Finite Group Databasebeta

← #15866313

Subgroups — #15866313

Order 3328.

982 conjugacy classes of subgroups.

The subgroup lattice has 630 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11718522C211yesmaximal, characteristic
2C2×#11718522C211yesmaximal, characteristic
3C2×#11718350C211yesmaximal, characteristic
4#13117458C211yesmaximal, characteristic
5#13117451C211yesmaximal, characteristic
6#13115580C211yesmaximal, characteristic, complemented
7#13115576C211yesmaximal, characteristic, complemented
8#13115572C211yesmaximal, characteristic, complemented
9#13115571C211yesmaximal, characteristic, complemented
10#13115569C211yesmaximal, characteristic, complemented
11#13115568C211yesmaximal, characteristic, complemented
12#13115563C211yesmaximal, characteristic, complemented
13#13114551C211yesmaximal, characteristic
14D13×#340C211yesmaximal, characteristic
15C13×#1941C211yesmaximal, characteristic, complemented
16C22×#6094C2211yescharacteristic
17C22×C8×D13C2211yescharacteristic
18C2×#88146C2211yescharacteristic
19C2×#88146C2211yescharacteristic
20C2×C4⋊C4×D13C2211yescharacteristic
21C2×C4⋊C4×D13C2211yescharacteristic
22C2×#88126C2211yescharacteristic
23C2×C13×#106C2211yescharacteristic
24C2×C13×#106C2211yescharacteristic
25C2×C4×C8×C13C2211yescharacteristic
26C13×#389C2211yescharacteristic
27C13×#389C2211yescharacteristic
28C2×#88058C2211yescharacteristic
29C2×#88058C2211yescharacteristic
30C2×#88055C2211yescharacteristic
31C2×C8×Dic13C2211yescharacteristic
32#11718650—21no—
33#11718648—21no—
34#11718646C2211yescharacteristic
35#11718646C2211yescharacteristic
36#11718637—21no—
37#11718634C2211yescharacteristic
38#11718634C2211yescharacteristic
39#11718633—21no—
40C2×#88048C2211yescharacteristic
41C2×#88048C2211yescharacteristic
42C2×#88047C2211yescharacteristic
43C2×#88047C2211yescharacteristic
44#11718589C2211yescharacteristic
45#11718589C2211yescharacteristic
46C2×#88043C2211yescharacteristic
47#11718522—24no—
48#11718522C418yescomplemented
49#11718350—24no—
50#11718329—21no—
51#11718328—21no—
52#11718311—21no—
53#11718308—21no—
54C13×#340C2211yescharacteristic
55C13×#340C2211yescharacteristic
56#11718143C2211yescharacteristic
57#11718143C2211yescharacteristic
58#11718121C2211yescharacteristic
59#11718121C2211yescharacteristic
60#11718121C2211yescharacteristic
61#11718121C2211yescharacteristic
62C22×C4×D13—21no—
63C22×C4×D13—21no—
64C22×C4×D13C2311yescharacteristic
65C22×C8×C13C2311yescharacteristic
66C22×C8×C13C2311yescharacteristic
67C2×C13×C4⋊C4—21no—
68C2×C13×C4⋊C4—21no—
69C2×C13×C4⋊C4C2311yescharacteristic
70C2×C13×C4⋊C4C2311yescharacteristic
71C2×C13×C4⋊C4C2311yescharacteristic
72C2×C13×C4⋊C4C2311yescharacteristic
73C2×C42×C13C2311yescharacteristic
74C2×#6103—21no—
75C2×#6103—21no—
76C2×#6103D411yescharacteristic
77C2×#6103D411yescharacteristic
78C2×C4⋊Dic13—21no—
79C2×C4⋊Dic13—21no—
80C2×C4⋊Dic13C2311yescharacteristic
81C2×C4⋊Dic13C2311yescharacteristic
82C2×C4⋊Dic13C2311yescharacteristic
83C2×C4⋊Dic13C2311yescharacteristic
84C2×#6101—21no—
85C2×#6101—21no—
86C2×#6101—21no—
87C2×#6101—21no—
88C2×#6101—21no—
89C2×#6101—21no—
90C2×#6101D411yescharacteristic
91C2×#6101D411yescharacteristic
92C2×C4×Dic13—21no—
93C2×C4×Dic13—21no—
94C2×C4×Dic13C2311yescharacteristic
95C22×#1055C2311yescharacteristic
96C22×#1055C2311yescharacteristic
97C2×#6094C2×C414yes—
98C2×#6094D414yes—
99C2×#6094Q814yes—
100C2×C8×D13—22no—
101C2×C8×D13—22no—
102C2×C8×D13Q814yes—
103#88146—22no—
104#88146—22no—
105#88146—22no—
106#88146C2×C414yes—
107C4⋊C4×D13—22no—
108C4⋊C4×D13—22no—
109C4⋊C4×D13—22no—
110C4⋊C4×D13C2×C414yes—
111#88126—22no—
112#88126—22no—
113C13×#106—22no—
114C13×#106C2×C414yescomplemented
115C4×C8×C13—22no—
116C13×#95—21no—
117C13×#95—21no—
118#88071—21no—
119#88071—21no—
120#88071—21no—
121#88071—21no—
122#88071—21no—
123#88071—21no—
124#88058—22no—
125#88058C2×C414yescomplemented
126#88055—22no—
127C8×Dic13—22no—
128#88048—22no—
129#88048C2×C414yescomplemented
130#88047—22no—
131#88047C2×C414yescomplemented
132#88043—22no—
133#21160—131nomaximal, Sylow
134C23×D13C2×D411yescharacteristic
135C22×C4×C13—21no—
136C22×C4×C13—21no—
137C22×C4×C13—21no—
138C22×C4×C13—21no—
139C22×C4×C13C2×D411yescharacteristic
140C22×C4×C13C2×D411yescharacteristic
141C22×C4×C13C2411yescharacteristic
142C22×Dic13—21no—
143C22×Dic13—21no—
144C22×Dic13—21no—
145C22×Dic13—21no—
146C22×Dic13C2×D411yescharacteristic
147C22×Dic13C2×D411yescharacteristic
148C22×Dic13C2×D411yescharacteristic
149C2×C4×D13—24no—
150C2×C4×D13—42no—
151C2×C4×D13—81no—
152C2×C4×D13—81no—
153C2×C4×D13—81no—
154C2×C4×D13—81no—
155C2×C4×D13C22×C412yes—
156C2×C4×D13C2×D412yes—
157C2×C4×D13C2×Q812yes—
158C2×C4×D13C2×Q812yes—
159C2×C4×D13#4112yes—
160C2×C4×D13#4112yes—
161C2×C8×C13—21no—
162C2×C8×C13—21no—
163C2×C8×C13C22×C412yes—
164C2×C8×C13C2×D412yes—
165C2×C8×C13C2×Q812yes—
166C2×C8×C13C2×Q812yes—
167C13×C4⋊C4—21no—
168C13×C4⋊C4—21no—
169C13×C4⋊C4—21no—
170C13×C4⋊C4—21no—
171C13×C4⋊C4—21no—
172C13×C4⋊C4—21no—
173C13×C4⋊C4—81no—
174C13×C4⋊C4—81no—
175C13×C4⋊C4C22×C412yes—
176C13×C4⋊C4C22×C412yes—
177C42×C13—21no—
178C42×C13—21no—
179#6103—24no—
180#6103—22no—
181#6103—22no—
182#6103—42no—
183C4⋊Dic13—21no—
184C4⋊Dic13—21no—
185C4⋊Dic13—21no—
186C4⋊Dic13—21no—
187C4⋊Dic13—21no—
188C4⋊Dic13—21no—
189C4⋊Dic13—81no—
190C4⋊Dic13—81no—
191C4⋊Dic13C22×C412yes—
192C4⋊Dic13C22×C412yes—
193#6101—24no—
194#6101—22no—
195#6101—22no—
196#6101—42no—
197#6101—81no—
198#6101—81no—
199#6101—81no—
200#6101—81no—
201C4×Dic13—21no—
202C4×Dic13—21no—
203C4×Dic13—24no—
204C4×Dic13—42no—
205C2×#1055—21no—
206C2×#1055—21no—
207C2×#1055C22×C412yes—
208C2×#1055C2×D412yes—
209C2×#1055C2×Q812yes—
210C2×#1055C2×Q812yes—
211#6094C4⋊C4116yescomplemented
212C8×D13—28no—
213C2×#428—131no—
214C2×#428—131no—
215C2×#405—131no—
216#2317—131no—
217#2316—131no—
218#1944—131no—
219#1944—131no—
220#1943—131no—
221#1943—131no—
222#1942—131no—
223#1942—131no—
224#1941—131no—
225#1941D1311yescharacteristic, complemented
226#1735—131no—
227C2×#340—131no—
228C23×C13C22×D411yescharacteristic
229C22×D13—21no—
230C22×D13—21no—
231C22×D13—21no—
232C22×D13—21no—
233C22×D13C4×D412yes—
234C22×D13#12112yes—
235C22×D13C4⋊Q812yes—
236C2×C4×C13—22no—
237C2×C4×C13—22no—
238C2×C4×C13—21no—
239C2×C4×C13—21no—
240C2×C4×C13—21no—
241C2×C4×C13—21no—
242C2×C4×C13—21no—
243C2×C4×C13—21no—
244C2×C4×C13—41no—
245C2×C4×C13—41no—
246C2×C4×C13—41no—
247C2×C4×C13—41no—
248C2×C4×C13—41no—
249C2×C4×C13—41no—
250C2×C4×C13—41no—
251C2×C4×C13—41no—
252C2×C4×C13—41no—
253C2×C4×C13—41no—
254C2×C4×C13C23×C411yescharacteristic
255C2×C4×C13C22×D411yescharacteristic
256C2×C4×C13C22×Q811yescharacteristic
257C2×C4×C13C22×Q811yescharacteristic
258C2×C4×C13C2×#4111yescharacteristic
259C2×C4×C13C2×#4111yescharacteristic
260C2×Dic13—22no—
261C2×Dic13—22no—
262C2×Dic13—21no—
263C2×Dic13—21no—
264C2×Dic13—21no—
265C2×Dic13—21no—
266C2×Dic13—21no—
267C2×Dic13—21no—
268C2×Dic13—41no—
269C2×Dic13—41no—
270C2×Dic13—41no—
271C2×Dic13—41no—
272C2×Dic13—41no—
273C2×Dic13—41no—
274C2×Dic13—41no—
275C2×Dic13—41no—
276C2×Dic13—41no—
277C2×Dic13—41no—
278C2×Dic13C4×D412yes—
279C2×Dic13#12112yes—
280C2×Dic13C4⋊Q812yes—
281C4×D13—84no—
282C4×D13—84no—
283C4×D13C2×C4⋊C414yes—
284C4×D13C4×Q814yes—
285C4×D13C22⋊Q814yes—
286C4×D13#12514yes—
287C8×C13—22no—
288C8×C13C2×C4⋊C414yes—
289#1055—22no—
290#1055C2×C4⋊C414yes—
291C22×M16—131no—
292C23×C8—131no—
293C2×#117—131no—
294C2×#117—131no—
295C2×#117—131no—
296C22×C4⋊C4—131no—
297C22×C4⋊C4—131no—
298#428—138no—
299#428—264no—
300C2×#107—131no—
301C2×#107—131no—
302C2×#107—131no—
303C2×#107—131no—
304C2×#106—131no—
305C2×#106—131no—
306C2×#106C2×D1311yescharacteristic
307C2×#106C2×D1311yescharacteristic
308#405—264no—
309C2×#97—131no—
310C2×#97—131no—
311C2×C4×C8—131no—
312C2×C4×C8C2×D1311yescharacteristic
313#389—131no—
314#389—131no—
315#389—131no—
316#389—131no—
317#389—131no—
318#389—131no—
319#389—261no—
320#389—261no—
321#389C2×D1311yescharacteristic
322#389C2×D1311yescharacteristic
323#385—261no—
324#385—261no—
325#384—261no—
326#382—261no—
327#380—261no—
328#380—261no—
329#340—131no—
330#340—131no—
331#340—131no—
332#340—131no—
333#340—131no—
334#340—131no—
335#340C2×D1311yescharacteristic
336#340C2×D1311yescharacteristic
337C22×C13C2×C4×D411yescharacteristic
338C22×C13C2×#12111yescharacteristic
339C22×C13C2×C4⋊Q811yescharacteristic
340C22×C13#57611yescharacteristic
341C22×C13#57711yescharacteristic
342C22×C13#57711yescharacteristic
343C22×C13#57811yescharacteristic
344C2×D13—22no—
345C2×D13—22no—
346C2×D13—22no—
347C2×D13—22no—
348C2×D13—22no—
349C2×D13—22no—
350C2×D13#38914yes—
351C4×C13—42no—
352C4×C13—42no—
353C4×C13—41no—
354C4×C13—41no—
355C4×C13—81no—
356C4×C13—81no—
357C4×C13C22×C4⋊C411yescharacteristic
358C4×C13C2×C4×Q811yescharacteristic
359C4×C13C2×C22⋊Q811yescharacteristic
360C4×C13C2×#12511yescharacteristic
361Dic13—42no—
362Dic13—42no—
363Dic13—41no—
364Dic13—41no—
365Dic13—81no—
366Dic13—81no—
367Dic13#38914yes—
368C23×C4—131no—
369C23×C4—261no—
370C23×C4—261no—
371C2×M16—134no—
372C2×M16—134no—
373C2×M16—134no—
374C22×C8—131no—
375C22×C8—131no—
376C22×C8—134no—
377C22×C8—262no—
378C22×C8—262no—
379C22×C8C22×D1311yescharacteristic
380C22×C8C22×D1311yescharacteristic
381#117—134no—
382#117—262no—
383#117—262no—
384#117—262no—
385#117—262no—
386#117—262no—
387C2×C4⋊C4—21no—
388C2×C4⋊C4—21no—
389C2×C4⋊C4—131no—
390C2×C4⋊C4—131no—
391C2×C4⋊C4—134no—
392C2×C4⋊C4—131no—
393C2×C4⋊C4—131no—
394C2×C4⋊C4—131no—
395C2×C4⋊C4—131no—
396C2×C4⋊C4—262no—
397C2×C4⋊C4—262no—
398C2×C4⋊C4—262no—
399C2×C4⋊C4—261no—
400C2×C4⋊C4—261no—
401C2×C4⋊C4—261no—
402C2×C4⋊C4—261no—
403C2×C4⋊C4—261no—
404C2×C4⋊C4—261no—
405C2×C4⋊C4—261no—
406C2×C4⋊C4—261no—
407C2×C4⋊C4C22×D1311yescharacteristic
408C2×C4⋊C4C22×D1311yescharacteristic
409C2×C4⋊C4C22×D1311yescharacteristic
410C2×C4⋊C4C22×D1311yescharacteristic
411C2×C22⋊C4—131no—
412C2×C22⋊C4—131no—
413C2×C22⋊C4—261no—
414C2×C22⋊C4—261no—
415C2×C42—131no—
416C2×C42—261no—
417C2×C42—261no—
418C2×C42C22×D1311yescharacteristic
419#107—134no—
420#107—134no—
421#107—262no—
422#107—262no—
423#106—22no—
424#106—134no—
425#106—262no—
426#106C4×D1314yescomplemented
427#97—262no—
428#97—262no—
429C4×C8—22no—
430C4×C8—262no—
431#95—21no—
432#95—21no—
433#95—261no—
434#95—261no—
435#95—261no—
436#95—261no—
437#95—261no—
438#95—261no—
439C2×C13C2×#38911yescharacteristic
440C2×C13#294611yescharacteristic
441C2×C13#294711yescharacteristic
442C2×C13#306211yescharacteristic
443C2×C13#306311yescharacteristic
444C2×C13#316311yescharacteristic
445C2×C13#316411yescharacteristic
446D13—24no—
447C24—131no—
448C22×C4—21no—
449C22×C4—21no—
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—131no—
453C22×C4—132no—
454C22×C4—132no—
455C22×C4—132no—
456C22×C4—132no—
457C22×C4—132no—
458C22×C4—132no—
459C22×C4—131no—
460C22×C4—131no—
461C22×C4—264no—
462C22×C4—261no—
463C22×C4—261no—
464C22×C4—261no—
465C22×C4—261no—
466C22×C4—522no—
467C22×C4—1041no—
468C22×C4—1041no—
469C22×C4—1041no—
470C22×C4—1041no—
471C22×C4D4×D1311yescharacteristic
472C22×C4D4×D1311yescharacteristic
473C22×C4C23×D1311yescharacteristic
474M16—1316no—
475C2×C8—21no—
476C2×C8—21no—
477C2×C8—132no—
478C2×C8—132no—
479C2×C8—132no—
480C2×C8—132no—
481C2×C8—268no—
482C2×C8—261no—
483C2×C8—261no—
484C2×C8C2×C4×D1312yes—
485C2×C8D4×D1312yes—
486C2×C8Q8×D1312yes—
487C2×C8Q8×D1312yes—
488C4⋊C4—21no—
489C4⋊C4—21no—
490C4⋊C4—21no—
491C4⋊C4—21no—
492C4⋊C4—21no—
493C4⋊C4—21no—
494C4⋊C4—81no—
495C4⋊C4—81no—
496C4⋊C4—132no—
497C4⋊C4—132no—
498C4⋊C4—264no—
499C4⋊C4—262no—
500C4⋊C4—262no—
501C4⋊C4—261no—
502C4⋊C4—261no—
503C4⋊C4—261no—
504C4⋊C4—261no—
505C4⋊C4—261no—
506C4⋊C4—261no—
507C4⋊C4—522no—
508C4⋊C4—1041no—
509C4⋊C4—1041no—
510C4⋊C4—1041no—
511C4⋊C4—1041no—
512C4⋊C4—1041no—
513C4⋊C4—1041no—
514C4⋊C4C2×C4×D1312yes—
515C4⋊C4C2×C4×D1312yes—
516C22⋊C4—262no—
517C22⋊C4—262no—
518C22⋊C4—264no—
519C22⋊C4—522no—
520C42—21no—
521C42—21no—
522C42—264no—
523C42—261no—
524C42—261no—
525C42—522no—
526C13#2116011yescharacteristic, complemented, Sylow
527C23—132no—
528C23—132no—
529C23—132no—
530C23—261no—
531C23—261no—
532C23—261no—
533C23—261no—
534C23C2×D4×D1311yescharacteristic
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—132no—
554C2×C4—132no—
555C2×C4—132no—
556C2×C4—134no—
557C2×C4—134no—
558C2×C4—134no—
559C2×C4—134no—
560C2×C4—261no—
561C2×C4—261no—
562C2×C4—261no—
563C2×C4—261no—
564C2×C4—262no—
565C2×C4—262no—
566C2×C4—261no—
567C2×C4—261no—
568C2×C4—521no—
569C2×C4—521no—
570C2×C4—521no—
571C2×C4—521no—
572C2×C4—521no—
573C2×C4—521no—
574C2×C4—521no—
575C2×C4—521no—
576C2×C4—521no—
577C2×C4—521no—
578C2×C4—1044no—
579C2×C4—1044no—
580C2×C4C22×C4×D1311yescharacteristic
581C2×C4C2×D4×D1311yescharacteristic
582C2×C4C2×Q8×D1311yescharacteristic
583C2×C4C2×Q8×D1311yescharacteristic
584C2×C4#41×D1311yescharacteristic
585C2×C4#41×D1311yescharacteristic
586C8—22no—
587C8—134no—
588C8—262no—
589C8C4⋊C4×D1314yes—
590C22—134no—
591C22—262no—
592C22—262no—
593C22—262no—
594C22—262no—
595C22—262no—
596C22—262no—
597C22C4×D4×D1311yescharacteristic
598C22D13×#12111yescharacteristic
599C22D13×C4⋊Q811yescharacteristic
600C22#1171946711yescharacteristic
601C22#1171946711yescharacteristic
602C22#1171947411yescharacteristic
603C22#1171947611yescharacteristic
604C4—42no—
605C4—42no—
606C4—41no—
607C4—41no—
608C4—81no—
609C4—81no—
610C4—134no—
611C4—522no—
612C4—522no—
613C4—521no—
614C4—521no—
615C4—1041no—
616C4—1041no—
617C4C2×C4⋊C4×D1311yescharacteristic
618C4C4×Q8×D1311yescharacteristic
619C4D13×C22⋊Q811yescharacteristic
620C4D13×#12511yescharacteristic
621C2—264no—
622C2D13×#38911yescharacteristic
623C2#1312366311yescharacteristic
624C2#1312367211yescharacteristic
625C2#1312450911yescharacteristic
626C2#1312452011yescharacteristic
627C2#1312536111yescharacteristic
628C2#1312537311yescharacteristic
subgroupG/H
centerC23C2×D4×D13
commutatorC2×C4×C13C23×C4
FrattiniC22×C4C23×D13
FittingC13×#1941C2
socleC23×C13C22×D4
prime radicalC23×D13C2×D4
cosocleC22×C4×C13C24
hypercenter#1941D13
hypocenterC13#21160
LCSG ▷ C2×C4×C13 ▷ C2×C13 ▷ C13
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1941

Complements

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

NH ≅ G/Ntype
6. #13115580C2semidirect
7. #13115576C2semidirect
8. #13115572C2semidirect
9. #13115571C2semidirect
10. #13115569C2semidirect
11. #13115568C2semidirect
12. #13115563C2semidirect
15. C13×#1941C2semidirect
48. #11718522C4semidirect
114. C13×#106C2×C4semidirect
125. #88058C2×C4semidirect
129. #88048C2×C4semidirect
131. #88047C2×C4semidirect
211. #6094C4⋊C4semidirect
225. #1941D13semidirect
426. #106C4×D13semidirect
526. C13#21160semidirect

Maximal subgroups

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

subgroupindex
C2×#117185222
C2×#117183502
#131174582
#131174512
#131155802
#131155762
#131155722
#131155712
#131155692
#131155682
#131155632
#131145512
D13×#3402
C13×#19412
#2116013