Finite Group Databasebeta

← #44560161

Subgroups — #44560161

Order 32128.

440 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 440 conjugacy classes of subgroups — too many to draw. Only its 135 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#30632240C211yesmaximal, characteristic, complemented
2C2×#22548677C211yesmaximal, characteristic
3#30631990C212yesmaximal, complemented
4#30631986C212yesmaximal, complemented
5#30631976C212yesmaximal, complemented
6#30631966C212yesmaximal, complemented
7#30631727C211yesmaximal, characteristic, complemented
8#30631710C211yesmaximal, characteristic, complemented
9#30631693C211yesmaximal, characteristic
10#30631691C211yesmaximal, characteristic
11#428×C251C211yesmaximal, characteristic, complemented
12C22×D1004C2211yescharacteristic
13C22×C4×D251—21no—
14C2×#16860794C2211yescharacteristic
15#22548714C2211yescharacteristic, complemented
16#22548714C2211yescharacteristic, complemented
17#22548706C2211yescharacteristic, complemented
18#22548706C2211yescharacteristic, complemented
19#22548692—21no—
20#22548692—21no—
21C4×Dic502—21no—
22C4×Dic502—21no—
23#22548677—21no—
24#22548677—21no—
25#22548677C2212yes—
26#22548677C2212yes—
27#22548674C2211yescharacteristic
28C2×(C2×D251)⋊C4—21no—
29C2×C4×Dic251C2211yescharacteristic
30#22548648C2212yescomplemented
31#22548646C2212yescomplemented
32Dic251⋊C8C2212yescomplemented
33Dic251⋊C8C2212yescomplemented
34#22548637C2212yes—
35#22548637C2212yes—
36#22548635C2212yes—
37#22548635C2212yes—
38C2×M16×C251C2211yescharacteristic
39#117×C251C2211yescharacteristic
40C2×C4⋊C4×C251C2211yescharacteristic
41#107×C251C2212yescomplemented
42#106×C251C2212yescomplemented
43C23×D251—21no—
44C2×Dic502—21no—
45C2×Dic502—21no—
46C2×D1004—21no—
47C2×D1004—22no—
48C2×D1004Q814yescomplemented
49C2×D1004C2312yes—
50C22×Dic251D411yescharacteristic
51C22×Dic251D411yescharacteristic
52C2×C4×D251—22no—
53C2×C4×D251—22no—
54C2×C4×D251—44no—
55#16860794—22no—
56C2×C251⋊C8C2312yes—
57C4⋊Dic251—21no—
58C4⋊Dic251—21no—
59C4⋊Dic251C2311yescharacteristic
60C4⋊Dic251C2311yescharacteristic
61Dic251⋊C4—21no—
62Dic251⋊C4—21no—
63Dic251⋊C4—21no—
64Dic251⋊C4—21no—
65C22⋊Dic251—21no—
66(C2×D251)⋊C4—22no—
67(C2×D251)⋊C4—22no—
68C4×Dic251—21no—
69C4×Dic251D411yescharacteristic
70C4×Dic251D411yescharacteristic
71C4×Dic251C2311yescharacteristic
72C4×Dic251C2311yescharacteristic
73C22×C4×C251—21no—
74C22×C4×C251C2311yescharacteristic
75M16×C251—22no—
76C2×C8×C251C2312yes—
77C4⋊C4×C251—21no—
78C4⋊C4×C251C2311yescharacteristic
79C4⋊C4×C251C2312yes—
80C4⋊C4×C251C2311yescharacteristic
81C22⋊C4×C251—21no—
82C42×C251—21no—
83C22×D251—24no—
84C22×D251—22no—
85C22×D251—42no—
86C22×D251—42no—
87Dic502—42no—
88Dic502—42no—
89D1004—24no—
90D1004—22no—
91D1004C2×Q814yes—
92C2×Dic251—22no—
93C2×Dic251—21no—
94C2×Dic251—21no—
95C2×Dic251—41no—
96C2×Dic251C2×D411yescharacteristic
97C2×Dic251C2×D411yescharacteristic
98C2×Dic251C2×D411yescharacteristic
99C2×Dic251C2×D411yescharacteristic
100C4×D251—44no—
101C4×D251—44no—
102C251⋊C8—22no—
103C23×C251C2×D411yescharacteristic
104C2×C4×C251—21no—
105C2×C4×C251—21no—
106C2×C4×C251—22no—
107C2×C4×C251—42no—
108C2×C4×C251C2×D411yescharacteristic
109C2×C4×C251C2×Q812yes—
110C2×C4×C251#4112yes—
111C2×C4×C251C2411yescharacteristic
112C8×C251—22no—
113C22×C251—21no—
114C22×C251—21no—
115C22×C251C22⋊Q812yes—
116C22×C251C22×D411yescharacteristic
117C2×D251—24no—
118C2×D251—48no—
119C2×D251—42no—
120C2×D251—42no—
121C4×C251—42no—
122C4×C251—41no—
123C4×C251—41no—
124C4×C251C22⋊Q812yes—
125C4×C251C22×Q811yescharacteristic
126C4×C251C2×#4111yescharacteristic
127Dic251—21no—
128Dic251—21no—
129Dic251—21no—
130Dic251—41no—
131Dic251—41no—
132Dic251AGL(1,ℤ/8)12yes—
133C2×C251—22no—
134C2×C251C2×C22⋊Q811yescharacteristic
135C2×C251C2×AGL(1,ℤ/8)11yescharacteristic
136C2×C251#57511yescharacteristic
137D251—44no—
138C251#307711yescharacteristic, complemented, Sylow
139#3077—2511nomaximal, Sylow
140#533—2511no—
141C2×C4×D4—2511no—
142#478—2512no—
143#478—2512no—
144#476—2512no—
145#474—2512no—
146#428—2511no—
147#428D25111yescharacteristic, complemented
148#423—2511no—
149#418—2511no—
150#418—2511no—
151C22×D4—2511no—
152C23×C4—5021no—
153C2×M16—2511no—
154C2×M16C2×D25111yescharacteristic
155C4⋊Q8—2511no—
156C4⋊Q8—2511no—
157#125—2511no—
158#125—2511no—
159C22⋊Q8—5021no—
160C22⋊Q8—5021no—
161C4×Q8—5021no—
162C4×Q8—5021no—
163C4×D4—5021no—
164C4×D4—5021no—
165C4×D4—2512no—
166C4×D4—2512no—
167#117—2511no—
168#117C2×D25111yescharacteristic
169C2×C4⋊C4C2×D25111yescharacteristic
170C2×C22⋊C4—5021no—
171C2×C42—2511no—
172#107—2512no—
173#107C2×D25112yescomplemented
174#106—2512no—
175#106C2×D25112yescomplemented
176C4⋊C8—2512no—
177C4⋊C8—2512no—
178#102—2512no—
179#102—2512no—
180#102—2512no—
181#102—2512no—
182C24—5021no—
183C2×Q8—5021no—
184C2×Q8—5021no—
185C2×D4—5021no—
186C2×D4—5022no—
187C2×D4—2514no—
188C2×D4—2512no—
189C22×C4—21no—
190C22×C4—10044no—
191C22×C4—5022no—
192C22×C4—5022no—
193C22×C4—2511no—
194C22×C4—2511no—
195C22×C4C22×D25111yescharacteristic
196M16—22no—
197M16—5022no—
198C2×C8—2512no—
199C2×C8C22×D25112yes—
200C4⋊C4—21no—
201C4⋊C4—5021no—
202C4⋊C4—5021no—
203C4⋊C4—5021no—
204C4⋊C4—5021no—
205C4⋊C4—5021no—
206C4⋊C4—5021no—
207C4⋊C4—2511no—
208C4⋊C4—2511no—
209C4⋊C4C22×D25111yescharacteristic
210C4⋊C4C22×D25111yescharacteristic
211C4⋊C4C22×D25112yes—
212C22⋊C4—21no—
213C22⋊C4—5022no—
214C22⋊C4—5022no—
215C22⋊C4—5021no—
216C42—21no—
217C42—5021no—
218C42—2511no—
219C42—2511no—
220C42—2511no—
221C42—2511no—
222C23—10042no—
223C23—10042no—
224C23—5022no—
225C23—5024no—
226C23D4×D25111yescharacteristic
227Q8—10042no—
228Q8—10042no—
229D4—5022no—
230D4—5024no—
231D4—2514no—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—22no—
235C2×C4—42no—
236C2×C4—10044no—
237C2×C4—10044no—
238C2×C4—10041no—
239C2×C4—5021no—
240C2×C4—5021no—
241C2×C4—5022no—
242C2×C4—2511no—
243C2×C4—2511no—
244C2×C4—2511no—
245C2×C4—2511no—
246C2×C4D4×D25111yescharacteristic
247C2×C4Q8×D25112yes—
248C2×C4#1686081012yes—
249C2×C4C23×D25111yescharacteristic
250C8—22no—
251C8—5022no—
252C22—21no—
253C22—21no—
254C22—10048no—
255C22—10042no—
256C22—10042no—
257C22—5024no—
258C22#2254869312yes—
259C22C2×D4×D25111yescharacteristic
260C4—42no—
261C4—41no—
262C4—41no—
263C4—10041no—
264C4—10041no—
265C4—5021no—
266C4—5021no—
267C4—5021no—
268C4—2512no—
269C4#2254869312yes—
270C4C2×Q8×D25111yescharacteristic
271C4C2×#1686081011yescharacteristic
272C2—22no—
273C2—10044no—
274C2C2×#2254869311yescharacteristic
275C2AGL(1,ℤ/8)×D25111yescharacteristic
276C2#3063248611yescharacteristic
subgroupG/H
centerC22C2×D4×D251
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
Fitting#428×C251C2
socleC22×C251C22×D4
prime radicalC22×D1004C22
hypercenter#428D251
hypocenterC251#3077
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C22×C4 ◁ #428

Complements

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

NH ≅ G/Ntype
1. #30632240C2semidirect
3. #30631990C2semidirect
4. #30631986C2semidirect
5. #30631976C2semidirect
6. #30631966C2semidirect
7. #30631727C2semidirect
8. #30631710C2semidirect
11. #428×C251C2semidirect
15. #22548714C22semidirect
16. #22548714C22semidirect
17. #22548706C22semidirect
18. #22548706C22semidirect
30. #22548648C22semidirect
31. #22548646C22semidirect
32. Dic251⋊C8C22semidirect
33. Dic251⋊C8C22semidirect
41. #107×C251C22semidirect
42. #106×C251C22semidirect
48. C2×D1004Q8semidirect
138. C251#3077semidirect
147. #428D251semidirect
173. #107C2×D251semidirect
175. #106C2×D251semidirect

Maximal subgroups

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

subgroupindex
#306322402
C2×#225486772
#306319902
#306319862
#306319762
#306319662
#306317272
#306317102
#306316932
#306316912
#428×C2512
#3077251