Finite Group Databasebeta

← #15880249

Subgroups — #15880249

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#13117719C211yesmaximal, characteristic
3#13115529C211yesmaximal, characteristic, complemented
4#13115515C211yesmaximal, characteristic, complemented
5#13115474C211yesmaximal, characteristic, complemented
6#13115467C211yesmaximal, characteristic, complemented
7#13115104C211yesmaximal, characteristic, complemented
8#13115090C211yesmaximal, characteristic, complemented
9#13114928C211yesmaximal, characteristic
10#13114867C211yesmaximal, characteristic
11#13114771C211yesmaximal, characteristic
12#13114666C211yesmaximal, characteristic, complemented
13C4×#88075C211yesmaximal, characteristic, complemented
14#13114620C211yesmaximal, characteristic
15Q8×C13×C22⋊C4C211yesmaximal, characteristic
16C23×Q8×C13C2211yescharacteristic
17C2×C4×Q8×C13C2211yescharacteristic
18C2×C4×Q8×C13C2211yescharacteristic
19C2×#88180C2211yescharacteristic
20C22×#6099C2211yescharacteristic
21C13×#391—21no—
22C13×#391—21no—
23C13×#391C2211yescharacteristic
24C13×#391C2211yescharacteristic
25C13×#380—21no—
26C13×#380C2211yescharacteristic
27C4×C13×C22⋊C4—21no—
28C4×C13×C22⋊C4C2211yescharacteristic
29#11718886C418yescomplemented
30C2×#88075C2211yescharacteristic
31C2×#88075C2211yescharacteristic
32C2×#88075C2211yescharacteristic
33C2×#88075C2211yescharacteristic
34C2×#88075C2211yescharacteristic
35C2×#88075C2211yescharacteristic
36C2×#88075C2211yescharacteristic
37C2×#88075C2211yescharacteristic
38#11718873C2211yescharacteristic
39C2×C22⋊#1055C2211yescharacteristic
40C2×#88048C2211yescharacteristic
41C2×#88047C2211yescharacteristic
42#11718605—21no—
43#11718605—21no—
44C4×C4⋊Dic13C2211yescharacteristic
45C4×C4⋊Dic13C2211yescharacteristic
46#11718589C2211yescharacteristic
47#11718589C2211yescharacteristic
48C2×#88043C2211yescharacteristic
49C2×C4×#1055C2211yescharacteristic
50#11718311—21no—
51#11718309—21no—
52#11718121C2211yescharacteristic
53#11718121C2211yescharacteristic
54#11718121C2211yescharacteristic
55#11718121C2211yescharacteristic
56#11718121C2211yescharacteristic
57#11718121C2211yescharacteristic
58#11718116C2211yescharacteristic
59#11718116C2211yescharacteristic
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
71C4×Q8×C13—22no—
72C4×Q8×C13—22no—
73C4×Q8×C13—41no—
74C4×Q8×C13—41no—
75C2×C13×C4⋊C4—21no—
76C2×C13×C4⋊C4—21no—
77C2×C13×C4⋊C4C2311yescharacteristic
78C2×C13×C4⋊C4C2311yescharacteristic
79C2×C13×C22⋊C4—21no—
80C2×C13×C22⋊C4D411yescharacteristic
81C2×C13×C22⋊C4D411yescharacteristic
82C2×C42×C13—21no—
83C2×C42×C13—21no—
84C2×C42×C13C2311yescharacteristic
85C2×C42×C13C2311yescharacteristic
86C2×C22⋊Dic13—21no—
87C2×C22⋊Dic13—21no—
88#88180—22no—
89#88180C2×C414yes—
90C2×C4⋊Dic13C2311yescharacteristic
91C2×C4⋊Dic13C2311yescharacteristic
92C2×C4⋊Dic13C2311yescharacteristic
93C2×C4⋊Dic13C2311yescharacteristic
94C2×#6101—21no—
95C2×#6101—21no—
96C2×C4×Dic13—21no—
97C2×C4×Dic13—21no—
98C2×C4×Dic13—21no—
99C2×#6099—41no—
100C2×#6099—41no—
101C2×#6099C2×C414yes—
102C22×#1055C2311yescharacteristic
103C22×#1055C2311yescharacteristic
104C22×#1055C2311yescharacteristic
105C22×#1055C2311yescharacteristic
106C13×#95—21no—
107C13×#95—21no—
108C13×#95—21no—
109C13×#95—21no—
110C13×#95—21no—
111C13×#95D411yescharacteristic
112C13×#95D411yescharacteristic
113#88075—22no—
114#88075—22no—
115#88075—41no—
116#88075—41no—
117#88075C2×C414yes—
118#88075C2×C414yes—
119#88075C2×C414yes—
120#88075C2×C414yes—
121#88071—21no—
122#88071—21no—
123#88071—21no—
124#88071—21no—
125#88071—21no—
126C22⋊#1055—22no—
127#88048—41no—
128#88047—41no—
129#88043—41no—
130C4×#1055—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×C13C2×D411yescharacteristic
166C22×C4×C13C2×D411yescharacteristic
167C22×C4×C13C2×D411yescharacteristic
168C22×C4×C13C2×D411yescharacteristic
169C22×C4×C13#4112yes—
170C22×C4×C13C2411yescharacteristic
171C22×Dic13—21no—
172C22×Dic13—21no—
173C22×Dic13—21no—
174C22×Dic13—21no—
175C13×C4⋊C4—21no—
176C13×C4⋊C4—21no—
177C13×C4⋊C4—21no—
178C13×C4⋊C4—21no—
179C13×C4⋊C4—42no—
180C13×C4⋊C4—42no—
181C13×C22⋊C4—22no—
182C13×C22⋊C4—41no—
183C13×C22⋊C4—42no—
184C42×C13—21no—
185C42×C13—21no—
186C42×C13—21no—
187C42×C13—21no—
188C42×C13—42no—
189C42×C13—42no—
190C22⋊Dic13—24no—
191C22⋊Dic13—81no—
192C4⋊Dic13—21no—
193C4⋊Dic13—21no—
194C4⋊Dic13—21no—
195C4⋊Dic13—21no—
196C4⋊Dic13—21no—
197C4⋊Dic13—21no—
198C4⋊Dic13C22×C412yes—
199C4⋊Dic13C22×C412yes—
200#6101—81no—
201#6101—81no—
202C4×Dic13—24no—
203C4×Dic13—42no—
204C4×Dic13—42no—
205#6099—44no—
206C2×#1055—21no—
207C2×#1055—21no—
208C2×#1055—21no—
209C2×#1055—21no—
210C2×#1055—21no—
211C2×#1055—21no—
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⋊C4D1311yescharacteristic
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—21no—
252C2×C4×C13—22no—
253C2×C4×C13—22no—
254C2×C4×C13—21no—
255C2×C4×C13—21no—
256C2×C4×C13—21no—
257C2×C4×C13—22no—
258C2×C4×C13—22no—
259C2×C4×C13—22no—
260C2×C4×C13—22no—
261C2×C4×C13—22no—
262C2×C4×C13—21no—
263C2×C4×C13—21no—
264C2×C4×C13—21no—
265C2×C4×C13—21no—
266C2×C4×C13—21no—
267C2×C4×C13—21no—
268C2×C4×C13—21no—
269C2×C4×C13—21no—
270C2×C4×C13—21no—
271C2×C4×C13—48no—
272C2×C4×C13—41no—
273C2×C4×C13—41no—
274C2×C4×C13—41no—
275C2×C4×C13—41no—
276C2×C4×C13—41no—
277C2×C4×C13—41no—
278C2×C4×C13—41no—
279C2×C4×C13—41no—
280C2×C4×C13C4×D412yes—
281C2×C4×C13#13712yes—
282C2×C4×C13C23×C411yescharacteristic
283C2×C4×C13C2×#4111yescharacteristic
284C2×C4×C13C2×#4111yescharacteristic
285C2×C4×C13C2×#4111yescharacteristic
286C2×C4×C13H(5,𝔽2)11yescharacteristic
287C2×C4×C13H(5,𝔽2)11yescharacteristic
288C2×Dic13—22no—
289C2×Dic13—22no—
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—41no—
300#1055—22no—
301#1055—22no—
302#1055—41no—
303#1055—41no—
304C23×Q8C2×D1311yescharacteristic
305C22×M16—131no—
306C2×C4×Q8C2×D1311yescharacteristic
307C2×C4×Q8C2×D1311yescharacteristic
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×#103—131no—
317C2×#103—131no—
318C2×#103—131no—
319C2×#103—131no—
320C2×C22⋊C8—131no—
321C2×#97—131no—
322C2×C4×C8—131no—
323#391—21no—
324#391—21no—
325#391C2×D1311yescharacteristic
326#391C2×D1311yescharacteristic
327#389—131no—
328#389—131no—
329#388—261no—
330#388—261no—
331#382—261no—
332#380—21no—
333#380—131no—
334#380C2×D1311yescharacteristic
335C4×C4⋊C4—131no—
336C4×C4⋊C4—131no—
337C4×C22⋊C4—21no—
338C4×C22⋊C4C2×D1311yescharacteristic
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—22no—
365C4×C13—41no—
366C4×C13—42no—
367C4×C13—41no—
368C4×C13—41no—
369C4×C13—81no—
370C4×C13—81no—
371C4×C13C2×#11711yescharacteristic
372C4×C13#51811yescharacteristic
373C4×C13#55111yescharacteristic
374C4×C13#55111yescharacteristic
375Dic13—42no—
376Dic13—42no—
377Dic13—81no—
378Dic13—81no—
379C22×Q8—22no—
380C22×Q8—21no—
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×C8—131no—
396C22×C8—131no—
397C4×Q8—22no—
398C4×Q8—22no—
399C4×Q8—41no—
400C4×Q8—41no—
401#117—134no—
402#117—262no—
403C2×C4⋊C4—21no—
404C2×C4⋊C4—21no—
405C2×C4⋊C4—131no—
406C2×C4⋊C4—131no—
407C2×C4⋊C4—131no—
408C2×C4⋊C4—131no—
409C2×C4⋊C4—261no—
410C2×C4⋊C4—261no—
411C2×C4⋊C4C22×D1311yescharacteristic
412C2×C4⋊C4C22×D1311yescharacteristic
413C2×C22⋊C4—21no—
414C2×C22⋊C4—261no—
415C2×C22⋊C4—261no—
416C2×C22⋊C4C22⋊D1311yescharacteristic
417C2×C22⋊C4C22⋊D1311yescharacteristic
418C2×C42—21no—
419C2×C42—21no—
420C2×C42—261no—
421C2×C42—261no—
422C2×C42—261no—
423C2×C42C22×D1311yescharacteristic
424C2×C42C22×D1311yescharacteristic
425#107—521no—
426#106—521no—
427#103—134no—
428#103—134no—
429#103—134no—
430#103—134no—
431#103—262no—
432#103—262no—
433#103—521no—
434#103—521no—
435C22⋊C8—262no—
436#97—521no—
437C4×C8—521no—
438#95—21no—
439#95—21no—
440#95—21no—
441#95—21no—
442#95—21no—
443#95—261no—
444#95—261no—
445#95—261no—
446#95—261no—
447#95—261no—
448#95C22⋊D1311yescharacteristic
449#95C22⋊D1311yescharacteristic
450C2×C13—42no—
451C2×C13#234211yescharacteristic
452C2×C13C4×#13711yescharacteristic
453C2×C13#294711yescharacteristic
454C2×C13#331211yescharacteristic
455C2×C13#332111yescharacteristic
456C2×C13#332711yescharacteristic
457C2×C13#332911yescharacteristic
458C24C2×C22⋊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—261no—
490C22×C4—261no—
491C22×C4—261no—
492C22×C4—261no—
493C22×C4C2×C4×D1312yes—
494C22×C4#612712yes—
495C22×C4C2×C22⋊D1311yescharacteristic
496C22×C4C2×C22⋊D1311yescharacteristic
497C22×C4C2×C22⋊D1311yescharacteristic
498C22×C4C2×C22⋊D1311yescharacteristic
499C22×C4C2×C22⋊D1311yescharacteristic
500C22×C4C23×D1311yescharacteristic
501M16—524no—
502C2×C8—132no—
503C2×C8—132no—
504C2×C8—261no—
505C2×C8—261no—
506C2×C8—261no—
507C2×C8—261no—
508C2×C8—261no—
509C2×C8—261no—
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—42no—
519C4⋊C4—42no—
520C4⋊C4—132no—
521C4⋊C4—132no—
522C4⋊C4—261no—
523C4⋊C4—261no—
524C4⋊C4—261no—
525C4⋊C4—261no—
526C4⋊C4—261no—
527C4⋊C4—261no—
528C4⋊C4—1041no—
529C4⋊C4—1041no—
530C22⋊C4—22no—
531C22⋊C4—41no—
532C22⋊C4—42no—
533C22⋊C4—264no—
534C22⋊C4—1041no—
535C42—21no—
536C42—21no—
537C42—21no—
538C42—21no—
539C42—42no—
540C42—42no—
541C42—264no—
542C42—522no—
543C42—522no—
544C13#2258511yescharacteristic, complemented, Sylow
545C23—21no—
546C23—21no—
547C23—21no—
548C23—21no—
549C23—21no—
550C23—21no—
551C23C4×C22⋊D1312yes—
552C23C22×C22⋊D1311yescharacteristic
553Q8—22no—
554Q8—22no—
555Q8—22no—
556Q8—22no—
557Q8—22no—
558Q8—48no—
559Q8—41no—
560Q8—41no—
561Q8#8813114yes—
562C2×C4—22no—
563C2×C4—21no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—48no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—262no—
593C2×C4—262no—
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×C4C4×C22⋊D1312yes—
605C2×C4#8819612yes—
606C2×C4C22×C4×D1311yescharacteristic
607C2×C4C2×#612711yescharacteristic
608C2×C4C2×#612911yescharacteristic
609C2×C4C2×#612911yescharacteristic
610C2×C4#8825111yescharacteristic
611C2×C4#8825111yescharacteristic
612C8—262no—
613C8—262no—
614C8—521no—
615C8—521no—
616C22—22no—
617C22—22no—
618C22—22no—
619C22—41no—
620C22—41no—
621C22—41no—
622C22—41no—
623C22C2×C4×C22⋊D1311yescharacteristic
624C22#1171951111yescharacteristic
625C22#1171951111yescharacteristic
626C22C2×#8819611yescharacteristic
627C22#1171952711yescharacteristic
628C22#1171954211yescharacteristic
629C22#1171954311yescharacteristic
630C4—22no—
631C4—22no—
632C4—24no—
633C4—41no—
634C4—41no—
635C4—41no—
636C4—42no—
637C4—81no—
638C4—81no—
639C4—522no—
640C4—522no—
641C4—1041no—
642C4—1041no—
643C4C2×#8813111yescharacteristic
644C4#1171919111yescharacteristic
645C4#1171920011yescharacteristic
646C4#1171920011yescharacteristic
647C2—42no—
648C2#1311768111yescharacteristic
649C2C4×#8819611yescharacteristic
650C2#1312367911yescharacteristic
651C2#1312688711yescharacteristic
652C2#1312702211yescharacteristic
653C2#1312710411yescharacteristic
654C2#1312713011yescharacteristic
subgroupG/H
centerC23C22×C22⋊D13
commutatorC2×C4×C13C23×C4
FrattiniC22×C4C23×D13
FittingQ8×C13×C22⋊C4C2
socleC23×C13C22×D4
prime radicalC24×C13C2×D4
cosocleC22×C4×C13C24
hypercenterQ8×C22⋊C4D13
hypocenterC13#22585
LCSG ▷ C2×C4×C13 ▷ C2×C13 ▷ C13
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ Q8×C22⋊C4

Complements

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

NH ≅ G/Ntype
3. #13115529C2semidirect
4. #13115515C2semidirect
5. #13115474C2semidirect
6. #13115467C2semidirect
7. #13115104C2semidirect
8. #13115090C2semidirect
12. #13114666C2semidirect
13. C4×#88075C2semidirect
29. #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
#131177192
#131155292
#131155152
#131154742
#131154672
#131151042
#131150902
#131149282
#131148672
#131147712
#131146662
C4×#880752
#131146202
Q8×C13×C22⋊C42
#2258513