Finite Group Databasebeta

← #24917091

Subgroups — #24917091

Order 10624.

496 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 496 conjugacy classes of subgroups — too many to draw. Only its 135 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×M16×D83C211yesmaximal, characteristic, complemented
3#19128213C211yesmaximal, characteristic
4C2×#14581768C211yesmaximal, characteristic, complemented
5#19127958C212yesmaximal, complemented
6#19127950C212yesmaximal, complemented
7#19127941C212yesmaximal, complemented
8#19127933C212yesmaximal, complemented
9#19127744C211yesmaximal, characteristic, complemented
10#19127703C211yesmaximal, characteristic, complemented
11#419×C83C211yesmaximal, characteristic, complemented
12C2×#13050590C2211yescharacteristic
13C22×C4×D83C2211yescharacteristic
14#14581823—24no—
15C2×#13050580C2212yescomplemented
16C2×#13050577C2212yescomplemented
17C2×#13050574C2211yescharacteristic
18C2×#13050573C2212yes—
19M16×D83—24no—
20C2×C8×D83C2212yes—
21#14581776—21no—
22#14581772C2211yescharacteristic
23#14581772C2211yescharacteristic
24#14581768—21no—
25#14581768—21no—
26#14581768C2212yes—
27#14581768C2212yes—
28#14581767C2211yescharacteristic
29#14581767C2211yescharacteristic
30#14581762—21no—
31C4×D332—21no—
32C4×D332—21no—
33C2×C4⋊Dic83C2211yescharacteristic
34C2×Dic83⋊C4—21no—
35C2×(C2×D83)⋊C4—21no—
36#14581725C2212yescomplemented
37#14581723C2212yescomplemented
38#14581717C2212yescomplemented
39#14581714C2212yescomplemented
40C22×Q8×C83C2211yescharacteristic
41C2×M16×C83C2211yescharacteristic
42#117×C83C2211yescharacteristic
43#103×C83C2212yescomplemented
44#103×C83C2212yescomplemented
45C23×D83D411yescharacteristic
46#13050590—22no—
47#13050590—22no—
48C2×Dic166C2311yescharacteristic
49C2×D332—21no—
50C2×D332—21no—
51C2×D332C2311yescharacteristic
52C2×C22⋊D83—21no—
53C22×Dic83—21no—
54C22×Dic83D411yescharacteristic
55C2×C4×D83—22no—
56C2×C4×D83—22no—
57C2×C4×D83—21no—
58C2×C4×D83D411yescharacteristic
59C2×C4×D83D411yescharacteristic
60C2×C4×D83C2311yescharacteristic
61C2×C4×D83C2311yescharacteristic
62#13050580—24no—
63#13050577—24no—
64#13050574D414yescomplemented
65#13050573—24no—
66C2×C83⋊C8C2312yes—
67C8×D83—24no—
68C4⋊Dic83—21no—
69C4⋊Dic83—21no—
70C4⋊Dic83—21no—
71C4⋊Dic83C2312yes—
72Dic83⋊C4—22no—
73Dic83⋊C4—22no—
74(C2×D83)⋊C4—21no—
75(C2×D83)⋊C4—21no—
76(C2×D83)⋊C4—22no—
77(C2×D83)⋊C4—22no—
78(C2×D83)⋊C4—21no—
79(C2×D83)⋊C4—21no—
80C2×Q8×C83—21no—
81C2×Q8×C83—22no—
82C2×Q8×C83—22no—
83C2×Q8×C83C2312yes—
84C22×C4×C83—21no—
85C22×C4×C83C2311yescharacteristic
86M16×C83—22no—
87C2×C8×C83C2312yes—
88C4⋊C4×C83C2311yescharacteristic
89C4⋊C4×C83C2311yescharacteristic
90C22⋊C4×C83—21no—
91C42×C83—21no—
92C23×C83C2×D411yescharacteristic
93C22×D83—22no—
94C22×D83—22no—
95C22×D83—21no—
96C22×D83—41no—
97C22×D83C2×D411yescharacteristic
98C22×D83C2×D411yescharacteristic
99Q8×C83—22no—
100Q8×C83—22no—
101Q8×C83—42no—
102C2×C4×C83—21no—
103C2×C4×C83—21no—
104C2×C4×C83—22no—
105C2×C4×C83—42no—
106C2×C4×C83C2×D411yescharacteristic
107C2×C4×C83C2×D412yes—
108C2×C4×C83#4112yes—
109C2×C4×C83C2411yescharacteristic
110C22⋊D83—42no—
111C2×Dic83—22no—
112C2×Dic83—22no—
113C2×Dic83—21no—
114C2×Dic83—42no—
115C2×Dic83C2×D411yescharacteristic
116C2×Dic83C2×D411yescharacteristic
117D332—21no—
118D332—21no—
119D332—42no—
120D332—42no—
121C4×D83—22no—
122C4×D83—22no—
123C4×D83—21no—
124C4×D83—21no—
125C4×D83—21no—
126C4×D83—21no—
127C4×D83—42no—
128Dic166—21no—
129Dic166—21no—
130C8×C83—22no—
131C83⋊C8C2×D414yes—
132C22×C83—21no—
133C22×C83—21no—
134C22×C83#12112yes—
135C22×C83C22×D411yescharacteristic
136C2×D83—21no—
137C2×D83—21no—
138C2×D83—21no—
139C2×D83—21no—
140C2×D83—21no—
141C2×D83—44no—
142C2×D83—41no—
143C2×D83—41no—
144C2×D83—41no—
145C2×D83#13712yescomplemented
146C4×C83—42no—
147C4×C83—41no—
148C4×C83—41no—
149C4×C83#12112yes—
150C4×C83C22×D411yescharacteristic
151C4×C83C2×#4111yescharacteristic
152Dic83—21no—
153Dic83—21no—
154Dic83—41no—
155Dic83—42no—
156C2×C83—22no—
157C2×C83C2×#12111yescharacteristic
158C2×C83C2×#13711yescharacteristic
159C2×C83#57511yescharacteristic
160D83—22no—
161D83—41no—
162D83—81no—
163#3056—831nomaximal, Sylow
164C83#305611yescharacteristic, complemented, Sylow
165C2×#137—831no—
166C22×M16—831no—
167#525—831no—
168C2×C22⋊Q8—831no—
169#470—832no—
170#468—832no—
171#467—832no—
172#465—832no—
173#428—831no—
174#419D8311yescharacteristic, complemented
175#417—831no—
176C2×#41—831no—
177C22×Q8C2×D8311yescharacteristic
178C23×C4—831no—
179#137—1664no—
180C2×Q16—832no—
181C2×SD16—832no—
182C2×M16—831no—
183C2×M16—832no—
184C2×M16—1664no—
185C2×M16C2×D8311yescharacteristic
186C22×C8—832no—
187#123—1661no—
188C22⋊Q8—832no—
189C22⋊Q8—832no—
190C22⋊Q8—831no—
191C22⋊Q8—831no—
192C22⋊Q8—1661no—
193C22⋊Q8—1661no—
194#121—831no—
195#121—831no—
196#120—1661no—
197C4×D4—1661no—
198C4×D4—1661no—
199#117C2×D8311yescharacteristic
200C2×C4⋊C4—831no—
201C2×C4⋊C4—1661no—
202C2×C22⋊C4—1661no—
203#107—832no—
204#106—832no—
205#103—832no—
206#103C2×D8312yescomplemented
207#103C2×D8312yescomplemented
208#102—832no—
209C24—831no—
210#41—1662no—
211#41—1662no—
212C2×Q8—21no—
213C2×Q8—22no—
214C2×Q8—22no—
215C2×Q8—831no—
216C2×Q8C22×D8312yes—
217C2×D4—831no—
218C2×D4—1661no—
219C2×D4—1661no—
220C2×D4—1661no—
221C22×C4—21no—
222C22×C4—831no—
223C22×C4—831no—
224C22×C4—831no—
225C22×C4—831no—
226C22×C4—831no—
227C22×C4—1661no—
228C22×C4—1661no—
229C22×C4—1662no—
230C22×C4—1662no—
231C22×C4C22×D8311yescharacteristic
232Q16—1664no—
233SD16—1664no—
234M16—22no—
235M16—834no—
236M16—1664no—
237C2×C8—832no—
238C2×C8—1664no—
239C2×C8C22×D8312yes—
240C4⋊C4—832no—
241C4⋊C4—1662no—
242C4⋊C4—1662no—
243C4⋊C4—1661no—
244C4⋊C4—1661no—
245C4⋊C4—1661no—
246C4⋊C4C22×D8311yescharacteristic
247C4⋊C4C22×D8311yescharacteristic
248C22⋊C4—21no—
249C22⋊C4—1661no—
250C22⋊C4—1661no—
251C22⋊C4—1662no—
252C22⋊C4—1662no—
253C22⋊C4—1661no—
254C22⋊C4—1661no—
255C42—21no—
256C23—3321no—
257C23—831no—
258C23—831no—
259C23—1662no—
260C23—1661no—
261C23—1662no—
262C23D4×D8311yescharacteristic
263Q8—22no—
264Q8—22no—
265Q8—42no—
266Q8—1661no—
267Q8—1661no—
268D4—3322no—
269D4—3322no—
270D4—3322no—
271D4—1661no—
272D4—1661no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—22no—
276C2×C4—42no—
277C2×C4—3322no—
278C2×C4—3322no—
279C2×C4—831no—
280C2×C4—831no—
281C2×C4—1662no—
282C2×C4—1661no—
283C2×C4—1662no—
284C2×C4—1662no—
285C2×C4—1662no—
286C2×C4—1661no—
287C2×C4—1661no—
288C2×C4—1661no—
289C2×C4—1661no—
290C2×C4D4×D8311yescharacteristic
291C2×C4D4×D8312yes—
292C2×C4#1305059012yes—
293C2×C4C23×D8311yescharacteristic
294C8—22no—
295C8—834no—
296C22—21no—
297C22—21no—
298C22—3324no—
299C22—3321no—
300C22—3321no—
301C22—3321no—
302C22—832no—
303C22—1661no—
304C22—1661no—
305C22—1661no—
306C22—1661no—
307C22—1661no—
308C22#1458176512yes—
309C22C2×D4×D8311yescharacteristic
310C4—42no—
311C4—41no—
312C4—41no—
313C4—3321no—
314C4—3322no—
315C4—1661no—
316C4—1661no—
317C4#1458176512yes—
318C4C2×D4×D8311yescharacteristic
319C4C2×#1305059011yescharacteristic
320C2—22no—
321C2—3321no—
322C2—6641no—
323C2—1662no—
324C2C2×#1458176511yescharacteristic
325C2#137×D8311yescharacteristic
326C2#1912850011yescharacteristic
subgroupG/H
centerC22C2×D4×D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting#419×C83C2
socleC22×C83C22×D4
prime radical#19128213C2
hypercenter#419D83
hypocenterC83#3056
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C22×C4 ◁ #419

Complements

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

NH ≅ G/Ntype
1. C2×#14581823C2semidirect
2. C2×M16×D83C2semidirect
4. C2×#14581768C2semidirect
5. #19127958C2semidirect
6. #19127950C2semidirect
7. #19127941C2semidirect
8. #19127933C2semidirect
9. #19127744C2semidirect
10. #19127703C2semidirect
11. #419×C83C2semidirect
15. C2×#13050580C22semidirect
16. C2×#13050577C22semidirect
36. #14581725C22semidirect
37. #14581723C22semidirect
38. #14581717C22semidirect
39. #14581714C22semidirect
43. #103×C83C22semidirect
44. #103×C83C22semidirect
64. #13050574D4semidirect
145. C2×D83#137semidirect
164. C83#3056semidirect
174. #419D83semidirect
206. #103C2×D83semidirect
207. #103C2×D83semidirect

Maximal subgroups

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

subgroupindex
C2×#145818232
C2×M16×D832
#191282132
C2×#145817682
#191279582
#191279502
#191279412
#191279332
#191277442
#191277032
#419×C832
#305683