Finite Group Databasebeta

← #44558644

Subgroups — #44558644

Order 32128.

1224 conjugacy classes of subgroups; only the 201 normal subgroups are shown.

normal subgroups only

This group has 1224 conjugacy classes of subgroups — too many to draw. Only its 201 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
1C23×C4×D251C211yesmaximal, characteristic, complemented
2C2×#22548686C212yesmaximal, complemented
3C2×#22548683C212yesmaximal, complemented
4#30631567C212yesmaximal, complemented
5#30631563C214yesmaximal, complemented
6#30631547C211yesmaximal, characteristic, complemented
7#30631544C212yesmaximal, complemented
8C2×#121×C251C211yesmaximal, characteristic, complemented
9C24×D251C2211yescharacteristic
10C22×C22⋊D251C2214yes—
11C23×Dic251C2211yescharacteristic
12C22×C4×D251—28no—
13C22×C4×D251—24no—
14C22×C4×D251C2212yes—
15C22×C4×D251C2212yes—
16#22548686—28no—
17#22548683—28no—
18C2×C4⋊Dic251C2212yescomplemented
19C2×C22⋊Dic251C2212yes—
20C2×C22⋊Dic251C2214yescomplemented
21C2×(C2×D251)⋊C4C2214yes—
22C251⋊#95C2214yescomplemented
23C251⋊#95C2212yescomplemented
24C22×D4×C251C2212yes—
25C22×D4×C251C2211yescharacteristic, complemented
26C23×C4×C251C2211yescharacteristic
27#121×C251—24no—
28C2×C4⋊C4×C251C2211yescharacteristic, complemented
29C2×C22⋊C4×C251C2212yescomplemented
30C23×D251—24no—
31C23×D251—24no—
32C23×D251—22no—
33C23×D251—22no—
34C23×D251D414yescomplemented
35C23×D251C2312yes—
36C2×C22⋊D251—28no—
37C2×C22⋊D251—28no—
38C2×C22⋊D251—28no—
39C22×Dic251—24no—
40C22×Dic251—22no—
41C22×Dic251C2314yes—
42C22×Dic251C2312yes—
43C2×C4×D251—232no—
44C2×C4×D251—24no—
45C2×C4×D251—24no—
46C2×C4×D251—24no—
47C2×C4×D251—24no—
48C2×C4×D251—24no—
49C2×C4×D251D418yescomplemented
50C4⋊Dic251—24no—
51C22⋊Dic251—28no—
52C22⋊Dic251—24no—
53(C2×D251)⋊C4—28no—
54C24×C251C2312yes—
55C24×C251C2311yescharacteristic
56C2×D4×C251—24no—
57C2×D4×C251—24no—
58C2×D4×C251—24no—
59C2×D4×C251—22no—
60C2×D4×C251—24no—
61C22×C4×C251—24no—
62C22×C4×C251D414yescomplemented
63C22×C4×C251C2312yes—
64C22×C4×C251C2311yescharacteristic
65C22×C4×C251C2311yescharacteristic
66C4⋊C4×C251—22no—
67C22⋊C4×C251—24no—
68C22×D251—28no—
69C22×D251—216no—
70C22×D251—28no—
71C22×D251—28no—
72C22×D251—22no—
73C22×D251—22no—
74C22×D251—22no—
75C22×D251—22no—
76C22×D251—216no—
77C22×D251C2×D414yes—
78C22×D251C2×D414yes—
79C22×D251#4114yes—
80C22⋊D251—416no—
81C2×Dic251—28no—
82C2×Dic251—22no—
83C2×Dic251—22no—
84C2×Dic251—24no—
85C2×Dic251—24no—
86C2×Dic251—24no—
87C2×Dic251C2×D414yes—
88C4×D251—216no—
89C4×D251—216no—
90C23×C251—22no—
91C23×C251—22no—
92C23×C251—22no—
93C23×C251—24no—
94C23×C251—22no—
95C23×C251—22no—
96C23×C251—24no—
97C23×C251C2×D412yes—
98C23×C251C2×D412yes—
99C23×C251#4112yes—
100C23×C251C2411yescharacteristic
101D4×C251—48no—
102D4×C251—44no—
103C2×C4×C251—22no—
104C2×C4×C251—28no—
105C2×C4×C251—22no—
106C2×C4×C251—22no—
107C2×C4×C251—22no—
108C2×C4×C251C2×D414yes—
109C2×C4×C251C2×D412yes—
110C2×C4×C251C2×D412yes—
111C22×C251—22no—
112C22×C251—24no—
113C22×C251—22no—
114C22×C251—24no—
115C22×C251—42no—
116C22×C251—42no—
117C22×C251—44no—
118C22×C251—42no—
119C22×C251—44no—
120C22×C251#12114yescomplemented
121C22×C251C22×D412yes—
122C22×C251C22×D411yescharacteristic
123C22×C251C22×D411yescharacteristic
124C22×C251C2×#4112yes—
125C22×C251C2×#4111yescharacteristic
126C2×D251—232no—
127C2×D251—24no—
128C2×D251—24no—
129C2×D251—24no—
130C2×D251—24no—
131C2×D251—24no—
132C2×D251—24no—
133C2×D251#12118yescomplemented
134C4×C251—22no—
135C4×C251—42no—
136C4×C251#12014yescomplemented
137Dic251—24no—
138Dic251—44no—
139C2×C251—24no—
140C2×C251—82no—
141C2×C251C2×#12011yescharacteristic
142C2×C251C2×#12112yes—
143C2×C251C2×#12111yescharacteristic
144C2×C251#52511yescharacteristic
145C2×C251#52512yes—
146D251—28no—
147C251#284711yescharacteristic, complemented, Sylow
148#2847—2511nomaximal, Sylow
149C24×C4—2511no—
150C2×#121—2512no—
151C2×#121D25111yescharacteristic, complemented
152C2×#120—2512no—
153#386—2512no—
154#386—2514no—
155#380—2511no—
156#380—2512no—
157C25—2511no—
158C22×D4—2514no—
159C22×D4C2×D25112yes—
160C22×D4C2×D25111yescharacteristic, complemented
161C23×C4—5028no—
162C23×C4—5024no—
163C23×C4—2511no—
164C23×C4—2512no—
165C23×C4—2512no—
166C23×C4C2×D25111yescharacteristic
167#121—24no—
168#121—5028no—
169#120—5028no—
170C2×C4⋊C4—2512no—
171C2×C4⋊C4C2×D25111yescharacteristic, complemented
172C2×C22⋊C4—2512no—
173C2×C22⋊C4—2514no—
174C2×C22⋊C4—2514no—
175C2×C22⋊C4C2×D25112yescomplemented
176#95—2512no—
177#95—2514no—
178C24—5024no—
179C24—5024no—
180C24—5022no—
181C24—5022no—
182C24—2514no—
183C24—2512no—
184C24C22×D25112yes—
185C24C22×D25111yescharacteristic
186C2×D4—24no—
187C2×D4—24no—
188C2×D4—24no—
189C2×D4—24no—
190C2×D4—22no—
191C2×D4—5028no—
192C2×D4—5028no—
193C2×D4—5028no—
194C22×C4—24no—
195C22×C4—50232no—
196C22×C4—5024no—
197C22×C4—5024no—
198C22×C4—5022no—
199C22×C4—5024no—
200C22×C4—5024no—
201C22×C4—5024no—
202C22×C4—5024no—
203C22×C4—2514no—
204C22×C4—2512no—
205C22×C4—2518no—
206C22×C4C22⋊D25114yescomplemented
207C22×C4C22×D25112yes—
208C22×C4C22×D25111yescharacteristic
209C22×C4C22×D25111yescharacteristic
210C4⋊C4—22no—
211C4⋊C4—5024no—
212C22⋊C4—24no—
213C22⋊C4—5028no—
214C22⋊C4—5024no—
215C22⋊C4—5028no—
216C23—22no—
217C23—22no—
218C23—24no—
219C23—22no—
220C23—22no—
221C23—24no—
222C23—22no—
223C23—5028no—
224C23—5028no—
225C23—50216no—
226C23—50216no—
227C23—5022no—
228C23—5022no—
229C23—5022no—
230C23—5022no—
231C23—5028no—
232C23—2514no—
233C23—2514no—
234C23—2514no—
235C23D4×D25112yes—
236C23C2×C22⋊D25112yes—
237C23#1686080912yes—
238C23C23×D25111yescharacteristic
239D4—48no—
240D4—44no—
241D4—100416no—
242C2×C4—22no—
243C2×C4—28no—
244C2×C4—22no—
245C2×C4—22no—
246C2×C4—22no—
247C2×C4—50216no—
248C2×C4—5024no—
249C2×C4—5024no—
250C2×C4—5024no—
251C2×C4—5028no—
252C2×C4—5022no—
253C2×C4—5022no—
254C2×C4—50216no—
255C2×C4—2514no—
256C2×C4D4×D25114yes—
257C2×C4C2×C22⋊D25112yes—
258C2×C4C2×C22⋊D25112yes—
259C22—22no—
260C22—24no—
261C22—22no—
262C22—24no—
263C22—42no—
264C22—44no—
265C22—42no—
266C22—44no—
267C22—42no—
268C22—50232no—
269C22—5024no—
270C22—5024no—
271C22—5024no—
272C22—5024no—
273C22—5024no—
274C22—5024no—
275C22—2518no—
276C22#2254868614yescomplemented
277C22C2×D4×D25112yes—
278C22C2×D4×D25111yescharacteristic
279C22C22×C22⋊D25111yescharacteristic
280C22#41×D25112yes—
281C22C2×#1686080911yescharacteristic
282C4—22no—
283C4—42no—
284C4—10044no—
285C4—5024no—
286C4#2254868314yescomplemented
287C2—24no—
288C2—82no—
289C2—5028no—
290C2C2×#2254868311yescharacteristic
291C2#121×D25112yes—
292C2C2×#2254868611yescharacteristic
293C2#3063219411yescharacteristic
294C2#3063219712yes—
subgroupG/H
center, FrattiniC23C23×D251
commutator, socle, cosocleC23×C251C24
FittingC2×#121×C251C2
hypercenterC2×#121D251
hypocenterC251#2847
LCSG ▷ C23×C251 ▷ C251
UCSC1 ◁ C23 ◁ C2×#121

Complements

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

NH ≅ G/Ntype
1. C23×C4×D251C2semidirect
2. C2×#22548686C2semidirect
3. C2×#22548683C2semidirect
4. #30631567C2semidirect
5. #30631563C2semidirect
6. #30631547C2semidirect
7. #30631544C2semidirect
8. C2×#121×C251C2semidirect
18. C2×C4⋊Dic251C22semidirect
20. C2×C22⋊Dic251C22semidirect
22. C251⋊#95C22semidirect
23. C251⋊#95C22semidirect
25. C22×D4×C251C22semidirect
28. C2×C4⋊C4×C251C22semidirect
29. C2×C22⋊C4×C251C22semidirect
34. C23×D251D4semidirect
49. C2×C4×D251D4semidirect
62. C22×C4×C251D4semidirect
120. C22×C251#121semidirect
133. C2×D251#121semidirect
136. C4×C251#120semidirect
147. C251#2847semidirect
151. C2×#121D251semidirect
160. C22×D4C2×D251semidirect
171. C2×C4⋊C4C2×D251semidirect
175. C2×C22⋊C4C2×D251semidirect
206. C22×C4C22⋊D251semidirect
276. C22#22548686semidirect
286. C4#22548683semidirect

Maximal subgroups

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

subgroupindex
C23×C4×D2512
C2×#225486862
C2×#225486832
#306315672
#306315632
#306315472
#306315442
C2×#121×C2512
#2847251