Finite Group Databasebeta

← #30619320

Subgroups — #30619320

Order 15616.

906 conjugacy classes of subgroups.

The subgroup lattice has 630 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4⋊Q8×#6923C211yesmaximal, characteristic, complemented
2#124×#6923C211yesmaximal, characteristic
3#21461950C211yesmaximal, characteristic, complemented
4#21461949C211yesmaximal, characteristic
5#21461815C212yesmaximal
6#21461813C212yesmaximal, complemented
7#21461813C212yesmaximal, complemented
8#21461812C212yesmaximal
9#21461742C211yesmaximal, characteristic, complemented
10#21461535C211yesmaximal, characteristic, complemented
11#497×D61C211yesmaximal, characteristic
12C2×Q8×#6923C2211yescharacteristic
13C2×Q8×#6923C2212yes—
14C2×D4×#6923C2211yescharacteristic
15C2×#14519840C2212yes—
16C2×#14519840C2212yes—
17C2×#14519840C2212yes—
18C2×#14519839C2212yes—
19#16840721C2211yescharacteristic
20(C2×#6923)⋊Q8C2211yescharacteristic
21(C2×#6923)⋊Q8C2212yes—
22#16840714C2211yescharacteristic
23#16840713—21no—
24#16840713—21no—
25#16840712—21no—
26#16840712—21no—
27#16840708C2211yescharacteristic
28#16840706C2211yescharacteristic
29C4⋊C4×#6923—21no—
30C4⋊C4×#6923—21no—
31C22⋊C4×#6923—21no—
32C22⋊C4×#6923—21no—
33C42×#6923C2211yescharacteristic
34#16840636C2212yes—
35#16840636C2212yes—
36(C2×#6923)⋊C8C2212yes—
37(C2×#6923)⋊C8C2212yes—
38C2×Q16×D61C2212yes—
39C2×SD16×D61C2212yes—
40C4⋊Q8×D61C2211yescharacteristic
41#124×D61C2211yescharacteristic
42#16840157C411yescharacteristic, complemented
43#16840156C411yescharacteristic, complemented
44#16840155C412yescomplemented
45#16840154C411yescharacteristic, complemented
46#16840153C412yescomplemented
47#97×D61C2211yescharacteristic
48C61×#497C411yescharacteristic, complemented
49C23×#6923—21no—
50C2×D61⋊C8C2312yes—
51Q8×#6923—42no—
52Q8×#6923—42no—
53Q8×#6923—42no—
54D4×#6923—42no—
55C2×C4⋊#6923—21no—
56C2×C4⋊#6923—22no—
57C2×C4⋊#6923—21no—
58C2×C4⋊#6923—21no—
59C2×C4⋊#6923C2311yescharacteristic
60C2×C4⋊#6923C2311yescharacteristic
61C2×C4⋊#6923C2312yes—
62C2×C22⋊#6923—21no—
63C2×C22⋊#6923—21no—
64C2×C22⋊#6923—21no—
65C2×C4×#6923—21no—
66C2×C4×#6923—22no—
67C2×C4×#6923D411yescharacteristic
68C2×C4×#6923D411yescharacteristic
69C2×C4×#6923D411yescharacteristic
70C2×C4×#6923D411yescharacteristic
71C2×C4×#6923C2311yescharacteristic
72C2×C4×#6923C2311yescharacteristic
73#14519840—42no—
74#14519840—42no—
75#14519840—42no—
76#14519839—42no—
77#14519832—21no—
78#14519832—21no—
79(C2×#6923)⋊C4—21no—
80(C2×#6923)⋊C4—21no—
81(C2×#6923)⋊C4—22no—
82(C2×#6923)⋊C4—22no—
83(C2×#6923)⋊C4—21no—
84(C2×#6923)⋊C4—21no—
85C2×Q8×D61C2311yescharacteristic
86C2×Q8×D61C2312yes—
87C2×D4×D61C2311yescharacteristic
88C2×Dic244C2×C412yescomplemented
89C2×#11758202C2×C412yescomplemented
90C2×#11758202C2×C412yescomplemented
91Q16×D61—42no—
92Q16×D61—42no—
93C2×#11758190C2×C412yes—
94C2×#11758200C2×C412yes—
95C2×#11758201C2×C412yes—
96SD16×D61—42no—
97SD16×D61—42no—
98C2×C8×D61C2312yes—
99#14519777C2×C411yescharacteristic, complemented
100#14519776—21no—
101#14519776—21no—
102#14519775C2×C412yescomplemented
103#14519766C2×C411yescharacteristic
104#14519765C2×C411yescharacteristic
105#14519764C2×C411yescharacteristic
106#14519763—21no—
107#14519763—21no—
108C4⋊C4×D61—21no—
109C4⋊C4×D61—21no—
110C22⋊C4×D61—21no—
111C22⋊C4×D61—21no—
112C42×D61C2311yescharacteristic
113#14519688C2×C411yescharacteristic, complemented
114#14519687C2×C412yescomplemented
115C2×Q16×C61C2×C412yescomplemented
116C2×SD16×C61C2×C412yes—
117C4⋊Q8×C61C2×C411yescharacteristic, complemented
118#124×C61C2×C411yescharacteristic
119#97×C61C2×C411yescharacteristic, complemented
120C23×D61—21no—
121C22×#6923—21no—
122C22×#6923—21no—
123C22×#6923—22no—
124C22×#6923—42no—
125C22×#6923—42no—
126C22×#6923—42no—
127C22×#6923C2×D411yescharacteristic
128C22×#6923C2×D411yescharacteristic
129C22×#6923C2×D411yescharacteristic
130C22×#6923C2×D411yescharacteristic
131C2×Q8×C61C22×C411yescharacteristic
132C2×Q8×C61C22×C412yes—
133C2×D4×C61C22×C411yescharacteristic
134C2×C22⋊D61—21no—
135Q8×D61—22no—
136Q8×D61—22no—
137Q8×D61—22no—
138Q8×D61—22no—
139Q8×D61—22no—
140Q8×D61—22no—
141D4×D61—22no—
142D4×D61—22no—
143C2×D244C22×C411yescharacteristic
144C2×C4×D61—21no—
145C2×C4×D61—22no—
146C2×C4×D61C2×D411yescharacteristic
147C2×C4×D61C2×D411yescharacteristic
148C2×C4×D61C2411yescharacteristic
149C2×Dic122—21no—
150C2×Dic122—22no—
151C2×Dic122C22×C411yescharacteristic
152C2×Dic122C22×C412yes—
153C22⋊#6923—42no—
154C22⋊#6923—42no—
155C22⋊#6923—42no—
156C2×#92420—22no—
157C4⋊#6923—22no—
158C4⋊#6923—22no—
159C4⋊#6923—22no—
160C4⋊#6923—22no—
161C4⋊#6923—42no—
162C4⋊#6923—42no—
163C4⋊#6923—44no—
164C4⋊#6923—42no—
165C4×#6923—22no—
166C4×#6923—22no—
167C4×#6923—22no—
168C4×#6923—22no—
169C4×#6923—41no—
170C4×#6923—41no—
171C4×#6923—44no—
172C4×#6923—42no—
173D61⋊C8—22no—
174D61⋊C8—22no—
175Q16×C61—42no—
176SD16×C61—42no—
177C2×C8×C61C22×C412yes—
178C4⋊C4×C61—21no—
179C4⋊C4×C61—21no—
180C22⋊C4×C61—21no—
181C22⋊C4×C61—21no—
182C42×C61C22×C411yescharacteristic
183C22⋊Dic61—21no—
184C22⋊Dic61—21no—
185#11758202—42no—
186#11758202—42no—
187#11758201—42no—
188#11758200—42no—
189#11758198—21no—
190#11758198—21no—
191#11758198—21no—
192#11758198—21no—
193C4⋊Dic61—21no—
194C4⋊Dic61—21no—
195#11758196—22no—
196#11758196—22no—
197C4×Dic61C22×C411yescharacteristic
198C4×Dic61#4111yescharacteristic
199C4×Dic61#4111yescharacteristic
200C2×#92418C22×C412yes—
201Dic244—42no—
202#11758190—42no—
203C8×D61—22no—
204C8×D61—22no—
205C23×C61—21no—
206C22×D61—21no—
207C22×D61—41no—
208C22×D61—41no—
209C22×D61—41no—
210C22×D61—41no—
211C22×D61—41no—
212C22×D61—41no—
213C22×D61C22×D411yescharacteristic
214C2×#6923—21no—
215C2×#6923—21no—
216C2×#6923—21no—
217C2×#6923—21no—
218C2×#6923—21no—
219C2×#6923—21no—
220C2×#6923—21no—
221C2×#6923—21no—
222C2×#6923—21no—
223C2×#6923—21no—
224C2×#6923—41no—
225C2×#6923—42no—
226C2×#6923—41no—
227C2×#6923—41no—
228C2×#6923—41no—
229C2×#6923—42no—
230C2×#6923—42no—
231C2×#6923—41no—
232C2×#6923—41no—
233C2×#6923—84no—
234C2×#6923#13712yes—
235C2×#6923#13712yes—
236Q8×C61—22no—
237Q8×C61—22no—
238Q8×C61—22no—
239D4×C61—22no—
240C2×C4×C61—22no—
241C2×C4×C61—21no—
242C2×C4×C61C4×D411yescharacteristic
243C2×C4×C61C4×D411yescharacteristic
244C2×C4×C61C23×C411yescharacteristic
245C22⋊D61—42no—
246C2×Dic61—22no—
247C2×Dic61—21no—
248C2×Dic61C4×D411yescharacteristic
249C2×Dic61C4×D411yescharacteristic
250C2×Dic61C2×#4111yescharacteristic
251D244—22no—
252C4×D61—21no—
253C4×D61—21no—
254C4×D61—21no—
255C4×D61—21no—
256C4×D61—41no—
257C4×D61—42no—
258C4×D61—42no—
259C4×D61—41no—
260C4×D61H(5,𝔽2)12yes—
261C4×D61H(5,𝔽2)12yes—
262Dic122—22no—
263Dic122—22no—
264Dic122—22no—
265Dic122—42no—
266Dic122—42no—
267Dic122—42no—
268#92420—42no—
269C8×C61—22no—
270#92418—22no—
271#22273—611nomaximal, Sylow
272C22×C61—41no—
273C22×C61—41no—
274C22×C61—41no—
275C22×C61C2×C4×D411yescharacteristic
276C2×D61—41no—
277C2×D61—41no—
278C2×D61—41no—
279C2×D61—81no—
280C2×D61—81no—
281C2×D61—81no—
282C2×D61—81no—
283C2×D61#53511yescharacteristic
284C2×D61#53511yescharacteristic
285C2×D61C2×#13711yescharacteristic
286C2×D61C2×#13711yescharacteristic
287C2×D61#57511yescharacteristic
288C2×D61#57511yescharacteristic
289#6923—22no—
290#6923—22no—
291#6923—41no—
292#6923—41no—
293#6923—81no—
294#6923—81no—
295#6923—82no—
296C4×C61—21no—
297C4×C61—21no—
298C4×C61—42no—
299C4×C61—41no—
300C4×C61#51812yes—
301Dic61—21no—
302Dic61—21no—
303Dic61—42no—
304Dic61—41no—
305Dic61#54012yes—
306C2×#497—611no—
307C4×C4⋊Q8—611no—
308C4×#124—611no—
309#1960—611no—
310#1959—611no—
311#1880—612no—
312#1878—612no—
313#1878—612no—
314#1877—612no—
315#1836—611no—
316#1745—611no—
317C2×C61—81no—
318C2×C61#232811yescharacteristic
319C2×C61C4×#13711yescharacteristic
320C2×C61C4×#13711yescharacteristic
321D61—81no—
322D61#321511yescharacteristic
323D61#321611yescharacteristic
324D61#321811yescharacteristic
325D61#321911yescharacteristic
326C22×Q16—612no—
327C22×SD16—612no—
328C2×C4⋊Q8—611no—
329C2×#124—611no—
330C2×C4×Q8—612no—
331C2×C4×Q8—611no—
332C2×C4×D4—611no—
333#497—612no—
334#497—612no—
335#497—611no—
336#497—611no—
337#497—611no—
338#497#692311yescharacteristic, complemented
339C2×#103—612no—
340C2×#103—612no—
341C2×#103—612no—
342C2×#102—612no—
343C2×#97—611no—
344C2×#97—611no—
345#391—612no—
346#391—611no—
347#390—611no—
348#389—1221no—
349#389—1221no—
350#388—1221no—
351#388—1221no—
352#384—611no—
353#382—611no—
354C4×C4⋊C4—1221no—
355C4×C4⋊C4—1221no—
356C4×C22⋊C4—1221no—
357C4×C22⋊C4—1221no—
358C43—611no—
359#340—612no—
360#340—612no—
361#336—612no—
362#336—612no—
363C61#2227311yescharacteristic, complemented, Sylow
364C22×Q8—611no—
365C22×Q8—612no—
366C22×D4—611no—
367C23×C4—1221no—
368C2×Q16—612no—
369C2×Q16—612no—
370C2×Q16—612no—
371C2×Q16—2442no—
372C2×Q16—2442no—
373C2×Q16C2×#692312yescomplemented
374C2×SD16—612no—
375C2×SD16—612no—
376C2×SD16—612no—
377C2×SD16—2442no—
378C2×SD16—2442no—
379C2×SD16C2×#692312yes—
380C22×C8—612no—
381C22×C8—612no—
382C4⋊Q8—612no—
383C4⋊Q8—611no—
384C4⋊Q8—1221no—
385C4⋊Q8—1221no—
386C4⋊Q8C2×#692311yescharacteristic, complemented
387#124—611no—
388#124—611no—
389#124—611no—
390#124—1221no—
391#124—1221no—
392#124C2×#692311yescharacteristic
393C4×Q8—2442no—
394C4×Q8—2442no—
395C4×Q8—2442no—
396C4×D4—2442no—
397C2×C4⋊C4—611no—
398C2×C4⋊C4—611no—
399C2×C4⋊C4—612no—
400C2×C4⋊C4—1221no—
401C2×C4⋊C4—1221no—
402C2×C4⋊C4—1221no—
403C2×C4⋊C4—1222no—
404C2×C4⋊C4—1221no—
405C2×C4⋊C4—1221no—
406C2×C22⋊C4—1221no—
407C2×C22⋊C4—1221no—
408C2×C22⋊C4—1221no—
409C2×C22⋊C4—1221no—
410C2×C22⋊C4—1221no—
411C2×C42—611no—
412C2×C42—611no—
413C2×C42—611no—
414C2×C42—611no—
415C2×C42—611no—
416C2×C42—611no—
417C2×C42—611no—
418C2×C42—1221no—
419C2×C42—1222no—
420#103—2442no—
421#103—2442no—
422#103—2442no—
423#102—2442no—
424#97—612no—
425#97—611no—
426#97—1221no—
427#97—1221no—
428#97C2×#692311yescharacteristic, complemented
429#95—1221no—
430#95—1221no—
431#95—1221no—
432#95—1221no—
433#95—1222no—
434#95—1222no—
435C24—1221no—
436C2×Q8—611no—
437C2×Q8—612no—
438C2×Q8—1221no—
439C2×Q8—1222no—
440C2×Q8—1222no—
441C2×Q8—1222no—
442C2×Q8—1222no—
443C2×Q8—1222no—
444C2×Q8—1222no—
445C2×Q8—1222no—
446C2×Q8C22×#692311yescharacteristic
447C2×Q8C22×#692312yes—
448C2×D4—611no—
449C2×D4—1222no—
450C2×D4—1222no—
451C2×D4—1221no—
452C2×D4C22×#692311yescharacteristic
453C22×C4—611no—
454C22×C4—611no—
455C22×C4—611no—
456C22×C4—611no—
457C22×C4—611no—
458C22×C4—611no—
459C22×C4—611no—
460C22×C4—1221no—
461C22×C4—1222no—
462C22×C4—1221no—
463C22×C4—1221no—
464C22×C4—1222no—
465C22×C4—2442no—
466C22×C4—2442no—
467C22×C4—2442no—
468Q16—42no—
469Q16—2442no—
470Q16—2442no—
471Q16—2442no—
472SD16—42no—
473SD16—2442no—
474SD16—2442no—
475SD16—2442no—
476C2×C8—612no—
477C2×C8—1222no—
478C2×C8—1222no—
479C2×C8—1222no—
480C2×C8—1222no—
481C2×C8—1222no—
482C2×C8C22×#692312yes—
483C4⋊C4—21no—
484C4⋊C4—21no—
485C4⋊C4—1222no—
486C4⋊C4—1222no—
487C4⋊C4—1222no—
488C4⋊C4—1222no—
489C4⋊C4—1222no—
490C4⋊C4—1222no—
491C4⋊C4—1221no—
492C4⋊C4—1221no—
493C4⋊C4—2442no—
494C4⋊C4—2444no—
495C4⋊C4—2442no—
496C4⋊C4—2442no—
497C22⋊C4—21no—
498C22⋊C4—21no—
499C22⋊C4—1221no—
500C22⋊C4—1221no—
501C22⋊C4—1221no—
502C22⋊C4—1221no—
503C22⋊C4—1221no—
504C22⋊C4—1221no—
505C22⋊C4—2442no—
506C22⋊C4—2442no—
507C22⋊C4—2442no—
508C42—611no—
509C42—611no—
510C42—611no—
511C42—1222no—
512C42—1222no—
513C42—1222no—
514C42—1222no—
515C42—2441no—
516C42—2441no—
517C42—2444no—
518C42—2442no—
519C42C22×#692311yescharacteristic
520C23—21no—
521C23—611no—
522C23—1221no—
523C23—2441no—
524C23—2441no—
525C23—2441no—
526C23—2441no—
527C23—2441no—
528C23—2441no—
529Q8—22no—
530Q8—22no—
531Q8—22no—
532Q8—1222no—
533Q8—1222no—
534Q8—1222no—
535Q8—2442no—
536Q8—2442no—
537Q8—2442no—
538D4—22no—
539D4—1222no—
540D4—2442no—
541C2×C4—21no—
542C2×C4—22no—
543C2×C4—611no—
544C2×C4—612no—
545C2×C4—612no—
546C2×C4—612no—
547C2×C4—612no—
548C2×C4—611no—
549C2×C4—611no—
550C2×C4—1221no—
551C2×C4—1221no—
552C2×C4—1221no—
553C2×C4—1221no—
554C2×C4—1221no—
555C2×C4—1221no—
556C2×C4—1221no—
557C2×C4—1221no—
558C2×C4—1221no—
559C2×C4—1221no—
560C2×C4—1221no—
561C2×C4—1221no—
562C2×C4—1221no—
563C2×C4—1221no—
564C2×C4—1221no—
565C2×C4—1222no—
566C2×C4—4884no—
567C2×C4—2441no—
568C2×C4—2442no—
569C2×C4—2441no—
570C2×C4—2442no—
571C2×C4—2442no—
572C2×C4—2441no—
573C2×C4—2441no—
574C2×C4—2441no—
575C2×C4—2441no—
576C2×C4—2442no—
577C2×C4—2442no—
578C2×C4—2441no—
579C2×C4—2441no—
580C2×C4D4×#692311yescharacteristic
581C2×C4D4×#692311yescharacteristic
582C2×C4C23×#692311yescharacteristic
583C8—22no—
584C8—1222no—
585C8—2442no—
586C22—41no—
587C22—41no—
588C22—41no—
589C22—611no—
590C22—611no—
591C22—611no—
592C22—611no—
593C22—611no—
594C22—611no—
595C22—4881no—
596C22—4881no—
597C22—4881no—
598C22—4881no—
599C22—2441no—
600C22—2441no—
601C22—2441no—
602C22C2×D4×#692311yescharacteristic
603C4—21no—
604C4—21no—
605C4—41no—
606C4—42no—
607C4—612no—
608C4—1222no—
609C4—1222no—
610C4—1221no—
611C4—1221no—
612C4—4881no—
613C4—4882no—
614C4—4881no—
615C4—2441no—
616C4—2441no—
617C4—2441no—
618C4—2442no—
619C4#1684082912yes—
620C2—81no—
621C2—611no—
622C2—611no—
623C2—611no—
624C2—611no—
625C2—4881no—
626C2#2146231111yescharacteristic
627C2#137×#692311yescharacteristic
628C2#137×#692311yescharacteristic
subgroupG/H
centerC22C2×D4×#6923
commutatorC2×C4×C61C23×C4
FrattiniC2×C4C23×#6923
FittingC61×#497C4
socleC22×C61C2×C4×D4
prime radicalC2×D4×D61C23
cosocleC2×C4×D61C24
hypercenter#497#6923
hypocenterC61#22273
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #497

Complements

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

NH ≅ G/Ntype
1. C4⋊Q8×#6923C2semidirect
3. #21461950C2semidirect
6. #21461813C2semidirect
7. #21461813C2semidirect
9. #21461742C2semidirect
10. #21461535C2semidirect
42. #16840157C4semidirect
43. #16840156C4semidirect
44. #16840155C4semidirect
45. #16840154C4semidirect
46. #16840153C4semidirect
48. C61×#497C4semidirect
88. C2×Dic244C2×C4semidirect
89. C2×#11758202C2×C4semidirect
90. C2×#11758202C2×C4semidirect
99. #14519777C2×C4semidirect
102. #14519775C2×C4semidirect
113. #14519688C2×C4semidirect
114. #14519687C2×C4semidirect
115. C2×Q16×C61C2×C4semidirect
117. C4⋊Q8×C61C2×C4semidirect
119. #97×C61C2×C4semidirect
338. #497#6923semidirect
363. C61#22273semidirect
373. C2×Q16C2×#6923semidirect
386. C4⋊Q8C2×#6923semidirect
428. #97C2×#6923semidirect

Maximal subgroups

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

subgroupindex
C4⋊Q8×#69232
#124×#69232
#214619502
#214619492
#214618152
#214618132
#214618122
#214617422
#214615352
#497×D612
#2227361