Finite Group Databasebeta

← #27447112

Subgroups — #27447112

Order 13568.

864 conjugacy classes of subgroups.

The subgroup lattice has 644 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810906C211yesmaximal, characteristic, complemented
2#20278559C211yesmaximal, characteristic
3Q8×#11719919C211yesmaximal, characteristic, complemented
4#20276246C211yesmaximal, characteristic
5#20276194C211yesmaximal, characteristic, complemented
6#20276087C211yesmaximal, characteristic
7#20276046C211yesmaximal, characteristic, complemented
8#20275443C211yesmaximal, characteristic, complemented
9#20275272C211yesmaximal, characteristic, complemented
10#20275072C211yesmaximal, characteristic
11#20275072C211yesmaximal, characteristic
12#20274985C211yesmaximal, characteristic, complemented
13#20274983C211yesmaximal, characteristic, complemented
14#20274976C211yesmaximal, characteristic
15C2×C4×SD16×C53C211yesmaximal, characteristic
16C2×#13133216C2211yescharacteristic
17C2×#13133196—21no—
18C2×Q8×Dic53C2211yescharacteristic
19C2×Q8×Dic53C2211yescharacteristic
20#16810906—24no—
21C2×#13133149C2211yescharacteristic
22C2×#13133149C2211yescharacteristic
23C2×#13133148C2211yescharacteristic
24C2×#13133148C2211yescharacteristic
25C2×#13133146C2211yescharacteristic
26C2×#13133141C2211yescharacteristic
27C2×#13133141C2211yescharacteristic
28C2×#13133138C2211yescharacteristic
29C2×#13133138C2211yescharacteristic
30#16810546—21no—
31#16810535—21no—
32#16810526C2211yescharacteristic
33#16810526C2211yescharacteristic
34#16810522C2211yescharacteristic
35#16810521—21no—
36#16810521C2211yescharacteristic
37#16810521C2211yescharacteristic
38#16810519—21no—
39#16810519C2211yescharacteristic
40#16810496—21no—
41#16810492—21no—
42#16810492C2211yescharacteristic
43C4×#11719919—21no—
44C4×#11719919C2211yescharacteristic
45#16810386C2211yescharacteristic
46#16810386C2211yescharacteristic
47#16810385C2211yescharacteristic
48#16810385C2211yescharacteristic
49#16810378C2211yescharacteristic
50#16810377C2211yescharacteristic
51#16810377C2211yescharacteristic
52#16810377C2211yescharacteristic
53C22×SD16×C53C2211yescharacteristic
54C2×C4×Q8×C53C2211yescharacteristic
55C2×C4×D4×C53C2211yescharacteristic
56C4×SD16×C53—24no—
57C2×#106×C53C2211yescharacteristic
58C2×#103×C53C2211yescharacteristic
59C2×#102×C53C2211yescharacteristic
60C2×C4×C8×C53C2211yescharacteristic
61C22×Dic106—21no—
62C22×#88322C2311yescharacteristic
63C22×#88322C2311yescharacteristic
64#13133216—22no—
65#13133216—41no—
66#13133196—44no—
67Q8×Dic53—22no—
68Q8×Dic53—22no—
69Q8×Dic53—22no—
70Q8×Dic53—41no—
71C2×C4⋊Dic53—21no—
72C2×C4⋊Dic53—21no—
73C2×C4⋊Dic53—21no—
74C2×C4⋊Dic53C2311yescharacteristic
75C2×C4⋊Dic53C2311yescharacteristic
76C2×C4⋊Dic53C2311yescharacteristic
77C2×C4⋊Dic53C2311yescharacteristic
78C2×#11719919—21no—
79C2×#11719919—21no—
80C2×#11719919—21no—
81C2×#11719919—21no—
82C2×#11719919—21no—
83C2×#11719919—21no—
84C2×#11719919D411yescharacteristic
85C2×#11719919D411yescharacteristic
86C2×C22⋊Dic53—21no—
87C2×C22⋊Dic53—21no—
88C2×C22⋊Dic53—21no—
89C2×C22⋊Dic53—21no—
90C2×C4×Dic53—21no—
91C2×C4×Dic53—21no—
92C2×C4×Dic53C2311yescharacteristic
93C2×C4×Dic53C2311yescharacteristic
94#13133149—22no—
95#13133149—22no—
96#13133148—22no—
97#13133148—41no—
98#13133146—22no—
99#13133141—22no—
100#13133141—22no—
101#13133138—22no—
102#13133138—41no—
103#13133124—21no—
104#13133124—21no—
105#13133124—21no—
106#13133124—21no—
107#13133124D411yescharacteristic
108#13133124D411yescharacteristic
109C22×Q8×C53C2311yescharacteristic
110C22×D4×C53C2311yescharacteristic
111C23×C4×C53—21no—
112C2×SD16×C53—22no—
113C2×SD16×C53—22no—
114C2×SD16×C53Q814yescomplemented
115C22×C8×C53C2311yescharacteristic
116C22×C8×C53C2311yescharacteristic
117C4×Q8×C53—22no—
118C4×Q8×C53—41no—
119C4×D4×C53—22no—
120C4×D4×C53—41no—
121C2×C4⋊C4×C53—21no—
122C2×C4⋊C4×C53C2311yescharacteristic
123C2×C4⋊C4×C53C2311yescharacteristic
124C2×C22⋊C4×C53—21no—
125C2×C42×C53—21no—
126C2×C42×C53C2311yescharacteristic
127#106×C53—22no—
128#103×C53D414yes—
129#102×C53—22no—
130C4×C8×C53—22no—
131C24×C53—21no—
132C2×Q8×C53—21no—
133C2×Q8×C53—21no—
134C2×Q8×C53—21no—
135C2×Q8×C53C2×D412yes—
136C2×Q8×C53C2×Q812yes—
137C2×Q8×C53#4112yes—
138C2×D4×C53—21no—
139C2×D4×C53—21no—
140C2×D4×C53—21no—
141C2×D4×C53—21no—
142C2×D4×C53—21no—
143C2×D4×C53C2×Q812yes—
144C22×C4×C53—21no—
145C22×C4×C53—21no—
146C22×C4×C53—21no—
147C22×C4×C53—42no—
148C22×C4×C53—42no—
149C22×C4×C53—81no—
150C22×C4×C53C2×D411yescharacteristic
151C22×C4×C53C2×D411yescharacteristic
152C22×C4×C53C2411yescharacteristic
153C22×Dic53—21no—
154C22×Dic53—21no—
155C22×Dic53—21no—
156C22×Dic53—21no—
157C22×Dic53C2×D411yescharacteristic
158C22×Dic53C2×D411yescharacteristic
159C22×Dic53C2×D411yescharacteristic
160C22×Dic53C2×D411yescharacteristic
161C2×Dic106—42no—
162C2×Dic106—42no—
163C2×Dic106—81no—
164SD16×C53—28no—
165C2×C8×C53—21no—
166C2×C8×C53—21no—
167C2×C8×C53—21no—
168C2×C8×C53—21no—
169C2×C8×C53C2×D412yes—
170C2×C8×C53C2×Q812yes—
171C4⋊C4×C53—21no—
172C4⋊C4×C53—21no—
173C4⋊C4×C53—21no—
174C4⋊C4×C53—42no—
175C4⋊C4×C53C2×D412yes—
176C22⋊C4×C53—42no—
177C42×C53—21no—
178C42×C53—21no—
179C42×C53—42no—
180C22⋊Dic53—42no—
181C22⋊Dic53—42no—
182C22⋊Dic53—42no—
183C22⋊Dic53—81no—
184C4⋊Dic53—24no—
185C4⋊Dic53—21no—
186C4⋊Dic53—21no—
187C4⋊Dic53—21no—
188C4⋊Dic53—21no—
189C4⋊Dic53—21no—
190C4⋊Dic53—21no—
191C4⋊Dic53—21no—
192C4⋊Dic53—42no—
193C4⋊Dic53—42no—
194C4⋊Dic53#4112yes—
195#11719919—41no—
196#11719919—42no—
197#11719919—42no—
198#11719919—42no—
199#11719919—81no—
200#11719919—81no—
201#11719919—81no—
202#11719919SD1614yescomplemented
203C4×Dic53—24no—
204C4×Dic53—21no—
205C4×Dic53—21no—
206C4×Dic53—21no—
207C4×Dic53—21no—
208C4×Dic53—42no—
209C2×#88322—21no—
210C2×#88322—21no—
211C2×#88322—21no—
212C2×#88322—21no—
213C2×#88322—21no—
214C2×#88322—21no—
215C23×C53—22no—
216C23×C53—41no—
217C23×C53—41no—
218C23×C53—41no—
219C23×C53—41no—
220C23×C53—41no—
221C23×C53—41no—
222C23×C53C22×D411yescharacteristic
223Q8×C53—22no—
224Q8×C53—22no—
225Q8×C53—22no—
226Q8×C53C22⋊Q814yescomplemented
227D4×C53—22no—
228D4×C53—22no—
229D4×C53—41no—
230D4×C53—41no—
231C2×C4×C53—22no—
232C2×C4×C53—22no—
233C2×C4×C53—21no—
234C2×C4×C53—22no—
235C2×C4×C53—21no—
236C2×C4×C53—22no—
237C2×C4×C53—21no—
238C2×C4×C53—21no—
239C2×C4×C53—21no—
240C2×C4×C53—41no—
241C2×C4×C53—41no—
242C2×C4×C53—41no—
243C2×C4×C53—41no—
244C2×C4×C53—41no—
245C2×C4×C53—84no—
246C2×C4×C53C2×SD1612yes—
247C2×C4×C53C22×D411yescharacteristic
248C2×C4×C53C22×Q811yescharacteristic
249C2×C4×C53C2×#4111yescharacteristic
250C2×C4×C53C2×#4111yescharacteristic
251C2×C4×C53H(5,𝔽2)11yescharacteristic
252C2×C4×C532−1+411yescharacteristic
253C2×Dic53—21no—
254C2×Dic53—21no—
255C2×Dic53—21no—
256C2×Dic53—21no—
257C2×Dic53—22no—
258C2×Dic53—21no—
259C2×Dic53—21no—
260C2×Dic53—21no—
261C2×Dic53—21no—
262C2×Dic53—21no—
263C2×Dic53—21no—
264C2×Dic53—41no—
265C2×Dic53—41no—
266C2×Dic53—41no—
267C2×Dic53—41no—
268C2×Dic53—41no—
269C2×Dic53—41no—
270C2×Dic53—41no—
271C2×Dic53—41no—
272C2×Dic53—41no—
273C2×Dic53—41no—
274C2×Dic53—41no—
275C2×Dic53C2×SD1612yes—
276C2×Dic53C2×SD1612yes—
277Dic106—84no—
278C8×C53—22no—
279C8×C53—22no—
280#88322—41no—
281#88322—41no—
282#26373—531nomaximal, Sylow
283C22×C53—42no—
284C22×C53—42no—
285C22×C53—42no—
286C22×C53—81no—
287C22×C53—81no—
288C22×C53—81no—
289C22×C53—81no—
290C22×C53#52511yescharacteristic
291C22×C53D4211yescharacteristic
292C22×C53D4×Q811yescharacteristic
293C22×C53C22×SD1611yescharacteristic
294C22×C53#57511yescharacteristic
295C22×C53#57711yescharacteristic
296C22×C53#57711yescharacteristic
297C4×C53—24no—
298C4×C53—22no—
299C4×C53—41no—
300C4×C53—42no—
301C4×C53—81no—
302C4×C53C2×C22⋊Q811yescharacteristic
303C4×C53#53911yescharacteristic
304C4×C53#54711yescharacteristic
305C4×C53#55411yescharacteristic
306Dic53—22no—
307Dic53—22no—
308Dic53—41no—
309Dic53—42no—
310Dic53—41no—
311Dic53—81no—
312Dic53—81no—
313Dic53—81no—
314C2×#475—531no—
315C2×C4×SD16D5311yescharacteristic
316#2452—531no—
317Q8×C4⋊C4—531no—
318#2094—531no—
319#2084—531no—
320#2065—531no—
321#2058—531no—
322#1941—531no—
323#1916—531no—
324#1880—531no—
325#1880—531no—
326#1865—531no—
327#1865—531no—
328#1864—531no—
329C2×C53—82no—
330C2×C53#244911yescharacteristic
331C2×C53#310311yescharacteristic
332C2×C53#312511yescharacteristic
333C2×C53#327111yescharacteristic
334C2×C53D4×SD1611yescharacteristic
335C2×C53Q8×SD1611yescharacteristic
336C2×C53#338811yescharacteristic
337C22×SD16C2×D5311yescharacteristic
338C2×C4⋊Q8—531no—
339C2×C22⋊Q8—1061no—
340C2×C4×Q8—531no—
341C2×C4×Q8—531no—
342C2×C4×Q8C2×D5311yescharacteristic
343C2×C4×D4C2×D5311yescharacteristic
344#475—1064no—
345C4×SD16—24no—
346C2×#106—531no—
347C2×#106—531no—
348C2×#106—531no—
349C2×#106—531no—
350C2×#106C2×D5311yescharacteristic
351C2×C4⋊C8—531no—
352C2×#103—531no—
353C2×#103—531no—
354C2×#103C2×D5311yescharacteristic
355C2×#102—531no—
356C2×#102—531no—
357C2×#102C2×D5311yescharacteristic
358C2×C4×C8C2×D5311yescharacteristic
359#396—1061no—
360#393—1061no—
361#390—531no—
362#390—531no—
363#389—531no—
364#389—531no—
365#389—531no—
366#389—531no—
367#389—1061no—
368#389—1061no—
369#385—1061no—
370#384—531no—
371#384—1061no—
372C4×C4⋊C4—531no—
373C4×C4⋊C4—1061no—
374#340—531no—
375#340—531no—
376#340—531no—
377#340—531no—
378#336—531no—
379#336—531no—
380#336—531no—
381#336—531no—
382C53#2637311yescharacteristic, complemented, Sylow
383C22×Q8—1061no—
384C22×Q8C22×D5311yescharacteristic
385C22×D4C22×D5311yescharacteristic
386C23×C4—21no—
387C2×SD16—22no—
388C2×SD16—22no—
389C2×SD16Dic10614yescomplemented
390C22×C8—531no—
391C22×C8—531no—
392C22×C8C22×D5311yescharacteristic
393C22×C8C22×D5311yescharacteristic
394C4⋊Q8—1062no—
395C4⋊Q8—2121no—
396C22⋊Q8—2124no—
397C4×Q8—22no—
398C4×Q8—41no—
399C4×Q8—1062no—
400C4×Q8—1062no—
401C4×Q8—1062no—
402C4×Q8—2121no—
403C4×D4—22no—
404C4×D4—41no—
405C2×C4⋊C4—21no—
406C2×C4⋊C4—531no—
407C2×C4⋊C4—531no—
408C2×C4⋊C4—531no—
409C2×C4⋊C4—531no—
410C2×C4⋊C4—531no—
411C2×C4⋊C4—531no—
412C2×C4⋊C4—1061no—
413C2×C4⋊C4—1061no—
414C2×C4⋊C4—1061no—
415C2×C4⋊C4—1061no—
416C2×C4⋊C4—1061no—
417C2×C4⋊C4—1061no—
418C2×C4⋊C4—1061no—
419C2×C4⋊C4—1061no—
420C2×C4⋊C4—1061no—
421C2×C4⋊C4C22×D5311yescharacteristic
422C2×C4⋊C4C22×D5311yescharacteristic
423C2×C22⋊C4—21no—
424C2×C22⋊C4—1061no—
425C2×C22⋊C4—1061no—
426C2×C22⋊C4—1061no—
427C2×C22⋊C4—1061no—
428C2×C42—21no—
429C2×C42—531no—
430C2×C42—531no—
431C2×C42—1061no—
432C2×C42—1061no—
433C2×C42C22×D5311yescharacteristic
434#106—22no—
435#106—1062no—
436#106—1062no—
437#106—1062no—
438#106—2121no—
439C4⋊C8—1062no—
440#103—1062no—
441#103—1062no—
442#103C22⋊D5314yes—
443#102—22no—
444#102—1062no—
445#102—2121no—
446C4×C8—22no—
447#95—531no—
448#95—531no—
449#95—1061no—
450#95—1061no—
451#95—1061no—
452#95—1061no—
453C24—21no—
454C2×Q8—21no—
455C2×Q8—21no—
456C2×Q8—21no—
457C2×Q8—4241no—
458C2×Q8—2122no—
459C2×Q8—2122no—
460C2×Q8C2×Dic10612yes—
461C2×Q8#1171994512yes—
462C2×Q8C2×C22⋊D5312yes—
463C2×D4—21no—
464C2×D4—21no—
465C2×D4—21no—
466C2×D4—21no—
467C2×D4—21no—
468C2×D4C2×Dic10612yes—
469C22×C4—21no—
470C22×C4—21no—
471C22×C4—21no—
472C22×C4—42no—
473C22×C4—42no—
474C22×C4—81no—
475C22×C4—531no—
476C22×C4—531no—
477C22×C4—531no—
478C22×C4—531no—
479C22×C4—1061no—
480C22×C4—1061no—
481C22×C4—1061no—
482C22×C4—1061no—
483C22×C4D4×D5311yescharacteristic
484C22×C4D4×D5311yescharacteristic
485C22×C4C23×D5311yescharacteristic
486SD16—28no—
487C2×C8—21no—
488C2×C8—21no—
489C2×C8—21no—
490C2×C8—21no—
491C2×C8—1061no—
492C2×C8—1061no—
493C2×C8—1061no—
494C2×C8—1061no—
495C2×C8—1061no—
496C2×C8—1061no—
497C2×C8C2×Dic10612yes—
498C2×C8C2×C22⋊D5312yes—
499C4⋊C4—21no—
500C4⋊C4—21no—
501C4⋊C4—21no—
502C4⋊C4—42no—
503C4⋊C4—534no—
504C4⋊C4—532no—
505C4⋊C4—1064no—
506C4⋊C4—1061no—
507C4⋊C4—1061no—
508C4⋊C4—1061no—
509C4⋊C4—1061no—
510C4⋊C4—1061no—
511C4⋊C4—1061no—
512C4⋊C4—1061no—
513C4⋊C4—4241no—
514C4⋊C4—4241no—
515C4⋊C4—4241no—
516C4⋊C4—2121no—
517C4⋊C4—2122no—
518C4⋊C4—2122no—
519C4⋊C4—2122no—
520C4⋊C4—2122no—
521C4⋊C4—2122no—
522C4⋊C4C2×C22⋊D5312yes—
523C22⋊C4—42no—
524C22⋊C4—4241no—
525C22⋊C4—2122no—
526C22⋊C4—2122no—
527C22⋊C4—2122no—
528C42—21no—
529C42—21no—
530C42—42no—
531C42—1064no—
532C42—1061no—
533C42—1061no—
534C42—1061no—
535C42—1061no—
536C42—2122no—
537C23—22no—
538C23—41no—
539C23—41no—
540C23—41no—
541C23—41no—
542C23—41no—
543C23—41no—
544C23C2×D4×D5311yescharacteristic
545Q8—22no—
546Q8—22no—
547Q8—22no—
548Q8—4244no—
549Q8#1313319614yescomplemented
550D4—22no—
551D4—22no—
552D4—41no—
553D4—41no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—22no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—22no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—84no—
569C2×C4—532no—
570C2×C4—532no—
571C2×C4—1061no—
572C2×C4—1061no—
573C2×C4—1061no—
574C2×C4—1061no—
575C2×C4—1062no—
576C2×C4—1061no—
577C2×C4—1061no—
578C2×C4—1061no—
579C2×C4—1061no—
580C2×C4—1061no—
581C2×C4—1061no—
582C2×C4—2121no—
583C2×C4—2121no—
584C2×C4—2121no—
585C2×C4—2121no—
586C2×C4—2121no—
587C2×C4—2121no—
588C2×C4—2121no—
589C2×C4—2121no—
590C2×C4—2121no—
591C2×C4—2121no—
592C2×C4—2121no—
593C2×C4SD16×D5312yes—
594C2×C4C22×C22⋊D5311yescharacteristic
595C2×C4C22×Dic10611yescharacteristic
596C2×C4#41×D5311yescharacteristic
597C2×C4C2×#1171994511yescharacteristic
598C2×C4#1313326211yescharacteristic
599C2×C4#1313326411yescharacteristic
600C8—22no—
601C8—22no—
602C8—2121no—
603C8—2121no—
604C22—42no—
605C22—42no—
606C22—42no—
607C22—81no—
608C22—81no—
609C22—81no—
610C22—81no—
611C22#1681113211yescharacteristic
612C22D4×C22⋊D5311yescharacteristic
613C22D4×Dic10611yescharacteristic
614C22C2×SD16×D5311yescharacteristic
615C22#1681142011yescharacteristic
616C22#1681142911yescharacteristic
617C22#1681143011yescharacteristic
618C4—22no—
619C4—24no—
620C4—41no—
621C4—42no—
622C4—81no—
623C4—1062no—
624C4—1062no—
625C4—4241no—
626C4—4241no—
627C4—4241no—
628C4—2121no—
629C4—2122no—
630C4—2121no—
631C4C2×#1313319611yescharacteristic
632C4#1681121511yescharacteristic
633C4#1681127211yescharacteristic
634C4#1681133211yescharacteristic
635C2—82no—
636C2#2027851811yescharacteristic
637C2#2028474411yescharacteristic
638C2#2028499811yescharacteristic
639C2#2028630011yescharacteristic
640C2SD16×C22⋊D5311yescharacteristic
641C2SD16×Dic10611yescharacteristic
642C2#2028765611yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC2×C4×SD16×C53C2
socleC23×C53C22×D4
prime radicalC22×D4×C53C23
hypercenterC2×C4×SD16D53
hypocenterC53#26373
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×C4×SD16

Complements

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

NH ≅ G/Ntype
1. C2×#16810906C2semidirect
3. Q8×#11719919C2semidirect
5. #20276194C2semidirect
7. #20276046C2semidirect
8. #20275443C2semidirect
9. #20275272C2semidirect
12. #20274985C2semidirect
13. #20274983C2semidirect
114. C2×SD16×C53Q8semidirect
202. #11719919SD16semidirect
226. Q8×C53C22⋊Q8semidirect
382. C53#26373semidirect
389. C2×SD16Dic106semidirect
549. Q8#13133196semidirect

Maximal subgroups

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

subgroupindex
C2×#168109062
#202785592
Q8×#117199192
#202762462
#202761942
#202760872
#202760462
#202754432
#202752722
#202750722
#202749852
#202749832
#202749762
C2×C4×SD16×C532
#2637353