Finite Group Databasebeta

← #15865327

Subgroups — #15865327

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×#11718782C211yesmaximal, characteristic, complemented
2C2×#11718794C211yesmaximal, characteristic
3C2×#11718794C211yesmaximal, characteristic
4#13117326C211yesmaximal, characteristic
5C2×C4×C4⋊Dic13C211yesmaximal, characteristic, complemented
6#13115478C212yesmaximal, complemented
7#13115471C212yesmaximal, complemented
8#13115464C212yesmaximal, complemented
9#13115457C212yesmaximal, complemented
10#13114556C211yesmaximal, characteristic, complemented
11C13×#1728C211yesmaximal, characteristic, complemented
12C22×#6127C2211yescharacteristic
13C22×C13×M16C2211yescharacteristic
14C22×C13×M16C2211yescharacteristic
15C22×C42×C13C2211yescharacteristic
16C2×#88180C2211yescharacteristic
17C22×C4⋊Dic13C2211yescharacteristic
18C22×C4⋊Dic13C2211yescharacteristic
19C22×C4×Dic13—21no—
20#11718874—21no—
21#11718874—21no—
22C2×#88071—21no—
23#11718794—24no—
24#11718794C418yescomplemented
25#11718782—24no—
26C2×#88061C2212yes—
27C2×#88061C2212yes—
28C2×#88058C2212yes—
29C2×#88057C2212yes—
30C2×#88056C2212yes—
31C2×#88056C2212yes—
32#11718600C2211yescharacteristic
33#11718600C2211yescharacteristic
34C4×#6103—21no—
35C4×#6103—21no—
36C4×C4⋊Dic13—21no—
37C4×C4⋊Dic13—21no—
38C4×C4⋊Dic13C2212yes—
39C4×C4⋊Dic13C2212yes—
40#11718590C2211yescharacteristic
41#11718590C2211yescharacteristic
42C13×#336C2212yes—
43C13×#336C2212yes—
44#11718143C2212yes—
45#11718143C2212yes—
46C23×C4×C13D411yescharacteristic
47C23×C4×C13D411yescharacteristic
48C23×C4×C13C2311yescharacteristic
49C22×C22⋊D13—21no—
50C23×Dic13—21no—
51C23×Dic13—21no—
52C2×#6127—22no—
53C2×#6127—22no—
54C2×#6127—22no—
55C2×#6127—22no—
56C2×#6127—22no—
57C2×#6127C2×C414yes—
58C22×D52C2311yescharacteristic
59C22×C4×D13—21no—
60C22×Dic26C2311yescharacteristic
61C2×C13×M16—22no—
62C2×C13×M16—22no—
63C2×C13×M16—22no—
64C2×C13×M16—22no—
65C2×C13×M16—22no—
66C2×C13×M16C2×C414yes—
67C22×C8×C13C2312yes—
68C22×C8×C13C2312yes—
69C2×C42×C13—22no—
70C2×C42×C13—22no—
71C2×C42×C13D411yescharacteristic
72C2×C42×C13D411yescharacteristic
73C2×C42×C13C2311yescharacteristic
74C2×C42×C13C2311yescharacteristic
75C2×C22⋊Dic13—21no—
76C2×#6103—21no—
77C2×#6103—21no—
78C2×#6103—21no—
79C2×#6103—21no—
80#88180—22no—
81#88180—22no—
82C2×C4⋊Dic13—21no—
83C2×C4⋊Dic13—21no—
84C2×C4⋊Dic13—22no—
85C2×C4⋊Dic13—22no—
86C2×C4⋊Dic13C2×C414yes—
87C2×C4⋊Dic13D414yes—
88C2×C4⋊Dic13C2311yescharacteristic
89C2×C4⋊Dic13C2311yescharacteristic
90C2×C4⋊Dic13C2312yes—
91C2×C4⋊Dic13C2312yes—
92C2×C4×Dic13—22no—
93C2×C4×Dic13—22no—
94C2×C4×Dic13—21no—
95C2×C4×Dic13—44no—
96#88071—21no—
97#88071—21no—
98#88071—21no—
99#88071—21no—
100#88071—22no—
101#88071—22no—
102#88061—24no—
103#88061C2×C418yescomplemented
104#88058—24no—
105#88057—24no—
106#88056—24no—
107#88056C2×C418yescomplemented
108#21059—131nomaximal, Sylow
109C24×C13C2×D411yescharacteristic
110C23×D13—21no—
111C22×C4×C13—22no—
112C22×C4×C13—22no—
113C22×C4×C13—22no—
114C22×C4×C13—22no—
115C22×C4×C13—41no—
116C22×C4×C13—41no—
117C22×C4×C13C22×C412yes—
118C22×C4×C13C2×D411yescharacteristic
119C22×C4×C13C2×D411yescharacteristic
120C22×C4×C13C2×D411yescharacteristic
121C22×C4×C13C2×D411yescharacteristic
122C22×C4×C13C2×D411yescharacteristic
123C22×C4×C13C2×D412yes—
124C22×C4×C13#4112yes—
125C22×C4×C13#4112yes—
126C22×C4×C13#4112yes—
127C22×C4×C13#4112yes—
128C22×C4×C13C2411yescharacteristic
129C2×C22⋊D13—24no—
130C2×C22⋊D13—42no—
131C2×C22⋊D13—42no—
132C22×Dic13—24no—
133C22×Dic13—24no—
134C22×Dic13—21no—
135C22×Dic13—22no—
136C22×Dic13—22no—
137C22×Dic13—21no—
138C22×Dic13—21no—
139C22×Dic13—42no—
140C22×Dic13—42no—
141C22×Dic13—42no—
142C22×Dic13—42no—
143#6127—28no—
144#6127—28no—
145#6127—44no—
146#6127—44no—
147C2×D52—21no—
148C2×D52—21no—
149C2×D52—21no—
150C2×D52—21no—
151C2×D52—21no—
152C2×D52C22×C412yes—
153C2×C4×D13—24no—
154C2×C4×D13—42no—
155C2×C4×D13—42no—
156C2×Dic26—21no—
157C2×Dic26—21no—
158C2×Dic26—21no—
159C2×Dic26—21no—
160C2×Dic26—21no—
161C2×Dic26C22×C412yes—
162C13×M16—28no—
163C13×M16—44no—
164C2×C8×C13—22no—
165C2×C8×C13—22no—
166C2×C8×C13—22no—
167C2×C8×C13—22no—
168C2×C8×C13—22no—
169C2×C8×C13C22×C414yes—
170C42×C13—22no—
171C42×C13—22no—
172C42×C13—21no—
173C42×C13—21no—
174C42×C13—21no—
175C42×C13—21no—
176C22⋊Dic13—42no—
177#6103—42no—
178#6103—42no—
179#6103—42no—
180#6103—42no—
181C4⋊Dic13—24no—
182C4⋊Dic13—24no—
183C4⋊Dic13—22no—
184C4⋊Dic13—22no—
185C4⋊Dic13—21no—
186C4⋊Dic13—21no—
187C4⋊Dic13—21no—
188C4⋊Dic13—21no—
189C4⋊Dic13C22×C414yes—
190C4⋊Dic13C2×D414yes—
191C4×Dic13—44no—
192C4×Dic13—42no—
193C4×Dic13—44no—
194C2×#428—131no—
195C2×#417—131no—
196C2×#417—131no—
197#2297—131no—
198C2×C4×C4⋊C4—131no—
199#1928—132no—
200#1927—132no—
201#1926—132no—
202#1925—132no—
203#1736—131no—
204#1728D1311yescharacteristic, complemented
205C23×C13—21no—
206C23×C13—21no—
207C23×C13—21no—
208C23×C13—21no—
209C23×C13C4×D412yes—
210C23×C13#12112yes—
211C23×C13#12312yes—
212C23×C13C22×D411yescharacteristic
213C22×D13—22no—
214C22×D13—41no—
215C22×D13—41no—
216C22×D13—41no—
217C22×D13—41no—
218C22×D13—41no—
219C22×D13—41no—
220C2×C4×C13—28no—
221C2×C4×C13—21no—
222C2×C4×C13—21no—
223C2×C4×C13—21no—
224C2×C4×C13—21no—
225C2×C4×C13—21no—
226C2×C4×C13—21no—
227C2×C4×C13—21no—
228C2×C4×C13—21no—
229C2×C4×C13—21no—
230C2×C4×C13—21no—
231C2×C4×C13—44no—
232C2×C4×C13#11714yes—
233C2×C4×C13C4×D414yes—
234C2×C4×C13C4×D412yes—
235C2×C4×C13#12112yes—
236C2×C4×C13#12312yes—
237C2×C4×C13#12414yes—
238C2×C4×C13#12614yes—
239C2×C4×C13AGL(1,ℤ/8)12yes—
240C2×C4×C13#13712yes—
241C2×C4×C13C23×C411yescharacteristic
242C2×C4×C13C22×D411yescharacteristic
243C2×C4×C13C2×#4111yescharacteristic
244C2×C4×C13C2×#4111yescharacteristic
245C2×C4×C13C2×#4111yescharacteristic
246C2×C4×C13C2×#4111yescharacteristic
247C22⋊D13—48no—
248C2×Dic13—24no—
249C2×Dic13—22no—
250C2×Dic13—24no—
251C2×Dic13—48no—
252C2×Dic13—48no—
253C2×Dic13—41no—
254C2×Dic13—41no—
255C2×Dic13—42no—
256C2×Dic13—42no—
257C2×Dic13—42no—
258C2×Dic13—42no—
259C2×Dic13—41no—
260C2×Dic13—41no—
261C2×Dic13—41no—
262C2×Dic13—41no—
263C2×Dic13—41no—
264C2×Dic13—41no—
265D52—22no—
266D52—22no—
267D52—41no—
268D52—41no—
269C4×D13—48no—
270Dic26—22no—
271Dic26—22no—
272Dic26—41no—
273Dic26—41no—
274C8×C13—24no—
275C8×C13—42no—
276C22×#41—131no—
277C22×M16C2×D1311yescharacteristic
278C22×M16C2×D1311yescharacteristic
279C2×#117—131no—
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#391—131no—
294#391—131no—
295#390—131no—
296#390—131no—
297#386—261no—
298#386—261no—
299C4×C4⋊C4—132no—
300C4×C4⋊C4—132no—
301C4×C4⋊C4—261no—
302C4×C4⋊C4—261no—
303C4×C22⋊C4—261no—
304C4×C22⋊C4—261no—
305C2×#95—261no—
306#340—132no—
307#340—132no—
308#336C2×D1312yes—
309#336C2×D1312yes—
310C22×C13—22no—
311C22×C13—22no—
312C22×C13—22no—
313C22×C13—22no—
314C22×C13—22no—
315C22×C13—22no—
316C22×C13#38814yes—
317C22×C13C2×C4×D411yescharacteristic
318C22×C13C2×#12111yescharacteristic
319C22×C13C2×#12311yescharacteristic
320C22×C13C2×AGL(1,ℤ/8)11yescharacteristic
321C22×C13C2×#13711yescharacteristic
322C22×C13#57511yescharacteristic
323C22×C13#57511yescharacteristic
324C2×D13—42no—
325C2×D13—42no—
326C2×D13—42no—
327C2×D13—81no—
328C2×D13—81no—
329C2×D13—81no—
330C2×D13—81no—
331C4×C13—22no—
332C4×C13—22no—
333C4×C13—41no—
334C4×C13—41no—
335C4×C13#38814yes—
336C4×C13C2×#11711yescharacteristic
337C4×C13C2×C4×D411yescharacteristic
338C4×C13C2×#12411yescharacteristic
339C4×C13C2×#12611yescharacteristic
340Dic13—44no—
341Dic13—44no—
342Dic13—42no—
343Dic13—81no—
344Dic13—81no—
345C2×#41—134no—
346C2×#41—262no—
347C2×#41—262no—
348C2×#41—262no—
349C2×#41—262no—
350C2×#41—262no—
351C22×Q8—131no—
352C22×D4—131no—
353C22×D4—261no—
354C23×C4—261no—
355C23×C4—261no—
356C23×C4—261no—
357C23×C4D5211yescharacteristic
358C23×C4D5211yescharacteristic
359C23×C4C22×D1311yescharacteristic
360C2×M16—22no—
361C2×M16—22no—
362C2×M16—22no—
363C2×M16—22no—
364C2×M16—22no—
365C2×M16C4×D1314yes—
366C22×C8C22×D1312yes—
367C22×C8C22×D1312yes—
368#117—262no—
369#117—262no—
370C2×C4⋊C4—134no—
371C2×C4⋊C4—134no—
372C2×C4⋊C4—132no—
373C2×C4⋊C4—132no—
374C2×C4⋊C4—131no—
375C2×C4⋊C4—131no—
376C2×C4⋊C4—261no—
377C2×C4⋊C4—261no—
378C2×C4⋊C4—262no—
379C2×C4⋊C4—262no—
380C2×C22⋊C4—261no—
381C2×C22⋊C4—261no—
382C2×C22⋊C4—261no—
383C2×C22⋊C4—261no—
384C2×C22⋊C4—261no—
385C2×C42—22no—
386C2×C42—22no—
387C2×C42—261no—
388C2×C42—262no—
389C2×C42—262no—
390C2×C42—524no—
391C2×C42D5211yescharacteristic
392C2×C42D5211yescharacteristic
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—262no—
402#95—262no—
403#95—261no—
404#95—261no—
405#95—261no—
406#95—261no—
407C2×C13—24no—
408C2×C13C2×#38811yescharacteristic
409C2×C13C4×AGL(1,ℤ/8)11yescharacteristic
410C2×C13C4×#13711yescharacteristic
411C2×C13#303611yescharacteristic
412C2×C13#303711yescharacteristic
413C2×C13#309211yescharacteristic
414C2×C13#309311yescharacteristic
415D13—82no—
416C24—261no—
417C24C2×D5211yescharacteristic
418#41—268no—
419#41—268no—
420#41—524no—
421#41—524no—
422C2×Q8—132no—
423C2×Q8—261no—
424C2×Q8—261no—
425C2×Q8—261no—
426C2×Q8—261no—
427C2×Q8—261no—
428C2×D4—132no—
429C2×D4—264no—
430C2×D4—261no—
431C2×D4—261no—
432C2×D4—261no—
433C2×D4—261no—
434C2×D4—261no—
435C2×D4—522no—
436C2×D4—522no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—41no—
442C22×C4—41no—
443C22×C4—264no—
444C22×C4—264no—
445C22×C4—261no—
446C22×C4—262no—
447C22×C4—262no—
448C22×C4—264no—
449C22×C4—261no—
450C22×C4—261no—
451C22×C4—522no—
452C22×C4—522no—
453C22×C4—522no—
454C22×C4—522no—
455C22×C4—522no—
456C22×C4—522no—
457C22×C4C2×C4×D1312yes—
458C22×C4C2×D5211yescharacteristic
459C22×C4C2×D5211yescharacteristic
460C22×C4C2×D5211yescharacteristic
461C22×C4C2×D5211yescharacteristic
462C22×C4C2×D5211yescharacteristic
463C22×C4#612712yes—
464C22×C4D4×D1312yes—
465C22×C4#612912yes—
466C22×C4#612912yes—
467C22×C4#613112yes—
468C22×C4C23×D1311yescharacteristic
469M16—28no—
470M16—44no—
471C2×C8—22no—
472C2×C8—22no—
473C2×C8—22no—
474C2×C8—22no—
475C2×C8—22no—
476C2×C8C2×C4×D1314yes—
477C4⋊C4—134no—
478C4⋊C4—134no—
479C4⋊C4—264no—
480C4⋊C4—264no—
481C4⋊C4—262no—
482C4⋊C4—262no—
483C4⋊C4—261no—
484C4⋊C4—261no—
485C4⋊C4—261no—
486C4⋊C4—261no—
487C22⋊C4—522no—
488C22⋊C4—522no—
489C22⋊C4—522no—
490C22⋊C4—522no—
491C22⋊C4—522no—
492C42—22no—
493C42—22no—
494C42—21no—
495C42—21no—
496C42—21no—
497C42—21no—
498C42—524no—
499C42—522no—
500C42—524no—
501C13#2105911yescharacteristic, complemented, Sylow
502C23—21no—
503C23—21no—
504C23—21no—
505C23—21no—
506C23—262no—
507C23—521no—
508C23—521no—
509C23—521no—
510C23—521no—
511C23—521no—
512C23—521no—
513C23C4×D5212yes—
514C23#8814012yes—
515C23#8814912yes—
516C23C22×D5211yescharacteristic
517Q8—262no—
518Q8—262no—
519Q8—521no—
520Q8—521no—
521D4—262no—
522D4—262no—
523D4—528no—
524D4—521no—
525D4—521no—
526C2×C4—28no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—44no—
538C2×C4—264no—
539C2×C4—262no—
540C2×C4—264no—
541C2×C4—528no—
542C2×C4—528no—
543C2×C4—522no—
544C2×C4—522no—
545C2×C4—521no—
546C2×C4—522no—
547C2×C4—522no—
548C2×C4—521no—
549C2×C4—528no—
550C2×C4—521no—
551C2×C4—521no—
552C2×C4—521no—
553C2×C4—521no—
554C2×C4—521no—
555C2×C4—521no—
556C2×C4C4×D5212yes—
557C2×C4#8813514yes—
558C2×C4#8813914yes—
559C2×C4#8814012yes—
560C2×C4#8814614yes—
561C2×C4#8814912yes—
562C2×C4#8815214yes—
563C2×C4#8816212yes—
564C2×C4#8816312yes—
565C2×C4C22×C4×D1311yescharacteristic
566C2×C4C2×#612711yescharacteristic
567C2×C4C2×D4×D1311yescharacteristic
568C2×C4C2×#612911yescharacteristic
569C2×C4C2×#612911yescharacteristic
570C2×C4C2×#613111yescharacteristic
571C8—24no—
572C8—42no—
573C22—22no—
574C22—22no—
575C22—22no—
576C22—22no—
577C22—22no—
578C22—22no—
579C22—522no—
580C22—522no—
581C22—522no—
582C22—1041no—
583C22—1041no—
584C22—1041no—
585C22—1041no—
586C22#1171833114yes—
587C22C2×C4×D5211yescharacteristic
588C22C2×#8814011yescharacteristic
589C22C2×#8814911yescharacteristic
590C22C2×#8816211yescharacteristic
591C22C2×#8816311yescharacteristic
592C22#1171945311yescharacteristic
593C22#1171945311yescharacteristic
594C4—22no—
595C4—22no—
596C4—41no—
597C4—41no—
598C4—524no—
599C4—522no—
600C4—524no—
601C4—1041no—
602C4—1041no—
603C4#1171833114yes—
604C4C2×#8813511yescharacteristic
605C4C2×#8813911yescharacteristic
606C4C2×#8814611yescharacteristic
607C4C2×#8815211yescharacteristic
608C2—24no—
609C2—1042no—
610C2C2×#1171833111yescharacteristic
611C2C4×#8816211yescharacteristic
612C2C4×#8816311yescharacteristic
613C2#1312430211yescharacteristic
614C2#1312431011yescharacteristic
615C2#1312476111yescharacteristic
616C2#1312476911yescharacteristic
subgroupG/H
centerC23C22×D52
commutatorC2×C4×C13C23×C4
FrattiniC22×C4C23×D13
FittingC13×#1728C2
socleC23×C13C22×D4
prime radicalC22×#6127C22
cosocleC22×C4×C13C24
hypercenter#1728D13
hypocenterC13#21059
LCSG ▷ C2×C4×C13 ▷ C2×C13 ▷ C13
UCSC1 ◁ C23 ◁ C22×C42 ◁ #1728

Complements

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

NH ≅ G/Ntype
1. C2×#11718782C2semidirect
5. C2×C4×C4⋊Dic13C2semidirect
6. #13115478C2semidirect
7. #13115471C2semidirect
8. #13115464C2semidirect
9. #13115457C2semidirect
10. #13114556C2semidirect
11. C13×#1728C2semidirect
24. #11718794C4semidirect
103. #88061C2×C4semidirect
107. #88056C2×C4semidirect
204. #1728D13semidirect
501. C13#21059semidirect

Maximal subgroups

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

subgroupindex
C2×#117187822
C2×#117187942
#131173262
C2×C4×C4⋊Dic132
#131154782
#131154712
#131154642
#131154572
#131145562
C13×#17282
#2105913