Finite Group Databasebeta

← #15921772

Subgroups — #15921772

Order 3328.

818 conjugacy classes of subgroups.

The subgroup lattice has 620 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11718542C211yesmaximal, characteristic
2#13118545C211yesmaximal, characteristic
3#13118534C211yesmaximal, characteristic
4#13116315C211yesmaximal, characteristic, complemented
5#13116310C211yesmaximal, characteristic, complemented
6#13116171C211yesmaximal, characteristic, complemented
7#13116166C211yesmaximal, characteristic, complemented
8#13115574C211yesmaximal, characteristic, complemented
9#13115570C211yesmaximal, characteristic, complemented
10#13115032C211yesmaximal, characteristic
11#13115027C211yesmaximal, characteristic
12#13114991C211yesmaximal, characteristic, complemented
13#13114866C211yesmaximal, characteristic
14#13114621C211yesmaximal, characteristic
15C2×C13×#501C211yesmaximal, characteristic, complemented
16C2×C13×C4⋊Q8C2211yescharacteristic
17C2×C13×#125C2211yescharacteristic
18C2×#88200—21no—
19C22×#6094C2211yescharacteristic
20C2×#88146C2211yescharacteristic
21C2×C4⋊C4×D13C2211yescharacteristic
22C13×#501—24no—
23C2×C13×#107C2211yescharacteristic
24C2×C13×#107C2211yescharacteristic
25C2×C13×#106C2211yescharacteristic
26C2×C13×#106C2211yescharacteristic
27C2×C13×#97C2211yescharacteristic
28#11718890—21no—
29C2×#88060C2211yescharacteristic
30C2×#88058C2211yescharacteristic
31C2×#88057C2211yescharacteristic
32C2×C8×Dic13C2211yescharacteristic
33#11718655—21no—
34#11718653—21no—
35#11718650—21no—
36#11718648C2211yescharacteristic
37#11718648C2211yescharacteristic
38#11718646C2211yescharacteristic
39#11718646C2211yescharacteristic
40#11718645—21no—
41#11718639—21no—
42#11718637C2211yescharacteristic
43#11718637C2211yescharacteristic
44#11718634—21no—
45C2×#88048C2211yescharacteristic
46C2×#88048C2211yescharacteristic
47C2×#88048C2211yescharacteristic
48C2×#88047C2211yescharacteristic
49C4×#6103C2211yescharacteristic
50#11718589C2211yescharacteristic
51#11718589C2211yescharacteristic
52#11718542—24no—
53#11718143C2211yescharacteristic
54#11718143C2211yescharacteristic
55#11718142C2211yescharacteristic
56#11718121C2211yescharacteristic
57#11718121C2211yescharacteristic
58#11718121C2211yescharacteristic
59#11718121C2211yescharacteristic
60#11718116C2211yescharacteristic
61C22×Q8×C13—21no—
62C22×C4×D13—21no—
63C22×C4×D13C2311yescharacteristic
64C22×C8×C13C2311yescharacteristic
65C22×C8×C13C2311yescharacteristic
66C13×C4⋊Q8—22no—
67C13×C4⋊Q8—41no—
68C13×#125—22no—
69C13×#125—41no—
70C2×C13×C4⋊C4—21no—
71C2×C13×C4⋊C4—21no—
72C2×C13×C4⋊C4—21no—
73C2×C13×C4⋊C4C2311yescharacteristic
74C2×C13×C4⋊C4C2311yescharacteristic
75C2×C13×C4⋊C4C2311yescharacteristic
76C2×C13×C4⋊C4C2311yescharacteristic
77C2×C42×C13C2311yescharacteristic
78#88200—44no—
79C2×#6103—21no—
80C2×#6103—21no—
81C2×#6103—21no—
82C2×#6103D411yescharacteristic
83C2×#6103D411yescharacteristic
84C2×C4⋊Dic13—21no—
85C2×C4⋊Dic13—21no—
86C2×C4⋊Dic13C2311yescharacteristic
87C2×C4⋊Dic13C2311yescharacteristic
88C2×C4⋊Dic13C2311yescharacteristic
89C2×C4⋊Dic13C2311yescharacteristic
90C2×#6101—21no—
91C2×#6101—21no—
92C2×#6101—21no—
93C2×#6101—21no—
94C2×#6101—21no—
95C2×#6101—21no—
96C2×#6101—21no—
97C2×C4×Dic13—21no—
98C2×C4×Dic13C2311yescharacteristic
99C22×#1055C2311yescharacteristic
100C22×#1055C2311yescharacteristic
101C2×#6094—22no—
102C2×#6094—22no—
103C2×#6094Q814yescomplemented
104#88146—22no—
105#88146—41no—
106C4⋊C4×D13—22no—
107C4⋊C4×D13—41no—
108C13×#107—22no—
109C13×#107D414yes—
110C13×#106—22no—
111C13×#106—22no—
112C13×#97—22no—
113#88071—21no—
114#88071—21no—
115#88071—21no—
116#88071—21no—
117#88071—21no—
118#88071—21no—
119#88071D411yescharacteristic
120#88071D411yescharacteristic
121#88060Q814yescomplemented
122#88058—22no—
123#88057—22no—
124C8×Dic13—22no—
125#88048—22no—
126#88048—41no—
127#88048—41no—
128#88047—22no—
129#26078—131nomaximal, Sylow
130C23×D13C2×D411yescharacteristic
131C2×Q8×C13—42no—
132C2×Q8×C13—42no—
133C2×Q8×C13—81no—
134C22×C4×C13—21no—
135C22×C4×C13—21no—
136C22×C4×C13—21no—
137C22×C4×C13—21no—
138C22×C4×C13C2×D411yescharacteristic
139C22×C4×C13C2×D411yescharacteristic
140C22×C4×C13C2411yescharacteristic
141C22×Dic13—21no—
142C22×Dic13—21no—
143C22×Dic13—21no—
144C22×Dic13—21no—
145C22×Dic13C2×D411yescharacteristic
146C22×Dic13C2×D411yescharacteristic
147C22×Dic13C2×D411yescharacteristic
148C2×C4×D13—21no—
149C2×C4×D13—21no—
150C2×C4×D13—21no—
151C2×C4×D13—21no—
152C2×C4×D13—42no—
153C2×C4×D13—42no—
154C2×C4×D13—81no—
155C2×C4×D13C2×Q812yes—
156C2×C4×D13C2×Q812yes—
157C2×C8×C13—21no—
158C2×C8×C13—21no—
159C2×C8×C13C2×D412yes—
160C2×C8×C13C2×Q812yes—
161C2×C8×C13C2×Q812yes—
162C2×C8×C13#4112yes—
163C13×C4⋊C4—21no—
164C13×C4⋊C4—21no—
165C13×C4⋊C4—21no—
166C13×C4⋊C4—21no—
167C13×C4⋊C4—21no—
168C13×C4⋊C4—21no—
169C13×C4⋊C4—42no—
170C13×C4⋊C4—42no—
171C13×C4⋊C4—42no—
172C13×C4⋊C4C2×D412yes—
173C13×C4⋊C4C2×D412yes—
174C42×C13—21no—
175C42×C13—21no—
176#6103—22no—
177#6103—22no—
178#6103—42no—
179#6103—42no—
180#6103—81no—
181C4⋊Dic13—21no—
182C4⋊Dic13—21no—
183C4⋊Dic13—21no—
184C4⋊Dic13—21no—
185C4⋊Dic13—21no—
186C4⋊Dic13—21no—
187C4⋊Dic13—21no—
188C4⋊Dic13—21no—
189C4⋊Dic13—42no—
190C4⋊Dic13—42no—
191#6101—42no—
192#6101—42no—
193#6101—42no—
194#6101—42no—
195#6101—81no—
196#6101—81no—
197#6101—81no—
198C4×Dic13—21no—
199C4×Dic13—21no—
200C4×Dic13—42no—
201C2×#1055—21no—
202C2×#1055—21no—
203C2×#1055—21no—
204C2×#1055—21no—
205C2×#1055C2×Q812yes—
206C2×#1055C2×Q812yes—
207#6094—28no—
208C2×#501D1311yescharacteristic, complemented
209C2×#428—131no—
210#2441—131no—
211#2440—131no—
212#2084—131no—
213#2083—131no—
214#2059—131no—
215#2058—131no—
216#1943—131no—
217#1942—131no—
218#1854—131no—
219#1853—131no—
220#1844—131no—
221#1809—131no—
222#1753—131no—
223C23×C13C22×D411yescharacteristic
224C22×D13—21no—
225C22×D13—21no—
226C22×D13—21no—
227C22×D13—21no—
228C22×D13—21no—
229C22×D13—21no—
230C22×D13C4⋊Q812yes—
231Q8×C13—84no—
232C2×C4×C13—22no—
233C2×C4×C13—22no—
234C2×C4×C13—21no—
235C2×C4×C13—21no—
236C2×C4×C13—21no—
237C2×C4×C13—21no—
238C2×C4×C13—21no—
239C2×C4×C13—21no—
240C2×C4×C13—41no—
241C2×C4×C13—41no—
242C2×C4×C13—41no—
243C2×C4×C13—41no—
244C2×C4×C13—41no—
245C2×C4×C13—41no—
246C2×C4×C13—41no—
247C2×C4×C13—41no—
248C2×C4×C13—41no—
249C2×C4×C13—41no—
250C2×C4×C13C22×D411yescharacteristic
251C2×C4×C13C22×Q811yescharacteristic
252C2×C4×C13C22×Q811yescharacteristic
253C2×C4×C13C2×#4111yescharacteristic
254C2×C4×C13H(5,𝔽2)11yescharacteristic
255C2×C4×C132−1+411yescharacteristic
256C2×Dic13—21no—
257C2×Dic13—21no—
258C2×Dic13—21no—
259C2×Dic13—21no—
260C2×Dic13—21no—
261C2×Dic13—21no—
262C2×Dic13—41no—
263C2×Dic13—41no—
264C2×Dic13—41no—
265C2×Dic13—41no—
266C2×Dic13—41no—
267C2×Dic13—41no—
268C2×Dic13—41no—
269C2×Dic13—41no—
270C2×Dic13—41no—
271C2×Dic13—41no—
272C2×Dic13—41no—
273C2×Dic13—41no—
274C2×Dic13C4⋊Q812yes—
275C2×Dic13AGL(1,ℤ/8)12yes—
276C2×Dic13#13712yes—
277C4×D13—22no—
278C4×D13—22no—
279C4×D13—22no—
280C4×D13—22no—
281C4×D13—84no—
282C8×C13—22no—
283C8×C13C22⋊Q814yescomplemented
284#1055—22no—
285#1055—22no—
286C22×M16—131no—
287C2×C4⋊Q8C2×D1311yescharacteristic
288C2×#125C2×D1311yescharacteristic
289C2×C22⋊Q8—261no—
290C2×#117—131no—
291C22×C4⋊C4—131no—
292#501—24no—
293#428—264no—
294C2×#107—131no—
295C2×#107—131no—
296C2×#107C2×D1311yescharacteristic
297C2×#107C2×D1311yescharacteristic
298C2×#106—131no—
299C2×#106—131no—
300C2×#106—131no—
301C2×#106—131no—
302C2×#106C2×D1311yescharacteristic
303C2×#106C2×D1311yescharacteristic
304C2×C22⋊C8—131no—
305C2×#97C2×D1311yescharacteristic
306C2×C4×C8—131no—
307#399—261no—
308#398—261no—
309#396—261no—
310#395—261no—
311#389—131no—
312#389—131no—
313#389—131no—
314#389—131no—
315#389—261no—
316#387—261no—
317#385—261no—
318#384—131no—
319#384—131no—
320#380—131no—
321#380—131no—
322C4×C22⋊C4—131no—
323#340—131no—
324#340—131no—
325#340—131no—
326#340—131no—
327#340—131no—
328#340—131no—
329#336—131no—
330#336—131no—
331C22×C13C2×C4⋊Q811yescharacteristic
332C22×C13D4211yescharacteristic
333C22×C13#54711yescharacteristic
334C22×C13C2×AGL(1,ℤ/8)11yescharacteristic
335C22×C13C2×#13711yescharacteristic
336C22×C13#57711yescharacteristic
337C22×C13#57711yescharacteristic
338C2×D13—22no—
339C2×D13—22no—
340C2×D13—22no—
341C2×D13—41no—
342C2×D13—41no—
343C2×D13—41no—
344C2×D13—41no—
345C4×C13—42no—
346C4×C13—41no—
347C4×C13—42no—
348C4×C13—41no—
349C4×C13—81no—
350C4×C13—81no—
351C4×C13C2×C22⋊Q811yescharacteristic
352C4×C13#54411yescharacteristic
353C4×C13D4×Q811yescharacteristic
354C4×C13#55411yescharacteristic
355Dic13—22no—
356Dic13—22no—
357Dic13—22no—
358Dic13—81no—
359Dic13—81no—
360Dic13—81no—
361Dic13—81no—
362C22×Q8—21no—
363C23×C4—131no—
364C23×C4—261no—
365C2×M16—134no—
366C2×M16—262no—
367C2×M16—262no—
368C22×C8—131no—
369C22×C8—131no—
370C22×C8C22×D1311yescharacteristic
371C22×C8C22×D1311yescharacteristic
372C4⋊Q8—22no—
373C4⋊Q8—41no—
374#125—22no—
375#125—41no—
376C22⋊Q8—524no—
377#117—262no—
378#117—521no—
379C2×C4⋊C4—21no—
380C2×C4⋊C4—21no—
381C2×C4⋊C4—21no—
382C2×C4⋊C4—131no—
383C2×C4⋊C4—131no—
384C2×C4⋊C4—131no—
385C2×C4⋊C4—131no—
386C2×C4⋊C4—262no—
387C2×C4⋊C4—261no—
388C2×C4⋊C4—261no—
389C2×C4⋊C4—261no—
390C2×C4⋊C4—261no—
391C2×C4⋊C4—261no—
392C2×C4⋊C4—261no—
393C2×C4⋊C4—261no—
394C2×C4⋊C4—261no—
395C2×C4⋊C4—261no—
396C2×C4⋊C4—521no—
397C2×C4⋊C4C22×D1311yescharacteristic
398C2×C4⋊C4C22×D1311yescharacteristic
399C2×C4⋊C4C22×D1311yescharacteristic
400C2×C4⋊C4C22×D1311yescharacteristic
401C2×C22⋊C4—131no—
402C2×C22⋊C4—131no—
403C2×C22⋊C4—261no—
404C2×C22⋊C4—261no—
405C2×C22⋊C4—261no—
406C2×C42—131no—
407C2×C42—261no—
408C2×C42C22×D1311yescharacteristic
409#107—22no—
410#107—262no—
411#107—262no—
412#107C22⋊D1314yes—
413#106—22no—
414#106—22no—
415#106—262no—
416#106—262no—
417#106—521no—
418#106—521no—
419C22⋊C8—134no—
420#97—22no—
421C4×C8—262no—
422#95—131no—
423#95—131no—
424#95—261no—
425#95—261no—
426#95—261no—
427#95—261no—
428#95—261no—
429#95—261no—
430C2×C13#244011yescharacteristic
431C2×C13#316311yescharacteristic
432C2×C13#316511yescharacteristic
433C2×C13#327211yescharacteristic
434C2×C13#327811yescharacteristic
435C2×C13#334411yescharacteristic
436C2×C13#334511yescharacteristic
437D13—42no—
438C24—131no—
439C2×Q8—42no—
440C2×Q8—42no—
441C2×Q8—81no—
442C22×C4—21no—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—131no—
447C22×C4—131no—
448C22×C4—131no—
449C22×C4—132no—
450C22×C4—132no—
451C22×C4—261no—
452C22×C4—261no—
453C22×C4—261no—
454C22×C4—261no—
455C22×C4—261no—
456C22×C4—261no—
457C22×C4—261no—
458C22×C4—261no—
459C22×C4—522no—
460C22×C4—522no—
461C22×C4—1041no—
462C22×C4D4×D1311yescharacteristic
463C22×C4D4×D1311yescharacteristic
464C22×C4C23×D1311yescharacteristic
465M16—268no—
466C2×C8—21no—
467C2×C8—21no—
468C2×C8—132no—
469C2×C8—132no—
470C2×C8—261no—
471C2×C8—261no—
472C2×C8—261no—
473C2×C8—261no—
474C2×C8Q8×D1312yes—
475C2×C8Q8×D1312yes—
476C2×C8#613112yes—
477C2×C8C2×C22⋊D1312yes—
478C4⋊C4—21no—
479C4⋊C4—21no—
480C4⋊C4—21no—
481C4⋊C4—21no—
482C4⋊C4—21no—
483C4⋊C4—21no—
484C4⋊C4—42no—
485C4⋊C4—42no—
486C4⋊C4—42no—
487C4⋊C4—261no—
488C4⋊C4—261no—
489C4⋊C4—261no—
490C4⋊C4—261no—
491C4⋊C4—261no—
492C4⋊C4—261no—
493C4⋊C4—261no—
494C4⋊C4—261no—
495C4⋊C4—522no—
496C4⋊C4—522no—
497C4⋊C4—522no—
498C4⋊C4—522no—
499C4⋊C4—522no—
500C4⋊C4—522no—
501C4⋊C4—1041no—
502C4⋊C4—1041no—
503C4⋊C4—1041no—
504C4⋊C4C2×C22⋊D1312yes—
505C4⋊C4C2×C22⋊D1312yes—
506C22⋊C4—262no—
507C22⋊C4—262no—
508C22⋊C4—522no—
509C22⋊C4—522no—
510C22⋊C4—1041no—
511C42—21no—
512C42—21no—
513C42—261no—
514C42—261no—
515C42—522no—
516C13#2607811yescharacteristic, complemented, Sylow
517C23—132no—
518C23—261no—
519C23—261no—
520C23—261no—
521C23—261no—
522C23—261no—
523C23—261no—
524C23C2×D4×D1311yescharacteristic
525Q8—84no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—132no—
545C2×C4—132no—
546C2×C4—132no—
547C2×C4—261no—
548C2×C4—261no—
549C2×C4—261no—
550C2×C4—261no—
551C2×C4—261no—
552C2×C4—261no—
553C2×C4—262no—
554C2×C4—262no—
555C2×C4—262no—
556C2×C4—262no—
557C2×C4—521no—
558C2×C4—521no—
559C2×C4—521no—
560C2×C4—521no—
561C2×C4—521no—
562C2×C4—521no—
563C2×C4—521no—
564C2×C4—521no—
565C2×C4—521no—
566C2×C4—521no—
567C2×C4—521no—
568C2×C4—521no—
569C2×C4—1044no—
570C2×C4#8825111yescharacteristic
571C2×C4C2×Q8×D1311yescharacteristic
572C2×C4C2×Q8×D1311yescharacteristic
573C2×C4C2×#613111yescharacteristic
574C2×C4#8825411yescharacteristic
575C2×C4C22×C22⋊D1311yescharacteristic
576C8—22no—
577C8—262no—
578C8—262no—
579C8#8820014yescomplemented
580C22—262no—
581C22—262no—
582C22—262no—
583C22—521no—
584C22—521no—
585C22—521no—
586C22—521no—
587C22D13×C4⋊Q811yescharacteristic
588C22#1171943111yescharacteristic
589C22D13×AGL(1,ℤ/8)11yescharacteristic
590C22#1171948011yescharacteristic
591C22D13×#13711yescharacteristic
592C22#1171948311yescharacteristic
593C22D4×C22⋊D1311yescharacteristic
594C4—41no—
595C4—42no—
596C4—42no—
597C4—41no—
598C4—81no—
599C4—81no—
600C4—262no—
601C4—262no—
602C4—262no—
603C4—1041no—
604C4—1041no—
605C4—1041no—
606C4—1041no—
607C4#1171943411yescharacteristic
608C4#1171943511yescharacteristic
609C4C2×#8820011yescharacteristic
610C4Q8×C22⋊D1311yescharacteristic
611C2—522no—
612C2#1311853411yescharacteristic
613C2#1312536311yescharacteristic
614C2#1312538111yescharacteristic
615C2#1312643011yescharacteristic
616C2#1312649611yescharacteristic
617C2#1312729011yescharacteristic
618C2#1312730511yescharacteristic
subgroupG/H
centerC23C2×D4×D13
commutator, cosocleC22×C4×C13C24
FrattiniC22×C4C23×D13
FittingC2×C13×#501C2
socleC23×C13C22×D4
prime radicalC23×D13C2×D4
hypercenterC2×#501D13
hypocenterC13#26078
LCSG ▷ C22×C4×C13 ▷ C2×C13 ▷ C13
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#501

Complements

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

NH ≅ G/Ntype
4. #13116315C2semidirect
5. #13116310C2semidirect
6. #13116171C2semidirect
7. #13116166C2semidirect
8. #13115574C2semidirect
9. #13115570C2semidirect
12. #13114991C2semidirect
15. C2×C13×#501C2semidirect
103. C2×#6094Q8semidirect
121. #88060Q8semidirect
208. C2×#501D13semidirect
283. C8×C13C22⋊Q8semidirect
516. C13#26078semidirect
579. C8#88200semidirect

Maximal subgroups

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

subgroupindex
C2×#117185422
#131185452
#131185342
#131163152
#131163102
#131161712
#131161662
#131155742
#131155702
#131150322
#131150272
#131149912
#131148662
#131146212
C2×C13×#5012
#2607813