Finite Group Databasebeta

← #44560048

Subgroups — #44560048

Order 32128.

504 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 504 conjugacy classes of subgroups — too many to draw. Only its 133 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×#22548740C211yesmaximal, characteristic, complemented
2C2×#22548721C211yesmaximal, characteristic, complemented
3#30632265C211yesmaximal, characteristic, complemented
4#30632256C211yesmaximal, characteristic, complemented
5#30631939C212yesmaximal, complemented
6#30631936C212yesmaximal, complemented
7#30631922C212yesmaximal, complemented
8#30631919C212yesmaximal, complemented
9#30631730C211yesmaximal, characteristic, complemented
10C2×#22548637C211yesmaximal, characteristic, complemented
11#416×C251C211yesmaximal, characteristic, complemented
12C2×#16860810C2211yescharacteristic
13C22×D1004C2211yescharacteristic
14#22548740—24no—
15C2×#16860797C2212yes—
16C2×#16860795C2212yes—
17#22548721—24no—
18C2×#16860794C2211yescharacteristic
19C2×#16860793C2212yescomplemented
20C2×C8×D251C2212yescomplemented
21#22548712—21no—
22#22548703—21no—
23#22548694—21no—
24#22548691C2212yescomplemented
25#22548690C2212yes—
26#22548690C2212yes—
27#22548689—21no—
28#22548687—22no—
29#22548686C2212yescomplemented
30#22548685—22no—
31C2×C4⋊Dic251C2211yescharacteristic
32#22548648C2212yescomplemented
33#22548646C2212yescomplemented
34#22548637C2212yes—
35#22548637C2212yes—
36C2×#41×C251C2211yescharacteristic
37C22×C8×C251C2211yescharacteristic
38#117×C251C2211yescharacteristic
39#103×C251C2211yescharacteristic, complemented
40#103×C251C2211yescharacteristic, complemented
41#102×C251C2211yescharacteristic, complemented
42#102×C251C2211yescharacteristic, complemented
43C23×D251—21no—
44#16860810—22no—
45#16860810—22no—
46C2×Dic502D411yescharacteristic
47C2×D1004—22no—
48C2×D1004—22no—
49C2×D1004—22no—
50C2×D1004—21no—
51C2×D1004—22no—
52C2×D1004D411yescharacteristic
53C2×D1004C2312yes—
54C2×C22⋊D251—22no—
55C2×C22⋊D251D412yes—
56C22×Dic251—21no—
57C2×C4×D251C2312yes—
58#16860797—24no—
59#16860795—24no—
60#16860794D414yescomplemented
61#16860793—24no—
62C2×C251⋊C8C2312yes—
63C8×D251—24no—
64C4⋊Dic251—21no—
65C4⋊Dic251C2312yes—
66Dic251⋊C4—22no—
67C22⋊Dic251—22no—
68(C2×D251)⋊C4—22no—
69(C2×D251)⋊C4—22no—
70(C2×D251)⋊C4—22no—
71#41×C251—22no—
72#41×C251—22no—
73C2×Q8×C251C2311yescharacteristic
74C2×D4×C251—21no—
75C2×D4×C251C2311yescharacteristic
76C22×C4×C251—21no—
77C22×C4×C251C2311yescharacteristic
78C2×C8×C251—22no—
79C2×C8×C251C2311yescharacteristic
80C2×C8×C251C2311yescharacteristic
81C4⋊C4×C251C2311yescharacteristic
82C4⋊C4×C251C2311yescharacteristic
83C22⋊C4×C251—21no—
84C42×C251—21no—
85C22×D251—22no—
86C22×D251—42no—
87C22×D251—81no—
88C22×D251—81no—
89C22×D251C2×D412yes—
90Dic502—22no—
91D1004—22no—
92D1004—22no—
93D1004—22no—
94D1004—42no—
95D1004—82no—
96D1004—82no—
97C22⋊D251—24no—
98C22⋊D251—82no—
99C2×Dic251—22no—
100C2×Dic251—81no—
101C2×Dic251C2×D412yes—
102C4×D251—22no—
103C4×D251—22no—
104C251⋊C8C2×D414yes—
105C23×C251—21no—
106C23×C251C2×D411yescharacteristic
107Q8×C251—21no—
108Q8×C251—21no—
109D4×C251—21no—
110D4×C251—21no—
111D4×C251—42no—
112C2×C4×C251—21no—
113C2×C4×C251—21no—
114C2×C4×C251—21no—
115C2×C4×C251—42no—
116C2×C4×C251C2×D411yescharacteristic
117C2×C4×C251C2×D412yes—
118C2×C4×C251#4112yes—
119C2×C4×C251C2411yescharacteristic
120C8×C251—21no—
121C8×C251—21no—
122C22×C251—21no—
123C22×C251—21no—
124C22×C251—41no—
125C22×C251—41no—
126C22×C251—41no—
127C22×C251#12112yes—
128C22×C251C22×D411yescharacteristic
129C2×D251—22no—
130C2×D251—22no—
131C2×D251—22no—
132C2×D251—42no—
133C2×D251—42no—
134C2×D251—42no—
135C2×D251—84no—
136C4×C251—41no—
137C4×C251—41no—
138C4×C251—41no—
139C4×C251#12112yes—
140C4×C251C22×D411yescharacteristic
141C4×C251C2×#4111yescharacteristic
142Dic251—22no—
143Dic251—42no—
144C2×C251—22no—
145C2×C251—81no—
146C2×C251C2×#12111yescharacteristic
147C2×C251#57611yescharacteristic
148C2×C251#57711yescharacteristic
149D251—42no—
150D251—82no—
151C251#306211yescharacteristic, complemented, Sylow
152#3062—2511nomaximal, Sylow
153C2×AGL(1,ℤ/8)—2511no—
154C2×#131—2511no—
155#537—2511no—
156#535—2511no—
157#469—2512no—
158#468—2512no—
159#466—2512no—
160#465—2512no—
161#428—2511no—
162#416D25111yescharacteristic, complemented
163C2×#102—2511no—
164C2×#41—2511no—
165C2×#41C2×D25111yescharacteristic
166C22×D4—2511no—
167AGL(1,ℤ/8)—5024no—
168C2×SD16—2512no—
169C2×D8—2512no—
170#131—5024no—
171C2×M16—2511no—
172C2×M16—2512no—
173C22×C8—2512no—
174C22×C8C2×D25111yescharacteristic
175#127—5021no—
176#124—5021no—
177C22⋊Q8—5021no—
178C22⋊Q8—2512no—
179#121—5021no—
180#121—5022no—
181#121—2512no—
182#121—2512no—
183#121—2512no—
184#120—5022no—
185#117C2×D25111yescharacteristic
186C2×C4⋊C4—2511no—
187#107—2512no—
188#106—2512no—
189#103C2×D25111yescharacteristic, complemented
190#103C2×D25111yescharacteristic, complemented
191#102—2512no—
192#102—2512no—
193#102C2×D25111yescharacteristic, complemented
194#102C2×D25111yescharacteristic, complemented
195C24—5021no—
196#41—22no—
197#41—22no—
198#41—5022no—
199#41—5022no—
200C2×Q8—2511no—
201C2×Q8C22×D25111yescharacteristic
202C2×D4—21no—
203C2×D4—5022no—
204C2×D4—5022no—
205C2×D4—5022no—
206C2×D4—5021no—
207C2×D4—5022no—
208C2×D4—5022no—
209C2×D4—2512no—
210C2×D4—2511no—
211C2×D4—2512no—
212C2×D4C22×D25111yescharacteristic
213C22×C4—21no—
214C22×C4—5021no—
215C22×C4—2512no—
216C22×C4C22×D25111yescharacteristic
217SD16—5024no—
218D8—5024no—
219M16—5024no—
220M16—2514no—
221C2×C8—22no—
222C2×C8—5024no—
223C2×C8—2512no—
224C2×C8C22×D25111yescharacteristic
225C2×C8C22×D25111yescharacteristic
226C4⋊C4—5022no—
227C4⋊C4—5021no—
228C4⋊C4—2512no—
229C4⋊C4C22×D25111yescharacteristic
230C4⋊C4C22×D25111yescharacteristic
231C22⋊C4—21no—
232C22⋊C4—5022no—
233C22⋊C4—5022no—
234C22⋊C4—5022no—
235C22⋊C4—5022no—
236C42—21no—
237C23—21no—
238C23—20081no—
239C23—20081no—
240C23—10042no—
241C23—5022no—
242C23—2512no—
243C23D4×D25111yescharacteristic
244Q8—21no—
245Q8—21no—
246Q8—5022no—
247D4—21no—
248D4—21no—
249D4—42no—
250D4—20082no—
251D4—20082no—
252D4—20082no—
253D4—10042no—
254D4—5024no—
255D4—5022no—
256D4—5022no—
257D4—5022no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—42no—
262C2×C4—20081no—
263C2×C4—5022no—
264C2×C4—5022no—
265C2×C4—5022no—
266C2×C4—2512no—
267C2×C4D4×D25112yes—
268C2×C4D4×D25111yescharacteristic
269C2×C4#1686081012yes—
270C2×C4C23×D25111yescharacteristic
271C8—21no—
272C8—21no—
273C8—2514no—
274C22—21no—
275C22—21no—
276C22—41no—
277C22—41no—
278C22—41no—
279C22—20084no—
280C22—10042no—
281C22—10042no—
282C22—10042no—
283C22—5022no—
284C22—5022no—
285C22—5022no—
286C22#2254868812yes—
287C22C2×D4×D25111yescharacteristic
288C4—41no—
289C4—41no—
290C4—41no—
291C4—10042no—
292C4—5022no—
293C4#2254868812yes—
294C4C2×D4×D25111yescharacteristic
295C4C2×#1686081011yescharacteristic
296C2—22no—
297C2—81no—
298C2—20082no—
299C2—10042no—
300C2C2×#2254868811yescharacteristic
301C2#3063248911yescharacteristic
302C2#3063249411yescharacteristic
subgroupG/H
centerC22C2×D4×D251
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
Fitting#416×C251C2
socleC22×C251C22×D4
hypercenter#416D251
hypocenterC251#3062
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C22×C4 ◁ #416

Complements

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

NH ≅ G/Ntype
1. C2×#22548740C2semidirect
2. C2×#22548721C2semidirect
3. #30632265C2semidirect
4. #30632256C2semidirect
5. #30631939C2semidirect
6. #30631936C2semidirect
7. #30631922C2semidirect
8. #30631919C2semidirect
9. #30631730C2semidirect
10. C2×#22548637C2semidirect
11. #416×C251C2semidirect
19. C2×#16860793C22semidirect
20. C2×C8×D251C22semidirect
24. #22548691C22semidirect
29. #22548686C22semidirect
32. #22548648C22semidirect
33. #22548646C22semidirect
39. #103×C251C22semidirect
40. #103×C251C22semidirect
41. #102×C251C22semidirect
42. #102×C251C22semidirect
60. #16860794D4semidirect
151. C251#3062semidirect
162. #416D251semidirect
189. #103C2×D251semidirect
190. #103C2×D251semidirect
193. #102C2×D251semidirect
194. #102C2×D251semidirect

Maximal subgroups

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

subgroupindex
C2×#225487402
C2×#225487212
#306322652
#306322562
#306319392
#306319362
#306319222
#306319192
#306317302
C2×#225486372
#416×C2512
#3062251