Finite Group Databasebeta

← #24917141

Subgroups — #24917141

Order 10624.

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

normal subgroups only

This group has 504 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
1C2×#14581818C211yesmaximal, characteristic, complemented
2C2×#14581798C211yesmaximal, characteristic, complemented
3#19128282C211yesmaximal, characteristic, complemented
4#19128268C211yesmaximal, characteristic, complemented
5#19127953C212yesmaximal, complemented
6#19127946C212yesmaximal, complemented
7#19127936C212yesmaximal, complemented
8#19127929C212yesmaximal, complemented
9#19127741C211yesmaximal, characteristic, complemented
10#19127698C211yesmaximal, characteristic, complemented
11C2×#102×C83C211yesmaximal, characteristic, complemented
12C2×#13050590C2211yescharacteristic
13C2×#13050590C2211yescharacteristic
14#14581818—24no—
15C2×#13050578C2212yes—
16C2×#13050575C2212yes—
17#14581798—24no—
18C2×#13050574C2211yescharacteristic
19C2×#13050573C2212yescomplemented
20C2×C8×D83C2212yescomplemented
21#14581788—21no—
22#14581779—21no—
23#14581772—21no—
24#14581772C2212yescomplemented
25#14581767—21no—
26#14581767C2212yescomplemented
27#14581766—22no—
28#14581763C2212yes—
29#14581763C2212yes—
30#14581761—22no—
31#14581751C2211yescharacteristic
32#14581725C2212yescomplemented
33#14581723C2212yescomplemented
34#14581717C2211yescharacteristic, complemented
35#14581717C2211yescharacteristic, complemented
36#14581714C2211yescharacteristic, complemented
37#14581714C2211yescharacteristic, complemented
38C22×D4×C83C2211yescharacteristic
39C22×C8×C83C2211yescharacteristic
40C2×C4⋊C4×C83C2211yescharacteristic
41#102×C83C2212yes—
42#102×C83C2212yes—
43#13050590—22no—
44#13050590—22no—
45#13050590—22no—
46#13050590—22no—
47C2×Dic166D411yescharacteristic
48C2×Dic166C2311yescharacteristic
49C2×D332—22no—
50C2×D332D411yescharacteristic
51C2×D332C2311yescharacteristic
52C2×C22⋊D83—21no—
53C2×C22⋊D83—22no—
54C2×C22⋊D83—22no—
55C2×C22⋊D83D412yes—
56C2×C4×D83—21no—
57C2×C4×D83C2312yes—
58#13050578—24no—
59#13050575—24no—
60#13050574D414yescomplemented
61#13050573—24no—
62C2×C83⋊C8C2312yes—
63C8×D83—24no—
64C4⋊Dic83—22no—
65C4⋊Dic83C2311yescharacteristic
66C4⋊Dic83C2311yescharacteristic
67C22⋊Dic83—21no—
68C22⋊Dic83—22no—
69C22⋊Dic83—22no—
70(C2×D83)⋊C4—22no—
71(C2×D83)⋊C4—22no—
72C4×Dic83—21no—
73C24×C83—21no—
74C2×D4×C83—22no—
75C2×D4×C83—21no—
76C2×D4×C83—22no—
77C2×D4×C83C2312yes—
78C22×C4×C83—21no—
79C22×C4×C83C2311yescharacteristic
80C2×C8×C83—22no—
81C2×C8×C83C2311yescharacteristic
82C2×C8×C83C2311yescharacteristic
83C4⋊C4×C83—21no—
84C4⋊C4×C83C2312yes—
85C23×C83—22no—
86C23×C83—42no—
87C23×C83—81no—
88C23×C83—81no—
89C23×C83C2×D411yescharacteristic
90C22×D83—21no—
91C22×D83C2×D412yes—
92D4×C83—22no—
93D4×C83—22no—
94D4×C83—42no—
95C2×C4×C83—22no—
96C2×C4×C83—81no—
97C2×C4×C83C2×D411yescharacteristic
98C2×C4×C83C2×D412yes—
99C2×C4×C83#4112yes—
100C2×C4×C83C2411yescharacteristic
101C22⋊D83—24no—
102C22⋊D83—42no—
103C22⋊D83—82no—
104C22⋊D83—82no—
105C2×Dic83—21no—
106C2×Dic83—21no—
107C2×Dic83—21no—
108C2×Dic83C2×D412yes—
109D332—22no—
110D332—21no—
111D332—21no—
112D332—82no—
113C4×D83—22no—
114C4×D83—22no—
115C4×D83—42no—
116Dic166—22no—
117Dic166—21no—
118Dic166—21no—
119C8×C83—21no—
120C8×C83—21no—
121C83⋊C8C2×D414yes—
122C22×C83—21no—
123C22×C83—21no—
124C22×C83—42no—
125C22×C83—42no—
126C22×C83—42no—
127C22×C83—84no—
128C22×C83#12112yes—
129C22×C83C22×D411yescharacteristic
130C2×D83—22no—
131C2×D83—22no—
132C2×D83—22no—
133C2×D83—41no—
134C2×D83—41no—
135C2×D83—41no—
136C4×C83—42no—
137C4×C83#12112yes—
138C4×C83C22×D411yescharacteristic
139C4×C83C2×#4111yescharacteristic
140Dic83—22no—
141Dic83—41no—
142Dic83—41no—
143Dic83—41no—
144C2×C83—22no—
145C2×C83—82no—
146C2×C83C2×#12111yescharacteristic
147C2×C83#57611yescharacteristic
148C2×C83#57711yescharacteristic
149D83—42no—
150D83—81no—
151#3062—831nomaximal, Sylow
152C83#306211yescharacteristic, complemented, Sylow
153C2×AGL(1,ℤ/8)—831no—
154C2×#131—831no—
155#537—831no—
156#535—831no—
157#469—832no—
158#468—832no—
159#466—832no—
160#465—832no—
161#428—831no—
162#416—831no—
163C2×#102D8311yescharacteristic, complemented
164C2×#41—831no—
165C2×#41—831no—
166C22×D4C2×D8311yescharacteristic
167AGL(1,ℤ/8)—1664no—
168C2×SD16—832no—
169C2×D8—832no—
170#131—1664no—
171C2×M16—831no—
172C2×M16—832no—
173C22×C8—832no—
174C22×C8C2×D8311yescharacteristic
175#127—1661no—
176#124—1661no—
177C22⋊Q8—832no—
178C22⋊Q8—1661no—
179#121—832no—
180#121—832no—
181#121—832no—
182#121—1661no—
183#121—1662no—
184#120—1662no—
185#117—831no—
186C2×C4⋊C4C2×D8311yescharacteristic
187#107—832no—
188#106—832no—
189#103—831no—
190#103—831no—
191#102—831no—
192#102—831no—
193#102C2×D8312yes—
194#102C2×D8312yes—
195C24—21no—
196#41—1662no—
197#41—1662no—
198#41—1662no—
199#41—1662no—
200C2×Q8—831no—
201C2×Q8—831no—
202C2×D4—22no—
203C2×D4—21no—
204C2×D4—22no—
205C2×D4—832no—
206C2×D4—831no—
207C2×D4—831no—
208C2×D4—1662no—
209C2×D4—1661no—
210C2×D4—1662no—
211C2×D4—1662no—
212C2×D4C22×D8312yes—
213C22×C4—21no—
214C22×C4—832no—
215C22×C4—1661no—
216C22×C4C22×D8311yescharacteristic
217SD16—1664no—
218D8—1664no—
219M16—834no—
220M16—1664no—
221C2×C8—22no—
222C2×C8—832no—
223C2×C8—1664no—
224C2×C8C22×D8311yescharacteristic
225C2×C8C22×D8311yescharacteristic
226C4⋊C4—21no—
227C4⋊C4—831no—
228C4⋊C4—831no—
229C4⋊C4—1662no—
230C4⋊C4C22×D8312yes—
231C22⋊C4—1662no—
232C22⋊C4—1662no—
233C22⋊C4—1662no—
234C22⋊C4—1662no—
235C22⋊C4—1661no—
236C42—1661no—
237C23—22no—
238C23—42no—
239C23—81no—
240C23—81no—
241C23—832no—
242C23—1661no—
243C23D4×D8311yescharacteristic
244Q8—1661no—
245Q8—1661no—
246Q8—1662no—
247D4—22no—
248D4—22no—
249D4—42no—
250D4—3322no—
251D4—6642no—
252D4—6642no—
253D4—6642no—
254D4—1664no—
255D4—1662no—
256D4—1661no—
257D4—1661no—
258C2×C4—22no—
259C2×C4—81no—
260C2×C4—3322no—
261C2×C4—832no—
262C2×C4—1661no—
263C2×C4—1661no—
264C2×C4—1661no—
265C2×C4—1662no—
266C2×C4—1662no—
267C2×C4D4×D8311yescharacteristic
268C2×C4D4×D8312yes—
269C2×C4#1305059012yes—
270C2×C4C23×D8311yescharacteristic
271C8—21no—
272C8—21no—
273C8—834no—
274C22—21no—
275C22—21no—
276C22—42no—
277C22—42no—
278C22—42no—
279C22—84no—
280C22—3321no—
281C22—3321no—
282C22—3321no—
283C22—1662no—
284C22—1662no—
285C22—1662no—
286C22#1458176512yes—
287C22C2×D4×D8311yescharacteristic
288C4—42no—
289C4—3321no—
290C4—3321no—
291C4—3321no—
292C4—1662no—
293C4#1458176512yes—
294C4C2×D4×D8311yescharacteristic
295C4C2×#1305059011yescharacteristic
296C2—22no—
297C2—82no—
298C2—3322no—
299C2—6641no—
300C2C2×#1458176511yescharacteristic
301C2#1912850211yescharacteristic
302C2#1912850711yescharacteristic
subgroupG/H
centerC22C2×D4×D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
FittingC2×#102×C83C2
socleC22×C83C22×D4
hypercenterC2×#102D83
hypocenterC83#3062
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#102

Complements

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

NH ≅ G/Ntype
1. C2×#14581818C2semidirect
2. C2×#14581798C2semidirect
3. #19128282C2semidirect
4. #19128268C2semidirect
5. #19127953C2semidirect
6. #19127946C2semidirect
7. #19127936C2semidirect
8. #19127929C2semidirect
9. #19127741C2semidirect
10. #19127698C2semidirect
11. C2×#102×C83C2semidirect
19. C2×#13050573C22semidirect
20. C2×C8×D83C22semidirect
24. #14581772C22semidirect
26. #14581767C22semidirect
32. #14581725C22semidirect
33. #14581723C22semidirect
34. #14581717C22semidirect
35. #14581717C22semidirect
36. #14581714C22semidirect
37. #14581714C22semidirect
60. #13050574D4semidirect
152. C83#3062semidirect
163. C2×#102D83semidirect

Maximal subgroups

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

subgroupindex
C2×#145818182
C2×#145817982
#191282822
#191282682
#191279532
#191279462
#191279362
#191279292
#191277412
#191276982
C2×#102×C832
#306283