Finite Group Databasebeta

← #44560041

Subgroups — #44560041

Order 32128.

504 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 504 conjugacy classes of subgroups — too many to draw. Only its 141 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×#22548741C211yesmaximal, characteristic, complemented
2C2×#22548722C211yesmaximal, characteristic, complemented
3#30632265C211yesmaximal, characteristic, complemented
4#30631938C212yesmaximal, complemented
5#30631931C212yesmaximal, complemented
6#30631921C212yesmaximal, complemented
7#30631914C212yesmaximal, complemented
8#30631730C211yesmaximal, characteristic, complemented
9#30631681C211yesmaximal, characteristic, complemented
10C2×#22548636C211yesmaximal, characteristic, complemented
11#535×C251C211yesmaximal, characteristic, complemented
12C2×#16860810C2211yescharacteristic
13#22548741—24no—
14C2×#16860798C2212yes—
15C2×#16860795C2212yes—
16#22548722—24no—
17C2×#16860794C2211yescharacteristic
18C2×#16860794C2212yescomplemented
19C22×C251⋊C8C2211yescharacteristic
20C22×C251⋊C8C2212yescomplemented
21#22548694C2212yescomplemented
22#22548691—21no—
23#22548689C2212yescomplemented
24#22548687—22no—
25#22548686—21no—
26C2×C4⋊Dic251C2211yescharacteristic
27#22548648C2212yescomplemented
28#22548646C2212yescomplemented
29#22548638C2211yescharacteristic, complemented
30#22548638C2211yescharacteristic, complemented
31#22548636C2212yes—
32#22548636C2212yes—
33#22548635C2211yescharacteristic, complemented
34#22548635C2211yescharacteristic, complemented
35C2×#41×C251C2211yescharacteristic
36C22×D4×C251C2211yescharacteristic
37#127×C251—21no—
38#124×C251—21no—
39#121×C251C2212yes—
40#121×C251C2212yes—
41#120×C251—22no—
42#117×C251C2211yescharacteristic
43#16860810—22no—
44#16860810—22no—
45C2×Dic502C2311yescharacteristic
46C2×D1004C2311yescharacteristic
47C2×C22⋊D251—21no—
48C2×C22⋊D251—22no—
49C22×Dic251—21no—
50C2×C4×D251—21no—
51#16860798—24no—
52#16860795—24no—
53#16860794—24no—
54#16860794D414yescomplemented
55C2×C251⋊C8—24no—
56C2×C251⋊C8—22no—
57C2×C251⋊C8C2311yescharacteristic
58C2×C251⋊C8C2311yescharacteristic
59C2×C251⋊C8C2312yes—
60C4⋊Dic251—21no—
61C4⋊Dic251C2312yes—
62Dic251⋊C4—22no—
63C22⋊Dic251—22no—
64(C2×D251)⋊C4—22no—
65C24×C251—21no—
66#41×C251—22no—
67#41×C251—22no—
68C2×Q8×C251D411yescharacteristic
69C2×D4×C251—22no—
70C2×D4×C251—22no—
71C2×D4×C251—22no—
72C2×D4×C251—21no—
73C2×D4×C251—22no—
74C2×D4×C251D412yes—
75C2×D4×C251D411yescharacteristic
76C2×D4×C251C2312yes—
77C22×C4×C251C2312yes—
78C22×C4×C251C2311yescharacteristic
79C4⋊C4×C251C2311yescharacteristic
80C4⋊C4×C251C2311yescharacteristic
81C22⋊C4×C251—21no—
82C22⋊C4×C251—22no—
83C22⋊C4×C251—22no—
84C42×C251—21no—
85C22×D251—21no—
86Dic502—21no—
87Dic502—21no—
88D1004—21no—
89D1004—21no—
90C22⋊D251—42no—
91C22⋊D251—82no—
92C2×Dic251—22no—
93C2×Dic251—21no—
94C2×Dic251—81no—
95C4×D251—42no—
96C251⋊C8—21no—
97C251⋊C8—21no—
98C251⋊C8C2×D414yes—
99C23×C251—22no—
100C23×C251—42no—
101C23×C251—81no—
102C23×C251—81no—
103C23×C251C2×D412yes—
104C23×C251C2×D411yescharacteristic
105Q8×C251—22no—
106D4×C251—24no—
107D4×C251—22no—
108D4×C251—22no—
109D4×C251—22no—
110D4×C251—42no—
111D4×C251—82no—
112D4×C251—82no—
113C2×C4×C251—22no—
114C2×C4×C251—22no—
115C2×C4×C251—21no—
116C2×C4×C251—21no—
117C2×C4×C251C2×D412yes—
118C2×C4×C251C2×D412yes—
119C2×C4×C251C2×D411yescharacteristic
120C2×C4×C251#4112yes—
121C2×C4×C251C2411yescharacteristic
122C22×C251—21no—
123C22×C251—21no—
124C22×C251—22no—
125C22×C251—22no—
126C22×C251—22no—
127C22×C251—42no—
128C22×C251—42no—
129C22×C251—42no—
130C22×C251—84no—
131C22×C251#12112yes—
132C22×C251C22×D411yescharacteristic
133C2×D251—41no—
134C2×D251—41no—
135C2×D251—41no—
136C4×C251—22no—
137C4×C251—41no—
138C4×C251—41no—
139C4×C251#12112yes—
140C4×C251C22×D411yescharacteristic
141C4×C251C2×#4111yescharacteristic
142Dic251—41no—
143Dic251—42no—
144C2×C251—22no—
145C2×C251—42no—
146C2×C251—82no—
147C2×C251C2×#12111yescharacteristic
148C2×C251#57611yescharacteristic
149C2×C251#57711yescharacteristic
150D251—81no—
151C251#306211yescharacteristic, complemented, Sylow
152#3062—2511nomaximal, Sylow
153C2×AGL(1,ℤ/8)—2511no—
154C2×#131—2511no—
155#537—2511no—
156#535D25111yescharacteristic, complemented
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×D4C2×D25111yescharacteristic
167AGL(1,ℤ/8)—5024no—
168C2×SD16—2512no—
169C2×D8—2512no—
170#131—5024no—
171C2×M16—2512no—
172C2×M16—2511no—
173C22×C8—2511no—
174C22×C8—2512no—
175#127—21no—
176#124—21no—
177C22⋊Q8—5021no—
178C22⋊Q8—2512no—
179#121—5021no—
180#121—5022no—
181#121—2512no—
182#121C2×D25112yes—
183#121C2×D25112yes—
184#120—22no—
185#117C2×D25111yescharacteristic
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—21no—
196#41—22no—
197#41—22no—
198#41—5022no—
199#41—5022no—
200C2×Q8—2511no—
201C2×Q8C22⋊D25111yescharacteristic
202C2×D4—22no—
203C2×D4—22no—
204C2×D4—22no—
205C2×D4—21no—
206C2×D4—22no—
207C2×D4—5022no—
208C2×D4—5021no—
209C2×D4—2511no—
210C2×D4C22⋊D25112yes—
211C2×D4C22⋊D25111yescharacteristic
212C2×D4C22×D25112yes—
213C22×C4—5021no—
214C22×C4—5021no—
215C22×C4C22×D25112yes—
216C22×C4C22×D25111yescharacteristic
217SD16—5024no—
218D8—5024no—
219M16—5024no—
220M16—2514no—
221C2×C8—5022no—
222C2×C8—5024no—
223C2×C8—2511no—
224C2×C8—2511no—
225C2×C8—2512no—
226C4⋊C4—5021no—
227C4⋊C4—5022no—
228C4⋊C4—2512no—
229C4⋊C4C22×D25111yescharacteristic
230C4⋊C4C22×D25111yescharacteristic
231C22⋊C4—21no—
232C22⋊C4—22no—
233C22⋊C4—22no—
234C22⋊C4—5022no—
235C22⋊C4—5022no—
236C42—21no—
237C23—22no—
238C23—42no—
239C23—81no—
240C23—81no—
241C23—5021no—
242C23C2×C22⋊D25112yes—
243C23C2×C22⋊D25111yescharacteristic
244Q8—22no—
245Q8—5021no—
246Q8—5021no—
247D4—24no—
248D4—22no—
249D4—22no—
250D4—22no—
251D4—42no—
252D4—82no—
253D4—82no—
254D4—20082no—
255D4—10042no—
256D4—5021no—
257D4—5021no—
258C2×C4—22no—
259C2×C4—22no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—20081no—
263C2×C4—10042no—
264C2×C4—5022no—
265C2×C4—5021no—
266C2×C4D4×D25112yes—
267C2×C4C2×C22⋊D25112yes—
268C2×C4C2×C22⋊D25111yescharacteristic
269C2×C4#1686080912yes—
270C2×C4C23×D25111yescharacteristic
271C8—5021no—
272C8—5021no—
273C8—2514no—
274C22—21no—
275C22—21no—
276C22—22no—
277C22—22no—
278C22—22no—
279C22—42no—
280C22—42no—
281C22—42no—
282C22—84no—
283C22—10041no—
284C22—10041no—
285C22—10041no—
286C22#2254868712yes—
287C22C22×C22⋊D25111yescharacteristic
288C4—22no—
289C4—41no—
290C4—41no—
291C4—10042no—
292C4—10041no—
293C4#2254868712yes—
294C4C2×D4×D25111yescharacteristic
295C4C2×#1686080911yescharacteristic
296C2—22no—
297C2—42no—
298C2—82no—
299C2—20081no—
300C2C2×#2254868711yescharacteristic
301C2#3063249011yescharacteristic
302C2#3063249811yescharacteristic
subgroupG/H
centerC22C22×C22⋊D251
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
Fitting#535×C251C2
socleC22×C251C22×D4
hypercenter#535D251
hypocenterC251#3062
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C2×#41 ◁ #535

Complements

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

NH ≅ G/Ntype
1. C2×#22548741C2semidirect
2. C2×#22548722C2semidirect
3. #30632265C2semidirect
4. #30631938C2semidirect
5. #30631931C2semidirect
6. #30631921C2semidirect
7. #30631914C2semidirect
8. #30631730C2semidirect
9. #30631681C2semidirect
10. C2×#22548636C2semidirect
11. #535×C251C2semidirect
18. C2×#16860794C22semidirect
20. C22×C251⋊C8C22semidirect
21. #22548694C22semidirect
23. #22548689C22semidirect
27. #22548648C22semidirect
28. #22548646C22semidirect
29. #22548638C22semidirect
30. #22548638C22semidirect
33. #22548635C22semidirect
34. #22548635C22semidirect
54. #16860794D4semidirect
151. C251#3062semidirect
156. #535D251semidirect

Maximal subgroups

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

subgroupindex
C2×#225487412
C2×#225487222
#306322652
#306319382
#306319312
#306319212
#306319142
#306317302
#306316812
C2×#225486362
#535×C2512
#3062251