Finite Group Databasebeta

← #44408962

Subgroups — #44408962

Order 29056.

994 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 994 conjugacy classes of subgroups — too many to draw. Only its 143 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×#20300315C212yesmaximal, complemented
3#28409545C211yesmaximal, characteristic, complemented
4#28409182C211yesmaximal, characteristic, complemented
5#28409181C212yesmaximal, complemented
6#28409168C211yesmaximal, characteristic, complemented
7#28409167C212yesmaximal, complemented
8#28409166C211yesmaximal, characteristic, complemented
9#28409035C212yesmaximal, complemented
10#28408994C211yesmaximal, characteristic, complemented
11#450×C227C211yesmaximal, characteristic, complemented
12C24×D227—21no—
13C2×#16813885C2211yescharacteristic, complemented
14C22×D908C2211yescharacteristic
15C22×C22⋊D227—21no—
16C2×D4×D227—28no—
17C2×D4×D227—21no—
18C2×D4×D227—21no—
19C2×D4×D227C2212yes—
20C2×D4×D227C2212yes—
21C22×C4×D227C2211yescharacteristic
22#20300315—28no—
23C2×#16813871C2212yescomplemented
24C2×D1816C2212yescomplemented
25C2×#16813869C2212yescomplemented
26C2×#16813867C2212yescomplemented
27#20300273—21no—
28#20300264C2211yescharacteristic, complemented
29#20300263C2211yescharacteristic, complemented
30#20300262C2211yescharacteristic, complemented
31#20300258—21no—
32#20300250—21no—
33(C22⋊D227)⋊C4—21no—
34#20300246C2211yescharacteristic, complemented
35#20300210C2212yescomplemented
36#20300209C2212yescomplemented
37#20300208C2212yescomplemented
38C22⋊(C227⋊C8)C2211yescharacteristic, complemented
39(C2×D227)⋊C8C2212yescomplemented
40C22×D4×C227C2211yescharacteristic
41C2×SD16×C227C2212yescomplemented
42C22⋊Q8×C227C2211yescharacteristic, complemented
43#102×C227C2212yescomplemented
44C22⋊C8×C227C2211yescharacteristic, complemented
45C23×D227—28no—
46C23×D227—22no—
47C23×D227—22no—
48C23×D227—21no—
49C23×D227—44no—
50C23×D227—44no—
51C23×D227D411yescharacteristic
52#16813885—24no—
53C2×D908—24no—
54C2×D908—21no—
55C2×D908—21no—
56C2×D908—21no—
57C2×D908C2312yes—
58C2×D908C2311yescharacteristic, complemented
59C2×C22⋊D227—28no—
60C2×C22⋊D227—22no—
61C2×C22⋊D227—22no—
62C2×C22⋊D227—21no—
63D4×D227—28no—
64D4×D227—28no—
65D4×D227—24no—
66D4×D227—24no—
67D4×D227—24no—
68D4×D227D418yescomplemented
69C22×Dic227D411yescharacteristic
70C2×C4×D227—24no—
71C2×C4×D227—21no—
72C2×C4×D227D411yescharacteristic
73C2×C4×D227D411yescharacteristic
74C2×C4×D227C2311yescharacteristic
75C2×C4×D227C2311yescharacteristic
76#16813871—24no—
77D1816—24no—
78#16813869—24no—
79#16813867—24no—
80C2×C227⋊C8C2312yescomplemented
81C4⋊Dic227C2311yescharacteristic, complemented
82Dic227⋊C4—21no—
83(C2×D227)⋊C4—21no—
84(C2×D227)⋊C4—21no—
85(C2×D227)⋊C4—21no—
86(C2×D227)⋊C4—21no—
87C4×Dic227—21no—
88C24×C227—21no—
89C2×Q8×C227C2311yescharacteristic, complemented
90C2×D4×C227—24no—
91C2×D4×C227—21no—
92C2×D4×C227C2312yes—
93C22×C4×C227C2311yescharacteristic
94SD16×C227—24no—
95C2×C8×C227C2312yescomplemented
96C4⋊C4×C227—21no—
97C4⋊C4×C227C2311yescharacteristic, complemented
98C22⋊C4×C227—21no—
99C22×D227—22no—
100C22×D227—22no—
101C22×D227—22no—
102C22×D227—28no—
103C22×D227—24no—
104C22×D227—22no—
105C22×D227—28no—
106C22×D227—21no—
107C22×D227—432no—
108C22×D227—44no—
109C22×D227—44no—
110C22×D227—44no—
111C22×D227—42no—
112C22×D227—42no—
113C22×D227—44no—
114C22×D227C2×D411yescharacteristic
115C22×D227C2×D411yescharacteristic
116D908—24no—
117D908—22no—
118D908—22no—
119D908—42no—
120D908—42no—
121D908C2×D414yes—
122C22⋊D227—28no—
123C22⋊D227—28no—
124C22⋊D227—42no—
125C2×Dic227—22no—
126C2×Dic227—21no—
127C2×Dic227C2×D411yescharacteristic
128C2×Dic227C2×D411yescharacteristic
129C4×D227—22no—
130C4×D227—22no—
131C4×D227—22no—
132C4×D227—42no—
133C4×D227C2×D414yes—
134C227⋊C8—22no—
135C23×C227—24no—
136C23×C227—22no—
137C23×C227—42no—
138C23×C227—42no—
139C23×C227C2×D411yescharacteristic
140Q8×C227—22no—
141D4×C227—24no—
142D4×C227—22no—
143D4×C227C2×D414yes—
144C2×C4×C227—22no—
145C2×C4×C227—21no—
146C2×C4×C227—21no—
147C2×C4×C227C2×D411yescharacteristic
148C2×C4×C227C2411yescharacteristic
149C8×C227—22no—
150C22×C227—21no—
151C22×C227—21no—
152C22×C227—24no—
153C22×C227—48no—
154C22×C227—42no—
155C22×C227—42no—
156C22×C227AGL(1,ℤ/8)12yescomplemented
157C22×C227C22×D411yescharacteristic
158C2×D227—28no—
159C2×D227—21no—
160C2×D227—21no—
161C2×D227—21no—
162C2×D227—21no—
163C2×D227—24no—
164C2×D227—416no—
165C2×D227—416no—
166C2×D227—48no—
167C2×D227—42no—
168C2×D227—42no—
169C2×D227—41no—
170C2×D227—41no—
171C2×D227—41no—
172C2×D227#12012yes—
173C2×D227AGL(1,ℤ/8)12yescomplemented
174C4×C227—21no—
175C4×C227—41no—
176C4×C227—41no—
177C4×C227C22×D412yes—
178Dic227—21no—
179Dic227—41no—
180Dic227#12012yes—
181C2×C227—22no—
182C2×C227—44no—
183C2×C227C2×#12011yescharacteristic
184C2×C227C2×AGL(1,ℤ/8)11yescharacteristic
185C2×C227C2×AGL(1,ℤ/8)11yescharacteristic
186D227—22no—
187D227—22no—
188D227—44no—
189D227—81no—
190C227#300311yescharacteristic, complemented, Sylow
191#3003—2271nomaximal, Sylow
192C23×D4—2271no—
193C2×AGL(1,ℤ/8)—2272no—
194#525—2271no—
195#450—2272no—
196#450—2271no—
197#450D22711yescharacteristic, complemented
198#447—2271no—
199#447—2272no—
200#447—2271no—
201#418—2272no—
202#407—2271no—
203C25—4541no—
204C2×#41—2271no—
205C22×D4—4548no—
206C22×D4—4541no—
207C22×D4—4541no—
208C22×D4—4541no—
209C22×D4—2272no—
210C22×D4—2272no—
211C22×D4—2271no—
212C22×D4C2×D22711yescharacteristic
213C23×C4—2271no—
214AGL(1,ℤ/8)—4548no—
215C2×SD16—2272no—
216C2×SD16C2×D22712yescomplemented
217C2×D8—2272no—
218C2×D8—2272no—
219C2×M16—2272no—
220#123—4541no—
221C22⋊Q8—2271no—
222C22⋊Q8C2×D22711yescharacteristic, complemented
223#121—2271no—
224#121—2271no—
225#120—4541no—
226C4×D4—4541no—
227C4×D4—4541no—
228#117—2271no—
229#102—2272no—
230#102—2272no—
231#102—2272no—
232#102C2×D22712yescomplemented
233C22⋊C8—2271no—
234C22⋊C8—2272no—
235C22⋊C8C2×D22711yescharacteristic, complemented
236C24—21no—
237C24—9084no—
238C24—9084no—
239C24—4548no—
240C24—4542no—
241C24—4542no—
242C24—4541no—
243C24—2271no—
244#41—4544no—
245C2×Q8C22×D22711yescharacteristic, complemented
246C2×D4—24no—
247C2×D4—21no—
248C2×D4—4541no—
249C2×D4—4541no—
250C2×D4—4544no—
251C2×D4—4548no—
252C2×D4—4548no—
253C2×D4—4544no—
254C2×D4—4541no—
255C2×D4—4548no—
256C2×D4—4544no—
257C2×D4—4544no—
258C2×D4—4542no—
259C2×D4—4542no—
260C2×D4—4541no—
261C2×D4—2278no—
262C2×D4—2271no—
263C2×D4—2272no—
264C2×D4C22×D22712yes—
265C22×C4—4541no—
266C22×C4—4544no—
267C22×C4—2271no—
268C22×C4—2271no—
269C22×C4—2271no—
270C22×C4—2271no—
271C22×C4—2271no—
272C22×C4C22×D22711yescharacteristic
273SD16—24no—
274SD16—4544no—
275D8—4544no—
276D8—4544no—
277M16—4544no—
278C2×C8—2272no—
279C2×C8C22×D22712yescomplemented
280C4⋊C4—21no—
281C4⋊C4—4541no—
282C4⋊C4—2271no—
283C4⋊C4C22×D22711yescharacteristic, complemented
284C22⋊C4—21no—
285C22⋊C4—4541no—
286C22⋊C4—4541no—
287C22⋊C4—4541no—
288C22⋊C4—4541no—
289C42—4541no—
290C23—24no—
291C23—22no—
292C23—42no—
293C23—42no—
294C23—9082no—
295C23—90832no—
296C23—9084no—
297C23—9084no—
298C23—9084no—
299C23—9084no—
300C23—9082no—
301C23—4544no—
302C23—4548no—
303C23—4548no—
304C23—4542no—
305C23—4542no—
306C23—4541no—
307C23—4542no—
308C23—4542no—
309C23—2271no—
310C23—2271no—
311C23D4×D22711yescharacteristic
312Q8—22no—
313D4—24no—
314D4—22no—
315D4—9082no—
316D4—9082no—
317D4—9082no—
318D4—4544no—
319D4—4548no—
320D4—4548no—
321D4—4542no—
322D4—4542no—
323D4—2274no—
324D4D4×D22714yes—
325C2×C4—22no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—9082no—
329C2×C4—4542no—
330C2×C4—4542no—
331C2×C4—4541no—
332C2×C4—4542no—
333C2×C4—4542no—
334C2×C4—2271no—
335C2×C4—2271no—
336C2×C4—2274no—
337C2×C4D4×D22711yescharacteristic
338C2×C4C23×D22711yescharacteristic
339C8—22no—
340C8—4542no—
341C22—21no—
342C22—21no—
343C22—24no—
344C22—48no—
345C22—42no—
346C22—42no—
347C22—9088no—
348C22—90816no—
349C22—90816no—
350C22—9081no—
351C22—9081no—
352C22—9081no—
353C22—9082no—
354C22—9082no—
355C22—4548no—
356C22—4544no—
357C22—4541no—
358C22—4541no—
359C22—4541no—
360C22—4541no—
361C22—2272no—
362C22—2272no—
363C22#2030031512yescomplemented
364C22C2×D4×D22711yescharacteristic
365C4—21no—
366C4—41no—
367C4—41no—
368C4—9081no—
369C4—4541no—
370C4—2272no—
371C4C2×D4×D22712yes—
372C2—22no—
373C2—44no—
374C2—18161no—
375C2—9084no—
376C2—4542no—
377C2—4542no—
378C2#120×D22711yescharacteristic
379C2AGL(1,ℤ/8)×D22711yescharacteristic
380C2C2×#2030031511yescharacteristic
subgroupG/H
centerC22C2×D4×D227
commutator, cosocleC2×C4×C227C24
FrattiniC2×C4C23×D227
Fitting#450×C227C2
socleC22×C227C22×D4
hypercenter#450D227
hypocenterC227#3003
LCSG ▷ C2×C4×C227 ▷ C2×C227 ▷ C227
UCSC1 ◁ C22 ◁ C22×C4 ◁ #450

Complements

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

NH ≅ G/Ntype
1. C22×D4×D227C2semidirect
2. C2×#20300315C2semidirect
3. #28409545C2semidirect
4. #28409182C2semidirect
5. #28409181C2semidirect
6. #28409168C2semidirect
7. #28409167C2semidirect
8. #28409166C2semidirect
9. #28409035C2semidirect
10. #28408994C2semidirect
11. #450×C227C2semidirect
13. C2×#16813885C22semidirect
23. C2×#16813871C22semidirect
24. C2×D1816C22semidirect
25. C2×#16813869C22semidirect
26. C2×#16813867C22semidirect
28. #20300264C22semidirect
29. #20300263C22semidirect
30. #20300262C22semidirect
34. #20300246C22semidirect
35. #20300210C22semidirect
36. #20300209C22semidirect
37. #20300208C22semidirect
38. C22⋊(C227⋊C8)C22semidirect
39. (C2×D227)⋊C8C22semidirect
41. C2×SD16×C227C22semidirect
42. C22⋊Q8×C227C22semidirect
43. #102×C227C22semidirect
44. C22⋊C8×C227C22semidirect
58. C2×D908C23semidirect
68. D4×D227D4semidirect
80. C2×C227⋊C8C23semidirect
81. C4⋊Dic227C23semidirect
89. C2×Q8×C227C23semidirect
95. C2×C8×C227C23semidirect
97. C4⋊C4×C227C23semidirect
156. C22×C227AGL(1,ℤ/8)semidirect
173. C2×D227AGL(1,ℤ/8)semidirect
190. C227#3003semidirect
197. #450D227semidirect
216. C2×SD16C2×D227semidirect
222. C22⋊Q8C2×D227semidirect
232. #102C2×D227semidirect
235. C22⋊C8C2×D227semidirect
245. C2×Q8C22×D227semidirect
279. C2×C8C22×D227semidirect
283. C4⋊C4C22×D227semidirect
363. C22#20300315semidirect

Maximal subgroups

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

subgroupindex
C22×D4×D2272
C2×#203003152
#284095452
#284091822
#284091812
#284091682
#284091672
#284091662
#284090352
#284089942
#450×C2272
#3003227