Finite Group Databasebeta

← #24916721

Subgroups — #24916721

Order 10624.

754 conjugacy classes of subgroups; only the 153 normal subgroups are shown.

normal subgroups only

This group has 754 conjugacy classes of subgroups — too many to draw. Only its 153 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
1C22×#13050590C211yesmaximal, characteristic
2C2×#14581823C212yesmaximal, complemented
3#19127861C214yesmaximal, complemented
4#19127856C212yesmaximal, complemented
5#19127851C212yesmaximal, complemented
6#19127700C212yesmaximal, complemented
7#19127665C211yesmaximal, characteristic, complemented
8C2×C22⋊Q8×C83C211yesmaximal, characteristic, complemented
9C2×#13050590—28no—
10C2×#13050590—21no—
11C2×#13050590—21no—
12C2×#13050590C2212yes—
13C2×#13050590C2212yes—
14C22×Dic166C2211yescharacteristic
15C22×D332C2211yescharacteristic
16C22×C22⋊D83—21no—
17C22×C4×D83—21no—
18#14581823—28no—
19C2×#13050580C2214yescomplemented
20C2×#13050577C2214yescomplemented
21C2×#13050574C2212yescomplemented
22#14581715C2214yescomplemented
23#14581712C2214yescomplemented
24C22⋊(C83⋊C8)C2212yes—
25C22⋊(C83⋊C8)C2212yescomplemented
26C22×Q8×C83C2211yescharacteristic, complemented
27C23×C4×C83C2211yescharacteristic
28C22⋊Q8×C83—21no—
29C22⋊Q8×C83—21no—
30C22⋊Q8×C83C2212yes—
31C22⋊Q8×C83C2212yescomplemented
32C2×C4⋊C4×C83—21no—
33C2×C4⋊C4×C83C2211yescharacteristic, complemented
34C2×C22⋊C4×C83—21no—
35C23×D83—21no—
36#13050590—28no—
37#13050590—28no—
38#13050590—24no—
39#13050590—24no—
40#13050590—24no—
41#13050590D418yes—
42C2×Dic166—21no—
43C2×Dic166—24no—
44C2×Dic166C2312yes—
45C2×D332—21no—
46C2×D332—24no—
47C2×D332C2312yes—
48C2×C22⋊D83—28no—
49C2×C22⋊D83—22no—
50C2×C22⋊D83—22no—
51C22×Dic83—21no—
52C2×C4×D83—28no—
53C2×C4×D83—22no—
54C2×C4×D83—22no—
55#13050580—28no—
56#13050577—28no—
57#13050574—24no—
58C2×C83⋊C8C2314yes—
59C24×C83D411yescharacteristic
60C2×Q8×C83—21no—
61C2×Q8×C83—24no—
62C2×Q8×C83C2312yes—
63C22×C4×C83—24no—
64C22×C4×C83—21no—
65C22×C4×C83—21no—
66C22×C4×C83D411yescharacteristic
67C22×C4×C83D411yescharacteristic
68C22×C4×C83D411yescharacteristic
69C22×C4×C83C2311yescharacteristic
70C22×C4×C83C2311yescharacteristic
71C22×C4×C83C2311yescharacteristic
72C4⋊C4×C83—22no—
73C4⋊C4×C83—22no—
74C4⋊C4×C83—21no—
75C4⋊C4×C83C2312yes—
76C22⋊C4×C83—22no—
77C22⋊C4×C83—22no—
78C23×C83—22no—
79C23×C83—22no—
80C23×C83—41no—
81C23×C83C2×D411yescharacteristic
82C23×C83C2×D411yescharacteristic
83C23×C83C2×D411yescharacteristic
84C22×D83—24no—
85C22×D83—22no—
86C22×D83—42no—
87C22×D83—42no—
88Q8×C83—24no—
89Q8×C83—42no—
90C2×C4×C83—22no—
91C2×C4×C83—22no—
92C2×C4×C83—22no—
93C2×C4×C83—22no—
94C2×C4×C83—22no—
95C2×C4×C83—22no—
96C2×C4×C83—22no—
97C2×C4×C83—42no—
98C2×C4×C83—42no—
99C2×C4×C83C2×D414yes—
100C2×C4×C83C2×D411yescharacteristic
101C2×C4×C83C2×D411yescharacteristic
102C2×C4×C83C2×D411yescharacteristic
103C2×C4×C83C2411yescharacteristic
104C22⋊D83—28no—
105C22⋊D83—28no—
106C2×Dic83—24no—
107C2×Dic83—22no—
108D332—24no—
109D332—22no—
110D332C2×D414yes—
111C4×D83—28no—
112C4×D83—28no—
113Dic166—24no—
114Dic166—22no—
115Dic166C2×D414yes—
116C83⋊C8—24no—
117C22×C83—21no—
118C22×C83—21no—
119C22×C83—21no—
120C22×C83—21no—
121C22×C83—21no—
122C22×C83—21no—
123C22×C83—21no—
124C22×C83—44no—
125C22×C83#12012yes—
126C22×C83#13712yescomplemented
127C22×C83C22×D411yescharacteristic
128C2×D83—24no—
129C2×D83—48no—
130C2×D83—42no—
131C2×D83—42no—
132C4×C83—21no—
133C4×C83—21no—
134C4×C83—42no—
135C4×C83—42no—
136C4×C83#12012yes—
137C4×C83C22×D412yes—
138Dic83—24no—
139C2×C83—22no—
140C2×C83—22no—
141C2×C83—41no—
142C2×C83C2×#12011yescharacteristic
143C2×C83C2×#13711yescharacteristic
144C2×C83#57511yescharacteristic
145D83—44no—
146#3007—831nomaximal, Sylow
147C83#300711yescharacteristic, complemented, Sylow
148C22×#41—831no—
149C2×#137—832no—
150C2×C22⋊Q8D8311yescharacteristic, complemented
151#452—834no—
152#451—832no—
153#450—832no—
154#417—832no—
155#407—831no—
156C2×#41—832no—
157C2×#41—832no—
158C2×#41—1668no—
159C2×#41—1661no—
160C2×#41—1661no—
161C22×Q8—831no—
162C22×Q8C2×D8311yescharacteristic, complemented
163C22×D4—831no—
164C22×D4—1661no—
165C23×C4—1661no—
166C23×C4C2×D8311yescharacteristic
167#137—1668no—
168C2×Q16—834no—
169C2×SD16—834no—
170C2×M16—832no—
171C22⋊Q8—21no—
172C22⋊Q8—21no—
173C22⋊Q8C2×D8312yes—
174C22⋊Q8C2×D8312yescomplemented
175C2×C4⋊C4—21no—
176C2×C4⋊C4C2×D8311yescharacteristic, complemented
177C2×C22⋊C4—21no—
178#103—834no—
179#102—834no—
180C22⋊C8—832no—
181C22⋊C8—832no—
182C24—1661no—
183C24C22⋊D8311yescharacteristic
184#41—838no—
185#41—1664no—
186#41—1664no—
187#41—1668no—
188#41—1668no—
189#41—1664no—
190C2×Q8—21no—
191C2×Q8—24no—
192C2×Q8—832no—
193C2×Q8—1664no—
194C2×Q8—1661no—
195C2×Q8C22×D8312yes—
196C2×D4—832no—
197C2×D4—1664no—
198C2×D4—1668no—
199C2×D4—1662no—
200C2×D4—1662no—
201C2×D4—1661no—
202C22×C4—24no—
203C22×C4—21no—
204C22×C4—21no—
205C22×C4—1668no—
206C22×C4—1661no—
207C22×C4—1662no—
208C22×C4—1662no—
209C22×C4C22⋊D8311yescharacteristic
210C22×C4C22⋊D8311yescharacteristic
211C22×C4C22⋊D8311yescharacteristic
212C22×C4C22×D8311yescharacteristic
213C22×C4C22×D8311yescharacteristic
214C22×C4C22×D8311yescharacteristic
215Q16—1668no—
216SD16—1668no—
217M16—1664no—
218C2×C8—834no—
219C4⋊C4—22no—
220C4⋊C4—22no—
221C4⋊C4—21no—
222C4⋊C4C22×D8312yes—
223C22⋊C4—22no—
224C22⋊C4—22no—
225C23—22no—
226C23—22no—
227C23—41no—
228C23—3322no—
229C23—3322no—
230C23—1664no—
231C23—1662no—
232C23C2×C22⋊D8311yescharacteristic
233C23C2×C22⋊D8311yescharacteristic
234C23C2×C22⋊D8311yescharacteristic
235Q8—24no—
236Q8—42no—
237Q8—834no—
238Q8—1664no—
239Q8—1662no—
240D4—834no—
241D4—1664no—
242D4—1668no—
243D4—1668no—
244D4—1662no—
245C2×C4—22no—
246C2×C4—22no—
247C2×C4—22no—
248C2×C4—22no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—42no—
253C2×C4—42no—
254C2×C4—1668no—
255C2×C4—1664no—
256C2×C4—1662no—
257C2×C4—1668no—
258C2×C4D4×D8314yes—
259C2×C4C2×C22⋊D8311yescharacteristic
260C2×C4C2×C22⋊D8311yescharacteristic
261C2×C4C2×C22⋊D8311yescharacteristic
262C2×C4C23×D8311yescharacteristic
263C8—1664no—
264C22—21no—
265C22—21no—
266C22—21no—
267C22—21no—
268C22—21no—
269C22—21no—
270C22—21no—
271C22—44no—
272C22—3328no—
273C22—3322no—
274C22—3322no—
275C22—1664no—
276C22#1458176012yes—
277C22#1458182312yescomplemented
278C22C22×C22⋊D8311yescharacteristic
279C4—21no—
280C4—21no—
281C4—42no—
282C4—42no—
283C4—1664no—
284C4#1458176012yes—
285C4C2×D4×D8312yes—
286C2—22no—
287C2—22no—
288C2—41no—
289C2—3324no—
290C2C2×#1458176011yescharacteristic
291C2C2×#1458182311yescharacteristic
292C2#1912849911yescharacteristic
subgroupG/H
centerC22C22×C22⋊D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
FittingC2×C22⋊Q8×C83C2
socleC22×C83C22×D4
prime radicalC22×#13050590C2
hypercenterC2×C22⋊Q8D83
hypocenterC83#3007
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×C22⋊Q8

Complements

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

NH ≅ G/Ntype
2. C2×#14581823C2semidirect
3. #19127861C2semidirect
4. #19127856C2semidirect
5. #19127851C2semidirect
6. #19127700C2semidirect
7. #19127665C2semidirect
8. C2×C22⋊Q8×C83C2semidirect
19. C2×#13050580C22semidirect
20. C2×#13050577C22semidirect
21. C2×#13050574C22semidirect
22. #14581715C22semidirect
23. #14581712C22semidirect
25. C22⋊(C83⋊C8)C22semidirect
26. C22×Q8×C83C22semidirect
31. C22⋊Q8×C83C22semidirect
33. C2×C4⋊C4×C83C22semidirect
126. C22×C83#137semidirect
147. C83#3007semidirect
150. C2×C22⋊Q8D83semidirect
162. C22×Q8C2×D83semidirect
174. C22⋊Q8C2×D83semidirect
176. C2×C4⋊C4C2×D83semidirect
277. C22#14581823semidirect

Maximal subgroups

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

subgroupindex
C22×#130505902
C2×#145818232
#191278612
#191278562
#191278512
#191277002
#191276652
C2×C22⋊Q8×C832
#300783