Finite Group Databasebeta

← #15874124

Subgroups — #15874124

Order 3328.

1296 conjugacy classes of subgroups.

The subgroup lattice has 624 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11718878C211yesmaximal, characteristic
2C4×#88184C211yesmaximal, characteristic, complemented
3#13115711C212yesmaximal, complemented
4#13115536C212yesmaximal, complemented
5#13115460C212yesmaximal, complemented
6#13115187C212yesmaximal, complemented
7#13115153C211yesmaximal, characteristic, complemented
8#13115029C211yesmaximal, characteristic, complemented
9#13114767C211yesmaximal, characteristic, complemented
10#13114598C211yesmaximal, characteristic, complemented
11C2×C13×#390C211yesmaximal, characteristic, complemented
12C23×D4×C13C2211yescharacteristic
13C22×C13×C4⋊C4C2211yescharacteristic
14C22×C13×C22⋊C4—21no—
15C22×C13×C22⋊C4—21no—
16C22×C42×C13C2211yescharacteristic
17C22×#6104C2212yes—
18C2×#88184C2211yescharacteristic
19C2×C4×C22⋊D13—21no—
20C2×C4×D52C2211yescharacteristic, complemented
21C13×#390—21no—
22C13×#390—21no—
23C13×#390C2212yes—
24C13×#390C2212yes—
25#11718878C418yescomplemented
26C2×#88072C2212yes—
27C2×#88072C2212yes—
28C2×#88072C2212yes—
29#11718874—21no—
30#11718873C2211yescharacteristic
31C2×C22⋊#1055C2211yescharacteristic
32C2×C22⋊#1055C2211yescharacteristic
33C2×#88049C2212yescomplemented
34C2×#88048C2212yescomplemented
35#11718601—21no—
36#11718600C2211yescharacteristic, complemented
37C4×#6103—21no—
38C4×C4⋊Dic13C2211yescharacteristic, complemented
39#11718589C2211yescharacteristic, complemented
40C2×#88043C2212yescomplemented
41#11718121C2212yescomplemented
42#11718121C2212yescomplemented
43#11718116C2212yescomplemented
44C25×C13—21no—
45C22×D4×C13—21no—
46C22×D4×C13—22no—
47C22×D4×C13—42no—
48C22×D4×C13—42no—
49C22×D4×C13C2×C414yescomplemented
50C22×D4×C13C2312yes—
51C23×C4×C13—21no—
52C23×C4×C13D411yescharacteristic
53C23×C4×C13D411yescharacteristic
54C23×C4×C13C2311yescharacteristic
55C22×C22⋊D13—21no—
56C22×D52C2311yescharacteristic
57C22×C4×D13—21no—
58C2×C13×C4⋊C4—21no—
59C2×C13×C4⋊C4—42no—
60C2×C13×C4⋊C4C2312yes—
61C2×C13×C22⋊C4—22no—
62C2×C13×C22⋊C4—22no—
63C2×C13×C22⋊C4—22no—
64C2×C13×C22⋊C4—22no—
65C2×C13×C22⋊C4—44no—
66C2×C13×C22⋊C4—44no—
67C2×C42×C13—42no—
68C2×C42×C13D411yescharacteristic
69C2×C42×C13D411yescharacteristic
70C2×C42×C13C2311yescharacteristic
71C2×C42×C13C2311yescharacteristic
72C2×C22⋊Dic13—21no—
73C2×#6104—42no—
74C2×#6104—42no—
75C2×#6104C2×C418yescomplemented
76#88184—22no—
77#88184C2×C414yes—
78C4×C22⋊D13—44no—
79C2×#6103—21no—
80C2×#6103—21no—
81C2×#6103—21no—
82C2×#6103—21no—
83C2×C4⋊Dic13C2311yescharacteristic
84C2×C4⋊Dic13C2311yescharacteristic
85C2×C4⋊Dic13C2311yescharacteristic
86C2×#6101—21no—
87C2×C4×Dic13—21no—
88C22×#1055C2312yes—
89C22×#1055C2312yes—
90C4×D52—42no—
91#88072—42no—
92#88072—42no—
93#88072C2×C418yescomplemented
94#88071—21no—
95#88071—21no—
96#88071—21no—
97C22⋊#1055—22no—
98C22⋊#1055C2×C414yescomplemented
99#88049—42no—
100#88048—42no—
101#88043—42no—
102#22046—131nomaximal, Sylow
103C24×C13—24no—
104C24×C13—22no—
105C24×C13—42no—
106C24×C13—42no—
107C24×C13—42no—
108C24×C13—42no—
109C24×C13—42no—
110C24×C13—42no—
111C24×C13C2×D411yescharacteristic
112C23×D13—21no—
113C2×D4×C13—24no—
114C2×D4×C13—22no—
115C2×D4×C13—22no—
116C2×D4×C13—22no—
117C2×D4×C13—22no—
118C2×D4×C13—22no—
119C2×D4×C13—48no—
120C2×D4×C13—42no—
121C2×D4×C13—42no—
122C2×D4×C13—48no—
123C2×D4×C13C22×C414yes—
124C22×C4×C13—22no—
125C22×C4×C13—22no—
126C22×C4×C13—22no—
127C22×C4×C13—22no—
128C22×C4×C13—22no—
129C22×C4×C13—22no—
130C22×C4×C13—41no—
131C22×C4×C13—41no—
132C22×C4×C13—41no—
133C22×C4×C13—42no—
134C22×C4×C13—42no—
135C22×C4×C13—42no—
136C22×C4×C13C22×C412yes—
137C22×C4×C13C2×D411yescharacteristic
138C22×C4×C13C2×D411yescharacteristic
139C22×C4×C13C2×D411yescharacteristic
140C22×C4×C13C2×D411yescharacteristic
141C22×C4×C13C2×D411yescharacteristic
142C22×C4×C13#4112yes—
143C22×C4×C13C2411yescharacteristic
144C2×C22⋊D13—24no—
145C2×C22⋊D13—42no—
146C2×C22⋊D13—42no—
147C22×Dic13—21no—
148C22×Dic13—21no—
149C22×Dic13—21no—
150C2×D52—22no—
151C2×D52—22no—
152C2×D52—21no—
153C2×D52C22×C412yes—
154C2×C4×D13—42no—
155C2×C4×D13—42no—
156C2×C4×D13—42no—
157C13×C4⋊C4—22no—
158C13×C4⋊C4—22no—
159C13×C4⋊C4—42no—
160C13×C22⋊C4—44no—
161C13×C22⋊C4—44no—
162C13×C22⋊C4—44no—
163C13×C22⋊C4—44no—
164C42×C13—22no—
165C42×C13—22no—
166C42×C13—41no—
167C42×C13—41no—
168C22⋊Dic13—42no—
169#6104—48no—
170#6103—24no—
171#6103—42no—
172#6103—42no—
173#6103—42no—
174C4⋊Dic13—21no—
175C4⋊Dic13—22no—
176C4⋊Dic13—22no—
177C4⋊Dic13C22×C412yes—
178#6101—42no—
179C4×Dic13—42no—
180C2×#1055—22no—
181C2×#1055—22no—
182C2×#1055—22no—
183C2×#1055—22no—
184C2×#1055—22no—
185C2×#1055C22×C414yes—
186C2×#447—131no—
187C4×#121—131no—
188C2×#390D1311yescharacteristic, complemented
189#1972—132no—
190#1936—132no—
191#1926—132no—
192#1879—132no—
193#1874—131no—
194#1853—131no—
195#1786—131no—
196#1749—131no—
197C23×C13—21no—
198C23×C13—21no—
199C23×C13—21no—
200C23×C13—21no—
201C23×C13—24no—
202C23×C13—48no—
203C23×C13—48no—
204C23×C13—42no—
205C23×C13—42no—
206C23×C13—42no—
207C23×C13—42no—
208C23×C13—42no—
209C23×C13—42no—
210C23×C13—48no—
211C23×C13—84no—
212C23×C13—84no—
213C23×C13—84no—
214C23×C13—84no—
215C23×C13C4×D412yes—
216C23×C13AGL(1,ℤ/8)12yes—
217C23×C13AGL(1,ℤ/8)12yes—
218C23×C13C22×D411yescharacteristic
219C22×D13—22no—
220C22×D13—41no—
221C22×D13—41no—
222C22×D13—41no—
223C22×D13—41no—
224C22×D13—41no—
225C22×D13—41no—
226D4×C13—24no—
227D4×C13—24no—
228D4×C13—48no—
229D4×C13—42no—
230D4×C13—42no—
231C2×C4×C13—21no—
232C2×C4×C13—21no—
233C2×C4×C13—21no—
234C2×C4×C13—21no—
235C2×C4×C13—24no—
236C2×C4×C13—24no—
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×C13—21no—
244C2×C4×C13—21no—
245C2×C4×C13—21no—
246C2×C4×C13—44no—
247C2×C4×C13—44no—
248C2×C4×C13—42no—
249C2×C4×C13—42no—
250C2×C4×C13—42no—
251C2×C4×C13—84no—
252C2×C4×C13C4×D412yes—
253C2×C4×C13AGL(1,ℤ/8)12yes—
254C2×C4×C13C23×C411yescharacteristic
255C2×C4×C13C2×#4111yescharacteristic
256C2×C4×C13H(5,𝔽2)12yes—
257C2×C4×C13H(5,𝔽2)12yes—
258C22⋊D13—48no—
259C2×Dic13—22no—
260C2×Dic13—41no—
261C2×Dic13—41no—
262C2×Dic13—41no—
263C2×Dic13—41no—
264C2×Dic13—41no—
265C2×Dic13—41no—
266C2×Dic13—41no—
267C2×Dic13—41no—
268D52—24no—
269D52—42no—
270C4×D13—84no—
271#1055—24no—
272#1055—42no—
273C23×D4C2×D1311yescharacteristic
274C22×D8—132no—
275C2×#121—131no—
276C2×C4×D4—131no—
277C2×C4×D4—261no—
278C22×C4⋊C4C2×D1311yescharacteristic
279C22×C22⋊C4—21no—
280C22×C22⋊C4—21no—
281C22×C42C2×D1311yescharacteristic
282#447—138no—
283C2×#106—132no—
284C2×#102—132no—
285C2×#102—132no—
286C2×#102—132no—
287C2×#102—132no—
288C2×C22⋊C8—131no—
289C2×C22⋊C8—131no—
290C2×#97—132no—
291#390—21no—
292#390—21no—
293#390—131no—
294#390C2×D1312yes—
295#390C2×D1312yes—
296#389—131no—
297#388—261no—
298#386—261no—
299#380—131no—
300C4×C4⋊C4—131no—
301C4×C22⋊C4—261no—
302#340—132no—
303#340—132no—
304#336—132no—
305C22×C13—22no—
306C22×C13—22no—
307C22×C13—22no—
308C22×C13—22no—
309C22×C13—22no—
310C22×C13—22no—
311C22×C13—44no—
312C22×C13—44no—
313C22×C13—44no—
314C22×C13—816no—
315C22×C13—82no—
316C22×C13—82no—
317C22×C13—82no—
318C22×C13—82no—
319C22×C13#44214yes—
320C22×C13C2×C4×D411yescharacteristic
321C22×C13#53411yescharacteristic
322C22×C13#53511yescharacteristic
323C22×C13C2×AGL(1,ℤ/8)11yescharacteristic
324C22×C13C2×AGL(1,ℤ/8)11yescharacteristic
325C22×C13C2×AGL(1,ℤ/8)11yescharacteristic
326C22×C13#57511yescharacteristic
327C2×D13—42no—
328C2×D13—42no—
329C2×D13—42no—
330C2×D13—81no—
331C2×D13—81no—
332C2×D13—81no—
333C2×D13—81no—
334C4×C13—22no—
335C4×C13—22no—
336C4×C13—41no—
337C4×C13—41no—
338C4×C13—41no—
339C4×C13—82no—
340C4×C13#51812yes—
341C4×C13#53812yes—
342Dic13—42no—
343Dic13—81no—
344Dic13—81no—
345C25—21no—
346C22×D4—22no—
347C22×D4—21no—
348C22×D4—42no—
349C22×D4—42no—
350C22×D4—131no—
351C22×D4—261no—
352C22×D4C4×D1314yescomplemented
353C22×D4C22×D1312yes—
354C23×C4—21no—
355C23×C4—261no—
356C23×C4C22⋊D1311yescharacteristic
357C23×C4C22⋊D1311yescharacteristic
358C23×C4C22×D1311yescharacteristic
359C2×D8—138no—
360C2×D8—522no—
361C2×D8—522no—
362C22×C8—132no—
363C22×C8—132no—
364#121—134no—
365#121—262no—
366C4×D4—522no—
367C4×D4—524no—
368C2×C4⋊C4—21no—
369C2×C4⋊C4—42no—
370C2×C4⋊C4—131no—
371C2×C4⋊C4—131no—
372C2×C4⋊C4—131no—
373C2×C4⋊C4—261no—
374C2×C4⋊C4C22×D1312yes—
375C2×C22⋊C4—22no—
376C2×C22⋊C4—22no—
377C2×C22⋊C4—22no—
378C2×C22⋊C4—22no—
379C2×C22⋊C4—44no—
380C2×C22⋊C4—44no—
381C2×C22⋊C4—261no—
382C2×C22⋊C4—261no—
383C2×C22⋊C4—261no—
384C2×C22⋊C4—261no—
385C2×C22⋊C4—261no—
386C2×C42—42no—
387C2×C42—261no—
388C2×C42C22⋊D1311yescharacteristic
389C2×C42C22⋊D1311yescharacteristic
390C2×C42C22×D1311yescharacteristic
391C2×C42C22×D1311yescharacteristic
392#106—522no—
393#102—138no—
394#102—522no—
395#102—522no—
396#102—522no—
397C22⋊C8—134no—
398C22⋊C8—262no—
399#97—522no—
400#95—261no—
401#95—261no—
402#95—261no—
403C2×C13—24no—
404C2×C13—84no—
405C2×C13#232111yescharacteristic
406C2×C13C2×#44211yescharacteristic
407C2×C13C4×AGL(1,ℤ/8)11yescharacteristic
408C2×C13#319211yescharacteristic
409C2×C13#319411yescharacteristic
410C2×C13#321311yescharacteristic
411C2×C13#321711yescharacteristic
412D13—82no—
413C24—24no—
414C24—22no—
415C24—42no—
416C24—42no—
417C24—42no—
418C24—42no—
419C24—42no—
420C24—42no—
421C24—261no—
422C24C2×C22⋊D1311yescharacteristic
423C2×D4—24no—
424C2×D4—22no—
425C2×D4—22no—
426C2×D4—22no—
427C2×D4—22no—
428C2×D4—22no—
429C2×D4—48no—
430C2×D4—42no—
431C2×D4—42no—
432C2×D4—48no—
433C2×D4—132no—
434C2×D4—264no—
435C2×D4—261no—
436C2×D4—262no—
437C2×D4—262no—
438C2×D4—522no—
439C2×D4—522no—
440C2×D4C2×C4×D1314yes—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—41no—
448C22×C4—41no—
449C22×C4—41no—
450C22×C4—42no—
451C22×C4—42no—
452C22×C4—42no—
453C22×C4—261no—
454C22×C4—261no—
455C22×C4—261no—
456C22×C4—522no—
457C22×C4—522no—
458C22×C4—522no—
459C22×C4C2×C4×D1312yes—
460C22×C4#612712yes—
461C22×C4C2×C22⋊D1311yescharacteristic
462C22×C4C2×C22⋊D1311yescharacteristic
463C22×C4C2×C22⋊D1311yescharacteristic
464C22×C4C2×C22⋊D1311yescharacteristic
465C22×C4C2×C22⋊D1311yescharacteristic
466C22×C4C23×D1311yescharacteristic
467D8—528no—
468C2×C8—134no—
469C2×C8—262no—
470C2×C8—262no—
471C2×C8—262no—
472C2×C8—262no—
473C2×C8—262no—
474C4⋊C4—22no—
475C4⋊C4—22no—
476C4⋊C4—42no—
477C4⋊C4—132no—
478C4⋊C4—261no—
479C4⋊C4—262no—
480C4⋊C4—262no—
481C4⋊C4—522no—
482C22⋊C4—44no—
483C22⋊C4—44no—
484C22⋊C4—44no—
485C22⋊C4—44no—
486C22⋊C4—264no—
487C22⋊C4—522no—
488C22⋊C4—522no—
489C22⋊C4—522no—
490C22⋊C4—522no—
491C42—22no—
492C42—22no—
493C42—41no—
494C42—41no—
495C42—522no—
496C13#2204611yescharacteristic, complemented, Sylow
497C23—21no—
498C23—21no—
499C23—21no—
500C23—21no—
501C23—24no—
502C23—48no—
503C23—48no—
504C23—48no—
505C23—42no—
506C23—42no—
507C23—42no—
508C23—42no—
509C23—42no—
510C23—42no—
511C23—84no—
512C23—84no—
513C23—84no—
514C23—84no—
515C23—262no—
516C23—521no—
517C23—521no—
518C23—521no—
519C23—521no—
520C23—521no—
521C23—521no—
522C23C4×C22⋊D1312yes—
523C23#8818612yes—
524C23#8820312yes—
525C23C22×C22⋊D1311yescharacteristic
526D4—24no—
527D4—24no—
528D4—48no—
529D4—42no—
530D4—42no—
531D4—264no—
532D4—528no—
533D4—522no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—24no—
539C2×C4—24no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—44no—
550C2×C4—44no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—84no—
555C2×C4—262no—
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×C22⋊D1312yes—
566C2×C4#8818612yes—
567C2×C4C22×C4×D1311yescharacteristic
568C2×C4C2×#612711yescharacteristic
569C2×C4#8825112yes—
570C2×C4#8825112yes—
571C8—264no—
572C8—522no—
573C22—22no—
574C22—22no—
575C22—22no—
576C22—22no—
577C22—22no—
578C22—22no—
579C22—44no—
580C22—44no—
581C22—44no—
582C22—816no—
583C22—82no—
584C22—82no—
585C22—82no—
586C22—82no—
587C22—522no—
588C22—522no—
589C22—522no—
590C22—1041no—
591C22—1041no—
592C22—1041no—
593C22—1041no—
594C22#1171867514yes—
595C22C2×C4×C22⋊D1311yescharacteristic
596C22C2×#8818611yescharacteristic
597C22C2×#8818611yescharacteristic
598C22#1171950911yescharacteristic
599C22#1171951011yescharacteristic
600C22C2×#8820311yescharacteristic
601C22#1171952711yescharacteristic
602C4—22no—
603C4—22no—
604C4—41no—
605C4—41no—
606C4—41no—
607C4—82no—
608C4—522no—
609C4—1041no—
610C4—1041no—
611C4#1171919112yes—
612C4#1171919912yes—
613C2—24no—
614C2—84no—
615C2—1042no—
616C2#1311749411yescharacteristic
617C2C2×#1171867511yescharacteristic
618C2C4×#8818611yescharacteristic
619C2#1312563311yescharacteristic
620C2#1312565011yescharacteristic
621C2#1312582311yescharacteristic
622C2#1312585411yescharacteristic
subgroupG/H
centerC23C22×C22⋊D13
commutatorC2×C4×C13C23×C4
FrattiniC22×C4C23×D13
FittingC2×C13×#390C2
socleC23×C13C22×D4
prime radicalC2×#11718878C2
cosocleC22×C4×C13C24
hypercenterC2×#390D13
hypocenterC13#22046
LCSG ▷ C2×C4×C13 ▷ C2×C13 ▷ C13
UCSC1 ◁ C23 ◁ C22×C42 ◁ C2×#390

Complements

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

NH ≅ G/Ntype
2. C4×#88184C2semidirect
3. #13115711C2semidirect
4. #13115536C2semidirect
5. #13115460C2semidirect
6. #13115187C2semidirect
7. #13115153C2semidirect
8. #13115029C2semidirect
9. #13114767C2semidirect
10. #13114598C2semidirect
11. C2×C13×#390C2semidirect
20. C2×C4×D52C22semidirect
25. #11718878C4semidirect
33. C2×#88049C22semidirect
34. C2×#88048C22semidirect
36. #11718600C22semidirect
38. C4×C4⋊Dic13C22semidirect
39. #11718589C22semidirect
40. C2×#88043C22semidirect
41. #11718121C22semidirect
42. #11718121C22semidirect
43. #11718116C22semidirect
49. C22×D4×C13C2×C4semidirect
75. C2×#6104C2×C4semidirect
93. #88072C2×C4semidirect
98. C22⋊#1055C2×C4semidirect
188. C2×#390D13semidirect
352. C22×D4C4×D13semidirect
496. C13#22046semidirect

Maximal subgroups

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

subgroupindex
C2×#117188782
C4×#881842
#131157112
#131155362
#131154602
#131151872
#131151532
#131150292
#131147672
#131145982
C2×C13×#3902
#2204613