Finite Group Databasebeta

← #15925639

Subgroups — #15925639

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×#11718681C211yesmaximal, characteristic, complemented
2C2×#11718656C211yesmaximal, characteristic
3#13118538C211yesmaximal, characteristic
4#13116374C211yesmaximal, characteristic, complemented
5#13116327C211yesmaximal, characteristic, complemented
6#13116231C211yesmaximal, characteristic, complemented
7#13116164C211yesmaximal, characteristic, complemented
8#13115599C211yesmaximal, characteristic, complemented
9#13115421C211yesmaximal, characteristic, complemented
10#13115270C211yesmaximal, characteristic
11#13115144C211yesmaximal, characteristic
12#13115104C211yesmaximal, characteristic, complemented
13#13115035C211yesmaximal, characteristic
14#13115021C211yesmaximal, characteristic
15C13×#2362C211yesmaximal, characteristic
16C2×C13×C22⋊Q8C2211yescharacteristic
17C2×C4×Q8×C13C2211yescharacteristic
18C2×C13×#117C2211yescharacteristic
19C22×C4⋊Dic13C2211yescharacteristic
20C22×#6099C2211yescharacteristic
21C2×#88132—21no—
22C2×#88123C2211yescharacteristic
23C13×#391C2211yescharacteristic
24C13×#389C2211yescharacteristic
25C13×#388—21no—
26C13×#388—21no—
27C13×#387—21no—
28C4×C13×C4⋊C4—21no—
29C4×C13×C4⋊C4C2211yescharacteristic
30C4×C13×C22⋊C4C2211yescharacteristic
31C2×#88075C2211yescharacteristic
32C2×#88075C2211yescharacteristic
33C2×#88075C2211yescharacteristic
34C2×#88075C2211yescharacteristic
35#11718873C2211yescharacteristic
36C2×C22⋊#1055C2211yescharacteristic
37#11718681—24no—
38#11718656—24no—
39C2×#88048C2211yescharacteristic
40C2×#88048C2211yescharacteristic
41C2×#88048C2211yescharacteristic
42C2×#88047C2211yescharacteristic
43C2×#88047C2211yescharacteristic
44C2×#88047C2211yescharacteristic
45#11718611—21no—
46#11718596C2211yescharacteristic
47#11718589C2211yescharacteristic
48#11718589C2211yescharacteristic
49C2×C4⋊#1055C2211yescharacteristic
50C2×C4⋊#1055C2211yescharacteristic
51#11718322—21no—
52#11718317—21no—
53#11718121C2211yescharacteristic
54#11718121C2211yescharacteristic
55#11718121C2211yescharacteristic
56#11718121C2211yescharacteristic
57#11718121C2211yescharacteristic
58#11718121C2211yescharacteristic
59#11718116C2211yescharacteristic
60#11718116C2211yescharacteristic
61C22×Q8×C13C2311yescharacteristic
62C23×C4×C13C2311yescharacteristic
63C23×Dic13—21no—
64C22×Dic26—21no—
65C13×C22⋊Q8—22no—
66C13×C22⋊Q8Q814yescomplemented
67C4×Q8×C13—22no—
68C4×Q8×C13—41no—
69C13×#117—22no—
70C13×#117—41no—
71C2×C13×C4⋊C4—21no—
72C2×C13×C4⋊C4—21no—
73C2×C13×C4⋊C4—21no—
74C2×C13×C4⋊C4C2311yescharacteristic
75C2×C13×C4⋊C4C2311yescharacteristic
76C2×C13×C4⋊C4C2311yescharacteristic
77C2×C13×C22⋊C4—21no—
78C2×C13×C22⋊C4—21no—
79C2×C13×C22⋊C4D411yescharacteristic
80C2×C13×C22⋊C4D411yescharacteristic
81C2×C42×C13—21no—
82C2×C42×C13—21no—
83C2×C42×C13—21no—
84C2×C42×C13C2311yescharacteristic
85C2×C42×C13C2311yescharacteristic
86C2×C22⋊Dic13—21no—
87C2×C4⋊Dic13—22no—
88C2×C4⋊Dic13—21no—
89C2×C4⋊Dic13—21no—
90C2×C4⋊Dic13—21no—
91C2×C4⋊Dic13—21no—
92C2×C4⋊Dic13—41no—
93C2×C4⋊Dic13C2311yescharacteristic
94C2×C4⋊Dic13C2311yescharacteristic
95C2×C4⋊Dic13C2311yescharacteristic
96C2×C4⋊Dic13C2311yescharacteristic
97C2×#6101—21no—
98C2×#6101—21no—
99C2×#6099—22no—
100C2×#6099—22no—
101C2×#6099—22no—
102C22×#1055C2311yescharacteristic
103C22×#1055C2311yescharacteristic
104C22×#1055C2311yescharacteristic
105C22×#1055C2311yescharacteristic
106#88132—44no—
107#88123—22no—
108#88123—41no—
109C13×#95—21no—
110C13×#95—21no—
111C13×#95—21no—
112C13×#95—21no—
113C13×#95D411yescharacteristic
114C13×#95D411yescharacteristic
115#88075—22no—
116#88075—22no—
117#88075—22no—
118#88075—41no—
119#88071—21no—
120#88071—21no—
121#88071—21no—
122C22⋊#1055—22no—
123#88048—22no—
124#88048—22no—
125#88048D414yes—
126#88047—22no—
127#88047—22no—
128#88047—41no—
129C4⋊#1055—22no—
130C4⋊#1055—22no—
131#26353—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×D411yescharacteristic
149C22×C4×C13C2×D411yescharacteristic
150C22×C4×C13C2×D411yescharacteristic
151C22×C4×C13C2×D411yescharacteristic
152C22×C4×C13C2×Q812yes—
153C22×C4×C13#4112yes—
154C22×C4×C13C2411yescharacteristic
155C22×Dic13—21no—
156C22×Dic13—21no—
157C22×Dic13—21no—
158C22×Dic13—21no—
159C22×Dic13—42no—
160C22×Dic13—42no—
161C22×Dic13—81no—
162C2×Dic26—42no—
163C2×Dic26—42no—
164C2×Dic26—81no—
165C13×C4⋊C4—24no—
166C13×C4⋊C4—21no—
167C13×C4⋊C4—21no—
168C13×C4⋊C4—21no—
169C13×C4⋊C4—21no—
170C13×C4⋊C4—42no—
171C13×C4⋊C4—42no—
172C13×C4⋊C4C2×D412yes—
173C13×C4⋊C4C2×Q812yes—
174C13×C22⋊C4—24no—
175C13×C22⋊C4—22no—
176C13×C22⋊C4—22no—
177C13×C22⋊C4—42no—
178C42×C13—21no—
179C42×C13—21no—
180C42×C13—21no—
181C42×C13—21no—
182C42×C13—42no—
183C42×C13—42no—
184C42×C13—42no—
185C22⋊Dic13—42no—
186C4⋊Dic13—21no—
187C4⋊Dic13—21no—
188C4⋊Dic13—21no—
189C4⋊Dic13—21no—
190C4⋊Dic13—21no—
191C4⋊Dic13—21no—
192C4⋊Dic13—21no—
193C4⋊Dic13—42no—
194C4⋊Dic13—42no—
195C4⋊Dic13—42no—
196C4⋊Dic13—81no—
197C4⋊Dic13C2×D412yes—
198#6101—42no—
199#6101—42no—
200#6099—28no—
201C2×#1055—21no—
202C2×#1055—21no—
203C2×#1055—21no—
204C2×#1055—21no—
205C2×#1055—21no—
206C2×#1055—21no—
207C2×#1055—21no—
208C2×#1055—21no—
209C2×#1055—21no—
210C2×#1055—21no—
211C2×#1055C2×D412yes—
212C2×#1055#4112yes—
213C2×#475—131no—
214C2×#428—131no—
215#2440—131no—
216#2362D1311yescharacteristic
217#2095—131no—
218#2086—131no—
219#2068—131no—
220#2057—131no—
221#1948—131no—
222#1920—131no—
223#1894—131no—
224#1872—131no—
225#1865—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—21no—
243C2×C4×C13—21no—
244C2×C4×C13—21no—
245C2×C4×C13—21no—
246C2×C4×C13—22no—
247C2×C4×C13—22no—
248C2×C4×C13—22no—
249C2×C4×C13—22no—
250C2×C4×C13—22no—
251C2×C4×C13—22no—
252C2×C4×C13—21no—
253C2×C4×C13—22no—
254C2×C4×C13—21no—
255C2×C4×C13—21no—
256C2×C4×C13—21no—
257C2×C4×C13—21no—
258C2×C4×C13—21no—
259C2×C4×C13—41no—
260C2×C4×C13—41no—
261C2×C4×C13—41no—
262C2×C4×C13—41no—
263C2×C4×C13—41no—
264C2×C4×C13—41no—
265C2×C4×C13—41no—
266C2×C4×C13—41no—
267C2×C4×C13—41no—
268C2×C4×C13C22⋊Q812yes—
269C2×C4×C13AGL(1,ℤ/8)12yes—
270C2×C4×C13#13712yes—
271C2×C4×C13C22×D411yescharacteristic
272C2×C4×C13C22×Q811yescharacteristic
273C2×C4×C13C2×#4111yescharacteristic
274C2×C4×C13C2×#4111yescharacteristic
275C2×C4×C13H(5,𝔽2)11yescharacteristic
276C2×C4×C132−1+411yescharacteristic
277C2×Dic13—22no—
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—41no—
288C2×Dic13—41no—
289C2×Dic13—84no—
290Dic26—84no—
291#1055—22no—
292#1055—22no—
293#1055—41no—
294#1055—41no—
295C22×M16—131no—
296C2×C4⋊Q8—131no—
297C2×C22⋊Q8—261no—
298C2×C22⋊Q8C2×D1311yescharacteristic
299C2×C4×Q8C2×D1311yescharacteristic
300C2×#117C2×D1311yescharacteristic
301C22×C4⋊C4—131no—
302#475—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×#103—131no—
315C2×#103—131no—
316C2×C22⋊C8—131no—
317#398—261no—
318#395—261no—
319#391C2×D1311yescharacteristic
320#389—131no—
321#389—131no—
322#389C2×D1311yescharacteristic
323#388—21no—
324#388—21no—
325#387—21no—
326#385—261no—
327#384—131no—
328#380—131no—
329C4×C4⋊C4—21no—
330C4×C4⋊C4C2×D1311yescharacteristic
331C4×C22⋊C4C2×D1311yescharacteristic
332#340—131no—
333#340—131no—
334#340—131no—
335#340—131no—
336#340—131no—
337#340—131no—
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—22no—
356C4×C13—22no—
357C4×C13—42no—
358C4×C13—41no—
359C4×C13—42no—
360C4×C13—41no—
361C4×C13—42no—
362C4×C13—81no—
363C4×C13#52511yescharacteristic
364C4×C13#53911yescharacteristic
365C4×C13D4×Q811yescharacteristic
366C4×C13#55411yescharacteristic
367Dic13—42no—
368Dic13—81no—
369Dic13—81no—
370Dic13—81no—
371C22×Q8—261no—
372C22×Q8C22×D1311yescharacteristic
373C23×C4—261no—
374C23×C4C22×D1311yescharacteristic
375C2×M16—262no—
376C2×M16—262no—
377C2×M16—262no—
378C22×C8—131no—
379C22×C8—131no—
380C22×C8—131no—
381C22×C8—131no—
382C4⋊Q8—262no—
383C4⋊Q8—521no—
384C22⋊Q8—22no—
385C22⋊Q8—524no—
386C22⋊Q8Dic2614yescomplemented
387C4×Q8—22no—
388C4×Q8—41no—
389#117—22no—
390#117—41no—
391C2×C4⋊C4—21no—
392C2×C4⋊C4—21no—
393C2×C4⋊C4—21no—
394C2×C4⋊C4—131no—
395C2×C4⋊C4—131no—
396C2×C4⋊C4—131no—
397C2×C4⋊C4—131no—
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—521no—
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⋊C4C22⋊D1311yescharacteristic
413C2×C22⋊C4C22⋊D1311yescharacteristic
414C2×C42—21no—
415C2×C42—21no—
416C2×C42—21no—
417C2×C42C22×D1311yescharacteristic
418C2×C42C22×D1311yescharacteristic
419#107—262no—
420#107—262no—
421#107—521no—
422#106—134no—
423#106—262no—
424#106—262no—
425C4⋊C8—262no—
426C4⋊C8—262no—
427#103—262no—
428#103—262no—
429#103—262no—
430#103—521no—
431C22⋊C8—262no—
432#95—21no—
433#95—21no—
434#95—21no—
435#95—21no—
436#95—261no—
437#95—261no—
438#95—261no—
439#95C22⋊D1311yescharacteristic
440#95C22⋊D1311yescharacteristic
441C2×C13—42no—
442C2×C13#244911yescharacteristic
443C2×C13#307911yescharacteristic
444C2×C13#308211yescharacteristic
445C2×C13#326511yescharacteristic
446C2×C13#327611yescharacteristic
447C2×C13#334411yescharacteristic
448C2×C13#335311yescharacteristic
449C24C2×C22⋊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—261no—
468C22×C4—261no—
469C22×C4—261no—
470C22×C4—261no—
471C22×C4—522no—
472C22×C4—522no—
473C22×C4—1041no—
474C22×C4C2×Dic2612yes—
475C22×C4#612712yes—
476C22×C4C2×C22⋊D1311yescharacteristic
477C22×C4C2×C22⋊D1311yescharacteristic
478C22×C4C2×C22⋊D1311yescharacteristic
479C22×C4C2×C22⋊D1311yescharacteristic
480C22×C4C2×C22⋊D1311yescharacteristic
481C22×C4C23×D1311yescharacteristic
482M16—268no—
483C2×C8—132no—
484C2×C8—132no—
485C2×C8—261no—
486C2×C8—261no—
487C2×C8—261no—
488C2×C8—261no—
489C2×C8—261no—
490C2×C8—261no—
491C2×C8—261no—
492C2×C8—261no—
493C2×C8—261no—
494C2×C8—261no—
495C4⋊C4—24no—
496C4⋊C4—21no—
497C4⋊C4—21no—
498C4⋊C4—21no—
499C4⋊C4—21no—
500C4⋊C4—42no—
501C4⋊C4—42no—
502C4⋊C4—132no—
503C4⋊C4—261no—
504C4⋊C4—261no—
505C4⋊C4—261no—
506C4⋊C4—261no—
507C4⋊C4—261no—
508C4⋊C4—261no—
509C4⋊C4—261no—
510C4⋊C4—522no—
511C4⋊C4—522no—
512C4⋊C4—522no—
513C4⋊C4—522no—
514C4⋊C4—522no—
515C4⋊C4—1041no—
516C4⋊C4C2×Dic2612yes—
517C4⋊C4D4×D1312yes—
518C22⋊C4—24no—
519C22⋊C4—22no—
520C22⋊C4—22no—
521C22⋊C4—42no—
522C22⋊C4—522no—
523C42—21no—
524C42—21no—
525C42—21no—
526C42—21no—
527C42—42no—
528C42—42no—
529C42—42no—
530C13#2635311yescharacteristic, complemented, Sylow
531C23—21no—
532C23—21no—
533C23—21no—
534C23—21no—
535C23—21no—
536C23—21no—
537C23#8817812yes—
538C23C22×C22⋊D1311yescharacteristic
539Q8—22no—
540Q8—22no—
541Q8—41no—
542Q8—41no—
543Q8—1044no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—21no—
557C2×C4—22no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—262no—
573C2×C4—521no—
574C2×C4—521no—
575C2×C4—521no—
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—1044no—
585C2×C4#8817812yes—
586C2×C4#8820312yes—
587C2×C4#8820512yes—
588C2×C4C22×Dic2611yescharacteristic
589C2×C4C2×#612711yescharacteristic
590C2×C4C2×D4×D1311yescharacteristic
591C2×C4#8825111yescharacteristic
592C2×C4#41×D1311yescharacteristic
593C2×C4#8825711yescharacteristic
594C8—262no—
595C8—262no—
596C8—521no—
597C8—521no—
598C22—22no—
599C22—22no—
600C22—22no—
601C22—41no—
602C22—41no—
603C22—41no—
604C22—41no—
605C22C2×#8817811yescharacteristic
606C22D4×C22⋊D1311yescharacteristic
607C22C2×#8820311yescharacteristic
608C22C2×#8820511yescharacteristic
609C22#1171952911yescharacteristic
610C22#1171954211yescharacteristic
611C22#1171954311yescharacteristic
612C4—22no—
613C4—22no—
614C4—22no—
615C4—42no—
616C4—41no—
617C4—42no—
618C4—42no—
619C4—41no—
620C4—81no—
621C4—522no—
622C4—1041no—
623C4—1041no—
624C4—1041no—
625C4D4×Dic2611yescharacteristic
626C4#1171924411yescharacteristic
627C4#1171925211yescharacteristic
628C4#1171926911yescharacteristic
629C2—42no—
630C2#1311863911yescharacteristic
631C2#1312465111yescharacteristic
632C2#1312467811yescharacteristic
633C2#1312633211yescharacteristic
634C2#1312647811yescharacteristic
635C2#1312729111yescharacteristic
636C2#1312741411yescharacteristic
subgroupG/H
centerC23C22×C22⋊D13
commutator, cosocleC22×C4×C13C24
FrattiniC22×C4C23×D13
FittingC13×#2362C2
socleC23×C13C22×D4
prime radicalC24×C13C2×D4
hypercenter#2362D13
hypocenterC13#26353
LCSG ▷ C22×C4×C13 ▷ C2×C13 ▷ C13
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2362

Complements

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

NH ≅ G/Ntype
1. C2×#11718681C2semidirect
4. #13116374C2semidirect
5. #13116327C2semidirect
6. #13116231C2semidirect
7. #13116164C2semidirect
8. #13115599C2semidirect
9. #13115421C2semidirect
12. #13115104C2semidirect
66. C13×C22⋊Q8Q8semidirect
386. C22⋊Q8Dic26semidirect
530. C13#26353semidirect

Maximal subgroups

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

subgroupindex
C2×#117186812
C2×#117186562
#131185382
#131163742
#131163272
#131162312
#131161642
#131155992
#131154212
#131152702
#131151442
#131151042
#131150352
#131150212
C13×#23622
#2635313