Finite Group Databasebeta

← #24916705

Subgroups — #24916705

Order 10624.

758 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 758 conjugacy classes of subgroups — too many to draw. Only its 165 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×#14581823C211yesmaximal, characteristic, complemented
2C2×#14581818C211yesmaximal, characteristic, complemented
3#19128212C211yesmaximal, characteristic, complemented
4#19127863C214yesmaximal, complemented
5#19127846C214yesmaximal, complemented
6#19127710C211yesmaximal, characteristic, complemented
7#19127706C211yesmaximal, characteristic, complemented
8#19127665C211yesmaximal, characteristic, complemented
9C22×#41×C83C211yesmaximal, characteristic, complemented
10C2×#13050590C2211yescharacteristic
11#14581823—24no—
12#14581818—24no—
13C2×#13050580C2212yescomplemented
14C2×#13050578C2212yescomplemented
15C2×#13050577C2212yescomplemented
16C2×#13050575C2212yescomplemented
17C2×#13050574C2211yescharacteristic
18C2×#13050574C2211yescharacteristic
19#14581775—21no—
20#14581771C2212yescomplemented
21#14581766C2212yescomplemented
22#14581761—21no—
23C4×C22⋊D83—22no—
24#14581751C2211yescharacteristic
25#14581716C2214yescomplemented
26#14581713C2214yescomplemented
27C22⋊(C83⋊C8)C2214yescomplemented
28C2×#41×C83—24no—
29C2×#41×C83—24no—
30C2×#41×C83—22no—
31C2×#41×C83C2214yes—
32C22×Q8×C83C2211yescharacteristic
33C22×D4×C83—21no—
34C22×D4×C83C2211yescharacteristic
35C23×C4×C83—21no—
36C23×C4×C83C2211yescharacteristic
37#13050590—22no—
38#13050590—22no—
39C2×Dic166C2311yescharacteristic
40C2×D332C2311yescharacteristic
41C2×C22⋊D83—22no—
42C2×C22⋊D83—21no—
43C2×C4×D83—21no—
44#13050580—24no—
45#13050578—24no—
46#13050577—24no—
47#13050575—24no—
48#13050574—22no—
49#13050574—22no—
50C2×C83⋊C8C2312yes—
51C2×C83⋊C8C2312yes—
52C4⋊Dic83C2312yes—
53Dic83⋊C4—22no—
54C22⋊Dic83—22no—
55C22⋊Dic83—21no—
56(C2×D83)⋊C4—22no—
57C4×Dic83—21no—
58C24×C83—21no—
59C24×C83D411yescharacteristic
60#41×C83—216no—
61#41×C83—28no—
62#41×C83—28no—
63C2×Q8×C83—22no—
64C2×Q8×C83—21no—
65C2×Q8×C83D414yescomplemented
66C2×Q8×C83C2312yes—
67C2×D4×C83—28no—
68C2×D4×C83—24no—
69C2×D4×C83—21no—
70C2×D4×C83—22no—
71C2×D4×C83D414yescomplemented
72C2×D4×C83C2312yes—
73C22×C4×C83—28no—
74C22×C4×C83—21no—
75C22×C4×C83—24no—
76C22×C4×C83—22no—
77C22×C4×C83—22no—
78C22×C4×C83D411yescharacteristic
79C22×C4×C83D412yes—
80C22×C4×C83C2312yes—
81C22×C4×C83C2311yescharacteristic
82C23×C83—22no—
83C23×C83—24no—
84C23×C83—22no—
85C23×C83—42no—
86C23×C83—44no—
87C23×C83C2×D412yes—
88C23×C83C2×D411yescharacteristic
89C22×D83—21no—
90Q8×C83—24no—
91Q8×C83—22no—
92Q8×C83C2×D414yes—
93D4×C83—216no—
94D4×C83—24no—
95D4×C83—22no—
96D4×C83C2×D414yes—
97C2×C4×C83—216no—
98C2×C4×C83—22no—
99C2×C4×C83—24no—
100C2×C4×C83—22no—
101C2×C4×C83—24no—
102C2×C4×C83—22no—
103C2×C4×C83—22no—
104C2×C4×C83C2×D412yes—
105C2×C4×C83C2×D411yescharacteristic
106C2×C4×C83C2×D412yes—
107C2×C4×C83C2×D412yes—
108C2×C4×C83C2411yescharacteristic
109C22⋊D83—44no—
110C22⋊D83—42no—
111C2×Dic83—22no—
112C2×Dic83—21no—
113D332—21no—
114D332—21no—
115C4×D83—42no—
116Dic166—21no—
117Dic166—21no—
118C83⋊C8—22no—
119C83⋊C8—22no—
120C22×C83—22no—
121C22×C83—22no—
122C22×C83—24no—
123C22×C83—22no—
124C22×C83—22no—
125C22×C83—44no—
126C22×C83—48no—
127C22×C83—44no—
128C22×C83#12012yes—
129C22×C83C22×D411yescharacteristic
130C2×D83—41no—
131C2×D83—42no—
132C4×C83—22no—
133C4×C83—24no—
134C4×C83#12012yes—
135C4×C83C22×D411yescharacteristic
136C4×C83C22×D411yescharacteristic
137Dic83—41no—
138Dic83—42no—
139C2×C83—22no—
140C2×C83—42no—
141C2×C83—44no—
142C2×C83C2×#12011yescharacteristic
143C2×C83#57512yes—
144D83—81no—
145#3005—831nomaximal, Sylow
146C83#300511yescharacteristic, complemented, Sylow
147C22×#41D8311yescharacteristic, complemented
148C2×#137—831no—
149C2×AGL(1,ℤ/8)—831no—
150#525—831no—
151#452—834no—
152#449—834no—
153#419—831no—
154#418—831no—
155#407—831no—
156C2×#41—24no—
157C2×#41—24no—
158C2×#41—22no—
159C2×#41—831no—
160C2×#41C2×D8314yes—
161C22×Q8C2×D8311yescharacteristic
162C22×D4—21no—
163C22×D4C2×D8311yescharacteristic
164C23×C4—21no—
165C23×C4C2×D8311yescharacteristic
166#137—1664no—
167AGL(1,ℤ/8)—1664no—
168C2×Q16—832no—
169C2×SD16—832no—
170C2×SD16—832no—
171C2×D8—832no—
172C2×M16—831no—
173C2×M16—831no—
174#123—1661no—
175C22⋊Q8—832no—
176#121—832no—
177#120—1661no—
178C4×D4—1662no—
179#117—831no—
180#103—834no—
181#102—834no—
182C22⋊C8—834no—
183C24—21no—
184C24C22⋊D8311yescharacteristic
185#41—216no—
186#41—28no—
187#41—28no—
188#41—1662no—
189#41—1662no—
190C2×Q8—22no—
191C2×Q8—21no—
192C2×Q8—831no—
193C2×Q8C22⋊D8314yescomplemented
194C2×Q8C22×D8312yes—
195C2×D4—28no—
196C2×D4—24no—
197C2×D4—21no—
198C2×D4—22no—
199C2×D4—831no—
200C2×D4—1662no—
201C2×D4—1661no—
202C2×D4C22⋊D8314yescomplemented
203C2×D4C22×D8312yes—
204C22×C4—28no—
205C22×C4—21no—
206C22×C4—24no—
207C22×C4—22no—
208C22×C4—22no—
209C22×C4—1661no—
210C22×C4C22⋊D8312yes—
211C22×C4C22⋊D8311yescharacteristic
212C22×C4C22×D8312yes—
213C22×C4C22×D8311yescharacteristic
214Q16—1664no—
215SD16—1664no—
216SD16—1664no—
217D8—1664no—
218M16—1662no—
219M16—1662no—
220C2×C8—832no—
221C2×C8—832no—
222C4⋊C4—832no—
223C4⋊C4—1662no—
224C22⋊C4—1661no—
225C22⋊C4—1662no—
226C22⋊C4—1662no—
227C42—1661no—
228C23—22no—
229C23—24no—
230C23—22no—
231C23—42no—
232C23—44no—
233C23—1661no—
234C23C2×C22⋊D8312yes—
235C23C2×C22⋊D8311yescharacteristic
236Q8—24no—
237Q8—22no—
238Q8—1661no—
239Q8—1661no—
240Q8C2×C22⋊D8314yes—
241D4—216no—
242D4—24no—
243D4—22no—
244D4—3324no—
245D4—3322no—
246D4—1661no—
247D4—1661no—
248D4C2×C22⋊D8314yes—
249C2×C4—216no—
250C2×C4—22no—
251C2×C4—24no—
252C2×C4—22no—
253C2×C4—24no—
254C2×C4—22no—
255C2×C4—22no—
256C2×C4—3322no—
257C2×C4—1662no—
258C2×C4—1661no—
259C2×C4C2×C22⋊D8312yes—
260C2×C4C2×C22⋊D8311yescharacteristic
261C2×C4C2×C22⋊D8312yes—
262C2×C4C2×C22⋊D8312yes—
263C2×C4C23×D8311yescharacteristic
264C8—1662no—
265C8—1662no—
266C22—22no—
267C22—22no—
268C22—24no—
269C22—22no—
270C22—22no—
271C22—44no—
272C22—48no—
273C22—44no—
274C22—3321no—
275C22—3322no—
276C22#1458176112yes—
277C22C22×C22⋊D8311yescharacteristic
278C4—22no—
279C4—24no—
280C4—3321no—
281C4—3322no—
282C4#1458176112yes—
283C4C22×C22⋊D8311yescharacteristic
284C4C22×C22⋊D8311yescharacteristic
285C2—22no—
286C2—42no—
287C2—44no—
288C2—6641no—
289C2C2×#1458176111yescharacteristic
290C2#1912849912yes—
subgroupG/H
centerC22C22×C22⋊D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
FittingC22×#41×C83C2
socleC22×C83C22×D4
hypercenterC22×#41D83
hypocenterC83#3005
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C23×C4 ◁ C22×#41

Complements

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

NH ≅ G/Ntype
1. C2×#14581823C2semidirect
2. C2×#14581818C2semidirect
3. #19128212C2semidirect
4. #19127863C2semidirect
5. #19127846C2semidirect
6. #19127710C2semidirect
7. #19127706C2semidirect
8. #19127665C2semidirect
9. C22×#41×C83C2semidirect
13. C2×#13050580C22semidirect
14. C2×#13050578C22semidirect
15. C2×#13050577C22semidirect
16. C2×#13050575C22semidirect
20. #14581771C22semidirect
21. #14581766C22semidirect
25. #14581716C22semidirect
26. #14581713C22semidirect
27. C22⋊(C83⋊C8)C22semidirect
65. C2×Q8×C83D4semidirect
71. C2×D4×C83D4semidirect
146. C83#3005semidirect
147. C22×#41D83semidirect
193. C2×Q8C22⋊D83semidirect
202. C2×D4C22⋊D83semidirect

Maximal subgroups

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

subgroupindex
C2×#145818232
C2×#145818182
#191282122
#191278632
#191278462
#191277102
#191277062
#191276652
C22×#41×C832
#300583