Finite Group Databasebeta

← #24919688

Subgroups — #24919688

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
1D4×D332C211yesmaximal, characteristic
2#19128049C211yesmaximal, characteristic, complemented
3#19128025C212yesmaximal, complemented
4#19127978C211yesmaximal, characteristic, complemented
5#19127949C212yesmaximal, complemented
6#19127901C211yesmaximal, characteristic, complemented
7#19127851C212yesmaximal, complemented
8C4×#13050577C211yesmaximal, characteristic, complemented
9#19127779C211yesmaximal, characteristic, complemented
10#19127705C212yesmaximal, complemented
11#556×C83C211yesmaximal, characteristic, complemented
12C22×D332C2212yes—
13C2×D4×D83—21no—
14C2×#13050577C2211yescharacteristic
15C2×#13050577C2212yescomplemented
16C2×#13050574C2212yescomplemented
17#14581789C2211yescharacteristic
18#14581767—21no—
19#14581766C2212yes—
20#14581762—22no—
21C4×D332C2211yescharacteristic
22#14581725C2212yescomplemented
23#14581723C2211yescharacteristic, complemented
24C4⋊(C83⋊C8)C2211yescharacteristic, complemented
25#14581716C2211yescharacteristic, complemented
26#14581712C2212yes—
27#14581712C2211yescharacteristic
28#14581712C2212yescomplemented
29#14581712C2212yescomplemented
30#14581712C2212yescomplemented
31C22⋊(C83⋊C8)C2212yescomplemented
32C4×C83⋊C8C2211yescharacteristic, complemented
33C4⋊Q8×C83C2211yescharacteristic, complemented
34#126×C83—22no—
35#124×C83—21no—
36C22⋊Q8×C83C2212yescomplemented
37C4×Q8×C83—21no—
38C4×Q8×C83C2211yescharacteristic, complemented
39C4×D4×C83C2211yescharacteristic
40#117×C83C2212yescomplemented
41C23×D83—22no—
42C2×D332—24no—
43C2×D332—22no—
44C2×D332—21no—
45C2×D332—21no—
46C2×D332—42no—
47C2×D332C2312yes—
48C2×D332C2311yescharacteristic
49C2×C22⋊D83—22no—
50D4×D83—44no—
51C2×C4×D83—21no—
52#13050577—22no—
53#13050577—42no—
54#13050574—42no—
55C2×C83⋊C8C2311yescharacteristic
56C2×C83⋊C8C2312yes—
57C2×C83⋊C8C2311yescharacteristic
58C4⋊Dic83C2311yescharacteristic
59(C2×D83)⋊C4—21no—
60(C2×D83)⋊C4—22no—
61C2×Q8×C83—21no—
62C2×Q8×C83C2311yescharacteristic
63C2×D4×C83D411yescharacteristic
64C22×C4×C83C2312yes—
65C4⋊C4×C83—22no—
66C4⋊C4×C83—21no—
67C4⋊C4×C83—21no—
68C4⋊C4×C83D411yescharacteristic
69C4⋊C4×C83C2311yescharacteristic
70C4⋊C4×C83C2312yes—
71C22⋊C4×C83—22no—
72C22⋊C4×C83—22no—
73C22⋊C4×C83D412yes—
74C42×C83—22no—
75C42×C83—21no—
76C42×C83C2311yescharacteristic
77C23×C83C2×D412yes—
78C22×D83—21no—
79C22×D83—22no—
80C22×D83—44no—
81C22×D83—44no—
82C22×D83—82no—
83Q8×C83—21no—
84Q8×C83—21no—
85Q8×C83—42no—
86D4×C83—22no—
87C2×C4×C83—22no—
88C2×C4×C83—22no—
89C2×C4×C83—21no—
90C2×C4×C83—21no—
91C2×C4×C83—22no—
92C2×C4×C83C2×D412yes—
93C2×C4×C83C2×D411yescharacteristic
94C2×C4×C83C2×D411yescharacteristic
95C2×C4×C83C2411yescharacteristic
96C22⋊D83—44no—
97C2×Dic83—21no—
98D332—22no—
99D332—21no—
100D332—22no—
101D332—44no—
102D332—42no—
103D332—81no—
104D332#4112yes—
105C4×D83—42no—
106C83⋊C8—21no—
107C83⋊C8—21no—
108C83⋊C8—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×C83—41no—
123C4×C83—42no—
124C4×C83—41no—
125C4×C83AGL(1,ℤ/8)12yes—
126C4×C83C2×#4111yescharacteristic
127C4×C83H(5,𝔽2)11yescharacteristic
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#556D8311yescharacteristic, complemented
138D42—831no—
139#486—831no—
140#482—832no—
141#474—831no—
142#468—832no—
143#460—831no—
144#450—832no—
145C4×SD16—831no—
146#436—831no—
147#418—832no—
148C22×D4—832no—
149C22×D4—1661no—
150C2×SD16—832no—
151C2×SD16—831no—
152C2×M16—832no—
153C4⋊Q8C2×D8311yescharacteristic, complemented
154#127—831no—
155#126—22no—
156#124—21no—
157C22⋊Q8C2×D8312yescomplemented
158#121—832no—
159#121—1661no—
160#120—1662no—
161C4×Q8—21no—
162C4×Q8C2×D8311yescharacteristic, complemented
163C4×D4—831no—
164C4×D4C2×D8311yescharacteristic
165#117C2×D8312yescomplemented
166#107—832no—
167#106—831no—
168C4⋊C8—831no—
169#103—831no—
170#102—831no—
171#102—832no—
172#102—832no—
173#102—832no—
174#102—832no—
175C22⋊C8—832no—
176C4×C8—831no—
177C24—1662no—
178C2×Q8—21no—
179C2×Q8C22×D8311yescharacteristic
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×D4C22⋊D8311yescharacteristic
190C22×C4—1661no—
191C22×C4C22×D8312yes—
192SD16—3322no—
193SD16—1662no—
194M16—3322no—
195C2×C8—831no—
196C2×C8—831no—
197C2×C8—832no—
198C4⋊C4—22no—
199C4⋊C4—21no—
200C4⋊C4—21no—
201C4⋊C4—831no—
202C4⋊C4C22⋊D8311yescharacteristic
203C4⋊C4C22×D8311yescharacteristic
204C4⋊C4C22×D8312yes—
205C22⋊C4—22no—
206C22⋊C4—22no—
207C22⋊C4—1661no—
208C22⋊C4—1662no—
209C22⋊C4C22⋊D8312yes—
210C42—22no—
211C42—21no—
212C42C22×D8311yescharacteristic
213C23—3324no—
214C23—3324no—
215C23—6642no—
216C23—1662no—
217C23—1661no—
218C23C2×C22⋊D8312yes—
219Q8—21no—
220Q8—21no—
221Q8—42no—
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—21no—
234C2×C4—21no—
235C2×C4—22no—
236C2×C4—3322no—
237C2×C4—1661no—
238C2×C4C2×C22⋊D8312yes—
239C2×C4C2×C22⋊D8311yescharacteristic
240C2×C4C2×C22⋊D8311yescharacteristic
241C2×C4C23×D8311yescharacteristic
242C8—1661no—
243C8—1661no—
244C8—1662no—
245C22—22no—
246C22—22no—
247C22—22no—
248C22—3321no—
249C22—3321no—
250C22—3322no—
251C22—3322no—
252C22—3322no—
253C22—6648no—
254C22—1662no—
255C22C22×C22⋊D8311yescharacteristic
256C4—22no—
257C4—21no—
258C4—41no—
259C4—42no—
260C4—41no—
261C4—3321no—
262C4#1458181712yes—
263C4#41×D8311yescharacteristic
264C4#1458183811yescharacteristic
265C2—42no—
266C2—3322no—
267C2—6642no—
268C2#1912833911yescharacteristic
269C2C2×#1458181711yescharacteristic
270C2#1912850911yescharacteristic
subgroupG/H
centerC22C22×C22⋊D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting#556×C83C2
socleC22×C83C22×D4
prime radicalD4×D332C2
hypercenter#556D83
hypocenterC83#3324
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C4×D4 ◁ #556

Complements

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

NH ≅ G/Ntype
2. #19128049C2semidirect
3. #19128025C2semidirect
4. #19127978C2semidirect
5. #19127949C2semidirect
6. #19127901C2semidirect
7. #19127851C2semidirect
8. C4×#13050577C2semidirect
9. #19127779C2semidirect
10. #19127705C2semidirect
11. #556×C83C2semidirect
15. C2×#13050577C22semidirect
16. C2×#13050574C22semidirect
22. #14581725C22semidirect
23. #14581723C22semidirect
24. C4⋊(C83⋊C8)C22semidirect
25. #14581716C22semidirect
28. #14581712C22semidirect
29. #14581712C22semidirect
30. #14581712C22semidirect
31. C22⋊(C83⋊C8)C22semidirect
32. C4×C83⋊C8C22semidirect
33. C4⋊Q8×C83C22semidirect
36. C22⋊Q8×C83C22semidirect
38. C4×Q8×C83C22semidirect
40. #117×C83C22semidirect
136. C83#3324semidirect
137. #556D83semidirect
153. C4⋊Q8C2×D83semidirect
157. C22⋊Q8C2×D83semidirect
162. C4×Q8C2×D83semidirect
165. #117C2×D83semidirect

Maximal subgroups

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

subgroupindex
D4×D3322
#191280492
#191280252
#191279782
#191279492
#191279012
#191278512
C4×#130505772
#191277792
#191277052
#556×C832
#332483