Finite Group Databasebeta

← #24918485

Subgroups — #24918485

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
1#127×D83C211yesmaximal, characteristic, complemented
2#19128238C211yesmaximal, characteristic, complemented
3#19128087C212yesmaximal, complemented
4#19128064C212yesmaximal, complemented
5#19127901C211yesmaximal, characteristic, complemented
6#19127897C211yesmaximal, characteristic, complemented
7#19127893C211yesmaximal, characteristic, complemented
8#19127850C212yesmaximal, complemented
9#19127836C212yesmaximal, complemented
10#19127757C211yesmaximal, characteristic, complemented
11#460×C83C211yesmaximal, characteristic, complemented
12C2×D4×D83—21no—
13C2×D4×D83—21no—
14C2×D4×D83C2212yes—
15C2×#13050577C2212yescomplemented
16C2×D664C2212yescomplemented
17C2×#13050575C2212yescomplemented
18#14581789C2211yescharacteristic
19#14581788—21no—
20#14581788—21no—
21#14581788C2212yes—
22#14581787—21no—
23#14581783C2211yescharacteristic, complemented
24#14581781C2211yescharacteristic, complemented
25#14581777—21no—
26#14581768C2211yescharacteristic
27#14581767C2211yescharacteristic
28C4×D332—21no—
29C4×D332C2211yescharacteristic, complemented
30#14581749—21no—
31C42×D83C2211yescharacteristic
32C4⋊(C83⋊C8)C2211yescharacteristic, complemented
33#14581714C2212yescomplemented
34#14581713C2212yescomplemented
35#14581712C2212yescomplemented
36(C2×D83)⋊C8C2212yes—
37#14581702C2212yescomplemented
38C2×SD16×C83C2212yescomplemented
39#127×C83C2211yescharacteristic
40C4×Q8×C83C2211yescharacteristic, complemented
41C4⋊C8×C83C2211yescharacteristic, complemented
42#102×C83C2212yescomplemented
43C23×D83—22no—
44C2×D332—21no—
45C2×D332—21no—
46C2×D332—21no—
47C2×D332C2311yescharacteristic
48C2×D332C2312yes—
49C2×C22⋊D83—22no—
50C2×C22⋊D83—22no—
51C2×C22⋊D83—22no—
52D4×D83—42no—
53D4×D83—42no—
54D4×D83—44no—
55D4×D83—44no—
56C2×C4×D83—21no—
57C2×C4×D83D411yescharacteristic
58C2×C4×D83D411yescharacteristic
59C2×C4×D83C2311yescharacteristic
60#13050577—42no—
61D664—42no—
62#13050575—42no—
63C2×C83⋊C8C2312yes—
64C4⋊Dic83—21no—
65C4⋊Dic83C2311yescharacteristic
66Dic83⋊C4—21no—
67Dic83⋊C4—21no—
68(C2×D83)⋊C4—21no—
69(C2×D83)⋊C4—21no—
70(C2×D83)⋊C4—21no—
71(C2×D83)⋊C4—21no—
72(C2×D83)⋊C4—21no—
73C4×Dic83D411yescharacteristic
74C4×Dic83D411yescharacteristic
75C4×Dic83C2311yescharacteristic
76C2×Q8×C83C2311yescharacteristic
77C2×D4×C83—21no—
78C2×D4×C83—21no—
79C2×D4×C83C2312yes—
80SD16×C83—42no—
81C2×C8×C83C2312yes—
82C4⋊C4×C83—21no—
83C4⋊C4×C83C2311yescharacteristic
84C42×C83—21no—
85C42×C83C2311yescharacteristic
86C23×C83—22no—
87C22×D83—21no—
88C22×D83—22no—
89C22×D83—44no—
90C22×D83—44no—
91C22×D83—44no—
92C22×D83C2×D411yescharacteristic
93Q8×C83—21no—
94Q8×C83—21no—
95D4×C83—22no—
96D4×C83—22no—
97D4×C83—42no—
98D4×C83—42no—
99C2×C4×C83—21no—
100C2×C4×C83—21no—
101C2×C4×C83C2×D411yescharacteristic
102C2×C4×C83C2×D411yescharacteristic
103C2×C4×C83C2411yescharacteristic
104C22⋊D83—44no—
105C22⋊D83—44no—
106C22⋊D83—44no—
107C2×Dic83—21no—
108C2×Dic83C2×D411yescharacteristic
109C2×Dic83C2×D411yescharacteristic
110C2×Dic83C2×D411yescharacteristic
111D332—21no—
112D332—21no—
113D332—22no—
114D332—22no—
115D332—42no—
116D332—42no—
117D332—42no—
118C4×D83—22no—
119C4×D83—22no—
120C4×D83—21no—
121C4×D83—21no—
122C4×D83—42no—
123C8×C83—22no—
124C83⋊C8—22no—
125C22×C83—42no—
126C22×C83—42no—
127C22×C83—42no—
128C22×C83C22×D411yescharacteristic
129C2×D83—21no—
130C2×D83—21no—
131C2×D83—42no—
132C2×D83—41no—
133C2×D83—41no—
134C2×D83—42no—
135C2×D83—42no—
136C2×D83—41no—
137C2×D83—88no—
138C2×D83AGL(1,ℤ/8)12yes—
139C4×C83—21no—
140C4×C83—41no—
141C4×C83—41no—
142C4×C83AGL(1,ℤ/8)12yes—
143C4×C83H(5,𝔽2)11yescharacteristic
144C4×C83H(5,𝔽2)11yescharacteristic
145Dic83—21no—
146Dic83—21no—
147Dic83—21no—
148Dic83—41no—
149C2×C83—82no—
150C2×C83#53511yescharacteristic
151C2×C83C2×AGL(1,ℤ/8)11yescharacteristic
152C2×C83C2×AGL(1,ℤ/8)11yescharacteristic
153D83—22no—
154D83—81no—
155D83—82no—
156#3217—831nomaximal, Sylow
157C83#321711yescharacteristic, complemented, Sylow
158C2×#127—831no—
159#529—831no—
160#496—832no—
161#490—832no—
162#460—831no—
163#460D8311yescharacteristic, complemented
164#459—831no—
165#459—831no—
166#450—832no—
167#447—832no—
168#432—831no—
169C22×D4—832no—
170C22×D4—1661no—
171C22×D4—1661no—
172C2×SD16—832no—
173C2×SD16C2×D8312yescomplemented
174C2×D8—832no—
175C2×D8—832no—
176#127—832no—
177#127—831no—
178#127—1661no—
179#127—1661no—
180#127C2×D8311yescharacteristic
181#126—1661no—
182#125—831no—
183#124—831no—
184#123—1661no—
185C22⋊Q8—831no—
186#121—831no—
187C4×Q8C2×D8311yescharacteristic, complemented
188C4×D4—831no—
189C4×D4—1661no—
190#117—1661no—
191C2×C42—831no—
192C4⋊C8—831no—
193C4⋊C8C2×D8311yescharacteristic, complemented
194#102—832no—
195#102—832no—
196#102—832no—
197#102C2×D8312yescomplemented
198C22⋊C8—832no—
199#97—832no—
200C24—1662no—
201C2×Q8C22×D8311yescharacteristic
202C2×D4—21no—
203C2×D4—21no—
204C2×D4—3322no—
205C2×D4—3322no—
206C2×D4—3324no—
207C2×D4—3324no—
208C2×D4—832no—
209C2×D4—831no—
210C2×D4—1661no—
211C2×D4—1662no—
212C2×D4—1662no—
213C2×D4—1662no—
214C2×D4—1661no—
215C2×D4—1661no—
216C2×D4C22×D8312yes—
217C22×C4—831no—
218C22×C4—831no—
219C22×C4—831no—
220C22×C4—1661no—
221SD16—42no—
222SD16—3322no—
223D8—3322no—
224D8—3322no—
225C2×C8—832no—
226C2×C8C22×D8312yes—
227C4⋊C4—21no—
228C4⋊C4—831no—
229C4⋊C4—1661no—
230C4⋊C4—1661no—
231C4⋊C4—1661no—
232C4⋊C4C22×D8311yescharacteristic
233C22⋊C4—1661no—
234C22⋊C4—1661no—
235C22⋊C4—1661no—
236C22⋊C4—1661no—
237C22⋊C4—1661no—
238C42—21no—
239C42—831no—
240C42—831no—
241C42—831no—
242C42C22×D8311yescharacteristic
243C23—22no—
244C23—3324no—
245C23—3324no—
246C23—3324no—
247C23—831no—
248C23—1661no—
249C23—1662no—
250Q8—21no—
251Q8—21no—
252D4—22no—
253D4—22no—
254D4—42no—
255D4—42no—
256D4—3322no—
257D4—3322no—
258D4—3324no—
259D4—3324no—
260D4—3324no—
261D4—3322no—
262D4—1661no—
263D4—1662no—
264D4—1661no—
265D4—1662no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—3322no—
269C2×C4—831no—
270C2×C4—831no—
271C2×C4—831no—
272C2×C4—1662no—
273C2×C4—1661no—
274C2×C4—1662no—
275C2×C4—1661no—
276C2×C4—1661no—
277C2×C4D4×D8311yescharacteristic
278C2×C4D4×D8311yescharacteristic
279C2×C4C23×D8311yescharacteristic
280C8—22no—
281C8—1662no—
282C22—42no—
283C22—42no—
284C22—42no—
285C22—3322no—
286C22—3321no—
287C22—3321no—
288C22—3322no—
289C22—3322no—
290C22—3321no—
291C22—832no—
292C22—6648no—
293C22—1661no—
294C22—1661no—
295C22C2×D4×D8311yescharacteristic
296C4—21no—
297C4—41no—
298C4—41no—
299C4—3321no—
300C4—1661no—
301C4—1661no—
302C4—1661no—
303C4#1458181912yes—
304C4#1458183711yescharacteristic
305C4#1458183811yescharacteristic
306C2—82no—
307C2—6642no—
308C2—6641no—
309C2—1662no—
310C2#1912826711yescharacteristic
311C2AGL(1,ℤ/8)×D8311yescharacteristic
312C2C2×#1458181911yescharacteristic
subgroupG/H
centerC22C2×D4×D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting#460×C83C2
socleC22×C83C22×D4
hypercenter#460D83
hypocenterC83#3217
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C42 ◁ #460

Complements

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

NH ≅ G/Ntype
1. #127×D83C2semidirect
2. #19128238C2semidirect
3. #19128087C2semidirect
4. #19128064C2semidirect
5. #19127901C2semidirect
6. #19127897C2semidirect
7. #19127893C2semidirect
8. #19127850C2semidirect
9. #19127836C2semidirect
10. #19127757C2semidirect
11. #460×C83C2semidirect
15. C2×#13050577C22semidirect
16. C2×D664C22semidirect
17. C2×#13050575C22semidirect
23. #14581783C22semidirect
24. #14581781C22semidirect
29. C4×D332C22semidirect
32. C4⋊(C83⋊C8)C22semidirect
33. #14581714C22semidirect
34. #14581713C22semidirect
35. #14581712C22semidirect
37. #14581702C22semidirect
38. C2×SD16×C83C22semidirect
40. C4×Q8×C83C22semidirect
41. C4⋊C8×C83C22semidirect
42. #102×C83C22semidirect
157. C83#3217semidirect
163. #460D83semidirect
173. C2×SD16C2×D83semidirect
187. C4×Q8C2×D83semidirect
193. C4⋊C8C2×D83semidirect
197. #102C2×D83semidirect

Maximal subgroups

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

subgroupindex
#127×D832
#191282382
#191280872
#191280642
#191279012
#191278972
#191278932
#191278502
#191278362
#191277572
#460×C832
#321783