Finite Group Databasebeta

← #15925791

Subgroups — #15925791

Order 3328.

860 conjugacy classes of subgroups.

The subgroup lattice has 664 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11718727C211yesmaximal, characteristic, complemented
2C2×#11718625C211yesmaximal, characteristic, complemented
3#13117872C211yesmaximal, characteristic
4#13116360C211yesmaximal, characteristic, complemented
5#13116338C211yesmaximal, characteristic, complemented
6#13116207C211yesmaximal, characteristic, complemented
7#13116171C211yesmaximal, characteristic, complemented
8#13115602C211yesmaximal, characteristic, complemented
9#13115404C211yesmaximal, characteristic, complemented
10#13115272C211yesmaximal, characteristic
11#13115153C211yesmaximal, characteristic
12#13115082C211yesmaximal, characteristic, complemented
13#13115033C211yesmaximal, characteristic, complemented
14#13115021C211yesmaximal, characteristic, complemented
15C13×#2442C211yesmaximal, characteristic, complemented
16C2×C13×#125C2211yescharacteristic
17C22×C13×C4⋊C4C2211yescharacteristic
18C2×#88184C2211yescharacteristic
19C2×#88180C2211yescharacteristic
20C22×#6099C2211yescharacteristic
21C2×C4×D52C2211yescharacteristic
22C13×#400—21no—
23C13×#398—21no—
24C13×#389C2211yescharacteristic
25C13×#389C2211yescharacteristic
26C13×#385—21no—
27C13×#382C2211yescharacteristic
28C13×#380—21no—
29C13×#380C2211yescharacteristic
30C4×C13×C22⋊C4C2211yescharacteristic
31C2×C22⋊#1055C2211yescharacteristic
32#11718727—24no—
33C2×#88049C2211yescharacteristic
34C2×#88049C2211yescharacteristic
35C2×#88049C2211yescharacteristic
36C2×#88049C2211yescharacteristic
37#11718625—24no—
38C2×#88048C2211yescharacteristic
39C2×#88048C2211yescharacteristic
40C2×#88048C2211yescharacteristic
41C2×#88047C2211yescharacteristic
42C2×#88047C2211yescharacteristic
43C2×#88047C2211yescharacteristic
44#11718618—21no—
45#11718617—21no—
46#11718600C2211yescharacteristic
47C4×C4⋊Dic13C2211yescharacteristic
48#11718589C2211yescharacteristic
49C2×C4⋊#1055C2211yescharacteristic
50C2×C4⋊#1055C2211yescharacteristic
51#11718330—21no—
52#11718327—21no—
53#11718121C2211yescharacteristic
54#11718121C2211yescharacteristic
55#11718121C2211yescharacteristic
56#11718121C2211yescharacteristic
57#11718121C2211yescharacteristic
58#11718121C2211yescharacteristic
59#11718116C2211yescharacteristic
60#11718116C2211yescharacteristic
61C23×C4×C13—21no—
62C23×C4×C13C2311yescharacteristic
63C22×C22⋊D13—21no—
64C22×D52C2311yescharacteristic
65C22×C4×D13—21no—
66C13×#125—22no—
67C13×#125—41no—
68C2×C13×C4⋊C4—22no—
69C2×C13×C4⋊C4—21no—
70C2×C13×C4⋊C4—21no—
71C2×C13×C4⋊C4—21no—
72C2×C13×C4⋊C4—21no—
73C2×C13×C4⋊C4—21no—
74C2×C13×C4⋊C4—41no—
75C2×C13×C4⋊C4C2311yescharacteristic
76C2×C13×C4⋊C4C2311yescharacteristic
77C2×C13×C4⋊C4C2311yescharacteristic
78C2×C13×C4⋊C4C2311yescharacteristic
79C2×C13×C22⋊C4—21no—
80C2×C13×C22⋊C4D411yescharacteristic
81C2×C13×C22⋊C4D411yescharacteristic
82C2×C42×C13C2311yescharacteristic
83C2×C42×C13C2311yescharacteristic
84C2×C22⋊Dic13—21no—
85#88184—22no—
86#88184Q814yes—
87C2×#6103—21no—
88C2×#6103—21no—
89C2×#6103—21no—
90C2×#6103—21no—
91#88180—22no—
92#88180—41no—
93C2×C4⋊Dic13C2311yescharacteristic
94C2×C4⋊Dic13C2311yescharacteristic
95C2×C4⋊Dic13C2311yescharacteristic
96C2×#6101—21no—
97C2×C4×Dic13—21no—
98C2×C4×Dic13—21no—
99C2×#6099—22no—
100C2×#6099—22no—
101C2×#6099—22no—
102C22×#1055C2311yescharacteristic
103C22×#1055C2311yescharacteristic
104C22×#1055C2311yescharacteristic
105C22×#1055C2311yescharacteristic
106C4×D52—22no—
107C4×D52—41no—
108C13×#95—21no—
109C13×#95—21no—
110C13×#95—21no—
111C13×#95—21no—
112C13×#95—21no—
113C13×#95D411yescharacteristic
114C13×#95D411yescharacteristic
115#88071—21no—
116#88071—21no—
117C22⋊#1055—22no—
118#88049—22no—
119#88049—22no—
120#88049—22no—
121#88049—41no—
122#88048—22no—
123#88048—22no—
124#88048—22no—
125#88047—22no—
126#88047—41no—
127#88047D414yescomplemented
128C4⋊#1055—22no—
129C4⋊#1055—22no—
130#26363—131nomaximal, Sylow
131C24×C13C2×D411yescharacteristic
132C23×D13—21no—
133C22×C4×C13—21no—
134C22×C4×C13—21no—
135C22×C4×C13—21no—
136C22×C4×C13—21no—
137C22×C4×C13—21no—
138C22×C4×C13—21no—
139C22×C4×C13—21no—
140C22×C4×C13—21no—
141C22×C4×C13—42no—
142C22×C4×C13—42no—
143C22×C4×C13—81no—
144C22×C4×C13C2×D411yescharacteristic
145C22×C4×C13C2×D411yescharacteristic
146C22×C4×C13C2×D411yescharacteristic
147C22×C4×C13C2×D411yescharacteristic
148C22×C4×C13C2×D411yescharacteristic
149C22×C4×C13C2×Q812yes—
150C22×C4×C13#4112yes—
151C22×C4×C13C2411yescharacteristic
152C2×C22⋊D13—24no—
153C2×C22⋊D13—81no—
154C2×C22⋊D13—81no—
155C22×Dic13—21no—
156C22×Dic13—21no—
157C22×Dic13—21no—
158C2×D52—21no—
159C2×D52—21no—
160C2×D52—21no—
161C2×D52—21no—
162C2×D52—21no—
163C2×D52C2×Q812yes—
164C2×C4×D13—42no—
165C2×C4×D13—42no—
166C2×C4×D13—81no—
167C13×C4⋊C4—21no—
168C13×C4⋊C4—21no—
169C13×C4⋊C4—21no—
170C13×C4⋊C4—21no—
171C13×C4⋊C4—21no—
172C13×C4⋊C4—21no—
173C13×C4⋊C4—21no—
174C13×C4⋊C4—42no—
175C13×C4⋊C4—42no—
176C13×C4⋊C4—42no—
177C13×C4⋊C4—81no—
178C13×C4⋊C4—81no—
179C13×C4⋊C4C2×D412yes—
180C13×C22⋊C4—41no—
181C13×C22⋊C4—41no—
182C13×C22⋊C4—42no—
183C42×C13—21no—
184C42×C13—21no—
185C42×C13—21no—
186C42×C13—21no—
187C22⋊Dic13—42no—
188#6103—24no—
189#6103—42no—
190#6103—42no—
191#6103—81no—
192C4⋊Dic13—21no—
193C4⋊Dic13—21no—
194C4⋊Dic13—21no—
195C4⋊Dic13—21no—
196C4⋊Dic13C2×D412yes—
197C4⋊Dic13C2×Q812yes—
198#6101—81no—
199C4×Dic13—42no—
200C4×Dic13—81no—
201#6099—28no—
202C2×#1055—21no—
203C2×#1055—21no—
204C2×#1055—21no—
205C2×#1055—21no—
206C2×#1055—21no—
207C2×#1055—21no—
208C2×#1055—21no—
209C2×#1055—21no—
210C2×#1055—21no—
211C2×#1055—21no—
212C2×#1055C2×D412yes—
213C2×#1055#4112yes—
214C2×#478—131no—
215C2×#428—131no—
216#2442D1311yescharacteristic, complemented
217#2363—131no—
218#2092—131no—
219#2088—131no—
220#2065—131no—
221#2059—131no—
222#1949—131no—
223#1918—131no—
224#1895—131no—
225#1874—131no—
226#1862—131no—
227#1854—131no—
228#1851—131no—
229C23×C13—21no—
230C23×C13—21no—
231C23×C13—21no—
232C23×C13—21no—
233C23×C13—21no—
234C23×C13—21no—
235C23×C13C22⋊Q812yes—
236C23×C13C22×D411yescharacteristic
237C22×D13—22no—
238C22×D13—41no—
239C22×D13—41no—
240C22×D13—41no—
241C22×D13—41no—
242C22×D13—41no—
243C22×D13—41no—
244C2×C4×C13—21no—
245C2×C4×C13—21no—
246C2×C4×C13—22no—
247C2×C4×C13—22no—
248C2×C4×C13—22no—
249C2×C4×C13—22no—
250C2×C4×C13—21no—
251C2×C4×C13—21no—
252C2×C4×C13—22no—
253C2×C4×C13—21no—
254C2×C4×C13—21no—
255C2×C4×C13—21no—
256C2×C4×C13—21no—
257C2×C4×C13—21no—
258C2×C4×C13—21no—
259C2×C4×C13—21no—
260C2×C4×C13—21no—
261C2×C4×C13—21no—
262C2×C4×C13—21no—
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×C13—41no—
270C2×C4×C13—41no—
271C2×C4×C13—41no—
272C2×C4×C13—41no—
273C2×C4×C13—41no—
274C2×C4×C13—84no—
275C2×C4×C13C22⋊Q812yes—
276C2×C4×C13C22×D411yescharacteristic
277C2×C4×C13C22×Q811yescharacteristic
278C2×C4×C13C2×#4111yescharacteristic
279C2×C4×C13C2×#4111yescharacteristic
280C2×C4×C13H(5,𝔽2)11yescharacteristic
281C2×C4×C132−1+411yescharacteristic
282C22⋊D13—84no—
283C2×Dic13—22no—
284C2×Dic13—22no—
285C2×Dic13—41no—
286C2×Dic13—41no—
287C2×Dic13—41no—
288C2×Dic13—41no—
289C2×Dic13—41no—
290C2×Dic13—41no—
291C2×Dic13—41no—
292D52—22no—
293D52—22no—
294D52—41no—
295D52—41no—
296C4×D13—84no—
297#1055—22no—
298#1055—22no—
299#1055—41no—
300#1055—41no—
301C22×M16—131no—
302C2×#125C2×D1311yescharacteristic
303C2×#121—131no—
304C2×C4×D4—131no—
305C2×#117—131no—
306C22×C4⋊C4C2×D1311yescharacteristic
307#478—264no—
308#428—264no—
309C2×#107—131no—
310C2×#107—131no—
311C2×#107—131no—
312C2×#106—131no—
313C2×#106—131no—
314C2×#106—131no—
315C2×C4⋊C8—131no—
316C2×C4⋊C8—131no—
317C2×#102—131no—
318C2×#102—131no—
319C2×#102—131no—
320C2×#102—131no—
321C2×C22⋊C8—131no—
322#400—21no—
323#398—21no—
324#390—131no—
325#390—261no—
326#389—131no—
327#389C2×D1311yescharacteristic
328#389C2×D1311yescharacteristic
329#388—261no—
330#386—261no—
331#385—21no—
332#382C2×D1311yescharacteristic
333#380—21no—
334#380C2×D1311yescharacteristic
335C4×C4⋊C4—131no—
336C4×C22⋊C4—261no—
337C4×C22⋊C4C2×D1311yescharacteristic
338#340—131no—
339#340—131no—
340#340—131no—
341#340—131no—
342#340—131no—
343#340—131no—
344#336—131no—
345#336—131no—
346C22×C13—22no—
347C22×C13—22no—
348C22×C13—22no—
349C22×C13—41no—
350C22×C13—41no—
351C22×C13—41no—
352C22×C13—41no—
353C22×C13C2×C22⋊Q811yescharacteristic
354C22×C13D4211yescharacteristic
355C22×C13#54711yescharacteristic
356C22×C13#57511yescharacteristic
357C22×C13#57511yescharacteristic
358C22×C13#57611yescharacteristic
359C22×C13#57711yescharacteristic
360C2×D13—42no—
361C2×D13—42no—
362C2×D13—42no—
363C2×D13—81no—
364C2×D13—81no—
365C2×D13—81no—
366C2×D13—81no—
367C4×C13—22no—
368C4×C13—41no—
369C4×C13—41no—
370C4×C13—41no—
371C4×C13—41no—
372C4×C13—42no—
373C4×C13—81no—
374C4×C13—81no—
375C4×C13—81no—
376C4×C13#52511yescharacteristic
377C4×C13#53911yescharacteristic
378C4×C13D4×Q811yescharacteristic
379C4×C13#55411yescharacteristic
380Dic13—42no—
381Dic13—42no—
382Dic13—81no—
383C22×D4—131no—
384C22×D4—261no—
385C23×C4—21no—
386C23×C4—261no—
387C23×C4C22×D1311yescharacteristic
388C2×M16—262no—
389C2×M16—262no—
390C2×M16—262no—
391C22×C8—131no—
392C22×C8—131no—
393C22×C8—131no—
394C22×C8—131no—
395#125—22no—
396#125—41no—
397#121—134no—
398#121—262no—
399C4×D4—262no—
400C4×D4—521no—
401#117—262no—
402#117—521no—
403C2×C4⋊C4—22no—
404C2×C4⋊C4—21no—
405C2×C4⋊C4—21no—
406C2×C4⋊C4—21no—
407C2×C4⋊C4—21no—
408C2×C4⋊C4—21no—
409C2×C4⋊C4—41no—
410C2×C4⋊C4—131no—
411C2×C4⋊C4—131no—
412C2×C4⋊C4—131no—
413C2×C4⋊C4—261no—
414C2×C4⋊C4C22×D1311yescharacteristic
415C2×C4⋊C4C22×D1311yescharacteristic
416C2×C4⋊C4C22×D1311yescharacteristic
417C2×C4⋊C4C22×D1311yescharacteristic
418C2×C22⋊C4—21no—
419C2×C22⋊C4—261no—
420C2×C22⋊C4—261no—
421C2×C22⋊C4—261no—
422C2×C22⋊C4—261no—
423C2×C22⋊C4—261no—
424C2×C22⋊C4C22⋊D1311yescharacteristic
425C2×C22⋊C4C22⋊D1311yescharacteristic
426C2×C42—261no—
427C2×C42—261no—
428C2×C42C22×D1311yescharacteristic
429C2×C42C22×D1311yescharacteristic
430#107—134no—
431#107—262no—
432#107—521no—
433#106—262no—
434#106—262no—
435#106—262no—
436C4⋊C8—262no—
437C4⋊C8—262no—
438#102—262no—
439#102—262no—
440#102—262no—
441#102—521no—
442C22⋊C8—262no—
443#95—21no—
444#95—21no—
445#95—21no—
446#95—21no—
447#95—21no—
448#95—261no—
449#95—261no—
450#95C22⋊D1311yescharacteristic
451#95C22⋊D1311yescharacteristic
452C2×C13—42no—
453C2×C13#244911yescharacteristic
454C2×C13#308011yescharacteristic
455C2×C13#308111yescharacteristic
456C2×C13#327111yescharacteristic
457C2×C13#327311yescharacteristic
458C2×C13#334811yescharacteristic
459C2×C13#335511yescharacteristic
460D13—82no—
461C24—261no—
462C24C2×C22⋊D1311yescharacteristic
463C2×D4—132no—
464C2×D4—264no—
465C2×D4—261no—
466C2×D4—261no—
467C2×D4—261no—
468C2×D4—261no—
469C2×D4—261no—
470C2×D4—1041no—
471C2×D4—1041no—
472C22×C4—21no—
473C22×C4—21no—
474C22×C4—21no—
475C22×C4—21no—
476C22×C4—21no—
477C22×C4—21no—
478C22×C4—21no—
479C22×C4—21no—
480C22×C4—42no—
481C22×C4—42no—
482C22×C4—81no—
483C22×C4—261no—
484C22×C4—261no—
485C22×C4—261no—
486C22×C4—522no—
487C22×C4—522no—
488C22×C4—1041no—
489C22×C4Q8×D1312yes—
490C22×C4#613112yes—
491C22×C4C2×C22⋊D1311yescharacteristic
492C22×C4C2×C22⋊D1311yescharacteristic
493C22×C4C2×C22⋊D1311yescharacteristic
494C22×C4C2×C22⋊D1311yescharacteristic
495C22×C4C2×C22⋊D1311yescharacteristic
496C22×C4C23×D1311yescharacteristic
497M16—268no—
498C2×C8—132no—
499C2×C8—132no—
500C2×C8—261no—
501C2×C8—261no—
502C2×C8—261no—
503C2×C8—261no—
504C2×C8—261no—
505C2×C8—261no—
506C2×C8—261no—
507C2×C8—261no—
508C2×C8—261no—
509C2×C8—261no—
510C4⋊C4—21no—
511C4⋊C4—21no—
512C4⋊C4—21no—
513C4⋊C4—21no—
514C4⋊C4—21no—
515C4⋊C4—21no—
516C4⋊C4—21no—
517C4⋊C4—42no—
518C4⋊C4—42no—
519C4⋊C4—42no—
520C4⋊C4—81no—
521C4⋊C4—81no—
522C4⋊C4—132no—
523C4⋊C4—132no—
524C4⋊C4—261no—
525C4⋊C4—261no—
526C4⋊C4—261no—
527C4⋊C4—261no—
528C4⋊C4—1041no—
529C4⋊C4D4×D1312yes—
530C22⋊C4—41no—
531C22⋊C4—41no—
532C22⋊C4—42no—
533C22⋊C4—264no—
534C22⋊C4—522no—
535C22⋊C4—522no—
536C22⋊C4—522no—
537C22⋊C4—1041no—
538C42—21no—
539C42—21no—
540C42—21no—
541C42—21no—
542C42—522no—
543C42—1041no—
544C13#2636311yescharacteristic, complemented, Sylow
545C23—21no—
546C23—21no—
547C23—21no—
548C23—21no—
549C23—21no—
550C23—21no—
551C23—262no—
552C23—521no—
553C23—521no—
554C23—521no—
555C23—521no—
556C23—521no—
557C23—521no—
558C23#8820012yes—
559C23C22×C22⋊D1311yescharacteristic
560D4—262no—
561D4—262no—
562D4—521no—
563D4—521no—
564D4—1044no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—21no—
572C2×C4—21no—
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—21no—
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—41no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—84no—
596C2×C4—262no—
597C2×C4—262no—
598C2×C4—521no—
599C2×C4—521no—
600C2×C4—521no—
601C2×C4—521no—
602C2×C4—521no—
603C2×C4—521no—
604C2×C4—521no—
605C2×C4—1044no—
606C2×C4#8820012yes—
607C2×C4C2×D4×D1311yescharacteristic
608C2×C4#8825111yescharacteristic
609C2×C4C2×Q8×D1311yescharacteristic
610C2×C4C2×#613111yescharacteristic
611C2×C4#41×D1311yescharacteristic
612C2×C4#8825711yescharacteristic
613C8—262no—
614C8—262no—
615C8—521no—
616C8—521no—
617C22—22no—
618C22—22no—
619C22—22no—
620C22—41no—
621C22—41no—
622C22—41no—
623C22—41no—
624C22—522no—
625C22—522no—
626C22—522no—
627C22—1041no—
628C22—1041no—
629C22—1041no—
630C22—1041no—
631C22D4×C22⋊D1311yescharacteristic
632C22C2×#8820011yescharacteristic
633C22#1171952711yescharacteristic
634C22#1171952711yescharacteristic
635C22#1171952911yescharacteristic
636C22#1171954011yescharacteristic
637C22#1171954211yescharacteristic
638C4—22no—
639C4—41no—
640C4—41no—
641C4—41no—
642C4—42no—
643C4—41no—
644C4—81no—
645C4—81no—
646C4—81no—
647C4—522no—
648C4—522no—
649C4—1041no—
650C4#1171933211yescharacteristic
651C4#1171933711yescharacteristic
652C4#1171934111yescharacteristic
653C4#1171935011yescharacteristic
654C2—42no—
655C2—1042no—
656C2#1311864111yescharacteristic
657C2#1312465411yescharacteristic
658C2#1312466111yescharacteristic
659C2#1312642211yescharacteristic
660C2#1312644311yescharacteristic
661C2#1312734511yescharacteristic
662C2#1312743611yescharacteristic
subgroupG/H
centerC23C22×C22⋊D13
commutator, cosocleC22×C4×C13C24
FrattiniC22×C4C23×D13
FittingC13×#2442C2
socleC23×C13C22×D4
prime radicalC2×#88184C22
hypercenter#2442D13
hypocenterC13#26363
LCSG ▷ C22×C4×C13 ▷ C2×C13 ▷ C13
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2442

Complements

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

NH ≅ G/Ntype
1. C2×#11718727C2semidirect
2. C2×#11718625C2semidirect
4. #13116360C2semidirect
5. #13116338C2semidirect
6. #13116207C2semidirect
7. #13116171C2semidirect
8. #13115602C2semidirect
9. #13115404C2semidirect
12. #13115082C2semidirect
13. #13115033C2semidirect
14. #13115021C2semidirect
15. C13×#2442C2semidirect
127. #88047D4semidirect
216. #2442D13semidirect
544. C13#26363semidirect

Maximal subgroups

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

subgroupindex
C2×#117187272
C2×#117186252
#131178722
#131163602
#131163382
#131162072
#131161712
#131156022
#131154042
#131152722
#131151532
#131150822
#131150332
#131150212
C13×#24422
#2636313