Finite Group Databasebeta

← #24918280

Subgroups — #24918280

Order 10624.

466 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 466 conjugacy classes of subgroups — too many to draw. Only its 113 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, complemented
2C2×#14581763C211yesmaximal, characteristic, complemented
3#19128029C211yesmaximal, characteristic, complemented
4#19128023C212yesmaximal, complemented
5#19128015C211yesmaximal, characteristic, complemented
6#19127953C212yesmaximal, complemented
7#19127944C212yesmaximal, complemented
8#19127845C212yesmaximal, complemented
9#19127835C211yesmaximal, characteristic, complemented
10#19127663C211yesmaximal, characteristic, complemented
11#450×C83C211yesmaximal, characteristic, complemented
12C2×#13050591C2211yescharacteristic, complemented
13C22×C22⋊D83—21no—
14C22×C4×D83C2211yescharacteristic
15C2×#13050575C2212yescomplemented
16C2×#13050573C2212yescomplemented
17#14581777—21no—
18#14581768C2211yescharacteristic, complemented
19#14581767C2211yescharacteristic, complemented
20#14581766C2211yescharacteristic
21#14581763—21no—
22#14581763—21no—
23#14581763C2212yes—
24#14581763C2212yes—
25#14581762—21no—
26#14581754—21no—
27(C22⋊D83)⋊C4—21no—
28#14581750C2211yescharacteristic, complemented
29C2×C4⋊Dic83C2211yescharacteristic
30C2×C22⋊Dic83—21no—
31#14581726C2212yescomplemented
32#14581723C2212yescomplemented
33#14581716C2212yescomplemented
34#14581714C2212yescomplemented
35#14581713C2212yescomplemented
36C22⋊(C83⋊C8)C2211yescharacteristic
37(C2×D83)⋊C8C2212yescomplemented
38C22×D4×C83C2211yescharacteristic
39C2×SD16×C83C2212yescomplemented
40C22⋊Q8×C83C2211yescharacteristic
41#102×C83C2212yescomplemented
42C22⋊C8×C83C2211yescharacteristic
43C23×D83D411yescharacteristic
44#13050591—42no—
45C2×D332—21no—
46C2×D332—21no—
47C2×D332C2311yescharacteristic
48C2×C22⋊D83—21no—
49C2×C22⋊D83—22no—
50C2×C22⋊D83—22no—
51C2×C22⋊D83—44no—
52C22×Dic83—21no—
53C22×Dic83D411yescharacteristic
54C2×C4×D83—21no—
55C2×C4×D83—42no—
56C2×C4×D83D411yescharacteristic
57C2×C4×D83D411yescharacteristic
58C2×C4×D83C2311yescharacteristic
59C2×C4×D83C2311yescharacteristic
60#13050575—42no—
61#13050573—42no—
62C2×C83⋊C8C2312yes—
63C4⋊Dic83—21no—
64C4⋊Dic83C2312yes—
65C4⋊Dic83C2311yescharacteristic
66Dic83⋊C4—21no—
67C22⋊Dic83—22no—
68C22⋊Dic83—22no—
69(C2×D83)⋊C4—21no—
70(C2×D83)⋊C4—21no—
71(C2×D83)⋊C4—21no—
72(C2×D83)⋊C4—21no—
73C4×Dic83—21no—
74C24×C83—21no—
75C2×Q8×C83C2311yescharacteristic
76C2×D4×C83—21no—
77C2×D4×C83—42no—
78C2×D4×C83C2312yes—
79C22×C4×C83C2311yescharacteristic
80SD16×C83—42no—
81C2×C8×C83C2312yes—
82C4⋊C4×C83—21no—
83C4⋊C4×C83C2311yescharacteristic
84C22⋊C4×C83—21no—
85C23×C83—22no—
86C23×C83—42no—
87C23×C83—42no—
88C23×C83—42no—
89C23×C83C2×D411yescharacteristic
90C22×D83—22no—
91C22×D83—21no—
92C22×D83—41no—
93C22×D83—41no—
94C22×D83C2×D411yescharacteristic
95C22×D83C2×D411yescharacteristic
96Q8×C83—22no—
97D4×C83—22no—
98D4×C83—22no—
99D4×C83—42no—
100C2×C4×C83—22no—
101C2×C4×C83—21no—
102C2×C4×C83—21no—
103C2×C4×C83C2×D411yescharacteristic
104C2×C4×C83C2411yescharacteristic
105C22⋊D83—44no—
106C22⋊D83—44no—
107C22⋊D83—42no—
108C2×Dic83—21no—
109C2×Dic83—22no—
110C2×Dic83—41no—
111C2×Dic83—42no—
112C2×Dic83C2×D411yescharacteristic
113C2×Dic83C2×D411yescharacteristic
114D332—22no—
115D332—42no—
116D332—42no—
117C4×D83—22no—
118C4×D83—22no—
119C4×D83—41no—
120C4×D83—41no—
121C4×D83—42no—
122C8×C83—22no—
123C83⋊C8—22no—
124C22×C83—21no—
125C22×C83—21no—
126C22×C83—42no—
127C22×C83—42no—
128C22×C83—42no—
129C22×C83—84no—
130C22×C83AGL(1,ℤ/8)12yes—
131C22×C83C22×D411yescharacteristic
132C2×D83—21no—
133C2×D83—21no—
134C2×D83—21no—
135C2×D83—21no—
136C2×D83—21no—
137C2×D83—21no—
138C2×D83—44no—
139C2×D83—41no—
140C2×D83—41no—
141C2×D83—41no—
142C4×C83—21no—
143C4×C83—41no—
144C4×C83—41no—
145C4×C83H(5,𝔽2)12yes—
146Dic83—21no—
147Dic83—21no—
148Dic83—42no—
149Dic83—41no—
150C2×C83—22no—
151C2×C83—82no—
152C2×C83#53411yescharacteristic
153C2×C83C2×AGL(1,ℤ/8)11yescharacteristic
154C2×C83#57511yescharacteristic
155D83—41no—
156D83—41no—
157D83—81no—
158#3194—831nomaximal, Sylow
159C83#319411yescharacteristic, complemented, Sylow
160#525—831no—
161C2×#121—831no—
162#483—831no—
163#482—832no—
164#480—831no—
165#469—832no—
166#468—832no—
167#450D8311yescharacteristic, complemented
168#449—832no—
169#447—831no—
170#407—831no—
171C2×#41—831no—
172C22×D4—1661no—
173C22×D4C2×D8311yescharacteristic
174C23×C4—831no—
175C2×SD16C2×D8312yescomplemented
176C2×D8—832no—
177C2×M16—832no—
178#123—1661no—
179C22⋊Q8—831no—
180C22⋊Q8C2×D8311yescharacteristic
181#121—832no—
182#121—832no—
183#121—831no—
184#121—831no—
185#121—1661no—
186#121—1661no—
187#120—1661no—
188C4×D4—1661no—
189C4×D4—1661no—
190#117—831no—
191C2×C4⋊C4—831no—
192C2×C22⋊C4—1661no—
193#107—832no—
194#106—832no—
195#103—832no—
196#102—832no—
197#102—832no—
198#102C2×D8312yescomplemented
199C22⋊C8—831no—
200C22⋊C8—832no—
201C22⋊C8C2×D8311yescharacteristic
202C24—21no—
203C24—831no—
204#41—3322no—
205C2×Q8C22×D8311yescharacteristic
206C2×D4—21no—
207C2×D4—42no—
208C2×D4—3324no—
209C2×D4—831no—
210C2×D4—1661no—
211C2×D4—1661no—
212C2×D4—1662no—
213C2×D4—1662no—
214C2×D4—1661no—
215C2×D4C22×D8312yes—
216C22×C4—3322no—
217C22×C4—831no—
218C22×C4—831no—
219C22×C4—831no—
220C22×C4—831no—
221C22×C4—831no—
222C22×C4—1661no—
223C22×C4—1661no—
224C22×C4C22×D8311yescharacteristic
225SD16—42no—
226D8—3322no—
227M16—3322no—
228C2×C8—832no—
229C2×C8C22×D8312yes—
230C4⋊C4—21no—
231C4⋊C4—831no—
232C4⋊C4—832no—
233C4⋊C4—1661no—
234C4⋊C4—1661no—
235C4⋊C4C22×D8311yescharacteristic
236C22⋊C4—21no—
237C22⋊C4—1661no—
238C22⋊C4—1661no—
239C22⋊C4—1661no—
240C22⋊C4—1662no—
241C22⋊C4—1662no—
242C22⋊C4—1661no—
243C42—1661no—
244C23—22no—
245C23—42no—
246C23—42no—
247C23—42no—
248C23—3321no—
249C23—3321no—
250C23—831no—
251C23—831no—
252C23—1661no—
253C23—1662no—
254C23D4×D8311yescharacteristic
255Q8—22no—
256D4—22no—
257D4—22no—
258D4—42no—
259D4—3322no—
260D4—3322no—
261D4—3324no—
262D4—3324no—
263D4—3322no—
264D4—1662no—
265C2×C4—22no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—3321no—
269C2×C4—3321no—
270C2×C4—3322no—
271C2×C4—3322no—
272C2×C4—3321no—
273C2×C4—831no—
274C2×C4—831no—
275C2×C4—1661no—
276C2×C4—1662no—
277C2×C4—1662no—
278C2×C4—1662no—
279C2×C4D4×D8311yescharacteristic
280C2×C4C23×D8311yescharacteristic
281C8—22no—
282C8—1662no—
283C22—21no—
284C22—21no—
285C22—42no—
286C22—42no—
287C22—42no—
288C22—84no—
289C22—3324no—
290C22—3321no—
291C22—3321no—
292C22—3321no—
293C22—1661no—
294C22—1661no—
295C22—1661no—
296C22—1661no—
297C22—1661no—
298C22—1661no—
299C22#1458181912yes—
300C22C2×D4×D8311yescharacteristic
301C4—21no—
302C4—41no—
303C4—41no—
304C4—3322no—
305C4—3321no—
306C4—1661no—
307C4—1661no—
308C4#1458183712yes—
309C2—22no—
310C2—82no—
311C2—3321no—
312C2—3321no—
313C2—6641no—
314C2#1912826111yescharacteristic
315C2C2×#1458181911yescharacteristic
316C2#1912849711yescharacteristic
subgroupG/H
centerC22C2×D4×D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting#450×C83C2
socleC22×C83C22×D4
hypercenter#450D83
hypocenterC83#3194
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C22×C4 ◁ #450

Complements

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

NH ≅ G/Ntype
1. #19128214C2semidirect
2. C2×#14581763C2semidirect
3. #19128029C2semidirect
4. #19128023C2semidirect
5. #19128015C2semidirect
6. #19127953C2semidirect
7. #19127944C2semidirect
8. #19127845C2semidirect
9. #19127835C2semidirect
10. #19127663C2semidirect
11. #450×C83C2semidirect
12. C2×#13050591C22semidirect
15. C2×#13050575C22semidirect
16. C2×#13050573C22semidirect
18. #14581768C22semidirect
19. #14581767C22semidirect
28. #14581750C22semidirect
31. #14581726C22semidirect
32. #14581723C22semidirect
33. #14581716C22semidirect
34. #14581714C22semidirect
35. #14581713C22semidirect
37. (C2×D83)⋊C8C22semidirect
39. C2×SD16×C83C22semidirect
41. #102×C83C22semidirect
159. C83#3194semidirect
167. #450D83semidirect
175. C2×SD16C2×D83semidirect
198. #102C2×D83semidirect

Maximal subgroups

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

subgroupindex
#191282142
C2×#145817632
#191280292
#191280232
#191280152
#191279532
#191279442
#191278452
#191278352
#191276632
#450×C832
#319483