Finite Group Databasebeta

← #24917948

Subgroups — #24917948

Order 10624.

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

normal subgroups only

This group has 596 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×#14581820C211yesmaximal, characteristic, complemented
2C2×#14581812C211yesmaximal, characteristic, complemented
3C22×#13050575C211yesmaximal, characteristic, complemented
4#19128268C211yesmaximal, characteristic, complemented
5#19128265C211yesmaximal, characteristic, complemented
6#19128089C212yesmaximal, complemented
7#19128087C212yesmaximal, complemented
8#19128082C212yesmaximal, complemented
9#19128076C212yesmaximal, complemented
10#19127659C211yesmaximal, characteristic, complemented
11C2×AGL(1,ℤ/8)×C83C211yesmaximal, characteristic, complemented
12C2×#13050590C2211yescharacteristic
13C22×D332C2211yescharacteristic
14#14581820—24no—
15#14581812—24no—
16C2×#13050580C2211yescharacteristic, complemented
17C2×#13050579C2212yescomplemented
18C2×#13050578C2211yescharacteristic, complemented
19C2×#13050577C2211yescharacteristic, complemented
20C2×D664C2212yescomplemented
21C2×#13050575—24no—
22C2×#13050575C2212yes—
23C2×#13050575C2212yes—
24C2×#13050575C2211yescharacteristic, complemented
25C22×C83⋊C8C2211yescharacteristic
26#14581788C2212yescomplemented
27#14581788C2212yescomplemented
28#14581781C2212yescomplemented
29#14581779C2212yescomplemented
30#14581766—22no—
31#14581763—22no—
32#14581761—22no—
33#14581760—22no—
34#14581751C2211yescharacteristic
35#14581702C2212yescomplemented
36C8×Dic83C2212yescomplemented
37C2×#41×C83C2211yescharacteristic
38C22×D4×C83C2211yescharacteristic
39AGL(1,ℤ/8)×C83—24no—
40C2×SD16×C83C2212yescomplemented
41C2×D8×C83C2212yescomplemented
42C2×M16×C83C2211yescharacteristic
43C23×D83—21no—
44#13050590—22no—
45#13050590—22no—
46C2×Dic166C2311yescharacteristic
47C2×D332—22no—
48C2×D332—22no—
49C2×D332—21no—
50C2×D332C2312yes—
51C2×D332C2311yescharacteristic
52C2×C22⋊D83—21no—
53C2×C22⋊D83—24no—
54C2×C22⋊D83—24no—
55C2×C4×D83—21no—
56#13050580—22no—
57#13050579—24no—
58#13050578—22no—
59#13050577—22no—
60D664—24no—
61#13050575—24no—
62#13050575—24no—
63#13050575—22no—
64C2×C83⋊C8D414yescomplemented
65C2×C83⋊C8C2311yescharacteristic
66C2×C83⋊C8C2311yescharacteristic
67C4⋊Dic83D412yes—
68C22⋊Dic83—24no—
69C22⋊Dic83D412yes—
70(C2×D83)⋊C4—24no—
71C4×Dic83C2312yes—
72C24×C83—21no—
73#41×C83—22no—
74#41×C83—22no—
75C2×Q8×C83C2311yescharacteristic
76C2×D4×C83—21no—
77C2×D4×C83—22no—
78C2×D4×C83—22no—
79C2×D4×C83—21no—
80C2×D4×C83C2312yes—
81C2×D4×C83C2311yescharacteristic
82C22×C4×C83—21no—
83C22×C4×C83C2311yescharacteristic
84SD16×C83—24no—
85D8×C83—24no—
86M16×C83D414yescomplemented
87C2×C8×C83C2312yes—
88C23×C83—21no—
89C23×C83—22no—
90C23×C83—42no—
91C23×C83—82no—
92C23×C83C2×D411yescharacteristic
93C22×D83—22no—
94C22×D83—21no—
95C22×D83—42no—
96C22×D83—82no—
97Q8×C83—21no—
98Q8×C83—21no—
99D4×C83—21no—
100D4×C83—22no—
101D4×C83—22no—
102D4×C83—21no—
103D4×C83—42no—
104D4×C83—42no—
105C2×C4×C83—21no—
106C2×C4×C83—42no—
107C2×C4×C83C2×D411yescharacteristic
108C2×C4×C83C2×D412yes—
109C2×C4×C83C2×D412yes—
110C2×C4×C83C2411yescharacteristic
111C22⋊D83—42no—
112C22⋊D83—84no—
113C22⋊D83—84no—
114C2×Dic83—21no—
115C2×Dic83C2×D414yes—
116D332—22no—
117D332—21no—
118D332—21no—
119D332—22no—
120D332—42no—
121C4×D83—42no—
122Dic166—21no—
123Dic166—21no—
124C8×C83C2×D414yes—
125C83⋊C8C2×D412yes—
126C83⋊C8C2×D412yes—
127C22×C83—22no—
128C22×C83—41no—
129C22×C83—42no—
130C22×C83—42no—
131C22×C83—44no—
132C22×C83—84no—
133C22×C83#12712yes—
134C22×C83C22×D411yescharacteristic
135C2×D83—41no—
136C2×D83—42no—
137C2×D83—42no—
138C2×D83—44no—
139C2×D83—84no—
140C4×C83—41no—
141C4×C83#12712yes—
142C4×C83C22×D411yescharacteristic
143C4×C83C22×D411yescharacteristic
144Dic83—24no—
145Dic83—41no—
146C2×C83—22no—
147C2×C83—81no—
148C2×C83—82no—
149C2×C83C2×#12711yescharacteristic
150C2×C83#57612yes—
151D83—81no—
152D83—82no—
153#3155—831nomaximal, Sylow
154C83#315511yescharacteristic, complemented, Sylow
155C2×AGL(1,ℤ/8)—831no—
156C2×AGL(1,ℤ/8)D8311yescharacteristic, complemented
157C2×#135—831no—
158C22×D8—831no—
159#535—831no—
160#535—831no—
161#496—832no—
162#496—832no—
163#495—832no—
164#493—832no—
165#405—831no—
166C2×#41—831no—
167C2×#41C2×D8311yescharacteristic
168C22×D4—831no—
169C22×D4C2×D8311yescharacteristic
170AGL(1,ℤ/8)—24no—
171AGL(1,ℤ/8)—1664no—
172#135—1664no—
173C2×Q16—831no—
174C2×SD16—831no—
175C2×SD16—831no—
176C2×SD16—832no—
177C2×SD16C2×D8312yescomplemented
178C2×D8—832no—
179C2×D8—832no—
180C2×D8—831no—
181C2×D8—832no—
182C2×D8—1664no—
183C2×D8C2×D8312yescomplemented
184C2×M16C2×D8311yescharacteristic
185C22×C8—831no—
186#127—832no—
187#127—832no—
188#124—832no—
189#124—832no—
190#121—1662no—
191#121—1662no—
192#120—1662no—
193#120—1662no—
194#117—831no—
195#97—832no—
196C4×C8—832no—
197C24—21no—
198C24—1661no—
199#41—22no—
200#41—22no—
201#41—1662no—
202#41—1662no—
203C2×Q8—831no—
204C2×Q8C22×D8311yescharacteristic
205C2×D4—21no—
206C2×D4—22no—
207C2×D4—21no—
208C2×D4—22no—
209C2×D4—832no—
210C2×D4—831no—
211C2×D4—1664no—
212C2×D4—1664no—
213C2×D4—1661no—
214C2×D4—1661no—
215C2×D4—1662no—
216C2×D4—1662no—
217C2×D4C22×D8312yes—
218C2×D4C22×D8311yescharacteristic
219C22×C4—21no—
220C22×C4—1661no—
221C22×C4C22×D8311yescharacteristic
222Q16—1662no—
223SD16—24no—
224SD16—1662no—
225SD16—1662no—
226SD16—1664no—
227D8—24no—
228D8—1664no—
229D8—1664no—
230D8—1662no—
231D8—1664no—
232M16C22⋊D8314yescomplemented
233C2×C8—834no—
234C2×C8—831no—
235C2×C8—831no—
236C2×C8C22×D8312yes—
237C4⋊C4—832no—
238C22⋊C4—832no—
239C22⋊C4—1664no—
240C22⋊C4—1664no—
241C42—832no—
242C23—22no—
243C23—21no—
244C23—42no—
245C23—82no—
246C23—3322no—
247C23—6642no—
248C23—1662no—
249C23—1661no—
250C23D4×D8311yescharacteristic
251Q8—21no—
252Q8—21no—
253Q8—1661no—
254Q8—1661no—
255D4—22no—
256D4—21no—
257D4—21no—
258D4—22no—
259D4—42no—
260D4—42no—
261D4—3322no—
262D4—3322no—
263D4—6644no—
264D4—6644no—
265D4—1662no—
266D4—1661no—
267D4—1661no—
268D4—1662no—
269C2×C4—21no—
270C2×C4—42no—
271C2×C4—3322no—
272C2×C4—834no—
273C2×C4—1661no—
274C2×C4D4×D8311yescharacteristic
275C2×C4D4×D8312yes—
276C2×C4C2×C22⋊D8312yes—
277C2×C4C23×D8311yescharacteristic
278C8—832no—
279C8—832no—
280C8C2×C22⋊D8314yes—
281C22—22no—
282C22—41no—
283C22—42no—
284C22—42no—
285C22—44no—
286C22—84no—
287C22—3321no—
288C22—3322no—
289C22—3322no—
290C22—3324no—
291C22—6644no—
292C22#1458178812yes—
293C22C2×D4×D8311yescharacteristic
294C4—41no—
295C4—3321no—
296C4—1664no—
297C4#1458178812yes—
298C4C2×D4×D8311yescharacteristic
299C4C22×C22⋊D8311yescharacteristic
300C2—22no—
301C2—82no—
302C2—81no—
303C2—6642no—
304C2—6641no—
305C2C2×#1458178811yescharacteristic
306C2#1912850512yes—
subgroupG/H
centerC22C2×D4×D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
FittingC2×AGL(1,ℤ/8)×C83C2
socleC22×C83C22×D4
hypercenterC2×AGL(1,ℤ/8)D83
hypocenterC83#3155
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
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×#14581820C2semidirect
2. C2×#14581812C2semidirect
3. C22×#13050575C2semidirect
4. #19128268C2semidirect
5. #19128265C2semidirect
6. #19128089C2semidirect
7. #19128087C2semidirect
8. #19128082C2semidirect
9. #19128076C2semidirect
10. #19127659C2semidirect
11. C2×AGL(1,ℤ/8)×C83C2semidirect
16. C2×#13050580C22semidirect
17. C2×#13050579C22semidirect
18. C2×#13050578C22semidirect
19. C2×#13050577C22semidirect
20. C2×D664C22semidirect
24. C2×#13050575C22semidirect
26. #14581788C22semidirect
27. #14581788C22semidirect
28. #14581781C22semidirect
29. #14581779C22semidirect
35. #14581702C22semidirect
36. C8×Dic83C22semidirect
40. C2×SD16×C83C22semidirect
41. C2×D8×C83C22semidirect
64. C2×C83⋊C8D4semidirect
86. M16×C83D4semidirect
154. C83#3155semidirect
156. C2×AGL(1,ℤ/8)D83semidirect
177. C2×SD16C2×D83semidirect
183. C2×D8C2×D83semidirect
232. M16C22⋊D83semidirect

Maximal subgroups

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

subgroupindex
C2×#145818202
C2×#145818122
C22×#130505752
#191282682
#191282652
#191280892
#191280872
#191280822
#191280762
#191276592
C2×AGL(1,ℤ/8)×C832
#315583