Finite Group Databasebeta

← #16730550

Subgroups — #16730550

Order 3328.

2038 conjugacy classes of subgroups.

The subgroup lattice has 698 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#13128241C211yesmaximal, characteristic, complemented
2C2×#11719209C211yesmaximal, characteristic
3#13124703C214yesmaximal, complemented
4#13124697C214yesmaximal, complemented
5#13124692C214yesmaximal, complemented
6#13124686C214yesmaximal, complemented
7#13124640C214yesmaximal, complemented
8#13124632C214yesmaximal, complemented
9C2×#11718625C211yesmaximal, characteristic, complemented
10#13123399C211yesmaximal, characteristic, complemented
11#13123356C211yesmaximal, characteristic
12#13123353C211yesmaximal, characteristic
13C2×C13×#428C211yesmaximal, characteristic, complemented
14C22×#6127C2211yescharacteristic
15C22×C13×M16C2211yescharacteristic
16C2×C13×#117C2211yescharacteristic
17C22×C13×C4⋊C4C2211yescharacteristic
18C2×C4×C22⋊D13—22no—
19C2×#88180C2211yescharacteristic
20C2×#88180C2211yescharacteristic
21C22×#6099C2211yescharacteristic
22#11719224—21no—
23#11719224—21no—
24#11719224C2212yescomplemented
25#11719224C2212yescomplemented
26#11719222—22no—
27#11719222C2214yes—
28#11719209—22no—
29#11719209—22no—
30#11719209C2214yes—
31#11719209C2214yes—
32C2×#88147C2212yes—
33C2×#88146—21no—
34C2×C4⋊C4×D13—21no—
35#11719205C2212yescomplemented
36#11719205C2212yescomplemented
37C2×#88141C2212yes—
38#11719188—22no—
39#11719175—22no—
40C13×#428C2214yes—
41C13×#428C2214yescomplemented
42C2×C13×#107C2212yescomplemented
43C2×C13×#106C2212yescomplemented
44#11718794C2212yes—
45#11718794C2212yes—
46#11718792C2212yes—
47#11718781C2212yes—
48#11718780C2214yescomplemented
49#11718779C2214yes—
50#11718774C2214yes—
51#11718763C2214yescomplemented
52#11718665C2212yes—
53#11718663C2214yes—
54#11718662C2212yes—
55#11718627C2212yes—
56#11718627C2212yes—
57#11718625C2214yes—
58#11718625C2214yescomplemented
59C2×#88048C2212yescomplemented
60C2×#88047C2212yescomplemented
61#11718552C2218yescomplemented
62#11718551C2218yescomplemented
63#11718538C2218yescomplemented
64#11718536C2218yescomplemented
65#11718532C2218yescomplemented
66#11718531C2218yescomplemented
67#11718518C2218yescomplemented
68#11718515C2218yescomplemented
69C23×C4×C13—21no—
70C23×C4×C13C2311yescharacteristic
71C22×C22⋊D13—21no—
72C2×#6127—21no—
73C2×#6127—21no—
74C2×#6127—24no—
75C2×#6127Q818yescomplemented
76C2×#6127C2312yes—
77C2×#6127C2312yes—
78C22×D52C2311yescharacteristic
79C22×C4×D13—21no—
80C22×C4×D13—22no—
81C22×Dic26C2311yescharacteristic
82C2×C13×M16—24no—
83C2×C13×M16C2312yes—
84C2×C13×M16C2312yes—
85C22×C8×C13C2312yes—
86C13×#117—21no—
87C13×#117—21no—
88C13×#117C2312yes—
89C13×#117C2312yes—
90C2×C13×C4⋊C4—21no—
91C2×C13×C4⋊C4—21no—
92C2×C13×C4⋊C4—21no—
93C2×C13×C4⋊C4C2312yes—
94C2×C13×C4⋊C4C2311yescharacteristic
95C2×C13×C4⋊C4C2311yescharacteristic
96C2×C13×C4⋊C4C2312yes—
97C2×C13×C4⋊C4C2312yes—
98C2×C13×C22⋊C4—21no—
99C2×C42×C13—21no—
100C2×C22⋊Dic13—21no—
101C2×C22⋊Dic13D412yes—
102C4×C22⋊D13—28no—
103C4×C22⋊D13—28no—
104C2×#6103—22no—
105#88180—21no—
106#88180—21no—
107#88180D412yes—
108#88180D412yes—
109#88180C2312yes—
110#88180C2312yes—
111#88180C2312yes—
112#88180C2312yes—
113C2×C4⋊Dic13D411yescharacteristic
114C2×C4⋊Dic13D411yescharacteristic
115C2×C4⋊Dic13C2311yescharacteristic
116C2×C4⋊Dic13C2311yescharacteristic
117#88178—24no—
118#88178—24no—
119C2×#6101—22no—
120C2×C4×Dic13—21no—
121C2×C4×Dic13—22no—
122C2×C4×Dic13C2312yes—
123C2×#6099—24no—
124C2×#6099C2312yes—
125C2×#6099C2312yes—
126C22×#1055C2312yes—
127#88147—24no—
128#88147C2318yes—
129#88146—24no—
130#88146—24no—
131C4⋊C4×D13—24no—
132C4⋊C4×D13—24no—
133#88144—22no—
134#88144—22no—
135#88144C2314yes—
136#88144C2314yes—
137#88142—22no—
138#88142—22no—
139#88142C2314yes—
140#88142C2314yes—
141#88141—24no—
142#88141C2318yes—
143#88132—24no—
144#88132—24no—
145#88132—24no—
146#88132—24no—
147#88124—22no—
148#88124—22no—
149#88123—22no—
150#88123—22no—
151C4×Dic26—24no—
152C4×Dic26—24no—
153C13×#107C2318yes—
154C13×#106C2318yes—
155#88061C2314yes—
156#88061C2314yes—
157#88056C2314yes—
158#88056C2314yes—
159#88054C2318yes—
160#88054C2318yes—
161#88050C2314yes—
162#88050C2314yes—
163#88049C2314yes—
164#88049C2314yes—
165#88048C2318yes—
166#88047C2318yes—
167#60492—131nomaximal, Sylow
168C24×C13C2×D411yescharacteristic
169C23×D13—21no—
170C22×C4×C13—22no—
171C22×C4×C13—21no—
172C22×C4×C13—21no—
173C22×C4×C13—22no—
174C22×C4×C13—22no—
175C22×C4×C13—82no—
176C22×C4×C13C2×D411yescharacteristic
177C22×C4×C13C2×D411yescharacteristic
178C22×C4×C13C2×D411yescharacteristic
179C22×C4×C13C2×Q814yes—
180C22×C4×C13#4114yes—
181C22×C4×C13C2411yescharacteristic
182C22×C4×C13C2411yescharacteristic
183C22×C4×C13C2411yescharacteristic
184C2×C22⋊D13—28no—
185C2×C22⋊D13—22no—
186C2×C22⋊D13—22no—
187C22×Dic13—21no—
188C22×Dic13—21no—
189C22×Dic13—21no—
190C22×Dic13C2×D412yes—
191C22×Dic13C2×D412yes—
192#6127—28no—
193#6127—28no—
194#6127—24no—
195#6127—24no—
196#6127C2×Q818yes—
197#6127C2×Q818yes—
198C2×D52—21no—
199C2×D52—22no—
200C2×D52C2×Q814yes—
201C2×D52C2412yes—
202C2×C4×D13—28no—
203C2×C4×D13—22no—
204C2×C4×D13—22no—
205C2×C4×D13—28no—
206C2×C4×D13—84no—
207C2×Dic26—21no—
208C2×Dic26—22no—
209C2×Dic26—24no—
210C2×Dic26—24no—
211C2×Dic26C2×Q814yes—
212C2×Dic26C2412yes—
213C13×M16—24no—
214C13×M16—24no—
215C2×C8×C13—24no—
216C2×C8×C13C2414yes—
217C13×C4⋊C4—22no—
218C13×C4⋊C4—21no—
219C13×C4⋊C4—21no—
220C13×C4⋊C4—22no—
221C13×C4⋊C4—22no—
222C13×C4⋊C4C2412yes—
223C13×C4⋊C4C2414yes—
224C13×C4⋊C4C2412yes—
225C13×C22⋊C4—22no—
226C13×C22⋊C4—22no—
227C42×C13—22no—
228C42×C13—22no—
229C22⋊Dic13—22no—
230C22⋊Dic13—22no—
231C22⋊Dic13C2×D414yes—
232C22⋊Dic13C2×D414yes—
233#6103—28no—
234C4⋊Dic13—21no—
235C4⋊Dic13—21no—
236C4⋊Dic13—24no—
237C4⋊Dic13—24no—
238C4⋊Dic13C2×D412yes—
239C4⋊Dic13C2×D412yes—
240C4⋊Dic13C2×D412yes—
241C4⋊Dic13C2×D412yes—
242C4⋊Dic13C2412yes—
243C4⋊Dic13C2412yes—
244#6101—24no—
245#6101—24no—
246#6101—24no—
247#6101—24no—
248#6101—28no—
249C4×Dic13—22no—
250C4×Dic13—22no—
251C4×Dic13—28no—
252C4×Dic13C2×D414yes—
253C4×Dic13C2414yes—
254#6099—24no—
255#6099—24no—
256C2×#1055—24no—
257C2×#1055C2414yes—
258#3458—131no—
259C2×#520—131no—
260#3084—134no—
261#3084—134no—
262#3083—134no—
263#3083—134no—
264#3078—134no—
265#3077—134no—
266C2×#428—131no—
267C2×#428D1311yescharacteristic, complemented
268#2905—131no—
269#2896—131no—
270#2896—131no—
271C23×C13—41no—
272C23×C13—41no—
273C23×C13C22⋊Q814yes—
274C23×C13C22×D411yescharacteristic
275C23×C13C22×D411yescharacteristic
276C23×C13C22×D411yescharacteristic
277C22×D13—24no—
278C22×D13—22no—
279C22×D13—42no—
280C22×D13—42no—
281C2×C4×C13—24no—
282C2×C4×C13—22no—
283C2×C4×C13—22no—
284C2×C4×C13—44no—
285C2×C4×C13—81no—
286C2×C4×C13—81no—
287C2×C4×C13—82no—
288C2×C4×C13—82no—
289C2×C4×C13C22⋊Q814yes—
290C2×C4×C13C22⋊Q814yes—
291C2×C4×C13C22⋊Q814yes—
292C2×C4×C13C22×D411yescharacteristic
293C2×C4×C13C22×D411yescharacteristic
294C2×C4×C13C22×D411yescharacteristic
295C2×C4×C13C22×Q812yes—
296C2×C4×C13C22×Q812yes—
297C2×C4×C13C22×Q812yes—
298C2×C4×C13C2×#4112yes—
299C2×C4×C13C2×#4112yes—
300C2×C4×C13C2×#4112yes—
301C2×C4×C13C2511yescharacteristic
302C22⋊D13—28no—
303C22⋊D13—28no—
304C2×Dic13—24no—
305C2×Dic13—22no—
306C2×Dic13—22no—
307C2×Dic13—22no—
308C2×Dic13—42no—
309C2×Dic13—42no—
310C2×Dic13—81no—
311C2×Dic13—81no—
312C2×Dic13C22×D414yes—
313C2×Dic13C22×D414yes—
314D52—24no—
315D52—22no—
316D52C22×Q814yes—
317C4×D13—28no—
318C4×D13—28no—
319C4×D13—88no—
320Dic26—24no—
321Dic26—22no—
322Dic26—84no—
323Dic26—84no—
324Dic26C22×Q814yes—
325C8×C13—24no—
326#1055—24no—
327C22×#41—131no—
328C22×M16—131no—
329C22×M16C2×D1311yescharacteristic
330#544—132no—
331#544—132no—
332#544—132no—
333#544—132no—
334#544—261no—
335#544—261no—
336#543—262no—
337#533—134no—
338#533—262no—
339#533—262no—
340#520—134no—
341#520—134no—
342#520—262no—
343#520—262no—
344C2×C4×Q8—132no—
345C2×C4×D4—132no—
346C2×C4×D4—262no—
347C2×#117—131no—
348C2×#117—131no—
349C2×#117—261no—
350C2×#117C2×D1311yescharacteristic
351C22×C4⋊C4—261no—
352C22×C4⋊C4C2×D1311yescharacteristic
353#479—138no—
354#479—138no—
355#478—138no—
356#478—138no—
357#477—138no—
358#476—138no—
359#475—138no—
360#474—138no—
361#428—134no—
362#428—134no—
363#428C2×D1314yes—
364#428C2×D1314yescomplemented
365C2×#107—132no—
366C2×#107C2×D1312yescomplemented
367C2×#106—132no—
368C2×#106C2×D1312yescomplemented
369#424—134no—
370#424—134no—
371#423—134no—
372#419—132no—
373#419—132no—
374#418—132no—
375#418—132no—
376#417—132no—
377#417—132no—
378#417—132no—
379#417—132no—
380#416—134no—
381#416—134no—
382C22×C13—21no—
383C22×C13—21no—
384C22×C13—21no—
385C22×C13—21no—
386C22×C13—21no—
387C22×C13—21no—
388C22×C13—44no—
389C22×C13C2×C22⋊Q812yes—
390C22×C13C2×C22⋊Q812yes—
391C22×C13C2×C22⋊Q812yes—
392C22×C13C23×D411yescharacteristic
393C2×D13—24no—
394C2×D13—48no—
395C2×D13—42no—
396C2×D13—42no—
397C4×C13—42no—
398C4×C13—44no—
399C4×C13—42no—
400C4×C13C2×C22⋊Q812yes—
401C4×C13C2×C22⋊Q812yes—
402C4×C13C2×C22⋊Q812yes—
403C4×C13C23×Q811yescharacteristic
404C4×C13C22×#4111yescharacteristic
405Dic13—24no—
406Dic13—24no—
407Dic13—24no—
408Dic13—42no—
409Dic13—42no—
410C2×#41—138no—
411C2×#41—132no—
412C2×#41—132no—
413C2×#41—264no—
414C2×#41—261no—
415C2×#41—261no—
416C22×Q8—131no—
417C22×D4—131no—
418C22×D4—261no—
419C23×C4—21no—
420C23×C4—262no—
421C23×C4—261no—
422C23×C4C22×D1311yescharacteristic
423C2×M16—24no—
424C2×M16—132no—
425C2×M16—132no—
426C2×M16—264no—
427C2×M16C22×D1312yes—
428C2×M16C22×D1312yes—
429C22×C8—132no—
430C22×C8C22×D1312yes—
431C4⋊Q8—134no—
432C4⋊Q8—134no—
433C4⋊Q8—262no—
434C4⋊Q8—262no—
435C4⋊Q8—262no—
436C4⋊Q8—262no—
437#125—134no—
438#125—134no—
439#125—262no—
440#125—262no—
441#125—262no—
442#125—262no—
443C22⋊Q8—264no—
444C22⋊Q8—264no—
445C22⋊Q8—264no—
446C22⋊Q8—264no—
447C22⋊Q8—264no—
448C22⋊Q8—264no—
449C4×Q8—138no—
450C4×Q8—264no—
451C4×Q8—264no—
452C4×Q8—264no—
453C4×D4—138no—
454C4×D4—264no—
455C4×D4—268no—
456C4×D4—268no—
457#117—21no—
458#117—21no—
459#117—132no—
460#117—132no—
461#117—132no—
462#117—132no—
463#117—132no—
464#117—132no—
465#117—264no—
466#117—264no—
467#117—261no—
468#117—261no—
469#117C22×D1312yes—
470#117C22×D1312yes—
471C2×C4⋊C4—21no—
472C2×C4⋊C4—21no—
473C2×C4⋊C4—21no—
474C2×C4⋊C4—131no—
475C2×C4⋊C4—131no—
476C2×C4⋊C4—131no—
477C2×C4⋊C4—131no—
478C2×C4⋊C4—264no—
479C2×C4⋊C4—264no—
480C2×C4⋊C4—262no—
481C2×C4⋊C4C22×D1312yes—
482C2×C4⋊C4C22×D1312yes—
483C2×C4⋊C4C22×D1312yes—
484C2×C4⋊C4C22×D1311yescharacteristic
485C2×C4⋊C4C22×D1311yescharacteristic
486C2×C22⋊C4—21no—
487C2×C22⋊C4—132no—
488C2×C22⋊C4—262no—
489C2×C22⋊C4—261no—
490C2×C42—21no—
491C2×C42—132no—
492C2×C42—262no—
493C2×C42—261no—
494#107—138no—
495#107C22×D1318yes—
496#106—138no—
497#106C22×D1318yes—
498C4⋊C8—138no—
499C4⋊C8—138no—
500#103—134no—
501#103—134no—
502#103—134no—
503#103—134no—
504#102—134no—
505#102—134no—
506#102—134no—
507#102—134no—
508C2×C13—22no—
509C2×C13—22no—
510C2×C13—22no—
511C2×C13C22×C22⋊Q811yescharacteristic
512C2×C13#358511yescharacteristic
513C2×C13#358611yescharacteristic
514D13—44no—
515C24—261no—
516C24D4×D1311yescharacteristic
517#41—138no—
518#41—138no—
519#41—268no—
520#41—268no—
521#41—264no—
522#41—264no—
523C2×Q8—134no—
524C2×Q8—132no—
525C2×Q8—264no—
526C2×Q8—264no—
527C2×Q8—262no—
528C2×Q8—261no—
529C2×D4—134no—
530C2×D4—132no—
531C2×D4—268no—
532C2×D4—262no—
533C2×D4—262no—
534C2×D4—262no—
535C2×D4—261no—
536C22×C4—22no—
537C22×C4—21no—
538C22×C4—21no—
539C22×C4—22no—
540C22×C4—22no—
541C22×C4—82no—
542C22×C4—132no—
543C22×C4—132no—
544C22×C4—268no—
545C22×C4—261no—
546C22×C4—268no—
547C22×C4—262no—
548C22×C4—262no—
549C22×C4—261no—
550C22×C4—261no—
551C22×C4—1044no—
552C22×C4#612714yes—
553C22×C4D4×D1311yescharacteristic
554C22×C4D4×D1311yescharacteristic
555C22×C4D4×D1311yescharacteristic
556C22×C4Q8×D1314yes—
557C22×C4C23×D1311yescharacteristic
558C22×C4C23×D1311yescharacteristic
559C22×C4C23×D1311yescharacteristic
560M16—24no—
561M16—24no—
562M16—264no—
563M16—264no—
564C2×C8—24no—
565C2×C8—134no—
566C2×C8—264no—
567C2×C8C23×D1314yes—
568C4⋊C4—22no—
569C4⋊C4—22no—
570C4⋊C4—22no—
571C4⋊C4—21no—
572C4⋊C4—21no—
573C4⋊C4—132no—
574C4⋊C4—132no—
575C4⋊C4—132no—
576C4⋊C4—132no—
577C4⋊C4—132no—
578C4⋊C4—132no—
579C4⋊C4—268no—
580C4⋊C4—264no—
581C4⋊C4—264no—
582C4⋊C4—264no—
583C4⋊C4—264no—
584C4⋊C4—264no—
585C4⋊C4—264no—
586C4⋊C4—261no—
587C4⋊C4—261no—
588C4⋊C4C23×D1312yes—
589C4⋊C4C23×D1312yes—
590C4⋊C4C23×D1314yes—
591C22⋊C4—22no—
592C22⋊C4—22no—
593C22⋊C4—134no—
594C22⋊C4—134no—
595C22⋊C4—268no—
596C22⋊C4—262no—
597C22⋊C4—262no—
598C42—22no—
599C42—22no—
600C42—134no—
601C42—134no—
602C42—268no—
603C42—262no—
604C42—262no—
605C13#6049211yescharacteristic, complemented, Sylow
606C23—41no—
607C23—41no—
608C23—264no—
609C23—262no—
610C23—522no—
611C23—522no—
612C23#8815014yes—
613C23C2×D4×D1311yescharacteristic
614C23C2×D4×D1311yescharacteristic
615C23C2×D4×D1311yescharacteristic
616Q8—134no—
617Q8—264no—
618Q8—262no—
619Q8—1044no—
620Q8—1044no—
621D4—134no—
622D4—268no—
623D4—268no—
624D4—264no—
625D4—262no—
626C2×C4—22no—
627C2×C4—22no—
628C2×C4—24no—
629C2×C4—44no—
630C2×C4—82no—
631C2×C4—82no—
632C2×C4—81no—
633C2×C4—81no—
634C2×C4—134no—
635C2×C4—134no—
636C2×C4—264no—
637C2×C4—262no—
638C2×C4—268no—
639C2×C4—268no—
640C2×C4—262no—
641C2×C4—262no—
642C2×C4—522no—
643C2×C4—522no—
644C2×C4—1048no—
645C2×C4—1041no—
646C2×C4—1041no—
647C2×C4#8815014yes—
648C2×C4#8815014yes—
649C2×C4#8815014yes—
650C2×C4C2×#612712yes—
651C2×C4C2×#612712yes—
652C2×C4C2×#612712yes—
653C2×C4C2×D4×D1311yescharacteristic
654C2×C4C2×D4×D1311yescharacteristic
655C2×C4C2×D4×D1311yescharacteristic
656C2×C4C2×Q8×D1312yes—
657C2×C4C2×Q8×D1312yes—
658C2×C4C2×Q8×D1312yes—
659C2×C4C24×D1311yescharacteristic
660C8—24no—
661C8—264no—
662C22—21no—
663C22—21no—
664C22—21no—
665C22—21no—
666C22—21no—
667C22—21no—
668C22—44no—
669C22—264no—
670C22—528no—
671C22—522no—
672C22—522no—
673C22C2×#8815012yes—
674C22C2×#8815012yes—
675C22C2×#8815012yes—
676C22C22×D4×D1311yescharacteristic
677C4—44no—
678C4—42no—
679C4—42no—
680C4—264no—
681C4—264no—
682C4—264no—
683C4—522no—
684C4—522no—
685C4C2×#8815012yes—
686C4C2×#8815012yes—
687C4C2×#8815012yes—
688C4C22×#612711yescharacteristic
689C4C22×Q8×D1311yescharacteristic
690C2—22no—
691C2—22no—
692C2—22no—
693C2—524no—
694C2C22×#8815011yescharacteristic
695C2#1312961211yescharacteristic
696C2#1312961911yescharacteristic
subgroupG/H
centerC22C22×D4×D13
commutator, cosocleC2×C4×C13C25
FrattiniC2×C4C24×D13
FittingC2×C13×#428C2
socleC22×C13C23×D4
prime radicalC22×#6127C22
hypercenterC2×#428D13
hypocenterC13#60492
LCSG ▷ C2×C4×C13 ▷ C2×C13 ▷ C13
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#428

Complements

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

NH ≅ G/Ntype
1. #13128241C2semidirect
3. #13124703C2semidirect
4. #13124697C2semidirect
5. #13124692C2semidirect
6. #13124686C2semidirect
7. #13124640C2semidirect
8. #13124632C2semidirect
9. C2×#11718625C2semidirect
10. #13123399C2semidirect
13. C2×C13×#428C2semidirect
24. #11719224C22semidirect
25. #11719224C22semidirect
35. #11719205C22semidirect
36. #11719205C22semidirect
41. C13×#428C22semidirect
42. C2×C13×#107C22semidirect
43. C2×C13×#106C22semidirect
48. #11718780C22semidirect
51. #11718763C22semidirect
58. #11718625C22semidirect
59. C2×#88048C22semidirect
60. C2×#88047C22semidirect
61. #11718552C22semidirect
62. #11718551C22semidirect
63. #11718538C22semidirect
64. #11718536C22semidirect
65. #11718532C22semidirect
66. #11718531C22semidirect
67. #11718518C22semidirect
68. #11718515C22semidirect
75. C2×#6127Q8semidirect
267. C2×#428D13semidirect
364. #428C2×D13semidirect
366. C2×#107C2×D13semidirect
368. C2×#106C2×D13semidirect
605. C13#60492semidirect

Maximal subgroups

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

subgroupindex
#131282412
C2×#117192092
#131247032
#131246972
#131246922
#131246862
#131246402
#131246322
C2×#117186252
#131233992
#131233562
#131233532
C2×C13×#4282
#6049213