Finite Group Databasebeta

← #44562181

Subgroups — #44562181

Order 32128.

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

normal subgroups only

This group has 494 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#30632342C211yesmaximal, characteristic, complemented
2#30632322C211yesmaximal, characteristic, complemented
3#30632033C211yesmaximal, characteristic, complemented
4#30631999C212yesmaximal, complemented
5#30631968C211yesmaximal, characteristic, complemented
6#30631920C212yesmaximal, complemented
7#30631819C212yesmaximal, complemented
8C4×#16860795C211yesmaximal, characteristic, complemented
9#30631752C211yesmaximal, characteristic, complemented
10#102×D251C212yesmaximal, complemented
11#459×C251C211yesmaximal, characteristic, complemented
12C2×#16860810—21no—
13C2×D4×D251—21no—
14C2×D4×D251C2212yes—
15C2×#16860795C2211yescharacteristic
16C2×C8×D251C2212yescomplemented
17#22548715C2211yescharacteristic, complemented
18#22548697—22no—
19#22548694—21no—
20#22548686—21no—
21#22548686C2212yes—
22#22548686C2212yescomplemented
23#22548683—22no—
24C4×D1004C2211yescharacteristic
25C4×D1004C2211yescharacteristic
26C4⋊C4×D251C2212yescomplemented
27#22548649C2212yescomplemented
28#22548648C2211yescharacteristic, complemented
29#22548637C2212yes—
30#22548636C2212yescomplemented
31#22548636C2211yescharacteristic, complemented
32#22548636C2212yescomplemented
33#22548635C2211yescharacteristic
34(C2×D251)⋊C8C2212yescomplemented
35C4×C251⋊C8C2211yescharacteristic, complemented
36C2×D8×C251C2212yescomplemented
37#127×C251C2211yescharacteristic
38C4×D4×C251C2211yescharacteristic, complemented
39C4⋊C8×C251C2211yescharacteristic, complemented
40#102×C251C2212yescomplemented
41C23×D251—22no—
42#16860810—44no—
43C2×Dic502—21no—
44C2×D1004D411yescharacteristic
45C2×D1004C2311yescharacteristic
46C2×C22⋊D251—22no—
47C2×C22⋊D251—22no—
48C2×C22⋊D251—22no—
49D4×D251—24no—
50D4×D251—42no—
51D4×D251—44no—
52C2×C4×D251—22no—
53C2×C4×D251—21no—
54C2×C4×D251C2312yes—
55#16860795—22no—
56C2×C251⋊C8C2311yescharacteristic
57C2×C251⋊C8C2311yescharacteristic
58C8×D251—42no—
59C4⋊Dic251—21no—
60C4⋊Dic251D411yescharacteristic
61C4⋊Dic251C2312yes—
62C4⋊Dic251C2311yescharacteristic
63Dic251⋊C4—22no—
64C22⋊Dic251—22no—
65C22⋊Dic251—22no—
66(C2×D251)⋊C4—21no—
67(C2×D251)⋊C4D412yes—
68C2×D4×C251—21no—
69C2×D4×C251—21no—
70C2×D4×C251C2312yes—
71C2×D4×C251C2311yescharacteristic
72C22×C4×C251—21no—
73D8×C251—42no—
74C2×C8×C251C2312yes—
75C4⋊C4×C251C2311yescharacteristic
76C22⋊C4×C251—21no—
77C42×C251C2311yescharacteristic
78C22×D251—21no—
79C22×D251—44no—
80C22×D251—44no—
81C22×D251—82no—
82C22×D251C2×D412yes—
83Dic502—42no—
84D1004—21no—
85D1004D814yescomplemented
86D1004#4112yescomplemented
87C22⋊D251—44no—
88C22⋊D251—44no—
89C22⋊D251—44no—
90C2×Dic251—21no—
91C2×Dic251—22no—
92C2×Dic251C2×D412yes—
93C4×D251—22no—
94C4×D251—22no—
95C4×D251—44no—
96C4×D251—42no—
97C251⋊C8—21no—
98C251⋊C8—21no—
99C23×C251—21no—
100C23×C251—22no—
101D4×C251—21no—
102D4×C251—22no—
103D4×C251—21no—
104D4×C251—22no—
105D4×C251—42no—
106D4×C251—81no—
107C2×C4×C251—21no—
108C2×C4×C251—42no—
109C2×C4×C251C2×D411yescharacteristic
110C2×C4×C251C2×D411yescharacteristic
111C2×C4×C251C2411yescharacteristic
112C8×C251—22no—
113C22×C251—41no—
114C22×C251—42no—
115C22×C251—41no—
116C22×C251—42no—
117C22×C251—42no—
118C22×C251—41no—
119C22×C251C22×D411yescharacteristic
120C2×D251—22no—
121C2×D251—22no—
122C2×D251—22no—
123C2×D251—41no—
124C2×D251—41no—
125C2×D251—88no—
126C2×D251C2×D814yes—
127C4×C251—21no—
128C4×C251—41no—
129C4×C251C2×D812yes—
130C4×C251C2×#4111yescharacteristic
131C4×C251H(5,𝔽2)11yescharacteristic
132Dic251—22no—
133Dic251—41no—
134Dic251—42no—
135C2×C251—82no—
136C2×C251—81no—
137C2×C251#54611yescharacteristic
138C2×C251C22×D811yescharacteristic
139C2×C251#57711yescharacteristic
140D251—24no—
141D251—42no—
142C251#329411yescharacteristic, complemented, Sylow
143#3294—2511nomaximal, Sylow
144#547—2511no—
145D42—2511no—
146#486—2511no—
147#480—2512no—
148#474—2511no—
149#466—2512no—
150#459D25111yescharacteristic, complemented
151#447—2512no—
152C4×D8—2511no—
153C8×D4—2511no—
154C2×#102—2512no—
155C2×#41—5021no—
156C22×D4—5021no—
157C22×D4—2512no—
158C2×D8—2511no—
159C2×D8C2×D25112yescomplemented
160C22×C8—2512no—
161C4⋊Q8—2511no—
162#127C2×D25111yescharacteristic
163#123—5022no—
164C22⋊Q8—5021no—
165#121—5021no—
166#121—2512no—
167#121—2512no—
168#120—5022no—
169C4×D4—2511no—
170C4×D4—2511no—
171C4×D4C2×D25111yescharacteristic, complemented
172C2×C4⋊C4—2512no—
173#107—2512no—
174#107—2511no—
175C4⋊C8C2×D25111yescharacteristic, complemented
176#102—2512no—
177#102—2511no—
178#102—2512no—
179#102—2512no—
180#102—2511no—
181#102C2×D25112yescomplemented
182C22⋊C8—2512no—
183C4×C8—2511no—
184C24—5022no—
185#41—10044no—
186C2×Q8—5021no—
187C2×D4—21no—
188C2×D4—21no—
189C2×D4—10044no—
190C2×D4—10042no—
191C2×D4—5024no—
192C2×D4—5022no—
193C2×D4—5022no—
194C2×D4—5022no—
195C2×D4—2511no—
196C2×D4—2511no—
197C2×D4C22×D25112yes—
198C2×D4C22×D25111yescharacteristic
199C22×C4—21no—
200C22×C4—5022no—
201C22×C4—5021no—
202C22×C4—2512no—
203D8—42no—
204D8—5022no—
205C2×C8—10042no—
206C2×C8—2511no—
207C2×C8—2511no—
208C2×C8C22×D25112yes—
209C4⋊C4—5022no—
210C4⋊C4—5021no—
211C4⋊C4—2512no—
212C4⋊C4—2511no—
213C4⋊C4—2511no—
214C4⋊C4C22×D25111yescharacteristic
215C22⋊C4—21no—
216C22⋊C4—5022no—
217C22⋊C4—5022no—
218C22⋊C4—5021no—
219C22⋊C4—2512no—
220C42C22×D25111yescharacteristic
221C23—21no—
222C23—22no—
223C23—20082no—
224C23—10044no—
225C23—10044no—
226C23—5021no—
227C23—2512no—
228Q8—10042no—
229D4—21no—
230D4—22no—
231D4—21no—
232D4—22no—
233D4—42no—
234D4—81no—
235D4—10044no—
236D4—10044no—
237D4—10044no—
238D4—5021no—
239D4—2512no—
240D4—2514no—
241C2×C4—21no—
242C2×C4—42no—
243C2×C4—10044no—
244C2×C4—10042no—
245C2×C4—5021no—
246C2×C4—5022no—
247C2×C4—5022no—
248C2×C4—5022no—
249C2×C4—2512no—
250C2×C4D4×D25111yescharacteristic
251C2×C4D4×D25111yescharacteristic
252C2×C4C23×D25111yescharacteristic
253C8—22no—
254C8—5021no—
255C8—5021no—
256C22—41no—
257C22—42no—
258C22—42no—
259C22—41no—
260C22—42no—
261C22—41no—
262C22—20088no—
263C22—10041no—
264C22—10041no—
265C22—5022no—
266C22—5022no—
267C22—5022no—
268C22—2514no—
269C22C2×D4×D25111yescharacteristic
270C4—21no—
271C4—41no—
272C4—10041no—
273C4—10042no—
274C4—5022no—
275C4D8×D25112yes—
276C4#41×D25111yescharacteristic
277C4#2254876011yescharacteristic
278C2—82no—
279C2—81no—
280C2—10042no—
281C2—5024no—
282C2#3063233411yescharacteristic
283C2C2×D8×D25111yescharacteristic
284C2#3063249711yescharacteristic
subgroupG/H
centerC22C2×D4×D251
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
Fitting#459×C251C2
socleC22×C251C22×D4
hypercenter#459D251
hypocenterC251#3294
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C42 ◁ #459

Complements

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

NH ≅ G/Ntype
1. #30632342C2semidirect
2. #30632322C2semidirect
3. #30632033C2semidirect
4. #30631999C2semidirect
5. #30631968C2semidirect
6. #30631920C2semidirect
7. #30631819C2semidirect
8. C4×#16860795C2semidirect
9. #30631752C2semidirect
10. #102×D251C2semidirect
11. #459×C251C2semidirect
16. C2×C8×D251C22semidirect
17. #22548715C22semidirect
22. #22548686C22semidirect
26. C4⋊C4×D251C22semidirect
27. #22548649C22semidirect
28. #22548648C22semidirect
30. #22548636C22semidirect
31. #22548636C22semidirect
32. #22548636C22semidirect
34. (C2×D251)⋊C8C22semidirect
35. C4×C251⋊C8C22semidirect
36. C2×D8×C251C22semidirect
38. C4×D4×C251C22semidirect
39. C4⋊C8×C251C22semidirect
40. #102×C251C22semidirect
85. D1004D8semidirect
86. D1004#41semidirect
142. C251#3294semidirect
150. #459D251semidirect
159. C2×D8C2×D251semidirect
171. C4×D4C2×D251semidirect
175. C4⋊C8C2×D251semidirect
181. #102C2×D251semidirect

Maximal subgroups

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

subgroupindex
#306323422
#306323222
#306320332
#306319992
#306319682
#306319202
#306318192
C4×#168607952
#306317522
#102×D2512
#459×C2512
#3294251