Finite Group Databasebeta

← #15865984

Subgroups — #15865984

Order 3328.

922 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
1#13117301C211yesmaximal, characteristic
2C4×Q8×Dic13C211yesmaximal, characteristic, complemented
3#13115465C211yesmaximal, characteristic
4#13115458C211yesmaximal, characteristic
5#13115430C211yesmaximal, characteristic, complemented
6#13115409C211yesmaximal, characteristic, complemented
7#13115193C211yesmaximal, characteristic
8#13115193C211yesmaximal, characteristic
9#13115106C211yesmaximal, characteristic, complemented
10#13115092C211yesmaximal, characteristic, complemented
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×Q8×Dic13C2211yescharacteristic
20C2×Q8×Dic13C2211yescharacteristic
21C2×#88180—21no—
22C13×#440C418yescomplemented
23C2×C13×#107C2211yescharacteristic
24C2×C13×#103C2211yescharacteristic
25C2×C13×#102C2211yescharacteristic
26C2×C13×#97C2211yescharacteristic
27C2×#88075C2211yescharacteristic
28C2×#88075C2211yescharacteristic
29#11718882C2211yescharacteristic
30#11718882C2211yescharacteristic
31C2×#88072C2211yescharacteristic
32C2×#88072C2211yescharacteristic
33C4×C22⋊Dic13—21no—
34C2×#88058C2211yescharacteristic
35C2×#88057C2211yescharacteristic
36#11718646C2211yescharacteristic
37#11718646C2211yescharacteristic
38C4⋊C4×Dic13—21no—
39C4⋊C4×Dic13—21no—
40C4⋊C4×Dic13C2211yescharacteristic
41C4⋊C4×Dic13C2211yescharacteristic
42#11718612—21no—
43#11718612—21no—
44C4×C4⋊Dic13—21no—
45C4×C4⋊Dic13C2211yescharacteristic
46C4×C4⋊Dic13C2211yescharacteristic
47C42×Dic13—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—
63C2×C13×SD16—22no—
64C2×C13×SD16—22no—
65C2×C13×SD16C2×C414yes—
66C22×C8×C13C2311yescharacteristic
67C22×C8×C13C2311yescharacteristic
68C4×Q8×C13—22no—
69C4×Q8×C13C2×C414yes—
70C4×D4×C13—22no—
71C4×D4×C13C2×C414yes—
72C2×C13×C4⋊C4—21no—
73C2×C13×C4⋊C4C2311yescharacteristic
74C2×C13×C4⋊C4C2311yescharacteristic
75C2×C13×C22⋊C4—21no—
76C2×C42×C13—21no—
77C2×C42×C13C2311yescharacteristic
78C2×C22⋊Dic13—21no—
79C2×C22⋊Dic13—21no—
80C2×C22⋊Dic13—21no—
81C2×C22⋊Dic13—21no—
82Q8×Dic13—22no—
83Q8×Dic13—22no—
84Q8×Dic13—22no—
85Q8×Dic13—22no—
86#88180—44no—
87C2×C4⋊Dic13—21no—
88C2×C4⋊Dic13—21no—
89C2×C4⋊Dic13—21no—
90C2×C4⋊Dic13—21no—
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×Dic13—21no—
100C2×C4×Dic13—21no—
101C2×C4×Dic13D411yescharacteristic
102C2×C4×Dic13D411yescharacteristic
103C2×C4×Dic13D411yescharacteristic
104C2×C4×Dic13D411yescharacteristic
105C2×C4×Dic13C2311yescharacteristic
106C2×C4×Dic13C2311yescharacteristic
107C22×#1055C2311yescharacteristic
108C22×#1055C2311yescharacteristic
109C13×#107C2×C414yes—
110C13×#103C2×C414yes—
111C13×#102C2×C414yes—
112C13×#97C2×C414yes—
113#88075—22no—
114#88075—22no—
115#88072—22no—
116#88072—22no—
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#21124—131nomaximal, Sylow
127C24×C13—21no—
128C2×Q8×C13—21no—
129C2×Q8×C13—21no—
130C2×Q8×C13—21no—
131C2×Q8×C13C22×C412yes—
132C2×Q8×C13#4112yes—
133C2×Q8×C13#4112yes—
134C2×D4×C13—21no—
135C2×D4×C13—21no—
136C2×D4×C13—21no—
137C2×D4×C13—21no—
138C2×D4×C13—21no—
139C2×D4×C13C22×C412yes—
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—21no—
150C22×Dic13—21no—
151C22×Dic13—21no—
152C22×Dic13—21no—
153C22×Dic13C2×D411yescharacteristic
154C22×Dic13C2×D411yescharacteristic
155C22×Dic13C2×D411yescharacteristic
156C22×Dic13C2×D411yescharacteristic
157C13×SD16—28no—
158C2×C8×C13—21no—
159C2×C8×C13—21no—
160C2×C8×C13—21no—
161C2×C8×C13—21no—
162C2×C8×C13C22×C412yes—
163C2×C8×C13C22×C412yes—
164C13×C4⋊C4—24no—
165C13×C4⋊C4—21no—
166C13×C4⋊C4—21no—
167C13×C4⋊C4C22×C412yes—
168C13×C4⋊C4C22×C412yes—
169C13×C22⋊C4—24no—
170C42×C13—24no—
171C42×C13C22×C412yes—
172C42×C13#4112yes—
173C22⋊Dic13—42no—
174C22⋊Dic13—42no—
175C22⋊Dic13—42no—
176C22⋊Dic13—42no—
177C4⋊Dic13—24no—
178C4⋊Dic13—24no—
179C4⋊Dic13—21no—
180C4⋊Dic13—21no—
181C4⋊Dic13—21no—
182C4⋊Dic13—21no—
183C4⋊Dic13—21no—
184C4⋊Dic13—21no—
185C4⋊Dic13—42no—
186C4⋊Dic13—42no—
187C4⋊Dic13#4112yes—
188C4⋊Dic13#4112yes—
189C4×Dic13—24no—
190C4×Dic13—24no—
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#2293—131no—
211C42×Q8—131no—
212#1926—131no—
213#1925—131no—
214#1921—131no—
215#1918—131no—
216#1880—131no—
217#1880—131no—
218#1865—131no—
219#1863—131no—
220#1776—131no—
221#1763—131no—
222#1762—131no—
223#1745—131no—
224C23×C13—22no—
225C23×C13—41no—
226C23×C13—41no—
227C23×C13—41no—
228C23×C13—41no—
229C23×C13—41no—
230C23×C13—41no—
231C23×C13C22×D411yescharacteristic
232Q8×C13—22no—
233Q8×C13—22no—
234Q8×C13—22no—
235Q8×C13#11714yescomplemented
236D4×C13—22no—
237D4×C13—22no—
238D4×C13—41no—
239D4×C13—41no—
240C2×C4×C13—22no—
241C2×C4×C13—22no—
242C2×C4×C13—22no—
243C2×C4×C13—21no—
244C2×C4×C13—21no—
245C2×C4×C13—21no—
246C2×C4×C13—21no—
247C2×C4×C13—22no—
248C2×C4×C13—22no—
249C2×C4×C13—48no—
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—41no—
275C2×Dic13—41no—
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×Dic13AGL(1,ℤ/8)12yes—
285C2×Dic13#13712yes—
286C8×C13—22no—
287C8×C13—22no—
288#1055—41no—
289#1055—41no—
290C22×SD16C2×D1311yescharacteristic
291C2×C4×Q8—131no—
292C2×C4×Q8—131no—
293C2×C4×Q8C2×D1311yescharacteristic
294C2×C4×D4C2×D1311yescharacteristic
295C2×#117—261no—
296#440Dic1318yescomplemented
297C2×#107—131no—
298C2×#107C2×D1311yescharacteristic
299C2×#106—131no—
300C2×#103—131no—
301C2×#103—131no—
302C2×#103C2×D1311yescharacteristic
303C2×#102—131no—
304C2×#102—131no—
305C2×#102C2×D1311yescharacteristic
306C2×#97—131no—
307C2×#97C2×D1311yescharacteristic
308#390—131no—
309#390—131no—
310#389—131no—
311#389—131no—
312#388—261no—
313#388—261no—
314C4×C4⋊C4—131no—
315C4×C4⋊C4—131no—
316C4×C4⋊C4—131no—
317C4×C4⋊C4—131no—
318C4×C4⋊C4—261no—
319C4×C4⋊C4—261no—
320C4×C4⋊C4—261no—
321C4×C22⋊C4—261no—
322C43—131no—
323C43—261no—
324#340—131no—
325#340—131no—
326#340—131no—
327#340—131no—
328#340—131no—
329#340—131no—
330#336—131no—
331#336—131no—
332#336—131no—
333#336—131no—
334C22×C13—42no—
335C22×C13—42no—
336C22×C13—42no—
337C22×C13—81no—
338C22×C13—81no—
339C22×C13—81no—
340C22×C13—81no—
341C22×C13C2×C4×D411yescharacteristic
342C22×C13#52511yescharacteristic
343C22×C13#52511yescharacteristic
344C22×C13C2×AGL(1,ℤ/8)11yescharacteristic
345C22×C13C2×#13711yescharacteristic
346C22×C13#57511yescharacteristic
347C22×C13#57511yescharacteristic
348C4×C13—22no—
349C4×C13—22no—
350C4×C13—24no—
351C4×C13—42no—
352C4×C13—42no—
353C4×C13C2×#11711yescharacteristic
354C4×C13C4×#4111yescharacteristic
355C4×C13#52911yescharacteristic
356C4×C13#52911yescharacteristic
357Dic13—22no—
358Dic13—22no—
359Dic13—41no—
360Dic13—42no—
361Dic13—41no—
362Dic13—42no—
363Dic13—81no—
364Dic13—81no—
365C22×Q8C22×D1311yescharacteristic
366C22×D4C22×D1311yescharacteristic
367C23×C4—21no—
368C2×SD16—22no—
369C2×SD16—22no—
370C2×SD16C2×Dic1314yes—
371C22×C8—131no—
372C22×C8—131no—
373C22×C8C22×D1311yescharacteristic
374C22×C8C22×D1311yescharacteristic
375C4×Q8—22no—
376C4×Q8—262no—
377C4×Q8—262no—
378C4×Q8—262no—
379C4×Q8—262no—
380C4×Q8C2×Dic1314yes—
381C4×D4—22no—
382C4×D4C2×Dic1314yes—
383#117—524no—
384C2×C4⋊C4—21no—
385C2×C4⋊C4—131no—
386C2×C4⋊C4—131no—
387C2×C4⋊C4—131no—
388C2×C4⋊C4—131no—
389C2×C4⋊C4—261no—
390C2×C4⋊C4—261no—
391C2×C4⋊C4—261no—
392C2×C4⋊C4—261no—
393C2×C4⋊C4C22×D1311yescharacteristic
394C2×C4⋊C4C22×D1311yescharacteristic
395C2×C22⋊C4—21no—
396C2×C22⋊C4—261no—
397C2×C22⋊C4—261no—
398C2×C22⋊C4—261no—
399C2×C22⋊C4—261no—
400C2×C42—21no—
401C2×C42—131no—
402C2×C42—131no—
403C2×C42—131no—
404C2×C42—131no—
405C2×C42—131no—
406C2×C42—131no—
407C2×C42—261no—
408C2×C42—261no—
409C2×C42—261no—
410C2×C42—261no—
411C2×C42—261no—
412C2×C42—261no—
413C2×C42C22×D1311yescharacteristic
414#107—262no—
415#107C2×Dic1314yes—
416#106—262no—
417#103—262no—
418#103—262no—
419#103C2×Dic1314yes—
420#102—262no—
421#102—262no—
422#102C2×Dic1314yes—
423#97—262no—
424#97C2×Dic1314yes—
425#95—261no—
426#95—261no—
427#95—261no—
428#95—261no—
429#95—261no—
430#95—261no—
431C2×C13—82no—
432C2×C13#229011yescharacteristic
433C2×C13C4×AGL(1,ℤ/8)11yescharacteristic
434C2×C13C4×#13711yescharacteristic
435C2×C13#311111yescharacteristic
436C2×C13#311211yescharacteristic
437C2×C13#311511yescharacteristic
438C2×C13#311711yescharacteristic
439C24—21no—
440C2×Q8—21no—
441C2×Q8—21no—
442C2×Q8—21no—
443C2×Q8#612712yes—
444C2×Q8#612712yes—
445C2×Q8C22×Dic1312yes—
446C2×D4—21no—
447C2×D4—21no—
448C2×D4—21no—
449C2×D4—21no—
450C2×D4—21no—
451C2×D4C22×Dic1312yes—
452C22×C4—21no—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—24no—
456C22×C4—42no—
457C22×C4—42no—
458C22×C4—131no—
459C22×C4—131no—
460C22×C4—131no—
461C22×C4—131no—
462C22×C4—261no—
463C22×C4—261no—
464C22×C4—261no—
465C22×C4—261no—
466C22×C4D4×D1311yescharacteristic
467C22×C4D4×D1311yescharacteristic
468C22×C4C23×D1311yescharacteristic
469SD16—28no—
470C2×C8—21no—
471C2×C8—21no—
472C2×C8—21no—
473C2×C8—21no—
474C2×C8—261no—
475C2×C8—261no—
476C2×C8—261no—
477C2×C8—261no—
478C2×C8—261no—
479C2×C8—261no—
480C2×C8C22×Dic1312yes—
481C2×C8C22×Dic1312yes—
482C4⋊C4—24no—
483C4⋊C4—21no—
484C4⋊C4—21no—
485C4⋊C4—132no—
486C4⋊C4—132no—
487C4⋊C4—264no—
488C4⋊C4—264no—
489C4⋊C4—261no—
490C4⋊C4—261no—
491C4⋊C4—261no—
492C4⋊C4—261no—
493C4⋊C4—261no—
494C4⋊C4—261no—
495C4⋊C4—522no—
496C4⋊C4—522no—
497C4⋊C4C22×Dic1312yes—
498C4⋊C4C22×Dic1312yes—
499C22⋊C4—24no—
500C22⋊C4—522no—
501C22⋊C4—522no—
502C22⋊C4—522no—
503C22⋊C4—522no—
504C42—24no—
505C42—262no—
506C42—262no—
507C42—264no—
508C42—264no—
509C42—261no—
510C42—261no—
511C42—261no—
512C42—261no—
513C42—521no—
514C42—521no—
515C42—522no—
516C42—522no—
517C42—522no—
518C42—522no—
519C42#612912yes—
520C42C22×Dic1312yes—
521C13#2112411yescharacteristic, complemented, Sylow
522C23—22no—
523C23—41no—
524C23—41no—
525C23—41no—
526C23—41no—
527C23—41no—
528C23—41no—
529C23C2×D4×D1311yescharacteristic
530Q8—22no—
531Q8—22no—
532Q8—22no—
533Q8#8818014yescomplemented
534D4—22no—
535D4—22no—
536D4—41no—
537D4—41no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—22no—
543C2×C4—22no—
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—132no—
553C2×C4—132no—
554C2×C4—261no—
555C2×C4—261no—
556C2×C4—261no—
557C2×C4—261no—
558C2×C4—261no—
559C2×C4—261no—
560C2×C4—261no—
561C2×C4—261no—
562C2×C4—262no—
563C2×C4—262no—
564C2×C4—261no—
565C2×C4—261no—
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×C4D4×Dic1312yes—
577C2×C4D4×Dic1312yes—
578C2×C4C2×#612711yescharacteristic
579C2×C4C2×#612711yescharacteristic
580C2×C4C2×#612911yescharacteristic
581C2×C4#41×D1311yescharacteristic
582C2×C4#41×D1311yescharacteristic
583C2×C4C23×Dic1311yescharacteristic
584C8—22no—
585C8—22no—
586C8—521no—
587C8—521no—
588C22—42no—
589C22—42no—
590C22—42no—
591C22—81no—
592C22—81no—
593C22—81no—
594C22—81no—
595C22#1171925211yescharacteristic
596C22#1171925211yescharacteristic
597C22D13×AGL(1,ℤ/8)11yescharacteristic
598C22#1171947811yescharacteristic
599C22D13×#13711yescharacteristic
600C22#1171948211yescharacteristic
601C22C2×D4×Dic1311yescharacteristic
602C4—22no—
603C4—22no—
604C4—24no—
605C4—42no—
606C4—42no—
607C4—262no—
608C4—262no—
609C4—521no—
610C4—522no—
611C4—521no—
612C4—522no—
613C4—1041no—
614C4—1041no—
615C4#1171924311yescharacteristic
616C4#1171924311yescharacteristic
617C4C2×#8818011yescharacteristic
618C4#41×Dic1311yescharacteristic
619C2—82no—
620C2#1311728111yescharacteristic
621C2AGL(1,ℤ/8)×Dic1311yescharacteristic
622C2#137×Dic1311yescharacteristic
623C2#1312494311yescharacteristic
624C2#1312495611yescharacteristic
625C2#1312498511yescharacteristic
626C2#1312500511yescharacteristic
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#21124
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
2. C4×Q8×Dic13C2semidirect
5. #13115430C2semidirect
6. #13115409C2semidirect
9. #13115106C2semidirect
10. #13115092C2semidirect
11. #13114733C2semidirect
12. #13114666C2semidirect
14. #13114584C2semidirect
22. C13×#440C4semidirect
235. Q8×C13#117semidirect
296. #440Dic13semidirect
521. C13#21124semidirect
533. Q8#88180semidirect

Maximal subgroups

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

subgroupindex
#131173012
C4×Q8×Dic132
#131154652
#131154582
#131154302
#131154092
#131151932
#131151062
#131150922
#131147332
#131146662
#131146612
#131145842
C2×C13×#4402
#2112413