Finite Group Databasebeta

← #44559601

Subgroups — #44559601

Order 32128.

738 conjugacy classes of subgroups; only the 153 normal subgroups are shown.

normal subgroups only

This group has 738 conjugacy classes of subgroups — too many to draw. Only its 153 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×Q8×D251C211yesmaximal, characteristic, complemented
2C2×#22548748C211yesmaximal, characteristic, complemented
3C2×#22548746C211yesmaximal, characteristic, complemented
4#30632199C211yesmaximal, characteristic
5#30631844C214yesmaximal, complemented
6#30631827C214yesmaximal, complemented
7#30631695C211yesmaximal, characteristic, complemented
8#30631650C211yesmaximal, characteristic, complemented
9#419×C251C211yesmaximal, characteristic, complemented
10C2×#16860810C2211yescharacteristic
11C22×Dic502—21no—
12C22×Dic502C2211yescharacteristic
13C2×Q8×D251—24no—
14C2×Q8×D251—22no—
15C2×Q8×D251—24no—
16C2×Q8×D251C2214yes—
17C22×C4×D251—21no—
18C22×C4×D251C2211yescharacteristic
19#22548748—24no—
20#22548746—24no—
21C2×Dic1004C2212yescomplemented
22C2×#16860800C2212yescomplemented
23C2×#16860799C2212yescomplemented
24C2×#16860797C2212yescomplemented
25C2×#16860794C2211yescharacteristic
26#22548699—21no—
27#22548695C2212yes—
28#22548690C2212yes—
29#22548685—21no—
30C4×D1004—22no—
31#22548638C2214yescomplemented
32(C2×D251)⋊C8C2214yes—
33C22×Q8×C251C2211yescharacteristic
34C2×M16×C251C2211yescharacteristic
35#117×C251C2211yescharacteristic
36#103×C251C2214yescomplemented
37C23×D251D411yescharacteristic
38#16860810—22no—
39#16860810—22no—
40C2×Dic502—28no—
41C2×Dic502—24no—
42C2×Dic502—21no—
43C2×Dic502—22no—
44C2×Dic502D414yescomplemented
45C2×Dic502C2311yescharacteristic
46C2×Dic502C2312yes—
47Q8×D251—216no—
48Q8×D251—28no—
49Q8×D251—28no—
50C2×D1004—22no—
51C2×D1004C2311yescharacteristic
52C2×C22⋊D251—21no—
53C22×Dic251—21no—
54C22×Dic251D411yescharacteristic
55C2×C4×D251—28no—
56C2×C4×D251—24no—
57C2×C4×D251—22no—
58C2×C4×D251—22no—
59C2×C4×D251—21no—
60C2×C4×D251D412yes—
61C2×C4×D251C2312yes—
62Dic1004—24no—
63#16860800—24no—
64#16860799—24no—
65#16860797—24no—
66#16860794—22no—
67C2×C251⋊C8C2312yes—
68C4⋊Dic251—22no—
69(C2×D251)⋊C4—22no—
70(C2×D251)⋊C4—22no—
71C2×Q8×C251—22no—
72C2×Q8×C251—21no—
73C2×Q8×C251D414yescomplemented
74C2×Q8×C251C2312yes—
75C22×C4×C251—21no—
76C22×C4×C251C2311yescharacteristic
77M16×C251—22no—
78C2×C8×C251C2312yes—
79C4⋊C4×C251C2312yes—
80C22⋊C4×C251—21no—
81C42×C251—21no—
82C22×D251—22no—
83C22×D251—24no—
84C22×D251—21no—
85C22×D251C2×D412yes—
86Dic502—216no—
87Dic502—24no—
88Dic502—22no—
89Dic502—21no—
90Dic502—21no—
91Dic502C2×D414yes—
92D1004—21no—
93D1004—21no—
94D1004—44no—
95C22⋊D251—42no—
96C2×Dic251—24no—
97C2×Dic251—22no—
98C2×Dic251—22no—
99C2×Dic251—21no—
100C2×Dic251C2×D412yes—
101C4×D251—216no—
102C4×D251—24no—
103C4×D251—22no—
104C4×D251—22no—
105C4×D251—42no—
106C251⋊C8—22no—
107C23×C251C2×D411yescharacteristic
108Q8×C251—24no—
109Q8×C251—22no—
110Q8×C251C2×D414yes—
111C2×C4×C251—24no—
112C2×C4×C251—22no—
113C2×C4×C251—22no—
114C2×C4×C251C2×D411yescharacteristic
115C2×C4×C251C2×D412yes—
116C2×C4×C251C2×D412yes—
117C2×C4×C251C2411yescharacteristic
118C8×C251—22no—
119C22×C251—22no—
120C22×C251#12012yes—
121C22×C251C22×D411yescharacteristic
122C2×D251—28no—
123C2×D251—22no—
124C2×D251—22no—
125C2×D251—41no—
126C2×D251—42no—
127C2×D251#13714yescomplemented
128C4×C251—24no—
129C4×C251—42no—
130C4×C251#12012yes—
131C4×C251C22×D411yescharacteristic
132C4×C251C22×D411yescharacteristic
133Dic251—24no—
134Dic251—22no—
135Dic251—41no—
136C2×C251—22no—
137C2×C251C2×#12011yescharacteristic
138C2×C251C2×#13712yes—
139D251—24no—
140D251—81no—
141C251#300411yescharacteristic, complemented, Sylow
142#3004—2511nomaximal, Sylow
143C23×Q8—2511no—
144C2×#137—2511no—
145C2×#137—2511no—
146#525—2511no—
147#451—2514no—
148#448—2514no—
149#419—2511no—
150#419D25111yescharacteristic, complemented
151#407—2511no—
152C2×#41—2511no—
153C22×Q8—5021no—
154C22×Q8—5024no—
155C22×Q8—5024no—
156C22×Q8—5022no—
157C22×Q8—2514no—
158C22×Q8—2511no—
159C22×Q8C2×D25111yescharacteristic
160C23×C4—5021no—
161C23×C4—2511no—
162#137—5024no—
163#137—5024no—
164C2×Q16—2512no—
165C2×Q16—2512no—
166C2×SD16—2512no—
167C2×SD16—2512no—
168C2×M16—2511no—
169C2×M16C2×D25111yescharacteristic
170#123—5021no—
171C22⋊Q8—2512no—
172#121—2512no—
173#120—5021no—
174C4×D4—5022no—
175#117C2×D25111yescharacteristic
176#103—2514no—
177#103C2×D25114yescomplemented
178C22⋊C8—2514no—
179C24—2511no—
180#41—5022no—
181#41—5022no—
182C2×Q8—21no—
183C2×Q8—22no—
184C2×Q8—5028no—
185C2×Q8—5024no—
186C2×Q8—50216no—
187C2×Q8—5028no—
188C2×Q8—5028no—
189C2×Q8—5021no—
190C2×Q8—5022no—
191C2×Q8—2514no—
192C2×Q8—2511no—
193C2×Q8—2512no—
194C2×Q8D100414yescomplemented
195C2×Q8C22×D25112yes—
196C2×D4—5022no—
197C2×D4—5021no—
198C2×D4—2511no—
199C22×C4—21no—
200C22×C4—5028no—
201C22×C4—5024no—
202C22×C4—5021no—
203C22×C4—5021no—
204C22×C4—5022no—
205C22×C4—5022no—
206C22×C4—2511no—
207C22×C4—2512no—
208C22×C4—2512no—
209C22×C4C22×D25111yescharacteristic
210Q16—5024no—
211Q16—5024no—
212SD16—5024no—
213SD16—5024no—
214M16—22no—
215M16—5022no—
216C2×C8—2512no—
217C2×C8C22×D25112yes—
218C4⋊C4—5022no—
219C4⋊C4C22×D25112yes—
220C22⋊C4—21no—
221C22⋊C4—5022no—
222C22⋊C4—5022no—
223C42—21no—
224C23—5024no—
225C23—5021no—
226C23—5022no—
227C23—2512no—
228C23D4×D25111yescharacteristic
229Q8—22no—
230Q8—24no—
231Q8—50216no—
232Q8—5021no—
233Q8—5021no—
234Q8—5022no—
235Q8—5024no—
236Q8—2514no—
237Q8C2×D100414yes—
238D4—10044no—
239D4—10042no—
240D4—5021no—
241D4—5021no—
242C2×C4—22no—
243C2×C4—24no—
244C2×C4—22no—
245C2×C4—10042no—
246C2×C4—50216no—
247C2×C4—5021no—
248C2×C4—5022no—
249C2×C4—5024no—
250C2×C4—5022no—
251C2×C4—5022no—
252C2×C4—5022no—
253C2×C4—5024no—
254C2×C4—2512no—
255C2×C4D4×D25112yes—
256C2×C4D4×D25111yescharacteristic
257C2×C4C2×D100412yes—
258C2×C4C23×D25111yescharacteristic
259C8—22no—
260C8—5022no—
261C22—22no—
262C22—10042no—
263C22—10041no—
264C22—5028no—
265C22—5022no—
266C22—5022no—
267C22—2514no—
268C22#2254868512yes—
269C22C2×D4×D25111yescharacteristic
270C4—24no—
271C4—42no—
272C4—10041no—
273C4—5024no—
274C4—5022no—
275C4#2254868512yes—
276C4C2×D4×D25111yescharacteristic
277C4C22×D100411yescharacteristic
278C2—22no—
279C2—20081no—
280C2—5024no—
281C2C2×#2254868511yescharacteristic
282C2#137×D25112yes—
subgroupG/H
centerC22C2×D4×D251
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
Fitting#419×C251C2
socleC22×C251C22×D4
prime radical#30632199C2
hypercenter#419D251
hypocenterC251#3004
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C22×C4 ◁ #419

Complements

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

NH ≅ G/Ntype
1. C22×Q8×D251C2semidirect
2. C2×#22548748C2semidirect
3. C2×#22548746C2semidirect
5. #30631844C2semidirect
6. #30631827C2semidirect
7. #30631695C2semidirect
8. #30631650C2semidirect
9. #419×C251C2semidirect
21. C2×Dic1004C22semidirect
22. C2×#16860800C22semidirect
23. C2×#16860799C22semidirect
24. C2×#16860797C22semidirect
31. #22548638C22semidirect
36. #103×C251C22semidirect
44. C2×Dic502D4semidirect
73. C2×Q8×C251D4semidirect
127. C2×D251#137semidirect
141. C251#3004semidirect
150. #419D251semidirect
177. #103C2×D251semidirect
194. C2×Q8D1004semidirect

Maximal subgroups

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

subgroupindex
C22×Q8×D2512
C2×#225487482
C2×#225487462
#306321992
#306318442
#306318272
#306316952
#306316502
#419×C2512
#3004251