Finite Group Databasebeta

← #15925681

Subgroups — #15925681

Order 3328.

848 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×#11718455C211yesmaximal, characteristic, complemented
2C2×#11718782C211yesmaximal, characteristic
3#13118539C211yesmaximal, characteristic
4#13117867C211yesmaximal, characteristic
5#13116377C211yesmaximal, characteristic, complemented
6#13116336C211yesmaximal, characteristic, complemented
7#13116215C211yesmaximal, characteristic, complemented
8#13116168C211yesmaximal, characteristic, complemented
9#13115593C211yesmaximal, characteristic, complemented
10#13115406C211yesmaximal, characteristic, complemented
11#13115144C211yesmaximal, characteristic
12#13115092C211yesmaximal, characteristic, complemented
13#13115033C211yesmaximal, characteristic
14#13115019C211yesmaximal, characteristic
15C13×#1894C211yesmaximal, characteristic
16C22×C13×M16C2211yescharacteristic
17C2×C13×C22⋊Q8C2211yescharacteristic
18C2×Q8×Dic13C2211yescharacteristic
19C2×#88180C2211yescharacteristic
20C22×C4⋊Dic13C2211yescharacteristic
21C2×#88142C2211yescharacteristic
22C2×#88132—21no—
23C2×C13×#103C2211yescharacteristic
24C2×C13×#103C2211yescharacteristic
25C13×#380C2211yescharacteristic
26#11718891C2211yescharacteristic
27C2×#88075C2211yescharacteristic
28C2×#88075C2211yescharacteristic
29C4×C22⋊Dic13C2211yescharacteristic
30C2×C22⋊#1055C2211yescharacteristic
31#11718782—24no—
32C2×#88058C2211yescharacteristic
33C2×#88058C2211yescharacteristic
34C2×#88057C2211yescharacteristic
35C2×#88057C2211yescharacteristic
36C2×#88054C2211yescharacteristic
37C2×#88054C2211yescharacteristic
38#11718646C2211yescharacteristic
39#11718646C2211yescharacteristic
40#11718642—21no—
41#11718637C2211yescharacteristic
42C4⋊C4×Dic13—21no—
43C4⋊C4×Dic13C2211yescharacteristic
44#11718634C2211yescharacteristic
45C2×#88048C2211yescharacteristic
46C2×#88047C2211yescharacteristic
47#11718612—21no—
48#11718612—21no—
49#11718607—21no—
50#11718455—24no—
51#11718318—21no—
52#11718317—21no—
53C13×#340C2211yescharacteristic
54C13×#340C2211yescharacteristic
55#11718142C2211yescharacteristic
56#11718142C2211yescharacteristic
57#11718121C2211yescharacteristic
58#11718121C2211yescharacteristic
59#11718121C2211yescharacteristic
60#11718121C2211yescharacteristic
61C22×Q8×C13C2311yescharacteristic
62C23×C4×C13C2311yescharacteristic
63C23×Dic13—21no—
64C22×Dic26—21no—
65C2×C13×M16—22no—
66C2×C13×M16—22no—
67C2×C13×M16—22no—
68C22×C8×C13C2311yescharacteristic
69C22×C8×C13C2311yescharacteristic
70C13×C22⋊Q8—22no—
71C13×C22⋊Q8Q814yescomplemented
72C2×C13×C4⋊C4—21no—
73C2×C13×C4⋊C4C2311yescharacteristic
74C2×C13×C4⋊C4C2311yescharacteristic
75C2×C13×C4⋊C4C2311yescharacteristic
76C2×C13×C22⋊C4—21no—
77C2×C13×C22⋊C4—21no—
78C2×C22⋊Dic13—21no—
79C2×C22⋊Dic13D411yescharacteristic
80C2×C22⋊Dic13D411yescharacteristic
81Q8×Dic13—22no—
82Q8×Dic13—41no—
83#88180—22no—
84#88180—41no—
85C2×C4⋊Dic13—22no—
86C2×C4⋊Dic13—21no—
87C2×C4⋊Dic13—21no—
88C2×C4⋊Dic13—21no—
89C2×C4⋊Dic13—21no—
90C2×C4⋊Dic13—41no—
91C2×C4⋊Dic13C2311yescharacteristic
92C2×C4⋊Dic13C2311yescharacteristic
93C2×C4⋊Dic13C2311yescharacteristic
94C2×C4⋊Dic13C2311yescharacteristic
95C2×#6101—21no—
96C2×#6101—21no—
97C2×#6101—21no—
98C2×#6101—21no—
99C2×C4×Dic13—21no—
100C2×C4×Dic13—21no—
101C2×C4×Dic13—21no—
102C2×C4×Dic13C2311yescharacteristic
103C2×C4×Dic13C2311yescharacteristic
104C22×#1055C2311yescharacteristic
105C22×#1055C2311yescharacteristic
106#88142—22no—
107#88142—41no—
108#88132—44no—
109C13×#103—22no—
110C13×#103—22no—
111C13×#95—21no—
112#88075—22no—
113#88075—41no—
114#88071—21no—
115#88071—21no—
116#88071—21no—
117#88071—21no—
118#88071—21no—
119#88071—21no—
120#88071D411yescharacteristic
121#88071D411yescharacteristic
122C22⋊#1055—22no—
123#88058—22no—
124#88058D414yes—
125#88057—22no—
126#88057—22no—
127#88054—22no—
128#88054—22no—
129#88048—41no—
130#88047—22no—
131#26355—131nomaximal, Sylow
132C24×C13C2×D411yescharacteristic
133C2×Q8×C13—21no—
134C2×Q8×C13—21no—
135C2×Q8×C13—21no—
136C2×Q8×C13—21no—
137C2×Q8×C13—21no—
138C2×Q8×C13C2×Q812yes—
139C22×C4×C13—21no—
140C22×C4×C13—21no—
141C22×C4×C13—21no—
142C22×C4×C13—21no—
143C22×C4×C13—21no—
144C22×C4×C13—21no—
145C22×C4×C13—21no—
146C22×C4×C13—21no—
147C22×C4×C13C2×D411yescharacteristic
148C22×C4×C13C2×Q812yes—
149C22×C4×C13#4112yes—
150C22×C4×C13C2411yescharacteristic
151C22×Dic13—21no—
152C22×Dic13—21no—
153C22×Dic13—21no—
154C22×Dic13—21no—
155C22×Dic13—42no—
156C22×Dic13—42no—
157C22×Dic13—81no—
158C22×Dic13C2×D411yescharacteristic
159C22×Dic13C2×D411yescharacteristic
160C22×Dic13C2×D411yescharacteristic
161C22×Dic13C2×D411yescharacteristic
162C2×Dic26—42no—
163C2×Dic26—42no—
164C2×Dic26—81no—
165C13×M16—28no—
166C2×C8×C13—21no—
167C2×C8×C13—21no—
168C2×C8×C13—21no—
169C2×C8×C13—21no—
170C2×C8×C13C2×D412yes—
171C2×C8×C13#4112yes—
172C13×C4⋊C4—24no—
173C13×C4⋊C4—21no—
174C13×C4⋊C4—21no—
175C13×C4⋊C4—21no—
176C13×C4⋊C4—21no—
177C13×C4⋊C4—21no—
178C13×C4⋊C4C2×Q812yes—
179C13×C22⋊C4—24no—
180C13×C22⋊C4—42no—
181C22⋊Dic13—22no—
182C22⋊Dic13—22no—
183C22⋊Dic13—42no—
184C4⋊Dic13—21no—
185C4⋊Dic13—21no—
186C4⋊Dic13—21no—
187C4⋊Dic13—21no—
188C4⋊Dic13—21no—
189C4⋊Dic13—21no—
190C4⋊Dic13—42no—
191C4⋊Dic13—42no—
192C4⋊Dic13—42no—
193C4⋊Dic13—42no—
194C4⋊Dic13C2×D412yes—
195C4⋊Dic13C2×D412yes—
196#6101—42no—
197#6101—42no—
198#6101—42no—
199#6101—81no—
200C4×Dic13—21no—
201C4×Dic13—21no—
202C4×Dic13—21no—
203C4×Dic13—21no—
204C4×Dic13—42no—
205C4×Dic13—42no—
206C4×Dic13—42no—
207C2×#1055—21no—
208C2×#1055—21no—
209C2×#1055—21no—
210C2×#1055—21no—
211C2×#1055—21no—
212C2×#1055—21no—
213C2×#477—131no—
214C2×#428—131no—
215#2440—131no—
216#2362—131no—
217#2096—131no—
218#2088—131no—
219#2066—131no—
220#2058—131no—
221#1947—131no—
222#1918—131no—
223#1894D1311yescharacteristic
224#1872—131no—
225#1863—131no—
226#1854—131no—
227#1851—131no—
228C23×C13—21no—
229C23×C13—21no—
230C23×C13—21no—
231C23×C13—21no—
232C23×C13—21no—
233C23×C13—21no—
234C23×C13C22⋊Q812yes—
235C23×C13C22×D411yescharacteristic
236Q8×C13—22no—
237Q8×C13—22no—
238Q8×C13—41no—
239Q8×C13—41no—
240C2×C4×C13—21no—
241C2×C4×C13—21no—
242C2×C4×C13—22no—
243C2×C4×C13—22no—
244C2×C4×C13—22no—
245C2×C4×C13—22no—
246C2×C4×C13—22no—
247C2×C4×C13—22no—
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×C13—41no—
255C2×C4×C13—41no—
256C2×C4×C13—41no—
257C2×C4×C13—41no—
258C2×C4×C13C22⋊Q812yes—
259C2×C4×C13C22×D411yescharacteristic
260C2×C4×C13C22×Q811yescharacteristic
261C2×C4×C13C2×#4111yescharacteristic
262C2×C4×C13C2×#4111yescharacteristic
263C2×C4×C13H(5,𝔽2)11yescharacteristic
264C2×C4×C132−1+411yescharacteristic
265C2×Dic13—21no—
266C2×Dic13—21no—
267C2×Dic13—21no—
268C2×Dic13—21no—
269C2×Dic13—21no—
270C2×Dic13—22no—
271C2×Dic13—22no—
272C2×Dic13—21no—
273C2×Dic13—21no—
274C2×Dic13—21no—
275C2×Dic13—21no—
276C2×Dic13—21no—
277C2×Dic13—41no—
278C2×Dic13—41no—
279C2×Dic13—41no—
280C2×Dic13—41no—
281C2×Dic13—41no—
282C2×Dic13—41no—
283C2×Dic13—41no—
284C2×Dic13—41no—
285C2×Dic13—41no—
286C2×Dic13—41no—
287C2×Dic13—84no—
288C2×Dic13AGL(1,ℤ/8)12yes—
289C2×Dic13#13712yes—
290Dic26—84no—
291C8×C13—22no—
292C8×C13—22no—
293#1055—41no—
294#1055—41no—
295C22×M16C2×D1311yescharacteristic
296C2×C4⋊Q8—131no—
297C2×C22⋊Q8—261no—
298C2×C22⋊Q8C2×D1311yescharacteristic
299C2×C4×Q8—131no—
300C2×#117—131no—
301C22×C4⋊C4—131no—
302#477—264no—
303#428—264no—
304C2×#107—131no—
305C2×#107—131no—
306C2×#107—131no—
307C2×#106—131no—
308C2×#106—131no—
309C2×#106—131no—
310C2×C4⋊C8—131no—
311C2×C4⋊C8—131no—
312C2×#103—131no—
313C2×#103—131no—
314C2×#103C2×D1311yescharacteristic
315C2×#103C2×D1311yescharacteristic
316C2×C22⋊C8—131no—
317#398—261no—
318#395—261no—
319#391—131no—
320#389—131no—
321#389—131no—
322#389—131no—
323#388—261no—
324#388—261no—
325#387—261no—
326#385—261no—
327#384—131no—
328#380C2×D1311yescharacteristic
329C4×C4⋊C4—131no—
330C4×C4⋊C4—261no—
331C4×C22⋊C4—131no—
332#340—131no—
333#340—131no—
334#340—131no—
335#340—131no—
336#340C2×D1311yescharacteristic
337#340C2×D1311yescharacteristic
338#336—131no—
339#336—131no—
340C22×C13—22no—
341C22×C13—22no—
342C22×C13—22no—
343C22×C13—41no—
344C22×C13—41no—
345C22×C13—41no—
346C22×C13—41no—
347C22×C13C2×C22⋊Q811yescharacteristic
348C22×C13D4211yescharacteristic
349C22×C13#54711yescharacteristic
350C22×C13C2×AGL(1,ℤ/8)11yescharacteristic
351C22×C13C2×#13711yescharacteristic
352C22×C13#57711yescharacteristic
353C22×C13#57811yescharacteristic
354C4×C13—22no—
355C4×C13—42no—
356C4×C13—42no—
357C4×C13—81no—
358C4×C13—81no—
359C4×C13#52511yescharacteristic
360C4×C13#53911yescharacteristic
361C4×C13D4×Q811yescharacteristic
362C4×C13#55411yescharacteristic
363Dic13—22no—
364Dic13—22no—
365Dic13—41no—
366Dic13—41no—
367Dic13—42no—
368Dic13—42no—
369Dic13—81no—
370Dic13—81no—
371C22×Q8—261no—
372C22×Q8C22×D1311yescharacteristic
373C23×C4—261no—
374C23×C4C22×D1311yescharacteristic
375C2×M16—22no—
376C2×M16—22no—
377C2×M16—22no—
378C22×C8—131no—
379C22×C8—131no—
380C22×C8C22×D1311yescharacteristic
381C22×C8C22×D1311yescharacteristic
382C4⋊Q8—262no—
383C4⋊Q8—521no—
384C22⋊Q8—22no—
385C22⋊Q8—524no—
386C22⋊Q8Dic2614yescomplemented
387C4×Q8—262no—
388C4×Q8—521no—
389#117—262no—
390#117—521no—
391C2×C4⋊C4—21no—
392C2×C4⋊C4—131no—
393C2×C4⋊C4—131no—
394C2×C4⋊C4—131no—
395C2×C4⋊C4—131no—
396C2×C4⋊C4—262no—
397C2×C4⋊C4—261no—
398C2×C4⋊C4—261no—
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—521no—
406C2×C4⋊C4C22×D1311yescharacteristic
407C2×C4⋊C4C22×D1311yescharacteristic
408C2×C4⋊C4C22×D1311yescharacteristic
409C2×C22⋊C4—21no—
410C2×C22⋊C4—21no—
411C2×C22⋊C4—131no—
412C2×C22⋊C4—131no—
413C2×C22⋊C4—261no—
414C2×C42—131no—
415C2×C42—131no—
416C2×C42—261no—
417C2×C42—261no—
418C2×C42—261no—
419#107—134no—
420#107—262no—
421#107—262no—
422#106—262no—
423#106—262no—
424#106—521no—
425C4⋊C8—262no—
426C4⋊C8—262no—
427#103—22no—
428#103—22no—
429#103—262no—
430#103—521no—
431C22⋊C8—262no—
432#95—21no—
433#95—131no—
434#95—131no—
435#95—261no—
436#95—261no—
437#95—261no—
438#95—261no—
439#95—261no—
440#95—261no—
441C2×C13—42no—
442C2×C13#244911yescharacteristic
443C2×C13#307911yescharacteristic
444C2×C13#308211yescharacteristic
445C2×C13#327011yescharacteristic
446C2×C13#327211yescharacteristic
447C2×C13#334311yescharacteristic
448C2×C13#335611yescharacteristic
449C24D4×D1311yescharacteristic
450C2×Q8—21no—
451C2×Q8—21no—
452C2×Q8—21no—
453C2×Q8—21no—
454C2×Q8—21no—
455C2×Q8—522no—
456C2×Q8—522no—
457C2×Q8—1041no—
458C2×Q8C2×Dic2612yes—
459C22×C4—21no—
460C22×C4—21no—
461C22×C4—21no—
462C22×C4—21no—
463C22×C4—21no—
464C22×C4—21no—
465C22×C4—21no—
466C22×C4—21no—
467C22×C4—131no—
468C22×C4—131no—
469C22×C4—131no—
470C22×C4—131no—
471C22×C4—261no—
472C22×C4—261no—
473C22×C4—261no—
474C22×C4—261no—
475C22×C4—522no—
476C22×C4—522no—
477C22×C4—1041no—
478C22×C4C2×Dic2612yes—
479C22×C4D4×D1311yescharacteristic
480C22×C4#612912yes—
481C22×C4C23×D1311yescharacteristic
482M16—28no—
483C2×C8—21no—
484C2×C8—21no—
485C2×C8—21no—
486C2×C8—21no—
487C2×C8—261no—
488C2×C8—261no—
489C2×C8—261no—
490C2×C8—261no—
491C2×C8—261no—
492C2×C8—261no—
493C2×C8D4×D1312yes—
494C2×C8#612912yes—
495C4⋊C4—24no—
496C4⋊C4—21no—
497C4⋊C4—21no—
498C4⋊C4—21no—
499C4⋊C4—21no—
500C4⋊C4—21no—
501C4⋊C4—132no—
502C4⋊C4—132no—
503C4⋊C4—261no—
504C4⋊C4—261no—
505C4⋊C4—261no—
506C4⋊C4—261no—
507C4⋊C4—261no—
508C4⋊C4—261no—
509C4⋊C4—522no—
510C4⋊C4—522no—
511C4⋊C4—522no—
512C4⋊C4—522no—
513C4⋊C4—522no—
514C4⋊C4—522no—
515C4⋊C4—522no—
516C4⋊C4—1041no—
517C4⋊C4C2×Dic2612yes—
518C22⋊C4—24no—
519C22⋊C4—42no—
520C22⋊C4—262no—
521C22⋊C4—262no—
522C22⋊C4—522no—
523C42—261no—
524C42—261no—
525C42—261no—
526C42—261no—
527C42—522no—
528C42—522no—
529C42—522no—
530C13#2635511yescharacteristic, complemented, Sylow
531C23—21no—
532C23—21no—
533C23—21no—
534C23—21no—
535C23—21no—
536C23—21no—
537C23#8813212yes—
538C23C2×D4×D1311yescharacteristic
539Q8—22no—
540Q8—22no—
541Q8—41no—
542Q8—41no—
543Q8—1044no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—132no—
563C2×C4—132no—
564C2×C4—261no—
565C2×C4—261no—
566C2×C4—261no—
567C2×C4—261no—
568C2×C4—261no—
569C2×C4—261no—
570C2×C4—261no—
571C2×C4—261no—
572C2×C4—261no—
573C2×C4—262no—
574C2×C4—262no—
575C2×C4—261no—
576C2×C4—521no—
577C2×C4—521no—
578C2×C4—521no—
579C2×C4—521no—
580C2×C4—521no—
581C2×C4—521no—
582C2×C4—521no—
583C2×C4—521no—
584C2×C4—521no—
585C2×C4—521no—
586C2×C4—1044no—
587C2×C4#8813212yes—
588C2×C4C22×Dic2611yescharacteristic
589C2×C4C2×D4×D1311yescharacteristic
590C2×C4C2×#612911yescharacteristic
591C2×C4C2×#612911yescharacteristic
592C2×C4#8825611yescharacteristic
593C2×C4#8825711yescharacteristic
594C8—22no—
595C8—22no—
596C8—521no—
597C8—521no—
598C22—22no—
599C22—22no—
600C22—22no—
601C22—41no—
602C22—41no—
603C22—41no—
604C22—41no—
605C22C2×#8813211yescharacteristic
606C22#1171931411yescharacteristic
607C22#1171931611yescharacteristic
608C22D13×AGL(1,ℤ/8)11yescharacteristic
609C22D13×#13711yescharacteristic
610C22#1171948311yescharacteristic
611C22#1171948411yescharacteristic
612C4—22no—
613C4—42no—
614C4—42no—
615C4—81no—
616C4—81no—
617C4—262no—
618C4—262no—
619C4—521no—
620C4—521no—
621C4—522no—
622C4—522no—
623C4—1041no—
624C4—1041no—
625C4D4×Dic2611yescharacteristic
626C4#1171924811yescharacteristic
627C4#1171929111yescharacteristic
628C4#1171937011yescharacteristic
629C2—42no—
630C2#1311865111yescharacteristic
631C2#1312464711yescharacteristic
632C2#1312468011yescharacteristic
633C2#1312641311yescharacteristic
634C2#1312643311yescharacteristic
635C2#1312727111yescharacteristic
636C2#1312744611yescharacteristic
subgroupG/H
centerC23C2×D4×D13
commutator, cosocleC22×C4×C13C24
FrattiniC22×C4C23×D13
FittingC13×#1894C2
socleC23×C13C22×D4
prime radicalC24×C13C2×D4
hypercenter#1894D13
hypocenterC13#26355
LCSG ▷ C22×C4×C13 ▷ C2×C13 ▷ C13
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1894

Complements

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

NH ≅ G/Ntype
1. C2×#11718455C2semidirect
5. #13116377C2semidirect
6. #13116336C2semidirect
7. #13116215C2semidirect
8. #13116168C2semidirect
9. #13115593C2semidirect
10. #13115406C2semidirect
12. #13115092C2semidirect
71. C13×C22⋊Q8Q8semidirect
386. C22⋊Q8Dic26semidirect
530. C13#26355semidirect

Maximal subgroups

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

subgroupindex
C2×#117184552
C2×#117187822
#131185392
#131178672
#131163772
#131163362
#131162152
#131161682
#131155932
#131154062
#131151442
#131150922
#131150332
#131150192
C13×#18942
#2635513