Finite Group Databasebeta

← #15878571

Subgroups — #15878571

Order 3328.

1530 conjugacy classes of subgroups.

The subgroup lattice has 592 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11718673C211yesmaximal, characteristic, complemented
2C2×#11718893C212yesmaximal, complemented
3D4×C4⋊Dic13C211yesmaximal, characteristic, complemented
4#13115760C211yesmaximal, characteristic, complemented
5#13115639C211yesmaximal, characteristic, complemented
6#13115270C212yesmaximal, complemented
7#13115144C212yesmaximal, complemented
8#13114836C211yesmaximal, characteristic, complemented
9#13114776C212yesmaximal, complemented
10C4×#88072C211yesmaximal, characteristic, complemented
11C2×C13×#547C211yesmaximal, characteristic
12C22×C13×#41C2212yes—
13C13×#547—24no—
14C13×#547C418yescomplemented
15C2×C13×C4⋊Q8C2211yescharacteristic
16C2×C13×#123—22no—
17C2×C13×C22⋊Q8C2212yes—
18C2×C13×#121—21no—
19C2×C4×D4×C13C2211yescharacteristic
20C2×C4×D4×C13C2211yescharacteristic
21C22×C13×C4⋊C4—21no—
22C2×D4×Dic13—21no—
23C22×C4⋊Dic13C2212yes—
24C22×#6099C2212yes—
25#11718893—28no—
26C2×#88075C2212yes—
27C2×#88075C2212yes—
28C2×#88075C2212yes—
29C2×#88075C2212yes—
30C2×#88072C2212yes—
31C2×#88072C2211yescharacteristic
32C2×#88072C2211yescharacteristic
33#11718873C2212yes—
34C2×C22⋊#1055C2212yes—
35#11718673—24no—
36#11718646—21no—
37#11718611—22no—
38#11718596C2211yescharacteristic
39C4×C4⋊Dic13C2211yescharacteristic
40C2×C4⋊#1055C2211yescharacteristic
41C2×C4×#1055C2211yescharacteristic
42#11718121C2211yescharacteristic
43#11718121C2212yes—
44#11718121C2211yescharacteristic
45C2×C13×#41—24no—
46C2×C13×#41—24no—
47C2×C13×#41—24no—
48C2×C13×#41—42no—
49C2×C13×#41—42no—
50C2×C13×#41C2×C418yescomplemented
51C22×Q8×C13C2312yes—
52C22×D4×C13—22no—
53C22×D4×C13D411yescharacteristic
54C22×D4×C13C2311yescharacteristic
55C23×C4×C13—21no—
56C23×C4×C13—22no—
57C23×C4×C13C2312yes—
58C23×Dic13—22no—
59C13×C4⋊Q8—22no—
60C13×C4⋊Q8C2×C414yescomplemented
61C13×#123—216no—
62C13×C22⋊Q8—24no—
63C13×C22⋊Q8C2×C418yescomplemented
64C13×#121—28no—
65C4×D4×C13—22no—
66C4×D4×C13C2×C414yes—
67C4×D4×C13C2×C414yescomplemented
68C4×D4×C13D414yes—
69C2×C13×C4⋊C4—21no—
70C2×C13×C4⋊C4—22no—
71C2×C13×C4⋊C4—21no—
72C2×C13×C4⋊C4—28no—
73C2×C13×C4⋊C4D411yescharacteristic
74C2×C13×C4⋊C4C2311yescharacteristic
75C2×C13×C22⋊C4—22no—
76C2×C13×C22⋊C4—21no—
77C2×C13×C22⋊C4D412yes—
78C2×C42×C13C2311yescharacteristic
79C2×C22⋊Dic13—22no—
80D4×Dic13—44no—
81C2×C4⋊Dic13—24no—
82C2×C4⋊Dic13—22no—
83C2×C4⋊Dic13—21no—
84C2×C4⋊Dic13—21no—
85C2×C4⋊Dic13—42no—
86C2×C4⋊Dic13C2312yes—
87C2×C4⋊Dic13C2311yescharacteristic
88C2×C4⋊Dic13C2311yescharacteristic
89C2×C4×Dic13—21no—
90C2×#6099—24no—
91C2×#6099—24no—
92C2×#6099—42no—
93C22×#1055C2312yes—
94C22×#1055C2311yescharacteristic
95C22×#1055C2311yescharacteristic
96#88075—24no—
97#88075—24no—
98#88075—42no—
99#88075—42no—
100#88072—24no—
101#88072—22no—
102#88072D414yescomplemented
103#88071—21no—
104#88071—22no—
105C22⋊#1055—24no—
106C4⋊#1055—22no—
107C4×#1055—22no—
108#22431—131nomaximal, Sylow
109C24×C13—21no—
110C24×C13C2×D412yes—
111C13×#41—216no—
112C13×#41—48no—
113C13×#41—48no—
114C13×#41—48no—
115C2×Q8×C13—22no—
116C2×Q8×C13—22no—
117C2×Q8×C13—22no—
118C2×Q8×C13—22no—
119C2×Q8×C13—22no—
120C2×Q8×C13C22×C414yes—
121C2×D4×C13—28no—
122C2×D4×C13—22no—
123C2×D4×C13—22no—
124C2×D4×C13—21no—
125C2×D4×C13—21no—
126C2×D4×C13—21no—
127C2×D4×C13—44no—
128C2×D4×C13—44no—
129C2×D4×C13C22⋊C414yes—
130C2×D4×C13C22×C412yes—
131C2×D4×C13C2×D412yes—
132C2×D4×C13#4112yes—
133C22×C4×C13—22no—
134C22×C4×C13—22no—
135C22×C4×C13—22no—
136C22×C4×C13—22no—
137C22×C4×C13—21no—
138C22×C4×C13—22no—
139C22×C4×C13—24no—
140C22×C4×C13—28no—
141C22×C4×C13—44no—
142C22×C4×C13—44no—
143C22×C4×C13—42no—
144C22×C4×C13—42no—
145C22×C4×C13C22×C414yes—
146C22×C4×C13C2×D411yescharacteristic
147C22×C4×C13C2×D412yes—
148C22×C4×C13C2×D414yes—
149C22×C4×C13C2×D411yescharacteristic
150C22×C4×C13C2411yescharacteristic
151C22×Dic13—21no—
152C22×Dic13—21no—
153C22×Dic13—22no—
154C22×Dic13—44no—
155C22×Dic13—44no—
156C22×Dic13—82no—
157C13×C4⋊C4—24no—
158C13×C4⋊C4—28no—
159C13×C4⋊C4—24no—
160C13×C4⋊C4—21no—
161C13×C4⋊C4C22⋊C414yes—
162C13×C4⋊C4C22×C412yes—
163C13×C22⋊C4—28no—
164C13×C22⋊C4—24no—
165C13×C22⋊C4C22⋊C418yes—
166C42×C13C22×C412yes—
167C42×C13C2×D412yes—
168C22⋊Dic13—44no—
169C4⋊Dic13—21no—
170C4⋊Dic13—22no—
171C4⋊Dic13—21no—
172C4⋊Dic13—21no—
173C4⋊Dic13—22no—
174C4⋊Dic13—44no—
175C4⋊Dic13—42no—
176C4⋊Dic13—81no—
177C4⋊Dic13C2×D412yes—
178C4×Dic13—42no—
179#6099—48no—
180C2×#1055—21no—
181C2×#1055—21no—
182C2×#1055—21no—
183C2×#1055—22no—
184C2×#1055—22no—
185C2×#1055—21no—
186C2×#1055—21no—
187C2×#1055—22no—
188C2×#1055C2×D412yes—
189C2×#547D1311yescharacteristic
190C2×#436—131no—
191C2×#417—132no—
192D4×C4⋊C4—131no—
193#1981—131no—
194#1956—131no—
195#1894—132no—
196#1872—132no—
197#1802—131no—
198#1788—132no—
199C4×#102—131no—
200C23×C13—22no—
201C23×C13—22no—
202C23×C13—22no—
203C23×C13—22no—
204C23×C13—22no—
205C23×C13—22no—
206C23×C13—22no—
207C23×C13—22no—
208C23×C13—22no—
209C23×C13—41no—
210C23×C13—41no—
211C23×C13—41no—
212C23×C13—41no—
213C23×C13C2×C22⋊C414yes—
214C23×C13C22×D411yescharacteristic
215Q8×C13—24no—
216Q8×C13—24no—
217Q8×C13—42no—
218Q8×C13—42no—
219D4×C13—22no—
220D4×C13—22no—
221D4×C13—22no—
222D4×C13—28no—
223D4×C13—416no—
224D4×C13C4×D414yescomplemented
225C2×C4×C13—22no—
226C2×C4×C13—22no—
227C2×C4×C13—21no—
228C2×C4×C13—21no—
229C2×C4×C13—24no—
230C2×C4×C13—24no—
231C2×C4×C13—24no—
232C2×C4×C13—24no—
233C2×C4×C13—22no—
234C2×C4×C13—24no—
235C2×C4×C13—416no—
236C2×C4×C13—42no—
237C2×C4×C13—42no—
238C2×C4×C13—41no—
239C2×C4×C13—41no—
240C2×C4×C13—48no—
241C2×C4×C13C2×C22⋊C412yes—
242C2×C4×C13C2×C22⋊C414yes—
243C2×C4×C13C2×C22⋊C412yes—
244C2×C4×C13AGL(1,ℤ/8)12yes—
245C2×C4×C13#13712yes—
246C2×C4×C13C23×C411yescharacteristic
247C2×C4×C13C22×D411yescharacteristic
248C2×C4×C13C22×D411yescharacteristic
249C2×C4×C13C2×#4111yescharacteristic
250C2×C4×C13H(5,𝔽2)11yescharacteristic
251C2×C4×C13H(5,𝔽2)11yescharacteristic
252C2×Dic13—22no—
253C2×Dic13—41no—
254C2×Dic13—41no—
255C2×Dic13—42no—
256C2×Dic13—41no—
257C2×Dic13—41no—
258C2×Dic13—42no—
259C2×Dic13—42no—
260C2×Dic13—41no—
261C2×Dic13—88no—
262#1055—22no—
263#1055—42no—
264#1055—41no—
265C22×#41C2×D1312yes—
266C22×M16—132no—
267#547—24no—
268#547Dic1318yescomplemented
269C2×C4⋊Q8C2×D1311yescharacteristic
270C2×#123—22no—
271C2×C22⋊Q8C2×D1312yes—
272C2×#121—21no—
273C2×C4×D4—261no—
274C2×C4×D4C2×D1311yescharacteristic
275C2×C4×D4C2×D1311yescharacteristic
276C22×C4⋊C4—21no—
277C22×C4⋊C4—132no—
278#436—264no—
279C2×C4⋊C8—131no—
280#417—268no—
281C2×#103—132no—
282C2×#103—132no—
283C2×#103—132no—
284C2×#103—132no—
285C2×#102—131no—
286C2×#102—132no—
287C2×#102—131no—
288C2×C22⋊C8—132no—
289C2×C4×C8—131no—
290#389—261no—
291#385—262no—
292#384—131no—
293#380—132no—
294C4×C4⋊C4—131no—
295#340—132no—
296#340—131no—
297#340—131no—
298C22×C13—24no—
299C22×C13—24no—
300C22×C13—24no—
301C22×C13—24no—
302C22×C13—42no—
303C22×C13—42no—
304C22×C13—42no—
305C22×C13—42no—
306C22×C13—42no—
307C22×C13—42no—
308C22×C13—42no—
309C22×C13—42no—
310C22×C13—42no—
311C22×C13—42no—
312C22×C13C22×C22⋊C411yescharacteristic
313C22×C13D4211yescharacteristic
314C22×C13#54611yescharacteristic
315C22×C13C2×AGL(1,ℤ/8)11yescharacteristic
316C22×C13C2×#13711yescharacteristic
317C22×C13#57711yescharacteristic
318C22×C13#57811yescharacteristic
319C4×C13—24no—
320C4×C13—22no—
321C4×C13—44no—
322C4×C13—42no—
323C4×C13#41714yescomplemented
324C4×C13C2×C4×D411yescharacteristic
325C4×C13#51811yescharacteristic
326C4×C13D4211yescharacteristic
327C4×C13#54611yescharacteristic
328Dic13—42no—
329Dic13—81no—
330Dic13—82no—
331C2×#41—24no—
332C2×#41—24no—
333C2×#41—24no—
334C2×#41—42no—
335C2×#41—42no—
336C2×#41C2×Dic1318yescomplemented
337C22×Q8C22×D1312yes—
338C22×D4—22no—
339C22×D4C22⋊D1311yescharacteristic
340C22×D4C22×D1311yescharacteristic
341C23×C4—21no—
342C23×C4—22no—
343C23×C4—262no—
344C23×C4C22×D1312yes—
345C2×M16—264no—
346C2×M16—264no—
347C2×M16—522no—
348C22×C8—131no—
349C22×C8—132no—
350C22×C8—131no—
351C4⋊Q8—22no—
352C4⋊Q8C2×Dic1314yescomplemented
353#123—216no—
354C22⋊Q8—24no—
355C22⋊Q8C2×Dic1318yescomplemented
356#121—28no—
357C4×D4—22no—
358C4×D4—524no—
359C4×D4C2×Dic1314yes—
360C4×D4C2×Dic1314yescomplemented
361C4×D4C22⋊D1314yes—
362C2×C4⋊C4—21no—
363C2×C4⋊C4—22no—
364C2×C4⋊C4—21no—
365C2×C4⋊C4—28no—
366C2×C4⋊C4—131no—
367C2×C4⋊C4—132no—
368C2×C4⋊C4—131no—
369C2×C4⋊C4—264no—
370C2×C4⋊C4—262no—
371C2×C4⋊C4—261no—
372C2×C4⋊C4—261no—
373C2×C4⋊C4—522no—
374C2×C4⋊C4C22⋊D1311yescharacteristic
375C2×C4⋊C4C22×D1311yescharacteristic
376C2×C22⋊C4—22no—
377C2×C22⋊C4—21no—
378C2×C22⋊C4—262no—
379C2×C22⋊C4C22⋊D1312yes—
380C2×C42—261no—
381C2×C42C22×D1311yescharacteristic
382C4⋊C8—262no—
383#103—264no—
384#103—264no—
385#103—522no—
386#103—522no—
387#102—134no—
388#102—264no—
389#102—262no—
390C22⋊C8—264no—
391C4×C8—262no—
392#95—261no—
393#95—262no—
394C2×C13—44no—
395C2×C13—44no—
396C2×C13D4×C22⋊C411yescharacteristic
397C2×C13C2×#41711yescharacteristic
398C2×C13#289811yescharacteristic
399C2×C13#327211yescharacteristic
400C2×C13#327611yescharacteristic
401C2×C13#333011yescharacteristic
402C2×C13#333111yescharacteristic
403C24—21no—
404C24C2×C22⋊D1312yes—
405#41—216no—
406#41—48no—
407#41—48no—
408#41—48no—
409C2×Q8—22no—
410C2×Q8—22no—
411C2×Q8—22no—
412C2×Q8—22no—
413C2×Q8—22no—
414C2×Q8C22×Dic1314yes—
415C2×D4—28no—
416C2×D4—22no—
417C2×D4—22no—
418C2×D4—21no—
419C2×D4—21no—
420C2×D4—21no—
421C2×D4—44no—
422C2×D4—44no—
423C2×D4C22⋊Dic1314yes—
424C2×D4D4×D1312yes—
425C2×D4#612912yes—
426C2×D4C22×Dic1312yes—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—21no—
432C22×C4—22no—
433C22×C4—24no—
434C22×C4—28no—
435C22×C4—44no—
436C22×C4—44no—
437C22×C4—42no—
438C22×C4—42no—
439C22×C4—261no—
440C22×C4—261no—
441C22×C4—262no—
442C22×C4—524no—
443C22×C4—524no—
444C22×C4—1042no—
445C22×C4C22×Dic1314yes—
446C22×C4C2×C22⋊D1311yescharacteristic
447C22×C4C2×C22⋊D1312yes—
448C22×C4C2×C22⋊D1314yes—
449C22×C4C2×C22⋊D1311yescharacteristic
450C22×C4C23×D1311yescharacteristic
451M16—528no—
452C2×C8—132no—
453C2×C8—261no—
454C2×C8—261no—
455C2×C8—262no—
456C2×C8—261no—
457C2×C8—261no—
458C2×C8—262no—
459C2×C8—261no—
460C2×C8—262no—
461C4⋊C4—24no—
462C4⋊C4—28no—
463C4⋊C4—24no—
464C4⋊C4—21no—
465C4⋊C4—132no—
466C4⋊C4—261no—
467C4⋊C4—262no—
468C4⋊C4—261no—
469C4⋊C4—261no—
470C4⋊C4—262no—
471C4⋊C4—524no—
472C4⋊C4—522no—
473C4⋊C4—1041no—
474C4⋊C4C22⋊Dic1314yes—
475C4⋊C4C22×Dic1312yes—
476C22⋊C4—28no—
477C22⋊C4—24no—
478C22⋊C4—524no—
479C22⋊C4C22⋊Dic1318yes—
480C42—522no—
481C42C22×Dic1312yes—
482C42C2×C22⋊D1312yes—
483C13#2243111yescharacteristic, complemented, Sylow
484C23—22no—
485C23—22no—
486C23—22no—
487C23—22no—
488C23—22no—
489C23—22no—
490C23—22no—
491C23—22no—
492C23—22no—
493C23—41no—
494C23—41no—
495C23—41no—
496C23—41no—
497C23C2×C22⋊Dic1314yes—
498C23C22×C22⋊D1311yescharacteristic
499Q8—24no—
500Q8—24no—
501Q8—42no—
502Q8—42no—
503D4—22no—
504D4—22no—
505D4—22no—
506D4—28no—
507D4—416no—
508D4D4×Dic1314yescomplemented
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—24no—
514C2×C4—24no—
515C2×C4—24no—
516C2×C4—24no—
517C2×C4—22no—
518C2×C4—24no—
519C2×C4—416no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—48no—
525C2×C4—262no—
526C2×C4—521no—
527C2×C4—521no—
528C2×C4—522no—
529C2×C4—521no—
530C2×C4—521no—
531C2×C4—522no—
532C2×C4—522no—
533C2×C4—521no—
534C2×C4—1048no—
535C2×C4#8820312yes—
536C2×C4#8820512yes—
537C2×C4C2×C22⋊Dic1312yes—
538C2×C4C2×C22⋊Dic1314yes—
539C2×C4C2×C22⋊Dic1312yes—
540C2×C4C2×D4×D1311yescharacteristic
541C2×C4C2×#612911yescharacteristic
542C2×C4#8825111yescharacteristic
543C2×C4#8825111yescharacteristic
544C2×C4C23×Dic1311yescharacteristic
545C2×C4C22×C22⋊D1311yescharacteristic
546C8—262no—
547C8—522no—
548C8—521no—
549C22—24no—
550C22—24no—
551C22—24no—
552C22—24no—
553C22—42no—
554C22—42no—
555C22—42no—
556C22—42no—
557C22—42no—
558C22—42no—
559C22—42no—
560C22—42no—
561C22—42no—
562C22—42no—
563C22D4×C22⋊D1311yescharacteristic
564C22#1171951111yescharacteristic
565C22C2×#8820311yescharacteristic
566C22C2×#8820511yescharacteristic
567C22#1171954211yescharacteristic
568C22#1171954311yescharacteristic
569C22C22×C22⋊Dic1311yescharacteristic
570C4—22no—
571C4—24no—
572C4—42no—
573C4—44no—
574C4—522no—
575C4—1042no—
576C4—1041no—
577C4#1171889314yescomplemented
578C4C2×D4×Dic1311yescharacteristic
579C4#1171950311yescharacteristic
580C4D4×C22⋊D1311yescharacteristic
581C4#1171951111yescharacteristic
582C2—44no—
583C2—44no—
584C2D4×C22⋊Dic1311yescharacteristic
585C2C2×#1171889311yescharacteristic
586C2#1312336911yescharacteristic
587C2#1312643611yescharacteristic
588C2#1312647811yescharacteristic
589C2#1312714111yescharacteristic
590C2#1312715211yescharacteristic
subgroupG/H
centerC23C22×C22⋊D13
commutatorC2×C4×C13C23×C4
FrattiniC22×C4C23×D13
Fitting, prime radicalC2×C13×#547C2
socleC23×C13C22×D4
cosocleC22×C4×C13C24
hypercenterC2×#547D13
hypocenterC13#22431
LCSG ▷ C2×C4×C13 ▷ C2×C13 ▷ C13
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ C2×#547

Complements

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

NH ≅ G/Ntype
1. C2×#11718673C2semidirect
2. C2×#11718893C2semidirect
3. D4×C4⋊Dic13C2semidirect
4. #13115760C2semidirect
5. #13115639C2semidirect
6. #13115270C2semidirect
7. #13115144C2semidirect
8. #13114836C2semidirect
9. #13114776C2semidirect
10. C4×#88072C2semidirect
14. C13×#547C4semidirect
50. C2×C13×#41C2×C4semidirect
60. C13×C4⋊Q8C2×C4semidirect
63. C13×C22⋊Q8C2×C4semidirect
67. C4×D4×C13C2×C4semidirect
102. #88072D4semidirect
224. D4×C13C4×D4semidirect
268. #547Dic13semidirect
323. C4×C13#417semidirect
336. C2×#41C2×Dic13semidirect
352. C4⋊Q8C2×Dic13semidirect
355. C22⋊Q8C2×Dic13semidirect
360. C4×D4C2×Dic13semidirect
483. C13#22431semidirect
508. D4D4×Dic13semidirect
577. C4#11718893semidirect

Maximal subgroups

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

subgroupindex
C2×#117186732
C2×#117188932
D4×C4⋊Dic132
#131157602
#131156392
#131152702
#131151442
#131148362
#131147762
C4×#880722
C2×C13×#5472
#2243113