Finite Group Databasebeta

← #44562186

Subgroups — #44562186

Order 32128.

462 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 462 conjugacy classes of subgroups — too many to draw. Only its 125 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#30632398C211yesmaximal, characteristic, complemented
2D4×D1004C211yesmaximal, characteristic
3#30632035C211yesmaximal, characteristic, complemented
4#30632007C212yesmaximal, complemented
5#30631917C212yesmaximal, complemented
6#30631888C211yesmaximal, characteristic, complemented
7#30631839C212yesmaximal, complemented
8C4×#16860798C211yesmaximal, characteristic, complemented
9#30631752C211yesmaximal, characteristic, complemented
10#102×D251C212yesmaximal, complemented
11#476×C251C211yesmaximal, characteristic, complemented
12C22×D1004—21no—
13C2×D4×D251C2212yes—
14C2×#16860799C2212yescomplemented
15C2×#16860798C2211yescharacteristic
16C2×C8×D251C2212yescomplemented
17#22548712C2211yescharacteristic
18#22548706—21no—
19#22548695C2212yescomplemented
20#22548691—22no—
21#22548690C2212yes—
22#22548689—21no—
23#22548685—22no—
24C4×Dic502C2211yescharacteristic, complemented
25C4×D1004C2211yescharacteristic
26#22548677—21no—
27C4⋊C4×D251C2212yescomplemented
28#22548646C2211yescharacteristic, complemented
29#22548638C2211yescharacteristic, complemented
30#22548637C2212yescomplemented
31#22548636C2211yescharacteristic
32#22548635C2212yescomplemented
33#22548635C2212yescomplemented
34(C2×D251)⋊C8C2212yescomplemented
35C4×C251⋊C8C2211yescharacteristic, complemented
36C4⋊Q8×C251C2211yescharacteristic, complemented
37C4×D4×C251C2211yescharacteristic
38#106×C251C2212yescomplemented
39C4⋊C8×C251C2211yescharacteristic, complemented
40#102×C251C2212yes—
41C23×D251—22no—
42C2×Dic502C2311yescharacteristic
43C2×D1004—22no—
44C2×D1004—21no—
45C2×D1004—21no—
46C2×D1004—44no—
47C2×D1004D411yescharacteristic
48C2×D1004C2312yes—
49C2×C22⋊D251—22no—
50D4×D251—24no—
51D4×D251—42no—
52C2×C4×D251—22no—
53C2×C4×D251C2312yes—
54#16860799—42no—
55#16860798—22no—
56C2×C251⋊C8C2311yescharacteristic
57C2×C251⋊C8C2311yescharacteristic
58C8×D251—42no—
59C4⋊Dic251—22no—
60C4⋊Dic251D411yescharacteristic
61C4⋊Dic251C2311yescharacteristic
62Dic251⋊C4—22no—
63Dic251⋊C4—21no—
64(C2×D251)⋊C4—22no—
65(C2×D251)⋊C4—22no—
66(C2×D251)⋊C4D412yes—
67C4×Dic251—21no—
68C2×Q8×C251—21no—
69C2×D4×C251C2311yescharacteristic
70C22×C4×C251—21no—
71C2×C8×C251C2312yes—
72C4⋊C4×C251—21no—
73C4⋊C4×C251C2312yes—
74C4⋊C4×C251C2311yescharacteristic
75C22⋊C4×C251—21no—
76C42×C251C2311yescharacteristic
77C22×D251—22no—
78C22×D251—44no—
79C22×D251—44no—
80C22×D251—82no—
81C22×D251C2×D412yes—
82Dic502—21no—
83Dic502—21no—
84D1004—22no—
85D1004—22no—
86D1004—44no—
87D1004—42no—
88D1004—81no—
89D1004SD1614yescomplemented
90C22⋊D251—44no—
91C2×Dic251—21no—
92C2×Dic251—21no—
93C2×Dic251C2×D412yes—
94C4×D251—22no—
95C4×D251—22no—
96C4×D251—82no—
97C251⋊C8—21no—
98C251⋊C8—21no—
99C23×C251—21no—
100Q8×C251—81no—
101D4×C251—21no—
102D4×C251#4112yes—
103C2×C4×C251—21no—
104C2×C4×C251—22no—
105C2×C4×C251—42no—
106C2×C4×C251C2×D411yescharacteristic
107C2×C4×C251C2×D411yescharacteristic
108C2×C4×C251C2411yescharacteristic
109C8×C251—22no—
110C22×C251—22no—
111C22×C251—41no—
112C22×C251—41no—
113C22×C251C22×D411yescharacteristic
114C2×D251—22no—
115C2×D251—22no—
116C2×D251—42no—
117C2×D251—42no—
118C2×D251—42no—
119C2×D251—88no—
120C2×D251C2×SD1614yes—
121C4×C251—21no—
122C4×C251—41no—
123C4×C251—42no—
124C4×C251C2×SD1612yes—
125C4×C251C2×#4111yescharacteristic
126C4×C251H(5,𝔽2)11yescharacteristic
127Dic251—22no—
128Dic251—41no—
129Dic251—41no—
130C2×C251—42no—
131C2×C251#54611yescharacteristic
132C2×C251C22×SD1611yescharacteristic
133C2×C251#57611yescharacteristic
134D251—24no—
135D251—82no—
136C251#329511yescharacteristic, complemented, Sylow
137#3295—2511nomaximal, Sylow
138#554—2511no—
139D42—2511no—
140#486—2511no—
141#481—2512no—
142#476D25111yescharacteristic, complemented
143#465—2512no—
144#460—2511no—
145#450—2512no—
146C4×SD16—2511no—
147C8×D4—2511no—
148C2×#102—2512no—
149C22×D4—5021no—
150C22×D4—2512no—
151C2×SD16—2511no—
152C2×SD16—2512no—
153C22×C8—2512no—
154C4⋊Q8C2×D25111yescharacteristic, complemented
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#106C2×D25112yescomplemented
169C4⋊C8C2×D25111yescharacteristic, complemented
170#103—2511no—
171#102—2511no—
172#102—2512no—
173#102—2512no—
174#102—2512no—
175#102C2×D25112yes—
176C22⋊C8—2512no—
177C4×C8—2511no—
178C24—5022no—
179C2×Q8—21no—
180C2×Q8—2511no—
181C2×D4—10044no—
182C2×D4—10042no—
183C2×D4—5022no—
184C2×D4—5024no—
185C2×D4—5022no—
186C2×D4—5021no—
187C2×D4—5021no—
188C2×D4—2511no—
189C2×D4—2512no—
190C2×D4C22×D25111yescharacteristic
191C22×C4—21no—
192C22×C4—5022no—
193C22×C4—2512no—
194SD16—10042no—
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—2511no—
205C4⋊C4—2511no—
206C4⋊C4C22×D25112yes—
207C4⋊C4C22×D25111yescharacteristic
208C22⋊C4—21no—
209C22⋊C4—5022no—
210C22⋊C4—5022no—
211C22⋊C4—2512no—
212C42—5021no—
213C42C22×D25111yescharacteristic
214C23—21no—
215C23—20082no—
216C23—10044no—
217C23—10044no—
218C23—5022no—
219C23—2512no—
220Q8—81no—
221Q8—5021no—
222Q8—5021no—
223D4—21no—
224D4—20081no—
225D4—10044no—
226D4—10044no—
227D4—10042no—
228D4—5022no—
229D4—5022no—
230D4—2514no—
231D4#1686081112yes—
232C2×C4—21no—
233C2×C4—22no—
234C2×C4—42no—
235C2×C4—20082no—
236C2×C4—5021no—
237C2×C4—5021no—
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—22no—
248C22—41no—
249C22—41no—
250C22—20088no—
251C22—10042no—
252C22—10042no—
253C22—10042no—
254C22—5022no—
255C22—5022no—
256C22—2514no—
257C22C2×D4×D25111yescharacteristic
258C4—21no—
259C4—41no—
260C4—42no—
261C4—10041no—
262C4—10041no—
263C4—5022no—
264C4SD16×D25112yes—
265C4C2×#1686081111yescharacteristic
266C4#2254876111yescharacteristic
267C2—42no—
268C2—20082no—
269C2—5024no—
270C2#3063232811yescharacteristic
271C2C2×SD16×D25111yescharacteristic
272C2#3063249111yescharacteristic
subgroupG/H
centerC22C2×D4×D251
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
Fitting#476×C251C2
socleC22×C251C22×D4
prime radicalD4×D1004C2
hypercenter#476D251
hypocenterC251#3295
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C42 ◁ #476

Complements

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

NH ≅ G/Ntype
1. #30632398C2semidirect
3. #30632035C2semidirect
4. #30632007C2semidirect
5. #30631917C2semidirect
6. #30631888C2semidirect
7. #30631839C2semidirect
8. C4×#16860798C2semidirect
9. #30631752C2semidirect
10. #102×D251C2semidirect
11. #476×C251C2semidirect
14. C2×#16860799C22semidirect
16. C2×C8×D251C22semidirect
19. #22548695C22semidirect
24. C4×Dic502C22semidirect
27. C4⋊C4×D251C22semidirect
28. #22548646C22semidirect
29. #22548638C22semidirect
30. #22548637C22semidirect
32. #22548635C22semidirect
33. #22548635C22semidirect
34. (C2×D251)⋊C8C22semidirect
35. C4×C251⋊C8C22semidirect
36. C4⋊Q8×C251C22semidirect
38. #106×C251C22semidirect
39. C4⋊C8×C251C22semidirect
89. D1004SD16semidirect
136. C251#3295semidirect
142. #476D251semidirect
154. C4⋊Q8C2×D251semidirect
168. #106C2×D251semidirect
169. C4⋊C8C2×D251semidirect

Maximal subgroups

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

subgroupindex
#306323982
D4×D10042
#306320352
#306320072
#306319172
#306318882
#306318392
C4×#168607982
#306317522
#102×D2512
#476×C2512
#3295251