Finite Group Databasebeta

← #30619326

Subgroups — #30619326

Order 15616.

986 conjugacy classes of subgroups.

The subgroup lattice has 572 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21462311C211yesmaximal, characteristic, complemented
2#21462309C211yesmaximal, characteristic, complemented
3#21461948C211yesmaximal, characteristic, complemented
4#21461947C211yesmaximal, characteristic, complemented
5#21461815C212yesmaximal, complemented
6#21461814C212yesmaximal, complemented
7#21461814C212yesmaximal, complemented
8#21461812C212yesmaximal, complemented
9#21461742C211yesmaximal, characteristic, complemented
10#21461539C211yesmaximal, characteristic, complemented
11#496×D61C211yesmaximal, characteristic
12C2×Q8×#6923C2211yescharacteristic
13C2×D4×#6923C2211yescharacteristic
14C2×D4×#6923C2212yes—
15C2×#14519840C2212yes—
16C2×#14519839C2212yes—
17C2×#14519839C2212yes—
18C2×#14519839C2212yes—
19C4×D61⋊C8C2211yescharacteristic
20(C2×#6923)⋊Q8C2211yescharacteristic
21#16840714C2211yescharacteristic
22#16840714C2212yes—
23#16840712—22no—
24#16840710—22no—
25#16840710—22no—
26#16840709—22no—
27#16840708C2211yescharacteristic
28#16840706C2211yescharacteristic
29#16840698C2211yescharacteristic
30#16840636C2212yes—
31#16840636C2212yes—
32(C2×#6923)⋊C8C2212yes—
33(C2×#6923)⋊C8C2212yes—
34C2×SD16×D61C2212yes—
35C2×D8×D61C2212yes—
36#127×D61C2211yescharacteristic
37#124×D61C2211yescharacteristic
38#16840152C411yescharacteristic, complemented
39#16840151C411yescharacteristic, complemented
40#16840150C412yescomplemented
41#16840149C411yescharacteristic, complemented
42#16840148C412yescomplemented
43#97×D61C2211yescharacteristic
44C61×#496C411yescharacteristic, complemented
45C23×#6923—21no—
46C23×#6923—22no—
47C2×D61⋊C8C2312yes—
48Q8×#6923—42no—
49D4×#6923—42no—
50D4×#6923—42no—
51D4×#6923—42no—
52C2×C4⋊#6923—21no—
53C2×C4⋊#6923—22no—
54C2×C4⋊#6923C2312yes—
55C2×C4⋊#6923C2311yescharacteristic
56C2×C4⋊#6923C2311yescharacteristic
57C2×C22⋊#6923—22no—
58C2×C22⋊#6923—21no—
59C2×C22⋊#6923—22no—
60C2×C22⋊#6923—22no—
61C2×C22⋊#6923—22no—
62C2×C4×#6923—21no—
63C2×C4×#6923C2311yescharacteristic
64C2×C4×#6923C2311yescharacteristic
65#14519840—42no—
66#14519839—42no—
67#14519839—42no—
68#14519839—42no—
69C4×#92420—22no—
70(C2×#6923)⋊C4—22no—
71(C2×#6923)⋊C4—22no—
72(C2×#6923)⋊C4D412yes—
73(C2×#6923)⋊C4D412yes—
74C2×Q8×D61C2311yescharacteristic
75C2×D4×D61—22no—
76C2×D4×D61C2311yescharacteristic
77C2×D4×D61C2312yes—
78C2×#11758190C2×C412yescomplemented
79C2×#11758200C2×C412yescomplemented
80C2×#11758201C2×C412yescomplemented
81SD16×D61—42no—
82SD16×D61—42no—
83C2×D488C2×C412yescomplemented
84C2×#11758199C2×C412yescomplemented
85C2×#11758199C2×C412yescomplemented
86D8×D61—42no—
87D8×D61—42no—
88C2×C8×D61C2312yes—
89#14519774C2×C411yescharacteristic, complemented
90#14519773—22no—
91#14519773C2×C412yescomplemented
92#14519766C2×C411yescharacteristic, complemented
93#14519765C2×C411yescharacteristic, complemented
94#14519764C2×C411yescharacteristic, complemented
95#14519763—22no—
96C22⋊C4×D61—22no—
97C42×D61C2311yescharacteristic
98#14519688C2×C411yescharacteristic, complemented
99#14519687C2×C412yescomplemented
100C2×SD16×C61C2×C412yescomplemented
101C2×D8×C61C2×C412yescomplemented
102#127×C61C2×C411yescharacteristic, complemented
103#124×C61C2×C411yescharacteristic, complemented
104#97×C61C2×C411yescharacteristic, complemented
105C23×D61—22no—
106C23×D61—21no—
107C22×#6923—22no—
108C22×#6923—21no—
109C22×#6923—21no—
110C22×#6923—42no—
111C22×#6923—44no—
112C22×#6923—82no—
113C22×#6923—82no—
114C22×#6923—82no—
115C22×#6923C2×D412yes—
116C22×#6923C2×D411yescharacteristic
117C22×#6923C2×D411yescharacteristic
118C2×Q8×C61C22×C411yescharacteristic
119C2×D4×C61—22no—
120C2×D4×C61C22×C411yescharacteristic
121C2×D4×C61C22×C412yes—
122C2×C22⋊D61—22no—
123C2×C22⋊D61—22no—
124C2×C22⋊D61—21no—
125C2×C22⋊D61—22no—
126Q8×D61—22no—
127Q8×D61—22no—
128D4×D61—22no—
129D4×D61—22no—
130D4×D61—22no—
131D4×D61—22no—
132D4×D61—22no—
133D4×D61—22no—
134D4×D61—82no—
135D4×D61—82no—
136C2×D244—22no—
137C2×D244C22×C411yescharacteristic
138C2×D244C22×C412yes—
139C2×C4×D61—21no—
140C2×C4×D61C2×D412yes—
141C2×C4×D61C2411yescharacteristic
142C2×Dic122—21no—
143C2×Dic122C22×C411yescharacteristic
144C22⋊#6923—42no—
145C22⋊#6923—44no—
146C22⋊#6923—82no—
147C22⋊#6923—82no—
148C22⋊#6923—82no—
149C2×#92420—22no—
150C4⋊#6923—22no—
151C4⋊#6923—22no—
152C4⋊#6923—22no—
153C4⋊#6923—22no—
154C4⋊#6923—42no—
155C4⋊#6923—82no—
156C4×#6923—22no—
157C4×#6923—22no—
158C4×#6923—42no—
159D61⋊C8—22no—
160D61⋊C8—22no—
161SD16×C61—42no—
162D8×C61—42no—
163C2×C8×C61C22×C412yes—
164C22⋊C4×C61—22no—
165C42×C61C22×C411yescharacteristic
166C22⋊Dic61—22no—
167#11758201—42no—
168#11758200—42no—
169#11758199—42no—
170#11758199—42no—
171#11758198—22no—
172#11758198—22no—
173C4×Dic61C22×C411yescharacteristic
174C4×Dic61#4112yes—
175C2×#92418C22×C412yes—
176D488—42no—
177#11758190—42no—
178C8×D61—22no—
179C8×D61—22no—
180C23×C61—21no—
181C23×C61—22no—
182C22×D61—21no—
183C22×D61—22no—
184C22×D61—41no—
185C22×D61—42no—
186C22×D61—42no—
187C22×D61—41no—
188C22×D61—42no—
189C22×D61—42no—
190C22×D61—42no—
191C22×D61—42no—
192C22×D61—42no—
193C22×D61—42no—
194C22×D61C22×D411yescharacteristic
195C2×#6923—21no—
196C2×#6923—21no—
197C2×#6923—22no—
198C2×#6923—22no—
199C2×#6923—22no—
200C2×#6923—22no—
201C2×#6923—22no—
202C2×#6923—41no—
203C2×#6923—41no—
204C2×#6923—42no—
205C2×#6923—42no—
206C2×#6923—42no—
207C2×#6923—42no—
208C2×#6923—42no—
209C2×#6923—84no—
210C2×#6923—88no—
211Q8×C61—22no—
212D4×C61—22no—
213D4×C61—22no—
214D4×C61—22no—
215D4×C61—82no—
216C2×C4×C61—21no—
217C2×C4×C61C4×D412yes—
218C2×C4×C61C23×C411yescharacteristic
219C22⋊D61—42no—
220C22⋊D61—42no—
221C22⋊D61—42no—
222C22⋊D61—82no—
223C22⋊D61—82no—
224C2×Dic61—21no—
225C2×Dic61C4×D412yes—
226C2×Dic61C2×#4111yescharacteristic
227D244—22no—
228D244—22no—
229D244—22no—
230D244—82no—
231C4×D61—22no—
232C4×D61—22no—
233C4×D61—41no—
234C4×D61—41no—
235C4×D61H(5,𝔽2)12yes—
236C4×D61H(5,𝔽2)12yes—
237Dic122—22no—
238Dic122—42no—
239#92420—42no—
240C8×C61—22no—
241#92418—22no—
242#22279—611nomaximal, Sylow
243C22×C61—41no—
244C22×C61—42no—
245C22×C61—42no—
246C22×C61—42no—
247C22×C61—42no—
248C22×C61C2×C4×D411yescharacteristic
249C2×D61—42no—
250C2×D61—41no—
251C2×D61—42no—
252C2×D61—42no—
253C2×D61—42no—
254C2×D61—81no—
255C2×D61—82no—
256C2×D61—81no—
257C2×D61—82no—
258C2×D61—82no—
259C2×D61—82no—
260C2×D61—82no—
261C2×D61#53511yescharacteristic
262C2×D61#53511yescharacteristic
263C2×D61#57612yes—
264C2×D61#57712yes—
265#6923—41no—
266#6923—41no—
267#6923—42no—
268#6923—81no—
269#6923—81no—
270#6923—82no—
271C4×C61—22no—
272C4×C61—41no—
273C4×C61#51812yes—
274Dic61—22no—
275Dic61—41no—
276Dic61#54012yes—
277C2×#496—611no—
278#2328—611no—
279#2326—611no—
280#1958—611no—
281#1957—611no—
282#1880—612no—
283#1879—612no—
284#1879—612no—
285#1877—612no—
286#1836—611no—
287#1746—611no—
288C2×C61—81no—
289C2×C61—82no—
290C2×C61#232811yescharacteristic
291C2×C61#294612yes—
292D61—82no—
293D61—81no—
294D61#322011yescharacteristic
295D61#322111yescharacteristic
296D61#322412yes—
297C22×SD16—612no—
298C22×D8—612no—
299C2×#127—611no—
300C2×#124—611no—
301C2×C4×Q8—611no—
302C2×C4×D4—611no—
303C2×C4×D4—612no—
304#496—612no—
305#496—612no—
306#496—611no—
307#496—611no—
308#496—611no—
309#496#692311yescharacteristic, complemented
310C2×#103—612no—
311C2×#102—612no—
312C2×#102—612no—
313C2×#102—612no—
314C2×#97—611no—
315C2×C4×C8—611no—
316#391—611no—
317#390—611no—
318#390—612no—
319#388—1222no—
320#386—1222no—
321#386—1222no—
322#385—1222no—
323#384—611no—
324#382—611no—
325#376—611no—
326#340—612no—
327#340—612no—
328#336—612no—
329#336—612no—
330C61#2227911yescharacteristic, complemented, Sylow
331C22×Q8—611no—
332C22×D4—611no—
333C22×D4—612no—
334C22×D4—1222no—
335C23×C4—1221no—
336C23×C4—1222no—
337C2×SD16—612no—
338C2×SD16—612no—
339C2×SD16—612no—
340C2×SD16—2442no—
341C2×SD16—2442no—
342C2×SD16C2×#692312yescomplemented
343C2×D8—612no—
344C2×D8—612no—
345C2×D8—612no—
346C2×D8—2442no—
347C2×D8—2442no—
348C2×D8C2×#692312yescomplemented
349C22×C8—612no—
350C22×C8—612no—
351#127—612no—
352#127—611no—
353#127—1222no—
354#127C2×#692311yescharacteristic, complemented
355#124—611no—
356#124—611no—
357#124—611no—
358#124—1222no—
359#124C2×#692311yescharacteristic, complemented
360C4×Q8—2442no—
361C4×D4—2442no—
362C4×D4—2442no—
363C4×D4—2442no—
364C2×C4⋊C4—612no—
365C2×C4⋊C4—611no—
366C2×C4⋊C4—611no—
367C2×C4⋊C4—1222no—
368C2×C4⋊C4—1221no—
369C2×C22⋊C4—1222no—
370C2×C22⋊C4—1222no—
371C2×C22⋊C4—1222no—
372C2×C22⋊C4—1222no—
373C2×C22⋊C4—1222no—
374C2×C22⋊C4—1221no—
375C2×C42—611no—
376C2×C42—611no—
377C2×C42—611no—
378C2×C42—1221no—
379#103—2442no—
380#102—2442no—
381#102—2442no—
382#102—2442no—
383#97—612no—
384#97—611no—
385#97C2×#692311yescharacteristic, complemented
386C4×C8—1222no—
387#95—612no—
388#95—612no—
389#95—1222no—
390#95—1222no—
391C24—1221no—
392C24—1222no—
393C2×Q8—611no—
394C2×Q8—1221no—
395C2×Q8—1222no—
396C2×Q8—1222no—
397C2×Q8C22×#692311yescharacteristic
398C2×D4—22no—
399C2×D4—611no—
400C2×D4—612no—
401C2×D4—1222no—
402C2×D4—1222no—
403C2×D4—1222no—
404C2×D4—1222no—
405C2×D4—1222no—
406C2×D4—1222no—
407C2×D4—1222no—
408C2×D4—1222no—
409C2×D4—1221no—
410C2×D4—1222no—
411C2×D4—1222no—
412C2×D4—4882no—
413C2×D4—4882no—
414C2×D4C22×#692311yescharacteristic
415C2×D4C22×#692312yes—
416C22×C4—611no—
417C22×C4—612no—
418C22×C4—612no—
419C22×C4—611no—
420C22×C4—611no—
421C22×C4—1221no—
422C22×C4—1222no—
423C22×C4—1221no—
424C22×C4—1221no—
425C22×C4—4882no—
426C22×C4—4882no—
427C22×C4—4882no—
428C22×C4—2444no—
429C22×C4—2442no—
430SD16—42no—
431SD16—2442no—
432SD16—2442no—
433SD16—2442no—
434D8—42no—
435D8—2442no—
436D8—2442no—
437D8—2442no—
438C2×C8—612no—
439C2×C8—1222no—
440C2×C8—1222no—
441C2×C8—1222no—
442C2×C8—1222no—
443C2×C8—1222no—
444C2×C8C22×#692312yes—
445C4⋊C4—1222no—
446C4⋊C4—1222no—
447C4⋊C4—1222no—
448C4⋊C4—1222no—
449C4⋊C4—4882no—
450C4⋊C4—2442no—
451C22⋊C4—22no—
452C22⋊C4—1222no—
453C22⋊C4—1222no—
454C22⋊C4—1222no—
455C22⋊C4—4882no—
456C22⋊C4—4882no—
457C22⋊C4—4882no—
458C22⋊C4—2442no—
459C22⋊C4—2444no—
460C42—612no—
461C42—611no—
462C42—1222no—
463C42—1222no—
464C42—2442no—
465C42C22×#692311yescharacteristic
466C23—21no—
467C23—22no—
468C23—611no—
469C23—1221no—
470C23—1222no—
471C23—2442no—
472C23—2442no—
473C23—2442no—
474C23—2441no—
475C23—2442no—
476C23—2442no—
477C23—2442no—
478C23—2441no—
479C23—2442no—
480C23—2442no—
481Q8—22no—
482Q8—1222no—
483Q8—2442no—
484D4—22no—
485D4—22no—
486D4—22no—
487D4—82no—
488D4—1222no—
489D4—1222no—
490D4—1222no—
491D4—4882no—
492D4—4882no—
493D4—4882no—
494D4—2442no—
495D4—2442no—
496D4—2442no—
497C2×C4—21no—
498C2×C4—611no—
499C2×C4—612no—
500C2×C4—612no—
501C2×C4—612no—
502C2×C4—1222no—
503C2×C4—1222no—
504C2×C4—1221no—
505C2×C4—1222no—
506C2×C4—1221no—
507C2×C4—1222no—
508C2×C4—1222no—
509C2×C4—1222no—
510C2×C4—1222no—
511C2×C4—1221no—
512C2×C4—4884no—
513C2×C4—4888no—
514C2×C4—2441no—
515C2×C4—2441no—
516C2×C4—2441no—
517C2×C4—2442no—
518C2×C4—2442no—
519C2×C4—2442no—
520C2×C4—2441no—
521C2×C4—2442no—
522C2×C4—2442no—
523C2×C4D4×#692312yes—
524C2×C4C23×#692311yescharacteristic
525C8—22no—
526C8—1222no—
527C8—2442no—
528C22—41no—
529C22—42no—
530C22—42no—
531C22—42no—
532C22—42no—
533C22—611no—
534C22—611no—
535C22—612no—
536C22—612no—
537C22—4881no—
538C22—4882no—
539C22—4882no—
540C22—4882no—
541C22—4882no—
542C22—4882no—
543C22—4881no—
544C22—2441no—
545C22—2442no—
546C22—2442no—
547C22—2442no—
548C22—2442no—
549C22C2×D4×#692311yescharacteristic
550C4—22no—
551C4—41no—
552C4—612no—
553C4—1222no—
554C4—4881no—
555C4—4882no—
556C4—4881no—
557C4—2441no—
558C4—2442no—
559C4—2441no—
560C4—2441no—
561C4#1684082912yes—
562C2—82no—
563C2—81no—
564C2—611no—
565C2—612no—
566C2—611no—
567C2—4882no—
568C2—4881no—
569C2#2146231111yescharacteristic
570C2#2146257612yes—
subgroupG/H
centerC22C2×D4×#6923
commutatorC2×C4×C61C23×C4
FrattiniC2×C4C23×#6923
FittingC61×#496C4
socleC22×C61C2×C4×D4
prime radical#496×D61C2
cosocleC2×C4×D61C24
hypercenter#496#6923
hypocenterC61#22279
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #496

Complements

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

NH ≅ G/Ntype
1. #21462311C2semidirect
2. #21462309C2semidirect
3. #21461948C2semidirect
4. #21461947C2semidirect
5. #21461815C2semidirect
6. #21461814C2semidirect
7. #21461814C2semidirect
8. #21461812C2semidirect
9. #21461742C2semidirect
10. #21461539C2semidirect
38. #16840152C4semidirect
39. #16840151C4semidirect
40. #16840150C4semidirect
41. #16840149C4semidirect
42. #16840148C4semidirect
44. C61×#496C4semidirect
78. C2×#11758190C2×C4semidirect
79. C2×#11758200C2×C4semidirect
80. C2×#11758201C2×C4semidirect
83. C2×D488C2×C4semidirect
84. C2×#11758199C2×C4semidirect
85. C2×#11758199C2×C4semidirect
89. #14519774C2×C4semidirect
91. #14519773C2×C4semidirect
92. #14519766C2×C4semidirect
93. #14519765C2×C4semidirect
94. #14519764C2×C4semidirect
98. #14519688C2×C4semidirect
99. #14519687C2×C4semidirect
100. C2×SD16×C61C2×C4semidirect
101. C2×D8×C61C2×C4semidirect
102. #127×C61C2×C4semidirect
103. #124×C61C2×C4semidirect
104. #97×C61C2×C4semidirect
309. #496#6923semidirect
330. C61#22279semidirect
342. C2×SD16C2×#6923semidirect
348. C2×D8C2×#6923semidirect
354. #127C2×#6923semidirect
359. #124C2×#6923semidirect
385. #97C2×#6923semidirect

Maximal subgroups

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

subgroupindex
#214623112
#214623092
#214619482
#214619472
#214618152
#214618142
#214618122
#214617422
#214615392
#496×D612
#2227961