Finite Group Databasebeta

← #27425617

Subgroups — #27425617

Order 13568.

1222 conjugacy classes of subgroups.

The subgroup lattice has 610 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810818C212yesmaximal, complemented
2C2×#16810818C211yesmaximal, characteristic, complemented
3C2×C4×#11719922C211yesmaximal, characteristic, complemented
4#20279657C211yesmaximal, characteristic, complemented
5#20276100C212yesmaximal, complemented
6#20275949C212yesmaximal, complemented
7#20275520C211yesmaximal, characteristic, complemented
8#20275434C211yesmaximal, characteristic, complemented
9#20275063C212yesmaximal, complemented
10#20274712C211yesmaximal, characteristic, complemented
11D4×C4⋊C4×C53C211yesmaximal, characteristic, complemented
12C22×#11719922C2212yes—
13C22×#11719922C2211yescharacteristic
14C2×#13133214C2211yescharacteristic
15C2×#13133214C2212yes—
16C2×#13133192—21no—
17C2×#13133192—21no—
18C2×C4×D212C2211yescharacteristic
19#16810818—28no—
20#16810818—24no—
21C4×#11719922—24no—
22C2×#13133150C2211yescharacteristic
23C2×C4⋊#88322C2211yescharacteristic
24C2×C4⋊#88322C2212yes—
25C2×#13133138C2211yescharacteristic
26C2×#13133137C2212yes—
27C2×#13133137C2212yes—
28C2×#13133137C2211yescharacteristic
29C2×#13133137C2212yes—
30C2×#13133137C2212yes—
31C2×C4×#88322C2211yescharacteristic
32#16810533—22no—
33#16810525C2212yes—
34#16810385C2212yes—
35#16810378C2212yes—
36C2×C4×D4×C53C2212yes—
37C2×C4×D4×C53C2211yescharacteristic
38C22×C4⋊C4×C53—21no—
39C53×#389C2212yes—
40C53×#385—22no—
41C53×#384C2211yescharacteristic
42C53×#380—21no—
43C4×C4⋊C4×C53C2211yescharacteristic
44C22×D212—21no—
45C22×D212—22no—
46C22×D212—22no—
47C22×D212—21no—
48C22×D212C2312yes—
49C22×D212C2311yescharacteristic
50C22×C4×D53—21no—
51C2×#11719922—22no—
52C2×#11719922—24no—
53C2×#11719922—24no—
54C2×#11719922—22no—
55C2×#11719922—24no—
56C2×#11719922D414yescomplemented
57C22×#88322C2311yescharacteristic
58C22×#88322C2312yes—
59C22×#88322C2311yescharacteristic
60#13133214—22no—
61#13133214—22no—
62#13133214—24no—
63#13133214—42no—
64#13133192—44no—
65#13133192—44no—
66C4×D212—22no—
67C4×D212—22no—
68C2×C4⋊Dic53C2311yescharacteristic
69C2×#11719921—22no—
70C2×#11719921—22no—
71C2×#11719921—21no—
72#13133150—22no—
73C4⋊#88322—22no—
74C4⋊#88322—24no—
75#13133138—22no—
76#13133137—24no—
77#13133137—22no—
78#13133137—24no—
79#13133137—24no—
80#13133137—42no—
81C4×#88322—22no—
82C22×D4×C53C2311yescharacteristic
83C23×C4×C53—21no—
84C23×C4×C53—22no—
85C4×D4×C53—22no—
86C4×D4×C53—24no—
87C4×D4×C53—42no—
88C4×D4×C53D414yescomplemented
89C2×C4⋊C4×C53—21no—
90C2×C4⋊C4×C53—22no—
91C2×C4⋊C4×C53—21no—
92C2×C4⋊C4×C53—44no—
93C2×C4⋊C4×C53D411yescharacteristic
94C2×C4⋊C4×C53D411yescharacteristic
95C2×C4⋊C4×C53C2312yes—
96C2×C4⋊C4×C53C2311yescharacteristic
97C2×C22⋊C4×C53—21no—
98C2×C22⋊C4×C53—22no—
99C2×C42×C53C2312yes—
100C2×C42×C53C2311yescharacteristic
101#95×C53—22no—
102#95×C53D412yes—
103C24×C53—21no—
104C23×D53—21no—
105C23×D53—22no—
106C2×D4×C53—21no—
107C2×D4×C53—21no—
108C2×D4×C53—21no—
109C2×D4×C53C2×D412yes—
110C2×D4×C53C2×D412yes—
111C2×D4×C53#4112yes—
112C22×C4×C53—21no—
113C22×C4×C53—24no—
114C22×C4×C53—22no—
115C22×C4×C53—44no—
116C22×C4×C53—42no—
117C22×C4×C53—42no—
118C22×C4×C53—44no—
119C22×C4×C53—82no—
120C22×C4×C53C2×D412yes—
121C22×C4×C53C2×D411yescharacteristic
122C22×C4×C53C2×D412yes—
123C22×C4×C53C2×D411yescharacteristic
124C22×C4×C53C2411yescharacteristic
125C22×Dic53—21no—
126C2×D212—21no—
127C2×D212—21no—
128C2×D212—21no—
129C2×D212—21no—
130C2×D212—22no—
131C2×D212—22no—
132C2×D212—22no—
133C2×D212—22no—
134C2×D212—21no—
135C2×D212—22no—
136C2×D212—22no—
137C2×D212—44no—
138C2×D212—44no—
139C2×D212—44no—
140C2×D212—42no—
141C2×D212—42no—
142C2×D212—44no—
143C2×D212—42no—
144C2×D212—42no—
145C2×D212—82no—
146C2×D212—81no—
147C2×D212—81no—
148C2×D212—82no—
149C2×D212C2×D412yes—
150C2×C4×D53—42no—
151C2×C4×D53—42no—
152C2×C4×D53—42no—
153C4⋊C4×C53—22no—
154C4⋊C4×C53—21no—
155C4⋊C4×C53—22no—
156C4⋊C4×C53—22no—
157C4⋊C4×C53—44no—
158C4⋊C4×C53—42no—
159C4⋊C4×C53—81no—
160C4⋊C4×C53D814yescomplemented
161C4⋊C4×C53C2×D412yes—
162C22⋊C4×C53—24no—
163C22⋊C4×C53—44no—
164C42×C53—22no—
165C42×C53—22no—
166C42×C53C2×D412yes—
167C42×C53#4112yes—
168#11719922—28no—
169#11719922—48no—
170#11719921—44no—
171#11719921—42no—
172#11719921—44no—
173C4⋊Dic53—21no—
174C4⋊Dic53—21no—
175C2×#88322—21no—
176C2×#88322—21no—
177C2×#88322—21no—
178C2×#88322—22no—
179C2×#88322—22no—
180C2×#88322—22no—
181C2×#88322—21no—
182C2×#88322—21no—
183C2×#88322C2×D412yes—
184C23×C53—22no—
185C23×C53—22no—
186C23×C53—22no—
187C23×C53—41no—
188C23×C53—41no—
189C23×C53—41no—
190C23×C53—41no—
191C23×C53C22×D411yescharacteristic
192C22×D53—22no—
193C22×D53—41no—
194C22×D53—41no—
195C22×D53—41no—
196C22×D53—41no—
197C22×D53—41no—
198C22×D53—41no—
199C22×D53—42no—
200C22×D53—42no—
201C22×D53—42no—
202C22×D53—42no—
203C22×D53—42no—
204C22×D53—42no—
205C22×D53—42no—
206D4×C53—22no—
207D4×C53—22no—
208D4×C53—22no—
209D4×C53#12114yescomplemented
210C2×C4×C53—22no—
211C2×C4×C53—22no—
212C2×C4×C53—21no—
213C2×C4×C53—21no—
214C2×C4×C53—22no—
215C2×C4×C53—22no—
216C2×C4×C53—22no—
217C2×C4×C53—22no—
218C2×C4×C53—41no—
219C2×C4×C53—41no—
220C2×C4×C53—48no—
221C2×C4×C53—42no—
222C2×C4×C53—42no—
223C2×C4×C53—42no—
224C2×C4×C53—88no—
225C2×C4×C53#12112yes—
226C2×C4×C53#12112yes—
227C2×C4×C53C2×D814yes—
228C2×C4×C53C2×D812yes—
229C2×C4×C53AGL(1,ℤ/8)12yes—
230C2×C4×C53C22×D411yescharacteristic
231C2×C4×C53C22×D411yescharacteristic
232C2×C4×C53C2×#4111yescharacteristic
233C2×C4×C53C2×#4111yescharacteristic
234C2×C4×C53H(5,𝔽2)11yescharacteristic
235C2×C4×C532−1+411yescharacteristic
236C2×Dic53—41no—
237C2×Dic53—41no—
238C2×Dic53—41no—
239D212—22no—
240D212—22no—
241D212—42no—
242D212—41no—
243D212—42no—
244D212—42no—
245D212—41no—
246D212—42no—
247D212—88no—
248D212—84no—
249D212—88no—
250D212—84no—
251C4×D53—84no—
252#88322—22no—
253#88322—42no—
254#88322—41no—
255#24726—531nomaximal, Sylow
256C22×C53—24no—
257C22×C53—42no—
258C22×C53—42no—
259C22×C53—42no—
260C22×C53—42no—
261C22×C53—42no—
262C22×C53—42no—
263C22×C53C2×#12111yescharacteristic
264C22×C53D4211yescharacteristic
265C22×C53#54811yescharacteristic
266C22×C53C22×D811yescharacteristic
267C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
268C22×C53#57611yescharacteristic
269C22×C53#57711yescharacteristic
270C2×D53—42no—
271C2×D53—42no—
272C2×D53—42no—
273C2×D53—81no—
274C2×D53—81no—
275C2×D53—81no—
276C2×D53—81no—
277C2×D53—82no—
278C2×D53—82no—
279C2×D53—82no—
280C2×D53—82no—
281C2×D53—82no—
282C2×D53—82no—
283C2×D53—82no—
284C4×C53—24no—
285C4×C53—22no—
286C4×C53—42no—
287C4×C53—42no—
288C4×C53—82no—
289C4×C53#45914yescomplemented
290C4×C53C2×#12111yescharacteristic
291C4×C53#53711yescharacteristic
292C4×C53#54711yescharacteristic
293C4×C53#55311yescharacteristic
294Dic53—81no—
295C2×#459—532no—
296C2×#459—531no—
297C2×C4×D8—531no—
298#2541—531no—
299D4×C4⋊C4D5311yescharacteristic, complemented
300#2067—532no—
301#2041—532no—
302#1957—531no—
303#1939—531no—
304#1879—532no—
305#1802—531no—
306C2×C53—44no—
307C2×C53#241611yescharacteristic
308C2×C53C2×#45911yescharacteristic
309C2×C53#304511yescharacteristic
310C2×C53#327211yescharacteristic
311C2×C53D4×D811yescharacteristic
312C2×C53#335811yescharacteristic
313C2×C53#336611yescharacteristic
314D53—82no—
315D53—162no—
316C22×D8—531no—
317C22×D8—532no—
318C2×#127—531no—
319C2×#127—532no—
320C2×#121—1061no—
321C2×#121—1061no—
322C2×C4×D4—531no—
323C2×C4×D4C2×D5312yes—
324C2×C4×D4C2×D5311yescharacteristic
325C22×C4⋊C4—21no—
326#459—1068no—
327#459—1064no—
328C4×D8—1064no—
329C2×#107—531no—
330C2×C4⋊C8—531no—
331C2×C4⋊C8—532no—
332C2×#102—531no—
333C2×#102—532no—
334C2×#102—532no—
335C2×#102—532no—
336C2×#102—531no—
337C2×#102—532no—
338C2×C4×C8—531no—
339#392—1062no—
340#390—532no—
341#389C2×D5312yes—
342#385—22no—
343#384C2×D5311yescharacteristic
344#380—21no—
345C4×C4⋊C4C2×D5311yescharacteristic
346#340—532no—
347#336—532no—
348C53#2472611yescharacteristic, complemented, Sylow
349C22×D4—531no—
350C22×D4—532no—
351C22×D4—1061no—
352C22×D4—1062no—
353C22×D4—1062no—
354C22×D4—1061no—
355C22×D4C22×D5311yescharacteristic
356C23×C4—21no—
357C23×C4—22no—
358C23×C4—1061no—
359C2×D8—534no—
360C2×D8—1062no—
361C2×D8—1062no—
362C2×D8—1064no—
363C2×D8—1064no—
364C2×D8—1064no—
365C22×C8—531no—
366C22×C8—532no—
367C22×C8—531no—
368#127—1064no—
369#127—1062no—
370#127—1062no—
371#127—2122no—
372#121—2124no—
373#121—2124no—
374C4×D4—24no—
375C4×D4—22no—
376C4×D4—42no—
377C4×D4—1062no—
378C4×D4—1062no—
379C4×D4D21214yescomplemented
380C2×C4⋊C4—21no—
381C2×C4⋊C4—22no—
382C2×C4⋊C4—21no—
383C2×C4⋊C4—44no—
384C2×C4⋊C4—531no—
385C2×C4⋊C4C22⋊D5311yescharacteristic
386C2×C4⋊C4C22⋊D5311yescharacteristic
387C2×C4⋊C4C22×D5312yes—
388C2×C4⋊C4C22×D5311yescharacteristic
389C2×C22⋊C4—21no—
390C2×C22⋊C4—22no—
391C2×C22⋊C4—1061no—
392C2×C22⋊C4—1062no—
393C2×C22⋊C4—1062no—
394C2×C42C22×D5312yes—
395C2×C42C22×D5311yescharacteristic
396#107—1062no—
397C4⋊C8—1064no—
398C4⋊C8—1062no—
399#102—1064no—
400#102—1064no—
401#102—1062no—
402#102—1062no—
403#102—1064no—
404#102—2122no—
405C4×C8—1062no—
406#95—22no—
407#95C22⋊D5312yes—
408C24—21no—
409C24—1061no—
410C24—1062no—
411C2×D4—21no—
412C2×D4—21no—
413C2×D4—21no—
414C2×D4—532no—
415C2×D4—1062no—
416C2×D4—1061no—
417C2×D4—1062no—
418C2×D4—1061no—
419C2×D4—1062no—
420C2×D4—1061no—
421C2×D4—1062no—
422C2×D4—1061no—
423C2×D4—1062no—
424C2×D4—1061no—
425C2×D4—1062no—
426C2×D4—4241no—
427C2×D4—4242no—
428C2×D4—4241no—
429C2×D4—4242no—
430C2×D4—2124no—
431C2×D4—2124no—
432C2×D4—2124no—
433C2×D4—2122no—
434C2×D4—2122no—
435C2×D4—2124no—
436C2×D4—2122no—
437C2×D4—2122no—
438C2×D4C2×D21212yes—
439C2×D4D4×D5312yes—
440C2×D4#1171994912yes—
441C22×C4—21no—
442C22×C4—24no—
443C22×C4—22no—
444C22×C4—44no—
445C22×C4—42no—
446C22×C4—42no—
447C22×C4—44no—
448C22×C4—82no—
449C22×C4—1061no—
450C22×C4—2122no—
451C22×C4—2122no—
452C22×C4—2122no—
453C22×C4C2×C22⋊D5312yes—
454C22×C4C2×C22⋊D5311yescharacteristic
455C22×C4C2×C22⋊D5312yes—
456C22×C4C2×C22⋊D5311yescharacteristic
457C22×C4C23×D5311yescharacteristic
458D8—1068no—
459D8—2128no—
460C2×C8—532no—
461C2×C8—1061no—
462C2×C8—1061no—
463C2×C8—1062no—
464C2×C8—1061no—
465C2×C8—1062no—
466C2×C8—1062no—
467C2×C8—1061no—
468C2×C8—1061no—
469C4⋊C4—22no—
470C4⋊C4—21no—
471C4⋊C4—22no—
472C4⋊C4—22no—
473C4⋊C4—44no—
474C4⋊C4—42no—
475C4⋊C4—81no—
476C4⋊C4—1061no—
477C4⋊C4—1061no—
478C4⋊C4#1171992214yescomplemented
479C4⋊C4C2×D21212yes—
480C22⋊C4—24no—
481C22⋊C4—44no—
482C22⋊C4—2124no—
483C22⋊C4—2122no—
484C22⋊C4—2124no—
485C42—22no—
486C42—22no—
487C42C2×D21212yes—
488C42#1171994512yes—
489C23—22no—
490C23—22no—
491C23—22no—
492C23—41no—
493C23—41no—
494C23—41no—
495C23—41no—
496C23—1062no—
497C23—2122no—
498C23—2122no—
499C23—2122no—
500C23—2122no—
501C23—2122no—
502C23—2122no—
503C23—2122no—
504C23—2121no—
505C23—2121no—
506C23—2121no—
507C23—2121no—
508C23—2121no—
509C23—2121no—
510C23C22×C22⋊D5311yescharacteristic
511D4—22no—
512D4—22no—
513D4—22no—
514D4—1062no—
515D4—1062no—
516D4—4244no—
517D4—4248no—
518D4—4248no—
519D4—4244no—
520D4—2122no—
521D4—2122no—
522D4—2121no—
523D4—2122no—
524D4—2122no—
525D4—2121no—
526D4#1313319214yescomplemented
527C2×C4—21no—
528C2×C4—22no—
529C2×C4—21no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—48no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—88no—
542C2×C4—4244no—
543C2×C4—2121no—
544C2×C4—2121no—
545C2×C4—2121no—
546C2×C4#1313319112yes—
547C2×C4#1313319112yes—
548C2×C4C2×#1171992214yes—
549C2×C4C2×#1171992212yes—
550C2×C4#1313324212yes—
551C2×C4C2×D4×D5311yescharacteristic
552C2×C4C22×D21211yescharacteristic
553C2×C4C2×#1171994511yescharacteristic
554C2×C4C2×#1171994911yescharacteristic
555C2×C4#1313326211yescharacteristic
556C2×C4#1313326511yescharacteristic
557C8—1062no—
558C8—2122no—
559C8—2121no—
560C22—24no—
561C22—42no—
562C22—42no—
563C22—42no—
564C22—42no—
565C22—42no—
566C22—42no—
567C22—4242no—
568C22—4242no—
569C22—4242no—
570C22—4242no—
571C22—4242no—
572C22—4242no—
573C22—4242no—
574C22—4241no—
575C22—4241no—
576C22—4241no—
577C22—4241no—
578C22—2122no—
579C22—2122no—
580C22—2122no—
581C22C2×#1313319111yescharacteristic
582C22D4×C22⋊D5311yescharacteristic
583C22#1681128411yescharacteristic
584C22C22×#1171992211yescharacteristic
585C22C2×#1313324211yescharacteristic
586C22#1681142611yescharacteristic
587C22#1681143111yescharacteristic
588C4—22no—
589C4—24no—
590C4—42no—
591C4—42no—
592C4—82no—
593C4—4241no—
594C4#1681081814yescomplemented
595C4C2×#1313319211yescharacteristic
596C4#1681120411yescharacteristic
597C4#1681127411yescharacteristic
598C4#1681132111yescharacteristic
599C2—44no—
600C2—8482no—
601C2—4242no—
602C2#2027814911yescharacteristic
603C2C2×#1681081811yescharacteristic
604C2#2028425411yescharacteristic
605C2#2028631511yescharacteristic
606C2D4×#1171992211yescharacteristic
607C2#2028735011yescharacteristic
608C2#2028742211yescharacteristic
subgroupG/H
centerC23C22×C22⋊D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingD4×C4⋊C4×C53C2
socleC23×C53C22×D4
hypercenterD4×C4⋊C4D53
hypocenterC53#24726
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ D4×C4⋊C4

Complements

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

NH ≅ G/Ntype
1. C2×#16810818C2semidirect
2. C2×#16810818C2semidirect
3. C2×C4×#11719922C2semidirect
4. #20279657C2semidirect
5. #20276100C2semidirect
6. #20275949C2semidirect
7. #20275520C2semidirect
8. #20275434C2semidirect
9. #20275063C2semidirect
10. #20274712C2semidirect
11. D4×C4⋊C4×C53C2semidirect
56. C2×#11719922D4semidirect
88. C4×D4×C53D4semidirect
160. C4⋊C4×C53D8semidirect
209. D4×C53#121semidirect
289. C4×C53#459semidirect
299. D4×C4⋊C4D53semidirect
348. C53#24726semidirect
379. C4×D4D212semidirect
478. C4⋊C4#11719922semidirect
526. D4#13133192semidirect
594. C4#16810818semidirect

Maximal subgroups

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

subgroupindex
C2×#168108182
C2×C4×#117199222
#202796572
#202761002
#202759492
#202755202
#202754342
#202750632
#202747122
D4×C4⋊C4×C532
#2472653