Finite Group Databasebeta

← #24910922

Subgroups — #24910922

Order 10624.

800 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×#14581760C211yesmaximal, characteristic
2C2×C4×C22⋊D83C211yesmaximal, characteristic, complemented
3C2×C22⋊C4×D83C211yesmaximal, characteristic, complemented
4#19127636C211yesmaximal, characteristic, complemented
5#19127629C211yesmaximal, characteristic, complemented
6#19127617C211yesmaximal, characteristic, complemented
7#19127615C211yesmaximal, characteristic, complemented
8#19127594C211yesmaximal, characteristic, complemented
9#19127591C211yesmaximal, characteristic, complemented
10#19127580C211yesmaximal, characteristic, complemented
11#19127577C211yesmaximal, characteristic, complemented
12#19127572C211yesmaximal, characteristic, complemented
13#19127558C211yesmaximal, characteristic, complemented
14#19127556C211yesmaximal, characteristic, complemented
15C2×#123×C83C211yesmaximal, characteristic, complemented
16C24×D83C2211yescharacteristic
17C22×C22⋊D83C2211yescharacteristic
18C22×C22⋊D83C2211yescharacteristic
19C22×C4×D83C2211yescharacteristic
20C22×C4×D83C2211yescharacteristic
21#14581760—24no—
22C4×C22⋊D83—24no—
23C2×C4⋊Dic83C2211yescharacteristic, complemented
24C2×Dic83⋊C4C2211yescharacteristic, complemented
25C2×Dic83⋊C4C2211yescharacteristic, complemented
26C2×C22⋊Dic83C2211yescharacteristic
27C2×C22⋊Dic83C2211yescharacteristic
28C2×C22⋊Dic83C2211yescharacteristic
29C2×C22⋊Dic83C2211yescharacteristic, complemented
30C2×C22⋊Dic83C2211yescharacteristic, complemented
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
38C2×(C2×D83)⋊C4C2211yescharacteristic
39C22⋊C4×D83—24no—
40C2×C4×Dic83C2211yescharacteristic, complemented
41C83⋊#95C2211yescharacteristic, complemented
42C83⋊#95C2211yescharacteristic, complemented
43C83⋊#95C2211yescharacteristic, complemented
44C83⋊#95C2211yescharacteristic, complemented
45C83⋊#95C2211yescharacteristic, complemented
46C83⋊#95C2211yescharacteristic, complemented
47C22×D4×C83C2211yescharacteristic
48C23×C4×C83C2211yescharacteristic, complemented
49#123×C83—24no—
50C2×C4⋊C4×C83C2211yescharacteristic, complemented
51C2×C4⋊C4×C83C2211yescharacteristic, complemented
52C2×C22⋊C4×C83C2211yescharacteristic
53C2×C22⋊C4×C83C2211yescharacteristic
54C2×C22⋊C4×C83C2211yescharacteristic, complemented
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
64C2×C22⋊D83—22no—
65C2×C22⋊D83—22no—
66C2×C22⋊D83—22no—
67C2×C22⋊D83—22no—
68C2×C22⋊D83—41no—
69C2×C22⋊D83D414yes—
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—41no—
83Dic83⋊C4—22no—
84Dic83⋊C4—41no—
85C22⋊Dic83—22no—
86C22⋊Dic83—22no—
87C22⋊Dic83—22no—
88C22⋊Dic83—41no—
89C22⋊Dic83—41no—
90(C2×D83)⋊C4—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—41no—
98C4×Dic83—22no—
99C24×C83C2311yescharacteristic
100C24×C83C2311yescharacteristic
101C2×D4×C83—22no—
102C2×D4×C83—22no—
103C2×D4×C83—41no—
104C22×C4×C83—22no—
105C22×C4×C83—22no—
106C22×C4×C83—41no—
107C22×C4×C83C2311yescharacteristic
108C22×C4×C83C2311yescharacteristic
109C22×C4×C83C2311yescharacteristic
110C22×C4×C83C2311yescharacteristic
111C22×C4×C83C2311yescharacteristic
112C4⋊C4×C83—22no—
113C4⋊C4×C83—22no—
114C22⋊C4×C83—22no—
115C22⋊C4×C83—22no—
116C22⋊C4×C83D414yescomplemented
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×C83C2×D412yes—
130C23×C83C2×D412yes—
131C23×C83C2411yescharacteristic
132C22×D83—21no—
133C22×D83—21no—
134C22×D83—21no—
135C22×D83—21no—
136C22×D83—28no—
137C22×D83—21no—
138C22×D83—21no—
139C22×D83—21no—
140C22×D83—21no—
141C22×D83—21no—
142C22×D83—21no—
143C22×D83—44no—
144C22×D83—44no—
145C22×D83—44no—
146C22×D83—44no—
147C22×D83—44no—
148C22×D83—44no—
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—44no—
168C2×C4×C83C2×D412yes—
169C2×C4×C83C2×D412yes—
170C22⋊D83—28no—
171C22⋊D83—44no—
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×Dic83—21no—
189C2×Dic83C2×D412yes—
190C4×D83—44no—
191C4×D83—44no—
192C22×C83—22no—
193C22×C83—22no—
194C22×C83—22no—
195C22×C83—22no—
196C22×C83—22no—
197C22×C83—22no—
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×C83C22×D411yescharacteristic
207C22×C83C22×D411yescharacteristic
208C22×C83C2×#4111yescharacteristic
209C22×C83C2×#4111yescharacteristic
210C22×C83C2×#4111yescharacteristic
211C22×C83H(5,𝔽2)11yescharacteristic
212C22×C83H(5,𝔽2)11yescharacteristic
213C2×D83—22no—
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—416no—
223C2×D83—41no—
224C2×D83—41no—
225C2×D83—41no—
226C2×D83—41no—
227C2×D83#12314yescomplemented
228C4×C83—22no—
229C4×C83—22no—
230C4×C83—41no—
231C4×C83—41no—
232C4×C83—41no—
233Dic83—22no—
234Dic83—41no—
235Dic83—41no—
236Dic83—41no—
237Dic83—41no—
238Dic83—41no—
239C2×C83—42no—
240C2×C83—42no—
241C2×C83C2×#12311yescharacteristic
242C2×C83#52511yescharacteristic
243C2×C83#53811yescharacteristic
244C2×C83D4211yescharacteristic
245C2×C83#54611yescharacteristic
246C2×C83#54611yescharacteristic
247C2×C83#55111yescharacteristic
248D83—24no—
249D83—42no—
250#2418—831nomaximal, Sylow
251C83#241811yescharacteristic, complemented, Sylow
252C2×#123D8311yescharacteristic, complemented
253C2×#120—831no—
254C2×C4×D4—831no—
255C22×C22⋊C4—831no—
256#399—831no—
257#397—831no—
258#394—831no—
259#394—831no—
260#388—831no—
261#388—831no—
262#386—831no—
263#386—831no—
264#385—831no—
265#380—831no—
266#379—831no—
267C25—831no—
268C22×D4—831no—
269C22×D4—831no—
270C22×D4C2×D8311yescharacteristic
271C23×C4—831no—
272C23×C4—831no—
273C23×C4C2×D8311yescharacteristic, complemented
274#123—24no—
275#120—1664no—
276C4×D4—1664no—
277C2×C4⋊C4—831no—
278C2×C4⋊C4—831no—
279C2×C4⋊C4—831no—
280C2×C4⋊C4C2×D8311yescharacteristic, complemented
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—831no—
294C2×C22⋊C4—831no—
295C2×C22⋊C4—1664no—
296C2×C22⋊C4C2×D8311yescharacteristic
297C2×C22⋊C4C2×D8311yescharacteristic
298C2×C22⋊C4C2×D8311yescharacteristic, complemented
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—1662no—
311C24—1662no—
312C24—1662no—
313C24—1662no—
314C24—1662no—
315C24C22×D8311yescharacteristic
316C24C22×D8311yescharacteristic
317C2×D4—22no—
318C2×D4—22no—
319C2×D4—41no—
320C2×D4—3321no—
321C2×D4—834no—
322C2×D4—1662no—
323C2×D4—1662no—
324C2×D4—1662no—
325C2×D4—1662no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—41no—
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×C4C22×D8311yescharacteristic
342C22×C4C22×D8311yescharacteristic
343C22×C4C22×D8311yescharacteristic
344C22×C4C22×D8311yescharacteristic
345C22×C4C22×D8311yescharacteristic
346C4⋊C4—22no—
347C4⋊C4—22no—
348C4⋊C4—3321no—
349C4⋊C4—3321no—
350C4⋊C4—1662no—
351C22⋊C4—22no—
352C22⋊C4—22no—
353C22⋊C4—3321no—
354C22⋊C4—3321no—
355C22⋊C4—3321no—
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⋊C4—1662no—
366C22⋊C4C22⋊D8314yescomplemented
367C42—1662no—
368C23—21no—
369C23—21no—
370C23—21no—
371C23—21no—
372C23—21no—
373C23—21no—
374C23—21no—
375C23—21no—
376C23—21no—
377C23—21no—
378C23—21no—
379C23—21no—
380C23—3324no—
381C23—3324no—
382C23—3324no—
383C23—3324no—
384C23—3324no—
385C23—3324no—
386C23—832no—
387C23—832no—
388C23—832no—
389C23—832no—
390C23—1668no—
391C23—1661no—
392C23—1661no—
393C23—1661no—
394C23—1661no—
395C23—1661no—
396C23—1661no—
397C23—1661no—
398C23—1661no—
399C23—1661no—
400C23—1661no—
401C23D4×D8312yes—
402C23C2×C22⋊D8312yes—
403C23C23×D8311yescharacteristic
404D4—44no—
405D4—3324no—
406D4—1668no—
407C2×C4—21no—
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—44no—
421C2×C4—3324no—
422C2×C4—3324no—
423C2×C4—832no—
424C2×C4—1661no—
425C2×C4—1661no—
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×C4—1661no—
441C2×C4C2×C22⋊D8312yes—
442C2×C4C2×C22⋊D8312yes—
443C22—22no—
444C22—22no—
445C22—22no—
446C22—22no—
447C22—22no—
448C22—22no—
449C22—41no—
450C22—41no—
451C22—41no—
452C22—41no—
453C22—41no—
454C22—41no—
455C22—41no—
456C22—41no—
457C22—33216no—
458C22—3321no—
459C22—3321no—
460C22—3321no—
461C22—3321no—
462C22—834no—
463C22—1662no—
464C22—1662no—
465C22—1662no—
466C22—1662no—
467C22—1662no—
468C22—1662no—
469C22—1662no—
470C22—1662no—
471C22—1662no—
472C22C2×D4×D8311yescharacteristic
473C22C22×C22⋊D8311yescharacteristic
474C22#41×D8311yescharacteristic
475C22#41×D8311yescharacteristic
476C22#41×D8311yescharacteristic
477C22#1458183711yescharacteristic
478C22#1458183811yescharacteristic
479C4—22no—
480C4—22no—
481C4—41no—
482C4—41no—
483C4—41no—
484C4—3321no—
485C4—3321no—
486C4—3321no—
487C4—3321no—
488C4—3321no—
489C4—1662no—
490C2—42no—
491C2—42no—
492C2—3322no—
493C2—1664no—
494C2#123×D8311yescharacteristic
495C2#1912820811yescharacteristic
496C2#1912828711yescharacteristic
497C2D4×C22⋊D8311yescharacteristic
498C2#1912833911yescharacteristic
499C2#1912834311yescharacteristic
500C2#1912837911yescharacteristic
subgroupG/H
center, FrattiniC23C23×D83
commutator, socle, cosocleC23×C83C24
FittingC2×#123×C83C2
prime radicalC2×#14581760C2
hypercenterC2×#123D83
hypocenterC83#2418
LCSG ▷ C23×C83 ▷ C83
UCSC1 ◁ C23 ◁ C2×#123

Complements

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

NH ≅ G/Ntype
2. C2×C4×C22⋊D83C2semidirect
3. C2×C22⋊C4×D83C2semidirect
4. #19127636C2semidirect
5. #19127629C2semidirect
6. #19127617C2semidirect
7. #19127615C2semidirect
8. #19127594C2semidirect
9. #19127591C2semidirect
10. #19127580C2semidirect
11. #19127577C2semidirect
12. #19127572C2semidirect
13. #19127558C2semidirect
14. #19127556C2semidirect
15. C2×#123×C83C2semidirect
23. C2×C4⋊Dic83C22semidirect
24. C2×Dic83⋊C4C22semidirect
25. C2×Dic83⋊C4C22semidirect
29. C2×C22⋊Dic83C22semidirect
30. C2×C22⋊Dic83C22semidirect
40. C2×C4×Dic83C22semidirect
41. C83⋊#95C22semidirect
42. C83⋊#95C22semidirect
43. C83⋊#95C22semidirect
44. C83⋊#95C22semidirect
45. C83⋊#95C22semidirect
46. C83⋊#95C22semidirect
48. C23×C4×C83C22semidirect
50. C2×C4⋊C4×C83C22semidirect
51. C2×C4⋊C4×C83C22semidirect
54. C2×C22⋊C4×C83C22semidirect
116. C22⋊C4×C83D4semidirect
227. C2×D83#123semidirect
251. C83#2418semidirect
252. C2×#123D83semidirect
273. C23×C4C2×D83semidirect
280. C2×C4⋊C4C2×D83semidirect
281. C2×C4⋊C4C2×D83semidirect
298. C2×C22⋊C4C2×D83semidirect
366. C22⋊C4C22⋊D83semidirect

Maximal subgroups

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

subgroupindex
C2×#145817602
C2×C4×C22⋊D832
C2×C22⋊C4×D832
#191276362
#191276292
#191276172
#191276152
#191275942
#191275912
#191275802
#191275772
#191275722
#191275582
#191275562
C2×#123×C832
#241883