Finite Group Databasebeta

← #15880253

Subgroups — #15880253

Order 3328.

934 conjugacy classes of subgroups.

The subgroup lattice has 656 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#13117721C211yesmaximal, characteristic
2Q8×C22⋊Dic13C211yesmaximal, characteristic
3#13115528C211yesmaximal, characteristic, complemented
4#13115514C211yesmaximal, characteristic, complemented
5#13115479C211yesmaximal, characteristic, complemented
6#13115472C211yesmaximal, characteristic, complemented
7#13115106C211yesmaximal, characteristic, complemented
8#13115092C211yesmaximal, characteristic, complemented
9#13114931C211yesmaximal, characteristic
10#13114869C211yesmaximal, characteristic
11#13114771C211yesmaximal, characteristic
12#13114668C211yesmaximal, characteristic, complemented
13#103×Dic13C211yesmaximal, characteristic, complemented
14#13114617C211yesmaximal, characteristic
15C2×C13×#419C211yesmaximal, characteristic
16C23×Q8×C13C2211yescharacteristic
17C22×C13×M16C2211yescharacteristic
18C2×C13×#117C2211yescharacteristic
19C2×Q8×Dic13C2211yescharacteristic
20C2×Q8×Dic13C2211yescharacteristic
21C13×#419C418yescomplemented
22C2×C13×#103C2211yescharacteristic
23C2×C13×#103C2211yescharacteristic
24C2×C13×#103C2211yescharacteristic
25C2×C13×#103C2211yescharacteristic
26#11718891—21no—
27#11718891—21no—
28#11718891C2211yescharacteristic
29#11718891C2211yescharacteristic
30C2×#88075C2211yescharacteristic
31C2×#88075C2211yescharacteristic
32C2×#88075C2211yescharacteristic
33C2×#88075C2211yescharacteristic
34#11718873—21no—
35#11718873C2211yescharacteristic
36#11718873C2211yescharacteristic
37C4×C22⋊Dic13—21no—
38C4×C22⋊Dic13C2211yescharacteristic
39C2×C22⋊#1055C2211yescharacteristic
40C2×#88058C2211yescharacteristic
41C2×#88057C2211yescharacteristic
42C2×#88055C2211yescharacteristic
43C2×C8×Dic13C2211yescharacteristic
44#11718646C2211yescharacteristic
45#11718646C2211yescharacteristic
46C4⋊C4×Dic13C2211yescharacteristic
47C4⋊C4×Dic13C2211yescharacteristic
48#11718612—21no—
49#11718612—21no—
50#11718597—21no—
51#11718593—21no—
52#11718143C2211yescharacteristic
53#11718143C2211yescharacteristic
54#11718142C2211yescharacteristic
55#11718142C2211yescharacteristic
56#11718121C2211yescharacteristic
57#11718121C2211yescharacteristic
58#11718121C2211yescharacteristic
59#11718121C2211yescharacteristic
60C22×Q8×C13—21no—
61C22×Q8×C13—22no—
62C22×Q8×C13—22no—
63C22×Q8×C13—22no—
64C22×Q8×C13—41no—
65C22×Q8×C13—41no—
66C22×Q8×C13C2×C414yes—
67C22×Q8×C13C2311yescharacteristic
68C22×Q8×C13C2311yescharacteristic
69C23×C4×C13—21no—
70C23×C4×C13C2311yescharacteristic
71C2×C13×M16—41no—
72C2×C13×M16—41no—
73C2×C13×M16C2×C414yes—
74C22×C8×C13C2311yescharacteristic
75C22×C8×C13C2311yescharacteristic
76C13×#117—22no—
77C13×#117C2×C414yes—
78C2×C13×C4⋊C4C2311yescharacteristic
79C2×C13×C4⋊C4C2311yescharacteristic
80C2×C13×C22⋊C4—21no—
81C2×C42×C13—21no—
82C2×C22⋊Dic13—21no—
83C2×C22⋊Dic13—21no—
84C2×C22⋊Dic13D411yescharacteristic
85C2×C22⋊Dic13D411yescharacteristic
86Q8×Dic13—22no—
87Q8×Dic13—22no—
88Q8×Dic13—41no—
89Q8×Dic13—41no—
90C2×C4⋊Dic13—21no—
91C2×C4⋊Dic13—21no—
92C2×C4⋊Dic13—21no—
93C2×C4⋊Dic13—21no—
94C2×C4⋊Dic13C2311yescharacteristic
95C2×C4⋊Dic13C2311yescharacteristic
96C2×C4⋊Dic13C2311yescharacteristic
97C2×C4⋊Dic13C2311yescharacteristic
98C2×C4×Dic13—21no—
99C2×C4×Dic13—21no—
100C2×C4×Dic13—21no—
101C2×C4×Dic13—21no—
102C2×C4×Dic13C2311yescharacteristic
103C2×C4×Dic13C2311yescharacteristic
104C22×#1055C2311yescharacteristic
105C22×#1055C2311yescharacteristic
106C13×#103C2×C414yes—
107C13×#103C2×C414yes—
108C13×#103C2×C414yes—
109C13×#103C2×C414yes—
110#88075—22no—
111#88075—22no—
112#88075—41no—
113#88075—41no—
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#88071—21no—
124#88071D411yescharacteristic
125#88071D411yescharacteristic
126C22⋊#1055—22no—
127#88058—41no—
128#88057—41no—
129#88055—41no—
130C8×Dic13—41no—
131#22585—131nomaximal, Sylow
132C24×C13C2×D411yescharacteristic
133C2×Q8×C13—24no—
134C2×Q8×C13—21no—
135C2×Q8×C13—21no—
136C2×Q8×C13—21no—
137C2×Q8×C13—21no—
138C2×Q8×C13—28no—
139C2×Q8×C13—21no—
140C2×Q8×C13—21no—
141C2×Q8×C13—21no—
142C2×Q8×C13—21no—
143C2×Q8×C13—42no—
144C2×Q8×C13—42no—
145C2×Q8×C13—44no—
146C2×Q8×C13—44no—
147C2×Q8×C13—44no—
148C2×Q8×C13C22×C412yes—
149C2×Q8×C13C22×C412yes—
150C2×Q8×C13#4112yes—
151C2×Q8×C13#4112yes—
152C22×C4×C13—21no—
153C22×C4×C13—21no—
154C22×C4×C13—21no—
155C22×C4×C13—21no—
156C22×C4×C13—21no—
157C22×C4×C13—24no—
158C22×C4×C13—21no—
159C22×C4×C13—21no—
160C22×C4×C13—21no—
161C22×C4×C13—42no—
162C22×C4×C13—42no—
163C22×C4×C13C22×C412yes—
164C22×C4×C13C2×D411yescharacteristic
165C22×C4×C13#4112yes—
166C22×C4×C13C2411yescharacteristic
167C22×Dic13—21no—
168C22×Dic13—21no—
169C22×Dic13—21no—
170C22×Dic13—21no—
171C22×Dic13C2×D411yescharacteristic
172C22×Dic13C2×D411yescharacteristic
173C22×Dic13C2×D411yescharacteristic
174C22×Dic13C2×D411yescharacteristic
175C13×M16—44no—
176C2×C8×C13—21no—
177C2×C8×C13—21no—
178C2×C8×C13—21no—
179C2×C8×C13—21no—
180C2×C8×C13C22×C412yes—
181C2×C8×C13C22×C412yes—
182C13×C4⋊C4—21no—
183C13×C4⋊C4—21no—
184C13×C4⋊C4C22×C412yes—
185C13×C4⋊C4C22×C412yes—
186C13×C22⋊C4—24no—
187C42×C13—24no—
188C22⋊Dic13—22no—
189C22⋊Dic13—41no—
190C22⋊Dic13—42no—
191C22⋊Dic13—81no—
192C4⋊Dic13—21no—
193C4⋊Dic13—21no—
194C4⋊Dic13—21no—
195C4⋊Dic13—21no—
196C4⋊Dic13—21no—
197C4⋊Dic13—21no—
198C4⋊Dic13—21no—
199C4⋊Dic13—21no—
200C4⋊Dic13—42no—
201C4⋊Dic13—42no—
202C4⋊Dic13—81no—
203C4⋊Dic13—81no—
204C4×Dic13—21no—
205C4×Dic13—21no—
206C4×Dic13—21no—
207C4×Dic13—21no—
208C4×Dic13—42no—
209C4×Dic13—42no—
210C4×Dic13—42no—
211C4×Dic13—42no—
212C2×#1055—21no—
213C2×#1055—21no—
214C2×#1055—21no—
215C2×#1055—21no—
216C2×#1055—21no—
217C2×#1055—21no—
218C2×#419D1311yescharacteristic
219#2346—131no—
220Q8×C22⋊C4—131no—
221#1935—131no—
222#1933—131no—
223#1928—131no—
224#1927—131no—
225#1865—131no—
226#1863—131no—
227#1825—131no—
228#1809—131no—
229#1787—131no—
230#1763—131no—
231C4×#103—131no—
232#1753—131no—
233C23×C13—21no—
234C23×C13—21no—
235C23×C13—21no—
236C23×C13—21no—
237C23×C13—21no—
238C23×C13—21no—
239C23×C13C4×D412yes—
240C23×C13C22×D411yescharacteristic
241Q8×C13—22no—
242Q8×C13—22no—
243Q8×C13—22no—
244Q8×C13—22no—
245Q8×C13—22no—
246Q8×C13—48no—
247Q8×C13—41no—
248Q8×C13—41no—
249Q8×C13#11714yes—
250C2×C4×C13—22no—
251C2×C4×C13—22no—
252C2×C4×C13—22no—
253C2×C4×C13—21no—
254C2×C4×C13—21no—
255C2×C4×C13—22no—
256C2×C4×C13—22no—
257C2×C4×C13—22no—
258C2×C4×C13—22no—
259C2×C4×C13—22no—
260C2×C4×C13—22no—
261C2×C4×C13—22no—
262C2×C4×C13—48no—
263C2×C4×C13—41no—
264C2×C4×C13—41no—
265C2×C4×C13—41no—
266C2×C4×C13—41no—
267C2×C4×C13—41no—
268C2×C4×C13—41no—
269C2×C4×C13C4×D412yes—
270C2×C4×C13C23×C411yescharacteristic
271C2×C4×C13C2×#4111yescharacteristic
272C2×C4×C13C2×#4111yescharacteristic
273C2×C4×C13C2×#4111yescharacteristic
274C2×C4×C13H(5,𝔽2)11yescharacteristic
275C2×C4×C13H(5,𝔽2)11yescharacteristic
276C2×Dic13—21no—
277C2×Dic13—21no—
278C2×Dic13—21no—
279C2×Dic13—21no—
280C2×Dic13—21no—
281C2×Dic13—21no—
282C2×Dic13—21no—
283C2×Dic13—21no—
284C2×Dic13—21no—
285C2×Dic13—21no—
286C2×Dic13—21no—
287C2×Dic13—41no—
288C2×Dic13—41no—
289C2×Dic13—41no—
290C2×Dic13—41no—
291C2×Dic13—41no—
292C2×Dic13—41no—
293C2×Dic13—41no—
294C2×Dic13—41no—
295C2×Dic13—41no—
296C2×Dic13—41no—
297C2×Dic13—41no—
298C2×Dic13—41no—
299C2×Dic13#13712yes—
300C8×C13—22no—
301C8×C13—22no—
302#1055—41no—
303#1055—41no—
304C23×Q8C2×D1311yescharacteristic
305C22×M16C2×D1311yescharacteristic
306C2×C4×Q8—131no—
307C2×C4×Q8—131no—
308C2×#117C2×D1311yescharacteristic
309C2×#107—131no—
310C2×#106—131no—
311#419Dic1318yescomplemented
312C2×#103—131no—
313C2×#103—131no—
314C2×#103—131no—
315C2×#103—131no—
316C2×#103C2×D1311yescharacteristic
317C2×#103C2×D1311yescharacteristic
318C2×#103C2×D1311yescharacteristic
319C2×#103C2×D1311yescharacteristic
320C2×C22⋊C8—131no—
321C2×#97—131no—
322C2×C4×C8—131no—
323#391—131no—
324#391—131no—
325#391—261no—
326#391—261no—
327#389—131no—
328#389—131no—
329#388—261no—
330#388—261no—
331#382—261no—
332#380—131no—
333#380—131no—
334#380—261no—
335C4×C4⋊C4—131no—
336C4×C4⋊C4—131no—
337C4×C22⋊C4—131no—
338C4×C22⋊C4—261no—
339#376—261no—
340#340—131no—
341#340—131no—
342#340—131no—
343#340—131no—
344#340—131no—
345#340—131no—
346#336—131no—
347#336—131no—
348C22×C13—22no—
349C22×C13—22no—
350C22×C13—22no—
351C22×C13—41no—
352C22×C13—41no—
353C22×C13—41no—
354C22×C13—41no—
355C22×C13C2×C4×D411yescharacteristic
356C22×C13#54611yescharacteristic
357C22×C13#54611yescharacteristic
358C22×C13C2×#13711yescharacteristic
359C22×C13#57511yescharacteristic
360C22×C13#57711yescharacteristic
361C22×C13#57811yescharacteristic
362C4×C13—22no—
363C4×C13—24no—
364C4×C13—42no—
365C4×C13—42no—
366C4×C13—42no—
367C4×C13C2×#11711yescharacteristic
368C4×C13#51811yescharacteristic
369C4×C13#55111yescharacteristic
370C4×C13#55111yescharacteristic
371Dic13—22no—
372Dic13—41no—
373Dic13—41no—
374Dic13—41no—
375Dic13—81no—
376Dic13—81no—
377Dic13—81no—
378Dic13—81no—
379C22×Q8—21no—
380C22×Q8—22no—
381C22×Q8—22no—
382C22×Q8—22no—
383C22×Q8—41no—
384C22×Q8—41no—
385C22×Q8C2×Dic1314yes—
386C22×Q8C22×D1311yescharacteristic
387C22×Q8C22×D1311yescharacteristic
388C23×C4—21no—
389C23×C4C22×D1311yescharacteristic
390C2×M16—41no—
391C2×M16—41no—
392C2×M16C2×Dic1314yes—
393C22×C8—131no—
394C22×C8—131no—
395C22×C8C22×D1311yescharacteristic
396C22×C8C22×D1311yescharacteristic
397C4×Q8—262no—
398C4×Q8—262no—
399C4×Q8—521no—
400C4×Q8—521no—
401#117—22no—
402#117C2×Dic1314yes—
403C2×C4⋊C4—131no—
404C2×C4⋊C4—131no—
405C2×C4⋊C4—131no—
406C2×C4⋊C4—131no—
407C2×C4⋊C4—261no—
408C2×C4⋊C4—261no—
409C2×C4⋊C4—261no—
410C2×C4⋊C4—261no—
411C2×C4⋊C4C22×D1311yescharacteristic
412C2×C4⋊C4C22×D1311yescharacteristic
413C2×C22⋊C4—21no—
414C2×C22⋊C4—131no—
415C2×C22⋊C4—131no—
416C2×C22⋊C4—261no—
417C2×C22⋊C4—261no—
418C2×C42—21no—
419C2×C42—131no—
420C2×C42—131no—
421C2×C42—261no—
422C2×C42—261no—
423C2×C42—261no—
424C2×C42—261no—
425#107—521no—
426#106—521no—
427#103—262no—
428#103—262no—
429#103—521no—
430#103—521no—
431#103C2×Dic1314yes—
432#103C2×Dic1314yes—
433#103C2×Dic1314yes—
434#103C2×Dic1314yes—
435C22⋊C8—262no—
436#97—521no—
437C4×C8—521no—
438#95—131no—
439#95—131no—
440#95—261no—
441#95—261no—
442#95—261no—
443#95—261no—
444#95—261no—
445#95—261no—
446#95—261no—
447#95—261no—
448#95—261no—
449#95—261no—
450C2×C13—42no—
451C2×C13#234211yescharacteristic
452C2×C13C4×#13711yescharacteristic
453C2×C13#294711yescharacteristic
454C2×C13#331211yescharacteristic
455C2×C13#332111yescharacteristic
456C2×C13#332711yescharacteristic
457C2×C13#332911yescharacteristic
458C24D4×D1311yescharacteristic
459C2×Q8—24no—
460C2×Q8—21no—
461C2×Q8—21no—
462C2×Q8—21no—
463C2×Q8—21no—
464C2×Q8—28no—
465C2×Q8—21no—
466C2×Q8—21no—
467C2×Q8—21no—
468C2×Q8—21no—
469C2×Q8—42no—
470C2×Q8—42no—
471C2×Q8—44no—
472C2×Q8—44no—
473C2×Q8—44no—
474C2×Q8#612712yes—
475C2×Q8#612712yes—
476C2×Q8C22×Dic1312yes—
477C2×Q8C22×Dic1312yes—
478C22×C4—21no—
479C22×C4—21no—
480C22×C4—21no—
481C22×C4—21no—
482C22×C4—21no—
483C22×C4—24no—
484C22×C4—21no—
485C22×C4—21no—
486C22×C4—21no—
487C22×C4—42no—
488C22×C4—42no—
489C22×C4—131no—
490C22×C4—131no—
491C22×C4—131no—
492C22×C4—131no—
493C22×C4—261no—
494C22×C4—261no—
495C22×C4—261no—
496C22×C4—261no—
497C22×C4D4×D1311yescharacteristic
498C22×C4#612912yes—
499C22×C4C22×Dic1312yes—
500C22×C4C23×D1311yescharacteristic
501M16—44no—
502C2×C8—21no—
503C2×C8—21no—
504C2×C8—21no—
505C2×C8—21no—
506C2×C8—261no—
507C2×C8—261no—
508C2×C8—261no—
509C2×C8—261no—
510C2×C8—261no—
511C2×C8—261no—
512C2×C8C22×Dic1312yes—
513C2×C8C22×Dic1312yes—
514C4⋊C4—21no—
515C4⋊C4—21no—
516C4⋊C4—261no—
517C4⋊C4—261no—
518C4⋊C4—261no—
519C4⋊C4—261no—
520C4⋊C4—261no—
521C4⋊C4—261no—
522C4⋊C4—261no—
523C4⋊C4—261no—
524C4⋊C4—522no—
525C4⋊C4—522no—
526C4⋊C4—1041no—
527C4⋊C4—1041no—
528C4⋊C4C22×Dic1312yes—
529C4⋊C4C22×Dic1312yes—
530C22⋊C4—24no—
531C22⋊C4—262no—
532C22⋊C4—521no—
533C22⋊C4—522no—
534C22⋊C4—1041no—
535C42—24no—
536C42—261no—
537C42—261no—
538C42—261no—
539C42—261no—
540C42—522no—
541C42—522no—
542C42—522no—
543C42—522no—
544C13#2258511yescharacteristic, complemented, Sylow
545C23—21no—
546C23—21no—
547C23—21no—
548C23—21no—
549C23—21no—
550C23—21no—
551C23D4×Dic1312yes—
552C23C2×D4×D1311yescharacteristic
553Q8—22no—
554Q8—22no—
555Q8—22no—
556Q8—22no—
557Q8—22no—
558Q8—48no—
559Q8—41no—
560Q8—41no—
561Q8#8818014yes—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—48no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—132no—
582C2×C4—261no—
583C2×C4—261no—
584C2×C4—261no—
585C2×C4—261no—
586C2×C4—261no—
587C2×C4—261no—
588C2×C4—261no—
589C2×C4—261no—
590C2×C4—261no—
591C2×C4—261no—
592C2×C4—261no—
593C2×C4—521no—
594C2×C4—521no—
595C2×C4—521no—
596C2×C4—521no—
597C2×C4—521no—
598C2×C4—521no—
599C2×C4—521no—
600C2×C4—521no—
601C2×C4—521no—
602C2×C4—521no—
603C2×C4—521no—
604C2×C4—521no—
605C2×C4D4×Dic1312yes—
606C2×C4C2×#612711yescharacteristic
607C2×C4C2×#612711yescharacteristic
608C2×C4C2×#612911yescharacteristic
609C2×C4#8825111yescharacteristic
610C2×C4#8825111yescharacteristic
611C2×C4C23×Dic1311yescharacteristic
612C8—22no—
613C8—22no—
614C8—521no—
615C8—521no—
616C22—22no—
617C22—22no—
618C22—22no—
619C22—41no—
620C22—41no—
621C22—41no—
622C22—41no—
623C22#1171919511yescharacteristic
624C22#1171919511yescharacteristic
625C22D13×#13711yescharacteristic
626C22#1171948211yescharacteristic
627C22#1171948311yescharacteristic
628C22#1171948411yescharacteristic
629C22C2×D4×Dic1311yescharacteristic
630C4—22no—
631C4—24no—
632C4—42no—
633C4—42no—
634C4—42no—
635C4—262no—
636C4—521no—
637C4—521no—
638C4—521no—
639C4—1041no—
640C4—1041no—
641C4—1041no—
642C4—1041no—
643C4#1171920011yescharacteristic
644C4#1171920011yescharacteristic
645C4C2×#8818011yescharacteristic
646C4#1171950311yescharacteristic
647C2—42no—
648C2#1311768511yescharacteristic
649C2#137×Dic1311yescharacteristic
650C2#1312368011yescharacteristic
651C2#1312688811yescharacteristic
652C2#1312702311yescharacteristic
653C2#1312710011yescharacteristic
654C2#1312712611yescharacteristic
subgroupG/H
centerC23C2×D4×D13
commutatorC2×C4×C13C23×C4
FrattiniC22×C4C23×D13
FittingC2×C13×#419C2
socleC23×C13C22×D4
prime radicalC24×C13C2×D4
cosocleC22×C4×C13C24
hypercenterC2×#419D13
hypocenterC13#22585
LCSG ▷ C2×C4×C13 ▷ C2×C13 ▷ C13
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#419

Complements

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

NH ≅ G/Ntype
3. #13115528C2semidirect
4. #13115514C2semidirect
5. #13115479C2semidirect
6. #13115472C2semidirect
7. #13115106C2semidirect
8. #13115092C2semidirect
12. #13114668C2semidirect
13. #103×Dic13C2semidirect
21. C13×#419C4semidirect
311. #419Dic13semidirect
544. C13#22585semidirect

Maximal subgroups

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

subgroupindex
#131177212
Q8×C22⋊Dic132
#131155282
#131155142
#131154792
#131154722
#131151062
#131150922
#131149312
#131148692
#131147712
#131146682
#103×Dic132
#131146172
C2×C13×#4192
#2258513