Finite Group Databasebeta

← #30615915

Subgroups — #30615915

Order 15616.

1238 conjugacy classes of subgroups.

The subgroup lattice has 584 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840773C211yesmaximal, characteristic, complemented
2C22×#14519839C211yesmaximal, characteristic
3#21462292C211yesmaximal, characteristic
4#21462256C211yesmaximal, characteristic, complemented
5#21461897C214yesmaximal, complemented
6#21461896C214yesmaximal, complemented
7#21461504C211yesmaximal, characteristic, complemented
8#21461487C211yesmaximal, characteristic, complemented
9#418×D61C211yesmaximal, characteristic
10C22×D61⋊C8C2211yescharacteristic
11C2×#14519856C2211yescharacteristic
12C2×#14519856C2211yescharacteristic
13C22×C4⋊#6923C2211yescharacteristic
14#16840773—24no—
15C2×#14519846C2212yescomplemented
16C2×#14519845C2212yescomplemented
17C2×#14519839—24no—
18C2×#14519839C2214yes—
19#16840714—22no—
20#16840714C2212yes—
21#16840714C2212yes—
22#16840703—22no—
23C4⋊C4×#6923C2214yescomplemented
24C4×C4⋊#6923—22no—
25C22⋊C4×#6923—22no—
26#16840635C2212yescomplemented
27#16840635C2212yescomplemented
28(C2×#6923)⋊C8C2214yescomplemented
29C22×D4×D61C2211yescharacteristic
30C2×M16×D61C2211yescharacteristic
31#117×D61C2211yescharacteristic
32#16839768C411yescharacteristic, complemented
33#16839767C411yescharacteristic, complemented
34#16839766C411yescharacteristic, complemented
35#16839765C414yescomplemented
36#102×D61C2214yes—
37C61×#418C411yescharacteristic, complemented
38C23×#6923—21no—
39C2×D61⋊C8—22no—
40C2×D61⋊C8—22no—
41C2×D61⋊C8—21no—
42C2×D61⋊C8C2312yes—
43C22×#92420—21no—
44#14519856—22no—
45#14519856—22no—
46#14519856—22no—
47#14519856—22no—
48C2×C4⋊#6923—21no—
49C2×C4⋊#6923—22no—
50C2×C4⋊#6923D412yes—
51C2×C4⋊#6923D414yes—
52C2×C4⋊#6923C2311yescharacteristic
53C2×C4⋊#6923C2311yescharacteristic
54C2×C4⋊#6923C2312yes—
55C2×C22⋊#6923—21no—
56C2×C22⋊#6923—24no—
57C2×C22⋊#6923—24no—
58C2×C22⋊#6923D412yes—
59C2×C4×#6923—21no—
60C2×C4×#6923—24no—
61C2×C4×#6923C2312yes—
62#14519846—24no—
63#14519845—24no—
64#14519839—28no—
65(C2×#6923)⋊C4—24no—
66C24×D61—21no—
67C22×D244C2×C411yescharacteristic
68C22×C22⋊D61—21no—
69C2×D4×D61—22no—
70C2×D4×D61—21no—
71C2×D4×D61—22no—
72C2×D4×D61—22no—
73C2×D4×D61—22no—
74C2×D4×D61—21no—
75C2×D4×D61C2×C412yes—
76C2×D4×D61C2312yes—
77C22×C4×D61C2311yescharacteristic
78C2×#11758194C2×C411yescharacteristic
79C2×#11758189C2×C412yes—
80M16×D61—22no—
81M16×D61—22no—
82C2×C8×D61C2312yes—
83#14519736C2×C411yescharacteristic
84#14519735C2×C412yes—
85#14519734—21no—
86#14519733—21no—
87C4⋊C4×D61C2312yes—
88C22⋊C4×D61—21no—
89C42×D61—21no—
90#14519699C2×C414yescomplemented
91#14519698C2×C414yescomplemented
92#14519697C2×C414yescomplemented
93C22×D4×C61C2×C411yescharacteristic
94C2×M16×C61C2×C411yescharacteristic
95#117×C61C2×C411yescharacteristic
96#102×C61C2×C414yescomplemented
97C24×C61—21no—
98C23×D61—21no—
99C23×D61—22no—
100C23×D61—22no—
101C23×D61—42no—
102C23×D61—42no—
103C23×D61—82no—
104C23×D61—82no—
105C23×D61C2×D411yescharacteristic
106C22×#6923—22no—
107C22×#6923—24no—
108C22×#6923—21no—
109C22×#6923—21no—
110C22×#6923—82no—
111C22×#6923C2×D414yes—
112C2×D4×C61—22no—
113C2×D4×C61—22no—
114C2×D4×C61—21no—
115C2×D4×C61C22×C412yes—
116C22×C4×C61C22×C411yescharacteristic
117C2×C22⋊D61—22no—
118C2×C22⋊D61—22no—
119C2×C22⋊D61—42no—
120C2×C22⋊D61—42no—
121C22×Dic61#4111yescharacteristic
122D4×D61—24no—
123D4×D61—24no—
124D4×D61—22no—
125D4×D61—22no—
126D4×D61—22no—
127D4×D61—22no—
128D4×D61—42no—
129D4×D61—42no—
130D4×D61—42no—
131D4×D61—42no—
132C2×D244—22no—
133C2×D244—22no—
134C2×D244—21no—
135C2×D244C22×C412yes—
136C2×C4×D61—22no—
137C2×C4×D61C22×C411yescharacteristic
138C2×C4×D61C2×D412yes—
139C2×C4×D61C2×D411yescharacteristic
140C2×C4×D61#4112yes—
141C2×C4×D61#4112yes—
142C2×C4×D61#4112yes—
143C2×C4×D61#4111yescharacteristic
144C2×C4×D61C2411yescharacteristic
145C22⋊#6923—24no—
146C22⋊#6923—42no—
147C22⋊#6923—84no—
148C22⋊#6923—84no—
149C2×#92420—22no—
150C2×#92420—42no—
151C4⋊#6923—24no—
152C4⋊#6923—24no—
153C4⋊#6923—21no—
154C4⋊#6923—21no—
155C4⋊#6923—22no—
156C4⋊#6923—21no—
157C4⋊#6923—21no—
158C4⋊#6923C2×D414yes—
159C4×#6923—22no—
160C4×#6923—22no—
161C4×#6923—42no—
162C4×#6923—84no—
163D61⋊C8—22no—
164D61⋊C8—22no—
165D61⋊C8—42no—
166M16×C61—22no—
167C2×C8×C61C22×C412yes—
168C4⋊C4×C61C22×C412yes—
169C22⋊C4×C61—21no—
170C42×C61—21no—
171C22⋊Dic61—21no—
172#11758198—22no—
173C4⋊Dic61C22×C412yes—
174#11758196—22no—
175C4×Dic61—21no—
176C4×Dic61—22no—
177#11758194—22no—
178C2×#92418C22×C412yes—
179#11758189—24no—
180C8×D61—22no—
181C8×D61—22no—
182C23×C61—22no—
183C23×C61—42no—
184C23×C61—82no—
185C23×C61C4×D411yescharacteristic
186C22×D61—22no—
187C22×D61—22no—
188C22×D61—22no—
189C22×D61—42no—
190C22×D61—42no—
191C22×D61—42no—
192C22×D61—44no—
193C22×D61—44no—
194C22×D61—44no—
195C22×D61—84no—
196C22×D61—88no—
197C22×D61—84no—
198C22×D61—84no—
199C22×D61—82no—
200C22×D61C4×D411yescharacteristic
201C22×D61#12112yes—
202C22×D61#12312yes—
203C22×D61C22×D411yescharacteristic
204C2×#6923—24no—
205C2×#6923—24no—
206C2×#6923—24no—
207C2×#6923—24no—
208C2×#6923—44no—
209C2×#6923—41no—
210C2×#6923—41no—
211C2×#6923—42no—
212C2×#6923—42no—
213C2×#6923—84no—
214D4×C61—22no—
215D4×C61—22no—
216D4×C61—42no—
217C2×C4×C61—22no—
218C2×C4×C61#11712yes—
219C2×C4×C61C4×D412yes—
220C2×C4×C61C4×D411yescharacteristic
221C2×C4×C61C23×C411yescharacteristic
222C22⋊D61—42no—
223C22⋊D61—42no—
224C22⋊D61—44no—
225C2×Dic61—22no—
226C2×Dic61C4×D411yescharacteristic
227C2×Dic61#12412yes—
228C2×Dic61#12612yes—
229C2×Dic61C2×#4111yescharacteristic
230D244—22no—
231D244—22no—
232D244—42no—
233C4×D61—42no—
234C4×D61—42no—
235C4×D61#11712yes—
236C4×D61C4×D412yes—
237C4×D61#12112yes—
238C4×D61#12312yes—
239C4×D61#12412yes—
240C4×D61#12612yes—
241C4×D61C22×D411yescharacteristic
242C4×D61C2×#4111yescharacteristic
243C4×D61C2×#4111yescharacteristic
244C4×D61C2×#4111yescharacteristic
245#92420—42no—
246C8×C61—22no—
247#92418—22no—
248#21063—611nomaximal, Sylow
249C22×C61—22no—
250C22×C61—44no—
251C22×C61—42no—
252C22×C61—84no—
253C22×C61#38812yes—
254C22×C61C2×C4×D411yescharacteristic
255C2×D61—22no—
256C2×D61—24no—
257C2×D61—22no—
258C2×D61—22no—
259C2×D61—44no—
260C2×D61—42no—
261C2×D61—88no—
262C2×D61—84no—
263C2×D61—84no—
264C2×D61—82no—
265C2×D61—82no—
266C2×D61#38812yes—
267C2×D61C2×#12111yescharacteristic
268C2×D61C2×#12311yescharacteristic
269C2×D61#57612yes—
270C2×D61#57712yes—
271#6923—44no—
272#6923—44no—
273#6923—81no—
274#6923—81no—
275C4×C61—42no—
276C4×C61#38812yes—
277C4×C61C2×#11711yescharacteristic
278C4×C61C2×C4×D411yescharacteristic
279Dic61—42no—
280Dic61#38812yes—
281Dic61C2×#12411yescharacteristic
282Dic61C2×#12611yescharacteristic
283C2×#428—611no—
284C2×#418—611no—
285C22×#102—611no—
286#2309—611no—
287#2274—611no—
288#1926—614no—
289#1925—614no—
290#1735—611no—
291#1729—611no—
292C2×C61—22no—
293C2×C61—82no—
294C2×C61C2×#38811yescharacteristic
295C2×C61#294612yes—
296D61—22no—
297D61—82no—
298D61#304111yescharacteristic
299D61#304311yescharacteristic
300D61#309612yes—
301C23×D4—611no—
302C22×M16—611no—
303C23×C8—611no—
304C2×#117—611no—
305C2×#117—611no—
306C2×#117—611no—
307C22×C4⋊C4—611no—
308#428—1224no—
309C2×#107—612no—
310C2×#106—612no—
311#418—611no—
312#418—611no—
313#418—611no—
314#418—614no—
315#418#692311yescharacteristic, complemented
316C2×#102—614no—
317C2×#102—614no—
318C2×#102—1224no—
319#390—612no—
320#390—612no—
321#390—1222no—
322#379—1222no—
323C4×C4⋊C4—614no—
324C4×C4⋊C4—1222no—
325C4×C22⋊C4—1222no—
326#340—612no—
327#340—612no—
328#336—614no—
329C61#2106311yescharacteristic, complemented, Sylow
330C25—1221no—
331C22×D4—612no—
332C22×D4—612no—
333C22×D4—611no—
334C22×D4—1222no—
335C22×D4—1222no—
336C22×D4—1222no—
337C22×D4—1222no—
338C22×D4—1221no—
339C22×D4—1221no—
340C22×D4—1221no—
341C22×D4C2×#692311yescharacteristic
342C23×C4—611no—
343C23×C4—1221no—
344C2×M16—611no—
345C2×M16—612no—
346C2×M16—1222no—
347C2×M16—1222no—
348C2×M16C2×#692311yescharacteristic
349C22×C8—612no—
350C22×C8—612no—
351C22×C8—1221no—
352C22×C8—1222no—
353C22×C8—1221no—
354C22×C8—1222no—
355#117—611no—
356#117—612no—
357#117—1221no—
358#117—1221no—
359#117—1222no—
360#117—1222no—
361#117—1222no—
362#117—1222no—
363#117C2×#692311yescharacteristic
364C2×C4⋊C4—614no—
365C2×C4⋊C4—612no—
366C2×C4⋊C4—612no—
367C2×C4⋊C4—612no—
368C2×C4⋊C4—611no—
369C2×C4⋊C4—611no—
370C2×C4⋊C4—1222no—
371C2×C4⋊C4—1221no—
372C2×C22⋊C4—612no—
373C2×C22⋊C4—1221no—
374C2×C22⋊C4—1224no—
375C2×C22⋊C4—1224no—
376C2×C22⋊C4—1221no—
377C2×C42—612no—
378C2×C42—1221no—
379C2×C42—1224no—
380C2×C42—1221no—
381#107—1224no—
382#106—1224no—
383#102—614no—
384#102—614no—
385#102—614no—
386#102—1228no—
387#102C2×#692314yescomplemented
388#95—1224no—
389C24—21no—
390C24—611no—
391C24—1222no—
392C24—1222no—
393C24—1221no—
394C24—4882no—
395C24—4882no—
396C24—2442no—
397C24—2442no—
398C2×D4—22no—
399C2×D4—22no—
400C2×D4—21no—
401C2×D4—612no—
402C2×D4—1222no—
403C2×D4—1224no—
404C2×D4—1222no—
405C2×D4—1222no—
406C2×D4—1224no—
407C2×D4—1222no—
408C2×D4—1222no—
409C2×D4—1222no—
410C2×D4—1222no—
411C2×D4—1222no—
412C2×D4—1221no—
413C2×D4—2442no—
414C2×D4—2442no—
415C2×D4—2442no—
416C2×D4—2442no—
417C2×D4—2442no—
418C2×D4—2442no—
419C2×D4C22×#692312yes—
420C22×C4—611no—
421C22×C4—612no—
422C22×C4—612no—
423C22×C4—611no—
424C22×C4—612no—
425C22×C4—611no—
426C22×C4—612no—
427C22×C4—611no—
428C22×C4—611no—
429C22×C4—614no—
430C22×C4—1222no—
431C22×C4—1224no—
432C22×C4—1222no—
433C22×C4—1221no—
434C22×C4—1221no—
435C22×C4—4882no—
436C22×C4C22×#692311yescharacteristic
437M16—22no—
438M16—1224no—
439M16—1222no—
440C2×C8—612no—
441C2×C8—1222no—
442C2×C8—1222no—
443C2×C8—1222no—
444C2×C8—1222no—
445C2×C8—1222no—
446C2×C8—2442no—
447C2×C8—2442no—
448C2×C8C22×#692312yes—
449C4⋊C4—614no—
450C4⋊C4—612no—
451C4⋊C4—1222no—
452C4⋊C4—1224no—
453C4⋊C4—1222no—
454C4⋊C4—1224no—
455C4⋊C4—1221no—
456C4⋊C4—1221no—
457C4⋊C4—1221no—
458C4⋊C4—1221no—
459C4⋊C4C22×#692312yes—
460C22⋊C4—21no—
461C22⋊C4—1222no—
462C22⋊C4—1224no—
463C22⋊C4—1221no—
464C22⋊C4—4884no—
465C22⋊C4—4884no—
466C22⋊C4—2442no—
467C42—21no—
468C42—1222no—
469C42—1222no—
470C42—1222no—
471C42—1221no—
472C42—4884no—
473C42—2442no—
474C23—22no—
475C23—42no—
476C23—82no—
477C23—611no—
478C23—612no—
479C23—612no—
480C23—611no—
481C23—1222no—
482C23—1222no—
483C23—1222no—
484C23—4882no—
485C23—4884no—
486C23—4884no—
487C23—4884no—
488C23—4888no—
489C23—2444no—
490C23—2444no—
491C23—2442no—
492C23—2442no—
493C23—2444no—
494C23—2442no—
495C23D4×#692311yescharacteristic
496D4—22no—
497D4—22no—
498D4—42no—
499D4—1222no—
500D4—1222no—
501D4—2442no—
502D4—2442no—
503D4—2444no—
504D4—2442no—
505C2×C4—22no—
506C2×C4—612no—
507C2×C4—612no—
508C2×C4—611no—
509C2×C4—611no—
510C2×C4—612no—
511C2×C4—612no—
512C2×C4—612no—
513C2×C4—612no—
514C2×C4—612no—
515C2×C4—611no—
516C2×C4—611no—
517C2×C4—612no—
518C2×C4—611no—
519C2×C4—611no—
520C2×C4—1224no—
521C2×C4—1224no—
522C2×C4—1224no—
523C2×C4—1224no—
524C2×C4—1222no—
525C2×C4—4884no—
526C2×C4—2442no—
527C2×C4—2442no—
528C2×C4—2441no—
529C2×C4—2441no—
530C2×C4—2442no—
531C2×C4—2442no—
532C2×C4—2444no—
533C2×C4#1451985612yes—
534C2×C4D4×#692311yescharacteristic
535C2×C4D4×#692312yes—
536C2×C4C23×#692311yescharacteristic
537C8—22no—
538C8—1222no—
539C8—2442no—
540C22—22no—
541C22—42no—
542C22—44no—
543C22—84no—
544C22—611no—
545C22—611no—
546C22—612no—
547C22—612no—
548C22—612no—
549C22—1222no—
550C22—1222no—
551C22—1224no—
552C22—1222no—
553C22—4884no—
554C22—4882no—
555C22—4882no—
556C22—4884no—
557C22—4888no—
558C22—2442no—
559C22—2444no—
560C22#1684071212yes—
561C22C2×D4×#692311yescharacteristic
562C4—42no—
563C4—611no—
564C4—611no—
565C4—612no—
566C4—4881no—
567C4—4881no—
568C4—2444no—
569C4—2444no—
570C4—2442no—
571C4#1684071212yes—
572C4C2×#1451985611yescharacteristic
573C4C2×D4×#692311yescharacteristic
574C2—22no—
575C2—82no—
576C2—611no—
577C2—611no—
578C2—612no—
579C2—1222no—
580C2—4882no—
581C2C2×#1684071211yescharacteristic
582C2#2146257612yes—
subgroupG/H
centerC22C2×D4×#6923
commutatorC2×C4×C61C23×C4
FrattiniC2×C4C23×#6923
FittingC61×#418C4
socleC22×C61C2×C4×D4
prime radicalC22×D4×D61C22
cosocleC2×C4×D61C24
hypercenter#418#6923
hypocenterC61#21063
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ #418

Complements

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

NH ≅ G/Ntype
1. C2×#16840773C2semidirect
4. #21462256C2semidirect
5. #21461897C2semidirect
6. #21461896C2semidirect
7. #21461504C2semidirect
8. #21461487C2semidirect
15. C2×#14519846C22semidirect
16. C2×#14519845C22semidirect
23. C4⋊C4×#6923C22semidirect
26. #16840635C22semidirect
27. #16840635C22semidirect
28. (C2×#6923)⋊C8C22semidirect
32. #16839768C4semidirect
33. #16839767C4semidirect
34. #16839766C4semidirect
35. #16839765C4semidirect
37. C61×#418C4semidirect
90. #14519699C2×C4semidirect
91. #14519698C2×C4semidirect
92. #14519697C2×C4semidirect
96. #102×C61C2×C4semidirect
315. #418#6923semidirect
329. C61#21063semidirect
387. #102C2×#6923semidirect

Maximal subgroups

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

subgroupindex
C2×#168407732
C22×#145198392
#214622922
#214622562
#214618972
#214618962
#214615042
#214614872
#418×D612
#2106361