Finite Group Databasebeta

← #15870283

Subgroups — #15870283

Order 3328.

844 conjugacy classes of subgroups.

The subgroup lattice has 618 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11718384C211yesmaximal, characteristic
2#13117518C211yesmaximal, characteristic
3#13117192C211yesmaximal, characteristic
4#13115471C211yesmaximal, characteristic, complemented
5#13115471C211yesmaximal, characteristic, complemented
6#13115429C211yesmaximal, characteristic, complemented
7#13115415C211yesmaximal, characteristic, complemented
8#13115141C211yesmaximal, characteristic
9#13115031C211yesmaximal, characteristic
10#13114932C211yesmaximal, characteristic
11#13114733C211yesmaximal, characteristic, complemented
12C4×#88057C211yesmaximal, characteristic, complemented
13#13114667C211yesmaximal, characteristic, complemented
14C4×#88056C211yesmaximal, characteristic, complemented
15C13×#1750C211yesmaximal, characteristic
16C2×C13×#117C2211yescharacteristic
17C2×#88180C2211yescharacteristic
18C22×C4⋊Dic13C2211yescharacteristic
19C2×#88178C2211yescharacteristic
20C2×C4×Dic26C2211yescharacteristic
21C2×C13×C22⋊C8C2211yescharacteristic
22C2×C13×C22⋊C8C2211yescharacteristic
23C2×C13×#97C2211yescharacteristic
24C2×C4×C8×C13C2211yescharacteristic
25#11718873C2211yescharacteristic
26C2×#88058C2211yescharacteristic
27C2×#88057C2211yescharacteristic
28C2×#88057C2211yescharacteristic
29C2×#88057C2211yescharacteristic
30C2×#88056C2211yescharacteristic
31C2×#88056C2211yescharacteristic
32C2×#88056C2211yescharacteristic
33C2×#88056C2211yescharacteristic
34#11718639—21no—
35C4⋊C4×Dic13—21no—
36C4⋊C4×Dic13—21no—
37#11718633—21no—
38#11718606—21no—
39#11718605—21no—
40C22⋊C4×Dic13—21no—
41C22⋊C4×Dic13—21no—
42C4×C4⋊Dic13C2211yescharacteristic
43C4×C4⋊Dic13C2211yescharacteristic
44C4×C4⋊Dic13C2211yescharacteristic
45C4×C4⋊Dic13C2211yescharacteristic
46C4×C4⋊Dic13C2211yescharacteristic
47#11718590C2211yescharacteristic
48#11718589C2211yescharacteristic
49#11718384C418yescomplemented
50C13×#336C2211yescharacteristic
51C13×#336C2211yescharacteristic
52#11718143C2211yescharacteristic
53#11718143C2211yescharacteristic
54#11718143C2211yescharacteristic
55#11718143C2211yescharacteristic
56#11718143C2211yescharacteristic
57#11718143C2211yescharacteristic
58#11718143C2211yescharacteristic
59#11718143C2211yescharacteristic
60C23×C4×C13C2311yescharacteristic
61C23×Dic13—21no—
62C22×Dic26C2311yescharacteristic
63C22×C8×C13C2311yescharacteristic
64C22×C8×C13C2311yescharacteristic
65C22×C8×C13C2311yescharacteristic
66C22×C8×C13C2311yescharacteristic
67C13×#117—22no—
68C13×#117—22no—
69C2×C13×C4⋊C4D411yescharacteristic
70C2×C13×C4⋊C4D411yescharacteristic
71C2×C13×C22⋊C4D411yescharacteristic
72C2×C13×C22⋊C4D411yescharacteristic
73C2×C42×C13C2311yescharacteristic
74C2×C42×C13C2311yescharacteristic
75C2×C22⋊Dic13—21no—
76C2×C22⋊Dic13—21no—
77C2×C22⋊Dic13—21no—
78#88180—22no—
79#88180C2×C414yes—
80C2×C4⋊Dic13—21no—
81C2×C4⋊Dic13—22no—
82C2×C4⋊Dic13—22no—
83C2×C4⋊Dic13C2311yescharacteristic
84C2×C4⋊Dic13C2311yescharacteristic
85C2×C4⋊Dic13C2311yescharacteristic
86C2×C4⋊Dic13C2311yescharacteristic
87C2×C4⋊Dic13C2311yescharacteristic
88C2×C4⋊Dic13C2311yescharacteristic
89C2×C4⋊Dic13C2311yescharacteristic
90#88178—22no—
91#88178C2×C414yes—
92C2×#6101—21no—
93C2×#6101—21no—
94C2×#6101—21no—
95C2×C4×Dic13—21no—
96C2×C4×Dic13—21no—
97C2×C4×Dic13—21no—
98C2×C4×Dic13—21no—
99C2×C4×Dic13—21no—
100C2×C4×Dic13—21no—
101C2×C4×Dic13—21no—
102C4×Dic26—22no—
103C4×Dic26—22no—
104C13×C22⋊C8—22no—
105C13×C22⋊C8C2×C414yes—
106C13×#97—22no—
107C4×C8×C13—22no—
108#88071—21no—
109#88071—21no—
110#88071—21no—
111#88071—21no—
112#88071—21no—
113#88071—21no—
114#88071—21no—
115#88071—21no—
116#88071—21no—
117#88058C2×C414yescomplemented
118#88057—22no—
119#88057—22no—
120#88057C2×C414yescomplemented
121#88056—22no—
122#88056—22no—
123#88056C2×C414yescomplemented
124#88056C2×C414yescomplemented
125#21591—131nomaximal, Sylow
126C24×C13C2×D411yescharacteristic
127C22×C4×C13—21no—
128C22×C4×C13—21no—
129C22×C4×C13—21no—
130C22×C4×C13—21no—
131C22×C4×C13C22×C412yes—
132C22×C4×C13C2×D411yescharacteristic
133C22×C4×C13C2×D411yescharacteristic
134C22×C4×C13C2×D411yescharacteristic
135C22×C4×C13C2×D411yescharacteristic
136C22×C4×C13C2×D411yescharacteristic
137C22×C4×C13#4112yes—
138C22×C4×C13C2411yescharacteristic
139C22×Dic13—21no—
140C22×Dic13—21no—
141C22×Dic13—21no—
142C22×Dic13—21no—
143C22×Dic13—21no—
144C22×Dic13—21no—
145C22×Dic13—24no—
146C22×Dic13—21no—
147C22×Dic13—21no—
148C22×Dic13—81no—
149C22×Dic13—81no—
150C2×Dic26—21no—
151C2×Dic26—21no—
152C2×Dic26—21no—
153C2×Dic26—21no—
154C2×Dic26—21no—
155C2×Dic26C22×C412yes—
156C2×C8×C13—21no—
157C2×C8×C13—21no—
158C2×C8×C13—21no—
159C2×C8×C13—21no—
160C2×C8×C13—21no—
161C2×C8×C13—21no—
162C2×C8×C13—21no—
163C2×C8×C13—21no—
164C2×C8×C13—21no—
165C2×C8×C13—21no—
166C2×C8×C13C22×C412yes—
167C2×C8×C13C22×C412yes—
168C13×C4⋊C4—22no—
169C13×C4⋊C4—22no—
170C13×C22⋊C4—22no—
171C13×C22⋊C4—22no—
172C42×C13—21no—
173C42×C13—21no—
174C42×C13#4112yes—
175C42×C13#4112yes—
176C22⋊Dic13—24no—
177C22⋊Dic13—24no—
178C22⋊Dic13—81no—
179C4⋊Dic13—24no—
180C4⋊Dic13—21no—
181C4⋊Dic13—21no—
182C4⋊Dic13—21no—
183C4⋊Dic13—21no—
184C4⋊Dic13—21no—
185C4⋊Dic13—21no—
186C4⋊Dic13—21no—
187C4⋊Dic13—21no—
188C4⋊Dic13—21no—
189C4⋊Dic13C22×C412yes—
190C4⋊Dic13C22×C412yes—
191C4⋊Dic13C22×C412yes—
192C4⋊Dic13#4112yes—
193C4⋊Dic13#4112yes—
194#6101—24no—
195#6101—42no—
196#6101—81no—
197C4×Dic13—24no—
198C4×Dic13—42no—
199C4×Dic13—81no—
200C4×Dic13—81no—
201C4×Dic13—81no—
202C4×Dic13—81no—
203C4×Dic13—81no—
204C2×#485—131no—
205#2323—131no—
206#2274—131no—
207#1927—131no—
208#1927—131no—
209#1921—131no—
210#1919—131no—
211#1872—131no—
212#1853—131no—
213#1825—131no—
214#1776—131no—
215C4×#106—131no—
216#1763—131no—
217C4×#103—131no—
218#1750D1311yescharacteristic
219C23×C13—21no—
220C23×C13—21no—
221C23×C13—21no—
222C23×C13—21no—
223C23×C13—21no—
224C23×C13—21no—
225C23×C13C4×D412yes—
226C23×C13C22×D411yescharacteristic
227C2×C4×C13—21no—
228C2×C4×C13—21no—
229C2×C4×C13—21no—
230C2×C4×C13—21no—
231C2×C4×C13—21no—
232C2×C4×C13—21no—
233C2×C4×C13—22no—
234C2×C4×C13—22no—
235C2×C4×C13—22no—
236C2×C4×C13—22no—
237C2×C4×C13—21no—
238C2×C4×C13—21no—
239C2×C4×C13—21no—
240C2×C4×C13—21no—
241C2×C4×C13—21no—
242C2×C4×C13—21no—
243C2×C4×C13C4×D412yes—
244C2×C4×C13#12312yes—
245C2×C4×C13#12312yes—
246C2×C4×C13C23×C411yescharacteristic
247C2×C4×C13C2×#4111yescharacteristic
248C2×C4×C13C2×#4111yescharacteristic
249C2×C4×C13C2×#4111yescharacteristic
250C2×C4×C13C2×#4111yescharacteristic
251C2×C4×C13C2×#4111yescharacteristic
252C2×Dic13—22no—
253C2×Dic13—22no—
254C2×Dic13—22no—
255C2×Dic13—22no—
256C2×Dic13—22no—
257C2×Dic13—22no—
258C2×Dic13—41no—
259C2×Dic13—41no—
260C2×Dic13—41no—
261C2×Dic13—41no—
262C2×Dic13—41no—
263C2×Dic13—41no—
264C2×Dic13—41no—
265C2×Dic13—41no—
266C2×Dic13—41no—
267C2×Dic13—41no—
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—84no—
277Dic26—22no—
278Dic26—22no—
279Dic26—41no—
280Dic26—41no—
281C8×C13—22no—
282C8×C13—22no—
283C8×C13—41no—
284C8×C13—41no—
285C2×C22⋊Q8—131no—
286C2×C4×Q8—131no—
287C2×#117—131no—
288C2×#117C2×D1311yescharacteristic
289C22×C4⋊C4—131no—
290#485—138no—
291C2×#107—131no—
292C2×#106—131no—
293C2×#106—131no—
294C2×#106—131no—
295C2×#103—131no—
296C2×#103—131no—
297C2×#103—131no—
298C2×#103—131no—
299C2×C22⋊C8C2×D1311yescharacteristic
300C2×C22⋊C8C2×D1311yescharacteristic
301C2×#97C2×D1311yescharacteristic
302C2×C4×C8C2×D1311yescharacteristic
303#391—131no—
304#389—131no—
305#389—261no—
306#388—261no—
307#388—261no—
308#387—261no—
309#380—131no—
310C4×C4⋊C4—131no—
311C4×C4⋊C4—131no—
312C4×C4⋊C4—131no—
313C4×C4⋊C4—131no—
314C4×C4⋊C4—131no—
315C4×C4⋊C4—261no—
316C4×C4⋊C4—261no—
317C4×C22⋊C4—261no—
318C4×C22⋊C4—261no—
319#340—131no—
320#340—131no—
321#340—131no—
322#340—131no—
323#340—131no—
324#340—131no—
325#340—131no—
326#340—131no—
327#336C2×D1311yescharacteristic
328#336C2×D1311yescharacteristic
329C22×C13—22no—
330C22×C13—22no—
331C22×C13—22no—
332C22×C13—41no—
333C22×C13—41no—
334C22×C13—41no—
335C22×C13—41no—
336C22×C13C2×C4×D411yescharacteristic
337C22×C13C2×#12311yescharacteristic
338C22×C13#52511yescharacteristic
339C22×C13#57711yescharacteristic
340C22×C13#57711yescharacteristic
341C22×C13#57711yescharacteristic
342C22×C13#57811yescharacteristic
343C4×C13—22no—
344C4×C13—22no—
345C4×C13—22no—
346C4×C13—41no—
347C4×C13—41no—
348C4×C13C4×#4111yescharacteristic
349C4×C13C4×#4111yescharacteristic
350C4×C13#52911yescharacteristic
351C4×C13#52911yescharacteristic
352Dic13—42no—
353Dic13—42no—
354Dic13—42no—
355Dic13—42no—
356Dic13—42no—
357Dic13—42no—
358Dic13—81no—
359Dic13—81no—
360C22×Q8—131no—
361C23×C4—261no—
362C23×C4C22×D1311yescharacteristic
363C22×C8C22×D1311yescharacteristic
364C22×C8C22×D1311yescharacteristic
365C22×C8C22×D1311yescharacteristic
366C22×C8C22×D1311yescharacteristic
367C22⋊Q8—134no—
368C22⋊Q8—262no—
369C4×Q8—262no—
370C4×Q8—262no—
371#117—22no—
372#117—22no—
373#117—134no—
374#117—262no—
375C2×C4⋊C4—131no—
376C2×C4⋊C4—131no—
377C2×C4⋊C4—131no—
378C2×C4⋊C4—131no—
379C2×C4⋊C4—131no—
380C2×C4⋊C4—131no—
381C2×C4⋊C4—131no—
382C2×C4⋊C4—262no—
383C2×C4⋊C4—262no—
384C2×C4⋊C4—261no—
385C2×C4⋊C4—261no—
386C2×C4⋊C4—261no—
387C2×C4⋊C4—261no—
388C2×C4⋊C4D5211yescharacteristic
389C2×C4⋊C4D5211yescharacteristic
390C2×C22⋊C4—261no—
391C2×C22⋊C4—261no—
392C2×C22⋊C4—261no—
393C2×C22⋊C4D5211yescharacteristic
394C2×C22⋊C4D5211yescharacteristic
395C2×C42—261no—
396C2×C42—261no—
397C2×C42—261no—
398C2×C42—261no—
399C2×C42—261no—
400C2×C42—261no—
401C2×C42—261no—
402C2×C42C22×D1311yescharacteristic
403C2×C42C22×D1311yescharacteristic
404#107—134no—
405#106—134no—
406#106—262no—
407#106—262no—
408#103—134no—
409#103—134no—
410#103—262no—
411#103—262no—
412C22⋊C8—22no—
413C22⋊C8C4×D1314yes—
414#97—22no—
415C4×C8—22no—
416#95—261no—
417#95—261no—
418#95—261no—
419#95—261no—
420#95—261no—
421#95—261no—
422#95—261no—
423#95—261no—
424#95—261no—
425C2×C13—42no—
426C2×C13C4×#12311yescharacteristic
427C2×C13#294711yescharacteristic
428C2×C13#295211yescharacteristic
429C2×C13#309711yescharacteristic
430C2×C13#309911yescharacteristic
431C2×C13#312111yescharacteristic
432C2×C13#312311yescharacteristic
433C24C2×D5211yescharacteristic
434C2×Q8—132no—
435C2×Q8—261no—
436C2×Q8—261no—
437C2×Q8—261no—
438C2×Q8—261no—
439C2×Q8—261no—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—21no—
444C22×C4—261no—
445C22×C4—261no—
446C22×C4—261no—
447C22×C4—261no—
448C22×C4—261no—
449C22×C4—261no—
450C22×C4—264no—
451C22×C4—261no—
452C22×C4—261no—
453C22×C4—1041no—
454C22×C4—1041no—
455C22×C4C2×C4×D1312yes—
456C22×C4C2×D5211yescharacteristic
457C22×C4C2×D5211yescharacteristic
458C22×C4C2×D5211yescharacteristic
459C22×C4C2×D5211yescharacteristic
460C22×C4C2×D5211yescharacteristic
461C22×C4#612712yes—
462C22×C4C23×D1311yescharacteristic
463C2×C8—21no—
464C2×C8—21no—
465C2×C8—21no—
466C2×C8—21no—
467C2×C8—21no—
468C2×C8—21no—
469C2×C8—21no—
470C2×C8—21no—
471C2×C8—21no—
472C2×C8—21no—
473C2×C8C2×C4×D1312yes—
474C2×C8C2×C4×D1312yes—
475C4⋊C4—22no—
476C4⋊C4—22no—
477C4⋊C4—132no—
478C4⋊C4—132no—
479C4⋊C4—132no—
480C4⋊C4—132no—
481C4⋊C4—132no—
482C4⋊C4—264no—
483C4⋊C4—264no—
484C4⋊C4—261no—
485C4⋊C4—261no—
486C4⋊C4—261no—
487C4⋊C4—261no—
488C4⋊C4—261no—
489C4⋊C4—261no—
490C4⋊C4—261no—
491C4⋊C4—261no—
492C4⋊C4—261no—
493C4⋊C4—522no—
494C4⋊C4—1041no—
495C22⋊C4—22no—
496C22⋊C4—22no—
497C22⋊C4—264no—
498C22⋊C4—264no—
499C22⋊C4—1041no—
500C42—21no—
501C42—21no—
502C42—264no—
503C42—522no—
504C42—1041no—
505C42—1041no—
506C42—1041no—
507C42—1041no—
508C42—1041no—
509C42#612912yes—
510C42#612912yes—
511C13#2159111yescharacteristic, complemented, Sylow
512C23—21no—
513C23—21no—
514C23—21no—
515C23—21no—
516C23—21no—
517C23—21no—
518C23C4×D5212yes—
519C23C22×D5211yescharacteristic
520Q8—262no—
521Q8—262no—
522Q8—521no—
523Q8—521no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—262no—
541C2×C4—262no—
542C2×C4—262no—
543C2×C4—262no—
544C2×C4—262no—
545C2×C4—262no—
546C2×C4—521no—
547C2×C4—521no—
548C2×C4—521no—
549C2×C4—521no—
550C2×C4—521no—
551C2×C4—521no—
552C2×C4—521no—
553C2×C4—521no—
554C2×C4—521no—
555C2×C4—521no—
556C2×C4—521no—
557C2×C4—521no—
558C2×C4—521no—
559C2×C4—521no—
560C2×C4—521no—
561C2×C4—521no—
562C2×C4—521no—
563C2×C4—521no—
564C2×C4—1044no—
565C2×C4C4×D5212yes—
566C2×C4#8814012yes—
567C2×C4#8814012yes—
568C2×C4C22×C4×D1311yescharacteristic
569C2×C4C2×#612711yescharacteristic
570C2×C4C2×#612911yescharacteristic
571C2×C4C2×#612911yescharacteristic
572C2×C4#41×D1311yescharacteristic
573C2×C4#41×D1311yescharacteristic
574C8—22no—
575C8—22no—
576C8—41no—
577C8—41no—
578C22—22no—
579C22—22no—
580C22—22no—
581C22—41no—
582C22—41no—
583C22—41no—
584C22—41no—
585C22C2×C4×D5211yescharacteristic
586C22C2×#8814011yescharacteristic
587C22#1171922911yescharacteristic
588C22#1171945611yescharacteristic
589C22#1171945611yescharacteristic
590C22#1171945611yescharacteristic
591C22#1171945811yescharacteristic
592C4—22no—
593C4—22no—
594C4—22no—
595C4—41no—
596C4—41no—
597C4—522no—
598C4—522no—
599C4—522no—
600C4—522no—
601C4—522no—
602C4—522no—
603C4—1041no—
604C4—1041no—
605C4C4×#612911yescharacteristic
606C4C4×#612911yescharacteristic
607C4#1171924311yescharacteristic
608C4#1171924311yescharacteristic
609C2—42no—
610C2C4×#8814011yescharacteristic
611C2#1312368111yescharacteristic
612C2#1312372111yescharacteristic
613C2#1312480611yescharacteristic
614C2#1312482711yescharacteristic
615C2#1312506711yescharacteristic
616C2#1312508911yescharacteristic
subgroupG/H
centerC23C22×D52
commutatorC2×C4×C13C23×C4
FrattiniC22×C4C23×D13
FittingC13×#1750C2
socleC23×C13C22×D4
prime radicalC24×C13C2×D4
cosocleC22×C4×C13C24
hypercenter#1750D13
hypocenterC13#21591
LCSG ▷ C2×C4×C13 ▷ C2×C13 ▷ C13
UCSC1 ◁ C23 ◁ C2×#117 ◁ #1750

Complements

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

NH ≅ G/Ntype
4. #13115471C2semidirect
5. #13115471C2semidirect
6. #13115429C2semidirect
7. #13115415C2semidirect
11. #13114733C2semidirect
12. C4×#88057C2semidirect
13. #13114667C2semidirect
14. C4×#88056C2semidirect
49. #11718384C4semidirect
117. #88058C2×C4semidirect
120. #88057C2×C4semidirect
123. #88056C2×C4semidirect
124. #88056C2×C4semidirect
511. C13#21591semidirect

Maximal subgroups

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

subgroupindex
C2×#117183842
#131175182
#131171922
#131154712
#131154292
#131154152
#131151412
#131150312
#131149322
#131147332
C4×#880572
#131146672
C4×#880562
C13×#17502
#2159113