Finite Group Databasebeta

← #24917007

Subgroups — #24917007

Order 10624.

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×#14581818C211yesmaximal, characteristic, complemented
2C2×#14581814C211yesmaximal, characteristic, complemented
3#19128268C211yesmaximal, characteristic, complemented
4#19128182C211yesmaximal, characteristic, complemented
5#19127922C212yesmaximal, complemented
6#19127915C212yesmaximal, complemented
7#19127900C212yesmaximal, complemented
8#19127895C212yesmaximal, complemented
9#19127727C211yesmaximal, characteristic, complemented
10#19127700C211yesmaximal, characteristic, complemented
11C2×#102×C83C211yesmaximal, characteristic, complemented
12C2×#13050590C2211yescharacteristic
13C2×#13050590C2211yescharacteristic
14#14581818—24no—
15#14581814—24no—
16C2×Dic332C2211yescharacteristic, complemented
17C2×#13050579C2211yescharacteristic, complemented
18C2×#13050579C2211yescharacteristic, complemented
19C2×#13050578C2212yes—
20C2×D664C2211yescharacteristic, complemented
21C2×#13050575C2212yes—
22C2×#13050574C2211yescharacteristic
23#14581788C2212yes—
24#14581779C2212yes—
25#14581763—21no—
26#14581763—21no—
27#14581761—22no—
28#14581757C2212yescomplemented
29#14581754C2212yescomplemented
30C4×C22⋊D83—22no—
31#14581751C2211yescharacteristic
32#14581750—21no—
33C4⋊C4×D83—21no—
34Dic83⋊C8C2212yescomplemented
35Dic83⋊C8C2212yescomplemented
36#14581715C2212yescomplemented
37#14581712C2212yescomplemented
38C22×D4×C83C2211yescharacteristic
39C22×C8×C83C2211yescharacteristic
40C2×C4⋊C4×C83C2211yescharacteristic
41#102×C83C2212yes—
42#102×C83C2212yes—
43#13050590—22no—
44#13050590—22no—
45#13050590—22no—
46#13050590D414yescomplemented
47C2×Dic166C2311yescharacteristic
48C2×Dic166C2311yescharacteristic
49C2×D332C2311yescharacteristic
50C2×D332C2311yescharacteristic
51C2×C22⋊D83—21no—
52C2×C22⋊D83—21no—
53C2×C22⋊D83—22no—
54C2×C22⋊D83—22no—
55C2×C4×D83—21no—
56C2×C4×D83—21no—
57C2×C4×D83—22no—
58Dic332—22no—
59#13050579—22no—
60#13050579—22no—
61#13050578—24no—
62D664—22no—
63#13050575—24no—
64#13050574—22no—
65C2×C83⋊C8C2312yes—
66C4⋊Dic83D411yescharacteristic
67C4⋊Dic83D411yescharacteristic
68Dic83⋊C4—22no—
69C22⋊Dic83—22no—
70C22⋊Dic83—22no—
71C22⋊Dic83D412yes—
72(C2×D83)⋊C4—22no—
73C4×Dic83—22no—
74C4×Dic83C2312yes—
75C24×C83—21no—
76C2×D4×C83—22no—
77C2×D4×C83—21no—
78C2×D4×C83—22no—
79C2×D4×C83C2312yes—
80C22×C4×C83—21no—
81C22×C4×C83C2311yescharacteristic
82C2×C8×C83—22no—
83C2×C8×C83C2311yescharacteristic
84C2×C8×C83C2311yescharacteristic
85C4⋊C4×C83—21no—
86C4⋊C4×C83C2312yes—
87C23×C83—22no—
88C23×C83—42no—
89C23×C83—81no—
90C23×C83—81no—
91C23×C83C2×D411yescharacteristic
92C22×D83—21no—
93C22×D83—21no—
94D4×C83—22no—
95D4×C83—22no—
96D4×C83—42no—
97C2×C4×C83—22no—
98C2×C4×C83—81no—
99C2×C4×C83C2×D411yescharacteristic
100C2×C4×C83C2×D412yes—
101C2×C4×C83#4112yes—
102C2×C4×C83C2411yescharacteristic
103C22⋊D83—24no—
104C22⋊D83—42no—
105C22⋊D83—82no—
106C22⋊D83—82no—
107C2×Dic83—21no—
108C2×Dic83—21no—
109C2×Dic83C2×D412yes—
110C2×Dic83C2×D412yes—
111D332—21no—
112D332—21no—
113D332—21no—
114D332C2×D412yes—
115C4×D83—24no—
116C4×D83—42no—
117C4×D83—82no—
118Dic166—21no—
119Dic166—21no—
120Dic166—21no—
121Dic166C2×D412yes—
122C8×C83—21no—
123C8×C83—21no—
124C83⋊C8—22no—
125C22×C83—21no—
126C22×C83—21no—
127C22×C83—42no—
128C22×C83—42no—
129C22×C83—42no—
130C22×C83—84no—
131C22×C83#12112yes—
132C22×C83C22×D411yescharacteristic
133C2×D83—22no—
134C2×D83—41no—
135C2×D83—41no—
136C2×D83—41no—
137C2×D83—41no—
138C2×D83—41no—
139C4×C83—42no—
140C4×C83#12112yes—
141C4×C83C22×D411yescharacteristic
142C4×C83C2×#4111yescharacteristic
143Dic83—22no—
144Dic83—22no—
145Dic83—22no—
146Dic83—41no—
147C2×C83—22no—
148C2×C83—82no—
149C2×C83C2×#12111yescharacteristic
150C2×C83#57611yescharacteristic
151C2×C83#57711yescharacteristic
152D83—42no—
153D83—81no—
154#3045—831nomaximal, Sylow
155C83#304511yescharacteristic, complemented, Sylow
156C2×AGL(1,ℤ/8)—831no—
157C2×#135—831no—
158#535—831no—
159#520—831no—
160#464—832no—
161#463—832no—
162#460—832no—
163#459—832no—
164#424—831no—
165#417—831no—
166C2×#102D8311yescharacteristic, complemented
167C2×#41—831no—
168C2×#41—831no—
169C22×D4C2×D8311yescharacteristic
170AGL(1,ℤ/8)—1664no—
171#135—1664no—
172C2×Q16—831no—
173C2×SD16—832no—
174C2×SD16—831no—
175C2×SD16—831no—
176C2×D8—832no—
177C2×D8—831no—
178C2×M16—831no—
179C22×C8C2×D8311yescharacteristic
180#127—832no—
181#124—832no—
182#121—1661no—
183#121—1661no—
184#120—1662no—
185C4×Q8—832no—
186C4×D4—832no—
187C4×D4—1662no—
188#117—831no—
189#117—1661no—
190C2×C4⋊C4—1661no—
191C2×C4⋊C4C2×D8311yescharacteristic
192C4⋊C8—832no—
193C4⋊C8—832no—
194#103—832no—
195#102—832no—
196#102C2×D8312yes—
197#102C2×D8312yes—
198C24—21no—
199#41—834no—
200#41—1662no—
201#41—1662no—
202#41—1662no—
203C2×Q8—831no—
204C2×Q8—831no—
205C2×D4—22no—
206C2×D4—21no—
207C2×D4—22no—
208C2×D4—831no—
209C2×D4—831no—
210C2×D4—1662no—
211C2×D4—1662no—
212C2×D4—1661no—
213C2×D4—1661no—
214C2×D4C22×D8312yes—
215C22×C4—21no—
216C22×C4—1662no—
217C22×C4—1661no—
218C22×C4—1661no—
219C22×C4C22×D8311yescharacteristic
220Q16—1662no—
221SD16—1662no—
222SD16—1662no—
223SD16—1664no—
224D8—1662no—
225D8—1664no—
226M16—1662no—
227C2×C8—22no—
228C2×C8—832no—
229C2×C8C22×D8311yescharacteristic
230C2×C8C22×D8311yescharacteristic
231C4⋊C4—21no—
232C4⋊C4—831no—
233C4⋊C4—831no—
234C4⋊C4—1662no—
235C4⋊C4C22×D8312yes—
236C22⋊C4—832no—
237C22⋊C4—1662no—
238C22⋊C4—1662no—
239C22⋊C4—1662no—
240C42—832no—
241C42—1662no—
242C23—22no—
243C23—42no—
244C23—81no—
245C23—81no—
246C23—1661no—
247C23—1661no—
248C23D4×D8311yescharacteristic
249Q8—832no—
250Q8—1661no—
251Q8—1661no—
252Q8—1661no—
253D4—22no—
254D4—22no—
255D4—42no—
256D4—3322no—
257D4—832no—
258D4—6642no—
259D4—6642no—
260D4—1664no—
261D4—1661no—
262D4—1661no—
263D4—1661no—
264C2×C4—22no—
265C2×C4—81no—
266C2×C4—3322no—
267C2×C4—832no—
268C2×C4—832no—
269C2×C4—6642no—
270C2×C4—1661no—
271C2×C4—1661no—
272C2×C4—1664no—
273C2×C4D4×D8311yescharacteristic
274C2×C4D4×D8312yes—
275C2×C4#1305059012yes—
276C2×C4C23×D8311yescharacteristic
277C8—21no—
278C8—21no—
279C8—1662no—
280C22—21no—
281C22—21no—
282C22—42no—
283C22—42no—
284C22—42no—
285C22—84no—
286C22—3321no—
287C22—3321no—
288C22—3321no—
289C22—3321no—
290C22—3321no—
291C22—1662no—
292C22#1458176512yes—
293C22C2×D4×D8311yescharacteristic
294C4—42no—
295C4—3321no—
296C4—1662no—
297C4—1662no—
298C4—1662no—
299C4#1458176512yes—
300C4C2×D4×D8311yescharacteristic
301C4C2×#1305059011yescharacteristic
302C2—22no—
303C2—82no—
304C2—3322no—
305C2—6641no—
306C2C2×#1458176511yescharacteristic
307C2#1912850211yescharacteristic
308C2#1912850711yescharacteristic
subgroupG/H
centerC22C2×D4×D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
FittingC2×#102×C83C2
socleC22×C83C22×D4
hypercenterC2×#102D83
hypocenterC83#3045
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#102

Complements

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

NH ≅ G/Ntype
1. C2×#14581818C2semidirect
2. C2×#14581814C2semidirect
3. #19128268C2semidirect
4. #19128182C2semidirect
5. #19127922C2semidirect
6. #19127915C2semidirect
7. #19127900C2semidirect
8. #19127895C2semidirect
9. #19127727C2semidirect
10. #19127700C2semidirect
11. C2×#102×C83C2semidirect
16. C2×Dic332C22semidirect
17. C2×#13050579C22semidirect
18. C2×#13050579C22semidirect
20. C2×D664C22semidirect
28. #14581757C22semidirect
29. #14581754C22semidirect
34. Dic83⋊C8C22semidirect
35. Dic83⋊C8C22semidirect
36. #14581715C22semidirect
37. #14581712C22semidirect
46. #13050590D4semidirect
155. C83#3045semidirect
166. C2×#102D83semidirect

Maximal subgroups

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

subgroupindex
C2×#145818182
C2×#145818142
#191282682
#191281822
#191279222
#191279152
#191279002
#191278952
#191277272
#191277002
C2×#102×C832
#304583