Finite Group Databasebeta

← #44560006

Subgroups — #44560006

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
1C22×#16860797C211yesmaximal, characteristic, complemented
2C2×#22548722C211yesmaximal, characteristic, complemented
3#30632251C211yesmaximal, characteristic
4#30631935C214yesmaximal, complemented
5#30631918C214yesmaximal, complemented
6#30631723C211yesmaximal, characteristic, complemented
7#30631696C211yesmaximal, characteristic, complemented
8#30631691C211yesmaximal, characteristic, complemented
9#536×C251C211yesmaximal, characteristic, complemented
10C22×D1004C2211yescharacteristic
11C2×#16860797—24no—
12C2×#16860797C2212yes—
13C2×#16860797C2212yes—
14#22548722—24no—
15C2×#16860794C2211yescharacteristic
16C2×#16860794C2212yescomplemented
17C22×C251⋊C8C2211yescharacteristic
18C22×C251⋊C8C2212yescomplemented
19#22548711—21no—
20#22548704—21no—
21#22548689C2214yes—
22#22548683—22no—
23#22548674C2211yescharacteristic
24#22548648C2212yescomplemented
25#22548646C2212yescomplemented
26#22548639C2214yescomplemented
27#22548635C2214yes—
28C2×#41×C251C2211yescharacteristic
29C22×Q8×C251C2211yescharacteristic
30C4⋊Q8×C251—21no—
31#124×C251—21no—
32#123×C251—22no—
33C22⋊Q8×C251C2214yescomplemented
34#117×C251C2211yescharacteristic
35C23×D251—21no—
36C2×D1004—22no—
37C2×D1004—21no—
38C2×D1004—22no—
39C2×D1004C2312yes—
40C2×C22⋊D251—24no—
41#16860797—24no—
42#16860797—24no—
43#16860794—24no—
44#16860794—22no—
45C2×C251⋊C8—24no—
46C2×C251⋊C8D414yescomplemented
47C2×C251⋊C8C2312yes—
48C2×C251⋊C8C2311yescharacteristic
49C2×C251⋊C8C2311yescharacteristic
50C4⋊Dic251C2312yes—
51C22⋊Dic251—21no—
52(C2×D251)⋊C4—24no—
53C4×Dic251—21no—
54#41×C251—22no—
55#41×C251—22no—
56C2×Q8×C251—21no—
57C2×Q8×C251—22no—
58C2×Q8×C251—22no—
59C2×Q8×C251D411yescharacteristic
60C2×Q8×C251C2312yes—
61C2×D4×C251D412yes—
62C2×D4×C251D411yescharacteristic
63C22×C4×C251—21no—
64C22×C4×C251C2312yes—
65C22×C4×C251C2311yescharacteristic
66C4⋊C4×C251—24no—
67C4⋊C4×C251C2312yes—
68C22⋊C4×C251—21no—
69C22⋊C4×C251—24no—
70C42×C251—21no—
71C22×D251—22no—
72C22×D251—42no—
73C22×D251—82no—
74D1004—22no—
75D1004—22no—
76D1004—42no—
77C22⋊D251—84no—
78C2×Dic251—22no—
79C251⋊C8—22no—
80C251⋊C8C2×D412yes—
81C251⋊C8C2×D412yes—
82C23×C251C2×D412yes—
83C23×C251C2×D411yescharacteristic
84Q8×C251—22no—
85Q8×C251—22no—
86Q8×C251—22no—
87Q8×C251—42no—
88D4×C251—24no—
89D4×C251—22no—
90C2×C4×C251—22no—
91C2×C4×C251—22no—
92C2×C4×C251—22no—
93C2×C4×C251—22no—
94C2×C4×C251—42no—
95C2×C4×C251C2×D412yes—
96C2×C4×C251C2×D412yes—
97C2×C4×C251C2×D411yescharacteristic
98C2×C4×C251#4112yes—
99C2×C4×C251C2411yescharacteristic
100C22×C251—22no—
101C22×C251—22no—
102C22×C251—24no—
103C22×C251#12112yes—
104C22×C251C22×D411yescharacteristic
105C2×D251—42no—
106C2×D251—44no—
107C2×D251—84no—
108C4×C251—22no—
109C4×C251—42no—
110C4×C251—42no—
111C4×C251#12112yes—
112C4×C251C22×D411yescharacteristic
113C4×C251C2×#4111yescharacteristic
114Dic251—42no—
115C2×C251—22no—
116C2×C251—42no—
117C2×C251C2×#12111yescharacteristic
118C2×C251#57712yes—
119D251—82no—
120C251#305711yescharacteristic, complemented, Sylow
121#3057—2511nomaximal, Sylow
122C22×SD16—2511no—
123C2×#131—2511no—
124#536D25111yescharacteristic, complemented
125#535—2511no—
126#468—2514no—
127#465—2514no—
128#427—2511no—
129#419—2511no—
130#418—2511no—
131C2×#41C2×D25111yescharacteristic
132C22×Q8C2×D25111yescharacteristic
133C22×D4—2511no—
134C2×SD16—5024no—
135C2×SD16—2512no—
136C2×SD16—2512no—
137#131—5024no—
138C2×M16—2511no—
139C2×M16—2512no—
140C22×C8—2512no—
141C22×C8—2511no—
142C4⋊Q8—21no—
143#127—5021no—
144#124—21no—
145#124—5021no—
146#123—22no—
147C22⋊Q8C2×D25114yescomplemented
148#121—2514no—
149#120—5022no—
150#117—2511no—
151#117C2×D25111yescharacteristic
152#107—2512no—
153#106—2512no—
154#103—2514no—
155#102—2514no—
156C24—5021no—
157#41—22no—
158#41—22no—
159C2×Q8—21no—
160C2×Q8—22no—
161C2×Q8—22no—
162C2×Q8C22⋊D25111yescharacteristic
163C2×Q8C22×D25112yes—
164C2×D4—5024no—
165C2×D4—5022no—
166C2×D4—5021no—
167C2×D4—5022no—
168C2×D4—2512no—
169C2×D4C22⋊D25112yes—
170C2×D4C22⋊D25111yescharacteristic
171C22×C4—21no—
172C22×C4C22×D25112yes—
173C22×C4C22×D25111yescharacteristic
174SD16—5024no—
175SD16—5024no—
176M16—5022no—
177M16—5024no—
178C2×C8—5024no—
179C2×C8—2512no—
180C2×C8—2514no—
181C2×C8—2511no—
182C2×C8—2511no—
183C4⋊C4—24no—
184C4⋊C4—2512no—
185C4⋊C4C22×D25112yes—
186C22⋊C4—21no—
187C22⋊C4—24no—
188C22⋊C4—5021no—
189C22⋊C4—5024no—
190C42—21no—
191C42—5021no—
192C23—20082no—
193C23—10042no—
194C23—5022no—
195C23C2×C22⋊D25112yes—
196C23C2×C22⋊D25111yescharacteristic
197Q8—22no—
198Q8—22no—
199Q8—22no—
200Q8—42no—
201D4—24no—
202D4—22no—
203D4—20084no—
204D4—10042no—
205D4—5022no—
206D4—5022no—
207C2×C4—22no—
208C2×C4—22no—
209C2×C4—22no—
210C2×C4—22no—
211C2×C4—42no—
212C2×C4—5022no—
213C2×C4D4×D25112yes—
214C2×C4C2×C22⋊D25112yes—
215C2×C4C2×C22⋊D25111yescharacteristic
216C2×C4#1686080912yes—
217C2×C4C23×D25111yescharacteristic
218C8—5022no—
219C8—2512no—
220C8—2512no—
221C22—22no—
222C22—24no—
223C22—22no—
224C22—20084no—
225C22—10044no—
226C22—10042no—
227C22#2254868712yes—
228C22C22×C22⋊D25111yescharacteristic
229C4—22no—
230C4—42no—
231C4—42no—
232C4—10042no—
233C4#2254868712yes—
234C4C2×D4×D25111yescharacteristic
235C4C2×#1686080911yescharacteristic
236C2—22no—
237C2—42no—
238C2—20082no—
239C2C2×#2254868711yescharacteristic
240C2#3063249512yes—
subgroupG/H
centerC22C22×C22⋊D251
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
Fitting#536×C251C2
socleC22×C251C22×D4
prime radical#30632251C2
hypercenter#536D251
hypocenterC251#3057
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C2×#41 ◁ #536

Complements

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

NH ≅ G/Ntype
1. C22×#16860797C2semidirect
2. C2×#22548722C2semidirect
4. #30631935C2semidirect
5. #30631918C2semidirect
6. #30631723C2semidirect
7. #30631696C2semidirect
8. #30631691C2semidirect
9. #536×C251C2semidirect
16. C2×#16860794C22semidirect
18. C22×C251⋊C8C22semidirect
24. #22548648C22semidirect
25. #22548646C22semidirect
26. #22548639C22semidirect
33. C22⋊Q8×C251C22semidirect
46. C2×C251⋊C8D4semidirect
120. C251#3057semidirect
124. #536D251semidirect
147. C22⋊Q8C2×D251semidirect

Maximal subgroups

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

subgroupindex
C22×#168607972
C2×#225487222
#306322512
#306319352
#306319182
#306317232
#306316962
#306316912
#536×C2512
#3057251