Finite Group Databasebeta

← #15793636

Subgroups — #15793636

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×#11718610C211yesmaximal, characteristic
2C2×#11718634C211yesmaximal, characteristic, complemented
3C2×#11718612C211yesmaximal, characteristic
4#13113103C211yesmaximal, characteristic, complemented
5#13113095C211yesmaximal, characteristic, complemented
6#13113090C212yesmaximal, complemented
7#13113087C212yesmaximal, complemented
8#13113083C212yesmaximal
9#13113079C212yesmaximal
10#13113061C211yesmaximal, characteristic, complemented
11C13×#1439C211yesmaximal, characteristic
12C22×C13×C4⋊C4C2211yescharacteristic
13C22×C13×C22⋊C4C2211yescharacteristic
14C22×C13×C22⋊C4C2212yes—
15C22×C22⋊Dic13C2211yescharacteristic
16C22×C22⋊Dic13C2211yescharacteristic
17C22×C22⋊Dic13C2212yes—
18C22×C4⋊Dic13C2211yescharacteristic
19C22×C4⋊Dic13C2211yescharacteristic
20C22×#6101C2212yes—
21C22×#6101C2212yes—
22C22×C4×Dic13C2211yescharacteristic
23C2×C13×#95C2211yescharacteristic
24C2×C13×#95C2212yes—
25C2×#88071C2212yes—
26C2×#88071C2211yescharacteristic
27C2×#88071C2211yescharacteristic
28C2×#88071C2212yes—
29C2×#88071C2212yes—
30C2×#88071C2212yes—
31C2×#88071C2212yes—
32C2×#88071C2212yes—
33C2×#88071C2212yes—
34C2×#88071C2211yescharacteristic
35#11718634—24no—
36#11718612—24no—
37#11718610C418yescomplemented
38C25×C13C2311yescharacteristic
39C23×C4×C13C2312yes—
40C23×C4×C13C2312yes—
41C23×C4×C13C2311yescharacteristic
42C23×C4×C13C2311yescharacteristic
43C23×Dic13C2311yescharacteristic
44C23×Dic13C2311yescharacteristic
45C23×Dic13C2312yes—
46C23×Dic13C2312yes—
47C23×Dic13C2311yescharacteristic
48C23×Dic13C2311yescharacteristic
49C2×C13×C4⋊C4—22no—
50C2×C13×C4⋊C4—42no—
51C2×C13×C22⋊C4—22no—
52C2×C13×C22⋊C4—22no—
53C2×C13×C22⋊C4—42no—
54C2×C13×C22⋊C4—42no—
55C2×C13×C22⋊C4—42no—
56C2×C13×C22⋊C4C2×C414yes—
57C2×C22⋊Dic13—22no—
58C2×C22⋊Dic13—22no—
59C2×C22⋊Dic13—42no—
60C2×C22⋊Dic13—42no—
61C2×C22⋊Dic13—42no—
62C2×C22⋊Dic13—42no—
63C2×C22⋊Dic13C2×C418yes—
64C2×C4⋊Dic13—22no—
65C2×C4⋊Dic13—22no—
66C2×C4⋊Dic13—22no—
67C2×C4⋊Dic13—42no—
68C2×C4⋊Dic13C2×C414yes—
69C2×#6101—24no—
70C2×#6101—42no—
71C2×#6101—42no—
72C2×#6101—42no—
73C2×#6101—42no—
74C2×#6101C2×C418yes—
75C2×C4×Dic13—42no—
76C2×C4×Dic13D414yes—
77C13×#95—22no—
78C13×#95—42no—
79#88071—22no—
80#88071—22no—
81#88071—24no—
82#88071—42no—
83#88071—42no—
84#88071—42no—
85#88071—42no—
86#88071—42no—
87#88071—42no—
88#88071C2×C414yes—
89#14628—131nomaximal, Sylow
90C24×C13—21no—
91C24×C13—21no—
92C24×C13—22no—
93C24×C13—22no—
94C24×C13—22no—
95C24×C13—21no—
96C24×C13—21no—
97C24×C13—21no—
98C24×C13—21no—
99C24×C13—22no—
100C24×C13C22×C412yes—
101C24×C13C2411yescharacteristic
102C22×C4×C13—22no—
103C22×C4×C13—22no—
104C22×C4×C13—21no—
105C22×C4×C13—21no—
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×C13C22×C412yes—
123C22×C4×C13C22×C412yes—
124C22×C4×C13C2×D412yes—
125C22×C4×C13#4112yes—
126C22×Dic13—22no—
127C22×Dic13—22no—
128C22×Dic13—22no—
129C22×Dic13—22no—
130C22×Dic13—21no—
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—21no—
144C22×Dic13—21no—
145C22×Dic13—21no—
146C22×Dic13—21no—
147C22×Dic13—21no—
148C22×Dic13—21no—
149C22×Dic13—22no—
150C22×Dic13—22no—
151C22×Dic13—22no—
152C22×Dic13—21no—
153C22×Dic13—21no—
154C22×Dic13—22no—
155C22×Dic13—21no—
156C22×Dic13C22×C414yes—
157C22×Dic13C22×C414yes—
158C22×Dic13C2×D412yes—
159C22×Dic13C2×D412yes—
160C13×C4⋊C4—44no—
161C13×C22⋊C4—28no—
162C13×C22⋊C4—48no—
163C22⋊Dic13—48no—
164C22⋊Dic13—44no—
165C22⋊Dic13—44no—
166C4⋊Dic13—28no—
167C4⋊Dic13—44no—
168#6101—48no—
169#6101—48no—
170C4×Dic13—44no—
171C2×#399—131no—
172C2×#389—131no—
173C2×#388—131no—
174#1445—131no—
175#1442—131no—
176#1441—132no—
177#1441—132no—
178#1440—132no—
179#1439—132no—
180#1439D1311yescharacteristic
181#1434—131no—
182C23×C13—22no—
183C23×C13—24no—
184C23×C13—22no—
185C23×C13—24no—
186C23×C13—22no—
187C23×C13—41no—
188C23×C13—41no—
189C23×C13—41no—
190C23×C13—41no—
191C23×C13—41no—
192C23×C13—41no—
193C23×C13—42no—
194C23×C13—42no—
195C23×C13—41no—
196C23×C13—41no—
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×C13C23×C411yescharacteristic
206C23×C13C22×D411yescharacteristic
207C23×C13C2×#4111yescharacteristic
208C23×C13H(5,𝔽2)12yes—
209C23×C13H(5,𝔽2)12yes—
210C23×C13H(5,𝔽2)12yes—
211C23×C13H(5,𝔽2)11yescharacteristic
212C23×C132−1+411yescharacteristic
213C23×C132−1+411yescharacteristic
214C23×C132−1+412yes—
215C23×C132−1+411yescharacteristic
216C2×C4×C13—24no—
217C2×C4×C13—22no—
218C2×C4×C13—22no—
219C2×C4×C13—24no—
220C2×C4×C13—22no—
221C2×C4×C13—24no—
222C2×C4×C13—41no—
223C2×C4×C13—41no—
224C2×C4×C13—42no—
225C2×C4×C13—42no—
226C2×C4×C13—42no—
227C2×C4×C13—42no—
228C2×C4×C13—42no—
229C2×C4×C13—42no—
230C2×C4×C13—42no—
231C2×C4×C13—42no—
232C2×C4×C13—42no—
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×C13C4×D414yes—
240C2×Dic13—24no—
241C2×Dic13—24no—
242C2×Dic13—24no—
243C2×Dic13—24no—
244C2×Dic13—24no—
245C2×Dic13—24no—
246C2×Dic13—22no—
247C2×Dic13—22no—
248C2×Dic13—22no—
249C2×Dic13—22no—
250C2×Dic13—22no—
251C2×Dic13—22no—
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—41no—
265C2×Dic13—41no—
266C2×Dic13—41no—
267C2×Dic13—41no—
268C2×Dic13—41no—
269C2×Dic13—41no—
270C2×Dic13—42no—
271C2×Dic13—42no—
272C2×Dic13—42no—
273C2×Dic13—42no—
274C22×C4⋊C4—131no—
275C22×C4⋊C4—132no—
276C22×C4⋊C4—132no—
277C22×C4⋊C4—131no—
278C22×C4⋊C4C2×D1311yescharacteristic
279C22×C22⋊C4—132no—
280C22×C22⋊C4—131no—
281C22×C22⋊C4—131no—
282C22×C22⋊C4C2×D1311yescharacteristic
283C22×C22⋊C4C2×D1312yes—
284C22×C42—131no—
285#399—138no—
286#389—264no—
287#388—264no—
288C2×#95—131no—
289C2×#95—132no—
290C2×#95—131no—
291C2×#95—131no—
292C2×#95—132no—
293C2×#95—132no—
294C2×#95—132no—
295C2×#95—132no—
296C2×#95—132no—
297C2×#95—132no—
298C2×#95C2×D1311yescharacteristic
299C2×#95C2×D1312yes—
300C22×C13—42no—
301C22×C13—42no—
302C22×C13—42no—
303C22×C13—44no—
304C22×C13—44no—
305C22×C13—81no—
306C22×C13—81no—
307C22×C13—81no—
308C22×C13—82no—
309C22×C13—82no—
310C22×C13—81no—
311C22×C13C2×C4×D411yescharacteristic
312C22×C13#51812yes—
313C22×C13#51911yescharacteristic
314C22×C13#52011yescharacteristic
315C22×C13#52012yes—
316C22×C13#53411yescharacteristic
317C22×C13#53511yescharacteristic
318C22×C13#53511yescharacteristic
319C22×C13#53611yescharacteristic
320C22×C13#53611yescharacteristic
321C22×C13#53711yescharacteristic
322C22×C13#53912yes—
323C22×C13#54012yes—
324C22×C13#54111yescharacteristic
325C22×C13#54211yescharacteristic
326C22×C13#56012yes—
327C22×C13#56212yes—
328C22×C13#56212yes—
329C22×C13#56212yes—
330C22×C13#56212yes—
331C22×C13#56312yes—
332C22×C13#56312yes—
333C22×C13#56312yes—
334C4×C13—24no—
335C4×C13—42no—
336C4×C13—82no—
337C4×C13—82no—
338Dic13—42no—
339Dic13—44no—
340Dic13—44no—
341Dic13—42no—
342Dic13—81no—
343Dic13—81no—
344C25C22×D1311yescharacteristic
345C23×C4—132no—
346C23×C4—131no—
347C23×C4—132no—
348C23×C4—131no—
349C23×C4—131no—
350C23×C4—131no—
351C23×C4C22×D1312yes—
352C23×C4C22×D1312yes—
353C23×C4C22×D1311yescharacteristic
354C23×C4C22×D1311yescharacteristic
355C2×C4⋊C4—22no—
356C2×C4⋊C4—42no—
357C2×C4⋊C4—134no—
358C2×C4⋊C4—138no—
359C2×C4⋊C4—262no—
360C2×C4⋊C4—262no—
361C2×C4⋊C4—264no—
362C2×C4⋊C4—262no—
363C2×C4⋊C4—522no—
364C2×C4⋊C4—522no—
365C2×C4⋊C4—522no—
366C2×C4⋊C4—522no—
367C2×C4⋊C4—522no—
368C2×C22⋊C4—22no—
369C2×C22⋊C4—22no—
370C2×C22⋊C4—42no—
371C2×C22⋊C4—42no—
372C2×C22⋊C4—42no—
373C2×C22⋊C4—138no—
374C2×C22⋊C4—262no—
375C2×C22⋊C4—262no—
376C2×C22⋊C4—522no—
377C2×C22⋊C4—522no—
378C2×C22⋊C4—522no—
379C2×C22⋊C4—522no—
380C2×C22⋊C4C4×D1314yes—
381C2×C42—134no—
382C2×C42—522no—
383#95—22no—
384#95—42no—
385#95—134no—
386#95—262no—
387#95—264no—
388#95—262no—
389#95—522no—
390#95—522no—
391#95—522no—
392#95—522no—
393#95—522no—
394#95—522no—
395C2×C13—82no—
396C2×C13#232211yescharacteristic
397C2×C13#232311yescharacteristic
398C2×C13#232811yescharacteristic
399C2×C13#237612yes—
400C2×C13#237812yes—
401C2×C13#281311yescharacteristic
402C2×C13#281611yescharacteristic
403C2×C13#282011yescharacteristic
404C2×C13#282111yescharacteristic
405C2×C13#283612yes—
406C2×C13#283712yes—
407C24—21no—
408C24—21no—
409C24—21no—
410C24—22no—
411C24—22no—
412C24—21no—
413C24—22no—
414C24—21no—
415C24—21no—
416C24—22no—
417C24C2×C4×D1312yes—
418C24C23×D1311yescharacteristic
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—21no—
422C22×C4—21no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
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—134no—
440C22×C4—134no—
441C22×C4—132no—
442C22×C4—132no—
443C22×C4—262no—
444C22×C4—262no—
445C22×C4—262no—
446C22×C4—262no—
447C22×C4—261no—
448C22×C4—262no—
449C22×C4—262no—
450C22×C4—262no—
451C22×C4—262no—
452C22×C4—262no—
453C22×C4—262no—
454C22×C4—262no—
455C22×C4—261no—
456C22×C4—262no—
457C22×C4—262no—
458C22×C4—262no—
459C22×C4—262no—
460C22×C4—262no—
461C22×C4—261no—
462C22×C4—261no—
463C22×C4—261no—
464C22×C4—261no—
465C22×C4—261no—
466C22×C4—262no—
467C22×C4—261no—
468C22×C4—262no—
469C22×C4—262no—
470C22×C4—261no—
471C22×C4—261no—
472C22×C4—262no—
473C22×C4C2×C4×D1312yes—
474C22×C4C2×C4×D1312yes—
475C22×C4D4×D1312yes—
476C22×C4#612912yes—
477C4⋊C4—44no—
478C4⋊C4—268no—
479C4⋊C4—528no—
480C4⋊C4—528no—
481C4⋊C4—524no—
482C22⋊C4—28no—
483C22⋊C4—48no—
484C22⋊C4—528no—
485C22⋊C4—524no—
486C22⋊C4—524no—
487C42—524no—
488C13#1462811yescharacteristic, complemented, Sylow
489C23—22no—
490C23—22no—
491C23—22no—
492C23—24no—
493C23—24no—
494C23—41no—
495C23—42no—
496C23—42no—
497C23—41no—
498C23—41no—
499C23—42no—
500C23—42no—
501C23—41no—
502C23—41no—
503C23—42no—
504C23—42no—
505C23—41no—
506C23—42no—
507C23—42no—
508C23—41no—
509C23—42no—
510C23—42no—
511C23—41no—
512C23C22×C4×D1311yescharacteristic
513C23C2×D4×D1311yescharacteristic
514C23C2×#612911yescharacteristic
515C23#8825111yescharacteristic
516C23#8825112yes—
517C23#8825112yes—
518C23#8825411yescharacteristic
519C23#8825612yes—
520C23#8825711yescharacteristic
521C23#8825711yescharacteristic
522C23#8825712yes—
523C2×C4—24no—
524C2×C4—22no—
525C2×C4—24no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—24no—
529C2×C4—42no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—264no—
547C2×C4—264no—
548C2×C4—264no—
549C2×C4—264no—
550C2×C4—264no—
551C2×C4—264no—
552C2×C4—262no—
553C2×C4—262no—
554C2×C4—262no—
555C2×C4—262no—
556C2×C4—262no—
557C2×C4—262no—
558C2×C4—522no—
559C2×C4—522no—
560C2×C4—522no—
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—521no—
571C2×C4—521no—
572C2×C4—521no—
573C2×C4—522no—
574C2×C4—521no—
575C2×C4—522no—
576C2×C4—521no—
577C2×C4—521no—
578C2×C4—522no—
579C2×C4—522no—
580C2×C4#8813514yes—
581C22—44no—
582C22—44no—
583C22—42no—
584C22—42no—
585C22—42no—
586C22—81no—
587C22—81no—
588C22—81no—
589C22—81no—
590C22—82no—
591C22—82no—
592C22C2×#8813511yescharacteristic
593C22#1171920911yescharacteristic
594C22#1171922111yescharacteristic
595C22#1171925012yes—
596C22#1171925112yes—
597C22#1171929411yescharacteristic
598C22#1171930012yes—
599C22#1171930712yes—
600C22#1171930811yescharacteristic
601C22#1171931311yescharacteristic
602C22#1171932412yes—
603C22#1171932612yes—
604C22#1171933011yescharacteristic
605C22#1171933311yescharacteristic
606C22#1171935112yes—
607C22#1171935212yes—
608C22#1171935311yescharacteristic
609C22#1171935612yes—
610C22#1171937212yes—
611C22#1171937911yescharacteristic
612C22#1171938912yes—
613C22#1171941712yes—
614C22#1171942011yescharacteristic
615C4—24no—
616C4—42no—
617C4—82no—
618C4—82no—
619C4—522no—
620C4—524no—
621C4—522no—
622C4—524no—
623C4—1041no—
624C4—1041no—
625C2—82no—
626C2#1311751011yescharacteristic
627C2#1311751411yescharacteristic
628C2#1311756011yescharacteristic
629C2#1311798512yes—
630C2#1311800312yes—
631C2#1312283511yescharacteristic
632C2#1312286211yescharacteristic
633C2#1312289211yescharacteristic
634C2#1312289911yescharacteristic
635C2#1312303712yes—
636C2#1312305212yes—
subgroupG/H
center, FrattiniC24C23×D13
commutatorC23×C13C23×C4
FittingC13×#1439C2
socle, cosocleC24×C13C24
prime radicalC25×C13C23
hypercenter#1439D13
hypocenterC13#14628
LCSG ▷ C23×C13 ▷ C13
UCSC1 ◁ C24 ◁ #1439

Complements

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

NH ≅ G/Ntype
2. C2×#11718634C2semidirect
4. #13113103C2semidirect
5. #13113095C2semidirect
6. #13113090C2semidirect
7. #13113087C2semidirect
10. #13113061C2semidirect
37. #11718610C4semidirect
488. C13#14628semidirect

Maximal subgroups

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

subgroupindex
C2×#117186102
C2×#117186342
C2×#117186122
#131131032
#131130952
#131130902
#131130872
#131130832
#131130792
#131130612
C13×#14392
#1462813