Finite Group Databasebeta

← #15865321

Subgroups — #15865321

Order 3328.

1218 conjugacy classes of subgroups.

The subgroup lattice has 618 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11718656C211yesmaximal, characteristic, complemented
2C2×#11718893C211yesmaximal, characteristic
3C2×#11718893C211yesmaximal, characteristic
4C2×C4×C4⋊Dic13C211yesmaximal, characteristic, complemented
5#13115474C212yesmaximal, complemented
6#13115467C212yesmaximal, complemented
7#13115460C212yesmaximal, complemented
8#13115453C212yesmaximal, complemented
9#13114557C211yesmaximal, characteristic, complemented
10#13114530C211yesmaximal, characteristic, complemented
11C13×#2297C211yesmaximal, characteristic
12C22×C13×#41C2211yescharacteristic
13C2×C13×#117C2211yescharacteristic
14C22×C42×C13C2211yescharacteristic
15C22×C4⋊Dic13C2211yescharacteristic
16C22×C4⋊Dic13C2211yescharacteristic
17C22×C4×Dic13—21no—
18C22×#6099C2211yescharacteristic
19C22×#6099C2211yescharacteristic
20C13×#391C2211yescharacteristic
21C13×#391C2211yescharacteristic
22C13×#390C2211yescharacteristic
23C13×#390C2211yescharacteristic
24C13×#386—21no—
25C13×#386—21no—
26C4×C13×C22⋊C4—21no—
27C4×C13×C22⋊C4—21no—
28#11718893—24no—
29#11718893C418yescomplemented
30C2×#88075C2212yes—
31C2×#88075C2212yes—
32C2×#88072C2212yes—
33C2×#88072C2212yes—
34C2×#88071—21no—
35#11718656—24no—
36C2×#88048C2212yes—
37C2×#88047C2212yes—
38C4×C4⋊Dic13—21no—
39C4×C4⋊Dic13—21no—
40C4×C4⋊Dic13C2212yes—
41C4×C4⋊Dic13C2212yes—
42#11718121C2212yes—
43#11718121C2212yes—
44#11718116C2212yes—
45#11718116C2212yes—
46C2×C13×#41—22no—
47C2×C13×#41—22no—
48C2×C13×#41—22no—
49C2×C13×#41—22no—
50C2×C13×#41—22no—
51C2×C13×#41C2×C414yes—
52C22×Q8×C13C2311yescharacteristic
53C22×D4×C13—21no—
54C22×D4×C13C2311yescharacteristic
55C23×C4×C13—21no—
56C23×C4×C13D411yescharacteristic
57C23×C4×C13D411yescharacteristic
58C23×C4×C13C2311yescharacteristic
59C23×Dic13—21no—
60C23×Dic13—21no—
61C13×#117—22no—
62C13×#117—22no—
63C2×C13×C4⋊C4C2311yescharacteristic
64C2×C13×C4⋊C4C2311yescharacteristic
65C2×C13×C22⋊C4—21no—
66C2×C13×C22⋊C4—21no—
67C2×C13×C22⋊C4—21no—
68C2×C13×C22⋊C4—21no—
69C2×C13×C22⋊C4—21no—
70C2×C42×C13—22no—
71C2×C42×C13—22no—
72C2×C42×C13—21no—
73C2×C42×C13D411yescharacteristic
74C2×C42×C13D411yescharacteristic
75C2×C42×C13C2311yescharacteristic
76C2×C42×C13C2311yescharacteristic
77C2×C4⋊Dic13—21no—
78C2×C4⋊Dic13—21no—
79C2×C4⋊Dic13—22no—
80C2×C4⋊Dic13—22no—
81C2×C4⋊Dic13C2×C414yes—
82C2×C4⋊Dic13D414yes—
83C2×C4⋊Dic13C2312yes—
84C2×C4⋊Dic13C2312yes—
85C2×C4×Dic13—22no—
86C2×C4×Dic13—22no—
87C2×C4×Dic13—44no—
88C2×#6099—22no—
89C2×#6099—22no—
90C2×#6099—22no—
91C2×#6099—22no—
92C2×#6099—22no—
93C2×#6099C2×C414yes—
94C22×#1055C2312yes—
95C22×#1055C2312yes—
96C13×#95—21no—
97C13×#95—21no—
98C13×#95—21no—
99C13×#95—21no—
100#88075—24no—
101#88075C2×C418yescomplemented
102#88072—24no—
103#88072C2×C418yescomplemented
104#88071—22no—
105#88071—22no—
106#88048—24no—
107#88047—24no—
108#21059—131nomaximal, Sylow
109C24×C13—21no—
110C24×C13C2×D411yescharacteristic
111C13×#41—28no—
112C13×#41—28no—
113C13×#41—44no—
114C13×#41—44no—
115C2×Q8×C13—21no—
116C2×Q8×C13—21no—
117C2×Q8×C13—21no—
118C2×Q8×C13—21no—
119C2×Q8×C13—21no—
120C2×Q8×C13C22×C412yes—
121C2×D4×C13—24no—
122C2×D4×C13—21no—
123C2×D4×C13—21no—
124C2×D4×C13—21no—
125C2×D4×C13—21no—
126C2×D4×C13—21no—
127C2×D4×C13—42no—
128C2×D4×C13—42no—
129C2×D4×C13C22×C412yes—
130C22×C4×C13—22no—
131C22×C4×C13—22no—
132C22×C4×C13—22no—
133C22×C4×C13—22no—
134C22×C4×C13—21no—
135C22×C4×C13—24no—
136C22×C4×C13—21no—
137C22×C4×C13—21no—
138C22×C4×C13—41no—
139C22×C4×C13—41no—
140C22×C4×C13—42no—
141C22×C4×C13—42no—
142C22×C4×C13C22×C412yes—
143C22×C4×C13C2×D412yes—
144C22×C4×C13C2×D411yescharacteristic
145C22×C4×C13C2×D411yescharacteristic
146C22×C4×C13C2×D411yescharacteristic
147C22×C4×C13C2×D411yescharacteristic
148C22×C4×C13C2×D411yescharacteristic
149C22×C4×C13#4112yes—
150C22×C4×C13#4112yes—
151C22×C4×C13#4112yes—
152C22×C4×C13#4112yes—
153C22×C4×C13C2411yescharacteristic
154C22×Dic13—24no—
155C22×Dic13—24no—
156C22×Dic13—22no—
157C22×Dic13—22no—
158C22×Dic13—42no—
159C22×Dic13—42no—
160C22×Dic13—42no—
161C22×Dic13—42no—
162C13×C4⋊C4—21no—
163C13×C4⋊C4—21no—
164C13×C4⋊C4—21no—
165C13×C4⋊C4—21no—
166C13×C22⋊C4—42no—
167C13×C22⋊C4—42no—
168C13×C22⋊C4—42no—
169C13×C22⋊C4—42no—
170C13×C22⋊C4—42no—
171C42×C13—22no—
172C42×C13—22no—
173C42×C13—21no—
174C42×C13—21no—
175C42×C13—21no—
176C42×C13—21no—
177C42×C13—42no—
178C4⋊Dic13—24no—
179C4⋊Dic13—24no—
180C4⋊Dic13—22no—
181C4⋊Dic13—22no—
182C4⋊Dic13C22×C414yes—
183C4⋊Dic13C2×D414yes—
184C4×Dic13—44no—
185C4×Dic13—44no—
186#6099—28no—
187#6099—44no—
188C2×#1055—22no—
189C2×#1055—22no—
190C2×#1055—22no—
191C2×#1055—22no—
192C2×#1055—22no—
193C2×#1055C22×C414yes—
194C2×#428—131no—
195C2×#417—131no—
196C2×#417—131no—
197#2297D1311yescharacteristic
198C2×C4×C4⋊C4—131no—
199#1928—132no—
200#1927—132no—
201#1926—132no—
202#1925—132no—
203#1736—131no—
204#1728—131no—
205C23×C13—21no—
206C23×C13—21no—
207C23×C13—21no—
208C23×C13—21no—
209C23×C13—22no—
210C23×C13—41no—
211C23×C13—41no—
212C23×C13—41no—
213C23×C13—41no—
214C23×C13—41no—
215C23×C13—41no—
216C23×C13C4×D412yes—
217C23×C13#12112yes—
218C23×C13#12312yes—
219C23×C13C22×D411yescharacteristic
220Q8×C13—22no—
221Q8×C13—22no—
222Q8×C13—41no—
223Q8×C13—41no—
224D4×C13—22no—
225D4×C13—22no—
226D4×C13—48no—
227D4×C13—41no—
228D4×C13—41no—
229C2×C4×C13—28no—
230C2×C4×C13—21no—
231C2×C4×C13—21no—
232C2×C4×C13—21no—
233C2×C4×C13—21no—
234C2×C4×C13—22no—
235C2×C4×C13—21no—
236C2×C4×C13—21no—
237C2×C4×C13—21no—
238C2×C4×C13—21no—
239C2×C4×C13—21no—
240C2×C4×C13—21no—
241C2×C4×C13—44no—
242C2×C4×C13—41no—
243C2×C4×C13—41no—
244C2×C4×C13—48no—
245C2×C4×C13—41no—
246C2×C4×C13—41no—
247C2×C4×C13—41no—
248C2×C4×C13—41no—
249C2×C4×C13—41no—
250C2×C4×C13—41no—
251C2×C4×C13#11714yes—
252C2×C4×C13C4×D414yes—
253C2×C4×C13C4×D412yes—
254C2×C4×C13#12112yes—
255C2×C4×C13#12312yes—
256C2×C4×C13#12414yes—
257C2×C4×C13#12614yes—
258C2×C4×C13AGL(1,ℤ/8)12yes—
259C2×C4×C13#13712yes—
260C2×C4×C13C23×C411yescharacteristic
261C2×C4×C13C22×D411yescharacteristic
262C2×C4×C13C2×#4111yescharacteristic
263C2×C4×C13C2×#4111yescharacteristic
264C2×C4×C13C2×#4111yescharacteristic
265C2×C4×C13C2×#4111yescharacteristic
266C2×Dic13—24no—
267C2×Dic13—24no—
268C2×Dic13—48no—
269C2×Dic13—48no—
270C2×Dic13—42no—
271C2×Dic13—42no—
272C2×Dic13—42no—
273C2×Dic13—42no—
274#1055—24no—
275#1055—42no—
276C22×#41C2×D1311yescharacteristic
277C22×M16—131no—
278C22×M16—131no—
279C2×#117C2×D1311yescharacteristic
280C22×C4⋊C4—131no—
281C22×C4⋊C4—131no—
282C22×C42—261no—
283C22×C42C2×D1311yescharacteristic
284#428—264no—
285C2×#107—132no—
286C2×#106—132no—
287#417—138no—
288#417—264no—
289C2×#103—132no—
290C2×#103—132no—
291C2×#102—132no—
292C2×#102—132no—
293#391C2×D1311yescharacteristic
294#391C2×D1311yescharacteristic
295#390C2×D1311yescharacteristic
296#390C2×D1311yescharacteristic
297#386—21no—
298#386—21no—
299C4×C4⋊C4—132no—
300C4×C4⋊C4—132no—
301C4×C4⋊C4—261no—
302C4×C4⋊C4—261no—
303C4×C22⋊C4—21no—
304C4×C22⋊C4—21no—
305C2×#95—261no—
306#340—132no—
307#340—132no—
308#336—132no—
309#336—132no—
310C22×C13—22no—
311C22×C13—22no—
312C22×C13—22no—
313C22×C13—22no—
314C22×C13—22no—
315C22×C13—22no—
316C22×C13—42no—
317C22×C13—42no—
318C22×C13—42no—
319C22×C13—81no—
320C22×C13—81no—
321C22×C13—81no—
322C22×C13—81no—
323C22×C13#38814yes—
324C22×C13C2×C4×D411yescharacteristic
325C22×C13C2×#12111yescharacteristic
326C22×C13C2×#12311yescharacteristic
327C22×C13C2×AGL(1,ℤ/8)11yescharacteristic
328C22×C13C2×#13711yescharacteristic
329C22×C13#57511yescharacteristic
330C22×C13#57511yescharacteristic
331C4×C13—22no—
332C4×C13—22no—
333C4×C13—41no—
334C4×C13—41no—
335C4×C13—42no—
336C4×C13—81no—
337C4×C13—81no—
338C4×C13#38814yes—
339C4×C13C2×#11711yescharacteristic
340C4×C13C2×C4×D411yescharacteristic
341C4×C13C2×#12411yescharacteristic
342C4×C13C2×#12611yescharacteristic
343Dic13—44no—
344Dic13—44no—
345C2×#41—22no—
346C2×#41—22no—
347C2×#41—22no—
348C2×#41—22no—
349C2×#41—22no—
350C2×#41C4×D1314yes—
351C22×Q8C22×D1311yescharacteristic
352C22×D4—21no—
353C22×D4C22×D1311yescharacteristic
354C23×C4—21no—
355C23×C4—261no—
356C23×C4—261no—
357C23×C4C22⋊D1311yescharacteristic
358C23×C4C22⋊D1311yescharacteristic
359C23×C4C22×D1311yescharacteristic
360C2×M16—134no—
361C2×M16—262no—
362C2×M16—262no—
363C2×M16—262no—
364C2×M16—262no—
365C2×M16—262no—
366C22×C8—132no—
367C22×C8—132no—
368#117—22no—
369#117—22no—
370C2×C4⋊C4—134no—
371C2×C4⋊C4—134no—
372C2×C4⋊C4—132no—
373C2×C4⋊C4—132no—
374C2×C4⋊C4—262no—
375C2×C4⋊C4—262no—
376C2×C4⋊C4—261no—
377C2×C4⋊C4—261no—
378C2×C4⋊C4C22×D1311yescharacteristic
379C2×C4⋊C4C22×D1311yescharacteristic
380C2×C22⋊C4—21no—
381C2×C22⋊C4—21no—
382C2×C22⋊C4—21no—
383C2×C22⋊C4—21no—
384C2×C22⋊C4—21no—
385C2×C42—22no—
386C2×C42—22no—
387C2×C42—21no—
388C2×C42—262no—
389C2×C42—262no—
390C2×C42—524no—
391C2×C42C22⋊D1311yescharacteristic
392C2×C42C22⋊D1311yescharacteristic
393C2×C42C22×D1311yescharacteristic
394C2×C42C22×D1311yescharacteristic
395#107—264no—
396#106—264no—
397#103—138no—
398#103—264no—
399#102—138no—
400#102—264no—
401#95—21no—
402#95—21no—
403#95—21no—
404#95—21no—
405#95—262no—
406#95—262no—
407C2×C13—24no—
408C2×C13—82no—
409C2×C13C2×#38811yescharacteristic
410C2×C13C4×AGL(1,ℤ/8)11yescharacteristic
411C2×C13C4×#13711yescharacteristic
412C2×C13#303611yescharacteristic
413C2×C13#303711yescharacteristic
414C2×C13#309211yescharacteristic
415C2×C13#309311yescharacteristic
416C24—21no—
417C24C2×C22⋊D1311yescharacteristic
418#41—28no—
419#41—28no—
420#41—44no—
421#41—44no—
422C2×Q8—21no—
423C2×Q8—21no—
424C2×Q8—21no—
425C2×Q8—21no—
426C2×Q8—21no—
427C2×Q8C2×C4×D1312yes—
428C2×D4—24no—
429C2×D4—21no—
430C2×D4—21no—
431C2×D4—21no—
432C2×D4—21no—
433C2×D4—21no—
434C2×D4—42no—
435C2×D4—42no—
436C2×D4C2×C4×D1312yes—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—21no—
442C22×C4—24no—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—41no—
446C22×C4—41no—
447C22×C4—42no—
448C22×C4—42no—
449C22×C4—264no—
450C22×C4—264no—
451C22×C4—262no—
452C22×C4—262no—
453C22×C4—522no—
454C22×C4—522no—
455C22×C4—522no—
456C22×C4—522no—
457C22×C4C2×C4×D1312yes—
458C22×C4#612712yes—
459C22×C4D4×D1312yes—
460C22×C4#612912yes—
461C22×C4#612912yes—
462C22×C4#612912yes—
463C22×C4C2×C22⋊D1311yescharacteristic
464C22×C4C2×C22⋊D1311yescharacteristic
465C22×C4C2×C22⋊D1311yescharacteristic
466C22×C4C2×C22⋊D1311yescharacteristic
467C22×C4C2×C22⋊D1311yescharacteristic
468C22×C4C23×D1311yescharacteristic
469M16—268no—
470M16—524no—
471C2×C8—134no—
472C2×C8—262no—
473C2×C8—262no—
474C2×C8—262no—
475C2×C8—262no—
476C2×C8—262no—
477C4⋊C4—21no—
478C4⋊C4—21no—
479C4⋊C4—21no—
480C4⋊C4—21no—
481C4⋊C4—134no—
482C4⋊C4—134no—
483C4⋊C4—264no—
484C4⋊C4—264no—
485C4⋊C4—262no—
486C4⋊C4—262no—
487C22⋊C4—42no—
488C22⋊C4—42no—
489C22⋊C4—42no—
490C22⋊C4—42no—
491C22⋊C4—42no—
492C42—22no—
493C42—22no—
494C42—21no—
495C42—21no—
496C42—21no—
497C42—21no—
498C42—42no—
499C42—524no—
500C42—524no—
501C13#2105911yescharacteristic, complemented, Sylow
502C23—21no—
503C23—21no—
504C23—21no—
505C23—21no—
506C23—22no—
507C23—41no—
508C23—41no—
509C23—41no—
510C23—41no—
511C23—41no—
512C23—41no—
513C23C4×C22⋊D1312yes—
514C23#8818912yes—
515C23#8819212yes—
516C23C22×C22⋊D1311yescharacteristic
517Q8—22no—
518Q8—22no—
519Q8—41no—
520Q8—41no—
521D4—22no—
522D4—22no—
523D4—48no—
524D4—41no—
525D4—41no—
526C2×C4—28no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—22no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—44no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—48no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—264no—
549C2×C4—264no—
550C2×C4—528no—
551C2×C4—528no—
552C2×C4—522no—
553C2×C4—522no—
554C2×C4—522no—
555C2×C4—522no—
556C2×C4#8813114yes—
557C2×C4#8813314yes—
558C2×C4#8813514yes—
559C2×C4#8813914yes—
560C2×C4C4×C22⋊D1312yes—
561C2×C4#8818912yes—
562C2×C4#8819212yes—
563C2×C4#8820312yes—
564C2×C4#8820512yes—
565C2×C4C22×C4×D1311yescharacteristic
566C2×C4C2×#612711yescharacteristic
567C2×C4C2×D4×D1311yescharacteristic
568C2×C4C2×#612911yescharacteristic
569C2×C4C2×#612911yescharacteristic
570C2×C4C2×#612911yescharacteristic
571C8—264no—
572C8—522no—
573C22—22no—
574C22—22no—
575C22—22no—
576C22—22no—
577C22—22no—
578C22—22no—
579C22—42no—
580C22—42no—
581C22—42no—
582C22—81no—
583C22—81no—
584C22—81no—
585C22—81no—
586C22#1171860514yes—
587C22C2×C4×C22⋊D1311yescharacteristic
588C22C2×#8818911yescharacteristic
589C22C2×#8819211yescharacteristic
590C22C2×#8820311yescharacteristic
591C22#1171952711yescharacteristic
592C22#1171952711yescharacteristic
593C22C2×#8820511yescharacteristic
594C4—22no—
595C4—22no—
596C4—42no—
597C4—41no—
598C4—41no—
599C4—81no—
600C4—81no—
601C4—524no—
602C4—524no—
603C4#1171860514yes—
604C4C2×#8813111yescharacteristic
605C4C2×#8813311yescharacteristic
606C4C2×#8813511yescharacteristic
607C4C2×#8813911yescharacteristic
608C2—24no—
609C2—82no—
610C2C2×#1171860511yescharacteristic
611C2C4×#8820311yescharacteristic
612C2C4×#8820511yescharacteristic
613C2#1312429611yescharacteristic
614C2#1312430411yescharacteristic
615C2#1312475611yescharacteristic
616C2#1312476511yescharacteristic
subgroupG/H
centerC23C22×C22⋊D13
commutatorC2×C4×C13C23×C4
FrattiniC22×C4C23×D13
FittingC13×#2297C2
socleC23×C13C22×D4
prime radicalC22×C13×#41C22
cosocleC22×C4×C13C24
hypercenter#2297D13
hypocenterC13#21059
LCSG ▷ C2×C4×C13 ▷ C2×C13 ▷ C13
UCSC1 ◁ C23 ◁ C22×C42 ◁ #2297

Complements

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

NH ≅ G/Ntype
1. C2×#11718656C2semidirect
4. C2×C4×C4⋊Dic13C2semidirect
5. #13115474C2semidirect
6. #13115467C2semidirect
7. #13115460C2semidirect
8. #13115453C2semidirect
9. #13114557C2semidirect
10. #13114530C2semidirect
29. #11718893C4semidirect
101. #88075C2×C4semidirect
103. #88072C2×C4semidirect
501. C13#21059semidirect

Maximal subgroups

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

subgroupindex
C2×#117186562
C2×#117188932
C2×C4×C4⋊Dic132
#131154742
#131154672
#131154602
#131154532
#131145572
#131145302
C13×#22972
#2105913