Finite Group Databasebeta

← #15870065

Subgroups — #15870065

Order 3328.

972 conjugacy classes of subgroups.

The subgroup lattice has 664 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11718664C211yesmaximal, characteristic
2#13117342C211yesmaximal, characteristic
3#117×Dic13C211yesmaximal, characteristic, complemented
4#13115461C211yesmaximal, characteristic, complemented
5#13115454C211yesmaximal, characteristic, complemented
6#13115426C211yesmaximal, characteristic, complemented
7#13115419C211yesmaximal, characteristic, complemented
8#13115156C211yesmaximal, characteristic
9#13115033C211yesmaximal, characteristic, complemented
10#13114728C211yesmaximal, characteristic, complemented
11#13114724C211yesmaximal, characteristic, complemented
12#13114650C211yesmaximal, characteristic, complemented
13#13114650C211yesmaximal, characteristic, complemented
14C22⋊C8×Dic13C211yesmaximal, characteristic, complemented
15C13×#1825C211yesmaximal, characteristic, complemented
16C2×C13×#117C2211yescharacteristic
17C2×#88184C2211yescharacteristic
18C2×#88180C2211yescharacteristic
19C22×C4×Dic13C2211yescharacteristic
20C2×#88147C2211yescharacteristic
21C2×#88135—21no—
22C2×C13×C22⋊C8C2211yescharacteristic
23C13×#380C2211yescharacteristic
24C2×C22⋊#1055C2211yescharacteristic
25C2×#88061C2211yescharacteristic
26C2×#88061C2211yescharacteristic
27C2×C8×Dic13C2211yescharacteristic
28C2×C8×Dic13C2211yescharacteristic
29#11718664C418yescomplemented
30#11718649C2211yescharacteristic
31C4⋊C4×Dic13C2211yescharacteristic
32C4⋊C4×Dic13C2211yescharacteristic
33C4⋊C4×Dic13C2211yescharacteristic
34C4⋊C4×Dic13C2211yescharacteristic
35C4⋊C4×Dic13C2211yescharacteristic
36#11718634C2211yescharacteristic
37C2×#88049C2211yescharacteristic
38C2×#88049C2211yescharacteristic
39C2×#88048C2211yescharacteristic
40C2×#88048C2211yescharacteristic
41C2×#88047C2211yescharacteristic
42C2×#88047C2211yescharacteristic
43#11718619—21no—
44C22⋊C4×Dic13—21no—
45C22⋊C4×Dic13—21no—
46#11718593—21no—
47C42×Dic13—21no—
48#11718330—21no—
49#11718327—21no—
50C13×#340C2211yescharacteristic
51C13×#340C2211yescharacteristic
52C13×#340C2211yescharacteristic
53C13×#340C2211yescharacteristic
54#11718143C2211yescharacteristic
55#11718143C2211yescharacteristic
56#11718142C2211yescharacteristic
57#11718142C2211yescharacteristic
58#11718121C2211yescharacteristic
59#11718121C2211yescharacteristic
60C23×C4×C13C2311yescharacteristic
61C22×C22⋊D13—21no—
62C23×Dic13D411yescharacteristic
63C23×Dic13D411yescharacteristic
64C22×D52C2311yescharacteristic
65C22×C4×D13—21no—
66C22×C8×C13C2311yescharacteristic
67C22×C8×C13C2311yescharacteristic
68C13×#117—22no—
69C13×#117C2×C414yescomplemented
70C2×C13×C4⋊C4C2311yescharacteristic
71C2×C13×C4⋊C4C2311yescharacteristic
72C2×C13×C4⋊C4C2311yescharacteristic
73C2×C13×C4⋊C4C2311yescharacteristic
74C2×C13×C22⋊C4—21no—
75C2×C13×C22⋊C4—21no—
76C2×C42×C13—21no—
77C2×C22⋊Dic13—21no—
78#88184—22no—
79#88184C2×C414yes—
80C2×#6103—21no—
81C2×#6103—21no—
82C2×#6103—21no—
83C2×#6103—21no—
84#88180—22no—
85#88180—22no—
86C2×C4⋊Dic13C2311yescharacteristic
87C2×C4⋊Dic13C2311yescharacteristic
88C2×C4⋊Dic13C2311yescharacteristic
89C2×#6101—21no—
90C2×C4×Dic13—21no—
91C2×C4×Dic13—22no—
92C2×C4×Dic13—22no—
93C2×C4×Dic13—21no—
94C2×C4×Dic13—21no—
95C2×C4×Dic13—21no—
96C2×C4×Dic13—21no—
97C2×C4×Dic13—21no—
98C2×C4×Dic13D411yescharacteristic
99C2×C4×Dic13D411yescharacteristic
100C2×C4×Dic13C2311yescharacteristic
101C2×C4×Dic13C2311yescharacteristic
102C22×#1055C2311yescharacteristic
103C22×#1055C2311yescharacteristic
104#88147—22no—
105#88147—22no—
106#88135—44no—
107C13×C22⋊C8—22no—
108C13×#95—21no—
109#88071—21no—
110#88071—21no—
111#88071—21no—
112#88071—21no—
113#88071—21no—
114#88071—21no—
115C22⋊#1055C2×C414yescomplemented
116#88061—22no—
117#88061—22no—
118C8×Dic13—22no—
119C8×Dic13—22no—
120#88049C2×C414yescomplemented
121#88049C2×C414yescomplemented
122#88048—22no—
123#88048C2×C414yescomplemented
124#88047—22no—
125#88047C2×C414yescomplemented
126C4×#482—131nomaximal, Sylow
127C24×C13C2×D411yescharacteristic
128C23×D13—21no—
129C22×C4×C13—21no—
130C22×C4×C13—21no—
131C22×C4×C13—21no—
132C22×C4×C13—21no—
133C22×C4×C13—21no—
134C22×C4×C13—21no—
135C22×C4×C13—21no—
136C22×C4×C13—21no—
137C22×C4×C13C22×C412yes—
138C22×C4×C13C2×D411yescharacteristic
139C22×C4×C13#4112yes—
140C22×C4×C13C2411yescharacteristic
141C2×C22⋊D13—24no—
142C2×C22⋊D13—42no—
143C2×C22⋊D13—42no—
144C22×Dic13—21no—
145C22×Dic13—21no—
146C22×Dic13—21no—
147C22×Dic13—22no—
148C22×Dic13—22no—
149C22×Dic13—22no—
150C22×Dic13—22no—
151C22×Dic13—22no—
152C22×Dic13—41no—
153C22×Dic13C2×D411yescharacteristic
154C22×Dic13C2×D411yescharacteristic
155C22×Dic13C2×D411yescharacteristic
156C22×Dic13C2×D411yescharacteristic
157C2×D52—21no—
158C2×D52—21no—
159C2×D52—21no—
160C2×D52—21no—
161C2×D52—21no—
162C2×D52C22×C412yes—
163C2×C4×D13—42no—
164C2×C4×D13—42no—
165C2×C4×D13—42no—
166C2×C8×C13—21no—
167C2×C8×C13—21no—
168C2×C8×C13—21no—
169C2×C8×C13—21no—
170C2×C8×C13—21no—
171C2×C8×C13—21no—
172C13×C4⋊C4—21no—
173C13×C4⋊C4—21no—
174C13×C4⋊C4—21no—
175C13×C4⋊C4—21no—
176C13×C4⋊C4—21no—
177C13×C4⋊C4—21no—
178C13×C4⋊C4C22×C412yes—
179C13×C4⋊C4C22×C412yes—
180C13×C22⋊C4—24no—
181C13×C22⋊C4—42no—
182C42×C13—24no—
183C22⋊Dic13—24no—
184#6103—24no—
185#6103—42no—
186#6103—42no—
187#6103—42no—
188C4⋊Dic13—21no—
189C4⋊Dic13—21no—
190C4⋊Dic13—21no—
191C4⋊Dic13C22×C412yes—
192C4⋊Dic13#4112yes—
193C4⋊Dic13#4112yes—
194#6101—42no—
195C4×Dic13—24no—
196C4×Dic13—21no—
197C4×Dic13—21no—
198C4×Dic13—22no—
199C4×Dic13—22no—
200C4×Dic13—42no—
201C4×Dic13—42no—
202C4×Dic13—42no—
203C4×Dic13—42no—
204C4×Dic13—42no—
205C4×Dic13#4112yes—
206C4×Dic13#4112yes—
207C2×#1055—21no—
208C2×#1055—21no—
209C2×#1055—21no—
210C2×#1055—21no—
211C2×#1055C22×C412yes—
212C2×#1055C22×C412yes—
213C2×#482—131no—
214C4×#121—131no—
215C4×#117—131no—
216#1926—131no—
217#1925—131no—
218#1921—131no—
219#1920—131no—
220#1875—131no—
221#1854—131no—
222#1825D1311yescharacteristic, complemented
223C4×#107—131no—
224C4×#106—131no—
225C4×#102—131no—
226C4×#102—131no—
227C4×C22⋊C8—131no—
228C23×C13—21no—
229C23×C13—21no—
230C23×C13—21no—
231C23×C13—21no—
232C23×C13—21no—
233C23×C13—21no—
234C23×C13C4×D412yes—
235C23×C13C22×D411yescharacteristic
236C22×D13—22no—
237C22×D13—41no—
238C22×D13—41no—
239C22×D13—41no—
240C22×D13—41no—
241C22×D13—41no—
242C22×D13—41no—
243C2×C4×C13—21no—
244C2×C4×C13—21no—
245C2×C4×C13—22no—
246C2×C4×C13—22no—
247C2×C4×C13—22no—
248C2×C4×C13—22no—
249C2×C4×C13—22no—
250C2×C4×C13—22no—
251C2×C4×C13—41no—
252C2×C4×C13—41no—
253C2×C4×C13—41no—
254C2×C4×C13—41no—
255C2×C4×C13—41no—
256C2×C4×C13—41no—
257C2×C4×C13—41no—
258C2×C4×C13—41no—
259C2×C4×C13—41no—
260C2×C4×C13—41no—
261C2×C4×C13C4×D412yes—
262C2×C4×C13C23×C411yescharacteristic
263C2×C4×C13C2×#4111yescharacteristic
264C2×C4×C13C2×#4111yescharacteristic
265C2×C4×C13C2×#4111yescharacteristic
266C2×C4×C13C2×#4111yescharacteristic
267C2×C4×C13C2×#4111yescharacteristic
268C22⋊D13—48no—
269C2×Dic13—21no—
270C2×Dic13—21no—
271C2×Dic13—21no—
272C2×Dic13—21no—
273C2×Dic13—22no—
274C2×Dic13—21no—
275C2×Dic13—22no—
276C2×Dic13—22no—
277C2×Dic13—21no—
278C2×Dic13—21no—
279C2×Dic13—44no—
280C2×Dic13—41no—
281C2×Dic13—41no—
282C2×Dic13—41no—
283C2×Dic13—41no—
284C2×Dic13—41no—
285C2×Dic13—41no—
286C2×Dic13—44no—
287C2×Dic13#12312yes—
288C2×Dic13#12312yes—
289C2×Dic13#13512yes—
290C2×Dic13#13512yes—
291C2×Dic13AGL(1,ℤ/8)12yes—
292D52—22no—
293D52—22no—
294D52—41no—
295D52—41no—
296C4×D13—84no—
297C8×C13—41no—
298C8×C13—41no—
299#1055—22no—
300#1055—22no—
301C2×#121—131no—
302C2×C4×D4—131no—
303C2×C4×D4—261no—
304C2×#117—131no—
305C2×#117C2×D1311yescharacteristic
306C22×C42—131no—
307#482—138no—
308C2×#107—131no—
309C2×#107—131no—
310C2×#106—131no—
311C2×#106—131no—
312C2×#102—131no—
313C2×#102—131no—
314C2×#102—131no—
315C2×#102—131no—
316C2×C22⋊C8—131no—
317C2×C22⋊C8C2×D1311yescharacteristic
318C2×C4×C8—131no—
319C2×C4×C8—131no—
320#390—131no—
321#389—131no—
322#388—261no—
323#386—261no—
324#380C2×D1311yescharacteristic
325C4×C4⋊C4—131no—
326C4×C4⋊C4—131no—
327C4×C4⋊C4—131no—
328C4×C4⋊C4—131no—
329C4×C4⋊C4—131no—
330C4×C22⋊C4—261no—
331C4×C22⋊C4—261no—
332C4×C22⋊C4—261no—
333#376—261no—
334C43—261no—
335#340—131no—
336#340—131no—
337#340—131no—
338#340—131no—
339#340C2×D1311yescharacteristic
340#340C2×D1311yescharacteristic
341#340C2×D1311yescharacteristic
342#340C2×D1311yescharacteristic
343#336—131no—
344#336—131no—
345C22×C13—22no—
346C22×C13—22no—
347C22×C13—22no—
348C22×C13—41no—
349C22×C13—41no—
350C22×C13—41no—
351C22×C13—41no—
352C22×C13C2×C4×D411yescharacteristic
353C22×C13C2×#12311yescharacteristic
354C22×C13#52511yescharacteristic
355C22×C13C2×#13511yescharacteristic
356C22×C13C2×#13511yescharacteristic
357C22×C13C2×AGL(1,ℤ/8)11yescharacteristic
358C22×C13#57511yescharacteristic
359C2×D13—42no—
360C2×D13—42no—
361C2×D13—42no—
362C2×D13—81no—
363C2×D13—81no—
364C2×D13—81no—
365C2×D13—81no—
366C4×C13—22no—
367C4×C13—42no—
368C4×C13—42no—
369C4×C13—81no—
370C4×C13—81no—
371C4×C13C4×#4111yescharacteristic
372C4×C13C4×#4111yescharacteristic
373C4×C13#52911yescharacteristic
374C4×C13#52911yescharacteristic
375Dic13—22no—
376Dic13—22no—
377Dic13—41no—
378Dic13—42no—
379Dic13—42no—
380Dic13—42no—
381Dic13#48214yescomplemented
382C22×D4—131no—
383C22×D4—261no—
384C23×C4—131no—
385C23×C4—131no—
386C23×C4—261no—
387C23×C4C22×D1311yescharacteristic
388C22×C8—131no—
389C22×C8—131no—
390C22×C8C22×D1311yescharacteristic
391C22×C8C22×D1311yescharacteristic
392#121—134no—
393#121—262no—
394C4×D4—262no—
395C4×D4—262no—
396C4×D4—524no—
397#117—22no—
398#117—262no—
399#117—262no—
400#117C4×D1314yescomplemented
401C2×C4⋊C4—131no—
402C2×C4⋊C4—131no—
403C2×C4⋊C4—131no—
404C2×C4⋊C4—261no—
405C2×C4⋊C4C22×D1311yescharacteristic
406C2×C4⋊C4C22×D1311yescharacteristic
407C2×C4⋊C4C22×D1311yescharacteristic
408C2×C4⋊C4C22×D1311yescharacteristic
409C2×C22⋊C4—21no—
410C2×C22⋊C4—21no—
411C2×C22⋊C4—261no—
412C2×C22⋊C4—261no—
413C2×C22⋊C4—261no—
414C2×C22⋊C4—261no—
415C2×C22⋊C4—261no—
416C2×C42—21no—
417C2×C42—131no—
418C2×C42—131no—
419C2×C42—131no—
420C2×C42—131no—
421C2×C42—262no—
422C2×C42—262no—
423C2×C42—261no—
424C2×C42—261no—
425C2×C42—261no—
426C2×C42—261no—
427C2×C42—261no—
428C2×C42—261no—
429#107—134no—
430#107—262no—
431#106—134no—
432#106—262no—
433#102—134no—
434#102—134no—
435#102—262no—
436#102—262no—
437C22⋊C8—22no—
438C22⋊C8—134no—
439C4×C8—262no—
440C4×C8—262no—
441#95—21no—
442#95—261no—
443#95—261no—
444#95—261no—
445#95—261no—
446#95—261no—
447#95—261no—
448C2×C13—42no—
449C2×C13C4×#12311yescharacteristic
450C2×C13C4×#13511yescharacteristic
451C2×C13C4×AGL(1,ℤ/8)11yescharacteristic
452C2×C13C2×#48211yescharacteristic
453C2×C13#309111yescharacteristic
454C2×C13#310211yescharacteristic
455C2×C13#310611yescharacteristic
456D13—82no—
457C24—261no—
458C24D4×D1311yescharacteristic
459C2×D4—132no—
460C2×D4—264no—
461C2×D4—261no—
462C2×D4—261no—
463C2×D4—261no—
464C2×D4—261no—
465C2×D4—261no—
466C2×D4—522no—
467C2×D4—522no—
468C22×C4—21no—
469C22×C4—21no—
470C22×C4—21no—
471C22×C4—21no—
472C22×C4—21no—
473C22×C4—21no—
474C22×C4—21no—
475C22×C4—21no—
476C22×C4—131no—
477C22×C4—131no—
478C22×C4—131no—
479C22×C4—131no—
480C22×C4—262no—
481C22×C4—262no—
482C22×C4—262no—
483C22×C4—262no—
484C22×C4—262no—
485C22×C4—261no—
486C22×C4—261no—
487C22×C4—261no—
488C22×C4—522no—
489C22×C4—522no—
490C22×C4—522no—
491C22×C4—521no—
492C22×C4C2×C4×D1312yes—
493C22×C4D4×D1311yescharacteristic
494C22×C4#612912yes—
495C22×C4C23×D1311yescharacteristic
496C2×C8—21no—
497C2×C8—21no—
498C2×C8—21no—
499C2×C8—21no—
500C2×C8—21no—
501C2×C8—21no—
502C2×C8—132no—
503C2×C8—132no—
504C2×C8—261no—
505C2×C8—261no—
506C2×C8—261no—
507C2×C8—261no—
508C4⋊C4—21no—
509C4⋊C4—21no—
510C4⋊C4—21no—
511C4⋊C4—21no—
512C4⋊C4—21no—
513C4⋊C4—21no—
514C4⋊C4—132no—
515C4⋊C4—132no—
516C4⋊C4—132no—
517C4⋊C4—261no—
518C4⋊C4—261no—
519C4⋊C4—261no—
520C4⋊C4—522no—
521C4⋊C4C2×C4×D1312yes—
522C4⋊C4C2×C4×D1312yes—
523C22⋊C4—24no—
524C22⋊C4—42no—
525C22⋊C4—264no—
526C22⋊C4—264no—
527C22⋊C4—522no—
528C22⋊C4—522no—
529C22⋊C4—522no—
530C42—24no—
531C42—132no—
532C42—132no—
533C42—262no—
534C42—262no—
535C42—264no—
536C42—261no—
537C42—261no—
538C42—522no—
539C42—522no—
540C42—522no—
541C42—522no—
542C42—522no—
543C13C4×#48211yescharacteristic, complemented, Sylow
544C23—21no—
545C23—21no—
546C23—21no—
547C23—21no—
548C23—21no—
549C23—21no—
550C23—262no—
551C23—521no—
552C23—521no—
553C23—521no—
554C23—521no—
555C23—521no—
556C23—521no—
557C23#8813512yes—
558C23C2×D4×D1311yescharacteristic
559D4—262no—
560D4—262no—
561D4—528no—
562D4—521no—
563D4—521no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—132no—
583C2×C4—132no—
584C2×C4—132no—
585C2×C4—132no—
586C2×C4—132no—
587C2×C4—261no—
588C2×C4—261no—
589C2×C4—261no—
590C2×C4—261no—
591C2×C4—262no—
592C2×C4—261no—
593C2×C4—261no—
594C2×C4—261no—
595C2×C4—262no—
596C2×C4—262no—
597C2×C4—524no—
598C2×C4—524no—
599C2×C4—521no—
600C2×C4—521no—
601C2×C4—521no—
602C2×C4—521no—
603C2×C4—521no—
604C2×C4—521no—
605C2×C4—1044no—
606C2×C4#8813512yes—
607C2×C4C22×C4×D1311yescharacteristic
608C2×C4C2×#612911yescharacteristic
609C2×C4#41×D1311yescharacteristic
610C2×C4#41×D1311yescharacteristic
611C2×C4#41×D1311yescharacteristic
612C2×C4#41×D1311yescharacteristic
613C8—41no—
614C8—41no—
615C8—262no—
616C8—262no—
617C22—22no—
618C22—22no—
619C22—22no—
620C22—41no—
621C22—41no—
622C22—41no—
623C22—41no—
624C22—522no—
625C22—522no—
626C22—522no—
627C22—1041no—
628C22—1041no—
629C22—1041no—
630C22—1041no—
631C22C2×#8813511yescharacteristic
632C22D13×#12311yescharacteristic
633C22#1171936111yescharacteristic
634C22D13×#13511yescharacteristic
635C22D13×#13511yescharacteristic
636C22D13×AGL(1,ℤ/8)11yescharacteristic
637C22#1171948211yescharacteristic
638C4—22no—
639C4—42no—
640C4—42no—
641C4—81no—
642C4—81no—
643C4—134no—
644C4—262no—
645C4—262no—
646C4—522no—
647C4—522no—
648C4—521no—
649C4—522no—
650C4#1171922711yescharacteristic
651C4#1171922711yescharacteristic
652C4#1171935411yescharacteristic
653C4#1171935411yescharacteristic
654C2—42no—
655C2—1042no—
656C2#1311735311yescharacteristic
657C2#1312360411yescharacteristic
658C2#1312364911yescharacteristic
659C2D13×#48211yescharacteristic
660C2#1312475411yescharacteristic
661C2#1312485711yescharacteristic
662C2#1312489111yescharacteristic
subgroupG/H
centerC23C2×D4×D13
commutatorC2×C4×C13C23×C4
FrattiniC22×C4C23×D13
FittingC13×#1825C2
socleC23×C13C22×D4
prime radicalC2×#88184C22
cosocleC22×C4×C13C24
hypercenter#1825D13
hypocenterC13C4×#482
LCSG ▷ C2×C4×C13 ▷ C2×C13 ▷ C13
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1825

Complements

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

NH ≅ G/Ntype
3. #117×Dic13C2semidirect
4. #13115461C2semidirect
5. #13115454C2semidirect
6. #13115426C2semidirect
7. #13115419C2semidirect
9. #13115033C2semidirect
10. #13114728C2semidirect
11. #13114724C2semidirect
12. #13114650C2semidirect
13. #13114650C2semidirect
14. C22⋊C8×Dic13C2semidirect
15. C13×#1825C2semidirect
29. #11718664C4semidirect
69. C13×#117C2×C4semidirect
115. C22⋊#1055C2×C4semidirect
120. #88049C2×C4semidirect
121. #88049C2×C4semidirect
123. #88048C2×C4semidirect
125. #88047C2×C4semidirect
222. #1825D13semidirect
381. Dic13#482semidirect
400. #117C4×D13semidirect
543. C13C4×#482semidirect

Maximal subgroups

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

subgroupindex
C2×#117186642
#131173422
#117×Dic132
#131154612
#131154542
#131154262
#131154192
#131151562
#131150332
#131147282
#131147242
#131146502
C22⋊C8×Dic132
C13×#18252
C4×#48213