Finite Group Databasebeta

← #24916692

Subgroups — #24916692

Order 10624.

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

normal subgroups only

This group has 994 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×D4×D83C211yesmaximal, characteristic, complemented
2C2×#14581820C211yesmaximal, characteristic, complemented
3C2×#14581818C211yesmaximal, characteristic, complemented
4#19128213C211yesmaximal, characteristic, complemented
5#19127853C214yesmaximal, complemented
6#19127836C214yesmaximal, complemented
7#19127705C211yesmaximal, characteristic, complemented
8#19127664C211yesmaximal, characteristic, complemented
9#418×C83C211yesmaximal, characteristic, complemented
10C24×D83—21no—
11C2×#13050590C2211yescharacteristic, complemented
12C22×D332C2211yescharacteristic
13C22×C22⋊D83—21no—
14C2×D4×D83—24no—
15C2×D4×D83—22no—
16C2×D4×D83—24no—
17C2×D4×D83C2214yes—
18C22×C4×D83C2211yescharacteristic
19#14581820—24no—
20#14581818—24no—
21C2×#13050579C2212yescomplemented
22C2×#13050578C2212yescomplemented
23C2×D664C2212yescomplemented
24C2×#13050575C2212yescomplemented
25C2×#13050574C2211yescharacteristic, complemented
26#14581776—21no—
27#14581772C2212yescomplemented
28#14581767C2212yescomplemented
29#14581762—21no—
30C4×D332—22no—
31#14581712C2214yescomplemented
32(C2×D83)⋊C8C2214yescomplemented
33C22×D4×C83C2211yescharacteristic
34C2×M16×C83C2211yescharacteristic, complemented
35#117×C83C2211yescharacteristic, complemented
36#102×C83C2214yescomplemented
37C23×D83—28no—
38C23×D83—21no—
39C23×D83—24no—
40C23×D83—48no—
41C23×D83D411yescharacteristic
42#13050590—22no—
43#13050590—22no—
44C2×Dic166C2311yescharacteristic, complemented
45C2×D332—21no—
46C2×D332—22no—
47C2×D332—22no—
48C2×D332D414yescomplemented
49C2×D332C2312yes—
50C2×D332C2311yescharacteristic, complemented
51C2×C22⋊D83—28no—
52C2×C22⋊D83—21no—
53C2×C22⋊D83—24no—
54D4×D83—216no—
55D4×D83—28no—
56D4×D83—28no—
57C22×Dic83D411yescharacteristic
58C2×C4×D83—22no—
59C2×C4×D83—22no—
60C2×C4×D83—21no—
61C2×C4×D83D412yes—
62C2×C4×D83C2312yes—
63#13050579—24no—
64#13050578—24no—
65D664—24no—
66#13050575—24no—
67#13050574—22no—
68C2×C83⋊C8C2312yescomplemented
69C4⋊Dic83—22no—
70(C2×D83)⋊C4—22no—
71(C2×D83)⋊C4—22no—
72C24×C83—21no—
73C2×D4×C83—21no—
74C2×D4×C83—22no—
75C2×D4×C83D414yescomplemented
76C2×D4×C83C2312yes—
77C22×C4×C83C2311yescharacteristic
78M16×C83—22no—
79C2×C8×C83C2312yescomplemented
80C4⋊C4×C83C2312yescomplemented
81C22⋊C4×C83—21no—
82C42×C83—21no—
83C23×C83—24no—
84C23×C83—22no—
85C23×C83—44no—
86C23×C83C2×D411yescharacteristic
87C22×D83—22no—
88C22×D83—24no—
89C22×D83—216no—
90C22×D83—24no—
91C22×D83—22no—
92C22×D83—21no—
93C22×D83—432no—
94C22×D83—48no—
95C22×D83—44no—
96C22×D83—48no—
97C22×D83C2×D412yes—
98D4×C83—24no—
99D4×C83—22no—
100D4×C83C2×D414yes—
101C2×C4×C83—22no—
102C2×C4×C83C2×D411yescharacteristic
103C2×C4×C83C2×D412yes—
104C2×C4×C83C2×D412yes—
105C2×C4×C83C2411yescharacteristic
106C22⋊D83—216no—
107C22⋊D83—42no—
108C2×Dic83—22no—
109C2×Dic83—21no—
110C2×Dic83C2×D412yes—
111D332—24no—
112D332—22no—
113D332—21no—
114D332—21no—
115D332—44no—
116D332C2×D414yes—
117C4×D83—24no—
118C4×D83—22no—
119C4×D83—22no—
120C4×D83—42no—
121Dic166—21no—
122Dic166—21no—
123C8×C83—22no—
124C83⋊C8—22no—
125C22×C83—22no—
126C22×C83—24no—
127C22×C83—48no—
128C22×C83—44no—
129C22×C83#12012yes—
130C22×C83C22×D411yescharacteristic
131C2×D83—28no—
132C2×D83—22no—
133C2×D83—22no—
134C2×D83—24no—
135C2×D83—432no—
136C2×D83—48no—
137C2×D83—44no—
138C2×D83—41no—
139C2×D83—42no—
140C2×D83AGL(1,ℤ/8)14yescomplemented
141C4×C83—42no—
142C4×C83#12012yes—
143C4×C83C22×D411yescharacteristic
144C4×C83C22×D411yescharacteristic
145Dic83—22no—
146Dic83—41no—
147C2×C83—22no—
148C2×C83—44no—
149C2×C83C2×#12011yescharacteristic
150C2×C83C2×AGL(1,ℤ/8)12yes—
151D83—24no—
152D83—44no—
153D83—81no—
154#3003—831nomaximal, Sylow
155C83#300311yescharacteristic, complemented, Sylow
156C23×D4—831no—
157C2×AGL(1,ℤ/8)—831no—
158C2×AGL(1,ℤ/8)—831no—
159#525—831no—
160#450—834no—
161#447—834no—
162#418—831no—
163#418D8311yescharacteristic, complemented
164#407—831no—
165C25—1661no—
166C2×#41—831no—
167C22×D4—834no—
168C22×D4—831no—
169C22×D4—1662no—
170C22×D4—1661no—
171C22×D4—1664no—
172C22×D4—1664no—
173C22×D4C2×D8311yescharacteristic
174C23×C4—831no—
175AGL(1,ℤ/8)—1664no—
176AGL(1,ℤ/8)—1664no—
177C2×SD16—832no—
178C2×SD16—832no—
179C2×D8—832no—
180C2×D8—832no—
181C2×M16—831no—
182C2×M16C2×D8311yescharacteristic, complemented
183#123—1661no—
184C22⋊Q8—832no—
185#121—832no—
186#120—1661no—
187C4×D4—1662no—
188#117C2×D8311yescharacteristic, complemented
189#102—834no—
190#102C2×D8314yescomplemented
191C22⋊C8—834no—
192C24—21no—
193C24—3328no—
194C24—831no—
195C24—1664no—
196C24—1668no—
197C24—1661no—
198#41—1662no—
199#41—1662no—
200C2×Q8—831no—
201C2×D4—21no—
202C2×D4—22no—
203C2×D4—834no—
204C2×D4—832no—
205C2×D4—831no—
206C2×D4—1662no—
207C2×D4—1664no—
208C2×D4—1668no—
209C2×D4—16616no—
210C2×D4—1661no—
211C2×D4—1668no—
212C2×D4—1668no—
213C2×D4—1661no—
214C2×D4—1662no—
215C2×D4D33214yescomplemented
216C2×D4C22×D8312yes—
217C22×C4—831no—
218C22×C4—832no—
219C22×C4—832no—
220C22×C4—1661no—
221C22×C4—1662no—
222C22×C4—1662no—
223C22×C4C22×D8311yescharacteristic
224SD16—1664no—
225SD16—1664no—
226D8—1664no—
227D8—1664no—
228M16—22no—
229M16—1662no—
230C2×C8—832no—
231C2×C8C22×D8312yescomplemented
232C4⋊C4—1662no—
233C4⋊C4C22×D8312yescomplemented
234C22⋊C4—21no—
235C22⋊C4—1662no—
236C22⋊C4—1662no—
237C42—21no—
238C23—24no—
239C23—22no—
240C23—44no—
241C23—3324no—
242C23—3328no—
243C23—33232no—
244C23—3328no—
245C23—832no—
246C23—16616no—
247C23—1664no—
248C23—1661no—
249C23—1664no—
250C23—1662no—
251C23—1662no—
252C23D4×D8311yescharacteristic
253Q8—1661no—
254Q8—1661no—
255D4—24no—
256D4—22no—
257D4—3324no—
258D4—3322no—
259D4—834no—
260D4—16616no—
261D4—1664no—
262D4—1661no—
263D4—1661no—
264D4—1662no—
265D4C2×D33214yes—
266C2×C4—22no—
267C2×C4—3322no—
268C2×C4—832no—
269C2×C4—1661no—
270C2×C4—1662no—
271C2×C4—1664no—
272C2×C4—1662no—
273C2×C4—1662no—
274C2×C4D4×D8311yescharacteristic
275C2×C4D4×D8312yes—
276C2×C4C2×D33212yes—
277C2×C4C23×D8311yescharacteristic
278C8—22no—
279C8—1662no—
280C22—22no—
281C22—24no—
282C22—48no—
283C22—44no—
284C22—3328no—
285C22—33232no—
286C22—3324no—
287C22—3321no—
288C22—3322no—
289C22—834no—
290C22—1668no—
291C22—1664no—
292C22—1662no—
293C22—1662no—
294C22#1458176212yes—
295C22C2×D4×D8311yescharacteristic
296C4—42no—
297C4—3321no—
298C4—1662no—
299C4#1458176212yes—
300C4C2×D4×D8311yescharacteristic
301C4C22×D33211yescharacteristic
302C2—22no—
303C2—44no—
304C2—3324no—
305C2—6641no—
306C2—1664no—
307C2C2×#1458176211yescharacteristic
308C2AGL(1,ℤ/8)×D8312yes—
subgroupG/H
centerC22C2×D4×D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting#418×C83C2
socleC22×C83C22×D4
hypercenter#418D83
hypocenterC83#3003
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C22×C4 ◁ #418

Complements

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

NH ≅ G/Ntype
1. C22×D4×D83C2semidirect
2. C2×#14581820C2semidirect
3. C2×#14581818C2semidirect
4. #19128213C2semidirect
5. #19127853C2semidirect
6. #19127836C2semidirect
7. #19127705C2semidirect
8. #19127664C2semidirect
9. #418×C83C2semidirect
11. C2×#13050590C22semidirect
21. C2×#13050579C22semidirect
22. C2×#13050578C22semidirect
23. C2×D664C22semidirect
24. C2×#13050575C22semidirect
25. C2×#13050574C22semidirect
27. #14581772C22semidirect
28. #14581767C22semidirect
31. #14581712C22semidirect
32. (C2×D83)⋊C8C22semidirect
34. C2×M16×C83C22semidirect
35. #117×C83C22semidirect
36. #102×C83C22semidirect
44. C2×Dic166C23semidirect
48. C2×D332D4semidirect
50. C2×D332C23semidirect
68. C2×C83⋊C8C23semidirect
75. C2×D4×C83D4semidirect
79. C2×C8×C83C23semidirect
80. C4⋊C4×C83C23semidirect
140. C2×D83AGL(1,ℤ/8)semidirect
155. C83#3003semidirect
163. #418D83semidirect
182. C2×M16C2×D83semidirect
188. #117C2×D83semidirect
190. #102C2×D83semidirect
215. C2×D4D332semidirect
231. C2×C8C22×D83semidirect
233. C4⋊C4C22×D83semidirect

Maximal subgroups

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

subgroupindex
C22×D4×D832
C2×#145818202
C2×#145818182
#191282132
#191278532
#191278362
#191277052
#191276642
#418×C832
#300383