Finite Group Databasebeta

← #24917390

Subgroups — #24917390

Order 10624.

458 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 458 conjugacy classes of subgroups — too many to draw. Only its 123 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#19128214C211yesmaximal, characteristic
2C2×C4⋊C4×D83C211yesmaximal, characteristic, complemented
3#19128034C211yesmaximal, characteristic, complemented
4#19128030C212yesmaximal, complemented
5#19128020C211yesmaximal, characteristic, complemented
6#19128019C211yesmaximal, characteristic, complemented
7#19128016C212yesmaximal, complemented
8#19127742C212yesmaximal, complemented
9#19127701C212yesmaximal, complemented
10#19127663C211yesmaximal, characteristic, complemented
11#484×C83C211yesmaximal, characteristic, complemented
12C2×#13050591C2211yescharacteristic
13C22×C4×D83—21no—
14C22×C4×D83C2211yescharacteristic
15C2×#13050573C2212yes—
16#14581777—21no—
17#14581768C2211yescharacteristic
18#14581767C2211yescharacteristic
19#14581766C2211yescharacteristic
20#14581762—21no—
21#14581754—21no—
22(C22⋊D83)⋊C4—21no—
23#14581750C2211yescharacteristic
24C2×C4⋊Dic83C2211yescharacteristic
25C2×Dic83⋊C4—21no—
26C4⋊C4×D83—21no—
27C4⋊C4×D83—21no—
28C4⋊C4×D83C2212yes—
29C4⋊C4×D83C2212yes—
30#14581725C2212yescomplemented
31#14581724C2212yescomplemented
32#14581723C2212yescomplemented
33#14581716C2212yescomplemented
34#14581714C2212yescomplemented
35#14581712C2212yescomplemented
36C22⋊(C83⋊C8)C2211yescharacteristic
37(C2×D83)⋊C8C2212yes—
38C22⋊Q8×C83C2211yescharacteristic
39C2×C4⋊C4×C83C2211yescharacteristic
40#107×C83C2212yescomplemented
41#103×C83C2212yescomplemented
42C22⋊C8×C83C2211yescharacteristic
43C23×D83D411yescharacteristic
44#13050591—24no—
45C2×D332—21no—
46C2×D332—21no—
47C2×D332C2311yescharacteristic
48C2×C22⋊D83—21no—
49C22×Dic83—21no—
50C22×Dic83D411yescharacteristic
51C2×C4×D83—24no—
52C2×C4×D83—22no—
53C2×C4×D83—22no—
54C2×C4×D83—21no—
55C2×C4×D83—44no—
56C2×C4×D83D411yescharacteristic
57C2×C4×D83D411yescharacteristic
58C2×C4×D83C2311yescharacteristic
59C2×C4×D83C2311yescharacteristic
60#13050573—24no—
61C2×C83⋊C8C2312yes—
62C4⋊Dic83—21no—
63C4⋊Dic83C2311yescharacteristic
64C4⋊Dic83C2312yes—
65Dic83⋊C4—21no—
66Dic83⋊C4—22no—
67Dic83⋊C4—22no—
68(C2×D83)⋊C4—21no—
69(C2×D83)⋊C4—21no—
70(C2×D83)⋊C4—21no—
71(C2×D83)⋊C4—21no—
72C4×Dic83—21no—
73C2×Q8×C83C2311yescharacteristic
74C22×C4×C83—21no—
75C22×C4×C83C2311yescharacteristic
76C2×C8×C83C2312yes—
77C4⋊C4×C83—21no—
78C4⋊C4×C83—21no—
79C4⋊C4×C83C2311yescharacteristic
80C4⋊C4×C83C2312yes—
81C22⋊C4×C83—21no—
82C23×C83C2×D411yescharacteristic
83C22×D83—22no—
84C22×D83—22no—
85C22×D83—22no—
86C22×D83—21no—
87C22×D83C2×D411yescharacteristic
88C22×D83C2×D411yescharacteristic
89Q8×C83—22no—
90C2×C4×C83—22no—
91C2×C4×C83—21no—
92C2×C4×C83—21no—
93C2×C4×C83—22no—
94C2×C4×C83—42no—
95C2×C4×C83C2×D411yescharacteristic
96C2×C4×C83C2411yescharacteristic
97C22⋊D83—42no—
98C2×Dic83—22no—
99C2×Dic83—21no—
100C2×Dic83—22no—
101C2×Dic83—42no—
102C2×Dic83C2×D411yescharacteristic
103C2×Dic83C2×D411yescharacteristic
104D332—22no—
105D332—42no—
106D332—42no—
107C4×D83—22no—
108C4×D83—22no—
109C4×D83—22no—
110C4×D83—44no—
111C4×D83—44no—
112C4×D83—42no—
113C4×D83#4114yes—
114C8×C83—22no—
115C83⋊C8—22no—
116C22×C83—21no—
117C22×C83—21no—
118C22×C83#13712yes—
119C22×C83C22×D411yescharacteristic
120C2×D83—28no—
121C2×D83—21no—
122C2×D83—21no—
123C2×D83—21no—
124C2×D83—21no—
125C2×D83—41no—
126C2×D83—41no—
127C2×D83—41no—
128C2×D83#12312yes—
129C2×D83AGL(1,ℤ/8)12yes—
130C4×C83—21no—
131C4×C83—41no—
132C4×C83—42no—
133C4×C83—41no—
134C4×C83C2×#4112yes—
135Dic83—21no—
136Dic83—42no—
137Dic83—41no—
138Dic83#12312yes—
139C2×C83—22no—
140C2×C83C2×#12311yescharacteristic
141C2×C83C2×AGL(1,ℤ/8)11yescharacteristic
142C2×C83C2×#13711yescharacteristic
143D83—22no—
144D83—22no—
145D83—81no—
146#3092—831nomaximal, Sylow
147C83#309211yescharacteristic, complemented, Sylow
148#525—831no—
149C22×C4⋊C4—831no—
150#484—831no—
151#484D8311yescharacteristic, complemented
152#483—832no—
153#481—831no—
154#481—831no—
155#480—832no—
156#428—832no—
157#417—832no—
158#407—831no—
159C2×#41—831no—
160C23×C4—831no—
161C23×C4—1661no—
162C2×M16—832no—
163#123—1661no—
164C22⋊Q8—831no—
165C22⋊Q8C2×D8311yescharacteristic
166#121—831no—
167#121—831no—
168#120—1661no—
169C4×D4—1661no—
170C4×D4—1661no—
171#117—831no—
172C2×C4⋊C4—831no—
173C2×C4⋊C4—832no—
174C2×C4⋊C4—832no—
175C2×C4⋊C4—1661no—
176C2×C4⋊C4—1661no—
177C2×C4⋊C4—1661no—
178C2×C4⋊C4C2×D8311yescharacteristic
179#107—832no—
180#107C2×D8312yescomplemented
181#106—832no—
182#106—832no—
183#103—832no—
184#103C2×D8312yescomplemented
185#102—832no—
186#102—832no—
187C22⋊C8—831no—
188C22⋊C8—832no—
189C22⋊C8C2×D8311yescharacteristic
190C24—831no—
191#41—1664no—
192C2×Q8C22×D8311yescharacteristic
193C2×D4—831no—
194C2×D4—1661no—
195C2×D4—1661no—
196C2×D4—1661no—
197C22×C4—21no—
198C22×C4—3324no—
199C22×C4—831no—
200C22×C4—831no—
201C22×C4—831no—
202C22×C4—831no—
203C22×C4—831no—
204C22×C4—1661no—
205C22×C4—1662no—
206C22×C4—1662no—
207C22×C4—1664no—
208C22×C4—1661no—
209C22×C4C22×D8311yescharacteristic
210M16—1664no—
211C2×C8—832no—
212C2×C8C22×D8312yes—
213C4⋊C4—21no—
214C4⋊C4—21no—
215C4⋊C4—832no—
216C4⋊C4—831no—
217C4⋊C4—1662no—
218C4⋊C4—1662no—
219C4⋊C4—1661no—
220C4⋊C4—1661no—
221C4⋊C4C22×D8312yes—
222C4⋊C4C22×D8311yescharacteristic
223C22⋊C4—21no—
224C22⋊C4—1661no—
225C22⋊C4—1661no—
226C22⋊C4—1661no—
227C22⋊C4—1661no—
228C42—1661no—
229C23—831no—
230C23—831no—
231C23—1662no—
232C23—1662no—
233C23—1661no—
234C23—1662no—
235C23D4×D8311yescharacteristic
236Q8—22no—
237D4—3322no—
238D4—3322no—
239D4—3322no—
240D4—1662no—
241C2×C4—22no—
242C2×C4—21no—
243C2×C4—21no—
244C2×C4—22no—
245C2×C4—42no—
246C2×C4—3324no—
247C2×C4—3324no—
248C2×C4—3322no—
249C2×C4—3322no—
250C2×C4—831no—
251C2×C4—834no—
252C2×C4—831no—
253C2×C4—1662no—
254C2×C4—1662no—
255C2×C4—1661no—
256C2×C4—1662no—
257C2×C4—1662no—
258C2×C4—1662no—
259C2×C4D4×D8311yescharacteristic
260C2×C4C23×D8311yescharacteristic
261C8—22no—
262C8—1662no—
263C22—21no—
264C22—21no—
265C22—3321no—
266C22—3321no—
267C22—3321no—
268C22—832no—
269C22—832no—
270C22—1668no—
271C22—1661no—
272C22—1661no—
273C22—1661no—
274C22—1661no—
275C22#1458182412yes—
276C22C2×D4×D8311yescharacteristic
277C4—21no—
278C4—41no—
279C4—42no—
280C4—41no—
281C4—3322no—
282C4—3321no—
283C4—832no—
284C4—1661no—
285C4#41×D8312yes—
286C2—22no—
287C2—6641no—
288C2—1662no—
289C2—1662no—
290C2#123×D8311yescharacteristic
291C2AGL(1,ℤ/8)×D8311yescharacteristic
292C2C2×#1458182411yescharacteristic
subgroupG/H
centerC22C2×D4×D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting#484×C83C2
socleC22×C83C22×D4
prime radical#19128214C2
hypercenter#484D83
hypocenterC83#3092
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C22×C4 ◁ #484

Complements

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

NH ≅ G/Ntype
2. C2×C4⋊C4×D83C2semidirect
3. #19128034C2semidirect
4. #19128030C2semidirect
5. #19128020C2semidirect
6. #19128019C2semidirect
7. #19128016C2semidirect
8. #19127742C2semidirect
9. #19127701C2semidirect
10. #19127663C2semidirect
11. #484×C83C2semidirect
30. #14581725C22semidirect
31. #14581724C22semidirect
32. #14581723C22semidirect
33. #14581716C22semidirect
34. #14581714C22semidirect
35. #14581712C22semidirect
40. #107×C83C22semidirect
41. #103×C83C22semidirect
147. C83#3092semidirect
151. #484D83semidirect
180. #107C2×D83semidirect
184. #103C2×D83semidirect

Maximal subgroups

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

subgroupindex
#191282142
C2×C4⋊C4×D832
#191280342
#191280302
#191280202
#191280192
#191280162
#191277422
#191277012
#191276632
#484×C832
#309283