Finite Group Databasebeta

← #44560157

Subgroups — #44560157

Order 32128.

440 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 440 conjugacy classes of subgroups — too many to draw. Only its 131 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#30632242C211yesmaximal, characteristic, complemented
2C4×D4×D251C211yesmaximal, characteristic
3#30631990C212yesmaximal, complemented
4#30631989C212yesmaximal, complemented
5#30631978C211yesmaximal, characteristic, complemented
6#30631977C211yesmaximal, characteristic, complemented
7#30631964C211yesmaximal, characteristic, complemented
8#30631728C212yesmaximal, complemented
9#30631709C211yesmaximal, characteristic, complemented
10#30631690C212yesmaximal
11#474×C251C211yesmaximal, characteristic, complemented
12C2×D4×D251C2211yescharacteristic
13C22×C4×D251—21no—
14C2×#16860793C2212yes—
15#22548715C2211yescharacteristic, complemented
16#22548706C2212yescomplemented
17#22548695—21no—
18#22548691—21no—
19Q8×Dic251—21no—
20#22548680—21no—
21C4×D1004C2211yescharacteristic
22D4×Dic251C2211yescharacteristic
23#22548677C2211yescharacteristic
24C4×C22⋊D251—21no—
25(C22⋊D251)⋊C4—21no—
26#22548673C2212yes—
27C4⋊C4×D251C2211yescharacteristic
28C22⋊C4×D251—21no—
29C42×D251C2211yescharacteristic
30#22548648C2212yescomplemented
31#22548647C2212yescomplemented
32#22548646C2212yescomplemented
33C4⋊(C251⋊C8)C2211yescharacteristic, complemented
34Dic251⋊C8C2212yescomplemented
35#22548637C2212yes—
36#22548636C2212yes—
37#22548635C2212yes—
38C4⋊Q8×C251C2211yescharacteristic, complemented
39C4×D4×C251C2211yescharacteristic
40#107×C251C2212yescomplemented
41C4⋊C8×C251C2211yescharacteristic, complemented
42#102×C251C2212yes—
43C23×D251—21no—
44C2×Dic502—21no—
45C2×D1004C2311yescharacteristic
46C2×C22⋊D251—21no—
47D4×D251—22no—
48D4×D251Q814yescomplemented
49C22×Dic251—21no—
50C2×C4×D251—21no—
51C2×C4×D251—21no—
52C2×C4×D251—21no—
53C2×C4×D251—44no—
54C2×C4×D251D411yescharacteristic
55C2×C4×D251D411yescharacteristic
56C2×C4×D251C2311yescharacteristic
57#16860793—24no—
58C2×C251⋊C8C2312yes—
59C4⋊Dic251—22no—
60C4⋊Dic251—21no—
61C4⋊Dic251C2312yes—
62C4⋊Dic251C2311yescharacteristic
63Dic251⋊C4—21no—
64Dic251⋊C4—22no—
65C22⋊Dic251—21no—
66(C2×D251)⋊C4—22no—
67(C2×D251)⋊C4—21no—
68(C2×D251)⋊C4—21no—
69C4×Dic251—22no—
70C4×Dic251D411yescharacteristic
71C4×Dic251D411yescharacteristic
72C4×Dic251C2311yescharacteristic
73C2×Q8×C251—21no—
74C2×D4×C251C2311yescharacteristic
75C22×C4×C251—21no—
76C2×C8×C251C2312yes—
77C4⋊C4×C251—21no—
78C4⋊C4×C251C2312yes—
79C4⋊C4×C251C2311yescharacteristic
80C22⋊C4×C251—21no—
81C42×C251C2311yescharacteristic
82C22×D251—21no—
83C22×D251—24no—
84C22×D251—42no—
85C22×D251—42no—
86C22×D251C2×D411yescharacteristic
87Dic502—42no—
88D1004—21no—
89D1004C2×Q812yes—
90C22⋊D251—24no—
91C2×Dic251—22no—
92C2×Dic251—21no—
93C2×Dic251—42no—
94C2×Dic251C2×D411yescharacteristic
95C2×Dic251C2×D411yescharacteristic
96C2×Dic251C2×D411yescharacteristic
97C4×D251—22no—
98C4×D251—41no—
99C4×D251—42no—
100C4×D251—42no—
101C4×D251—42no—
102C4×D251C2×Q812yes—
103C4×D251#4112yes—
104C251⋊C8—22no—
105C23×C251—21no—
106Q8×C251—42no—
107D4×C251—21no—
108D4×C251C2×Q812yes—
109C2×C4×C251—21no—
110C2×C4×C251—22no—
111C2×C4×C251—42no—
112C2×C4×C251C2×D411yescharacteristic
113C2×C4×C251C2×D411yescharacteristic
114C2×C4×C251C2411yescharacteristic
115C8×C251—22no—
116C22×C251—22no—
117C22×C251—41no—
118C22×C251—41no—
119C22×C251C22×D411yescharacteristic
120C2×D251—21no—
121C2×D251—21no—
122C2×D251—22no—
123C2×D251—48no—
124C2×D251—41no—
125C2×D251—41no—
126C2×D251C22⋊Q812yes—
127C4×C251—21no—
128C4×C251—41no—
129C4×C251—42no—
130C4×C251AGL(1,ℤ/8)12yes—
131C4×C251C22×Q811yescharacteristic
132C4×C251C2×#4111yescharacteristic
133Dic251—21no—
134Dic251—21no—
135Dic251—41no—
136Dic251—42no—
137Dic251C22⋊Q812yes—
138C2×C251—42no—
139C2×C251C2×C22⋊Q811yescharacteristic
140C2×C251C2×AGL(1,ℤ/8)11yescharacteristic
141C2×C251#57511yescharacteristic
142D251—22no—
143D251—42no—
144C251#307711yescharacteristic, complemented, Sylow
145#3077—2511nomaximal, Sylow
146#533—2511no—
147C2×C4×D4—2511no—
148#478—2512no—
149#478—2512no—
150#476—2511no—
151#476—2511no—
152#474—2511no—
153#474D25111yescharacteristic, complemented
154#428—2512no—
155#423—2511no—
156#418—2512no—
157C22×D4—2511no—
158C23×C4—5021no—
159C2×M16—2512no—
160C4⋊Q8—2511no—
161C4⋊Q8C2×D25111yescharacteristic, complemented
162#125—2512no—
163C22⋊Q8—5021no—
164C22⋊Q8—5021no—
165C4×Q8—5021no—
166C4×Q8—5021no—
167C4×D4—5021no—
168C4×D4—5021no—
169C4×D4—2511no—
170C4×D4—2511no—
171C4×D4—2511no—
172C4×D4C2×D25111yescharacteristic
173#117—2512no—
174C2×C4⋊C4—2511no—
175C2×C22⋊C4—5021no—
176C2×C42—2511no—
177#107—2512no—
178#107C2×D25112yescomplemented
179#106—2512no—
180#106—2512no—
181C4⋊C8—2512no—
182C4⋊C8—2511no—
183C4⋊C8C2×D25111yescharacteristic, complemented
184#102—2512no—
185#102—2512no—
186#102—2512no—
187#102C2×D25112yes—
188C24—5021no—
189C2×Q8—21no—
190C2×Q8—5021no—
191C2×D4—5022no—
192C2×D4—5021no—
193C2×D4—2514no—
194C2×D4—2511no—
195C2×D4C22×D25111yescharacteristic
196C22×C4—21no—
197C22×C4—10044no—
198C22×C4—5021no—
199C22×C4—5021no—
200C22×C4—5021no—
201C22×C4—5021no—
202C22×C4—2511no—
203C22×C4—2511no—
204C22×C4—2511no—
205M16—5024no—
206C2×C8—2512no—
207C2×C8C22×D25112yes—
208C4⋊C4—21no—
209C4⋊C4—5021no—
210C4⋊C4—5022no—
211C4⋊C4—5022no—
212C4⋊C4—5021no—
213C4⋊C4—2512no—
214C4⋊C4—2511no—
215C4⋊C4C22×D25112yes—
216C4⋊C4C22×D25111yescharacteristic
217C22⋊C4—21no—
218C22⋊C4—5022no—
219C22⋊C4—5021no—
220C22⋊C4—5021no—
221C22⋊C4—5021no—
222C42—5022no—
223C42—2511no—
224C42—2511no—
225C42—2511no—
226C42C22×D25111yescharacteristic
227C23—21no—
228C23—10042no—
229C23—10042no—
230C23—5024no—
231C23—5021no—
232C23—2511no—
233Q8—42no—
234Q8—10042no—
235D4—21no—
236D4—5024no—
237D4—5021no—
238D4—2512no—
239D4Q8×D25112yes—
240C2×C4—22no—
241C2×C4—21no—
242C2×C4—42no—
243C2×C4—10041no—
244C2×C4—10042no—
245C2×C4—10042no—
246C2×C4—10042no—
247C2×C4—10042no—
248C2×C4—5022no—
249C2×C4—5021no—
250C2×C4—5022no—
251C2×C4—2511no—
252C2×C4—2511no—
253C2×C4—2511no—
254C2×C4—2512no—
255C2×C4—2512no—
256C2×C4D4×D25111yescharacteristic
257C2×C4D4×D25111yescharacteristic
258C2×C4C23×D25111yescharacteristic
259C8—22no—
260C8—5022no—
261C22—22no—
262C22—41no—
263C22—41no—
264C22—10048no—
265C22—10041no—
266C22—10041no—
267C22—5022no—
268C22—5021no—
269C22—5021no—
270C22—2512no—
271C22C2×D4×D25111yescharacteristic
272C4—21no—
273C4—41no—
274C4—42no—
275C4—10041no—
276C4—10042no—
277C4—5021no—
278C4—5021no—
279C4—2512no—
280C4#2254873912yes—
281C4C2×Q8×D25111yescharacteristic
282C4#41×D25111yescharacteristic
283C2—42no—
284C2—10042no—
285C2—5022no—
286C2C22⋊Q8×D25111yescharacteristic
287C2C2×#2254873911yescharacteristic
288C2#3063248611yescharacteristic
subgroupG/H
centerC22C2×D4×D251
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
Fitting#474×C251C2
socleC22×C251C22×D4
prime radicalC2×D4×D251C22
hypercenter#474D251
hypocenterC251#3077
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. #30632242C2semidirect
3. #30631990C2semidirect
4. #30631989C2semidirect
5. #30631978C2semidirect
6. #30631977C2semidirect
7. #30631964C2semidirect
8. #30631728C2semidirect
9. #30631709C2semidirect
11. #474×C251C2semidirect
15. #22548715C22semidirect
16. #22548706C22semidirect
30. #22548648C22semidirect
31. #22548647C22semidirect
32. #22548646C22semidirect
33. C4⋊(C251⋊C8)C22semidirect
34. Dic251⋊C8C22semidirect
38. C4⋊Q8×C251C22semidirect
40. #107×C251C22semidirect
41. C4⋊C8×C251C22semidirect
48. D4×D251Q8semidirect
144. C251#3077semidirect
153. #474D251semidirect
161. C4⋊Q8C2×D251semidirect
178. #107C2×D251semidirect
183. C4⋊C8C2×D251semidirect

Maximal subgroups

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

subgroupindex
#306322422
C4×D4×D2512
#306319902
#306319892
#306319782
#306319772
#306319642
#306317282
#306317092
#306316902
#474×C2512
#3077251