Finite Group Databasebeta

← #30617056

Subgroups — #30617056

Order 15616.

1066 conjugacy classes of subgroups.

The subgroup lattice has 604 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840803C211yesmaximal, characteristic, complemented
2C2×#16840725C211yesmaximal, characteristic, complemented
3#21462307C212yesmaximal, complemented
4#21461968C211yesmaximal, characteristic, complemented
5#21461924C211yesmaximal, characteristic, complemented
6#21461909C212yesmaximal, complemented
7#21461835C212yesmaximal, complemented
8#21461695C211yesmaximal, characteristic, complemented
9#21461634C211yesmaximal, characteristic, complemented
10#102×#6923C212yesmaximal, complemented
11#466×D61C211yesmaximal, characteristic
12C2×D61⋊M16C2211yescharacteristic
13C2×D4×#6923C2212yes—
14C2×#14519856C2211yescharacteristic
15#16840803—24no—
16C2×#14519845C2211yescharacteristic
17C2×#14519845C2211yescharacteristic
18C2×#14519839C2212yes—
19C2×#14519839C2212yes—
20#16840725—24no—
21C2×#14519831C2211yescharacteristic
22C2×#14519831C2211yescharacteristic
23C2×C8×#6923C2211yescharacteristic
24C2×C8×#6923C2211yescharacteristic
25#16840714—22no—
26#16840714C2212yes—
27#16840713C2212yes—
28#16840712—22no—
29#16840710—22no—
30#16840709—22no—
31#16840704C2212yes—
32C4⋊C4×#6923C2212yes—
33#16840636C2212yes—
34#16840635C2212yes—
35#16840635C2212yes—
36C2×D8×D61C2211yescharacteristic
37C22×C8×D61C2211yescharacteristic
38#121×D61C2212yes—
39#16839999C411yescharacteristic, complemented
40#16839998C411yescharacteristic, complemented
41#16839997C412yescomplemented
42#16839996C412yescomplemented
43#16839995C411yescharacteristic, complemented
44#107×D61C2211yescharacteristic
45#102×D61C2212yes—
46C61×#466C411yescharacteristic, complemented
47C23×#6923—22no—
48C2×#11758213—21no—
49D61⋊M16—22no—
50D61⋊M16—22no—
51C2×#11758217—21no—
52C2×D61⋊C8C2311yescharacteristic
53C2×D61⋊C8C2311yescharacteristic
54D4×#6923—24no—
55D4×#6923—24no—
56#14519856—22no—
57#14519856—22no—
58C2×C4⋊#6923—22no—
59C2×C4⋊#6923D412yes—
60C2×C4⋊#6923C2312yes—
61C2×C4⋊#6923C2312yes—
62C2×C22⋊#6923—22no—
63C2×C22⋊#6923—22no—
64C2×C22⋊#6923—22no—
65C2×C22⋊#6923—22no—
66C2×C22⋊#6923D412yes—
67C2×C4×#6923—22no—
68C2×C4×#6923C2311yescharacteristic
69C2×C4×#6923C2311yescharacteristic
70#14519845—22no—
71#14519845—22no—
72#14519839—24no—
73#14519839—24no—
74#14519831—22no—
75#14519831D414yescomplemented
76C8×#6923—22no—
77C8×#6923—22no—
78(C2×#6923)⋊C4—22no—
79(C2×#6923)⋊C4—22no—
80(C2×#6923)⋊C4—22no—
81C2×D4×D61—22no—
82C2×D4×D61C2312yes—
83C22×C4×D61C2311yescharacteristic
84C2×D488C2×C411yescharacteristic
85C2×#11758199C2×C412yes—
86D8×D61—22no—
87D8×D61—22no—
88C22×#92418C2×C411yescharacteristic, complemented
89C2×C8×D61—22no—
90C2×C8×D61—22no—
91C2×C8×D61C2×C411yescharacteristic, complemented
92C2×C8×D61C2×C411yescharacteristic, complemented
93C2×C8×D61C2311yescharacteristic
94C2×C8×D61C2311yescharacteristic
95#14519752C2×C412yescomplemented
96#14519751C2×C412yescomplemented
97#14519750—22no—
98#14519749—22no—
99#14519748C2×C412yescomplemented
100C4⋊C4×D61C2312yes—
101C22⋊C4×D61—22no—
102#14519711C2×C411yescharacteristic, complemented
103#14519710C2×C412yescomplemented
104#14519699C2×C412yescomplemented
105#14519698C2×C412yescomplemented
106#14519697C2×C412yescomplemented
107C2×D8×C61C2×C411yescharacteristic
108C22×C8×C61C2×C411yescharacteristic, complemented
109#121×C61C2×C412yescomplemented
110#107×C61C2×C411yescharacteristic, complemented
111#102×C61C2×C412yescomplemented
112C23×D61—22no—
113C23×D61C2×D411yescharacteristic
114C22×#6923—22no—
115C22×#6923—22no—
116C22×#6923—44no—
117C22×#6923—82no—
118C22×#6923—82no—
119C22×#6923C2×D412yes—
120C22×#6923C2×D412yes—
121C2×D4×C61—22no—
122C2×D4×C61C22×C412yes—
123C22×C4×C61C22×C411yescharacteristic
124C2×C22⋊D61—22no—
125C2×C22⋊D61—22no—
126C2×C22⋊D61—22no—
127C22×Dic61#4111yescharacteristic
128D4×D61—22no—
129D4×D61—22no—
130D4×D61—22no—
131D4×D61—22no—
132D4×D61—82no—
133D4×D61—82no—
134C2×D244—22no—
135C2×D244C22×C412yes—
136C2×C4×D61—21no—
137C2×C4×D61—21no—
138C2×C4×D61—21no—
139C2×C4×D61—21no—
140C2×C4×D61—22no—
141C2×C4×D61C22×C411yescharacteristic
142C2×C4×D61C2×D411yescharacteristic
143C2×C4×D61#4111yescharacteristic
144C2×C4×D61C2411yescharacteristic
145C22⋊#6923—24no—
146C22⋊#6923—44no—
147C22⋊#6923—82no—
148C22⋊#6923—82no—
149C22⋊#6923—82no—
150#11758217—24no—
151C2×#92420—21no—
152C2×#92420—21no—
153C4⋊#6923—24no—
154C4⋊#6923—22no—
155C4⋊#6923—22no—
156C4⋊#6923—22no—
157C4⋊#6923—22no—
158C4⋊#6923—82no—
159C4×#6923—21no—
160C4×#6923—21no—
161C4×#6923—82no—
162C4×#6923C2×D412yes—
163C4×#6923#4112yes—
164#11758213—24no—
165D61⋊C8—21no—
166D61⋊C8—21no—
167D61⋊C8C2×D412yes—
168D61⋊C8#4112yes—
169D8×C61—22no—
170C2×C8×C61—22no—
171C2×C8×C61C22×C411yescharacteristic
172C2×C8×C61C22×C411yescharacteristic
173C4⋊C4×C61C22×C412yes—
174C22⋊C4×C61—22no—
175C22⋊Dic61—22no—
176#11758199—24no—
177#11758198—22no—
178#11758198—22no—
179C4⋊Dic61C22×C412yes—
180#11758196—22no—
181C2×#92418—22no—
182C2×#92418C22×C411yescharacteristic
183C2×#92418C22×C411yescharacteristic
184D488—22no—
185C8×D61—21no—
186C8×D61—21no—
187C8×D61—22no—
188C8×D61—22no—
189C8×D61C2×D412yes—
190C8×D61#4112yes—
191C23×C61—22no—
192C23×C61C4×D411yescharacteristic
193C22×D61—21no—
194C22×D61—21no—
195C22×D61—22no—
196C22×D61—22no—
197C22×D61—22no—
198C22×D61—42no—
199C22×D61—42no—
200C22×D61—42no—
201C22×D61—42no—
202C22×D61—42no—
203C22×D61—42no—
204C22×D61C4×D411yescharacteristic
205C22×D61C22×D411yescharacteristic
206C2×#6923—22no—
207C2×#6923—22no—
208C2×#6923—22no—
209C2×#6923—22no—
210C2×#6923—22no—
211C2×#6923—42no—
212C2×#6923—42no—
213C2×#6923—42no—
214C2×#6923—42no—
215C2×#6923—42no—
216C2×#6923—42no—
217C2×#6923—88no—
218C2×#6923#12114yes—
219D4×C61—22no—
220D4×C61—22no—
221D4×C61—82no—
222C2×C4×C61—21no—
223C2×C4×C61—21no—
224C2×C4×C61—22no—
225C2×C4×C61C4×D411yescharacteristic
226C2×C4×C61C23×C411yescharacteristic
227C22⋊D61—42no—
228C22⋊D61—42no—
229C22⋊D61—82no—
230C22⋊D61—82no—
231C2×Dic61—21no—
232C2×Dic61—21no—
233C2×Dic61—22no—
234C2×Dic61C4×D411yescharacteristic
235C2×Dic61C2×#4111yescharacteristic
236D244—22no—
237D244—22no—
238D244—82no—
239C4×D61—21no—
240C4×D61—21no—
241C4×D61—21no—
242C4×D61—21no—
243C4×D61—21no—
244C4×D61—21no—
245C4×D61—42no—
246C4×D61—42no—
247C4×D61C22×D411yescharacteristic
248C4×D61C2×#4111yescharacteristic
249C4×D61C2×#4111yescharacteristic
250C4×D61C2×#4111yescharacteristic
251#92420—22no—
252#92420—22no—
253C8×C61—21no—
254C8×C61C4×D412yes—
255#92418—21no—
256#92418C4×D412yes—
257#21529—611nomaximal, Sylow
258C22×C61—21no—
259C22×C61—21no—
260C22×C61—21no—
261C22×C61—42no—
262C22×C61—42no—
263C22×C61—42no—
264C22×C61C2×C4×D411yescharacteristic
265C2×D61—21no—
266C2×D61—21no—
267C2×D61—21no—
268C2×D61—42no—
269C2×D61—42no—
270C2×D61—42no—
271C2×D61—42no—
272C2×D61—42no—
273C2×D61—82no—
274C2×D61—82no—
275C2×D61—82no—
276C2×D61—82no—
277C2×D61C2×#12111yescharacteristic
278C2×D61#53211yescharacteristic
279C2×D61#57612yes—
280C2×D61#57712yes—
281#6923—24no—
282#6923—42no—
283#6923—82no—
284#6923—82no—
285C4×C61—21no—
286C4×C61—42no—
287C4×C61C2×C4×D411yescharacteristic
288C4×C61C4×#4111yescharacteristic
289Dic61—21no—
290Dic61—42no—
291Dic61#52511yescharacteristic
292Dic61#52911yescharacteristic
293C2×#466—611no—
294C2×#442—611no—
295C2×#405—611no—
296#2324—612no—
297#1972—611no—
298#1943—611no—
299#1934—612no—
300#1895—612no—
301#1807—611no—
302#1776—611no—
303C4×#102—612no—
304C2×C61—41no—
305C2×C61—82no—
306C2×C61C4×#12111yescharacteristic
307C2×C61#294811yescharacteristic
308C2×C61#295211yescharacteristic
309D61—41no—
310D61—82no—
311D61#306212yes—
312D61#317912yes—
313C22×D8—611no—
314C22×M16—611no—
315C23×C8—611no—
316C2×#121—612no—
317C2×C4×D4—612no—
318C2×#117—611no—
319#466—611no—
320#466—611no—
321#466—612no—
322#466—611no—
323#466—612no—
324#466#692311yescharacteristic, complemented
325#442—1224no—
326C2×#107—611no—
327C2×#106—611no—
328C2×#106—611no—
329C2×#102—612no—
330C2×#102—612no—
331C2×#102—612no—
332#405—1224no—
333C2×#97—611no—
334C2×#97—611no—
335C2×C4×C8—611no—
336C2×C4×C8—611no—
337#390—612no—
338#390—1222no—
339#389—612no—
340#388—1222no—
341#386—1222no—
342#385—1222no—
343#380—612no—
344C4×C4⋊C4—612no—
345#340—612no—
346#340—612no—
347#340—612no—
348C61#2152911yescharacteristic, complemented, Sylow
349C22×D4—612no—
350C22×D4—1222no—
351C23×C4—611no—
352C23×C4—1222no—
353C2×D8—612no—
354C2×D8—611no—
355C2×D8—1222no—
356C2×D8—1222no—
357C2×D8C2×#692311yescharacteristic
358C2×M16—1222no—
359C2×M16—1222no—
360C2×M16—1221no—
361C2×M16—1221no—
362C22×C8—611no—
363C22×C8—611no—
364C22×C8—611no—
365C22×C8—611no—
366C22×C8—611no—
367C22×C8—611no—
368C22×C8—611no—
369C22×C8—1222no—
370C22×C8—1222no—
371C22×C8C2×#692311yescharacteristic, complemented
372#121—612no—
373#121—612no—
374#121—612no—
375#121—1222no—
376#121—1222no—
377#121C2×#692312yescomplemented
378C4×D4—1224no—
379C4×D4—1224no—
380#117—1222no—
381#117—1222no—
382C2×C4⋊C4—612no—
383C2×C4⋊C4—612no—
384C2×C4⋊C4—612no—
385C2×C4⋊C4—612no—
386C2×C4⋊C4—1222no—
387C2×C22⋊C4—612no—
388C2×C22⋊C4—1222no—
389C2×C22⋊C4—1222no—
390C2×C22⋊C4—1222no—
391C2×C22⋊C4—1222no—
392C2×C22⋊C4—1222no—
393C2×C42—611no—
394C2×C42—611no—
395C2×C42—1222no—
396#107—611no—
397#107—612no—
398#107C2×#692311yescharacteristic, complemented
399#106—1222no—
400#106—1222no—
401#102—612no—
402#102—612no—
403#102—612no—
404#102—1224no—
405#102—1224no—
406#102C2×#692312yescomplemented
407#97—614no—
408#97—1222no—
409C4×C8—1222no—
410C4×C8—1222no—
411#95—1222no—
412#95—1222no—
413#95—1222no—
414C24—611no—
415C24—1222no—
416C2×D4—22no—
417C2×D4—612no—
418C2×D4—1222no—
419C2×D4—1222no—
420C2×D4—1222no—
421C2×D4—1222no—
422C2×D4—1222no—
423C2×D4—1222no—
424C2×D4—1222no—
425C2×D4—1222no—
426C2×D4—4882no—
427C2×D4—4882no—
428C2×D4C22×#692312yes—
429C22×C4—611no—
430C22×C4—612no—
431C22×C4—611no—
432C22×C4—611no—
433C22×C4—611no—
434C22×C4—611no—
435C22×C4—612no—
436C22×C4—1221no—
437C22×C4—1221no—
438C22×C4—1221no—
439C22×C4—1221no—
440C22×C4—1222no—
441C22×C4—1222no—
442C22×C4—1222no—
443C22×C4—4882no—
444C22×C4—4882no—
445C22×C4—2444no—
446C22×C4C22×#692311yescharacteristic
447D8—22no—
448D8—1224no—
449D8—1222no—
450M16—1224no—
451M16—1224no—
452C2×C8—22no—
453C2×C8—611no—
454C2×C8—612no—
455C2×C8—612no—
456C2×C8—612no—
457C2×C8—612no—
458C2×C8—611no—
459C2×C8—1222no—
460C2×C8—1221no—
461C2×C8—1221no—
462C2×C8—1222no—
463C2×C8—1221no—
464C2×C8—1221no—
465C2×C8—1221no—
466C2×C8—1221no—
467C2×C8—1222no—
468C2×C8C22×#692311yescharacteristic
469C2×C8C22×#692311yescharacteristic
470C4⋊C4—612no—
471C4⋊C4—1222no—
472C4⋊C4—1224no—
473C4⋊C4—1222no—
474C4⋊C4—1222no—
475C4⋊C4—1222no—
476C4⋊C4—1222no—
477C4⋊C4—4882no—
478C4⋊C4C22×#692312yes—
479C22⋊C4—22no—
480C22⋊C4—1222no—
481C22⋊C4—1222no—
482C22⋊C4—1224no—
483C22⋊C4—1222no—
484C22⋊C4—4882no—
485C22⋊C4—4882no—
486C22⋊C4—4882no—
487C22⋊C4—2444no—
488C42—612no—
489C42—612no—
490C42—1221no—
491C42—1221no—
492C42—4882no—
493C23—22no—
494C23—611no—
495C23—611no—
496C23—1221no—
497C23—1222no—
498C23—1221no—
499C23—1222no—
500C23—1222no—
501C23—2442no—
502C23—2442no—
503C23—2442no—
504C23—2442no—
505C23—2442no—
506C23—2442no—
507C23D4×#692311yescharacteristic
508D4—22no—
509D4—22no—
510D4—82no—
511D4—1222no—
512D4—1222no—
513D4—4882no—
514D4—4882no—
515D4—4882no—
516D4—2442no—
517D4—2442no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—22no—
521C2×C4—611no—
522C2×C4—611no—
523C2×C4—611no—
524C2×C4—611no—
525C2×C4—611no—
526C2×C4—614no—
527C2×C4—611no—
528C2×C4—1221no—
529C2×C4—1221no—
530C2×C4—1221no—
531C2×C4—1221no—
532C2×C4—1221no—
533C2×C4—1221no—
534C2×C4—1222no—
535C2×C4—1222no—
536C2×C4—1221no—
537C2×C4—1221no—
538C2×C4—1222no—
539C2×C4—1222no—
540C2×C4—1222no—
541C2×C4—1222no—
542C2×C4—4888no—
543C2×C4—2442no—
544C2×C4—2442no—
545C2×C4—2442no—
546C2×C4—2442no—
547C2×C4—2442no—
548C2×C4—2442no—
549C2×C4—2442no—
550C2×C4—2442no—
551C2×C4D4×#692311yescharacteristic
552C2×C4C23×#692311yescharacteristic
553C8—21no—
554C8—612no—
555C8—1222no—
556C8—1222no—
557C8—1221no—
558C8D4×#692312yes—
559C22—21no—
560C22—21no—
561C22—21no—
562C22—42no—
563C22—42no—
564C22—42no—
565C22—611no—
566C22—611no—
567C22—612no—
568C22—612no—
569C22—1221no—
570C22—1221no—
571C22—1221no—
572C22—4882no—
573C22—4882no—
574C22—4882no—
575C22—4882no—
576C22—2442no—
577C22—2442no—
578C22—2442no—
579C22—2442no—
580C22—2442no—
581C22C2×D4×#692311yescharacteristic
582C4—21no—
583C4—42no—
584C4—611no—
585C4—611no—
586C4—1224no—
587C4—1221no—
588C4—4882no—
589C4—4882no—
590C4—2442no—
591C4—2442no—
592C4C2×D4×#692311yescharacteristic
593C4#41×#692311yescharacteristic
594C2—41no—
595C2—82no—
596C2—612no—
597C2—612no—
598C2—4882no—
599C2—2441no—
600C2#121×#692311yescharacteristic
601C2#2146257811yescharacteristic
602C2#2146258211yescharacteristic
subgroupG/H
centerC22C2×D4×#6923
commutatorC2×C4×C61C23×C4
FrattiniC2×C4C23×#6923
FittingC61×#466C4
socleC22×C61C2×C4×D4
prime radical#466×D61C2
cosocleC2×C4×D61C24
hypercenter#466#6923
hypocenterC61#21529
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ #466

Complements

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

NH ≅ G/Ntype
1. C2×#16840803C2semidirect
2. C2×#16840725C2semidirect
3. #21462307C2semidirect
4. #21461968C2semidirect
5. #21461924C2semidirect
6. #21461909C2semidirect
7. #21461835C2semidirect
8. #21461695C2semidirect
9. #21461634C2semidirect
10. #102×#6923C2semidirect
39. #16839999C4semidirect
40. #16839998C4semidirect
41. #16839997C4semidirect
42. #16839996C4semidirect
43. #16839995C4semidirect
46. C61×#466C4semidirect
75. #14519831D4semidirect
88. C22×#92418C2×C4semidirect
91. C2×C8×D61C2×C4semidirect
92. C2×C8×D61C2×C4semidirect
95. #14519752C2×C4semidirect
96. #14519751C2×C4semidirect
99. #14519748C2×C4semidirect
102. #14519711C2×C4semidirect
103. #14519710C2×C4semidirect
104. #14519699C2×C4semidirect
105. #14519698C2×C4semidirect
106. #14519697C2×C4semidirect
108. C22×C8×C61C2×C4semidirect
109. #121×C61C2×C4semidirect
110. #107×C61C2×C4semidirect
111. #102×C61C2×C4semidirect
324. #466#6923semidirect
348. C61#21529semidirect
371. C22×C8C2×#6923semidirect
377. #121C2×#6923semidirect
398. #107C2×#6923semidirect
406. #102C2×#6923semidirect

Maximal subgroups

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

subgroupindex
C2×#168408032
C2×#168407252
#214623072
#214619682
#214619242
#214619092
#214618352
#214616952
#214616342
#102×#69232
#466×D612
#2152961