Finite Group Databasebeta

← #24918484

Subgroups — #24918484

Order 10624.

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

normal subgroups only

This group has 490 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×#14581788C211yesmaximal, characteristic, complemented
2#19128237C211yesmaximal, characteristic, complemented
3#19128087C212yesmaximal, complemented
4#19128064C212yesmaximal, complemented
5#19127899C212yesmaximal, complemented
6#19127895C212yesmaximal, complemented
7#19127851C211yesmaximal, characteristic, complemented
8#19127837C211yesmaximal, characteristic, complemented
9#19127835C211yesmaximal, characteristic, complemented
10#19127755C211yesmaximal, characteristic, complemented
11#450×C83C211yesmaximal, characteristic, complemented
12C22×D332C2211yescharacteristic
13C22×C22⋊D83—21no—
14C22×C22⋊D83—21no—
15C2×#13050577C2212yescomplemented
16C2×D664C2212yescomplemented
17C2×#13050575C2212yescomplemented
18#14581788—21no—
19#14581788—21no—
20#14581788C2212yes—
21#14581788C2212yes—
22#14581787—21no—
23#14581783C2211yescharacteristic, complemented
24#14581781C2211yescharacteristic, complemented
25#14581776—21no—
26#14581766C2211yescharacteristic
27Q8×Dic83C2211yescharacteristic, complemented
28#14581754C2211yescharacteristic, complemented
29(C22⋊D83)⋊C4—21no—
30#14581750—21no—
31C2×C4×Dic83C2211yescharacteristic
32Dic83⋊C8C2212yescomplemented
33#14581714C2212yescomplemented
34#14581713C2212yescomplemented
35#14581712C2212yescomplemented
36C22⋊(C83⋊C8)C2211yescharacteristic
37#14581702C2212yescomplemented
38C22×D4×C83C2211yescharacteristic
39C2×SD16×C83C2212yescomplemented
40C22⋊Q8×C83C2211yescharacteristic
41#102×C83C2212yescomplemented
42C22⋊C8×C83C2211yescharacteristic
43C23×D83—21no—
44C2×D332—21no—
45C2×D332—42no—
46C2×D332C2311yescharacteristic
47C2×D332C2312yes—
48C2×C22⋊D83—21no—
49C2×C22⋊D83—22no—
50C2×C22⋊D83—22no—
51C2×C22⋊D83—22no—
52C2×C22⋊D83—22no—
53C2×C22⋊D83—44no—
54C2×C22⋊D83—44no—
55C22×Dic83D411yescharacteristic
56C22×Dic83D411yescharacteristic
57C2×C4×D83—21no—
58#13050577—42no—
59D664—42no—
60#13050575—42no—
61C2×C83⋊C8C2312yes—
62C4⋊Dic83—21no—
63C4⋊Dic83—21no—
64C4⋊Dic83C2311yescharacteristic
65Dic83⋊C4—21no—
66(C2×D83)⋊C4—21no—
67(C2×D83)⋊C4—21no—
68(C2×D83)⋊C4—21no—
69(C2×D83)⋊C4—21no—
70C4×Dic83—21no—
71C4×Dic83D411yescharacteristic
72C4×Dic83D411yescharacteristic
73C4×Dic83C2311yescharacteristic
74C4×Dic83C2311yescharacteristic
75C24×C83—21no—
76C2×Q8×C83C2311yescharacteristic
77C2×D4×C83—21no—
78C2×D4×C83—42no—
79C2×D4×C83C2312yes—
80C22×C4×C83C2311yescharacteristic
81SD16×C83—42no—
82C2×C8×C83C2312yes—
83C4⋊C4×C83—21no—
84C4⋊C4×C83C2311yescharacteristic
85C22⋊C4×C83—21no—
86C23×C83—22no—
87C23×C83—42no—
88C23×C83—42no—
89C23×C83—42no—
90C23×C83C2×D411yescharacteristic
91C22×D83—21no—
92C22×D83—22no—
93C22×D83—42no—
94C22×D83—42no—
95C22×D83—42no—
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—44no—
108C22⋊D83—42no—
109C22⋊D83—44no—
110C2×Dic83—22no—
111C2×Dic83—22no—
112C2×Dic83—21no—
113C2×Dic83C2×D411yescharacteristic
114C2×Dic83C2×D411yescharacteristic
115C2×Dic83C2×D411yescharacteristic
116C2×Dic83C2×D411yescharacteristic
117D332—22no—
118D332—22no—
119D332—22no—
120D332—42no—
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—42no—
133C2×D83—41no—
134C2×D83—41no—
135C2×D83—42no—
136C2×D83—42no—
137C2×D83—41no—
138C2×D83—84no—
139C4×C83—21no—
140C4×C83—41no—
141C4×C83—41no—
142C4×C83H(5,𝔽2)12yes—
143Dic83—21no—
144Dic83—21no—
145Dic83—21no—
146Dic83—41no—
147Dic83AGL(1,ℤ/8)12yes—
148C2×C83—22no—
149C2×C83—82no—
150C2×C83#53511yescharacteristic
151C2×C83C2×AGL(1,ℤ/8)11yescharacteristic
152C2×C83C2×AGL(1,ℤ/8)11yescharacteristic
153D83—81no—
154D83—82no—
155#3217—831nomaximal, Sylow
156C83#321711yescharacteristic, complemented, Sylow
157C2×#127—831no—
158#529—831no—
159#496—832no—
160#490—832no—
161#460—832no—
162#459—832no—
163#450—831no—
164#450D8311yescharacteristic, complemented
165#447—831no—
166#447—831no—
167#432—831no—
168C22×D4—831no—
169C22×D4—1661no—
170C22×D4—1661no—
171C22×D4C2×D8311yescharacteristic
172C2×SD16—832no—
173C2×SD16C2×D8312yescomplemented
174C2×D8—832no—
175C2×D8—832no—
176#127—832no—
177#127—832no—
178#127—1661no—
179#127—1661no—
180#126—1661no—
181#125—831no—
182#124—831no—
183#123—1661no—
184C22⋊Q8C2×D8311yescharacteristic
185#121—831no—
186C4×Q8—831no—
187C4×D4—831no—
188C4×D4—1661no—
189#117—1661no—
190C2×C42—831no—
191C4⋊C8—832no—
192#102—832no—
193#102—832no—
194#102—832no—
195#102C2×D8312yescomplemented
196C22⋊C8—831no—
197C22⋊C8C2×D8311yescharacteristic
198#97—832no—
199C24—21no—
200C24—1661no—
201C2×Q8C22×D8311yescharacteristic
202C2×D4—21no—
203C2×D4—42no—
204C2×D4—3322no—
205C2×D4—3324no—
206C2×D4—3324no—
207C2×D4—832no—
208C2×D4—831no—
209C2×D4—1662no—
210C2×D4—1662no—
211C2×D4—1662no—
212C2×D4—1662no—
213C2×D4—1661no—
214C2×D4—1661no—
215C2×D4C22×D8312yes—
216C22×C4—831no—
217C22×C4—831no—
218C22×C4—1661no—
219C22×C4C22×D8311yescharacteristic
220SD16—42no—
221SD16—3322no—
222D8—3322no—
223D8—3322no—
224C2×C8—832no—
225C2×C8C22×D8312yes—
226C4⋊C4—21no—
227C4⋊C4—831no—
228C4⋊C4—1661no—
229C4⋊C4—1661no—
230C4⋊C4—1661no—
231C4⋊C4C22×D8311yescharacteristic
232C22⋊C4—21no—
233C22⋊C4—1661no—
234C22⋊C4—1661no—
235C22⋊C4—1661no—
236C22⋊C4—1661no—
237C42—831no—
238C42—831no—
239C42—831no—
240C42—831no—
241C42—1661no—
242C23—22no—
243C23—42no—
244C23—42no—
245C23—42no—
246C23—3322no—
247C23—3322no—
248C23—3322no—
249C23—1661no—
250C23—1662no—
251C23D4×D8311yescharacteristic
252Q8—22no—
253D4—22no—
254D4—22no—
255D4—42no—
256D4—3322no—
257D4—3322no—
258D4—3324no—
259D4—3324no—
260D4—3324no—
261D4—3324no—
262D4—1662no—
263D4—1662no—
264D4—1662no—
265C2×C4—22no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—3322no—
269C2×C4—831no—
270C2×C4—831no—
271C2×C4—831no—
272C2×C4—831no—
273C2×C4—1661no—
274C2×C4—1662no—
275C2×C4—1662no—
276C2×C4D4×D8311yescharacteristic
277C2×C4C23×D8311yescharacteristic
278C8—22no—
279C8—1662no—
280C22—21no—
281C22—21no—
282C22—42no—
283C22—42no—
284C22—42no—
285C22—84no—
286C22—3321no—
287C22—3322no—
288C22—3322no—
289C22—3321no—
290C22—3322no—
291C22—3321no—
292C22—6644no—
293C22#1458181912yes—
294C22C2×D4×D8311yescharacteristic
295C4—21no—
296C4—41no—
297C4—41no—
298C4—3321no—
299C4—832no—
300C4—1661no—
301C4—1661no—
302C4—1661no—
303C4#1458183712yes—
304C2—22no—
305C2—82no—
306C2—6642no—
307C2—6641no—
308C2#1912826611yescharacteristic
309C2AGL(1,ℤ/8)×D8311yescharacteristic
310C2C2×#1458181911yescharacteristic
subgroupG/H
centerC22C2×D4×D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting#450×C83C2
socleC22×C83C22×D4
hypercenter#450D83
hypocenterC83#3217
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. C2×#14581788C2semidirect
2. #19128237C2semidirect
3. #19128087C2semidirect
4. #19128064C2semidirect
5. #19127899C2semidirect
6. #19127895C2semidirect
7. #19127851C2semidirect
8. #19127837C2semidirect
9. #19127835C2semidirect
10. #19127755C2semidirect
11. #450×C83C2semidirect
15. C2×#13050577C22semidirect
16. C2×D664C22semidirect
17. C2×#13050575C22semidirect
23. #14581783C22semidirect
24. #14581781C22semidirect
27. Q8×Dic83C22semidirect
28. #14581754C22semidirect
32. Dic83⋊C8C22semidirect
33. #14581714C22semidirect
34. #14581713C22semidirect
35. #14581712C22semidirect
37. #14581702C22semidirect
39. C2×SD16×C83C22semidirect
41. #102×C83C22semidirect
156. C83#3217semidirect
164. #450D83semidirect
173. C2×SD16C2×D83semidirect
195. #102C2×D83semidirect

Maximal subgroups

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

subgroupindex
C2×#145817882
#191282372
#191280872
#191280642
#191278992
#191278952
#191278512
#191278372
#191278352
#191277552
#450×C832
#321783