Finite Group Databasebeta

← #24917006

Subgroups — #24917006

Order 10624.

504 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 504 conjugacy classes of subgroups — too many to draw. Only its 141 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×#14581812C211yesmaximal, characteristic, complemented
2#19128268C211yesmaximal, characteristic, complemented
3#19128184C211yesmaximal, characteristic, complemented
4#19127921C212yesmaximal, complemented
5#19127914C212yesmaximal, complemented
6#19127899C212yesmaximal, complemented
7#19127894C212yesmaximal, complemented
8#19127729C211yesmaximal, characteristic, complemented
9#19127703C211yesmaximal, characteristic, complemented
10C2×#14581713C211yesmaximal, characteristic, complemented
11C2×AGL(1,ℤ/8)×C83C211yesmaximal, characteristic, complemented
12C2×#13050590C2211yescharacteristic
13#14581812—24no—
14C2×#13050580C2211yescharacteristic, complemented
15C2×#13050578C2211yescharacteristic, complemented
16C2×#13050577C2211yescharacteristic, complemented
17C2×#13050575C2211yescharacteristic, complemented
18C22×C83⋊C8C2211yescharacteristic
19#14581788C2212yes—
20#14581779C2212yes—
21#14581763—21no—
22#14581763—21no—
23#14581761—22no—
24Q8×Dic83C2212yescomplemented
25D4×Dic83—22no—
26D4×Dic83C2212yescomplemented
27#14581751—21no—
28#14581751C2211yescharacteristic
29C2×C4⋊Dic83—21no—
30C2×C4⋊Dic83C2211yescharacteristic
31Dic83⋊C8C2212yescomplemented
32Dic83⋊C8C2212yescomplemented
33#14581717C2212yescomplemented
34#14581714C2212yescomplemented
35#14581713C2212yes—
36#14581713C2212yes—
37C2×#41×C83C2211yescharacteristic
38C22×D4×C83C2211yescharacteristic
39AGL(1,ℤ/8)×C83—24no—
40C2×SD16×C83C2212yes—
41C2×D8×C83C2212yes—
42C2×M16×C83C2211yescharacteristic
43#13050590—22no—
44#13050590—22no—
45C2×Dic166C2311yescharacteristic
46C2×D332C2311yescharacteristic
47C2×C22⋊D83—21no—
48C2×C22⋊D83—22no—
49C2×C22⋊D83—22no—
50C22×Dic83—21no—
51C22×Dic83—22no—
52C2×C4×D83—21no—
53#13050580—22no—
54#13050578—22no—
55#13050577—22no—
56#13050575—22no—
57C2×C83⋊C8—22no—
58C2×C83⋊C8C2311yescharacteristic
59C2×C83⋊C8C2311yescharacteristic
60C4⋊Dic83—21no—
61C4⋊Dic83—22no—
62C4⋊Dic83D411yescharacteristic
63C4⋊Dic83D411yescharacteristic
64C4⋊Dic83C2312yes—
65C22⋊Dic83—22no—
66C22⋊Dic83—22no—
67C22⋊Dic83—22no—
68C22⋊Dic83D412yes—
69C4×Dic83—22no—
70C4×Dic83C2312yes—
71C24×C83—21no—
72#41×C83—22no—
73#41×C83D414yescomplemented
74C2×Q8×C83C2311yescharacteristic
75C2×D4×C83—21no—
76C2×D4×C83—22no—
77C2×D4×C83—21no—
78C2×D4×C83—22no—
79C2×D4×C83C2312yes—
80C2×D4×C83C2311yescharacteristic
81C22×C4×C83—21no—
82C22×C4×C83C2311yescharacteristic
83SD16×C83—24no—
84D8×C83—24no—
85M16×C83—22no—
86C2×C8×C83C2312yes—
87C23×C83—21no—
88C23×C83—22no—
89C23×C83—42no—
90C23×C83—81no—
91C23×C83—81no—
92C23×C83C2×D411yescharacteristic
93C22×D83—21no—
94Q8×C83—21no—
95Q8×C83C2×D412yes—
96D4×C83—24no—
97D4×C83—22no—
98D4×C83—21no—
99D4×C83—22no—
100D4×C83—42no—
101D4×C83C2×D412yes—
102C2×C4×C83—21no—
103C2×C4×C83—24no—
104C2×C4×C83C2×D411yescharacteristic
105C2×C4×C83C2×D412yes—
106C2×C4×C83#4112yes—
107C2×C4×C83C2411yescharacteristic
108C22⋊D83—42no—
109C22⋊D83—82no—
110C22⋊D83—82no—
111C2×Dic83—22no—
112C2×Dic83—21no—
113C2×Dic83—82no—
114C2×Dic83—81no—
115C2×Dic83C2×D412yes—
116C2×Dic83C2×D412yes—
117D332—21no—
118D332—21no—
119C4×D83—42no—
120Dic166—21no—
121Dic166—21no—
122C8×C83—22no—
123C83⋊C8—21no—
124C83⋊C8—21no—
125C22×C83—21no—
126C22×C83—21no—
127C22×C83—22no—
128C22×C83—41no—
129C22×C83—41no—
130C22×C83—42no—
131C22×C83—42no—
132C22×C83—42no—
133C22×C83—84no—
134C22×C83#12112yes—
135C22×C83C22×D411yescharacteristic
136C2×D83—41no—
137C2×D83—41no—
138C2×D83—41no—
139C4×C83—22no—
140C4×C83#12112yes—
141C4×C83C22×D411yescharacteristic
142C4×C83C2×#4111yescharacteristic
143Dic83—22no—
144Dic83—22no—
145Dic83—41no—
146Dic83—42no—
147C2×C83—22no—
148C2×C83—42no—
149C2×C83—82no—
150C2×C83C2×#12111yescharacteristic
151C2×C83#57611yescharacteristic
152C2×C83#57711yescharacteristic
153D83—81no—
154#3045—831nomaximal, Sylow
155C83#304511yescharacteristic, complemented, Sylow
156C2×AGL(1,ℤ/8)D8311yescharacteristic, complemented
157C2×#135—831no—
158#535—831no—
159#520—831no—
160#464—832no—
161#463—832no—
162#460—832no—
163#459—832no—
164#424—831no—
165#417—831no—
166C2×#102—831no—
167C2×#41—831no—
168C2×#41C2×D8311yescharacteristic
169C22×D4C2×D8311yescharacteristic
170AGL(1,ℤ/8)—24no—
171#135—1664no—
172C2×Q16—831no—
173C2×SD16—831no—
174C2×SD16—831no—
175C2×SD16C2×D8312yes—
176C2×D8—831no—
177C2×D8C2×D8312yes—
178C2×M16C2×D8311yescharacteristic
179C22×C8—831no—
180#127—832no—
181#124—832no—
182#121—1661no—
183#121—1661no—
184#120—1662no—
185C4×Q8—832no—
186C4×D4—832no—
187C4×D4—1662no—
188#117—831no—
189#117—1661no—
190C2×C4⋊C4—831no—
191C2×C4⋊C4—1661no—
192C4⋊C8—832no—
193C4⋊C8—832no—
194#103—832no—
195#102—832no—
196#102—832no—
197#102—832no—
198C24—21no—
199#41—22no—
200#41—1662no—
201#41—1662no—
202#41C22⋊D8314yescomplemented
203C2×Q8—831no—
204C2×Q8C22×D8311yescharacteristic
205C2×D4—21no—
206C2×D4—21no—
207C2×D4—22no—
208C2×D4—22no—
209C2×D4—831no—
210C2×D4—1662no—
211C2×D4—1662no—
212C2×D4—1661no—
213C2×D4C22×D8312yes—
214C2×D4C22×D8311yescharacteristic
215C22×C4—21no—
216C22×C4—1661no—
217C22×C4—1662no—
218C22×C4—1661no—
219C22×C4C22×D8311yescharacteristic
220Q16—1662no—
221SD16—24no—
222SD16—1662no—
223SD16—1662no—
224D8—24no—
225D8—1662no—
226M16—22no—
227C2×C8—831no—
228C2×C8—831no—
229C2×C8—1662no—
230C2×C8C22×D8312yes—
231C4⋊C4—831no—
232C4⋊C4—831no—
233C4⋊C4—832no—
234C4⋊C4—1662no—
235C4⋊C4—1661no—
236C22⋊C4—832no—
237C22⋊C4—1662no—
238C22⋊C4—1662no—
239C22⋊C4—1662no—
240C42—832no—
241C42—1662no—
242C23—21no—
243C23—22no—
244C23—42no—
245C23—81no—
246C23—81no—
247C23—1661no—
248C23D4×D8311yescharacteristic
249Q8—21no—
250Q8—1661no—
251Q8—1661no—
252Q8C2×C22⋊D8312yes—
253D4—24no—
254D4—22no—
255D4—21no—
256D4—22no—
257D4—42no—
258D4—3322no—
259D4—6642no—
260D4—6642no—
261D4—1661no—
262D4—1661no—
263D4C2×C22⋊D8312yes—
264C2×C4—21no—
265C2×C4—24no—
266C2×C4—3322no—
267C2×C4—832no—
268C2×C4—832no—
269C2×C4—6641no—
270C2×C4—6642no—
271C2×C4—1662no—
272C2×C4—1661no—
273C2×C4D4×D8311yescharacteristic
274C2×C4C2×C22⋊D8312yes—
275C2×C4#1305058912yes—
276C2×C4C23×D8311yescharacteristic
277C8—22no—
278C8—1661no—
279C8—1661no—
280C22—21no—
281C22—21no—
282C22—22no—
283C22—41no—
284C22—41no—
285C22—42no—
286C22—42no—
287C22—42no—
288C22—84no—
289C22—3321no—
290C22—3321no—
291C22—3321no—
292C22#1458176412yes—
293C22C2×D4×D8311yescharacteristic
294C4—22no—
295C4—3321no—
296C4—3322no—
297C4—1662no—
298C4—1662no—
299C4#1458176412yes—
300C4C22×C22⋊D8311yescharacteristic
301C4C2×#1305058911yescharacteristic
302C2—22no—
303C2—42no—
304C2—82no—
305C2—6641no—
306C2C2×#1458176411yescharacteristic
307C2#1912850511yescharacteristic
308C2#1912851111yescharacteristic
subgroupG/H
centerC22C2×D4×D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
FittingC2×AGL(1,ℤ/8)×C83C2
socleC22×C83C22×D4
hypercenterC2×AGL(1,ℤ/8)D83
hypocenterC83#3045
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. C2×#14581812C2semidirect
2. #19128268C2semidirect
3. #19128184C2semidirect
4. #19127921C2semidirect
5. #19127914C2semidirect
6. #19127899C2semidirect
7. #19127894C2semidirect
8. #19127729C2semidirect
9. #19127703C2semidirect
10. C2×#14581713C2semidirect
11. C2×AGL(1,ℤ/8)×C83C2semidirect
14. C2×#13050580C22semidirect
15. C2×#13050578C22semidirect
16. C2×#13050577C22semidirect
17. C2×#13050575C22semidirect
24. Q8×Dic83C22semidirect
26. D4×Dic83C22semidirect
31. Dic83⋊C8C22semidirect
32. Dic83⋊C8C22semidirect
33. #14581717C22semidirect
34. #14581714C22semidirect
73. #41×C83D4semidirect
155. C83#3045semidirect
156. C2×AGL(1,ℤ/8)D83semidirect
202. #41C22⋊D83semidirect

Maximal subgroups

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

subgroupindex
C2×#145818122
#191282682
#191281842
#191279212
#191279142
#191278992
#191278942
#191277292
#191277032
C2×#145817132
C2×AGL(1,ℤ/8)×C832
#304583