Finite Group Databasebeta

← #15882753

Subgroups — #15882753

Order 3328.

886 conjugacy classes of subgroups.

The subgroup lattice has 652 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#13117809C211yesmaximal, characteristic, complemented
2#13117785C211yesmaximal, characteristic, complemented
3#13115713C211yesmaximal, characteristic
4#13115703C211yesmaximal, characteristic
5#13115465C211yesmaximal, characteristic, complemented
6#13115465C211yesmaximal, characteristic, complemented
7#13115458C211yesmaximal, characteristic, complemented
8#13115458C211yesmaximal, characteristic, complemented
9#13115004C211yesmaximal, characteristic, complemented
10#13114846C211yesmaximal, characteristic, complemented
11#13114836C211yesmaximal, characteristic, complemented
12#13114736C211yesmaximal, characteristic, complemented
13#106×Dic13C211yesmaximal, characteristic, complemented
14#13114708C211yesmaximal, characteristic, complemented
15C2×C13×#442C211yesmaximal, characteristic
16C22×C13×D8C2211yescharacteristic
17C2×C4×D4×C13C2211yescharacteristic
18C2×C4×D4×C13C2211yescharacteristic
19C2×#88180—21no—
20C13×#442C418yescomplemented
21C2×C13×#106C2211yescharacteristic
22C2×C13×#102C2211yescharacteristic
23C2×C13×#102C2211yescharacteristic
24C2×C13×#97C2211yescharacteristic
25#11718882C2211yescharacteristic
26#11718882C2211yescharacteristic
27#11718882C2211yescharacteristic
28#11718882C2211yescharacteristic
29C2×#88072C2211yescharacteristic
30C2×#88072C2211yescharacteristic
31C2×#88072C2211yescharacteristic
32C2×#88072C2211yescharacteristic
33#11718873—21no—
34#11718872—21no—
35C4×C22⋊Dic13—21no—
36C2×#88058C2211yescharacteristic
37C2×#88058C2211yescharacteristic
38C2×#88057C2211yescharacteristic
39C2×#88057C2211yescharacteristic
40C2×#88055C2211yescharacteristic
41C2×C8×Dic13C2211yescharacteristic
42C4⋊C4×Dic13C2211yescharacteristic
43C4⋊C4×Dic13C2211yescharacteristic
44C4⋊C4×Dic13C2211yescharacteristic
45C4⋊C4×Dic13C2211yescharacteristic
46#11718612—21no—
47#11718612—21no—
48#11718612—21no—
49#11718612—21no—
50#11718597C2211yescharacteristic
51#11718596C2211yescharacteristic
52C4×C4⋊Dic13C2211yescharacteristic
53C2×C4⋊#1055C2211yescharacteristic
54#11718142C2211yescharacteristic
55#11718142C2211yescharacteristic
56#11718121C2211yescharacteristic
57#11718121C2211yescharacteristic
58#11718121C2211yescharacteristic
59#11718121C2211yescharacteristic
60C22×D4×C13C2311yescharacteristic
61C22×D4×C13C2311yescharacteristic
62C23×C4×C13—21no—
63C23×C4×C13—21no—
64C2×C13×D8—22no—
65C2×C13×D8—22no—
66C2×C13×D8C2×C414yes—
67C22×C8×C13C2311yescharacteristic
68C22×C8×C13C2311yescharacteristic
69C4×D4×C13—22no—
70C4×D4×C13—22no—
71C4×D4×C13C2×C414yes—
72C4×D4×C13C2×C414yes—
73C2×C13×C4⋊C4C2311yescharacteristic
74C2×C13×C4⋊C4C2311yescharacteristic
75C2×C13×C22⋊C4—21no—
76C2×C13×C22⋊C4—21no—
77C2×C42×C13C2311yescharacteristic
78C2×C22⋊Dic13—21no—
79C2×C22⋊Dic13—21no—
80C2×C22⋊Dic13—21no—
81C2×C22⋊Dic13—21no—
82C2×C22⋊Dic13—21no—
83C2×C22⋊Dic13—21no—
84C2×C22⋊Dic13—21no—
85C2×C22⋊Dic13—21no—
86#88180—44no—
87C2×C4⋊Dic13—21no—
88C2×C4⋊Dic13—21no—
89C2×C4⋊Dic13D411yescharacteristic
90C2×C4⋊Dic13D411yescharacteristic
91C2×C4⋊Dic13C2311yescharacteristic
92C2×C4⋊Dic13C2311yescharacteristic
93C2×C4⋊Dic13C2311yescharacteristic
94C2×C4⋊Dic13C2311yescharacteristic
95C2×C4×Dic13—21no—
96C2×C4×Dic13—21no—
97C2×C4×Dic13—21no—
98C2×C4×Dic13—21no—
99C2×C4×Dic13C2311yescharacteristic
100C2×C4×Dic13C2311yescharacteristic
101C22×#1055C2311yescharacteristic
102C22×#1055C2311yescharacteristic
103C13×#106C2×C414yes—
104C13×#102C2×C414yes—
105C13×#102C2×C414yes—
106C13×#97C2×C414yes—
107#88072—41no—
108#88072—41no—
109#88072—41no—
110#88072—41no—
111#88071—21no—
112#88071—21no—
113#88071—21no—
114#88071—21no—
115#88071—21no—
116#88071—21no—
117#88071D411yescharacteristic
118#88071D411yescharacteristic
119#88058—22no—
120#88058—41no—
121#88057—22no—
122#88057—41no—
123#88055—22no—
124C8×Dic13—22no—
125C4⋊#1055—22no—
126#22817—131nomaximal, Sylow
127C24×C13—21no—
128C24×C13—21no—
129C2×D4×C13—21no—
130C2×D4×C13—21no—
131C2×D4×C13—21no—
132C2×D4×C13—21no—
133C2×D4×C13—21no—
134C2×D4×C13—21no—
135C2×D4×C13—21no—
136C2×D4×C13—21no—
137C2×D4×C13—21no—
138C2×D4×C13—21no—
139C2×D4×C13C22×C412yes—
140C2×D4×C13C22×C412yes—
141C22×C4×C13—21no—
142C22×C4×C13—21no—
143C22×C4×C13—24no—
144C22×C4×C13—24no—
145C22×C4×C13—42no—
146C22×C4×C13—42no—
147C22×C4×C13—81no—
148C22×C4×C13—81no—
149C22×C4×C13C2×D411yescharacteristic
150C22×C4×C13C2×D411yescharacteristic
151C22×C4×C13C2411yescharacteristic
152C22×Dic13—21no—
153C22×Dic13—21no—
154C22×Dic13—21no—
155C22×Dic13—21no—
156C22×Dic13C2×D411yescharacteristic
157C22×Dic13C2×D411yescharacteristic
158C22×Dic13C2×D411yescharacteristic
159C22×Dic13C2×D411yescharacteristic
160C13×D8—28no—
161C2×C8×C13—21no—
162C2×C8×C13—21no—
163C2×C8×C13C22×C412yes—
164C2×C8×C13C22×C412yes—
165C2×C8×C13#4112yes—
166C2×C8×C13#4112yes—
167C13×C4⋊C4—21no—
168C13×C4⋊C4—21no—
169C13×C4⋊C4C22×C412yes—
170C13×C4⋊C4C22×C412yes—
171C13×C22⋊C4—24no—
172C13×C22⋊C4—24no—
173C42×C13C22×C412yes—
174C42×C13#4112yes—
175C22⋊Dic13—42no—
176C22⋊Dic13—42no—
177C22⋊Dic13—42no—
178C22⋊Dic13—42no—
179C22⋊Dic13—81no—
180C22⋊Dic13—81no—
181C22⋊Dic13—81no—
182C22⋊Dic13—81no—
183C4⋊Dic13—22no—
184C4⋊Dic13—21no—
185C4⋊Dic13—21no—
186C4⋊Dic13—21no—
187C4⋊Dic13—21no—
188C4⋊Dic13—21no—
189C4⋊Dic13—21no—
190C4⋊Dic13—21no—
191C4⋊Dic13—21no—
192C4⋊Dic13—41no—
193C4⋊Dic13—42no—
194C4⋊Dic13—81no—
195C4×Dic13—21no—
196C4×Dic13—21no—
197C4×Dic13—21no—
198C4×Dic13—21no—
199C4×Dic13—42no—
200C4×Dic13—42no—
201C4×Dic13—81no—
202C4×Dic13—81no—
203C2×#1055—21no—
204C2×#1055—21no—
205C2×#1055—21no—
206C2×#1055—21no—
207C2×#1055—21no—
208C2×#1055—21no—
209C2×#442D1311yescharacteristic
210#2357—131no—
211#2354—131no—
212#1972—131no—
213#1970—131no—
214#1926—131no—
215#1926—131no—
216#1925—131no—
217#1925—131no—
218#1847—131no—
219#1804—131no—
220#1802—131no—
221#1776—131no—
222C4×#106—131no—
223#1771—131no—
224C23×C13—22no—
225C23×C13—22no—
226C23×C13—41no—
227C23×C13—41no—
228C23×C13—41no—
229C23×C13—41no—
230C23×C13—41no—
231C23×C13—41no—
232C23×C13—41no—
233C23×C13—41no—
234C23×C13—41no—
235C23×C13—41no—
236C23×C13—41no—
237C23×C13—41no—
238C23×C13C22×D411yescharacteristic
239D4×C13—22no—
240D4×C13—22no—
241D4×C13—22no—
242D4×C13—22no—
243D4×C13—41no—
244D4×C13—41no—
245D4×C13—41no—
246D4×C13—41no—
247C2×C4×C13—21no—
248C2×C4×C13—21no—
249C2×C4×C13—21no—
250C2×C4×C13—21no—
251C2×C4×C13—22no—
252C2×C4×C13—22no—
253C2×C4×C13—48no—
254C2×C4×C13—41no—
255C2×C4×C13—41no—
256C2×C4×C13—41no—
257C2×C4×C13—41no—
258C2×C4×C13—84no—
259C2×C4×C13C4×D412yes—
260C2×C4×C13C4×D412yes—
261C2×C4×C13C23×C411yescharacteristic
262C2×C4×C13C2×#4111yescharacteristic
263C2×C4×C13C2×#4111yescharacteristic
264C2×C4×C13C2×#4111yescharacteristic
265C2×C4×C13H(5,𝔽2)11yescharacteristic
266C2×C4×C132−1+411yescharacteristic
267C2×Dic13—21no—
268C2×Dic13—21no—
269C2×Dic13—21no—
270C2×Dic13—21no—
271C2×Dic13—21no—
272C2×Dic13—21no—
273C2×Dic13—21no—
274C2×Dic13—21no—
275C2×Dic13—21no—
276C2×Dic13—21no—
277C2×Dic13—21no—
278C2×Dic13—41no—
279C2×Dic13—41no—
280C2×Dic13—41no—
281C2×Dic13—41no—
282C2×Dic13—41no—
283C2×Dic13—41no—
284C2×Dic13—41no—
285C2×Dic13—41no—
286C2×Dic13—41no—
287C2×Dic13—41no—
288C2×Dic13—41no—
289C2×Dic13—41no—
290C2×Dic13AGL(1,ℤ/8)12yes—
291C8×C13—22no—
292C8×C13#11714yescomplemented
293#1055—41no—
294#1055—41no—
295C22×D8C2×D1311yescharacteristic
296C2×C4×D4C2×D1311yescharacteristic
297C2×C4×D4C2×D1311yescharacteristic
298C2×#117—261no—
299#442Dic1318yescomplemented
300C2×#107—131no—
301C2×#107—131no—
302C2×#106—131no—
303C2×#106—131no—
304C2×#106C2×D1311yescharacteristic
305C2×C4⋊C8—131no—
306C2×#102—131no—
307C2×#102—131no—
308C2×#102—131no—
309C2×#102—131no—
310C2×#102C2×D1311yescharacteristic
311C2×#102C2×D1311yescharacteristic
312C2×#97—131no—
313C2×#97C2×D1311yescharacteristic
314C2×C4×C8—131no—
315#390—131no—
316#390—131no—
317#390—131no—
318#390—131no—
319#388—261no—
320#388—261no—
321#388—261no—
322#388—261no—
323#384—131no—
324#382—131no—
325#381—261no—
326#380—261no—
327C4×C4⋊C4—131no—
328C4×C4⋊C4—131no—
329C4×C4⋊C4—131no—
330C4×C4⋊C4—131no—
331C4×C4⋊C4—131no—
332C4×C22⋊C4—261no—
333#340—131no—
334#340—131no—
335#340—131no—
336#340—131no—
337#336—131no—
338#336—131no—
339C22×C13—42no—
340C22×C13—42no—
341C22×C13—42no—
342C22×C13—42no—
343C22×C13—42no—
344C22×C13—42no—
345C22×C13—81no—
346C22×C13—81no—
347C22×C13—81no—
348C22×C13—81no—
349C22×C13—81no—
350C22×C13—81no—
351C22×C13—81no—
352C22×C13—81no—
353C22×C13C2×C4×D411yescharacteristic
354C22×C13#54711yescharacteristic
355C22×C13#55011yescharacteristic
356C22×C13C2×AGL(1,ℤ/8)11yescharacteristic
357C22×C13#57511yescharacteristic
358C22×C13#57611yescharacteristic
359C22×C13#57711yescharacteristic
360C4×C13—22no—
361C4×C13—22no—
362C4×C13—42no—
363C4×C13—42no—
364C4×C13C2×#11711yescharacteristic
365C4×C13#52011yescharacteristic
366C4×C13#55511yescharacteristic
367C4×C13#55611yescharacteristic
368Dic13—22no—
369Dic13—41no—
370Dic13—41no—
371Dic13—41no—
372Dic13—81no—
373Dic13—81no—
374Dic13—81no—
375Dic13—81no—
376C22×D4C22×D1311yescharacteristic
377C22×D4C22×D1311yescharacteristic
378C23×C4—21no—
379C23×C4—21no—
380C2×D8—22no—
381C2×D8—22no—
382C2×D8C2×Dic1314yes—
383C22×C8—131no—
384C22×C8—131no—
385C22×C8C22×D1311yescharacteristic
386C22×C8C22×D1311yescharacteristic
387C4×D4—22no—
388C4×D4—22no—
389C4×D4C2×Dic1314yes—
390C4×D4C2×Dic1314yes—
391#117—524no—
392C2×C4⋊C4—131no—
393C2×C4⋊C4—131no—
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⋊C4C22×D1311yescharacteristic
401C2×C4⋊C4C22×D1311yescharacteristic
402C2×C22⋊C4—21no—
403C2×C22⋊C4—21no—
404C2×C22⋊C4—261no—
405C2×C22⋊C4—261no—
406C2×C22⋊C4—261no—
407C2×C22⋊C4—261no—
408C2×C22⋊C4—261no—
409C2×C22⋊C4—261no—
410C2×C22⋊C4—261no—
411C2×C22⋊C4—261no—
412C2×C42—131no—
413C2×C42—131no—
414C2×C42—261no—
415C2×C42—261no—
416C2×C42—261no—
417C2×C42—261no—
418C2×C42C22×D1311yescharacteristic
419#107—262no—
420#107—521no—
421#106—262no—
422#106—521no—
423#106C2×Dic1314yes—
424C4⋊C8—262no—
425#102—521no—
426#102—521no—
427#102—521no—
428#102—521no—
429#102C2×Dic1314yes—
430#102C2×Dic1314yes—
431#97—262no—
432#97C2×Dic1314yes—
433C4×C8—262no—
434#95—131no—
435#95—131no—
436#95—261no—
437#95—261no—
438#95—261no—
439#95—261no—
440#95—261no—
441#95—261no—
442C2×C13—82no—
443C2×C13—82no—
444C2×C13#235411yescharacteristic
445C2×C13C4×AGL(1,ℤ/8)11yescharacteristic
446C2×C13#294611yescharacteristic
447C2×C13#334511yescharacteristic
448C2×C13#334711yescharacteristic
449C2×C13#339911yescharacteristic
450C2×C13#340111yescharacteristic
451C24—21no—
452C24—21no—
453C2×D4—21no—
454C2×D4—21no—
455C2×D4—21no—
456C2×D4—21no—
457C2×D4—21no—
458C2×D4—21no—
459C2×D4—21no—
460C2×D4—21no—
461C2×D4—21no—
462C2×D4—21no—
463C2×D4C22×Dic1312yes—
464C2×D4C22×Dic1312yes—
465C22×C4—21no—
466C22×C4—21no—
467C22×C4—24no—
468C22×C4—24no—
469C22×C4—42no—
470C22×C4—42no—
471C22×C4—81no—
472C22×C4—81no—
473C22×C4—131no—
474C22×C4—131no—
475C22×C4—131no—
476C22×C4—131no—
477C22×C4—261no—
478C22×C4—261no—
479C22×C4—261no—
480C22×C4—261no—
481C22×C4D4×D1311yescharacteristic
482C22×C4D4×D1311yescharacteristic
483C22×C4C23×D1311yescharacteristic
484D8—28no—
485C2×C8—21no—
486C2×C8—21no—
487C2×C8—261no—
488C2×C8—261no—
489C2×C8—261no—
490C2×C8—261no—
491C2×C8—261no—
492C2×C8—261no—
493C2×C8#612712yes—
494C2×C8#612712yes—
495C2×C8C22×Dic1312yes—
496C2×C8C22×Dic1312yes—
497C4⋊C4—21no—
498C4⋊C4—21no—
499C4⋊C4—262no—
500C4⋊C4—261no—
501C4⋊C4—261no—
502C4⋊C4—261no—
503C4⋊C4—261no—
504C4⋊C4—261no—
505C4⋊C4—261no—
506C4⋊C4—261no—
507C4⋊C4—261no—
508C4⋊C4—521no—
509C4⋊C4—522no—
510C4⋊C4—1041no—
511C4⋊C4C22×Dic1312yes—
512C4⋊C4C22×Dic1312yes—
513C22⋊C4—24no—
514C22⋊C4—24no—
515C22⋊C4—522no—
516C22⋊C4—522no—
517C22⋊C4—522no—
518C22⋊C4—522no—
519C22⋊C4—1041no—
520C22⋊C4—1041no—
521C22⋊C4—1041no—
522C22⋊C4—1041no—
523C42—261no—
524C42—261no—
525C42—261no—
526C42—261no—
527C42—522no—
528C42—522no—
529C42—1041no—
530C42—1041no—
531C42#612912yes—
532C42C22×Dic1312yes—
533C13#2281711yescharacteristic, complemented, Sylow
534C23—22no—
535C23—22no—
536C23—41no—
537C23—41no—
538C23—41no—
539C23—41no—
540C23—41no—
541C23—41no—
542C23—41no—
543C23—41no—
544C23—41no—
545C23—41no—
546C23—41no—
547C23—41no—
548C23C2×D4×D1311yescharacteristic
549D4—22no—
550D4—22no—
551D4—22no—
552D4—22no—
553D4—41no—
554D4—41no—
555D4—41no—
556D4—41no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—48no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—84no—
569C2×C4—132no—
570C2×C4—261no—
571C2×C4—261no—
572C2×C4—261no—
573C2×C4—261no—
574C2×C4—261no—
575C2×C4—261no—
576C2×C4—261no—
577C2×C4—261no—
578C2×C4—261no—
579C2×C4—261no—
580C2×C4—261no—
581C2×C4—521no—
582C2×C4—521no—
583C2×C4—521no—
584C2×C4—521no—
585C2×C4—521no—
586C2×C4—521no—
587C2×C4—521no—
588C2×C4—521no—
589C2×C4—521no—
590C2×C4—521no—
591C2×C4—521no—
592C2×C4—521no—
593C2×C4D4×Dic1312yes—
594C2×C4D4×Dic1312yes—
595C2×C4C2×#612711yescharacteristic
596C2×C4C2×#612711yescharacteristic
597C2×C4C2×#612911yescharacteristic
598C2×C4#8825611yescharacteristic
599C2×C4#8825711yescharacteristic
600C2×C4C23×Dic1311yescharacteristic
601C8—22no—
602C8—521no—
603C8—521no—
604C8#8818014yescomplemented
605C22—42no—
606C22—42no—
607C22—42no—
608C22—42no—
609C22—42no—
610C22—42no—
611C22—81no—
612C22—81no—
613C22—81no—
614C22—81no—
615C22—81no—
616C22—81no—
617C22—81no—
618C22—81no—
619C22#1171925611yescharacteristic
620C22#1171925811yescharacteristic
621C22D13×AGL(1,ℤ/8)11yescharacteristic
622C22#1171947811yescharacteristic
623C22#1171947911yescharacteristic
624C22#1171948011yescharacteristic
625C22C2×D4×Dic1311yescharacteristic
626C4—22no—
627C4—22no—
628C4—42no—
629C4—42no—
630C4—262no—
631C4—521no—
632C4—521no—
633C4—521no—
634C4—1041no—
635C4—1041no—
636C4—1041no—
637C4—1041no—
638C4#1171924711yescharacteristic
639C4#1171926811yescharacteristic
640C4C2×#8818011yescharacteristic
641C4#1171953211yescharacteristic
642C2—82no—
643C2—82no—
644C2#1311778511yescharacteristic
645C2AGL(1,ℤ/8)×Dic1311yescharacteristic
646C2#1312366911yescharacteristic
647C2#1312731111yescharacteristic
648C2#1312734111yescharacteristic
649C2#1312787911yescharacteristic
650C2#1312790711yescharacteristic
subgroupG/H
centerC23C2×D4×D13
commutatorC2×C4×C13C23×C4
FrattiniC22×C4C23×D13
FittingC2×C13×#442C2
socleC23×C13C22×D4
prime radicalC22×C13×D8C22
cosocleC22×C4×C13C24
hypercenterC2×#442D13
hypocenterC13#22817
LCSG ▷ C2×C4×C13 ▷ C2×C13 ▷ C13
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#442

Complements

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

NH ≅ G/Ntype
1. #13117809C2semidirect
2. #13117785C2semidirect
5. #13115465C2semidirect
6. #13115465C2semidirect
7. #13115458C2semidirect
8. #13115458C2semidirect
9. #13115004C2semidirect
10. #13114846C2semidirect
11. #13114836C2semidirect
12. #13114736C2semidirect
13. #106×Dic13C2semidirect
14. #13114708C2semidirect
20. C13×#442C4semidirect
292. C8×C13#117semidirect
299. #442Dic13semidirect
533. C13#22817semidirect
604. C8#88180semidirect

Maximal subgroups

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

subgroupindex
#131178092
#131177852
#131157132
#131157032
#131154652
#131154582
#131150042
#131148462
#131148362
#131147362
#106×Dic132
#131147082
C2×C13×#4422
#2281713