Finite Group Databasebeta

← #24913845

Subgroups — #24913845

Order 10624.

688 conjugacy classes of subgroups.

The subgroup lattice has 488 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#14581778C211yesmaximal, characteristic, complemented
2C2×#14581762C211yesmaximal, characteristic
3#19127629C211yesmaximal, characteristic, complemented
4#19127619C211yesmaximal, characteristic, complemented
5#19127618C211yesmaximal, characteristic, complemented
6#19127614C211yesmaximal, characteristic, complemented
7#19127614C211yesmaximal, characteristic, complemented
8#19127591C211yesmaximal, characteristic, complemented
9#19127590C211yesmaximal, characteristic, complemented
10#19127579C211yesmaximal, characteristic, complemented
11#19127579C211yesmaximal, characteristic, complemented
12#19127577C211yesmaximal, characteristic, complemented
13#19127556C211yesmaximal, characteristic, complemented
14#19127556C211yesmaximal, characteristic, complemented
15#397×C83C211yesmaximal, characteristic, complemented
16C24×D83C2211yescharacteristic
17C22×Dic166C2211yescharacteristic, complemented
18C22×D332C2211yescharacteristic
19C22×D332C2211yescharacteristic
20C22×C22⋊D83C2211yescharacteristic
21C22×C4×D83C2211yescharacteristic, complemented
22#14581778—24no—
23#14581762—24no—
24C2×Dic83⋊C4C2211yescharacteristic, complemented
25C2×C22⋊Dic83C2211yescharacteristic
26C2×C22⋊Dic83C2211yescharacteristic
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
36C2×(C2×D83)⋊C4C2211yescharacteristic
37C2×(C2×D83)⋊C4C2211yescharacteristic, complemented
38C2×(C2×D83)⋊C4C2211yescharacteristic, complemented
39C2×(C2×D83)⋊C4C2211yescharacteristic, complemented
40C2×C4×Dic83C2211yescharacteristic, complemented
41C83⋊#95C2211yescharacteristic, complemented
42C83⋊#95C2211yescharacteristic, complemented
43C83⋊#95C2211yescharacteristic, complemented
44C83⋊#95C2211yescharacteristic, complemented
45C83⋊#95C2211yescharacteristic, complemented
46C2×C4⋊C4×C83C2211yescharacteristic, complemented
47C2×C22⋊C4×C83C2211yescharacteristic
48C2×C22⋊C4×C83C2211yescharacteristic
49C2×C22⋊C4×C83C2211yescharacteristic
50#95×C83C2211yescharacteristic, complemented
51#95×C83C2211yescharacteristic, complemented
52#95×C83C2211yescharacteristic, complemented
53C23×D83—22no—
54C23×D83—22no—
55C23×D83—22no—
56C23×D83—41no—
57C23×D83—41no—
58C23×D83—41no—
59C23×D83—41no—
60C23×D83C2311yescharacteristic
61C23×D83C2311yescharacteristic
62C23×D83C2311yescharacteristic
63C2×Dic166—22no—
64C2×Dic166—41no—
65C2×Dic166—41no—
66C2×D332—22no—
67C2×D332—22no—
68C2×D332—22no—
69C2×D332—22no—
70C2×D332—41no—
71C2×D332—41no—
72C2×C22⋊D83—41no—
73C2×C22⋊D83—41no—
74C2×C22⋊D83D414yes—
75C22×Dic83C2311yescharacteristic
76C22×Dic83C2311yescharacteristic
77C22×Dic83C2311yescharacteristic
78C22×Dic83C2311yescharacteristic
79C22×Dic83C2311yescharacteristic
80C2×C4×D83—41no—
81C2×C4×D83—41no—
82C2×C4×D83—41no—
83Dic83⋊C4—41no—
84C22⋊Dic83—22no—
85C22⋊Dic83—41no—
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)⋊C4—41no—
97(C2×D83)⋊C4—41no—
98(C2×D83)⋊C4—41no—
99C4×Dic83—22no—
100C24×C83C2311yescharacteristic
101C22×C4×C83C2311yescharacteristic
102C22×C4×C83C2311yescharacteristic
103C22×C4×C83C2311yescharacteristic
104C22×C4×C83C2311yescharacteristic
105C22×C4×C83C2311yescharacteristic
106C22×C4×C83C2311yescharacteristic
107C4⋊C4×C83—41no—
108C22⋊C4×C83—22no—
109C22⋊C4×C83—22no—
110C22⋊C4×C83—41no—
111C23×C83—21no—
112C23×C83—21no—
113C23×C83—21no—
114C23×C83—21no—
115C23×C83—21no—
116C23×C83—21no—
117C23×C83C2×D412yes—
118C23×C83C2411yescharacteristic
119C22×D83—21no—
120C22×D83—21no—
121C22×D83—21no—
122C22×D83—21no—
123C22×D83—21no—
124C22×D83—21no—
125C22×D83—21no—
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—44no—
138C22×D83—44no—
139C22×D83—44no—
140C22×D83—44no—
141C22×D83—44no—
142C22×D83—44no—
143C22×D83—44no—
144C22×D83C2×D412yes—
145C22×D83#4112yes—
146C22×D83#4112yes—
147C2×C4×C83—21no—
148C2×C4×C83—21no—
149C2×C4×C83—21no—
150C2×C4×C83—21no—
151C2×C4×C83—21no—
152C2×C4×C83—21no—
153C2×C4×C83—21no—
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—
165C22⋊D83—44no—
166C2×Dic83—21no—
167C2×Dic83—21no—
168C2×Dic83—21no—
169C2×Dic83—21no—
170C2×Dic83—21no—
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×Dic83C2×D412yes—
181D332—44no—
182D332—44no—
183C4×D83—44no—
184Dic166—44no—
185C22×C83—22no—
186C22×C83—22no—
187C22×C83—22no—
188C22×C83—41no—
189C22×C83—41no—
190C22×C83—41no—
191C22×C83—41no—
192C22×C83C22×D411yescharacteristic
193C22×C83C2×#4111yescharacteristic
194C22×C83C2×#4111yescharacteristic
195C22×C83H(5,𝔽2)11yescharacteristic
196C22×C83H(5,𝔽2)11yescharacteristic
197C22×C83H(5,𝔽2)11yescharacteristic
198C22×C83H(5,𝔽2)11yescharacteristic
199C2×D83—22no—
200C2×D83—22no—
201C2×D83—22no—
202C2×D83—22no—
203C2×D83—22no—
204C2×D83—22no—
205C2×D83—22no—
206C2×D83—22no—
207C2×D83—22no—
208C2×D83—416no—
209C2×D83—41no—
210C2×D83—41no—
211C2×D83—41no—
212C2×D83—41no—
213C2×D83—41no—
214C2×D83—41no—
215C2×D83—41no—
216C2×D83—41no—
217C2×D83—41no—
218C2×D83—41no—
219C2×D83—41no—
220C2×D83—41no—
221C4×C83—41no—
222C4×C83—41no—
223C4×C83—41no—
224C4×C83—41no—
225C4×C83—41no—
226C4×C83—41no—
227Dic83—22no—
228Dic83—41no—
229Dic83—41no—
230Dic83—41no—
231Dic83—41no—
232C2×C83—42no—
233C2×C83#53411yescharacteristic
234C2×C83#53811yescharacteristic
235C2×C83#53811yescharacteristic
236C2×C83#54611yescharacteristic
237C2×C83#54611yescharacteristic
238C2×C83#55111yescharacteristic
239C2×C83H(3,𝔽4)11yescharacteristic
240D83—42no—
241D83—42no—
242D83—42no—
243#2684—831nomaximal, Sylow
244C83#268411yescharacteristic, complemented, Sylow
245C2×#124—831no—
246C2×#120—831no—
247#397—831no—
248#397D8311yescharacteristic, complemented
249#394—831no—
250#394—831no—
251#393—831no—
252#393—831no—
253#388—831no—
254#388—831no—
255#386—831no—
256#386—831no—
257#386—831no—
258#379—831no—
259#379—831no—
260C25—831no—
261C22×Q8—831no—
262C22×D4—831no—
263C22×D4—831no—
264C22×D4—831no—
265C23×C4—831no—
266#124—1664no—
267#120—1664no—
268C2×C4⋊C4—831no—
269C2×C4⋊C4C2×D8311yescharacteristic, complemented
270C2×C22⋊C4—831no—
271C2×C22⋊C4—831no—
272C2×C22⋊C4—831no—
273C2×C22⋊C4—831no—
274C2×C22⋊C4—831no—
275C2×C22⋊C4—831no—
276C2×C22⋊C4—831no—
277C2×C22⋊C4—831no—
278C2×C22⋊C4—831no—
279C2×C22⋊C4—831no—
280C2×C22⋊C4—831no—
281C2×C22⋊C4—831no—
282C2×C22⋊C4—831no—
283C2×C22⋊C4—831no—
284C2×C22⋊C4—831no—
285C2×C22⋊C4C2×D8311yescharacteristic
286C2×C22⋊C4C2×D8311yescharacteristic
287C2×C22⋊C4C2×D8311yescharacteristic
288C2×C42—831no—
289#95—831no—
290#95—831no—
291#95—831no—
292#95—831no—
293#95—831no—
294#95C2×D8311yescharacteristic, complemented
295#95C2×D8311yescharacteristic, complemented
296#95C2×D8311yescharacteristic, complemented
297C24—3321no—
298C24—3321no—
299C24—3321no—
300C24—3321no—
301C24—831no—
302C24—831no—
303C24—831no—
304C24—1662no—
305C24—1662no—
306C24—1662no—
307C24C22×D8311yescharacteristic
308C2×Q8—3321no—
309C2×Q8—3321no—
310C2×Q8—1662no—
311C2×D4—3321no—
312C2×D4—3321no—
313C2×D4—3321no—
314C2×D4—3321no—
315C2×D4—834no—
316C2×D4—1662no—
317C2×D4—1662no—
318C2×D4—1662no—
319C2×D4—1662no—
320C22×C4—3321no—
321C22×C4—3321no—
322C22×C4—3321no—
323C22×C4—831no—
324C22×C4—831no—
325C22×C4—831no—
326C22×C4—831no—
327C22×C4—831no—
328C22×C4C22×D8311yescharacteristic
329C22×C4C22×D8311yescharacteristic
330C22×C4C22×D8311yescharacteristic
331C22×C4C22×D8311yescharacteristic
332C22×C4C22×D8311yescharacteristic
333C22×C4C22×D8311yescharacteristic
334C4⋊C4—41no—
335C4⋊C4—3321no—
336C22⋊C4—22no—
337C22⋊C4—22no—
338C22⋊C4—41no—
339C22⋊C4—3321no—
340C22⋊C4—3321no—
341C22⋊C4—3321no—
342C22⋊C4—3321no—
343C22⋊C4—3321no—
344C22⋊C4—3321no—
345C22⋊C4—3321no—
346C22⋊C4—3321no—
347C22⋊C4—1662no—
348C22⋊C4—1662no—
349C22⋊C4—1662no—
350C22⋊C4—1662no—
351C22⋊C4—1662no—
352C22⋊C4—1662no—
353C22⋊C4—1662no—
354C42—1662no—
355C23—21no—
356C23—21no—
357C23—21no—
358C23—21no—
359C23—21no—
360C23—21no—
361C23—3324no—
362C23—3324no—
363C23—3324no—
364C23—3324no—
365C23—3324no—
366C23—3324no—
367C23—3324no—
368C23—832no—
369C23—832no—
370C23—832no—
371C23—1661no—
372C23—1661no—
373C23—1661no—
374C23—1661no—
375C23—1661no—
376C23—1661no—
377C23—1661no—
378C23—1661no—
379C23—1661no—
380C23—1661no—
381C23—1661no—
382C23—1661no—
383C23—1661no—
384C23—1661no—
385C23—1661no—
386C23—1661no—
387C23—1661no—
388C23—1661no—
389C23D4×D8312yes—
390C23C23×D8311yescharacteristic
391Q8—3324no—
392D4—3324no—
393D4—3324no—
394D4—3324no—
395C2×C4—21no—
396C2×C4—21no—
397C2×C4—21no—
398C2×C4—21no—
399C2×C4—21no—
400C2×C4—21no—
401C2×C4—21no—
402C2×C4—21no—
403C2×C4—21no—
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—21no—
408C2×C4—21no—
409C2×C4—21no—
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—3324no—
414C2×C4—832no—
415C2×C4—1661no—
416C2×C4—1661no—
417C2×C4—1661no—
418C2×C4—1661no—
419C2×C4—1661no—
420C2×C4—1661no—
421C2×C4—1661no—
422C2×C4—1661no—
423C2×C4—1661no—
424C2×C4—1661no—
425C2×C4—1661no—
426C2×C4—1661no—
427C2×C4—1661no—
428C2×C4—1661no—
429C22—22no—
430C22—22no—
431C22—22no—
432C22—41no—
433C22—41no—
434C22—41no—
435C22—41no—
436C22—33216no—
437C22—3321no—
438C22—3321no—
439C22—3321no—
440C22—3321no—
441C22—3321no—
442C22—3321no—
443C22—3321no—
444C22—3321no—
445C22—3321no—
446C22—3321no—
447C22—3321no—
448C22—3321no—
449C22—1662no—
450C22—1662no—
451C22—1662no—
452C22—1662no—
453C22—1662no—
454C22—1662no—
455C22—1662no—
456C22—1662no—
457C22—1662no—
458C22C2×D4×D8311yescharacteristic
459C22#41×D8311yescharacteristic
460C22#41×D8311yescharacteristic
461C22#1458183711yescharacteristic
462C22#1458183811yescharacteristic
463C22#1458183811yescharacteristic
464C22#1458183811yescharacteristic
465C4—41no—
466C4—41no—
467C4—41no—
468C4—41no—
469C4—41no—
470C4—41no—
471C4—3321no—
472C4—3321no—
473C4—3321no—
474C4—3321no—
475C4—1662no—
476C2—42no—
477C2—3322no—
478C2—3322no—
479C2—3322no—
480C2#1912826211yescharacteristic
481C2#1912828711yescharacteristic
482C2#1912829011yescharacteristic
483C2#1912834011yescharacteristic
484C2#1912834311yescharacteristic
485C2#1912837611yescharacteristic
486C2#1912844311yescharacteristic
subgroupG/H
center, FrattiniC23C23×D83
commutator, socle, cosocleC23×C83C24
Fitting#397×C83C2
prime radicalC2×#14581762C2
hypercenter#397D83
hypocenterC83#2684
LCSG ▷ C23×C83 ▷ C83
UCSC1 ◁ C23 ◁ #397

Complements

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

NH ≅ G/Ntype
1. C2×#14581778C2semidirect
3. #19127629C2semidirect
4. #19127619C2semidirect
5. #19127618C2semidirect
6. #19127614C2semidirect
7. #19127614C2semidirect
8. #19127591C2semidirect
9. #19127590C2semidirect
10. #19127579C2semidirect
11. #19127579C2semidirect
12. #19127577C2semidirect
13. #19127556C2semidirect
14. #19127556C2semidirect
15. #397×C83C2semidirect
17. C22×Dic166C22semidirect
21. C22×C4×D83C22semidirect
24. C2×Dic83⋊C4C22semidirect
37. C2×(C2×D83)⋊C4C22semidirect
38. C2×(C2×D83)⋊C4C22semidirect
39. C2×(C2×D83)⋊C4C22semidirect
40. C2×C4×Dic83C22semidirect
41. C83⋊#95C22semidirect
42. C83⋊#95C22semidirect
43. C83⋊#95C22semidirect
44. C83⋊#95C22semidirect
45. C83⋊#95C22semidirect
46. C2×C4⋊C4×C83C22semidirect
50. #95×C83C22semidirect
51. #95×C83C22semidirect
52. #95×C83C22semidirect
244. C83#2684semidirect
248. #397D83semidirect
269. C2×C4⋊C4C2×D83semidirect
294. #95C2×D83semidirect
295. #95C2×D83semidirect
296. #95C2×D83semidirect

Maximal subgroups

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

subgroupindex
C2×#145817782
C2×#145817622
#191276292
#191276192
#191276182
#191276142
#191275912
#191275902
#191275792
#191275772
#191275562
#397×C832
#268483