Finite Group Databasebeta

← #27366359

Subgroups — #27366359

Order 13568.

694 conjugacy classes of subgroups.

The subgroup lattice has 456 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20285225C211yesmaximal, characteristic, complemented
2#20284723C211yesmaximal, characteristic
3C2×D4×#88322C211yesmaximal, characteristic
4#20274112C212yesmaximal, complemented
5#20274102C212yesmaximal, complemented
6#20274092C212yesmaximal, complemented
7#20274088C212yesmaximal, complemented
8#20273461C211yesmaximal, characteristic, complemented
9#20273372C211yesmaximal, characteristic, complemented
10#20273370C211yesmaximal, characteristic, complemented
11C53×#3101C211yesmaximal, characteristic
12C23×#88322—21no—
13#16811180C2211yescharacteristic
14#16811026C2211yescharacteristic, complemented
15#16811026C2211yescharacteristic, complemented
16#16811022—21no—
17#16811022—21no—
18#16811019C2211yescharacteristic, complemented
19#16811019C2211yescharacteristic, complemented
20#16810953—21no—
21#16810913C2212yes—
22#16810828C2212yes—
23#16810774—21no—
24C4×#11719923—22no—
25D4×#88322—21no—
26D4×#88322—21no—
27D4×#88322C2212yes—
28D4×#88322C2212yes—
29#16810676C2211yescharacteristic
30#16810659—21no—
31#16810659—21no—
32C2×C4⋊#88322C2211yescharacteristic
33C2×C22⋊#88322—21no—
34C2×C4×#88322C2211yescharacteristic
35#16810375C2212yescomplemented
36#16810373C2212yescomplemented
37#16810372C2212yescomplemented
38#16810371C2212yescomplemented
39#16810370C2212yescomplemented
40#16810369C2212yescomplemented
41C53×#533C2211yescharacteristic
42C2×C4×D4×C53C2211yescharacteristic
43C53×#483—21no—
44C53×#476C2212yes—
45C53×#461C2212yes—
46C53×#450—21no—
47C4×SD16×C53—22no—
48C53×#431C2211yescharacteristic
49C2×#11719923—22no—
50C22×#88322—22no—
51C22×#88322—22no—
52C22×#88322—21no—
53C22×#88322—44no—
54C22×#88322D411yescharacteristic
55C22×#88322D411yescharacteristic
56#13133217C2311yescharacteristic
57#13133217C2311yescharacteristic
58#13133209—21no—
59#13133196—21no—
60#13133196—21no—
61C4×Dic106—21no—
62C4×Dic106—21no—
63#13133176—21no—
64#13133150—21no—
65#13133150—21no—
66#13133150—21no—
67#13133150—21no—
68#13133148—22no—
69#13133148—21no—
70#13133148—21no—
71#13133148—21no—
72#13133148—21no—
73C4⋊#88322—21no—
74C4⋊#88322C2312yes—
75C4⋊#88322C2312yes—
76#13133140—22no—
77#13133138—22no—
78C22⋊#88322—22no—
79C22⋊#88322—22no—
80C22⋊#88322—21no—
81C4×#88322—21no—
82C4×#88322D411yescharacteristic
83C4×#88322D411yescharacteristic
84C4×#88322C2311yescharacteristic
85C4×#88322C2311yescharacteristic
86C22×D4×C53D411yescharacteristic
87C23×C4×C53—21no—
88C2×SD16×C53—22no—
89C4⋊Q8×C53C2311yescharacteristic
90C4⋊Q8×C53C2311yescharacteristic
91#125×C53—21no—
92C22⋊Q8×C53—21no—
93C22⋊Q8×C53—21no—
94C4×Q8×C53—21no—
95C4×Q8×C53—21no—
96C4×D4×C53—21no—
97C4×D4×C53—21no—
98C4×D4×C53D412yes—
99C4×D4×C53C2312yes—
100#117×C53—21no—
101C2×C4⋊C4×C53D411yescharacteristic
102C2×C22⋊C4×C53—21no—
103C2×C42×C53C2311yescharacteristic
104#106×C53—22no—
105C4⋊C8×C53C2312yes—
106#103×C53—22no—
107#102×C53—22no—
108C22⋊C8×C53—21no—
109C4×C8×C53—21no—
110C24×C53—21no—
111C2×Q8×C53—21no—
112C2×Q8×C53—21no—
113C2×D4×C53—24no—
114C2×D4×C53—21no—
115C2×D4×C53C2×D412yes—
116C22×C4×C53—21no—
117C22×C4×C53—22no—
118C22×C4×C53—22no—
119C22×C4×C53—42no—
120C22×C4×C53—42no—
121C22×C4×C53C2×D411yescharacteristic
122C22×C4×C53C2×D411yescharacteristic
123C22×C4×C53#4111yescharacteristic
124C2×Dic106—21no—
125C2×Dic106—21no—
126SD16×C53—44no—
127C2×C8×C53—22no—
128C4⋊C4×C53—21no—
129C4⋊C4×C53—21no—
130C4⋊C4×C53—21no—
131C4⋊C4×C53—21no—
132C4⋊C4×C53—21no—
133C4⋊C4×C53—41no—
134C4⋊C4×C53—41no—
135C4⋊C4×C53C2×D412yes—
136C22⋊C4×C53—22no—
137C22⋊C4×C53—22no—
138C22⋊C4×C53—41no—
139C42×C53—41no—
140C42×C53C2×D411yescharacteristic
141C42×C53C2×D411yescharacteristic
142C42×C53#4111yescharacteristic
143C42×C53C2411yescharacteristic
144C22⋊Dic53—41no—
145#11719923—44no—
146C4⋊Dic53—21no—
147C4⋊Dic53—21no—
148C4⋊Dic53—21no—
149C4⋊Dic53—21no—
150#11719919—41no—
151#11719919—41no—
152C4×Dic53—41no—
153C2×#88322—22no—
154C2×#88322—22no—
155C2×#88322—22no—
156C2×#88322—42no—
157C2×#88322—44no—
158C2×#88322—44no—
159C2×#88322SD1614yescomplemented
160C2×#88322C2×D411yescharacteristic
161C2×#88322C2×D411yescharacteristic
162C2×#88322C2×D411yescharacteristic
163C2×#88322C2×D411yescharacteristic
164C23×C53—24no—
165C23×C53—22no—
166C23×C53—42no—
167C23×C53—42no—
168C23×C53#12111yescharacteristic
169Q8×C53—42no—
170Q8×C53—81no—
171D4×C53—24no—
172D4×C53—22no—
173D4×C53#13514yes—
174C2×C4×C53—22no—
175C2×C4×C53—22no—
176C2×C4×C53—41no—
177C2×C4×C53—42no—
178C2×C4×C53—42no—
179C2×C4×C53—42no—
180C2×C4×C53—44no—
181C2×C4×C53—41no—
182C2×C4×C53—41no—
183C2×C4×C53#12111yescharacteristic
184C2×C4×C53#12111yescharacteristic
185C2×C4×C53#12111yescharacteristic
186C2×C4×C53C2×SD1612yes—
187C2×C4×C53#13712yes—
188C2×C4×C53C22×D411yescharacteristic
189C2×C4×C53C22×D411yescharacteristic
190C2×C4×C53C2×#4111yescharacteristic
191C2×Dic53—41no—
192C2×Dic53—41no—
193Dic106—42no—
194Dic106—81no—
195C8×C53—42no—
196#88322—21no—
197#88322—21no—
198#88322—42no—
199#88322—42no—
200#88322C2×SD1612yes—
201#88322C2×SD1612yes—
202#19105—531nomaximal, Sylow
203C22×C53—21no—
204C22×C53—21no—
205C22×C53—24no—
206C22×C53—48no—
207C22×C53—42no—
208C22×C53—42no—
209C22×C53#46112yes—
210C22×C53C2×#12111yescharacteristic
211C4×C53—21no—
212C4×C53—21no—
213C4×C53—21no—
214C4×C53—42no—
215C4×C53—81no—
216C4×C53—81no—
217C4×C53#46112yes—
218C4×C53C22×SD1611yescharacteristic
219C4×C53C2×#13512yes—
220C4×C53C2×#13711yescharacteristic
221Dic53—81no—
222Dic53—81no—
223#3160—531no—
224#3101—531no—
225#3101D5311yescharacteristic
226C2×C8×D4—531no—
227#1675—532no—
228#1673—532no—
229#1671—532no—
230#1671—532no—
231#1564—531no—
232#1542—531no—
233#1542—531no—
234C2×C53—22no—
235C2×C53—44no—
236C2×C53C2×#46111yescharacteristic
237C2×C53#305011yescharacteristic
238C2×C53#306711yescharacteristic
239C23×C8—1061no—
240#533—531no—
241#533C2×D5311yescharacteristic
242C2×C4×D4C2×D5311yescharacteristic
243#500—531no—
244#500—531no—
245#499—1061no—
246#499—1061no—
247#498—531no—
248#498—531no—
249#483—21no—
250#483—1061no—
251#476—532no—
252#476C2×D5312yes—
253#461—532no—
254#461C2×D5312yes—
255#450—21no—
256#450—1061no—
257C4×SD16—22no—
258C4×SD16—1062no—
259C8×D4—532no—
260C8×D4—532no—
261C8×D4—1061no—
262C8×D4—1061no—
263#431—531no—
264#431C2×D5311yescharacteristic
265#427—1061no—
266#427—1061no—
267C2×C4⋊C8—531no—
268C2×C22⋊C8—1061no—
269C2×C4×C8—531no—
270#335—532no—
271#334—532no—
272#333—532no—
273#333—532no—
274#332—532no—
275#332—532no—
276C53#1910511yescharacteristic, complemented, Sylow
277C22×D4C22⋊D5311yescharacteristic
278C23×C4—21no—
279C2×SD16—22no—
280C2×SD16—1062no—
281C22×C8—531no—
282C22×C8—531no—
283C22×C8—1062no—
284C22×C8—1062no—
285C22×C8—1061no—
286C22×C8—2124no—
287C4⋊Q8—531no—
288C4⋊Q8—531no—
289C4⋊Q8C22×D5311yescharacteristic
290C4⋊Q8C22×D5311yescharacteristic
291#125—21no—
292#125—1061no—
293C22⋊Q8—21no—
294C22⋊Q8—21no—
295C22⋊Q8—1061no—
296C22⋊Q8—1061no—
297C4×Q8—21no—
298C4×Q8—21no—
299C4×Q8—1061no—
300C4×Q8—1061no—
301C4×D4—21no—
302C4×D4—21no—
303C4×D4C22⋊D5312yes—
304C4×D4C22×D5312yes—
305#117—21no—
306#117—1061no—
307C2×C4⋊C4C22⋊D5311yescharacteristic
308C2×C22⋊C4—21no—
309C2×C42C22×D5311yescharacteristic
310#107—1061no—
311#107—1061no—
312#107—1061no—
313#107—1061no—
314#106—22no—
315#106—1061no—
316#106—1061no—
317#106—1061no—
318#106—1061no—
319#106—1062no—
320C4⋊C8—532no—
321C4⋊C8—532no—
322C4⋊C8—1061no—
323C4⋊C8C22×D5312yes—
324#103—22no—
325#103—1062no—
326#102—22no—
327#102—1062no—
328C22⋊C8—21no—
329C22⋊C8—1062no—
330C22⋊C8—1062no—
331C22⋊C8—1061no—
332C4×C8—21no—
333C4×C8—531no—
334C4×C8—531no—
335C4×C8—531no—
336C4×C8—531no—
337C4×C8—1061no—
338C24—21no—
339C2×Q8—21no—
340C2×Q8—21no—
341C2×Q8—1061no—
342C2×Q8—1061no—
343C2×D4—24no—
344C2×D4—21no—
345C2×D4C2×C22⋊D5312yes—
346C22×C4—21no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—42no—
350C22×C4—42no—
351C22×C4D4×D5311yescharacteristic
352C22×C4#1171994711yescharacteristic
353C22×C4C2×C22⋊D5311yescharacteristic
354SD16—44no—
355SD16—2124no—
356C2×C8—22no—
357C2×C8—534no—
358C2×C8—531no—
359C2×C8—531no—
360C2×C8—531no—
361C2×C8—531no—
362C2×C8—1062no—
363C2×C8—1062no—
364C2×C8—1062no—
365C2×C8—2124no—
366C2×C8—2124no—
367C2×C8—2122no—
368C4⋊C4—21no—
369C4⋊C4—21no—
370C4⋊C4—21no—
371C4⋊C4—21no—
372C4⋊C4—21no—
373C4⋊C4—41no—
374C4⋊C4—41no—
375C4⋊C4—1061no—
376C4⋊C4—1061no—
377C4⋊C4—1061no—
378C4⋊C4—1061no—
379C4⋊C4—2121no—
380C4⋊C4—2121no—
381C4⋊C4C2×C22⋊D5312yes—
382C22⋊C4—22no—
383C22⋊C4—22no—
384C22⋊C4—41no—
385C22⋊C4—2121no—
386C42—41no—
387C42—2121no—
388C42D4×D5311yescharacteristic
389C42#1171994711yescharacteristic
390C42C2×C22⋊D5311yescharacteristic
391C42C23×D5311yescharacteristic
392C23—24no—
393C23—22no—
394C23—42no—
395C23—42no—
396C23#1313318911yescharacteristic
397Q8—42no—
398Q8—81no—
399Q8—4241no—
400Q8—2122no—
401D4—24no—
402D4—22no—
403D4#1313323714yes—
404C2×C4—22no—
405C2×C4—22no—
406C2×C4—41no—
407C2×C4—42no—
408C2×C4—44no—
409C2×C4—42no—
410C2×C4—42no—
411C2×C4—41no—
412C2×C4—41no—
413C2×C4—2121no—
414C2×C4—2121no—
415C2×C4#1313318911yescharacteristic
416C2×C4#1313318911yescharacteristic
417C2×C4#1313318911yescharacteristic
418C2×C4SD16×D5312yes—
419C2×C4#1313324612yes—
420C2×C4C2×D4×D5311yescharacteristic
421C2×C4C22×C22⋊D5311yescharacteristic
422C2×C4C2×#1171994711yescharacteristic
423C8—42no—
424C8—532no—
425C8—532no—
426C8—1061no—
427C8—1061no—
428C8—2122no—
429C8—2122no—
430C22—21no—
431C22—21no—
432C22—24no—
433C22—48no—
434C22—42no—
435C22—42no—
436C22#1681082612yes—
437C22C2×#1313318911yescharacteristic
438C4—21no—
439C4—21no—
440C4—21no—
441C4—42no—
442C4—81no—
443C4—81no—
444C4—4241no—
445C4—4241no—
446C4#1681082612yes—
447C4C2×SD16×D5311yescharacteristic
448C4C2×#1313323712yes—
449C4C2×#1313324611yescharacteristic
450C2—22no—
451C2—44no—
452C2C2×#1681082611yescharacteristic
453C2#2028429311yescharacteristic
454C2#2028443211yescharacteristic
subgroupG/H
centerC2×C4#13133189
commutator, cosocleC42×C53C24
FrattiniC42C23×D53
FittingC53×#3101C2
socleC22×C53C2×#121
prime radicalC22×D4×C53D4
hypercenter#3101D53
hypocenterC53#19105
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #3101

Complements

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

NH ≅ G/Ntype
1. #20285225C2semidirect
4. #20274112C2semidirect
5. #20274102C2semidirect
6. #20274092C2semidirect
7. #20274088C2semidirect
8. #20273461C2semidirect
9. #20273372C2semidirect
10. #20273370C2semidirect
14. #16811026C22semidirect
15. #16811026C22semidirect
18. #16811019C22semidirect
19. #16811019C22semidirect
35. #16810375C22semidirect
36. #16810373C22semidirect
37. #16810372C22semidirect
38. #16810371C22semidirect
39. #16810370C22semidirect
40. #16810369C22semidirect
159. C2×#88322SD16semidirect
276. C53#19105semidirect

Maximal subgroups

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

subgroupindex
#202852252
#202847232
C2×D4×#883222
#202741122
#202741022
#202740922
#202740882
#202734612
#202733722
#202733702
C53×#31012
#1910553