Finite Group Databasebeta

← #15785540

Subgroups — #15785540

Order 3328.

1630 conjugacy classes of subgroups.

The subgroup lattice has 650 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11718651C212yesmaximal
2C2×C4×#6103C211yesmaximal, characteristic
3#13113095C212yesmaximal, complemented
4#13113082C212yesmaximal
5#13113070C212yesmaximal, complemented
6#13113066C211yesmaximal, characteristic, complemented
7#13113061C211yesmaximal, characteristic, complemented
8#13113057C211yesmaximal, characteristic
9#13113057C211yesmaximal, characteristic
10#95×Dic13C211yesmaximal, characteristic, complemented
11C13×C4⋊#95C211yesmaximal, characteristic, complemented
12C23×C4×D13C2211yescharacteristic
13C22×C13×C4⋊C4C2212yes—
14C22×C42×C13C2211yescharacteristic
15C22×#6103C2212yes—
16C22×#6103C2212yes—
17C22×#6103C2211yescharacteristic
18C22×#6103C2211yescharacteristic
19C22×C4⋊Dic13C2212yes—
20C22×C4×Dic13C2212yes—
21C22×C4×Dic13C2211yescharacteristic
22C2×C13×#95C2211yescharacteristic
23C2×C13×#95C2211yescharacteristic
24C2×C13×#95C2212yes—
25C2×#88071C2212yes—
26C2×#88071C2212yes—
27C2×#88071C2212yes—
28C2×#88071C2211yescharacteristic
29C2×#88071C2211yescharacteristic
30C2×#88071C2212yes—
31C2×#88071C2212yes—
32C2×#88071C2212yes—
33C2×#88071C2211yescharacteristic
34C2×#88071C2211yescharacteristic
35#11718651—28no—
36C4×#6103C418yescomplemented
37C24×D13C2311yescharacteristic
38C23×C4×C13C2312yes—
39C23×C4×C13C2312yes—
40C23×C4×C13C2311yescharacteristic
41C23×C4×C13C2311yescharacteristic
42C23×C4×C13C2311yescharacteristic
43C23×Dic13C2312yes—
44C23×Dic13C2311yescharacteristic
45C23×Dic13C2311yescharacteristic
46C23×Dic13C2312yes—
47C23×Dic13C2311yescharacteristic
48C22×C4×D13—22no—
49C22×C4×D13—22no—
50C22×C4×D13—42no—
51C22×C4×D13—42no—
52C22×C4×D13C2×C414yes—
53C2×C13×C4⋊C4—24no—
54C2×C13×C4⋊C4—24no—
55C2×C13×C4⋊C4—42no—
56C2×C42×C13—42no—
57C2×C42×C13C2×C414yes—
58C2×#6103—22no—
59C2×#6103—22no—
60C2×#6103—24no—
61C2×#6103—24no—
62C2×#6103—24no—
63C2×#6103—24no—
64C2×#6103—42no—
65C2×#6103—42no—
66C2×#6103C2×C414yes—
67C2×#6103C2×C414yes—
68C2×#6103D414yes—
69C2×#6103Q814yes—
70C2×C4⋊Dic13—24no—
71C2×C4⋊Dic13—24no—
72C2×C4⋊Dic13—42no—
73C2×C4×Dic13—24no—
74C2×C4×Dic13—24no—
75C2×C4×Dic13—42no—
76C2×C4×Dic13—42no—
77C2×C4×Dic13C2×C414yes—
78C13×#95—22no—
79C13×#95—24no—
80C13×#95—22no—
81#88071—24no—
82#88071—24no—
83#88071—22no—
84#88071—22no—
85#88071—24no—
86#88071—42no—
87#88071—42no—
88#88071—42no—
89#88071C2×C414yes—
90#88071C2×C414yes—
91#13948—131nomaximal, Sylow
92C24×C13C2411yescharacteristic
93C23×D13—22no—
94C23×D13—21no—
95C23×D13—21no—
96C23×D13—22no—
97C23×D13—21no—
98C23×D13—21no—
99C23×D13—22no—
100C23×D13—22no—
101C23×D13C22×C412yes—
102C23×D13C2×D412yes—
103C23×D13C2×Q812yes—
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—22no—
111C22×C4×C13—22no—
112C22×C4×C13—22no—
113C22×C4×C13—22no—
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×C13C22×C412yes—
124C22×C4×C13C22×C412yes—
125C22×C4×C13C22×C412yes—
126C22×C4×C13C2×D412yes—
127C22×C4×C13C2×Q812yes—
128C22×C4×C13#4114yes—
129C22×C4×C13#4112yes—
130C22×C4×C13#4112yes—
131C22×C4×C13#4112yes—
132C22×C4×C13#4112yes—
133C22×Dic13—22no—
134C22×Dic13—22no—
135C22×Dic13—22no—
136C22×Dic13—22no—
137C22×Dic13—22no—
138C22×Dic13—22no—
139C22×Dic13—22no—
140C22×Dic13—21no—
141C22×Dic13—21no—
142C22×Dic13—21no—
143C22×Dic13—21no—
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×Dic13C22×C412yes—
157C22×Dic13C22×C412yes—
158C22×Dic13C22×C412yes—
159C22×Dic13C2×D412yes—
160C22×Dic13C2×Q812yes—
161C22×Dic13#4114yes—
162C2×C4×D13—28no—
163C2×C4×D13—44no—
164C2×C4×D13—44no—
165C2×C4×D13—48no—
166C2×C4×D13—48no—
167C13×C4⋊C4—48no—
168C42×C13—44no—
169#6103—28no—
170#6103—48no—
171#6103—48no—
172#6103C4⋊C4116yescomplemented
173C4⋊Dic13—48no—
174C4×Dic13—44no—
175C4×Dic13—48no—
176C2×#388—132no—
177C2×C4×C22⋊C4—131no—
178#1442—132no—
179#1439—132no—
180#1436—132no—
181C4⋊#95—131no—
182C4⋊#95D1311yescharacteristic, complemented
183#1434—131no—
184C22⋊#95—131no—
185C22⋊#95—131no—
186C4×#95—131no—
187C23×C13C23×C411yescharacteristic
188C23×C13C22×D411yescharacteristic
189C23×C13C22×Q811yescharacteristic
190C23×C13C2×#4111yescharacteristic
191C23×C13C2×#4111yescharacteristic
192C23×C13C2×#4111yescharacteristic
193C23×C13C2×#4111yescharacteristic
194C23×C13C2×#4112yes—
195C23×C13C2×#4112yes—
196C23×C13H(5,𝔽2)12yes—
197C23×C13H(5,𝔽2)12yes—
198C22×D13—22no—
199C22×D13—22no—
200C22×D13—22no—
201C22×D13—22no—
202C22×D13—22no—
203C22×D13—22no—
204C22×D13—22no—
205C22×D13—22no—
206C22×D13—22no—
207C22×D13—22no—
208C22×D13—24no—
209C22×D13—24no—
210C22×D13—24no—
211C22×D13—24no—
212C22×D13—42no—
213C22×D13—42no—
214C22×D13—42no—
215C22×D13—42no—
216C22×D13—42no—
217C22×D13—42no—
218C22×D13—42no—
219C22×D13—42no—
220C22×D13C2×C4⋊C414yes—
221C2×C4×C13—22no—
222C2×C4×C13—22no—
223C2×C4×C13—22no—
224C2×C4×C13—22no—
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—24no—
232C2×C4×C13—24no—
233C2×C4×C13—24no—
234C2×C4×C13—24no—
235C2×C4×C13—24no—
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×C13—42no—
245C2×C4×C13—42no—
246C2×C4×C13—42no—
247C2×C4×C13C2×C4⋊C414yes—
248C2×C4×C13#11714yes—
249C2×C4×C13#11714yes—
250C2×Dic13—22no—
251C2×Dic13—22no—
252C2×Dic13—24no—
253C2×Dic13—24no—
254C2×Dic13—22no—
255C2×Dic13—22no—
256C2×Dic13—22no—
257C2×Dic13—22no—
258C2×Dic13—22no—
259C2×Dic13—22no—
260C2×Dic13—24no—
261C2×Dic13—24no—
262C2×Dic13—24no—
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—
274C2×Dic13—42no—
275C2×Dic13—42no—
276C2×Dic13—42no—
277C2×Dic13—42no—
278C2×Dic13C2×C4⋊C414yes—
279C4×D13—416no—
280C24×C4—131no—
281C22×C4⋊C4—132no—
282C22×C4⋊C4C2×D1312yes—
283C22×C22⋊C4—131no—
284C22×C22⋊C4—131no—
285C22×C22⋊C4—132no—
286C22×C22⋊C4—132no—
287C22×C42—131no—
288C22×C42—132no—
289C22×C42C2×D1311yescharacteristic
290#388—268no—
291C4×C22⋊C4—138no—
292C2×#95—131no—
293C2×#95—131no—
294C2×#95—131no—
295C2×#95—132no—
296C2×#95—131no—
297C2×#95—132no—
298C2×#95—132no—
299C2×#95—132no—
300C2×#95—132no—
301C2×#95—132no—
302C2×#95C2×D1311yescharacteristic
303C2×#95C2×D1311yescharacteristic
304C2×#95C2×D1312yes—
305C22×C13C22×C4⋊C411yescharacteristic
306C22×C13C2×#11711yescharacteristic
307C22×C13C2×#11711yescharacteristic
308C22×C13C4×#4111yescharacteristic
309C22×C13C4×#4111yescharacteristic
310C22×C13#51811yescharacteristic
311C22×C13#51811yescharacteristic
312C22×C13#52912yes—
313C22×C13#52912yes—
314C22×C13#53211yescharacteristic
315C22×C13#53211yescharacteristic
316C22×C13#53311yescharacteristic
317C22×C13#53311yescharacteristic
318C22×C13#53812yes—
319C22×C13#53812yes—
320C22×C13#54612yes—
321C22×C13#54612yes—
322C22×C13#55112yes—
323C22×C13#55312yes—
324C22×C13#55312yes—
325C22×C13#55412yes—
326C22×C13#55412yes—
327C22×C13#55912yes—
328C2×D13—24no—
329C2×D13—24no—
330C2×D13—24no—
331C2×D13—42no—
332C2×D13—44no—
333C2×D13—44no—
334C2×D13—42no—
335C2×D13—42no—
336C2×D13—42no—
337C2×D13—44no—
338C2×D13—44no—
339C4×C13—24no—
340C4×C13—24no—
341C4×C13—24no—
342C4×C13—44no—
343C4×C13—82no—
344Dic13—24no—
345Dic13—42no—
346Dic13—42no—
347Dic13—44no—
348Dic13—82no—
349C25—131no—
350C23×C4—131no—
351C23×C4—132no—
352C23×C4—131no—
353C23×C4—131no—
354C23×C4—134no—
355C23×C4—132no—
356C23×C4—262no—
357C23×C4—262no—
358C23×C4—522no—
359C23×C4—522no—
360C23×C4C22×D1312yes—
361C23×C4C22×D1312yes—
362C23×C4C22×D1311yescharacteristic
363C23×C4C22×D1311yescharacteristic
364C23×C4C22×D1311yescharacteristic
365C2×C4⋊C4—24no—
366C2×C4⋊C4—24no—
367C2×C4⋊C4—42no—
368C2×C4⋊C4—264no—
369C2×C4⋊C4—264no—
370C2×C4⋊C4—522no—
371C2×C22⋊C4—134no—
372C2×C22⋊C4—134no—
373C2×C22⋊C4—134no—
374C2×C22⋊C4—134no—
375C2×C22⋊C4—262no—
376C2×C22⋊C4—262no—
377C2×C22⋊C4—264no—
378C2×C22⋊C4—264no—
379C2×C22⋊C4—264no—
380C2×C22⋊C4—264no—
381C2×C22⋊C4—522no—
382C2×C22⋊C4—522no—
383C2×C42—42no—
384C2×C42—134no—
385C2×C42—264no—
386C2×C42—264no—
387C2×C42—522no—
388C2×C42—522no—
389C2×C42C4×D1314yes—
390#95—22no—
391#95—24no—
392#95—22no—
393#95—134no—
394#95—134no—
395#95—264no—
396#95—264no—
397#95—264no—
398#95—262no—
399#95—262no—
400#95—522no—
401#95—522no—
402#95—522no—
403C2×C13#228511yescharacteristic
404C2×C13#229811yescharacteristic
405C2×C13#231211yescharacteristic
406C2×C13#234911yescharacteristic
407C2×C13#234911yescharacteristic
408C2×C13#235311yescharacteristic
409C2×C13#235711yescharacteristic
410C2×C13#255712yes—
411C2×C13#255912yes—
412C2×C13#258712yes—
413C2×C13#258912yes—
414D13—44no—
415C24—132no—
416C24—132no—
417C24—132no—
418C24—262no—
419C24—261no—
420C24—261no—
421C24—262no—
422C24—261no—
423C24—261no—
424C24—262no—
425C24—262no—
426C24C23×D1311yescharacteristic
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—132no—
447C22×C4—132no—
448C22×C4—132no—
449C22×C4—132no—
450C22×C4—132no—
451C22×C4—134no—
452C22×C4—262no—
453C22×C4—262no—
454C22×C4—262no—
455C22×C4—262no—
456C22×C4—262no—
457C22×C4—261no—
458C22×C4—261no—
459C22×C4—262no—
460C22×C4—262no—
461C22×C4—262no—
462C22×C4—261no—
463C22×C4—262no—
464C22×C4—261no—
465C22×C4—262no—
466C22×C4—262no—
467C22×C4—262no—
468C22×C4—268no—
469C22×C4—262no—
470C22×C4—262no—
471C22×C4—262no—
472C22×C4—262no—
473C22×C4—262no—
474C22×C4—262no—
475C22×C4—262no—
476C22×C4—524no—
477C22×C4—524no—
478C22×C4—528no—
479C22×C4—528no—
480C22×C4C2×C4×D1312yes—
481C22×C4C2×C4×D1312yes—
482C22×C4C2×C4×D1312yes—
483C22×C4#612714yes—
484C22×C4D4×D1312yes—
485C22×C4#612912yes—
486C22×C4#612912yes—
487C22×C4Q8×D1312yes—
488C22×C4#613112yes—
489C22×C4#613112yes—
490C4⋊C4—48no—
491C4⋊C4—528no—
492C22⋊C4—1316no—
493C22⋊C4—268no—
494C22⋊C4—528no—
495C22⋊C4—528no—
496C42—44no—
497C42—524no—
498C42—528no—
499C13#1394811yescharacteristic, complemented, Sylow
500C23—134no—
501C23—262no—
502C23—262no—
503C23—262no—
504C23—262no—
505C23—262no—
506C23—264no—
507C23—262no—
508C23—262no—
509C23—264no—
510C23—262no—
511C23—262no—
512C23—264no—
513C23—264no—
514C23—262no—
515C23—522no—
516C23—522no—
517C23—522no—
518C23—522no—
519C23—522no—
520C23—522no—
521C23—522no—
522C23—522no—
523C23C22×C4×D1311yescharacteristic
524C23C2×#612712yes—
525C23C2×D4×D1311yescharacteristic
526C23C2×#612911yescharacteristic
527C23C2×#612911yescharacteristic
528C23#8825112yes—
529C23C2×Q8×D1311yescharacteristic
530C23C2×#613111yescharacteristic
531C23C2×#613111yescharacteristic
532C23#41×D1312yes—
533C23#8825612yes—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—24no—
539C2×C4—24no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—24no—
543C2×C4—24no—
544C2×C4—24no—
545C2×C4—24no—
546C2×C4—24no—
547C2×C4—24no—
548C2×C4—24no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—134no—
561C2×C4—262no—
562C2×C4—262no—
563C2×C4—264no—
564C2×C4—264no—
565C2×C4—262no—
566C2×C4—262no—
567C2×C4—262no—
568C2×C4—262no—
569C2×C4—262no—
570C2×C4—262no—
571C2×C4—264no—
572C2×C4—264no—
573C2×C4—264no—
574C2×C4—5216no—
575C2×C4—522no—
576C2×C4—522no—
577C2×C4—522no—
578C2×C4—522no—
579C2×C4—522no—
580C2×C4—522no—
581C2×C4—522no—
582C2×C4—522no—
583C2×C4—522no—
584C2×C4—522no—
585C2×C4—522no—
586C2×C4—522no—
587C2×C4—522no—
588C2×C4—522no—
589C2×C4—522no—
590C2×C4C4⋊C4×D1314yes—
591C2×C4#8814614yes—
592C2×C4#8814614yes—
593C22—264no—
594C22—264no—
595C22—264no—
596C22—524no—
597C22—524no—
598C22—522no—
599C22—522no—
600C22—522no—
601C22—524no—
602C22—524no—
603C22—522no—
604C22C4×#612711yescharacteristic
605C22#1171919111yescharacteristic
606C22#1171919512yes—
607C22C2×C4⋊C4×D1311yescharacteristic
608C22C2×#8814611yescharacteristic
609C22C2×#8814611yescharacteristic
610C22#1171921612yes—
611C22#1171922611yescharacteristic
612C22#1171922711yescharacteristic
613C22#1171924312yes—
614C22#1171927012yes—
615C22#1171928512yes—
616C22#1171929012yes—
617C22#1171930612yes—
618C22#1171932212yes—
619C22#1171934212yes—
620C22#1171934912yes—
621C22#1171938112yes—
622C22#1171939512yes—
623C22#1171942111yescharacteristic
624C22#1171943011yescharacteristic
625C22#1171943311yescharacteristic
626C22#1171943311yescharacteristic
627C4—24no—
628C4—24no—
629C4—24no—
630C4—44no—
631C4—82no—
632C4—264no—
633C4—522no—
634C4—524no—
635C4—522no—
636C4—1042no—
637C2—524no—
638C2#1311724111yescharacteristic
639C2#1311732911yescharacteristic
640C2#1311743311yescharacteristic
641C2#1311774311yescharacteristic
642C2#1311774411yescharacteristic
643C2#1311777811yescharacteristic
644C2#1311780711yescharacteristic
645C2#1311994212yes—
646C2#1311997012yes—
647C2#1312026412yes—
648C2#1312029212yes—
subgroupG/H
center, FrattiniC24C23×D13
commutatorC23×C13C23×C4
FittingC13×C4⋊#95C2
socle, cosocleC24×C13C24
prime radicalC24×D13C23
hypercenterC4⋊#95D13
hypocenterC13#13948
LCSG ▷ C23×C13 ▷ C13
UCSC1 ◁ C24 ◁ C4⋊#95

Complements

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

NH ≅ G/Ntype
3. #13113095C2semidirect
5. #13113070C2semidirect
6. #13113066C2semidirect
7. #13113061C2semidirect
10. #95×Dic13C2semidirect
11. C13×C4⋊#95C2semidirect
36. C4×#6103C4semidirect
172. #6103C4⋊C4semidirect
182. C4⋊#95D13semidirect
499. C13#13948semidirect

Maximal subgroups

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

subgroupindex
C2×#117186512
C2×C4×#61032
#131130952
#131130822
#131130702
#131130662
#131130612
#131130572
#95×Dic132
C13×C4⋊#952
#1394813