Finite Group Databasebeta

← #44559609

Subgroups — #44559609

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×#16860809C211yesmaximal, characteristic, complemented
2C2×#22548748C211yesmaximal, characteristic, complemented
3C2×#22548741C211yesmaximal, characteristic, complemented
4#30632199C211yesmaximal, characteristic, complemented
5#30631850C214yesmaximal, complemented
6#30631833C214yesmaximal, complemented
7#30631695C211yesmaximal, characteristic, complemented
8#30631650C211yesmaximal, characteristic, complemented
9#418×C251C211yesmaximal, characteristic, complemented
10C2×#16860810C2211yescharacteristic
11C2×#16860809—24no—
12C2×#16860809—22no—
13C2×#16860809—24no—
14C2×#16860809C2214yes—
15C22×Dic502C2211yescharacteristic
16C22×C22⋊D251—21no—
17C23×Dic251—21no—
18C22×C4×D251C2211yescharacteristic
19#22548748—24no—
20#22548741—24no—
21C2×Dic1004C2212yescomplemented
22C2×#16860799C2212yescomplemented
23C2×#16860798C2212yescomplemented
24C2×#16860795C2212yescomplemented
25C2×#16860794C2211yescharacteristic
26#22548699—21no—
27#22548695C2212yescomplemented
28#22548690C2212yescomplemented
29#22548685—21no—
30C4×D1004—22no—
31#22548638C2214yescomplemented
32(C2×D251)⋊C8C2214yescomplemented
33C22×D4×C251C2211yescharacteristic
34C2×M16×C251C2211yescharacteristic
35#117×C251C2211yescharacteristic
36#102×C251C2214yescomplemented
37C23×D251D411yescharacteristic
38#16860810—22no—
39#16860810—22no—
40#16860809—216no—
41#16860809—28no—
42#16860809—28no—
43C2×Dic502—21no—
44C2×Dic502—22no—
45C2×Dic502D414yescomplemented
46C2×Dic502C2311yescharacteristic
47C2×Dic502C2312yes—
48C2×D1004—22no—
49C2×D1004C2311yescharacteristic
50C2×C22⋊D251—28no—
51C2×C22⋊D251—24no—
52C2×C22⋊D251—21no—
53C22×Dic251—28no—
54C22×Dic251—24no—
55C22×Dic251—21no—
56C22×Dic251D411yescharacteristic
57C2×C4×D251—22no—
58C2×C4×D251—22no—
59C2×C4×D251—21no—
60C2×C4×D251D412yes—
61C2×C4×D251C2312yes—
62Dic1004—24no—
63#16860799—24no—
64#16860798—24no—
65#16860795—24no—
66#16860794—22no—
67C2×C251⋊C8C2312yes—
68C4⋊Dic251—22no—
69(C2×D251)⋊C4—22no—
70(C2×D251)⋊C4—22no—
71C24×C251—21no—
72C2×D4×C251—21no—
73C2×D4×C251—22no—
74C2×D4×C251D414yescomplemented
75C2×D4×C251C2312yes—
76C22×C4×C251C2311yescharacteristic
77M16×C251—22no—
78C2×C8×C251C2312yes—
79C4⋊C4×C251C2312yes—
80C22⋊C4×C251—21no—
81C42×C251—21no—
82C22×D251—22no—
83C22×D251—21no—
84C22×D251—42no—
85C22×D251C2×D412yes—
86Dic502—24no—
87Dic502—22no—
88Dic502—21no—
89Dic502—21no—
90Dic502C2×D414yes—
91D1004—21no—
92D1004—21no—
93D1004—44no—
94C22⋊D251—216no—
95C22⋊D251—42no—
96C2×Dic251—216no—
97C2×Dic251—22no—
98C2×Dic251—24no—
99C2×Dic251—22no—
100C2×Dic251—21no—
101C2×Dic251C2×D412yes—
102C4×D251—24no—
103C4×D251—22no—
104C4×D251—22no—
105C4×D251—42no—
106C251⋊C8—22no—
107C23×C251—24no—
108C23×C251—22no—
109C23×C251—44no—
110C23×C251C2×D411yescharacteristic
111D4×C251—24no—
112D4×C251—22no—
113D4×C251C2×D414yes—
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—24no—
122C22×C251—48no—
123C22×C251—44no—
124C22×C251#12012yes—
125C22×C251C22×D411yescharacteristic
126C2×D251—22no—
127C2×D251—22no—
128C2×D251—22no—
129C2×D251—44no—
130C2×D251—41no—
131C2×D251—42no—
132C4×C251—42no—
133C4×C251#12012yes—
134C4×C251C22×D411yescharacteristic
135C4×C251C22×D411yescharacteristic
136Dic251—22no—
137Dic251—24no—
138Dic251—41no—
139C2×C251—22no—
140C2×C251—44no—
141C2×C251C2×#12011yescharacteristic
142C2×C251#57512yes—
143D251—42no—
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#419—2511no—
154#418D25111yescharacteristic, complemented
155#407—2511no—
156C2×#41—5024no—
157C2×#41—5024no—
158C2×#41—5022no—
159C2×#41—2514no—
160C2×#41—2511no—
161C22×Q8—2511no—
162C22×D4—5021no—
163C22×D4C2×D25111yescharacteristic
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#103—2514no—
181#102C2×D25114yescomplemented
182C22⋊C8—2514no—
183C24—21no—
184C24—2511no—
185#41—5022no—
186#41—5022no—
187#41—50216no—
188#41—5028no—
189#41—5028no—
190C2×Q8—5021no—
191C2×Q8—5022no—
192C2×Q8—2514no—
193C2×Q8—2511no—
194C2×Q8—2512no—
195C2×D4—22no—
196C2×D4—21no—
197C2×D4—5022no—
198C2×D4—5021no—
199C2×D4—5028no—
200C2×D4—5024no—
201C2×D4—2511no—
202C2×D4D100414yescomplemented
203C2×D4C22×D25112yes—
204C22×C4—5021no—
205C22×C4—5028no—
206C22×C4—5021no—
207C22×C4—5024no—
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—22no—
229C23—24no—
230C23—44no—
231C23—10042no—
232C23—5021no—
233C23—5022no—
234C23—2512no—
235C23D4×D25111yescharacteristic
236Q8—5021no—
237Q8—5021no—
238Q8—5022no—
239Q8—5024no—
240Q8—2514no—
241D4—22no—
242D4—24no—
243D4—10044no—
244D4—10042no—
245D4—50216no—
246D4—5021no—
247D4—5021no—
248D4C2×D100414yes—
249C2×C4—22no—
250C2×C4—10042no—
251C2×C4—5021no—
252C2×C4—50216no—
253C2×C4—5022no—
254C2×C4—5022no—
255C2×C4—5024no—
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—24no—
268C22—48no—
269C22—44no—
270C22—10044no—
271C22—10042no—
272C22—10041no—
273C22—5022no—
274C22—5022no—
275C22—5022no—
276C22#2254868512yes—
277C22C2×D4×D25111yescharacteristic
278C4—42no—
279C4—10041no—
280C4—5022no—
281C4—5024no—
282C4#2254868512yes—
283C4C2×D4×D25111yescharacteristic
284C4C22×D100411yescharacteristic
285C2—22no—
286C2—44no—
287C2—20081no—
288C2—10042no—
289C2C2×#2254868511yescharacteristic
290C2#3063248312yes—
subgroupG/H
centerC22C2×D4×D251
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
Fitting#418×C251C2
socleC22×C251C22×D4
hypercenter#418D251
hypocenterC251#3005
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C22×C4 ◁ #418

Complements

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

NH ≅ G/Ntype
1. C22×#16860809C2semidirect
2. C2×#22548748C2semidirect
3. C2×#22548741C2semidirect
4. #30632199C2semidirect
5. #30631850C2semidirect
6. #30631833C2semidirect
7. #30631695C2semidirect
8. #30631650C2semidirect
9. #418×C251C2semidirect
21. C2×Dic1004C22semidirect
22. C2×#16860799C22semidirect
23. C2×#16860798C22semidirect
24. C2×#16860795C22semidirect
27. #22548695C22semidirect
28. #22548690C22semidirect
31. #22548638C22semidirect
32. (C2×D251)⋊C8C22semidirect
36. #102×C251C22semidirect
45. C2×Dic502D4semidirect
74. C2×D4×C251D4semidirect
145. C251#3005semidirect
154. #418D251semidirect
181. #102C2×D251semidirect
202. C2×D4D1004semidirect

Maximal subgroups

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

subgroupindex
C22×#168608092
C2×#225487482
C2×#225487412
#306321992
#306318502
#306318332
#306316952
#306316502
#418×C2512
#3005251