Finite Group Databasebeta

← #24911127

Subgroups — #24911127

Order 10624.

780 conjugacy classes of subgroups.

The subgroup lattice has 474 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#14581772C211yesmaximal, characteristic, complemented
2C2×#14581769C211yesmaximal, characteristic, complemented
3C22×(C2×D83)⋊C4C211yesmaximal, characteristic
4C2×C22⋊C4×D83C211yesmaximal, characteristic
5#19127638C211yesmaximal, characteristic, complemented
6#19127623C211yesmaximal, characteristic, complemented
7#19127623C211yesmaximal, characteristic, complemented
8#19127614C211yesmaximal, characteristic, complemented
9#19127576C211yesmaximal, characteristic, complemented
10#19127575C211yesmaximal, characteristic, complemented
11#19127575C211yesmaximal, characteristic, complemented
12#19127561C211yesmaximal, characteristic, complemented
13#19127558C211yesmaximal, characteristic, complemented
14#19127556C211yesmaximal, characteristic
15#385×C83C211yesmaximal, characteristic, complemented
16C24×D83C2211yescharacteristic
17C22×Dic166C2211yescharacteristic, complemented
18C23×Dic83C2211yescharacteristic
19C22×C4×D83C2211yescharacteristic
20C22×C4×D83C2211yescharacteristic
21#14581772—24no—
22#14581769—24no—
23C2×C4⋊Dic83C2211yescharacteristic, complemented
24C2×C4⋊Dic83C2211yescharacteristic, complemented
25C2×C4⋊Dic83C2211yescharacteristic, complemented
26C2×C4⋊Dic83C2211yescharacteristic, complemented
27C2×C4⋊Dic83C2211yescharacteristic, complemented
28C2×Dic83⋊C4C2211yescharacteristic, complemented
29C2×Dic83⋊C4C2211yescharacteristic, complemented
30C2×C22⋊Dic83C2211yescharacteristic
31C2×C22⋊Dic83C2211yescharacteristic
32C2×(C2×D83)⋊C4—24no—
33C2×(C2×D83)⋊C4C2211yescharacteristic
34C2×(C2×D83)⋊C4C2211yescharacteristic
35C2×(C2×D83)⋊C4C2211yescharacteristic
36C2×(C2×D83)⋊C4C2211yescharacteristic
37C2×(C2×D83)⋊C4C2211yescharacteristic
38C2×(C2×D83)⋊C4C2211yescharacteristic
39C2×(C2×D83)⋊C4C2211yescharacteristic
40C2×(C2×D83)⋊C4C2211yescharacteristic
41C2×(C2×D83)⋊C4C2211yescharacteristic
42C2×(C2×D83)⋊C4C2211yescharacteristic
43C22⋊C4×D83—24no—
44C83⋊#95C2211yescharacteristic, complemented
45C83⋊#95C2211yescharacteristic, complemented
46C83⋊#95C2211yescharacteristic, complemented
47C83⋊#95C2211yescharacteristic, complemented
48C23×C4×C83C2211yescharacteristic
49C2×C4⋊C4×C83C2211yescharacteristic, complemented
50C2×C4⋊C4×C83C2211yescharacteristic, complemented
51C2×C22⋊C4×C83C2211yescharacteristic
52C2×C22⋊C4×C83C2211yescharacteristic
53#95×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×D83Q814yescomplemented
62C23×D83C2311yescharacteristic
63C23×D83C2311yescharacteristic
64C2×Dic166—22no—
65C2×Dic166—22no—
66C2×Dic166—41no—
67C22×Dic83—22no—
68C22×Dic83—22no—
69C22×Dic83—41no—
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—41no—
81C2×C4×D83—41no—
82C4⋊Dic83—22no—
83C4⋊Dic83—22no—
84C4⋊Dic83—22no—
85C4⋊Dic83—41no—
86C4⋊Dic83—41no—
87Dic83⋊C4—22no—
88Dic83⋊C4—22no—
89C22⋊Dic83—22no—
90C22⋊Dic83—22no—
91(C2×D83)⋊C4—22no—
92(C2×D83)⋊C4—22no—
93(C2×D83)⋊C4—22no—
94(C2×D83)⋊C4—22no—
95(C2×D83)⋊C4—22no—
96(C2×D83)⋊C4—22no—
97(C2×D83)⋊C4—22no—
98(C2×D83)⋊C4—41no—
99(C2×D83)⋊C4—41no—
100(C2×D83)⋊C4D414yes—
101C24×C83C2311yescharacteristic
102C22×C4×C83—22no—
103C22×C4×C83—22no—
104C22×C4×C83—41no—
105C22×C4×C83C2311yescharacteristic
106C22×C4×C83C2311yescharacteristic
107C22×C4×C83C2311yescharacteristic
108C22×C4×C83C2311yescharacteristic
109C22×C4×C83C2311yescharacteristic
110C22×C4×C83C2311yescharacteristic
111C4⋊C4×C83—22no—
112C4⋊C4×C83—41no—
113C22⋊C4×C83—22no—
114C22⋊C4×C83D414yes—
115C23×C83—21no—
116C23×C83—21no—
117C23×C83—21no—
118C23×C83—21no—
119C23×C83C2×D412yes—
120C23×C83C2×Q812yes—
121C23×C83#4112yes—
122C23×C83C2411yescharacteristic
123C22×D83—21no—
124C22×D83—21no—
125C22×D83—21no—
126C22×D83—21no—
127C22×D83—28no—
128C22×D83—21no—
129C22×D83—21no—
130C22×D83—21no—
131C22×D83—21no—
132C22×D83—21no—
133C22×D83—21no—
134C22×D83—28no—
135C22×D83—44no—
136C22×D83—44no—
137C22×D83—44no—
138C22×D83—44no—
139C22×D83—44no—
140C22×D83C2×D412yes—
141C22×D83C2×Q812yes—
142C22×D83C2×Q812yes—
143C22×D83#4112yes—
144C2×C4×C83—21no—
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—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—44no—
160C2×C4×C83C2×D412yes—
161C2×C4×C83C2×D412yes—
162C2×C4×C83C2×D412yes—
163C2×Dic83—21no—
164C2×Dic83—21no—
165C2×Dic83—21no—
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×Dic83—44no—
181C2×Dic83C2×D412yes—
182C4×D83—44no—
183C4×D83—44no—
184Dic166—44no—
185C22×C83—22no—
186C22×C83—22no—
187C22×C83—22no—
188C22×C83—22no—
189C22×C83—22no—
190C22×C83—22no—
191C22×C83C22⋊Q814yescomplemented
192C22×C83C22×D411yescharacteristic
193C22×C83C22×D411yescharacteristic
194C22×C83C22×Q811yescharacteristic
195C22×C83C2×#4111yescharacteristic
196C22×C83C2×#4111yescharacteristic
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×D83C22⋊Q814yescomplemented
214C4×C83—22no—
215C4×C83—22no—
216C4×C83—22no—
217C4×C83—41no—
218C4×C83—41no—
219C4×C83—41no—
220Dic83—22no—
221Dic83—41no—
222Dic83—41no—
223Dic83—41no—
224Dic83—41no—
225Dic83—41no—
226C2×C83—24no—
227C2×C83C2×C22⋊Q811yescharacteristic
228C2×C83C2×C22⋊Q811yescharacteristic
229C2×C83#54311yescharacteristic
230C2×C83D4211yescharacteristic
231C2×C83#54611yescharacteristic
232C2×C83#54611yescharacteristic
233C2×C83#55111yescharacteristic
234D83—24no—
235D83—42no—
236#2437—831nomaximal, Sylow
237C83#243711yescharacteristic, complemented, Sylow
238C2×C22⋊Q8—831no—
239C2×C22⋊Q8—831no—
240C22×C22⋊C4—831no—
241C22×C22⋊C4—831no—
242#399—831no—
243#396—831no—
244#396—831no—
245#393—831no—
246#385—831no—
247#385—831no—
248#385—831no—
249#385D8311yescharacteristic, complemented
250#380—831no—
251#380—831no—
252#379—831no—
253C25—831no—
254C22×Q8—831no—
255C23×C4—831no—
256C23×C4—831no—
257C23×C4—831no—
258C23×C4C2×D8311yescharacteristic
259C22⋊Q8—1664no—
260C22⋊Q8—1664no—
261C2×C4⋊C4—831no—
262C2×C4⋊C4—831no—
263C2×C4⋊C4—831no—
264C2×C4⋊C4—831no—
265C2×C4⋊C4—831no—
266C2×C4⋊C4—831no—
267C2×C4⋊C4—831no—
268C2×C4⋊C4C2×D8311yescharacteristic, complemented
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—1664no—
283C2×C22⋊C4—1664no—
284C2×C22⋊C4C2×D8311yescharacteristic
285C2×C22⋊C4C2×D8311yescharacteristic
286#95—831no—
287#95—831no—
288#95—831no—
289#95—831no—
290#95C2×D8311yescharacteristic, complemented
291#95C2×D8311yescharacteristic, complemented
292C24—3321no—
293C24—3321no—
294C24—834no—
295C24—831no—
296C24—831no—
297C24—1662no—
298C24—1662no—
299C24—1662no—
300C24—1662no—
301C24C22×D8311yescharacteristic
302C2×Q8—3321no—
303C2×Q8—1662no—
304C2×Q8—1662no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—41no—
308C22×C4—3321no—
309C22×C4—3321no—
310C22×C4—3321no—
311C22×C4—831no—
312C22×C4—831no—
313C22×C4—831no—
314C22×C4—831no—
315C22×C4—831no—
316C22×C4—831no—
317C22×C4—1662no—
318C22×C4—1662no—
319C22×C4—1662no—
320C22×C4—1662no—
321C22×C4—1662no—
322C22×C4—1662no—
323C22×C4C22×D8311yescharacteristic
324C22×C4C22×D8311yescharacteristic
325C22×C4C22×D8311yescharacteristic
326C22×C4C22×D8311yescharacteristic
327C22×C4C22×D8311yescharacteristic
328C22×C4C22×D8311yescharacteristic
329C4⋊C4—22no—
330C4⋊C4—41no—
331C4⋊C4—3321no—
332C4⋊C4—3321no—
333C4⋊C4—1662no—
334C4⋊C4—1662no—
335C4⋊C4—1662no—
336C4⋊C4—1662no—
337C4⋊C4—1662no—
338C22⋊C4—22no—
339C22⋊C4—3321no—
340C22⋊C4—3321no—
341C22⋊C4—834no—
342C22⋊C4—1662no—
343C22⋊C4—1662no—
344C22⋊C4—1662no—
345C22⋊C4—1662no—
346C22⋊C4—1662no—
347C22⋊C4—1662no—
348C22⋊C4—1662no—
349C22⋊C4—1662no—
350C22⋊C4—1662no—
351C22⋊C4D33214yes—
352C23—21no—
353C23—21no—
354C23—21no—
355C23—21no—
356C23—3324no—
357C23—3324no—
358C23—3324no—
359C23—3324no—
360C23—3324no—
361C23—832no—
362C23—832no—
363C23—832no—
364C23—832no—
365C23—1668no—
366C23—1668no—
367C23—1661no—
368C23—1661no—
369C23—1661no—
370C23—1661no—
371C23—1661no—
372C23—1661no—
373C23—1661no—
374C23—1661no—
375C23—1661no—
376C23—1661no—
377C23C2×D33212yes—
378C23Q8×D8312yes—
379C23#1305058912yes—
380C23C23×D8311yescharacteristic
381Q8—3324no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—21no—
386C2×C4—21no—
387C2×C4—21no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—21no—
395C2×C4—21no—
396C2×C4—21no—
397C2×C4—44no—
398C2×C4—3324no—
399C2×C4—3324no—
400C2×C4—3324no—
401C2×C4—832no—
402C2×C4—1661no—
403C2×C4—1661no—
404C2×C4—1661no—
405C2×C4—1661no—
406C2×C4—1661no—
407C2×C4—1661no—
408C2×C4—1661no—
409C2×C4—1661no—
410C2×C4—1661no—
411C2×C4—1661no—
412C2×C4—1661no—
413C2×C4—1661no—
414C2×C4—1661no—
415C2×C4—1661no—
416C2×C4—1661no—
417C2×C4—1661no—
418C2×C4—1661no—
419C2×C4D4×D8312yes—
420C2×C4C2×D33212yes—
421C2×C4C2×D33212yes—
422C22—22no—
423C22—22no—
424C22—22no—
425C22—22no—
426C22—22no—
427C22—22no—
428C22—33216no—
429C22—3321no—
430C22—3321no—
431C22—3321no—
432C22—3321no—
433C22—834no—
434C22—1662no—
435C22—1662no—
436C22—1662no—
437C22—1662no—
438C22—1662no—
439C22—1662no—
440C22—1662no—
441C22—1662no—
442C22—1662no—
443C22#1458177214yescomplemented
444C22C2×D4×D8311yescharacteristic
445C22C22×D33211yescharacteristic
446C22C2×Q8×D8311yescharacteristic
447C22#41×D8311yescharacteristic
448C22C2×#1305058911yescharacteristic
449C22#1458183711yescharacteristic
450C22#1458183811yescharacteristic
451C4—22no—
452C4—22no—
453C4—22no—
454C4—41no—
455C4—41no—
456C4—41no—
457C4—3321no—
458C4—3321no—
459C4—3321no—
460C4—3321no—
461C4—3321no—
462C4—1662no—
463C2—24no—
464C2—3322no—
465C2—1664no—
466C2C22⋊Q8×D8311yescharacteristic
467C2C2×#1458177211yescharacteristic
468C2#1912832611yescharacteristic
469C2D4×D33211yescharacteristic
470C2#1912833811yescharacteristic
471C2#1912834611yescharacteristic
472C2#1912837411yescharacteristic
subgroupG/H
center, FrattiniC23C23×D83
commutator, socle, cosocleC23×C83C24
Fitting#385×C83C2
prime radicalC24×D83C22
hypercenter#385D83
hypocenterC83#2437
LCSG ▷ C23×C83 ▷ C83
UCSC1 ◁ C23 ◁ #385

Complements

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

NH ≅ G/Ntype
1. C2×#14581772C2semidirect
2. C2×#14581769C2semidirect
5. #19127638C2semidirect
6. #19127623C2semidirect
7. #19127623C2semidirect
8. #19127614C2semidirect
9. #19127576C2semidirect
10. #19127575C2semidirect
11. #19127575C2semidirect
12. #19127561C2semidirect
13. #19127558C2semidirect
15. #385×C83C2semidirect
17. C22×Dic166C22semidirect
23. C2×C4⋊Dic83C22semidirect
24. C2×C4⋊Dic83C22semidirect
25. C2×C4⋊Dic83C22semidirect
26. C2×C4⋊Dic83C22semidirect
27. C2×C4⋊Dic83C22semidirect
28. C2×Dic83⋊C4C22semidirect
29. C2×Dic83⋊C4C22semidirect
44. C83⋊#95C22semidirect
45. C83⋊#95C22semidirect
46. C83⋊#95C22semidirect
47. C83⋊#95C22semidirect
49. C2×C4⋊C4×C83C22semidirect
50. C2×C4⋊C4×C83C22semidirect
53. #95×C83C22semidirect
54. #95×C83C22semidirect
61. C23×D83Q8semidirect
191. C22×C83C22⋊Q8semidirect
213. C2×D83C22⋊Q8semidirect
237. C83#2437semidirect
249. #385D83semidirect
268. C2×C4⋊C4C2×D83semidirect
269. C2×C4⋊C4C2×D83semidirect
290. #95C2×D83semidirect
291. #95C2×D83semidirect
443. C22#14581772semidirect

Maximal subgroups

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

subgroupindex
C2×#145817722
C2×#145817692
C22×(C2×D83)⋊C42
C2×C22⋊C4×D832
#191276382
#191276232
#191276142
#191275762
#191275752
#191275612
#191275582
#191275562
#385×C832
#243783