Finite Group Databasebeta

← #44559604

Subgroups — #44559604

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×#22548745C211yesmaximal, characteristic, complemented
3#30632194C211yesmaximal, characteristic, complemented
4#30631846C214yesmaximal, complemented
5#30631829C214yesmaximal, complemented
6#30631697C211yesmaximal, characteristic, complemented
7#30631692C211yesmaximal, characteristic, complemented
8#30631650C211yesmaximal, characteristic, complemented
9C2×AGL(1,ℤ/8)×C251C211yesmaximal, characteristic, complemented
10C2×#16860809—24no—
11C2×#16860809—22no—
12C2×#16860809—24no—
13C2×#16860809C2214yes—
14C22×Dic502C2211yescharacteristic
15C22×C22⋊D251—21no—
16C23×Dic251—21no—
17C22×C4×D251C2211yescharacteristic
18#22548745—24no—
19C2×#16860800C2212yescomplemented
20C2×#16860798C2212yescomplemented
21C2×#16860794C2211yescharacteristic
22#22548696—21no—
23#22548691C2212yescomplemented
24#22548686C2212yescomplemented
25#22548683—21no—
26C4×C22⋊D251—22no—
27#22548674C2211yescharacteristic
28#22548640C2214yescomplemented
29#22548636C2214yescomplemented
30(C2×D251)⋊C8C2214yescomplemented
31C2×#41×C251C2211yescharacteristic
32C22×D4×C251C2211yescharacteristic
33AGL(1,ℤ/8)×C251—24no—
34C2×SD16×C251C2212yescomplemented
35C2×D8×C251C2212yescomplemented
36C2×M16×C251C2211yescharacteristic
37C23×D251D411yescharacteristic
38#16860809—216no—
39#16860809—28no—
40#16860809—28no—
41C2×Dic502—21no—
42C2×Dic502—22no—
43C2×Dic502D414yescomplemented
44C2×Dic502C2312yes—
45C2×C22⋊D251—28no—
46C2×C22⋊D251—24no—
47C2×C22⋊D251—22no—
48C22×Dic251—28no—
49C22×Dic251—24no—
50C22×Dic251—21no—
51C22×Dic251D411yescharacteristic
52C2×C4×D251—22no—
53C2×C4×D251—22no—
54C2×C4×D251D412yes—
55C2×C4×D251C2312yes—
56#16860800—24no—
57#16860798—24no—
58#16860794—22no—
59C2×C251⋊C8C2312yes—
60C4⋊Dic251C2312yes—
61Dic251⋊C4—22no—
62C22⋊Dic251—22no—
63C22⋊Dic251—21no—
64(C2×D251)⋊C4—22no—
65C4×Dic251—21no—
66C24×C251—21no—
67#41×C251—22no—
68#41×C251—22no—
69C2×Q8×C251C2311yescharacteristic
70C2×D4×C251—21no—
71C2×D4×C251—21no—
72C2×D4×C251—22no—
73C2×D4×C251D414yescomplemented
74C2×D4×C251C2311yescharacteristic
75C2×D4×C251C2312yes—
76C22×C4×C251—21no—
77C22×C4×C251C2311yescharacteristic
78SD16×C251—24no—
79D8×C251—24no—
80M16×C251—22no—
81C2×C8×C251C2312yes—
82C22×D251—22no—
83C22×D251—42no—
84C22×D251C2×D412yes—
85Dic502—24no—
86Dic502—22no—
87Dic502C2×D414yes—
88C22⋊D251—216no—
89C22⋊D251—44no—
90C2×Dic251—216no—
91C2×Dic251—22no—
92C2×Dic251—24no—
93C2×Dic251—22no—
94C2×Dic251—22no—
95C2×Dic251C2×D412yes—
96C4×D251—24no—
97C4×D251—22no—
98C4×D251—22no—
99C251⋊C8—22no—
100C23×C251—24no—
101C23×C251—22no—
102C23×C251—21no—
103C23×C251—44no—
104C23×C251C2×D411yescharacteristic
105Q8×C251—21no—
106Q8×C251—21no—
107D4×C251—24no—
108D4×C251—22no—
109D4×C251—21no—
110D4×C251—21no—
111D4×C251—42no—
112D4×C251C2×D414yes—
113C2×C4×C251—21no—
114C2×C4×C251—42no—
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—41no—
125C22×C251—42no—
126C22×C251#12012yes—
127C22×C251C22×D411yescharacteristic
128C2×D251—22no—
129C2×D251—22no—
130C2×D251—22no—
131C2×D251—44no—
132C4×C251—41no—
133C4×C251#12012yes—
134C4×C251C22×D411yescharacteristic
135C4×C251C22×D411yescharacteristic
136Dic251—22no—
137Dic251—24no—
138Dic251—42no—
139C2×C251—22no—
140C2×C251—44no—
141C2×C251—81no—
142C2×C251C2×#12011yescharacteristic
143C2×C251#57512yes—
144D251—42no—
145C251#300511yescharacteristic, complemented, Sylow
146#3005—2511nomaximal, Sylow
147C22×#41—2511no—
148C2×#137—2511no—
149C2×AGL(1,ℤ/8)D25111yescharacteristic, complemented
150#525—2511no—
151#452—2514no—
152#449—2514no—
153#419—2511no—
154#418—2511no—
155#407—2511no—
156C2×#41—5024no—
157C2×#41—5024no—
158C2×#41—5022no—
159C2×#41—2514no—
160C2×#41C2×D25111yescharacteristic
161C22×Q8—2511no—
162C22×D4—5021no—
163C22×D4C2×D25111yescharacteristic
164C23×C4—5021no—
165C23×C4—2511no—
166#137—5024no—
167AGL(1,ℤ/8)—24no—
168C2×Q16—2512no—
169C2×SD16—2512no—
170C2×SD16C2×D25112yescomplemented
171C2×D8C2×D25112yescomplemented
172C2×M16—2511no—
173C2×M16C2×D25111yescharacteristic
174#123—5021no—
175C22⋊Q8—2512no—
176#121—2512no—
177#120—5021no—
178C4×D4—5022no—
179#117—2511no—
180#103—2514no—
181#102—2514no—
182C22⋊C8—2514no—
183C24—21no—
184C24—2511no—
185#41—22no—
186#41—22no—
187#41—50216no—
188#41—5028no—
189#41—5028no—
190C2×Q8—5021no—
191C2×Q8—5022no—
192C2×Q8—2514no—
193C2×Q8—2512no—
194C2×Q8C22×D25111yescharacteristic
195C2×D4—21no—
196C2×D4—22no—
197C2×D4—21no—
198C2×D4—5022no—
199C2×D4—5028no—
200C2×D4—5024no—
201C2×D4C22⋊D25114yescomplemented
202C2×D4C22×D25111yescharacteristic
203C2×D4C22×D25112yes—
204C22×C4—21no—
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—24no—
216SD16—5024no—
217D8—24no—
218M16—22no—
219M16—5022no—
220C2×C8—2512no—
221C2×C8C22×D25112yes—
222C4⋊C4—5022no—
223C4⋊C4—2512no—
224C22⋊C4—5022no—
225C22⋊C4—5021no—
226C22⋊C4—5022no—
227C42—5021no—
228C23—22no—
229C23—24no—
230C23—21no—
231C23—44no—
232C23—10042no—
233C23—5022no—
234C23—2512no—
235C23D4×D25111yescharacteristic
236Q8—21no—
237Q8—21no—
238Q8—5022no—
239Q8—5024no—
240Q8—2514no—
241D4—22no—
242D4—21no—
243D4—21no—
244D4—24no—
245D4—42no—
246D4—10044no—
247D4—50216no—
248D4C2×C22⋊D25114yes—
249C2×C4—21no—
250C2×C4—42no—
251C2×C4—5022no—
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×C22⋊D25112yes—
263C2×C4C23×D25111yescharacteristic
264C8—22no—
265C8—5022no—
266C22—22no—
267C22—24no—
268C22—48no—
269C22—44no—
270C22—42no—
271C22—41no—
272C22—10044no—
273C22—5022no—
274C22—5022no—
275C22—5022no—
276C22#2254868312yes—
277C22C2×D4×D25111yescharacteristic
278C4—41no—
279C4—10042no—
280C4—5022no—
281C4—5024no—
282C4#2254868312yes—
283C4C2×D4×D25111yescharacteristic
284C4C22×C22⋊D25111yescharacteristic
285C2—22no—
286C2—44no—
287C2—81no—
288C2—10042no—
289C2C2×#2254868311yescharacteristic
290C2#3063248312yes—
subgroupG/H
centerC22C2×D4×D251
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
FittingC2×AGL(1,ℤ/8)×C251C2
socleC22×C251C22×D4
hypercenterC2×AGL(1,ℤ/8)D251
hypocenterC251#3005
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. C22×#16860809C2semidirect
2. C2×#22548745C2semidirect
3. #30632194C2semidirect
4. #30631846C2semidirect
5. #30631829C2semidirect
6. #30631697C2semidirect
7. #30631692C2semidirect
8. #30631650C2semidirect
9. C2×AGL(1,ℤ/8)×C251C2semidirect
19. C2×#16860800C22semidirect
20. C2×#16860798C22semidirect
23. #22548691C22semidirect
24. #22548686C22semidirect
28. #22548640C22semidirect
29. #22548636C22semidirect
30. (C2×D251)⋊C8C22semidirect
34. C2×SD16×C251C22semidirect
35. C2×D8×C251C22semidirect
43. C2×Dic502D4semidirect
73. C2×D4×C251D4semidirect
145. C251#3005semidirect
149. C2×AGL(1,ℤ/8)D251semidirect
170. C2×SD16C2×D251semidirect
171. C2×D8C2×D251semidirect
201. C2×D4C22⋊D251semidirect

Maximal subgroups

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

subgroupindex
C22×#168608092
C2×#225487452
#306321942
#306318462
#306318292
#306316972
#306316922
#306316502
C2×AGL(1,ℤ/8)×C2512
#3005251