Finite Group Databasebeta

← #24918267

Subgroups — #24918267

Order 10624.

462 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 462 conjugacy classes of subgroups — too many to draw. Only its 115 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×#14581768C211yesmaximal, characteristic, complemented
2#121×D83C211yesmaximal, characteristic
3#19128035C212yesmaximal, complemented
4#19128015C211yesmaximal, characteristic, complemented
5#19128012C211yesmaximal, characteristic, complemented
6#19127950C212yesmaximal, complemented
7#19127944C212yesmaximal, complemented
8#19127850C212yesmaximal, complemented
9#19127840C211yesmaximal, characteristic, complemented
10#19127663C211yesmaximal, characteristic, complemented
11#448×C83C211yesmaximal, characteristic, complemented
12C2×D4×D83—21no—
13C2×D4×D83C2211yescharacteristic
14C22×C4×D83C2211yescharacteristic
15C2×#13050577C2212yescomplemented
16C2×#13050573C2212yescomplemented
17#14581768—21no—
18#14581768—21no—
19#14581768C2212yes—
20#14581768C2212yescomplemented
21#14581767C2211yescharacteristic
22#14581766C2211yescharacteristic
23#14581765—21no—
24#14581764—21no—
25#14581763C2211yescharacteristic
26C2×C4⋊Dic83C2211yescharacteristic, complemented
27C2×Dic83⋊C4—21no—
28C4⋊C4×D83C2211yescharacteristic
29C2×(C2×D83)⋊C4—21no—
30C22⋊C4×D83—21no—
31#14581726C2212yescomplemented
32#14581725C2212yescomplemented
33#14581716C2212yescomplemented
34#14581714C2212yescomplemented
35#14581713C2212yescomplemented
36C22⋊(C83⋊C8)C2211yescharacteristic, complemented
37(C2×D83)⋊C8C2212yes—
38C22×Q8×C83C2211yescharacteristic, complemented
39C2×SD16×C83C2212yescomplemented
40#121×C83C2211yescharacteristic
41#103×C83C2212yescomplemented
42C22⋊C8×C83C2211yescharacteristic, complemented
43C23×D83—21no—
44C23×D83D411yescharacteristic
45C2×D332—21no—
46C2×D332C2311yescharacteristic
47C2×C22⋊D83—21no—
48C2×C22⋊D83—21no—
49C2×C22⋊D83—21no—
50D4×D83—42no—
51D4×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#13050577—42no—
61#13050573—42no—
62C2×C83⋊C8C2312yes—
63C4⋊Dic83—21no—
64C4⋊Dic83C2311yescharacteristic
65C4⋊Dic83C2312yes—
66Dic83⋊C4—21no—
67Dic83⋊C4—22no—
68Dic83⋊C4—22no—
69C22⋊Dic83—21no—
70(C2×D83)⋊C4—22no—
71(C2×D83)⋊C4—22no—
72(C2×D83)⋊C4—21no—
73(C2×D83)⋊C4—21no—
74C2×Q8×C83—21no—
75C2×Q8×C83—42no—
76C2×Q8×C83C2312yes—
77C2×D4×C83—21no—
78C2×D4×C83C2311yescharacteristic
79C22×C4×C83—21no—
80C22×C4×C83C2311yescharacteristic
81SD16×C83—42no—
82C2×C8×C83C2312yes—
83C4⋊C4×C83C2311yescharacteristic
84C22⋊C4×C83—21no—
85C23×C83—21no—
86C23×C83C2×D411yescharacteristic
87C22×D83—22no—
88C22×D83—22no—
89C22×D83—21no—
90C22×D83—41no—
91C22×D83—42no—
92C22×D83—42no—
93C22×D83—42no—
94C22×D83C2×D411yescharacteristic
95C22×D83C2×D411yescharacteristic
96Q8×C83—22no—
97Q8×C83—22no—
98Q8×C83—42no—
99D4×C83—22no—
100D4×C83—42no—
101C2×C4×C83—22no—
102C2×C4×C83—21no—
103C2×C4×C83—22no—
104C2×C4×C83—42no—
105C2×C4×C83C2×D411yescharacteristic
106C2×C4×C83C2411yescharacteristic
107C22⋊D83—42no—
108C22⋊D83—42no—
109C22⋊D83—42no—
110C2×Dic83—22no—
111C2×Dic83—21no—
112C2×Dic83—41no—
113C2×Dic83—42no—
114C2×Dic83C2×D411yescharacteristic
115C2×Dic83C2×D411yescharacteristic
116D332—22no—
117D332—42no—
118C4×D83—22no—
119C4×D83—22no—
120C4×D83—41no—
121C4×D83—41no—
122C4×D83—42no—
123C8×C83—22no—
124C83⋊C8—22no—
125C22×C83—21no—
126C22×C83—21no—
127C22×C83—41no—
128C22×C83—41no—
129C22×C83—41no—
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—44no—
138C2×D83—41no—
139C2×D83—41no—
140C2×D83—41no—
141C2×D83—84no—
142C2×D83#13712yes—
143C4×C83—21no—
144C4×C83—42no—
145C4×C83—41no—
146C4×C83H(5,𝔽2)12yes—
147Dic83—21no—
148Dic83—21no—
149Dic83—41no—
150Dic83—42no—
151C2×C83—22no—
152C2×C83—81no—
153C2×C83#53411yescharacteristic
154C2×C83C2×AGL(1,ℤ/8)11yescharacteristic
155C2×C83C2×#13711yescharacteristic
156D83—22no—
157D83—41no—
158D83—81no—
159#3193—831nomaximal, Sylow
160C83#319311yescharacteristic, complemented, Sylow
161C2×C22⋊Q8—831no—
162C2×#121—831no—
163#484—832no—
164#480—831no—
165#480—831no—
166#468—832no—
167#468—832no—
168#450—832no—
169#448—831no—
170#448D8311yescharacteristic, complemented
171#407—831no—
172C22×Q8C2×D8311yescharacteristic, complemented
173C22×D4—831no—
174C22×D4—1661no—
175C23×C4—831no—
176C2×SD16—832no—
177C2×SD16C2×D8312yescomplemented
178C2×M16—832no—
179C22⋊Q8—832no—
180C22⋊Q8—832no—
181C22⋊Q8—1661no—
182C22⋊Q8—1661no—
183#121—831no—
184#121—831no—
185#121—831no—
186#121—1661no—
187#121—1661no—
188#121C2×D8311yescharacteristic
189C2×C4⋊C4—831no—
190C2×C4⋊C4—831no—
191C2×C4⋊C4—1661no—
192C2×C22⋊C4—1661no—
193C2×C22⋊C4—1661no—
194#107—832no—
195#107—832no—
196#103—832no—
197#103C2×D8312yescomplemented
198#102—832no—
199#102—832no—
200C22⋊C8—831no—
201C22⋊C8—832no—
202C22⋊C8C2×D8311yescharacteristic, complemented
203C24—831no—
204C24—1661no—
205C2×Q8—21no—
206C2×Q8—42no—
207C2×Q8C22×D8312yes—
208C2×D4—21no—
209C2×D4—3324no—
210C2×D4—3322no—
211C2×D4—831no—
212C2×D4—1661no—
213C2×D4—1661no—
214C2×D4—1661no—
215C2×D4—1661no—
216C2×D4C22×D8311yescharacteristic
217C22×C4—21no—
218C22×C4—3322no—
219C22×C4—831no—
220C22×C4—831no—
221C22×C4—831no—
222C22×C4—831no—
223C22×C4—831no—
224C22×C4—1661no—
225C22×C4—1661no—
226C22×C4C22×D8311yescharacteristic
227SD16—42no—
228SD16—3322no—
229M16—3322no—
230C2×C8—832no—
231C2×C8C22×D8312yes—
232C4⋊C4—832no—
233C4⋊C4—831no—
234C4⋊C4—1661no—
235C4⋊C4—1662no—
236C4⋊C4—1662no—
237C4⋊C4—1661no—
238C4⋊C4C22×D8311yescharacteristic
239C22⋊C4—21no—
240C22⋊C4—1661no—
241C22⋊C4—1662no—
242C22⋊C4—1662no—
243C22⋊C4—1661no—
244C22⋊C4—1661no—
245C23—21no—
246C23—3321no—
247C23—3322no—
248C23—3322no—
249C23—3322no—
250C23—831no—
251C23—831no—
252C23—1662no—
253C23—1661no—
254C23—1662no—
255C23D4×D8311yescharacteristic
256Q8—22no—
257Q8—22no—
258Q8—42no—
259D4—22no—
260D4—42no—
261D4—3322no—
262D4—3322no—
263D4—3322no—
264D4—3322no—
265D4—1662no—
266C2×C4—22no—
267C2×C4—21no—
268C2×C4—22no—
269C2×C4—42no—
270C2×C4—3321no—
271C2×C4—3321no—
272C2×C4—3322no—
273C2×C4—3322no—
274C2×C4—3321no—
275C2×C4—831no—
276C2×C4—831no—
277C2×C4—1662no—
278C2×C4—1661no—
279C2×C4—1662no—
280C2×C4—1662no—
281C2×C4D4×D8311yescharacteristic
282C2×C4C23×D8311yescharacteristic
283C8—22no—
284C8—1662no—
285C22—21no—
286C22—21no—
287C22—41no—
288C22—41no—
289C22—41no—
290C22—3324no—
291C22—3321no—
292C22—3321no—
293C22—3321no—
294C22—832no—
295C22—6644no—
296C22—1661no—
297C22—1661no—
298C22—1661no—
299C22—1661no—
300C22—1661no—
301C22#1458181912yes—
302C22C2×D4×D8311yescharacteristic
303C4—21no—
304C4—41no—
305C4—42no—
306C4—3321no—
307C4—3322no—
308C4—1661no—
309C4—1661no—
310C4#1458183712yes—
311C2—22no—
312C2—81no—
313C2—3321no—
314C2—6641no—
315C2—1662no—
316C2#1912826111yescharacteristic
317C2C2×#1458181911yescharacteristic
318C2#137×D8311yescharacteristic
subgroupG/H
centerC22C2×D4×D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting#448×C83C2
socleC22×C83C22×D4
prime radical#121×D83C2
hypercenter#448D83
hypocenterC83#3193
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C22×C4 ◁ #448

Complements

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

NH ≅ G/Ntype
1. C2×#14581768C2semidirect
3. #19128035C2semidirect
4. #19128015C2semidirect
5. #19128012C2semidirect
6. #19127950C2semidirect
7. #19127944C2semidirect
8. #19127850C2semidirect
9. #19127840C2semidirect
10. #19127663C2semidirect
11. #448×C83C2semidirect
15. C2×#13050577C22semidirect
16. C2×#13050573C22semidirect
20. #14581768C22semidirect
26. C2×C4⋊Dic83C22semidirect
31. #14581726C22semidirect
32. #14581725C22semidirect
33. #14581716C22semidirect
34. #14581714C22semidirect
35. #14581713C22semidirect
36. C22⋊(C83⋊C8)C22semidirect
38. C22×Q8×C83C22semidirect
39. C2×SD16×C83C22semidirect
41. #103×C83C22semidirect
42. C22⋊C8×C83C22semidirect
160. C83#3193semidirect
170. #448D83semidirect
172. C22×Q8C2×D83semidirect
177. C2×SD16C2×D83semidirect
197. #103C2×D83semidirect
202. C22⋊C8C2×D83semidirect

Maximal subgroups

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

subgroupindex
C2×#145817682
#121×D832
#191280352
#191280152
#191280122
#191279502
#191279442
#191278502
#191278402
#191276632
#448×C832
#319383