Finite Group Databasebeta

← #44560008

Subgroups — #44560008

Order 32128.

524 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 524 conjugacy classes of subgroups — too many to draw. Only its 127 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
1C2×D8×D251C211yesmaximal, characteristic, complemented
2C2×#22548721C211yesmaximal, characteristic, complemented
3#30632253C212yesmaximal, complemented
4#30631939C212yesmaximal, complemented
5#30631938C212yesmaximal, complemented
6#30631924C211yesmaximal, characteristic, complemented
7#30631923C211yesmaximal, characteristic, complemented
8#30631920C211yesmaximal, characteristic, complemented
9#30631724C211yesmaximal, characteristic, complemented
10#30631690C212yesmaximal, complemented
11#466×C251C211yesmaximal, characteristic, complemented
12C2×#16860810C2211yescharacteristic
13C2×D4×D251C2212yes—
14C2×D2008C2211yescharacteristic
15C2×#16860795C2212yes—
16D8×D251—24no—
17#22548721—24no—
18C2×#16860794C2211yescharacteristic, complemented
19C2×#16860793C2211yescharacteristic
20C2×#16860793C2211yescharacteristic, complemented
21C2×C8×D251C2211yescharacteristic
22C2×C8×D251C2211yescharacteristic, complemented
23#22548711—22no—
24#22548701—22no—
25#22548690C2212yescomplemented
26#22548689C2212yescomplemented
27#22548686C2212yescomplemented
28#22548685—22no—
29#22548683—22no—
30#22548673C2212yes—
31#22548649C2211yescharacteristic, complemented
32#22548646C2212yescomplemented
33#22548637C2212yescomplemented
34#22548636C2212yescomplemented
35#22548635C2212yescomplemented
36C2×D8×C251C2211yescharacteristic
37C22×C8×C251C2211yescharacteristic, complemented
38#121×C251C2212yescomplemented
39#107×C251C2211yescharacteristic, complemented
40#102×C251C2212yescomplemented
41C23×D251—22no—
42#16860810—22no—
43#16860810—22no—
44C2×Dic502D411yescharacteristic
45C2×D1004—22no—
46C2×D1004D411yescharacteristic
47C2×D1004C2312yes—
48C2×C22⋊D251—22no—
49C2×C22⋊D251—22no—
50C2×C22⋊D251—22no—
51C2×C22⋊D251D411yescharacteristic
52C2×C22⋊D251D411yescharacteristic
53D4×D251—24no—
54D4×D251—24no—
55C2×C4×D251C2311yescharacteristic
56C2×C4×D251C2311yescharacteristic
57D2008—22no—
58#16860795—24no—
59#16860794—22no—
60#16860793—22no—
61#16860793—22no—
62C2×C251⋊C8C2311yescharacteristic
63C2×C251⋊C8C2311yescharacteristic
64C8×D251—22no—
65C8×D251D414yescomplemented
66C4⋊Dic251C2312yes—
67C22⋊Dic251—22no—
68(C2×D251)⋊C4—22no—
69(C2×D251)⋊C4—22no—
70(C2×D251)⋊C4—22no—
71C4×Dic251—22no—
72C2×D4×C251—22no—
73C2×D4×C251C2312yes—
74C22×C4×C251C2311yescharacteristic
75D8×C251—22no—
76C2×C8×C251—22no—
77C2×C8×C251C2311yescharacteristic
78C2×C8×C251C2311yescharacteristic
79C4⋊C4×C251C2312yes—
80C22⋊C4×C251—22no—
81C22×D251—22no—
82C22×D251—44no—
83C22×D251—82no—
84C22×D251—82no—
85C22×D251C2×D411yescharacteristic
86C22×D251C2×D411yescharacteristic
87Dic502—22no—
88D1004—22no—
89D1004—22no—
90D1004—22no—
91D1004—82no—
92C22⋊D251—22no—
93C22⋊D251—22no—
94C22⋊D251—44no—
95C22⋊D251—82no—
96C22⋊D251—82no—
97C2×Dic251—22no—
98C2×Dic251C2×D411yescharacteristic
99C2×Dic251C2×D411yescharacteristic
100C4×D251—21no—
101C4×D251—21no—
102C4×D251C2×D412yes—
103C4×D251#4112yes—
104C251⋊C8—21no—
105C251⋊C8C2×D412yes—
106C23×C251—22no—
107C23×C251C2×D411yescharacteristic
108D4×C251—22no—
109D4×C251—22no—
110D4×C251—82no—
111C2×C4×C251—21no—
112C2×C4×C251—21no—
113C2×C4×C251—22no—
114C2×C4×C251C2×D411yescharacteristic
115C2×C4×C251C2411yescharacteristic
116C8×C251—21no—
117C8×C251C2×D412yes—
118C22×C251—21no—
119C22×C251—21no—
120C22×C251—21no—
121C22×C251—42no—
122C22×C251—42no—
123C22×C251—42no—
124C22×C251C22×D411yescharacteristic
125C2×D251—21no—
126C2×D251—21no—
127C2×D251—21no—
128C2×D251—21no—
129C2×D251—21no—
130C2×D251—42no—
131C2×D251—42no—
132C2×D251—42no—
133C2×D251—88no—
134C2×D251#12112yes—
135C4×C251—21no—
136C4×C251—42no—
137C4×C251C22×D411yescharacteristic
138C4×C251C2×#4111yescharacteristic
139Dic251—21no—
140Dic251—42no—
141Dic251#12112yes—
142C2×C251—41no—
143C2×C251—82no—
144C2×C251C2×#12111yescharacteristic
145C2×C251#57611yescharacteristic
146C2×C251#57611yescharacteristic
147D251—22no—
148D251—41no—
149D251—82no—
150C251#305811yescharacteristic, complemented, Sylow
151#3058—2511nomaximal, Sylow
152C22×D8—2511no—
153C2×#131—2511no—
154#535—2512no—
155#469—2512no—
156#469—2512no—
157#466—2511no—
158#466—2511no—
159#466—2511no—
160#466D25111yescharacteristic, complemented
161#427—2511no—
162#418—2512no—
163C2×#41—2511no—
164C22×D4—2512no—
165C2×D8—5024no—
166C2×D8—2512no—
167C2×D8—2511no—
168C2×D8C2×D25111yescharacteristic
169#131—5024no—
170C2×M16—2511no—
171C2×M16—2511no—
172C2×M16—2511no—
173C22×C8—2511no—
174C22×C8—2511no—
175C22×C8C2×D25111yescharacteristic, complemented
176#127—5022no—
177#124—5022no—
178#121—2512no—
179#121—2512no—
180#121—2512no—
181#121C2×D25112yescomplemented
182#120—5022no—
183#120—5022no—
184#117—2512no—
185#107—2511no—
186#107C2×D25111yescharacteristic, complemented
187#106—2512no—
188#102—2512no—
189#102—2512no—
190#102—2512no—
191#102C2×D25112yescomplemented
192C24—5022no—
193#41—5022no—
194#41—5022no—
195C2×Q8—2511no—
196C2×D4—22no—
197C2×D4—5022no—
198C2×D4—5022no—
199C2×D4—5022no—
200C2×D4—5024no—
201C2×D4—5024no—
202C2×D4—5022no—
203C2×D4—2511no—
204C2×D4—2511no—
205C2×D4—2511no—
206C2×D4—2512no—
207C2×D4C22×D25112yes—
208C22×C4—2511no—
209C22×C4—2511no—
210C22×C4C22×D25111yescharacteristic
211D8—22no—
212D8—5024no—
213D8—5022no—
214M16—5022no—
215M16—5022no—
216M16—5022no—
217C2×C8—22no—
218C2×C8—5022no—
219C2×C8—2511no—
220C2×C8—2511no—
221C2×C8—2514no—
222C2×C8C22×D25111yescharacteristic
223C2×C8C22×D25111yescharacteristic
224C4⋊C4—2512no—
225C4⋊C4C22×D25112yes—
226C22⋊C4—22no—
227C22⋊C4—5022no—
228C22⋊C4—5022no—
229C22⋊C4—5022no—
230C22⋊C4—5022no—
231C42—5022no—
232C23—22no—
233C23—20082no—
234C23—20082no—
235C23—10044no—
236C23—5022no—
237C23—2511no—
238C23—2511no—
239C23D4×D25111yescharacteristic
240Q8—5022no—
241D4—22no—
242D4—22no—
243D4—82no—
244D4—20082no—
245D4—20082no—
246D4—20082no—
247D4—10044no—
248D4—5022no—
249D4—5022no—
250D4—5022no—
251D4—5022no—
252D4—5022no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—22no—
256C2×C4—5022no—
257C2×C4—5021no—
258C2×C4—5021no—
259C2×C4—2511no—
260C2×C4—2511no—
261C2×C4—2512no—
262C2×C4—2512no—
263C2×C4D4×D25111yescharacteristic
264C2×C4C23×D25111yescharacteristic
265C8—21no—
266C8—5021no—
267C8—2512no—
268C8D4×D25112yes—
269C22—21no—
270C22—21no—
271C22—21no—
272C22—42no—
273C22—42no—
274C22—42no—
275C22—20088no—
276C22—10042no—
277C22—10042no—
278C22—10042no—
279C22—5021no—
280C22—5021no—
281C22—5021no—
282C22—5021no—
283C22—5021no—
284C22—2512no—
285C22C2×D4×D25111yescharacteristic
286C4—21no—
287C4—42no—
288C4—10042no—
289C4—5021no—
290C4—2512no—
291C4C2×D4×D25111yescharacteristic
292C4#41×D25111yescharacteristic
293C2—41no—
294C2—82no—
295C2—20082no—
296C2—10041no—
297C2—5022no—
298C2#121×D25111yescharacteristic
299C2#3063248811yescharacteristic
300C2#3063248911yescharacteristic
subgroupG/H
centerC22C2×D4×D251
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
Fitting#466×C251C2
socleC22×C251C22×D4
hypercenter#466D251
hypocenterC251#3058
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C22×C4 ◁ #466

Complements

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

NH ≅ G/Ntype
1. C2×D8×D251C2semidirect
2. C2×#22548721C2semidirect
3. #30632253C2semidirect
4. #30631939C2semidirect
5. #30631938C2semidirect
6. #30631924C2semidirect
7. #30631923C2semidirect
8. #30631920C2semidirect
9. #30631724C2semidirect
10. #30631690C2semidirect
11. #466×C251C2semidirect
18. C2×#16860794C22semidirect
20. C2×#16860793C22semidirect
22. C2×C8×D251C22semidirect
25. #22548690C22semidirect
26. #22548689C22semidirect
27. #22548686C22semidirect
31. #22548649C22semidirect
32. #22548646C22semidirect
33. #22548637C22semidirect
34. #22548636C22semidirect
35. #22548635C22semidirect
37. C22×C8×C251C22semidirect
38. #121×C251C22semidirect
39. #107×C251C22semidirect
40. #102×C251C22semidirect
65. C8×D251D4semidirect
150. C251#3058semidirect
160. #466D251semidirect
175. C22×C8C2×D251semidirect
181. #121C2×D251semidirect
186. #107C2×D251semidirect
191. #102C2×D251semidirect

Maximal subgroups

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

subgroupindex
C2×D8×D2512
C2×#225487212
#306322532
#306319392
#306319382
#306319242
#306319232
#306319202
#306317242
#306316902
#466×C2512
#3058251