Finite Group Databasebeta

← #44408951

Subgroups — #44408951

Order 29056.

764 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 764 conjugacy classes of subgroups — too many to draw. Only its 141 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×#16813885C211yesmaximal, characteristic, complemented
2C2×#20300309C212yesmaximal, complemented
3#28409542C211yesmaximal, characteristic, complemented
4#28409197C212yesmaximal, complemented
5#28409188C211yesmaximal, characteristic, complemented
6#28409185C211yesmaximal, characteristic, complemented
7#28409176C212yesmaximal, complemented
8#28409165C211yesmaximal, characteristic, complemented
9#28409027C212yesmaximal, complemented
10C22⋊C8×D227C211yesmaximal, characteristic, complemented
11#448×C227C211yesmaximal, characteristic, complemented
12C2×#16813885—28no—
13C2×#16813885—21no—
14C2×#16813885—21no—
15C2×#16813885C2212yes—
16C2×#16813885C2212yes—
17C2×#16813883C2211yescharacteristic, complemented
18C22×D908—21no—
19C22×D908C2211yescharacteristic
20C22×C4×D227—21no—
21C22×C4×D227C2211yescharacteristic
22#20300309—28no—
23C2×#16813874C2212yescomplemented
24C2×#16813873C2212yescomplemented
25C2×#16813869C2212yescomplemented
26C2×C8×D227C2212yescomplemented
27#20300270—21no—
28#20300268C2211yescharacteristic, complemented
29#20300267C2211yescharacteristic
30#20300259C2211yescharacteristic, complemented
31#20300256—21no—
32D4×Dic227—21no—
33(C22⋊D227)⋊C4—21no—
34#20300246C2211yescharacteristic, complemented
35#20300212C2212yescomplemented
36#20300210C2212yescomplemented
37#20300208C2212yescomplemented
38C22⋊(C227⋊C8)C2211yescharacteristic
39(C2×D227)⋊C8C2212yescomplemented
40C22×Q8×C227C2211yescharacteristic
41C2×SD16×C227C2212yescomplemented
42#121×C227C2211yescharacteristic
43#103×C227C2212yescomplemented
44C22⋊C8×C227C2211yescharacteristic
45C23×D227—21no—
46C23×D227D411yescharacteristic
47#16813885—28no—
48#16813885—28no—
49#16813885—24no—
50#16813885—24no—
51#16813885—24no—
52#16813885D418yescomplemented
53#16813883—24no—
54C2×Dic454C2311yescharacteristic
55C2×D908—28no—
56C2×D908—22no—
57C2×D908—22no—
58C2×D908—24no—
59C2×D908—21no—
60C2×D908C2312yes—
61C2×C22⋊D227—21no—
62C2×C22⋊D227—21no—
63C22×Dic227—21no—
64C22×Dic227D411yescharacteristic
65C2×C4×D227—28no—
66C2×C4×D227—22no—
67C2×C4×D227—22no—
68C2×C4×D227—24no—
69C2×C4×D227D411yescharacteristic
70C2×C4×D227D411yescharacteristic
71C2×C4×D227C2311yescharacteristic
72C2×C4×D227C2311yescharacteristic
73#16813874—24no—
74#16813873—24no—
75#16813869—24no—
76C2×C227⋊C8C2312yes—
77C8×D227—24no—
78C4⋊Dic227—21no—
79C4⋊Dic227C2311yescharacteristic
80Dic227⋊C4—21no—
81C22⋊Dic227—21no—
82C22⋊Dic227—21no—
83(C2×D227)⋊C4—21no—
84(C2×D227)⋊C4—21no—
85C4×Dic227—21no—
86C2×Q8×C227—24no—
87C2×Q8×C227—21no—
88C2×Q8×C227C2312yes—
89C2×D4×C227—21no—
90C2×D4×C227C2311yescharacteristic
91C22×C4×C227—21no—
92C22×C4×C227C2311yescharacteristic
93SD16×C227—24no—
94C2×C8×C227C2312yes—
95C4⋊C4×C227C2311yescharacteristic
96C22⋊C4×C227—21no—
97C22×D227—24no—
98C22×D227—22no—
99C22×D227—22no—
100C22×D227—22no—
101C22×D227—41no—
102C22×D227—42no—
103C22×D227—42no—
104C22×D227C2×D411yescharacteristic
105C22×D227C2×D411yescharacteristic
106Dic454—22no—
107D908—28no—
108D908—24no—
109D908—28no—
110D908—22no—
111D908C2×D414yes—
112C22⋊D227—42no—
113C22⋊D227—42no—
114C2×Dic227—22no—
115C2×Dic227—21no—
116C2×Dic227—21no—
117C2×Dic227—42no—
118C2×Dic227C2×D411yescharacteristic
119C2×Dic227C2×D411yescharacteristic
120C4×D227—28no—
121C4×D227—22no—
122C4×D227—22no—
123C4×D227—28no—
124C4×D227—22no—
125C4×D227C2×D414yes—
126C227⋊C8—22no—
127C23×C227—21no—
128C23×C227C2×D411yescharacteristic
129Q8×C227—24no—
130Q8×C227—22no—
131Q8×C227C2×D414yes—
132D4×C227—22no—
133D4×C227—42no—
134C2×C4×C227—24no—
135C2×C4×C227—22no—
136C2×C4×C227—22no—
137C2×C4×C227—21no—
138C2×C4×C227C2×D411yescharacteristic
139C2×C4×C227C2411yescharacteristic
140C8×C227—22no—
141C22×C227—21no—
142C22×C227—21no—
143C22×C227—41no—
144C22×C227—41no—
145C22×C227—41no—
146C22×C227#13512yescomplemented
147C22×C227C22×D411yescharacteristic
148C2×D227—21no—
149C2×D227—21no—
150C2×D227—21no—
151C2×D227—24no—
152C2×D227—21no—
153C2×D227—21no—
154C2×D227—44no—
155C2×D227—48no—
156C2×D227—42no—
157C2×D227—42no—
158C2×D227#12012yes—
159C4×C227—24no—
160C4×C227—21no—
161C4×C227—41no—
162C4×C227C22×D412yes—
163Dic227—21no—
164Dic227—41no—
165Dic227—41no—
166Dic227#12012yes—
167C2×C227—22no—
168C2×C227—81no—
169C2×C227C2×#12011yescharacteristic
170C2×C227C2×#13511yescharacteristic
171C2×C227#57511yescharacteristic
172D227—22no—
173D227—41no—
174D227—44no—
175C227#300211yescharacteristic, complemented, Sylow
176#3002—2271nomaximal, Sylow
177C22×#41—2271no—
178C2×#135—2272no—
179#525—2271no—
180#452—2272no—
181#451—2271no—
182#450—2271no—
183#449—2272no—
184#448D22711yescharacteristic, complemented
185#447—2271no—
186#416—2272no—
187C2×C22⋊C8—2271no—
188C2×#41—4548no—
189C2×#41—4541no—
190C2×#41—4541no—
191C2×#41—2272no—
192C2×#41—2272no—
193C2×#41—2271no—
194C22×Q8C2×D22711yescharacteristic
195C22×D4—4541no—
196C22×D4—2271no—
197C23×C4—4541no—
198C23×C4—2271no—
199#135—4548no—
200C2×Q16—2272no—
201C2×SD16—2272no—
202C2×SD16C2×D22712yescomplemented
203C2×D8—2272no—
204C22×C8—2272no—
205#123—4541no—
206C22⋊Q8—2271no—
207C22⋊Q8—2271no—
208#121—2271no—
209#121C2×D22711yescharacteristic
210#120—4541no—
211C4×D4—4541no—
212C4×D4—4541no—
213#117—2271no—
214#103—2272no—
215#103C2×D22712yescomplemented
216#102—2272no—
217#102—2272no—
218C22⋊C8—2271no—
219C22⋊C8—2272no—
220C22⋊C8C2×D22711yescharacteristic
221C24—4541no—
222C24—2271no—
223#41—4544no—
224#41—4548no—
225#41—4548no—
226#41—4544no—
227#41—4544no—
228#41—4544no—
229#41—2278no—
230C2×Q8—24no—
231C2×Q8—21no—
232C2×Q8—2271no—
233C2×Q8C22×D22712yes—
234C2×D4—21no—
235C2×D4—4548no—
236C2×D4—4542no—
237C2×D4—4542no—
238C2×D4—4544no—
239C2×D4—4541no—
240C2×D4—4541no—
241C2×D4—4541no—
242C2×D4—2272no—
243C2×D4C22×D22711yescharacteristic
244C22×C4—21no—
245C22×C4—4548no—
246C22×C4—4542no—
247C22×C4—4542no—
248C22×C4—4544no—
249C22×C4—4541no—
250C22×C4—2271no—
251C22×C4—2271no—
252C22×C4—2271no—
253C22×C4—2271no—
254C22×C4—2271no—
255C22×C4C22×D22711yescharacteristic
256Q16—4544no—
257SD16—24no—
258SD16—4544no—
259D8—4544no—
260C2×C8—4544no—
261C2×C8—2272no—
262C2×C8C22×D22712yes—
263C4⋊C4—4541no—
264C4⋊C4—4541no—
265C4⋊C4—2271no—
266C4⋊C4C22×D22711yescharacteristic
267C22⋊C4—21no—
268C22⋊C4—4541no—
269C22⋊C4—4541no—
270C22⋊C4—4541no—
271C22⋊C4—4541no—
272C42—4541no—
273C23—21no—
274C23—9082no—
275C23—9081no—
276C23—9082no—
277C23—4544no—
278C23—4542no—
279C23—4542no—
280C23—4542no—
281C23—2271no—
282C23—2271no—
283C23D4×D22711yescharacteristic
284Q8—24no—
285Q8—22no—
286Q8—4542no—
287Q8D4×D22714yes—
288D4—22no—
289D4—42no—
290D4—9082no—
291D4—9082no—
292D4—4548no—
293D4—4544no—
294D4—4548no—
295D4—4542no—
296D4—2274no—
297C2×C4—24no—
298C2×C4—22no—
299C2×C4—22no—
300C2×C4—21no—
301C2×C4—9082no—
302C2×C4—4548no—
303C2×C4—4542no—
304C2×C4—4548no—
305C2×C4—4542no—
306C2×C4—4541no—
307C2×C4—4541no—
308C2×C4—4542no—
309C2×C4—4542no—
310C2×C4—2271no—
311C2×C4—2271no—
312C2×C4—2274no—
313C2×C4D4×D22711yescharacteristic
314C2×C4C23×D22711yescharacteristic
315C8—22no—
316C8—4542no—
317C22—21no—
318C22—21no—
319C22—41no—
320C22—41no—
321C22—41no—
322C22—9088no—
323C22—9084no—
324C22—9082no—
325C22—9082no—
326C22—4541no—
327C22—4544no—
328C22—4541no—
329C22—4541no—
330C22—4541no—
331C22—4541no—
332C22—2272no—
333C22#2030030912yescomplemented
334C22C2×D4×D22711yescharacteristic
335C4—24no—
336C4—21no—
337C4—41no—
338C4—9081no—
339C4—9081no—
340C4—4541no—
341C4—2272no—
342C4C2×D4×D22712yes—
343C2—22no—
344C2—81no—
345C2—9084no—
346C2—9081no—
347C2—4542no—
348C2#120×D22711yescharacteristic
349C2C2×#2030030911yescharacteristic
350C2#2840983111yescharacteristic
subgroupG/H
centerC22C2×D4×D227
commutator, cosocleC2×C4×C227C24
FrattiniC2×C4C23×D227
Fitting#448×C227C2
socleC22×C227C22×D4
hypercenter#448D227
hypocenterC227#3002
LCSG ▷ C2×C4×C227 ▷ C2×C227 ▷ C227
UCSC1 ◁ C22 ◁ C22×C4 ◁ #448

Complements

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

NH ≅ G/Ntype
1. C22×#16813885C2semidirect
2. C2×#20300309C2semidirect
3. #28409542C2semidirect
4. #28409197C2semidirect
5. #28409188C2semidirect
6. #28409185C2semidirect
7. #28409176C2semidirect
8. #28409165C2semidirect
9. #28409027C2semidirect
10. C22⋊C8×D227C2semidirect
11. #448×C227C2semidirect
17. C2×#16813883C22semidirect
23. C2×#16813874C22semidirect
24. C2×#16813873C22semidirect
25. C2×#16813869C22semidirect
26. C2×C8×D227C22semidirect
28. #20300268C22semidirect
30. #20300259C22semidirect
34. #20300246C22semidirect
35. #20300212C22semidirect
36. #20300210C22semidirect
37. #20300208C22semidirect
39. (C2×D227)⋊C8C22semidirect
41. C2×SD16×C227C22semidirect
43. #103×C227C22semidirect
52. #16813885D4semidirect
146. C22×C227#135semidirect
175. C227#3002semidirect
184. #448D227semidirect
202. C2×SD16C2×D227semidirect
215. #103C2×D227semidirect
333. C22#20300309semidirect

Maximal subgroups

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

subgroupindex
C22×#168138852
C2×#203003092
#284095422
#284091972
#284091882
#284091852
#284091762
#284091652
#284090272
C22⋊C8×D2272
#448×C2272
#3002227