Finite Group Databasebeta

← #15833566

Subgroups — #15833566

Order 3328.

582 conjugacy classes of subgroups.

The subgroup lattice has 562 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#13123985C211yesmaximal, characteristic
2#13123931C211yesmaximal, characteristic, complemented
3#13113745C211yesmaximal, characteristic, complemented
4#13113739C211yesmaximal, characteristic, complemented
5#13113733C211yesmaximal, characteristic, complemented
6#13113727C211yesmaximal, characteristic, complemented
7#13113655C211yesmaximal, characteristic, complemented
8SD16⋊#1055C211yesmaximal, characteristic, complemented
9Q16×#1055C211yesmaximal, characteristic, complemented
10SD16×#1055C211yesmaximal, characteristic, complemented
11#13113603C211yesmaximal, characteristic, complemented
12#13113296C211yesmaximal, characteristic
13#13113291C211yesmaximal, characteristic, complemented
14M16×#1055C211yesmaximal, characteristic, complemented
15C13×#3172C211yesmaximal, characteristic
16C13×#533C2211yescharacteristic
17C13×#529C2211yescharacteristic
18C13×#501C411yescharacteristic
19C13×#497C411yescharacteristic
20C13×#491C411yescharacteristic
21C13×#489C411yescharacteristic
22C13×#470C411yescharacteristic
23C13×#470C411yescharacteristic
24C13×#468C411yescharacteristic
25C13×#468C411yescharacteristic
26C4×C13×Q16C2211yescharacteristic
27C4×C13×Q16C2211yescharacteristic
28C4×C13×SD16C2211yescharacteristic
29C4×C13×SD16C2211yescharacteristic
30C4×C13×M16C2211yescharacteristic
31#11718685—21no—
32#11718685—21no—
33#11718685C2211yescharacteristic
34#11718685C2211yescharacteristic
35#11718685C2211yescharacteristic
36Q8×#1055C2211yescharacteristic
37Q8×#1055C2211yescharacteristic
38Q8×#1055C2211yescharacteristic
39#11718673—21no—
40#11718673C2211yescharacteristic
41#11718672—21no—
42D4×#1055C2211yescharacteristic
43#11718661—21no—
44#11718661—21no—
45#11718661—21no—
46#11718660—21no—
47#11718587C2211yescharacteristic
48#11718586C2211yescharacteristic
49#11718585C2211yescharacteristic
50C2×C4×#1055C2211yescharacteristic
51Q8⋊#1055C2211yescharacteristic
52Q8⋊#1055C2211yescharacteristic
53Q8⋊#1055C2211yescharacteristic
54Q8⋊#1055C2211yescharacteristic
55Q8⋊#1055C2211yescharacteristic
56Q8⋊#1055C2211yescharacteristic
57D4⋊#1055C2211yescharacteristic
58D4⋊#1055C2211yescharacteristic
59#11718050C2211yescharacteristic
60#11718050C2211yescharacteristic
61#11718049C2211yescharacteristic
62#11718049C2211yescharacteristic
63#11718047C2211yescharacteristic
64#11718047C2211yescharacteristic
65C8×#1055C2211yescharacteristic
66C8×#1055C2211yescharacteristic
67C2×C13×Q16C2×C411yescharacteristic
68C2×C13×Q16C2×C411yescharacteristic
69C2×C13×SD16C2×C411yescharacteristic
70C2×C13×SD16C2×C411yescharacteristic
71C2×C13×M16C2×C411yescharacteristic
72C2×C13×M16C2×C411yescharacteristic
73C13×C4⋊Q8C2×C411yescharacteristic
74C13×C4⋊Q8C2×C411yescharacteristic
75C13×#126—21no—
76C13×#125—21no—
77C13×#125C2×C411yescharacteristic
78C13×#124C2×C411yescharacteristic
79C13×#123—21no—
80C13×C22⋊Q8—21no—
81C13×C22⋊Q8C2×C411yescharacteristic
82C13×C22⋊Q8C2×C411yescharacteristic
83C13×C22⋊Q8C2×C411yescharacteristic
84C13×#121C2×C411yescharacteristic
85C4×Q8×C13—21no—
86C4×Q8×C13C2311yescharacteristic
87C4×Q8×C13C2311yescharacteristic
88C4×Q8×C13C2311yescharacteristic
89C4×D4×C13—21no—
90C4×D4×C13C2311yescharacteristic
91C13×#117—21no—
92C13×#117—21no—
93C2×C42×C13C2311yescharacteristic
94C2×#6099—21no—
95C2×#6099—21no—
96C22×#1055—21no—
97C22×#1055—21no—
98C13×#107C2×C411yescharacteristic
99C13×#107C2×C411yescharacteristic
100C13×#106C2×C411yescharacteristic
101C13×#106C2×C411yescharacteristic
102C13×#103C2×C411yescharacteristic
103C13×#103C2×C411yescharacteristic
104C13×#103C2×C411yescharacteristic
105C13×#103C2×C411yescharacteristic
106C13×#103C2×C411yescharacteristic
107C13×#103C2×C411yescharacteristic
108C13×#102C2×C411yescharacteristic
109C13×#102C2×C411yescharacteristic
110C13×#97C2×C411yescharacteristic
111C13×#97C2×C411yescharacteristic
112C4×C8×C13C2311yescharacteristic
113C4×C8×C13C2311yescharacteristic
114C22⋊#1055—21no—
115C22⋊#1055—21no—
116C22⋊#1055—21no—
117C22⋊#1055D411yescharacteristic
118C22⋊#1055D411yescharacteristic
119C4⋊#1055—21no—
120C4⋊#1055—21no—
121C4⋊#1055—21no—
122C4⋊#1055—21no—
123C4⋊#1055—21no—
124C4⋊#1055—21no—
125C4⋊#1055—21no—
126C4⋊#1055D411yescharacteristic
127C4⋊#1055D411yescharacteristic
128C4⋊#1055C2311yescharacteristic
129C4⋊#1055C2311yescharacteristic
130C4⋊#1055C2311yescharacteristic
131C4⋊#1055C2311yescharacteristic
132#88043—21no—
133#88043—21no—
134#88043—21no—
135#88043—21no—
136C4×#1055—21no—
137C4×#1055—21no—
138C4×#1055—21no—
139C4×#1055—21no—
140C4×#1055C2311yescharacteristic
141C4×#1055C2311yescharacteristic
142C4×#1055C2311yescharacteristic
143C4×#1055C2311yescharacteristic
144#17969—131nomaximal, Sylow
145C2×Q8×C13—21no—
146C2×Q8×C13C22×C411yescharacteristic
147C2×Q8×C13C22×C411yescharacteristic
148C2×Q8×C13C22×C411yescharacteristic
149C2×D4×C13—21no—
150C2×D4×C13C22×C411yescharacteristic
151C22×C4×C13—21no—
152C22×C4×C13C22×C411yescharacteristic
153C22×C4×C13C2×D411yescharacteristic
154C22×C4×C13#4111yescharacteristic
155C13×Q16—22no—
156C13×Q16—22no—
157C13×SD16—22no—
158C13×SD16—22no—
159C13×M16—22no—
160C13×M16—22no—
161C2×C8×C13C22×C411yescharacteristic
162C2×C8×C13C22×C411yescharacteristic
163C2×C8×C13C22×C411yescharacteristic
164C2×C8×C13C22×C411yescharacteristic
165C13×C4⋊C4—21no—
166C13×C4⋊C4—21no—
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—21no—
175C13×C4⋊C4—21no—
176C13×C4⋊C4C22×C411yescharacteristic
177C13×C4⋊C4C22×C411yescharacteristic
178C13×C4⋊C4C22×C411yescharacteristic
179C13×C4⋊C4C22×C411yescharacteristic
180C13×C22⋊C4—21no—
181C13×C22⋊C4—21no—
182C13×C22⋊C4—21no—
183C13×C22⋊C4—21no—
184C13×C22⋊C4—21no—
185C13×C22⋊C4—21no—
186C13×C22⋊C4—21no—
187C42×C13—21no—
188C42×C13—21no—
189C42×C13—21no—
190C42×C13C22×C411yescharacteristic
191C42×C13C2×D411yescharacteristic
192C42×C13#4111yescharacteristic
193C42×C13C2411yescharacteristic
194#6099—81no—
195#6099—81no—
196C2×#1055—21no—
197C2×#1055—21no—
198C2×#1055—21no—
199C2×#1055—21no—
200C2×#1055—21no—
201C2×#1055—21no—
202C2×#1055—42no—
203C2×#1055—81no—
204C2×#1055C2×D411yescharacteristic
205C2×#1055C2×D411yescharacteristic
206C2×#1055C2×D411yescharacteristic
207C2×#1055C2×D411yescharacteristic
208#3172D1311yescharacteristic
209#2993—131no—
210#2985—131no—
211#1593—131no—
212#1592—131no—
213#1591—131no—
214#1590—131no—
215#1579—131no—
216SD16⋊C8—131no—
217C8×Q16—131no—
218C8×SD16—131no—
219#1569—131no—
220#1494—131no—
221#1493—131no—
222C8×M16—131no—
223C23×C13—21no—
224C23×C13C4×D411yescharacteristic
225Q8×C13—21no—
226Q8×C13—21no—
227Q8×C13—21no—
228Q8×C13—21no—
229Q8×C13—21no—
230Q8×C13—21no—
231Q8×C13—81no—
232D4×C13—21no—
233D4×C13—21no—
234D4×C13—81no—
235C2×C4×C13—21no—
236C2×C4×C13—21no—
237C2×C4×C13—21no—
238C2×C4×C13—21no—
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—21no—
246C2×C4×C13—41no—
247C2×C4×C13—41no—
248C2×C4×C13—41no—
249C2×C4×C13C4×D411yescharacteristic
250C2×C4×C13C4×D411yescharacteristic
251C2×C4×C13C4×D411yescharacteristic
252C2×C4×C13C23×C411yescharacteristic
253C2×C4×C13C22×D411yescharacteristic
254C2×C4×C13C2×#4111yescharacteristic
255C8×C13—21no—
256C8×C13—21no—
257C8×C13#13112yes—
258C8×C13#13112yes—
259#1055—22no—
260#1055—21no—
261#1055—21no—
262#1055—21no—
263#1055—21no—
264#1055—41no—
265#1055—41no—
266#1055—41no—
267#1055—41no—
268#1055—41no—
269#533C2×D1311yescharacteristic
270#529C2×D1311yescharacteristic
271#501Dic1311yescharacteristic
272#497Dic1311yescharacteristic
273#491Dic1311yescharacteristic
274#489Dic1311yescharacteristic
275#470Dic1311yescharacteristic
276#470Dic1311yescharacteristic
277#468Dic1311yescharacteristic
278#468Dic1311yescharacteristic
279#446—131no—
280#446—131no—
281#446—131no—
282#446—261no—
283#446—261no—
284C8×Q8—131no—
285C8×Q8—131no—
286C8×Q8—131no—
287C4×Q16C2×D1311yescharacteristic
288C4×Q16C2×D1311yescharacteristic
289C4×SD16C2×D1311yescharacteristic
290C4×SD16C2×D1311yescharacteristic
291#436—131no—
292#436—261no—
293#435—261no—
294C8×D4—131no—
295#433—261no—
296#433—261no—
297#433—261no—
298#432—131no—
299#431—131no—
300#424—261no—
301#423—131no—
302C4×M16C2×D1311yescharacteristic
303C2×C4×C8—131no—
304#335—131no—
305#335—131no—
306#334—131no—
307#334—131no—
308Q8⋊C8—131no—
309Q8⋊C8—131no—
310Q8⋊C8—131no—
311Q8⋊C8—131no—
312Q8⋊C8—131no—
313Q8⋊C8—131no—
314D4⋊C8—131no—
315D4⋊C8—131no—
316#322—131no—
317#322—131no—
318C82—131no—
319C82—131no—
320C22×C13—21no—
321C22×C13—21no—
322C22×C13—21no—
323C22×C13—41no—
324C22×C13—41no—
325C22×C13—41no—
326C22×C13C2×C4×D411yescharacteristic
327C4×C13—21no—
328C4×C13—21no—
329C4×C13—21no—
330C4×C13—21no—
331C4×C13—41no—
332C4×C13—41no—
333C4×C13—41no—
334C4×C13—41no—
335C4×C13—41no—
336C4×C13—41no—
337C4×C13—41no—
338C4×C13C2×#13111yescharacteristic
339C4×C13C2×#13111yescharacteristic
340C4×C13#57711yescharacteristic
341C4×C13#57811yescharacteristic
342C2×Q16C2×Dic1311yescharacteristic
343C2×Q16C2×Dic1311yescharacteristic
344C2×SD16C2×Dic1311yescharacteristic
345C2×SD16C2×Dic1311yescharacteristic
346C2×M16—261no—
347C2×M16—261no—
348C2×M16C2×Dic1311yescharacteristic
349C2×M16C2×Dic1311yescharacteristic
350C22×C8—261no—
351C22×C8—261no—
352C4⋊Q8C2×Dic1311yescharacteristic
353C4⋊Q8C2×Dic1311yescharacteristic
354#126—21no—
355#125—21no—
356#125C2×Dic1311yescharacteristic
357#124C2×Dic1311yescharacteristic
358#123—21no—
359C22⋊Q8—21no—
360C22⋊Q8C2×Dic1311yescharacteristic
361C22⋊Q8C2×Dic1311yescharacteristic
362C22⋊Q8C2×Dic1311yescharacteristic
363#121C2×Dic1311yescharacteristic
364C4×Q8—21no—
365C4×Q8C22×D1311yescharacteristic
366C4×Q8C22×D1311yescharacteristic
367C4×Q8C22×D1311yescharacteristic
368C4×D4—21no—
369C4×D4C22×D1311yescharacteristic
370#117—21no—
371#117—21no—
372C2×C42C22×D1311yescharacteristic
373#107C2×Dic1311yescharacteristic
374#107C2×Dic1311yescharacteristic
375#106C2×Dic1311yescharacteristic
376#106C2×Dic1311yescharacteristic
377C4⋊C8—131no—
378C4⋊C8—131no—
379C4⋊C8—131no—
380C4⋊C8—131no—
381C4⋊C8—131no—
382C4⋊C8—131no—
383C4⋊C8—261no—
384C4⋊C8—261no—
385C4⋊C8—261no—
386C4⋊C8—261no—
387C4⋊C8—261no—
388C4⋊C8—261no—
389C4⋊C8—261no—
390#103C2×Dic1311yescharacteristic
391#103C2×Dic1311yescharacteristic
392#103C2×Dic1311yescharacteristic
393#103C2×Dic1311yescharacteristic
394#103C2×Dic1311yescharacteristic
395#103C2×Dic1311yescharacteristic
396#102C2×Dic1311yescharacteristic
397#102C2×Dic1311yescharacteristic
398C22⋊C8—131no—
399C22⋊C8—131no—
400C22⋊C8—261no—
401C22⋊C8—261no—
402C22⋊C8—261no—
403#97—261no—
404#97—261no—
405#97—261no—
406#97—261no—
407#97C2×Dic1311yescharacteristic
408#97C2×Dic1311yescharacteristic
409C4×C8—131no—
410C4×C8—131no—
411C4×C8—131no—
412C4×C8—131no—
413C4×C8—261no—
414C4×C8—261no—
415C4×C8—261no—
416C4×C8—261no—
417C4×C8C22×D1311yescharacteristic
418C4×C8C22×D1311yescharacteristic
419C2×C13—41no—
420C2×C13—81no—
421C2×C13#293111yescharacteristic
422C2×C13#294711yescharacteristic
423C2×C13#295611yescharacteristic
424C2×Q8—21no—
425C2×Q8C22×Dic1311yescharacteristic
426C2×Q8C22×Dic1311yescharacteristic
427C2×Q8C22×Dic1311yescharacteristic
428C2×D4—21no—
429C2×D4C22×Dic1311yescharacteristic
430C22×C4—21no—
431C22×C4D4×D1311yescharacteristic
432C22×C4#612911yescharacteristic
433C22×C4C22×Dic1311yescharacteristic
434Q16—22no—
435Q16—22no—
436SD16—22no—
437SD16—22no—
438M16—22no—
439M16—22no—
440M16—1041no—
441M16—1041no—
442C2×C8—131no—
443C2×C8—131no—
444C2×C8—131no—
445C2×C8—131no—
446C2×C8—261no—
447C2×C8—261no—
448C2×C8—261no—
449C2×C8—261no—
450C2×C8—261no—
451C2×C8—261no—
452C2×C8—522no—
453C2×C8—1041no—
454C2×C8C22×Dic1311yescharacteristic
455C2×C8C22×Dic1311yescharacteristic
456C2×C8C22×Dic1311yescharacteristic
457C2×C8C22×Dic1311yescharacteristic
458C4⋊C4—21no—
459C4⋊C4—21no—
460C4⋊C4—21no—
461C4⋊C4—21no—
462C4⋊C4—21no—
463C4⋊C4—21no—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—21no—
468C4⋊C4—21no—
469C4⋊C4C22×Dic1311yescharacteristic
470C4⋊C4C22×Dic1311yescharacteristic
471C4⋊C4C22×Dic1311yescharacteristic
472C4⋊C4C22×Dic1311yescharacteristic
473C22⋊C4—21no—
474C22⋊C4—21no—
475C22⋊C4—21no—
476C22⋊C4—21no—
477C22⋊C4—21no—
478C22⋊C4—21no—
479C22⋊C4—21no—
480C42—21no—
481C42—21no—
482C42—21no—
483C42D4×D1311yescharacteristic
484C42#612911yescharacteristic
485C42C22×Dic1311yescharacteristic
486C42C23×D1311yescharacteristic
487C13#1796911yescharacteristic, complemented, Sylow
488C23—21no—
489C23D4×Dic1311yescharacteristic
490Q8—21no—
491Q8—21no—
492Q8—21no—
493Q8—21no—
494Q8—21no—
495Q8—21no—
496Q8—81no—
497D4—21no—
498D4—21no—
499D4—81no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4D4×Dic1311yescharacteristic
515C2×C4D4×Dic1311yescharacteristic
516C2×C4D4×Dic1311yescharacteristic
517C2×C4C2×D4×D1311yescharacteristic
518C2×C4C2×#612911yescharacteristic
519C2×C4C23×Dic1311yescharacteristic
520C8—21no—
521C8—21no—
522C8—262no—
523C8—261no—
524C8—261no—
525C8—261no—
526C8—261no—
527C8—521no—
528C8—521no—
529C8—521no—
530C8—521no—
531C8—521no—
532C8#8820212yes—
533C8#8820212yes—
534C22—21no—
535C22—21no—
536C22—21no—
537C22—41no—
538C22—41no—
539C22—41no—
540C22C2×D4×Dic1311yescharacteristic
541C4—21no—
542C4—21no—
543C4—21no—
544C4—21no—
545C4—41no—
546C4—41no—
547C4—41no—
548C4—41no—
549C4—41no—
550C4—41no—
551C4—41no—
552C4#1171948311yescharacteristic
553C4#1171948411yescharacteristic
554C4C2×#8820211yescharacteristic
555C4C2×#8820211yescharacteristic
556C2—41no—
557C2—81no—
558C2#1312354811yescharacteristic
559C2#1312368011yescharacteristic
560C2#1312374611yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D13
commutatorC2×C4×C13C23×C4
FrattiniC42C23×D13
FittingC13×#3172C2
socleC22×C13C2×C4×D4
prime radicalC13×#121C2×C4
cosocleC42×C13C24
hypercenter#3172D13
hypocenterC13#17969
LCSG ▷ C2×C4×C13 ▷ C2×C13 ▷ C13
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #3172

Complements

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

NH ≅ G/Ntype
2. #13123931C2semidirect
3. #13113745C2semidirect
4. #13113739C2semidirect
5. #13113733C2semidirect
6. #13113727C2semidirect
7. #13113655C2semidirect
8. SD16⋊#1055C2semidirect
9. Q16×#1055C2semidirect
10. SD16×#1055C2semidirect
11. #13113603C2semidirect
13. #13113291C2semidirect
14. M16×#1055C2semidirect
487. C13#17969semidirect

Maximal subgroups

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

subgroupindex
#131239852
#131239312
#131137452
#131137392
#131137332
#131137272
#131136552
SD16⋊#10552
Q16×#10552
SD16×#10552
#131136032
#131132962
#131132912
M16×#10552
C13×#31722
#1796913