Finite Group Databasebeta

← #15935660

Subgroups — #15935660

Order 3328.

708 conjugacy classes of subgroups.

The subgroup lattice has 604 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#13119269C211yesmaximal, characteristic
2#13116244C211yesmaximal, characteristic, complemented
3#13116231C211yesmaximal, characteristic, complemented
4#13116036C211yesmaximal, characteristic
5#13116036C211yesmaximal, characteristic
6#13115594C211yesmaximal, characteristic, complemented
7#13115566C211yesmaximal, characteristic, complemented
8#13115444C211yesmaximal, characteristic
9#13115428C211yesmaximal, characteristic, complemented
10#13115425C211yesmaximal, characteristic, complemented
11#13115090C211yesmaximal, characteristic, complemented
12#13115024C211yesmaximal, characteristic
13#13114926C211yesmaximal, characteristic
14C4×#88075C211yesmaximal, characteristic, complemented
15C13×#2360C211yesmaximal, characteristic
16C2×C13×C22⋊Q8C2211yescharacteristic
17C2×C4×Q8×C13C2211yescharacteristic
18C22×C13×C4⋊C4C2211yescharacteristic
19C13×#391C2211yescharacteristic
20C13×#389—21no—
21C13×#389C2211yescharacteristic
22C13×#387—21no—
23C13×#385—21no—
24C4×C13×C4⋊C4C2211yescharacteristic
25C4×C13×C22⋊C4—21no—
26C4×C13×C22⋊C4C2211yescharacteristic
27C2×#88075C2211yescharacteristic
28C2×#88075C2211yescharacteristic
29C2×#88075C2211yescharacteristic
30C2×#88075C2211yescharacteristic
31#11718873C2211yescharacteristic
32#11718873C2211yescharacteristic
33C2×C22⋊#1055C2211yescharacteristic
34C2×C22⋊#1055C2211yescharacteristic
35C2×#88047C2211yescharacteristic
36C2×#88047C2211yescharacteristic
37C2×#88047C2211yescharacteristic
38C2×#88047C2211yescharacteristic
39#11718612—21no—
40#11718609—21no—
41#11718596C2211yescharacteristic
42C4×C4⋊Dic13C2211yescharacteristic
43#11718589C2211yescharacteristic
44#11718589C2211yescharacteristic
45C2×C4⋊#1055C2211yescharacteristic
46C2×C4×#1055C2211yescharacteristic
47#11718321—21no—
48#11718316—21no—
49#11718121C2211yescharacteristic
50#11718121C2211yescharacteristic
51#11718121C2211yescharacteristic
52#11718121C2211yescharacteristic
53#11718121C2211yescharacteristic
54#11718121C2211yescharacteristic
55#11718121C2211yescharacteristic
56#11718121C2211yescharacteristic
57#11718116C2211yescharacteristic
58#11718116C2211yescharacteristic
59C22×Q8×C13C2311yescharacteristic
60C23×C4×C13—21no—
61C23×C4×C13C2311yescharacteristic
62C13×C22⋊Q8—22no—
63C13×C22⋊Q8—22no—
64C4×Q8×C13—22no—
65C4×Q8×C13—41no—
66C2×C13×C4⋊C4—21no—
67C2×C13×C4⋊C4—22no—
68C2×C13×C4⋊C4—21no—
69C2×C13×C4⋊C4—21no—
70C2×C13×C4⋊C4—21no—
71C2×C13×C4⋊C4—41no—
72C2×C13×C4⋊C4C2311yescharacteristic
73C2×C13×C4⋊C4C2311yescharacteristic
74C2×C13×C4⋊C4C2311yescharacteristic
75C2×C13×C22⋊C4—21no—
76C2×C13×C22⋊C4D411yescharacteristic
77C2×C13×C22⋊C4D411yescharacteristic
78C2×C42×C13—21no—
79C2×C42×C13—21no—
80C2×C42×C13C2311yescharacteristic
81C2×C42×C13C2311yescharacteristic
82C2×C22⋊Dic13—21no—
83C2×C22⋊Dic13—21no—
84C2×C4⋊Dic13—21no—
85C2×C4⋊Dic13—21no—
86C2×C4⋊Dic13C2311yescharacteristic
87C2×C4⋊Dic13C2311yescharacteristic
88C2×C4⋊Dic13C2311yescharacteristic
89C2×C4⋊Dic13C2311yescharacteristic
90C2×#6101—21no—
91C2×#6101—21no—
92C2×C4×Dic13—21no—
93C22×#1055C2311yescharacteristic
94C22×#1055C2311yescharacteristic
95C22×#1055C2311yescharacteristic
96C22×#1055C2311yescharacteristic
97C13×#95—21no—
98C13×#95—21no—
99C13×#95—21no—
100C13×#95—21no—
101C13×#95D411yescharacteristic
102C13×#95D411yescharacteristic
103#88075—22no—
104#88075—22no—
105#88075—22no—
106#88075—41no—
107#88071—21no—
108#88071—21no—
109#88071—21no—
110#88071—21no—
111#88071—21no—
112C22⋊#1055—22no—
113C22⋊#1055—41no—
114#88047—22no—
115#88047—22no—
116#88047—22no—
117#88047—41no—
118C4⋊#1055—22no—
119C4×#1055—41no—
120#27036—131nomaximal, Sylow
121C24×C13C2×D411yescharacteristic
122C2×Q8×C13—21no—
123C2×Q8×C13—21no—
124C2×Q8×C13—21no—
125C2×Q8×C13—21no—
126C2×Q8×C13—21no—
127C2×Q8×C13#4112yes—
128C22×C4×C13—21no—
129C22×C4×C13—21no—
130C22×C4×C13—21no—
131C22×C4×C13—21no—
132C22×C4×C13—21no—
133C22×C4×C13—21no—
134C22×C4×C13—21no—
135C22×C4×C13—21no—
136C22×C4×C13—21no—
137C22×C4×C13—21no—
138C22×C4×C13—42no—
139C22×C4×C13—42no—
140C22×C4×C13—81no—
141C22×C4×C13C2×D411yescharacteristic
142C22×C4×C13C2×D411yescharacteristic
143C22×C4×C13C2×D411yescharacteristic
144C22×C4×C13C2×D411yescharacteristic
145C22×C4×C13C2×D411yescharacteristic
146C22×C4×C13C2411yescharacteristic
147C22×Dic13—21no—
148C22×Dic13—21no—
149C22×Dic13—21no—
150C22×Dic13—21no—
151C13×C4⋊C4—24no—
152C13×C4⋊C4—21no—
153C13×C4⋊C4—21no—
154C13×C4⋊C4—21no—
155C13×C4⋊C4—21no—
156C13×C4⋊C4—42no—
157C13×C4⋊C4—42no—
158C13×C4⋊C4—81no—
159C13×C4⋊C4#4112yes—
160C13×C4⋊C4#4112yes—
161C13×C22⋊C4—24no—
162C13×C22⋊C4—41no—
163C13×C22⋊C4Q1614yes—
164C42×C13—21no—
165C42×C13—21no—
166C42×C13—21no—
167C42×C13—21no—
168C42×C13—42no—
169C42×C13—81no—
170C22⋊Dic13—81no—
171C22⋊Dic13—81no—
172C4⋊Dic13—21no—
173C4⋊Dic13—21no—
174C4⋊Dic13—21no—
175C4⋊Dic13—21no—
176C4⋊Dic13—21no—
177C4⋊Dic13—21no—
178C4⋊Dic13—21no—
179C4⋊Dic13—21no—
180C4⋊Dic13—81no—
181C4⋊Dic13—81no—
182#6101—42no—
183#6101—81no—
184C4×Dic13—81no—
185C2×#1055—21no—
186C2×#1055—21no—
187C2×#1055—21no—
188C2×#1055—21no—
189C2×#1055—21no—
190C2×#1055—21no—
191C2×#1055—21no—
192C2×#1055—21no—
193C2×#1055—21no—
194C2×#1055—21no—
195C2×#1055—21no—
196C2×#1055#4112yes—
197#2497—131no—
198#2360D1311yescharacteristic
199#2072—131no—
200#2068—131no—
201#2036—131no—
202#2036—131no—
203#1947—131no—
204#1942—131no—
205#1923—131no—
206#1921—131no—
207#1921—131no—
208#1863—131no—
209#1852—131no—
210#1824—131no—
211C4×#103—131no—
212C23×C13—21no—
213C23×C13—21no—
214C23×C13—21no—
215C23×C13—21no—
216C23×C13—21no—
217C23×C13—21no—
218C23×C13C2×Q1612yes—
219C23×C13C22×D411yescharacteristic
220Q8×C13—22no—
221Q8×C13—22no—
222Q8×C13—41no—
223Q8×C13—41no—
224C2×C4×C13—21no—
225C2×C4×C13—21no—
226C2×C4×C13—21no—
227C2×C4×C13—21no—
228C2×C4×C13—22no—
229C2×C4×C13—22no—
230C2×C4×C13—21no—
231C2×C4×C13—21no—
232C2×C4×C13—22no—
233C2×C4×C13—21no—
234C2×C4×C13—21no—
235C2×C4×C13—21no—
236C2×C4×C13—21no—
237C2×C4×C13—21no—
238C2×C4×C13—21no—
239C2×C4×C13—21no—
240C2×C4×C13—41no—
241C2×C4×C13—41no—
242C2×C4×C13—41no—
243C2×C4×C13—41no—
244C2×C4×C13—41no—
245C2×C4×C13—41no—
246C2×C4×C13—41no—
247C2×C4×C13—41no—
248C2×C4×C13—41no—
249C2×C4×C13—41no—
250C2×C4×C13—41no—
251C2×C4×C13—41no—
252C2×C4×C13—41no—
253C2×C4×C13—84no—
254C2×C4×C13C2×Q1612yes—
255C2×C4×C13C2×Q1612yes—
256C2×C4×C13C2×#4111yescharacteristic
257C2×C4×C13C2×#4111yescharacteristic
258C2×C4×C13C2×#4111yescharacteristic
259C2×C4×C13C2×#4111yescharacteristic
260C2×C4×C13H(5,𝔽2)11yescharacteristic
261C2×C4×C132−1+411yescharacteristic
262C2×Dic13—41no—
263C2×Dic13—41no—
264C2×Dic13—41no—
265C2×Dic13—41no—
266C2×Dic13—41no—
267C2×Dic13—41no—
268C2×Dic13—41no—
269C2×Dic13—41no—
270C2×Dic13—41no—
271C2×Dic13—41no—
272C2×Dic13—41no—
273C2×Dic13—41no—
274#1055—22no—
275#1055—41no—
276#1055—41no—
277#1055—41no—
278C2×C22⋊Q8C2×D1311yescharacteristic
279C2×C4×Q8C2×D1311yescharacteristic
280C22×C4⋊C4C2×D1311yescharacteristic
281C2×#107—131no—
282C2×#107—131no—
283C2×#107—131no—
284C2×#107—131no—
285C2×C4⋊C8—131no—
286C2×#103—131no—
287C2×#103—131no—
288C2×#103—131no—
289C2×#103—131no—
290C2×C22⋊C8—131no—
291C2×C22⋊C8—131no—
292C2×C4×C8—131no—
293#400—261no—
294#396—261no—
295#391C2×D1311yescharacteristic
296#389—21no—
297#389—131no—
298#389—131no—
299#389C2×D1311yescharacteristic
300#388—261no—
301#387—21no—
302#387—261no—
303#385—21no—
304#384—131no—
305#380—131no—
306#380—131no—
307C4×C4⋊C4—131no—
308C4×C4⋊C4C2×D1311yescharacteristic
309C4×C22⋊C4—21no—
310C4×C22⋊C4C2×D1311yescharacteristic
311#340—131no—
312#340—131no—
313#340—131no—
314#340—131no—
315#340—131no—
316#340—131no—
317#340—131no—
318#340—131no—
319#336—131no—
320#336—131no—
321C22×C13—22no—
322C22×C13—22no—
323C22×C13—22no—
324C22×C13—41no—
325C22×C13—41no—
326C22×C13—41no—
327C22×C13—41no—
328C22×C13#52511yescharacteristic
329C22×C13#54611yescharacteristic
330C22×C13#54711yescharacteristic
331C22×C13C22×Q1611yescharacteristic
332C22×C13#57511yescharacteristic
333C22×C13#57611yescharacteristic
334C22×C13#57711yescharacteristic
335C4×C13—22no—
336C4×C13—22no—
337C4×C13—41no—
338C4×C13—42no—
339C4×C13—41no—
340C4×C13—42no—
341C4×C13—41no—
342C4×C13—42no—
343C4×C13—81no—
344C4×C13#52911yescharacteristic
345C4×C13#53911yescharacteristic
346C4×C13#55311yescharacteristic
347C4×C13#55611yescharacteristic
348Dic13—81no—
349Dic13—81no—
350Dic13—81no—
351Dic13—81no—
352C22×Q8C22×D1311yescharacteristic
353C23×C4—21no—
354C23×C4C22×D1311yescharacteristic
355C22×C8—131no—
356C22×C8—131no—
357C22×C8—131no—
358C22×C8—131no—
359C22⋊Q8—22no—
360C22⋊Q8—22no—
361C4×Q8—22no—
362C4×Q8—41no—
363C2×C4⋊C4—21no—
364C2×C4⋊C4—22no—
365C2×C4⋊C4—21no—
366C2×C4⋊C4—21no—
367C2×C4⋊C4—21no—
368C2×C4⋊C4—41no—
369C2×C4⋊C4—131no—
370C2×C4⋊C4—131no—
371C2×C4⋊C4—131no—
372C2×C4⋊C4—131no—
373C2×C4⋊C4—261no—
374C2×C4⋊C4—261no—
375C2×C4⋊C4—261no—
376C2×C4⋊C4—261no—
377C2×C4⋊C4C22×D1311yescharacteristic
378C2×C4⋊C4C22×D1311yescharacteristic
379C2×C4⋊C4C22×D1311yescharacteristic
380C2×C22⋊C4—21no—
381C2×C22⋊C4—261no—
382C2×C22⋊C4—261no—
383C2×C22⋊C4C22⋊D1311yescharacteristic
384C2×C22⋊C4C22⋊D1311yescharacteristic
385C2×C42—21no—
386C2×C42—21no—
387C2×C42—261no—
388C2×C42C22×D1311yescharacteristic
389C2×C42C22×D1311yescharacteristic
390#107—262no—
391#107—262no—
392#107—262no—
393#107—521no—
394C4⋊C8—262no—
395#103—262no—
396#103—262no—
397#103—262no—
398#103—521no—
399C22⋊C8—262no—
400C22⋊C8—521no—
401C4×C8—521no—
402#95—21no—
403#95—21no—
404#95—21no—
405#95—21no—
406#95—261no—
407#95—261no—
408#95—261no—
409#95—261no—
410#95—261no—
411#95C22⋊D1311yescharacteristic
412#95C22⋊D1311yescharacteristic
413C2×C13—42no—
414C2×C13#246611yescharacteristic
415C2×C13#310411yescharacteristic
416C2×C13#311911yescharacteristic
417C2×C13#329911yescharacteristic
418C2×C13#331911yescharacteristic
419C2×C13#333811yescharacteristic
420C2×C13#335111yescharacteristic
421C24C2×C22⋊D1311yescharacteristic
422C2×Q8—21no—
423C2×Q8—21no—
424C2×Q8—21no—
425C2×Q8—21no—
426C2×Q8—21no—
427C2×Q8#612712yes—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—21no—
432C22×C4—21no—
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—21no—
438C22×C4—42no—
439C22×C4—42no—
440C22×C4—81no—
441C22×C4—261no—
442C22×C4—261no—
443C22×C4—261no—
444C22×C4—261no—
445C22×C4C2×C22⋊D1311yescharacteristic
446C22×C4C2×C22⋊D1311yescharacteristic
447C22×C4C2×C22⋊D1311yescharacteristic
448C22×C4C2×C22⋊D1311yescharacteristic
449C22×C4C2×C22⋊D1311yescharacteristic
450C22×C4C23×D1311yescharacteristic
451C2×C8—132no—
452C2×C8—261no—
453C2×C8—261no—
454C2×C8—261no—
455C2×C8—261no—
456C2×C8—261no—
457C2×C8—261no—
458C2×C8—261no—
459C2×C8—261no—
460C2×C8—261no—
461C2×C8—261no—
462C2×C8—261no—
463C4⋊C4—24no—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—21no—
468C4⋊C4—42no—
469C4⋊C4—42no—
470C4⋊C4—81no—
471C4⋊C4—261no—
472C4⋊C4—261no—
473C4⋊C4—261no—
474C4⋊C4—261no—
475C4⋊C4—261no—
476C4⋊C4—261no—
477C4⋊C4—261no—
478C4⋊C4—261no—
479C4⋊C4—522no—
480C4⋊C4—1041no—
481C4⋊C4—1041no—
482C4⋊C4—1041no—
483C4⋊C4#612712yes—
484C4⋊C4#612912yes—
485C22⋊C4—24no—
486C22⋊C4—41no—
487C22⋊C4—1041no—
488C22⋊C4—1041no—
489C22⋊C4#610714yes—
490C42—21no—
491C42—21no—
492C42—21no—
493C42—21no—
494C42—42no—
495C42—81no—
496C42—1041no—
497C13#2703611yescharacteristic, complemented, Sylow
498C23—21no—
499C23—21no—
500C23—21no—
501C23—21no—
502C23—21no—
503C23—21no—
504C23C2×#610712yes—
505C23C22×C22⋊D1311yescharacteristic
506Q8—22no—
507Q8—22no—
508Q8—41no—
509Q8—41no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—22no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—84no—
540C2×C4—521no—
541C2×C4—521no—
542C2×C4—521no—
543C2×C4—521no—
544C2×C4—521no—
545C2×C4—521no—
546C2×C4—521no—
547C2×C4—521no—
548C2×C4—521no—
549C2×C4—521no—
550C2×C4—521no—
551C2×C4—521no—
552C2×C4C2×#610712yes—
553C2×C4C2×#610712yes—
554C2×C4C2×#612711yescharacteristic
555C2×C4C2×#612711yescharacteristic
556C2×C4C2×#612911yescharacteristic
557C2×C4#8825111yescharacteristic
558C2×C4#41×D1311yescharacteristic
559C2×C4#8825711yescharacteristic
560C8—262no—
561C8—521no—
562C8—521no—
563C8—521no—
564C22—22no—
565C22—22no—
566C22—22no—
567C22—41no—
568C22—41no—
569C22—41no—
570C22—41no—
571C22#1171949611yescharacteristic
572C22#1171951111yescharacteristic
573C22C22×#610711yescharacteristic
574C22#1171952711yescharacteristic
575C22#1171952911yescharacteristic
576C22#1171954011yescharacteristic
577C22#1171954211yescharacteristic
578C4—22no—
579C4—22no—
580C4—41no—
581C4—42no—
582C4—41no—
583C4—42no—
584C4—42no—
585C4—41no—
586C4—81no—
587C4—1041no—
588C4—1041no—
589C4—1041no—
590C4—1041no—
591C4#1171924311yescharacteristic
592C4#1171924711yescharacteristic
593C4#1171926311yescharacteristic
594C4#1171926911yescharacteristic
595C2—42no—
596C2#1311886611yescharacteristic
597C2#1312487511yescharacteristic
598C2#1312503511yescharacteristic
599C2#1312671211yescharacteristic
600C2#1312699211yescharacteristic
601C2#1312722411yescharacteristic
602C2#1312739611yescharacteristic
subgroupG/H
centerC23C22×C22⋊D13
commutator, cosocleC22×C4×C13C24
FrattiniC22×C4C23×D13
FittingC13×#2360C2
socleC23×C13C22×D4
prime radicalC24×C13C2×D4
hypercenter#2360D13
hypocenterC13#27036
LCSG ▷ C22×C4×C13 ▷ C2×C13 ▷ C13
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2360

Complements

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

NH ≅ G/Ntype
2. #13116244C2semidirect
3. #13116231C2semidirect
6. #13115594C2semidirect
7. #13115566C2semidirect
9. #13115428C2semidirect
10. #13115425C2semidirect
11. #13115090C2semidirect
14. C4×#88075C2semidirect
497. C13#27036semidirect

Maximal subgroups

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

subgroupindex
#131192692
#131162442
#131162312
#131160362
#131155942
#131155662
#131154442
#131154282
#131154252
#131150902
#131150242
#131149262
C4×#880752
C13×#23602
#2703613