Finite Group Databasebeta

← #15926005

Subgroups — #15926005

Order 3328.

856 conjugacy classes of subgroups.

The subgroup lattice has 640 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11718682C211yesmaximal, characteristic, complemented
2C2×C4×#6106C211yesmaximal, characteristic
3#13118686C211yesmaximal, characteristic
4#13116360C211yesmaximal, characteristic
5#13116318C211yesmaximal, characteristic, complemented
6#13116221C211yesmaximal, characteristic
7#13116169C211yesmaximal, characteristic, complemented
8#13115566C211yesmaximal, characteristic, complemented
9#13115395C211yesmaximal, characteristic, complemented
10#13115191C211yesmaximal, characteristic
11#13115191C211yesmaximal, characteristic
12#13115104C211yesmaximal, characteristic, complemented
13#13115090C211yesmaximal, characteristic, complemented
14#13115082C211yesmaximal, characteristic
15C13×#2351C211yesmaximal, characteristic, complemented
16C2×C4×Q8×C13C2211yescharacteristic
17C2×C4×Q8×C13C2211yescharacteristic
18C2×C4×Q8×C13C2211yescharacteristic
19C22×#6106C2211yescharacteristic
20C2×#88150—21no—
21C2×C4×D52C2211yescharacteristic
22C2×#88123C2211yescharacteristic
23C13×#389—21no—
24C13×#389—21no—
25C13×#389C2211yescharacteristic
26C13×#389C2211yescharacteristic
27C13×#382—21no—
28C13×#382C2211yescharacteristic
29C4×C13×C4⋊C4—21no—
30C4×C13×C4⋊C4C2211yescharacteristic
31C2×#88075C2211yescharacteristic
32C2×#88075C2211yescharacteristic
33C2×#88075C2211yescharacteristic
34C4×#6106—24no—
35#11718682—24no—
36#11718655—21no—
37C2×#88049C2211yescharacteristic
38C2×#88049C2211yescharacteristic
39C2×#88049C2211yescharacteristic
40C2×#88048C2211yescharacteristic
41C2×#88048C2211yescharacteristic
42C2×#88047C2211yescharacteristic
43C2×#88047C2211yescharacteristic
44C2×#88047C2211yescharacteristic
45#11718600C2211yescharacteristic
46#11718600C2211yescharacteristic
47#11718589C2211yescharacteristic
48#11718589C2211yescharacteristic
49C2×C4⋊#1055C2211yescharacteristic
50C2×C4×#1055C2211yescharacteristic
51#11718334—21no—
52#11718329—21no—
53#11718121C2211yescharacteristic
54#11718121C2211yescharacteristic
55#11718121C2211yescharacteristic
56#11718121C2211yescharacteristic
57#11718116C2211yescharacteristic
58#11718116C2211yescharacteristic
59#11718116C2211yescharacteristic
60#11718116C2211yescharacteristic
61C22×Q8×C13C2311yescharacteristic
62C22×D52C2311yescharacteristic
63C22×C4×D13—21no—
64C22×Dic26—21no—
65C4×Q8×C13—22no—
66C4×Q8×C13—22no—
67C4×Q8×C13—22no—
68C4×Q8×C13—22no—
69C4×Q8×C13—41no—
70C4×Q8×C13—41no—
71C2×C13×C4⋊C4—21no—
72C2×C13×C4⋊C4—21no—
73C2×C13×C4⋊C4—21no—
74C2×C13×C4⋊C4—21no—
75C2×C13×C4⋊C4—21no—
76C2×C13×C4⋊C4D411yescharacteristic
77C2×C13×C4⋊C4D411yescharacteristic
78C2×C13×C4⋊C4C2311yescharacteristic
79C2×C13×C4⋊C4C2311yescharacteristic
80C2×C13×C4⋊C4C2311yescharacteristic
81C2×C42×C13—21no—
82C2×C42×C13—21no—
83C2×C42×C13—21no—
84C2×C42×C13C2311yescharacteristic
85C2×C42×C13C2311yescharacteristic
86C2×C42×C13C2311yescharacteristic
87C2×#6106—22no—
88C2×#6106—22no—
89C2×#6106Q814yescomplemented
90C2×#6103—21no—
91C2×#6103—21no—
92C2×#6103—21no—
93C2×#6103—21no—
94C2×#6103—21no—
95C2×C4⋊Dic13—21no—
96C2×C4⋊Dic13C2311yescharacteristic
97C2×C4⋊Dic13C2311yescharacteristic
98C2×C4⋊Dic13C2311yescharacteristic
99C2×#6101—21no—
100C2×#6101—21no—
101C22×#1055C2311yescharacteristic
102C22×#1055C2311yescharacteristic
103C22×#1055C2311yescharacteristic
104C22×#1055C2311yescharacteristic
105#88150—44no—
106C4×D52—22no—
107C4×D52—41no—
108#88123—22no—
109#88123—41no—
110C13×#95—21no—
111C13×#95—21no—
112C13×#95—21no—
113C13×#95—21no—
114C13×#95D411yescharacteristic
115C13×#95D411yescharacteristic
116#88075—22no—
117#88075—22no—
118#88075D414yes—
119#88071—21no—
120#88071—21no—
121#88049—22no—
122#88049—22no—
123#88049—41no—
124#88048—22no—
125#88048—41no—
126#88047—22no—
127#88047—22no—
128#88047—22no—
129C4⋊#1055—22no—
130C4×#1055—22no—
131#26377—131nomaximal, Sylow
132C23×D13—21no—
133C2×Q8×C13—21no—
134C2×Q8×C13—21no—
135C2×Q8×C13—21no—
136C2×Q8×C13C2×D412yes—
137C2×Q8×C13C2×Q812yes—
138C2×Q8×C13#4112yes—
139C22×C4×C13—21no—
140C22×C4×C13—21no—
141C22×C4×C13—21no—
142C22×C4×C13—21no—
143C22×C4×C13C2×D411yescharacteristic
144C22×C4×C13C2×D411yescharacteristic
145C22×C4×C13C2×D411yescharacteristic
146C22×C4×C13C2×D411yescharacteristic
147C22×C4×C13C2×D411yescharacteristic
148C22×C4×C13C2×D411yescharacteristic
149C22×C4×C13C2411yescharacteristic
150C22×Dic13—21no—
151C22×Dic13—21no—
152C22×Dic13—21no—
153C2×D52—21no—
154C2×D52—21no—
155C2×D52—21no—
156C2×D52—21no—
157C2×D52—21no—
158C2×D52C2×Q812yes—
159C2×C4×D13—42no—
160C2×C4×D13—42no—
161C2×C4×D13—81no—
162C2×Dic26—42no—
163C2×Dic26—42no—
164C2×Dic26—81no—
165C13×C4⋊C4—24no—
166C13×C4⋊C4—22no—
167C13×C4⋊C4—22no—
168C13×C4⋊C4—21no—
169C13×C4⋊C4—21no—
170C13×C4⋊C4—21no—
171C13×C4⋊C4—21no—
172C13×C4⋊C4—21no—
173C13×C4⋊C4—42no—
174C13×C4⋊C4—42no—
175C13×C4⋊C4—42no—
176C13×C4⋊C4—42no—
177C13×C4⋊C4#4112yes—
178C42×C13—24no—
179C42×C13—21no—
180C42×C13—21no—
181C42×C13—21no—
182C42×C13—21no—
183C42×C13—21no—
184C42×C13—21no—
185C42×C13—42no—
186C42×C13—42no—
187#6106—28no—
188#6103—42no—
189#6103—42no—
190#6103—42no—
191#6103—42no—
192#6103—81no—
193C4⋊Dic13—21no—
194C4⋊Dic13—21no—
195C4⋊Dic13—21no—
196C4⋊Dic13—21no—
197C4⋊Dic13—21no—
198C4⋊Dic13—42no—
199C4⋊Dic13C2×D412yes—
200#6101—42no—
201#6101—42no—
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×#1055—21no—
212C2×#1055C2×D412yes—
213C2×#1055C2×Q812yes—
214C2×#477—131no—
215C2×C4×SD16—131no—
216#2452—131no—
217#2351D1311yescharacteristic, complemented
218#2092—131no—
219#2085—131no—
220#2067—131no—
221#2058—131no—
222#1942—131no—
223#1916—131no—
224#1880—131no—
225#1880—131no—
226#1865—131no—
227#1863—131no—
228#1862—131no—
229C23×C13C22×D411yescharacteristic
230C22×D13—22no—
231C22×D13—41no—
232C22×D13—41no—
233C22×D13—41no—
234C22×D13—41no—
235C22×D13—41no—
236C22×D13—41no—
237Q8×C13—22no—
238Q8×C13—22no—
239Q8×C13—22no—
240Q8×C13C22⋊Q814yescomplemented
241C2×C4×C13—22no—
242C2×C4×C13—21no—
243C2×C4×C13—21no—
244C2×C4×C13—22no—
245C2×C4×C13—22no—
246C2×C4×C13—21no—
247C2×C4×C13—21no—
248C2×C4×C13—21no—
249C2×C4×C13—21no—
250C2×C4×C13—22no—
251C2×C4×C13—21no—
252C2×C4×C13—21no—
253C2×C4×C13—21no—
254C2×C4×C13—21no—
255C2×C4×C13—21no—
256C2×C4×C13—21no—
257C2×C4×C13—21no—
258C2×C4×C13—21no—
259C2×C4×C13—21no—
260C2×C4×C13—21no—
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×C13—41no—
269C2×C4×C13#13512yes—
270C2×C4×C13AGL(1,ℤ/8)12yes—
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—
286D52—22no—
287D52—22no—
288D52—41no—
289D52—41no—
290C4×D13—84no—
291Dic26—84no—
292#1055—22no—
293#1055—22no—
294#1055—41no—
295#1055—41no—
296C22×SD16—131no—
297C2×C4⋊Q8—131no—
298C2×C22⋊Q8—261no—
299C2×C4×Q8C2×D1311yescharacteristic
300C2×C4×Q8C2×D1311yescharacteristic
301C2×C4×Q8C2×D1311yescharacteristic
302C2×C4×D4—131no—
303#477—264no—
304C4×SD16—264no—
305C2×#107—131no—
306C2×#107—131no—
307C2×#107—131no—
308C2×#106—131no—
309C2×#106—131no—
310C2×C4⋊C8—131no—
311C2×#103—131no—
312C2×#103—131no—
313C2×#103—131no—
314C2×#102—131no—
315C2×#102—131no—
316C2×#102—131no—
317C2×C4×C8—131no—
318#396—261no—
319#393—261no—
320#390—131no—
321#390—131no—
322#389—21no—
323#389—21no—
324#389—131no—
325#389—131no—
326#389C2×D1311yescharacteristic
327#389C2×D1311yescharacteristic
328#385—261no—
329#382—21no—
330#382C2×D1311yescharacteristic
331C4×C4⋊C4—21no—
332C4×C4⋊C4C2×D1311yescharacteristic
333#340—131no—
334#340—131no—
335#340—131no—
336#340—131no—
337#336—131no—
338#336—131no—
339#336—131no—
340#336—131no—
341C22×C13#52511yescharacteristic
342C22×C13D4211yescharacteristic
343C22×C13D4×Q811yescharacteristic
344C22×C13C2×#13511yescharacteristic
345C22×C13C2×AGL(1,ℤ/8)11yescharacteristic
346C22×C13#57711yescharacteristic
347C22×C13#57811yescharacteristic
348C2×D13—42no—
349C2×D13—42no—
350C2×D13—42no—
351C2×D13—81no—
352C2×D13—81no—
353C2×D13—81no—
354C2×D13—81no—
355C4×C13—24no—
356C4×C13—22no—
357C4×C13—22no—
358C4×C13—41no—
359C4×C13—42no—
360C4×C13—41no—
361C4×C13—41no—
362C4×C13—41no—
363C4×C13—81no—
364C4×C13—81no—
365C4×C13C2×C22⋊Q811yescharacteristic
366C4×C13#53911yescharacteristic
367C4×C13#54711yescharacteristic
368C4×C13#55411yescharacteristic
369Dic13—42no—
370Dic13—81no—
371Dic13—81no—
372C22×Q8—261no—
373C22×Q8C22×D1311yescharacteristic
374C22×D4—131no—
375C23×C4—261no—
376C2×SD16—134no—
377C2×SD16—262no—
378C2×SD16—262no—
379C22×C8—131no—
380C22×C8—131no—
381C22×C8—131no—
382C22×C8—131no—
383C4⋊Q8—262no—
384C4⋊Q8—521no—
385C22⋊Q8—524no—
386C4×Q8—22no—
387C4×Q8—22no—
388C4×Q8—22no—
389C4×Q8—22no—
390C4×Q8—41no—
391C4×Q8—41no—
392C4×D4—262no—
393C4×D4—521no—
394C2×C4⋊C4—21no—
395C2×C4⋊C4—21no—
396C2×C4⋊C4—21no—
397C2×C4⋊C4—21no—
398C2×C4⋊C4—21no—
399C2×C4⋊C4—131no—
400C2×C4⋊C4—131no—
401C2×C4⋊C4—131no—
402C2×C4⋊C4—261no—
403C2×C4⋊C4—261no—
404C2×C4⋊C4—261no—
405C2×C4⋊C4C22⋊D1311yescharacteristic
406C2×C4⋊C4C22⋊D1311yescharacteristic
407C2×C4⋊C4C22×D1311yescharacteristic
408C2×C4⋊C4C22×D1311yescharacteristic
409C2×C4⋊C4C22×D1311yescharacteristic
410C2×C22⋊C4—261no—
411C2×C22⋊C4—261no—
412C2×C22⋊C4—261no—
413C2×C22⋊C4—261no—
414C2×C22⋊C4—261no—
415C2×C42—21no—
416C2×C42—21no—
417C2×C42—21no—
418C2×C42C22×D1311yescharacteristic
419C2×C42C22×D1311yescharacteristic
420C2×C42C22×D1311yescharacteristic
421#107—262no—
422#107—262no—
423#107—262no—
424#106—262no—
425#106—521no—
426C4⋊C8—262no—
427#103—134no—
428#103—262no—
429#103—262no—
430#102—262no—
431#102—262no—
432#102—521no—
433C4×C8—262no—
434#95—21no—
435#95—21no—
436#95—21no—
437#95—21no—
438#95—261no—
439#95—261no—
440#95C22⋊D1311yescharacteristic
441#95C22⋊D1311yescharacteristic
442C2×C13#244911yescharacteristic
443C2×C13#310611yescharacteristic
444C2×C13#312111yescharacteristic
445C2×C13#326511yescharacteristic
446C2×C13#328511yescharacteristic
447C2×C13#338111yescharacteristic
448C2×C13#338611yescharacteristic
449D13—82no—
450C24—261no—
451C2×Q8—21no—
452C2×Q8—21no—
453C2×Q8—21no—
454C2×Q8—522no—
455C2×Q8—522no—
456C2×Q8—1041no—
457C2×Q8#612712yes—
458C2×Q8D4×D1312yes—
459C2×Q8Q8×D1312yes—
460C2×D4—132no—
461C2×D4—261no—
462C2×D4—261no—
463C2×D4—261no—
464C2×D4—261no—
465C2×D4—261no—
466C22×C4—21no—
467C22×C4—21no—
468C22×C4—21no—
469C22×C4—21no—
470C22×C4—261no—
471C22×C4—261no—
472C22×C4—261no—
473C22×C4—522no—
474C22×C4—522no—
475C22×C4—1041no—
476C22×C4C2×C22⋊D1311yescharacteristic
477C22×C4C2×C22⋊D1311yescharacteristic
478C22×C4C2×C22⋊D1311yescharacteristic
479C22×C4C2×C22⋊D1311yescharacteristic
480C22×C4C2×C22⋊D1311yescharacteristic
481C22×C4C2×C22⋊D1311yescharacteristic
482C22×C4C23×D1311yescharacteristic
483SD16—268no—
484C2×C8—132no—
485C2×C8—132no—
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—
495C2×C8—261no—
496C4⋊C4—24no—
497C4⋊C4—22no—
498C4⋊C4—22no—
499C4⋊C4—21no—
500C4⋊C4—21no—
501C4⋊C4—21no—
502C4⋊C4—21no—
503C4⋊C4—21no—
504C4⋊C4—42no—
505C4⋊C4—42no—
506C4⋊C4—42no—
507C4⋊C4—42no—
508C4⋊C4—132no—
509C4⋊C4—261no—
510C4⋊C4—261no—
511C4⋊C4—261no—
512C4⋊C4—261no—
513C4⋊C4—261no—
514C4⋊C4—522no—
515C4⋊C4—522no—
516C4⋊C4—522no—
517C4⋊C4#612712yes—
518C22⋊C4—522no—
519C22⋊C4—522no—
520C22⋊C4—522no—
521C22⋊C4—522no—
522C22⋊C4—1041no—
523C42—24no—
524C42—21no—
525C42—21no—
526C42—21no—
527C42—21no—
528C42—21no—
529C42—21no—
530C42—42no—
531C42—42no—
532C13#2637711yescharacteristic, complemented, Sylow
533C23—262no—
534C23—521no—
535C23—521no—
536C23—521no—
537C23—521no—
538C23—521no—
539C23—521no—
540C23C22×C22⋊D1311yescharacteristic
541Q8—22no—
542Q8—22no—
543Q8—22no—
544Q8—1044no—
545Q8#8815014yescomplemented
546D4—262no—
547D4—262no—
548D4—521no—
549D4—521no—
550C2×C4—22no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—22no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—262no—
579C2×C4—521no—
580C2×C4—521no—
581C2×C4—521no—
582C2×C4—521no—
583C2×C4—521no—
584C2×C4—521no—
585C2×C4—521no—
586C2×C4—521no—
587C2×C4—1044no—
588C2×C4#8820312yes—
589C2×C4#8820412yes—
590C2×C4C2×#612711yescharacteristic
591C2×C4C2×#612711yescharacteristic
592C2×C4C2×D4×D1311yescharacteristic
593C2×C4#8825111yescharacteristic
594C2×C4C2×Q8×D1311yescharacteristic
595C2×C4#8825411yescharacteristic
596C8—262no—
597C8—262no—
598C8—521no—
599C8—521no—
600C22—522no—
601C22—522no—
602C22—522no—
603C22—1041no—
604C22—1041no—
605C22—1041no—
606C22—1041no—
607C22#1171949611yescharacteristic
608C22D4×C22⋊D1311yescharacteristic
609C22Q8×C22⋊D1311yescharacteristic
610C22C2×#8820311yescharacteristic
611C22C2×#8820411yescharacteristic
612C22#1171954211yescharacteristic
613C22#1171954311yescharacteristic
614C4—22no—
615C4—24no—
616C4—22no—
617C4—41no—
618C4—42no—
619C4—41no—
620C4—41no—
621C4—41no—
622C4—81no—
623C4—81no—
624C4—522no—
625C4—1041no—
626C4—1041no—
627C4#1171921211yescharacteristic
628C4C2×#8815011yescharacteristic
629C4#1171921611yescharacteristic
630C4#1171922011yescharacteristic
631C2—1042no—
632C2#1311864511yescharacteristic
633C2#1312489611yescharacteristic
634C2#1312506511yescharacteristic
635C2#1312633211yescharacteristic
636C2#1312657311yescharacteristic
637C2#1312770711yescharacteristic
638C2#1312775311yescharacteristic
subgroupG/H
centerC23C22×C22⋊D13
commutator, cosocleC22×C4×C13C24
FrattiniC22×C4C23×D13
FittingC13×#2351C2
socleC23×C13C22×D4
prime radicalC22×D52C23
hypercenter#2351D13
hypocenterC13#26377
LCSG ▷ C22×C4×C13 ▷ C2×C13 ▷ C13
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #2351

Complements

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

NH ≅ G/Ntype
1. C2×#11718682C2semidirect
5. #13116318C2semidirect
7. #13116169C2semidirect
8. #13115566C2semidirect
9. #13115395C2semidirect
12. #13115104C2semidirect
13. #13115090C2semidirect
15. C13×#2351C2semidirect
89. C2×#6106Q8semidirect
217. #2351D13semidirect
240. Q8×C13C22⋊Q8semidirect
532. C13#26377semidirect
545. Q8#88150semidirect

Maximal subgroups

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

subgroupindex
C2×#117186822
C2×C4×#61062
#131186862
#131163602
#131163182
#131162212
#131161692
#131155662
#131153952
#131151912
#131151042
#131150902
#131150822
C13×#23512
#2637713