Finite Group Databasebeta

← #24916713

Subgroups — #24916713

Order 10624.

806 conjugacy classes of subgroups; only the 153 normal subgroups are shown.

normal subgroups only

This group has 806 conjugacy classes of subgroups — too many to draw. Only its 153 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
1C22×#13050590C211yesmaximal, characteristic, complemented
2C2×#14581818C212yesmaximal, complemented
3#19127844C214yesmaximal, complemented
4#19127839C212yesmaximal, complemented
5#19127834C212yesmaximal, complemented
6#19127700C212yesmaximal, complemented
7#19127665C211yesmaximal, characteristic, complemented
8C2×#121×C83C211yesmaximal, characteristic, complemented
9C2×#13050590—28no—
10C2×#13050590—21no—
11C2×#13050590—21no—
12C2×#13050590C2212yes—
13C2×#13050590C2212yescomplemented
14C22×Dic166C2211yescharacteristic, complemented
15C22×D332C2211yescharacteristic, complemented
16C22×C22⋊D83—21no—
17C22×C4×D83—21no—
18#14581818—28no—
19C2×#13050578C2214yescomplemented
20C2×#13050575C2214yescomplemented
21C2×#13050574C2212yescomplemented
22#14581715C2214yescomplemented
23#14581712C2214yescomplemented
24C22⋊(C83⋊C8)C2212yescomplemented
25C22⋊(C83⋊C8)C2212yescomplemented
26C22×D4×C83—21no—
27C22×D4×C83C2211yescharacteristic, complemented
28C23×C4×C83C2211yescharacteristic
29#121×C83—21no—
30#121×C83—21no—
31#121×C83C2212yes—
32#121×C83C2212yescomplemented
33C2×C4⋊C4×C83C2211yescharacteristic, complemented
34C2×C22⋊C4×C83—21no—
35C23×D83—21no—
36#13050590—28no—
37#13050590—28no—
38#13050590—24no—
39#13050590—24no—
40#13050590—24no—
41#13050590D418yescomplemented
42C2×Dic166—21no—
43C2×Dic166—24no—
44C2×Dic166C2312yes—
45C2×D332—21no—
46C2×D332—24no—
47C2×D332C2312yes—
48C2×C22⋊D83—28no—
49C2×C22⋊D83—22no—
50C2×C22⋊D83—22no—
51C22×Dic83—21no—
52C2×C4×D83—28no—
53C2×C4×D83—22no—
54C2×C4×D83—22no—
55#13050578—28no—
56#13050575—28no—
57#13050574—24no—
58C2×C83⋊C8C2314yes—
59C24×C83—21no—
60C24×C83D411yescharacteristic
61C2×D4×C83—22no—
62C2×D4×C83—22no—
63C2×D4×C83—24no—
64C2×D4×C83—21no—
65C2×D4×C83—44no—
66C2×D4×C83C2312yes—
67C22×C4×C83—24no—
68C22×C4×C83—21no—
69C22×C4×C83D411yescharacteristic
70C22×C4×C83D411yescharacteristic
71C22×C4×C83D411yescharacteristic
72C22×C4×C83C2311yescharacteristic
73C22×C4×C83C2311yescharacteristic
74C22×C4×C83C2311yescharacteristic
75C4⋊C4×C83—21no—
76C4⋊C4×C83C2312yes—
77C22⋊C4×C83—22no—
78C22⋊C4×C83—22no—
79C23×C83—22no—
80C23×C83—22no—
81C23×C83—22no—
82C23×C83—41no—
83C23×C83—42no—
84C23×C83—42no—
85C23×C83—42no—
86C23×C83C2×D411yescharacteristic
87C23×C83C2×D411yescharacteristic
88C23×C83C2×D411yescharacteristic
89C22×D83—24no—
90C22×D83—22no—
91C22×D83—42no—
92C22×D83—42no—
93D4×C83—24no—
94D4×C83—44no—
95D4×C83—44no—
96D4×C83—42no—
97C2×C4×C83—22no—
98C2×C4×C83—22no—
99C2×C4×C83—22no—
100C2×C4×C83—22no—
101C2×C4×C83—22no—
102C2×C4×C83—22no—
103C2×C4×C83—42no—
104C2×C4×C83C2×D414yes—
105C2×C4×C83C2×D411yescharacteristic
106C2×C4×C83C2×D411yescharacteristic
107C2×C4×C83C2×D411yescharacteristic
108C2×C4×C83C2411yescharacteristic
109C22⋊D83—28no—
110C22⋊D83—28no—
111C2×Dic83—24no—
112C2×Dic83—22no—
113D332—24no—
114D332—22no—
115D332C2×D414yescomplemented
116C4×D83—28no—
117C4×D83—28no—
118Dic166—24no—
119Dic166—22no—
120Dic166C2×D414yescomplemented
121C83⋊C8—24no—
122C22×C83—21no—
123C22×C83—21no—
124C22×C83—21no—
125C22×C83—21no—
126C22×C83—21no—
127C22×C83—21no—
128C22×C83—21no—
129C22×C83—44no—
130C22×C83—42no—
131C22×C83—42no—
132C22×C83—42no—
133C22×C83—84no—
134C22×C83#12012yes—
135C22×C83AGL(1,ℤ/8)12yescomplemented
136C22×C83C22×D411yescharacteristic
137C2×D83—24no—
138C2×D83—48no—
139C2×D83—42no—
140C2×D83—42no—
141C4×C83—21no—
142C4×C83—21no—
143C4×C83—42no—
144C4×C83#12012yes—
145C4×C83C22×D412yes—
146Dic83—24no—
147C2×C83—22no—
148C2×C83—22no—
149C2×C83—41no—
150C2×C83—82no—
151C2×C83C2×#12011yescharacteristic
152C2×C83C2×AGL(1,ℤ/8)11yescharacteristic
153C2×C83#57511yescharacteristic
154D83—44no—
155#3006—831nomaximal, Sylow
156C83#300611yescharacteristic, complemented, Sylow
157C22×#41—831no—
158C2×AGL(1,ℤ/8)—832no—
159C2×#121D8311yescharacteristic, complemented
160#449—834no—
161#448—832no—
162#447—832no—
163#417—832no—
164#407—831no—
165C2×#41—832no—
166C2×#41—832no—
167C2×#41—1668no—
168C2×#41—1661no—
169C2×#41—1661no—
170C22×Q8—831no—
171C22×D4—21no—
172C22×D4—831no—
173C22×D4—1661no—
174C22×D4C2×D8311yescharacteristic, complemented
175C23×C4—1661no—
176C23×C4C2×D8311yescharacteristic
177AGL(1,ℤ/8)—1668no—
178C2×SD16—834no—
179C2×D8—834no—
180C2×M16—832no—
181#121—21no—
182#121—21no—
183#121C2×D8312yes—
184#121C2×D8312yescomplemented
185C2×C4⋊C4C2×D8311yescharacteristic, complemented
186C2×C22⋊C4—21no—
187#103—834no—
188#102—834no—
189C22⋊C8—832no—
190C22⋊C8—832no—
191C24—21no—
192C24—1661no—
193C24C22⋊D8311yescharacteristic
194#41—838no—
195#41—1664no—
196#41—1664no—
197#41—1668no—
198#41—1668no—
199#41—1664no—
200C2×Q8—832no—
201C2×Q8—1664no—
202C2×Q8—1661no—
203C2×D4—22no—
204C2×D4—22no—
205C2×D4—24no—
206C2×D4—21no—
207C2×D4—44no—
208C2×D4—832no—
209C2×D4—1664no—
210C2×D4—1668no—
211C2×D4—1662no—
212C2×D4—1662no—
213C2×D4—1661no—
214C2×D4C22×D8312yes—
215C22×C4—24no—
216C22×C4—21no—
217C22×C4—1668no—
218C22×C4—1661no—
219C22×C4—1662no—
220C22×C4—1662no—
221C22×C4C22⋊D8311yescharacteristic
222C22×C4C22⋊D8311yescharacteristic
223C22×C4C22⋊D8311yescharacteristic
224C22×C4C22×D8311yescharacteristic
225C22×C4C22×D8311yescharacteristic
226C22×C4C22×D8311yescharacteristic
227SD16—1668no—
228D8—1668no—
229M16—1664no—
230C2×C8—834no—
231C4⋊C4—21no—
232C4⋊C4C22×D8312yes—
233C22⋊C4—22no—
234C22⋊C4—22no—
235C23—22no—
236C23—22no—
237C23—22no—
238C23—41no—
239C23—42no—
240C23—42no—
241C23—42no—
242C23—3322no—
243C23—3322no—
244C23—1664no—
245C23—1662no—
246C23C2×C22⋊D8311yescharacteristic
247C23C2×C22⋊D8311yescharacteristic
248C23C2×C22⋊D8311yescharacteristic
249Q8—834no—
250Q8—1664no—
251Q8—1662no—
252D4—24no—
253D4—44no—
254D4—44no—
255D4—42no—
256D4—834no—
257D4—1664no—
258D4—1668no—
259D4—1668no—
260D4—1662no—
261C2×C4—22no—
262C2×C4—22no—
263C2×C4—22no—
264C2×C4—22no—
265C2×C4—22no—
266C2×C4—22no—
267C2×C4—42no—
268C2×C4—1668no—
269C2×C4—1664no—
270C2×C4—1662no—
271C2×C4—1668no—
272C2×C4D4×D8314yes—
273C2×C4C2×C22⋊D8311yescharacteristic
274C2×C4C2×C22⋊D8311yescharacteristic
275C2×C4C2×C22⋊D8311yescharacteristic
276C2×C4C23×D8311yescharacteristic
277C8—1664no—
278C22—21no—
279C22—21no—
280C22—21no—
281C22—21no—
282C22—21no—
283C22—21no—
284C22—21no—
285C22—44no—
286C22—42no—
287C22—42no—
288C22—42no—
289C22—84no—
290C22—3328no—
291C22—3322no—
292C22—3322no—
293C22—1664no—
294C22#1458176012yes—
295C22#1458181812yescomplemented
296C22C22×C22⋊D8311yescharacteristic
297C4—21no—
298C4—21no—
299C4—42no—
300C4—1664no—
301C4#1458176012yes—
302C4C2×D4×D8312yes—
303C2—22no—
304C2—22no—
305C2—41no—
306C2—82no—
307C2—3324no—
308C2C2×#1458176011yescharacteristic
309C2C2×#1458181811yescharacteristic
310C2#1912849911yescharacteristic
subgroupG/H
centerC22C22×C22⋊D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
FittingC2×#121×C83C2
socleC22×C83C22×D4
hypercenterC2×#121D83
hypocenterC83#3006
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#121

Complements

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

NH ≅ G/Ntype
1. C22×#13050590C2semidirect
2. C2×#14581818C2semidirect
3. #19127844C2semidirect
4. #19127839C2semidirect
5. #19127834C2semidirect
6. #19127700C2semidirect
7. #19127665C2semidirect
8. C2×#121×C83C2semidirect
13. C2×#13050590C22semidirect
14. C22×Dic166C22semidirect
15. C22×D332C22semidirect
19. C2×#13050578C22semidirect
20. C2×#13050575C22semidirect
21. C2×#13050574C22semidirect
22. #14581715C22semidirect
23. #14581712C22semidirect
24. C22⋊(C83⋊C8)C22semidirect
25. C22⋊(C83⋊C8)C22semidirect
27. C22×D4×C83C22semidirect
32. #121×C83C22semidirect
33. C2×C4⋊C4×C83C22semidirect
41. #13050590D4semidirect
115. D332C2×D4semidirect
120. Dic166C2×D4semidirect
135. C22×C83AGL(1,ℤ/8)semidirect
156. C83#3006semidirect
159. C2×#121D83semidirect
174. C22×D4C2×D83semidirect
184. #121C2×D83semidirect
185. C2×C4⋊C4C2×D83semidirect
295. C22#14581818semidirect

Maximal subgroups

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

subgroupindex
C22×#130505902
C2×#145818182
#191278442
#191278392
#191278342
#191277002
#191276652
C2×#121×C832
#300683