Finite Group Databasebeta

← #44409082

Subgroups — #44409082

Order 29056.

724 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 724 conjugacy classes of subgroups — too many to draw. Only its 145 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
1#28409868C211yesmaximal, characteristic, complemented
2#28409830C212yesmaximal, complemented
3#28409214C212yesmaximal, complemented
4#28409209C212yesmaximal, complemented
5#28409204C212yesmaximal, complemented
6#28409199C212yesmaximal, complemented
7C2×#20300216C212yesmaximal, complemented
8#28409016C211yesmaximal, characteristic, complemented
9#532×C227C211yesmaximal, characteristic, complemented
10#20300336C2212yes—
11#20300335—22no—
12#20300334—22no—
13#20300333—22no—
14#20300333C2212yes—
15C2×#16813884—22no—
16C2×#16813884C2212yes—
17#41×D227—24no—
18#20300319C2214yescomplemented
19#20300318—22no—
20#20300314C2214yescomplemented
21#20300313—22no—
22#20300308—24no—
23C2×#16813868C2212yescomplemented
24#20300216C2214yescomplemented
25#20300216C2214yescomplemented
26#20300207C2212yescomplemented
27#20300205C2212yescomplemented
28C2×#41×C227C2211yescharacteristic
29C2×#41×C227C2211yescharacteristic, complemented
30C4⋊Q8×C227C2211yescharacteristic, complemented
31#127×C227C2211yescharacteristic, complemented
32#124×C227C2211yescharacteristic, complemented
33#124×C227C2211yescharacteristic, complemented
34#121×C227—22no—
35C4×D4×C227—22no—
36C2×C42×C227C2211yescharacteristic, complemented
37#16813885—24no—
38#16813884—24no—
39#16813884—24no—
40#16813884—22no—
41#16813884—24no—
42#16813884—22no—
43#16813884D414yescomplemented
44#16813884C2314yes—
45#16813883—28no—
46#16813883—24no—
47C2×Dic454—22no—
48C2×Dic454D412yescomplemented
49Q8×D227—24no—
50C2×D908—22no—
51C2×D908D412yescomplemented
52C2×C22⋊D227—24no—
53C2×C22⋊D227—22no—
54D4×D227—28no—
55D4×D227—24no—
56C2×C4×D227—24no—
57C2×C4×D227—22no—
58#16813874—24no—
59#16813872—24no—
60#16813871—24no—
61#16813869—24no—
62#16813868—22no—
63#16813868C2314yes—
64C2×C227⋊C8—22no—
65#41×C227—22no—
66#41×C227—22no—
67#41×C227—22no—
68#41×C227—22no—
69C2×Q8×C227C2311yescharacteristic
70C2×Q8×C227C2311yescharacteristic
71C2×D4×C227—22no—
72C2×D4×C227—21no—
73C2×D4×C227D412yes—
74C2×D4×C227C2311yescharacteristic
75C2×D4×C227C2311yescharacteristic
76C22×C4×C227—21no—
77C22×C4×C227—21no—
78C22×C4×C227D412yes—
79C22×C4×C227C2311yescharacteristic
80C4⋊C4×C227—22no—
81C22⋊C4×C227—22no—
82C22⋊C4×C227—22no—
83C42×C227—21no—
84C42×C227C2311yescharacteristic
85C42×C227C2311yescharacteristic
86C22×D227—24no—
87C22×D227—22no—
88Dic454—24no—
89Dic454—22no—
90Dic454C2×D414yes—
91D908—24no—
92D908—22no—
93D908C2×D414yes—
94C22⋊D227—28no—
95C22⋊D227—24no—
96C22⋊D227—24no—
97C2×Dic227—24no—
98C2×Dic227—22no—
99C4×D227—28no—
100C4×D227—24no—
101C4×D227—24no—
102C227⋊C8—24no—
103C23×C227—21no—
104C23×C227C2×D412yes—
105C23×C227C2×D411yescharacteristic
106Q8×C227—22no—
107Q8×C227—22no—
108D4×C227—24no—
109D4×C227—22no—
110D4×C227—22no—
111D4×C227—44no—
112D4×C227—42no—
113C2×C4×C227—21no—
114C2×C4×C227—22no—
115C2×C4×C227—22no—
116C2×C4×C227—21no—
117C2×C4×C227—21no—
118C2×C4×C227—42no—
119C2×C4×C227—41no—
120C2×C4×C227—41no—
121C2×C4×C227C2×D412yes—
122C2×C4×C227C2×D411yescharacteristic
123C2×C4×C227C2×D412yes—
124C2×C4×C227C2×D412yes—
125C2×C4×C227C2411yescharacteristic
126C22×C227—22no—
127C22×C227—41no—
128C22×C227—42no—
129C22×C227—42no—
130C22×C227—41no—
131C22×C227#12012yes—
132C22×C227C22×D411yescharacteristic
133C2×D227—24no—
134C2×D227—82no—
135C2×D227—84no—
136C4×C227—22no—
137C4×C227—22no—
138C4×C227—41no—
139C4×C227—41no—
140C4×C227#12011yescharacteristic
141C4×C227#12011yescharacteristic
142C4×C227C22×D412yes—
143Dic227—24no—
144C2×C227—21no—
145C2×C227—22no—
146C2×C227—42no—
147C2×C227—81no—
148C2×C227C2×#12011yescharacteristic
149D227—44no—
150C227#301811yescharacteristic, complemented, Sylow
151#3018—2271nomaximal, Sylow
152#585—2271no—
153#575—2272no—
154#532D22711yescharacteristic, complemented
155#456—2272no—
156#455—2272no—
157#454—2272no—
158#453—2272no—
159C2×#104—2272no—
160#413—2271no—
1612−1+4—4542no—
1622−1+4—2272no—
163H(5,𝔽2)—4542no—
164H(5,𝔽2)—4542no—
165H(5,𝔽2)—2272no—
166C2×#41—4544no—
167C2×#41—4542no—
168C2×#41—2272no—
169C2×#41C2×D22711yescharacteristic
170C2×#41C2×D22711yescharacteristic, complemented
171#137—4542no—
172#137—2274no—
173AGL(1,ℤ/8)—4542no—
174AGL(1,ℤ/8)—2274no—
175#135—4544no—
176C2×M16—2272no—
177C4⋊Q8C2×D22711yescharacteristic, complemented
178#127C2×D22711yescharacteristic, complemented
179#124C2×D22711yescharacteristic, complemented
180#124C2×D22711yescharacteristic, complemented
181#121—22no—
182C4×D4—22no—
183C2×C42C2×D22711yescharacteristic, complemented
184#104—2274no—
185#104—2274no—
186#101—2272no—
187#100—2272no—
188#41—22no—
189#41—22no—
190#41—22no—
191#41—22no—
192#41—4544no—
193#41—4544no—
194#41—4548no—
195#41—4544no—
196#41—4544no—
197#41—4542no—
198#41—4542no—
199#41—4544no—
200#41—2274no—
201#41—2274no—
202C2×Q8—4542no—
203C2×Q8—4544no—
204C2×Q8—2272no—
205C2×Q8C22×D22711yescharacteristic
206C2×Q8C22×D22711yescharacteristic
207C2×D4—22no—
208C2×D4—21no—
209C2×D4—4542no—
210C2×D4—4548no—
211C2×D4—4544no—
212C2×D4—4544no—
213C2×D4—4542no—
214C2×D4—2272no—
215C2×D4C22⋊D22712yes—
216C2×D4C22×D22711yescharacteristic
217C2×D4C22×D22711yescharacteristic
218C22×C4—21no—
219C22×C4—21no—
220C22×C4—4544no—
221C22×C4—4542no—
222C22×C4C22⋊D22712yes—
223C22×C4C22×D22711yescharacteristic
224Q16—4544no—
225SD16—4544no—
226SD16—4544no—
227D8—4544no—
228M16—4542no—
229M16—2274no—
230C2×C8—4542no—
231C4⋊C4—22no—
232C22⋊C4—22no—
233C22⋊C4—22no—
234C42—21no—
235C42C22×D22711yescharacteristic
236C42C22×D22711yescharacteristic
237C23—21no—
238C23—4544no—
239C23—4542no—
240C23C2×C22⋊D22712yes—
241C23C2×C22⋊D22711yescharacteristic
242Q8—22no—
243Q8—22no—
244Q8—4544no—
245Q8—4542no—
246Q8—2274no—
247D4—24no—
248D4—22no—
249D4—22no—
250D4—44no—
251D4—42no—
252D4—4544no—
253D4—4548no—
254D4—4544no—
255D4—4544no—
256D4—4542no—
257D4—2274no—
258C2×C4—21no—
259C2×C4—22no—
260C2×C4—22no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—42no—
264C2×C4—41no—
265C2×C4—41no—
266C2×C4—4548no—
267C2×C4—4544no—
268C2×C4—4542no—
269C2×C4—4544no—
270C2×C4—4544no—
271C2×C4D4×D22712yes—
272C2×C4D4×D22712yes—
273C2×C4C2×C22⋊D22712yes—
274C2×C4C2×C22⋊D22711yescharacteristic
275C2×C4C23×D22711yescharacteristic
276C8—4544no—
277C22—22no—
278C22—41no—
279C22—42no—
280C22—42no—
281C22—41no—
282C22—18162no—
283C22—18164no—
284C22—4544no—
285C22#2030025612yes—
286C22C22×C22⋊D22711yescharacteristic
287C4—22no—
288C4—22no—
289C4—41no—
290C4—41no—
291C4—4544no—
292C4#2030025611yescharacteristic
293C4#2030025611yescharacteristic
294C4C2×D4×D22712yes—
295C2—21no—
296C2—22no—
297C2—42no—
298C2—81no—
299C2—9084no—
300C2C2×#2030025611yescharacteristic
subgroupG/H
centerC4#20300256
commutator, cosocleC2×C4×C227C24
FrattiniC2×C4C23×D227
Fitting#532×C227C2
socleC2×C227C2×#120
hypercenter#532D227
hypocenterC227#3018
LCSG ▷ C2×C4×C227 ▷ C2×C227 ▷ C227
UCSC1 ◁ C4 ◁ C22×C4 ◁ #532

Complements

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

NH ≅ G/Ntype
1. #28409868C2semidirect
2. #28409830C2semidirect
3. #28409214C2semidirect
4. #28409209C2semidirect
5. #28409204C2semidirect
6. #28409199C2semidirect
7. C2×#20300216C2semidirect
8. #28409016C2semidirect
9. #532×C227C2semidirect
18. #20300319C22semidirect
20. #20300314C22semidirect
23. C2×#16813868C22semidirect
24. #20300216C22semidirect
25. #20300216C22semidirect
26. #20300207C22semidirect
27. #20300205C22semidirect
29. C2×#41×C227C22semidirect
30. C4⋊Q8×C227C22semidirect
31. #127×C227C22semidirect
32. #124×C227C22semidirect
33. #124×C227C22semidirect
36. C2×C42×C227C22semidirect
43. #16813884D4semidirect
48. C2×Dic454D4semidirect
51. C2×D908D4semidirect
150. C227#3018semidirect
154. #532D227semidirect
170. C2×#41C2×D227semidirect
177. C4⋊Q8C2×D227semidirect
178. #127C2×D227semidirect
179. #124C2×D227semidirect
180. #124C2×D227semidirect
183. C2×C42C2×D227semidirect

Maximal subgroups

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

subgroupindex
#284098682
#284098302
#284092142
#284092092
#284092042
#284091992
C2×#203002162
#284090162
#532×C2272
#3018227