Finite Group Databasebeta

← #15789888

Subgroups — #15789888

Order 3328.

1558 conjugacy classes of subgroups.

The subgroup lattice has 656 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11718338C211yesmaximal, characteristic
2C2×#11718634C211yesmaximal, characteristic, complemented
3C2×#11718327C211yesmaximal, characteristic
4#13113105C211yesmaximal, characteristic, complemented
5#13113088C212yesmaximal, complemented
6#13113080C212yesmaximal
7#13113078C211yesmaximal, characteristic
8#13113070C212yesmaximal, complemented
9#13113061C211yesmaximal, characteristic, complemented
10#13113057C212yesmaximal
11C13×#1442C211yesmaximal, characteristic, complemented
12C23×C4×D13C2211yescharacteristic
13C22×C13×C4⋊C4C2211yescharacteristic
14C22×#6103C2211yescharacteristic
15C22×#6103C2211yescharacteristic
16C22×#6103C2212yes—
17C22×#6103C2212yes—
18C22×C4⋊Dic13C2211yescharacteristic
19C22×C4⋊Dic13C2212yes—
20C22×#6101C2212yes—
21C22×C4×Dic13C2211yescharacteristic
22C22×C4×Dic13C2211yescharacteristic
23C2×C13×#95C2211yescharacteristic
24C2×C13×#95C2212yes—
25C2×C13×#95C2212yes—
26C2×C13×#95C2211yescharacteristic
27C2×#88071C2211yescharacteristic
28C2×#88071C2211yescharacteristic
29C2×#88071C2212yes—
30C2×#88071C2212yes—
31C2×#88071C2212yes—
32C2×#88071C2212yes—
33C2×#88071C2212yes—
34C2×#88071C2212yes—
35#11718634—24no—
36#11718338C418yescomplemented
37#11718327—24no—
38C24×D13C2311yescharacteristic
39C23×C4×C13C2312yes—
40C23×C4×C13C2312yes—
41C23×C4×C13C2311yescharacteristic
42C23×C4×C13C2311yescharacteristic
43C23×C4×C13C2311yescharacteristic
44C23×Dic13C2312yes—
45C23×Dic13C2311yescharacteristic
46C23×Dic13C2312yes—
47C23×Dic13C2311yescharacteristic
48C23×Dic13C2311yescharacteristic
49C22×C4×D13—22no—
50C22×C4×D13—22no—
51C22×C4×D13—42no—
52C22×C4×D13—42no—
53C22×C4×D13Q814yes—
54C2×C13×C4⋊C4—22no—
55C2×C13×C4⋊C4—42no—
56C2×#6103—22no—
57C2×#6103—22no—
58C2×#6103—22no—
59C2×#6103—22no—
60C2×#6103—22no—
61C2×#6103—24no—
62C2×#6103—42no—
63C2×#6103—42no—
64C2×#6103—42no—
65C2×#6103—42no—
66C2×#6103C2×C414yes—
67C2×#6103C2×C418yes—
68C2×C4⋊Dic13—22no—
69C2×C4⋊Dic13—22no—
70C2×C4⋊Dic13—42no—
71C2×C4⋊Dic13—42no—
72C2×C4⋊Dic13C2×C414yescomplemented
73C2×C4⋊Dic13C2×C418yescomplemented
74C2×#6101—24no—
75C2×#6101—42no—
76C2×#6101—42no—
77C2×C4×Dic13—22no—
78C2×C4×Dic13—42no—
79C2×C4×Dic13—42no—
80C2×C4×Dic13D414yes—
81C13×#95—22no—
82C13×#95—24no—
83C13×#95—42no—
84C13×#95C2×C414yescomplemented
85#88071—22no—
86#88071—22no—
87#88071—24no—
88#88071—24no—
89#88071—42no—
90#88071—42no—
91#88071—42no—
92#88071—42no—
93#14317—131nomaximal, Sylow
94C24×C13C2411yescharacteristic
95C23×D13—21no—
96C23×D13—21no—
97C23×D13—21no—
98C23×D13—21no—
99C23×D13—22no—
100C23×D13—22no—
101C23×D13—22no—
102C23×D13—22no—
103C23×D13C22×C412yes—
104C23×D13C2×Q812yes—
105C23×D13#4112yes—
106C22×C4×C13—22no—
107C22×C4×C13—22no—
108C22×C4×C13—21no—
109C22×C4×C13—21no—
110C22×C4×C13—22no—
111C22×C4×C13—22no—
112C22×C4×C13—21no—
113C22×C4×C13—21no—
114C22×C4×C13—22no—
115C22×C4×C13—22no—
116C22×C4×C13—22no—
117C22×C4×C13—22no—
118C22×C4×C13—22no—
119C22×C4×C13—22no—
120C22×C4×C13—22no—
121C22×C4×C13—22no—
122C22×C4×C13—22no—
123C22×C4×C13—22no—
124C22×C4×C13—22no—
125C22×C4×C13—22no—
126C22×C4×C13—22no—
127C22×C4×C13—22no—
128C22×C4×C13—22no—
129C22×C4×C13C22×C414yes—
130C22×C4×C13C22×C412yes—
131C22×C4×C13C2×D412yes—
132C22×C4×C13C2×Q812yes—
133C22×C4×C13#4112yes—
134C22×C4×C13#4112yes—
135C22×Dic13—22no—
136C22×Dic13—22no—
137C22×Dic13—21no—
138C22×Dic13—22no—
139C22×Dic13—22no—
140C22×Dic13—21no—
141C22×Dic13—21no—
142C22×Dic13—21no—
143C22×Dic13—22no—
144C22×Dic13—22no—
145C22×Dic13—22no—
146C22×Dic13—22no—
147C22×Dic13—22no—
148C22×Dic13—22no—
149C22×Dic13—22no—
150C22×Dic13—22no—
151C22×Dic13—22no—
152C22×Dic13—22no—
153C22×Dic13—22no—
154C22×Dic13—22no—
155C22×Dic13—22no—
156C22×Dic13—22no—
157C22×Dic13—22no—
158C22×Dic13C22×C414yes—
159C22×Dic13C22×C412yes—
160C22×Dic13C2×D412yes—
161C22×Dic13C2×D412yes—
162C22×Dic13C2×Q812yes—
163C22×Dic13#4112yes—
164C2×C4×D13—28no—
165C2×C4×D13—48no—
166C2×C4×D13—44no—
167C2×C4×D13—44no—
168C2×C4×D13—48no—
169C13×C4⋊C4—44no—
170#6103—28no—
171#6103—28no—
172#6103—48no—
173#6103—48no—
174C4⋊Dic13—28no—
175C4⋊Dic13—48no—
176#6101—48no—
177C4×Dic13—44no—
178C4×Dic13—44no—
179C2×#399—131no—
180C2×#389—131no—
181C2×C4×C22⋊C4—131no—
182#1445—131no—
183#1442D1311yescharacteristic, complemented
184#1441—132no—
185#1439—132no—
186#1439—131no—
187#1436—132no—
188#1434—131no—
189C22⋊#95—132no—
190C23×C13C23×C411yescharacteristic
191C23×C13C22×D411yescharacteristic
192C23×C13C22×Q811yescharacteristic
193C23×C13C2×#4111yescharacteristic
194C23×C13C2×#4111yescharacteristic
195C23×C13C2×#4111yescharacteristic
196C23×C13C2×#4111yescharacteristic
197C23×C13H(5,𝔽2)12yes—
198C23×C13H(5,𝔽2)12yes—
199C23×C13H(5,𝔽2)12yes—
200C23×C132−1+412yes—
201C22×D13—22no—
202C22×D13—22no—
203C22×D13—24no—
204C22×D13—24no—
205C22×D13—22no—
206C22×D13—22no—
207C22×D13—22no—
208C22×D13—22no—
209C22×D13—24no—
210C22×D13—24no—
211C22×D13—24no—
212C22×D13—24no—
213C22×D13—41no—
214C22×D13—41no—
215C22×D13—41no—
216C22×D13—41no—
217C22×D13—42no—
218C22×D13—42no—
219C22×D13—42no—
220C22×D13—42no—
221C22×D13—42no—
222C22×D13—42no—
223C22×D13C4×Q814yes—
224C2×C4×C13—24no—
225C2×C4×C13—22no—
226C2×C4×C13—22no—
227C2×C4×C13—24no—
228C2×C4×C13—24no—
229C2×C4×C13—24no—
230C2×C4×C13—24no—
231C2×C4×C13—22no—
232C2×C4×C13—22no—
233C2×C4×C13—24no—
234C2×C4×C13—24no—
235C2×C4×C13—41no—
236C2×C4×C13—41no—
237C2×C4×C13—41no—
238C2×C4×C13—41no—
239C2×C4×C13—42no—
240C2×C4×C13—42no—
241C2×C4×C13—42no—
242C2×C4×C13—42no—
243C2×C4×C13—42no—
244C2×C4×C13—42no—
245C2×C4×C13—42no—
246C2×C4×C13—42no—
247C2×C4×C13—42no—
248C2×C4×C13—42no—
249C2×C4×C13—42no—
250C2×C4×C13—42no—
251C2×C4×C13—42no—
252C2×C4×C13C4×D414yes—
253C2×Dic13—22no—
254C2×Dic13—22no—
255C2×Dic13—24no—
256C2×Dic13—24no—
257C2×Dic13—24no—
258C2×Dic13—22no—
259C2×Dic13—24no—
260C2×Dic13—24no—
261C2×Dic13—24no—
262C2×Dic13—22no—
263C2×Dic13—22no—
264C2×Dic13—22no—
265C2×Dic13—41no—
266C2×Dic13—42no—
267C2×Dic13—42no—
268C2×Dic13—41no—
269C2×Dic13—42no—
270C2×Dic13—42no—
271C2×Dic13—42no—
272C2×Dic13—42no—
273C2×Dic13—42no—
274C2×Dic13—42no—
275C2×Dic13—42no—
276C2×Dic13—42no—
277C2×Dic13—42no—
278C2×Dic13—42no—
279C2×Dic13—42no—
280C2×Dic13—42no—
281C2×Dic13C22⋊Q814yes—
282C4×D13—416no—
283C24×C4—131no—
284C22×C4⋊C4—131no—
285C22×C4⋊C4—132no—
286C22×C4⋊C4—132no—
287C22×C4⋊C4C2×D1311yescharacteristic
288C22×C22⋊C4—131no—
289C22×C22⋊C4—132no—
290C22×C22⋊C4—131no—
291C22×C22⋊C4—132no—
292C22×C42—131no—
293C22×C42—131no—
294#399—138no—
295#389—264no—
296C4×C22⋊C4—264no—
297C2×#95—132no—
298C2×#95—132no—
299C2×#95—131no—
300C2×#95—131no—
301C2×#95—132no—
302C2×#95—132no—
303C2×#95—132no—
304C2×#95—132no—
305C2×#95C2×D1312yes—
306C2×#95C2×D1311yescharacteristic
307C2×#95C2×D1312yes—
308C2×#95C2×D1311yescharacteristic
309C22×C13C2×C4×D411yescharacteristic
310C22×C13C2×C4×Q811yescharacteristic
311C22×C13C4×#4111yescharacteristic
312C22×C13#51812yes—
313C22×C13#52012yes—
314C22×C13C2×C22⋊Q811yescharacteristic
315C22×C13#52511yescharacteristic
316C22×C13#52911yescharacteristic
317C22×C13#53311yescharacteristic
318C22×C13#53411yescharacteristic
319C22×C13#53511yescharacteristic
320C22×C13#53511yescharacteristic
321C22×C13#53611yescharacteristic
322C22×C13#53812yes—
323C22×C13#53812yes—
324C22×C13#53912yes—
325C22×C13#53912yes—
326C22×C13#53912yes—
327C22×C13#54012yes—
328C22×C13#54012yes—
329C22×C13#54212yes—
330C22×C13#54312yes—
331C22×C13#54412yes—
332C2×D13—28no—
333C2×D13—24no—
334C2×D13—42no—
335C2×D13—42no—
336C2×D13—44no—
337C2×D13—42no—
338C2×D13—42no—
339C2×D13—42no—
340C2×D13—42no—
341C2×D13—44no—
342C2×D13—44no—
343C4×C13—24no—
344C4×C13—44no—
345C4×C13—42no—
346C4×C13—42no—
347C4×C13—82no—
348Dic13—24no—
349Dic13—44no—
350Dic13—42no—
351Dic13—42no—
352Dic13—82no—
353C25—131no—
354C23×C4—131no—
355C23×C4—132no—
356C23×C4—131no—
357C23×C4—131no—
358C23×C4—134no—
359C23×C4—132no—
360C23×C4—262no—
361C23×C4—262no—
362C23×C4—522no—
363C23×C4—522no—
364C23×C4C22×D1312yes—
365C23×C4C22×D1312yes—
366C23×C4C22×D1311yescharacteristic
367C23×C4C22×D1311yescharacteristic
368C23×C4C22×D1311yescharacteristic
369C2×C4⋊C4—22no—
370C2×C4⋊C4—42no—
371C2×C4⋊C4—134no—
372C2×C4⋊C4—138no—
373C2×C4⋊C4—262no—
374C2×C4⋊C4—262no—
375C2×C4⋊C4—264no—
376C2×C4⋊C4—522no—
377C2×C4⋊C4—522no—
378C2×C4⋊C4—522no—
379C2×C4⋊C4—522no—
380C2×C22⋊C4—134no—
381C2×C22⋊C4—138no—
382C2×C22⋊C4—262no—
383C2×C22⋊C4—262no—
384C2×C22⋊C4—262no—
385C2×C22⋊C4—262no—
386C2×C22⋊C4—262no—
387C2×C22⋊C4—264no—
388C2×C22⋊C4—522no—
389C2×C22⋊C4—522no—
390C2×C22⋊C4—522no—
391C2×C22⋊C4—522no—
392C2×C42—134no—
393C2×C42—262no—
394C2×C42—522no—
395C2×C42—522no—
396#95—22no—
397#95—24no—
398#95—42no—
399#95—264no—
400#95—262no—
401#95—264no—
402#95—262no—
403#95—522no—
404#95—522no—
405#95—522no—
406#95—522no—
407#95C4×D1314yescomplemented
408C2×C13C4×C22⋊Q811yescharacteristic
409C2×C13#232211yescharacteristic
410C2×C13#232811yescharacteristic
411C2×C13#232912yes—
412C2×C13#233512yes—
413C2×C13#262611yescharacteristic
414C2×C13#262811yescharacteristic
415C2×C13#263311yescharacteristic
416C2×C13#263511yescharacteristic
417C2×C13#264912yes—
418C2×C13#266712yes—
419D13—44no—
420C24—132no—
421C24—132no—
422C24—132no—
423C24—261no—
424C24—261no—
425C24—262no—
426C24—261no—
427C24—261no—
428C24—262no—
429C24—262no—
430C24—262no—
431C24C23×D1311yescharacteristic
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—132no—
456C22×C4—132no—
457C22×C4—132no—
458C22×C4—132no—
459C22×C4—134no—
460C22×C4—132no—
461C22×C4—268no—
462C22×C4—262no—
463C22×C4—262no—
464C22×C4—261no—
465C22×C4—262no—
466C22×C4—261no—
467C22×C4—262no—
468C22×C4—261no—
469C22×C4—262no—
470C22×C4—261no—
471C22×C4—262no—
472C22×C4—262no—
473C22×C4—262no—
474C22×C4—262no—
475C22×C4—262no—
476C22×C4—262no—
477C22×C4—262no—
478C22×C4—262no—
479C22×C4—262no—
480C22×C4—262no—
481C22×C4—262no—
482C22×C4—262no—
483C22×C4—262no—
484C22×C4—262no—
485C22×C4—528no—
486C22×C4—524no—
487C22×C4—524no—
488C22×C4—528no—
489C22×C4C2×C4×D1314yes—
490C22×C4C2×C4×D1312yes—
491C22×C4D4×D1312yes—
492C22×C4#612912yes—
493C22×C4Q8×D1312yes—
494C22×C4#613112yes—
495C4⋊C4—44no—
496C4⋊C4—268no—
497C4⋊C4—528no—
498C4⋊C4—528no—
499C22⋊C4—268no—
500C22⋊C4—268no—
501C22⋊C4—528no—
502C22⋊C4—528no—
503C42—524no—
504C42—524no—
505C13#1431711yescharacteristic, complemented, Sylow
506C23—134no—
507C23—262no—
508C23—262no—
509C23—264no—
510C23—262no—
511C23—264no—
512C23—262no—
513C23—262no—
514C23—264no—
515C23—264no—
516C23—264no—
517C23—262no—
518C23—264no—
519C23—521no—
520C23—522no—
521C23—522no—
522C23—521no—
523C23—522no—
524C23—521no—
525C23—521no—
526C23—522no—
527C23—522no—
528C23—522no—
529C23C22×C4×D1311yescharacteristic
530C23C2×D4×D1311yescharacteristic
531C23C2×#612911yescharacteristic
532C23#8825112yes—
533C23C2×Q8×D1311yescharacteristic
534C23C2×#613111yescharacteristic
535C23#41×D1311yescharacteristic
536C23#41×D1311yescharacteristic
537C23#8825612yes—
538C23#8825612yes—
539C23#8825712yes—
540C2×C4—24no—
541C2×C4—22no—
542C2×C4—24no—
543C2×C4—22no—
544C2×C4—24no—
545C2×C4—24no—
546C2×C4—24no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—24no—
550C2×C4—24no—
551C2×C4—41no—
552C2×C4—42no—
553C2×C4—41no—
554C2×C4—42no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—134no—
569C2×C4—262no—
570C2×C4—264no—
571C2×C4—262no—
572C2×C4—264no—
573C2×C4—262no—
574C2×C4—264no—
575C2×C4—264no—
576C2×C4—264no—
577C2×C4—264no—
578C2×C4—262no—
579C2×C4—262no—
580C2×C4—262no—
581C2×C4—5216no—
582C2×C4—522no—
583C2×C4—522no—
584C2×C4—521no—
585C2×C4—522no—
586C2×C4—522no—
587C2×C4—521no—
588C2×C4—522no—
589C2×C4—522no—
590C2×C4—522no—
591C2×C4—522no—
592C2×C4—522no—
593C2×C4—522no—
594C2×C4—522no—
595C2×C4—522no—
596C2×C4—522no—
597C2×C4—522no—
598C2×C4#8814714yes—
599C22—268no—
600C22—264no—
601C22—524no—
602C22—522no—
603C22—524no—
604C22—522no—
605C22—522no—
606C22—522no—
607C22—524no—
608C22—522no—
609C22—522no—
610C22C2×#8814711yescharacteristic
611C22#1171922612yes—
612C22C4×#612911yescharacteristic
613C22#1171925112yes—
614C22C4×Q8×D1311yescharacteristic
615C22#1171929411yescharacteristic
616C22#1171931111yescharacteristic
617C22#1171932412yes—
618C22D13×C22⋊Q811yescharacteristic
619C22#1171933912yes—
620C22#1171935012yes—
621C22#1171935311yescharacteristic
622C22#1171936212yes—
623C22#1171937012yes—
624C22#1171937911yescharacteristic
625C22#1171937911yescharacteristic
626C22#1171938312yes—
627C22#1171939412yes—
628C22#1171939611yescharacteristic
629C22#1171940312yes—
630C22#1171941212yes—
631C22#1171941512yes—
632C22#1171943311yescharacteristic
633C4—24no—
634C4—42no—
635C4—44no—
636C4—42no—
637C4—82no—
638C4—264no—
639C4—522no—
640C4—524no—
641C4—522no—
642C4—1042no—
643C2—524no—
644C2#1311736611yescharacteristic
645C2#1311750411yescharacteristic
646C2#1311755911yescharacteristic
647C2#1311756512yes—
648C2#1311763112yes—
649C2#1312064011yescharacteristic
650C2#1312066411yescharacteristic
651C2#1312070911yescharacteristic
652C2#1312072811yescharacteristic
653C2#1312083812yes—
654C2#1312103512yes—
subgroupG/H
center, FrattiniC24C23×D13
commutatorC23×C13C23×C4
FittingC13×#1442C2
socle, cosocleC24×C13C24
prime radicalC24×D13C23
hypercenter#1442D13
hypocenterC13#14317
LCSG ▷ C23×C13 ▷ C13
UCSC1 ◁ C24 ◁ #1442

Complements

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

NH ≅ G/Ntype
2. C2×#11718634C2semidirect
4. #13113105C2semidirect
5. #13113088C2semidirect
8. #13113070C2semidirect
9. #13113061C2semidirect
11. C13×#1442C2semidirect
36. #11718338C4semidirect
72. C2×C4⋊Dic13C2×C4semidirect
73. C2×C4⋊Dic13C2×C4semidirect
84. C13×#95C2×C4semidirect
183. #1442D13semidirect
407. #95C4×D13semidirect
505. C13#14317semidirect

Maximal subgroups

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

subgroupindex
C2×#117183382
C2×#117186342
C2×#117183272
#131131052
#131130882
#131130802
#131130782
#131130702
#131130612
#131130572
C13×#14422
#1431713