Finite Group Databasebeta

← #44560827

Subgroups — #44560827

Order 32128.

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

normal subgroups only

This group has 572 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×#22548748C211yesmaximal, characteristic, complemented
2C2×#22548745C211yesmaximal, characteristic, complemented
3C2×#22548741C211yesmaximal, characteristic, complemented
4#30632233C211yesmaximal, characteristic, complemented
5C2×#22548702C211yesmaximal, characteristic, complemented
6#30632078C212yesmaximal, complemented
7#30632075C212yesmaximal, complemented
8#30632070C212yesmaximal, complemented
9#30632068C212yesmaximal, complemented
10M16×Dic251C211yesmaximal, characteristic, complemented
11C2×AGL(1,ℤ/8)×C251C211yesmaximal, characteristic, complemented
12C2×#16860810C2211yescharacteristic
13C22×Dic502C2211yescharacteristic
14#22548748—24no—
15#22548745—24no—
16#22548741—24no—
17C2×Dic1004C2212yescomplemented
18C2×#16860800C2212yescomplemented
19C2×#16860799C2212yescomplemented
20C2×#16860798C2212yes—
21C2×#16860798C2212yescomplemented
22C2×#16860795C2212yes—
23C2×#16860794C2211yescharacteristic
24#22548713C2211yescharacteristic, complemented
25#22548711C2211yescharacteristic, complemented
26#22548704C2211yescharacteristic, complemented
27#22548702—22no—
28#22548702C2212yes—
29#22548702C2212yes—
30#22548702C2211yescharacteristic, complemented
31#22548687—22no—
32C4×C22⋊D251—22no—
33C2×C22⋊Dic251—22no—
34C2×C4×Dic251C2211yescharacteristic
35#22548625C2212yescomplemented
36C8×Dic251C2212yescomplemented
37C2×#41×C251C2211yescharacteristic
38C22×D4×C251C2211yescharacteristic
39AGL(1,ℤ/8)×C251—24no—
40C2×SD16×C251C2212yes—
41C2×D8×C251C2212yes—
42C2×M16×C251C2211yescharacteristic
43#16860810—22no—
44#16860810—22no—
45C2×Dic502—21no—
46C2×Dic502—22no—
47C2×Dic502—22no—
48C2×Dic502C2312yes—
49C2×Dic502C2311yescharacteristic
50C2×D1004C2311yescharacteristic
51C2×C22⋊D251—22no—
52C2×C22⋊D251—21no—
53C22×Dic251—21no—
54C22×Dic251D412yes—
55C2×C4×D251—21no—
56Dic1004—24no—
57#16860800—24no—
58#16860799—24no—
59#16860798—24no—
60#16860798—24no—
61#16860795—24no—
62#16860794D414yescomplemented
63C2×C251⋊C8C2312yes—
64Dic251⋊C4—22no—
65C22⋊Dic251—24no—
66C22⋊Dic251—24no—
67C22⋊Dic251—22no—
68(C2×D251)⋊C4—22no—
69C4×Dic251D412yes—
70C4×Dic251C2311yescharacteristic
71C4×Dic251C2311yescharacteristic
72C24×C251—21no—
73#41×C251—22no—
74#41×C251—22no—
75C2×Q8×C251C2311yescharacteristic
76C2×D4×C251—21no—
77C2×D4×C251—22no—
78C2×D4×C251—22no—
79C2×D4×C251—21no—
80C2×D4×C251C2311yescharacteristic
81C2×D4×C251C2312yes—
82C22×C4×C251—21no—
83C22×C4×C251C2311yescharacteristic
84SD16×C251—24no—
85D8×C251—24no—
86M16×C251D414yescomplemented
87C2×C8×C251C2312yes—
88C22×D251—21no—
89Dic502—21no—
90Dic502—22no—
91Dic502—22no—
92Dic502—21no—
93Dic502—42no—
94D1004—21no—
95D1004—21no—
96C22⋊D251—44no—
97C22⋊D251—42no—
98C2×Dic251—22no—
99C2×Dic251—21no—
100C2×Dic251—42no—
101C2×Dic251—42no—
102C2×Dic251C2×D412yes—
103C2×Dic251C2×D412yes—
104C4×D251—42no—
105C251⋊C8C2×D414yes—
106C23×C251—21no—
107C23×C251—22no—
108C23×C251—44no—
109C23×C251—42no—
110C23×C251C2×D411yescharacteristic
111Q8×C251—21no—
112Q8×C251—21no—
113D4×C251—21no—
114D4×C251—22no—
115D4×C251—22no—
116D4×C251—21no—
117D4×C251—42no—
118D4×C251—42no—
119C2×C4×C251—21no—
120C2×C4×C251—42no—
121C2×C4×C251C2×D411yescharacteristic
122C2×C4×C251C2×D412yes—
123C2×C4×C251C2×D412yes—
124C2×C4×C251C2411yescharacteristic
125C8×C251C2×D414yes—
126C22×C251—22no—
127C22×C251—41no—
128C22×C251—42no—
129C22×C251—44no—
130C22×C251—42no—
131C22×C251—84no—
132C22×C251#12712yes—
133C22×C251C22×D411yescharacteristic
134C2×D251—41no—
135C2×D251—42no—
136C4×C251—41no—
137C4×C251#12712yes—
138C4×C251C22×D411yescharacteristic
139C4×C251C22×D411yescharacteristic
140Dic251—22no—
141Dic251—22no—
142Dic251—41no—
143Dic251—42no—
144C2×C251—22no—
145C2×C251—81no—
146C2×C251—82no—
147C2×C251C2×#12711yescharacteristic
148C2×C251#57512yes—
149D251—81no—
150C251#315311yescharacteristic, complemented, Sylow
151#3153—2511nomaximal, Sylow
152C2×#137—2511no—
153C2×#137—2511no—
154C2×AGL(1,ℤ/8)—2511no—
155C2×AGL(1,ℤ/8)D25111yescharacteristic, complemented
156#532—2511no—
157C2×#124—2511no—
158#497—2512no—
159#496—2512no—
160#495—2512no—
161#495—2512no—
162C4×M16—2511no—
163C2×#41—2511no—
164C2×#41C2×D25111yescharacteristic
165C22×Q8—2511no—
166C22×D4C2×D25111yescharacteristic
167#137—5024no—
168#137—5024no—
169AGL(1,ℤ/8)—24no—
170AGL(1,ℤ/8)—5024no—
171C2×Q16—2512no—
172C2×Q16—2512no—
173C2×SD16—2512no—
174C2×SD16—2512no—
175C2×SD16—2512no—
176C2×SD16C2×D25112yes—
177C2×D8—2512no—
178C2×D8C2×D25112yes—
179C2×M16—2511no—
180C2×M16C2×D25111yescharacteristic
181C4⋊Q8—2511no—
182#127—2511no—
183#124—5022no—
184#124—2511no—
185#124—2512no—
186#124—2512no—
187#124—2511no—
188#121—5022no—
189C4×D4—5022no—
190C2×C22⋊C4—5022no—
191C2×C42—2511no—
192#97—2512no—
193C4×C8—2512no—
194C24—21no—
195#41—22no—
196#41—22no—
197#41—5022no—
198#41—5022no—
199C2×Q8—5022no—
200C2×Q8—5022no—
201C2×Q8—5021no—
202C2×Q8—2512no—
203C2×Q8—2511no—
204C2×Q8C22×D25111yescharacteristic
205C2×D4—21no—
206C2×D4—22no—
207C2×D4—21no—
208C2×D4—22no—
209C2×D4—5021no—
210C2×D4—5022no—
211C2×D4—2511no—
212C2×D4C22×D25112yes—
213C2×D4C22×D25111yescharacteristic
214C22×C4—21no—
215C22×C4—5021no—
216C22×C4—5021no—
217C22×C4—2512no—
218C22×C4C22×D25111yescharacteristic
219Q16—5024no—
220Q16—5024no—
221SD16—24no—
222SD16—5024no—
223SD16—5024no—
224SD16—5024no—
225D8—24no—
226D8—5024no—
227M16—2514no—
228M16C22⋊D25114yescomplemented
229C2×C8—2512no—
230C2×C8C22×D25112yes—
231C4⋊C4—5022no—
232C22⋊C4—5022no—
233C22⋊C4—5024no—
234C22⋊C4—5024no—
235C22⋊C4—5022no—
236C42—2512no—
237C42—2511no—
238C42—2511no—
239C23—22no—
240C23—21no—
241C23—44no—
242C23—42no—
243C23—5021no—
244C23D4×D25111yescharacteristic
245Q8—21no—
246Q8—21no—
247Q8—10042no—
248Q8—5021no—
249Q8—5022no—
250Q8—5022no—
251Q8—5021no—
252D4—22no—
253D4—21no—
254D4—21no—
255D4—22no—
256D4—42no—
257D4—42no—
258D4—10042no—
259D4—10044no—
260D4—5021no—
261D4—5021no—
262C2×C4—21no—
263C2×C4—42no—
264C2×C4—10042no—
265C2×C4—10042no—
266C2×C4—10042no—
267C2×C4—5022no—
268C2×C4—5021no—
269C2×C4—2512no—
270C2×C4—2512no—
271C2×C4D4×D25112yes—
272C2×C4D4×D25111yescharacteristic
273C2×C4C2×C22⋊D25112yes—
274C2×C4C23×D25111yescharacteristic
275C8—2514no—
276C8C2×C22⋊D25114yes—
277C22—22no—
278C22—41no—
279C22—42no—
280C22—42no—
281C22—44no—
282C22—84no—
283C22—10041no—
284C22—10042no—
285C22#2254871112yes—
286C22C2×D4×D25111yescharacteristic
287C4—41no—
288C4—10041no—
289C4—10042no—
290C4—5022no—
291C4—5022no—
292C4#2254871112yes—
293C4C2×D4×D25111yescharacteristic
294C4C22×C22⋊D25111yescharacteristic
295C2—22no—
296C2—82no—
297C2—81no—
298C2—20081no—
299C2C2×#2254871111yescharacteristic
300C2#3063248312yes—
subgroupG/H
centerC22C2×D4×D251
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
FittingC2×AGL(1,ℤ/8)×C251C2
socleC22×C251C22×D4
hypercenterC2×AGL(1,ℤ/8)D251
hypocenterC251#3153
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. C2×#22548748C2semidirect
2. C2×#22548745C2semidirect
3. C2×#22548741C2semidirect
4. #30632233C2semidirect
5. C2×#22548702C2semidirect
6. #30632078C2semidirect
7. #30632075C2semidirect
8. #30632070C2semidirect
9. #30632068C2semidirect
10. M16×Dic251C2semidirect
11. C2×AGL(1,ℤ/8)×C251C2semidirect
17. C2×Dic1004C22semidirect
18. C2×#16860800C22semidirect
19. C2×#16860799C22semidirect
21. C2×#16860798C22semidirect
24. #22548713C22semidirect
25. #22548711C22semidirect
26. #22548704C22semidirect
30. #22548702C22semidirect
35. #22548625C22semidirect
36. C8×Dic251C22semidirect
62. #16860794D4semidirect
86. M16×C251D4semidirect
150. C251#3153semidirect
155. C2×AGL(1,ℤ/8)D251semidirect
228. M16C22⋊D251semidirect

Maximal subgroups

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

subgroupindex
C2×#225487482
C2×#225487452
C2×#225487412
#306322332
C2×#225487022
#306320782
#306320752
#306320702
#306320682
M16×Dic2512
C2×AGL(1,ℤ/8)×C2512
#3153251