Finite Group Databasebeta

← #24919695

Subgroups — #24919695

Order 10624.

448 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 448 conjugacy classes of subgroups — too many to draw. Only its 123 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#19128421C211yesmaximal, characteristic, complemented
2D4×D332C211yesmaximal, characteristic
3#19128048C211yesmaximal, characteristic, complemented
4#19128026C212yesmaximal, complemented
5#19127982C211yesmaximal, characteristic, complemented
6#19127947C212yesmaximal, complemented
7#19127902C211yesmaximal, characteristic, complemented
8#19127854C212yesmaximal, complemented
9C4×#13050579C211yesmaximal, characteristic, complemented
10#19127707C212yesmaximal, complemented
11#436×C83C211yesmaximal, characteristic, complemented
12C22×D332C2212yes—
13C2×D4×D83—21no—
14C2×#13050579C2211yescharacteristic
15C2×#13050579C2212yescomplemented
16#14581792C2211yescharacteristic, complemented
17#14581789C2211yescharacteristic
18#14581785—22no—
19#14581779—21no—
20#14581771C2212yescomplemented
21#14581767—21no—
22#14581766C2212yes—
23#14581762—22no—
24C4×Dic166C2211yescharacteristic, complemented
25#14581757—21no—
26C4×D332C2211yescharacteristic
27#14581751C2212yescomplemented
28#14581726C2212yescomplemented
29#14581724C2211yescharacteristic, complemented
30#14581717C2211yescharacteristic, complemented
31#14581714C2212yes—
32#14581714C2211yescharacteristic
33#14581714C2212yescomplemented
34#14581714C2212yescomplemented
35#14581714C2212yescomplemented
36C2×M16×C83C2212yescomplemented
37C4×D4×C83C2211yescharacteristic
38C4⋊C8×C83C2211yescharacteristic, complemented
39C22⋊C8×C83C2212yescomplemented
40C4×C8×C83C2211yescharacteristic, complemented
41C23×D83—22no—
42C2×Dic166—21no—
43C2×Dic166C2311yescharacteristic
44C2×D332—24no—
45C2×D332—22no—
46C2×D332—21no—
47C2×D332—21no—
48C2×D332—42no—
49C2×D332C2312yes—
50C2×D332C2311yescharacteristic
51C2×C22⋊D83—22no—
52D4×D83—44no—
53C2×C4×D83—21no—
54#13050579—22no—
55#13050579—42no—
56C4⋊Dic83—21no—
57C4⋊Dic83C2311yescharacteristic
58C4⋊Dic83C2312yes—
59C4⋊Dic83C2311yescharacteristic
60Dic83⋊C4—22no—
61Dic83⋊C4—21no—
62C22⋊Dic83—22no—
63C22⋊Dic83—22no—
64(C2×D83)⋊C4—21no—
65(C2×D83)⋊C4—22no—
66C4×Dic83—21no—
67C4×Dic83—22no—
68C2×D4×C83D411yescharacteristic
69C22×C4×C83C2312yes—
70M16×C83—42no—
71C2×C8×C83C2311yescharacteristic
72C2×C8×C83C2312yes—
73C2×C8×C83C2311yescharacteristic
74C4⋊C4×C83D411yescharacteristic
75C22⋊C4×C83D412yes—
76C42×C83C2311yescharacteristic
77C23×C83C2×D412yes—
78C22×D83—21no—
79C22×D83—22no—
80C22×D83—44no—
81C22×D83—44no—
82C22×D83—82no—
83D4×C83—22no—
84C2×C4×C83—22no—
85C2×C4×C83—22no—
86C2×C4×C83C2×D411yescharacteristic
87C2×C4×C83C2×D412yes—
88C2×C4×C83C2×D411yescharacteristic
89C2×C4×C83C2411yescharacteristic
90C22⋊D83—44no—
91C2×Dic83—21no—
92C2×Dic83—21no—
93C2×Dic83—22no—
94C2×Dic83—21no—
95D332—22no—
96D332—21no—
97D332—22no—
98D332—44no—
99D332—42no—
100D332—81no—
101D332#4112yes—
102C4×D83—42no—
103Dic166—21no—
104Dic166—21no—
105Dic166—42no—
106C8×C83—21no—
107C8×C83—21no—
108C8×C83—22no—
109C22×C83—22no—
110C22×C83—22no—
111C22×C83—22no—
112C22×C83C22×D411yescharacteristic
113C2×D83—22no—
114C2×D83—41no—
115C2×D83—41no—
116C2×D83—42no—
117C2×D83—42no—
118C2×D83—42no—
119C2×D83—88no—
120C4×C83—22no—
121C4×C83—21no—
122C4×C83AGL(1,ℤ/8)12yes—
123C4×C83C2×#4111yescharacteristic
124C4×C83H(5,𝔽2)11yescharacteristic
125Dic83—41no—
126Dic83—42no—
127Dic83—41no—
128Dic83—41no—
129C2×C83—42no—
130C2×C83#54611yescharacteristic
131C2×C83C2×AGL(1,ℤ/8)11yescharacteristic
132C2×C83#57711yescharacteristic
133D83—42no—
134D83—82no—
135#3324—831nomaximal, Sylow
136C83#332411yescharacteristic, complemented, Sylow
137#556—831no—
138D42—831no—
139#486—831no—
140#482—832no—
141#474—831no—
142#468—832no—
143#460—831no—
144#450—832no—
145C4×SD16—831no—
146#436D8311yescharacteristic, complemented
147#418—832no—
148C22×D4—832no—
149C22×D4—1661no—
150C2×SD16—832no—
151C2×SD16—831no—
152C2×M16C2×D8312yescomplemented
153C4⋊Q8—831no—
154#127—831no—
155#126—1662no—
156#124—1661no—
157C22⋊Q8—832no—
158#121—832no—
159#121—1661no—
160#120—1662no—
161C4×Q8—831no—
162C4×Q8—1661no—
163C4×D4—831no—
164C4×D4C2×D8311yescharacteristic
165#117—832no—
166#107—832no—
167#106—831no—
168C4⋊C8C2×D8311yescharacteristic, complemented
169#103—831no—
170#102—831no—
171#102—832no—
172#102—832no—
173#102—832no—
174#102—832no—
175C22⋊C8C2×D8312yescomplemented
176C4×C8C2×D8311yescharacteristic, complemented
177C24—1662no—
178C2×Q8—831no—
179C2×Q8—1661no—
180C2×D4—3322no—
181C2×D4—3324no—
182C2×D4—831no—
183C2×D4—832no—
184C2×D4—1664no—
185C2×D4—1662no—
186C2×D4—1662no—
187C2×D4—1661no—
188C2×D4—1661no—
189C2×D4D33211yescharacteristic
190C22×C4—1661no—
191C22×C4C22×D8312yes—
192SD16—3322no—
193SD16—1662no—
194M16—42no—
195C2×C8C22×D8311yescharacteristic
196C2×C8C22×D8311yescharacteristic
197C2×C8C22×D8312yes—
198C4⋊C4—831no—
199C4⋊C4—832no—
200C4⋊C4—831no—
201C4⋊C4—1661no—
202C4⋊C4—1662no—
203C4⋊C4—1661no—
204C4⋊C4D33211yescharacteristic
205C22⋊C4—1662no—
206C22⋊C4—1662no—
207C22⋊C4—1661no—
208C22⋊C4—1662no—
209C22⋊C4D33212yes—
210C42—1662no—
211C42—1661no—
212C42C22×D8311yescharacteristic
213C23—3324no—
214C23—3324no—
215C23—6642no—
216C23—1662no—
217C23—1661no—
218C23C2×D33212yes—
219Q8—3322no—
220Q8—1661no—
221Q8—1661no—
222D4—22no—
223D4—3324no—
224D4—3324no—
225D4—3322no—
226D4—832no—
227D4—6641no—
228D4—1662no—
229D4—1662no—
230D4—1661no—
231C2×C4—22no—
232C2×C4—22no—
233C2×C4—3322no—
234C2×C4—1661no—
235C2×C4—1662no—
236C2×C4—1661no—
237C2×C4—1661no—
238C2×C4C2×D33212yes—
239C2×C4C2×D33211yescharacteristic
240C2×C4C2×D33211yescharacteristic
241C2×C4C23×D8311yescharacteristic
242C8—21no—
243C8—21no—
244C8—22no—
245C22—22no—
246C22—22no—
247C22—22no—
248C22—3321no—
249C22—3321no—
250C22—3322no—
251C22—3322no—
252C22—3322no—
253C22—6648no—
254C22—1662no—
255C22C22×D33211yescharacteristic
256C4—22no—
257C4—21no—
258C4—3321no—
259C4—3321no—
260C4—3322no—
261C4—3321no—
262C4#1458182012yes—
263C4#41×D8311yescharacteristic
264C4#1458183711yescharacteristic
265C2—42no—
266C2—3322no—
267C2—6642no—
268C2#1912834611yescharacteristic
269C2C2×#1458182011yescharacteristic
270C2#1912851311yescharacteristic
subgroupG/H
centerC22C22×D332
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting#436×C83C2
socleC22×C83C22×D4
prime radicalD4×D332C2
hypercenter#436D83
hypocenterC83#3324
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C4×D4 ◁ #436

Complements

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

NH ≅ G/Ntype
1. #19128421C2semidirect
3. #19128048C2semidirect
4. #19128026C2semidirect
5. #19127982C2semidirect
6. #19127947C2semidirect
7. #19127902C2semidirect
8. #19127854C2semidirect
9. C4×#13050579C2semidirect
10. #19127707C2semidirect
11. #436×C83C2semidirect
15. C2×#13050579C22semidirect
16. #14581792C22semidirect
20. #14581771C22semidirect
24. C4×Dic166C22semidirect
27. #14581751C22semidirect
28. #14581726C22semidirect
29. #14581724C22semidirect
30. #14581717C22semidirect
33. #14581714C22semidirect
34. #14581714C22semidirect
35. #14581714C22semidirect
36. C2×M16×C83C22semidirect
38. C4⋊C8×C83C22semidirect
39. C22⋊C8×C83C22semidirect
40. C4×C8×C83C22semidirect
136. C83#3324semidirect
146. #436D83semidirect
152. C2×M16C2×D83semidirect
168. C4⋊C8C2×D83semidirect
175. C22⋊C8C2×D83semidirect
176. C4×C8C2×D83semidirect

Maximal subgroups

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

subgroupindex
#191284212
D4×D3322
#191280482
#191280262
#191279822
#191279472
#191279022
#191278542
C4×#130505792
#191277072
#436×C832
#332483