Finite Group Databasebeta

← #44562192

Subgroups — #44562192

Order 32128.

462 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 462 conjugacy classes of subgroups — too many to draw. Only its 123 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#30632397C211yesmaximal, characteristic, complemented
2#30632322C211yesmaximal, characteristic
3#30632033C211yesmaximal, characteristic, complemented
4#30632004C212yesmaximal, complemented
5#30631978C211yesmaximal, characteristic, complemented
6#30631913C212yesmaximal, complemented
7#30631836C212yesmaximal, complemented
8C4×#16860797C211yesmaximal, characteristic, complemented
9#30631752C211yesmaximal, characteristic, complemented
10#102×D251C212yesmaximal, complemented
11#460×C251C211yesmaximal, characteristic, complemented
12C2×D4×D251—21no—
13C2×D4×D251C2212yes—
14C2×#16860797C2211yescharacteristic
15C2×C8×D251C2212yescomplemented
16#22548715C2211yescharacteristic, complemented
17#22548707—21no—
18#22548693—22no—
19#22548691C2212yescomplemented
20#22548686—21no—
21#22548686C2212yes—
22#22548683—22no—
23C4×D1004—21no—
24C4×D1004C2211yescharacteristic
25C4×D1004C2211yescharacteristic
26C4⋊C4×D251C2212yescomplemented
27#22548647C2212yescomplemented
28#22548646C2211yescharacteristic, complemented
29#22548639C2211yescharacteristic, complemented
30#22548637C2212yes—
31#22548636C2212yescomplemented
32#22548636C2212yescomplemented
33#22548635C2211yescharacteristic
34(C2×D251)⋊C8C2212yescomplemented
35C4×C251⋊C8C2211yescharacteristic, complemented
36C2×SD16×C251C2212yescomplemented
37#127×C251C2211yescharacteristic
38C4×Q8×C251C2211yescharacteristic, complemented
39C4⋊C8×C251C2211yescharacteristic, complemented
40#102×C251C2212yescomplemented
41C23×D251—22no—
42C2×Dic502—21no—
43C2×D1004D411yescharacteristic
44C2×D1004C2311yescharacteristic
45C2×C22⋊D251—22no—
46C2×C22⋊D251—22no—
47D4×D251—24no—
48D4×D251—42no—
49D4×D251—44no—
50C2×C4×D251—22no—
51C2×C4×D251—21no—
52C2×C4×D251C2312yes—
53#16860797—22no—
54C2×C251⋊C8C2311yescharacteristic
55C2×C251⋊C8C2311yescharacteristic
56C8×D251—42no—
57C4⋊Dic251—21no—
58C4⋊Dic251D411yescharacteristic
59C4⋊Dic251C2312yes—
60C4⋊Dic251C2311yescharacteristic
61Dic251⋊C4—22no—
62Dic251⋊C4—22no—
63C22⋊Dic251—22no—
64(C2×D251)⋊C4—22no—
65(C2×D251)⋊C4—21no—
66(C2×D251)⋊C4D412yes—
67C2×Q8×C251C2311yescharacteristic
68C2×D4×C251—21no—
69C2×D4×C251—21no—
70C2×D4×C251C2312yes—
71SD16×C251—42no—
72C2×C8×C251C2312yes—
73C4⋊C4×C251—21no—
74C4⋊C4×C251C2311yescharacteristic
75C42×C251—21no—
76C42×C251C2311yescharacteristic
77C22×D251—21no—
78C22×D251—44no—
79C22×D251—44no—
80C22×D251—82no—
81C22×D251C2×D412yes—
82Dic502—81no—
83D1004—21no—
84D1004SD1614yescomplemented
85D1004#4112yes—
86C22⋊D251—44no—
87C22⋊D251—44no—
88C2×Dic251—21no—
89C2×Dic251—22no—
90C2×Dic251C2×D412yes—
91C4×D251—22no—
92C4×D251—22no—
93C4×D251—42no—
94C4×D251—82no—
95C251⋊C8—21no—
96C251⋊C8—21no—
97C23×C251—22no—
98Q8×C251—21no—
99Q8×C251—21no—
100D4×C251—22no—
101D4×C251—22no—
102D4×C251—42no—
103D4×C251—81no—
104C2×C4×C251—21no—
105C2×C4×C251—21no—
106C2×C4×C251C2×D411yescharacteristic
107C2×C4×C251C2×D411yescharacteristic
108C2×C4×C251C2411yescharacteristic
109C8×C251—22no—
110C22×C251—42no—
111C22×C251—42no—
112C22×C251—42no—
113C22×C251C22×D411yescharacteristic
114C2×D251—22no—
115C2×D251—22no—
116C2×D251—22no—
117C2×D251—41no—
118C2×D251—41no—
119C2×D251—88no—
120C2×D251C2×SD1614yes—
121C4×C251—21no—
122C4×C251—41no—
123C4×C251—41no—
124C4×C251C2×SD1612yes—
125C4×C251C2×#4111yescharacteristic
126C4×C251H(5,𝔽2)11yescharacteristic
127Dic251—22no—
128Dic251—41no—
129Dic251—42no—
130C2×C251—82no—
131C2×C251#54611yescharacteristic
132C2×C251C22×SD1611yescharacteristic
133C2×C251#57611yescharacteristic
134D251—24no—
135D251—42no—
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#460D25111yescharacteristic, complemented
145#450—2512no—
146C4×SD16—2511no—
147C8×D4—2511no—
148C2×#102—2512no—
149C22×D4—5021no—
150C22×D4—2512no—
151C2×SD16—2511no—
152C2×SD16C2×D25112yescomplemented
153C22×C8—2512no—
154C4⋊Q8—2511no—
155#127C2×D25111yescharacteristic
156#125—5021no—
157C22⋊Q8—5022no—
158C22⋊Q8—2512no—
159#121—5021no—
160#121—2512no—
161#120—5022no—
162C4×Q8C2×D25111yescharacteristic, complemented
163C4×D4—5021no—
164C4×D4—2511no—
165C4×D4—2511no—
166C2×C4⋊C4—2512no—
167#106—2512no—
168#106—2511no—
169C4⋊C8C2×D25111yescharacteristic, complemented
170#103—2511no—
171#102—2512no—
172#102—2512no—
173#102—2512no—
174#102—2511no—
175#102C2×D25112yescomplemented
176C22⋊C8—2512no—
177C4×C8—2511no—
178C24—5022no—
179C2×Q8—5021no—
180C2×Q8C22×D25111yescharacteristic
181C2×D4—21no—
182C2×D4—21no—
183C2×D4—10044no—
184C2×D4—10042no—
185C2×D4—5024no—
186C2×D4—5022no—
187C2×D4—5022no—
188C2×D4—2511no—
189C2×D4—2511no—
190C2×D4C22×D25112yes—
191C22×C4—5022no—
192C22×C4—5021no—
193C22×C4—2512no—
194SD16—42no—
195SD16—5022no—
196C2×C8—10042no—
197C2×C8—2511no—
198C2×C8—2511no—
199C2×C8C22×D25112yes—
200C4⋊C4—21no—
201C4⋊C4—5022no—
202C4⋊C4—5022no—
203C4⋊C4—5021no—
204C4⋊C4—2512no—
205C4⋊C4—2511no—
206C4⋊C4—2511no—
207C4⋊C4C22×D25111yescharacteristic
208C22⋊C4—5022no—
209C22⋊C4—5022no—
210C22⋊C4—5021no—
211C22⋊C4—2512no—
212C42—21no—
213C42C22×D25111yescharacteristic
214C23—22no—
215C23—20082no—
216C23—10044no—
217C23—10044no—
218C23—5021no—
219C23—2512no—
220Q8—21no—
221Q8—21no—
222Q8—20081no—
223D4—22no—
224D4—22no—
225D4—42no—
226D4—81no—
227D4—10044no—
228D4—10044no—
229D4—5021no—
230D4—2512no—
231D4—2514no—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—20082no—
235C2×C4—10042no—
236C2×C4—5021no—
237C2×C4—5022no—
238C2×C4—5022no—
239C2×C4—5022no—
240C2×C4—2512no—
241C2×C4D4×D25111yescharacteristic
242C2×C4D4×D25111yescharacteristic
243C2×C4C23×D25111yescharacteristic
244C8—22no—
245C8—5021no—
246C8—5021no—
247C22—42no—
248C22—42no—
249C22—42no—
250C22—20088no—
251C22—10041no—
252C22—10041no—
253C22—5022no—
254C22—5022no—
255C22—5022no—
256C22—2514no—
257C22C2×D4×D25111yescharacteristic
258C4—21no—
259C4—41no—
260C4—41no—
261C4—10041no—
262C4—10042no—
263C4—5022no—
264C4SD16×D25112yes—
265C4#41×D25111yescharacteristic
266C4#2254876011yescharacteristic
267C2—82no—
268C2—10042no—
269C2—5024no—
270C2#3063233411yescharacteristic
271C2C2×SD16×D25111yescharacteristic
272C2#3063249111yescharacteristic
subgroupG/H
centerC22C2×D4×D251
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
Fitting#460×C251C2
socleC22×C251C22×D4
prime radical#30632322C2
hypercenter#460D251
hypocenterC251#3295
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C42 ◁ #460

Complements

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

NH ≅ G/Ntype
1. #30632397C2semidirect
3. #30632033C2semidirect
4. #30632004C2semidirect
5. #30631978C2semidirect
6. #30631913C2semidirect
7. #30631836C2semidirect
8. C4×#16860797C2semidirect
9. #30631752C2semidirect
10. #102×D251C2semidirect
11. #460×C251C2semidirect
15. C2×C8×D251C22semidirect
16. #22548715C22semidirect
19. #22548691C22semidirect
26. C4⋊C4×D251C22semidirect
27. #22548647C22semidirect
28. #22548646C22semidirect
29. #22548639C22semidirect
31. #22548636C22semidirect
32. #22548636C22semidirect
34. (C2×D251)⋊C8C22semidirect
35. C4×C251⋊C8C22semidirect
36. C2×SD16×C251C22semidirect
38. C4×Q8×C251C22semidirect
39. C4⋊C8×C251C22semidirect
40. #102×C251C22semidirect
84. D1004SD16semidirect
136. C251#3295semidirect
144. #460D251semidirect
152. C2×SD16C2×D251semidirect
162. C4×Q8C2×D251semidirect
169. C4⋊C8C2×D251semidirect
175. #102C2×D251semidirect

Maximal subgroups

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

subgroupindex
#306323972
#306323222
#306320332
#306320042
#306319782
#306319132
#306318362
C4×#168607972
#306317522
#102×D2512
#460×C2512
#3295251