Finite Group Databasebeta

← #15884724

Subgroups — #15884724

Order 3328.

898 conjugacy classes of subgroups.

The subgroup lattice has 618 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11718422C211yesmaximal, characteristic
2#13117922C211yesmaximal, characteristic, complemented
3D4×#6101C211yesmaximal, characteristic
4#13115795C211yesmaximal, characteristic, complemented
5#13115596C211yesmaximal, characteristic, complemented
6#13115562C211yesmaximal, characteristic, complemented
7#13115561C211yesmaximal, characteristic, complemented
8#13115415C211yesmaximal, characteristic, complemented
9#13115412C211yesmaximal, characteristic, complemented
10#13115095C211yesmaximal, characteristic
11#13114697C211yesmaximal, characteristic, complemented
12#102×Dic13C211yesmaximal, characteristic
13#102×Dic13C211yesmaximal, characteristic
14C4×#88072C211yesmaximal, characteristic
15C13×#1864C211yesmaximal, characteristic
16C2×C4×D4×C13C2211yescharacteristic
17C2×D4×Dic13C2211yescharacteristic
18C2×D4×Dic13C2211yescharacteristic
19C22×#6101—21no—
20C2×#88142C2211yescharacteristic
21C2×C13×#106C2211yescharacteristic
22C2×C13×#102C2211yescharacteristic
23C2×C13×#102C2211yescharacteristic
24C13×#389C2211yescharacteristic
25C2×#88072C2211yescharacteristic
26C2×#88072C2211yescharacteristic
27#11718871—21no—
28C2×#88057C2211yescharacteristic
29C2×#88054C2211yescharacteristic
30C2×C8×Dic13C2211yescharacteristic
31C2×C8×Dic13C2211yescharacteristic
32#11718637C2211yescharacteristic
33C4⋊C4×Dic13C2211yescharacteristic
34C4⋊C4×Dic13C2211yescharacteristic
35#11718635—21no—
36#11718634C2211yescharacteristic
37#11718634C2211yescharacteristic
38#11718633—21no—
39C2×#88048C2211yescharacteristic
40C2×#88048C2211yescharacteristic
41#11718607—21no—
42#11718603—21no—
43C4×C4⋊Dic13C2211yescharacteristic
44C4×#6101C2211yescharacteristic
45#11718589C2211yescharacteristic
46C2×C4×#1055C2211yescharacteristic
47#11718422C418yescomplemented
48#11718310—21no—
49#11718307—21no—
50C13×#340C2211yescharacteristic
51C13×#336C2211yescharacteristic
52#11718143C2211yescharacteristic
53#11718143C2211yescharacteristic
54#11718143C2211yescharacteristic
55#11718142C2211yescharacteristic
56#11718121C2211yescharacteristic
57#11718121C2211yescharacteristic
58#11718121C2211yescharacteristic
59#11718121C2211yescharacteristic
60C22×D4×C13C2311yescharacteristic
61C23×C4×C13—21no—
62C23×Dic13—21no—
63C23×Dic13—21no—
64C22×Dic26—21no—
65C22×C8×C13C2311yescharacteristic
66C22×C8×C13C2311yescharacteristic
67C4×D4×C13—22no—
68C4×D4×C13—41no—
69C2×C13×C4⋊C4—21no—
70C2×C13×C4⋊C4C2311yescharacteristic
71C2×C13×C4⋊C4C2311yescharacteristic
72C2×C13×C4⋊C4C2311yescharacteristic
73C2×C13×C22⋊C4—21no—
74C2×C42×C13C2311yescharacteristic
75C2×C22⋊Dic13—21no—
76C2×C22⋊Dic13—21no—
77D4×Dic13—22no—
78D4×Dic13—22no—
79D4×Dic13—41no—
80D4×Dic13C2×C414yes—
81C2×C4⋊Dic13C2311yescharacteristic
82C2×C4⋊Dic13C2311yescharacteristic
83C2×C4⋊Dic13C2311yescharacteristic
84C2×C4⋊Dic13C2311yescharacteristic
85C2×#6101—21no—
86C2×#6101—21no—
87C2×#6101—21no—
88C2×#6101—21no—
89C2×#6101—21no—
90C2×#6101—21no—
91C2×#6101—44no—
92C2×#6101D411yescharacteristic
93C2×#6101D411yescharacteristic
94C2×C4×Dic13—21no—
95C2×C4×Dic13—21no—
96C2×C4×Dic13—21no—
97C2×C4×Dic13C2311yescharacteristic
98C2×C4×Dic13C2311yescharacteristic
99C22×#1055C2311yescharacteristic
100C22×#1055C2311yescharacteristic
101#88142—22no—
102#88142C2×C414yescomplemented
103C13×#106—41no—
104C13×#102—22no—
105C13×#102C2×C414yes—
106C13×#95—21no—
107#88072C2×C414yes—
108#88072Q814yescomplemented
109#88071—21no—
110#88071—21no—
111#88071—21no—
112#88071—21no—
113#88071—21no—
114#88071—21no—
115#88071D411yescharacteristic
116#88071D411yescharacteristic
117#88057C2×C414yescomplemented
118#88054C2×C414yescomplemented
119C8×Dic13—22no—
120C8×Dic13—41no—
121#88048—41no—
122#88048C2×C414yescomplemented
123C4×#1055—41no—
124#22963—131nomaximal, Sylow
125C24×C13—21no—
126C2×D4×C13—21no—
127C2×D4×C13—21no—
128C2×D4×C13—21no—
129C2×D4×C13C22×C412yes—
130C2×D4×C13C2×Q812yes—
131C2×D4×C13#4112yes—
132C22×C4×C13—21no—
133C22×C4×C13—21no—
134C22×C4×C13—21no—
135C22×C4×C13—42no—
136C22×C4×C13—42no—
137C22×C4×C13—81no—
138C22×C4×C13C2×D411yescharacteristic
139C22×C4×C13C2×D411yescharacteristic
140C22×C4×C13C2411yescharacteristic
141C22×Dic13—21no—
142C22×Dic13—21no—
143C22×Dic13—21no—
144C22×Dic13—24no—
145C22×Dic13—21no—
146C22×Dic13—42no—
147C22×Dic13—42no—
148C22×Dic13—42no—
149C22×Dic13—42no—
150C22×Dic13—81no—
151C22×Dic13C2×D411yescharacteristic
152C22×Dic13C2×D411yescharacteristic
153C22×Dic13C2×D411yescharacteristic
154C22×Dic13C2×D411yescharacteristic
155C2×Dic26—24no—
156C2×Dic26—81no—
157C2×Dic26—81no—
158C2×C8×C13—21no—
159C2×C8×C13—21no—
160C2×C8×C13—21no—
161C2×C8×C13—21no—
162C2×C8×C13—21no—
163C2×C8×C13C22×C412yes—
164C13×C4⋊C4—21no—
165C13×C4⋊C4—21no—
166C13×C4⋊C4—21no—
167C13×C4⋊C4—21no—
168C13×C4⋊C4—21no—
169C13×C4⋊C4—81no—
170C13×C4⋊C4C22×C412yes—
171C13×C22⋊C4—42no—
172C42×C13—21no—
173C42×C13—21no—
174C22⋊Dic13—24no—
175C22⋊Dic13—42no—
176C4⋊Dic13—21no—
177C4⋊Dic13—21no—
178C4⋊Dic13—21no—
179C4⋊Dic13—21no—
180C4⋊Dic13—21no—
181C4⋊Dic13C22×C412yes—
182C4⋊Dic13C22×C412yes—
183C4⋊Dic13C2×Q812yes—
184#6101—24no—
185#6101—22no—
186#6101—42no—
187#6101—42no—
188#6101—42no—
189#6101—81no—
190#6101—81no—
191#6101SD1614yes—
192C4×Dic13—21no—
193C4×Dic13—21no—
194C4×Dic13—42no—
195C4×Dic13—81no—
196C4×Dic13—81no—
197C4×Dic13C22×C412yes—
198C4×Dic13#4112yes—
199C2×#1055—21no—
200C2×#1055—21no—
201C2×#1055—21no—
202C2×#1055—21no—
203C2×#1055C22×C412yes—
204C2×#1055C2×Q812yes—
205C2×#476—131no—
206#2369—131no—
207D4×C4⋊C4—131no—
208#1990—131no—
209#1948—131no—
210#1941—131no—
211#1941—131no—
212#1919—131no—
213#1919—131no—
214#1864—131no—
215#1864D1311yescharacteristic
216C8×C4⋊C4—131no—
217C4×#102—131no—
218C4×#102—131no—
219C4×#102—131no—
220C23×C13—22no—
221C23×C13—22no—
222C23×C13—22no—
223C23×C13—41no—
224C23×C13—41no—
225C23×C13—41no—
226C23×C13—41no—
227C23×C13C22×D411yescharacteristic
228D4×C13—22no—
229D4×C13—22no—
230D4×C13—22no—
231D4×C13C4×Q814yes—
232C2×C4×C13—21no—
233C2×C4×C13—21no—
234C2×C4×C13—22no—
235C2×C4×C13—21no—
236C2×C4×C13—21no—
237C2×C4×C13—21no—
238C2×C4×C13—41no—
239C2×C4×C13—41no—
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—84no—
247C2×C4×C13C2×SD1612yes—
248C2×C4×C13C23×C411yescharacteristic
249C2×C4×C13C22×Q811yescharacteristic
250C2×C4×C13C2×#4111yescharacteristic
251C2×C4×C13C2×#4111yescharacteristic
252C2×C4×C13H(5,𝔽2)11yescharacteristic
253C2×C4×C132−1+411yescharacteristic
254C2×Dic13—21no—
255C2×Dic13—21no—
256C2×Dic13—21no—
257C2×Dic13—21no—
258C2×Dic13—22no—
259C2×Dic13—22no—
260C2×Dic13—21no—
261C2×Dic13—22no—
262C2×Dic13—21no—
263C2×Dic13—21no—
264C2×Dic13—41no—
265C2×Dic13—41no—
266C2×Dic13—41no—
267C2×Dic13—41no—
268C2×Dic13—41no—
269C2×Dic13—41no—
270C2×Dic13—48no—
271C2×Dic13—41no—
272C2×Dic13—41no—
273C2×Dic13—41no—
274C2×Dic13—84no—
275C2×Dic13C4×D412yes—
276C2×Dic13C4×D412yes—
277C2×Dic13C2×SD1612yes—
278C2×Dic13C2×SD1612yes—
279C2×Dic13#13512yes—
280Dic26—84no—
281C8×C13—22no—
282C8×C13—41no—
283#1055—22no—
284#1055—22no—
285C2×C4⋊Q8—131no—
286C2×C4×D4—131no—
287C2×C4×D4—131no—
288C2×C4×D4C2×D1311yescharacteristic
289C22×C4⋊C4—261no—
290#476—138no—
291C2×#106—131no—
292C2×#106—131no—
293C2×#106—131no—
294C2×#106C2×D1311yescharacteristic
295C2×C4⋊C8—131no—
296C2×#102—131no—
297C2×#102—131no—
298C2×#102C2×D1311yescharacteristic
299C2×#102C2×D1311yescharacteristic
300C2×C4×C8—131no—
301C2×C4×C8—131no—
302C2×C4×C8—131no—
303#391—261no—
304#391—261no—
305#389—131no—
306#389—131no—
307#389—131no—
308#389—261no—
309#389C2×D1311yescharacteristic
310#385—261no—
311#385—261no—
312#384—131no—
313#380—261no—
314C4×C4⋊C4—131no—
315C4×C4⋊C4—131no—
316C4×C4⋊C4—131no—
317C4×C4⋊C4—131no—
318C4×C4⋊C4—261no—
319#340—131no—
320#340—131no—
321#340—131no—
322#340—131no—
323#340—131no—
324#340—131no—
325#340—131no—
326#340C2×D1311yescharacteristic
327#336—131no—
328#336C2×D1311yescharacteristic
329C22×C13—24no—
330C22×C13—42no—
331C22×C13—42no—
332C22×C13—42no—
333C22×C13—42no—
334C22×C13—42no—
335C22×C13—42no—
336C22×C13C2×C4×D411yescharacteristic
337C22×C13#54611yescharacteristic
338C22×C13D4×Q811yescharacteristic
339C22×C13C22×SD1611yescharacteristic
340C22×C13C2×#13511yescharacteristic
341C22×C13#57611yescharacteristic
342C22×C13#57711yescharacteristic
343C4×C13—22no—
344C4×C13—42no—
345C4×C13—41no—
346C4×C13—81no—
347C4×C13—81no—
348C4×C13C2×C4×Q811yescharacteristic
349C4×C13#52011yescharacteristic
350C4×C13#55211yescharacteristic
351C4×C13#55411yescharacteristic
352Dic13—22no—
353Dic13—22no—
354Dic13—42no—
355Dic13—42no—
356Dic13—42no—
357Dic13—41no—
358Dic13—81no—
359Dic13C4×SD1614yes—
360C22×Q8—261no—
361C22×D4C22×D1311yescharacteristic
362C23×C4—21no—
363C23×C4—261no—
364C23×C4—261no—
365C22×C8—131no—
366C22×C8—131no—
367C22×C8C22×D1311yescharacteristic
368C22×C8C22×D1311yescharacteristic
369C4⋊Q8—134no—
370C4⋊Q8—262no—
371C4×D4—22no—
372C4×D4—41no—
373C4×D4—134no—
374C4×D4—262no—
375C4×D4—262no—
376C4×D4—521no—
377C2×C4⋊C4—21no—
378C2×C4⋊C4—131no—
379C2×C4⋊C4—131no—
380C2×C4⋊C4—131no—
381C2×C4⋊C4—131no—
382C2×C4⋊C4—131no—
383C2×C4⋊C4—131no—
384C2×C4⋊C4—261no—
385C2×C4⋊C4—261no—
386C2×C4⋊C4—261no—
387C2×C4⋊C4—261no—
388C2×C4⋊C4—261no—
389C2×C4⋊C4—261no—
390C2×C4⋊C4—524no—
391C2×C4⋊C4C22×D1311yescharacteristic
392C2×C4⋊C4C22×D1311yescharacteristic
393C2×C4⋊C4C22×D1311yescharacteristic
394C2×C22⋊C4—21no—
395C2×C22⋊C4—261no—
396C2×C22⋊C4—261no—
397C2×C42—131no—
398C2×C42—131no—
399C2×C42—261no—
400C2×C42—261no—
401C2×C42—261no—
402C2×C42C22×D1311yescharacteristic
403#106—41no—
404#106—134no—
405#106—134no—
406#106—521no—
407C4⋊C8—134no—
408#102—22no—
409#102—134no—
410#102—134no—
411#102C4×D1314yes—
412C4×C8—262no—
413C4×C8—521no—
414C4×C8—521no—
415#95—21no—
416#95—131no—
417#95—131no—
418#95—261no—
419#95—261no—
420#95—261no—
421#95—261no—
422#95—261no—
423#95—261no—
424C2×C13—44no—
425C2×C13#236011yescharacteristic
426C2×C13C2×C4×SD1611yescharacteristic
427C2×C13#294611yescharacteristic
428C2×C13#329511yescharacteristic
429C2×C13#330311yescharacteristic
430C2×C13Q8×SD1611yescharacteristic
431C2×C13#338011yescharacteristic
432C24—21no—
433C2×Q8—264no—
434C2×Q8—1041no—
435C2×Q8—1041no—
436C2×D4—21no—
437C2×D4—21no—
438C2×D4—21no—
439C2×D4C2×C4×D1312yes—
440C2×D4#612912yes—
441C2×D4Q8×D1312yes—
442C22×C4—21no—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—42no—
446C22×C4—42no—
447C22×C4—81no—
448C22×C4—131no—
449C22×C4—131no—
450C22×C4—131no—
451C22×C4—131no—
452C22×C4—264no—
453C22×C4—261no—
454C22×C4—261no—
455C22×C4—261no—
456C22×C4—261no—
457C22×C4—522no—
458C22×C4—522no—
459C22×C4—522no—
460C22×C4—522no—
461C22×C4—1041no—
462C22×C4D4×D1311yescharacteristic
463C22×C4D4×D1311yescharacteristic
464C22×C4C23×D1311yescharacteristic
465C2×C8—21no—
466C2×C8—21no—
467C2×C8—21no—
468C2×C8—21no—
469C2×C8—21no—
470C2×C8—132no—
471C2×C8—132no—
472C2×C8—261no—
473C2×C8—261no—
474C2×C8—261no—
475C2×C8—261no—
476C2×C8C2×C4×D1312yes—
477C4⋊C4—21no—
478C4⋊C4—21no—
479C4⋊C4—21no—
480C4⋊C4—21no—
481C4⋊C4—21no—
482C4⋊C4—81no—
483C4⋊C4—134no—
484C4⋊C4—132no—
485C4⋊C4—132no—
486C4⋊C4—132no—
487C4⋊C4—262no—
488C4⋊C4—264no—
489C4⋊C4—261no—
490C4⋊C4—261no—
491C4⋊C4—261no—
492C4⋊C4—261no—
493C4⋊C4—261no—
494C4⋊C4—522no—
495C4⋊C4—522no—
496C4⋊C4—522no—
497C4⋊C4—1041no—
498C4⋊C4—1041no—
499C4⋊C4C2×C4×D1312yes—
500C22⋊C4—42no—
501C22⋊C4—264no—
502C22⋊C4—522no—
503C42—21no—
504C42—21no—
505C42—132no—
506C42—132no—
507C42—261no—
508C42—261no—
509C42—522no—
510C42—1041no—
511C42—1041no—
512C13#2296311yescharacteristic, complemented, Sylow
513C23—22no—
514C23—22no—
515C23—22no—
516C23—41no—
517C23—41no—
518C23—41no—
519C23—41no—
520C23C2×D4×D1311yescharacteristic
521Q8—1044no—
522D4—22no—
523D4—22no—
524D4—22no—
525D4#8814114yes—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—22no—
531C2×C4—21no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—84no—
541C2×C4—132no—
542C2×C4—132no—
543C2×C4—132no—
544C2×C4—132no—
545C2×C4—132no—
546C2×C4—261no—
547C2×C4—261no—
548C2×C4—261no—
549C2×C4—261no—
550C2×C4—261no—
551C2×C4—261no—
552C2×C4—262no—
553C2×C4—262no—
554C2×C4—261no—
555C2×C4—262no—
556C2×C4—528no—
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—1044no—
567C2×C4SD16×D1312yes—
568C2×C4C22×C4×D1311yescharacteristic
569C2×C4C2×#612911yescharacteristic
570C2×C4#8825111yescharacteristic
571C2×C4C2×Q8×D1311yescharacteristic
572C2×C4#41×D1311yescharacteristic
573C2×C4#8825711yescharacteristic
574C8—22no—
575C8—41no—
576C8—262no—
577C8—262no—
578C22—24no—
579C22—42no—
580C22—42no—
581C22—42no—
582C22—42no—
583C22—42no—
584C22—42no—
585C22C4×D4×D1311yescharacteristic
586C22#1171929611yescharacteristic
587C22#1171933711yescharacteristic
588C22C2×SD16×D1311yescharacteristic
589C22D13×#13511yescharacteristic
590C22#1171947911yescharacteristic
591C22#1171948011yescharacteristic
592C4—22no—
593C4—42no—
594C4—41no—
595C4—81no—
596C4—81no—
597C4—134no—
598C4—262no—
599C4—262no—
600C4—522no—
601C4—522no—
602C4—522no—
603C4—521no—
604C4—1041no—
605C4C2×#8814111yescharacteristic
606C4#1171925111yescharacteristic
607C4#1171934111yescharacteristic
608C4#1171935511yescharacteristic
609C2—44no—
610C2#1311783611yescharacteristic
611C2C4×SD16×D1311yescharacteristic
612C2#1312366511yescharacteristic
613C2#1312665311yescharacteristic
614C2#1312675711yescharacteristic
615C2#1312769111yescharacteristic
616C2#1312769811yescharacteristic
subgroupG/H
centerC23C2×D4×D13
commutatorC2×C4×C13C23×C4
FrattiniC22×C4C23×D13
FittingC13×#1864C2
socleC23×C13C22×D4
prime radicalC22×D4×C13C23
cosocleC22×C4×C13C24
hypercenter#1864D13
hypocenterC13#22963
LCSG ▷ C2×C4×C13 ▷ C2×C13 ▷ C13
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1864

Complements

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

NH ≅ G/Ntype
2. #13117922C2semidirect
4. #13115795C2semidirect
5. #13115596C2semidirect
6. #13115562C2semidirect
7. #13115561C2semidirect
8. #13115415C2semidirect
9. #13115412C2semidirect
11. #13114697C2semidirect
47. #11718422C4semidirect
102. #88142C2×C4semidirect
108. #88072Q8semidirect
117. #88057C2×C4semidirect
118. #88054C2×C4semidirect
122. #88048C2×C4semidirect
512. C13#22963semidirect

Maximal subgroups

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

subgroupindex
C2×#117184222
#131179222
D4×#61012
#131157952
#131155962
#131155622
#131155612
#131154152
#131154122
#131150952
#131146972
#102×Dic132
C4×#880722
C13×#18642
#2296313