Finite Group Databasebeta

← #15904361

Subgroups — #15904361

Order 3328.

738 conjugacy classes of subgroups.

The subgroup lattice has 614 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11718683C211yesmaximal, characteristic, complemented
2C2×#11718661C211yesmaximal, characteristic
3#13120666C211yesmaximal, characteristic
4#13116272C211yesmaximal, characteristic
5#13116267C211yesmaximal, characteristic
6#13116029C211yesmaximal, characteristic
7#13116029C211yesmaximal, characteristic
8#13115793C211yesmaximal, characteristic, complemented
9#13115599C211yesmaximal, characteristic, complemented
10#13115576C211yesmaximal, characteristic, complemented
11#13115561C211yesmaximal, characteristic, complemented
12#13115529C211yesmaximal, characteristic, complemented
13#13115515C211yesmaximal, characteristic, complemented
14#13114841C211yesmaximal, characteristic, complemented
15C13×#2323C211yesmaximal, characteristic
16C2×C13×C22⋊Q8C2211yescharacteristic
17C2×C4×Q8×C13C2211yescharacteristic
18C2×C13×#117C2211yescharacteristic
19C2×#88124C2211yescharacteristic
20C13×#391C2211yescharacteristic
21C13×#389—21no—
22C13×#389C2211yescharacteristic
23C13×#388—21no—
24C13×#388—21no—
25C13×#387—21no—
26C13×#380C2211yescharacteristic
27C4×C13×C4⋊C4C2211yescharacteristic
28C2×#88075C2211yescharacteristic
29C2×#88075C2211yescharacteristic
30C2×#88075C2211yescharacteristic
31C2×#88075C2211yescharacteristic
32#11718873C2211yescharacteristic
33#11718873C2211yescharacteristic
34C2×C22⋊#1055C2211yescharacteristic
35C2×C22⋊#1055C2211yescharacteristic
36#11718683—24no—
37#11718661—24no—
38#11718645—21no—
39#11718645—21no—
40C2×#88048C2211yescharacteristic
41C2×#88048C2211yescharacteristic
42C2×#88048C2211yescharacteristic
43C2×#88047C2211yescharacteristic
44#11718609—21no—
45#11718609—21no—
46#11718596C2211yescharacteristic
47#11718589C2211yescharacteristic
48#11718589C2211yescharacteristic
49C2×C4⋊#1055C2211yescharacteristic
50C2×C4⋊#1055C2211yescharacteristic
51C2×#88043C2211yescharacteristic
52C2×C4×#1055C2211yescharacteristic
53#11718121C2211yescharacteristic
54#11718121C2211yescharacteristic
55#11718121C2211yescharacteristic
56#11718121C2211yescharacteristic
57#11718121C2211yescharacteristic
58#11718121C2211yescharacteristic
59#11718121C2211yescharacteristic
60#11718121C2211yescharacteristic
61C22×Q8×C13C2311yescharacteristic
62C23×C4×C13C2311yescharacteristic
63C13×C22⋊Q8—22no—
64C13×C22⋊Q8—22no—
65C4×Q8×C13—22no—
66C4×Q8×C13—22no—
67C13×#117—22no—
68C13×#117—22no—
69C2×C13×C4⋊C4—21no—
70C2×C13×C4⋊C4—21no—
71C2×C13×C4⋊C4—21no—
72C2×C13×C4⋊C4D411yescharacteristic
73C2×C13×C4⋊C4D411yescharacteristic
74C2×C13×C4⋊C4C2311yescharacteristic
75C2×C13×C4⋊C4C2311yescharacteristic
76C2×C13×C4⋊C4C2311yescharacteristic
77C2×C13×C22⋊C4—21no—
78C2×C13×C22⋊C4—21no—
79C2×C13×C22⋊C4D411yescharacteristic
80C2×C13×C22⋊C4D411yescharacteristic
81C2×C42×C13—21no—
82C2×C42×C13—21no—
83C2×C42×C13C2311yescharacteristic
84C2×C42×C13C2311yescharacteristic
85C2×C22⋊Dic13—21no—
86C2×C22⋊Dic13—21no—
87C2×C4⋊Dic13—21no—
88C2×C4⋊Dic13—21no—
89C2×C4⋊Dic13C2311yescharacteristic
90C2×C4⋊Dic13C2311yescharacteristic
91C2×C4⋊Dic13C2311yescharacteristic
92C2×C4⋊Dic13C2311yescharacteristic
93C2×#6101—21no—
94C2×#6101—21no—
95C2×#6101—21no—
96C2×#6101—21no—
97C22×#1055C2311yescharacteristic
98C22×#1055C2311yescharacteristic
99C22×#1055C2311yescharacteristic
100C22×#1055C2311yescharacteristic
101#88124—22no—
102#88124—22no—
103C13×#95—21no—
104C13×#95—21no—
105C13×#95—21no—
106C13×#95—21no—
107C13×#95—21no—
108#88075—22no—
109#88075—22no—
110#88075—22no—
111#88075—22no—
112#88071—21no—
113#88071—21no—
114#88071—21no—
115#88071—21no—
116C22⋊#1055—22no—
117C22⋊#1055—22no—
118#88048—22no—
119#88048—22no—
120#88048—22no—
121#88047—22no—
122C4⋊#1055—22no—
123C4⋊#1055D414yes—
124#88043—22no—
125C4×#1055—22no—
126#24719—131nomaximal, Sylow
127C24×C13C2×D411yescharacteristic
128C2×Q8×C13—21no—
129C2×Q8×C13—21no—
130C2×Q8×C13—21no—
131C2×Q8×C13—21no—
132C2×Q8×C13—21no—
133C2×Q8×C13#4112yes—
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—21no—
142C22×C4×C13—21no—
143C22×C4×C13—21no—
144C22×C4×C13C2×D411yescharacteristic
145C22×C4×C13C2×D411yescharacteristic
146C22×C4×C13C2×D411yescharacteristic
147C22×C4×C13C2×D411yescharacteristic
148C22×C4×C13C2×D411yescharacteristic
149C22×C4×C13C2411yescharacteristic
150C22×Dic13—21no—
151C22×Dic13—21no—
152C22×Dic13—21no—
153C22×Dic13—21no—
154C13×C4⋊C4—24no—
155C13×C4⋊C4—22no—
156C13×C4⋊C4—22no—
157C13×C4⋊C4—21no—
158C13×C4⋊C4—21no—
159C13×C4⋊C4—21no—
160C13×C4⋊C4—21no—
161C13×C4⋊C4—21no—
162C13×C4⋊C4—42no—
163C13×C4⋊C4—81no—
164C13×C4⋊C4#4112yes—
165C13×C22⋊C4—24no—
166C13×C22⋊C4—22no—
167C13×C22⋊C4—22no—
168C13×C22⋊C4—81no—
169C42×C13—21no—
170C42×C13—21no—
171C42×C13—42no—
172C42×C13—81no—
173C42×C13C2×D412yes—
174C42×C13#4112yes—
175C22⋊Dic13—81no—
176C22⋊Dic13—81no—
177C4⋊Dic13—21no—
178C4⋊Dic13—21no—
179C4⋊Dic13—21no—
180C4⋊Dic13—21no—
181C4⋊Dic13—21no—
182C4⋊Dic13—21no—
183C4⋊Dic13—21no—
184C4⋊Dic13—21no—
185C4⋊Dic13—81no—
186C4⋊Dic13—81no—
187#6101—42no—
188#6101—42no—
189#6101—81no—
190#6101—81no—
191C2×#1055—21no—
192C2×#1055—21no—
193C2×#1055—21no—
194C2×#1055—21no—
195C2×#1055—21no—
196C2×#1055—21no—
197C2×#1055—21no—
198C2×#1055—21no—
199C2×#1055—21no—
200C2×#1055—21no—
201C2×#1055C2×D412yes—
202C2×#1055C2×D412yes—
203C2×#479—131no—
204C2×#433—131no—
205#2629—131no—
206#2323D1311yescharacteristic
207#2077—131no—
208#2076—131no—
209#2035—131no—
210#2035—131no—
211#1989—131no—
212#1948—131no—
213#1944—131no—
214#1941—131no—
215#1935—131no—
216#1933—131no—
217#1803—131no—
218C23×C13—21no—
219C23×C13—21no—
220C23×C13—21no—
221C23×C13—21no—
222C23×C13—21no—
223C23×C13—21no—
224C23×C13—21no—
225C23×C13C22×D411yescharacteristic
226Q8×C13—22no—
227Q8×C13—22no—
228Q8×C13—41no—
229Q8×C13—41no—
230C2×C4×C13—21no—
231C2×C4×C13—21no—
232C2×C4×C13—21no—
233C2×C4×C13—21no—
234C2×C4×C13—21no—
235C2×C4×C13—21no—
236C2×C4×C13—21no—
237C2×C4×C13—22no—
238C2×C4×C13—22no—
239C2×C4×C13—21no—
240C2×C4×C13—21no—
241C2×C4×C13—21no—
242C2×C4×C13—21no—
243C2×C4×C13—21no—
244C2×C4×C13—21no—
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—41no—
254C2×C4×C13—41no—
255C2×C4×C13—41no—
256C2×C4×C13—41no—
257C2×C4×C13—41no—
258C2×C4×C13—41no—
259C2×C4×C13#12112yes—
260C2×C4×C13#12112yes—
261C2×C4×C13C22×D411yescharacteristic
262C2×C4×C13C2×#4111yescharacteristic
263C2×C4×C13C2×#4111yescharacteristic
264C2×C4×C13C2×#4111yescharacteristic
265C2×C4×C132−1+411yescharacteristic
266C2×C4×C132−1+411yescharacteristic
267C2×Dic13—41no—
268C2×Dic13—41no—
269C2×Dic13—41no—
270C2×Dic13—41no—
271C2×Dic13—41no—
272C2×Dic13—41no—
273C2×Dic13—41no—
274C2×Dic13—41no—
275C2×Dic13—41no—
276C2×Dic13—41no—
277C2×Dic13—41no—
278C2×Dic13—41no—
279#1055—22no—
280#1055—22no—
281#1055—41no—
282#1055—41no—
283C2×#125—131no—
284C2×C22⋊Q8C2×D1311yescharacteristic
285C2×C4×Q8C2×D1311yescharacteristic
286C2×#117C2×D1311yescharacteristic
287#479—264no—
288#433—264no—
289C2×#107—131no—
290C2×#106—131no—
291C2×#106—131no—
292C2×#106—131no—
293C2×C4⋊C8—131no—
294C2×C4⋊C8—131no—
295C2×#103—131no—
296C2×#103—131no—
297C2×#103—131no—
298C2×#103—131no—
299C2×C22⋊C8—131no—
300C2×C22⋊C8—131no—
301C2×#97—131no—
302C2×C4×C8—131no—
303#398—261no—
304#398—261no—
305#396—261no—
306#396—261no—
307#391C2×D1311yescharacteristic
308#389—21no—
309#389—131no—
310#389—131no—
311#389C2×D1311yescharacteristic
312#388—21no—
313#388—21no—
314#387—21no—
315#384—131no—
316#380—131no—
317#380—131no—
318#380C2×D1311yescharacteristic
319C4×C4⋊C4C2×D1311yescharacteristic
320#340—131no—
321#340—131no—
322#340—131no—
323#340—131no—
324#340—131no—
325#340—131no—
326#340—131no—
327#340—131no—
328C22×C13—41no—
329C22×C13—41no—
330C22×C13—41no—
331C22×C13—41no—
332C22×C13—41no—
333C22×C13—41no—
334C22×C13—41no—
335C22×C13C2×#12111yescharacteristic
336C22×C13#52511yescharacteristic
337C22×C13#53611yescharacteristic
338C22×C13#57711yescharacteristic
339C22×C13#57711yescharacteristic
340C22×C13#57711yescharacteristic
341C22×C13#57811yescharacteristic
342C4×C13—22no—
343C4×C13—22no—
344C4×C13—42no—
345C4×C13—41no—
346C4×C13—42no—
347C4×C13—41no—
348C4×C13—41no—
349C4×C13—81no—
350C4×C13—81no—
351C4×C13#54711yescharacteristic
352C4×C13#54811yescharacteristic
353C4×C13#55211yescharacteristic
354C4×C13#55611yescharacteristic
355Dic13—81no—
356Dic13—81no—
357Dic13—81no—
358Dic13—81no—
359C22×Q8C22×D1311yescharacteristic
360C23×C4C22×D1311yescharacteristic
361C22×C8—131no—
362C22×C8—131no—
363C22×C8—131no—
364C22×C8—131no—
365#125—262no—
366#125—262no—
367C22⋊Q8—22no—
368C22⋊Q8—22no—
369C4×Q8—22no—
370C4×Q8—22no—
371#117—22no—
372#117—22no—
373C2×C4⋊C4—21no—
374C2×C4⋊C4—21no—
375C2×C4⋊C4—21no—
376C2×C4⋊C4—131no—
377C2×C4⋊C4—131no—
378C2×C4⋊C4—131no—
379C2×C4⋊C4—131no—
380C2×C4⋊C4—261no—
381C2×C4⋊C4—261no—
382C2×C4⋊C4—261no—
383C2×C4⋊C4—261no—
384C2×C4⋊C4—261no—
385C2×C4⋊C4—261no—
386C2×C4⋊C4C22⋊D1311yescharacteristic
387C2×C4⋊C4C22⋊D1311yescharacteristic
388C2×C4⋊C4C22×D1311yescharacteristic
389C2×C4⋊C4C22×D1311yescharacteristic
390C2×C4⋊C4C22×D1311yescharacteristic
391C2×C22⋊C4—21no—
392C2×C22⋊C4—21no—
393C2×C22⋊C4—261no—
394C2×C22⋊C4—261no—
395C2×C22⋊C4C22⋊D1311yescharacteristic
396C2×C22⋊C4C22⋊D1311yescharacteristic
397C2×C42—21no—
398C2×C42—21no—
399C2×C42C22×D1311yescharacteristic
400C2×C42C22×D1311yescharacteristic
401#107—262no—
402#106—262no—
403#106—262no—
404#106—262no—
405C4⋊C8—134no—
406C4⋊C8—262no—
407#103—262no—
408#103—262no—
409#103—262no—
410#103—262no—
411C22⋊C8—262no—
412C22⋊C8—262no—
413#97—262no—
414C4×C8—262no—
415#95—21no—
416#95—21no—
417#95—21no—
418#95—21no—
419#95—21no—
420#95—261no—
421#95—261no—
422#95—261no—
423#95—261no—
424C2×C13—81no—
425C2×C13#241511yescharacteristic
426C2×C13#306011yescharacteristic
427C2×C13#306311yescharacteristic
428C2×C13#312011yescharacteristic
429C2×C13#312611yescharacteristic
430C2×C13#324211yescharacteristic
431C2×C13#324411yescharacteristic
432C24C2×C22⋊D1311yescharacteristic
433C2×Q8—21no—
434C2×Q8—21no—
435C2×Q8—21no—
436C2×Q8—21no—
437C2×Q8—21no—
438C2×Q8#612712yes—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—261no—
450C22×C4—261no—
451C22×C4—261no—
452C22×C4—261no—
453C22×C4C2×C22⋊D1311yescharacteristic
454C22×C4C2×C22⋊D1311yescharacteristic
455C22×C4C2×C22⋊D1311yescharacteristic
456C22×C4C2×C22⋊D1311yescharacteristic
457C22×C4C2×C22⋊D1311yescharacteristic
458C22×C4C23×D1311yescharacteristic
459C2×C8—132no—
460C2×C8—132no—
461C2×C8—261no—
462C2×C8—261no—
463C2×C8—261no—
464C2×C8—261no—
465C2×C8—261no—
466C2×C8—261no—
467C2×C8—261no—
468C2×C8—261no—
469C2×C8—261no—
470C2×C8—261no—
471C4⋊C4—24no—
472C4⋊C4—22no—
473C4⋊C4—22no—
474C4⋊C4—21no—
475C4⋊C4—21no—
476C4⋊C4—21no—
477C4⋊C4—21no—
478C4⋊C4—21no—
479C4⋊C4—42no—
480C4⋊C4—81no—
481C4⋊C4—261no—
482C4⋊C4—261no—
483C4⋊C4—261no—
484C4⋊C4—261no—
485C4⋊C4—261no—
486C4⋊C4—261no—
487C4⋊C4—261no—
488C4⋊C4—261no—
489C4⋊C4—522no—
490C4⋊C4—522no—
491C4⋊C4—1041no—
492C4⋊C4—1041no—
493C4⋊C4—1041no—
494C4⋊C4—1041no—
495C4⋊C4#612712yes—
496C22⋊C4—24no—
497C22⋊C4—22no—
498C22⋊C4—22no—
499C22⋊C4—81no—
500C22⋊C4—1041no—
501C22⋊C4—1041no—
502C42—21no—
503C42—21no—
504C42—42no—
505C42—81no—
506C42D4×D1312yes—
507C42#612912yes—
508C13#2471911yescharacteristic, complemented, Sylow
509C23—21no—
510C23—21no—
511C23—21no—
512C23—21no—
513C23—21no—
514C23—21no—
515C23—21no—
516C23C22×C22⋊D1311yescharacteristic
517Q8—22no—
518Q8—22no—
519Q8—41no—
520Q8—41no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—521no—
551C2×C4—521no—
552C2×C4—521no—
553C2×C4—521no—
554C2×C4—521no—
555C2×C4—521no—
556C2×C4—521no—
557C2×C4—521no—
558C2×C4—521no—
559C2×C4—521no—
560C2×C4—521no—
561C2×C4—521no—
562C2×C4#8819312yes—
563C2×C4#8819312yes—
564C2×C4C2×#612711yescharacteristic
565C2×C4C2×#612711yescharacteristic
566C2×C4C2×D4×D1311yescharacteristic
567C2×C4C2×#612911yescharacteristic
568C2×C4#8825711yescharacteristic
569C2×C4#8825711yescharacteristic
570C8—262no—
571C8—262no—
572C8—521no—
573C8—521no—
574C22—41no—
575C22—41no—
576C22—41no—
577C22—41no—
578C22—41no—
579C22—41no—
580C22—41no—
581C22#1171949611yescharacteristic
582C22C2×#8819311yescharacteristic
583C22#1171953011yescharacteristic
584C22#1171954111yescharacteristic
585C22#1171954211yescharacteristic
586C22#1171954211yescharacteristic
587C22#1171954311yescharacteristic
588C4—22no—
589C4—22no—
590C4—41no—
591C4—42no—
592C4—41no—
593C4—41no—
594C4—42no—
595C4—81no—
596C4—81no—
597C4—1041no—
598C4—1041no—
599C4—1041no—
600C4—1041no—
601C4#1171924611yescharacteristic
602C4#1171924711yescharacteristic
603C4#1171925611yescharacteristic
604C4#1171925711yescharacteristic
605C2—81no—
606C2#1311826511yescharacteristic
607C2#1312449511yescharacteristic
608C2#1312452311yescharacteristic
609C2#1312505011yescharacteristic
610C2#1312512711yescharacteristic
611C2#1312609811yescharacteristic
612C2#1312612811yescharacteristic
subgroupG/H
centerC23C22×C22⋊D13
commutator, cosocleC22×C4×C13C24
FrattiniC22×C4C23×D13
FittingC13×#2323C2
socleC23×C13C22×D4
prime radicalC24×C13C2×D4
hypercenter#2323D13
hypocenterC13#24719
LCSG ▷ C22×C4×C13 ▷ C2×C13 ▷ C13
UCSC1 ◁ C23 ◁ C2×#117 ◁ #2323

Complements

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

NH ≅ G/Ntype
1. C2×#11718683C2semidirect
8. #13115793C2semidirect
9. #13115599C2semidirect
10. #13115576C2semidirect
11. #13115561C2semidirect
12. #13115529C2semidirect
13. #13115515C2semidirect
14. #13114841C2semidirect
508. C13#24719semidirect

Maximal subgroups

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

subgroupindex
C2×#117186832
C2×#117186612
#131206662
#131162722
#131162672
#131160292
#131157932
#131155992
#131155762
#131155612
#131155292
#131155152
#131148412
C13×#23232
#2471913