Finite Group Databasebeta

← #44559611

Subgroups — #44559611

Order 32128.

758 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 758 conjugacy classes of subgroups — too many to draw. Only its 149 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×#16860811C211yesmaximal, characteristic, complemented
2C2×#22548746C211yesmaximal, characteristic, complemented
3C2×#22548743C211yesmaximal, characteristic, complemented
4#30632199C211yesmaximal, characteristic, complemented
5#30631852C214yesmaximal, complemented
6#30631835C214yesmaximal, complemented
7#30631691C211yesmaximal, characteristic, complemented
8#30631650C211yesmaximal, characteristic, complemented
9#419×C251C211yesmaximal, characteristic, complemented
10C2×#16860811—24no—
11C2×#16860811—22no—
12C2×#16860811—24no—
13C2×#16860811C2214yes—
14C2×#16860810C2211yescharacteristic
15C22×D1004—21no—
16C22×D1004C2211yescharacteristic
17C22×C4×D251—21no—
18C22×C4×D251C2211yescharacteristic
19#22548746—24no—
20#22548743—24no—
21C2×#16860800C2212yescomplemented
22C2×#16860799C2212yescomplemented
23C2×#16860797C2212yescomplemented
24C2×D2008C2212yescomplemented
25C2×#16860794C2211yescharacteristic
26#22548699—21no—
27#22548695C2212yescomplemented
28#22548690C2212yescomplemented
29#22548685—21no—
30C4×D1004—22no—
31#22548635C2214yescomplemented
32(C2×D251)⋊C8C2214yescomplemented
33C22×Q8×C251C2211yescharacteristic
34C2×M16×C251C2211yescharacteristic
35#117×C251C2211yescharacteristic
36#103×C251C2214yescomplemented
37C23×D251—21no—
38C23×D251D411yescharacteristic
39#16860811—216no—
40#16860811—28no—
41#16860811—28no—
42#16860810—22no—
43#16860810—22no—
44C2×Dic502C2311yescharacteristic
45C2×D1004—28no—
46C2×D1004—24no—
47C2×D1004—22no—
48C2×D1004—21no—
49C2×D1004—22no—
50C2×D1004D414yescomplemented
51C2×D1004C2311yescharacteristic
52C2×D1004C2312yes—
53C2×C22⋊D251—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—
62#16860800—24no—
63#16860799—24no—
64#16860797—24no—
65D2008—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—22no—
85C22×D251—21no—
86C22×D251—42no—
87C22×D251—44no—
88C22×D251C2×D412yes—
89Dic502—21no—
90Dic502—21no—
91D1004—216no—
92D1004—24no—
93D1004—22no—
94D1004—21no—
95D1004—21no—
96D1004—44no—
97D1004C2×D414yes—
98C22⋊D251—42no—
99C2×Dic251—22no—
100C2×Dic251—21no—
101C2×Dic251C2×D412yes—
102C4×D251—216no—
103C4×D251—24no—
104C4×D251—22no—
105C4×D251—22no—
106C4×D251—42no—
107C251⋊C8—22no—
108C23×C251C2×D411yescharacteristic
109Q8×C251—24no—
110Q8×C251—22no—
111Q8×C251C2×D414yes—
112C2×C4×C251—24no—
113C2×C4×C251—22no—
114C2×C4×C251—22no—
115C2×C4×C251C2×D411yescharacteristic
116C2×C4×C251C2×D412yes—
117C2×C4×C251C2×D412yes—
118C2×C4×C251C2411yescharacteristic
119C8×C251—22no—
120C22×C251—22no—
121C22×C251#12012yes—
122C22×C251C22×D411yescharacteristic
123C2×D251—22no—
124C2×D251—24no—
125C2×D251—22no—
126C2×D251—22no—
127C2×D251—44no—
128C2×D251—48no—
129C2×D251—44no—
130C2×D251—41no—
131C2×D251—42no—
132C4×C251—24no—
133C4×C251—42no—
134C4×C251#12012yes—
135C4×C251C22×D411yescharacteristic
136C4×C251C22×D411yescharacteristic
137Dic251—22no—
138Dic251—41no—
139C2×C251—22no—
140C2×C251C2×#12011yescharacteristic
141C2×C251#57512yes—
142D251—42no—
143D251—44no—
144D251—81no—
145C251#300511yescharacteristic, complemented, Sylow
146#3005—2511nomaximal, Sylow
147C22×#41—2511no—
148C2×#137—2511no—
149C2×AGL(1,ℤ/8)—2511no—
150#525—2511no—
151#452—2514no—
152#449—2514no—
153#419D25111yescharacteristic, complemented
154#418—2511no—
155#407—2511no—
156C2×#41—5024no—
157C2×#41—5024no—
158C2×#41—5022no—
159C2×#41—2514no—
160C2×#41—2511no—
161C22×Q8C2×D25111yescharacteristic
162C22×D4—5021no—
163C22×D4—2511no—
164C23×C4—5021no—
165C23×C4—2511no—
166#137—5024no—
167AGL(1,ℤ/8)—5024no—
168C2×Q16—2512no—
169C2×SD16—2512no—
170C2×SD16—2512no—
171C2×D8—2512no—
172C2×M16—2511no—
173C2×M16C2×D25111yescharacteristic
174#123—5021no—
175C22⋊Q8—2512no—
176#121—2512no—
177#120—5021no—
178C4×D4—5022no—
179#117C2×D25111yescharacteristic
180#103C2×D25114yescomplemented
181#102—2514no—
182C22⋊C8—2514no—
183C24—5021no—
184C24—2511no—
185#41—5022no—
186#41—5022no—
187#41—50216no—
188#41—5028no—
189#41—5028no—
190C2×Q8—21no—
191C2×Q8—22no—
192C2×Q8—2511no—
193C2×Q8D100414yescomplemented
194C2×Q8C22×D25112yes—
195C2×D4—5022no—
196C2×D4—5028no—
197C2×D4—5024no—
198C2×D4—5021no—
199C2×D4—5021no—
200C2×D4—5022no—
201C2×D4—2514no—
202C2×D4—2511no—
203C2×D4—2512no—
204C22×C4—21no—
205C22×C4—5028no—
206C22×C4—5024no—
207C22×C4—5021no—
208C22×C4—5022no—
209C22×C4—5022no—
210C22×C4—2511no—
211C22×C4—2512no—
212C22×C4—2512no—
213C22×C4C22×D25111yescharacteristic
214Q16—5024no—
215SD16—5024no—
216SD16—5024no—
217D8—5024no—
218M16—22no—
219M16—5022no—
220C2×C8—2512no—
221C2×C8C22×D25112yes—
222C4⋊C4—5022no—
223C4⋊C4C22×D25112yes—
224C22⋊C4—21no—
225C22⋊C4—5022no—
226C22⋊C4—5022no—
227C42—21no—
228C23—10044no—
229C23—10042no—
230C23—5021no—
231C23—5022no—
232C23—5022no—
233C23—5024no—
234C23—2512no—
235C23D4×D25111yescharacteristic
236Q8—22no—
237Q8—24no—
238Q8—5021no—
239Q8—5021no—
240Q8C2×D100414yes—
241D4—10044no—
242D4—10042no—
243D4—50216no—
244D4—5021no—
245D4—5021no—
246D4—5022no—
247D4—5024no—
248D4—2514no—
249C2×C4—22no—
250C2×C4—24no—
251C2×C4—22no—
252C2×C4—10042no—
253C2×C4—50216no—
254C2×C4—5021no—
255C2×C4—5022no—
256C2×C4—5022no—
257C2×C4—5022no—
258C2×C4—5024no—
259C2×C4—2512no—
260C2×C4D4×D25112yes—
261C2×C4D4×D25111yescharacteristic
262C2×C4C2×D100412yes—
263C2×C4C23×D25111yescharacteristic
264C8—22no—
265C8—5022no—
266C22—22no—
267C22—10048no—
268C22—10044no—
269C22—10042no—
270C22—10041no—
271C22—10044no—
272C22—5022no—
273C22—5022no—
274C22—5022no—
275C22—5024no—
276C22#2254868512yes—
277C22C2×D4×D25111yescharacteristic
278C4—24no—
279C4—42no—
280C4—10041no—
281C4—5022no—
282C4#2254868512yes—
283C4C2×D4×D25111yescharacteristic
284C4C22×D100411yescharacteristic
285C2—22no—
286C2—20081no—
287C2—10044no—
288C2—10042no—
289C2C2×#2254868511yescharacteristic
290C2#3063248612yes—
subgroupG/H
centerC22C2×D4×D251
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
Fitting#419×C251C2
socleC22×C251C22×D4
hypercenter#419D251
hypocenterC251#3005
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×#16860811C2semidirect
2. C2×#22548746C2semidirect
3. C2×#22548743C2semidirect
4. #30632199C2semidirect
5. #30631852C2semidirect
6. #30631835C2semidirect
7. #30631691C2semidirect
8. #30631650C2semidirect
9. #419×C251C2semidirect
21. C2×#16860800C22semidirect
22. C2×#16860799C22semidirect
23. C2×#16860797C22semidirect
24. C2×D2008C22semidirect
27. #22548695C22semidirect
28. #22548690C22semidirect
31. #22548635C22semidirect
32. (C2×D251)⋊C8C22semidirect
36. #103×C251C22semidirect
50. C2×D1004D4semidirect
73. C2×Q8×C251D4semidirect
145. C251#3005semidirect
153. #419D251semidirect
180. #103C2×D251semidirect
193. C2×Q8D1004semidirect

Maximal subgroups

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

subgroupindex
C22×#168608112
C2×#225487462
C2×#225487432
#306321992
#306318522
#306318352
#306316912
#306316502
#419×C2512
#3005251