Finite Group Databasebeta

← #15919468

Subgroups — #15919468

Order 3328.

1322 conjugacy classes of subgroups.

The subgroup lattice has 618 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11718716C211yesmaximal, characteristic, complemented
2C2×#11718682C211yesmaximal, characteristic, complemented
3#13119828C211yesmaximal, characteristic, complemented
4#13116036C212yesmaximal, complemented
5#13116015C212yesmaximal, complemented
6#13115769C211yesmaximal, characteristic, complemented
7#13115635C211yesmaximal, characteristic, complemented
8#13114776C212yesmaximal, complemented
9#13114771C212yesmaximal, complemented
10#13114666C211yesmaximal, characteristic, complemented
11C2×C13×#548C211yesmaximal, characteristic
12C22×C13×#41C2211yescharacteristic
13C23×Q8×C13C2211yescharacteristic
14C13×#548—24no—
15C13×#548—24no—
16C2×C13×#124—21no—
17C2×C13×#124C2211yescharacteristic
18C2×C13×C22⋊Q8—21no—
19C2×C13×C22⋊Q8C2211yescharacteristic
20C2×C13×#121—21no—
21C2×C13×#121C2211yescharacteristic
22C2×C4×Q8×C13C2211yescharacteristic
23C2×C4×D4×C13—21no—
24C2×C4×D4×C13C2211yescharacteristic
25C2×#88190—21no—
26C2×#88123C2211yescharacteristic
27C2×#88075C2212yes—
28C2×#88075C2212yes—
29C2×#88075C2212yes—
30C2×#88075C2212yes—
31C2×#88072C2212yes—
32C2×#88072C2212yes—
33#11718873C2212yes—
34#11718873C2212yes—
35C2×C22⋊#1055C2212yes—
36C2×C22⋊#1055C2212yes—
37#11718716—24no—
38#11718682—24no—
39#11718635—21no—
40C2×#88047C2212yes—
41#11718614—21no—
42#11718614—21no—
43C4×C4⋊Dic13C2211yescharacteristic
44C2×C4⋊#1055C2211yescharacteristic
45C2×#88043C2211yescharacteristic
46#11718121C2212yes—
47C2×C13×#41—22no—
48C2×C13×#41—22no—
49C2×C13×#41—24no—
50C2×C13×#41—42no—
51C22×Q8×C13—22no—
52C22×Q8×C13—22no—
53C22×Q8×C13—21no—
54C22×Q8×C13—24no—
55C22×Q8×C13—42no—
56C22×Q8×C13C2311yescharacteristic
57C22×Q8×C13C2311yescharacteristic
58C22×D4×C13—21no—
59C22×D4×C13—21no—
60C22×D4×C13D411yescharacteristic
61C22×D4×C13C2311yescharacteristic
62C23×C4×C13—21no—
63C23×C4×C13—21no—
64C23×C4×C13C2311yescharacteristic
65C23×C4×C13C2311yescharacteristic
66C22×Dic26—21no—
67C13×#124—28no—
68C13×#124—22no—
69C13×#124D414yes—
70C13×C22⋊Q8—28no—
71C13×C22⋊Q8—22no—
72C13×C22⋊Q8—22no—
73C13×#121—28no—
74C13×#121—22no—
75C13×#121—22no—
76C4×Q8×C13—22no—
77C4×Q8×C13D414yes—
78C4×D4×C13—28no—
79C4×D4×C13—22no—
80C4×D4×C13—22no—
81C2×C13×C4⋊C4—21no—
82C2×C13×C4⋊C4—21no—
83C2×C13×C4⋊C4D411yescharacteristic
84C2×C13×C4⋊C4C2311yescharacteristic
85C2×C13×C22⋊C4—21no—
86C2×C13×C22⋊C4—21no—
87C2×C13×C22⋊C4—21no—
88C2×C13×C22⋊C4—21no—
89C2×C13×C22⋊C4D411yescharacteristic
90C2×C13×C22⋊C4D411yescharacteristic
91C2×C42×C13—21no—
92C2×C42×C13C2311yescharacteristic
93C2×C22⋊Dic13—22no—
94C2×C22⋊Dic13—22no—
95#88190—44no—
96C2×C4⋊Dic13—21no—
97C2×C4⋊Dic13C2312yes—
98C2×C4⋊Dic13C2312yes—
99C2×C4×Dic13—21no—
100C22×#1055C2312yes—
101C22×#1055C2312yes—
102#88123—22no—
103#88123—22no—
104#88075—24no—
105#88075—24no—
106#88075—24no—
107#88075—42no—
108#88072—24no—
109#88072—42no—
110#88071—21no—
111#88071—22no—
112#88071—22no—
113C22⋊#1055—42no—
114C22⋊#1055—42no—
115#88047—24no—
116C4⋊#1055—22no—
117#88043—22no—
118#25896—131nomaximal, Sylow
119C24×C13—21no—
120C24×C13C2×D411yescharacteristic
121C24×C13C2×D411yescharacteristic
122C13×#41—48no—
123C13×#41—44no—
124C13×#41—44no—
125C2×Q8×C13—24no—
126C2×Q8×C13—21no—
127C2×Q8×C13—21no—
128C2×Q8×C13—22no—
129C2×Q8×C13—22no—
130C2×Q8×C13—22no—
131C2×Q8×C13—44no—
132C2×Q8×C13—48no—
133C2×Q8×C13—44no—
134C2×Q8×C13—44no—
135C2×Q8×C13C2×D412yes—
136C2×Q8×C13C2×D412yes—
137C2×Q8×C13#4114yes—
138C2×D4×C13—24no—
139C2×D4×C13—24no—
140C2×D4×C13—22no—
141C2×D4×C13—22no—
142C2×D4×C13—21no—
143C2×D4×C13—22no—
144C2×D4×C13—22no—
145C2×D4×C13—41no—
146C2×D4×C13—44no—
147C2×D4×C13—42no—
148C2×D4×C13—42no—
149C2×D4×C13C2×D412yes—
150C22×C4×C13—21no—
151C22×C4×C13—24no—
152C22×C4×C13—24no—
153C22×C4×C13—21no—
154C22×C4×C13—21no—
155C22×C4×C13—21no—
156C22×C4×C13—21no—
157C22×C4×C13—21no—
158C22×C4×C13—21no—
159C22×C4×C13—22no—
160C22×C4×C13—22no—
161C22×C4×C13—22no—
162C22×C4×C13—22no—
163C22×C4×C13—44no—
164C22×C4×C13—44no—
165C22×C4×C13C2×D411yescharacteristic
166C22×C4×C13C2×D411yescharacteristic
167C22×C4×C13C2×D411yescharacteristic
168C22×C4×C13C2×D411yescharacteristic
169C22×C4×C13C2411yescharacteristic
170C22×Dic13—22no—
171C22×Dic13—22no—
172C2×Dic26—42no—
173C2×Dic26—42no—
174C2×Dic26—42no—
175C13×C4⋊C4—24no—
176C13×C4⋊C4—24no—
177C13×C4⋊C4—22no—
178C13×C4⋊C4—21no—
179C13×C4⋊C4C2×D412yes—
180C13×C22⋊C4—24no—
181C13×C22⋊C4—24no—
182C13×C22⋊C4—24no—
183C13×C22⋊C4—24no—
184C13×C22⋊C4—22no—
185C13×C22⋊C4—22no—
186C42×C13—24no—
187C42×C13C2×D412yes—
188C42×C13C2×D412yes—
189C22⋊Dic13—44no—
190C22⋊Dic13—44no—
191C4⋊Dic13—22no—
192C4⋊Dic13—22no—
193C4⋊Dic13—22no—
194C4⋊Dic13—22no—
195C4⋊Dic13—42no—
196C4×Dic13—42no—
197C2×#1055—22no—
198C2×#1055—22no—
199C2×#1055—22no—
200C2×#1055—22no—
201C2×#1055—22no—
202C2×#1055—22no—
203C2×#548D1311yescharacteristic
204C2×#489—131no—
205C2×#477—131no—
206#2547—131no—
207#2036—132no—
208#2033—132no—
209#1983—131no—
210#1955—131no—
211#1788—132no—
212#1787—132no—
213#1763—131no—
214C23×C13—21no—
215C23×C13—22no—
216C23×C13—21no—
217C23×C13—22no—
218C23×C13—21no—
219C23×C13—22no—
220C23×C13—22no—
221C23×C13—21no—
222C23×C13—21no—
223C23×C13—22no—
224C23×C13—42no—
225C23×C13—41no—
226C23×C13—41no—
227C23×C13—42no—
228C23×C13AGL(1,ℤ/8)12yes—
229C23×C13C22×D411yescharacteristic
230Q8×C13—22no—
231Q8×C13—24no—
232Q8×C13—24no—
233Q8×C13—48no—
234Q8×C13—41no—
235Q8×C13—41no—
236Q8×C13#12414yescomplemented
237D4×C13—24no—
238D4×C13—44no—
239D4×C13—48no—
240D4×C13—48no—
241D4×C13—41no—
242D4×C13—41no—
243C2×C4×C13—22no—
244C2×C4×C13—24no—
245C2×C4×C13—21no—
246C2×C4×C13—21no—
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—22no—
254C2×C4×C13—48no—
255C2×C4×C13—48no—
256C2×C4×C13—42no—
257C2×C4×C13—41no—
258C2×C4×C13—41no—
259C2×C4×C13—41no—
260C2×C4×C13—41no—
261C2×C4×C13—41no—
262C2×C4×C13—41no—
263C2×C4×C13—42no—
264C2×C4×C13—42no—
265C2×C4×C13#12012yes—
266C2×C4×C13#12012yes—
267C2×C4×C13C22×D411yescharacteristic
268C2×C4×C13C22×D411yescharacteristic
269C2×C4×C13C2×#4112yes—
270C2×C4×C13H(5,𝔽2)12yes—
271C2×Dic13—42no—
272C2×Dic13—42no—
273C2×Dic13—42no—
274C2×Dic13—42no—
275C2×Dic13—42no—
276C2×Dic13—42no—
277Dic26—84no—
278#1055—42no—
279#1055—42no—
280C22×#41C2×D1311yescharacteristic
281C23×Q8C2×D1311yescharacteristic
282#548—24no—
283#548—24no—
284C2×C4⋊Q8—131no—
285C2×#124—21no—
286C2×#124—261no—
287C2×#124C2×D1311yescharacteristic
288C2×C22⋊Q8—21no—
289C2×C22⋊Q8C2×D1311yescharacteristic
290C2×#121—21no—
291C2×#121C2×D1311yescharacteristic
292C2×C4×Q8C2×D1311yescharacteristic
293C2×C4×D4—21no—
294C2×C4×D4C2×D1311yescharacteristic
295#489—264no—
296#477—264no—
297C2×#107—132no—
298C2×C4⋊C8—131no—
299C2×#103—132no—
300C2×#103—132no—
301C2×#103—132no—
302C2×#103—132no—
303C2×#102—132no—
304C2×#102—132no—
305C2×C22⋊C8—132no—
306C2×C22⋊C8—132no—
307C2×#97—131no—
308#397—261no—
309#397—261no—
310#391—261no—
311#380—132no—
312#380—132no—
313C4×C4⋊C4—131no—
314#340—132no—
315C22×C13—22no—
316C22×C13—24no—
317C22×C13—41no—
318C22×C13—41no—
319C22×C13—42no—
320C22×C13—42no—
321C22×C13—41no—
322C22×C13—41no—
323C22×C13—42no—
324C22×C13—41no—
325C22×C13—42no—
326C22×C13—42no—
327C22×C13—42no—
328C22×C13—82no—
329C22×C13—82no—
330C22×C13C2×#12011yescharacteristic
331C22×C13#54612yes—
332C22×C13C2×AGL(1,ℤ/8)11yescharacteristic
333C22×C13#57511yescharacteristic
334C22×C13#57812yes—
335C4×C13—24no—
336C4×C13—22no—
337C4×C13—22no—
338C4×C13—41no—
339C4×C13—41no—
340C4×C13—42no—
341C4×C13—42no—
342C4×C13C2×#12411yescharacteristic
343C4×C13#53511yescharacteristic
344C4×C13#54612yes—
345Dic13—82no—
346Dic13—82no—
347C2×#41—22no—
348C2×#41—22no—
349C2×#41—24no—
350C2×#41—42no—
351C22×Q8—22no—
352C22×Q8—22no—
353C22×Q8—21no—
354C22×Q8—24no—
355C22×Q8—42no—
356C22×Q8—261no—
357C22×Q8C22×D1311yescharacteristic
358C22×Q8C22×D1311yescharacteristic
359C22×D4—21no—
360C22×D4—21no—
361C22×D4C22⋊D1311yescharacteristic
362C22×D4C22×D1311yescharacteristic
363C23×C4—21no—
364C23×C4—21no—
365C23×C4C22×D1311yescharacteristic
366C23×C4C22×D1311yescharacteristic
367C22×C8—132no—
368C22×C8—132no—
369C4⋊Q8—262no—
370C4⋊Q8—262no—
371#124—28no—
372#124—22no—
373#124—524no—
374#124C22⋊D1314yes—
375C22⋊Q8—28no—
376C22⋊Q8—22no—
377C22⋊Q8—22no—
378#121—28no—
379#121—22no—
380#121—22no—
381C4×Q8—22no—
382C4×Q8C22⋊D1314yes—
383C4×D4—28no—
384C4×D4—22no—
385C4×D4—22no—
386C2×C4⋊C4—21no—
387C2×C4⋊C4—21no—
388C2×C4⋊C4—132no—
389C2×C4⋊C4—132no—
390C2×C4⋊C4—261no—
391C2×C4⋊C4C22⋊D1311yescharacteristic
392C2×C4⋊C4C22×D1311yescharacteristic
393C2×C22⋊C4—21no—
394C2×C22⋊C4—21no—
395C2×C22⋊C4—21no—
396C2×C22⋊C4—21no—
397C2×C22⋊C4—262no—
398C2×C22⋊C4—262no—
399C2×C22⋊C4C22⋊D1311yescharacteristic
400C2×C22⋊C4C22⋊D1311yescharacteristic
401C2×C42—21no—
402C2×C42—261no—
403C2×C42C22×D1311yescharacteristic
404#107—264no—
405C4⋊C8—262no—
406#103—264no—
407#103—264no—
408#103—264no—
409#103—522no—
410#102—264no—
411#102—522no—
412C22⋊C8—522no—
413C22⋊C8—522no—
414#97—262no—
415#95—261no—
416#95—262no—
417#95—262no—
418C2×C13—42no—
419C2×C13—81no—
420C2×C13—82no—
421C2×C13#243511yescharacteristic
422C2×C13#300611yescharacteristic
423C2×C13#301111yescharacteristic
424C2×C13#330812yes—
425C2×C13#331312yes—
426C24—21no—
427C24C2×C22⋊D1311yescharacteristic
428C24C2×C22⋊D1311yescharacteristic
429#41—48no—
430#41—44no—
431#41—44no—
432C2×Q8—24no—
433C2×Q8—22no—
434C2×Q8—21no—
435C2×Q8—21no—
436C2×Q8—22no—
437C2×Q8—22no—
438C2×Q8—44no—
439C2×Q8—48no—
440C2×Q8—44no—
441C2×Q8—44no—
442C2×Q8—522no—
443C2×Q8—522no—
444C2×Q8—522no—
445C2×Q8#612914yes—
446C2×Q8C2×C22⋊D1312yes—
447C2×Q8C2×C22⋊D1312yes—
448C2×D4—24no—
449C2×D4—24no—
450C2×D4—22no—
451C2×D4—22no—
452C2×D4—22no—
453C2×D4—22no—
454C2×D4—21no—
455C2×D4—41no—
456C2×D4—44no—
457C2×D4—42no—
458C2×D4—42no—
459C2×D4C2×C22⋊D1312yes—
460C22×C4—21no—
461C22×C4—24no—
462C22×C4—24no—
463C22×C4—21no—
464C22×C4—21no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—21no—
470C22×C4—21no—
471C22×C4—21no—
472C22×C4—21no—
473C22×C4—44no—
474C22×C4—44no—
475C22×C4—262no—
476C22×C4—262no—
477C22×C4C2×C22⋊D1311yescharacteristic
478C22×C4C2×C22⋊D1311yescharacteristic
479C22×C4C2×C22⋊D1311yescharacteristic
480C22×C4C2×C22⋊D1311yescharacteristic
481C22×C4C23×D1311yescharacteristic
482C2×C8—262no—
483C2×C8—262no—
484C2×C8—262no—
485C2×C8—262no—
486C2×C8—262no—
487C2×C8—262no—
488C4⋊C4—24no—
489C4⋊C4—24no—
490C4⋊C4—22no—
491C4⋊C4—21no—
492C4⋊C4—262no—
493C4⋊C4—262no—
494C4⋊C4—262no—
495C4⋊C4—262no—
496C4⋊C4—522no—
497C4⋊C4C2×C22⋊D1312yes—
498C22⋊C4—24no—
499C22⋊C4—24no—
500C22⋊C4—24no—
501C22⋊C4—24no—
502C22⋊C4—22no—
503C22⋊C4—22no—
504C22⋊C4—524no—
505C22⋊C4—524no—
506C42—24no—
507C42—522no—
508C42C2×C22⋊D1312yes—
509C42C2×C22⋊D1312yes—
510C13#2589611yescharacteristic, complemented, Sylow
511C23—22no—
512C23—21no—
513C23—22no—
514C23—21no—
515C23—22no—
516C23—21no—
517C23—21no—
518C23—22no—
519C23—21no—
520C23—22no—
521C23—42no—
522C23—41no—
523C23—41no—
524C23—42no—
525C23#8820312yes—
526C23C22×C22⋊D1311yescharacteristic
527Q8—22no—
528Q8—24no—
529Q8—24no—
530Q8—48no—
531Q8—41no—
532Q8—41no—
533Q8—1044no—
534Q8#8819014yescomplemented
535D4—24no—
536D4—44no—
537D4—48no—
538D4—48no—
539D4—41no—
540D4—41no—
541C2×C4—24no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—48no—
553C2×C4—48no—
554C2×C4—42no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—522no—
564C2×C4—522no—
565C2×C4—522no—
566C2×C4—522no—
567C2×C4—522no—
568C2×C4—522no—
569C2×C4#8820712yes—
570C2×C4#8820712yes—
571C2×C4C2×#612912yes—
572C2×C4#8825112yes—
573C2×C4C22×C22⋊D1311yescharacteristic
574C2×C4C22×C22⋊D1311yescharacteristic
575C8—522no—
576C8—522no—
577C22—24no—
578C22—22no—
579C22—41no—
580C22—41no—
581C22—42no—
582C22—42no—
583C22—41no—
584C22—41no—
585C22—42no—
586C22—42no—
587C22—42no—
588C22—41no—
589C22—42no—
590C22—82no—
591C22—82no—
592C22#1171951112yes—
593C22C2×#8820311yescharacteristic
594C22#1171952711yescharacteristic
595C22#1171954312yes—
596C22C2×#8820711yescharacteristic
597C4—24no—
598C4—22no—
599C4—22no—
600C4—41no—
601C4—41no—
602C4—42no—
603C4—42no—
604C4—1042no—
605C4—1042no—
606C4C2×#8819011yescharacteristic
607C4#1171951011yescharacteristic
608C4#1171951112yes—
609C2—42no—
610C2—81no—
611C2—82no—
612C2#1311848311yescharacteristic
613C2#1312408811yescharacteristic
614C2#1312413111yescharacteristic
615C2#1312682712yes—
616C2#1312690212yes—
subgroupG/H
centerC23C22×C22⋊D13
commutator, cosocleC22×C4×C13C24
FrattiniC22×C4C23×D13
Fitting, prime radicalC2×C13×#548C2
socleC23×C13C22×D4
hypercenterC2×#548D13
hypocenterC13#25896
LCSG ▷ C22×C4×C13 ▷ C2×C13 ▷ C13
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ C2×#548

Complements

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

NH ≅ G/Ntype
1. C2×#11718716C2semidirect
2. C2×#11718682C2semidirect
3. #13119828C2semidirect
4. #13116036C2semidirect
5. #13116015C2semidirect
6. #13115769C2semidirect
7. #13115635C2semidirect
8. #13114776C2semidirect
9. #13114771C2semidirect
10. #13114666C2semidirect
236. Q8×C13#124semidirect
510. C13#25896semidirect
534. Q8#88190semidirect

Maximal subgroups

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

subgroupindex
C2×#117187162
C2×#117186822
#131198282
#131160362
#131160152
#131157692
#131156352
#131147762
#131147712
#131146662
C2×C13×#5482
#2589613