Finite Group Databasebeta

← #44560045

Subgroups — #44560045

Order 32128.

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

normal subgroups only

This group has 504 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×#22548743C211yesmaximal, characteristic, complemented
2#30632265C211yesmaximal, characteristic, complemented
3#30632251C211yesmaximal, characteristic, complemented
4#30631940C212yesmaximal, complemented
5#30631933C212yesmaximal, complemented
6#30631923C212yesmaximal, complemented
7#30631916C212yesmaximal, complemented
8#30631729C211yesmaximal, characteristic, complemented
9#30631684C211yesmaximal, characteristic, complemented
10C2×#22548637C211yesmaximal, characteristic, complemented
11C2×#131×C251C211yesmaximal, characteristic, complemented
12C2×#16860810C2211yescharacteristic
13C22×D1004C2211yescharacteristic
14#22548743—24no—
15C2×#16860799C2212yes—
16C2×D2008C2212yes—
17#22548711—21no—
18#22548704—21no—
19#22548694C2212yescomplemented
20#22548691—21no—
21#22548689C2212yes—
22#22548689C2212yes—
23#22548689C2212yescomplemented
24#22548687—22no—
25#22548686—21no—
26#22548683—22no—
27#22548674C2211yescharacteristic
28C2×C4⋊Dic251C2211yescharacteristic
29#22548649C2212yescomplemented
30#22548647C2212yescomplemented
31#22548640C2211yescharacteristic, complemented
32#22548640C2211yescharacteristic, complemented
33#22548637C2212yes—
34#22548637C2212yes—
35#22548637C2211yescharacteristic, complemented
36#22548637C2211yescharacteristic, complemented
37C2×#41×C251C2211yescharacteristic
38#131×C251—24no—
39C2×M16×C251C2211yescharacteristic
40C2×M16×C251C2212yescomplemented
41C22×C8×C251C2211yescharacteristic
42C22×C8×C251C2212yescomplemented
43C23×D251—21no—
44#16860810—22no—
45#16860810—22no—
46C2×Dic502C2311yescharacteristic
47C2×D1004—22no—
48C2×D1004—21no—
49C2×D1004—22no—
50C2×D1004C2312yes—
51C2×D1004C2311yescharacteristic
52C2×C22⋊D251—21no—
53C2×C22⋊D251—22no—
54C2×C22⋊D251—22no—
55C2×C22⋊D251—22no—
56C22×Dic251—21no—
57C2×C4×D251—21no—
58#16860799—24no—
59D2008—24no—
60C4⋊Dic251—21no—
61C4⋊Dic251C2312yes—
62C4⋊Dic251C2311yescharacteristic
63C4⋊Dic251C2311yescharacteristic
64Dic251⋊C4—22no—
65C22⋊Dic251—21no—
66C22⋊Dic251—22no—
67(C2×D251)⋊C4—22no—
68(C2×D251)⋊C4—22no—
69(C2×D251)⋊C4—22no—
70C4×Dic251—21no—
71#41×C251—22no—
72#41×C251—22no—
73C2×Q8×C251D411yescharacteristic
74C2×D4×C251D412yes—
75C2×D4×C251D411yescharacteristic
76C22×C4×C251C2312yes—
77C22×C4×C251C2311yescharacteristic
78M16×C251—24no—
79M16×C251D414yescomplemented
80C2×C8×C251—22no—
81C2×C8×C251—24no—
82C2×C8×C251C2311yescharacteristic
83C2×C8×C251C2311yescharacteristic
84C2×C8×C251C2312yes—
85C22×D251—21no—
86C22×D251—22no—
87C22×D251—42no—
88C22×D251—81no—
89C22×D251—81no—
90Dic502—21no—
91Dic502—21no—
92D1004—21no—
93D1004—22no—
94D1004—22no—
95D1004—21no—
96D1004—42no—
97C22⋊D251—42no—
98C22⋊D251—82no—
99C22⋊D251—82no—
100C22⋊D251—82no—
101C2×Dic251—21no—
102C2×Dic251—21no—
103C2×Dic251—22no—
104C2×Dic251—21no—
105C2×Dic251—81no—
106C4×D251—42no—
107C23×C251C2×D412yes—
108C23×C251C2×D411yescharacteristic
109Q8×C251—22no—
110D4×C251—24no—
111D4×C251—22no—
112C2×C4×C251—22no—
113C2×C4×C251—22no—
114C2×C4×C251C2×D412yes—
115C2×C4×C251C2×D412yes—
116C2×C4×C251C2×D411yescharacteristic
117C2×C4×C251#4112yes—
118C2×C4×C251C2411yescharacteristic
119C8×C251—21no—
120C8×C251—21no—
121C8×C251C2×D414yes—
122C22×C251—21no—
123C22×C251—21no—
124C22×C251—22no—
125C22×C251—22no—
126C22×C251—22no—
127C22×C251#12112yes—
128C22×C251C22×D411yescharacteristic
129C2×D251—41no—
130C2×D251—42no—
131C2×D251—41no—
132C2×D251—42no—
133C2×D251—42no—
134C2×D251—41no—
135C2×D251—84no—
136C4×C251—22no—
137C4×C251#12112yes—
138C4×C251C22×D411yescharacteristic
139C4×C251C2×#4111yescharacteristic
140Dic251—41no—
141Dic251—42no—
142Dic251—41no—
143Dic251—41no—
144C2×C251—22no—
145C2×C251—42no—
146C2×C251C2×#12111yescharacteristic
147C2×C251#57611yescharacteristic
148C2×C251#57711yescharacteristic
149D251—81no—
150D251—82no—
151C251#306211yescharacteristic, complemented, Sylow
152#3062—2511nomaximal, Sylow
153C2×AGL(1,ℤ/8)—2511no—
154C2×#131D25111yescharacteristic, complemented
155#537—2511no—
156#535—2511no—
157#469—2512no—
158#468—2512no—
159#466—2512no—
160#465—2512no—
161#428—2511no—
162#416—2511no—
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—24no—
171C2×M16C2×D25111yescharacteristic
172C2×M16C2×D25112yescomplemented
173C22×C8C2×D25111yescharacteristic
174C22×C8C2×D25112yescomplemented
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#117—2511no—
186C2×C4⋊C4—2511no—
187#107—2512no—
188#106—2512no—
189#103—2511no—
190#103—2511no—
191#102—2512no—
192#102—2512no—
193#102—2511no—
194#102—2511no—
195C24—5021no—
196#41—22no—
197#41—22no—
198#41—5022no—
199#41—5022no—
200C2×Q8—2511no—
201C2×Q8D100411yescharacteristic
202C2×D4—5022no—
203C2×D4—5022no—
204C2×D4—5022no—
205C2×D4—5021no—
206C2×D4—5022no—
207C2×D4—5021no—
208C2×D4—5022no—
209C2×D4—2511no—
210C2×D4—2512no—
211C2×D4D100412yes—
212C2×D4D100411yescharacteristic
213C22×C4—5021no—
214C22×C4—5021no—
215C22×C4C22×D25112yes—
216C22×C4C22×D25111yescharacteristic
217SD16—5024no—
218D8—5024no—
219M16—24no—
220M16C22⋊D25114yescomplemented
221C2×C8—22no—
222C2×C8—24no—
223C2×C8C22×D25111yescharacteristic
224C2×C8C22×D25111yescharacteristic
225C2×C8C22×D25112yes—
226C4⋊C4—5021no—
227C4⋊C4—5022no—
228C4⋊C4—2512no—
229C4⋊C4—2511no—
230C4⋊C4—2511no—
231C22⋊C4—5021no—
232C22⋊C4—5022no—
233C22⋊C4—5022no—
234C22⋊C4—5022no—
235C22⋊C4—5022no—
236C42—5021no—
237C23—20081no—
238C23—20081no—
239C23—10042no—
240C23—5021no—
241C23—5022no—
242C23C2×D100412yes—
243C23C2×D100411yescharacteristic
244Q8—22no—
245Q8—5021no—
246Q8—5021no—
247D4—24no—
248D4—22no—
249D4—20082no—
250D4—20082no—
251D4—20082no—
252D4—10042no—
253D4—10042no—
254D4—5021no—
255D4—5022no—
256D4—5022no—
257D4—5021no—
258C2×C4—22no—
259C2×C4—22no—
260C2×C4—20081no—
261C2×C4—10042no—
262C2×C4—5021no—
263C2×C4—5021no—
264C2×C4—5022no—
265C2×C4—5021no—
266C2×C4C2×C22⋊D25112yes—
267C2×C4C2×D100412yes—
268C2×C4C2×D100411yescharacteristic
269C2×C4#1686081012yes—
270C2×C4C23×D25111yescharacteristic
271C8—21no—
272C8—21no—
273C8C2×C22⋊D25114yes—
274C22—21no—
275C22—21no—
276C22—22no—
277C22—22no—
278C22—22no—
279C22—20084no—
280C22—10041no—
281C22—10042no—
282C22—10042no—
283C22—10041no—
284C22—10041no—
285C22—10042no—
286C22#2254868912yes—
287C22C22×D100411yescharacteristic
288C4—22no—
289C4—10042no—
290C4—10041no—
291C4—10041no—
292C4—10041no—
293C4#2254868912yes—
294C4C22×C22⋊D25111yescharacteristic
295C4C2×#1686081011yescharacteristic
296C2—22no—
297C2—42no—
298C2—20082no—
299C2—20081no—
300C2C2×#2254868911yescharacteristic
301C2#3063249211yescharacteristic
302C2#3063249911yescharacteristic
subgroupG/H
centerC22C22×D1004
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
FittingC2×#131×C251C2
socleC22×C251C22×D4
hypercenterC2×#131D251
hypocenterC251#3062
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C2×#41 ◁ C2×#131

Complements

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

NH ≅ G/Ntype
1. C2×#22548743C2semidirect
2. #30632265C2semidirect
3. #30632251C2semidirect
4. #30631940C2semidirect
5. #30631933C2semidirect
6. #30631923C2semidirect
7. #30631916C2semidirect
8. #30631729C2semidirect
9. #30631684C2semidirect
10. C2×#22548637C2semidirect
11. C2×#131×C251C2semidirect
19. #22548694C22semidirect
23. #22548689C22semidirect
29. #22548649C22semidirect
30. #22548647C22semidirect
31. #22548640C22semidirect
32. #22548640C22semidirect
35. #22548637C22semidirect
36. #22548637C22semidirect
40. C2×M16×C251C22semidirect
42. C22×C8×C251C22semidirect
79. M16×C251D4semidirect
151. C251#3062semidirect
154. C2×#131D251semidirect
172. C2×M16C2×D251semidirect
174. C22×C8C2×D251semidirect
220. M16C22⋊D251semidirect

Maximal subgroups

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

subgroupindex
C2×#225487432
#306322652
#306322512
#306319402
#306319332
#306319232
#306319162
#306317292
#306316842
C2×#225486372
C2×#131×C2512
#3062251