Finite Group Databasebeta

← #15866777

Subgroups — #15866777

Order 3328.

1566 conjugacy classes of subgroups.

The subgroup lattice has 626 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#88204C211yesmaximal, characteristic
2C22×#88048C211yesmaximal, characteristic, complemented
3C2×#11718893C211yesmaximal, characteristic, complemented
4C2×#11718627C211yesmaximal, characteristic, complemented
5C2×#11718658C211yesmaximal, characteristic, complemented
6#13117424C211yesmaximal, characteristic, complemented
7#13115711C212yesmaximal, complemented
8#13115706C212yesmaximal, complemented
9#13115693C212yesmaximal, complemented
10#13115689C212yesmaximal, complemented
11C13×#2311C211yesmaximal, characteristic, complemented
12C22×C13×#41C2211yescharacteristic
13C22×#6127C2211yescharacteristic
14C2×C13×#117C2211yescharacteristic
15C22×C13×C4⋊C4C2211yescharacteristic
16C2×#88204—24no—
17C2×#88204—24no—
18C2×#88204C418yescomplemented
19C22×#6107C2211yescharacteristic
20C22×#6106C2211yescharacteristic
21C22×#6105C2211yescharacteristic
22C22×#6104C2211yescharacteristic
23C22×C4⋊Dic13C2211yescharacteristic
24C22×#6099C2211yescharacteristic
25C23×#1055C2211yescharacteristic
26C13×#391C2212yes—
27C13×#390C2212yes—
28C13×#386—22no—
29C13×#380—22no—
30#11718893—24no—
31C2×#88075C2212yes—
32C2×#88072C2212yes—
33#11718658—24no—
34#11718648—22no—
35C2×#88050C2212yes—
36#11718627—24no—
37C2×#88049C2212yes—
38C2×#88048—24no—
39C2×#88048C2212yes—
40C2×#88048C2212yes—
41#11718619—22no—
42#11718600C2212yes—
43#11718590C2212yes—
44C2×#88043C2212yes—
45C2×C4×#1055C2212yes—
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×C13—21no—
57C23×C4×C13C2311yescharacteristic
58C22×C22⋊D13—21no—
59C23×Dic13—21no—
60C2×#6127—22no—
61C2×#6127—22no—
62C2×#6127—22no—
63C2×#6127—22no—
64C2×#6127—22no—
65C2×#6127C2×C414yes—
66C22×D52C2311yescharacteristic
67C22×C4×D13—21no—
68C22×Dic26C2311yescharacteristic
69C13×#117—22no—
70C13×#117—22no—
71C2×C13×C4⋊C4—22no—
72C2×C13×C4⋊C4—22no—
73C2×C13×C4⋊C4—21no—
74C2×C13×C4⋊C4D412yes—
75C2×C13×C4⋊C4C2312yes—
76C2×C13×C22⋊C4—24no—
77C2×C13×C22⋊C4D412yes—
78C2×C42×C13C2312yes—
79#88204—232no—
80C2×#6107—22no—
81C2×#6107—22no—
82C2×#6107C2×C414yescomplemented
83C2×#6106—22no—
84C2×#6106—22no—
85C2×#6106C2×C414yescomplemented
86C2×#6105—22no—
87C2×#6105—22no—
88C2×#6105C2×C414yescomplemented
89C2×#6104—22no—
90C2×#6104—22no—
91C2×#6104C2×C414yescomplemented
92C2×#6103—24no—
93C2×C4⋊Dic13—22no—
94C2×C4⋊Dic13—22no—
95C2×C4⋊Dic13—21no—
96C2×C4⋊Dic13C2312yes—
97C2×#6099—22no—
98C2×#6099—22no—
99C2×#6099D414yes—
100C22×#1055C2×C414yes—
101C22×#1055D414yes—
102C22×#1055D414yes—
103C22×#1055C2312yes—
104C22×#1055C2311yescharacteristic
105C22×#1055C2311yescharacteristic
106C13×#95—24no—
107#88075—24no—
108#88072—24no—
109#88071—24no—
110#88050—24no—
111#88049—24no—
112#88048—24no—
113#88048—24no—
114#88043—24no—
115C4×#1055—24no—
116#21229—131nomaximal, Sylow
117C24×C13—21no—
118C24×C13C2×D411yescharacteristic
119C23×D13—21no—
120C13×#41—28no—
121C13×#41—28no—
122C13×#41—44no—
123C13×#41—44no—
124C2×Q8×C13—21no—
125C2×Q8×C13—21no—
126C2×Q8×C13—21no—
127C2×Q8×C13—21no—
128C2×Q8×C13—21no—
129C2×Q8×C13C22×C412yes—
130C2×D4×C13—24no—
131C2×D4×C13—21no—
132C2×D4×C13—21no—
133C2×D4×C13—21no—
134C2×D4×C13—21no—
135C2×D4×C13—21no—
136C2×D4×C13—42no—
137C2×D4×C13—42no—
138C2×D4×C13C22×C412yes—
139C22×C4×C13—21no—
140C22×C4×C13—24no—
141C22×C4×C13—22no—
142C22×C4×C13—42no—
143C22×C4×C13—42no—
144C22×C4×C13—42no—
145C22×C4×C13—82no—
146C22×C4×C13C22×C412yes—
147C22×C4×C13C2×D412yes—
148C22×C4×C13C2×D412yes—
149C22×C4×C13C2×D412yes—
150C22×C4×C13C2×D414yes—
151C22×C4×C13C2×D411yescharacteristic
152C22×C4×C13#4112yes—
153C22×C4×C13#4112yes—
154C22×C4×C13C2411yescharacteristic
155C2×C22⋊D13—24no—
156C2×C22⋊D13—42no—
157C2×C22⋊D13—42no—
158C22×Dic13—21no—
159C22×Dic13—22no—
160C22×Dic13—42no—
161C22×Dic13—82no—
162#6127—28no—
163#6127—28no—
164#6127—44no—
165#6127—44no—
166C2×D52—21no—
167C2×D52—21no—
168C2×D52—21no—
169C2×D52—21no—
170C2×D52—21no—
171C2×D52C22×C412yes—
172C2×C4×D13—24no—
173C2×C4×D13—42no—
174C2×C4×D13—42no—
175C2×Dic26—21no—
176C2×Dic26—21no—
177C2×Dic26—21no—
178C2×Dic26—21no—
179C2×Dic26—21no—
180C2×Dic26C22×C412yes—
181C13×C4⋊C4—24no—
182C13×C4⋊C4—22no—
183C13×C4⋊C4—22no—
184C13×C4⋊C4—42no—
185C13×C22⋊C4—24no—
186C13×C22⋊C4—84no—
187C42×C13—22no—
188C42×C13—22no—
189#6107—28no—
190#6106—28no—
191#6105—28no—
192#6104—28no—
193#6103—84no—
194C4⋊Dic13—22no—
195C4⋊Dic13—22no—
196C4⋊Dic13—42no—
197#6099—28no—
198C2×#1055—22no—
199C2×#1055—22no—
200C2×#1055C22⋊C4116yescomplemented
201C2×#1055C22×C412yes—
202C2×#1055C22×C412yes—
203C2×#1055C2×D414yes—
204C2×#1055C2×D412yes—
205C2×#1055C2×D412yes—
206C2×#1055C2×D412yes—
207C2×#1055C2×D412yes—
208C22×#135—131no—
209C22×#106—131no—
210C2×#417—131no—
211C2×#417—131no—
212C2×#405—131no—
213#2311—131no—
214#2311D1311yescharacteristic, complemented
215#1972—132no—
216#1971—132no—
217#1968—132no—
218#1968—132no—
219C23×C13—22no—
220C23×C13—22no—
221C23×C13—22no—
222C23×C13—41no—
223C23×C13—41no—
224C23×C13—42no—
225C23×C13—42no—
226C23×C13C4×D412yes—
227C23×C13#12112yes—
228C23×C13#12712yes—
229C23×C13C22×D411yescharacteristic
230C22×D13—22no—
231C22×D13—41no—
232C22×D13—41no—
233C22×D13—42no—
234C22×D13—42no—
235Q8×C13—22no—
236Q8×C13—22no—
237Q8×C13—41no—
238Q8×C13—41no—
239D4×C13—22no—
240D4×C13—22no—
241D4×C13—48no—
242D4×C13—41no—
243D4×C13—41no—
244C2×C4×C13—22no—
245C2×C4×C13—24no—
246C2×C4×C13—24no—
247C2×C4×C13—24no—
248C2×C4×C13—41no—
249C2×C4×C13—41no—
250C2×C4×C13—48no—
251C2×C4×C13—42no—
252C2×C4×C13—44no—
253C2×C4×C13—84no—
254C2×C4×C13C2×C22⋊C414yes—
255C2×C4×C13C4×D414yes—
256C2×C4×C13C4×D412yes—
257C2×C4×C13#12114yes—
258C2×C4×C13#12112yes—
259C2×C4×C13#12414yes—
260C2×C4×C13#12712yes—
261C2×C4×C13C23×C411yescharacteristic
262C2×C4×C13C22×D411yescharacteristic
263C2×C4×C13C22×D411yescharacteristic
264C2×C4×C13C22×D411yescharacteristic
265C2×C4×C13C2×#4111yescharacteristic
266C2×C4×C13C2×#4111yescharacteristic
267C22⋊D13—48no—
268C2×Dic13—22no—
269C2×Dic13—41no—
270C2×Dic13—41no—
271C2×Dic13—42no—
272C2×Dic13—44no—
273C2×Dic13—84no—
274D52—22no—
275D52—22no—
276D52—41no—
277D52—41no—
278C4×D13—48no—
279Dic26—22no—
280Dic26—22no—
281Dic26—41no—
282Dic26—41no—
283#1055—24no—
284#1055C2×C22⋊C414yes—
285#1055C2×C22⋊C414yes—
286C22×#41—131no—
287C22×#41C2×D1311yescharacteristic
288C2×#135—138no—
289C2×#135—264no—
290C2×#135—264no—
291C22×Q16—131no—
292C22×SD16—131no—
293C22×SD16—131no—
294C22×D8—131no—
295C22×M16—131no—
296C23×C8—131no—
297C2×#117C2×D1311yescharacteristic
298C22×C4⋊C4—131no—
299C22×C4⋊C4C2×D1311yescharacteristic
300C2×#106—132no—
301C2×#106—132no—
302C2×#106—264no—
303#417—264no—
304#417—264no—
305C2×#103—132no—
306C2×#103—132no—
307C2×#102—132no—
308C2×#102—132no—
309#405—264no—
310C2×#97—132no—
311C2×C4×C8—132no—
312#391—132no—
313#391C2×D1312yes—
314#390—132no—
315#390C2×D1312yes—
316#386—22no—
317#386—262no—
318#380—22no—
319#380—262no—
320C22×C13—22no—
321C22×C13—22no—
322C22×C13—24no—
323C22×C13—24no—
324C22×C13—44no—
325C22×C13—42no—
326C22×C13—81no—
327C22×C13—81no—
328C22×C13—82no—
329C22×C13#39014yes—
330C22×C13C2×C4×D411yescharacteristic
331C22×C13C2×#12111yescharacteristic
332C22×C13C2×#12711yescharacteristic
333C22×C13#57712yes—
334C22×C13#57712yes—
335C2×D13—44no—
336C2×D13—42no—
337C2×D13—81no—
338C2×D13—81no—
339C2×D13—82no—
340C4×C13—42no—
341C4×C13—44no—
342C4×C13—82no—
343C4×C13#39014yes—
344C4×C13C22×C22⋊C411yescharacteristic
345C4×C13C2×C4×D411yescharacteristic
346C4×C13C2×#12111yescharacteristic
347C4×C13C2×#12411yescharacteristic
348Dic13—42no—
349Dic13—82no—
350C2×#41—22no—
351C2×#41—22no—
352C2×#41—22no—
353C2×#41—22no—
354C2×#41—22no—
355C2×#41—134no—
356C2×#41—262no—
357C2×#41—262no—
358C2×#41—262no—
359C2×#41—262no—
360C2×#41—262no—
361C2×#41C4×D1314yes—
362C22×Q8—131no—
363C22×Q8C22×D1311yescharacteristic
364C22×D4—21no—
365C22×D4—131no—
366C22×D4—261no—
367C22×D4C22×D1311yescharacteristic
368C23×C4—21no—
369C23×C4—21no—
370C23×C4—261no—
371C23×C4—261no—
372C23×C4C22×D1311yescharacteristic
373#135—2632no—
374C2×Q16—134no—
375C2×Q16—262no—
376C2×Q16—262no—
377C2×SD16—134no—
378C2×SD16—134no—
379C2×SD16—262no—
380C2×SD16—262no—
381C2×SD16—262no—
382C2×SD16—262no—
383C2×D8—134no—
384C2×D8—262no—
385C2×D8—262no—
386C2×M16—134no—
387C2×M16—262no—
388C2×M16—262no—
389C22×C8—134no—
390C22×C8—134no—
391C22×C8—134no—
392C22×C8—131no—
393C22×C8—131no—
394C22×C8—132no—
395#117—22no—
396#117—22no—
397C2×C4⋊C4—22no—
398C2×C4⋊C4—22no—
399C2×C4⋊C4—21no—
400C2×C4⋊C4—132no—
401C2×C4⋊C4—262no—
402C2×C4⋊C4—262no—
403C2×C4⋊C4—261no—
404C2×C4⋊C4C22⋊D1312yes—
405C2×C4⋊C4C22×D1312yes—
406C2×C22⋊C4—24no—
407C2×C22⋊C4—264no—
408C2×C22⋊C4C22⋊D1312yes—
409C2×C42C22×D1312yes—
410#106—264no—
411#106—264no—
412#103—264no—
413#103—264no—
414#102—264no—
415#102—264no—
416#97—264no—
417C4×C8—264no—
418#95—24no—
419#95—264no—
420C2×C13—24no—
421C2×C13—82no—
422C2×C13C2×#39011yescharacteristic
423C2×C13#295212yes—
424C2×C13#306012yes—
425C2×C13#315411yescharacteristic
426C2×C13#315411yescharacteristic
427D13—82no—
428C24—21no—
429C24—261no—
430C24C2×C22⋊D1311yescharacteristic
431#41—28no—
432#41—28no—
433#41—44no—
434#41—44no—
435#41—268no—
436#41—268no—
437#41—524no—
438#41—524no—
439C2×Q8—21no—
440C2×Q8—21no—
441C2×Q8—21no—
442C2×Q8—21no—
443C2×Q8—21no—
444C2×Q8—132no—
445C2×Q8—261no—
446C2×Q8—261no—
447C2×Q8—261no—
448C2×Q8—261no—
449C2×Q8—261no—
450C2×Q8C2×C4×D1312yes—
451C2×D4—24no—
452C2×D4—21no—
453C2×D4—21no—
454C2×D4—21no—
455C2×D4—21no—
456C2×D4—21no—
457C2×D4—42no—
458C2×D4—42no—
459C2×D4—132no—
460C2×D4—264no—
461C2×D4—261no—
462C2×D4—261no—
463C2×D4—261no—
464C2×D4—261no—
465C2×D4—261no—
466C2×D4—522no—
467C2×D4—522no—
468C2×D4C2×C4×D1312yes—
469C22×C4—21no—
470C22×C4—24no—
471C22×C4—22no—
472C22×C4—42no—
473C22×C4—42no—
474C22×C4—42no—
475C22×C4—82no—
476C22×C4—261no—
477C22×C4—264no—
478C22×C4—262no—
479C22×C4—522no—
480C22×C4—522no—
481C22×C4—522no—
482C22×C4—1042no—
483C22×C4C2×C4×D1312yes—
484C22×C4#612712yes—
485C22×C4D4×D1312yes—
486C22×C4D4×D1312yes—
487C22×C4D4×D1312yes—
488C22×C4#612912yes—
489C22×C4C2×C22⋊D1314yes—
490C22×C4C2×C22⋊D1311yescharacteristic
491C22×C4C23×D1311yescharacteristic
492Q16—268no—
493SD16—268no—
494SD16—268no—
495D8—268no—
496M16—268no—
497C2×C8—1316no—
498C2×C8—132no—
499C2×C8—132no—
500C2×C8—132no—
501C2×C8—132no—
502C2×C8—132no—
503C2×C8—132no—
504C2×C8—134no—
505C2×C8—262no—
506C2×C8—262no—
507C4⋊C4—24no—
508C4⋊C4—22no—
509C4⋊C4—22no—
510C4⋊C4—42no—
511C4⋊C4—262no—
512C4⋊C4—262no—
513C4⋊C4—522no—
514C22⋊C4—24no—
515C22⋊C4—84no—
516C22⋊C4—1044no—
517C42—22no—
518C42—22no—
519C13#2122911yescharacteristic, complemented, Sylow
520C23—22no—
521C23—22no—
522C23—22no—
523C23—41no—
524C23—41no—
525C23—42no—
526C23—42no—
527C23—262no—
528C23—521no—
529C23—521no—
530C23—522no—
531C23—522no—
532C23C4×C22⋊D1312yes—
533C23#8819312yes—
534C23#8819412yes—
535C23C22×C22⋊D1311yescharacteristic
536Q8—22no—
537Q8—22no—
538Q8—41no—
539Q8—41no—
540Q8—262no—
541Q8—262no—
542Q8—521no—
543Q8—521no—
544D4—22no—
545D4—22no—
546D4—48no—
547D4—41no—
548D4—41no—
549D4—262no—
550D4—262no—
551D4—528no—
552D4—521no—
553D4—521no—
554C2×C4—22no—
555C2×C4—24no—
556C2×C4—24no—
557C2×C4—24no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—48no—
561C2×C4—42no—
562C2×C4—44no—
563C2×C4—84no—
564C2×C4—262no—
565C2×C4—521no—
566C2×C4—521no—
567C2×C4—528no—
568C2×C4—522no—
569C2×C4—524no—
570C2×C4—1044no—
571C2×C4C22⋊C4×D1314yes—
572C2×C4#8813514yes—
573C2×C4#8813814yes—
574C2×C4#8813914yes—
575C2×C4C4×C22⋊D1312yes—
576C2×C4#8819312yes—
577C2×C4#8819412yes—
578C2×C4C22×C4×D1311yescharacteristic
579C2×C4C2×#612711yescharacteristic
580C2×C4C2×D4×D1311yescharacteristic
581C2×C4C2×D4×D1311yescharacteristic
582C2×C4C2×D4×D1311yescharacteristic
583C2×C4C2×#612911yescharacteristic
584C8—134no—
585C8—134no—
586C8—264no—
587C22—24no—
588C22—24no—
589C22—22no—
590C22—22no—
591C22—42no—
592C22—44no—
593C22—82no—
594C22—81no—
595C22—81no—
596C22—522no—
597C22—524no—
598C22—1042no—
599C22—1041no—
600C22—1041no—
601C22#1171861814yes—
602C22C2×C4×C22⋊D1311yescharacteristic
603C22C2×#8819311yescharacteristic
604C22C2×#8819411yescharacteristic
605C22#1171954112yes—
606C22#1171954212yes—
607C4—42no—
608C4—44no—
609C4—82no—
610C4—522no—
611C4—1042no—
612C4#1171861814yes—
613C4C2×C22⋊C4×D1311yescharacteristic
614C4C2×#8813511yescharacteristic
615C4C2×#8813811yescharacteristic
616C4C2×#8813911yescharacteristic
617C2—24no—
618C2—82no—
619C2—1042no—
620C2C2×#1171861811yescharacteristic
621C2#1312371712yes—
622C2#1312449512yes—
623C2#1312530811yescharacteristic
624C2#1312531411yescharacteristic
subgroupG/H
centerC23C22×C22⋊D13
commutatorC2×C4×C13C23×C4
FrattiniC22×C4C23×D13
FittingC13×#2311C2
socleC23×C13C22×D4
prime radicalC22×#88204C2
cosocleC22×C4×C13C24
hypercenter#2311D13
hypocenterC13#21229
LCSG ▷ C2×C4×C13 ▷ C2×C13 ▷ C13
UCSC1 ◁ C23 ◁ C2×#117 ◁ #2311

Complements

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

NH ≅ G/Ntype
2. C22×#88048C2semidirect
3. C2×#11718893C2semidirect
4. C2×#11718627C2semidirect
5. C2×#11718658C2semidirect
6. #13117424C2semidirect
7. #13115711C2semidirect
8. #13115706C2semidirect
9. #13115693C2semidirect
10. #13115689C2semidirect
11. C13×#2311C2semidirect
18. C2×#88204C4semidirect
82. C2×#6107C2×C4semidirect
85. C2×#6106C2×C4semidirect
88. C2×#6105C2×C4semidirect
91. C2×#6104C2×C4semidirect
200. C2×#1055C22⋊C4semidirect
214. #2311D13semidirect
519. C13#21229semidirect

Maximal subgroups

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

subgroupindex
C22×#882042
C22×#880482
C2×#117188932
C2×#117186272
C2×#117186582
#131174242
#131157112
#131157062
#131156932
#131156892
C13×#23112
#2122913