Finite Group Databasebeta

← #24919694

Subgroups — #24919694

Order 10624.

448 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 448 conjugacy classes of subgroups — too many to draw. Only its 125 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
1#19128421C211yesmaximal, characteristic, complemented
2#19128047C211yesmaximal, characteristic, complemented
3#19128022C212yesmaximal, complemented
4#19127981C211yesmaximal, characteristic, complemented
5#19127945C212yesmaximal, complemented
6#19127898C211yesmaximal, characteristic, complemented
7#19127852C212yesmaximal, complemented
8C4×#13050578C211yesmaximal, characteristic, complemented
9#19127779C211yesmaximal, characteristic, complemented
10#19127706C212yesmaximal, complemented
11D42×C83C211yesmaximal, characteristic
12C2×#13050578C2211yescharacteristic
13C2×#13050578C2212yescomplemented
14C2×#13050574C2212yescomplemented
15#14581792C2211yescharacteristic, complemented
16#14581785—22no—
17#14581779—21no—
18#14581771C2212yescomplemented
19C4×Dic166C2211yescharacteristic, complemented
20#14581757—21no—
21#14581751C2212yescomplemented
22#14581725C2212yescomplemented
23#14581723C2211yescharacteristic, complemented
24C4⋊(C83⋊C8)C2211yescharacteristic, complemented
25#14581715C2211yescharacteristic, complemented
26#14581713C2212yes—
27#14581713C2211yescharacteristic
28#14581713C2212yescomplemented
29#14581713C2212yescomplemented
30#14581713C2212yescomplemented
31C22⋊(C83⋊C8)C2212yescomplemented
32C4×C83⋊C8C2211yescharacteristic, complemented
33C22×D4×C83—21no—
34C22×D4×C83C2212yes—
35#127×C83C2211yescharacteristic
36#121×C83—21no—
37#121×C83C2212yes—
38#120×C83—22no—
39C4×D4×C83C2211yescharacteristic
40C4×D4×C83C2211yescharacteristic
41C2×Dic166—21no—
42C2×Dic166C2311yescharacteristic
43#13050578—22no—
44#13050578—42no—
45#13050574—42no—
46C2×C83⋊C8C2311yescharacteristic
47C2×C83⋊C8C2312yes—
48C2×C83⋊C8C2311yescharacteristic
49C4⋊Dic83—21no—
50C4⋊Dic83C2312yes—
51C4⋊Dic83C2311yescharacteristic
52Dic83⋊C4—22no—
53Dic83⋊C4—21no—
54C22⋊Dic83—22no—
55C22⋊Dic83—22no—
56C4×Dic83—21no—
57C4×Dic83—22no—
58C24×C83—22no—
59C2×D4×C83—24no—
60C2×D4×C83—22no—
61C2×D4×C83—22no—
62C2×D4×C83—21no—
63C2×D4×C83—21no—
64C2×D4×C83—42no—
65C2×D4×C83—44no—
66C2×D4×C83D411yescharacteristic
67C2×D4×C83C2312yes—
68C2×D4×C83C2311yescharacteristic
69C22×C4×C83—21no—
70C22×C4×C83C2312yes—
71C4⋊C4×C83D411yescharacteristic
72C4⋊C4×C83C2311yescharacteristic
73C22⋊C4×C83—21no—
74C22⋊C4×C83—22no—
75C22⋊C4×C83D412yes—
76C42×C83C2311yescharacteristic
77C23×C83—21no—
78C23×C83—22no—
79C23×C83—44no—
80C23×C83—44no—
81C23×C83—82no—
82C23×C83C2×D412yes—
83D4×C83—22no—
84D4×C83—22no—
85D4×C83—21no—
86D4×C83—22no—
87D4×C83—44no—
88D4×C83—44no—
89D4×C83—42no—
90D4×C83—81no—
91D4×C83#4112yes—
92C2×C4×C83—22no—
93C2×C4×C83—22no—
94C2×C4×C83—21no—
95C2×C4×C83—42no—
96C2×C4×C83C2×D411yescharacteristic
97C2×C4×C83C2×D412yes—
98C2×C4×C83C2×D411yescharacteristic
99C2×C4×C83C2411yescharacteristic
100C2×Dic83—21no—
101C2×Dic83—21no—
102C2×Dic83—22no—
103Dic166—21no—
104Dic166—21no—
105Dic166—42no—
106C83⋊C8—21no—
107C83⋊C8—21no—
108C83⋊C8—22no—
109C22×C83—22no—
110C22×C83—22no—
111C22×C83—22no—
112C22×C83—22no—
113C22×C83—41no—
114C22×C83—41no—
115C22×C83—42no—
116C22×C83—42no—
117C22×C83—42no—
118C22×C83—88no—
119C22×C83C22×D411yescharacteristic
120C4×C83—22no—
121C4×C83—21no—
122C4×C83—41no—
123C4×C83AGL(1,ℤ/8)12yes—
124C4×C83C2×#4111yescharacteristic
125C4×C83H(5,𝔽2)11yescharacteristic
126Dic83—41no—
127Dic83—42no—
128Dic83—41no—
129C2×C83—42no—
130C2×C83—42no—
131C2×C83—82no—
132C2×C83#54611yescharacteristic
133C2×C83C2×AGL(1,ℤ/8)11yescharacteristic
134C2×C83#57711yescharacteristic
135#3324—831nomaximal, Sylow
136C83#332411yescharacteristic, complemented, Sylow
137#556—831no—
138D42D8311yescharacteristic
139#486—831no—
140#482—832no—
141#474—831no—
142#468—832no—
143#460—831no—
144#450—832no—
145C4×SD16—831no—
146#436—831no—
147#418—832no—
148C22×D4—21no—
149C22×D4C2×D8312yes—
150C2×SD16—832no—
151C2×SD16—831no—
152C2×M16—832no—
153C4⋊Q8—831no—
154#127C2×D8311yescharacteristic
155#126—1662no—
156#124—1661no—
157C22⋊Q8—832no—
158#121—21no—
159#121C2×D8312yes—
160#120—22no—
161C4×Q8—831no—
162C4×Q8—1661no—
163C4×D4C2×D8311yescharacteristic
164C4×D4C2×D8311yescharacteristic
165#117—832no—
166#107—832no—
167#106—831no—
168C4⋊C8—831no—
169#103—831no—
170#102—831no—
171#102—832no—
172#102—832no—
173#102—832no—
174#102—832no—
175C22⋊C8—832no—
176C4×C8—831no—
177C24—22no—
178C2×Q8—831no—
179C2×Q8—1661no—
180C2×D4—24no—
181C2×D4—22no—
182C2×D4—22no—
183C2×D4—21no—
184C2×D4—21no—
185C2×D4—42no—
186C2×D4—44no—
187C2×D4C22⋊D8311yescharacteristic
188C2×D4C22×D8312yes—
189C2×D4C22×D8311yescharacteristic
190C22×C4—21no—
191C22×C4C22×D8312yes—
192SD16—3322no—
193SD16—1662no—
194M16—3322no—
195C2×C8—831no—
196C2×C8—831no—
197C2×C8—832no—
198C4⋊C4—831no—
199C4⋊C4—832no—
200C4⋊C4—1661no—
201C4⋊C4—1662no—
202C4⋊C4—1661no—
203C4⋊C4C22⋊D8311yescharacteristic
204C4⋊C4C22×D8311yescharacteristic
205C22⋊C4—21no—
206C22⋊C4—22no—
207C22⋊C4—1662no—
208C22⋊C4—1662no—
209C22⋊C4C22⋊D8312yes—
210C42—1662no—
211C42—1661no—
212C42C22×D8311yescharacteristic
213C23—21no—
214C23—22no—
215C23—44no—
216C23—44no—
217C23—82no—
218C23C2×C22⋊D8312yes—
219Q8—3322no—
220Q8—1661no—
221Q8—1661no—
222D4—22no—
223D4—22no—
224D4—21no—
225D4—22no—
226D4—44no—
227D4—44no—
228D4—42no—
229D4—81no—
230D4#1305058912yes—
231C2×C4—22no—
232C2×C4—22no—
233C2×C4—21no—
234C2×C4—42no—
235C2×C4—1661no—
236C2×C4—1662no—
237C2×C4—1661no—
238C2×C4C2×C22⋊D8312yes—
239C2×C4C2×C22⋊D8311yescharacteristic
240C2×C4C2×C22⋊D8311yescharacteristic
241C2×C4C23×D8311yescharacteristic
242C8—1661no—
243C8—1661no—
244C8—1662no—
245C22—22no—
246C22—22no—
247C22—22no—
248C22—22no—
249C22—41no—
250C22—41no—
251C22—42no—
252C22—42no—
253C22—42no—
254C22—88no—
255C22C22×C22⋊D8311yescharacteristic
256C4—22no—
257C4—21no—
258C4—41no—
259C4—3321no—
260C4—3322no—
261C4—3321no—
262C4#1458181812yes—
263C4C2×#1305058911yescharacteristic
264C4#1458183711yescharacteristic
265C2—42no—
266C2—42no—
267C2—82no—
268C2#1912834511yescharacteristic
269C2C2×#1458181811yescharacteristic
270C2#1912851211yescharacteristic
subgroupG/H
centerC22C22×C22⋊D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting, prime radicalD42×C83C2
socleC22×C83C22×D4
hypercenterD42D83
hypocenterC83#3324
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C4×D4 ◁ D42

Complements

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

NH ≅ G/Ntype
1. #19128421C2semidirect
2. #19128047C2semidirect
3. #19128022C2semidirect
4. #19127981C2semidirect
5. #19127945C2semidirect
6. #19127898C2semidirect
7. #19127852C2semidirect
8. C4×#13050578C2semidirect
9. #19127779C2semidirect
10. #19127706C2semidirect
13. C2×#13050578C22semidirect
14. C2×#13050574C22semidirect
15. #14581792C22semidirect
18. #14581771C22semidirect
19. C4×Dic166C22semidirect
21. #14581751C22semidirect
22. #14581725C22semidirect
23. #14581723C22semidirect
24. C4⋊(C83⋊C8)C22semidirect
25. #14581715C22semidirect
28. #14581713C22semidirect
29. #14581713C22semidirect
30. #14581713C22semidirect
31. C22⋊(C83⋊C8)C22semidirect
32. C4×C83⋊C8C22semidirect
136. C83#3324semidirect

Maximal subgroups

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

subgroupindex
#191284212
#191280472
#191280222
#191279812
#191279452
#191278982
#191278522
C4×#130505782
#191277792
#191277062
D42×C832
#332483