Finite Group Databasebeta

← #15870064

Subgroups — #15870064

Order 3328.

976 conjugacy classes of subgroups.

The subgroup lattice has 638 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11718470C211yesmaximal, characteristic
2#13117374C211yesmaximal, characteristic
3C4×C4⋊C4×D13C211yesmaximal, characteristic
4#13115474C211yesmaximal, characteristic, complemented
5#13115429C211yesmaximal, characteristic, complemented
6#13115428C211yesmaximal, characteristic, complemented
7#13115140C211yesmaximal, characteristic
8#13115022C211yesmaximal, characteristic
9#13114925C211yesmaximal, characteristic
10#107×Dic13C211yesmaximal, characteristic, complemented
11#13114728C211yesmaximal, characteristic, complemented
12#103×Dic13C211yesmaximal, characteristic, complemented
13#103×Dic13C211yesmaximal, characteristic, complemented
14#13114593C211yesmaximal, characteristic
15C13×#1928C211yesmaximal, characteristic, complemented
16C2×#88200C2211yescharacteristic
17C2×Q8×Dic13C2211yescharacteristic
18C2×C4⋊C4×D13C2211yescharacteristic
19C2×C4⋊C4×D13C2211yescharacteristic
20C2×C42×D13—21no—
21C2×C42×D13C2211yescharacteristic
22C2×C13×#107C2211yescharacteristic
23C2×C13×#103C2211yescharacteristic
24C2×C13×#103C2211yescharacteristic
25C13×#391C2211yescharacteristic
26C4×C13×C4⋊C4C2211yescharacteristic
27C2×#88075C2211yescharacteristic
28C2×#88075C2211yescharacteristic
29C2×#88060C2211yescharacteristic
30C2×#88060C2211yescharacteristic
31C2×#88058C2211yescharacteristic
32C2×C8×Dic13C2211yescharacteristic
33C2×C8×Dic13C2211yescharacteristic
34#11718648C2211yescharacteristic
35C4⋊C4×Dic13C2211yescharacteristic
36C4⋊C4×Dic13C2211yescharacteristic
37C4⋊C4×Dic13C2211yescharacteristic
38C2×#88047C2211yescharacteristic
39C2×#88047C2211yescharacteristic
40C4×C4⋊Dic13C2211yescharacteristic
41C4×#6101—21no—
42#11718589C2211yescharacteristic
43#11718470C418yescomplemented
44#11718329—21no—
45#11718327—21no—
46D13×#95—21no—
47#11718316—21no—
48#11718310—21no—
49#11718310—21no—
50C13×#340C2211yescharacteristic
51C13×#336C2211yescharacteristic
52#11718143C2211yescharacteristic
53#11718143C2211yescharacteristic
54#11718143C2211yescharacteristic
55#11718121C2211yescharacteristic
56#11718121C2211yescharacteristic
57#11718121C2211yescharacteristic
58#11718121C2211yescharacteristic
59#11718116C2211yescharacteristic
60C22×Q8×C13C2311yescharacteristic
61C22×C4×D13—21no—
62C22×C4×D13—21no—
63C22×C4×D13D411yescharacteristic
64C22×C4×D13D411yescharacteristic
65C22×C4×D13C2311yescharacteristic
66C22×C8×C13C2311yescharacteristic
67C22×C8×C13C2311yescharacteristic
68C2×C13×C4⋊C4C2311yescharacteristic
69C2×C13×C4⋊C4C2311yescharacteristic
70C2×C13×C4⋊C4C2311yescharacteristic
71C2×C42×C13—21no—
72C2×C42×C13C2311yescharacteristic
73#88200—22no—
74#88200C2×C414yes—
75Q8×Dic13—22no—
76Q8×Dic13—22no—
77C2×#6103—21no—
78C2×#6103—21no—
79C2×C4⋊Dic13—21no—
80C2×C4⋊Dic13C2311yescharacteristic
81C2×C4⋊Dic13C2311yescharacteristic
82C2×C4⋊Dic13C2311yescharacteristic
83C2×C4⋊Dic13C2311yescharacteristic
84C2×#6101—21no—
85C2×#6101—21no—
86C2×#6101—21no—
87C2×#6101—21no—
88C2×C4×Dic13—21no—
89C2×C4×Dic13—21no—
90C2×C4×Dic13—21no—
91C2×C4×Dic13—21no—
92C2×C4×Dic13—21no—
93C2×C4×Dic13D411yescharacteristic
94C2×C4×Dic13D411yescharacteristic
95C2×C4×Dic13C2311yescharacteristic
96C22×#1055C2311yescharacteristic
97C22×#1055C2311yescharacteristic
98C4⋊C4×D13—22no—
99C4⋊C4×D13—22no—
100C4⋊C4×D13—22no—
101C4⋊C4×D13C2×C414yes—
102C42×D13—22no—
103C42×D13—22no—
104C42×D13—44no—
105C13×#107—22no—
106C13×#103—22no—
107C13×#103C2×C414yescomplemented
108C13×#95—21no—
109C13×#95—21no—
110C13×#95—21no—
111#88075—22no—
112#88075C2×C414yescomplemented
113#88071—21no—
114#88071—21no—
115#88071—21no—
116#88071—21no—
117#88071—21no—
118#88071—21no—
119#88060—22no—
120#88060C2×C414yes—
121#88058C2×C414yescomplemented
122C8×Dic13—22no—
123C8×Dic13—22no—
124#88047—22no—
125#88047C2×C414yescomplemented
126C4×#484—131nomaximal, Sylow
127C23×D13C2×D411yescharacteristic
128C2×Q8×C13—21no—
129C2×Q8×C13—21no—
130C2×Q8×C13—21no—
131C2×Q8×C13—21no—
132C2×Q8×C13—21no—
133C2×Q8×C13C22×C412yes—
134C22×C4×C13—21no—
135C22×C4×C13—21no—
136C22×C4×C13—21no—
137C22×C4×C13—21no—
138C22×C4×C13C2×D411yescharacteristic
139C22×C4×C13C2×D411yescharacteristic
140C22×C4×C13C2411yescharacteristic
141C22×Dic13—21no—
142C22×Dic13—21no—
143C22×Dic13—21no—
144C22×Dic13—21no—
145C22×Dic13C2×D411yescharacteristic
146C22×Dic13C2×D411yescharacteristic
147C22×Dic13C2×D411yescharacteristic
148C2×C4×D13—21no—
149C2×C4×D13—21no—
150C2×C4×D13—21no—
151C2×C4×D13—21no—
152C2×C4×D13—22no—
153C2×C4×D13—22no—
154C2×C4×D13—24no—
155C2×C4×D13—24no—
156C2×C4×D13—22no—
157C2×C4×D13—22no—
158C2×C4×D13—41no—
159C2×C4×D13—42no—
160C2×C4×D13—42no—
161C2×C4×D13—42no—
162C2×C4×D13—42no—
163C2×C4×D13Q1614yes—
164C2×C4×D13C22×C412yes—
165C2×C4×D13#4112yes—
166C2×C8×C13—21no—
167C2×C8×C13—21no—
168C2×C8×C13—21no—
169C2×C8×C13—21no—
170C2×C8×C13—21no—
171C2×C8×C13C22×C412yes—
172C13×C4⋊C4—21no—
173C13×C4⋊C4—21no—
174C13×C4⋊C4—21no—
175C13×C4⋊C4—21no—
176C13×C4⋊C4C22×C412yes—
177C13×C4⋊C4#4112yes—
178C42×C13—21no—
179C42×C13—21no—
180C42×C13—42no—
181#6103—24no—
182#6103—42no—
183C4⋊Dic13—21no—
184C4⋊Dic13—21no—
185C4⋊Dic13—21no—
186C4⋊Dic13—21no—
187C4⋊Dic13—21no—
188C4⋊Dic13—42no—
189C4⋊Dic13C22×C412yes—
190C4⋊Dic13C22×C412yes—
191C4⋊Dic13#4112yes—
192#6101—24no—
193#6101—24no—
194#6101—24no—
195#6101—42no—
196C4×Dic13—22no—
197C4×Dic13—22no—
198C4×Dic13—42no—
199C4×Dic13—42no—
200C4×Dic13—42no—
201C4×Dic13—42no—
202C4×Dic13—42no—
203C4×Dic13#4112yes—
204C4×Dic13#4112yes—
205C2×#1055—21no—
206C2×#1055—21no—
207C2×#1055—21no—
208C2×#1055—21no—
209C2×#1055—21no—
210C2×#1055C22×C412yes—
211C2×#484—131no—
212C4×C22⋊Q8—131no—
213C2×C4×C4⋊C4—131no—
214#1928—131no—
215#1928D1311yescharacteristic, complemented
216#1921—131no—
217#1921—131no—
218#1872—131no—
219#1852—131no—
220#1824—131no—
221C4×#107—131no—
222C4×#107—131no—
223C4×#103—131no—
224C4×#103—131no—
225C4×C22⋊C8—131no—
226C23×C13C22×D411yescharacteristic
227C22×D13—21no—
228C22×D13—21no—
229C22×D13—21no—
230C22×D13—21no—
231C22×D13C4×D412yes—
232C22×D13C2×Q1612yes—
233C22×D13#13512yes—
234Q8×C13—22no—
235Q8×C13—22no—
236Q8×C13—41no—
237Q8×C13—41no—
238C2×C4×C13—21no—
239C2×C4×C13—21no—
240C2×C4×C13—22no—
241C2×C4×C13—22no—
242C2×C4×C13—21no—
243C2×C4×C13—22no—
244C2×C4×C13—21no—
245C2×C4×C13—21no—
246C2×C4×C13—41no—
247C2×C4×C13—41no—
248C2×C4×C13—41no—
249C2×C4×C13—41no—
250C2×C4×C13—41no—
251C2×C4×C13—41no—
252C2×C4×C13—41no—
253C2×C4×C13—41no—
254C2×C4×C13—41no—
255C2×C4×C13C2×Q1612yes—
256C2×C4×C13C23×C411yescharacteristic
257C2×C4×C13C2×#4111yescharacteristic
258C2×C4×C13C2×#4111yescharacteristic
259C2×C4×C13C2×#4111yescharacteristic
260C2×C4×C13C2×#4111yescharacteristic
261C2×C4×C13C2×#4111yescharacteristic
262C2×Dic13—21no—
263C2×Dic13—21no—
264C2×Dic13—21no—
265C2×Dic13—21no—
266C2×Dic13—22no—
267C2×Dic13—22no—
268C2×Dic13—22no—
269C2×Dic13—41no—
270C2×Dic13—41no—
271C2×Dic13—41no—
272C2×Dic13—41no—
273C2×Dic13—41no—
274C2×Dic13—41no—
275C2×Dic13—41no—
276C2×Dic13—41no—
277C2×Dic13—41no—
278C2×Dic13C4×D412yes—
279C2×Dic13#12312yes—
280C2×Dic13#12312yes—
281C2×Dic13C2×Q1612yes—
282C2×Dic13AGL(1,ℤ/8)12yes—
283C4×D13—22no—
284C4×D13—22no—
285C4×D13—22no—
286C4×D13—22no—
287C4×D13—28no—
288C4×D13—44no—
289C4×D13—48no—
290C4×D13—48no—
291C8×C13—22no—
292C8×C13—41no—
293#1055—22no—
294#1055—41no—
295C2×C22⋊Q8—131no—
296C2×C4×Q8—131no—
297C22×C4⋊C4—131no—
298C22×C4⋊C4—131no—
299C22×C42—131no—
300C22×C42—261no—
301#484—138no—
302C2×#107—131no—
303C2×#107—131no—
304C2×#107—131no—
305C2×#107C2×D1311yescharacteristic
306C2×#103—131no—
307C2×#103—131no—
308C2×#103C2×D1311yescharacteristic
309C2×#103C2×D1311yescharacteristic
310C2×C22⋊C8—131no—
311C2×C22⋊C8—131no—
312C2×C4×C8—131no—
313C2×C4×C8—131no—
314#391C2×D1311yescharacteristic
315#389—131no—
316#387—261no—
317#385—261no—
318#380—131no—
319C4×C4⋊C4—131no—
320C4×C4⋊C4—131no—
321C4×C4⋊C4—131no—
322C4×C4⋊C4—131no—
323C4×C4⋊C4—261no—
324C4×C4⋊C4—261no—
325C4×C4⋊C4—261no—
326C4×C4⋊C4C2×D1311yescharacteristic
327C4×C22⋊C4—261no—
328C2×#95—261no—
329#340—131no—
330#340—131no—
331#340—131no—
332#340—131no—
333#340—131no—
334#340—131no—
335#340—131no—
336#340C2×D1311yescharacteristic
337#336—131no—
338#336C2×D1311yescharacteristic
339C22×C13C2×C4×D411yescharacteristic
340C22×C13C2×#12311yescharacteristic
341C22×C13#52511yescharacteristic
342C22×C13C22×Q1611yescharacteristic
343C22×C13C2×#13511yescharacteristic
344C22×C13C2×AGL(1,ℤ/8)11yescharacteristic
345C22×C13#57511yescharacteristic
346C2×D13—22no—
347C2×D13—22no—
348C2×D13—22no—
349C2×D13—22no—
350C2×D13—22no—
351C2×D13—22no—
352C2×D13C4×Q1614yes—
353C4×C13—22no—
354C4×C13—41no—
355C4×C13—42no—
356C4×C13—42no—
357C4×C13—42no—
358C4×C13—81no—
359C4×C13C4×#4111yescharacteristic
360C4×C13C4×#4111yescharacteristic
361C4×C13#52911yescharacteristic
362C4×C13#52911yescharacteristic
363Dic13—22no—
364Dic13—22no—
365Dic13—42no—
366Dic13—42no—
367Dic13—42no—
368Dic13—81no—
369Dic13#48414yescomplemented
370C22×Q8C22×D1311yescharacteristic
371C23×C4—131no—
372C23×C4—131no—
373C23×C4—131no—
374C23×C4—261no—
375C23×C4—261no—
376C22×C8—131no—
377C22×C8—131no—
378C22×C8C22×D1311yescharacteristic
379C22×C8C22×D1311yescharacteristic
380C22⋊Q8—134no—
381C22⋊Q8—262no—
382C4×Q8—262no—
383C4×Q8—262no—
384C2×C4⋊C4—131no—
385C2×C4⋊C4—131no—
386C2×C4⋊C4—131no—
387C2×C4⋊C4—134no—
388C2×C4⋊C4—131no—
389C2×C4⋊C4—262no—
390C2×C4⋊C4—262no—
391C2×C4⋊C4—261no—
392C2×C4⋊C4—261no—
393C2×C4⋊C4—261no—
394C2×C4⋊C4—262no—
395C2×C4⋊C4—261no—
396C2×C4⋊C4—261no—
397C2×C4⋊C4C22×D1311yescharacteristic
398C2×C4⋊C4C22×D1311yescharacteristic
399C2×C4⋊C4C22×D1311yescharacteristic
400C2×C22⋊C4—261no—
401C2×C22⋊C4—261no—
402C2×C42—21no—
403C2×C42—131no—
404C2×C42—131no—
405C2×C42—131no—
406C2×C42—262no—
407C2×C42—262no—
408C2×C42—261no—
409C2×C42—261no—
410C2×C42—261no—
411C2×C42—261no—
412C2×C42—261no—
413C2×C42—524no—
414C2×C42C22×D1311yescharacteristic
415#107—22no—
416#107—134no—
417#107—134no—
418#107—262no—
419#103—22no—
420#103—134no—
421#103—262no—
422#103C4×D1314yescomplemented
423C22⋊C8—134no—
424C22⋊C8—262no—
425C4×C8—262no—
426C4×C8—262no—
427#95—21no—
428#95—21no—
429#95—21no—
430#95—261no—
431#95—261no—
432#95—261no—
433#95—261no—
434#95—261no—
435#95—261no—
436C2×C13C4×#12311yescharacteristic
437C2×C13C2×C4×Q1611yescharacteristic
438C2×C13C4×AGL(1,ℤ/8)11yescharacteristic
439C2×C13C2×#48411yescharacteristic
440C2×C13#309111yescharacteristic
441C2×C13#310411yescharacteristic
442C2×C13#310511yescharacteristic
443D13—24no—
444C24—131no—
445C2×Q8—21no—
446C2×Q8—21no—
447C2×Q8—21no—
448C2×Q8—21no—
449C2×Q8—21no—
450C2×Q8C2×C4×D1312yes—
451C22×C4—21no—
452C22×C4—21no—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—131no—
456C22×C4—131no—
457C22×C4—131no—
458C22×C4—134no—
459C22×C4—132no—
460C22×C4—132no—
461C22×C4—262no—
462C22×C4—262no—
463C22×C4—262no—
464C22×C4—264no—
465C22×C4—262no—
466C22×C4—264no—
467C22×C4—261no—
468C22×C4—261no—
469C22×C4—261no—
470C22×C4—261no—
471C22×C4—261no—
472C22×C4—261no—
473C22×C4—261no—
474C22×C4—261no—
475C22×C4—521no—
476C22×C4—522no—
477C22×C4—522no—
478C22×C4—522no—
479C22×C4—522no—
480C22×C4D4×D1311yescharacteristic
481C22×C4D4×D1311yescharacteristic
482C22×C4C23×D1311yescharacteristic
483C2×C8—21no—
484C2×C8—21no—
485C2×C8—21no—
486C2×C8—21no—
487C2×C8—21no—
488C2×C8—132no—
489C2×C8—261no—
490C2×C8—261no—
491C2×C8—261no—
492C2×C8—261no—
493C2×C8—261no—
494C2×C8C2×C4×D1312yes—
495C4⋊C4—21no—
496C4⋊C4—21no—
497C4⋊C4—21no—
498C4⋊C4—21no—
499C4⋊C4—132no—
500C4⋊C4—132no—
501C4⋊C4—132no—
502C4⋊C4—264no—
503C4⋊C4—264no—
504C4⋊C4—264no—
505C4⋊C4—261no—
506C4⋊C4—261no—
507C4⋊C4—261no—
508C4⋊C4—261no—
509C4⋊C4—261no—
510C4⋊C4—522no—
511C4⋊C4—522no—
512C4⋊C4C2×C4×D1312yes—
513C4⋊C4#612912yes—
514C22⋊C4—264no—
515C22⋊C4—522no—
516C42—21no—
517C42—21no—
518C42—42no—
519C42—132no—
520C42—132no—
521C42—262no—
522C42—262no—
523C42—522no—
524C42—522no—
525C42—522no—
526C42—522no—
527C42—522no—
528C13C4×#48411yescharacteristic, complemented, Sylow
529C23—132no—
530C23—132no—
531C23—132no—
532C23—261no—
533C23—261no—
534C23—261no—
535C23—261no—
536C23C2×D4×D1311yescharacteristic
537Q8—22no—
538Q8—22no—
539Q8—41no—
540Q8—41no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—21no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—132no—
559C2×C4—132no—
560C2×C4—132no—
561C2×C4—132no—
562C2×C4—132no—
563C2×C4—268no—
564C2×C4—261no—
565C2×C4—261no—
566C2×C4—261no—
567C2×C4—261no—
568C2×C4—262no—
569C2×C4—262no—
570C2×C4—262no—
571C2×C4—262no—
572C2×C4—262no—
573C2×C4—262no—
574C2×C4—262no—
575C2×C4—524no—
576C2×C4—528no—
577C2×C4—528no—
578C2×C4—521no—
579C2×C4—521no—
580C2×C4—521no—
581C2×C4—521no—
582C2×C4—521no—
583C2×C4—521no—
584C2×C4—521no—
585C2×C4—521no—
586C2×C4—521no—
587C2×C4Q16×D1312yes—
588C2×C4C22×C4×D1311yescharacteristic
589C2×C4C2×#612911yescharacteristic
590C2×C4#41×D1311yescharacteristic
591C2×C4#41×D1311yescharacteristic
592C2×C4#41×D1311yescharacteristic
593C2×C4#41×D1311yescharacteristic
594C8—22no—
595C8—41no—
596C8—262no—
597C8—521no—
598C22—134no—
599C22—262no—
600C22—262no—
601C22—262no—
602C22—262no—
603C22—262no—
604C22—262no—
605C22C4×D4×D1311yescharacteristic
606C22#1171931111yescharacteristic
607C22D13×#12311yescharacteristic
608C22C2×Q16×D1311yescharacteristic
609C22D13×#13511yescharacteristic
610C22D13×AGL(1,ℤ/8)11yescharacteristic
611C22#1171948211yescharacteristic
612C4—22no—
613C4—42no—
614C4—42no—
615C4—41no—
616C4—42no—
617C4—81no—
618C4—134no—
619C4—262no—
620C4—262no—
621C4—522no—
622C4—522no—
623C4—522no—
624C4—1041no—
625C4#1171922711yescharacteristic
626C4C4×#612911yescharacteristic
627C4#1171938511yescharacteristic
628C4#1171941111yescharacteristic
629C2—264no—
630C2#1311736311yescharacteristic
631C2C4×Q16×D1311yescharacteristic
632C2#1312364111yescharacteristic
633C2D13×#48411yescharacteristic
634C2#1312475411yescharacteristic
635C2#1312487411yescharacteristic
636C2#1312488211yescharacteristic
subgroupG/H
centerC23C2×D4×D13
commutatorC2×C4×C13C23×C4
FrattiniC22×C4C23×D13
FittingC13×#1928C2
socleC23×C13C22×D4
prime radicalC23×D13C2×D4
cosocleC22×C4×C13C24
hypercenter#1928D13
hypocenterC13C4×#484
LCSG ▷ C2×C4×C13 ▷ C2×C13 ▷ C13
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1928

Complements

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

NH ≅ G/Ntype
4. #13115474C2semidirect
5. #13115429C2semidirect
6. #13115428C2semidirect
10. #107×Dic13C2semidirect
11. #13114728C2semidirect
12. #103×Dic13C2semidirect
13. #103×Dic13C2semidirect
15. C13×#1928C2semidirect
43. #11718470C4semidirect
107. C13×#103C2×C4semidirect
112. #88075C2×C4semidirect
121. #88058C2×C4semidirect
125. #88047C2×C4semidirect
215. #1928D13semidirect
369. Dic13#484semidirect
422. #103C4×D13semidirect
528. C13C4×#484semidirect

Maximal subgroups

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

subgroupindex
C2×#117184702
#131173742
C4×C4⋊C4×D132
#131154742
#131154292
#131154282
#131151402
#131150222
#131149252
#107×Dic132
#131147282
#103×Dic132
#131145932
C13×#19282
C4×#48413