Finite Group Databasebeta

← #24910854

Subgroups — #24910854

Order 10624.

814 conjugacy classes of subgroups.

The subgroup lattice has 504 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#14581774C211yesmaximal, characteristic, complemented
2C2×#14581760C211yesmaximal, characteristic
3C2×C22⋊C4×D83C211yesmaximal, characteristic, complemented
4C2×C22⋊C4×D83C211yesmaximal, characteristic, complemented
5#19127629C211yesmaximal, characteristic, complemented
6#19127626C211yesmaximal, characteristic, complemented
7#19127615C211yesmaximal, characteristic, complemented
8#19127577C211yesmaximal, characteristic, complemented
9#19127577C211yesmaximal, characteristic, complemented
10#19127577C211yesmaximal, characteristic, complemented
11#19127572C211yesmaximal, characteristic, complemented
12#19127563C211yesmaximal, characteristic, complemented
13#19127560C211yesmaximal, characteristic, complemented
14#19127558C211yesmaximal, characteristic, complemented
15#394×C83C211yesmaximal, characteristic, complemented
16C24×D83C2211yescharacteristic
17C22×C22⋊D83C2211yescharacteristic
18C22×C22⋊D83C2211yescharacteristic
19C22×C4×D83C2211yescharacteristic
20C22×C4×D83C2211yescharacteristic
21C22×C4×D83C2211yescharacteristic
22C22×C4×D83C2211yescharacteristic
23#14581774—24no—
24#14581760—24no—
25C2×C4⋊Dic83C2211yescharacteristic, complemented
26C2×Dic83⋊C4C2211yescharacteristic, complemented
27C2×Dic83⋊C4C2211yescharacteristic, complemented
28C2×C22⋊Dic83C2211yescharacteristic
29C2×C22⋊Dic83C2211yescharacteristic
30C2×C22⋊Dic83C2211yescharacteristic
31C2×C22⋊Dic83C2211yescharacteristic, complemented
32C2×C22⋊Dic83C2211yescharacteristic, complemented
33C2×(C2×D83)⋊C4C2211yescharacteristic
34C2×(C2×D83)⋊C4C2211yescharacteristic
35C2×(C2×D83)⋊C4C2211yescharacteristic
36C2×(C2×D83)⋊C4C2211yescharacteristic
37C2×(C2×D83)⋊C4C2211yescharacteristic
38C2×(C2×D83)⋊C4C2211yescharacteristic
39C22⋊C4×D83—24no—
40C22⋊C4×D83—24no—
41C83⋊#95C2211yescharacteristic, complemented
42C83⋊#95C2211yescharacteristic, complemented
43C83⋊#95C2211yescharacteristic, complemented
44C83⋊#95C2211yescharacteristic, complemented
45C83⋊#95C2211yescharacteristic, complemented
46C83⋊#95C2211yescharacteristic, complemented
47C83⋊#95C2211yescharacteristic, complemented
48C22×D4×C83C2211yescharacteristic
49C2×C4⋊C4×C83C2211yescharacteristic, complemented
50C2×C22⋊C4×C83C2211yescharacteristic
51C2×C22⋊C4×C83C2211yescharacteristic
52C2×C22⋊C4×C83C2211yescharacteristic, complemented
53C2×C22⋊C4×C83C2211yescharacteristic, complemented
54#95×C83C2211yescharacteristic, complemented
55C23×D83—22no—
56C23×D83—22no—
57C23×D83—22no—
58C23×D83—22no—
59C23×D83—41no—
60C23×D83—41no—
61C23×D83D414yes—
62C23×D83C2311yescharacteristic
63C23×D83C2311yescharacteristic
64C2×C22⋊D83—22no—
65C2×C22⋊D83—22no—
66C2×C22⋊D83—22no—
67C2×C22⋊D83—22no—
68C2×C22⋊D83—22no—
69C2×C22⋊D83—22no—
70C22×Dic83C2311yescharacteristic
71C22×Dic83C2311yescharacteristic
72C22×Dic83C2311yescharacteristic
73C22×Dic83C2311yescharacteristic
74C22×Dic83C2311yescharacteristic
75C22×Dic83C2311yescharacteristic
76C2×C4×D83—22no—
77C2×C4×D83—22no—
78C2×C4×D83—22no—
79C2×C4×D83—22no—
80C2×C4×D83—22no—
81C2×C4×D83—22no—
82C2×C4×D83—22no—
83C2×C4×D83—22no—
84C2×C4×D83—41no—
85C2×C4×D83—41no—
86C2×C4×D83—41no—
87C2×C4×D83—41no—
88C4⋊Dic83—22no—
89Dic83⋊C4—22no—
90Dic83⋊C4—41no—
91C22⋊Dic83—22no—
92C22⋊Dic83—22no—
93C22⋊Dic83—22no—
94C22⋊Dic83—22no—
95C22⋊Dic83—41no—
96(C2×D83)⋊C4—22no—
97(C2×D83)⋊C4—22no—
98(C2×D83)⋊C4—22no—
99(C2×D83)⋊C4—22no—
100(C2×D83)⋊C4—22no—
101(C2×D83)⋊C4D414yescomplemented
102C24×C83C2311yescharacteristic
103C24×C83C2311yescharacteristic
104C2×D4×C83—22no—
105C2×D4×C83—41no—
106C2×D4×C83—41no—
107C22×C4×C83C2311yescharacteristic
108C22×C4×C83C2311yescharacteristic
109C22×C4×C83C2311yescharacteristic
110C22×C4×C83C2311yescharacteristic
111C22×C4×C83C2311yescharacteristic
112C4⋊C4×C83—41no—
113C22⋊C4×C83—22no—
114C22⋊C4×C83—22no—
115C22⋊C4×C83—22no—
116C22⋊C4×C83—41no—
117C23×C83—21no—
118C23×C83—21no—
119C23×C83—21no—
120C23×C83—21no—
121C23×C83—21no—
122C23×C83—21no—
123C23×C83—21no—
124C23×C83—21no—
125C23×C83—21no—
126C23×C83—21no—
127C23×C83—21no—
128C23×C83—21no—
129C23×C83—21no—
130C23×C83C2×D412yes—
131C23×C83C2411yescharacteristic
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—28no—
141C22×D83—28no—
142C22×D83—44no—
143C22×D83—44no—
144C22×D83—44no—
145C22×D83—44no—
146C22×D83—44no—
147C22×D83C2×D412yes—
148C22×D83C2×D412yes—
149C22×D83C2×D412yes—
150C22×D83#4112yes—
151C22×D83#4112yes—
152C22×D83#4112yes—
153D4×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×C83C2×D412yes—
169C22⋊D83—44no—
170C22⋊D83—44no—
171C2×Dic83—21no—
172C2×Dic83—21no—
173C2×Dic83—21no—
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×Dic83C2×D412yes—
189C4×D83—44no—
190C4×D83—44no—
191C4×D83—44no—
192C4×D83—44no—
193C22×C83—22no—
194C22×C83—22no—
195C22×C83—22no—
196C22×C83—41no—
197C22×C83—41no—
198C22×C83—41no—
199C22×C83—41no—
200C22×C83—41no—
201C22×C83—41no—
202C22×C83—41no—
203C22×C83—41no—
204C22×C83—41no—
205C22×C83—41no—
206C22×C83—41no—
207C22×C83C22×D411yescharacteristic
208C22×C83C22×D411yescharacteristic
209C22×C83C2×#4111yescharacteristic
210C22×C83C2×#4111yescharacteristic
211C22×C83C2×#4111yescharacteristic
212C22×C83H(5,𝔽2)11yescharacteristic
213C22×C83H(5,𝔽2)11yescharacteristic
214C2×D83—22no—
215C2×D83—22no—
216C2×D83—22no—
217C2×D83—22no—
218C2×D83—22no—
219C2×D83—22no—
220C2×D83—22no—
221C2×D83—22no—
222C2×D83—22no—
223C2×D83—22no—
224C2×D83—22no—
225C2×D83—22no—
226C2×D83—416no—
227C2×D83#12114yes—
228C2×D83#12314yescomplemented
229C4×C83—22no—
230C4×C83—41no—
231C4×C83—41no—
232C4×C83—41no—
233C4×C83—41no—
234Dic83—22no—
235Dic83—41no—
236Dic83—41no—
237Dic83—41no—
238Dic83—41no—
239Dic83—41no—
240C2×C83—42no—
241C2×C83—81no—
242C2×C83C2×#12111yescharacteristic
243C2×C83C2×#12311yescharacteristic
244C2×C83#53411yescharacteristic
245C2×C83#54611yescharacteristic
246C2×C83#54611yescharacteristic
247C2×C83#55111yescharacteristic
248C2×C83#55111yescharacteristic
249D83—24no—
250D83—24no—
251#2412—831nomaximal, Sylow
252C83#241211yescharacteristic, complemented, Sylow
253C2×#123—831no—
254C2×#120—831no—
255C22×C22⋊C4—831no—
256C22×C22⋊C4—831no—
257#397—831no—
258#397—831no—
259#394—831no—
260#394D8311yescharacteristic, complemented
261#386—831no—
262#386—831no—
263#386—831no—
264#385—831no—
265#381—831no—
266#380—831no—
267#380—831no—
268C25—831no—
269C22×D4—831no—
270C22×D4—831no—
271C22×D4C2×D8311yescharacteristic
272C23×C4—831no—
273C23×C4—831no—
274C23×C4—831no—
275C23×C4—831no—
276#123—1664no—
277#120—1664no—
278C2×C4⋊C4—831no—
279C2×C4⋊C4—831no—
280C2×C4⋊C4—831no—
281C2×C4⋊C4C2×D8311yescharacteristic, complemented
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—1664no—
294C2×C22⋊C4—1664no—
295C2×C22⋊C4C2×D8311yescharacteristic
296C2×C22⋊C4C2×D8311yescharacteristic
297C2×C22⋊C4C2×D8311yescharacteristic, complemented
298C2×C22⋊C4C2×D8311yescharacteristic, complemented
299#95—831no—
300#95—831no—
301#95—831no—
302#95—831no—
303#95—831no—
304#95—831no—
305#95—831no—
306#95C2×D8311yescharacteristic, complemented
307C24—3321no—
308C24—3321no—
309C24—834no—
310C24—831no—
311C24—831no—
312C24—1662no—
313C24—1662no—
314C24—1662no—
315C24—1662no—
316C24C22×D8311yescharacteristic
317C24C22×D8311yescharacteristic
318C2×D4—22no—
319C2×D4—41no—
320C2×D4—41no—
321C2×D4—1662no—
322C2×D4—1662no—
323C2×D4—1662no—
324C2×D4—1662no—
325C2×D4—1662no—
326C2×D4—1662no—
327C22×C4—3321no—
328C22×C4—3321no—
329C22×C4—3321no—
330C22×C4—3321no—
331C22×C4—831no—
332C22×C4—831no—
333C22×C4—831no—
334C22×C4—831no—
335C22×C4—831no—
336C22×C4—831no—
337C22×C4—1662no—
338C22×C4—1662no—
339C22×C4—1662no—
340C22×C4—1662no—
341C22×C4—1662no—
342C22×C4—1662no—
343C22×C4—1662no—
344C22×C4—1662no—
345C22×C4C22×D8311yescharacteristic
346C22×C4C22×D8311yescharacteristic
347C22×C4C22×D8311yescharacteristic
348C22×C4C22×D8311yescharacteristic
349C22×C4C22×D8311yescharacteristic
350C4⋊C4—41no—
351C4⋊C4—3321no—
352C4⋊C4—1662no—
353C4⋊C4—1662no—
354C22⋊C4—22no—
355C22⋊C4—22no—
356C22⋊C4—22no—
357C22⋊C4—41no—
358C22⋊C4—3321no—
359C22⋊C4—834no—
360C22⋊C4—1662no—
361C22⋊C4—1662no—
362C22⋊C4—1662no—
363C22⋊C4—1662no—
364C22⋊C4—1662no—
365C22⋊C4—1662no—
366C22⋊C4—1662no—
367C22⋊C4—1662no—
368C22⋊C4—1662no—
369C23—21no—
370C23—21no—
371C23—21no—
372C23—21no—
373C23—21no—
374C23—21no—
375C23—21no—
376C23—21no—
377C23—21no—
378C23—21no—
379C23—21no—
380C23—21no—
381C23—21no—
382C23—3324no—
383C23—3324no—
384C23—3324no—
385C23—3324no—
386C23—3324no—
387C23—832no—
388C23—832no—
389C23—832no—
390C23—832no—
391C23—832no—
392C23—832no—
393C23—1668no—
394C23—1668no—
395C23—1661no—
396C23—1661no—
397C23—1661no—
398C23—1661no—
399C23—1661no—
400C23—1661no—
401C23—1661no—
402C23—1661no—
403C23D4×D8312yes—
404C23C23×D8311yescharacteristic
405D4—44no—
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—3324no—
423C2×C4—3324no—
424C2×C4—3324no—
425C2×C4—3324no—
426C2×C4—832no—
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×C4—1661no—
441C2×C4—1661no—
442C2×C4—1661no—
443C2×C4—1661no—
444C2×C4D4×D8312yes—
445C22—22no—
446C22—22no—
447C22—22no—
448C22—41no—
449C22—41no—
450C22—41no—
451C22—41no—
452C22—41no—
453C22—41no—
454C22—41no—
455C22—41no—
456C22—41no—
457C22—41no—
458C22—41no—
459C22—33216no—
460C22—834no—
461C22—834no—
462C22—1662no—
463C22—1662no—
464C22—1662no—
465C22—1662no—
466C22—1662no—
467C22—1662no—
468C22—1662no—
469C22—1662no—
470C22—1662no—
471C22—1662no—
472C22—1662no—
473C22—1662no—
474C22C2×D4×D8311yescharacteristic
475C22C2×D4×D8311yescharacteristic
476C22#41×D8311yescharacteristic
477C22#41×D8311yescharacteristic
478C22#41×D8311yescharacteristic
479C22#1458183711yescharacteristic
480C22#1458183711yescharacteristic
481C4—22no—
482C4—41no—
483C4—41no—
484C4—41no—
485C4—41no—
486C4—3321no—
487C4—3321no—
488C4—3321no—
489C4—3321no—
490C4—3321no—
491C4—1662no—
492C2—42no—
493C2—81no—
494C2—1664no—
495C2—1664no—
496C2#121×D8311yescharacteristic
497C2#123×D8311yescharacteristic
498C2#1912826111yescharacteristic
499C2#1912834011yescharacteristic
500C2#1912834811yescharacteristic
501C2#1912837911yescharacteristic
502C2#1912838111yescharacteristic
subgroupG/H
center, FrattiniC23C23×D83
commutator, socle, cosocleC23×C83C24
Fitting#394×C83C2
prime radicalC2×#14581760C2
hypercenter#394D83
hypocenterC83#2412
LCSG ▷ C23×C83 ▷ C83
UCSC1 ◁ C23 ◁ #394

Complements

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

NH ≅ G/Ntype
1. C2×#14581774C2semidirect
3. C2×C22⋊C4×D83C2semidirect
4. C2×C22⋊C4×D83C2semidirect
5. #19127629C2semidirect
6. #19127626C2semidirect
7. #19127615C2semidirect
8. #19127577C2semidirect
9. #19127577C2semidirect
10. #19127577C2semidirect
11. #19127572C2semidirect
12. #19127563C2semidirect
13. #19127560C2semidirect
14. #19127558C2semidirect
15. #394×C83C2semidirect
25. C2×C4⋊Dic83C22semidirect
26. C2×Dic83⋊C4C22semidirect
27. C2×Dic83⋊C4C22semidirect
31. C2×C22⋊Dic83C22semidirect
32. C2×C22⋊Dic83C22semidirect
41. C83⋊#95C22semidirect
42. C83⋊#95C22semidirect
43. C83⋊#95C22semidirect
44. C83⋊#95C22semidirect
45. C83⋊#95C22semidirect
46. C83⋊#95C22semidirect
47. C83⋊#95C22semidirect
49. C2×C4⋊C4×C83C22semidirect
52. C2×C22⋊C4×C83C22semidirect
53. C2×C22⋊C4×C83C22semidirect
54. #95×C83C22semidirect
101. (C2×D83)⋊C4D4semidirect
228. C2×D83#123semidirect
252. C83#2412semidirect
260. #394D83semidirect
281. C2×C4⋊C4C2×D83semidirect
297. C2×C22⋊C4C2×D83semidirect
298. C2×C22⋊C4C2×D83semidirect
306. #95C2×D83semidirect

Maximal subgroups

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

subgroupindex
C2×#145817742
C2×#145817602
C2×C22⋊C4×D832
#191276292
#191276262
#191276152
#191275772
#191275722
#191275632
#191275602
#191275582
#394×C832
#241283