Finite Group Databasebeta

← #30615849

Subgroups — #30615849

Order 15616.

898 conjugacy classes of subgroups.

The subgroup lattice has 594 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21462300C211yesmaximal, characteristic
2#21462257C211yesmaximal, characteristic
3#21461883C212yesmaximal, complemented
4#21461882C212yesmaximal, complemented
5#21461881C212yesmaximal, complemented
6#21461880C212yesmaximal, complemented
7#21461507C211yesmaximal, characteristic
8#21461504C211yesmaximal, characteristic
9C2×#16840635C211yesmaximal, characteristic
10#21461487C211yesmaximal, characteristic
11#428×D61C211yesmaximal, characteristic
12C22×D61⋊C8C2211yescharacteristic
13C2×#14519856C2211yescharacteristic
14C2×#14519856C2211yescharacteristic
15C22×C4⋊#6923C2211yescharacteristic
16#16840713C2212yes—
17#16840713C2212yes—
18#16840709—22no—
19#16840708—21no—
20#16840706—21no—
21C4⋊C4×#6923—21no—
22C4⋊C4×#6923—21no—
23C4⋊C4×#6923C2212yes—
24C4⋊C4×#6923C2212yes—
25C4×C22⋊#6923—22no—
26#16840636C2212yes—
27#16840636C2212yes—
28#16840635C2211yescharacteristic
29#16840635C2211yescharacteristic
30#16840635C2211yescharacteristic
31#16840635C2211yescharacteristic
32#16840635C2212yes—
33#16840635C2212yes—
34(C2×#6923)⋊C8C2212yes—
35(C2×#6923)⋊C8C2212yes—
36C2×M16×D61C2211yescharacteristic
37#117×D61C2211yescharacteristic
38C2×C4⋊C4×D61C2211yescharacteristic
39#16839805C411yescharacteristic, complemented
40#16839804C411yescharacteristic, complemented
41#16839803C412yescomplemented
42#16839802C411yescharacteristic, complemented
43#16839801C412yescomplemented
44#107×D61C2212yes—
45#106×D61C2212yes—
46C61×#428C411yescharacteristic, complemented
47C23×#6923—21no—
48C2×D61⋊C8—22no—
49C2×D61⋊C8—22no—
50C2×D61⋊C8—21no—
51C2×D61⋊C8C2311yescharacteristic
52C2×D61⋊C8C2311yescharacteristic
53C22×#92420—21no—
54#14519856—22no—
55#14519856—22no—
56#14519856—22no—
57#14519856Q814yes—
58C2×C4⋊#6923—22no—
59C2×C4⋊#6923—22no—
60C2×C4⋊#6923—21no—
61C2×C4⋊#6923—22no—
62C2×C4⋊#6923—22no—
63C2×C4⋊#6923D411yescharacteristic
64C2×C4⋊#6923D411yescharacteristic
65C2×C4⋊#6923C2312yes—
66C2×C4⋊#6923C2311yescharacteristic
67C2×C4⋊#6923C2311yescharacteristic
68C2×C22⋊#6923—21no—
69C2×C22⋊#6923D412yes—
70C2×C4×#6923—21no—
71C2×C4×#6923—22no—
72C2×C4×#6923—22no—
73C2×C4×#6923C2312yes—
74(C2×#6923)⋊C4—22no—
75(C2×#6923)⋊C4—22no—
76(C2×#6923)⋊C4—22no—
77(C2×#6923)⋊C4—22no—
78C22×C4×D61—21no—
79C22×C4×D61C2311yescharacteristic
80C2×#11758194C2×C411yescharacteristic
81C2×#11758189C2×C412yes—
82M16×D61—22no—
83M16×D61—22no—
84C2×C8×D61C2312yes—
85#14519736C2×C411yescharacteristic
86#14519735C2×C411yescharacteristic
87#14519735C2×C411yescharacteristic
88#14519734—21no—
89#14519733—21no—
90C2×C4⋊Dic61C2×C411yescharacteristic
91C2×#11758196—21no—
92C4⋊C4×D61—21no—
93C4⋊C4×D61—21no—
94C4⋊C4×D61C2×C412yes—
95C4⋊C4×D61C2311yescharacteristic
96C4⋊C4×D61C2311yescharacteristic
97C4⋊C4×D61C2312yes—
98C22⋊C4×D61—21no—
99C42×D61—21no—
100#14519711C2×C412yescomplemented
101#14519710C2×C412yescomplemented
102#14519710C2×C412yescomplemented
103#14519709C2×C412yescomplemented
104#14519708C2×C412yescomplemented
105#14519708C2×C412yescomplemented
106C2×M16×C61C2×C411yescharacteristic
107#117×C61C2×C411yescharacteristic
108C2×C4⋊C4×C61C2×C411yescharacteristic
109#107×C61C2×C412yescomplemented
110#106×C61C2×C412yescomplemented
111C23×D61C2×D411yescharacteristic
112C22×#6923—21no—
113C22×#6923—21no—
114C22×#6923—22no—
115C22×#6923—42no—
116C22×#6923—81no—
117C22×#6923—81no—
118C22×#6923C2×D412yes—
119C22×#6923C2×D412yes—
120C22×C4×C61—21no—
121C22×C4×C61C22×C411yescharacteristic
122C22×Dic61—21no—
123C22×Dic61#4111yescharacteristic
124C2×C4×D61—22no—
125C2×C4×D61—21no—
126C2×C4×D61—21no—
127C2×C4×D61—22no—
128C2×C4×D61—81no—
129C2×C4×D61—81no—
130C2×C4×D61C22×C411yescharacteristic
131C2×C4×D61C2×D411yescharacteristic
132C2×C4×D61C2×Q812yes—
133C2×C4×D61#4112yes—
134C2×C4×D61#4112yes—
135C2×C4×D61#4112yes—
136C2×C4×D61#4111yescharacteristic
137C2×C4×D61C2411yescharacteristic
138C22⋊#6923—24no—
139C22⋊#6923—24no—
140C2×#92420—21no—
141C2×#92420—21no—
142C2×#92420—42no—
143C4⋊#6923—22no—
144C4⋊#6923—22no—
145C4⋊#6923—21no—
146C4⋊#6923—21no—
147C4⋊#6923—22no—
148C4⋊#6923—22no—
149C4⋊#6923—42no—
150C4⋊#6923—82no—
151C4⋊#6923—82no—
152C4⋊#6923C2×Q812yes—
153C4⋊#6923C2×Q812yes—
154C4×#6923—24no—
155C4×#6923—22no—
156C4×#6923—22no—
157C4×#6923—82no—
158C4×#6923—82no—
159D61⋊C8—21no—
160D61⋊C8—21no—
161D61⋊C8—21no—
162D61⋊C8—21no—
163D61⋊C8—42no—
164M16×C61—22no—
165C2×C8×C61C22×C412yes—
166C4⋊C4×C61—21no—
167C4⋊C4×C61C22×C411yescharacteristic
168C4⋊C4×C61C22×C412yes—
169C4⋊C4×C61C22×C411yescharacteristic
170C22⋊C4×C61—21no—
171C42×C61—21no—
172C22⋊Dic61—21no—
173#11758198—21no—
174#11758198—21no—
175C4⋊Dic61—21no—
176C4⋊Dic61C22×C411yescharacteristic
177C4⋊Dic61C22×C412yes—
178C4⋊Dic61C22×C411yescharacteristic
179#11758196—22no—
180#11758196—22no—
181#11758196—21no—
182#11758196—21no—
183C4×Dic61—21no—
184C4×Dic61—21no—
185C4×Dic61—21no—
186#11758194—22no—
187C2×#92418C22×C412yes—
188#11758189—24no—
189C8×D61—22no—
190C8×D61—22no—
191C23×C61C4×D411yescharacteristic
192C22×D61—21no—
193C22×D61—21no—
194C22×D61—21no—
195C22×D61—21no—
196C22×D61C4×D411yescharacteristic
197C22×D61C22⋊Q812yes—
198C22×D61#12312yes—
199C22×D61C22×D411yescharacteristic
200C2×#6923—22no—
201C2×#6923—22no—
202C2×#6923—22no—
203C2×#6923—22no—
204C2×#6923—22no—
205C2×#6923—22no—
206C2×#6923—22no—
207C2×#6923—22no—
208C2×#6923—41no—
209C2×#6923—41no—
210C2×#6923—41no—
211C2×#6923—41no—
212C2×#6923—42no—
213C2×#6923—42no—
214C2×#6923—42no—
215C2×#6923—84no—
216C2×C4×C61—21no—
217C2×C4×C61—21no—
218C2×C4×C61—22no—
219C2×C4×C61—81no—
220C2×C4×C61#11712yes—
221C2×C4×C61C4×D411yescharacteristic
222C2×C4×C61C4×Q812yes—
223C2×C4×C61C23×C411yescharacteristic
224C2×Dic61—21no—
225C2×Dic61—21no—
226C2×Dic61—22no—
227C2×Dic61—81no—
228C2×Dic61C4×D411yescharacteristic
229C2×Dic61#12512yes—
230C2×Dic61#12612yes—
231C2×Dic61C2×#4111yescharacteristic
232C4×D61—42no—
233C4×D61—41no—
234C4×D61—42no—
235C4×D61—41no—
236C4×D61—41no—
237C4×D61—41no—
238C4×D61—82no—
239C4×D61#11712yes—
240C4×D61C4×Q812yes—
241C4×D61C22⋊Q812yes—
242C4×D61#12312yes—
243C4×D61#12512yes—
244C4×D61#12612yes—
245C4×D61C22×Q811yescharacteristic
246C4×D61C2×#4111yescharacteristic
247C4×D61C2×#4111yescharacteristic
248C4×D61C2×#4111yescharacteristic
249#92420—41no—
250#92420—41no—
251C8×C61—22no—
252#92418—22no—
253#21035—611nomaximal, Sylow
254C22×C61—21no—
255C22×C61—21no—
256C22×C61#38712yes—
257C22×C61C2×C4×D411yescharacteristic
258C2×D61—21no—
259C2×D61—21no—
260C2×D61—22no—
261C2×D61—22no—
262C2×D61—22no—
263C2×D61—22no—
264C2×D61#38712yes—
265C2×D61C2×C22⋊Q811yescharacteristic
266C2×D61C2×#12311yescharacteristic
267C2×D61#57611yescharacteristic
268C2×D61#57711yescharacteristic
269C2×D61#57711yescharacteristic
270C2×D61#57811yescharacteristic
271#6923—42no—
272#6923—42no—
273#6923—42no—
274#6923—42no—
275#6923—82no—
276C4×C61—42no—
277C4×C61—41no—
278C4×C61—41no—
279C4×C61#38712yes—
280C4×C61C2×#11711yescharacteristic
281C4×C61C2×C4×Q811yescharacteristic
282Dic61—42no—
283Dic61—41no—
284Dic61—41no—
285Dic61#38712yes—
286Dic61C2×#12511yescharacteristic
287Dic61C2×#12611yescharacteristic
288C2×#428—611no—
289#2317—611no—
290#2274—611no—
291#1921—612no—
292#1920—612no—
293#1919—612no—
294#1918—612no—
295#1736—611no—
296#1735—611no—
297C2×#340—611no—
298#1729—611no—
299C2×C61—22no—
300C2×C61C2×#38711yescharacteristic
301C2×C61#294611yescharacteristic
302C2×C61#294711yescharacteristic
303D61—22no—
304D61#308311yescharacteristic
305D61#308411yescharacteristic
306D61#309611yescharacteristic
307D61#309711yescharacteristic
308C22×M16—611no—
309C23×C8—611no—
310C2×#117—611no—
311C2×#117—611no—
312C2×#117—611no—
313C22×C4⋊C4—611no—
314C22×C4⋊C4—611no—
315#428—611no—
316#428—611no—
317#428—611no—
318#428—612no—
319#428—612no—
320#428#692311yescharacteristic, complemented
321C2×#107—612no—
322C2×#106—612no—
323#389—612no—
324#389—612no—
325#385—1222no—
326#384—1221no—
327#382—1221no—
328C4×C4⋊C4—612no—
329C4×C4⋊C4—612no—
330C4×C4⋊C4—1221no—
331C4×C4⋊C4—1221no—
332C4×C22⋊C4—1222no—
333#340—611no—
334#340—612no—
335#340—612no—
336#340—611no—
337#340—612no—
338#340—611no—
339#340—612no—
340#340—611no—
341#336—612no—
342#336—612no—
343C61#2103511yescharacteristic, complemented, Sylow
344C23×C4—611no—
345C23×C4—1221no—
346C23×C4—1221no—
347C2×M16—611no—
348C2×M16—612no—
349C2×M16—1222no—
350C2×M16—1222no—
351C2×M16C2×#692311yescharacteristic
352C22×C8—612no—
353C22×C8—611no—
354C22×C8—611no—
355C22×C8—1221no—
356C22×C8—1222no—
357C22×C8—1222no—
358C22×C8—1221no—
359#117—614no—
360#117—611no—
361#117—611no—
362#117—611no—
363#117—1221no—
364#117—1221no—
365#117—1222no—
366#117—1222no—
367#117—1222no—
368#117C2×#692311yescharacteristic
369C2×C4⋊C4—611no—
370C2×C4⋊C4—611no—
371C2×C4⋊C4—611no—
372C2×C4⋊C4—612no—
373C2×C4⋊C4—612no—
374C2×C4⋊C4—611no—
375C2×C4⋊C4—611no—
376C2×C4⋊C4—611no—
377C2×C4⋊C4—611no—
378C2×C4⋊C4—612no—
379C2×C4⋊C4—1221no—
380C2×C4⋊C4—1221no—
381C2×C4⋊C4—1221no—
382C2×C4⋊C4—1222no—
383C2×C4⋊C4—1222no—
384C2×C4⋊C4—1222no—
385C2×C4⋊C4—1222no—
386C2×C4⋊C4—1221no—
387C2×C4⋊C4C2×#692311yescharacteristic
388C2×C22⋊C4—612no—
389C2×C22⋊C4—1221no—
390C2×C22⋊C4—1221no—
391C2×C42—612no—
392C2×C42—1221no—
393C2×C42—1221no—
394C2×C42—1222no—
395C2×C42—1222no—
396#107—612no—
397#107—612no—
398#107—612no—
399#107C2×#692312yescomplemented
400#106—612no—
401#106—612no—
402#106—612no—
403#106C2×#692312yescomplemented
404#95—1222no—
405#95—1222no—
406#95—1222no—
407#95—1222no—
408C24—611no—
409C22×C4—21no—
410C22×C4—611no—
411C22×C4—612no—
412C22×C4—612no—
413C22×C4—611no—
414C22×C4—611no—
415C22×C4—612no—
416C22×C4—612no—
417C22×C4—611no—
418C22×C4—611no—
419C22×C4—612no—
420C22×C4—612no—
421C22×C4—1222no—
422C22×C4—1222no—
423C22×C4—1221no—
424C22×C4—1221no—
425C22×C4—1221no—
426C22×C4—1221no—
427C22×C4—1221no—
428C22×C4—1222no—
429C22×C4—4881no—
430C22×C4—4881no—
431C22×C4—4881no—
432C22×C4—4881no—
433C22×C4—2442no—
434C22×C4C22×#692311yescharacteristic
435M16—22no—
436M16—1224no—
437M16—1222no—
438C2×C8—612no—
439C2×C8—1222no—
440C2×C8—1222no—
441C2×C8—1221no—
442C2×C8—1221no—
443C2×C8—1221no—
444C2×C8—1221no—
445C2×C8—1221no—
446C2×C8—1221no—
447C2×C8—2442no—
448C2×C8—2442no—
449C2×C8C22×#692312yes—
450C4⋊C4—21no—
451C4⋊C4—612no—
452C4⋊C4—612no—
453C4⋊C4—611no—
454C4⋊C4—612no—
455C4⋊C4—611no—
456C4⋊C4—1222no—
457C4⋊C4—1222no—
458C4⋊C4—1221no—
459C4⋊C4—1221no—
460C4⋊C4—1222no—
461C4⋊C4—1222no—
462C4⋊C4—1221no—
463C4⋊C4—1221no—
464C4⋊C4—1222no—
465C4⋊C4—1222no—
466C4⋊C4—1221no—
467C4⋊C4—4882no—
468C4⋊C4—4882no—
469C4⋊C4—2442no—
470C4⋊C4C22×#692311yescharacteristic
471C4⋊C4C22×#692312yes—
472C4⋊C4C22×#692311yescharacteristic
473C22⋊C4—21no—
474C22⋊C4—1221no—
475C22⋊C4—1221no—
476C22⋊C4—1224no—
477C22⋊C4—1224no—
478C22⋊C4—1221no—
479C42—21no—
480C42—1221no—
481C42—1221no—
482C42—1222no—
483C42—1222no—
484C42—1224no—
485C42—1221no—
486C42—4882no—
487C42—4882no—
488C23—611no—
489C23—612no—
490C23—612no—
491C23—611no—
492C23—1221no—
493C23—1221no—
494C23—1221no—
495C23—1221no—
496C23D4×#692311yescharacteristic
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—22no—
500C2×C4—81no—
501C2×C4—612no—
502C2×C4—612no—
503C2×C4—611no—
504C2×C4—611no—
505C2×C4—612no—
506C2×C4—612no—
507C2×C4—612no—
508C2×C4—612no—
509C2×C4—612no—
510C2×C4—611no—
511C2×C4—611no—
512C2×C4—612no—
513C2×C4—611no—
514C2×C4—611no—
515C2×C4—1222no—
516C2×C4—1222no—
517C2×C4—1222no—
518C2×C4—1222no—
519C2×C4—1222no—
520C2×C4—1222no—
521C2×C4—1222no—
522C2×C4—1222no—
523C2×C4—1221no—
524C2×C4—1221no—
525C2×C4—1222no—
526C2×C4—4882no—
527C2×C4—4884no—
528C2×C4—4881no—
529C2×C4—2442no—
530C2×C4—2442no—
531C2×C4—2441no—
532C2×C4—2441no—
533C2×C4—2441no—
534C2×C4—2441no—
535C2×C4—2442no—
536C2×C4—2442no—
537C2×C4—2442no—
538C2×C4—2441no—
539C2×C4—2441no—
540C2×C4—2441no—
541C2×C4—2441no—
542C2×C4#1451985612yes—
543C2×C4D4×#692311yescharacteristic
544C2×C4Q8×#692312yes—
545C2×C4C23×#692311yescharacteristic
546C8—22no—
547C8—1222no—
548C8—2441no—
549C8—2441no—
550C22—21no—
551C22—21no—
552C22—611no—
553C22—611no—
554C22—611no—
555C22—611no—
556C22—611no—
557C22—611no—
558C22—612no—
559C22—1222no—
560C22—1222no—
561C22—1222no—
562C22—1222no—
563C22—1221no—
564C22—1221no—
565C22#1684071112yes—
566C22C2×D4×#692311yescharacteristic
567C4—42no—
568C4—41no—
569C4—41no—
570C4—611no—
571C4—611no—
572C4—612no—
573C4—4882no—
574C4—2442no—
575C4—2442no—
576C4—2442no—
577C4—2442no—
578C4—2442no—
579C4—2441no—
580C4—2441no—
581C4#1684071112yes—
582C4C2×#1451985611yescharacteristic
583C4C2×Q8×#692311yescharacteristic
584C2—22no—
585C2—611no—
586C2—611no—
587C2—611no—
588C2—611no—
589C2—1222no—
590C2C2×#1684071111yescharacteristic
591C2#2146257611yescharacteristic
592C2#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#21035
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
3. #21461883C2semidirect
4. #21461882C2semidirect
5. #21461881C2semidirect
6. #21461880C2semidirect
39. #16839805C4semidirect
40. #16839804C4semidirect
41. #16839803C4semidirect
42. #16839802C4semidirect
43. #16839801C4semidirect
46. C61×#428C4semidirect
100. #14519711C2×C4semidirect
101. #14519710C2×C4semidirect
102. #14519710C2×C4semidirect
103. #14519709C2×C4semidirect
104. #14519708C2×C4semidirect
105. #14519708C2×C4semidirect
109. #107×C61C2×C4semidirect
110. #106×C61C2×C4semidirect
320. #428#6923semidirect
343. C61#21035semidirect
399. #107C2×#6923semidirect
403. #106C2×#6923semidirect

Maximal subgroups

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

subgroupindex
#214623002
#214622572
#214618832
#214618822
#214618812
#214618802
#214615072
#214615042
C2×#168406352
#214614872
#428×D612
#2103561