Finite Group Databasebeta

← #27362783

Subgroups — #27362783

Order 13568.

976 conjugacy classes of subgroups.

The subgroup lattice has 482 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20285640C211yesmaximal, characteristic
2#20284746C211yesmaximal, characteristic
3#20284187C211yesmaximal, characteristic
4#20273936C214yesmaximal, complemented
5#20273793C214yesmaximal, complemented
6#20273396C211yesmaximal, characteristic, complemented
7#20273144C211yesmaximal, characteristic, complemented
8#20273110C211yesmaximal, characteristic, complemented
9C2×Q82×C53C211yesmaximal, characteristic, complemented
10C2×#13133248—21no—
11#16811173C2211yescharacteristic
12#16810997C2212yes—
13#16810973C2212yes—
14#16810941—21no—
15#16810908C2212yes—
16#16810845—22no—
17#16810833C2212yes—
18#16810825—22no—
19#16810823C2212yes—
20#16810823C2212yes—
21#16810762—21no—
22#16810762—22no—
23C4×#11719924—22no—
24#16810676C2211yescharacteristic
25#16810676C2211yescharacteristic
26#16810647C2211yescharacteristic
27#16810622—21no—
28#16810372C2214yes—
29Q8⋊#88322C2214yes—
30Q8⋊#88322C2214yes—
31Q82×C53—22no—
32Q82×C53—22no—
33Q82×C53—22no—
34Q82×C53C2214yes—
35C2×C4⋊Q8×C53—22no—
36C2×C4⋊Q8×C53—21no—
37C2×C4⋊Q8×C53—21no—
38C2×C4⋊Q8×C53C2211yescharacteristic
39C2×C4×Q8×C53—21no—
40C2×C4×Q8×C53—21no—
41C2×C4×Q8×C53C2211yescharacteristic
42C2×C4×Q8×C53C2211yescharacteristic
43C2×#11719945—21no—
44#13133248—44no—
45C2×#11719925—22no—
46C2×#11719924—22no—
47C2×#11719924—22no—
48C2×#11719924—24no—
49C2×#11719917—21no—
50#13133217C2311yescharacteristic
51#13133214C2311yescharacteristic
52#13133205—21no—
53#13133191—21no—
54#13133191—21no—
55C4×D212—21no—
56C4×D212—21no—
57#13133148—22no—
58C4⋊#88322—21no—
59C4⋊#88322—21no—
60C4⋊#88322C2312yes—
61C4⋊#88322C2312yes—
62#13133141—24no—
63#13133141—22no—
64#13133140—22no—
65#13133137—22no—
66#13133137—22no—
67C22⋊#88322—21no—
68C22⋊#88322—21no—
69C4×#88322—21no—
70C4×#88322C2312yes—
71C22×Q8×C53—22no—
72C22×Q8×C53D411yescharacteristic
73C22×Q8×C53D411yescharacteristic
74C22×Q8×C53D412yes—
75C4⋊Q8×C53—24no—
76C4⋊Q8×C53—24no—
77C4⋊Q8×C53—28no—
78C4⋊Q8×C53—24no—
79C4⋊Q8×C53—24no—
80C4⋊Q8×C53—21no—
81C4⋊Q8×C53—21no—
82C4⋊Q8×C53—28no—
83C4⋊Q8×C53D412yes—
84C4⋊Q8×C53C2312yes—
85C4×Q8×C53—24no—
86C4×Q8×C53—24no—
87C4×Q8×C53—21no—
88C4×Q8×C53—21no—
89C4×Q8×C53—21no—
90C4×Q8×C53—21no—
91C4×Q8×C53—24no—
92C4×Q8×C53—24no—
93C4×Q8×C53D412yes—
94C4×Q8×C53D412yes—
95C4×Q8×C53C2312yes—
96C4×Q8×C53C2312yes—
97C2×C4⋊C4×C53—22no—
98C2×C4⋊C4×C53—22no—
99C2×C4⋊C4×C53—21no—
100C2×C4⋊C4×C53—21no—
101C2×C4⋊C4×C53—21no—
102C2×C4⋊C4×C53—21no—
103C2×C4⋊C4×C53D411yescharacteristic
104C2×C4⋊C4×C53D411yescharacteristic
105C2×C42×C53—22no—
106C2×C42×C53—21no—
107C2×C42×C53—21no—
108C2×C42×C53C2311yescharacteristic
109C2×Q8×C53—24no—
110C2×Q8×C53—28no—
111C2×Q8×C53—21no—
112C2×Q8×C53—21no—
113C2×Q8×C53—22no—
114C2×Q8×C53—22no—
115C2×Q8×C53—82no—
116C2×Q8×C53—84no—
117C2×Q8×C53SD1614yescomplemented
118C2×Q8×C53C2×D412yes—
119C2×Q8×C53C2×D412yes—
120C2×Q8×C53C2×D414yes—
121C22×C4×C53—21no—
122C22×C4×C53—21no—
123C22×C4×C53—22no—
124C22×C4×C53—21no—
125C22×C4×C53—21no—
126C22×C4×C53C2×D411yescharacteristic
127C22×C4×C53C2×D411yescharacteristic
128C22×C4×C53C2×D411yescharacteristic
129C2×C22⋊D53—41no—
130#11719945—42no—
131#11719945—42no—
132C2×D212—21no—
133C2×D212—21no—
134C2×D212—21no—
135C2×C4×D53—41no—
136C2×Dic106—21no—
137C4⋊C4×C53—28no—
138C4⋊C4×C53—28no—
139C4⋊C4×C53—22no—
140C4⋊C4×C53—22no—
141C4⋊C4×C53—22no—
142C4⋊C4×C53—22no—
143C4⋊C4×C53—21no—
144C4⋊C4×C53—21no—
145C4⋊C4×C53—22no—
146C4⋊C4×C53—22no—
147C4⋊C4×C53—22no—
148C4⋊C4×C53—22no—
149C4⋊C4×C53C2×D412yes—
150C4⋊C4×C53C2×D412yes—
151C42×C53—28no—
152C42×C53—22no—
153C42×C53—22no—
154C42×C53—22no—
155C42×C53—22no—
156C42×C53C2×D411yescharacteristic
157C42×C53C2×D411yescharacteristic
158C42×C53C2×D411yescharacteristic
159C42×C53C2411yescharacteristic
160#11719925—44no—
161#11719924—44no—
162#11719924—44no—
163#11719924—84no—
164#11719921—41no—
165#11719921—41no—
166C4⋊Dic53—21no—
167C4⋊Dic53—21no—
168#11719917—42no—
169C2×#88322—22no—
170C2×#88322—22no—
171C2×#88322—22no—
172C23×C53#12011yescharacteristic
173C22×D53—41no—
174Q8×C53—24no—
175Q8×C53—22no—
176Q8×C53—22no—
177Q8×C53—24no—
178Q8×C53—44no—
179Q8×C53—82no—
180Q8×C53—88no—
181Q8×C53C2×SD1614yes—
182Q8×C53#13714yescomplemented
183C2×C4×C53—24no—
184C2×C4×C53—22no—
185C2×C4×C53—22no—
186C2×C4×C53—24no—
187C2×C4×C53—22no—
188C2×C4×C53—22no—
189C2×C4×C53—22no—
190C2×C4×C53—24no—
191C2×C4×C53—41no—
192C2×C4×C53—44no—
193C2×C4×C53—42no—
194C2×C4×C53—42no—
195C2×C4×C53#12011yescharacteristic
196C2×C4×C53#12011yescharacteristic
197C2×C4×C53#12011yescharacteristic
198C2×C4×C53C2×SD1612yes—
199C2×C4×C53#13712yes—
200C2×C4×C53C22×D411yescharacteristic
201C2×C4×C53C22×D411yescharacteristic
202C2×C4×C53C22×D411yescharacteristic
203C22⋊D53—82no—
204C2×Dic53—41no—
205D212—42no—
206D212—41no—
207D212—41no—
208D212—42no—
209C4×D53—82no—
210Dic106—41no—
211Dic106—41no—
212#88322—42no—
213#88322—42no—
214#88322—42no—
215#18768—531nomaximal, Sylow
216C22×C53—21no—
217C22×C53—21no—
218C22×C53#44812yes—
219C22×C53C2×#12011yescharacteristic
220C2×D53—81no—
221C2×D53—81no—
222C2×D53—81no—
223C4×C53—24no—
224C4×C53—24no—
225C4×C53—21no—
226C4×C53—21no—
227C4×C53—21no—
228C4×C53—44no—
229C4×C53—42no—
230C4×C53—42no—
231C4×C53#44812yes—
232C4×C53C22×SD1611yescharacteristic
233C4×C53C2×#13711yescharacteristic
234C4×C53C2×#13712yes—
235Dic53—81no—
236C2×Q82D5311yescharacteristic, complemented
237#3206—531no—
238#3103—531no—
239#3037—531no—
240#1645—534no—
241#1623—534no—
242#1548—531no—
243#1487—531no—
244#1479—531no—
245C2×C53—22no—
246C2×C53C2×#44811yescharacteristic
247C2×C53#300411yescharacteristic
248C2×C53#301911yescharacteristic
249D53—161no—
250C2×#137—1061no—
251Q82—22no—
252Q82—22no—
253Q82—22no—
254Q82C2×D5314yes—
255#532—531no—
256C2×C4⋊Q8—22no—
257C2×C4⋊Q8—21no—
258C2×C4⋊Q8—21no—
259C2×C4⋊Q8C2×D5311yescharacteristic
260C2×C4×Q8—21no—
261C2×C4×Q8—21no—
262C2×C4×Q8C2×D5311yescharacteristic
263C2×C4×Q8C2×D5311yescharacteristic
264#492—532no—
265#487—532no—
266#481—1061no—
267#475—532no—
268#464—1062no—
269#462—532no—
270#461—1062no—
271#460—532no—
272#460—532no—
273#448—1061no—
274#448—1062no—
275C4×SD16—1062no—
276#431—531no—
277#431—531no—
278#423—531no—
279#417—1061no—
280#333—534no—
281Q8⋊C8—534no—
282Q8⋊C8—534no—
283C53#1876811yescharacteristic, complemented, Sylow
284C2×#41—1061no—
285C22×Q8—22no—
286C22×Q8C22⋊D5311yescharacteristic
287C22×Q8C22⋊D5311yescharacteristic
288C22×Q8C22⋊D5312yes—
289#137—2124no—
290C2×Q16—1062no—
291C2×SD16—1064no—
292C2×SD16—1062no—
293C2×SD16—1062no—
294C2×M16—1061no—
295C4⋊Q8—28no—
296C4⋊Q8—24no—
297C4⋊Q8—24no—
298C4⋊Q8—28no—
299C4⋊Q8—24no—
300C4⋊Q8—24no—
301C4⋊Q8—21no—
302C4⋊Q8—21no—
303C4⋊Q8—531no—
304C4⋊Q8C22⋊D5312yes—
305C4⋊Q8C22×D5312yes—
306#127—531no—
307#124—1061no—
308#121—1061no—
309#121—1061no—
310C4×Q8—24no—
311C4×Q8—24no—
312C4×Q8—24no—
313C4×Q8—24no—
314C4×Q8—21no—
315C4×Q8—21no—
316C4×Q8—21no—
317C4×Q8—21no—
318C4×Q8C22⋊D5312yes—
319C4×Q8C22⋊D5312yes—
320C4×Q8C22×D5312yes—
321C4×Q8C22×D5312yes—
322C4×D4—1061no—
323C4×D4—1061no—
324C2×C4⋊C4—22no—
325C2×C4⋊C4—22no—
326C2×C4⋊C4—21no—
327C2×C4⋊C4—21no—
328C2×C4⋊C4—21no—
329C2×C4⋊C4—21no—
330C2×C4⋊C4C22⋊D5311yescharacteristic
331C2×C4⋊C4C22⋊D5311yescharacteristic
332C2×C42—22no—
333C2×C42—21no—
334C2×C42—21no—
335C2×C42C22×D5311yescharacteristic
336#106—1062no—
337C4⋊C8—532no—
338C4⋊C8—532no—
339C4⋊C8—1061no—
340C4⋊C8—1061no—
341#103—1062no—
342#103—1064no—
343#103—1062no—
344#102—1062no—
345#102—1062no—
346C22⋊C8—1061no—
347C22⋊C8—1061no—
348C4×C8—532no—
349C4×C8—1061no—
350#41—2122no—
351#41—2122no—
352C2×Q8—24no—
353C2×Q8—28no—
354C2×Q8—21no—
355C2×Q8—21no—
356C2×Q8—22no—
357C2×Q8—22no—
358C2×Q8—82no—
359C2×Q8—84no—
360C2×Q8—1061no—
361C2×Q8#1171992414yescomplemented
362C2×Q8C2×C22⋊D5312yes—
363C2×Q8C2×C22⋊D5312yes—
364C2×Q8C2×C22⋊D5314yes—
365C2×D4—1061no—
366C2×D4—1061no—
367C2×D4—1061no—
368C2×D4—2121no—
369C22×C4—21no—
370C22×C4—21no—
371C22×C4—22no—
372C22×C4—21no—
373C22×C4—21no—
374C22×C4—2121no—
375C22×C4C2×C22⋊D5311yescharacteristic
376C22×C4C2×C22⋊D5311yescharacteristic
377C22×C4C2×C22⋊D5311yescharacteristic
378Q16—2124no—
379SD16—4244no—
380SD16—2124no—
381SD16—2124no—
382M16—2122no—
383C2×C8—1062no—
384C2×C8—1062no—
385C2×C8—1062no—
386C4⋊C4—28no—
387C4⋊C4—28no—
388C4⋊C4—22no—
389C4⋊C4—22no—
390C4⋊C4—22no—
391C4⋊C4—22no—
392C4⋊C4—21no—
393C4⋊C4—21no—
394C4⋊C4—22no—
395C4⋊C4—22no—
396C4⋊C4—22no—
397C4⋊C4—22no—
398C4⋊C4—1061no—
399C4⋊C4—1061no—
400C4⋊C4C2×C22⋊D5312yes—
401C4⋊C4C2×C22⋊D5312yes—
402C22⋊C4—2121no—
403C22⋊C4—2121no—
404C42—28no—
405C42—22no—
406C42—22no—
407C42—22no—
408C42—22no—
409C42C2×C22⋊D5311yescharacteristic
410C42C2×C22⋊D5311yescharacteristic
411C42C2×C22⋊D5311yescharacteristic
412C42C23×D5311yescharacteristic
413C23—2121no—
414C23#1313318611yescharacteristic
415Q8—24no—
416Q8—24no—
417Q8—22no—
418Q8—22no—
419Q8—44no—
420Q8—82no—
421Q8—88no—
422Q8—2121no—
423Q8—2121no—
424Q8C2×#1171992414yes—
425Q8#1313324814yescomplemented
426D4—4242no—
427D4—2122no—
428D4—2121no—
429D4—2122no—
430D4—2121no—
431C2×C4—24no—
432C2×C4—24no—
433C2×C4—22no—
434C2×C4—22no—
435C2×C4—22no—
436C2×C4—24no—
437C2×C4—22no—
438C2×C4—22no—
439C2×C4—41no—
440C2×C4—44no—
441C2×C4—42no—
442C2×C4—42no—
443C2×C4—4242no—
444C2×C4—2121no—
445C2×C4#1313318611yescharacteristic
446C2×C4#1313318611yescharacteristic
447C2×C4#1313318611yescharacteristic
448C2×C4C2×#1171992412yes—
449C2×C4#1313324812yes—
450C2×C4C22×C22⋊D5311yescharacteristic
451C2×C4C22×C22⋊D5311yescharacteristic
452C2×C4C22×C22⋊D5311yescharacteristic
453C8—2122no—
454C8—2122no—
455C8—2122no—
456C22—21no—
457C22—21no—
458C22—4241no—
459C22—4241no—
460C22—4241no—
461C22#1681076212yes—
462C22C2×#1313318611yescharacteristic
463C4—24no—
464C4—21no—
465C4—21no—
466C4—21no—
467C4—24no—
468C4—44no—
469C4—42no—
470C4—42no—
471C4—4241no—
472C4#1681076212yes—
473C4C22×#1171992411yescharacteristic
474C4C2×#1313324812yes—
475C4C2×#1313324811yescharacteristic
476C2—22no—
477C2—8481no—
478C2C2×#1681076211yescharacteristic
479C2#2028394911yescharacteristic
480C2#2028407211yescharacteristic
subgroupG/H
centerC22C2×#13133186
commutator, cosocleC42×C53C24
FrattiniC42C23×D53
FittingC2×Q82×C53C2
socleC22×C53C2×#120
prime radical#16811173C22
hypercenterC2×Q82D53
hypocenterC53#18768
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×Q82

Complements

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

NH ≅ G/Ntype
4. #20273936C2semidirect
5. #20273793C2semidirect
6. #20273396C2semidirect
7. #20273144C2semidirect
8. #20273110C2semidirect
9. C2×Q82×C53C2semidirect
117. C2×Q8×C53SD16semidirect
182. Q8×C53#137semidirect
236. C2×Q82D53semidirect
283. C53#18768semidirect
361. C2×Q8#11719924semidirect
425. Q8#13133248semidirect

Maximal subgroups

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

subgroupindex
#202856402
#202847462
#202841872
#202739362
#202737932
#202733962
#202731442
#202731102
C2×Q82×C532
#1876853