Finite Group Databasebeta

← #15791376

Subgroups — #15791376

Order 3328.

1658 conjugacy classes of subgroups.

The subgroup lattice has 644 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11718612C212yesmaximal, complemented
2#13113085C212yesmaximal, complemented
3#13113081C212yesmaximal, complemented
4#13113077C212yesmaximal
5#13113073C212yesmaximal, complemented
6#13113068C212yesmaximal, complemented
7#13113066C211yesmaximal, characteristic, complemented
8#13113063C211yesmaximal, characteristic, complemented
9C2×C13×#390C211yesmaximal, characteristic
10C23×D4×C13C2211yescharacteristic
11C22×C13×C4⋊C4C2211yescharacteristic
12C22×C13×C22⋊C4C2212yes—
13C22×C13×C22⋊C4C2212yes—
14C22×C42×C13C2211yescharacteristic
15C22×C22⋊Dic13C2212yes—
16C22×C22⋊Dic13C2212yes—
17C22×C22⋊Dic13C2212yes—
18C22×C22⋊Dic13C2212yes—
19C22×C4⋊Dic13C2212yes—
20C22×C4×Dic13C2212yes—
21C13×#390C418yescomplemented
22C2×#88071C2212yes—
23C2×#88071C2212yes—
24C2×#88071C2211yescharacteristic
25C2×#88071C2212yes—
26C2×#88071C2212yes—
27C2×#88071C2211yescharacteristic
28C2×#88071C2211yescharacteristic
29C2×#88071C2212yes—
30C2×#88071C2212yes—
31C2×#88071C2211yescharacteristic
32#11718612—28no—
33C25×C13C2312yes—
34C22×D4×C13—22no—
35C22×D4×C13—22no—
36C22×D4×C13—42no—
37C22×D4×C13—42no—
38C22×D4×C13C2×C414yes—
39C23×C4×C13C2312yes—
40C23×C4×C13C2311yescharacteristic
41C23×C4×C13C2311yescharacteristic
42C23×C4×C13C2311yescharacteristic
43C23×Dic13C2312yes—
44C23×Dic13C2312yes—
45C23×Dic13C2312yes—
46C23×Dic13C2312yes—
47C2×C13×C4⋊C4—42no—
48C2×C13×C4⋊C4C2×C414yes—
49C2×C13×C22⋊C4—24no—
50C2×C13×C22⋊C4—24no—
51C2×C13×C22⋊C4—42no—
52C2×C13×C22⋊C4—42no—
53C2×C13×C22⋊C4C2×C418yes—
54C2×C13×C22⋊C4C2×C418yes—
55C2×C42×C13—42no—
56C2×C42×C13C2×C414yes—
57C2×C22⋊Dic13—24no—
58C2×C22⋊Dic13—24no—
59C2×C22⋊Dic13—24no—
60C2×C22⋊Dic13—24no—
61C2×C22⋊Dic13—42no—
62C2×C22⋊Dic13—42no—
63C2×C22⋊Dic13—42no—
64C2×C22⋊Dic13—42no—
65C2×C22⋊Dic13—42no—
66C2×C22⋊Dic13—42no—
67C2×C22⋊Dic13—42no—
68C2×C22⋊Dic13—42no—
69C2×C4⋊Dic13—24no—
70C2×C4⋊Dic13—24no—
71C2×C4⋊Dic13—42no—
72C2×C4×Dic13—24no—
73C2×C4×Dic13—42no—
74C2×C4×Dic13—42no—
75#88071—24no—
76#88071—24no—
77#88071—22no—
78#88071—24no—
79#88071—42no—
80#88071—41no—
81#88071—42no—
82#88071—41no—
83#88071—42no—
84#88071D414yescomplemented
85#14444—131nomaximal, Sylow
86C24×C13—22no—
87C24×C13—22no—
88C24×C13—22no—
89C24×C13—22no—
90C24×C13—22no—
91C24×C13—22no—
92C24×C13—22no—
93C24×C13—22no—
94C24×C13—22no—
95C24×C13—22no—
96C24×C13—22no—
97C24×C13—22no—
98C24×C13—22no—
99C24×C13—22no—
100C24×C13C22×C414yes—
101C24×C13C2411yescharacteristic
102C2×D4×C13—28no—
103C2×D4×C13—44no—
104C2×D4×C13—44no—
105C2×D4×C13—48no—
106C2×D4×C13—48no—
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—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—21no—
121C22×C4×C13C22×C412yes—
122C22×C4×C13C22×C414yes—
123C22×C4×C13C22×C412yes—
124C22×C4×C13C22×C412yes—
125C22×C4×C13C2×D412yes—
126C22×C4×C13#4112yes—
127C22×C4×C13#4112yes—
128C22×C4×C13#4112yes—
129C22×Dic13—22no—
130C22×Dic13—22no—
131C22×Dic13—22no—
132C22×Dic13—22no—
133C22×Dic13—22no—
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—22no—
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×Dic13C2×D414yes—
156C22×Dic13#4114yes—
157C13×C4⋊C4—44no—
158C13×C22⋊C4—216no—
159C13×C22⋊C4—48no—
160C42×C13—44no—
161C22⋊Dic13—48no—
162C22⋊Dic13—48no—
163C22⋊Dic13—48no—
164C22⋊Dic13—48no—
165C4⋊Dic13—48no—
166C4×Dic13—48no—
167C2×#390D1311yescharacteristic
168C2×#388—132no—
169#1440—132no—
170#1439—132no—
171#1438—132no—
172#1437—132no—
173#1436—132no—
174C4⋊#95—131no—
175#1434—131no—
176C23×C13—24no—
177C23×C13—24no—
178C23×C13—24no—
179C23×C13—24no—
180C23×C13—24no—
181C23×C13—24no—
182C23×C13—24no—
183C23×C13—42no—
184C23×C13—42no—
185C23×C13—42no—
186C23×C13—42no—
187C23×C13—42no—
188C23×C13—42no—
189C23×C13—42no—
190C23×C13—42no—
191C23×C13—42no—
192C23×C13—42no—
193C23×C13—42no—
194C23×C13—42no—
195C23×C13—42no—
196C23×C13—42no—
197C23×C13—42no—
198C23×C13—42no—
199C23×C13—42no—
200C23×C13—42no—
201C23×C13—42no—
202C23×C13—42no—
203C23×C13—42no—
204C23×C13—42no—
205C23×C13—42no—
206C23×C13—42no—
207C23×C13—42no—
208C23×C13—42no—
209C23×C13—42no—
210C23×C13—42no—
211C23×C13C23×C411yescharacteristic
212C23×C13C22×D411yescharacteristic
213C23×C13C2×#4111yescharacteristic
214C23×C13C2×#4111yescharacteristic
215C23×C13C2×#4111yescharacteristic
216C23×C13C2×#4112yes—
217C23×C13H(5,𝔽2)12yes—
218C23×C13H(5,𝔽2)12yes—
219C23×C13H(5,𝔽2)12yes—
220C23×C13H(5,𝔽2)11yescharacteristic
221C23×C132−1+411yescharacteristic
222D4×C13—416no—
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—22no—
233C2×C4×C13—22no—
234C2×C4×C13—24no—
235C2×C4×C13—24no—
236C2×C4×C13—22no—
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#11714yes—
244C2×C4×C13C4×D414yes—
245C2×Dic13—24no—
246C2×Dic13—24no—
247C2×Dic13—24no—
248C2×Dic13—24no—
249C2×Dic13—24no—
250C2×Dic13—24no—
251C2×Dic13—42no—
252C2×Dic13—42no—
253C2×Dic13—42no—
254C2×Dic13—42no—
255C2×Dic13—42no—
256C2×Dic13—42no—
257C2×Dic13—42no—
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—
273C23×D4C2×D1311yescharacteristic
274C22×C4⋊C4—132no—
275C22×C4⋊C4C2×D1311yescharacteristic
276C22×C22⋊C4—132no—
277C22×C22⋊C4—132no—
278C22×C22⋊C4—132no—
279C22×C22⋊C4—132no—
280C22×C22⋊C4C2×D1312yes—
281C22×C22⋊C4C2×D1312yes—
282C22×C42—132no—
283C22×C42C2×D1311yescharacteristic
284#390Dic1318yescomplemented
285#388—268no—
286C2×#95—131no—
287C2×#95—132no—
288C2×#95—131no—
289C2×#95—132no—
290C2×#95—132no—
291C2×#95—132no—
292C2×#95—131no—
293C2×#95—131no—
294C2×#95—132no—
295C2×#95—132no—
296C22×C13—44no—
297C22×C13—44no—
298C22×C13—44no—
299C22×C13—44no—
300C22×C13—44no—
301C22×C13—44no—
302C22×C13—44no—
303C22×C13—82no—
304C22×C13—82no—
305C22×C13—82no—
306C22×C13—82no—
307C22×C13—82no—
308C22×C13—82no—
309C22×C13—82no—
310C22×C13—82no—
311C22×C13C2×#11711yescharacteristic
312C22×C13C2×C4×D411yescharacteristic
313C22×C13C4×#4111yescharacteristic
314C22×C13#51811yescharacteristic
315C22×C13#51811yescharacteristic
316C22×C13#51811yescharacteristic
317C22×C13#52011yescharacteristic
318C22×C13#53512yes—
319C22×C13#53812yes—
320C22×C13#53812yes—
321C22×C13#53812yes—
322C22×C13#54012yes—
323C22×C13#54212yes—
324C22×C13#54612yes—
325C22×C13#54711yescharacteristic
326C22×C13#55011yescharacteristic
327C22×C13#55312yes—
328C22×C13#55312yes—
329C22×C13#55511yescharacteristic
330C22×C13#55611yescharacteristic
331C22×C13#55912yes—
332C22×C13#56012yes—
333C22×C13#56212yes—
334C4×C13—24no—
335C4×C13—24no—
336C4×C13—44no—
337C4×C13—42no—
338Dic13—44no—
339Dic13—44no—
340Dic13—82no—
341Dic13—82no—
342C25C22×D1312yes—
343C22×D4—22no—
344C22×D4—22no—
345C22×D4—42no—
346C22×D4—42no—
347C22×D4C2×Dic1314yes—
348C23×C4—132no—
349C23×C4—132no—
350C23×C4—132no—
351C23×C4—132no—
352C23×C4C22×D1312yes—
353C23×C4C22×D1311yescharacteristic
354C23×C4C22×D1311yescharacteristic
355C23×C4C22×D1311yescharacteristic
356C2×C4⋊C4—42no—
357C2×C4⋊C4—264no—
358C2×C4⋊C4—264no—
359C2×C4⋊C4—522no—
360C2×C4⋊C4C2×Dic1314yes—
361C2×C22⋊C4—24no—
362C2×C22⋊C4—24no—
363C2×C22⋊C4—42no—
364C2×C22⋊C4—42no—
365C2×C22⋊C4—264no—
366C2×C22⋊C4—264no—
367C2×C22⋊C4—264no—
368C2×C22⋊C4—264no—
369C2×C22⋊C4—522no—
370C2×C22⋊C4—522no—
371C2×C22⋊C4—522no—
372C2×C22⋊C4—522no—
373C2×C22⋊C4—522no—
374C2×C22⋊C4—522no—
375C2×C22⋊C4—522no—
376C2×C22⋊C4—522no—
377C2×C22⋊C4C2×Dic1318yes—
378C2×C22⋊C4C2×Dic1318yes—
379C2×C42—42no—
380C2×C42—264no—
381C2×C42—522no—
382C2×C42—522no—
383C2×C42C2×Dic1314yes—
384#95—134no—
385#95—264no—
386#95—264no—
387#95—264no—
388#95—262no—
389#95—522no—
390#95—522no—
391#95—522no—
392#95—521no—
393#95—521no—
394C2×C13—84no—
395C2×C13#231211yescharacteristic
396C2×C13#232811yescharacteristic
397C2×C13#233711yescharacteristic
398C2×C13#235411yescharacteristic
399C2×C13#235711yescharacteristic
400C2×C13#236411yescharacteristic
401C2×C13#237711yescharacteristic
402C2×C13#271012yes—
403C2×C13#271412yes—
404C2×C13#280012yes—
405C2×C13#280312yes—
406C24—22no—
407C24—22no—
408C24—22no—
409C24—22no—
410C24—22no—
411C24—22no—
412C24—22no—
413C24—22no—
414C24—22no—
415C24—22no—
416C24—22no—
417C24—22no—
418C24—22no—
419C24—22no—
420C24C22×Dic1314yes—
421C24C23×D1311yescharacteristic
422C2×D4—28no—
423C2×D4—44no—
424C2×D4—44no—
425C2×D4—48no—
426C2×D4—48no—
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—21no—
440C22×C4—21no—
441C22×C4—134no—
442C22×C4—134no—
443C22×C4—262no—
444C22×C4—262no—
445C22×C4—262no—
446C22×C4—262no—
447C22×C4—262no—
448C22×C4—262no—
449C22×C4—262no—
450C22×C4—262no—
451C22×C4—262no—
452C22×C4—262no—
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—262no—
464C22×C4—262no—
465C22×C4—262no—
466C22×C4—262no—
467C22×C4—262no—
468C22×C4—262no—
469C22×C4#612712yes—
470C22×C4#612712yes—
471C22×C4D4×D1312yes—
472C22×C4#612912yes—
473C22×C4C22×Dic1312yes—
474C22×C4C22×Dic1314yes—
475C22×C4C22×Dic1312yes—
476C22×C4C22×Dic1312yes—
477C4⋊C4—44no—
478C4⋊C4—528no—
479C22⋊C4—216no—
480C22⋊C4—48no—
481C22⋊C4—528no—
482C22⋊C4—528no—
483C22⋊C4—528no—
484C22⋊C4—528no—
485C42—44no—
486C42—528no—
487C13#1444411yescharacteristic, complemented, Sylow
488C23—24no—
489C23—24no—
490C23—24no—
491C23—24no—
492C23—24no—
493C23—24no—
494C23—24no—
495C23—42no—
496C23—42no—
497C23—42no—
498C23—42no—
499C23—42no—
500C23—42no—
501C23—42no—
502C23—42no—
503C23—42no—
504C23—42no—
505C23—42no—
506C23—42no—
507C23—42no—
508C23—42no—
509C23—42no—
510C23—42no—
511C23—42no—
512C23—42no—
513C23—42no—
514C23—42no—
515C23—42no—
516C23—42no—
517C23—42no—
518C23—42no—
519C23—42no—
520C23—42no—
521C23—42no—
522C23—42no—
523C23C2×#612711yescharacteristic
524C23C2×#612711yescharacteristic
525C23C2×D4×D1311yescharacteristic
526C23C2×#612911yescharacteristic
527C23#8825112yes—
528C23#8825112yes—
529C23#8825112yes—
530C23#41×D1312yes—
531C23#8825611yescharacteristic
532C23#8825711yescharacteristic
533C23C23×Dic1311yescharacteristic
534D4—416no—
535C2×C4—24no—
536C2×C4—24no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—24no—
542C2×C4—24no—
543C2×C4—24no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—24no—
547C2×C4—24no—
548C2×C4—22no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—264no—
556C2×C4—264no—
557C2×C4—264no—
558C2×C4—264no—
559C2×C4—264no—
560C2×C4—264no—
561C2×C4—522no—
562C2×C4—522no—
563C2×C4—522no—
564C2×C4—522no—
565C2×C4—522no—
566C2×C4—522no—
567C2×C4—522no—
568C2×C4—522no—
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—522no—
581C2×C4—522no—
582C2×C4—522no—
583C2×C4#8818014yes—
584C2×C4D4×Dic1314yes—
585C22—44no—
586C22—44no—
587C22—44no—
588C22—44no—
589C22—44no—
590C22—44no—
591C22—44no—
592C22—82no—
593C22—82no—
594C22—82no—
595C22—82no—
596C22—82no—
597C22—82no—
598C22—82no—
599C22—82no—
600C22#1171919912yes—
601C22#1171919912yes—
602C22#1171924711yescharacteristic
603C22#1171925611yescharacteristic
604C22#1171925811yescharacteristic
605C22#1171926312yes—
606C22#1171926412yes—
607C22#1171926811yescharacteristic
608C22#1171930612yes—
609C22#1171931512yes—
610C22#1171932512yes—
611C22#1171932612yes—
612C22#1171937412yes—
613C22#1171938312yes—
614C22#1171942012yes—
615C22#1171942412yes—
616C22C2×#8818011yescharacteristic
617C22C2×D4×Dic1311yescharacteristic
618C22#1171950311yescharacteristic
619C22#1171950311yescharacteristic
620C22#1171950311yescharacteristic
621C22#41×Dic1311yescharacteristic
622C22#1171953211yescharacteristic
623C4—24no—
624C4—24no—
625C4—42no—
626C4—44no—
627C4—524no—
628C4—524no—
629C4—1042no—
630C4—1042no—
631C2—84no—
632C2#1311743411yescharacteristic
633C2#1311756211yescharacteristic
634C2#1311765111yescharacteristic
635C2#1311778511yescharacteristic
636C2#1311780911yescharacteristic
637C2#1311788811yescharacteristic
638C2#1311799311yescharacteristic
639C2#1312157512yes—
640C2#1312162912yes—
641C2#1312273312yes—
642C2#1312277012yes—
subgroupG/H
center, FrattiniC24C23×D13
commutatorC23×C13C23×C4
FittingC2×C13×#390C2
socle, cosocleC24×C13C24
prime radicalC23×D4×C13C22
hypercenterC2×#390D13
hypocenterC13#14444
LCSG ▷ C23×C13 ▷ C13
UCSC1 ◁ C24 ◁ C2×#390

Complements

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

NH ≅ G/Ntype
1. C2×#11718612C2semidirect
2. #13113085C2semidirect
3. #13113081C2semidirect
5. #13113073C2semidirect
6. #13113068C2semidirect
7. #13113066C2semidirect
8. #13113063C2semidirect
21. C13×#390C4semidirect
84. #88071D4semidirect
284. #390Dic13semidirect
487. C13#14444semidirect

Maximal subgroups

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

subgroupindex
C2×#117186122
#131130852
#131130812
#131130772
#131130732
#131130682
#131130662
#131130632
C2×C13×#3902
#1444413