic
159D227—22no—
160D227—42no—
161D227—82no—
162C227#303811yescharacteristic, complemented, Sylow
163#3038—2271nomaximal, Sylow
164C2×AGL(1,ℤ/8)—2272no—
165C2×#127—2271no—
166C4×#41—2271no—
167#460—2272no—
168#460—2271no—
169#460D22711yescharacteristic, complemented
170#459—2271no—
171#459—2271no—
172#459—2272no—
173#423—2271no—
174#418—2272no—
175C2×#41—2271no—
176C22×D4—4541no—
177C22×D4—4541no—
178C22×D4—2272no—
179AGL(1,ℤ/8)—4548no—
180C2×SD16—2272no—
181C2×SD16C2×D22712yescomplemented
182C2×D8—2272no—
183C2×D8—2272no—
184C2×M16—2272no—
185#127—4541no—
186#127—4541no—
187#127—2272no—
188#127—2271no—
189#127C2×D22711yescharacteristic
190C4×Q8—2271no—
191C4×Q8C2×D22711yescharacteristic, complemented
192C4×D4—4541no—
193C4×D4—4541no—
194C4×D4—2271no—
195C4×D4—2271no—
196#117—4541no—
197#117—2271no—
198C2×C42—4541no—
199C2×C42—2271no—
200C4⋊C8—2272no—
201C4⋊C8—2271no—
202C4⋊C8C2×D22711yescharacteristic, complemented
203#102—2272no—
204#102—2272no—
205#102—2272no—
206#102C2×D22712yescomplemented
207C24—4542no—
208#41—4542no—
209#41—2274no—
210C2×Q8C22×D22711yescharacteristic
211C2×D4—21no—
212C2×D4—21no—
213C2×D4—9084no—
214C2×D4—9084no—
215C2×D4—4541no—
216C2×D4—4544no—
217C2×D4—4544no—
218C2×D4—4542no—
219C2×D4—4542no—
220C2×D4—4542no—
221C2×D4—4541no—
222C2×D4—4541no—
223C2×D4—2272no—
224C2×D4—2271no—
225C2×D4C22×D22712yes—
226C22×C4—4541no—
227C22×C4—4541no—
228C22×C4—4541no—
229C22×C4—2271no—
230C22×C4—2271no—
231C22×C4—2271no—
232SD16—24no—
233SD16—4544no—
234D8—4544no—
235D8—4544no—
236M16—4544no—
237C2×C8—2272no—
238C2×C8C22×D22712yes—
239C4⋊C4—21no—
240C4⋊C4—4541no—
241C4⋊C4—2271no—
242C4⋊C4C22×D22711yescharacteristic
243C22⋊C4—4541no—
244C22⋊C4—4541no—
245C22⋊C4—4541no—
246C42—21no—
247C42—4541no—
248C42—4541no—
249C42—2271no—
250C42—2271no—
251C42—2271no—
252C42C22×D22711yescharacteristic
253C23—22no—
254C23—9084no—
255C23—9084no—
256C23—9084no—
257C23—4542no—
258C23—4541no—
259C23—2271no—
260Q8—21no—
261Q8D4×D22712yescomplemented
262D4—22no—
263D4—22no—
264D4—42no—
265D4—42no—
266D4—9082no—
267D4—9084no—
268D4—9084no—
269D4—9084no—
270D4—9082no—
271D4—4544no—
272D4—4542no—
273D4—4542no—
274D4—4541no—
275D4—2272no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—9082no—
279C2×C4—9082no—
280C2×C4—4544no—
281C2×C4—4542no—
282C2×C4—4541no—
283C2×C4—4542no—
284C2×C4—2271no—
285C2×C4—2271no—
286C2×C4—2271no—
287C2×C4—2272no—
288C2×C4—2272no—
289C2×C4D4×D22711yescharacteristic
290C2×C4D4×D22711yescharacteristic
291C2×C4C23×D22711yescharacteristic
292C8—22no—
293C8—4542no—
294C22—42no—
295C22—42no—
296C22—42no—
297C22—18168no—
298C22—9082no—
299C22—9082no—
300C22—9082no—
301C22—9081no—
302C22—9081no—
303C22—4542no—
304C22—4541no—
305C22—4541no—
306C22—2272no—
307C22C2×D4×D22711yescharacteristic
308C4—22no—
309C4—21no—
310C4—41no—
311C4—9081no—
312C4—4541no—
313C4—2272no—
314C4—2272no—
315C4#2030031512yescomplemented
316C4C2×D4×D22711yescharacteristic
317C4#41×D22711yescharacteristic
318C2—82no—
319C2—18162no—
320C2—9082no—
321C2—4542no—
322C2#121×D22711yescharacteristic
323C2AGL(1,ℤ/8)×D22711yescharacteristic
324C2C2×#2030031511yescharacteristic
subgroupG/H
centerC22C2×D4×D227
commutator, cosocleC2×C4×C227C24
FrattiniC2×C4C23×D227
Fitting#460×C227C2
socleC22×C227C22×D4
hypercenter#460D227
hypocenterC227#3038
LCSG ▷ C2×C4×C227 ▷ C2×C227 ▷ C227
UCSC1 ◁ C22 ◁ C42 ◁ #460

Complements

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

NH ≅ G/Ntype
1. C2×#20300315C2semidirect
2. #127×D227C2semidirect
3. C4×#16813885C2semidirect
4. #28409232C2semidirect
5. #28409230C2semidirect
6. #28409228C2semidirect
7. #28409226C2semidirect
8. #28409224C2semidirect
9. #28409054C2semidirect
10. #28409035C2semidirect
11. #460×C227C2semidirect
17. C2×#16813871C22semidirect
18. C2×D1816C22semidirect
19. C2×#16813869C22semidirect
25. Q8×Dic227C22semidirect
27. C4×D908C22semidirect
29. #20300250C22semidirect
34. C4⋊(C227⋊C8)C22semidirect
35. Dic227⋊C8C22semidirect
36. #20300210C22semidirect
37. #20300209C22semidirect
38. #20300208C22semidirect
39. C2×SD16×C227C22semidirect
41. C4×Q8×C227C22semidirect
42. C4⋊C8×C227C22semidirect
43. #102×C227C22semidirect
46. #16813885D4semidirect
102. D908C2×D4semidirect
120. Q8×C227C2×D4semidirect
148. C4×C227AGL(1,ℤ/8)semidirect
154. Dic227AGL(1,ℤ/8)semidirect
162. C227#3038semidirect
169. #460D227semidirect
181. C2×SD16C2×D227semidirect
191. C4×Q8C2×D227semidirect
202. C4⋊C8C2×D227semidirect
206. #102C2×D227semidirect
261. Q8D4×D227semidirect
315. C4#20300315semidirect

Maximal subgroups

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

subgroupindex
C2×#203003152
#127×D2272
C4×#168138852
#284092322
#284092302
#284092282
#284092262
#284092242
#284090542
#284090352
#460×C2272
#3038227