Finite Group Databasebeta

← #44562175

Subgroups — #44562175

Order 32128.

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

normal subgroups only

This group has 494 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#30632340C211yesmaximal, characteristic, complemented
2D4×D1004C211yesmaximal, characteristic, complemented
3#30632035C211yesmaximal, characteristic, complemented
4#30632002C212yesmaximal, complemented
5#30631924C212yesmaximal, complemented
6#30631883C211yesmaximal, characteristic, complemented
7#30631822C212yesmaximal, complemented
8C4×#16860795C211yesmaximal, characteristic, complemented
9#30631752C211yesmaximal, characteristic, complemented
10#102×D251C212yesmaximal, complemented
11#474×C251C211yesmaximal, characteristic, complemented
12C2×#16860811—21no—
13C22×D1004—21no—
14C2×D4×D251C2212yes—
15C2×D2008C2212yescomplemented
16C2×#16860795C2211yescharacteristic
17C2×C8×D251C2212yescomplemented
18#22548712C2211yescharacteristic
19#22548700—22no—
20#22548691—21no—
21#22548690C2212yes—
22#22548690C2212yescomplemented
23#22548689—21no—
24#22548685—22no—
25C4×D1004C2211yescharacteristic
26C4×D1004C2211yescharacteristic, complemented
27C4⋊C4×D251C2212yescomplemented
28#22548648C2211yescharacteristic, complemented
29#22548637C2212yescomplemented
30#22548636C2211yescharacteristic
31#22548635C2212yescomplemented
32#22548635C2211yescharacteristic, complemented
33#22548635C2212yescomplemented
34(C2×D251)⋊C8C2212yescomplemented
35C4×C251⋊C8C2211yescharacteristic, complemented
36C4⋊Q8×C251C2211yescharacteristic, complemented
37C4×D4×C251C2211yescharacteristic
38#107×C251C2212yescomplemented
39C4⋊C8×C251C2211yescharacteristic, complemented
40#102×C251C2212yes—
41C23×D251—22no—
42#16860811—44no—
43C2×D1004—22no—
44C2×D1004—22no—
45C2×D1004—21no—
46C2×D1004—21no—
47C2×D1004—44no—
48C2×D1004D411yescharacteristic
49C2×D1004C2312yes—
50C2×D1004C2311yescharacteristic
51C2×C22⋊D251—22no—
52D4×D251—24no—
53D4×D251—42no—
54C2×C4×D251—22no—
55C2×C4×D251—21no—
56C2×C4×D251C2312yes—
57D2008—42no—
58#16860795—22no—
59C2×C251⋊C8C2311yescharacteristic
60C2×C251⋊C8C2311yescharacteristic
61C8×D251—42no—
62C4⋊Dic251D411yescharacteristic
63C4⋊Dic251C2311yescharacteristic
64Dic251⋊C4—22no—
65(C2×D251)⋊C4—22no—
66(C2×D251)⋊C4—22no—
67(C2×D251)⋊C4—21no—
68(C2×D251)⋊C4D412yes—
69C2×Q8×C251—21no—
70C2×D4×C251C2311yescharacteristic
71C22×C4×C251—21no—
72C2×C8×C251C2312yes—
73C4⋊C4×C251—21no—
74C4⋊C4×C251C2312yes—
75C4⋊C4×C251C2311yescharacteristic
76C22⋊C4×C251—21no—
77C42×C251C2311yescharacteristic
78C22×D251—21no—
79C22×D251—22no—
80C22×D251—44no—
81C22×D251—44no—
82C22×D251—82no—
83C22×D251C2×D412yes—
84D1004—21no—
85D1004—22no—
86D1004—21no—
87D1004—22no—
88D1004—44no—
89D1004—44no—
90D1004—42no—
91D1004—81no—
92D1004D814yescomplemented
93C22⋊D251—44no—
94C2×Dic251—21no—
95C2×Dic251C2×D412yes—
96C4×D251—22no—
97C4×D251—22no—
98C4×D251—44no—
99C4×D251—42no—
100C251⋊C8—21no—
101C251⋊C8—21no—
102C23×C251—21no—
103Q8×C251—42no—
104D4×C251—21no—
105D4×C251#4112yescomplemented
106C2×C4×C251—21no—
107C2×C4×C251—22no—
108C2×C4×C251—42no—
109C2×C4×C251C2×D411yescharacteristic
110C2×C4×C251C2×D411yescharacteristic
111C2×C4×C251C2411yescharacteristic
112C8×C251—22no—
113C22×C251—22no—
114C22×C251—41no—
115C22×C251—41no—
116C22×C251C22×D411yescharacteristic
117C2×D251—22no—
118C2×D251—22no—
119C2×D251—42no—
120C2×D251—41no—
121C2×D251—41no—
122C2×D251—42no—
123C2×D251—42no—
124C2×D251—41no—
125C2×D251—88no—
126C2×D251C2×D814yes—
127C4×C251—21no—
128C4×C251—41no—
129C4×C251—42no—
130C4×C251C2×D812yes—
131C4×C251C2×#4111yescharacteristic
132C4×C251H(5,𝔽2)11yescharacteristic
133Dic251—22no—
134Dic251—41no—
135C2×C251—42no—
136C2×C251#54611yescharacteristic
137C2×C251C22×D811yescharacteristic
138C2×C251#57711yescharacteristic
139D251—24no—
140D251—81no—
141D251—82no—
142C251#329411yescharacteristic, complemented, Sylow
143#3294—2511nomaximal, Sylow
144#547—2511no—
145D42—2511no—
146#486—2511no—
147#480—2512no—
148#474D25111yescharacteristic, complemented
149#466—2512no—
150#459—2511no—
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×D8—2512no—
160C22×C8—2512no—
161C4⋊Q8C2×D25111yescharacteristic, complemented
162#127—2511no—
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
172C2×C4⋊C4—2512no—
173#107—2511no—
174#107C2×D25112yescomplemented
175C4⋊C8C2×D25111yescharacteristic, complemented
176#102—2511no—
177#102—2512no—
178#102—2512no—
179#102—2511no—
180#102—2512no—
181#102C2×D25112yes—
182C22⋊C8—2512no—
183C4×C8—2511no—
184C24—5022no—
185#41—10044no—
186C2×Q8—21no—
187C2×D4—10044no—
188C2×D4—10042no—
189C2×D4—5022no—
190C2×D4—5022no—
191C2×D4—5024no—
192C2×D4—5022no—
193C2×D4—5021no—
194C2×D4—5021no—
195C2×D4—2511no—
196C2×D4—2512no—
197C2×D4—2511no—
198C2×D4C22×D25111yescharacteristic
199C22×C4—21no—
200C22×C4—5022no—
201C22×C4—5021no—
202C22×C4—2512no—
203D8—10042no—
204D8—5022no—
205C2×C8—10042no—
206C2×C8—2511no—
207C2×C8—2511no—
208C2×C8C22×D25112yes—
209C4⋊C4—21no—
210C4⋊C4—5022no—
211C4⋊C4—2511no—
212C4⋊C4—2511no—
213C4⋊C4C22×D25112yes—
214C4⋊C4C22×D25111yescharacteristic
215C22⋊C4—21no—
216C22⋊C4—5022no—
217C22⋊C4—5022no—
218C22⋊C4—5021no—
219C22⋊C4—2512no—
220C42C22×D25111yescharacteristic
221C23—21no—
222C23—20082no—
223C23—10044no—
224C23—10044no—
225C23—5021no—
226C23—5022no—
227C23—2512no—
228Q8—42no—
229D4—21no—
230D4—20081no—
231D4—10044no—
232D4—10044no—
233D4—10044no—
234D4—10042no—
235D4—5021no—
236D4—5022no—
237D4—5021no—
238D4—5022no—
239D4—2514no—
240D4#1686081112yescomplemented
241C2×C4—21no—
242C2×C4—22no—
243C2×C4—42no—
244C2×C4—10044no—
245C2×C4—10042no—
246C2×C4—5021no—
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—22no—
257C22—41no—
258C22—41no—
259C22—20088no—
260C22—10041no—
261C22—10042no—
262C22—10042no—
263C22—10041no—
264C22—10042no—
265C22—10041no—
266C22—5022no—
267C22—5022no—
268C22—2514no—
269C22C2×D4×D25111yescharacteristic
270C4—21no—
271C4—41no—
272C4—42no—
273C4—10041no—
274C4—5022no—
275C4D8×D25112yes—
276C4C2×#1686081111yescharacteristic
277C4#2254876111yescharacteristic
278C2—42no—
279C2—20082no—
280C2—20081no—
281C2—5024no—
282C2#3063232811yescharacteristic
283C2C2×D8×D25111yescharacteristic
284C2#3063249611yescharacteristic
subgroupG/H
centerC22C2×D4×D251
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
Fitting#474×C251C2
socleC22×C251C22×D4
hypercenter#474D251
hypocenterC251#3294
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C42 ◁ #474

Complements

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

NH ≅ G/Ntype
1. #30632340C2semidirect
2. D4×D1004C2semidirect
3. #30632035C2semidirect
4. #30632002C2semidirect
5. #30631924C2semidirect
6. #30631883C2semidirect
7. #30631822C2semidirect
8. C4×#16860795C2semidirect
9. #30631752C2semidirect
10. #102×D251C2semidirect
11. #474×C251C2semidirect
15. C2×D2008C22semidirect
17. C2×C8×D251C22semidirect
22. #22548690C22semidirect
26. C4×D1004C22semidirect
27. C4⋊C4×D251C22semidirect
28. #22548648C22semidirect
29. #22548637C22semidirect
31. #22548635C22semidirect
32. #22548635C22semidirect
33. #22548635C22semidirect
34. (C2×D251)⋊C8C22semidirect
35. C4×C251⋊C8C22semidirect
36. C4⋊Q8×C251C22semidirect
38. #107×C251C22semidirect
39. C4⋊C8×C251C22semidirect
92. D1004D8semidirect
105. D4×C251#41semidirect
142. C251#3294semidirect
148. #474D251semidirect
161. C4⋊Q8C2×D251semidirect
174. #107C2×D251semidirect
175. C4⋊C8C2×D251semidirect
240. D4#16860811semidirect

Maximal subgroups

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

subgroupindex
#306323402
D4×D10042
#306320352
#306320022
#306319242
#306318832
#306318222
C4×#168607952
#306317522
#102×D2512
#474×C2512
#3294251