Finite Group Databasebeta

← #30616147

Subgroups — #30616147

Order 15616.

982 conjugacy classes of subgroups.

The subgroup lattice has 606 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840773C211yesmaximal, characteristic
2C2×#16840728C211yesmaximal, characteristic
3#21462300C211yesmaximal, characteristic
4#21462299C211yesmaximal, characteristic
5#21461925C212yesmaximal, complemented
6#21461924C212yesmaximal, complemented
7#21461923C212yesmaximal, complemented
8#21461922C212yesmaximal, complemented
9#21461504C211yesmaximal, characteristic
10C2×#16840635C211yesmaximal, characteristic
11#428×D61C211yesmaximal, characteristic
12C22×D61⋊C8C2211yescharacteristic
13C2×#14519856C2211yescharacteristic
14C2×#14519856C2211yescharacteristic
15C22×C4⋊#6923C2211yescharacteristic
16#16840773—24no—
17C2×#14519846C2212yes—
18C2×#14519845C2212yes—
19#16840728—24no—
20C2×#14519831C2212yes—
21C2×C8×#6923C2212yes—
22#16840713—21no—
23#16840713—21no—
24#16840713C2212yes—
25#16840713C2212yes—
26#16840713C2212yes—
27#16840713C2212yes—
28#16840709—22no—
29#16840708—21no—
30#16840706—21no—
31#16840704—22no—
32#16840635C2211yescharacteristic
33#16840635C2211yescharacteristic
34#16840635C2211yescharacteristic
35#16840635C2211yescharacteristic
36#16840635C2212yes—
37#16840635C2212yes—
38C2×M16×D61C2211yescharacteristic
39#117×D61C2211yescharacteristic
40C2×C4⋊C4×D61C2211yescharacteristic
41#16839805C411yescharacteristic, complemented
42#16839804C411yescharacteristic, complemented
43#16839803C412yescomplemented
44#16839802C411yescharacteristic, complemented
45#16839801C412yescomplemented
46#107×D61C2212yes—
47#106×D61C2212yes—
48C61×#428C411yescharacteristic, complemented
49C23×#6923—21no—
50C2×D61⋊C8—22no—
51C2×D61⋊C8—21no—
52C2×D61⋊C8Q814yes—
53C2×D61⋊C8C2311yescharacteristic
54C2×D61⋊C8C2311yescharacteristic
55C22×#92420—21no—
56#14519856—22no—
57#14519856—22no—
58#14519856—22no—
59#14519856—22no—
60C2×C4⋊#6923—22no—
61C2×C4⋊#6923—22no—
62C2×C4⋊#6923—21no—
63C2×C4⋊#6923—22no—
64C2×C4⋊#6923—22no—
65C2×C4⋊#6923—22no—
66C2×C4⋊#6923—22no—
67C2×C4⋊#6923D411yescharacteristic
68C2×C4⋊#6923D411yescharacteristic
69C2×C4⋊#6923C2311yescharacteristic
70C2×C4⋊#6923C2312yes—
71C2×C4⋊#6923C2311yescharacteristic
72C2×C22⋊#6923—21no—
73C2×C22⋊#6923D412yes—
74C2×C4×#6923—21no—
75C2×C4×#6923C2312yes—
76#14519846—24no—
77#14519845—24no—
78#14519831—24no—
79C8×#6923—24no—
80(C2×#6923)⋊C4—22no—
81(C2×#6923)⋊C4—22no—
82(C2×#6923)⋊C4—22no—
83(C2×#6923)⋊C4—22no—
84C22×C4×D61—21no—
85C22×C4×D61C2311yescharacteristic
86C2×#11758194C2×C411yescharacteristic
87C2×#11758189C2×C412yes—
88M16×D61D414yes—
89M16×D61Q814yes—
90C2×C8×D61C2312yes—
91#14519736C2×C411yescharacteristic
92#14519735C2×C411yescharacteristic
93#14519735C2×C411yescharacteristic
94#14519734—21no—
95#14519733—21no—
96C2×C4⋊Dic61C2×C411yescharacteristic
97C2×#11758196—21no—
98C4⋊C4×D61—21no—
99C4⋊C4×D61—21no—
100C4⋊C4×D61C2×C412yes—
101C4⋊C4×D61C2312yes—
102C4⋊C4×D61C2311yescharacteristic
103C4⋊C4×D61C2311yescharacteristic
104C22⋊C4×D61—21no—
105C42×D61—21no—
106#14519711C2×C412yescomplemented
107#14519710C2×C412yescomplemented
108#14519710C2×C412yescomplemented
109#14519709C2×C412yescomplemented
110#14519708C2×C412yescomplemented
111#14519708C2×C412yescomplemented
112C2×M16×C61C2×C411yescharacteristic
113#117×C61C2×C411yescharacteristic
114C2×C4⋊C4×C61C2×C411yescharacteristic
115#107×C61C2×C412yescomplemented
116#106×C61C2×C412yescomplemented
117C23×D61C2×D411yescharacteristic
118C22×#6923—21no—
119C22×#6923—21no—
120C22×#6923—22no—
121C22×#6923—42no—
122C22×#6923—81no—
123C22×#6923—81no—
124C22×#6923C2×D412yes—
125C22×#6923C2×D412yes—
126C22×C4×C61—21no—
127C22×C4×C61C22×C411yescharacteristic
128C22×Dic61—21no—
129C22×Dic61#4111yescharacteristic
130C2×C4×D61—22no—
131C2×C4×D61—21no—
132C2×C4×D61—21no—
133C2×C4×D61—22no—
134C2×C4×D61—81no—
135C2×C4×D61—81no—
136C2×C4×D61C22×C411yescharacteristic
137C2×C4×D61C2×D412yes—
138C2×C4×D61C2×D411yescharacteristic
139C2×C4×D61C2×Q812yes—
140C2×C4×D61C2×Q812yes—
141C2×C4×D61#4112yes—
142C2×C4×D61#4111yescharacteristic
143C2×C4×D61C2411yescharacteristic
144C22⋊#6923—24no—
145C22⋊#6923—42no—
146C2×#92420—24no—
147C2×#92420—21no—
148C2×#92420—21no—
149C4⋊#6923—22no—
150C4⋊#6923—22no—
151C4⋊#6923—21no—
152C4⋊#6923—22no—
153C4⋊#6923—21no—
154C4⋊#6923—22no—
155C4⋊#6923—21no—
156C4⋊#6923—21no—
157C4⋊#6923—42no—
158C4⋊#6923—82no—
159C4⋊#6923—82no—
160C4⋊#6923—82no—
161C4⋊#6923—82no—
162C4×#6923—22no—
163C4×#6923—22no—
164C4×#6923—42no—
165D61⋊C8—24no—
166D61⋊C8—21no—
167D61⋊C8—21no—
168D61⋊C8C2×Q812yes—
169D61⋊C8C2×Q812yes—
170M16×C61C4⋊C414yescomplemented
171C2×C8×C61C22×C412yes—
172C4⋊C4×C61—21no—
173C4⋊C4×C61C22×C411yescharacteristic
174C4⋊C4×C61C22×C412yes—
175C4⋊C4×C61C22×C411yescharacteristic
176C22⋊C4×C61—21no—
177C42×C61—21no—
178C22⋊Dic61—21no—
179#11758198—21no—
180#11758198—21no—
181C4⋊Dic61—21no—
182C4⋊Dic61C22×C411yescharacteristic
183C4⋊Dic61C22×C412yes—
184C4⋊Dic61C22×C411yescharacteristic
185#11758196—22no—
186#11758196—22no—
187#11758196—21no—
188#11758196—21no—
189C4×Dic61—21no—
190C4×Dic61—21no—
191C4×Dic61—21no—
192#11758194C4⋊C414yescomplemented
193C2×#92418C22×C412yes—
194#11758189C4⋊C418yescomplemented
195C8×D61C2×D414yes—
196C8×D61C2×Q814yes—
197C23×C61C4×D411yescharacteristic
198C22×D61—21no—
199C22×D61—21no—
200C22×D61—21no—
201C22×D61—21no—
202C22×D61C4×D411yescharacteristic
203C22×D61#12112yes—
204C22×D61C4⋊Q812yes—
205C22×D61C22×D411yescharacteristic
206C2×#6923—22no—
207C2×#6923—22no—
208C2×#6923—22no—
209C2×#6923—22no—
210C2×#6923—22no—
211C2×#6923—22no—
212C2×#6923—41no—
213C2×#6923—41no—
214C2×#6923—42no—
215C2×#6923—41no—
216C2×#6923—42no—
217C2×#6923—41no—
218C2×#6923—42no—
219C2×#6923—41no—
220C2×#6923—41no—
221C2×#6923—84no—
222C2×C4×C61—21no—
223C2×C4×C61—21no—
224C2×C4×C61—22no—
225C2×C4×C61—81no—
226C2×C4×C61C2×C4⋊C412yes—
227C2×C4×C61C4×D411yescharacteristic
228C2×C4×C61C4×Q812yes—
229C2×C4×C61C23×C411yescharacteristic
230C2×Dic61—21no—
231C2×Dic61—21no—
232C2×Dic61—22no—
233C2×Dic61—81no—
234C2×Dic61C4×D411yescharacteristic
235C2×Dic61C22⋊Q812yes—
236C2×Dic61#12512yes—
237C2×Dic61C2×#4111yescharacteristic
238C4×D61—41no—
239C4×D61—42no—
240C4×D61—41no—
241C4×D61—42no—
242C4×D61—41no—
243C4×D61—41no—
244C4×D61—82no—
245C4×D61C2×C4⋊C412yes—
246C4×D61C4×Q812yes—
247C4×D61#12112yes—
248C4×D61C22⋊Q812yes—
249C4×D61#12512yes—
250C4×D61C4⋊Q812yes—
251C4×D61C22×D411yescharacteristic
252C4×D61C22×Q811yescharacteristic
253C4×D61C22×Q811yescharacteristic
254C4×D61C2×#4111yescharacteristic
255#92420—22no—
256#92420—22no—
257C8×C61C2×C4⋊C414yes—
258#92418C2×C4⋊C414yes—
259#21160—611nomaximal, Sylow
260C22×C61—21no—
261C22×C61—21no—
262C22×C61#38912yes—
263C22×C61C2×C4×D411yescharacteristic
264C2×D61—21no—
265C2×D61—21no—
266C2×D61—22no—
267C2×D61—22no—
268C2×D61—22no—
269C2×D61—22no—
270C2×D61#38912yes—
271C2×D61C2×#12111yescharacteristic
272C2×D61C2×C4⋊Q811yescharacteristic
273C2×D61#57611yescharacteristic
274C2×D61#57711yescharacteristic
275C2×D61#57711yescharacteristic
276C2×D61#57811yescharacteristic
277#6923—42no—
278#6923—42no—
279#6923—81no—
280#6923—82no—
281#6923—81no—
282C4×C61—42no—
283C4×C61—41no—
284C4×C61—41no—
285C4×C61#38912yes—
286C4×C61C22×C4⋊C411yescharacteristic
287C4×C61C2×C4×Q811yescharacteristic
288Dic61—42no—
289Dic61—41no—
290Dic61—41no—
291Dic61#38912yes—
292Dic61C2×C22⋊Q811yescharacteristic
293Dic61C2×#12511yescharacteristic
294C2×#428—611no—
295C2×#428—611no—
296C2×#405—611no—
297#2317—611no—
298#2316—611no—
299#1944—612no—
300#1943—612no—
301#1942—612no—
302#1941—612no—
303#1735—611no—
304C2×#340—611no—
305C2×C61—22no—
306C2×C61C2×#38911yescharacteristic
307C2×C61#294611yescharacteristic
308C2×C61#294711yescharacteristic
309D61—22no—
310D61#306211yescharacteristic
311D61#306311yescharacteristic
312D61#316311yescharacteristic
313D61#316411yescharacteristic
314C22×M16—611no—
315C23×C8—611no—
316C2×#117—611no—
317C2×#117—611no—
318C2×#117—611no—
319C22×C4⋊C4—611no—
320C22×C4⋊C4—611no—
321#428—611no—
322#428—611no—
323#428—611no—
324#428—612no—
325#428—612no—
326#428—1224no—
327#428#692311yescharacteristic, complemented
328C2×#107—612no—
329C2×#107—612no—
330C2×#106—612no—
331C2×#106—612no—
332#405—1224no—
333C2×#97—612no—
334C2×C4×C8—612no—
335#389—612no—
336#389—612no—
337#389—612no—
338#389—612no—
339#389—1221no—
340#389—1221no—
341#385—1222no—
342#384—1221no—
343#382—1221no—
344#380—1222no—
345#340—611no—
346#340—612no—
347#340—611no—
348#340—611no—
349#340—612no—
350#340—611no—
351C61#2116011yescharacteristic, complemented, Sylow
352C23×C4—611no—
353C23×C4—1221no—
354C23×C4—1221no—
355C2×M16—611no—
356C2×M16—614no—
357C2×M16—614no—
358C2×M16—612no—
359C2×M16C2×#692311yescharacteristic
360C22×C8—612no—
361C22×C8—614no—
362C22×C8—611no—
363C22×C8—611no—
364C22×C8—1221no—
365C22×C8—1221no—
366C22×C8—1222no—
367#117—611no—
368#117—611no—
369#117—611no—
370#117—1221no—
371#117—1221no—
372#117—1222no—
373#117—1222no—
374#117—1222no—
375#117—1222no—
376#117C2×#692311yescharacteristic
377C2×C4⋊C4—611no—
378C2×C4⋊C4—612no—
379C2×C4⋊C4—612no—
380C2×C4⋊C4—611no—
381C2×C4⋊C4—611no—
382C2×C4⋊C4—611no—
383C2×C4⋊C4—611no—
384C2×C4⋊C4—612no—
385C2×C4⋊C4—611no—
386C2×C4⋊C4—611no—
387C2×C4⋊C4—1221no—
388C2×C4⋊C4—1221no—
389C2×C4⋊C4—1221no—
390C2×C4⋊C4—1222no—
391C2×C4⋊C4—1222no—
392C2×C4⋊C4—1222no—
393C2×C4⋊C4—1222no—
394C2×C4⋊C4—1222no—
395C2×C4⋊C4—1222no—
396C2×C4⋊C4—1221no—
397C2×C4⋊C4C2×#692311yescharacteristic
398C2×C22⋊C4—612no—
399C2×C22⋊C4—1221no—
400C2×C22⋊C4—1221no—
401C2×C42—612no—
402C2×C42—1221no—
403C2×C42—1221no—
404#107—612no—
405#107—612no—
406#107—612no—
407#107—1224no—
408#107C2×#692312yescomplemented
409#106—612no—
410#106—612no—
411#106—612no—
412#106—1224no—
413#106C2×#692312yescomplemented
414#97—1224no—
415C4×C8—1224no—
416#95—1222no—
417#95—1222no—
418#95—1222no—
419#95—1222no—
420C24—611no—
421C22×C4—21no—
422C22×C4—611no—
423C22×C4—612no—
424C22×C4—612no—
425C22×C4—611no—
426C22×C4—611no—
427C22×C4—612no—
428C22×C4—612no—
429C22×C4—611no—
430C22×C4—611no—
431C22×C4—612no—
432C22×C4—612no—
433C22×C4—1222no—
434C22×C4—1221no—
435C22×C4—1221no—
436C22×C4—1222no—
437C22×C4—1221no—
438C22×C4—1221no—
439C22×C4—1221no—
440C22×C4—1222no—
441C22×C4—4881no—
442C22×C4—4881no—
443C22×C4—4881no—
444C22×C4—4881no—
445C22×C4—2442no—
446C22×C4C22×#692311yescharacteristic
447M16—614no—
448M16—618no—
449M16C4⋊#692314yescomplemented
450C2×C8—612no—
451C2×C8—614no—
452C2×C8—614no—
453C2×C8—612no—
454C2×C8—612no—
455C2×C8—1221no—
456C2×C8—1221no—
457C2×C8—1224no—
458C2×C8—1224no—
459C2×C8—1221no—
460C2×C8—1221no—
461C2×C8C22×#692312yes—
462C4⋊C4—21no—
463C4⋊C4—611no—
464C4⋊C4—612no—
465C4⋊C4—611no—
466C4⋊C4—1222no—
467C4⋊C4—1222no—
468C4⋊C4—1221no—
469C4⋊C4—1221no—
470C4⋊C4—1222no—
471C4⋊C4—1222no—
472C4⋊C4—1221no—
473C4⋊C4—1222no—
474C4⋊C4—1221no—
475C4⋊C4—1222no—
476C4⋊C4—1221no—
477C4⋊C4—1221no—
478C4⋊C4—1221no—
479C4⋊C4—4882no—
480C4⋊C4—4882no—
481C4⋊C4—4882no—
482C4⋊C4—4882no—
483C4⋊C4—2442no—
484C4⋊C4C22×#692311yescharacteristic
485C4⋊C4C22×#692312yes—
486C4⋊C4C22×#692311yescharacteristic
487C22⋊C4—21no—
488C22⋊C4—1221no—
489C22⋊C4—1221no—
490C22⋊C4—1224no—
491C22⋊C4—1221no—
492C22⋊C4—2442no—
493C42—21no—
494C42—1221no—
495C42—1221no—
496C42—1222no—
497C42—1222no—
498C42—1221no—
499C42—2442no—
500C23—611no—
501C23—612no—
502C23—612no—
503C23—611no—
504C23—1221no—
505C23—1221no—
506C23—1221no—
507C23—1221no—
508C23D4×#692311yescharacteristic
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—22no—
512C2×C4—81no—
513C2×C4—612no—
514C2×C4—612no—
515C2×C4—611no—
516C2×C4—611no—
517C2×C4—612no—
518C2×C4—612no—
519C2×C4—612no—
520C2×C4—612no—
521C2×C4—612no—
522C2×C4—611no—
523C2×C4—611no—
524C2×C4—612no—
525C2×C4—611no—
526C2×C4—611no—
527C2×C4—1222no—
528C2×C4—1222no—
529C2×C4—1222no—
530C2×C4—1222no—
531C2×C4—1222no—
532C2×C4—1222no—
533C2×C4—1221no—
534C2×C4—1221no—
535C2×C4—1222no—
536C2×C4—4882no—
537C2×C4—4884no—
538C2×C4—4881no—
539C2×C4—2441no—
540C2×C4—2441no—
541C2×C4—2442no—
542C2×C4—2442no—
543C2×C4—2441no—
544C2×C4—2441no—
545C2×C4—2441no—
546C2×C4—2441no—
547C2×C4—2442no—
548C2×C4—2441no—
549C2×C4—2441no—
550C2×C4—2442no—
551C2×C4—2442no—
552C2×C4—2441no—
553C2×C4—2441no—
554C2×C4C2×C4⋊#692312yes—
555C2×C4D4×#692311yescharacteristic
556C2×C4Q8×#692312yes—
557C2×C4C23×#692311yescharacteristic
558C8—614no—
559C8—1222no—
560C8—1222no—
561C8C2×C4⋊#692314yes—
562C22—21no—
563C22—21no—
564C22—611no—
565C22—611no—
566C22—611no—
567C22—611no—
568C22—611no—
569C22—611no—
570C22—612no—
571C22—1222no—
572C22—1222no—
573C22—1222no—
574C22—1222no—
575C22—1221no—
576C22—1221no—
577C22#1684071312yes—
578C22C2×D4×#692311yescharacteristic
579C4—42no—
580C4—41no—
581C4—41no—
582C4—611no—
583C4—611no—
584C4—612no—
585C4—4882no—
586C4—4881no—
587C4—4881no—
588C4—2442no—
589C4—2442no—
590C4—2442no—
591C4—2441no—
592C4—2441no—
593C4#1684071312yes—
594C4C22×C4⋊#692311yescharacteristic
595C4C2×Q8×#692311yescharacteristic
596C2—22no—
597C2—611no—
598C2—611no—
599C2—611no—
600C2—611no—
601C2—1222no—
602C2C2×#1684071311yescharacteristic
603C2#2146257611yescharacteristic
604C2#2146257711yescharacteristic
subgroupG/H
centerC22C2×D4×#6923
commutatorC2×C4×C61C23×C4
FrattiniC2×C4C23×#6923
FittingC61×#428C4
socleC22×C61C2×C4×D4
prime radicalC23×D61C2×D4
cosocleC2×C4×D61C24
hypercenter#428#6923
hypocenterC61#21160
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ #428

Complements

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

NH ≅ G/Ntype
5. #21461925C2semidirect
6. #21461924C2semidirect
7. #21461923C2semidirect
8. #21461922C2semidirect
41. #16839805C4semidirect
42. #16839804C4semidirect
43. #16839803C4semidirect
44. #16839802C4semidirect
45. #16839801C4semidirect
48. C61×#428C4semidirect
106. #14519711C2×C4semidirect
107. #14519710C2×C4semidirect
108. #14519710C2×C4semidirect
109. #14519709C2×C4semidirect
110. #14519708C2×C4semidirect
111. #14519708C2×C4semidirect
115. #107×C61C2×C4semidirect
116. #106×C61C2×C4semidirect
170. M16×C61C4⋊C4semidirect
192. #11758194C4⋊C4semidirect
194. #11758189C4⋊C4semidirect
327. #428#6923semidirect
351. C61#21160semidirect
408. #107C2×#6923semidirect
413. #106C2×#6923semidirect
449. M16C4⋊#6923semidirect

Maximal subgroups

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

subgroupindex
C2×#168407732
C2×#168407282
#214623002
#214622992
#214619252
#214619242
#214619232
#214619222
#214615042
C2×#168406352
#428×D612
#2116061