Finite Group Databasebeta

← #44559903

Subgroups — #44559903

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×#22548743C211yesmaximal, characteristic, complemented
2C2×#22548735C211yesmaximal, characteristic, complemented
3#30632251C211yesmaximal, characteristic, complemented
4#30632168C211yesmaximal, characteristic, complemented
5#30631912C212yesmaximal, complemented
6#30631905C212yesmaximal, complemented
7#30631886C212yesmaximal, complemented
8#30631881C212yesmaximal, complemented
9#30631715C211yesmaximal, characteristic, complemented
10C2×#22548635C211yesmaximal, characteristic, complemented
11#417×C251C211yesmaximal, characteristic, complemented
12C2×#16860810C2211yescharacteristic
13C22×D1004C2211yescharacteristic
14#22548743—24no—
15#22548735—24no—
16C2×#16860800C2211yescharacteristic, complemented
17C2×#16860799C2212yes—
18C2×#16860798C2211yescharacteristic, complemented
19C2×#16860797C2211yescharacteristic, complemented
20C2×D2008C2212yes—
21C2×#16860795C2211yescharacteristic, complemented
22C22×C251⋊C8C2211yescharacteristic
23#22548711C2212yes—
24#22548704C2212yes—
25#22548689—21no—
26#22548689—21no—
27#22548683—22no—
28#22548680C2212yescomplemented
29#22548677C2212yescomplemented
30C4×C22⋊D251—22no—
31#22548674C2211yescharacteristic
32#22548673—21no—
33C4⋊C4×D251—21no—
34Dic251⋊C8C2212yescomplemented
35Dic251⋊C8C2212yescomplemented
36#22548635C2212yes—
37#22548635C2212yes—
38C2×#41×C251C2211yescharacteristic
39C2×M16×C251C2211yescharacteristic
40C2×C4⋊C4×C251C2211yescharacteristic
41#103×C251C2212yescomplemented
42#102×C251C2212yescomplemented
43C23×D251—21no—
44#16860810—22no—
45#16860810D414yescomplemented
46C2×Dic502C2311yescharacteristic
47C2×D1004—22no—
48C2×D1004—21no—
49C2×D1004—22no—
50C2×D1004C2312yes—
51C2×D1004C2311yescharacteristic
52C2×C22⋊D251—22no—
53C2×C22⋊D251—22no—
54C2×C22⋊D251—21no—
55C2×C4×D251—22no—
56C2×C4×D251—21no—
57#16860800—22no—
58#16860799—24no—
59#16860798—22no—
60#16860797—22no—
61D2008—24no—
62#16860795—22no—
63C2×C251⋊C8—22no—
64C2×C251⋊C8C2311yescharacteristic
65C2×C251⋊C8C2311yescharacteristic
66C4⋊Dic251D411yescharacteristic
67C4⋊Dic251D411yescharacteristic
68Dic251⋊C4—22no—
69C22⋊Dic251D412yes—
70(C2×D251)⋊C4—22no—
71(C2×D251)⋊C4—22no—
72(C2×D251)⋊C4—22no—
73C4×Dic251—22no—
74C4×Dic251C2312yes—
75#41×C251—22no—
76#41×C251—22no—
77C2×Q8×C251C2311yescharacteristic
78C2×D4×C251—21no—
79C2×D4×C251C2311yescharacteristic
80C22×C4×C251—21no—
81C22×C4×C251—21no—
82C22×C4×C251C2311yescharacteristic
83M16×C251—22no—
84C2×C8×C251C2312yes—
85C4⋊C4×C251—21no—
86C4⋊C4×C251C2312yes—
87C22×D251—21no—
88C22×D251—22no—
89C22×D251—42no—
90C22×D251—81no—
91C22×D251—81no—
92Dic502—21no—
93Dic502C2×D412yes—
94D1004—22no—
95D1004—21no—
96D1004—22no—
97D1004—42no—
98D1004C2×D412yes—
99C22⋊D251—24no—
100C22⋊D251—82no—
101C22⋊D251—82no—
102C2×Dic251—21no—
103C2×Dic251C2×D412yes—
104C2×Dic251C2×D412yes—
105C4×D251—24no—
106C4×D251—82no—
107C251⋊C8—21no—
108C251⋊C8—21no—
109C23×C251—21no—
110C23×C251C2×D411yescharacteristic
111Q8×C251—21no—
112Q8×C251—21no—
113D4×C251—21no—
114D4×C251—21no—
115D4×C251—42no—
116C2×C4×C251—22no—
117C2×C4×C251—21no—
118C2×C4×C251—42no—
119C2×C4×C251—81no—
120C2×C4×C251C2×D411yescharacteristic
121C2×C4×C251C2×D412yes—
122C2×C4×C251#4112yes—
123C2×C4×C251C2411yescharacteristic
124C8×C251—22no—
125C22×C251—21no—
126C22×C251—21no—
127C22×C251—41no—
128C22×C251—41no—
129C22×C251—41no—
130C22×C251#12112yes—
131C22×C251C22×D411yescharacteristic
132C2×D251—22no—
133C2×D251—41no—
134C2×D251—41no—
135C2×D251—42no—
136C2×D251—42no—
137C2×D251—42no—
138C2×D251—84no—
139C4×C251—41no—
140C4×C251—42no—
141C4×C251#12112yes—
142C4×C251C22×D411yescharacteristic
143C4×C251C2×#4111yescharacteristic
144Dic251—22no—
145Dic251—22no—
146Dic251—22no—
147C2×C251—22no—
148C2×C251—81no—
149C2×C251C2×#12111yescharacteristic
150C2×C251#57611yescharacteristic
151C2×C251#57711yescharacteristic
152D251—42no—
153D251—82no—
154C251#304511yescharacteristic, complemented, Sylow
155#3045—2511nomaximal, Sylow
156C2×AGL(1,ℤ/8)—2511no—
157C2×#135—2511no—
158#535—2511no—
159#520—2511no—
160#464—2512no—
161#463—2512no—
162#460—2512no—
163#459—2512no—
164#424—2511no—
165#417D25111yescharacteristic, complemented
166C2×#102—2511no—
167C2×#41—2511no—
168C2×#41C2×D25111yescharacteristic
169C22×D4—2511no—
170AGL(1,ℤ/8)—5024no—
171#135—5024no—
172C2×Q16—2511no—
173C2×SD16—2511no—
174C2×SD16—2511no—
175C2×SD16—2512no—
176C2×D8—2511no—
177C2×D8—2512no—
178C2×M16C2×D25111yescharacteristic
179C22×C8—2511no—
180#127—2512no—
181#124—2512no—
182#121—5021no—
183#121—5021no—
184#120—5022no—
185C4×Q8—2512no—
186C4×D4—5022no—
187C4×D4—2512no—
188#117—5021no—
189#117—2511no—
190C2×C4⋊C4—5021no—
191C2×C4⋊C4C2×D25111yescharacteristic
192C4⋊C8—2512no—
193C4⋊C8—2512no—
194#103C2×D25112yescomplemented
195#102—2512no—
196#102—2512no—
197#102C2×D25112yescomplemented
198C24—5021no—
199#41—22no—
200#41—22no—
201#41—5022no—
202#41—2514no—
203C2×Q8—2511no—
204C2×Q8C22×D25111yescharacteristic
205C2×D4—21no—
206C2×D4—5021no—
207C2×D4—5022no—
208C2×D4—5022no—
209C2×D4—5021no—
210C2×D4—5022no—
211C2×D4—5022no—
212C2×D4—2512no—
213C2×D4—2511no—
214C2×D4C22×D25111yescharacteristic
215C22×C4—21no—
216C22×C4—21no—
217C22×C4—5022no—
218C22×C4—5021no—
219C22×C4C22×D25111yescharacteristic
220Q16—5022no—
221SD16—5024no—
222SD16—5022no—
223SD16—5022no—
224D8—5024no—
225D8—5022no—
226M16—22no—
227C2×C8—5022no—
228C2×C8—2511no—
229C2×C8—2511no—
230C2×C8C22×D25112yes—
231C4⋊C4—21no—
232C4⋊C4—5022no—
233C4⋊C4—2511no—
234C4⋊C4—2511no—
235C4⋊C4C22×D25112yes—
236C22⋊C4—5022no—
237C22⋊C4—5022no—
238C22⋊C4—5022no—
239C22⋊C4—2512no—
240C42—5022no—
241C42—2512no—
242C23—21no—
243C23—20081no—
244C23—20081no—
245C23—10042no—
246C23—5022no—
247C23—5021no—
248C23D4×D25111yescharacteristic
249Q8—21no—
250Q8—21no—
251Q8—5021no—
252Q8—2512no—
253D4—21no—
254D4—21no—
255D4—42no—
256D4—20082no—
257D4—20082no—
258D4—10042no—
259D4—5024no—
260D4—5022no—
261D4—5022no—
262D4—5021no—
263D4—2512no—
264C2×C4—22no—
265C2×C4—21no—
266C2×C4—42no—
267C2×C4—81no—
268C2×C4—20082no—
269C2×C4—5021no—
270C2×C4—5024no—
271C2×C4—2512no—
272C2×C4—2512no—
273C2×C4D4×D25112yes—
274C2×C4D4×D25111yescharacteristic
275C2×C4#1686081012yes—
276C2×C4C23×D25111yescharacteristic
277C8—22no—
278C8—5021no—
279C8—5021no—
280C22—21no—
281C22—21no—
282C22—41no—
283C22—41no—
284C22—41no—
285C22—20084no—
286C22—10042no—
287C22—10042no—
288C22—10042no—
289C22—10041no—
290C22—10041no—
291C22—5022no—
292C22#2254868812yes—
293C22C2×D4×D25111yescharacteristic
294C4—42no—
295C4—41no—
296C4—5022no—
297C4—5022no—
298C4—5022no—
299C4#2254868812yes—
300C4C2×D4×D25111yescharacteristic
301C4C2×#1686081011yescharacteristic
302C2—22no—
303C2—81no—
304C2—20082no—
305C2—10042no—
306C2C2×#2254868811yescharacteristic
307C2#3063249111yescharacteristic
308C2#3063249611yescharacteristic
subgroupG/H
centerC22C2×D4×D251
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
Fitting#417×C251C2
socleC22×C251C22×D4
hypercenter#417D251
hypocenterC251#3045
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C22×C4 ◁ #417

Complements

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

NH ≅ G/Ntype
1. C2×#22548743C2semidirect
2. C2×#22548735C2semidirect
3. #30632251C2semidirect
4. #30632168C2semidirect
5. #30631912C2semidirect
6. #30631905C2semidirect
7. #30631886C2semidirect
8. #30631881C2semidirect
9. #30631715C2semidirect
10. C2×#22548635C2semidirect
11. #417×C251C2semidirect
16. C2×#16860800C22semidirect
18. C2×#16860798C22semidirect
19. C2×#16860797C22semidirect
21. C2×#16860795C22semidirect
28. #22548680C22semidirect
29. #22548677C22semidirect
34. Dic251⋊C8C22semidirect
35. Dic251⋊C8C22semidirect
41. #103×C251C22semidirect
42. #102×C251C22semidirect
45. #16860810D4semidirect
154. C251#3045semidirect
165. #417D251semidirect
194. #103C2×D251semidirect
197. #102C2×D251semidirect

Maximal subgroups

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

subgroupindex
C2×#225487432
C2×#225487352
#306322512
#306321682
#306319122
#306319052
#306318862
#306318812
#306317152
C2×#225486352
#417×C2512
#3045251