Finite Group Databasebeta

← #44560290

Subgroups — #44560290

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
2C22×C4⋊Dic251C211yesmaximal, characteristic, complemented
3#30632023C212yesmaximal, complemented
4#30632018C212yesmaximal, complemented
5#30632006C212yesmaximal, complemented
6#30632001C212yesmaximal, complemented
7#30631729C212yesmaximal, complemented
8#30631689C212yesmaximal, complemented
9#407×C251C211yesmaximal, characteristic, complemented
10C2×#16860810C2211yescharacteristic
11C23×Dic251—21no—
12#22548698—21no—
13#22548694C2211yescharacteristic
14#22548694C2211yescharacteristic
15#22548689C2211yescharacteristic
16#22548689C2211yescharacteristic
17#22548684—21no—
18C4×C22⋊D251—21no—
19C4×C22⋊D251—21no—
20#22548674C2211yescharacteristic
21C2×C4⋊Dic251—21no—
22C2×C4⋊Dic251—21no—
23C2×C4⋊Dic251—21no—
24C2×C4⋊Dic251C2212yes—
25C2×C4⋊Dic251C2212yes—
26C2×C4⋊Dic251C2212yes—
27#22548649C2214yescomplemented
28#22548647C2214yescomplemented
29#22548640C2214yescomplemented
30#22548637C2214yescomplemented
31C23×C4×C251C2211yescharacteristic
32C2×M16×C251C2212yes—
33C22⋊C8×C251C2212yes—
34C22⋊C8×C251C2212yes—
35#16860810—24no—
36C2×Dic502C2311yescharacteristic
37C2×D1004C2311yescharacteristic
38C2×C22⋊D251—21no—
39C2×C22⋊D251—21no—
40C2×C22⋊D251—21no—
41C22×Dic251—22no—
42C22×Dic251—22no—
43C22×Dic251—22no—
44C22×Dic251—44no—
45C2×C4×D251—21no—
46C4⋊Dic251—22no—
47C4⋊Dic251—22no—
48C4⋊Dic251—22no—
49C4⋊Dic251C2314yes—
50C4⋊Dic251C2311yescharacteristic
51C4⋊Dic251C2311yescharacteristic
52Dic251⋊C4—21no—
53Dic251⋊C4—21no—
54C22⋊Dic251—21no—
55C22⋊Dic251—21no—
56C22⋊Dic251—21no—
57(C2×D251)⋊C4—21no—
58(C2×D251)⋊C4—21no—
59C4×Dic251—21no—
60C24×C251D411yescharacteristic
61C22×C4×C251—24no—
62C22×C4×C251D411yescharacteristic
63C22×C4×C251D411yescharacteristic
64C22×C4×C251D411yescharacteristic
65C22×C4×C251C2311yescharacteristic
66C22×C4×C251C2311yescharacteristic
67C22×C4×C251C2311yescharacteristic
68M16×C251—24no—
69C2×C8×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—
79C2×Dic251—24no—
80C2×Dic251—44no—
81C2×Dic251—44no—
82C2×Dic251—44no—
83C4×D251—42no—
84C23×C251—22no—
85C23×C251—22no—
86C23×C251—22no—
87C23×C251C2×D411yescharacteristic
88C23×C251C2×D411yescharacteristic
89C23×C251C2×D411yescharacteristic
90C2×C4×C251—22no—
91C2×C4×C251—22no—
92C2×C4×C251—22no—
93C2×C4×C251—22no—
94C2×C4×C251—22no—
95C2×C4×C251C2×D411yescharacteristic
96C2×C4×C251C2×D411yescharacteristic
97C2×C4×C251C2×D411yescharacteristic
98C2×C4×C251#4114yes—
99C2×C4×C251C2411yescharacteristic
100C8×C251—24no—
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#12312yes—
118C4×C251C2×#4112yes—
119Dic251—41no—
120Dic251—44no—
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⋊C4—2511no—
134#484—2512no—
135#483—2512no—
136#481—2512no—
137#480—2512no—
138#428—2512no—
139#417—2512no—
140#407D25111yescharacteristic, complemented
141C2×#41—2511no—
142C23×C4—5021no—
143C23×C4C2×D25111yescharacteristic
144C2×M16C2×D25112yes—
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—5021no—
155C2×C4⋊C4—5021no—
156C2×C4⋊C4—5021no—
157C2×C4⋊C4—2512no—
158C2×C4⋊C4—2512no—
159C2×C4⋊C4—2512no—
160#107—2514no—
161#106—2514no—
162#103—2514no—
163#102—2514no—
164C22⋊C8C2×D25112yes—
165C22⋊C8C2×D25112yes—
166C24D100411yescharacteristic
167#41—5024no—
168C2×Q8—2511no—
169C2×D4—5021no—
170C2×D4—5021no—
171C2×D4—5021no—
172C2×D4—2511no—
173C22×C4—24no—
174C22×C4—10044no—
175C22×C4—5022no—
176C22×C4—5022no—
177C22×C4—5022no—
178C22×C4—5021no—
179C22×C4D100411yescharacteristic
180C22×C4D100411yescharacteristic
181C22×C4D100411yescharacteristic
182C22×C4C22×D25111yescharacteristic
183C22×C4C22×D25111yescharacteristic
184C22×C4C22×D25111yescharacteristic
185M16—24no—
186C2×C8C22×D25114yes—
187C4⋊C4—5022no—
188C4⋊C4—5022no—
189C4⋊C4—5022no—
190C4⋊C4—5021no—
191C4⋊C4—5021no—
192C4⋊C4—2514no—
193C4⋊C4—2511no—
194C4⋊C4—2511no—
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×D100411yescharacteristic
206C23C2×D100411yescharacteristic
207C23C2×D100411yescharacteristic
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—10044no—
219C2×C4—10044no—
220C2×C4—10044no—
221C2×C4—10042no—
222C2×C4—5021no—
223C2×C4—5021no—
224C2×C4—5021no—
225C2×C4—5024no—
226C2×C4C2×D100411yescharacteristic
227C2×C4C2×D100411yescharacteristic
228C2×C4C2×D100411yescharacteristic
229C2×C4#1686080914yes—
230C2×C4C23×D25111yescharacteristic
231C8—24no—
232C22—28no—
233C22—21no—
234C22—21no—
235C22—21no—
236C22—21no—
237C22—21no—
238C22—21no—
239C22—10041no—
240C22—10041no—
241C22—10041no—
242C22#2254869912yes—
243C22#2254874312yes—
244C22#2254874812yes—
245C22C22×D100411yescharacteristic
246C4—21no—
247C4—21no—
248C4—10044no—
249C4—10041no—
250C4—10041no—
251C4—10041no—
252C4#2254869912yes—
253C4C2×#1686080912yes—
254C2—22no—
255C2—22no—
256C2—22no—
257C2—20081no—
258C2C2×#2254869911yescharacteristic
259C2C2×#2254874311yescharacteristic
260C2C2×#2254874811yescharacteristic
subgroupG/H
centerC22C22×D1004
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
Fitting#407×C251C2
socleC22×C251C22×D4
prime radical#30632198C2
hypercenter#407D251
hypocenterC251#3092
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C23×C4 ◁ #407

Complements

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

NH ≅ G/Ntype
2. C22×C4⋊Dic251C2semidirect
3. #30632023C2semidirect
4. #30632018C2semidirect
5. #30632006C2semidirect
6. #30632001C2semidirect
7. #30631729C2semidirect
8. #30631689C2semidirect
9. #407×C251C2semidirect
27. #22548649C22semidirect
28. #22548647C22semidirect
29. #22548640C22semidirect
30. #22548637C22semidirect
130. C251#3092semidirect
140. #407D251semidirect

Maximal subgroups

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

subgroupindex
#306321982
C22×C4⋊Dic2512
#306320232
#306320182
#306320062
#306320012
#306317292
#306316892
#407×C2512
#3092251