Finite Group Databasebeta

← #16726602

Subgroups — #16726602

Order 3328.

2694 conjugacy classes of subgroups.

The subgroup lattice has 658 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11719455C211yesmaximal, characteristic
2#13128036C211yesmaximal, characteristic
3D13×#520C211yesmaximal, characteristic
4#13123435C213yesmaximal, complemented
5#13123434C211yesmaximal, characteristic, complemented
6#13123432C213yesmaximal, complemented
7#13123431C211yesmaximal, characteristic, complemented
8#13123430C213yesmaximal, complemented
9#13123429C211yesmaximal, characteristic, complemented
10#13123428C213yesmaximal, complemented
11#13123424C213yesmaximal, complemented
12#13123423C213yesmaximal, complemented
13C2×#11718542C213yesmaximal, complemented
14D13×#428C213yesmaximal, complemented
15C13×#2913C211yesmaximal, characteristic, complemented
16C2×#41×D13C2211yescharacteristic
17C2×C13×#131C2211yescharacteristic
18C13×#520C2211yescharacteristic
19C13×#520C2211yescharacteristic
20#11719532C2211yescharacteristic
21#11719532C2211yescharacteristic
22C2×#88202C2211yescharacteristic
23#11719455C4116yescomplemented
24C2×#88161C2213yes—
25C2×M16×D13C2213yes—
26C2×#88155C2213yes—
27C22×#6094C2213yes—
28#11719279—21no—
29C4×#6131—21no—
30#11719277—21no—
31C4×Q8×D13—21no—
32#11719251—23no—
33#11719250—23no—
34C4×D4×D13—23no—
35C4×#6129—23no—
36#11719228C2213yes—
37#11719228C2213yes—
38D13×#117C2213yes—
39D13×#117C2213yes—
40#11719209C2213yes—
41#11719209C2213yes—
42C2×#88146C2213yes—
43C2×C4⋊C4×D13C2213yes—
44C13×#428C2213yescomplemented
45C13×#428C2213yescomplemented
46C13×#427C2213yescomplemented
47C2×C13×#107C2213yescomplemented
48#11718782C2213yescomplemented
49#11718782C2213yescomplemented
50#11718767C2213yescomplemented
51C2×#88057C2213yescomplemented
52#11718659C2213yescomplemented
53#11718659C2213yescomplemented
54#11718656C2213yescomplemented
55#11718656C2213yescomplemented
56#11718625C2213yescomplemented
57#11718625C2213yescomplemented
58C2×#88048C2213yescomplemented
59C2×#88047C2213yescomplemented
60#11718542C2216yes—
61#11718542C2213yes—
62#11718542C2211yescharacteristic
63#11718542C2216yescomplemented
64#11718542C2216yescomplemented
65#11718542C2213yescomplemented
66#11718542C2213yescomplemented
67#11718541C2216yescomplemented
68D13×#107C2216yescomplemented
69#11718522C2213yes—
70#11718522C2213yes—
71#11718522C2213yescomplemented
72#11718522C2213yescomplemented
73#11718521C2216yescomplemented
74D13×#106C2216yescomplemented
75C2×C13×#41C2311yescharacteristic
76#41×D13C2×C418yes—
77#41×D13Q818yes—
78C2×#6131D411yescharacteristic
79C2×Q8×D13D411yescharacteristic
80C2×#6129D413yes—
81C2×D4×D13D413yes—
82C2×#6127C2313yes—
83C22×C4×D13—23no—
84C22×C4×D13C2313yes—
85C13×#131C2×C418yescomplemented
86C2×C13×M16C2313yes—
87C22×C8×C13C2313yes—
88C4×Q8×C13—21no—
89C4×Q8×C13—21no—
90C4×D4×C13—23no—
91C4×D4×C13—23no—
92C13×#117C2313yes—
93C13×#117C2313yes—
94C2×C13×C4⋊C4C2313yes—
95C2×C13×C4⋊C4C2313yes—
96#88202C2×C418yescomplemented
97Q8×Dic13—21no—
98Q8×Dic13—21no—
99D4×Dic13—23no—
100D4×Dic13—23no—
101C4×C22⋊D13—26no—
102C4×C22⋊D13—26no—
103C2×#6103—23no—
104#88180C2313yes—
105#88180C2313yes—
106C2×C4⋊Dic13C2313yes—
107C2×C4⋊Dic13C2313yes—
108C2×#6101—23no—
109C2×C4×Dic13—23no—
110C2×#6099C2313yes—
111C22×#1055C2313yes—
112#88161C2×C4124yescomplemented
113M16×D13C2×C4124yescomplemented
114#88155C2×C4124yescomplemented
115C2×#6094C2×C4124yescomplemented
116C2×#6094C2311yescharacteristic
117C2×#6094C2313yes—
118C2×#6094C2313yes—
119C2×#6094C2313yes—
120C2×C8×D13C2316yes—
121#88147—23no—
122#88147—23no—
123#88146—23no—
124#88146—23no—
125#88146C2313yes—
126#88146C2316yes—
127#88146C2313yes—
128#88146C2311yescharacteristic
129#88146C2313yes—
130C4⋊C4×D13—23no—
131C4⋊C4×D13—23no—
132C4⋊C4×D13C2313yes—
133C4⋊C4×D13C2316yes—
134C4⋊C4×D13C2313yes—
135C4⋊C4×D13C2311yescharacteristic
136C4⋊C4×D13C2313yes—
137#88141—23no—
138#88141—23no—
139#88135—26no—
140#88135—26no—
141C22⋊C4×D13—23no—
142C22⋊C4×D13—23no—
143#88131—23no—
144#88131—23no—
145C4×D52—23no—
146C4×D52—23no—
147#88126—23no—
148#88126—23no—
149C42×D13—23no—
150C42×D13—23no—
151C4×Dic26—23no—
152C4×Dic26—23no—
153C13×#107C2313yes—
154C13×#107C2316yes—
155C13×#107C2311yescharacteristic
156C13×#106C2313yes—
157C13×#106C2313yes—
158#88058C2313yes—
159#88058C2313yes—
160#88057C2313yes—
161#88057C2316yes—
162#88057C2311yescharacteristic
163#88048C2313yes—
164#88048C2316yes—
165#88048C2311yescharacteristic
166#88048C2313yes—
167#88048C2313yes—
168#88047C2313yes—
169#88047C2316yes—
170#88047C2311yescharacteristic
171#88047C2313yes—
172#88047C2313yes—
173#60109—131nomaximal, Sylow
174C23×D13C2×D413yes—
175C13×#41C22×C414yes—
176C13×#41C2×Q814yes—
177C2×Q8×C13C2×D411yescharacteristic
178C2×D4×C13C2×D413yes—
179C22×C4×C13—23no—
180C22×C4×C13—23no—
181C22×C4×C13C2413yes—
182C2×C22⋊D13C2×D416yes—
183C22×Dic13—23no—
184C22×Dic13—23no—
185C22×Dic13C2×D413yes—
186#6131C4⋊C418yes—
187Q8×D13C4⋊C418yes—
188#6129C4⋊C4124yes—
189D4×D13C4⋊C4124yes—
190#6127C22×C4112yes—
191#6127C2×Q8112yes—
192C2×D52C2×D413yes—
193C2×C4×D13—21no—
194C2×C4×D13—23no—
195C2×C4×D13—23no—
196C2×C4×D13—23no—
197C2×C4×D13—26no—
198C2×C4×D13—86no—
199C2×C4×D13C22×C4112yes—
200C2×C4×D13C2×D413yes—
201C2×C4×D13C2×D413yes—
202C2×C4×D13C2×Q8112yes—
203C2×C4×D13C2413yes—
204C2×C4×D13C2411yescharacteristic
205C2×Dic26C2×D413yes—
206C13×M16C22×C4112yescomplemented
207C2×C8×C13C22×C4112yescomplemented
208C2×C8×C13C2413yes—
209C2×C8×C13C2411yescharacteristic
210C13×C4⋊C4—23no—
211C13×C4⋊C4—23no—
212C13×C4⋊C4C2411yescharacteristic
213C13×C4⋊C4C2411yescharacteristic
214C13×C4⋊C4C2413yes—
215C13×C4⋊C4C2413yes—
216C13×C22⋊C4—23no—
217C13×C22⋊C4—23no—
218C42×C13—23no—
219C42×C13—23no—
220C22⋊Dic13—23no—
221C22⋊Dic13—23no—
222#6103—21no—
223#6103—23no—
224#6103—23no—
225#6103—23no—
226#6103—26no—
227C4⋊Dic13—23no—
228C4⋊Dic13—23no—
229C4⋊Dic13C2413yes—
230C4⋊Dic13C2411yescharacteristic
231C4⋊Dic13C2413yes—
232C4⋊Dic13C2411yescharacteristic
233#6101—23no—
234#6101—26no—
235#6101—21no—
236#6101—23no—
237#6101—23no—
238C4×Dic13—23no—
239C4×Dic13—23no—
240C4×Dic13—23no—
241C4×Dic13—26no—
242C4×Dic13—21no—
243C4×Dic13—23no—
244C4×Dic13—23no—
245#6099C22×C4112yescomplemented
246C2×#1055C22×C4112yescomplemented
247C2×#1055C2413yes—
248C2×#1055C2411yescharacteristic
249#6094C22×C414yes—
250#6094C22×C4112yes—
251#6094C22×C4112yescomplemented
252#6094C22×C4112yescomplemented
253C8×D13C22×C4124yescomplemented
254C2×#568—131no—
255#3428—131no—
256C2×#520—131no—
257#2913—133no—
258#2913—133no—
259#2913—131no—
260#2913D1311yescharacteristic, complemented
261#2912—133no—
262#2912—133no—
263#2912—131no—
264#2912—131no—
265#2911—133no—
266#2911—133no—
267C2×#428—133no—
268C2×#428—133no—
269C23×C13C22×D413yes—
270C22×D13—43no—
271C22×D13—43no—
272C22×D13C2×C4⋊C4112yes—
273C22×D13C22×D411yescharacteristic
274C22×D13C22×D413yes—
275Q8×C13C2×C4⋊C414yes—
276D4×C13C2×C4⋊C4112yes—
277C2×C4×C13—21no—
278C2×C4×C13—23no—
279C2×C4×C13—21no—
280C2×C4×C13—23no—
281C2×C4×C13—83no—
282C2×C4×C13—83no—
283C2×C4×C13C23×C416yes—
284C2×C4×C13C22×D413yes—
285C2×C4×C13C22×Q816yes—
286C2×C4×C13C2511yescharacteristic
287C22⋊D13C2×C4⋊C4124yes—
288C2×Dic13—21no—
289C2×Dic13—21no—
290C2×Dic13—23no—
291C2×Dic13—23no—
292C2×Dic13—83no—
293C2×Dic13—83no—
294C2×Dic13C2×C4⋊C4112yes—
295C2×Dic13C22×D413yes—
296C2×Dic13C22×D411yescharacteristic
297D52C2×C4⋊C4112yes—
298C4×D13—42no—
299C4×D13—46no—
300C4×D13—83no—
301C4×D13—83no—
302C4×D13—86no—
303C4×D13C2×C4⋊C4112yes—
304C4×D13C2×C4⋊C4112yes—
305C4×D13C23×C416yes—
306C4×D13C23×C412yes—
307C4×D13C22×Q812yes—
308C4×D13C22×Q816yes—
309Dic26C2×C4⋊C4112yes—
310C8×C13C23×C416yes—
311C8×C13C23×C412yes—
312#1055C23×C416yes—
313#1055C23×C412yes—
314C22×#41—131no—
315#568—1316no—
316C2×#131—131no—
317C2×#131—133no—
318C2×#131—133no—
319C2×#131C2×D1311yescharacteristic
320C22×M16—133no—
321C22×M16—133no—
322#520—131no—
323#520—131no—
324#520—133no—
325#520—133no—
326#520—133no—
327#520—133no—
328#520—263no—
329#520—261no—
330#520C2×D1311yescharacteristic
331#520C2×D1311yescharacteristic
332#519—261no—
333#518—263no—
334C4×#41—263no—
335C4×#41—261no—
336C2×C4×Q8—261no—
337C2×C4×D4—263no—
338C2×#117—133no—
339C2×#117—133no—
340C2×#117—133no—
341C22×C4⋊C4—133no—
342#428—133no—
343#428—133no—
344#428—133no—
345#428—133no—
346#428—133no—
347#428—133no—
348#428—136no—
349#428—133no—
350#428—136no—
351#428—136no—
352#428—131no—
353#428—133no—
354#428—133no—
355#428—133no—
356#428—133no—
357#428—133no—
358#428—133no—
359#428C2×D1313yescomplemented
360#428C2×D1313yescomplemented
361#427—133no—
362#427—133no—
363#427—133no—
364#427—136no—
365#427—136no—
366#427C2×D1313yescomplemented
367C2×#107—133no—
368C2×#107—136no—
369C2×#107C2×D1313yescomplemented
370C2×#106—133no—
371C2×#106—133no—
372C2×#106—136no—
373C22×C13—23no—
374C22×C13—23no—
375C22×C13C22×C4⋊C416yes—
376C22×C13C23×D411yescharacteristic
377C2×D13—21no—
378C2×D13—23no—
379C2×D13—21no—
380C2×D13—23no—
381C2×D13—412no—
382C2×D13C22×C4⋊C412yes—
383C2×D13C22×C4⋊C416yes—
384C4×C13—43no—
385C4×C13—43no—
386C4×C13—41no—
387C4×C13—41no—
388C4×C13C22×C4⋊C416yes—
389C4×C13C24×C411yescharacteristic
390C4×C13C23×Q811yescharacteristic
391Dic13—43no—
392Dic13—41no—
393Dic13—41no—
394Dic13—43no—
395Dic13C22×C4⋊C416yes—
396Dic13C22×C4⋊C412yes—
397C2×#41—138no—
398C2×#41—138no—
399C2×#41—131no—
400C2×#41—133no—
401C2×#41—133no—
402C2×#41C22×D1311yescharacteristic
403C22×Q8—131no—
404C22×D4—133no—
405C23×C4—133no—
406C23×C4—263no—
407#131—138no—
408#131—1324no—
409#131—1324no—
410#131C4×D1318yescomplemented
411C2×M16—133no—
412C2×M16—1324no—
413C2×M16—1324no—
414C2×M16—131no—
415C2×M16—133no—
416C2×M16—133no—
417C2×M16—133no—
418C2×M16C22×D1313yes—
419C22×C8—133no—
420C22×C8—136no—
421C22×C8C22×D1313yes—
422C4×Q8—21no—
423C4×Q8—21no—
424C4×Q8—263no—
425C4×Q8—263no—
426C4×Q8—263no—
427C4×Q8—263no—
428C4×Q8—261no—
429C4×Q8—261no—
430C4×D4—23no—
431C4×D4—23no—
432C4×D4—263no—
433C4×D4—263no—
434C4×D4—266no—
435C4×D4—266no—
436C4×D4—263no—
437C4×D4—263no—
438C4×D4—266no—
439C4×D4—266no—
440C4×D4—263no—
441C4×D4—263no—
442#117—133no—
443#117—133no—
444#117—133no—
445#117—131no—
446#117—136no—
447#117—133no—
448#117—133no—
449#117—263no—
450#117—263no—
451#117—263no—
452#117—263no—
453#117—263no—
454#117—263no—
455#117C22×D1313yes—
456#117C22×D1313yes—
457C2×C4⋊C4—133no—
458C2×C4⋊C4—133no—
459C2×C4⋊C4—133no—
460C2×C4⋊C4—136no—
461C2×C4⋊C4—131no—
462C2×C4⋊C4—133no—
463C2×C4⋊C4—133no—
464C2×C4⋊C4—263no—
465C2×C4⋊C4—263no—
466C2×C4⋊C4—263no—
467C2×C4⋊C4C22×D1313yes—
468C2×C4⋊C4C22×D1313yes—
469C2×C22⋊C4—263no—
470C2×C22⋊C4—263no—
471C2×C22⋊C4—263no—
472C2×C42—263no—
473C2×C42—263no—
474C2×C42—263no—
475#107—133no—
476#107—133no—
477#107—133no—
478#107—136no—
479#107—133no—
480#107—133no—
481#107—131no—
482#107C22×D1316yes—
483#107C22×D1313yes—
484#107C22×D1311yescharacteristic
485#106—133no—
486#106—136no—
487#106—131no—
488#106—133no—
489#106—136no—
490#106—133no—
491#106—133no—
492#106—131no—
493#106C22×D1313yes—
494#106C22×D1313yes—
495C2×C13—26no—
496C2×C13C23×C4⋊C411yescharacteristic
497C2×C13#358511yescharacteristic
498C2×C13#358611yescharacteristic
499D13—22no—
500D13—26no—
501C24—133no—
502#41—138no—
503#41—1324no—
504#41—1312no—
505#41—1312no—
506#41C2×C4×D1314yes—
507#41Q8×D1314yes—
508C2×Q8—138no—
509C2×Q8—133no—
510C2×Q8D4×D1311yescharacteristic
511C2×D4—136no—
512C2×D4—1324no—
513C2×D4—133no—
514C2×D4D4×D1313yes—
515C22×C4—23no—
516C22×C4—23no—
517C22×C4—133no—
518C22×C4—1312no—
519C22×C4—133no—
520C22×C4—133no—
521C22×C4—1312no—
522C22×C4—133no—
523C22×C4—131no—
524C22×C4—261no—
525C22×C4—263no—
526C22×C4—263no—
527C22×C4—263no—
528C22×C4—266no—
529C22×C4—263no—
530C22×C4—263no—
531C22×C4—1046no—
532C22×C4C23×D1313yes—
533M16—1312no—
534M16—1312no—
535M16—134no—
536M16—1312no—
537M16—1312no—
538M16C2×C4×D13112yescomplemented
539C2×C8—1312no—
540C2×C8—131no—
541C2×C8—133no—
542C2×C8—1324no—
543C2×C8C2×C4×D13112yescomplemented
544C2×C8C23×D1311yescharacteristic
545C2×C8C23×D1313yes—
546C4⋊C4—23no—
547C4⋊C4—23no—
548C4⋊C4—133no—
549C4⋊C4—133no—
550C4⋊C4—131no—
551C4⋊C4—131no—
552C4⋊C4—263no—
553C4⋊C4—266no—
554C4⋊C4—261no—
555C4⋊C4—263no—
556C4⋊C4—263no—
557C4⋊C4—263no—
558C4⋊C4—263no—
559C4⋊C4C23×D1311yescharacteristic
560C4⋊C4C23×D1313yes—
561C4⋊C4C23×D1311yescharacteristic
562C4⋊C4C23×D1313yes—
563C22⋊C4—23no—
564C22⋊C4—23no—
565C22⋊C4—261no—
566C22⋊C4—263no—
567C22⋊C4—263no—
568C22⋊C4—266no—
569C22⋊C4—263no—
570C22⋊C4—263no—
571C22⋊C4—263no—
572C42—23no—
573C42—23no—
574C42—261no—
575C42—263no—
576C42—263no—
577C42—266no—
578C42—263no—
579C42—263no—
580C42—263no—
581C13#6010911yescharacteristic, complemented, Sylow
582C23—131no—
583C23—133no—
584C23—1312no—
585C23—523no—
586C23—523no—
587C23C2×D4×D1313yes—
588Q8—1312no—
589Q8C4⋊C4×D1314yes—
590D4—1324no—
591D4—1312no—
592D4C4⋊C4×D13112yes—
593C2×C4—21no—
594C2×C4—21no—
595C2×C4—23no—
596C2×C4—23no—
597C2×C4—83no—
598C2×C4—83no—
599C2×C4—1312no—
600C2×C4—131no—
601C2×C4—133no—
602C2×C4—1312no—
603C2×C4—1312no—
604C2×C4—136no—
605C2×C4—132no—
606C2×C4—132no—
607C2×C4—136no—
608C2×C4—261no—
609C2×C4—261no—
610C2×C4—263no—
611C2×C4—263no—
612C2×C4—522no—
613C2×C4—526no—
614C2×C4—1043no—
615C2×C4—1043no—
616C2×C4—1046no—
617C2×C4—1043no—
618C2×C4—1043no—
619C2×C4C22×C4×D1316yes—
620C2×C4C2×D4×D1313yes—
621C2×C4C2×Q8×D1316yes—
622C2×C4C24×D1311yescharacteristic
623C8—136no—
624C8—132no—
625C8C22×C4×D1316yes—
626C8C22×C4×D1312yes—
627C22—23no—
628C22—23no—
629C22—132no—
630C22—136no—
631C22—261no—
632C22—263no—
633C22—261no—
634C22—263no—
635C22—5212no—
636C22C2×C4⋊C4×D1316yes—
637C22C22×D4×D1311yescharacteristic
638C4—43no—
639C4—43no—
640C4—41no—
641C4—41no—
642C4—132no—
643C4—136no—
644C4—523no—
645C4—521no—
646C4—521no—
647C4—523no—
648C4C2×C4⋊C4×D1316yes—
649C4C23×C4×D1311yescharacteristic
650C4C22×Q8×D1311yescharacteristic
651C2—26no—
652C2—262no—
653C2—266no—
654C2C22×C4⋊C4×D1311yescharacteristic
655C2#1312961111yescharacteristic
656C2#1312962111yescharacteristic
subgroupG/H
centerC22C22×D4×D13
commutatorC4×C13C24×C4
FrattiniC2×C4C24×D13
FittingC13×#2913C2
socleC22×C13C23×D4
prime radicalC2×#41×D13C22
cosocleC2×C4×C13C25
hypercenter#2913D13
hypocenterC13#60109
LCSG ▷ C4×C13 ▷ C2×C13 ▷ C13
UCSC1 ◁ C22 ◁ C2×#41 ◁ #2913

Complements

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

NH ≅ G/Ntype
4. #13123435C2semidirect
5. #13123434C2semidirect
6. #13123432C2semidirect
7. #13123431C2semidirect
8. #13123430C2semidirect
9. #13123429C2semidirect
10. #13123428C2semidirect
11. #13123424C2semidirect
12. #13123423C2semidirect
13. C2×#11718542C2semidirect
14. D13×#428C2semidirect
15. C13×#2913C2semidirect
23. #11719455C4semidirect
44. C13×#428C22semidirect
45. C13×#428C22semidirect
46. C13×#427C22semidirect
47. C2×C13×#107C22semidirect
48. #11718782C22semidirect
49. #11718782C22semidirect
50. #11718767C22semidirect
51. C2×#88057C22semidirect
52. #11718659C22semidirect
53. #11718659C22semidirect
54. #11718656C22semidirect
55. #11718656C22semidirect
56. #11718625C22semidirect
57. #11718625C22semidirect
58. C2×#88048C22semidirect
59. C2×#88047C22semidirect
63. #11718542C22semidirect
64. #11718542C22semidirect
65. #11718542C22semidirect
66. #11718542C22semidirect
67. #11718541C22semidirect
68. D13×#107C22semidirect
71. #11718522C22semidirect
72. #11718522C22semidirect
73. #11718521C22semidirect
74. D13×#106C22semidirect
85. C13×#131C2×C4semidirect
96. #88202C2×C4semidirect
112. #88161C2×C4semidirect
113. M16×D13C2×C4semidirect
114. #88155C2×C4semidirect
115. C2×#6094C2×C4semidirect
206. C13×M16C22×C4semidirect
207. C2×C8×C13C22×C4semidirect
245. #6099C22×C4semidirect
246. C2×#1055C22×C4semidirect
251. #6094C22×C4semidirect
252. #6094C22×C4semidirect
253. C8×D13C22×C4semidirect
260. #2913D13semidirect
359. #428C2×D13semidirect
360. #428C2×D13semidirect
366. #427C2×D13semidirect
369. C2×#107C2×D13semidirect
410. #131C4×D13semidirect
538. M16C2×C4×D13semidirect
543. C2×C8C2×C4×D13semidirect
581. C13#60109semidirect

Maximal subgroups

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

subgroupindex
C2×#117194552
#131280362
D13×#5202
#131234352
#131234342
#131234322
#131234312
#131234302
#131234292
#131234282
#131234242
#131234232
C2×#117185422
D13×#4282
C13×#29132
#6010913