Finite Group Databasebeta

← #27447160

Subgroups — #27447160

Order 13568.

900 conjugacy classes of subgroups.

The subgroup lattice has 648 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810910C211yesmaximal, characteristic
2C2×C4×#11719924C211yesmaximal, characteristic
3C4⋊C4×D212C211yesmaximal, characteristic
4#20276258C211yesmaximal, characteristic, complemented
5#20276191C211yesmaximal, characteristic, complemented
6#20276096C211yesmaximal, characteristic, complemented
7#20276045C211yesmaximal, characteristic, complemented
8#20275440C211yesmaximal, characteristic, complemented
9#20275274C211yesmaximal, characteristic, complemented
10#20275052C211yesmaximal, characteristic
11#20275052C211yesmaximal, characteristic
12#20274983C211yesmaximal, characteristic, complemented
13#20274975C211yesmaximal, characteristic
14#20274975C211yesmaximal, characteristic
15C53×#2451C211yesmaximal, characteristic, complemented
16C22×#11719924C2211yescharacteristic
17C2×C4×D212C2211yescharacteristic
18C2×C4×D212C2211yescharacteristic
19C2×C4×D212C2211yescharacteristic
20C2×C4⋊C4×D53—21no—
21#16810910—24no—
22C4×#11719924—24no—
23C2×#13133148C2211yescharacteristic
24C2×#13133148C2211yescharacteristic
25C2×#13133148C2211yescharacteristic
26C2×#13133148C2211yescharacteristic
27C2×#13133148C2211yescharacteristic
28C2×C4⋊#88322C2211yescharacteristic
29C2×#13133141C2211yescharacteristic
30C2×#13133141C2211yescharacteristic
31C2×#13133141C2211yescharacteristic
32C2×#13133137C2211yescharacteristic
33C2×#13133137C2211yescharacteristic
34C2×#13133137C2211yescharacteristic
35C2×C4×#88322C2211yescharacteristic
36#16810521C2211yescharacteristic
37#16810521C2211yescharacteristic
38#16810497—21no—
39#16810497—21no—
40#16810493C2211yescharacteristic
41#16810480—21no—
42#16810385C2211yescharacteristic
43#16810385C2211yescharacteristic
44#16810385C2211yescharacteristic
45#16810385C2211yescharacteristic
46#16810378C2211yescharacteristic
47#16810378C2211yescharacteristic
48#16810378C2211yescharacteristic
49#16810378C2211yescharacteristic
50C2×C4⋊Q8×C53C2211yescharacteristic
51C2×C4×Q8×C53C2211yescharacteristic
52C53×#398—21no—
53C53×#395—21no—
54C53×#391—21no—
55C53×#391C2211yescharacteristic
56C53×#391C2211yescharacteristic
57C53×#389C2211yescharacteristic
58C53×#389C2211yescharacteristic
59C53×#387—21no—
60C4×C4⋊C4×C53C2211yescharacteristic
61C22×D212C2311yescharacteristic
62C22×C4×D53—21no—
63C22×C4×D53—21no—
64C22×C4×D53—21no—
65C2×#11719924—22no—
66C2×#11719924—22no—
67C2×#11719924Q814yescomplemented
68C22×#88322C2311yescharacteristic
69C22×#88322C2311yescharacteristic
70C22×#88322C2311yescharacteristic
71C22×#88322C2311yescharacteristic
72C4×D212—22no—
73C4×D212—22no—
74C4×D212—22no—
75C4×D212—22no—
76C4×D212—41no—
77C4×D212—41no—
78C2×C4⋊Dic53—21no—
79C2×C4⋊Dic53—21no—
80C2×C4⋊Dic53C2311yescharacteristic
81C2×C4⋊Dic53C2311yescharacteristic
82C2×C4⋊Dic53C2311yescharacteristic
83C2×#11719919—21no—
84C2×#11719919—21no—
85C4⋊C4×D53—44no—
86C2×#11719921—21no—
87C2×#11719921—21no—
88C2×#11719921—21no—
89#13133148—22no—
90#13133148—22no—
91#13133148—22no—
92#13133148—22no—
93#13133148—41no—
94C4⋊#88322—22no—
95#13133141—22no—
96#13133141—22no—
97#13133141—41no—
98#13133137—22no—
99#13133137—22no—
100#13133137D414yes—
101C4×#88322—22no—
102#13133124—21no—
103#13133124—21no—
104C22×Q8×C53—21no—
105C22×Q8×C53C2311yescharacteristic
106C4⋊Q8×C53—22no—
107C4⋊Q8×C53—41no—
108C4×Q8×C53—22no—
109C4×Q8×C53—41no—
110C2×C4⋊C4×C53—21no—
111C2×C4⋊C4×C53—21no—
112C2×C4⋊C4×C53—21no—
113C2×C4⋊C4×C53—21no—
114C2×C4⋊C4×C53D411yescharacteristic
115C2×C4⋊C4×C53D411yescharacteristic
116C2×C4⋊C4×C53C2311yescharacteristic
117C2×C4⋊C4×C53C2311yescharacteristic
118C2×C4⋊C4×C53C2311yescharacteristic
119C2×C42×C53—21no—
120C2×C42×C53C2311yescharacteristic
121C2×C42×C53C2311yescharacteristic
122C2×C42×C53C2311yescharacteristic
123#95×C53—21no—
124#95×C53—21no—
125#95×C53—21no—
126#95×C53—21no—
127#95×C53—21no—
128#95×C53—21no—
129#95×C53D411yescharacteristic
130#95×C53D411yescharacteristic
131C23×D53—21no—
132C2×Q8×C53—21no—
133C2×Q8×C53—21no—
134C2×Q8×C53—21no—
135C2×Q8×C53—21no—
136C2×Q8×C53—21no—
137C2×Q8×C53—42no—
138C2×Q8×C53—42no—
139C2×Q8×C53—81no—
140C2×Q8×C53C2×Q812yes—
141C22×C4×C53—21no—
142C22×C4×C53—21no—
143C22×C4×C53—21no—
144C22×C4×C53—21no—
145C22×C4×C53C2×D411yescharacteristic
146C22×C4×C53C2×D411yescharacteristic
147C22×C4×C53C2×D411yescharacteristic
148C22×C4×C53C2×D411yescharacteristic
149C22×C4×C53C2×D411yescharacteristic
150C22×C4×C53C2×D411yescharacteristic
151C22×C4×C53C2411yescharacteristic
152C22×Dic53—21no—
153C22×Dic53—21no—
154C22×Dic53—21no—
155C2×D212—21no—
156C2×D212—21no—
157C2×D212—21no—
158C2×D212C2×D412yes—
159C2×D212C2×Q812yes—
160C2×D212#4112yes—
161C2×C4×D53—24no—
162C2×C4×D53—42no—
163C2×C4×D53—42no—
164C2×C4×D53—42no—
165C2×C4×D53—42no—
166C2×C4×D53—42no—
167C2×C4×D53—42no—
168C2×C4×D53—81no—
169C2×C4×D53—81no—
170C4⋊C4×C53—21no—
171C4⋊C4×C53—21no—
172C4⋊C4×C53—21no—
173C4⋊C4×C53—21no—
174C4⋊C4×C53—21no—
175C4⋊C4×C53—41no—
176C4⋊C4×C53—42no—
177C4⋊C4×C53—42no—
178C4⋊C4×C53—81no—
179C4⋊C4×C53—81no—
180C4⋊C4×C53SD1614yescomplemented
181C4⋊C4×C53C2×D412yes—
182C42×C53—21no—
183C42×C53—21no—
184C42×C53—21no—
185C42×C53—21no—
186C42×C53—21no—
187C42×C53—21no—
188C42×C53—42no—
189#11719924—28no—
190#11719921—24no—
191#11719921—42no—
192#11719921—42no—
193C4⋊Dic53—21no—
194C4⋊Dic53—21no—
195C4⋊Dic53—21no—
196C4⋊Dic53—21no—
197C4⋊Dic53—21no—
198C4⋊Dic53—42no—
199C4⋊Dic53—81no—
200C4⋊Dic53#4112yes—
201#11719919—42no—
202#11719919—42no—
203C2×#88322—21no—
204C2×#88322—21no—
205C2×#88322—21no—
206C2×#88322—21no—
207C2×#88322—21no—
208C2×#88322—21no—
209C2×#88322—21no—
210C2×#88322—21no—
211C2×#88322—21no—
212C2×#88322—21no—
213C2×#88322C2×D412yes—
214C2×#88322C2×Q812yes—
215C23×C53C22×D411yescharacteristic
216C22×D53—22no—
217C22×D53—22no—
218C22×D53—22no—
219C22×D53—41no—
220C22×D53—41no—
221C22×D53—41no—
222C22×D53—41no—
223Q8×C53—22no—
224Q8×C53—22no—
225Q8×C53—41no—
226Q8×C53—41no—
227Q8×C53—84no—
228C2×C4×C53—21no—
229C2×C4×C53—21no—
230C2×C4×C53—21no—
231C2×C4×C53—21no—
232C2×C4×C53—21no—
233C2×C4×C53—21no—
234C2×C4×C53—21no—
235C2×C4×C53—22no—
236C2×C4×C53—21no—
237C2×C4×C53—21no—
238C2×C4×C53—22no—
239C2×C4×C53—21no—
240C2×C4×C53—21no—
241C2×C4×C53—21no—
242C2×C4×C53—21no—
243C2×C4×C53—21no—
244C2×C4×C53—21no—
245C2×C4×C53—41no—
246C2×C4×C53—41no—
247C2×C4×C53—41no—
248C2×C4×C53—41no—
249C2×C4×C53—41no—
250C2×C4×C53—41no—
251C2×C4×C53—41no—
252C2×C4×C53—41no—
253C2×C4×C53—41no—
254C2×C4×C53—41no—
255C2×C4×C53C2×SD1612yes—
256C2×C4×C53C2×SD1612yes—
257C2×C4×C53C2×SD1612yes—
258C2×C4×C53C22×D411yescharacteristic
259C2×C4×C53C22×Q811yescharacteristic
260C2×C4×C53C2×#4111yescharacteristic
261C2×C4×C53C2×#4111yescharacteristic
262C2×C4×C53H(5,𝔽2)11yescharacteristic
263C2×C4×C532−1+411yescharacteristic
264C2×Dic53—22no—
265C2×Dic53—41no—
266C2×Dic53—41no—
267C2×Dic53—41no—
268C2×Dic53—41no—
269C2×Dic53—41no—
270C2×Dic53—41no—
271C2×Dic53—41no—
272C2×Dic53—41no—
273D212—22no—
274D212—22no—
275D212—22no—
276D212C22⋊Q814yes—
277C4×D53—48no—
278C4×D53—84no—
279C4×D53—84no—
280#88322—22no—
281#88322—22no—
282#88322—41no—
283#88322—41no—
284#26376—531nomaximal, Sylow
285C22×C53#52511yescharacteristic
286C22×C53D4211yescharacteristic
287C22×C53D4×Q811yescharacteristic
288C22×C53C22×SD1611yescharacteristic
289C22×C53#57511yescharacteristic
290C22×C53#57711yescharacteristic
291C22×C53#57711yescharacteristic
292C2×D53—24no—
293C2×D53—42no—
294C2×D53—42no—
295C2×D53—42no—
296C2×D53—42no—
297C2×D53—42no—
298C2×D53—42no—
299C4×C53—22no—
300C4×C53—22no—
301C4×C53—22no—
302C4×C53—41no—
303C4×C53—42no—
304C4×C53—41no—
305C4×C53—41no—
306C4×C53—42no—
307C4×C53—81no—
308C4×C53—81no—
309C4×C53C2×C22⋊Q811yescharacteristic
310C4×C53#53911yescharacteristic
311C4×C53#54711yescharacteristic
312C4×C53#55411yescharacteristic
313Dic53—42no—
314Dic53—81no—
315Dic53—81no—
316C2×#476—531no—
317C2×C4×SD16—531no—
318#2451D5311yescharacteristic, complemented
319D4×C4⋊C4—531no—
320#2096—531no—
321#2084—531no—
322#2066—531no—
323#2058—531no—
324#1941—531no—
325#1916—531no—
326#1877—531no—
327#1877—531no—
328#1865—531no—
329#1864—531no—
330#1864—531no—
331C2×C53#244911yescharacteristic
332C2×C53#310111yescharacteristic
333C2×C53#312511yescharacteristic
334C2×C53#326711yescharacteristic
335C2×C53D4×SD1611yescharacteristic
336C2×C53Q8×SD1611yescharacteristic
337C2×C53#338811yescharacteristic
338D53—44no—
339C22×SD16—531no—
340C2×C4⋊Q8C2×D5311yescharacteristic
341C2×C4×Q8C2×D5311yescharacteristic
342C2×C4×D4—531no—
343C2×C4×D4—531no—
344C2×C4×D4—531no—
345C22×C4⋊C4—1061no—
346#476—1064no—
347C4×SD16—1064no—
348C2×#106—531no—
349C2×#106—531no—
350C2×#106—531no—
351C2×#106—531no—
352C2×#106—531no—
353C2×C4⋊C8—531no—
354C2×#103—531no—
355C2×#103—531no—
356C2×#103—531no—
357C2×#102—531no—
358C2×#102—531no—
359C2×#102—531no—
360C2×C4×C8—531no—
361#398—21no—
362#395—21no—
363#391—21no—
364#391C2×D5311yescharacteristic
365#391C2×D5311yescharacteristic
366#389—531no—
367#389—531no—
368#389C2×D5311yescharacteristic
369#389C2×D5311yescharacteristic
370#387—21no—
371#385—1061no—
372#385—1061no—
373#384—531no—
374#380—1061no—
375C4×C4⋊C4C2×D5311yescharacteristic
376#340—531no—
377#340—531no—
378#340—531no—
379#340—531no—
380#336—531no—
381#336—531no—
382#336—531no—
383#336—531no—
384C53#2637611yescharacteristic, complemented, Sylow
385C22×Q8—21no—
386C22×Q8C22×D5311yescharacteristic
387C22×D4—531no—
388C23×C4—1061no—
389C23×C4—1061no—
390C23×C4—1061no—
391C2×SD16—534no—
392C2×SD16—1062no—
393C2×SD16—1062no—
394C22×C8—531no—
395C22×C8—531no—
396C22×C8—531no—
397C22×C8—531no—
398C4⋊Q8—22no—
399C4⋊Q8—41no—
400C4×Q8—22no—
401C4×Q8—41no—
402C4×D4—1062no—
403C4×D4—1062no—
404C4×D4—1062no—
405C4×D4—1062no—
406C4×D4—2121no—
407C4×D4—2121no—
408C2×C4⋊C4—21no—
409C2×C4⋊C4—21no—
410C2×C4⋊C4—21no—
411C2×C4⋊C4—21no—
412C2×C4⋊C4—531no—
413C2×C4⋊C4—531no—
414C2×C4⋊C4—531no—
415C2×C4⋊C4—1061no—
416C2×C4⋊C4—1061no—
417C2×C4⋊C4—1061no—
418C2×C4⋊C4—1061no—
419C2×C4⋊C4—2124no—
420C2×C4⋊C4C22⋊D5311yescharacteristic
421C2×C4⋊C4C22⋊D5311yescharacteristic
422C2×C4⋊C4C22×D5311yescharacteristic
423C2×C4⋊C4C22×D5311yescharacteristic
424C2×C4⋊C4C22×D5311yescharacteristic
425C2×C22⋊C4—1061no—
426C2×C22⋊C4—1061no—
427C2×C22⋊C4—1061no—
428C2×C42—21no—
429C2×C42C22×D5311yescharacteristic
430C2×C42C22×D5311yescharacteristic
431C2×C42C22×D5311yescharacteristic
432#106—1062no—
433#106—1062no—
434#106—1062no—
435#106—1062no—
436#106—2121no—
437C4⋊C8—1062no—
438#103—1062no—
439#103—1062no—
440#103—2121no—
441#102—534no—
442#102—1062no—
443#102—1062no—
444C4×C8—1062no—
445#95—21no—
446#95—21no—
447#95—21no—
448#95—21no—
449#95—21no—
450#95—21no—
451#95—1061no—
452#95—1061no—
453#95C22⋊D5311yescharacteristic
454#95C22⋊D5311yescharacteristic
455C24—1061no—
456C2×Q8—21no—
457C2×Q8—21no—
458C2×Q8—21no—
459C2×Q8—21no—
460C2×Q8—21no—
461C2×Q8—42no—
462C2×Q8—42no—
463C2×Q8—81no—
464C2×Q8Q8×D5312yes—
465C2×D4—532no—
466C2×D4—532no—
467C2×D4—532no—
468C2×D4—1061no—
469C2×D4—1061no—
470C2×D4—1061no—
471C22×C4—21no—
472C22×C4—21no—
473C22×C4—21no—
474C22×C4—21no—
475C22×C4—1064no—
476C22×C4—1061no—
477C22×C4—1061no—
478C22×C4—1061no—
479C22×C4—4241no—
480C22×C4—4241no—
481C22×C4—2122no—
482C22×C4—2122no—
483C22×C4—2122no—
484C22×C4—2122no—
485C22×C4—2122no—
486C22×C4—2122no—
487C22×C4C2×C22⋊D5311yescharacteristic
488C22×C4C2×C22⋊D5311yescharacteristic
489C22×C4C2×C22⋊D5311yescharacteristic
490C22×C4C2×C22⋊D5311yescharacteristic
491C22×C4C2×C22⋊D5311yescharacteristic
492C22×C4C2×C22⋊D5311yescharacteristic
493C22×C4C23×D5311yescharacteristic
494SD16—1068no—
495C2×C8—532no—
496C2×C8—532no—
497C2×C8—1061no—
498C2×C8—1061no—
499C2×C8—1061no—
500C2×C8—1061no—
501C2×C8—1061no—
502C2×C8—1061no—
503C2×C8—1061no—
504C2×C8—1061no—
505C2×C8—1061no—
506C2×C8—1061no—
507C4⋊C4—21no—
508C4⋊C4—21no—
509C4⋊C4—21no—
510C4⋊C4—21no—
511C4⋊C4—21no—
512C4⋊C4—41no—
513C4⋊C4—42no—
514C4⋊C4—42no—
515C4⋊C4—81no—
516C4⋊C4—81no—
517C4⋊C4—532no—
518C4⋊C4—1061no—
519C4⋊C4—1061no—
520C4⋊C4—1061no—
521C4⋊C4—1061no—
522C4⋊C4—1061no—
523C4⋊C4—4241no—
524C4⋊C4—2122no—
525C4⋊C4—2122no—
526C4⋊C4—2122no—
527C4⋊C4#1171992414yescomplemented
528C4⋊C4D4×D5312yes—
529C22⋊C4—1064no—
530C22⋊C4—2122no—
531C22⋊C4—2122no—
532C42—21no—
533C42—21no—
534C42—21no—
535C42—21no—
536C42—21no—
537C42—21no—
538C42—42no—
539C23—1062no—
540C23—1062no—
541C23—1062no—
542C23—2121no—
543C23—2121no—
544C23—2121no—
545C23—2121no—
546C23C22×C22⋊D5311yescharacteristic
547Q8—22no—
548Q8—22no—
549Q8—41no—
550Q8—41no—
551Q8—84no—
552D4—534no—
553D4—1062no—
554D4—1062no—
555D4—1062no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—22no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—22no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—1062no—
584C2×C4—4244no—
585C2×C4—4244no—
586C2×C4—2128no—
587C2×C4—2121no—
588C2×C4—2121no—
589C2×C4—2121no—
590C2×C4—2121no—
591C2×C4—2121no—
592C2×C4—2121no—
593C2×C4—2121no—
594C2×C4—2121no—
595C2×C4C2×#1171992412yes—
596C2×C4C2×#1171992412yes—
597C2×C4C2×#1171992412yes—
598C2×C4C2×D4×D5311yescharacteristic
599C2×C4C2×Q8×D5311yescharacteristic
600C2×C4#41×D5311yescharacteristic
601C2×C4#41×D5311yescharacteristic
602C2×C4#1313326211yescharacteristic
603C2×C4#1313326511yescharacteristic
604C8—1062no—
605C8—1062no—
606C8—2121no—
607C8—2121no—
608C22—1064no—
609C22—2122no—
610C22—2122no—
611C22—2122no—
612C22—2122no—
613C22—2122no—
614C22—2122no—
615C22#1681113011yescharacteristic
616C22D4×C22⋊D5311yescharacteristic
617C22Q8×C22⋊D5311yescharacteristic
618C22C22×#1171992411yescharacteristic
619C22#1681142111yescharacteristic
620C22#1681143111yescharacteristic
621C22#1681143411yescharacteristic
622C4—22no—
623C4—22no—
624C4—22no—
625C4—41no—
626C4—41no—
627C4—42no—
628C4—41no—
629C4—42no—
630C4—81no—
631C4—81no—
632C4—4241no—
633C4—4241no—
634C4—2122no—
635C4C22⋊Q8×D5311yescharacteristic
636C4#1681121911yescharacteristic
637C4#1681127511yescharacteristic
638C4#1681132711yescharacteristic
639C2—2124no—
640C2#2027852111yescharacteristic
641C2#2028471911yescharacteristic
642C2#2028498611yescharacteristic
643C2#2028623611yescharacteristic
644C2D4×#1171992411yescharacteristic
645C2Q8×#1171992411yescharacteristic
646C2#2028765411yescharacteristic
subgroupG/H
centerC23C22×C22⋊D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#2451C2
socleC23×C53C22×D4
prime radicalC22×D212C23
hypercenter#2451D53
hypocenterC53#26376
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #2451

Complements

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

NH ≅ G/Ntype
4. #20276258C2semidirect
5. #20276191C2semidirect
6. #20276096C2semidirect
7. #20276045C2semidirect
8. #20275440C2semidirect
9. #20275274C2semidirect
12. #20274983C2semidirect
15. C53×#2451C2semidirect
67. C2×#11719924Q8semidirect
180. C4⋊C4×C53SD16semidirect
318. #2451D53semidirect
384. C53#26376semidirect
527. C4⋊C4#11719924semidirect

Maximal subgroups

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

subgroupindex
C2×#168109102
C2×C4×#117199242
C4⋊C4×D2122
#202762582
#202761912
#202760962
#202760452
#202754402
#202752742
#202750522
#202749832
#202749752
C53×#24512
#2637653