Finite Group Databasebeta

← #44559629

Subgroups — #44559629

Order 32128.

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

normal subgroups only

This group has 784 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
1C2×#22548760C211yesmaximal, characteristic, complemented
2C2×#22548741C211yesmaximal, characteristic, complemented
3#30632254C211yesmaximal, characteristic, complemented
4#30631829C214yesmaximal, complemented
5#30631819C214yesmaximal, complemented
6#30631692C211yesmaximal, characteristic, complemented
7#30631684C211yesmaximal, characteristic, complemented
8#30631653C211yesmaximal, characteristic, complemented
9C22×D8×C251C211yesmaximal, characteristic, complemented
10#22548760—24no—
11#22548760—24no—
12C2×#16860810C2211yescharacteristic
13C2×#16860810C2211yescharacteristic
14C2×#16860809—21no—
15C2×#16860809C2212yescomplemented
16C22×C22⋊D251—22no—
17C2×D4×D251—21no—
18C2×D4×D251C2212yescomplemented
19#22548741—24no—
20C2×#16860798C2212yescomplemented
21C2×#16860795C2212yescomplemented
22C2×#16860794C2211yescharacteristic, complemented
23#22548711—21no—
24#22548702—21no—
25#22548686C2214yescomplemented
26#22548684—22no—
27#22548674C2211yescharacteristic, complemented
28#22548640C2212yescomplemented
29#22548637C2212yescomplemented
30#22548636C2214yescomplemented
31(C2×D251)⋊C8C2214yescomplemented
32C22×D4×C251C2211yescharacteristic
33C22×D4×C251C2211yescharacteristic, complemented
34C2×D8×C251—24no—
35C2×D8×C251C2214yescomplemented
36C22×C8×C251C2211yescharacteristic, complemented
37C23×D251—22no—
38#16860810—22no—
39#16860810—22no—
40#16860810—22no—
41#16860810D414yescomplemented
42#16860809—28no—
43#16860809—24no—
44#16860809—24no—
45C2×Dic502D411yescharacteristic
46C2×Dic502C2311yescharacteristic
47C2×D1004D411yescharacteristic
48C2×D1004C2311yescharacteristic
49C2×C22⋊D251—216no—
50C2×C22⋊D251—22no—
51C2×C22⋊D251—24no—
52C2×C22⋊D251—22no—
53C2×C22⋊D251—21no—
54C2×C22⋊D251D412yes—
55D4×D251—28no—
56D4×D251—24no—
57D4×D251—24no—
58C22×Dic251—22no—
59C2×C4×D251—21no—
60C2×C4×D251C2312yes—
61#16860798—24no—
62#16860795—24no—
63#16860794—22no—
64C2×C251⋊C8C2312yes—
65C4⋊Dic251C2312yes—
66C22⋊Dic251—24no—
67C22⋊Dic251—21no—
68C4×Dic251—21no—
69C24×C251—21no—
70C24×C251—21no—
71C2×D4×C251—21no—
72C2×D4×C251—22no—
73C2×D4×C251—22no—
74C2×D4×C251—22no—
75C2×D4×C251—21no—
76C2×D4×C251D414yescomplemented
77C2×D4×C251C2312yes—
78C2×D4×C251C2312yes—
79C22×C4×C251C2311yescharacteristic
80D8×C251—28no—
81C2×C8×C251—22no—
82C2×C8×C251C2312yes—
83C22×D251—21no—
84C22×D251—28no—
85C22×D251—84no—
86C22×D251C2×D412yes—
87Dic502—22no—
88Dic502—21no—
89Dic502C2×D412yes—
90D1004—22no—
91D1004—21no—
92D1004C2×D412yes—
93C22⋊D251—28no—
94C22⋊D251—28no—
95C22⋊D251—24no—
96C22⋊D251—24no—
97C22⋊D251—84no—
98C2×Dic251—28no—
99C2×Dic251—21no—
100C2×Dic251—22no—
101C2×Dic251C2×D412yes—
102C4×D251—24no—
103C4×D251—22no—
104C4×D251—22no—
105C251⋊C8—22no—
106C23×C251—22no—
107C23×C251—24no—
108C23×C251—22no—
109C23×C251—42no—
110C23×C251—82no—
111C23×C251—82no—
112C23×C251C2×D411yescharacteristic
113D4×C251—24no—
114D4×C251—22no—
115D4×C251—22no—
116D4×C251—22no—
117D4×C251—42no—
118D4×C251C2×D414yes—
119C2×C4×C251C2×D411yescharacteristic
120C2×C4×C251C2×D412yes—
121C2×C4×C251C2×D412yes—
122C2×C4×C251C2411yescharacteristic
123C8×C251—22no—
124C22×C251—22no—
125C22×C251—24no—
126C22×C251—44no—
127C22×C251—42no—
128C22×C251—44no—
129C22×C251—84no—
130C22×C251—84no—
131C22×C251#12012yes—
132C22×C251C22×D411yescharacteristic
133C2×D251—22no—
134C2×D251—22no—
135C2×D251—24no—
136C2×D251—42no—
137C2×D251—88no—
138C4×C251#12012yes—
139C4×C251C22×D411yescharacteristic
140C4×C251C22×D411yescharacteristic
141Dic251—22no—
142Dic251—22no—
143Dic251—42no—
144C2×C251—22no—
145C2×C251—44no—
146C2×C251—82no—
147C2×C251C2×#12011yescharacteristic
148C2×C251#57612yes—
149D251—42no—
150D251—42no—
151C251#300811yescharacteristic, complemented, Sylow
152#3008—2511nomaximal, Sylow
153C2×H(5,𝔽2)—2511no—
154C2×AGL(1,ℤ/8)—2511no—
155C22×D8D25111yescharacteristic, complemented
156#535—2511no—
157#449—2514no—
158#447—2514no—
159#418—2511no—
160#416—2511no—
161#408—2511no—
162H(5,𝔽2)—5024no—
163H(5,𝔽2)—5024no—
164C2×#41—5021no—
165C2×#41—2512no—
166C2×#41—2511no—
167C2×#41—2511no—
168C22×D4—5021no—
169C22×D4—5022no—
170C22×D4—2512no—
171C22×D4C2×D25111yescharacteristic
172C22×D4C2×D25111yescharacteristic, complemented
173AGL(1,ℤ/8)—5024no—
174C2×SD16—2512no—
175C2×D8—24no—
176C2×D8—2512no—
177C2×D8C2×D25114yescomplemented
178C2×M16—2511no—
179C22×C8C2×D25111yescharacteristic, complemented
180#127—5021no—
181#124—5021no—
182#121—2514no—
183#120—5022no—
184#117—2511no—
185#103—2512no—
186#102—2514no—
187#102—2512no—
188C22⋊C8—2514no—
189C24—21no—
190C24—21no—
191C24—5022no—
192#41—5028no—
193#41—5024no—
194#41—5024no—
195#41—5022no—
196#41—5022no—
197#41—5022no—
198#41—2514no—
199C2×Q8—2511no—
200C2×Q8—2511no—
201C2×D4—21no—
202C2×D4—22no—
203C2×D4—22no—
204C2×D4—21no—
205C2×D4—22no—
206C2×D4—5028no—
207C2×D4—5024no—
208C2×D4—5024no—
209C2×D4—50216no—
210C2×D4—5022no—
211C2×D4—5021no—
212C2×D4—5024no—
213C2×D4—5022no—
214C2×D4—2512no—
215C2×D4—2511no—
216C2×D4—2511no—
217C2×D4C22⋊D25114yescomplemented
218C2×D4C22×D25112yes—
219C2×D4C22×D25112yes—
220C22×C4—5022no—
221C22×C4—5021no—
222C22×C4—2512no—
223C22×C4C22×D25111yescharacteristic
224SD16—5024no—
225D8—28no—
226D8—5024no—
227M16—5022no—
228C2×C8—22no—
229C2×C8—2512no—
230C2×C8C22×D25112yes—
231C4⋊C4—2512no—
232C22⋊C4—5021no—
233C22⋊C4—5024no—
234C42—5021no—
235C23—22no—
236C23—24no—
237C23—22no—
238C23—42no—
239C23—82no—
240C23—82no—
241C23—20084no—
242C23—5028no—
243C23—5021no—
244C23—2512no—
245C23D4×D25111yescharacteristic
246Q8—5021no—
247Q8—5022no—
248Q8—2512no—
249D4—22no—
250D4—22no—
251D4—22no—
252D4—24no—
253D4—42no—
254D4—20084no—
255D4—5028no—
256D4—5028no—
257D4—5024no—
258D4—5024no—
259D4—5021no—
260D4—5022no—
261D4—2512no—
262D4C2×C22⋊D25114yes—
263C2×C4—5022no—
264C2×C4—5028no—
265C2×C4—5021no—
266C2×C4—5022no—
267C2×C4—5022no—
268C2×C4—5024no—
269C2×C4—2512no—
270C2×C4D4×D25112yes—
271C2×C4D4×D25111yescharacteristic
272C2×C4C2×C22⋊D25112yes—
273C2×C4C23×D25111yescharacteristic
274C8—22no—
275C8—5022no—
276C22—22no—
277C22—24no—
278C22—44no—
279C22—44no—
280C22—42no—
281C22—84no—
282C22—84no—
283C22—20088no—
284C22—10042no—
285C22—5024no—
286C22—5022no—
287C22—5022no—
288C22#2254868312yes—
289C22C2×D4×D25111yescharacteristic
290C4—10042no—
291C4—5022no—
292C4—5022no—
293C4#2254868312yes—
294C4C2×D4×D25111yescharacteristic
295C4C22×C22⋊D25111yescharacteristic
296C2—22no—
297C2—44no—
298C2—82no—
299C2—10042no—
300C2—10042no—
301C2C2×#2254868311yescharacteristic
302C2#3063248812yes—
subgroupG/H
centerC22C2×D4×D251
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
FittingC22×D8×C251C2
socleC22×C251C22×D4
hypercenterC22×D8D251
hypocenterC251#3008
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×D8

Complements

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

NH ≅ G/Ntype
1. C2×#22548760C2semidirect
2. C2×#22548741C2semidirect
3. #30632254C2semidirect
4. #30631829C2semidirect
5. #30631819C2semidirect
6. #30631692C2semidirect
7. #30631684C2semidirect
8. #30631653C2semidirect
9. C22×D8×C251C2semidirect
15. C2×#16860809C22semidirect
18. C2×D4×D251C22semidirect
20. C2×#16860798C22semidirect
21. C2×#16860795C22semidirect
22. C2×#16860794C22semidirect
25. #22548686C22semidirect
27. #22548674C22semidirect
28. #22548640C22semidirect
29. #22548637C22semidirect
30. #22548636C22semidirect
31. (C2×D251)⋊C8C22semidirect
33. C22×D4×C251C22semidirect
35. C2×D8×C251C22semidirect
36. C22×C8×C251C22semidirect
41. #16860810D4semidirect
76. C2×D4×C251D4semidirect
151. C251#3008semidirect
155. C22×D8D251semidirect
172. C22×D4C2×D251semidirect
177. C2×D8C2×D251semidirect
179. C22×C8C2×D251semidirect
217. C2×D4C22⋊D251semidirect

Maximal subgroups

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

subgroupindex
C2×#225487602
C2×#225487412
#306322542
#306318292
#306318192
#306316922
#306316842
#306316532
C22×D8×C2512
#3008251