Finite Group Databasebeta

← #24914023

Subgroups — #24914023

Order 10624.

680 conjugacy classes of subgroups.

The subgroup lattice has 486 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#14581777C211yesmaximal, characteristic, complemented
2C2×#14581762C211yesmaximal, characteristic
3#19127630C211yesmaximal, characteristic, complemented
4#19127630C211yesmaximal, characteristic, complemented
5#19127624C211yesmaximal, characteristic, complemented
6#19127623C211yesmaximal, characteristic, complemented
7#19127619C211yesmaximal, characteristic, complemented
8#19127616C211yesmaximal, characteristic, complemented
9#19127603C211yesmaximal, characteristic, complemented
10#19127592C211yesmaximal, characteristic, complemented
11#19127579C211yesmaximal, characteristic, complemented
12#19127577C211yesmaximal, characteristic, complemented
13#19127556C211yesmaximal, characteristic, complemented
14#19127556C211yesmaximal, characteristic, complemented
15#388×C83C211yesmaximal, characteristic, complemented
16C24×D83C2211yescharacteristic
17C22×D332C2211yescharacteristic
18C22×D332C2211yescharacteristic
19C22×C22⋊D83C2211yescharacteristic
20C22×C4×D83C2211yescharacteristic, complemented
21#14581777—24no—
22#14581762—24no—
23C2×C4⋊Dic83C2211yescharacteristic, complemented
24C2×Dic83⋊C4C2211yescharacteristic, complemented
25C2×Dic83⋊C4C2211yescharacteristic, complemented
26C2×Dic83⋊C4C2211yescharacteristic, complemented
27C2×C22⋊Dic83C2211yescharacteristic
28C2×C22⋊Dic83C2211yescharacteristic
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
38C2×(C2×D83)⋊C4C2211yescharacteristic
39C2×(C2×D83)⋊C4C2211yescharacteristic, complemented
40C2×(C2×D83)⋊C4C2211yescharacteristic, complemented
41C2×(C2×D83)⋊C4C2211yescharacteristic, 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
50C2×C42×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×D332—22no—
64C2×D332—22no—
65C2×D332—41no—
66C2×D332—41no—
67C2×D332—41no—
68C2×D332D414yes—
69C2×C22⋊D83—22no—
70C2×C22⋊D83—22no—
71C2×C22⋊D83—41no—
72C22×Dic83C2311yescharacteristic
73C22×Dic83C2311yescharacteristic
74C22×Dic83C2311yescharacteristic
75C22×Dic83C2311yescharacteristic
76C22×Dic83C2311yescharacteristic
77C2×C4×D83—22no—
78C2×C4×D83—41no—
79C2×C4×D83—41no—
80C4⋊Dic83—41no—
81Dic83⋊C4—22no—
82Dic83⋊C4—41no—
83Dic83⋊C4—41no—
84C22⋊Dic83—41no—
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—
99C24×C83C2311yescharacteristic
100C22×C4×C83C2311yescharacteristic
101C22×C4×C83C2311yescharacteristic
102C22×C4×C83C2311yescharacteristic
103C22×C4×C83C2311yescharacteristic
104C22×C4×C83C2311yescharacteristic
105C22×C4×C83C2311yescharacteristic
106C4⋊C4×C83—22no—
107C22⋊C4×C83—22no—
108C22⋊C4×C83—22no—
109C22⋊C4×C83—41no—
110C42×C83—41no—
111C23×C83—21no—
112C23×C83—21no—
113C23×C83—21no—
114C23×C83—21no—
115C23×C83—21no—
116C23×C83—21no—
117C23×C83#4112yes—
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×D83C2×D412yes—
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×C83C2×D412yes—
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×Dic83—21no—
181D332—44no—
182D332—44no—
183C4×D83—44no—
184C22×C83—22no—
185C22×C83—22no—
186C22×C83—22no—
187C22×C83—41no—
188C22×C83—41no—
189C22×C83—41no—
190C22×C83—41no—
191C22×C83C22×D411yescharacteristic
192C22×C83C2×#4111yescharacteristic
193C22×C83C2×#4111yescharacteristic
194C22×C83H(5,𝔽2)11yescharacteristic
195C22×C83H(5,𝔽2)11yescharacteristic
196C22×C83H(5,𝔽2)11yescharacteristic
197C22×C83H(5,𝔽2)11yescharacteristic
198C2×D83—22no—
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—416no—
208C2×D83—41no—
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—
220C4×C83—22no—
221C4×C83—41no—
222C4×C83—41no—
223C4×C83—41no—
224C4×C83—41no—
225C4×C83—41no—
226Dic83—41no—
227Dic83—41no—
228Dic83—41no—
229Dic83—41no—
230Dic83—41no—
231C2×C83—42no—
232C2×C83#53411yescharacteristic
233C2×C83#53811yescharacteristic
234C2×C83#53811yescharacteristic
235C2×C83#54611yescharacteristic
236C2×C83#54611yescharacteristic
237C2×C83#55111yescharacteristic
238C2×C83#56011yescharacteristic
239D83—42no—
240D83—42no—
241D83—42no—
242#2697—831nomaximal, Sylow
243C83#269711yescharacteristic, complemented, Sylow
244C2×#123—831no—
245C2×#120—831no—
246#397—831no—
247#397—831no—
248#396—831no—
249#396—831no—
250#394—831no—
251#394—831no—
252#390—831no—
253#388—831no—
254#388D8311yescharacteristic, complemented
255#386—831no—
256#386—831no—
257#379—831no—
258#379—831no—
259C25—831no—
260C22×D4—831no—
261C22×D4—831no—
262C22×D4—831no—
263C23×C4—831no—
264#123—1664no—
265#120—1664no—
266C2×C4⋊C4—831no—
267C2×C4⋊C4—831no—
268C2×C4⋊C4—831no—
269C2×C4⋊C4—831no—
270C2×C4⋊C4C2×D8311yescharacteristic, complemented
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⋊C4—831no—
286C2×C22⋊C4C2×D8311yescharacteristic
287C2×C22⋊C4C2×D8311yescharacteristic
288C2×C22⋊C4C2×D8311yescharacteristic
289C2×C42C2×D8311yescharacteristic, complemented
290#95—831no—
291#95—831no—
292#95—831no—
293#95—831no—
294#95C2×D8311yescharacteristic, complemented
295#95C2×D8311yescharacteristic, complemented
296C24—3321no—
297C24—3321no—
298C24—3321no—
299C24—3321no—
300C24—831no—
301C24—831no—
302C24—831no—
303C24—1662no—
304C24—1662no—
305C24—1662no—
306C24C22×D8311yescharacteristic
307C2×D4—3321no—
308C2×D4—3321no—
309C2×D4—3321no—
310C2×D4—3321no—
311C2×D4—834no—
312C2×D4—1662no—
313C2×D4—1662no—
314C2×D4—1662no—
315C2×D4—1662no—
316C22×C4—3321no—
317C22×C4—3321no—
318C22×C4—831no—
319C22×C4—831no—
320C22×C4—831no—
321C22×C4—831no—
322C22×C4—831no—
323C22×C4—1662no—
324C22×C4C22×D8311yescharacteristic
325C22×C4C22×D8311yescharacteristic
326C22×C4C22×D8311yescharacteristic
327C22×C4C22×D8311yescharacteristic
328C22×C4C22×D8311yescharacteristic
329C22×C4C22×D8311yescharacteristic
330C4⋊C4—22no—
331C4⋊C4—3321no—
332C4⋊C4—3321no—
333C4⋊C4—3321no—
334C4⋊C4—1662no—
335C22⋊C4—22no—
336C22⋊C4—22no—
337C22⋊C4—41no—
338C22⋊C4—3321no—
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—
353C42—41no—
354C23—21no—
355C23—21no—
356C23—21no—
357C23—21no—
358C23—21no—
359C23—21no—
360C23—3324no—
361C23—3324no—
362C23—3324no—
363C23—3324no—
364C23—3324no—
365C23—3324no—
366C23—3324no—
367C23—832no—
368C23—832no—
369C23—832no—
370C23—1661no—
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#1305059012yes—
389C23C23×D8311yescharacteristic
390D4—3324no—
391D4—3324no—
392D4—3324no—
393C2×C4—21no—
394C2×C4—21no—
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—3324no—
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×C4—1661no—
420C2×C4—1661no—
421C2×C4—1661no—
422C2×C4—1661no—
423C2×C4—1661no—
424C2×C4—1661no—
425C2×C4—1661no—
426C2×C4D4×D8312yes—
427C22—22no—
428C22—22no—
429C22—22no—
430C22—41no—
431C22—41no—
432C22—41no—
433C22—41no—
434C22—33216no—
435C22—3321no—
436C22—3321no—
437C22—3321no—
438C22—3321no—
439C22—3321no—
440C22—3321no—
441C22—3321no—
442C22—3321no—
443C22—3321no—
444C22—3321no—
445C22—3321no—
446C22—3321no—
447C22—1662no—
448C22—1662no—
449C22—1662no—
450C22—1662no—
451C22—1662no—
452C22—1662no—
453C22—1662no—
454C22—1662no—
455C22—1662no—
456C22C2×D4×D8311yescharacteristic
457C22#41×D8311yescharacteristic
458C22C2×#1305059011yescharacteristic
459C22#1458183711yescharacteristic
460C22#1458183811yescharacteristic
461C22#1458183811yescharacteristic
462C22#1458183811yescharacteristic
463C4—22no—
464C4—41no—
465C4—41no—
466C4—41no—
467C4—41no—
468C4—41no—
469C4—3321no—
470C4—3321no—
471C4—3321no—
472C4—3321no—
473C4—3321no—
474C2—42no—
475C2—3322no—
476C2—3322no—
477C2—3322no—
478C2#1912825911yescharacteristic
479C2#1912828411yescharacteristic
480C2#1912828711yescharacteristic
481C2#1912834411yescharacteristic
482C2#1912834711yescharacteristic
483C2#1912837511yescharacteristic
484C2#1912844211yescharacteristic
subgroupG/H
center, FrattiniC23C23×D83
commutator, socle, cosocleC23×C83C24
Fitting#388×C83C2
prime radicalC2×#14581762C2
hypercenter#388D83
hypocenterC83#2697
LCSG ▷ C23×C83 ▷ C83
UCSC1 ◁ C23 ◁ #388

Complements

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

NH ≅ G/Ntype
1. C2×#14581777C2semidirect
3. #19127630C2semidirect
4. #19127630C2semidirect
5. #19127624C2semidirect
6. #19127623C2semidirect
7. #19127619C2semidirect
8. #19127616C2semidirect
9. #19127603C2semidirect
10. #19127592C2semidirect
11. #19127579C2semidirect
12. #19127577C2semidirect
13. #19127556C2semidirect
14. #19127556C2semidirect
15. #388×C83C2semidirect
20. C22×C4×D83C22semidirect
23. C2×C4⋊Dic83C22semidirect
24. C2×Dic83⋊C4C22semidirect
25. C2×Dic83⋊C4C22semidirect
26. C2×Dic83⋊C4C22semidirect
39. C2×(C2×D83)⋊C4C22semidirect
40. C2×(C2×D83)⋊C4C22semidirect
41. C2×(C2×D83)⋊C4C22semidirect
42. C83⋊#95C22semidirect
43. C83⋊#95C22semidirect
44. C83⋊#95C22semidirect
45. C83⋊#95C22semidirect
46. C2×C4⋊C4×C83C22semidirect
50. C2×C42×C83C22semidirect
51. #95×C83C22semidirect
52. #95×C83C22semidirect
243. C83#2697semidirect
254. #388D83semidirect
270. C2×C4⋊C4C2×D83semidirect
289. C2×C42C2×D83semidirect
294. #95C2×D83semidirect
295. #95C2×D83semidirect

Maximal subgroups

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

subgroupindex
C2×#145817772
C2×#145817622
#191276302
#191276242
#191276232
#191276192
#191276162
#191276032
#191275922
#191275792
#191275772
#191275562
#388×C832
#269783