Finite Group Databasebeta

← #44560288

Subgroups — #44560288

Order 32128.

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

normal subgroups only

This group has 458 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
1#30632198C211yesmaximal, characteristic
2#30632022C212yesmaximal, complemented
3#30632017C212yesmaximal, complemented
4#30632005C212yesmaximal, complemented
5#30632000C212yesmaximal, complemented
6#30631727C212yesmaximal, complemented
7#30631686C212yesmaximal, complemented
8#30631651C211yesmaximal, characteristic, complemented
9C22×C4⋊C4×C251C211yesmaximal, characteristic, complemented
10C2×#16860810C2211yescharacteristic
11C2×#16860794C2212yes—
12#22548698—21no—
13#22548694C2211yescharacteristic
14#22548694C2211yescharacteristic
15#22548689C2211yescharacteristic
16#22548689C2211yescharacteristic
17#22548684—21no—
18C4×C22⋊D251—21no—
19C4×C22⋊D251—21no—
20#22548674C2211yescharacteristic
21#22548648C2214yescomplemented
22#22548646C2214yescomplemented
23#22548638C2214yescomplemented
24#22548635C2214yescomplemented
25C22⋊(C251⋊C8)C2212yes—
26C22⋊(C251⋊C8)C2212yes—
27C23×C4×C251—21no—
28C23×C4×C251C2211yescharacteristic
29C2×C4⋊C4×C251—21no—
30C2×C4⋊C4×C251—21no—
31C2×C4⋊C4×C251—21no—
32C2×C4⋊C4×C251C2212yes—
33C2×C4⋊C4×C251C2212yes—
34C2×C4⋊C4×C251C2212yes—
35#16860810—24no—
36C2×Dic502C2311yescharacteristic
37C2×D1004C2311yescharacteristic
38C2×C22⋊D251—21no—
39C2×C22⋊D251—21no—
40C2×C22⋊D251—21no—
41C2×C4×D251—21no—
42#16860794—24no—
43C2×C251⋊C8C2314yes—
44C4⋊Dic251C2311yescharacteristic
45C4⋊Dic251C2311yescharacteristic
46Dic251⋊C4—21no—
47Dic251⋊C4—21no—
48C22⋊Dic251—21no—
49C22⋊Dic251—21no—
50C22⋊Dic251—21no—
51(C2×D251)⋊C4—21no—
52(C2×D251)⋊C4—21no—
53C4×Dic251—21no—
54C24×C251D411yescharacteristic
55C22×C4×C251—24no—
56C22×C4×C251—22no—
57C22×C4×C251—22no—
58C22×C4×C251—22no—
59C22×C4×C251—44no—
60C22×C4×C251D411yescharacteristic
61C22×C4×C251D411yescharacteristic
62C22×C4×C251D411yescharacteristic
63C22×C4×C251C2311yescharacteristic
64C22×C4×C251C2311yescharacteristic
65C22×C4×C251C2311yescharacteristic
66C4⋊C4×C251—22no—
67C4⋊C4×C251—22no—
68C4⋊C4×C251—22no—
69C4⋊C4×C251C2314yes—
70C22×D251—21no—
71Dic502—22no—
72D1004—22no—
73C22⋊D251—42no—
74C22⋊D251—42no—
75C22⋊D251—42no—
76C2×Dic251—21no—
77C2×Dic251—21no—
78C2×Dic251—21no—
79C4×D251—42no—
80C251⋊C8—24no—
81C23×C251—22no—
82C23×C251—22no—
83C23×C251—22no—
84C23×C251C2×D411yescharacteristic
85C23×C251C2×D411yescharacteristic
86C23×C251C2×D411yescharacteristic
87C2×C4×C251—22no—
88C2×C4×C251—22no—
89C2×C4×C251—22no—
90C2×C4×C251—22no—
91C2×C4×C251—22no—
92C2×C4×C251—24no—
93C2×C4×C251—44no—
94C2×C4×C251—44no—
95C2×C4×C251—44no—
96C2×C4×C251C2×D411yescharacteristic
97C2×C4×C251C2×D411yescharacteristic
98C2×C4×C251C2×D411yescharacteristic
99C2×C4×C251#4114yes—
100C2×C4×C251C2411yescharacteristic
101C22×C251—28no—
102C22×C251—21no—
103C22×C251—21no—
104C22×C251—21no—
105C22×C251—21no—
106C22×C251—21no—
107C22×C251—21no—
108C22×C251#12312yes—
109C22×C251AGL(1,ℤ/8)12yes—
110C22×C251#13712yes—
111C22×C251C22×D411yescharacteristic
112C2×D251—41no—
113C2×D251—41no—
114C2×D251—41no—
115C4×C251—21no—
116C4×C251—21no—
117C4×C251—44no—
118C4×C251#12312yes—
119C4×C251C2×#4112yes—
120Dic251—41no—
121Dic251—41no—
122Dic251—41no—
123C2×C251—22no—
124C2×C251—22no—
125C2×C251—22no—
126C2×C251C2×#12311yescharacteristic
127C2×C251C2×AGL(1,ℤ/8)11yescharacteristic
128C2×C251C2×#13711yescharacteristic
129D251—81no—
130C251#309211yescharacteristic, complemented, Sylow
131#3092—2511nomaximal, Sylow
132#525—2511no—
133C22×C4⋊C4D25111yescharacteristic, complemented
134#484—2512no—
135#483—2512no—
136#481—2512no—
137#480—2512no—
138#428—2512no—
139#417—2512no—
140#407—2511no—
141C2×#41—2511no—
142C23×C4—21no—
143C23×C4C2×D25111yescharacteristic
144C2×M16—2512no—
145#123—5021no—
146C22⋊Q8—2511no—
147C22⋊Q8—2511no—
148#121—2511no—
149#121—2511no—
150#120—5021no—
151C4×D4—5021no—
152C4×D4—5021no—
153#117—2511no—
154C2×C4⋊C4—21no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4C2×D25112yes—
158C2×C4⋊C4C2×D25112yes—
159C2×C4⋊C4C2×D25112yes—
160#107—2514no—
161#106—2514no—
162#103—2514no—
163#102—2514no—
164C22⋊C8—2512no—
165C22⋊C8—2512no—
166C24C22⋊D25111yescharacteristic
167#41—5024no—
168C2×Q8—2511no—
169C2×D4—5021no—
170C2×D4—5021no—
171C2×D4—5021no—
172C2×D4—2511no—
173C22×C4—24no—
174C22×C4—22no—
175C22×C4—22no—
176C22×C4—22no—
177C22×C4—44no—
178C22×C4—5021no—
179C22×C4C22⋊D25111yescharacteristic
180C22×C4C22⋊D25111yescharacteristic
181C22×C4C22⋊D25111yescharacteristic
182C22×C4C22×D25111yescharacteristic
183C22×C4C22×D25111yescharacteristic
184C22×C4C22×D25111yescharacteristic
185M16—5024no—
186C2×C8—2514no—
187C4⋊C4—22no—
188C4⋊C4—22no—
189C4⋊C4—22no—
190C4⋊C4—5021no—
191C4⋊C4—5021no—
192C4⋊C4—2511no—
193C4⋊C4—2511no—
194C4⋊C4C22×D25114yes—
195C22⋊C4—5021no—
196C22⋊C4—5021no—
197C22⋊C4—5021no—
198C22⋊C4—5021no—
199C22⋊C4—5021no—
200C42—5021no—
201C23—22no—
202C23—22no—
203C23—22no—
204C23—5021no—
205C23C2×C22⋊D25111yescharacteristic
206C23C2×C22⋊D25111yescharacteristic
207C23C2×C22⋊D25111yescharacteristic
208Q8—5022no—
209D4—10042no—
210D4—10042no—
211D4—10042no—
212D4—5022no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—22no—
216C2×C4—22no—
217C2×C4—22no—
218C2×C4—24no—
219C2×C4—44no—
220C2×C4—44no—
221C2×C4—44no—
222C2×C4—10042no—
223C2×C4—5021no—
224C2×C4—5021no—
225C2×C4—5021no—
226C2×C4C2×C22⋊D25111yescharacteristic
227C2×C4C2×C22⋊D25111yescharacteristic
228C2×C4C2×C22⋊D25111yescharacteristic
229C2×C4#1686081014yes—
230C2×C4C23×D25111yescharacteristic
231C8—5024no—
232C22—28no—
233C22—21no—
234C22—21no—
235C22—21no—
236C22—21no—
237C22—21no—
238C22—21no—
239C22—10041no—
240C22—10041no—
241C22—10041no—
242C22#2254869812yes—
243C22#2254874112yes—
244C22#2254874612yes—
245C22C22×C22⋊D25111yescharacteristic
246C4—21no—
247C4—21no—
248C4—44no—
249C4—10041no—
250C4—10041no—
251C4—10041no—
252C4#2254869812yes—
253C4C2×#1686081012yes—
254C2—22no—
255C2—22no—
256C2—22no—
257C2—20081no—
258C2C2×#2254869811yescharacteristic
259C2C2×#2254874111yescharacteristic
260C2C2×#2254874611yescharacteristic
subgroupG/H
centerC22C22×C22⋊D251
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
FittingC22×C4⋊C4×C251C2
socleC22×C251C22×D4
prime radical#30632198C2
hypercenterC22×C4⋊C4D251
hypocenterC251#3092
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C23×C4 ◁ C22×C4⋊C4

Complements

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

NH ≅ G/Ntype
2. #30632022C2semidirect
3. #30632017C2semidirect
4. #30632005C2semidirect
5. #30632000C2semidirect
6. #30631727C2semidirect
7. #30631686C2semidirect
8. #30631651C2semidirect
9. C22×C4⋊C4×C251C2semidirect
21. #22548648C22semidirect
22. #22548646C22semidirect
23. #22548638C22semidirect
24. #22548635C22semidirect
130. C251#3092semidirect
133. C22×C4⋊C4D251semidirect

Maximal subgroups

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

subgroupindex
#306321982
#306320222
#306320172
#306320052
#306320002
#306317272
#306316862
#306316512
C22×C4⋊C4×C2512
#3092251