Finite Group Databasebeta

← #24917848

Subgroups — #24917848

Order 10624.

400 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 400 conjugacy classes of subgroups — too many to draw. Only its 117 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#19128256C211yesmaximal, characteristic, complemented
2#124×D83C211yesmaximal, characteristic
3#19128063C211yesmaximal, characteristic, complemented
4#19128061C211yesmaximal, characteristic, complemented
5#19128060C211yesmaximal, characteristic, complemented
6#19128057C212yesmaximal, complemented
7#19128054C212yesmaximal, complemented
8#19127708C212yesmaximal, complemented
9#19127704C212yesmaximal
10#19127653C211yesmaximal, characteristic, complemented
11#489×C83C211yesmaximal, characteristic, complemented
12C2×Q8×D83C2211yescharacteristic
13C2×D4×D83C2211yescharacteristic
14C2×#13050573C2212yes—
15#14581792C2211yescharacteristic, complemented
16#14581783C2212yescomplemented
17#14581781C2211yescharacteristic
18#14581780C2211yescharacteristic
19#14581779C2211yescharacteristic
20#14581778—21no—
21#14581778—21no—
22#14581772—21no—
23#14581768—21no—
24Q8×Dic83—21no—
25#14581757—21no—
26#14581750C2212yes—
27C22⋊C4×D83—21no—
28C22⋊C4×D83—21no—
29C42×D83C2211yescharacteristic
30#14581717C2212yescomplemented
31#14581716C2212yescomplemented
32#14581715C2212yescomplemented
33#14581714C2212yes—
34#14581713C2212yes—
35#14581712C2212yes—
36#14581703C2211yescharacteristic, complemented
37C8×Dic83C2212yescomplemented
38C4⋊Q8×C83C2211yescharacteristic, complemented
39#124×C83C2211yescharacteristic
40#103×C83C2212yescomplemented
41#102×C83C2212yes—
42#97×C83C2211yescharacteristic, complemented
43C23×D83—21no—
44C2×Dic166—21no—
45C2×Dic166—21no—
46C2×Dic166C2311yescharacteristic
47Q8×D83—24no—
48C2×D332C2311yescharacteristic
49C2×C22⋊D83—21no—
50D4×D83—24no—
51C2×C4×D83—21no—
52C2×C4×D83D411yescharacteristic
53C2×C4×D83D411yescharacteristic
54C2×C4×D83C2311yescharacteristic
55#13050573—24no—
56C2×C83⋊C8C2312yes—
57C4⋊Dic83—21no—
58C4⋊Dic83—22no—
59C4⋊Dic83C2312yes—
60Dic83⋊C4—22no—
61C22⋊Dic83—21no—
62C22⋊Dic83—21no—
63(C2×D83)⋊C4—22no—
64(C2×D83)⋊C4—21no—
65(C2×D83)⋊C4—21no—
66(C2×D83)⋊C4—21no—
67(C2×D83)⋊C4—21no—
68C4×Dic83—22no—
69C4×Dic83D411yescharacteristic
70C4×Dic83D411yescharacteristic
71C4×Dic83C2311yescharacteristic
72C2×Q8×C83—21no—
73C2×Q8×C83C2311yescharacteristic
74C2×D4×C83C2311yescharacteristic
75C2×C8×C83C2312yes—
76C4⋊C4×C83—21no—
77C4⋊C4×C83C2312yes—
78C22⋊C4×C83—21no—
79C22⋊C4×C83—21no—
80C42×C83C2311yescharacteristic
81C23×C83—21no—
82C22×D83—21no—
83C22×D83—42no—
84C22×D83—42no—
85C22×D83—42no—
86C22×D83C2×D411yescharacteristic
87Q8×C83—22no—
88Q8×C83—42no—
89D4×C83—22no—
90C2×C4×C83—21no—
91C2×C4×C83—22no—
92C2×C4×C83C2×D411yescharacteristic
93C2×C4×C83C2×D411yescharacteristic
94C2×C4×C83C2411yescharacteristic
95C22⋊D83—42no—
96C2×Dic83—22no—
97C2×Dic83—21no—
98C2×Dic83C2×D411yescharacteristic
99C2×Dic83C2×D411yescharacteristic
100C2×Dic83C2×D411yescharacteristic
101D332—22no—
102C4×D83—41no—
103C4×D83—41no—
104C4×D83—42no—
105C4×D83#4114yes—
106Dic166—22no—
107Dic166—42no—
108Dic166—42no—
109C8×C83—22no—
110C83⋊C8—22no—
111C22×C83—41no—
112C22×C83—41no—
113C22×C83—41no—
114C22×C83C22×D411yescharacteristic
115C2×D83—21no—
116C2×D83—21no—
117C2×D83—41no—
118C2×D83—41no—
119C2×D83—41no—
120C2×D83—84no—
121C2×D83#12412yes—
122C4×C83—21no—
123C4×C83—21no—
124C4×C83—41no—
125C4×C83—42no—
126C4×C83C2×#4112yes—
127Dic83—21no—
128Dic83—21no—
129Dic83—41no—
130Dic83—42no—
131Dic83#12412yes—
132C2×C83—81no—
133C2×C83C2×#12411yescharacteristic
134C2×C83#57511yescharacteristic
135C2×C83#57511yescharacteristic
136D83—22no—
137D83—81no—
138#3140—831nomaximal, Sylow
139C83#314011yescharacteristic, complemented, Sylow
140#533—831no—
141C2×#124—831no—
142#489—831no—
143#489—831no—
144#489—831no—
145#489D8311yescharacteristic, complemented
146#488—832no—
147#488—832no—
148#419—832no—
149#418—832no—
150C4×M16—831no—
151C22×Q8—831no—
152C22×D4—831no—
153C2×M16—832no—
154C4⋊Q8—831no—
155C4⋊Q8C2×D8311yescharacteristic, complemented
156#125—832no—
157#124—831no—
158#124—831no—
159#124—831no—
160#124—1661no—
161#124—1661no—
162#124C2×D8311yescharacteristic
163C22⋊Q8—1661no—
164C22⋊Q8—1661no—
165C4×Q8—1661no—
166C4×Q8—1661no—
167#117—832no—
168C2×C22⋊C4—1661no—
169C2×C22⋊C4—1661no—
170C2×C42—831no—
171#103—832no—
172#103—832no—
173#103—832no—
174#103C2×D8312yescomplemented
175#102—832no—
176#102—832no—
177#102—832no—
178#102C2×D8312yes—
179#97—831no—
180#97C2×D8311yescharacteristic, complemented
181C4×C8—832no—
182C24—1661no—
183C2×Q8—21no—
184C2×Q8—831no—
185C2×Q8—1661no—
186C2×Q8—1664no—
187C2×Q8—1661no—
188C2×Q8C22×D8311yescharacteristic
189C2×D4—831no—
190C2×D4—1664no—
191C2×D4—1661no—
192C2×D4C22×D8311yescharacteristic
193C22×C4—831no—
194C22×C4—831no—
195C22×C4—831no—
196C22×C4—1661no—
197M16—1664no—
198C2×C8—832no—
199C2×C8C22×D8312yes—
200C4⋊C4—21no—
201C4⋊C4—832no—
202C4⋊C4—1662no—
203C4⋊C4—1662no—
204C4⋊C4—1661no—
205C4⋊C4C22×D8312yes—
206C22⋊C4—21no—
207C22⋊C4—21no—
208C22⋊C4—1661no—
209C22⋊C4—1661no—
210C22⋊C4—1662no—
211C22⋊C4—1661no—
212C22⋊C4—1661no—
213C22⋊C4—1661no—
214C22⋊C4—1661no—
215C42—831no—
216C42—831no—
217C42—831no—
218C42—1662no—
219C42C22×D8311yescharacteristic
220C23—21no—
221C23—3322no—
222C23—3322no—
223C23—3322no—
224C23—831no—
225C23—1661no—
226Q8—22no—
227Q8—42no—
228Q8—3322no—
229Q8—3322no—
230Q8—1662no—
231D4—22no—
232D4—3322no—
233D4—1662no—
234C2×C4—22no—
235C2×C4—21no—
236C2×C4—3321no—
237C2×C4—3321no—
238C2×C4—3322no—
239C2×C4—831no—
240C2×C4—834no—
241C2×C4—831no—
242C2×C4—831no—
243C2×C4—1662no—
244C2×C4—1661no—
245C2×C4D4×D8311yescharacteristic
246C2×C4D4×D8311yescharacteristic
247C2×C4C23×D8311yescharacteristic
248C8—22no—
249C8—1662no—
250C22—41no—
251C22—41no—
252C22—41no—
253C22—3321no—
254C22—3321no—
255C22—3321no—
256C22—832no—
257C22—6644no—
258C22—1661no—
259C22—1661no—
260C22C2×D4×D8311yescharacteristic
261C4—21no—
262C4—21no—
263C4—41no—
264C4—42no—
265C4—3321no—
266C4—3322no—
267C4—832no—
268C4—1661no—
269C4—1661no—
270C4#41×D8312yes—
271C2—81no—
272C2—6641no—
273C2—1662no—
274C2#124×D8311yescharacteristic
275C2#1912849711yescharacteristic
276C2#1912850011yescharacteristic
subgroupG/H
centerC22C2×D4×D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting#489×C83C2
socleC22×C83C22×D4
prime radicalC2×D4×D83C22
hypercenter#489D83
hypocenterC83#3140
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C42 ◁ #489

Complements

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

NH ≅ G/Ntype
1. #19128256C2semidirect
3. #19128063C2semidirect
4. #19128061C2semidirect
5. #19128060C2semidirect
6. #19128057C2semidirect
7. #19128054C2semidirect
8. #19127708C2semidirect
10. #19127653C2semidirect
11. #489×C83C2semidirect
15. #14581792C22semidirect
16. #14581783C22semidirect
30. #14581717C22semidirect
31. #14581716C22semidirect
32. #14581715C22semidirect
36. #14581703C22semidirect
37. C8×Dic83C22semidirect
38. C4⋊Q8×C83C22semidirect
40. #103×C83C22semidirect
42. #97×C83C22semidirect
139. C83#3140semidirect
145. #489D83semidirect
155. C4⋊Q8C2×D83semidirect
174. #103C2×D83semidirect
180. #97C2×D83semidirect

Maximal subgroups

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

subgroupindex
#191282562
#124×D832
#191280632
#191280612
#191280602
#191280572
#191280542
#191277082
#191277042
#191276532
#489×C832
#314083