Finite Group Databasebeta

← #15880257

Subgroups — #15880257

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
1C2×#11718886C211yesmaximal, characteristic
2#13117723C211yesmaximal, characteristic
3Q8×C22⋊Dic13C211yesmaximal, characteristic
4#13115523C211yesmaximal, characteristic, complemented
5#13115509C211yesmaximal, characteristic, complemented
6#13115480C211yesmaximal, characteristic, complemented
7#13115473C211yesmaximal, characteristic, complemented
8#13115102C211yesmaximal, characteristic, complemented
9#13115088C211yesmaximal, characteristic, complemented
10#13114932C211yesmaximal, characteristic
11#13114867C211yesmaximal, characteristic
12#13114668C211yesmaximal, characteristic, complemented
13#103×Dic13C211yesmaximal, characteristic, complemented
14#13114622C211yesmaximal, characteristic
15C13×#1787C211yesmaximal, characteristic
16C23×Q8×C13C2211yescharacteristic
17C2×Q8×Dic13C2211yescharacteristic
18C2×Q8×Dic13C2211yescharacteristic
19C2×#88180C2211yescharacteristic
20C22×#6099C2211yescharacteristic
21C2×C13×#103C2211yescharacteristic
22C2×C13×#103C2211yescharacteristic
23C2×C13×#103C2211yescharacteristic
24C2×C13×#103C2211yescharacteristic
25C2×C13×C22⋊C8C2211yescharacteristic
26C13×#380C2211yescharacteristic
27#11718891—21no—
28#11718891—21no—
29#11718891C2211yescharacteristic
30#11718891C2211yescharacteristic
31#11718886C418yescomplemented
32C2×#88075C2211yescharacteristic
33C2×#88075C2211yescharacteristic
34C2×#88075C2211yescharacteristic
35C2×#88075C2211yescharacteristic
36#11718873—21no—
37#11718873C2211yescharacteristic
38C4×C22⋊Dic13—21no—
39C4×C22⋊Dic13C2211yescharacteristic
40C2×#88055C2211yescharacteristic
41C2×C8×Dic13C2211yescharacteristic
42C4⋊C4×Dic13C2211yescharacteristic
43C4⋊C4×Dic13C2211yescharacteristic
44#11718634C2211yescharacteristic
45#11718634C2211yescharacteristic
46C2×#88048C2211yescharacteristic
47C2×#88047C2211yescharacteristic
48#11718606—21no—
49#11718606—21no—
50#11718311—21no—
51#11718309—21no—
52#11718143C2211yescharacteristic
53#11718143C2211yescharacteristic
54#11718143C2211yescharacteristic
55#11718143C2211yescharacteristic
56#11718142C2211yescharacteristic
57#11718142C2211yescharacteristic
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
71C22×C8×C13C2311yescharacteristic
72C22×C8×C13C2311yescharacteristic
73C2×C13×C4⋊C4C2311yescharacteristic
74C2×C13×C4⋊C4C2311yescharacteristic
75C2×C13×C22⋊C4—21no—
76C2×C22⋊Dic13—21no—
77C2×C22⋊Dic13—21no—
78C2×C22⋊Dic13D411yescharacteristic
79C2×C22⋊Dic13D411yescharacteristic
80Q8×Dic13—22no—
81Q8×Dic13—22no—
82Q8×Dic13—41no—
83Q8×Dic13—41no—
84#88180—22no—
85#88180C2×C414yes—
86C2×C4⋊Dic13—21no—
87C2×C4⋊Dic13—21no—
88C2×C4⋊Dic13C2311yescharacteristic
89C2×C4⋊Dic13C2311yescharacteristic
90C2×C4⋊Dic13C2311yescharacteristic
91C2×C4⋊Dic13C2311yescharacteristic
92C2×#6101—21no—
93C2×#6101—21no—
94C2×C4×Dic13—21no—
95C2×C4×Dic13—21no—
96C2×C4×Dic13—21no—
97C2×C4×Dic13—21no—
98C2×C4×Dic13—21no—
99C2×C4×Dic13C2311yescharacteristic
100C2×C4×Dic13C2311yescharacteristic
101C2×#6099—41no—
102C2×#6099—41no—
103C2×#6099C2×C414yes—
104C22×#1055C2311yescharacteristic
105C22×#1055C2311yescharacteristic
106C13×#103—22no—
107C13×#103—22no—
108C13×#103—41no—
109C13×#103—41no—
110C13×C22⋊C8—22no—
111C13×#95—21no—
112#88075C2×C414yes—
113#88075C2×C414yes—
114#88075C2×C414yes—
115#88075C2×C414yes—
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#88071—21no—
125#88071D411yescharacteristic
126#88071D411yescharacteristic
127#88055—41no—
128C8×Dic13—41no—
129#88048—41no—
130#88047—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
175C2×C8×C13—21no—
176C2×C8×C13—21no—
177C2×C8×C13—21no—
178C2×C8×C13—21no—
179C2×C8×C13—21no—
180C2×C8×C13—21no—
181C13×C4⋊C4—21no—
182C13×C4⋊C4—21no—
183C13×C4⋊C4—21no—
184C13×C4⋊C4—21no—
185C13×C22⋊C4—81no—
186C22⋊Dic13—24no—
187C22⋊Dic13—22no—
188C22⋊Dic13—41no—
189C22⋊Dic13—42no—
190C4⋊Dic13—21no—
191C4⋊Dic13—21no—
192C4⋊Dic13—21no—
193C4⋊Dic13—21no—
194C4⋊Dic13—21no—
195C4⋊Dic13—21no—
196C4⋊Dic13—42no—
197C4⋊Dic13—42no—
198C4⋊Dic13C22×C412yes—
199C4⋊Dic13C22×C412yes—
200#6101—81no—
201#6101—81no—
202C4×Dic13—24no—
203C4×Dic13—21no—
204C4×Dic13—21no—
205C4×Dic13—21no—
206C4×Dic13—21no—
207C4×Dic13—42no—
208C4×Dic13—42no—
209C4×Dic13—42no—
210C4×Dic13—42no—
211#6099—44no—
212C2×#1055—21no—
213C2×#1055—21no—
214C2×#1055—21no—
215C2×#1055—21no—
216C2×#1055C22×C412yes—
217C2×#1055C22×C412yes—
218C2×#419—131no—
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#1787D1311yescharacteristic
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—48no—
261C2×C4×C13—41no—
262C2×C4×C13—41no—
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—22no—
279C2×Dic13—22no—
280C2×Dic13—21no—
281C2×Dic13—21no—
282C2×Dic13—21no—
283C2×Dic13—21no—
284C2×Dic13—21no—
285C2×Dic13—21no—
286C2×Dic13—21no—
287C2×Dic13—21no—
288C2×Dic13—21no—
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—41no—
301C8×C13—41no—
302#1055—22no—
303#1055—22no—
304C23×Q8C2×D1311yescharacteristic
305C22×M16—131no—
306C2×C4×Q8—131no—
307C2×C4×Q8—131no—
308C2×#117—131no—
309C2×#107—131no—
310C2×#106—131no—
311#419—138no—
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⋊C8C2×D1311yescharacteristic
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—261no—
334#380C2×D1311yescharacteristic
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—81no—
366C4×C13—81no—
367C4×C13C2×#11711yescharacteristic
368C4×C13#51811yescharacteristic
369C4×C13#55111yescharacteristic
370C4×C13#55111yescharacteristic
371Dic13—22no—
372Dic13—41no—
373Dic13—42no—
374Dic13—42no—
375Dic13—41no—
376Dic13—41no—
377Dic13—81no—
378Dic13—81no—
379C22×Q8—21no—
380C22×Q8—22no—
381C22×Q8—22no—
382C22×Q8—22no—
383C22×Q8—41no—
384C22×Q8—41no—
385C22×Q8C4×D1314yes—
386C22×Q8C22×D1311yescharacteristic
387C22×Q8C22×D1311yescharacteristic
388C23×C4—21no—
389C23×C4C22×D1311yescharacteristic
390C2×M16—134no—
391C2×M16—521no—
392C2×M16—521no—
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—134no—
402#117—262no—
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—131no—
419C2×C42—131no—
420C2×C42—261no—
421C2×C42—261no—
422C2×C42—261no—
423C2×C42—261no—
424C2×C42—261no—
425#107—521no—
426#106—521no—
427#103—22no—
428#103—22no—
429#103—41no—
430#103—41no—
431#103—134no—
432#103—134no—
433#103—134no—
434#103—134no—
435C22⋊C8—22no—
436#97—521no—
437C4×C8—521no—
438#95—21no—
439#95—131no—
440#95—131no—
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×Q8C2×C4×D1312yes—
475C2×Q8C2×C4×D1312yes—
476C2×Q8#612912yes—
477C2×Q8#612912yes—
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×C4C2×C4×D1312yes—
498C22×C4D4×D1311yescharacteristic
499C22×C4#612912yes—
500C22×C4C23×D1311yescharacteristic
501M16—524no—
502C2×C8—21no—
503C2×C8—21no—
504C2×C8—21no—
505C2×C8—21no—
506C2×C8—21no—
507C2×C8—21no—
508C2×C8—132no—
509C2×C8—132no—
510C2×C8—261no—
511C2×C8—261no—
512C2×C8—261no—
513C2×C8—261no—
514C4⋊C4—21no—
515C4⋊C4—21no—
516C4⋊C4—21no—
517C4⋊C4—21no—
518C4⋊C4—132no—
519C4⋊C4—132no—
520C4⋊C4—261no—
521C4⋊C4—261no—
522C4⋊C4—261no—
523C4⋊C4—261no—
524C4⋊C4—261no—
525C4⋊C4—261no—
526C4⋊C4—522no—
527C4⋊C4—522no—
528C4⋊C4—1041no—
529C4⋊C4—1041no—
530C22⋊C4—81no—
531C22⋊C4—262no—
532C22⋊C4—264no—
533C22⋊C4—521no—
534C22⋊C4—522no—
535C42—264no—
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—
551C23#8813512yes—
552C23C2×D4×D1311yescharacteristic
553Q8—22no—
554Q8—22no—
555Q8—22no—
556Q8—22no—
557Q8—22no—
558Q8—48no—
559Q8—41no—
560Q8—41no—
561Q8#8813114yes—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—48no—
573C2×C4—41no—
574C2×C4—41no—
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—262no—
585C2×C4—262no—
586C2×C4—261no—
587C2×C4—261no—
588C2×C4—261no—
589C2×C4—261no—
590C2×C4—261no—
591C2×C4—261no—
592C2×C4—261no—
593C2×C4—261no—
594C2×C4—261no—
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×C4#8813512yes—
606C2×C4C22×C4×D1311yescharacteristic
607C2×C4C2×#612911yescharacteristic
608C2×C4C2×#612911yescharacteristic
609C2×C4C2×#612911yescharacteristic
610C2×C4#8825111yescharacteristic
611C2×C4#8825111yescharacteristic
612C8—41no—
613C8—41no—
614C8—262no—
615C8—262no—
616C22—22no—
617C22—22no—
618C22—22no—
619C22—41no—
620C22—41no—
621C22—41no—
622C22—41no—
623C22C2×#8813511yescharacteristic
624C22#1171929611yescharacteristic
625C22#1171929611yescharacteristic
626C22#1171946711yescharacteristic
627C22#1171947111yescharacteristic
628C22D13×#13711yescharacteristic
629C22#1171948211yescharacteristic
630C4—22no—
631C4—24no—
632C4—42no—
633C4—81no—
634C4—81no—
635C4—262no—
636C4—522no—
637C4—522no—
638C4—521no—
639C4—521no—
640C4—521no—
641C4—1041no—
642C4—1041no—
643C4C2×#8813111yescharacteristic
644C4#1171919111yescharacteristic
645C4#1171929211yescharacteristic
646C4#1171929211yescharacteristic
647C2—42no—
648C2#1311768711yescharacteristic
649C2#1312365111yescharacteristic
650C2#1312367311yescharacteristic
651C2#1312688011yescharacteristic
652C2#1312701511yescharacteristic
653C2#1312710811yescharacteristic
654C2#1312712311yescharacteristic
subgroupG/H
centerC23C2×D4×D13
commutatorC2×C4×C13C23×C4
FrattiniC22×C4C23×D13
FittingC13×#1787C2
socleC23×C13C22×D4
prime radicalC24×C13C2×D4
cosocleC22×C4×C13C24
hypercenter#1787D13
hypocenterC13#22585
LCSG ▷ C2×C4×C13 ▷ C2×C13 ▷ C13
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1787

Complements

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

NH ≅ G/Ntype
4. #13115523C2semidirect
5. #13115509C2semidirect
6. #13115480C2semidirect
7. #13115473C2semidirect
8. #13115102C2semidirect
9. #13115088C2semidirect
12. #13114668C2semidirect
13. #103×Dic13C2semidirect
31. #11718886C4semidirect
544. C13#22585semidirect

Maximal subgroups

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

subgroupindex
C2×#117188862
#131177232
Q8×C22⋊Dic132
#131155232
#131155092
#131154802
#131154732
#131151022
#131150882
#131149322
#131148672
#131146682
#103×Dic132
#131146222
C13×#17872
#2258513