Finite Group Databasebeta

← #24911596

Subgroups — #24911596

Order 10624.

786 conjugacy classes of subgroups.

The subgroup lattice has 502 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#14581781C211yesmaximal, characteristic, complemented
2C2×#14581762C211yesmaximal, characteristic
3C22×(C2×D83)⋊C4C211yesmaximal, characteristic, complemented
4#19127627C211yesmaximal, characteristic, complemented
5#19127618C211yesmaximal, characteristic, complemented
6#19127618C211yesmaximal, characteristic, complemented
7#19127612C211yesmaximal, characteristic, complemented
8#19127602C211yesmaximal, characteristic, complemented
9#19127591C211yesmaximal, characteristic, complemented
10#19127591C211yesmaximal, characteristic, complemented
11#19127580C211yesmaximal, characteristic, complemented
12#19127578C211yesmaximal, characteristic, complemented
13#19127556C211yesmaximal, characteristic, complemented
14C22⋊C4×Dic83C211yesmaximal, characteristic, complemented
15C2×C22⋊Q8×C83C211yesmaximal, characteristic, complemented
16C24×D83C2211yescharacteristic
17C22×D332C2211yescharacteristic
18C22×D332C2211yescharacteristic
19C22×C22⋊D83C2211yescharacteristic
20C23×Dic83C2211yescharacteristic
21#14581781—24no—
22#14581762—24no—
23C2×C22⋊Dic83C2211yescharacteristic
24C2×(C2×D83)⋊C4—24no—
25C2×(C2×D83)⋊C4C2211yescharacteristic
26C2×(C2×D83)⋊C4C2211yescharacteristic
27C2×(C2×D83)⋊C4C2211yescharacteristic
28C2×(C2×D83)⋊C4C2211yescharacteristic
29C2×(C2×D83)⋊C4C2211yescharacteristic
30C2×(C2×D83)⋊C4C2211yescharacteristic
31C2×(C2×D83)⋊C4C2211yescharacteristic
32C2×(C2×D83)⋊C4C2211yescharacteristic
33C2×(C2×D83)⋊C4C2211yescharacteristic
34C2×(C2×D83)⋊C4C2211yescharacteristic
35C2×(C2×D83)⋊C4C2211yescharacteristic, complemented
36C2×(C2×D83)⋊C4C2211yescharacteristic, complemented
37C2×(C2×D83)⋊C4C2211yescharacteristic, complemented
38C2×(C2×D83)⋊C4C2211yescharacteristic, complemented
39C2×C4×Dic83C2211yescharacteristic, complemented
40C2×C4×Dic83C2211yescharacteristic, complemented
41C83⋊#95C2211yescharacteristic, complemented
42C83⋊#95C2211yescharacteristic, complemented
43C83⋊#95C2211yescharacteristic, complemented
44C83⋊#95C2211yescharacteristic, complemented
45C83⋊#95C2211yescharacteristic, complemented
46C83⋊#95C2211yescharacteristic, complemented
47C22×Q8×C83C2211yescharacteristic, complemented
48C23×C4×C83C2211yescharacteristic
49C22⋊Q8×C83—24no—
50C2×C4⋊C4×C83C2211yescharacteristic, complemented
51C2×C4⋊C4×C83C2211yescharacteristic, complemented
52C2×C4⋊C4×C83C2211yescharacteristic, complemented
53C2×C22⋊C4×C83C2211yescharacteristic
54C2×C22⋊C4×C83C2211yescharacteristic
55C23×D83—22no—
56C23×D83—22no—
57C23×D83—22no—
58C23×D83—22no—
59C23×D83—22no—
60C23×D83—41no—
61C23×D83—41no—
62C23×D83C2311yescharacteristic
63C23×D83C2311yescharacteristic
64C23×D83C2311yescharacteristic
65C2×D332—22no—
66C2×D332—22no—
67C2×D332—22no—
68C2×D332—22no—
69C2×D332—41no—
70C2×D332—41no—
71C2×C22⋊D83—22no—
72C2×C22⋊D83—22no—
73C2×C22⋊D83—22no—
74C22×Dic83—22no—
75C22×Dic83—22no—
76C22×Dic83—41no—
77C22×Dic83C2311yescharacteristic
78C22×Dic83C2311yescharacteristic
79C22×Dic83C2311yescharacteristic
80C22×Dic83C2311yescharacteristic
81C22×Dic83C2311yescharacteristic
82C22⋊Dic83—22no—
83(C2×D83)⋊C4—22no—
84(C2×D83)⋊C4—22no—
85(C2×D83)⋊C4—22no—
86(C2×D83)⋊C4—22no—
87(C2×D83)⋊C4—22no—
88(C2×D83)⋊C4—22no—
89(C2×D83)⋊C4—22no—
90(C2×D83)⋊C4—22no—
91(C2×D83)⋊C4—22no—
92(C2×D83)⋊C4—41no—
93(C2×D83)⋊C4—41no—
94(C2×D83)⋊C4—41no—
95(C2×D83)⋊C4—41no—
96(C2×D83)⋊C4D414yescomplemented
97C4×Dic83—22no—
98C4×Dic83—41no—
99C24×C83C2311yescharacteristic
100C2×Q8×C83—22no—
101C2×Q8×C83—22no—
102C2×Q8×C83—41no—
103C22×C4×C83—22no—
104C22×C4×C83—22no—
105C22×C4×C83—41no—
106C22×C4×C83C2311yescharacteristic
107C22×C4×C83C2311yescharacteristic
108C22×C4×C83C2311yescharacteristic
109C22×C4×C83C2311yescharacteristic
110C22×C4×C83C2311yescharacteristic
111C22×C4×C83C2311yescharacteristic
112C4⋊C4×C83—22no—
113C4⋊C4×C83—22no—
114C4⋊C4×C83—22no—
115C22⋊C4×C83—22no—
116C22⋊C4×C83D414yes—
117C23×C83—21no—
118C23×C83—21no—
119C23×C83—21no—
120C23×C83—21no—
121C23×C83C2×D412yes—
122C23×C83#4112yes—
123C23×C83#4112yes—
124C23×C83C2411yescharacteristic
125C22×D83—28no—
126C22×D83—21no—
127C22×D83—21no—
128C22×D83—21no—
129C22×D83—21no—
130C22×D83—21no—
131C22×D83—21no—
132C22×D83—21no—
133C22×D83—21no—
134C22×D83—21no—
135C22×D83—21no—
136C22×D83—21no—
137C22×D83—21no—
138C22×D83—21no—
139C22×D83—21no—
140C22×D83—21no—
141C22×D83—21no—
142C22×D83—21no—
143C22×D83—21no—
144C22×D83—21no—
145C22×D83—44no—
146C22×D83—44no—
147C22×D83—44no—
148C22×D83—44no—
149C22×D83—44no—
150C22×D83—44no—
151C22×D83C2×D412yes—
152C22×D83#4112yes—
153Q8×C83—44no—
154C2×C4×C83—21no—
155C2×C4×C83—21no—
156C2×C4×C83—21no—
157C2×C4×C83—21no—
158C2×C4×C83—21no—
159C2×C4×C83—21no—
160C2×C4×C83—21no—
161C2×C4×C83—21no—
162C2×C4×C83—21no—
163C2×C4×C83—21no—
164C2×C4×C83—21no—
165C2×C4×C83—21no—
166C2×C4×C83—21no—
167C2×C4×C83—21no—
168C2×C4×C83—21no—
169C2×C4×C83—44no—
170C2×C4×C83C2×D412yes—
171C2×C4×C83C2×D412yes—
172C2×C4×C83C2×D412yes—
173C22⋊D83—44no—
174C2×Dic83—21no—
175C2×Dic83—21no—
176C2×Dic83—21no—
177C2×Dic83—21no—
178C2×Dic83—21no—
179C2×Dic83—21no—
180C2×Dic83—21no—
181C2×Dic83—21no—
182C2×Dic83—21no—
183C2×Dic83—21no—
184C2×Dic83—21no—
185C2×Dic83—21no—
186C2×Dic83—21no—
187C2×Dic83—21no—
188C2×Dic83—44no—
189C2×Dic83C2×D412yes—
190D332—44no—
191D332—44no—
192C22×C83—22no—
193C22×C83—22no—
194C22×C83—22no—
195C22×C83—22no—
196C22×C83—22no—
197C22×C83—22no—
198C22×C83#12414yescomplemented
199C22×C83C22×D411yescharacteristic
200C22×C83C22×D411yescharacteristic
201C22×C83C2×#4111yescharacteristic
202C22×C83C2×#4111yescharacteristic
203C22×C83C2×#4111yescharacteristic
204C22×C83H(5,𝔽2)11yescharacteristic
205C22×C83H(5,𝔽2)11yescharacteristic
206C2×D83—22no—
207C2×D83—22no—
208C2×D83—22no—
209C2×D83—22no—
210C2×D83—22no—
211C2×D83—22no—
212C2×D83—416no—
213C2×D83—41no—
214C2×D83—41no—
215C2×D83—41no—
216C2×D83—41no—
217C2×D83—41no—
218C2×D83—41no—
219C2×D83—41no—
220C2×D83—41no—
221C2×D83—41no—
222C2×D83—41no—
223C2×D83—41no—
224C2×D83—41no—
225C2×D83—41no—
226C2×D83—41no—
227C2×D83—41no—
228C4×C83—22no—
229C4×C83—22no—
230C4×C83—22no—
231C4×C83—41no—
232C4×C83—41no—
233C4×C83—41no—
234Dic83—22no—
235Dic83—41no—
236Dic83—41no—
237Dic83—41no—
238Dic83—41no—
239C2×C83—24no—
240C2×C83#52511yescharacteristic
241C2×C83C2×#12411yescharacteristic
242C2×C83#53811yescharacteristic
243C2×C83D4211yescharacteristic
244C2×C83#54611yescharacteristic
245C2×C83#54611yescharacteristic
246C2×C83#55111yescharacteristic
247D83—42no—
248D83—42no—
249D83—81no—
250#2472—831nomaximal, Sylow
251C83#247211yescharacteristic, complemented, Sylow
252C2×#124—831no—
253C2×C22⋊Q8D8311yescharacteristic, complemented
254C2×#120—831no—
255C22×C22⋊C4—831no—
256#397—831no—
257#394—831no—
258#394—831no—
259#393—831no—
260#390—831no—
261#388—831no—
262#388—831no—
263#386—831no—
264#386—831no—
265#379—831no—
266C4×C22⋊C4—831no—
267C25—831no—
268C22×Q8C2×D8311yescharacteristic, complemented
269C22×D4—831no—
270C22×D4—831no—
271C22×D4—831no—
272C23×C4—831no—
273C23×C4C2×D8311yescharacteristic
274#124—1664no—
275C22⋊Q8—24no—
276#120—1664no—
277C2×C4⋊C4C2×D8311yescharacteristic, complemented
278C2×C4⋊C4C2×D8311yescharacteristic, complemented
279C2×C4⋊C4C2×D8311yescharacteristic, complemented
280C2×C22⋊C4—831no—
281C2×C22⋊C4—831no—
282C2×C22⋊C4—831no—
283C2×C22⋊C4—831no—
284C2×C22⋊C4—831no—
285C2×C22⋊C4—831no—
286C2×C22⋊C4—831no—
287C2×C22⋊C4—831no—
288C2×C22⋊C4—831no—
289C2×C22⋊C4—831no—
290C2×C22⋊C4—831no—
291C2×C22⋊C4—831no—
292C2×C22⋊C4—831no—
293C2×C22⋊C4—831no—
294C2×C22⋊C4—831no—
295C2×C22⋊C4—1664no—
296C2×C22⋊C4C2×D8311yescharacteristic
297C2×C22⋊C4C2×D8311yescharacteristic
298C2×C42—831no—
299C2×C42—831no—
300#95—831no—
301#95—831no—
302#95—831no—
303#95—831no—
304#95—831no—
305#95—831no—
306C24—3321no—
307C24—3321no—
308C24—831no—
309C24—831no—
310C24—831no—
311C24—1662no—
312C24—1662no—
313C24—1662no—
314C24—1662no—
315C24—1662no—
316C24C22×D8311yescharacteristic
317C2×Q8—22no—
318C2×Q8—22no—
319C2×Q8—41no—
320C2×D4—3321no—
321C2×D4—3321no—
322C2×D4—1662no—
323C2×D4—1662no—
324C2×D4—1662no—
325C2×D4—1662no—
326C2×D4—1662no—
327C2×D4—1662no—
328C2×D4—1662no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—41no—
332C22×C4—3321no—
333C22×C4—831no—
334C22×C4—831no—
335C22×C4—831no—
336C22×C4—831no—
337C22×C4—831no—
338C22×C4—1662no—
339C22×C4—1662no—
340C22×C4C22×D8311yescharacteristic
341C22×C4C22×D8311yescharacteristic
342C22×C4C22×D8311yescharacteristic
343C22×C4C22×D8311yescharacteristic
344C22×C4C22×D8311yescharacteristic
345C22×C4C22×D8311yescharacteristic
346C4⋊C4—22no—
347C4⋊C4—22no—
348C4⋊C4—22no—
349C22⋊C4—22no—
350C22⋊C4—3321no—
351C22⋊C4—3321no—
352C22⋊C4—3321no—
353C22⋊C4—3321no—
354C22⋊C4—834no—
355C22⋊C4—1662no—
356C22⋊C4—1662no—
357C22⋊C4—1662no—
358C22⋊C4—1662no—
359C22⋊C4—1662no—
360C22⋊C4—1662no—
361C22⋊C4—1662no—
362C22⋊C4—1662no—
363C22⋊C4—1662no—
364C22⋊C4—1662no—
365C22⋊C4C22⋊D8314yes—
366C42—3321no—
367C42—1662no—
368C23—21no—
369C23—21no—
370C23—21no—
371C23—21no—
372C23—3324no—
373C23—3324no—
374C23—3324no—
375C23—3324no—
376C23—3324no—
377C23—3324no—
378C23—832no—
379C23—832no—
380C23—1668no—
381C23—1661no—
382C23—1661no—
383C23—1661no—
384C23—1661no—
385C23—1661no—
386C23—1661no—
387C23—1661no—
388C23—1661no—
389C23—1661no—
390C23—1661no—
391C23—1661no—
392C23—1661no—
393C23—1661no—
394C23—1661no—
395C23—1661no—
396C23—1661no—
397C23—1661no—
398C23—1661no—
399C23—1661no—
400C23C2×C22⋊D8312yes—
401C23#1305059112yes—
402C23#1305059112yes—
403C23C23×D8311yescharacteristic
404Q8—44no—
405D4—3324no—
406D4—3324no—
407D4—3324no—
408C2×C4—21no—
409C2×C4—21no—
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—21no—
414C2×C4—21no—
415C2×C4—21no—
416C2×C4—21no—
417C2×C4—21no—
418C2×C4—21no—
419C2×C4—21no—
420C2×C4—21no—
421C2×C4—21no—
422C2×C4—21no—
423C2×C4—44no—
424C2×C4—3324no—
425C2×C4—832no—
426C2×C4—1661no—
427C2×C4—1661no—
428C2×C4—1661no—
429C2×C4—1661no—
430C2×C4—1661no—
431C2×C4—1661no—
432C2×C4—1661no—
433C2×C4—1661no—
434C2×C4—1661no—
435C2×C4—1661no—
436C2×C4—1661no—
437C2×C4—1661no—
438C2×C4—1661no—
439C2×C4—1661no—
440C2×C4D4×D8312yes—
441C2×C4C2×C22⋊D8312yes—
442C2×C4C2×C22⋊D8312yes—
443C22—22no—
444C22—22no—
445C22—22no—
446C22—22no—
447C22—22no—
448C22—22no—
449C22—33216no—
450C22—3321no—
451C22—3321no—
452C22—3321no—
453C22—3321no—
454C22—3321no—
455C22—3321no—
456C22—3321no—
457C22—3321no—
458C22—3321no—
459C22—3321no—
460C22—3321no—
461C22—3321no—
462C22—3321no—
463C22—3321no—
464C22—3321no—
465C22—1662no—
466C22—1662no—
467C22—1662no—
468C22—1662no—
469C22—1662no—
470C22—1662no—
471C22#1458178114yescomplemented
472C22C2×D4×D8311yescharacteristic
473C22C22×C22⋊D8311yescharacteristic
474C22#41×D8311yescharacteristic
475C22C2×#1305059111yescharacteristic
476C22C2×#1305059111yescharacteristic
477C22#1458183711yescharacteristic
478C22#1458183811yescharacteristic
479C4—22no—
480C4—22no—
481C4—22no—
482C4—41no—
483C4—41no—
484C4—41no—
485C4—3321no—
486C4—3321no—
487C4—3321no—
488C4—3321no—
489C4—1662no—
490C2—24no—
491C2—3322no—
492C2—3322no—
493C2—6641no—
494C2#1912821411yescharacteristic
495C2C2×#1458178111yescharacteristic
496C2#1912829111yescharacteristic
497C2D4×C22⋊D8311yescharacteristic
498C2#1912833911yescharacteristic
499C2#1912834211yescharacteristic
500C2#1912837811yescharacteristic
subgroupG/H
center, FrattiniC23C23×D83
commutator, socle, cosocleC23×C83C24
FittingC2×C22⋊Q8×C83C2
prime radicalC2×#14581762C2
hypercenterC2×C22⋊Q8D83
hypocenterC83#2472
LCSG ▷ C23×C83 ▷ C83
UCSC1 ◁ C23 ◁ C2×C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. C2×#14581781C2semidirect
3. C22×(C2×D83)⋊C4C2semidirect
4. #19127627C2semidirect
5. #19127618C2semidirect
6. #19127618C2semidirect
7. #19127612C2semidirect
8. #19127602C2semidirect
9. #19127591C2semidirect
10. #19127591C2semidirect
11. #19127580C2semidirect
12. #19127578C2semidirect
13. #19127556C2semidirect
14. C22⋊C4×Dic83C2semidirect
15. C2×C22⋊Q8×C83C2semidirect
35. C2×(C2×D83)⋊C4C22semidirect
36. C2×(C2×D83)⋊C4C22semidirect
37. C2×(C2×D83)⋊C4C22semidirect
38. C2×(C2×D83)⋊C4C22semidirect
39. C2×C4×Dic83C22semidirect
40. C2×C4×Dic83C22semidirect
41. C83⋊#95C22semidirect
42. C83⋊#95C22semidirect
43. C83⋊#95C22semidirect
44. C83⋊#95C22semidirect
45. C83⋊#95C22semidirect
46. C83⋊#95C22semidirect
47. C22×Q8×C83C22semidirect
50. C2×C4⋊C4×C83C22semidirect
51. C2×C4⋊C4×C83C22semidirect
52. C2×C4⋊C4×C83C22semidirect
96. (C2×D83)⋊C4D4semidirect
198. C22×C83#124semidirect
251. C83#2472semidirect
253. C2×C22⋊Q8D83semidirect
268. C22×Q8C2×D83semidirect
277. C2×C4⋊C4C2×D83semidirect
278. C2×C4⋊C4C2×D83semidirect
279. C2×C4⋊C4C2×D83semidirect
471. C22#14581781semidirect

Maximal subgroups

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

subgroupindex
C2×#145817812
C2×#145817622
C22×(C2×D83)⋊C42
#191276272
#191276182
#191276122
#191276022
#191275912
#191275802
#191275782
#191275562
C22⋊C4×Dic832
C2×C22⋊Q8×C832
#247283