Finite Group Databasebeta

← #24917422

Subgroups — #24917422

Order 10624.

418 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 418 conjugacy classes of subgroups — too many to draw. Only its 133 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#19128265C211yesmaximal, characteristic
2#19128184C211yesmaximal, characteristic, complemented
3#19128026C212yesmaximal, complemented
4#19128026C212yesmaximal, complemented
5#19128020C212yesmaximal, complemented
6#19128015C212yesmaximal, complemented
7#19127743C211yesmaximal, characteristic, complemented
8#19127739C211yesmaximal, characteristic, complemented
9#19127707C211yesmaximal, characteristic, complemented
10C2×#14581714C211yesmaximal, characteristic, complemented
11#408×C83C211yesmaximal, characteristic, complemented
12C22×D332C2211yescharacteristic
13#14581788—21no—
14#14581781—21no—
15#14581766C2212yes—
16#14581766C2212yes—
17#14581760—22no—
18Q8×Dic83—21no—
19D4×Dic83—22no—
20D4×Dic83—21no—
21#14581751C2211yescharacteristic
22#14581751C2211yescharacteristic
23#14581751C2212yescomplemented
24C2×C4⋊Dic83C2211yescharacteristic
25C2×C4⋊Dic83C2212yescomplemented
26#14581726C2211yescharacteristic, complemented
27#14581726C2211yescharacteristic, complemented
28#14581726C2212yescomplemented
29#14581724C2211yescharacteristic, complemented
30#14581724C2211yescharacteristic, complemented
31#14581724C2212yescomplemented
32#14581714C2212yes—
33#14581714C2212yes—
34#14581714C2212yes—
35#14581714C2212yes—
36C2×#41×C83C2211yescharacteristic
37C2×M16×C83C2211yescharacteristic
38C22×C8×C83C2211yescharacteristic
39C22⋊C8×C83C2212yescomplemented
40C22⋊C8×C83C2212yescomplemented
41C23×D83—21no—
42C2×D332—22no—
43C2×D332—22no—
44C2×D332—21no—
45C2×D332C2312yes—
46C2×C22⋊D83—22no—
47C2×C22⋊D83—22no—
48C22×Dic83—21no—
49C22×Dic83—22no—
50C4⋊Dic83—21no—
51C4⋊Dic83—22no—
52C4⋊Dic83C2311yescharacteristic
53C4⋊Dic83C2311yescharacteristic
54C4⋊Dic83C2312yes—
55C4⋊Dic83C2311yescharacteristic
56C4⋊Dic83C2311yescharacteristic
57C22⋊Dic83—21no—
58C22⋊Dic83—21no—
59C22⋊Dic83—22no—
60(C2×D83)⋊C4—22no—
61(C2×D83)⋊C4—22no—
62C4×Dic83—21no—
63C4×Dic83—21no—
64C4×Dic83—22no—
65#41×C83—22no—
66#41×C83—22no—
67C2×Q8×C83D411yescharacteristic
68C2×D4×C83D412yes—
69C2×D4×C83D411yescharacteristic
70C22×C4×C83C2312yes—
71C22×C4×C83C2311yescharacteristic
72M16×C83—22no—
73C2×C8×C83—22no—
74C2×C8×C83C2311yescharacteristic
75C2×C8×C83C2311yescharacteristic
76C2×C8×C83C2312yes—
77C23×C83C2×D412yes—
78C23×C83C2×D411yescharacteristic
79C22×D83—22no—
80C22×D83—42no—
81C22×D83—81no—
82C22×D83—81no—
83Q8×C83—22no—
84D4×C83—24no—
85D4×C83—22no—
86C2×C4×C83—22no—
87C2×C4×C83—22no—
88C2×C4×C83C2×D412yes—
89C2×C4×C83C2×D411yescharacteristic
90C2×C4×C83#4112yes—
91C2×C4×C83#4112yes—
92C2×C4×C83C2411yescharacteristic
93C22⋊D83—82no—
94C22⋊D83—82no—
95C2×Dic83—21no—
96C2×Dic83—21no—
97C2×Dic83—21no—
98C2×Dic83—21no—
99C2×Dic83—22no—
100C2×Dic83—81no—
101C2×Dic83—82no—
102D332—22no—
103D332—22no—
104D332—42no—
105C8×C83—21no—
106C8×C83—21no—
107C8×C83—22no—
108C22×C83—21no—
109C22×C83—21no—
110C22×C83—22no—
111C22×C83—22no—
112C22×C83—22no—
113C22×C83#12312yes—
114C22×C83C22×D411yescharacteristic
115C2×D83—42no—
116C2×D83—42no—
117C2×D83—42no—
118C2×D83—84no—
119C4×C83—22no—
120C4×C83#12312yes—
121C4×C83C2×#4111yescharacteristic
122C4×C83C2×#4111yescharacteristic
123Dic83—41no—
124Dic83—42no—
125Dic83—41no—
126Dic83—41no—
127Dic83—41no—
128C2×C83—22no—
129C2×C83—42no—
130C2×C83C2×#12311yescharacteristic
131C2×C83#57611yescharacteristic
132C2×C83#57711yescharacteristic
133D83—82no—
134#3096—831nomaximal, Sylow
135C83#309611yescharacteristic, complemented, Sylow
136#535—831no—
137#520—831no—
138#482—832no—
139#482—832no—
140#481—832no—
141#480—832no—
142#428—831no—
143#427—831no—
144#418—831no—
145C2×#102—831no—
146#408D8311yescharacteristic, complemented
147C2×#41C2×D8311yescharacteristic
148C22×D4—831no—
149C2×M16C2×D8311yescharacteristic
150C22×C8C2×D8311yescharacteristic
151#127—1661no—
152#124—1661no—
153#121—832no—
154#121—832no—
155#120—1662no—
156C4×Q8—1661no—
157C4×D4—1662no—
158C4×D4—1661no—
159#117—831no—
160#117—832no—
161#117—831no—
162C2×C4⋊C4—831no—
163C2×C4⋊C4—832no—
164#107—831no—
165#107—831no—
166#107—832no—
167#106—831no—
168#106—831no—
169#106—832no—
170#102—832no—
171#102—832no—
172#102—832no—
173#102—832no—
174C22⋊C8C2×D8312yescomplemented
175C22⋊C8C2×D8312yescomplemented
176C24—1661no—
177#41—22no—
178#41—22no—
179C2×Q8D33211yescharacteristic
180C2×D4—832no—
181C2×D4—1662no—
182C2×D4—1662no—
183C2×D4—1661no—
184C2×D4—1662no—
185C2×D4—1662no—
186C2×D4D33212yes—
187C2×D4D33211yescharacteristic
188C22×C4—1662no—
189C22×C4—1661no—
190C22×C4C22×D8312yes—
191C22×C4C22×D8311yescharacteristic
192M16—22no—
193C2×C8—22no—
194C2×C8C22×D8311yescharacteristic
195C2×C8C22×D8312yes—
196C2×C8C22×D8311yescharacteristic
197C4⋊C4—832no—
198C4⋊C4—831no—
199C4⋊C4—831no—
200C4⋊C4—831no—
201C4⋊C4—831no—
202C4⋊C4—1662no—
203C4⋊C4—1661no—
204C22⋊C4—1662no—
205C22⋊C4—1661no—
206C22⋊C4—1661no—
207C22⋊C4—1662no—
208C22⋊C4—1662no—
209C42—1662no—
210C42—1661no—
211C42—1661no—
212C23—3322no—
213C23—6641no—
214C23—6641no—
215C23—1662no—
216C23C2×D33212yes—
217C23C2×D33211yescharacteristic
218Q8—22no—
219D4—24no—
220D4—22no—
221D4—3322no—
222D4—6642no—
223D4—6642no—
224D4—1662no—
225D4—1662no—
226C2×C4—22no—
227C2×C4—22no—
228C2×C4—6641no—
229C2×C4—6642no—
230C2×C4—1661no—
231C2×C4—1661no—
232C2×C4—1661no—
233C2×C4—1662no—
234C2×C4—1661no—
235C2×C4C2×D33212yes—
236C2×C4C2×D33211yescharacteristic
237C2×C4#1305058912yes—
238C2×C4#1305058912yes—
239C2×C4C23×D8311yescharacteristic
240C8—21no—
241C8—21no—
242C8—22no—
243C22—21no—
244C22—21no—
245C22—22no—
246C22—22no—
247C22—22no—
248C22—3322no—
249C22—3322no—
250C22—3322no—
251C22—6644no—
252C22#1458177612yes—
253C22C22×D33211yescharacteristic
254C4—22no—
255C4—3322no—
256C4—3321no—
257C4—3321no—
258C4—3321no—
259C4—3321no—
260C4#1458177612yes—
261C4C2×#1305058911yescharacteristic
262C4C2×#1305058911yescharacteristic
263C2—22no—
264C2—42no—
265C2—6642no—
266C2C2×#1458177611yescharacteristic
267C2#1912850611yescharacteristic
268C2#1912851311yescharacteristic
subgroupG/H
centerC22C22×D332
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting#408×C83C2
socleC22×C83C22×D4
prime radical#19128265C2
hypercenter#408D83
hypocenterC83#3096
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C2×#41 ◁ #408

Complements

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

NH ≅ G/Ntype
2. #19128184C2semidirect
3. #19128026C2semidirect
4. #19128026C2semidirect
5. #19128020C2semidirect
6. #19128015C2semidirect
7. #19127743C2semidirect
8. #19127739C2semidirect
9. #19127707C2semidirect
10. C2×#14581714C2semidirect
11. #408×C83C2semidirect
23. #14581751C22semidirect
25. C2×C4⋊Dic83C22semidirect
26. #14581726C22semidirect
27. #14581726C22semidirect
28. #14581726C22semidirect
29. #14581724C22semidirect
30. #14581724C22semidirect
31. #14581724C22semidirect
39. C22⋊C8×C83C22semidirect
40. C22⋊C8×C83C22semidirect
135. C83#3096semidirect
146. #408D83semidirect
174. C22⋊C8C2×D83semidirect
175. C22⋊C8C2×D83semidirect

Maximal subgroups

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

subgroupindex
#191282652
#191281842
#191280262
#191280202
#191280152
#191277432
#191277392
#191277072
C2×#145817142
#408×C832
#309683