Finite Group Databasebeta

← #24916733

Subgroups — #24916733

Order 10624.

784 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 784 conjugacy classes of subgroups — too many to draw. Only its 149 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
1C2×#14581838C211yesmaximal, characteristic, complemented
2C2×#14581820C211yesmaximal, characteristic, complemented
3C22×D664C211yesmaximal, characteristic, complemented
4#19128265C211yesmaximal, characteristic, complemented
5#19127848C214yesmaximal, complemented
6#19127837C214yesmaximal, complemented
7#19127707C211yesmaximal, characteristic, complemented
8#19127698C211yesmaximal, characteristic, complemented
9#408×C83C211yesmaximal, characteristic, complemented
10#14581838—24no—
11#14581838—24no—
12C2×#13050591—21no—
13C2×#13050590C2211yescharacteristic
14C2×#13050590C2212yescomplemented
15C22×D332C2211yescharacteristic
16C22×D332C2212yescomplemented
17C22×D332C2211yescharacteristic, complemented
18C2×D4×D83—22no—
19C2×D4×D83—21no—
20#14581820—24no—
21C2×#13050579C2212yescomplemented
22C2×D664—24no—
23C2×D664C2212yescomplemented
24C2×D664C2214yescomplemented
25#14581788—21no—
26#14581781—21no—
27#14581766C2214yescomplemented
28#14581760—22no—
29#14581751C2211yescharacteristic, complemented
30#14581717C2212yescomplemented
31#14581714C2212yescomplemented
32#14581714C2214yescomplemented
33C2×#41×C83C2211yescharacteristic
34C2×M16×C83C2211yescharacteristic, complemented
35C22×C8×C83C2211yescharacteristic, complemented
36C22⋊C8×C83C2214yescomplemented
37C23×D83—21no—
38C23×D83—22no—
39C23×D83—21no—
40#13050591—28no—
41#13050590—22no—
42#13050590—24no—
43#13050590—24no—
44#13050590D414yescomplemented
45C2×Dic166C2311yescharacteristic
46C2×D332—21no—
47C2×D332—22no—
48C2×D332—22no—
49C2×D332—24no—
50C2×D332—24no—
51C2×D332—22no—
52C2×D332—22no—
53C2×D332—21no—
54C2×D332D414yescomplemented
55C2×D332C2312yes—
56C2×D332C2312yes—
57C2×D332C2311yescharacteristic
58C2×C22⋊D83—21no—
59C2×C22⋊D83—22no—
60C2×C22⋊D83—24no—
61D4×D83—216no—
62D4×D83—28no—
63C2×C4×D83—21no—
64C2×C4×D83—22no—
65#13050579—24no—
66D664—28no—
67D664—24no—
68C4⋊Dic83C2312yes—
69C22⋊Dic83—21no—
70(C2×D83)⋊C4—24no—
71C4×Dic83—21no—
72#41×C83—22no—
73#41×C83—22no—
74C2×Q8×C83D411yescharacteristic
75C2×D4×C83D412yes—
76C2×D4×C83D411yescharacteristic
77C22×C4×C83C2312yes—
78C22×C4×C83C2311yescharacteristic
79M16×C83—22no—
80C2×C8×C83—22no—
81C2×C8×C83C2312yes—
82C2×C8×C83C2312yes—
83C23×C83C2×D412yes—
84C23×C83C2×D411yescharacteristic
85C22×D83—21no—
86C22×D83—22no—
87C22×D83—28no—
88C22×D83—24no—
89C22×D83—22no—
90C22×D83—42no—
91C22×D83—82no—
92C22×D83—84no—
93C22×D83—82no—
94Q8×C83—22no—
95D4×C83—24no—
96D4×C83—22no—
97C2×C4×C83—22no—
98C2×C4×C83—22no—
99C2×C4×C83C2×D412yes—
100C2×C4×C83C2×D412yes—
101C2×C4×C83C2×D412yes—
102C2×C4×C83C2×D411yescharacteristic
103C2×C4×C83C2411yescharacteristic
104C22⋊D83—28no—
105C22⋊D83—24no—
106C22⋊D83—84no—
107C2×Dic83—21no—
108C2×Dic83—22no—
109D332—28no—
110D332—24no—
111D332—21no—
112D332—22no—
113D332—22no—
114D332—22no—
115D332—42no—
116D332C2×D412yes—
117D332C2×D414yes—
118C4×D83—28no—
119C4×D83—24no—
120Dic166—21no—
121Dic166C2×D412yes—
122C8×C83—22no—
123C8×C83—22no—
124C22×C83—22no—
125C22×C83—22no—
126C22×C83—24no—
127C22×C83#12012yes—
128C22×C83C22×D411yescharacteristic
129C2×D83—24no—
130C2×D83—22no—
131C2×D83—44no—
132C2×D83—42no—
133C2×D83—42no—
134C2×D83—44no—
135C2×D83—84no—
136C2×D83—88no—
137C2×D83—84no—
138C4×C83—22no—
139C4×C83#12012yes—
140C4×C83C22×D411yescharacteristic
141C4×C83C22×D411yescharacteristic
142Dic83—22no—
143Dic83—42no—
144C2×C83—22no—
145C2×C83—42no—
146C2×C83C2×#12011yescharacteristic
147C2×C83#57612yes—
148D83—44no—
149D83—42no—
150D83—82no—
151#3008—831nomaximal, Sylow
152C83#300811yescharacteristic, complemented, Sylow
153C2×H(5,𝔽2)—831no—
154C2×AGL(1,ℤ/8)—831no—
155C22×D8—831no—
156#535—831no—
157#449—834no—
158#447—834no—
159#418—831no—
160#416—831no—
161#408D8311yescharacteristic, complemented
162H(5,𝔽2)—1664no—
163H(5,𝔽2)—1664no—
164C2×#41—831no—
165C2×#41—832no—
166C2×#41—1661no—
167C2×#41C2×D8311yescharacteristic
168C22×D4—831no—
169C22×D4—831no—
170C22×D4—832no—
171C22×D4—1662no—
172C22×D4—1661no—
173AGL(1,ℤ/8)—1664no—
174C2×SD16—832no—
175C2×D8—834no—
176C2×D8—832no—
177C2×D8—1664no—
178C2×M16C2×D8311yescharacteristic, complemented
179C22×C8C2×D8311yescharacteristic, complemented
180#127—1661no—
181#124—1661no—
182#121—834no—
183#120—1662no—
184#117—831no—
185#103—832no—
186#102—832no—
187#102—834no—
188C22⋊C8C2×D8314yescomplemented
189C24—1661no—
190C24—1662no—
191C24—1661no—
192#41—22no—
193#41—22no—
194#41—834no—
195#41—1664no—
196#41—1664no—
197#41—1668no—
198#41—1662no—
199C2×Q8—831no—
200C2×Q8D33211yescharacteristic
201C2×D4—834no—
202C2×D4—832no—
203C2×D4—831no—
204C2×D4—832no—
205C2×D4—1664no—
206C2×D4—1664no—
207C2×D4—16616no—
208C2×D4—1668no—
209C2×D4—1664no—
210C2×D4—1662no—
211C2×D4—1661no—
212C2×D4—1662no—
213C2×D4—1662no—
214C2×D4—1662no—
215C2×D4—1661no—
216C2×D4—1662no—
217C2×D4—1661no—
218C2×D4D33212yes—
219C2×D4D33211yescharacteristic
220C22×C4—1662no—
221C22×C4—1661no—
222C22×C4C22×D8312yes—
223C22×C4C22×D8311yescharacteristic
224SD16—1664no—
225D8—1668no—
226D8—1664no—
227M16—22no—
228C2×C8—22no—
229C2×C8C22×D8312yes—
230C2×C8C22×D8312yes—
231C4⋊C4—832no—
232C22⋊C4—1661no—
233C22⋊C4—1664no—
234C42—1661no—
235C23—3322no—
236C23—6642no—
237C23—6642no—
238C23—6644no—
239C23—1668no—
240C23—1662no—
241C23—1662no—
242C23—1661no—
243C23—1664no—
244C23C2×D33212yes—
245C23C2×D33211yescharacteristic
246Q8—22no—
247Q8—832no—
248Q8—1661no—
249D4—24no—
250D4—22no—
251D4—3322no—
252D4—834no—
253D4—832no—
254D4—6644no—
255D4—1668no—
256D4—1668no—
257D4—1664no—
258D4—1664no—
259D4—1662no—
260D4—1662no—
261D4—1661no—
262D4—1662no—
263C2×C4—22no—
264C2×C4—22no—
265C2×C4—1668no—
266C2×C4—1662no—
267C2×C4—1661no—
268C2×C4—1664no—
269C2×C4D4×D8312yes—
270C2×C4D4×D8312yes—
271C2×C4C2×D33212yes—
272C2×C4C2×D33211yescharacteristic
273C2×C4C23×D8311yescharacteristic
274C8—22no—
275C8—22no—
276C22—22no—
277C22—22no—
278C22—24no—
279C22—3324no—
280C22—3322no—
281C22—3322no—
282C22—3324no—
283C22—6644no—
284C22—6644no—
285C22—6648no—
286C22—1664no—
287C22—1662no—
288C22#1458176212yes—
289C22C22×D33211yescharacteristic
290C4—22no—
291C4—3322no—
292C4—1662no—
293C4#1458176212yes—
294C4C2×D4×D8311yescharacteristic
295C4C2×D4×D8311yescharacteristic
296C2—22no—
297C2—42no—
298C2—3324no—
299C2—3322no—
300C2—6642no—
301C2C2×#1458176211yescharacteristic
302C2#1912850612yes—
subgroupG/H
centerC22C22×D332
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting#408×C83C2
socleC22×C83C22×D4
hypercenter#408D83
hypocenterC83#3008
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
1. C2×#14581838C2semidirect
2. C2×#14581820C2semidirect
3. C22×D664C2semidirect
4. #19128265C2semidirect
5. #19127848C2semidirect
6. #19127837C2semidirect
7. #19127707C2semidirect
8. #19127698C2semidirect
9. #408×C83C2semidirect
14. C2×#13050590C22semidirect
16. C22×D332C22semidirect
17. C22×D332C22semidirect
21. C2×#13050579C22semidirect
23. C2×D664C22semidirect
24. C2×D664C22semidirect
27. #14581766C22semidirect
29. #14581751C22semidirect
30. #14581717C22semidirect
31. #14581714C22semidirect
32. #14581714C22semidirect
34. C2×M16×C83C22semidirect
35. C22×C8×C83C22semidirect
36. C22⋊C8×C83C22semidirect
44. #13050590D4semidirect
54. C2×D332D4semidirect
152. C83#3008semidirect
161. #408D83semidirect
178. C2×M16C2×D83semidirect
179. C22×C8C2×D83semidirect
188. C22⋊C8C2×D83semidirect

Maximal subgroups

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

subgroupindex
C2×#145818382
C2×#145818202
C22×D6642
#191282652
#191278482
#191278372
#191277072
#191276982
#408×C832
#300883