Finite Group Databasebeta

← #15793659

Subgroups — #15793659

Order 3328.

1326 conjugacy classes of subgroups.

The subgroup lattice has 638 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11718654C211yesmaximal, characteristic
2C2×#11718646C211yesmaximal, characteristic, complemented
3C2×#11718330C211yesmaximal, characteristic
4#13113100C212yesmaximal, complemented
5#13113100C211yesmaximal, characteristic, complemented
6#13113097C212yesmaximal, complemented
7#13113082C211yesmaximal, characteristic
8#13113082C212yesmaximal
9#13113080C212yesmaximal
10#13113061C211yesmaximal, characteristic, complemented
11C13×#1445C211yesmaximal, characteristic, complemented
12C22×C13×C4⋊C4C2211yescharacteristic
13C22×#6103C2211yescharacteristic
14C22×#6103C2211yescharacteristic
15C22×#6103C2211yescharacteristic
16C22×#6103C2212yes—
17C22×#6103C2212yes—
18C22×C4⋊Dic13C2211yescharacteristic
19C22×C4⋊Dic13C2212yes—
20C22×#6101C2211yescharacteristic
21C22×#6101C2212yes—
22C22×C4×Dic13C2211yescharacteristic
23C2×C13×#95C2211yescharacteristic
24C2×C13×#95C2211yescharacteristic
25C2×C13×#95C2212yes—
26C2×C13×#95C2212yes—
27C2×#88071C2211yescharacteristic
28C2×#88071C2212yes—
29C2×#88071C2212yes—
30C2×#88071C2212yes—
31C2×#88071C2212yes—
32C2×#88071C2212yes—
33C2×#88071C2212yes—
34C2×#88071C2211yescharacteristic
35#11718654C418yescomplemented
36#11718646—24no—
37#11718330—24no—
38C24×D13C2311yescharacteristic
39C23×C4×C13C2311yescharacteristic
40C23×C4×C13C2311yescharacteristic
41C23×C4×C13C2312yes—
42C23×C4×C13C2312yes—
43C23×C4×C13C2311yescharacteristic
44C23×Dic13C2312yes—
45C23×Dic13C2311yescharacteristic
46C23×Dic13C2312yes—
47C23×Dic13C2311yescharacteristic
48C23×Dic13C2311yescharacteristic
49C2×C13×C4⋊C4—22no—
50C2×C13×C4⋊C4—22no—
51C2×C13×C4⋊C4C2×C414yes—
52C2×#6103—22no—
53C2×#6103—22no—
54C2×#6103—22no—
55C2×#6103—22no—
56C2×#6103—42no—
57C2×#6103—42no—
58C2×#6103—42no—
59C2×#6103—42no—
60C2×#6103—42no—
61C2×#6103—42no—
62C2×#6103—42no—
63C2×#6103C2×C414yes—
64C2×#6103C2×C418yes—
65C2×C4⋊Dic13—22no—
66C2×C4⋊Dic13—24no—
67C2×C4⋊Dic13—42no—
68C2×C4⋊Dic13—42no—
69C2×C4⋊Dic13—42no—
70C2×#6101—22no—
71C2×#6101—42no—
72C2×#6101—42no—
73C2×#6101—42no—
74C2×#6101—42no—
75C2×C4×Dic13—42no—
76C2×C4×Dic13D414yes—
77C13×#95—22no—
78C13×#95—22no—
79C13×#95—42no—
80C13×#95—42no—
81#88071—22no—
82#88071—24no—
83#88071—42no—
84#88071—42no—
85#88071—42no—
86#88071—42no—
87#88071C2×C418yes—
88#88071C2×C414yes—
89#14630—131nomaximal, Sylow
90C24×C13C2411yescharacteristic
91C23×D13—21no—
92C23×D13—21no—
93C23×D13—21no—
94C23×D13—21no—
95C23×D13—21no—
96C23×D13—22no—
97C23×D13—22no—
98C23×D13—22no—
99C23×D13—22no—
100C23×D13—21no—
101C23×D13C22×C412yes—
102C22×C4×C13—22no—
103C22×C4×C13—22no—
104C22×C4×C13—22no—
105C22×C4×C13—22no—
106C22×C4×C13—22no—
107C22×C4×C13—22no—
108C22×C4×C13—22no—
109C22×C4×C13—22no—
110C22×C4×C13—21no—
111C22×C4×C13—21no—
112C22×C4×C13—22no—
113C22×C4×C13—22no—
114C22×C4×C13—22no—
115C22×C4×C13—22no—
116C22×C4×C13—22no—
117C22×C4×C13—21no—
118C22×C4×C13—21no—
119C22×C4×C13—22no—
120C22×C4×C13—22no—
121C22×C4×C13—21no—
122C22×C4×C13—22no—
123C22×C4×C13—22no—
124C22×C4×C13—22no—
125C22×C4×C13—21no—
126C22×C4×C13—22no—
127C22×C4×C13C22×C414yes—
128C22×C4×C13C22×C412yes—
129C22×C4×C13C2×D412yes—
130C22×C4×C13#4112yes—
131C22×Dic13—21no—
132C22×Dic13—21no—
133C22×Dic13—21no—
134C22×Dic13—22no—
135C22×Dic13—22no—
136C22×Dic13—22no—
137C22×Dic13—22no—
138C22×Dic13—22no—
139C22×Dic13—22no—
140C22×Dic13—22no—
141C22×Dic13—22no—
142C22×Dic13—21no—
143C22×Dic13—21no—
144C22×Dic13—22no—
145C22×Dic13—22no—
146C22×Dic13—22no—
147C22×Dic13—22no—
148C22×Dic13—21no—
149C22×Dic13—22no—
150C22×Dic13—22no—
151C22×Dic13—22no—
152C22×Dic13—22no—
153C22×Dic13—22no—
154C22×Dic13—22no—
155C22×Dic13—22no—
156C22×Dic13C22×C412yes—
157C22×Dic13C22×C414yes—
158C22×Dic13C2×D412yes—
159C22×Dic13C2×D412yes—
160C13×C4⋊C4—28no—
161#6103—28no—
162#6103—48no—
163#6103—44no—
164#6103—44no—
165#6103—48no—
166C4⋊Dic13—48no—
167C4⋊Dic13—44no—
168#6101—44no—
169#6101—48no—
170C4×Dic13—44no—
171C2×#397—131no—
172C2×#389—131no—
173C2×#388—131no—
174#1445D1311yescharacteristic, complemented
175#1444—132no—
176#1444—131no—
177#1443—132no—
178#1439—132no—
179#1439—131no—
180#1439—132no—
181#1434—131no—
182C23×C13C23×C411yescharacteristic
183C23×C13C22×D411yescharacteristic
184C23×C13C2×#4111yescharacteristic
185C23×C13H(5,𝔽2)11yescharacteristic
186C23×C13H(5,𝔽2)11yescharacteristic
187C23×C13H(5,𝔽2)12yes—
188C23×C13H(5,𝔽2)11yescharacteristic
189C23×C13H(5,𝔽2)12yes—
190C23×C132−1+412yes—
191C23×C132−1+412yes—
192C23×C132−1+411yescharacteristic
193C22×D13—22no—
194C22×D13—22no—
195C22×D13—24no—
196C22×D13—22no—
197C22×D13—24no—
198C22×D13—41no—
199C22×D13—41no—
200C22×D13—41no—
201C22×D13—41no—
202C22×D13—41no—
203C22×D13—41no—
204C22×D13—42no—
205C22×D13—41no—
206C22×D13—41no—
207C22×D13—42no—
208C22×D13—42no—
209C22×D13—42no—
210C22×D13—42no—
211C22×D13—42no—
212C22×D13—42no—
213C22×D13—42no—
214C22×D13—42no—
215C22×D13—42no—
216C2×C4×C13—24no—
217C2×C4×C13—22no—
218C2×C4×C13—22no—
219C2×C4×C13—24no—
220C2×C4×C13—24no—
221C2×C4×C13—24no—
222C2×C4×C13—24no—
223C2×C4×C13—42no—
224C2×C4×C13—42no—
225C2×C4×C13—42no—
226C2×C4×C13—42no—
227C2×C4×C13—41no—
228C2×C4×C13—41no—
229C2×C4×C13—41no—
230C2×C4×C13—41no—
231C2×C4×C13—41no—
232C2×C4×C13—41no—
233C2×C4×C13—42no—
234C2×C4×C13—42no—
235C2×C4×C13—42no—
236C2×C4×C13—42no—
237C2×C4×C13—42no—
238C2×C4×C13—42no—
239C2×C4×C13—42no—
240C2×C4×C13—42no—
241C2×C4×C13—42no—
242C2×C4×C13—42no—
243C2×C4×C13—42no—
244C2×C4×C13C4×D414yes—
245C2×Dic13—22no—
246C2×Dic13—24no—
247C2×Dic13—24no—
248C2×Dic13—24no—
249C2×Dic13—24no—
250C2×Dic13—22no—
251C2×Dic13—22no—
252C2×Dic13—22no—
253C2×Dic13—22no—
254C2×Dic13—22no—
255C2×Dic13—22no—
256C2×Dic13—41no—
257C2×Dic13—41no—
258C2×Dic13—42no—
259C2×Dic13—42no—
260C2×Dic13—42no—
261C2×Dic13—42no—
262C2×Dic13—42no—
263C2×Dic13—42no—
264C2×Dic13—42no—
265C2×Dic13—42no—
266C2×Dic13—42no—
267C2×Dic13—42no—
268C2×Dic13—42no—
269C2×Dic13—42no—
270C2×Dic13—42no—
271C2×Dic13—42no—
272C2×Dic13—42no—
273C2×Dic13—42no—
274C22×C4⋊C4—131no—
275C22×C4⋊C4—132no—
276C22×C4⋊C4—131no—
277C22×C4⋊C4—132no—
278C22×C4⋊C4C2×D1311yescharacteristic
279C22×C22⋊C4—131no—
280C22×C22⋊C4—132no—
281C22×C22⋊C4—131no—
282C22×C22⋊C4—131no—
283C22×C22⋊C4—132no—
284C22×C42—131no—
285#397—138no—
286#389—264no—
287#388—264no—
288C2×#95—131no—
289C2×#95—132no—
290C2×#95—132no—
291C2×#95—131no—
292C2×#95—132no—
293C2×#95—132no—
294C2×#95—132no—
295C2×#95—132no—
296C2×#95C2×D1311yescharacteristic
297C2×#95C2×D1311yescharacteristic
298C2×#95C2×D1312yes—
299C2×#95C2×D1312yes—
300C22×C13C2×C4×D411yescharacteristic
301C22×C13#51811yescharacteristic
302C22×C13#52012yes—
303C22×C13#52011yescharacteristic
304C22×C13#52012yes—
305C22×C13#53411yescharacteristic
306C22×C13#53511yescharacteristic
307C22×C13#53511yescharacteristic
308C22×C13#53611yescharacteristic
309C22×C13#53611yescharacteristic
310C22×C13#53711yescharacteristic
311C22×C13#53912yes—
312C22×C13#54011yescharacteristic
313C22×C13#54212yes—
314C22×C13#54211yescharacteristic
315C22×C13#56012yes—
316C22×C13#56012yes—
317C22×C13#56212yes—
318C22×C13#56212yes—
319C22×C13#56312yes—
320C22×C13#56312yes—
321C22×C13#56312yes—
322C22×C13#56312yes—
323C2×D13—42no—
324C2×D13—42no—
325C2×D13—42no—
326C2×D13—44no—
327C2×D13—44no—
328C2×D13—81no—
329C2×D13—81no—
330C2×D13—82no—
331C2×D13—82no—
332C2×D13—81no—
333C2×D13—81no—
334C4×C13—24no—
335C4×C13—44no—
336C4×C13—81no—
337C4×C13—82no—
338C4×C13—81no—
339Dic13—42no—
340Dic13—44no—
341Dic13—42no—
342Dic13—42no—
343Dic13—82no—
344C25—131no—
345C23×C4—131no—
346C23×C4—131no—
347C23×C4—132no—
348C23×C4—131no—
349C23×C4—132no—
350C23×C4C22×D1311yescharacteristic
351C23×C4C22×D1311yescharacteristic
352C23×C4C22×D1312yes—
353C23×C4C22×D1312yes—
354C23×C4C22×D1311yescharacteristic
355C2×C4⋊C4—22no—
356C2×C4⋊C4—22no—
357C2×C4⋊C4—262no—
358C2×C4⋊C4—264no—
359C2×C4⋊C4—262no—
360C2×C4⋊C4—522no—
361C2×C4⋊C4—522no—
362C2×C4⋊C4—522no—
363C2×C4⋊C4—522no—
364C2×C4⋊C4—522no—
365C2×C4⋊C4—522no—
366C2×C4⋊C4—522no—
367C2×C4⋊C4C4×D1314yes—
368C2×C22⋊C4—134no—
369C2×C22⋊C4—138no—
370C2×C22⋊C4—262no—
371C2×C22⋊C4—262no—
372C2×C22⋊C4—262no—
373C2×C22⋊C4—262no—
374C2×C22⋊C4—522no—
375C2×C22⋊C4—522no—
376C2×C22⋊C4—522no—
377C2×C22⋊C4—522no—
378C2×C22⋊C4—522no—
379C2×C22⋊C4—522no—
380C2×C22⋊C4—522no—
381C2×C42—134no—
382C2×C42—522no—
383#95—22no—
384#95—22no—
385#95—42no—
386#95—42no—
387#95—138no—
388#95—134no—
389#95—262no—
390#95—264no—
391#95—522no—
392#95—522no—
393#95—522no—
394#95—522no—
395C2×C13#232211yescharacteristic
396C2×C13#232411yescharacteristic
397C2×C13#232611yescharacteristic
398C2×C13#237612yes—
399C2×C13#238312yes—
400C2×C13#281411yescharacteristic
401C2×C13#281511yescharacteristic
402C2×C13#282011yescharacteristic
403C2×C13#282111yescharacteristic
404C2×C13#283712yes—
405C2×C13#283812yes—
406D13—82no—
407C24—132no—
408C24—261no—
409C24—261no—
410C24—261no—
411C24—261no—
412C24—262no—
413C24—262no—
414C24—261no—
415C24—262no—
416C24—262no—
417C24—261no—
418C24C23×D1311yescharacteristic
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—22no—
430C22×C4—21no—
431C22×C4—21no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—21no—
440C22×C4—22no—
441C22×C4—21no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—132no—
445C22×C4—134no—
446C22×C4—132no—
447C22×C4—132no—
448C22×C4—261no—
449C22×C4—261no—
450C22×C4—261no—
451C22×C4—262no—
452C22×C4—261no—
453C22×C4—262no—
454C22×C4—262no—
455C22×C4—262no—
456C22×C4—262no—
457C22×C4—262no—
458C22×C4—262no—
459C22×C4—262no—
460C22×C4—262no—
461C22×C4—262no—
462C22×C4—262no—
463C22×C4—261no—
464C22×C4—262no—
465C22×C4—261no—
466C22×C4—262no—
467C22×C4—262no—
468C22×C4—262no—
469C22×C4—262no—
470C22×C4—262no—
471C22×C4—262no—
472C22×C4—262no—
473C22×C4C2×C4×D1314yes—
474C22×C4C2×C4×D1312yes—
475C22×C4D4×D1312yes—
476C22×C4#612912yes—
477C4⋊C4—28no—
478C4⋊C4—524no—
479C4⋊C4—524no—
480C4⋊C4—528no—
481C4⋊C4—528no—
482C22⋊C4—268no—
483C22⋊C4—528no—
484C22⋊C4—524no—
485C22⋊C4—524no—
486C22⋊C4—528no—
487C42—524no—
488C13#1463011yescharacteristic, complemented, Sylow
489C23—262no—
490C23—264no—
491C23—262no—
492C23—262no—
493C23—264no—
494C23—521no—
495C23—521no—
496C23—521no—
497C23—521no—
498C23—522no—
499C23—522no—
500C23—522no—
501C23—521no—
502C23—521no—
503C23—522no—
504C23—522no—
505C23—522no—
506C23—521no—
507C23—521no—
508C23—522no—
509C23—522no—
510C23—522no—
511C23—522no—
512C23C22×C4×D1311yescharacteristic
513C23C2×D4×D1311yescharacteristic
514C23C2×#612911yescharacteristic
515C23#8825111yescharacteristic
516C23#8825111yescharacteristic
517C23#8825412yes—
518C23#8825612yes—
519C23#8825611yescharacteristic
520C23#8825612yes—
521C23#8825712yes—
522C23#8825711yescharacteristic
523C2×C4—24no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—24no—
527C2×C4—24no—
528C2×C4—24no—
529C2×C4—24no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—262no—
552C2×C4—262no—
553C2×C4—264no—
554C2×C4—264no—
555C2×C4—264no—
556C2×C4—264no—
557C2×C4—262no—
558C2×C4—262no—
559C2×C4—262no—
560C2×C4—262no—
561C2×C4—262no—
562C2×C4—521no—
563C2×C4—522no—
564C2×C4—522no—
565C2×C4—522no—
566C2×C4—522no—
567C2×C4—522no—
568C2×C4—521no—
569C2×C4—522no—
570C2×C4—522no—
571C2×C4—522no—
572C2×C4—522no—
573C2×C4—522no—
574C2×C4—522no—
575C2×C4—522no—
576C2×C4—522no—
577C2×C4—522no—
578C2×C4—522no—
579C2×C4—522no—
580C2×C4#8813514yes—
581C22—524no—
582C22—522no—
583C22—524no—
584C22—522no—
585C22—522no—
586C22—1041no—
587C22—1041no—
588C22—1041no—
589C22—1041no—
590C22—1042no—
591C22—1042no—
592C22C2×#8813511yescharacteristic
593C22#1171922812yes—
594C22#1171925011yescharacteristic
595C22#1171925111yescharacteristic
596C22#1171927912yes—
597C22#1171931211yescharacteristic
598C22#1171931311yescharacteristic
599C22#1171934812yes—
600C22#1171935112yes—
601C22#1171935212yes—
602C22#1171935311yescharacteristic
603C22#1171935811yescharacteristic
604C22#1171937012yes—
605C22#1171937112yes—
606C22#1171937212yes—
607C22#1171937911yescharacteristic
608C22#1171938711yescharacteristic
609C22#1171940412yes—
610C22#1171940412yes—
611C22#1171941612yes—
612C22#1171941712yes—
613C22#1171942411yescharacteristic
614C22#1171942911yescharacteristic
615C4—24no—
616C4—44no—
617C4—81no—
618C4—81no—
619C4—82no—
620C4—524no—
621C4—522no—
622C4—522no—
623C4—522no—
624C4—1042no—
625C2—1042no—
626C2#1311750211yescharacteristic
627C2#1311753211yescharacteristic
628C2#1311755011yescharacteristic
629C2#1311798312yes—
630C2#1311802612yes—
631C2#1312285211yescharacteristic
632C2#1312285911yescharacteristic
633C2#1312289211yescharacteristic
634C2#1312289611yescharacteristic
635C2#1312305012yes—
636C2#1312306512yes—
subgroupG/H
center, FrattiniC24C23×D13
commutatorC23×C13C23×C4
FittingC13×#1445C2
socle, cosocleC24×C13C24
prime radicalC24×D13C23
hypercenter#1445D13
hypocenterC13#14630
LCSG ▷ C23×C13 ▷ C13
UCSC1 ◁ C24 ◁ #1445

Complements

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

NH ≅ G/Ntype
2. C2×#11718646C2semidirect
4. #13113100C2semidirect
5. #13113100C2semidirect
6. #13113097C2semidirect
10. #13113061C2semidirect
11. C13×#1445C2semidirect
35. #11718654C4semidirect
174. #1445D13semidirect
488. C13#14630semidirect

Maximal subgroups

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

subgroupindex
C2×#117186542
C2×#117186462
C2×#117183302
#131131002
#131130972
#131130822
#131130802
#131130612
C13×#14452
#1463013