Finite Group Databasebeta

← #15868562

Subgroups — #15868562

Order 3328.

1952 conjugacy classes of subgroups.

The subgroup lattice has 658 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11718878C211yesmaximal, characteristic
2C2×#11718675C212yesmaximal, complemented
3C4×#88184C211yesmaximal, characteristic, complemented
4#13115184C212yesmaximal, complemented
5#13115153C211yesmaximal, characteristic, complemented
6#13115097C212yesmaximal, complemented
7#13115029C211yesmaximal, characteristic, complemented
8#13114767C211yesmaximal, characteristic, complemented
9#13114661C212yesmaximal, complemented
10#13114598C211yesmaximal, characteristic, complemented
11C22×C4×D4×C13C211yesmaximal, characteristic, complemented
12C23×D4×C13C2211yescharacteristic
13C24×C4×C13—21no—
14C2×C4×D4×C13—28no—
15C2×C4×D4×C13—21no—
16C2×C4×D4×C13—21no—
17C2×C4×D4×C13C2212yes—
18C2×C4×D4×C13C2212yes—
19C22×C13×C4⋊C4C2211yescharacteristic
20C22×C13×C22⋊C4—21no—
21C22×C42×C13C2211yescharacteristic
22C22×#6104C2212yes—
23C2×#88184C2211yescharacteristic
24C2×C4×C22⋊D13—21no—
25C2×C4×D52C2211yescharacteristic, complemented
26#11718878C418yescomplemented
27C2×#88072C2212yes—
28C2×#88072C2212yes—
29C2×#88072C2212yes—
30#11718874—21no—
31#11718873C2211yescharacteristic
32C2×C22⋊#1055C2211yescharacteristic
33C2×C22⋊#1055C2211yescharacteristic
34#11718675—28no—
35C2×#88049C2212yescomplemented
36C2×#88048C2212yescomplemented
37#11718601—21no—
38#11718600C2211yescharacteristic, complemented
39C4×#6103—21no—
40C4×C4⋊Dic13C2211yescharacteristic, complemented
41#11718589C2211yescharacteristic, complemented
42C2×#88043C2212yescomplemented
43#11718121C2212yescomplemented
44#11718121C2212yescomplemented
45#11718116C2212yescomplemented
46C25×C13—21no—
47C22×D4×C13—24no—
48C22×D4×C13—24no—
49C22×D4×C13—21no—
50C22×D4×C13—22no—
51C22×D4×C13C2×C414yescomplemented
52C22×D4×C13C2312yes—
53C23×C4×C13—28no—
54C23×C4×C13—21no—
55C23×C4×C13—22no—
56C23×C4×C13—22no—
57C23×C4×C13—44no—
58C23×C4×C13—44no—
59C23×C4×C13D411yescharacteristic
60C23×C4×C13D411yescharacteristic
61C23×C4×C13C2311yescharacteristic
62C22×C22⋊D13—21no—
63C22×D52C2311yescharacteristic
64C22×C4×D13—21no—
65C4×D4×C13—28no—
66C4×D4×C13—28no—
67C4×D4×C13—24no—
68C4×D4×C13—24no—
69C4×D4×C13—24no—
70C4×D4×C13D418yescomplemented
71C2×C13×C4⋊C4—21no—
72C2×C13×C4⋊C4—24no—
73C2×C13×C4⋊C4C2312yes—
74C2×C13×C22⋊C4—28no—
75C2×C13×C22⋊C4—22no—
76C2×C13×C22⋊C4—22no—
77C2×C42×C13—24no—
78C2×C42×C13D411yescharacteristic
79C2×C42×C13D411yescharacteristic
80C2×C42×C13C2311yescharacteristic
81C2×C42×C13C2311yescharacteristic
82C2×C22⋊Dic13—21no—
83C2×#6104—24no—
84C2×#6104—24no—
85C2×#6104C2×C418yescomplemented
86#88184—22no—
87#88184C2×C414yescomplemented
88C4×C22⋊D13—44no—
89C2×#6103—21no—
90C2×#6103—21no—
91C2×#6103—21no—
92C2×#6103—21no—
93C2×C4⋊Dic13C2311yescharacteristic
94C2×C4⋊Dic13C2311yescharacteristic
95C2×C4⋊Dic13C2311yescharacteristic
96C2×#6101—21no—
97C2×C4×Dic13—21no—
98C22×#1055C2312yes—
99C22×#1055C2312yes—
100C4×D52—24no—
101#88072—24no—
102#88072—24no—
103#88072C2×C418yescomplemented
104#88071—21no—
105#88071—21no—
106#88071—21no—
107C22⋊#1055—22no—
108C22⋊#1055C2×C414yescomplemented
109#88049—24no—
110#88048—24no—
111#88043—24no—
112#21452—131nomaximal, Sylow
113C24×C13—24no—
114C24×C13—24no—
115C24×C13—24no—
116C24×C13—22no—
117C24×C13—42no—
118C24×C13—42no—
119C24×C13—42no—
120C24×C13—42no—
121C24×C13C2×D411yescharacteristic
122C23×D13—21no—
123C2×D4×C13—24no—
124C2×D4×C13—24no—
125C2×D4×C13—216no—
126C2×D4×C13—216no—
127C2×D4×C13—24no—
128C2×D4×C13—22no—
129C2×D4×C13—22no—
130C2×D4×C13—22no—
131C2×D4×C13C22×C414yes—
132C2×D4×C13C2×D414yes—
133C2×D4×C13#4114yes—
134C22×C4×C13—22no—
135C22×C4×C13—22no—
136C22×C4×C13—22no—
137C22×C4×C13—22no—
138C22×C4×C13—22no—
139C22×C4×C13—24no—
140C22×C4×C13—28no—
141C22×C4×C13—22no—
142C22×C4×C13—22no—
143C22×C4×C13—22no—
144C22×C4×C13—28no—
145C22×C4×C13—41no—
146C22×C4×C13—432no—
147C22×C4×C13—44no—
148C22×C4×C13—44no—
149C22×C4×C13—44no—
150C22×C4×C13—42no—
151C22×C4×C13—42no—
152C22×C4×C13—44no—
153C22×C4×C13C22×C412yes—
154C22×C4×C13C2×D411yescharacteristic
155C22×C4×C13C2×D411yescharacteristic
156C22×C4×C13C2×D411yescharacteristic
157C22×C4×C13C2×D411yescharacteristic
158C22×C4×C13C2×D411yescharacteristic
159C22×C4×C13#4112yes—
160C22×C4×C13C2411yescharacteristic
161C2×C22⋊D13—24no—
162C2×C22⋊D13—42no—
163C2×C22⋊D13—42no—
164C22×Dic13—21no—
165C22×Dic13—21no—
166C22×Dic13—21no—
167C2×D52—22no—
168C2×D52—22no—
169C2×D52—21no—
170C2×D52C22×C412yescomplemented
171C2×C4×D13—42no—
172C2×C4×D13—42no—
173C2×C4×D13—42no—
174C13×C4⋊C4—24no—
175C13×C4⋊C4—22no—
176C13×C4⋊C4C2×D414yes—
177C13×C22⋊C4—28no—
178C13×C22⋊C4—28no—
179C42×C13—22no—
180C42×C13—22no—
181C42×C13—22no—
182C42×C13C2×D414yes—
183C22⋊Dic13—42no—
184#6104—216no—
185#6103—24no—
186#6103—42no—
187#6103—42no—
188#6103—42no—
189C4⋊Dic13—21no—
190C4⋊Dic13—22no—
191C4⋊Dic13—22no—
192C4⋊Dic13C22×C412yescomplemented
193#6101—42no—
194C4×Dic13—42no—
195C2×#1055—22no—
196C2×#1055—22no—
197C2×#1055—22no—
198C2×#1055—22no—
199C2×#1055—22no—
200C2×#1055C22×C414yescomplemented
201C22×C4×D4D1311yescharacteristic, complemented
202C2×#447—131no—
203C2×#442—132no—
204C4×#121—131no—
205#1879—132no—
206#1874—131no—
207#1864—132no—
208#1853—131no—
209#1786—131no—
210#1762—132no—
211#1749—131no—
212C23×C13—21no—
213C23×C13—21no—
214C23×C13—21no—
215C23×C13—21no—
216C23×C13—216no—
217C23×C13—24no—
218C23×C13—24no—
219C23×C13—24no—
220C23×C13—48no—
221C23×C13—48no—
222C23×C13—48no—
223C23×C13—48no—
224C23×C13—48no—
225C23×C13—42no—
226C23×C13—42no—
227C23×C13—42no—
228C23×C13—42no—
229C23×C13—48no—
230C23×C13C4×D412yes—
231C23×C13AGL(1,ℤ/8)12yes—
232C23×C13AGL(1,ℤ/8)12yes—
233C23×C13C22×D411yescharacteristic
234C22×D13—22no—
235C22×D13—41no—
236C22×D13—41no—
237C22×D13—41no—
238C22×D13—41no—
239C22×D13—41no—
240C22×D13—41no—
241D4×C13—216no—
242D4×C13—24no—
243D4×C13—24no—
244D4×C13—24no—
245D4×C13C4×D418yescomplemented
246C2×C4×C13—28no—
247C2×C4×C13—21no—
248C2×C4×C13—21no—
249C2×C4×C13—21no—
250C2×C4×C13—21no—
251C2×C4×C13—21no—
252C2×C4×C13—21no—
253C2×C4×C13—21no—
254C2×C4×C13—21no—
255C2×C4×C13—24no—
256C2×C4×C13—24no—
257C2×C4×C13—21no—
258C2×C4×C13—24no—
259C2×C4×C13—21no—
260C2×C4×C13—21no—
261C2×C4×C13—44no—
262C2×C4×C13—416no—
263C2×C4×C13—416no—
264C2×C4×C13—48no—
265C2×C4×C13—42no—
266C2×C4×C13—42no—
267C2×C4×C13C4×D412yes—
268C2×C4×C13#12012yes—
269C2×C4×C13#12012yes—
270C2×C4×C13AGL(1,ℤ/8)12yes—
271C2×C4×C13C23×C411yescharacteristic
272C2×C4×C13C22×D412yes—
273C2×C4×C13C2×#4112yes—
274C2×C4×C13C2×#4111yescharacteristic
275C22⋊D13—48no—
276C2×Dic13—22no—
277C2×Dic13—41no—
278C2×Dic13—41no—
279C2×Dic13—41no—
280C2×Dic13—41no—
281C2×Dic13—41no—
282C2×Dic13—41no—
283C2×Dic13—41no—
284C2×Dic13—41no—
285D52—24no—
286D52—42no—
287C4×D13—84no—
288#1055—24no—
289#1055—42no—
290C23×D4C2×D1311yescharacteristic
291C24×C4—21no—
292C22×D8—132no—
293C2×#121—131no—
294C2×C4×D4—28no—
295C2×C4×D4—21no—
296C2×C4×D4—21no—
297C2×C4×D4—131no—
298C2×C4×D4—261no—
299C2×C4×D4C2×D1312yes—
300C2×C4×D4C2×D1312yes—
301C22×C4⋊C4C2×D1311yescharacteristic
302C22×C22⋊C4—21no—
303C22×C42C2×D1311yescharacteristic
304#447—138no—
305#442—268no—
306C2×#106—132no—
307C2×#102—132no—
308C2×#102—132no—
309C2×#102—132no—
310C2×#102—132no—
311C2×C22⋊C8—131no—
312C2×C22⋊C8—131no—
313C2×#97—132no—
314#390—131no—
315#389—131no—
316#388—261no—
317#386—261no—
318#380—131no—
319C4×C4⋊C4—131no—
320C4×C22⋊C4—261no—
321#340—132no—
322#340—132no—
323#336—132no—
324C22×C13—22no—
325C22×C13—22no—
326C22×C13—22no—
327C22×C13—22no—
328C22×C13—22no—
329C22×C13—22no—
330C22×C13—28no—
331C22×C13—432no—
332C22×C13—44no—
333C22×C13—44no—
334C22×C13—44no—
335C22×C13—44no—
336C22×C13—44no—
337C22×C13—44no—
338C22×C13#44214yescomplemented
339C22×C13C2×C4×D411yescharacteristic
340C22×C13C2×#12011yescharacteristic
341C22×C13#52511yescharacteristic
342C22×C13C2×AGL(1,ℤ/8)11yescharacteristic
343C22×C13C2×AGL(1,ℤ/8)11yescharacteristic
344C22×C13C2×AGL(1,ℤ/8)11yescharacteristic
345C22×C13#57511yescharacteristic
346C2×D13—42no—
347C2×D13—42no—
348C2×D13—42no—
349C2×D13—81no—
350C2×D13—81no—
351C2×D13—81no—
352C2×D13—81no—
353C4×C13—22no—
354C4×C13—22no—
355C4×C13—22no—
356C4×C13—22no—
357C4×C13—44no—
358C4×C13—41no—
359C4×C13C2×C4×D412yes—
360C4×C13#52512yes—
361Dic13—42no—
362Dic13—81no—
363Dic13—81no—
364C25—21no—
365C22×D4—24no—
366C22×D4—24no—
367C22×D4—22no—
368C22×D4—21no—
369C22×D4—131no—
370C22×D4—261no—
371C22×D4C4×D1314yescomplemented
372C22×D4C22×D1312yes—
373C23×C4—28no—
374C23×C4—21no—
375C23×C4—22no—
376C23×C4—22no—
377C23×C4—44no—
378C23×C4—44no—
379C23×C4—261no—
380C23×C4C22⋊D1311yescharacteristic
381C23×C4C22⋊D1311yescharacteristic
382C23×C4C22×D1311yescharacteristic
383C2×D8—138no—
384C2×D8—264no—
385C2×D8—264no—
386C22×C8—132no—
387C22×C8—132no—
388#121—134no—
389#121—262no—
390C4×D4—28no—
391C4×D4—28no—
392C4×D4—24no—
393C4×D4—24no—
394C4×D4—24no—
395C4×D4—264no—
396C4×D4—524no—
397C4×D4C22⋊D1318yescomplemented
398C2×C4⋊C4—21no—
399C2×C4⋊C4—24no—
400C2×C4⋊C4—131no—
401C2×C4⋊C4—131no—
402C2×C4⋊C4—131no—
403C2×C4⋊C4—261no—
404C2×C4⋊C4C22×D1312yes—
405C2×C22⋊C4—28no—
406C2×C22⋊C4—22no—
407C2×C22⋊C4—22no—
408C2×C22⋊C4—261no—
409C2×C22⋊C4—261no—
410C2×C22⋊C4—261no—
411C2×C22⋊C4—261no—
412C2×C22⋊C4—261no—
413C2×C42—24no—
414C2×C42—261no—
415C2×C42C22⋊D1311yescharacteristic
416C2×C42C22⋊D1311yescharacteristic
417C2×C42C22×D1311yescharacteristic
418C2×C42C22×D1311yescharacteristic
419#106—264no—
420#102—138no—
421#102—264no—
422#102—264no—
423#102—264no—
424C22⋊C8—134no—
425C22⋊C8—262no—
426#97—264no—
427#95—261no—
428#95—261no—
429#95—261no—
430C2×C13—24no—
431C2×C13—48no—
432C2×C13C4×#12011yescharacteristic
433C2×C13C2×#44211yescharacteristic
434C2×C13C4×AGL(1,ℤ/8)11yescharacteristic
435C2×C13#300311yescharacteristic
436C2×C13#300611yescharacteristic
437C2×C13#310911yescharacteristic
438C2×C13#311411yescharacteristic
439D13—82no—
440C24—24no—
441C24—24no—
442C24—24no—
443C24—22no—
444C24—42no—
445C24—42no—
446C24—42no—
447C24—42no—
448C24—261no—
449C24C2×C22⋊D1311yescharacteristic
450C2×D4—24no—
451C2×D4—24no—
452C2×D4—216no—
453C2×D4—216no—
454C2×D4—24no—
455C2×D4—22no—
456C2×D4—22no—
457C2×D4—22no—
458C2×D4—132no—
459C2×D4—264no—
460C2×D4—262no—
461C2×D4—261no—
462C2×D4—262no—
463C2×D4—522no—
464C2×D4—522no—
465C2×D4C2×C4×D1314yes—
466C2×D4#612714yes—
467C2×D4C2×C22⋊D1314yes—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—24no—
474C22×C4—28no—
475C22×C4—22no—
476C22×C4—22no—
477C22×C4—22no—
478C22×C4—28no—
479C22×C4—41no—
480C22×C4—432no—
481C22×C4—44no—
482C22×C4—44no—
483C22×C4—44no—
484C22×C4—42no—
485C22×C4—42no—
486C22×C4—44no—
487C22×C4—261no—
488C22×C4—261no—
489C22×C4—261no—
490C22×C4—522no—
491C22×C4—522no—
492C22×C4—522no—
493C22×C4C2×C4×D1312yes—
494C22×C4#612712yes—
495C22×C4C2×C22⋊D1311yescharacteristic
496C22×C4C2×C22⋊D1311yescharacteristic
497C22×C4C2×C22⋊D1311yescharacteristic
498C22×C4C2×C22⋊D1311yescharacteristic
499C22×C4C2×C22⋊D1311yescharacteristic
500C22×C4C23×D1311yescharacteristic
501D8—2616no—
502C2×C8—134no—
503C2×C8—262no—
504C2×C8—262no—
505C2×C8—262no—
506C2×C8—262no—
507C2×C8—262no—
508C4⋊C4—24no—
509C4⋊C4—22no—
510C4⋊C4—132no—
511C4⋊C4—261no—
512C4⋊C4—262no—
513C4⋊C4—262no—
514C4⋊C4—522no—
515C4⋊C4C2×C22⋊D1314yes—
516C22⋊C4—28no—
517C22⋊C4—28no—
518C22⋊C4—264no—
519C22⋊C4—522no—
520C22⋊C4—522no—
521C22⋊C4—522no—
522C22⋊C4—522no—
523C42—22no—
524C42—22no—
525C42—22no—
526C42—522no—
527C42C2×C22⋊D1314yes—
528C13#2145211yescharacteristic, complemented, Sylow
529C23—21no—
530C23—21no—
531C23—21no—
532C23—21no—
533C23—216no—
534C23—24no—
535C23—24no—
536C23—24no—
537C23—48no—
538C23—48no—
539C23—48no—
540C23—48no—
541C23—48no—
542C23—48no—
543C23—42no—
544C23—42no—
545C23—42no—
546C23—42no—
547C23—262no—
548C23—521no—
549C23—521no—
550C23—521no—
551C23—521no—
552C23—521no—
553C23—521no—
554C23C4×C22⋊D1312yes—
555C23#8818612yes—
556C23#8820312yes—
557C23C22×C22⋊D1311yescharacteristic
558D4—216no—
559D4—24no—
560D4—24no—
561D4—24no—
562D4—264no—
563D4—528no—
564D4—522no—
565D4C4×C22⋊D1318yescomplemented
566C2×C4—28no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—24no—
576C2×C4—24no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—24no—
581C2×C4—44no—
582C2×C4—416no—
583C2×C4—416no—
584C2×C4—48no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—262no—
588C2×C4—521no—
589C2×C4—521no—
590C2×C4—521no—
591C2×C4—521no—
592C2×C4—521no—
593C2×C4—521no—
594C2×C4—521no—
595C2×C4—521no—
596C2×C4—1044no—
597C2×C4C4×C22⋊D1312yes—
598C2×C4#8818612yes—
599C2×C4#8820712yes—
600C2×C4#8820712yes—
601C2×C4C22×C4×D1311yescharacteristic
602C2×C4C2×#612712yes—
603C2×C4C2×#612711yescharacteristic
604C2×C4C22×C22⋊D1312yes—
605C8—264no—
606C8—522no—
607C22—22no—
608C22—22no—
609C22—22no—
610C22—22no—
611C22—22no—
612C22—22no—
613C22—28no—
614C22—432no—
615C22—44no—
616C22—44no—
617C22—44no—
618C22—44no—
619C22—44no—
620C22—44no—
621C22—522no—
622C22—522no—
623C22—522no—
624C22—1041no—
625C22—1041no—
626C22—1041no—
627C22—1041no—
628C22#1171867514yescomplemented
629C22C2×C4×C22⋊D1311yescharacteristic
630C22#1171949611yescharacteristic
631C22C2×#8818611yescharacteristic
632C22C2×#8818611yescharacteristic
633C22C2×#8820311yescharacteristic
634C22#1171952711yescharacteristic
635C22C2×#8820711yescharacteristic
636C4—22no—
637C4—22no—
638C4—22no—
639C4—22no—
640C4—44no—
641C4—41no—
642C4—522no—
643C4—1041no—
644C4—1041no—
645C4C2×C4×C22⋊D1312yes—
646C4#1171949612yes—
647C2—24no—
648C2—48no—
649C2—1042no—
650C2C4×#8820711yescharacteristic
651C2C2×#1171867511yescharacteristic
652C2C4×#8818611yescharacteristic
653C2#1312406311yescharacteristic
654C2#1312408811yescharacteristic
655C2#1312492711yescharacteristic
656C2#1312497411yescharacteristic
subgroupG/H
centerC23C22×C22⋊D13
commutatorC2×C4×C13C23×C4
FrattiniC22×C4C23×D13
FittingC22×C4×D4×C13C2
socleC23×C13C22×D4
prime radicalC2×#11718878C2
cosocleC22×C4×C13C24
hypercenterC22×C4×D4D13
hypocenterC13#21452
LCSG ▷ C2×C4×C13 ▷ C2×C13 ▷ C13
UCSC1 ◁ C23 ◁ C22×C42 ◁ C22×C4×D4

Complements

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

NH ≅ G/Ntype
2. C2×#11718675C2semidirect
3. C4×#88184C2semidirect
4. #13115184C2semidirect
5. #13115153C2semidirect
6. #13115097C2semidirect
7. #13115029C2semidirect
8. #13114767C2semidirect
9. #13114661C2semidirect
10. #13114598C2semidirect
11. C22×C4×D4×C13C2semidirect
25. C2×C4×D52C22semidirect
26. #11718878C4semidirect
35. C2×#88049C22semidirect
36. C2×#88048C22semidirect
38. #11718600C22semidirect
40. C4×C4⋊Dic13C22semidirect
41. #11718589C22semidirect
42. C2×#88043C22semidirect
43. #11718121C22semidirect
44. #11718121C22semidirect
45. #11718116C22semidirect
51. C22×D4×C13C2×C4semidirect
70. C4×D4×C13D4semidirect
85. C2×#6104C2×C4semidirect
87. #88184C2×C4semidirect
103. #88072C2×C4semidirect
108. C22⋊#1055C2×C4semidirect
170. C2×D52C22×C4semidirect
192. C4⋊Dic13C22×C4semidirect
200. C2×#1055C22×C4semidirect
201. C22×C4×D4D13semidirect
245. D4×C13C4×D4semidirect
338. C22×C13#442semidirect
371. C22×D4C4×D13semidirect
397. C4×D4C22⋊D13semidirect
528. C13#21452semidirect
565. D4C4×C22⋊D13semidirect
628. C22#11718675semidirect

Maximal subgroups

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

subgroupindex
C2×#117188782
C2×#117186752
C4×#881842
#131151842
#131151532
#131150972
#131150292
#131147672
#131146612
#131145982
C22×C4×D4×C132
#2145213