Finite Group Databasebeta

← #15865939

Subgroups — #15865939

Order 3328.

966 conjugacy classes of subgroups.

The subgroup lattice has 632 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#13117314C211yesmaximal, characteristic, complemented
2C4×D4×Dic13C211yesmaximal, characteristic
3#13115479C211yesmaximal, characteristic, complemented
4#13115472C211yesmaximal, characteristic, complemented
5#13115430C211yesmaximal, characteristic, complemented
6#13115409C211yesmaximal, characteristic, complemented
7#13115172C211yesmaximal, characteristic
8#13115172C211yesmaximal, characteristic
9#13115099C211yesmaximal, characteristic
10#13115085C211yesmaximal, characteristic
11#13114733C211yesmaximal, characteristic, complemented
12#13114666C211yesmaximal, characteristic, complemented
13#13114661C211yesmaximal, characteristic
14#13114584C211yesmaximal, characteristic, complemented
15C2×C13×#440C211yesmaximal, characteristic
16C22×C13×SD16C2211yescharacteristic
17C2×C4×Q8×C13C2211yescharacteristic
18C2×C4×D4×C13C2211yescharacteristic
19C2×D4×Dic13C2211yescharacteristic
20C2×D4×Dic13C2211yescharacteristic
21C22×C4×Dic13—21no—
22C13×#440C418yescomplemented
23C2×C13×#107C2211yescharacteristic
24C2×C13×#103C2211yescharacteristic
25C2×C13×#102C2211yescharacteristic
26C2×C13×#97C2211yescharacteristic
27#11718891C2211yescharacteristic
28#11718891C2211yescharacteristic
29C2×#88075C2211yescharacteristic
30C2×#88075C2211yescharacteristic
31C2×#88072C2211yescharacteristic
32C2×#88072C2211yescharacteristic
33C4×C22⋊Dic13—21no—
34C2×#88058C2211yescharacteristic
35C2×#88057C2211yescharacteristic
36#11718646C2211yescharacteristic
37#11718646C2211yescharacteristic
38#11718642—21no—
39#11718642—21no—
40C4⋊C4×Dic13C2211yescharacteristic
41C4⋊C4×Dic13C2211yescharacteristic
42C22⋊C4×Dic13—21no—
43C22⋊C4×Dic13—21no—
44C4×C4⋊Dic13—21no—
45C4×C4⋊Dic13C2211yescharacteristic
46C4×C4⋊Dic13C2211yescharacteristic
47#11718593—21no—
48C42×Dic13C2211yescharacteristic
49C2×#88043C2211yescharacteristic
50#11718143C2211yescharacteristic
51#11718143C2211yescharacteristic
52#11718142C2211yescharacteristic
53#11718142C2211yescharacteristic
54#11718142C2211yescharacteristic
55#11718142C2211yescharacteristic
56#11718121C2211yescharacteristic
57#11718121C2211yescharacteristic
58#11718121C2211yescharacteristic
59#11718121C2211yescharacteristic
60C22×Q8×C13C2311yescharacteristic
61C22×D4×C13C2311yescharacteristic
62C23×C4×C13—21no—
63C23×Dic13—21no—
64C23×Dic13—21no—
65C2×C13×SD16—22no—
66C2×C13×SD16—22no—
67C2×C13×SD16C2×C414yes—
68C22×C8×C13C2311yescharacteristic
69C22×C8×C13C2311yescharacteristic
70C4×Q8×C13—22no—
71C4×Q8×C13C2×C414yescomplemented
72C4×D4×C13—22no—
73C4×D4×C13C2×C414yes—
74C2×C13×C4⋊C4—21no—
75C2×C13×C4⋊C4C2311yescharacteristic
76C2×C13×C4⋊C4C2311yescharacteristic
77C2×C13×C22⋊C4—21no—
78C2×C42×C13—21no—
79C2×C42×C13C2311yescharacteristic
80C2×C22⋊Dic13—21no—
81C2×C22⋊Dic13—21no—
82D4×Dic13—22no—
83D4×Dic13—22no—
84D4×Dic13—22no—
85D4×Dic13—22no—
86C2×C4⋊Dic13—21no—
87C2×C4⋊Dic13—21no—
88C2×C4⋊Dic13C2311yescharacteristic
89C2×C4⋊Dic13C2311yescharacteristic
90C2×C4⋊Dic13C2311yescharacteristic
91C2×C4⋊Dic13C2311yescharacteristic
92C2×C4×Dic13—21no—
93C2×C4×Dic13—21no—
94C2×C4×Dic13—21no—
95C2×C4×Dic13—21no—
96C2×C4×Dic13—44no—
97C2×C4×Dic13D411yescharacteristic
98C2×C4×Dic13D411yescharacteristic
99C2×C4×Dic13D411yescharacteristic
100C2×C4×Dic13D411yescharacteristic
101C2×C4×Dic13C2311yescharacteristic
102C2×C4×Dic13C2311yescharacteristic
103C22×#1055C2311yescharacteristic
104C22×#1055C2311yescharacteristic
105C13×#107C2×C414yescomplemented
106C13×#103C2×C414yescomplemented
107C13×#102C2×C414yes—
108C13×#97C2×C414yescomplemented
109#88075—22no—
110#88075—22no—
111#88072—22no—
112#88072—22no—
113#88071—21no—
114#88071—21no—
115#88071—21no—
116#88071—21no—
117#88071—21no—
118#88071—21no—
119#88071—21no—
120#88071—21no—
121#88071—21no—
122#88071—21no—
123#88058—22no—
124#88057—22no—
125#88043—22no—
126#21121—131nomaximal, Sylow
127C24×C13—21no—
128C2×Q8×C13—21no—
129C2×Q8×C13—21no—
130C2×Q8×C13—21no—
131C2×Q8×C13—21no—
132C2×Q8×C13—21no—
133C2×Q8×C13C22×C412yes—
134C2×D4×C13—21no—
135C2×D4×C13—21no—
136C2×D4×C13—21no—
137C2×D4×C13C22×C412yes—
138C2×D4×C13#4112yes—
139C2×D4×C13#4112yes—
140C22×C4×C13—21no—
141C22×C4×C13—21no—
142C22×C4×C13—21no—
143C22×C4×C13—24no—
144C22×C4×C13—42no—
145C22×C4×C13—42no—
146C22×C4×C13C2×D411yescharacteristic
147C22×C4×C13C2×D411yescharacteristic
148C22×C4×C13C2411yescharacteristic
149C22×Dic13—24no—
150C22×Dic13—24no—
151C22×Dic13—21no—
152C22×Dic13—21no—
153C22×Dic13—21no—
154C22×Dic13—21no—
155C22×Dic13—42no—
156C22×Dic13—42no—
157C22×Dic13—42no—
158C22×Dic13—42no—
159C22×Dic13C2×D411yescharacteristic
160C22×Dic13C2×D411yescharacteristic
161C22×Dic13C2×D411yescharacteristic
162C22×Dic13C2×D411yescharacteristic
163C13×SD16—28no—
164C2×C8×C13—21no—
165C2×C8×C13—21no—
166C2×C8×C13—21no—
167C2×C8×C13—21no—
168C2×C8×C13C22×C412yes—
169C2×C8×C13C22×C412yes—
170C13×C4⋊C4—24no—
171C13×C4⋊C4—21no—
172C13×C4⋊C4—21no—
173C13×C4⋊C4C22×C412yes—
174C13×C4⋊C4C22×C412yes—
175C13×C22⋊C4—24no—
176C42×C13—24no—
177C42×C13C22×C412yes—
178C42×C13#4112yes—
179C22⋊Dic13—24no—
180C22⋊Dic13—24no—
181C4⋊Dic13—21no—
182C4⋊Dic13—21no—
183C4⋊Dic13—21no—
184C4⋊Dic13—21no—
185C4⋊Dic13—21no—
186C4⋊Dic13—21no—
187C4⋊Dic13—42no—
188C4⋊Dic13—42no—
189C4⋊Dic13#4112yes—
190C4⋊Dic13#4112yes—
191C4×Dic13—21no—
192C4×Dic13—21no—
193C4×Dic13—21no—
194C4×Dic13—21no—
195C4×Dic13—22no—
196C4×Dic13—22no—
197C4×Dic13—41no—
198C4×Dic13—41no—
199C4×Dic13—42no—
200C4×Dic13—42no—
201C4×Dic13—42no—
202C4×Dic13—42no—
203C2×#1055—21no—
204C2×#1055—21no—
205C2×#1055—21no—
206C2×#1055—21no—
207C2×#1055—21no—
208C2×#1055—21no—
209C2×#440D1311yescharacteristic
210#2295—131no—
211C42×D4—131no—
212#1928—131no—
213#1927—131no—
214#1921—131no—
215#1918—131no—
216#1877—131no—
217#1877—131no—
218#1864—131no—
219#1862—131no—
220#1776—131no—
221#1763—131no—
222#1762—131no—
223#1745—131no—
224C23×C13—22no—
225C23×C13—22no—
226C23×C13—22no—
227C23×C13—41no—
228C23×C13—41no—
229C23×C13—41no—
230C23×C13—41no—
231C23×C13C22×D411yescharacteristic
232Q8×C13—22no—
233Q8×C13—22no—
234Q8×C13—41no—
235Q8×C13—41no—
236D4×C13—22no—
237D4×C13—22no—
238D4×C13—22no—
239D4×C13#11714yes—
240C2×C4×C13—21no—
241C2×C4×C13—21no—
242C2×C4×C13—21no—
243C2×C4×C13—21no—
244C2×C4×C13—22no—
245C2×C4×C13—22no—
246C2×C4×C13—22no—
247C2×C4×C13—48no—
248C2×C4×C13—41no—
249C2×C4×C13—41no—
250C2×C4×C13—41no—
251C2×C4×C13—41no—
252C2×C4×C13—41no—
253C2×C4×C13—41no—
254C2×C4×C13C4×D412yes—
255C2×C4×C13C4×D412yes—
256C2×C4×C13C23×C411yescharacteristic
257C2×C4×C13C2×#4111yescharacteristic
258C2×C4×C13C2×#4111yescharacteristic
259C2×C4×C13C2×#4111yescharacteristic
260C2×C4×C13C2×#4111yescharacteristic
261C2×C4×C13C2×#4111yescharacteristic
262C2×Dic13—21no—
263C2×Dic13—21no—
264C2×Dic13—21no—
265C2×Dic13—21no—
266C2×Dic13—21no—
267C2×Dic13—21no—
268C2×Dic13—22no—
269C2×Dic13—22no—
270C2×Dic13—21no—
271C2×Dic13—21no—
272C2×Dic13—21no—
273C2×Dic13—21no—
274C2×Dic13—48no—
275C2×Dic13—48no—
276C2×Dic13—41no—
277C2×Dic13—41no—
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×Dic13AGL(1,ℤ/8)12yes—
287C2×Dic13#13712yes—
288C8×C13—22no—
289C8×C13—22no—
290#1055—41no—
291#1055—41no—
292C22×SD16C2×D1311yescharacteristic
293C2×C4×Q8C2×D1311yescharacteristic
294C2×C4×D4—131no—
295C2×C4×D4—131no—
296C2×C4×D4C2×D1311yescharacteristic
297C22×C42—261no—
298#440Dic1318yescomplemented
299C2×#107—131no—
300C2×#107C2×D1311yescharacteristic
301C2×#106—131no—
302C2×#103—131no—
303C2×#103—131no—
304C2×#103C2×D1311yescharacteristic
305C2×#102—131no—
306C2×#102—131no—
307C2×#102C2×D1311yescharacteristic
308C2×#97—131no—
309C2×#97C2×D1311yescharacteristic
310#391—131no—
311#391—131no—
312#389—131no—
313#389—131no—
314#387—261no—
315#387—261no—
316C4×C4⋊C4—131no—
317C4×C4⋊C4—131no—
318C4×C4⋊C4—131no—
319C4×C4⋊C4—131no—
320C4×C4⋊C4—261no—
321C4×C22⋊C4—261no—
322C4×C22⋊C4—261no—
323C4×C22⋊C4—261no—
324#376—261no—
325C43—131no—
326#340—131no—
327#340—131no—
328#340—131no—
329#340—131no—
330#340—131no—
331#340—131no—
332#336—131no—
333#336—131no—
334#336—131no—
335#336—131no—
336C22×C13—24no—
337C22×C13—42no—
338C22×C13—42no—
339C22×C13—42no—
340C22×C13—42no—
341C22×C13—42no—
342C22×C13—42no—
343C22×C13C2×C4×D411yescharacteristic
344C22×C13#52511yescharacteristic
345C22×C13#52511yescharacteristic
346C22×C13C2×AGL(1,ℤ/8)11yescharacteristic
347C22×C13C2×#13711yescharacteristic
348C22×C13#57511yescharacteristic
349C22×C13#57511yescharacteristic
350C4×C13—22no—
351C4×C13—22no—
352C4×C13—42no—
353C4×C13—42no—
354C4×C13—42no—
355C4×C13C2×#11711yescharacteristic
356C4×C13C4×#4111yescharacteristic
357C4×C13#52911yescharacteristic
358C4×C13#52911yescharacteristic
359Dic13—22no—
360Dic13—22no—
361Dic13—42no—
362Dic13—41no—
363Dic13—42no—
364Dic13—41no—
365Dic13—81no—
366Dic13—81no—
367C22×Q8C22×D1311yescharacteristic
368C22×D4C22×D1311yescharacteristic
369C23×C4—21no—
370C23×C4—261no—
371C23×C4—261no—
372C2×SD16—22no—
373C2×SD16—22no—
374C2×SD16C2×Dic1314yes—
375C22×C8—131no—
376C22×C8—131no—
377C22×C8C22×D1311yescharacteristic
378C22×C8C22×D1311yescharacteristic
379C4×Q8—22no—
380C4×Q8C2×Dic1314yescomplemented
381C4×D4—22no—
382C4×D4—262no—
383C4×D4—262no—
384C4×D4—262no—
385C4×D4—262no—
386C4×D4C2×Dic1314yes—
387C2×C4⋊C4—21no—
388C2×C4⋊C4—131no—
389C2×C4⋊C4—131no—
390C2×C4⋊C4—131no—
391C2×C4⋊C4—131no—
392C2×C4⋊C4—261no—
393C2×C4⋊C4—261no—
394C2×C4⋊C4C22×D1311yescharacteristic
395C2×C4⋊C4C22×D1311yescharacteristic
396C2×C22⋊C4—21no—
397C2×C22⋊C4—261no—
398C2×C22⋊C4—261no—
399C2×C42—21no—
400C2×C42—131no—
401C2×C42—131no—
402C2×C42—131no—
403C2×C42—131no—
404C2×C42—131no—
405C2×C42—131no—
406C2×C42—261no—
407C2×C42—261no—
408C2×C42—261no—
409C2×C42—261no—
410C2×C42—524no—
411C2×C42C22×D1311yescharacteristic
412#107—262no—
413#107C2×Dic1314yescomplemented
414#106—262no—
415#103—262no—
416#103—262no—
417#103C2×Dic1314yescomplemented
418#102—262no—
419#102—262no—
420#102C2×Dic1314yes—
421#97—262no—
422#97C2×Dic1314yescomplemented
423#95—261no—
424#95—261no—
425#95—261no—
426#95—261no—
427#95—261no—
428#95—261no—
429#95—261no—
430#95—261no—
431#95—261no—
432#95—261no—
433C2×C13—44no—
434C2×C13#229011yescharacteristic
435C2×C13C4×AGL(1,ℤ/8)11yescharacteristic
436C2×C13C4×#13711yescharacteristic
437C2×C13#311011yescharacteristic
438C2×C13#311111yescharacteristic
439C2×C13#311411yescharacteristic
440C2×C13#311611yescharacteristic
441C24—21no—
442C2×Q8—21no—
443C2×Q8—21no—
444C2×Q8—21no—
445C2×Q8—21no—
446C2×Q8—21no—
447C2×Q8C22×Dic1312yes—
448C2×D4—21no—
449C2×D4—21no—
450C2×D4—21no—
451C2×D4#612712yes—
452C2×D4#612712yes—
453C2×D4C22×Dic1312yes—
454C22×C4—21no—
455C22×C4—21no—
456C22×C4—21no—
457C22×C4—24no—
458C22×C4—42no—
459C22×C4—42no—
460C22×C4—131no—
461C22×C4—131no—
462C22×C4—131no—
463C22×C4—131no—
464C22×C4—264no—
465C22×C4—264no—
466C22×C4—261no—
467C22×C4—261no—
468C22×C4—261no—
469C22×C4—261no—
470C22×C4—522no—
471C22×C4—522no—
472C22×C4—522no—
473C22×C4—522no—
474C22×C4D4×D1311yescharacteristic
475C22×C4D4×D1311yescharacteristic
476C22×C4C23×D1311yescharacteristic
477SD16—28no—
478C2×C8—21no—
479C2×C8—21no—
480C2×C8—21no—
481C2×C8—21no—
482C2×C8—261no—
483C2×C8—261no—
484C2×C8—261no—
485C2×C8—261no—
486C2×C8—261no—
487C2×C8—261no—
488C2×C8C22×Dic1312yes—
489C2×C8C22×Dic1312yes—
490C4⋊C4—24no—
491C4⋊C4—21no—
492C4⋊C4—21no—
493C4⋊C4—132no—
494C4⋊C4—132no—
495C4⋊C4—261no—
496C4⋊C4—261no—
497C4⋊C4—261no—
498C4⋊C4—261no—
499C4⋊C4—261no—
500C4⋊C4—261no—
501C4⋊C4—522no—
502C4⋊C4—522no—
503C4⋊C4C22×Dic1312yes—
504C4⋊C4C22×Dic1312yes—
505C22⋊C4—24no—
506C22⋊C4—264no—
507C22⋊C4—264no—
508C42—24no—
509C42—262no—
510C42—262no—
511C42—261no—
512C42—261no—
513C42—261no—
514C42—261no—
515C42—521no—
516C42—521no—
517C42—522no—
518C42—522no—
519C42—522no—
520C42—522no—
521C42#612912yes—
522C42C22×Dic1312yes—
523C13#2112111yescharacteristic, complemented, Sylow
524C23—22no—
525C23—22no—
526C23—22no—
527C23—41no—
528C23—41no—
529C23—41no—
530C23—41no—
531C23C2×D4×D1311yescharacteristic
532Q8—22no—
533Q8—22no—
534Q8—41no—
535Q8—41no—
536D4—22no—
537D4—22no—
538D4—22no—
539D4#8818014yes—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—48no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
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—261no—
565C2×C4—261no—
566C2×C4—262no—
567C2×C4—262no—
568C2×C4—528no—
569C2×C4—528no—
570C2×C4—521no—
571C2×C4—521no—
572C2×C4—521no—
573C2×C4—521no—
574C2×C4—521no—
575C2×C4—521no—
576C2×C4—521no—
577C2×C4—521no—
578C2×C4—521no—
579C2×C4—521no—
580C2×C4D4×Dic1312yes—
581C2×C4D4×Dic1312yes—
582C2×C4C2×#612711yescharacteristic
583C2×C4C2×#612711yescharacteristic
584C2×C4C2×#612911yescharacteristic
585C2×C4#41×D1311yescharacteristic
586C2×C4#41×D1311yescharacteristic
587C2×C4C23×Dic1311yescharacteristic
588C8—22no—
589C8—22no—
590C8—521no—
591C8—521no—
592C22—24no—
593C22—42no—
594C22—42no—
595C22—42no—
596C22—42no—
597C22—42no—
598C22—42no—
599C22#1171925211yescharacteristic
600C22#1171925211yescharacteristic
601C22D13×AGL(1,ℤ/8)11yescharacteristic
602C22#1171947811yescharacteristic
603C22D13×#13711yescharacteristic
604C22#1171948211yescharacteristic
605C22C2×D4×Dic1311yescharacteristic
606C4—22no—
607C4—22no—
608C4—42no—
609C4—42no—
610C4—42no—
611C4—262no—
612C4—262no—
613C4—522no—
614C4—521no—
615C4—522no—
616C4—521no—
617C4—1041no—
618C4—1041no—
619C4#1171924311yescharacteristic
620C4#1171924311yescharacteristic
621C4C2×#8818011yescharacteristic
622C4#41×Dic1311yescharacteristic
623C2—44no—
624C2#1311728111yescharacteristic
625C2AGL(1,ℤ/8)×Dic1311yescharacteristic
626C2#137×Dic1311yescharacteristic
627C2#1312493311yescharacteristic
628C2#1312494911yescharacteristic
629C2#1312497811yescharacteristic
630C2#1312499411yescharacteristic
subgroupG/H
centerC23C2×D4×D13
commutatorC2×C4×C13C23×C4
FrattiniC22×C4C23×D13
FittingC2×C13×#440C2
socleC23×C13C22×D4
prime radicalC22×D4×C13C23
cosocleC22×C4×C13C24
hypercenterC2×#440D13
hypocenterC13#21121
LCSG ▷ C2×C4×C13 ▷ C2×C13 ▷ C13
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#440

Complements

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

NH ≅ G/Ntype
1. #13117314C2semidirect
3. #13115479C2semidirect
4. #13115472C2semidirect
5. #13115430C2semidirect
6. #13115409C2semidirect
11. #13114733C2semidirect
12. #13114666C2semidirect
14. #13114584C2semidirect
22. C13×#440C4semidirect
71. C4×Q8×C13C2×C4semidirect
105. C13×#107C2×C4semidirect
106. C13×#103C2×C4semidirect
108. C13×#97C2×C4semidirect
298. #440Dic13semidirect
380. C4×Q8C2×Dic13semidirect
413. #107C2×Dic13semidirect
417. #103C2×Dic13semidirect
422. #97C2×Dic13semidirect
523. C13#21121semidirect

Maximal subgroups

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

subgroupindex
#131173142
C4×D4×Dic132
#131154792
#131154722
#131154302
#131154092
#131151722
#131150992
#131150852
#131147332
#131146662
#131146612
#131145842
C2×C13×#4402
#2112113