Finite Group Databasebeta

← #44409209

Subgroups — #44409209

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×#20300312C212yesmaximal, complemented
2#127×D227C211yesmaximal, characteristic, complemented
3C4×#16813883C211yesmaximal, characteristic, complemented
4#28409233C211yesmaximal, characteristic, complemented
5#28409231C212yesmaximal, complemented
6#28409229C211yesmaximal, characteristic, complemented
7#28409227C211yesmaximal, characteristic, complemented
8#28409225C212yesmaximal, complemented
9#28409054C211yesmaximal, characteristic, complemented
10#28409035C212yesmaximal, complemented
11#459×C227C211yesmaximal, characteristic, complemented
12C2×#16813883C2211yescharacteristic
13C2×D4×D227—21no—
14C2×D4×D227—21no—
15C2×D4×D227C2212yes—
16#20300312—28no—
17C2×#16813873C2212yescomplemented
18C2×#16813872C2212yescomplemented
19C2×#16813869C2212yescomplemented
20C2×#16813867C2212yes—
21#20300285C2211yescharacteristic
22#20300284—21no—
23#20300284—21no—
24#20300284C2212yes—
25C4×Dic454C2211yescharacteristic, complemented
26#20300253C2211yescharacteristic, complemented
27D4×Dic227C2211yescharacteristic, complemented
28C4×C22⋊D227—21no—
29(C22⋊D227)⋊C4—21no—
30#20300246C2211yescharacteristic
31#20300244—21no—
32C2×C4×Dic227—21no—
33C42×D227C2211yescharacteristic
34C4⋊(C227⋊C8)C2211yescharacteristic, complemented
35Dic227⋊C8C2212yescomplemented
36#20300210C2212yescomplemented
37#20300209C2212yescomplemented
38#20300208C2212yescomplemented
39C2×D8×C227C2212yescomplemented
40#127×C227C2211yescharacteristic
41C4×D4×C227C2211yescharacteristic, complemented
42C4⋊C8×C227C2211yescharacteristic, complemented
43#102×C227C2212yescomplemented
44C23×D227—22no—
45#16813883—22no—
46#16813883D414yescomplemented
47C2×Dic454C2311yescharacteristic
48C2×D908—21no—
49C2×D908—21no—
50C2×D908C2312yes—
51C2×C22⋊D227—21no—
52C2×C22⋊D227—22no—
53C2×C22⋊D227—22no—
54C2×C22⋊D227—22no—
55D4×D227—24no—
56D4×D227—24no—
57D4×D227—44no—
58D4×D227—44no—
59C22×Dic227—21no—
60C22×Dic227—21no—
61C2×C4×D227D411yescharacteristic
62C2×C4×D227D411yescharacteristic
63C2×C4×D227C2311yescharacteristic
64#16813873—24no—
65#16813872—24no—
66#16813869—24no—
67#16813867—24no—
68C2×C227⋊C8C2312yes—
69C4⋊Dic227C2311yescharacteristic
70Dic227⋊C4—21no—
71Dic227⋊C4—21no—
72C22⋊Dic227—21no—
73(C2×D227)⋊C4—21no—
74C4×Dic227—21no—
75C4×Dic227—21no—
76C4×Dic227—21no—
77C4×Dic227D411yescharacteristic
78C4×Dic227D411yescharacteristic
79C4×Dic227C2311yescharacteristic
80C2×D4×C227—21no—
81C2×D4×C227—21no—
82C2×D4×C227C2312yes—
83C2×D4×C227C2311yescharacteristic
84C22×C4×C227—21no—
85D8×C227—24no—
86C2×C8×C227C2312yes—
87C4⋊C4×C227C2311yescharacteristic
88C22⋊C4×C227—21no—
89C42×C227C2311yescharacteristic
90C22×D227—22no—
91C22×D227—44no—
92C22×D227—44no—
93C22×D227—44no—
94C22×D227C2×D411yescharacteristic
95Dic454—21no—
96Dic454C2×D412yescomplemented
97D908—22no—
98D908—22no—
99D908—42no—
100D908—42no—
101C22⋊D227—24no—
102C22⋊D227—44no—
103C22⋊D227—44no—
104C22⋊D227—44no—
105C2×Dic227—21no—
106C2×Dic227—24no—
107C2×Dic227—21no—
108C2×Dic227—42no—
109C2×Dic227C2×D411yescharacteristic
110C2×Dic227C2×D411yescharacteristic
111C2×Dic227C2×D411yescharacteristic
112C4×D227—22no—
113C4×D227—22no—
114C4×D227C2×D412yes—
115C4×D227#4112yes—
116C227⋊C8—22no—
117C23×C227—21no—
118C23×C227—22no—
119D4×C227—22no—
120D4×C227—21no—
121D4×C227—22no—
122D4×C227—42no—
123D4×C227—42no—
124D4×C227C2×D412yescomplemented
125C2×C4×C227—21no—
126C2×C4×C227—42no—
127C2×C4×C227C2×D411yescharacteristic
128C2×C4×C227C2×D411yescharacteristic
129C2×C4×C227C2411yescharacteristic
130C8×C227—22no—
131C22×C227—22no—
132C22×C227—41no—
133C22×C227—41no—
134C22×C227—42no—
135C22×C227—42no—
136C22×C227—42no—
137C22×C227C22×D411yescharacteristic
138C2×D227—21no—
139C2×D227—21no—
140C2×D227—42no—
141C2×D227—42no—
142C2×D227—42no—
143C2×D227—88no—
144C2×D227#12112yes—
145C4×C227—21no—
146C4×C227—41no—
147C4×C227AGL(1,ℤ/8)12yescomplemented
148C4×C227C22×D411yescharacteristic
149C4×C227C2×#4111yescharacteristic
150Dic227—22no—
151Dic227—21no—
152Dic227—41no—
153Dic227#12112yes—
154Dic227AGL(1,ℤ/8)12yescomplemented
155C2×C227—42no—
156C2×C227—82no—
157C2×C227C2×#12111yescharacteristic
158C2×C227C2×AGL(1,ℤ/8)11yescharacteristic
159C2×C227C2×AGL(1,ℤ/8)11yescharacteristic
160D227—22no—
161D227—82no—
162C227#303811yescharacteristic, complemented, Sylow
163#3038—2271nomaximal, Sylow
164C2×AGL(1,ℤ/8)—2272no—
165C2×#127—2271no—
166C4×#41—2271no—
167#460—2271no—
168#460—2271no—
169#460—2272no—
170#459—2272no—
171#459—2271no—
172#459D22711yescharacteristic, complemented
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×SD16—2272no—
182C2×D8—2272no—
183C2×D8C2×D22712yescomplemented
184C2×M16—2272no—
185#127—4541no—
186#127—4541no—
187#127—2272no—
188#127—2271no—
189#127C2×D22711yescharacteristic
190C4×Q8—2271no—
191C4×Q8—2271no—
192C4×D4—4541no—
193C4×D4—4541no—
194C4×D4—2271no—
195C4×D4C2×D22711yescharacteristic, complemented
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×Q8—2271no—
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×D4C22×D22712yes—
225C2×D4C22×D22711yescharacteristic
226C22×C4—21no—
227C22×C4—4541no—
228C22×C4—4541no—
229C22×C4—2271no—
230C22×C4—2271no—
231C22×C4—2271no—
232SD16—4544no—
233SD16—4544no—
234D8—24no—
235D8—4544no—
236M16—4544no—
237C2×C8—2272no—
238C2×C8C22×D22712yes—
239C4⋊C4—4541no—
240C4⋊C4—4541no—
241C4⋊C4—2271no—
242C4⋊C4C22×D22711yescharacteristic
243C22⋊C4—21no—
244C22⋊C4—4541no—
245C22⋊C4—4541no—
246C42—4541no—
247C42—4541no—
248C42—4541no—
249C42—2271no—
250C42—2271no—
251C42—2271no—
252C42C22×D22711yescharacteristic
253C23—21no—
254C23—22no—
255C23—9084no—
256C23—9084no—
257C23—9084no—
258C23—4542no—
259C23—2271no—
260Q8—4541no—
261Q8—2272no—
262D4—22no—
263D4—21no—
264D4—22no—
265D4—42no—
266D4—42no—
267D4—9082no—
268D4—9084no—
269D4—9084no—
270D4—9084no—
271D4—9082no—
272D4—4544no—
273D4—4542no—
274D4—4542no—
275D4D4×D22712yescomplemented
276C2×C4—21no—
277C2×C4—42no—
278C2×C4—9082no—
279C2×C4—4542no—
280C2×C4—4541no—
281C2×C4—4541no—
282C2×C4—4544no—
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—22no—
295C22—41no—
296C22—41no—
297C22—42no—
298C22—42no—
299C22—42no—
300C22—18168no—
301C22—9082no—
302C22—9082no—
303C22—9082no—
304C22—4541no—
305C22—4541no—
306C22—2272no—
307C22C2×D4×D22711yescharacteristic
308C4—21no—
309C4—41no—
310C4—9081no—
311C4—4542no—
312C4—4541no—
313C4—2272no—
314C4—2272no—
315C4#2030031212yescomplemented
316C4C2×D4×D22711yescharacteristic
317C4#41×D22711yescharacteristic
318C2—42no—
319C2—82no—
320C2—18162no—
321C2—4542no—
322C2#121×D22711yescharacteristic
323C2AGL(1,ℤ/8)×D22711yescharacteristic
324C2C2×#2030031211yescharacteristic
subgroupG/H
centerC22C2×D4×D227
commutator, cosocleC2×C4×C227C24
FrattiniC2×C4C23×D227
Fitting#459×C227C2
socleC22×C227C22×D4
hypercenter#459D227
hypocenterC227#3038
LCSG ▷ C2×C4×C227 ▷ C2×C227 ▷ C227
UCSC1 ◁ C22 ◁ C42 ◁ #459

Complements

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

NH ≅ G/Ntype
1. C2×#20300312C2semidirect
2. #127×D227C2semidirect
3. C4×#16813883C2semidirect
4. #28409233C2semidirect
5. #28409231C2semidirect
6. #28409229C2semidirect
7. #28409227C2semidirect
8. #28409225C2semidirect
9. #28409054C2semidirect
10. #28409035C2semidirect
11. #459×C227C2semidirect
17. C2×#16813873C22semidirect
18. C2×#16813872C22semidirect
19. C2×#16813869C22semidirect
25. C4×Dic454C22semidirect
26. #20300253C22semidirect
27. D4×Dic227C22semidirect
34. C4⋊(C227⋊C8)C22semidirect
35. Dic227⋊C8C22semidirect
36. #20300210C22semidirect
37. #20300209C22semidirect
38. #20300208C22semidirect
39. C2×D8×C227C22semidirect
41. C4×D4×C227C22semidirect
42. C4⋊C8×C227C22semidirect
43. #102×C227C22semidirect
46. #16813883D4semidirect
96. Dic454C2×D4semidirect
124. D4×C227C2×D4semidirect
147. C4×C227AGL(1,ℤ/8)semidirect
154. Dic227AGL(1,ℤ/8)semidirect
162. C227#3038semidirect
172. #459D227semidirect
183. C2×D8C2×D227semidirect
195. C4×D4C2×D227semidirect
202. C4⋊C8C2×D227semidirect
206. #102C2×D227semidirect
275. D4D4×D227semidirect
315. C4#20300312semidirect

Maximal subgroups

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

subgroupindex
C2×#203003122
#127×D2272
C4×#168138832
#284092332
#284092312
#284092292
#284092272
#284092252
#284090542
#284090352
#459×C2272
#3038227