Finite Group Databasebeta

← #44560000

Subgroups — #44560000

Order 32128.

492 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 492 conjugacy classes of subgroups — too many to draw. Only its 141 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×#22548721C211yesmaximal, characteristic, complemented
2#30632261C211yesmaximal, characteristic, complemented
3#30632254C211yesmaximal, characteristic
4#30631930C214yesmaximal, complemented
5#30631913C214yesmaximal, complemented
6#30631725C211yesmaximal, characteristic, complemented
7#30631696C211yesmaximal, characteristic, complemented
8#30631692C211yesmaximal, characteristic, complemented
9C22×SD16×C251C211yesmaximal, characteristic, complemented
10C2×#16860810C2211yescharacteristic
11#22548721—24no—
12C2×#16860794C2211yescharacteristic
13C2×#16860793C2212yescomplemented
14C2×C8×D251C2212yescomplemented
15#22548713—21no—
16#22548711—21no—
17#22548704—21no—
18#22548702—21no—
19#22548698—22no—
20#22548691C2214yescomplemented
21#22548686C2214yes—
22#22548684—22no—
23#22548674C2211yescharacteristic
24#22548674C2211yescharacteristic
25#22548649C2212yescomplemented
26#22548647C2212yescomplemented
27#22548639C2214yescomplemented
28#22548636C2214yes—
29C22×Q8×C251C2211yescharacteristic
30C22×D4×C251C2211yescharacteristic
31C2×SD16×C251—24no—
32C2×SD16×C251C2212yes—
33C2×SD16×C251C2212yes—
34C22×C8×C251C2211yescharacteristic
35#16860810—22no—
36#16860810—22no—
37C2×Dic502D411yescharacteristic
38C2×D1004D411yescharacteristic
39C2×C22⋊D251—24no—
40C2×C22⋊D251D412yes—
41C2×C4×D251C2312yes—
42#16860794—22no—
43#16860793—24no—
44C2×C251⋊C8C2312yes—
45C8×D251—24no—
46C4⋊Dic251C2312yes—
47C4⋊Dic251C2312yes—
48Dic251⋊C4—24no—
49C22⋊Dic251—24no—
50C22⋊Dic251—21no—
51C22⋊Dic251—21no—
52(C2×D251)⋊C4—24no—
53C4×Dic251—21no—
54C4×Dic251—21no—
55C24×C251—21no—
56C2×Q8×C251—21no—
57C2×Q8×C251—22no—
58C2×Q8×C251—22no—
59C2×Q8×C251C2312yes—
60C2×D4×C251—22no—
61C2×D4×C251—21no—
62C2×D4×C251—22no—
63C2×D4×C251C2312yes—
64C22×C4×C251—21no—
65C22×C4×C251C2311yescharacteristic
66SD16×C251—24no—
67SD16×C251—24no—
68C2×C8×C251D414yescomplemented
69C2×C8×C251C2311yescharacteristic
70C2×C8×C251C2311yescharacteristic
71C22×D251C2×D412yes—
72Dic502—22no—
73D1004—22no—
74C22⋊D251—24no—
75C22⋊D251—84no—
76C2×Dic251—22no—
77C2×Dic251—22no—
78C2×Dic251C2×D412yes—
79C4×D251—22no—
80C4×D251—22no—
81C251⋊C8—22no—
82C23×C251—22no—
83C23×C251—42no—
84C23×C251—82no—
85C23×C251C2×D411yescharacteristic
86Q8×C251—22no—
87Q8×C251—22no—
88Q8×C251—42no—
89D4×C251—22no—
90D4×C251—22no—
91D4×C251—42no—
92C2×C4×C251—22no—
93C2×C4×C251—42no—
94C2×C4×C251C2×D411yescharacteristic
95C2×C4×C251C2×D412yes—
96C2×C4×C251#4112yes—
97C2×C4×C251C2411yescharacteristic
98C8×C251C2×D412yes—
99C8×C251C2×D412yes—
100C22×C251—22no—
101C22×C251—42no—
102C22×C251—44no—
103C22×C251—84no—
104C22×C251#12112yes—
105C22×C251C22×D411yescharacteristic
106C2×D251—22no—
107C2×D251—24no—
108C4×C251—42no—
109C4×C251#12112yes—
110C4×C251C22×D411yescharacteristic
111C4×C251C2×#4111yescharacteristic
112Dic251—22no—
113Dic251—42no—
114Dic251—42no—
115C2×C251—22no—
116C2×C251—82no—
117C2×C251C2×#12111yescharacteristic
118C2×C251#57712yes—
119D251—42no—
120C251#305711yescharacteristic, complemented, Sylow
121#3057—2511nomaximal, Sylow
122C22×SD16D25111yescharacteristic, complemented
123C2×#131—2511no—
124#536—2511no—
125#535—2511no—
126#468—2514no—
127#465—2514no—
128#427—2511no—
129#419—2511no—
130#418—2511no—
131C2×#41—2511no—
132C22×Q8C2×D25111yescharacteristic
133C22×D4C2×D25111yescharacteristic
134C2×SD16—24no—
135C2×SD16C2×D25112yes—
136C2×SD16C2×D25112yes—
137#131—5024no—
138C2×M16—2511no—
139C2×M16—2512no—
140C22×C8—2512no—
141C22×C8C2×D25111yescharacteristic
142C4⋊Q8—5021no—
143#127—5021no—
144#124—5021no—
145#124—5021no—
146#123—5022no—
147C22⋊Q8—2514no—
148#121—2514no—
149#120—5022no—
150#117—2511no—
151#117—2511no—
152#107—2512no—
153#106—2512no—
154#103—2514no—
155#102—2514no—
156C24—21no—
157#41—5022no—
158#41—5022no—
159C2×Q8—21no—
160C2×Q8—22no—
161C2×Q8—22no—
162C2×Q8—2511no—
163C2×Q8C22×D25112yes—
164C2×D4—21no—
165C2×D4—22no—
166C2×D4—22no—
167C2×D4—5024no—
168C2×D4—2512no—
169C2×D4—2511no—
170C2×D4C22×D25112yes—
171C22×C4—21no—
172C22×C4—2512no—
173C22×C4C22×D25111yescharacteristic
174SD16—24no—
175SD16—24no—
176M16—5022no—
177M16—5024no—
178C2×C8—5024no—
179C2×C8—2512no—
180C2×C8C22⋊D25114yescomplemented
181C2×C8C22×D25111yescharacteristic
182C2×C8C22×D25111yescharacteristic
183C4⋊C4—5024no—
184C4⋊C4—2512no—
185C4⋊C4—2512no—
186C22⋊C4—5024no—
187C22⋊C4—5021no—
188C22⋊C4—5021no—
189C22⋊C4—5024no—
190C42—5021no—
191C42—5021no—
192C23—22no—
193C23—42no—
194C23—82no—
195C23—2512no—
196C23D4×D25111yescharacteristic
197Q8—22no—
198Q8—22no—
199Q8—42no—
200Q8—5022no—
201D4—22no—
202D4—22no—
203D4—42no—
204D4—20084no—
205D4—5024no—
206D4—5022no—
207C2×C4—22no—
208C2×C4—42no—
209C2×C4—5022no—
210C2×C4—5022no—
211C2×C4—5022no—
212C2×C4—5022no—
213C2×C4—2512no—
214C2×C4D4×D25111yescharacteristic
215C2×C4C2×C22⋊D25112yes—
216C2×C4#1686080912yes—
217C2×C4C23×D25111yescharacteristic
218C8—5022no—
219C8C2×C22⋊D25112yes—
220C8C2×C22⋊D25112yes—
221C22—22no—
222C22—44no—
223C22—42no—
224C22—84no—
225C22—5024no—
226C22—5022no—
227C22#2254868612yes—
228C22C2×D4×D25111yescharacteristic
229C4—42no—
230C4—10042no—
231C4—10042no—
232C4—5022no—
233C4#2254868612yes—
234C4C22×C22⋊D25111yescharacteristic
235C4C2×#1686080911yescharacteristic
236C2—22no—
237C2—82no—
238C2—10042no—
239C2C2×#2254868611yescharacteristic
240C2#3063249312yes—
subgroupG/H
centerC22C2×D4×D251
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
FittingC22×SD16×C251C2
socleC22×C251C22×D4
prime radical#30632254C2
hypercenterC22×SD16D251
hypocenterC251#3057
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×SD16

Complements

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

NH ≅ G/Ntype
1. C2×#22548721C2semidirect
2. #30632261C2semidirect
4. #30631930C2semidirect
5. #30631913C2semidirect
6. #30631725C2semidirect
7. #30631696C2semidirect
8. #30631692C2semidirect
9. C22×SD16×C251C2semidirect
13. C2×#16860793C22semidirect
14. C2×C8×D251C22semidirect
20. #22548691C22semidirect
25. #22548649C22semidirect
26. #22548647C22semidirect
27. #22548639C22semidirect
68. C2×C8×C251D4semidirect
120. C251#3057semidirect
122. C22×SD16D251semidirect
180. C2×C8C22⋊D251semidirect

Maximal subgroups

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

subgroupindex
C2×#225487212
#306322612
#306322542
#306319302
#306319132
#306317252
#306316962
#306316922
C22×SD16×C2512
#3057251