Finite Group Databasebeta

← #24918449

Subgroups — #24918449

Order 10624.

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

normal subgroups only

This group has 444 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#19128246C211yesmaximal, characteristic, complemented
2C2×#14581781C211yesmaximal, characteristic, complemented
3#19128087C212yesmaximal, complemented
4#19128062C212yesmaximal, complemented
5#19127919C212yesmaximal, complemented
6#19127918C212yesmaximal, complemented
7#19127840C211yesmaximal, characteristic, complemented
8#19127837C211yesmaximal, characteristic, complemented
9#19127834C211yesmaximal, characteristic, complemented
10#19127755C211yesmaximal, characteristic, complemented
11#448×C83C211yesmaximal, characteristic, complemented
12C2×#13050589—21no—
13C22×D332C2211yescharacteristic
14C2×#13050577C2212yescomplemented
15C2×D664C2212yes—
16C2×#13050575C2212yes—
17#14581791C2211yescharacteristic, complemented
18#14581788C2211yescharacteristic, complemented
19#14581781—21no—
20#14581781—21no—
21#14581781C2212yes—
22#14581781C2212yes—
23#14581778—21no—
24#14581766C2211yescharacteristic
25#14581764—21no—
26D4×Dic83C2211yescharacteristic, complemented
27#14581754C2211yescharacteristic, complemented
28(C22⋊D83)⋊C4—21no—
29C2×(C2×D83)⋊C4—21no—
30C2×(C2×D83)⋊C4—21no—
31C2×C4×Dic83C2211yescharacteristic
32Dic83⋊C8C2212yescomplemented
33#14581716C2212yescomplemented
34#14581714C2212yes—
35#14581712C2212yes—
36C22⋊(C83⋊C8)C2211yescharacteristic
37#14581702C2212yescomplemented
38C22×Q8×C83C2211yescharacteristic
39C2×SD16×C83C2212yescomplemented
40#121×C83C2211yescharacteristic
41#103×C83C2212yescomplemented
42C22⋊C8×C83C2211yescharacteristic
43C23×D83—21no—
44#13050589—44no—
45C2×Dic166—21no—
46C2×D332—21no—
47C2×D332—42no—
48C2×D332C2311yescharacteristic
49C2×D332C2312yes—
50C2×C22⋊D83—21no—
51C2×C22⋊D83—21no—
52C2×C22⋊D83—21no—
53C22×Dic83—21no—
54C22×Dic83D411yescharacteristic
55C22×Dic83D411yescharacteristic
56C2×C4×D83—21no—
57#13050577—42no—
58D664—42no—
59#13050575—42no—
60C2×C83⋊C8C2312yes—
61C4⋊Dic83C2311yescharacteristic
62Dic83⋊C4—21no—
63C22⋊Dic83—21no—
64(C2×D83)⋊C4—21no—
65(C2×D83)⋊C4—21no—
66(C2×D83)⋊C4—22no—
67(C2×D83)⋊C4—22no—
68(C2×D83)⋊C4—22no—
69(C2×D83)⋊C4—22no—
70C4×Dic83D411yescharacteristic
71C4×Dic83D411yescharacteristic
72C4×Dic83C2311yescharacteristic
73C4×Dic83C2311yescharacteristic
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—21no—
89C22×D83—42no—
90C22×D83—42no—
91C22×D83—42no—
92Q8×C83—22no—
93Q8×C83—22no—
94Q8×C83—42no—
95D4×C83—22no—
96D4×C83—42no—
97C2×C4×C83—22no—
98C2×C4×C83—21no—
99C2×C4×C83—22no—
100C2×C4×C83—42no—
101C2×C4×C83C2×D411yescharacteristic
102C2×C4×C83C2411yescharacteristic
103C22⋊D83—42no—
104C22⋊D83—42no—
105C22⋊D83—42no—
106C2×Dic83—22no—
107C2×Dic83—21no—
108C2×Dic83—41no—
109C2×Dic83—42no—
110C2×Dic83C2×D411yescharacteristic
111C2×Dic83C2×D411yescharacteristic
112C2×Dic83C2×D411yescharacteristic
113C2×Dic83C2×D411yescharacteristic
114D332—22no—
115D332—22no—
116D332—22no—
117D332—42no—
118C4×D83—42no—
119Dic166—42no—
120C8×C83—22no—
121C83⋊C8—22no—
122C22×C83—21no—
123C22×C83—21no—
124C22×C83—41no—
125C22×C83—41no—
126C22×C83—41no—
127C22×C83AGL(1,ℤ/8)12yes—
128C22×C83C22×D411yescharacteristic
129C2×D83—42no—
130C2×D83—41no—
131C2×D83—41no—
132C2×D83—42no—
133C2×D83—41no—
134C2×D83—42no—
135C2×D83—84no—
136C4×C83—21no—
137C4×C83—42no—
138C4×C83—41no—
139C4×C83H(5,𝔽2)12yes—
140Dic83—21no—
141Dic83—21no—
142Dic83—21no—
143Dic83—21no—
144Dic83—41no—
145C2×C83—22no—
146C2×C83—81no—
147C2×C83#53511yescharacteristic
148C2×C83C2×AGL(1,ℤ/8)11yescharacteristic
149C2×C83#57511yescharacteristic
150D83—81no—
151D83—82no—
152#3213—831nomaximal, Sylow
153C83#321311yescharacteristic, complemented, Sylow
154#532—831no—
155C2×#124—831no—
156#496—832no—
157#489—832no—
158#463—832no—
159#463—832no—
160#448—831no—
161#448D8311yescharacteristic, complemented
162#447—831no—
163#447—831no—
164#432—831no—
165C2×#41—1661no—
166C22×Q8C2×D8311yescharacteristic
167C22×D4—831no—
168C2×SD16—832no—
169C2×SD16C2×D8312yescomplemented
170C2×D8—832no—
171C2×D8—832no—
172C4⋊Q8—831no—
173#127—831no—
174#124—832no—
175#124—832no—
176#124—1661no—
177#124—1661no—
178#124—1661no—
179#121—831no—
180#121—1661no—
181#121C2×D8311yescharacteristic
182C4×D4—831no—
183C4×D4—831no—
184C4×D4—1661no—
185C2×C22⋊C4—1661no—
186C2×C22⋊C4—1661no—
187C2×C42—831no—
188C4⋊C8—832no—
189#103—832no—
190#103C2×D8312yescomplemented
191#102—832no—
192#102—832no—
193C22⋊C8—831no—
194C22⋊C8C2×D8311yescharacteristic
195#97—832no—
196C24—1661no—
197#41—3324no—
198C2×Q8—21no—
199C2×Q8—42no—
200C2×Q8—1661no—
201C2×Q8C22×D8312yes—
202C2×D4—21no—
203C2×D4—3322no—
204C2×D4—832no—
205C2×D4—831no—
206C2×D4—1661no—
207C2×D4—1661no—
208C2×D4—1661no—
209C2×D4—1661no—
210C2×D4C22×D8311yescharacteristic
211C22×C4—21no—
212C22×C4—831no—
213C22×C4—831no—
214C22×C4—1661no—
215C22×C4—1661no—
216C22×C4C22×D8311yescharacteristic
217SD16—42no—
218SD16—3322no—
219D8—3322no—
220D8—3322no—
221C2×C8—832no—
222C2×C8C22×D8312yes—
223C4⋊C4—831no—
224C4⋊C4—1661no—
225C4⋊C4C22×D8311yescharacteristic
226C22⋊C4—21no—
227C22⋊C4—1662no—
228C22⋊C4—1662no—
229C22⋊C4—1662no—
230C22⋊C4—1662no—
231C22⋊C4—1661no—
232C22⋊C4—1661no—
233C22⋊C4—1661no—
234C42—831no—
235C42—831no—
236C42—831no—
237C42—831no—
238C23—21no—
239C23—3322no—
240C23—3322no—
241C23—3322no—
242C23—1662no—
243C23—1661no—
244C23D4×D8311yescharacteristic
245Q8—22no—
246Q8—22no—
247Q8—42no—
248Q8—3322no—
249D4—22no—
250D4—42no—
251D4—3322no—
252D4—3322no—
253D4—3322no—
254D4—3322no—
255D4—1662no—
256D4—1662no—
257D4—1662no—
258C2×C4—22no—
259C2×C4—21no—
260C2×C4—22no—
261C2×C4—42no—
262C2×C4—3322no—
263C2×C4—3322no—
264C2×C4—3321no—
265C2×C4—831no—
266C2×C4—831no—
267C2×C4—831no—
268C2×C4—831no—
269C2×C4—1661no—
270C2×C4—1662no—
271C2×C4D4×D8311yescharacteristic
272C2×C4C23×D8311yescharacteristic
273C8—22no—
274C8—1662no—
275C22—21no—
276C22—21no—
277C22—41no—
278C22—41no—
279C22—41no—
280C22—3321no—
281C22—3322no—
282C22—3321no—
283C22—3322no—
284C22—3322no—
285C22—3321no—
286C22—6644no—
287C22#1458181912yes—
288C22C2×D4×D8311yescharacteristic
289C4—21no—
290C4—41no—
291C4—42no—
292C4—3321no—
293C4—1661no—
294C4—1661no—
295C4—1661no—
296C4—1661no—
297C4#1458183712yes—
298C2—22no—
299C2—81no—
300C2—6642no—
301C2—6641no—
302C2#1912826611yescharacteristic
303C2C2×#1458181911yescharacteristic
304C2#1912850011yescharacteristic
subgroupG/H
centerC22C2×D4×D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting#448×C83C2
socleC22×C83C22×D4
hypercenter#448D83
hypocenterC83#3213
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. #19128246C2semidirect
2. C2×#14581781C2semidirect
3. #19128087C2semidirect
4. #19128062C2semidirect
5. #19127919C2semidirect
6. #19127918C2semidirect
7. #19127840C2semidirect
8. #19127837C2semidirect
9. #19127834C2semidirect
10. #19127755C2semidirect
11. #448×C83C2semidirect
14. C2×#13050577C22semidirect
17. #14581791C22semidirect
18. #14581788C22semidirect
26. D4×Dic83C22semidirect
27. #14581754C22semidirect
32. Dic83⋊C8C22semidirect
33. #14581716C22semidirect
37. #14581702C22semidirect
39. C2×SD16×C83C22semidirect
41. #103×C83C22semidirect
153. C83#3213semidirect
161. #448D83semidirect
169. C2×SD16C2×D83semidirect
190. #103C2×D83semidirect

Maximal subgroups

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

subgroupindex
#191282462
C2×#145817812
#191280872
#191280622
#191279192
#191279182
#191278402
#191278372
#191278342
#191277552
#448×C832
#321383