Finite Group Databasebeta

← #15880696

Subgroups — #15880696

Order 3328.

810 conjugacy classes of subgroups.

The subgroup lattice has 588 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#13117721C211yesmaximal, characteristic
2#13117705C211yesmaximal, characteristic
3#13115580C211yesmaximal, characteristic, complemented
4#13115565C211yesmaximal, characteristic, complemented
5#13115430C211yesmaximal, characteristic, complemented
6#13115423C211yesmaximal, characteristic, complemented
7#13115416C211yesmaximal, characteristic, complemented
8#13115409C211yesmaximal, characteristic, complemented
9#13114931C211yesmaximal, characteristic
10#13114926C211yesmaximal, characteristic
11#13114869C211yesmaximal, characteristic
12#13114735C211yesmaximal, characteristic, complemented
13#107×Dic13C211yesmaximal, characteristic, complemented
14#13114617C211yesmaximal, characteristic
15C2×C13×#428C211yesmaximal, characteristic
16C22×C13×M16C2211yescharacteristic
17C2×C13×#117C2211yescharacteristic
18C22×C13×C4⋊C4C2211yescharacteristic
19C2×#88180—21no—
20C13×#428C418yescomplemented
21C2×C13×#107C2211yescharacteristic
22C2×C13×#107C2211yescharacteristic
23C2×C13×#106C2211yescharacteristic
24C2×C13×#106C2211yescharacteristic
25#11718873C2211yescharacteristic
26#11718873C2211yescharacteristic
27#11718872—21no—
28C4×C22⋊Dic13C2211yescharacteristic
29C2×C22⋊#1055C2211yescharacteristic
30C2×#88058C2211yescharacteristic
31C2×#88057C2211yescharacteristic
32C2×#88055C2211yescharacteristic
33C2×C8×Dic13C2211yescharacteristic
34#11718646C2211yescharacteristic
35#11718646C2211yescharacteristic
36#11718646C2211yescharacteristic
37#11718646C2211yescharacteristic
38#11718642—21no—
39#11718642—21no—
40C4⋊C4×Dic13C2211yescharacteristic
41C4⋊C4×Dic13C2211yescharacteristic
42C4⋊C4×Dic13C2211yescharacteristic
43C4⋊C4×Dic13C2211yescharacteristic
44#11718612—21no—
45#11718612—21no—
46#11718597—21no—
47#11718593—21no—
48#11718143C2211yescharacteristic
49#11718143C2211yescharacteristic
50#11718142C2211yescharacteristic
51#11718142C2211yescharacteristic
52#11718121C2211yescharacteristic
53#11718121C2211yescharacteristic
54#11718121C2211yescharacteristic
55#11718121C2211yescharacteristic
56#11718121C2211yescharacteristic
57#11718121C2211yescharacteristic
58#11718121C2211yescharacteristic
59#11718121C2211yescharacteristic
60C23×C4×C13—21no—
61C23×C4×C13C2311yescharacteristic
62C2×C13×M16—22no—
63C2×C13×M16—22no—
64C2×C13×M16C2×C414yes—
65C22×C8×C13C2311yescharacteristic
66C22×C8×C13C2311yescharacteristic
67C13×#117—22no—
68C13×#117C2×C414yes—
69C2×C13×C4⋊C4—21no—
70C2×C13×C4⋊C4—22no—
71C2×C13×C4⋊C4C2×C414yes—
72C2×C13×C4⋊C4C2311yescharacteristic
73C2×C13×C4⋊C4C2311yescharacteristic
74C2×C13×C4⋊C4C2311yescharacteristic
75C2×C13×C4⋊C4C2311yescharacteristic
76C2×C13×C22⋊C4—21no—
77C2×C42×C13—21no—
78C2×C22⋊Dic13—21no—
79C2×C22⋊Dic13—21no—
80C2×C22⋊Dic13D411yescharacteristic
81C2×C22⋊Dic13D411yescharacteristic
82#88180—44no—
83C2×C4⋊Dic13—21no—
84C2×C4⋊Dic13—21no—
85C2×C4⋊Dic13—21no—
86C2×C4⋊Dic13—21no—
87C2×C4⋊Dic13C2311yescharacteristic
88C2×C4⋊Dic13C2311yescharacteristic
89C2×C4⋊Dic13C2311yescharacteristic
90C2×C4⋊Dic13C2311yescharacteristic
91C2×C4×Dic13—21no—
92C2×C4×Dic13—21no—
93C2×C4×Dic13—21no—
94C2×C4×Dic13—21no—
95C2×C4×Dic13C2311yescharacteristic
96C2×C4×Dic13C2311yescharacteristic
97C22×#1055C2311yescharacteristic
98C22×#1055C2311yescharacteristic
99C13×#107C2×C414yes—
100C13×#107C2×C414yes—
101C13×#106C2×C414yes—
102C13×#106C2×C414yes—
103#88071—21no—
104#88071—21no—
105#88071—21no—
106#88071—21no—
107#88071—21no—
108#88071—21no—
109#88071—21no—
110#88071—21no—
111#88071—21no—
112#88071—21no—
113#88071D411yescharacteristic
114#88071D411yescharacteristic
115C22⋊#1055—22no—
116#88058—22no—
117#88057—22no—
118#88055—22no—
119C8×Dic13—22no—
120#22624—131nomaximal, Sylow
121C24×C13C2×D411yescharacteristic
122C22×C4×C13—21no—
123C22×C4×C13—21no—
124C22×C4×C13—21no—
125C22×C4×C13—21no—
126C22×C4×C13—24no—
127C22×C4×C13—21no—
128C22×C4×C13—21no—
129C22×C4×C13—21no—
130C22×C4×C13—21no—
131C22×C4×C13—42no—
132C22×C4×C13—42no—
133C22×C4×C13C22×C412yes—
134C22×C4×C13C2×D411yescharacteristic
135C22×C4×C13#4112yes—
136C22×C4×C13C2411yescharacteristic
137C22×Dic13—21no—
138C22×Dic13—21no—
139C22×Dic13—21no—
140C22×Dic13—21no—
141C22×Dic13C2×D411yescharacteristic
142C22×Dic13C2×D411yescharacteristic
143C22×Dic13C2×D411yescharacteristic
144C22×Dic13C2×D411yescharacteristic
145C13×M16—28no—
146C2×C8×C13—21no—
147C2×C8×C13—21no—
148C2×C8×C13C22×C412yes—
149C2×C8×C13C22×C412yes—
150C2×C8×C13#4112yes—
151C2×C8×C13#4112yes—
152C13×C4⋊C4—24no—
153C13×C4⋊C4—21no—
154C13×C4⋊C4—21no—
155C13×C4⋊C4—21no—
156C13×C4⋊C4—21no—
157C13×C4⋊C4C22×C412yes—
158C13×C4⋊C4C22×C412yes—
159C13×C4⋊C4C22×C412yes—
160C13×C4⋊C4C22×C412yes—
161C13×C22⋊C4—24no—
162C42×C13—24no—
163C22⋊Dic13—22no—
164C22⋊Dic13—41no—
165C22⋊Dic13—42no—
166C22⋊Dic13—81no—
167C4⋊Dic13—21no—
168C4⋊Dic13—21no—
169C4⋊Dic13—21no—
170C4⋊Dic13—21no—
171C4⋊Dic13—21no—
172C4⋊Dic13—21no—
173C4⋊Dic13—21no—
174C4⋊Dic13—21no—
175C4⋊Dic13—42no—
176C4⋊Dic13—42no—
177C4⋊Dic13—81no—
178C4⋊Dic13—81no—
179C4×Dic13—21no—
180C4×Dic13—21no—
181C4×Dic13—21no—
182C4×Dic13—21no—
183C4×Dic13—42no—
184C4×Dic13—42no—
185C4×Dic13—42no—
186C4×Dic13—42no—
187C2×#1055—21no—
188C2×#1055—21no—
189C2×#1055—21no—
190C2×#1055—21no—
191C2×#1055—21no—
192C2×#1055—21no—
193C2×#428D1311yescharacteristic
194#2346—131no—
195#2344—131no—
196#1944—131no—
197#1941—131no—
198#1921—131no—
199#1920—131no—
200#1919—131no—
201#1918—131no—
202#1825—131no—
203#1824—131no—
204#1809—131no—
205#1776—131no—
206C4×#107—131no—
207#1753—131no—
208C23×C13—21no—
209C23×C13—21no—
210C23×C13—21no—
211C23×C13—21no—
212C23×C13—21no—
213C23×C13—21no—
214C23×C13C4×D412yes—
215C23×C13C22×D411yescharacteristic
216C2×C4×C13—21no—
217C2×C4×C13—21no—
218C2×C4×C13—22no—
219C2×C4×C13—22no—
220C2×C4×C13—22no—
221C2×C4×C13—22no—
222C2×C4×C13—22no—
223C2×C4×C13—22no—
224C2×C4×C13—22no—
225C2×C4×C13—22no—
226C2×C4×C13—48no—
227C2×C4×C13—41no—
228C2×C4×C13—41no—
229C2×C4×C13—41no—
230C2×C4×C13—41no—
231C2×C4×C13—41no—
232C2×C4×C13—41no—
233C2×C4×C13—41no—
234C2×C4×C13—41no—
235C2×C4×C13C4×D412yes—
236C2×C4×C13C23×C411yescharacteristic
237C2×C4×C13C2×#4111yescharacteristic
238C2×C4×C13C2×#4111yescharacteristic
239C2×C4×C13C2×#4111yescharacteristic
240C2×C4×C132−1+411yescharacteristic
241C2×C4×C132−1+411yescharacteristic
242C2×Dic13—21no—
243C2×Dic13—21no—
244C2×Dic13—21no—
245C2×Dic13—21no—
246C2×Dic13—21no—
247C2×Dic13—21no—
248C2×Dic13—21no—
249C2×Dic13—21no—
250C2×Dic13—21no—
251C2×Dic13—21no—
252C2×Dic13—21no—
253C2×Dic13—41no—
254C2×Dic13—41no—
255C2×Dic13—41no—
256C2×Dic13—41no—
257C2×Dic13—41no—
258C2×Dic13—41no—
259C2×Dic13—41no—
260C2×Dic13—41no—
261C2×Dic13—41no—
262C2×Dic13—41no—
263C2×Dic13—41no—
264C2×Dic13—41no—
265C2×Dic13AGL(1,ℤ/8)12yes—
266C8×C13—22no—
267C8×C13#11714yescomplemented
268#1055—41no—
269#1055—41no—
270C22×M16C2×D1311yescharacteristic
271C2×#117—261no—
272C2×#117C2×D1311yescharacteristic
273C22×C4⋊C4C2×D1311yescharacteristic
274#428Dic1318yescomplemented
275C2×#107—131no—
276C2×#107C2×D1311yescharacteristic
277C2×#107C2×D1311yescharacteristic
278C2×#106—131no—
279C2×#106C2×D1311yescharacteristic
280C2×#106C2×D1311yescharacteristic
281C2×C22⋊C8—131no—
282C2×#97—131no—
283C2×C4×C8—131no—
284#389—131no—
285#389—131no—
286#389—131no—
287#389—131no—
288#388—261no—
289#388—261no—
290#387—261no—
291#387—261no—
292#382—261no—
293#381—261no—
294#380—131no—
295#380—131no—
296C4×C4⋊C4—131no—
297C4×C4⋊C4—131no—
298C4×C4⋊C4—131no—
299C4×C4⋊C4—131no—
300C4×C22⋊C4—131no—
301#376—261no—
302#340—131no—
303#340—131no—
304#340—131no—
305#340—131no—
306#340—131no—
307#340—131no—
308#340—131no—
309#340—131no—
310#340—131no—
311#340—131no—
312#336—131no—
313#336—131no—
314C22×C13—22no—
315C22×C13—22no—
316C22×C13—22no—
317C22×C13—41no—
318C22×C13—41no—
319C22×C13—41no—
320C22×C13—41no—
321C22×C13C2×C4×D411yescharacteristic
322C22×C13#54711yescharacteristic
323C22×C13#54711yescharacteristic
324C22×C13C2×AGL(1,ℤ/8)11yescharacteristic
325C22×C13#57511yescharacteristic
326C22×C13#57711yescharacteristic
327C22×C13#57811yescharacteristic
328C4×C13—22no—
329C4×C13—42no—
330C4×C13—42no—
331C4×C13—42no—
332C4×C13—42no—
333C4×C13C2×#11711yescharacteristic
334C4×C13#51911yescharacteristic
335C4×C13#55211yescharacteristic
336C4×C13#55211yescharacteristic
337Dic13—22no—
338Dic13—41no—
339Dic13—41no—
340Dic13—41no—
341Dic13—81no—
342Dic13—81no—
343Dic13—81no—
344Dic13—81no—
345C23×C4—21no—
346C23×C4C22×D1311yescharacteristic
347C2×M16—22no—
348C2×M16—22no—
349C2×M16C2×Dic1314yes—
350C22×C8—131no—
351C22×C8—131no—
352C22×C8C22×D1311yescharacteristic
353C22×C8C22×D1311yescharacteristic
354#117—22no—
355#117—524no—
356#117C2×Dic1314yes—
357C2×C4⋊C4—21no—
358C2×C4⋊C4—22no—
359C2×C4⋊C4—131no—
360C2×C4⋊C4—131no—
361C2×C4⋊C4—131no—
362C2×C4⋊C4—131no—
363C2×C4⋊C4—261no—
364C2×C4⋊C4—261no—
365C2×C4⋊C4—261no—
366C2×C4⋊C4—261no—
367C2×C4⋊C4C2×Dic1314yes—
368C2×C4⋊C4C22×D1311yescharacteristic
369C2×C4⋊C4C22×D1311yescharacteristic
370C2×C4⋊C4C22×D1311yescharacteristic
371C2×C4⋊C4C22×D1311yescharacteristic
372C2×C22⋊C4—21no—
373C2×C22⋊C4—131no—
374C2×C22⋊C4—131no—
375C2×C22⋊C4—261no—
376C2×C22⋊C4—261no—
377C2×C42—21no—
378C2×C42—131no—
379C2×C42—131no—
380C2×C42—261no—
381C2×C42—261no—
382C2×C42—261no—
383C2×C42—261no—
384#107—262no—
385#107C2×Dic1314yes—
386#107C2×Dic1314yes—
387#106—262no—
388#106C2×Dic1314yes—
389#106C2×Dic1314yes—
390C22⋊C8—262no—
391#97—262no—
392C4×C8—262no—
393#95—131no—
394#95—131no—
395#95—261no—
396#95—261no—
397#95—261no—
398#95—261no—
399#95—261no—
400#95—261no—
401#95—261no—
402#95—261no—
403#95—261no—
404#95—261no—
405C2×C13—42no—
406C2×C13#234411yescharacteristic
407C2×C13C4×AGL(1,ℤ/8)11yescharacteristic
408C2×C13#294711yescharacteristic
409C2×C13#334311yescharacteristic
410C2×C13#334811yescharacteristic
411C2×C13#335211yescharacteristic
412C2×C13#335411yescharacteristic
413C24D4×D1311yescharacteristic
414C22×C4—21no—
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—24no—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—21no—
423C22×C4—42no—
424C22×C4—42no—
425C22×C4—131no—
426C22×C4—131no—
427C22×C4—131no—
428C22×C4—131no—
429C22×C4—261no—
430C22×C4—261no—
431C22×C4—261no—
432C22×C4—261no—
433C22×C4D4×D1311yescharacteristic
434C22×C4#612912yes—
435C22×C4C22×Dic1312yes—
436C22×C4C23×D1311yescharacteristic
437M16—28no—
438C2×C8—21no—
439C2×C8—21no—
440C2×C8—261no—
441C2×C8—261no—
442C2×C8—261no—
443C2×C8—261no—
444C2×C8—261no—
445C2×C8—261no—
446C2×C8#612712yes—
447C2×C8#612712yes—
448C2×C8C22×Dic1312yes—
449C2×C8C22×Dic1312yes—
450C4⋊C4—24no—
451C4⋊C4—21no—
452C4⋊C4—21no—
453C4⋊C4—21no—
454C4⋊C4—21no—
455C4⋊C4—261no—
456C4⋊C4—261no—
457C4⋊C4—261no—
458C4⋊C4—261no—
459C4⋊C4—261no—
460C4⋊C4—261no—
461C4⋊C4—261no—
462C4⋊C4—261no—
463C4⋊C4—522no—
464C4⋊C4—522no—
465C4⋊C4—1041no—
466C4⋊C4—1041no—
467C4⋊C4C22×Dic1312yes—
468C4⋊C4C22×Dic1312yes—
469C4⋊C4C22×Dic1312yes—
470C4⋊C4C22×Dic1312yes—
471C22⋊C4—24no—
472C22⋊C4—262no—
473C22⋊C4—521no—
474C22⋊C4—522no—
475C22⋊C4—1041no—
476C42—24no—
477C42—261no—
478C42—261no—
479C42—261no—
480C42—261no—
481C42—522no—
482C42—522no—
483C42—522no—
484C42—522no—
485C13#2262411yescharacteristic, complemented, Sylow
486C23—21no—
487C23—21no—
488C23—21no—
489C23—21no—
490C23—21no—
491C23—21no—
492C23D4×Dic1312yes—
493C23C2×D4×D1311yescharacteristic
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—48no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—132no—
514C2×C4—261no—
515C2×C4—261no—
516C2×C4—261no—
517C2×C4—261no—
518C2×C4—261no—
519C2×C4—261no—
520C2×C4—261no—
521C2×C4—261no—
522C2×C4—261no—
523C2×C4—261no—
524C2×C4—261no—
525C2×C4—521no—
526C2×C4—521no—
527C2×C4—521no—
528C2×C4—521no—
529C2×C4—521no—
530C2×C4—521no—
531C2×C4—521no—
532C2×C4—521no—
533C2×C4—521no—
534C2×C4—521no—
535C2×C4—521no—
536C2×C4—521no—
537C2×C4D4×Dic1312yes—
538C2×C4C2×#612711yescharacteristic
539C2×C4C2×#612711yescharacteristic
540C2×C4C2×#612911yescharacteristic
541C2×C4#8825411yescharacteristic
542C2×C4#8825411yescharacteristic
543C2×C4C23×Dic1311yescharacteristic
544C8—22no—
545C8—521no—
546C8—521no—
547C8#8818014yescomplemented
548C22—22no—
549C22—22no—
550C22—22no—
551C22—41no—
552C22—41no—
553C22—41no—
554C22—41no—
555C22#1171921211yescharacteristic
556C22#1171921211yescharacteristic
557C22D13×AGL(1,ℤ/8)11yescharacteristic
558C22#1171947811yescharacteristic
559C22#1171948311yescharacteristic
560C22#1171948411yescharacteristic
561C22C2×D4×Dic1311yescharacteristic
562C4—22no—
563C4—42no—
564C4—42no—
565C4—42no—
566C4—42no—
567C4—262no—
568C4—521no—
569C4—521no—
570C4—521no—
571C4—1041no—
572C4—1041no—
573C4—1041no—
574C4—1041no—
575C4#1171921811yescharacteristic
576C4#1171921811yescharacteristic
577C4C2×#8818011yescharacteristic
578C4#1171951711yescharacteristic
579C2—42no—
580C2#1311770511yescharacteristic
581C2AGL(1,ℤ/8)×Dic1311yescharacteristic
582C2#1312368011yescharacteristic
583C2#1312727711yescharacteristic
584C2#1312735211yescharacteristic
585C2#1312740711yescharacteristic
586C2#1312742711yescharacteristic
subgroupG/H
centerC23C2×D4×D13
commutatorC2×C4×C13C23×C4
FrattiniC22×C4C23×D13
FittingC2×C13×#428C2
socleC23×C13C22×D4
prime radicalC24×C13C2×D4
cosocleC22×C4×C13C24
hypercenterC2×#428D13
hypocenterC13#22624
LCSG ▷ C2×C4×C13 ▷ C2×C13 ▷ C13
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#428

Complements

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

NH ≅ G/Ntype
3. #13115580C2semidirect
4. #13115565C2semidirect
5. #13115430C2semidirect
6. #13115423C2semidirect
7. #13115416C2semidirect
8. #13115409C2semidirect
12. #13114735C2semidirect
13. #107×Dic13C2semidirect
20. C13×#428C4semidirect
267. C8×C13#117semidirect
274. #428Dic13semidirect
485. C13#22624semidirect
547. C8#88180semidirect

Maximal subgroups

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

subgroupindex
#131177212
#131177052
#131155802
#131155652
#131154302
#131154232
#131154162
#131154092
#131149312
#131149262
#131148692
#131147352
#107×Dic132
#131146172
C2×C13×#4282
#2262413