Finite Group Databasebeta

← #15872476

Subgroups — #15872476

Order 3328.

970 conjugacy classes of subgroups.

The subgroup lattice has 628 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11719224C211yesmaximal, characteristic
2C2×#11718381C211yesmaximal, characteristic
3#13117443C211yesmaximal, characteristic
4#13115807C211yesmaximal, characteristic, complemented
5#13115803C211yesmaximal, characteristic, complemented
6#13115799C211yesmaximal, characteristic, complemented
7#13115795C211yesmaximal, characteristic, complemented
8#13115792C211yesmaximal, characteristic, complemented
9#13115790C211yesmaximal, characteristic, complemented
10#13115786C211yesmaximal, characteristic, complemented
11#13115782C211yesmaximal, characteristic, complemented
12#13114931C211yesmaximal, characteristic
13#13114924C211yesmaximal, characteristic
14#13114923C211yesmaximal, characteristic
15C13×#1825C211yesmaximal, characteristic
16C2×C13×#117C2211yescharacteristic
17C2×#88180C2211yescharacteristic
18C22×C4⋊Dic13C2211yescharacteristic
19#11719224—24no—
20#11719224C418yescomplemented
21C2×#88144C2211yescharacteristic
22C2×#88144C2211yescharacteristic
23C2×#88142C2211yescharacteristic
24C2×#88142C2211yescharacteristic
25C2×#88132—21no—
26C2×#88132—21no—
27C2×#88124—21no—
28C2×#88123—21no—
29C2×C13×C22⋊C8C2211yescharacteristic
30C13×#380C2211yescharacteristic
31#11718873C2211yescharacteristic
32C2×C22⋊#1055C2211yescharacteristic
33C2×#88055C2211yescharacteristic
34C2×#88054C2211yescharacteristic
35C2×#88054C2211yescharacteristic
36C2×C8×Dic13C2211yescharacteristic
37#11718638C2211yescharacteristic
38#11718637C2211yescharacteristic
39C4⋊C4×Dic13C2211yescharacteristic
40C4⋊C4×Dic13C2211yescharacteristic
41#11718606—21no—
42#11718605—21no—
43#11718381—24no—
44#11718315—21no—
45#11718314—21no—
46C13×#340C2211yescharacteristic
47C13×#340C2211yescharacteristic
48C13×#340C2211yescharacteristic
49C13×#340C2211yescharacteristic
50#11718143C2211yescharacteristic
51#11718143C2211yescharacteristic
52#11718143C2211yescharacteristic
53#11718143C2211yescharacteristic
54#11718121C2211yescharacteristic
55#11718121C2211yescharacteristic
56#11718121C2211yescharacteristic
57#11718121C2211yescharacteristic
58#11718121C2211yescharacteristic
59#11718121C2211yescharacteristic
60#11718121C2211yescharacteristic
61#11718121C2211yescharacteristic
62C23×C4×C13C2311yescharacteristic
63C23×Dic13—21no—
64C22×Dic26—21no—
65C22×Dic26—21no—
66C22×C8×C13C2311yescharacteristic
67C22×C8×C13C2311yescharacteristic
68C13×#117—22no—
69C13×#117C2×C414yes—
70C2×C13×C4⋊C4C2311yescharacteristic
71C2×C13×C4⋊C4C2311yescharacteristic
72C2×C13×C4⋊C4C2311yescharacteristic
73C2×C13×C4⋊C4C2311yescharacteristic
74C2×C13×C22⋊C4—21no—
75C2×C13×C22⋊C4—21no—
76C2×C42×C13—21no—
77C2×C22⋊Dic13—21no—
78C2×C22⋊Dic13D411yescharacteristic
79C2×C22⋊Dic13D411yescharacteristic
80#88180C2×C414yes—
81#88180D414yes—
82C2×C4⋊Dic13—21no—
83C2×C4⋊Dic13—22no—
84C2×C4⋊Dic13—21no—
85C2×C4⋊Dic13—21no—
86C2×C4⋊Dic13C2×C414yes—
87C2×C4⋊Dic13D411yescharacteristic
88C2×C4⋊Dic13D411yescharacteristic
89C2×C4⋊Dic13C2311yescharacteristic
90C2×C4⋊Dic13C2311yescharacteristic
91C2×C4⋊Dic13C2311yescharacteristic
92C2×C4⋊Dic13C2311yescharacteristic
93C2×#6101—21no—
94C2×#6101—21no—
95C2×#6101—21no—
96C2×#6101—21no—
97C2×#6101—21no—
98C2×#6101—21no—
99C2×C4×Dic13—21no—
100C2×C4×Dic13—21no—
101C2×C4×Dic13C2311yescharacteristic
102C2×C4×Dic13C2311yescharacteristic
103C22×#1055C2311yescharacteristic
104C22×#1055C2311yescharacteristic
105#88144—22no—
106#88144—22no—
107#88144C2×C414yes—
108#88144C2×C414yes—
109#88142—22no—
110#88142—22no—
111#88142C2×C414yes—
112#88142C2×C414yes—
113#88132—28no—
114#88132—28no—
115#88124—28no—
116#88123—28no—
117C13×C22⋊C8—22no—
118C13×#95—21no—
119#88071—21no—
120#88071—21no—
121#88071—21no—
122#88071—21no—
123#88071—21no—
124C22⋊#1055—22no—
125#88055—22no—
126#88054—22no—
127#88054—22no—
128C8×Dic13—22no—
129#21843—131nomaximal, Sylow
130C24×C13C2×D411yescharacteristic
131C22×C4×C13—21no—
132C22×C4×C13—21no—
133C22×C4×C13—21no—
134C22×C4×C13—21no—
135C22×C4×C13—21no—
136C22×C4×C13—21no—
137C22×C4×C13—21no—
138C22×C4×C13—21no—
139C22×C4×C13C22×C412yes—
140C22×C4×C13C2×D411yescharacteristic
141C22×C4×C13C2×D412yes—
142C22×C4×C13C2411yescharacteristic
143C22×Dic13—21no—
144C22×Dic13—21no—
145C22×Dic13—24no—
146C22×Dic13—21no—
147C22×Dic13—21no—
148C22×Dic13—81no—
149C22×Dic13—81no—
150C22×Dic13C2×D411yescharacteristic
151C22×Dic13C2×D411yescharacteristic
152C22×Dic13C2×D411yescharacteristic
153C22×Dic13C2×D411yescharacteristic
154C2×Dic26—24no—
155C2×Dic26—24no—
156C2×Dic26—81no—
157C2×Dic26—81no—
158C2×Dic26—81no—
159C2×Dic26—81no—
160C2×C8×C13—21no—
161C2×C8×C13—21no—
162C2×C8×C13—21no—
163C2×C8×C13—21no—
164C2×C8×C13—21no—
165C2×C8×C13—21no—
166C13×C4⋊C4—21no—
167C13×C4⋊C4—21no—
168C13×C4⋊C4—21no—
169C13×C4⋊C4—21no—
170C13×C4⋊C4—21no—
171C13×C4⋊C4—21no—
172C13×C4⋊C4C22×C412yes—
173C13×C4⋊C4C22×C412yes—
174C13×C22⋊C4—24no—
175C13×C22⋊C4—81no—
176C42×C13—24no—
177C22⋊Dic13—81no—
178C22⋊Dic13C22⋊C414yes—
179C22⋊Dic13C22⋊C414yes—
180C4⋊Dic13—24no—
181C4⋊Dic13—21no—
182C4⋊Dic13—21no—
183C4⋊Dic13—24no—
184C4⋊Dic13—24no—
185C4⋊Dic13—21no—
186C4⋊Dic13—21no—
187C4⋊Dic13—21no—
188C4⋊Dic13—21no—
189C4⋊Dic13C22⋊C414yes—
190C4⋊Dic13C22⋊C414yes—
191C4⋊Dic13C22×C412yes—
192C4⋊Dic13C22×C412yes—
193#6101—24no—
194#6101—24no—
195#6101—24no—
196#6101—24no—
197#6101—81no—
198#6101—81no—
199C4×Dic13—81no—
200C4×Dic13—81no—
201C4×Dic13C22×C412yes—
202C4×Dic13C22×C412yes—
203C4×Dic13C2×D412yes—
204C4×Dic13C2×D412yes—
205C2×#1055—21no—
206C2×#1055—21no—
207C2×#1055—21no—
208C2×#1055—21no—
209C2×#1055—21no—
210C2×#1055—21no—
211C2×#544—131no—
212C2×#433—131no—
213#2314—131no—
214#1993—131no—
215#1992—131no—
216#1991—131no—
217#1990—131no—
218#1989—131no—
219#1989—131no—
220#1988—131no—
221#1987—131no—
222#1825—131no—
223#1825D1311yescharacteristic
224#1824—131no—
225#1823—131no—
226C23×C13—21no—
227C23×C13—21no—
228C23×C13—21no—
229C23×C13—21no—
230C23×C13—21no—
231C23×C13—21no—
232C23×C13C2×C22⋊C412yes—
233C23×C13C22×D411yescharacteristic
234C2×C4×C13—21no—
235C2×C4×C13—21no—
236C2×C4×C13—22no—
237C2×C4×C13—22no—
238C2×C4×C13—22no—
239C2×C4×C13—22no—
240C2×C4×C13—22no—
241C2×C4×C13—22no—
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×C13—41no—
251C2×C4×C13—41no—
252C2×C4×C13C2×C22⋊C412yes—
253C2×C4×C13C23×C411yescharacteristic
254C2×C4×C13C22×D411yescharacteristic
255C2×C4×C132−1+411yescharacteristic
256C2×C4×C132−1+411yescharacteristic
257C2×C4×C132−1+411yescharacteristic
258C2×C4×C132−1+411yescharacteristic
259C2×Dic13—21no—
260C2×Dic13—21no—
261C2×Dic13—21no—
262C2×Dic13—21no—
263C2×Dic13—21no—
264C2×Dic13—21no—
265C2×Dic13—21no—
266C2×Dic13—21no—
267C2×Dic13—22no—
268C2×Dic13—22no—
269C2×Dic13—41no—
270C2×Dic13—41no—
271C2×Dic13—41no—
272C2×Dic13—41no—
273C2×Dic13—41no—
274C2×Dic13—41no—
275C2×Dic13—41no—
276C2×Dic13—41no—
277C2×Dic13—41no—
278C2×Dic13—41no—
279C2×Dic13—84no—
280C2×Dic13C2×C22⋊C412yes—
281C2×Dic13C2×C22⋊C412yes—
282C2×Dic13C2×C22⋊C412yes—
283C2×Dic13C2×C22⋊C412yes—
284Dic26—84no—
285Dic26—84no—
286C8×C13—41no—
287C8×C13—41no—
288#1055—41no—
289#1055—41no—
290#544—138no—
291#544—264no—
292C2×C4⋊Q8—131no—
293C2×C4⋊Q8—131no—
294C2×C4⋊Q8—261no—
295C2×#125—131no—
296C2×#125—131no—
297C2×#125—261no—
298C2×C22⋊Q8—261no—
299C2×C22⋊Q8—261no—
300C2×#117—131no—
301C2×#117C2×D1311yescharacteristic
302C22×C4⋊C4—131no—
303#433—264no—
304C2×C4⋊C8—131no—
305C2×C4⋊C8—131no—
306C2×C22⋊C8—131no—
307C2×C22⋊C8C2×D1311yescharacteristic
308C2×#97—131no—
309C2×C4×C8—131no—
310#388—261no—
311#388—261no—
312#387—261no—
313#387—261no—
314#384—131no—
315#382—131no—
316#380—131no—
317#380C2×D1311yescharacteristic
318C4×C4⋊C4—131no—
319C4×C4⋊C4—131no—
320#340—131no—
321#340—131no—
322#340—131no—
323#340—131no—
324#340—131no—
325#340—131no—
326#340—131no—
327#340—131no—
328#340—131no—
329#340—131no—
330#340—131no—
331#340—131no—
332#340C2×D1311yescharacteristic
333#340C2×D1311yescharacteristic
334#340C2×D1311yescharacteristic
335#340C2×D1311yescharacteristic
336C22×C13—22no—
337C22×C13—22no—
338C22×C13—22no—
339C22×C13—41no—
340C22×C13—41no—
341C22×C13—41no—
342C22×C13—41no—
343C22×C13C22×C22⋊C411yescharacteristic
344C22×C13#53611yescharacteristic
345C22×C13#53611yescharacteristic
346C22×C13#57611yescharacteristic
347C22×C13#57711yescharacteristic
348C22×C13#57711yescharacteristic
349C22×C13#57811yescharacteristic
350C4×C13—22no—
351C4×C13—42no—
352C4×C13—42no—
353C4×C13—81no—
354C4×C13—81no—
355C4×C13#51911yescharacteristic
356C4×C13#51911yescharacteristic
357C4×C13#53611yescharacteristic
358C4×C13#53611yescharacteristic
359Dic13—22no—
360Dic13—22no—
361Dic13—22no—
362Dic13—22no—
363Dic13—42no—
364Dic13—42no—
365Dic13—81no—
366Dic13—81no—
367C22×Q8—261no—
368C22×Q8—261no—
369C23×C4—261no—
370C23×C4C22×D1311yescharacteristic
371C22×C8—131no—
372C22×C8—131no—
373C22×C8C22×D1311yescharacteristic
374C22×C8C22×D1311yescharacteristic
375C4⋊Q8—134no—
376C4⋊Q8—134no—
377C4⋊Q8—268no—
378C4⋊Q8—262no—
379C4⋊Q8—262no—
380#125—134no—
381#125—134no—
382#125—268no—
383#125—262no—
384#125—262no—
385C22⋊Q8—268no—
386C22⋊Q8—268no—
387#117—22no—
388#117—134no—
389#117—134no—
390#117C4×D1314yes—
391C2×C4⋊C4—131no—
392C2×C4⋊C4—131no—
393C2×C4⋊C4—134no—
394C2×C4⋊C4—131no—
395C2×C4⋊C4—131no—
396C2×C4⋊C4—131no—
397C2×C4⋊C4—131no—
398C2×C4⋊C4—261no—
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—262no—
406C2×C4⋊C4—261no—
407C2×C4⋊C4—261no—
408C2×C4⋊C4C22×D1311yescharacteristic
409C2×C4⋊C4C22×D1311yescharacteristic
410C2×C4⋊C4C22×D1311yescharacteristic
411C2×C4⋊C4C22×D1311yescharacteristic
412C2×C22⋊C4—21no—
413C2×C22⋊C4—21no—
414C2×C22⋊C4—131no—
415C2×C22⋊C4—131no—
416C2×C22⋊C4—261no—
417C2×C42—21no—
418C2×C42—131no—
419C2×C42—131no—
420C2×C42—261no—
421C2×C42—261no—
422C4⋊C8—262no—
423C4⋊C8—262no—
424C22⋊C8—22no—
425C22⋊C8—262no—
426#97—262no—
427C4×C8—262no—
428#95—21no—
429#95—261no—
430#95—261no—
431#95—261no—
432#95—261no—
433#95—261no—
434C2×C13—42no—
435C2×C13#231011yescharacteristic
436C2×C13#289711yescharacteristic
437C2×C13#289811yescharacteristic
438C2×C13#324111yescharacteristic
439C2×C13#324211yescharacteristic
440C2×C13#324311yescharacteristic
441C2×C13#324411yescharacteristic
442C24D4×D1311yescharacteristic
443C2×Q8—264no—
444C2×Q8—264no—
445C2×Q8—1041no—
446C2×Q8—1041no—
447C2×Q8—1041no—
448C2×Q8—1041no—
449C22×C4—21no—
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—21no—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—21no—
456C22×C4—21no—
457C22×C4—131no—
458C22×C4—131no—
459C22×C4—131no—
460C22×C4—131no—
461C22×C4—261no—
462C22×C4—261no—
463C22×C4—264no—
464C22×C4—261no—
465C22×C4—261no—
466C22×C4—1041no—
467C22×C4—1041no—
468C22×C4C2×C4×D1312yes—
469C22×C4D4×D1311yescharacteristic
470C22×C4D4×D1312yes—
471C22×C4C23×D1311yescharacteristic
472C2×C8—21no—
473C2×C8—21no—
474C2×C8—21no—
475C2×C8—21no—
476C2×C8—21no—
477C2×C8—21no—
478C2×C8—261no—
479C2×C8—261no—
480C2×C8—261no—
481C2×C8—261no—
482C2×C8—261no—
483C2×C8—261no—
484C4⋊C4—21no—
485C4⋊C4—21no—
486C4⋊C4—21no—
487C4⋊C4—21no—
488C4⋊C4—21no—
489C4⋊C4—21no—
490C4⋊C4—134no—
491C4⋊C4—134no—
492C4⋊C4—132no—
493C4⋊C4—132no—
494C4⋊C4—264no—
495C4⋊C4—264no—
496C4⋊C4—264no—
497C4⋊C4—264no—
498C4⋊C4—264no—
499C4⋊C4—264no—
500C4⋊C4—264no—
501C4⋊C4—261no—
502C4⋊C4—261no—
503C4⋊C4—261no—
504C4⋊C4—261no—
505C4⋊C4—261no—
506C4⋊C4—261no—
507C4⋊C4—1041no—
508C4⋊C4—1041no—
509C4⋊C4C2×C4×D1312yes—
510C4⋊C4C2×C4×D1312yes—
511C22⋊C4—24no—
512C22⋊C4—81no—
513C22⋊C4—134no—
514C22⋊C4—134no—
515C22⋊C4—1041no—
516C42—24no—
517C42—132no—
518C42—132no—
519C42—132no—
520C42—132no—
521C42—1041no—
522C42—1041no—
523C13#2184311yescharacteristic, complemented, Sylow
524C23—21no—
525C23—21no—
526C23—21no—
527C23—21no—
528C23—21no—
529C23—21no—
530C23C22⋊C4×D1312yes—
531C23C2×D4×D1311yescharacteristic
532Q8—1044no—
533Q8—1044no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—132no—
553C2×C4—132no—
554C2×C4—132no—
555C2×C4—132no—
556C2×C4—261no—
557C2×C4—261no—
558C2×C4—261no—
559C2×C4—261no—
560C2×C4—261no—
561C2×C4—261no—
562C2×C4—261no—
563C2×C4—261no—
564C2×C4—262no—
565C2×C4—262no—
566C2×C4—521no—
567C2×C4—521no—
568C2×C4—521no—
569C2×C4—521no—
570C2×C4—521no—
571C2×C4—521no—
572C2×C4—521no—
573C2×C4—521no—
574C2×C4—521no—
575C2×C4—521no—
576C2×C4—1044no—
577C2×C4C22⋊C4×D1312yes—
578C2×C4C22×C4×D1311yescharacteristic
579C2×C4C2×D4×D1311yescharacteristic
580C2×C4#8825711yescharacteristic
581C2×C4#8825711yescharacteristic
582C2×C4#8825711yescharacteristic
583C2×C4#8825711yescharacteristic
584C8—41no—
585C8—41no—
586C8—521no—
587C8—521no—
588C22—22no—
589C22—22no—
590C22—22no—
591C22—41no—
592C22—41no—
593C22—41no—
594C22—41no—
595C22C2×C22⋊C4×D1311yescharacteristic
596C22#1171935311yescharacteristic
597C22#1171935311yescharacteristic
598C22#1171947411yescharacteristic
599C22#1171947511yescharacteristic
600C22#1171948011yescharacteristic
601C22#1171948411yescharacteristic
602C4—22no—
603C4—42no—
604C4—42no—
605C4—81no—
606C4—81no—
607C4—262no—
608C4—262no—
609C4—262no—
610C4—262no—
611C4—522no—
612C4—522no—
613C4—1041no—
614C4—1041no—
615C4#1171922111yescharacteristic
616C4#1171922111yescharacteristic
617C4#1171935311yescharacteristic
618C4#1171935311yescharacteristic
619C2—42no—
620C2#1311742111yescharacteristic
621C2#1312335811yescharacteristic
622C2#1312336711yescharacteristic
623C2#1312609511yescharacteristic
624C2#1312611011yescharacteristic
625C2#1312612511yescharacteristic
626C2#1312613211yescharacteristic
subgroupG/H
centerC23C2×D4×D13
commutatorC2×C4×C13C23×C4
FrattiniC22×C4C23×D13
FittingC13×#1825C2
socleC23×C13C22×D4
prime radicalC24×C13C2×D4
cosocleC22×C4×C13C24
hypercenter#1825D13
hypocenterC13#21843
LCSG ▷ C2×C4×C13 ▷ C2×C13 ▷ C13
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1825

Complements

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

NH ≅ G/Ntype
4. #13115807C2semidirect
5. #13115803C2semidirect
6. #13115799C2semidirect
7. #13115795C2semidirect
8. #13115792C2semidirect
9. #13115790C2semidirect
10. #13115786C2semidirect
11. #13115782C2semidirect
20. #11719224C4semidirect
523. C13#21843semidirect

Maximal subgroups

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

subgroupindex
C2×#117192242
C2×#117183812
#131174432
#131158072
#131158032
#131157992
#131157952
#131157922
#131157902
#131157862
#131157822
#131149312
#131149242
#131149232
C13×#18252
#2184313