Finite Group Databasebeta

← #24917805

Subgroups — #24917805

Order 10624.

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

normal subgroups only

This group has 392 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#19128250C211yesmaximal, characteristic
3#19128068C212yesmaximal, complemented
4#19128064C212yesmaximal, complemented
5#19128063C211yesmaximal, characteristic, complemented
6#19128061C211yesmaximal, characteristic, complemented
7#19128060C211yesmaximal, characteristic, complemented
8#19127696C212yesmaximal, complemented
9#19127694C212yesmaximal, complemented
10#97×D83C211yesmaximal, characteristic, complemented
11#489×C83C211yesmaximal, characteristic, complemented
12C2×#13050591C2211yescharacteristic
13C2×#13050589C2211yescharacteristic
14C2×C8×D83C2212yes—
15#14581792C2211yescharacteristic
16#14581790C2211yescharacteristic
17#14581788C2211yescharacteristic
18#14581783C2212yes—
19#14581780C2211yescharacteristic
20#14581772—21no—
21#14581768—21no—
22#14581765—21no—
23#14581765—21no—
24Q8×Dic83—21no—
25#14581757—21no—
26(C22⋊D83)⋊C4—21no—
27(C22⋊D83)⋊C4—21no—
28#14581750C2212yes—
29C42×D83C2211yescharacteristic
30#14581717C2212yes—
31#14581716C2212yes—
32#14581715C2212yes—
33#14581714C2212yes—
34#14581713C2212yes—
35#14581712C2212yes—
36#14581703C2211yescharacteristic
37#14581702C2212yes—
38C4⋊Q8×C83C2211yescharacteristic
39#124×C83C2211yescharacteristic
40#103×C83C2212yes—
41#102×C83C2212yes—
42#97×C83C2211yescharacteristic
43#13050591—24no—
44#13050589—24no—
45C2×Dic166—21no—
46C2×Dic166C2311yescharacteristic
47C2×D332—21no—
48C2×D332C2311yescharacteristic
49C2×C22⋊D83—21no—
50C2×C22⋊D83—21no—
51C2×C22⋊D83—21no—
52C22×Dic83—21no—
53C2×C4×D83—21no—
54C2×C4×D83D411yescharacteristic
55C2×C4×D83D411yescharacteristic
56C2×C4×D83C2311yescharacteristic
57C2×C83⋊C8C2312yes—
58C8×D83—24no—
59C4⋊Dic83—21no—
60C4⋊Dic83—22no—
61C4⋊Dic83C2312yes—
62Dic83⋊C4—21no—
63Dic83⋊C4—21no—
64Dic83⋊C4—22no—
65(C2×D83)⋊C4—22no—
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×D83C2×D411yescharacteristic
84Q8×C83—22no—
85Q8×C83—42no—
86D4×C83—22no—
87C2×C4×C83—21no—
88C2×C4×C83—22no—
89C2×C4×C83C2×D411yescharacteristic
90C2×C4×C83C2×D411yescharacteristic
91C2×C4×C83C2411yescharacteristic
92C22⋊D83—42no—
93C22⋊D83—42no—
94C22⋊D83—42no—
95C2×Dic83—22no—
96C2×Dic83—21no—
97C2×Dic83—42no—
98C2×Dic83C2×D411yescharacteristic
99C2×Dic83C2×D411yescharacteristic
100C2×Dic83C2×D411yescharacteristic
101D332—22no—
102D332—42no—
103C4×D83—41no—
104C4×D83—41no—
105C4×D83—42no—
106C4×D83#4114yes—
107Dic166—22no—
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#12412yes—
121C4×C83—21no—
122C4×C83—21no—
123C4×C83—41no—
124C4×C83—42no—
125C4×C83C2×#4112yes—
126Dic83—21no—
127Dic83—21no—
128Dic83—41no—
129Dic83—42no—
130Dic83#12412yes—
131C2×C83—81no—
132C2×C83C2×#12411yescharacteristic
133C2×C83#57511yescharacteristic
134C2×C83#57511yescharacteristic
135D83—22no—
136D83—81no—
137#3135—831nomaximal, Sylow
138C83#313511yescharacteristic, complemented, Sylow
139#533—831no—
140#532—831no—
141#491—832no—
142#490—832no—
143#489—831no—
144#489—831no—
145#489—831no—
146#489D8311yescharacteristic, complemented
147#416—832no—
148#416—832no—
149C2×#97—831no—
150C2×#41—831no—
151C2×#41—831no—
152C22×C8—832no—
153C4⋊Q8—831no—
154C4⋊Q8—831no—
155C4⋊Q8C2×D8311yescharacteristic
156#127—831no—
157#125—832no—
158#124—831no—
159#124C2×D8311yescharacteristic
160C22⋊Q8—1661no—
161C22⋊Q8—1661no—
162#121—1661no—
163#121—1661no—
164C4×Q8—1661no—
165C4×Q8—1661no—
166C4×D4—1661no—
167C4×D4—1661no—
168#117—832no—
169C2×C42—831no—
170#103—832no—
171#103—832no—
172#103—832no—
173#103C2×D8312yes—
174#102—832no—
175#102—832no—
176#102—832no—
177#102C2×D8312yes—
178#97—832no—
179#97—831no—
180#97C2×D8311yescharacteristic
181#41—1664no—
182#41—1664no—
183C2×Q8—21no—
184C2×Q8—831no—
185C2×Q8—1661no—
186C2×Q8C22×D8311yescharacteristic
187C2×D4—831no—
188C2×D4—1661no—
189C2×D4—1661no—
190C2×D4—1661no—
191C2×D4—1661no—
192C2×D4C22×D8311yescharacteristic
193C22×C4—831no—
194C22×C4—831no—
195C22×C4—831no—
196C22×C4—1661no—
197C22×C4—1661no—
198C2×C8—832no—
199C2×C8—1664no—
200C2×C8C22×D8312yes—
201C4⋊C4—21no—
202C4⋊C4—832no—
203C4⋊C4—1661no—
204C4⋊C4—1661no—
205C4⋊C4—1662no—
206C4⋊C4—1662no—
207C4⋊C4—1661no—
208C4⋊C4C22×D8312yes—
209C22⋊C4—21no—
210C22⋊C4—21no—
211C22⋊C4—1662no—
212C22⋊C4—1661no—
213C22⋊C4—1661no—
214C42—831no—
215C42—831no—
216C42—831no—
217C42—1662no—
218C42C22×D8311yescharacteristic
219C23—21no—
220C23—831no—
221C23—1661no—
222Q8—22no—
223Q8—42no—
224Q8—3322no—
225Q8—1662no—
226D4—22no—
227D4—3322no—
228D4—3322no—
229D4—3322no—
230D4—3322no—
231D4—1662no—
232C2×C4—22no—
233C2×C4—21no—
234C2×C4—3321no—
235C2×C4—3321no—
236C2×C4—3322no—
237C2×C4—3322no—
238C2×C4—831no—
239C2×C4—834no—
240C2×C4—831no—
241C2×C4—831no—
242C2×C4—1662no—
243C2×C4—1661no—
244C2×C4D4×D8311yescharacteristic
245C2×C4D4×D8311yescharacteristic
246C2×C4C23×D8311yescharacteristic
247C8—22no—
248C8—1662no—
249C22—41no—
250C22—41no—
251C22—41no—
252C22—3321no—
253C22—3321no—
254C22—3321no—
255C22—832no—
256C22—1661no—
257C22—1661no—
258C22C2×D4×D8311yescharacteristic
259C4—21no—
260C4—21no—
261C4—41no—
262C4—42no—
263C4—3321no—
264C4—3322no—
265C4—832no—
266C4—1661no—
267C4—1661no—
268C4#41×D8312yes—
269C2—81no—
270C2—6641no—
271C2—1662no—
272C2#124×D8311yescharacteristic
273C2#1912849711yescharacteristic
274C2#1912850011yescharacteristic
subgroupG/H
centerC22C2×D4×D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting#489×C83C2
socleC22×C83C22×D4
prime radical#19128250C2
hypercenter#489D83
hypocenterC83#3135
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. #19128068C2semidirect
4. #19128064C2semidirect
5. #19128063C2semidirect
6. #19128061C2semidirect
7. #19128060C2semidirect
8. #19127696C2semidirect
9. #19127694C2semidirect
10. #97×D83C2semidirect
11. #489×C83C2semidirect
138. C83#3135semidirect
146. #489D83semidirect

Maximal subgroups

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

subgroupindex
#191282562
#191282502
#191280682
#191280642
#191280632
#191280612
#191280602
#191276962
#191276942
#97×D832
#489×C832
#313583