Finite Group Databasebeta

← #44562190

Subgroups — #44562190

Order 32128.

462 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 462 conjugacy classes of subgroups — too many to draw. Only its 139 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#30632394C211yesmaximal, characteristic, complemented
2D4×D1004C211yesmaximal, characteristic
3#30632034C211yesmaximal, characteristic, complemented
4#30632006C212yesmaximal, complemented
5#30631978C211yesmaximal, characteristic, complemented
6#30631916C212yesmaximal, complemented
7#30631888C211yesmaximal, characteristic, complemented
8#30631840C212yesmaximal, complemented
9C4×#16860799C211yesmaximal, characteristic, complemented
10C2×#22548637C212yesmaximal, complemented
11C8×D4×C251C211yesmaximal, characteristic, complemented
12C22×D1004C2212yes—
13C2×D4×D251—21no—
14C2×#16860799C2211yescharacteristic
15C2×#16860799C2212yescomplemented
16#22548715C2211yescharacteristic, complemented
17#22548712C2211yescharacteristic
18#22548706—21no—
19#22548694C2212yescomplemented
20#22548692—22no—
21#22548690—21no—
22#22548689C2212yes—
23#22548685—22no—
24C4×Dic502C2211yescharacteristic, complemented
25C4×D1004C2211yescharacteristic
26D4×Dic251—21no—
27C2×C4⋊Dic251C2212yescomplemented
28#22548647C2212yescomplemented
29#22548647C2211yescharacteristic, complemented
30#22548640C2211yescharacteristic, complemented
31#22548637C2212yes—
32#22548637C2211yescharacteristic
33#22548637C2212yescomplemented
34#22548637C2212yescomplemented
35#22548637C2212yescomplemented
36C22×C8×C251C2212yescomplemented
37C4×D4×C251C2211yescharacteristic
38C4⋊C8×C251C2211yescharacteristic, complemented
39C22⋊C8×C251C2212yescomplemented
40C4×C8×C251C2211yescharacteristic, complemented
41C23×D251—22no—
42C2×Dic502—21no—
43C2×Dic502C2311yescharacteristic
44C2×D1004—24no—
45C2×D1004—21no—
46C2×D1004—22no—
47C2×D1004—21no—
48C2×D1004—42no—
49C2×D1004C2312yes—
50C2×D1004C2311yescharacteristic
51C2×C22⋊D251—22no—
52D4×D251—44no—
53C22×Dic251—22no—
54C2×C4×D251—21no—
55#16860799—22no—
56#16860799—42no—
57C4⋊Dic251—22no—
58C4⋊Dic251—21no—
59C4⋊Dic251C2311yescharacteristic
60C4⋊Dic251C2312yes—
61C4⋊Dic251C2311yescharacteristic
62Dic251⋊C4—21no—
63Dic251⋊C4—22no—
64C22⋊Dic251—22no—
65(C2×D251)⋊C4—21no—
66(C2×D251)⋊C4—22no—
67C4×Dic251—21no—
68C2×D4×C251D411yescharacteristic
69C22×C4×C251C2312yes—
70C2×C8×C251—42no—
71C2×C8×C251C2311yescharacteristic
72C2×C8×C251C2311yescharacteristic
73C2×C8×C251C2312yes—
74C4⋊C4×C251D411yescharacteristic
75C22⋊C4×C251D412yes—
76C42×C251C2311yescharacteristic
77C22×D251—21no—
78C22×D251—22no—
79C22×D251—44no—
80C22×D251—44no—
81C22×D251—82no—
82Dic502—21no—
83Dic502—21no—
84Dic502—81no—
85D1004—22no—
86D1004—21no—
87D1004—22no—
88D1004—44no—
89D1004—42no—
90D1004—81no—
91D1004#4112yes—
92C22⋊D251—44no—
93C2×Dic251—21no—
94C2×Dic251—21no—
95C2×Dic251—21no—
96C2×Dic251—22no—
97C2×Dic251—82no—
98C4×D251—42no—
99C23×C251C2×D412yes—
100D4×C251SD1614yescomplemented
101C2×C4×C251—22no—
102C2×C4×C251—22no—
103C2×C4×C251C2×D411yescharacteristic
104C2×C4×C251C2×D412yes—
105C2×C4×C251C2×D411yescharacteristic
106C2×C4×C251C2411yescharacteristic
107C8×C251—21no—
108C8×C251—21no—
109C8×C251—22no—
110C22×C251—22no—
111C22×C251—22no—
112C22×C251C2×SD1614yes—
113C22×C251C22×D411yescharacteristic
114C2×D251—22no—
115C2×D251—41no—
116C2×D251—41no—
117C2×D251—42no—
118C2×D251—42no—
119C2×D251—42no—
120C2×D251—88no—
121C4×C251—22no—
122C4×C251—21no—
123C4×C251C2×SD1612yes—
124C4×C251C2×#4111yescharacteristic
125C4×C251H(5,𝔽2)11yescharacteristic
126Dic251—41no—
127Dic251—41no—
128Dic251—41no—
129Dic251—42no—
130C2×C251—24no—
131C2×C251#54611yescharacteristic
132C2×C251C22×SD1611yescharacteristic
133C2×C251#57611yescharacteristic
134D251—42no—
135D251—82no—
136C251#329511yescharacteristic, complemented, Sylow
137#3295—2511nomaximal, Sylow
138#554—2511no—
139D42—2511no—
140#486—2511no—
141#481—2512no—
142#476—2511no—
143#465—2512no—
144#460—2511no—
145#450—2512no—
146C4×SD16—2511no—
147C8×D4D25111yescharacteristic, complemented
148C2×#102—2512no—
149C22×D4—5021no—
150C22×D4—2512no—
151C2×SD16—2512no—
152C2×SD16—2511no—
153C22×C8C2×D25112yescomplemented
154C4⋊Q8—2511no—
155#127—2511no—
156#125—5021no—
157C22⋊Q8—5022no—
158C22⋊Q8—2512no—
159#121—5021no—
160#121—2512no—
161#120—5022no—
162C4×Q8—2511no—
163C4×D4—5021no—
164C4×D4—2511no—
165C4×D4C2×D25111yescharacteristic
166C2×C4⋊C4—2512no—
167#106—2511no—
168#106—2512no—
169C4⋊C8C2×D25111yescharacteristic, complemented
170#103—2511no—
171#102—2511no—
172#102—2512no—
173#102—2512no—
174#102—2512no—
175#102—2512no—
176C22⋊C8C2×D25112yescomplemented
177C4×C8C2×D25111yescharacteristic, complemented
178C24—5022no—
179C2×Q8—5021no—
180C2×Q8—2511no—
181C2×D4—10042no—
182C2×D4—10044no—
183C2×D4—5024no—
184C2×D4—5022no—
185C2×D4—5021no—
186C2×D4—5022no—
187C2×D4—5021no—
188C2×D4—2512no—
189C2×D4—2511no—
190C2×D4D100411yescharacteristic
191C22×C4—5022no—
192C22×C4—5021no—
193C22×C4C22×D25112yes—
194SD16—10042no—
195SD16—5022no—
196C2×C8—42no—
197C2×C8C22×D25111yescharacteristic
198C2×C8C22×D25111yescharacteristic
199C2×C8C22×D25112yes—
200C4⋊C4—5022no—
201C4⋊C4—5021no—
202C4⋊C4—5022no—
203C4⋊C4—5021no—
204C4⋊C4—2511no—
205C4⋊C4—2512no—
206C4⋊C4—2511no—
207C4⋊C4D100411yescharacteristic
208C22⋊C4—5022no—
209C22⋊C4—5021no—
210C22⋊C4—5022no—
211C22⋊C4D100412yes—
212C42—5021no—
213C42C22×D25111yescharacteristic
214C23—20082no—
215C23—10044no—
216C23—10044no—
217C23—5022no—
218C23—5021no—
219C23C2×D100412yes—
220Q8—20081no—
221Q8—5021no—
222Q8—5021no—
223D4—20081no—
224D4—10044no—
225D4—10044no—
226D4—10042no—
227D4—5022no—
228D4—5022no—
229D4—5021no—
230D4—2512no—
231D4#1686079914yescomplemented
232C2×C4—22no—
233C2×C4—22no—
234C2×C4—20082no—
235C2×C4—10042no—
236C2×C4—5021no—
237C2×C4—5021no—
238C2×C4—5021no—
239C2×C4—5022no—
240C2×C4C2×D100411yescharacteristic
241C2×C4C2×D100412yes—
242C2×C4C2×D100411yescharacteristic
243C2×C4C23×D25111yescharacteristic
244C8—21no—
245C8—21no—
246C8—22no—
247C22—22no—
248C22—22no—
249C22—20088no—
250C22—10042no—
251C22—10042no—
252C22—10042no—
253C22—10041no—
254C22—10041no—
255C22—5022no—
256C22C2×#1686079914yes—
257C22C22×D100411yescharacteristic
258C4—21no—
259C4—22no—
260C4—10041no—
261C4—10041no—
262C4—10042no—
263C4—10041no—
264C4C2×#1686079912yes—
265C4#41×D25111yescharacteristic
266C4#2254876011yescharacteristic
267C2—24no—
268C2—20082no—
269C2—10042no—
270C2#3063233211yescharacteristic
271C2C22×#1686079911yescharacteristic
272C2#3063249211yescharacteristic
subgroupG/H
centerC22C22×D1004
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
FittingC8×D4×C251C2
socleC22×C251C22×D4
prime radicalD4×D1004C2
hypercenterC8×D4D251
hypocenterC251#3295
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C4×D4 ◁ C8×D4

Complements

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

NH ≅ G/Ntype
1. #30632394C2semidirect
3. #30632034C2semidirect
4. #30632006C2semidirect
5. #30631978C2semidirect
6. #30631916C2semidirect
7. #30631888C2semidirect
8. #30631840C2semidirect
9. C4×#16860799C2semidirect
10. C2×#22548637C2semidirect
11. C8×D4×C251C2semidirect
15. C2×#16860799C22semidirect
16. #22548715C22semidirect
19. #22548694C22semidirect
24. C4×Dic502C22semidirect
27. C2×C4⋊Dic251C22semidirect
28. #22548647C22semidirect
29. #22548647C22semidirect
30. #22548640C22semidirect
33. #22548637C22semidirect
34. #22548637C22semidirect
35. #22548637C22semidirect
36. C22×C8×C251C22semidirect
38. C4⋊C8×C251C22semidirect
39. C22⋊C8×C251C22semidirect
40. C4×C8×C251C22semidirect
100. D4×C251SD16semidirect
136. C251#3295semidirect
147. C8×D4D251semidirect
153. C22×C8C2×D251semidirect
169. C4⋊C8C2×D251semidirect
176. C22⋊C8C2×D251semidirect
177. C4×C8C2×D251semidirect
231. D4#16860799semidirect

Maximal subgroups

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

subgroupindex
#306323942
D4×D10042
#306320342
#306320062
#306319782
#306319162
#306318882
#306318402
C4×#168607992
C2×#225486372
C8×D4×C2512
#3295251