Finite Group Databasebeta

← #24919070

Subgroups — #24919070

Order 10624.

520 conjugacy classes of subgroups.

The subgroup lattice has 432 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#14581818C211yesmaximal, characteristic, complemented
2C2×#14581817C211yesmaximal, characteristic, complemented
3#19128347C211yesmaximal, characteristic, complemented
4#19128083C211yesmaximal, characteristic, complemented
5#19127949C211yesmaximal, characteristic, complemented
6#19127945C211yesmaximal, characteristic, complemented
7#19127916C211yesmaximal, characteristic, complemented
8#19127913C211yesmaximal, characteristic, complemented
9#19127863C211yesmaximal, characteristic, complemented
10#19127861C211yesmaximal, characteristic, complemented
11#19127835C211yesmaximal, characteristic, complemented
12#19127834C211yesmaximal, characteristic, complemented
13C4×#13050575C211yesmaximal, characteristic, complemented
14#19127779C211yesmaximal, characteristic, complemented
15#546×C83C211yesmaximal, characteristic, complemented
16C2×#13050590C2211yescharacteristic
17C22×D332C2211yescharacteristic, complemented
18#14581818—24no—
19#14581817—24no—
20C2×#13050580C2211yescharacteristic, complemented
21C2×#13050578C2211yescharacteristic, complemented
22C2×#13050578C2211yescharacteristic, complemented
23C2×#13050577C2211yescharacteristic, complemented
24C2×#13050577C2211yescharacteristic, complemented
25C2×#13050575C2211yescharacteristic
26C2×#13050575C2211yescharacteristic
27C2×#13050575C2211yescharacteristic
28C2×#13050575C2211yescharacteristic
29C2×#13050574C2211yescharacteristic, complemented
30C2×#13050574C2211yescharacteristic, complemented
31#14581780C2211yescharacteristic, complemented
32#14581777—21no—
33#14581771C2211yescharacteristic, complemented
34#14581766C2211yescharacteristic
35#14581765—21no—
36#14581760—21no—
37C4×D332C2211yescharacteristic, complemented
38C22⋊C4×D83—21no—
39#14581725C2211yescharacteristic, complemented
40C4⋊(C83⋊C8)C2211yescharacteristic, complemented
41#14581716C2211yescharacteristic, complemented
42#14581715C2211yescharacteristic, complemented
43#14581713C2211yescharacteristic, complemented
44#14581713C2211yescharacteristic, complemented
45#14581712C2211yescharacteristic, complemented
46#14581712C2211yescharacteristic, complemented
47C22⋊(C83⋊C8)C2211yescharacteristic, complemented
48C22⋊(C83⋊C8)C2211yescharacteristic, complemented
49C4×C83⋊C8C2211yescharacteristic, complemented
50C2×#41×C83C2211yescharacteristic
51C22×D4×C83C2211yescharacteristic, complemented
52#124×C83C2211yescharacteristic, complemented
53#123×C83—21no—
54C22⋊Q8×C83C2211yescharacteristic, complemented
55#121×C83—21no—
56#121×C83C2211yescharacteristic
57#120×C83—21no—
58C4×D4×C83C2211yescharacteristic
59C4×D4×C83C2211yescharacteristic, complemented
60C2×C22⋊C4×C83—21no—
61C23×D83—21no—
62#13050590—22no—
63#13050590—41no—
64C2×Dic166C2311yescharacteristic
65C2×D332—22no—
66C2×D332—21no—
67C2×D332—41no—
68C2×D332C2311yescharacteristic
69C2×D332C2311yescharacteristic
70C2×C22⋊D83—21no—
71C2×C22⋊D83—21no—
72C2×C4×D83—21no—
73C2×C4×D83—21no—
74#13050580—41no—
75#13050578—22no—
76#13050578—22no—
77#13050577—22no—
78#13050577—22no—
79#13050575—22no—
80#13050575—22no—
81#13050575—41no—
82#13050575D414yescomplemented
83#13050574—22no—
84#13050574—22no—
85C2×C83⋊C8C2311yescharacteristic
86C2×C83⋊C8C2311yescharacteristic
87C2×C83⋊C8C2311yescharacteristic
88C2×C83⋊C8C2311yescharacteristic
89C4⋊Dic83C2311yescharacteristic
90Dic83⋊C4—21no—
91C22⋊Dic83—21no—
92(C2×D83)⋊C4—21no—
93(C2×D83)⋊C4—21no—
94(C2×D83)⋊C4—21no—
95(C2×D83)⋊C4—21no—
96C24×C83—21no—
97#41×C83—22no—
98#41×C83—41no—
99C2×Q8×C83C2311yescharacteristic
100C2×D4×C83—22no—
101C2×D4×C83—21no—
102C2×D4×C83—21no—
103C2×D4×C83—21no—
104C2×D4×C83—41no—
105C2×D4×C83D411yescharacteristic
106C2×D4×C83C2311yescharacteristic
107C2×D4×C83C2311yescharacteristic
108C22×C4×C83—21no—
109C22×C4×C83—21no—
110C22×C4×C83C2311yescharacteristic
111C22×C4×C83C2311yescharacteristic
112C4⋊C4×C83—21no—
113C4⋊C4×C83D411yescharacteristic
114C4⋊C4×C83C2311yescharacteristic
115C22⋊C4×C83—21no—
116C22⋊C4×C83—21no—
117C22⋊C4×C83—21no—
118C22⋊C4×C83—21no—
119C22⋊C4×C83—21no—
120C22⋊C4×C83D411yescharacteristic
121C22⋊C4×C83D411yescharacteristic
122C42×C83C2311yescharacteristic
123C23×C83—21no—
124C23×C83—21no—
125C23×C83—42no—
126C23×C83—42no—
127C23×C83—81no—
128C23×C83C2×D411yescharacteristic
129C23×C83C2×D411yescharacteristic
130C22×D83—21no—
131C22×D83—21no—
132C22×D83—42no—
133C22×D83—42no—
134C22×D83—81no—
135Q8×C83—21no—
136Q8×C83—21no—
137D4×C83—21no—
138D4×C83—21no—
139D4×C83—21no—
140D4×C83—41no—
141D4×C83—42no—
142D4×C83—42no—
143D4×C83—81no—
144D4×C83C2×D412yes—
145C2×C4×C83—21no—
146C2×C4×C83—21no—
147C2×C4×C83—21no—
148C2×C4×C83—21no—
149C2×C4×C83—21no—
150C2×C4×C83—21no—
151C2×C4×C83—42no—
152C2×C4×C83—81no—
153C2×C4×C83C2×D411yescharacteristic
154C2×C4×C83C2×D411yescharacteristic
155C2×C4×C83C2×D411yescharacteristic
156C2×C4×C83C2×D411yescharacteristic
157C2×C4×C83C2411yescharacteristic
158C22⋊D83—42no—
159C22⋊D83—81no—
160C2×Dic83—21no—
161C2×Dic83—21no—
162D332—21no—
163D332—21no—
164D332—21no—
165D332—42no—
166D332C2×D412yes—
167C4×D83—42no—
168C4×D83—81no—
169Dic166—21no—
170Dic166—21no—
171C83⋊C8—21no—
172C83⋊C8—21no—
173C83⋊C8—21no—
174C83⋊C8C2×D412yes—
175C22×C83—21no—
176C22×C83—21no—
177C22×C83—22no—
178C22×C83—21no—
179C22×C83—21no—
180C22×C83—21no—
181C22×C83—21no—
182C22×C83—41no—
183C22×C83—41no—
184C22×C83—41no—
185C22×C83—41no—
186C22×C83—41no—
187C22×C83—84no—
188C22×C83C22×D411yescharacteristic
189C2×D83—22no—
190C2×D83—41no—
191C2×D83—41no—
192C2×D83—41no—
193C2×D83—41no—
194C2×D83—41no—
195C2×D83—84no—
196C4×C83—21no—
197C4×C83—21no—
198C4×C83—21no—
199C4×C83—21no—
200C4×C83—41no—
201C4×C83—41no—
202C4×C83C22×D411yescharacteristic
203C4×C83H(5,𝔽2)11yescharacteristic
204Dic83—41no—
205Dic83—41no—
206C2×C83—41no—
207C2×C83—41no—
208C2×C83—42no—
209C2×C83—81no—
210C2×C83D4211yescharacteristic
211C2×C83#57511yescharacteristic
212C2×C83#57611yescharacteristic
213D83—42no—
214D83—81no—
215#3273—831nomaximal, Sylow
216C83#327311yescharacteristic, complemented, Sylow
217C2×AGL(1,ℤ/8)—831no—
218C2×AGL(1,ℤ/8)—831no—
219#546—831no—
220#546D8311yescharacteristic, complemented
221#495—831no—
222#468—831no—
223#468—831no—
224#463—831no—
225#463—831no—
226#452—831no—
227#452—831no—
228#447—831no—
229#447—831no—
230C4×D8—831no—
231#436—831no—
232C2×#41—831no—
233C2×#41C2×D8311yescharacteristic
234C22×D4—831no—
235C22×D4C2×D8311yescharacteristic, complemented
236AGL(1,ℤ/8)—1664no—
237AGL(1,ℤ/8)—1664no—
238C2×Q16—831no—
239C2×SD16—831no—
240C2×SD16—831no—
241C2×SD16—831no—
242C2×SD16—831no—
243C2×D8—831no—
244C2×D8—831no—
245C2×D8—831no—
246C2×D8—831no—
247C2×M16—831no—
248C2×M16—831no—
249#124—831no—
250#124C2×D8311yescharacteristic, complemented
251#123—21no—
252#123—1661no—
253C22⋊Q8—831no—
254C22⋊Q8C2×D8311yescharacteristic, complemented
255#121—21no—
256#121—831no—
257#121—1661no—
258#121C2×D8311yescharacteristic
259#120—21no—
260#120—1661no—
261C4×D4—831no—
262C4×D4C2×D8311yescharacteristic
263C4×D4C2×D8311yescharacteristic, complemented
264C2×C22⋊C4—21no—
265C2×C22⋊C4—1661no—
266#107—831no—
267C4⋊C8—831no—
268#103—831no—
269#103—831no—
270#102—831no—
271#102—831no—
272#102—831no—
273#102—831no—
274C22⋊C8—831no—
275C22⋊C8—831no—
276C4×C8—831no—
277C24—21no—
278C24—1661no—
279#41—22no—
280#41—41no—
281#41—3321no—
282#41—1662no—
283C2×Q8—831no—
284C2×Q8C22×D8311yescharacteristic
285C2×D4—22no—
286C2×D4—21no—
287C2×D4—21no—
288C2×D4—21no—
289C2×D4—41no—
290C2×D4—3321no—
291C2×D4—831no—
292C2×D4—831no—
293C2×D4—1662no—
294C2×D4—1661no—
295C2×D4—1661no—
296C2×D4—1661no—
297C2×D4C22⋊D8311yescharacteristic
298C2×D4C22×D8311yescharacteristic
299C2×D4C22×D8311yescharacteristic
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—1661no—
303C22×C4—1661no—
304C22×C4C22×D8311yescharacteristic
305C22×C4C22×D8311yescharacteristic
306Q16—3321no—
307SD16—1662no—
308SD16—1662no—
309SD16—1662no—
310SD16—1662no—
311D8—3321no—
312D8—834no—
313D8—1662no—
314D8—1662no—
315M16—1662no—
316M16—1662no—
317C2×C8—831no—
318C2×C8—831no—
319C2×C8—831no—
320C2×C8—831no—
321C4⋊C4—21no—
322C4⋊C4—831no—
323C4⋊C4—1661no—
324C4⋊C4C22⋊D8311yescharacteristic
325C4⋊C4C22×D8311yescharacteristic
326C22⋊C4—21no—
327C22⋊C4—21no—
328C22⋊C4—21no—
329C22⋊C4—21no—
330C22⋊C4—21no—
331C22⋊C4—1661no—
332C22⋊C4—1661no—
333C22⋊C4—1661no—
334C22⋊C4—1661no—
335C22⋊C4—1661no—
336C22⋊C4C22⋊D8311yescharacteristic
337C22⋊C4C22⋊D8311yescharacteristic
338C42C22×D8311yescharacteristic
339C23—21no—
340C23—21no—
341C23—42no—
342C23—42no—
343C23—81no—
344C23—3322no—
345C23—3322no—
346C23—6641no—
347C23—1661no—
348C23—1661no—
349C23C2×C22⋊D8311yescharacteristic
350C23C2×C22⋊D8311yescharacteristic
351Q8—21no—
352Q8—21no—
353Q8—1661no—
354Q8—1661no—
355D4—21no—
356D4—21no—
357D4—21no—
358D4—41no—
359D4—42no—
360D4—42no—
361D4—81no—
362D4—3322no—
363D4—3322no—
364D4—832no—
365D4—6641no—
366D4—1661no—
367D4—1661no—
368D4—1661no—
369D4D4×D8312yes—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—42no—
377C2×C4—81no—
378C2×C4—3322no—
379C2×C4—6641no—
380C2×C4—1661no—
381C2×C4—1661no—
382C2×C4C2×C22⋊D8311yescharacteristic
383C2×C4C2×C22⋊D8311yescharacteristic
384C2×C4C2×C22⋊D8311yescharacteristic
385C2×C4C2×C22⋊D8311yescharacteristic
386C2×C4C23×D8311yescharacteristic
387C8—832no—
388C8—1661no—
389C8—1661no—
390C8—1661no—
391C22—21no—
392C22—21no—
393C22—22no—
394C22—21no—
395C22—21no—
396C22—21no—
397C22—21no—
398C22—41no—
399C22—41no—
400C22—41no—
401C22—41no—
402C22—41no—
403C22—84no—
404C22—3321no—
405C22—3321no—
406C22—3321no—
407C22—3321no—
408C22—3321no—
409C22—6644no—
410C22—1662no—
411C22C22×C22⋊D8311yescharacteristic
412C4—21no—
413C4—21no—
414C4—21no—
415C4—21no—
416C4—41no—
417C4—41no—
418C4—3321no—
419C4—3321no—
420C4C2×D4×D8311yescharacteristic
421C4#1458183711yescharacteristic
422C2—41no—
423C2—41no—
424C2—42no—
425C2—81no—
426C2—3322no—
427C2—6641no—
428C2D4×C22⋊D8311yescharacteristic
429C2#1912849911yescharacteristic
430C2#1912850411yescharacteristic
subgroupG/H
centerC22C22×C22⋊D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting#546×C83C2
socleC22×C83C22×D4
hypercenter#546D83
hypocenterC83#3273
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C4×D4 ◁ #546

Complements

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

NH ≅ G/Ntype
1. C2×#14581818C2semidirect
2. C2×#14581817C2semidirect
3. #19128347C2semidirect
4. #19128083C2semidirect
5. #19127949C2semidirect
6. #19127945C2semidirect
7. #19127916C2semidirect
8. #19127913C2semidirect
9. #19127863C2semidirect
10. #19127861C2semidirect
11. #19127835C2semidirect
12. #19127834C2semidirect
13. C4×#13050575C2semidirect
14. #19127779C2semidirect
15. #546×C83C2semidirect
17. C22×D332C22semidirect
20. C2×#13050580C22semidirect
21. C2×#13050578C22semidirect
22. C2×#13050578C22semidirect
23. C2×#13050577C22semidirect
24. C2×#13050577C22semidirect
29. C2×#13050574C22semidirect
30. C2×#13050574C22semidirect
31. #14581780C22semidirect
33. #14581771C22semidirect
37. C4×D332C22semidirect
39. #14581725C22semidirect
40. C4⋊(C83⋊C8)C22semidirect
41. #14581716C22semidirect
42. #14581715C22semidirect
43. #14581713C22semidirect
44. #14581713C22semidirect
45. #14581712C22semidirect
46. #14581712C22semidirect
47. C22⋊(C83⋊C8)C22semidirect
48. C22⋊(C83⋊C8)C22semidirect
49. C4×C83⋊C8C22semidirect
51. C22×D4×C83C22semidirect
52. #124×C83C22semidirect
54. C22⋊Q8×C83C22semidirect
59. C4×D4×C83C22semidirect
82. #13050575D4semidirect
216. C83#3273semidirect
220. #546D83semidirect
235. C22×D4C2×D83semidirect
250. #124C2×D83semidirect
254. C22⋊Q8C2×D83semidirect
263. C4×D4C2×D83semidirect

Maximal subgroups

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

subgroupindex
C2×#145818182
C2×#145818172
#191283472
#191280832
#191279492
#191279452
#191279162
#191279132
#191278632
#191278612
#191278352
#191278342
C4×#130505752
#191277792
#546×C832
#327383