Finite Group Databasebeta

← #24916769

Subgroups — #24916769

Order 10624.

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

normal subgroups only

This group has 764 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
3C2×#14581812C211yesmaximal, characteristic, complemented
4#19128282C211yesmaximal, characteristic, complemented
5#19127866C212yesmaximal, complemented
6#19127853C212yesmaximal, complemented
7#19127849C212yesmaximal, complemented
8#19127836C212yesmaximal, complemented
9C2×#14581712C211yesmaximal, characteristic, complemented
10#19127669C211yesmaximal, characteristic, complemented
11#417×C83C211yesmaximal, characteristic, complemented
12#14581838—24no—
13#14581838—24no—
14C2×#13050591C2212yes—
15C2×#13050590—21no—
16C2×#13050590C2211yescharacteristic
17C2×#13050590C2211yescharacteristic, complemented
18C22×D332—21no—
19C22×D332C2211yescharacteristic
20C2×D4×D83—22no—
21C2×D4×D83C2212yes—
22#14581820—24no—
23#14581812—24no—
24C2×#13050580C2211yescharacteristic, complemented
25C2×#13050579C2212yescomplemented
26C2×#13050578C2211yescharacteristic, complemented
27C2×#13050577C2211yescharacteristic, complemented
28C2×D664C2212yescomplemented
29C2×#13050575C2211yescharacteristic, complemented
30C22×C83⋊C8C2211yescharacteristic, complemented
31#14581772—21no—
32#14581772C2212yescomplemented
33#14581767—21no—
34#14581767C2212yescomplemented
35#14581766—22no—
36#14581712C2212yescomplemented
37#14581712C2212yescomplemented
38(C2×D83)⋊C8C2212yescomplemented
39(C2×D83)⋊C8C2212yescomplemented
40C2×#41×C83C2211yescharacteristic
41C2×M16×C83C2211yescharacteristic, complemented
42C2×C4⋊C4×C83C2211yescharacteristic, complemented
43#103×C83C2212yescomplemented
44#102×C83C2212yescomplemented
45C23×D83—21no—
46C23×D83—22no—
47#13050591—24no—
48#13050591—24no—
49#13050590—28no—
50#13050590—22no—
51#13050590—22no—
52#13050590—22no—
53#13050590—22no—
54C2×Dic166D411yescharacteristic
55C2×Dic166C2311yescharacteristic
56C2×D332—28no—
57C2×D332—22no—
58C2×D332—21no—
59C2×D332—22no—
60C2×D332—22no—
61C2×D332D411yescharacteristic
62C2×D332D414yescomplemented
63C2×D332C2311yescharacteristic
64C2×D332C2312yes—
65C2×C22⋊D83—21no—
66C2×C22⋊D83—22no—
67C2×C22⋊D83D412yes—
68D4×D83—216no—
69D4×D83—24no—
70D4×D83—24no—
71C2×C4×D83—22no—
72C2×C4×D83—21no—
73C2×C4×D83C2312yes—
74#13050580—22no—
75#13050579—24no—
76#13050578—22no—
77#13050577—22no—
78D664—24no—
79#13050575—22no—
80C2×C83⋊C8—22no—
81C2×C83⋊C8C2311yescharacteristic
82C2×C83⋊C8C2311yescharacteristic
83C4⋊Dic83—22no—
84(C2×D83)⋊C4—22no—
85(C2×D83)⋊C4—22no—
86#41×C83—22no—
87#41×C83D414yescomplemented
88C2×Q8×C83C2311yescharacteristic
89C2×D4×C83—21no—
90C2×D4×C83C2311yescharacteristic
91C22×C4×C83—21no—
92C22×C4×C83—21no—
93C22×C4×C83C2311yescharacteristic
94M16×C83—22no—
95C2×C8×C83C2312yes—
96C4⋊C4×C83—21no—
97C4⋊C4×C83C2312yes—
98C23×C83—21no—
99C23×C83C2×D411yescharacteristic
100C22×D83—22no—
101C22×D83—28no—
102C22×D83—24no—
103C22×D83—21no—
104C22×D83—82no—
105C22×D83—82no—
106C22×D83—81no—
107C22×D83—81no—
108C22×D83C2×D412yes—
109Q8×C83—21no—
110Q8×C83C2×D412yes—
111D4×C83—24no—
112D4×C83—21no—
113D4×C83C2×D412yes—
114C2×C4×C83—21no—
115C2×C4×C83—24no—
116C2×C4×C83—22no—
117C2×C4×C83—81no—
118C2×C4×C83C2×D411yescharacteristic
119C2×C4×C83C2×D412yes—
120C2×C4×C83C2×D412yes—
121C2×C4×C83C2411yescharacteristic
122C22⋊D83—28no—
123C22⋊D83—24no—
124C22⋊D83—42no—
125C2×Dic83—21no—
126C2×Dic83C2×D412yes—
127D332—28no—
128D332—22no—
129D332—24no—
130D332—22no—
131D332—21no—
132D332—21no—
133D332—82no—
134D332C2×D414yes—
135C4×D83—28no—
136C4×D83—22no—
137C4×D83—22no—
138C4×D83—42no—
139Dic166—22no—
140Dic166—21no—
141Dic166—21no—
142C8×C83—22no—
143C83⋊C8—21no—
144C83⋊C8—21no—
145C22×C83—21no—
146C22×C83—21no—
147C22×C83—22no—
148C22×C83—41no—
149C22×C83—41no—
150C22×C83#12012yes—
151C22×C83C22×D411yescharacteristic
152C2×D83—22no—
153C2×D83—24no—
154C2×D83—22no—
155C2×D83—22no—
156C2×D83—42no—
157C2×D83—42no—
158C2×D83—41no—
159C2×D83—41no—
160C2×D83—41no—
161C2×D83—84no—
162C2×D83—88no—
163C4×C83—22no—
164C4×C83—42no—
165C4×C83#12012yes—
166C4×C83C22×D411yescharacteristic
167C4×C83C22×D411yescharacteristic
168Dic83—22no—
169Dic83—41no—
170C2×C83—22no—
171C2×C83—42no—
172C2×C83C2×#12011yescharacteristic
173C2×C83#57611yescharacteristic
174C2×C83#57711yescharacteristic
175D83—42no—
176D83—44no—
177D83—81no—
178#3012—831nomaximal, Sylow
179C83#301211yescharacteristic, complemented, Sylow
180C2×H(5,𝔽2)—831no—
181C2×AGL(1,ℤ/8)—831no—
182C2×#135—831no—
183#537—831no—
184#452—832no—
185#450—832no—
186#449—832no—
187#447—832no—
188#417D8311yescharacteristic, complemented
189C2×#102—831no—
190#408—831no—
191H(5,𝔽2)—1664no—
192H(5,𝔽2)—1664no—
193C2×#41—831no—
194C2×#41—832no—
195C2×#41—831no—
196C2×#41—1661no—
197C2×#41C2×D8311yescharacteristic
198C22×D4—832no—
199C22×D4—831no—
200C22×D4—1661no—
201C22×D4—1662no—
202AGL(1,ℤ/8)—1664no—
203#135—1664no—
204C2×Q16—831no—
205C2×SD16—831no—
206C2×SD16—831no—
207C2×SD16—832no—
208C2×D8—831no—
209C2×D8—832no—
210C2×M16C2×D8311yescharacteristic, complemented
211C22×C8—831no—
212C22⋊Q8—832no—
213C22⋊Q8—1661no—
214#121—832no—
215#121—1661no—
216#121—1662no—
217C2×C4⋊C4C2×D8311yescharacteristic, complemented
218#103C2×D8312yescomplemented
219#102—832no—
220#102—832no—
221#102C2×D8312yescomplemented
222C22⋊C8—832no—
223C22⋊C8—832no—
224C24—1662no—
225C24—1661no—
226#41—22no—
227#41—1668no—
228#41—1662no—
229#41—1662no—
230#41—1664no—
231#41—1664no—
232#41—1662no—
233#41—1662no—
234#41D33214yescomplemented
235C2×Q8—831no—
236C2×Q8—831no—
237C2×Q8C22×D8311yescharacteristic
238C2×D4—21no—
239C2×D4—832no—
240C2×D4—834no—
241C2×D4—831no—
242C2×D4—832no—
243C2×D4—831no—
244C2×D4—1662no—
245C2×D4—1662no—
246C2×D4—1668no—
247C2×D4—1662no—
248C2×D4—1664no—
249C2×D4—16616no—
250C2×D4—1661no—
251C2×D4—1664no—
252C2×D4—1661no—
253C2×D4—1662no—
254C2×D4C22×D8311yescharacteristic
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—832no—
258C22×C4—1662no—
259C22×C4—1661no—
260C22×C4C22×D8311yescharacteristic
261Q16—1662no—
262SD16—1664no—
263SD16—1662no—
264SD16—1662no—
265D8—1664no—
266D8—1662no—
267M16—22no—
268C2×C8—831no—
269C2×C8—831no—
270C2×C8—1662no—
271C2×C8C22×D8312yes—
272C4⋊C4—21no—
273C4⋊C4—1662no—
274C4⋊C4C22×D8312yes—
275C22⋊C4—1662no—
276C22⋊C4—1662no—
277C23—21no—
278C23—832no—
279C23—6641no—
280C23—6641no—
281C23—6642no—
282C23—6642no—
283C23—1668no—
284C23—1661no—
285C23—1662no—
286C23—1664no—
287C23D4×D8311yescharacteristic
288Q8—21no—
289Q8—1661no—
290Q8—1661no—
291Q8—1662no—
292Q8C2×D33212yes—
293D4—24no—
294D4—21no—
295D4—3322no—
296D4—834no—
297D4—6642no—
298D4—1668no—
299D4—1668no—
300D4—1664no—
301D4—1664no—
302D4—1662no—
303D4—1661no—
304D4—1661no—
305D4—1662no—
306D4C2×D33212yes—
307C2×C4—21no—
308C2×C4—24no—
309C2×C4—22no—
310C2×C4—81no—
311C2×C4—3322no—
312C2×C4—832no—
313C2×C4—1668no—
314C2×C4—1661no—
315C2×C4—1662no—
316C2×C4—1662no—
317C2×C4D4×D8311yescharacteristic
318C2×C4D4×D8312yes—
319C2×C4C2×D33212yes—
320C2×C4C23×D8311yescharacteristic
321C8—22no—
322C8—1661no—
323C8—1661no—
324C22—21no—
325C22—21no—
326C22—22no—
327C22—41no—
328C22—41no—
329C22—3322no—
330C22—3322no—
331C22—3321no—
332C22—3321no—
333C22—3321no—
334C22—6644no—
335C22—6648no—
336C22—1664no—
337C22—1662no—
338C22—1662no—
339C22—1662no—
340C22#1458176212yes—
341C22C2×D4×D8311yescharacteristic
342C4—22no—
343C4—42no—
344C4—3321no—
345C4—1662no—
346C4#1458176212yes—
347C4C2×D4×D8311yescharacteristic
348C4C22×D33211yescharacteristic
349C2—22no—
350C2—42no—
351C2—3324no—
352C2—3322no—
353C2—6641no—
354C2C2×#1458176211yescharacteristic
355C2#1912850511yescharacteristic
356C2#1912851011yescharacteristic
subgroupG/H
centerC22C2×D4×D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting#417×C83C2
socleC22×C83C22×D4
hypercenter#417D83
hypocenterC83#3012
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C22×C4 ◁ #417

Complements

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

NH ≅ G/Ntype
1. C2×#14581838C2semidirect
2. C2×#14581820C2semidirect
3. C2×#14581812C2semidirect
4. #19128282C2semidirect
5. #19127866C2semidirect
6. #19127853C2semidirect
7. #19127849C2semidirect
8. #19127836C2semidirect
9. C2×#14581712C2semidirect
10. #19127669C2semidirect
11. #417×C83C2semidirect
17. C2×#13050590C22semidirect
24. C2×#13050580C22semidirect
25. C2×#13050579C22semidirect
26. C2×#13050578C22semidirect
27. C2×#13050577C22semidirect
28. C2×D664C22semidirect
29. C2×#13050575C22semidirect
30. C22×C83⋊C8C22semidirect
32. #14581772C22semidirect
34. #14581767C22semidirect
36. #14581712C22semidirect
37. #14581712C22semidirect
38. (C2×D83)⋊C8C22semidirect
39. (C2×D83)⋊C8C22semidirect
41. C2×M16×C83C22semidirect
42. C2×C4⋊C4×C83C22semidirect
43. #103×C83C22semidirect
44. #102×C83C22semidirect
62. C2×D332D4semidirect
87. #41×C83D4semidirect
179. C83#3012semidirect
188. #417D83semidirect
210. C2×M16C2×D83semidirect
217. C2×C4⋊C4C2×D83semidirect
218. #103C2×D83semidirect
221. #102C2×D83semidirect
234. #41D332semidirect

Maximal subgroups

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

subgroupindex
C2×#145818382
C2×#145818202
C2×#145818122
#191282822
#191278662
#191278532
#191278492
#191278362
C2×#145817122
#191276692
#417×C832
#301283