Finite Group Databasebeta

← #30622829

Subgroups — #30622829

Order 15616.

934 conjugacy classes of subgroups.

The subgroup lattice has 588 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21462352C211yesmaximal, characteristic, complemented
2#21462349C211yesmaximal, characteristic
3#21461997C211yesmaximal, characteristic, complemented
4#21461986C211yesmaximal, characteristic
5#21461909C212yesmaximal, complemented
6#21461897C212yesmaximal, complemented
7#21461784C212yesmaximal, complemented
8C4×#14519845C211yesmaximal, characteristic, complemented
9C8×C4⋊#6923C211yesmaximal, characteristic, complemented
10#102×#6923C212yesmaximal, complemented
11#486×D61C211yesmaximal, characteristic
12C2×D4×#6923—21no—
13C2×D4×#6923C2212yes—
14C2×#14519845C2211yescharacteristic
15C2×#14519845C2211yescharacteristic
16D61⋊(C4⋊C8)C2211yescharacteristic
17C2×#14519839C2212yes—
18C2×#14519839C2212yes—
19C2×C8×#6923C2212yes—
20(C2×#6923)⋊Q8—22no—
21#16840714—21no—
22#16840714C2212yes—
23#16840713—21no—
24#16840713C2212yes—
25#16840710—22no—
26C4⋊C4×#6923—21no—
27C4⋊C4×#6923C2212yes—
28C4×C4⋊#6923C2211yescharacteristic
29C4×C4⋊#6923C2211yescharacteristic
30C4×C4⋊#6923C2211yescharacteristic
31#16840636C2211yescharacteristic
32#16840636C2211yescharacteristic
33#16840635C2212yes—
34#16840635C2212yes—
35(C2×#6923)⋊C8C2212yes—
36C4⋊Q8×D61C2211yescharacteristic
37#127×D61C2211yescharacteristic
38#16840110C411yescharacteristic, complemented
39#16840109C411yescharacteristic, complemented
40#16840108C411yescharacteristic, complemented
41#16840107C412yescomplemented
42#16840107C412yescomplemented
43#102×D61C2212yes—
44#102×D61C2212yes—
45C4×C8×D61C2211yescharacteristic
46C61×#486C411yescharacteristic, complemented
47C23×#6923—22no—
48C2×D61⋊C8C2312yes—
49D4×#6923—24no—
50D4×#6923—42no—
51D4×#6923—44no—
52C2×C4⋊#6923—22no—
53C2×C4⋊#6923D411yescharacteristic
54C2×C4⋊#6923D411yescharacteristic
55C2×C4⋊#6923C2311yescharacteristic
56C2×C4⋊#6923C2311yescharacteristic
57C2×C4⋊#6923C2311yescharacteristic
58C2×C4⋊#6923C2311yescharacteristic
59C2×C22⋊#6923—22no—
60C2×C22⋊#6923—22no—
61C2×C22⋊#6923—22no—
62C2×C4×#6923—21no—
63C2×C4×#6923—21no—
64C2×C4×#6923—22no—
65C2×C4×#6923C2312yes—
66#14519845—22no—
67#14519845—22no—
68#14519844—21no—
69C4⋊#92420—21no—
70#14519839—24no—
71#14519839—42no—
72C8×#6923—42no—
73(C2×#6923)⋊C4—22no—
74(C2×#6923)⋊C4—22no—
75(C2×#6923)⋊C4—21no—
76(C2×#6923)⋊C4—21no—
77(C2×#6923)⋊C4D412yes—
78C2×Q8×D61—21no—
79C2×D4×D61—21no—
80C2×D4×D61—21no—
81C2×D4×D61C2312yes—
82C2×C8×D61C2311yescharacteristic
83C2×C8×D61C2311yescharacteristic
84#14519777C2×C411yescharacteristic, complemented
85#14519776—21no—
86#14519776C2×C412yescomplemented
87#14519775—21no—
88#14519774C2×C411yescharacteristic
89#14519773—21no—
90#14519773—21no—
91#14519773C2×C412yes—
92C4⋊C4×D61—21no—
93C4⋊C4×D61C2312yes—
94C42×D61C2311yescharacteristic
95#14519699C2×C412yescomplemented
96#14519699C2×C412yescomplemented
97#14519698C2×C412yescomplemented
98#14519698C2×C412yescomplemented
99#14519697C2×C412yescomplemented
100#14519697C2×C412yescomplemented
101C4×#92418C2×C411yescharacteristic, complemented
102C8×Dic61C2×C411yescharacteristic, complemented
103C8×Dic61C2×C411yescharacteristic, complemented
104C4⋊Q8×C61C2×C411yescharacteristic, complemented
105#127×C61C2×C411yescharacteristic
106#102×C61C2×C412yescomplemented
107#102×C61C2×C412yescomplemented
108C4×C8×C61C2×C411yescharacteristic, complemented
109C23×D61—22no—
110C22×#6923—21no—
111C22×#6923—21no—
112C22×#6923—21no—
113C22×#6923—21no—
114C22×#6923—44no—
115C22×#6923—44no—
116C22×#6923—82no—
117C22×#6923C2×D412yes—
118C22×#6923C2×D412yes—
119C2×Q8×C61—21no—
120C2×D4×C61—21no—
121C2×D4×C61—21no—
122C2×D4×C61C22×C412yes—
123C2×C22⋊D61—22no—
124C2×C22⋊D61—22no—
125C2×C22⋊D61—22no—
126Q8×D61—81no—
127Q8×D61—81no—
128D4×D61—22no—
129D4×D61—22no—
130D4×D61—22no—
131D4×D61—22no—
132D4×D61—42no—
133D4×D61—42no—
134D4×D61—81no—
135D4×D61—81no—
136C2×D244—21no—
137C2×D244—21no—
138C2×D244C22×C412yes—
139C2×C4×D61—22no—
140C2×C4×D61C2×D411yescharacteristic
141C2×C4×D61C2×D411yescharacteristic
142C2×C4×D61C2411yescharacteristic
143C2×Dic122—21no—
144C2×Dic122—22no—
145C22⋊#6923—44no—
146C22⋊#6923—44no—
147C22⋊#6923—82no—
148C2×#92420—22no—
149C4⋊#6923—22no—
150C4⋊#6923—21no—
151C4⋊#6923—21no—
152C4⋊#6923—21no—
153C4⋊#6923—21no—
154C4⋊#6923—21no—
155C4⋊#6923—21no—
156C4⋊#6923—82no—
157C4⋊#6923SD1614yes—
158C4⋊#6923#4112yes—
159C4⋊#6923#4112yes—
160C4×#6923—22no—
161C4×#6923—22no—
162C4×#6923—42no—
163C4×#6923—42no—
164C4×#6923—82no—
165D61⋊C8—22no—
166D61⋊C8—22no—
167C2×C8×C61C22×C411yescharacteristic
168C2×C8×C61C22×C411yescharacteristic
169C4⋊C4×C61—21no—
170C4⋊C4×C61C22×C412yes—
171C42×C61C22×C411yescharacteristic
172C4⋊Dic61—21no—
173C4⋊Dic61C22×C412yes—
174#11758196—22no—
175#11758196—22no—
176C4×Dic61C22×C411yescharacteristic
177C4×Dic61#4111yescharacteristic
178C4×Dic61#4111yescharacteristic
179C2×#92418C22×C411yescharacteristic
180C2×#92418C22×C411yescharacteristic
181C8×D61—21no—
182C8×D61—21no—
183C8×D61—21no—
184C8×D61—21no—
185C23×C61—22no—
186C22×D61—22no—
187C22×D61—42no—
188C22×D61—42no—
189C22×D61—42no—
190C22×D61—42no—
191C22×D61—42no—
192C22×D61—42no—
193C22×D61C22×D411yescharacteristic
194C2×#6923—22no—
195C2×#6923—22no—
196C2×#6923—22no—
197C2×#6923—22no—
198C2×#6923—22no—
199C2×#6923—22no—
200C2×#6923—22no—
201C2×#6923—41no—
202C2×#6923—41no—
203C2×#6923—41no—
204C2×#6923—41no—
205C2×#6923—41no—
206C2×#6923—41no—
207C2×#6923—41no—
208C2×#6923—41no—
209C2×#6923—41no—
210C2×#6923—41no—
211C2×#6923—88no—
212C2×#6923C2×SD1614yes—
213Q8×C61—81no—
214D4×C61—22no—
215D4×C61—22no—
216D4×C61—42no—
217D4×C61—81no—
218C2×C4×C61—22no—
219C2×C4×C61C4×D411yescharacteristic
220C2×C4×C61C4×D411yescharacteristic
221C2×C4×C61C23×C411yescharacteristic
222C22⋊D61—42no—
223C22⋊D61—42no—
224C22⋊D61—44no—
225C22⋊D61—82no—
226C2×Dic61—22no—
227C2×Dic61C4×D411yescharacteristic
228C2×Dic61C4×D411yescharacteristic
229C2×Dic61C2×#4111yescharacteristic
230D244—22no—
231D244—22no—
232D244—42no—
233D244—81no—
234C4×D61—21no—
235C4×D61—21no—
236C4×D61—42no—
237C4×D61—42no—
238C4×D61C2×SD1612yes—
239C4×D61#13512yes—
240C4×D61C2×#4111yescharacteristic
241C4×D61C2×#4111yescharacteristic
242C4×D61H(5,𝔽2)11yescharacteristic
243C4×D612−1+411yescharacteristic
244Dic122—82no—
245Dic122—81no—
246#92420—42no—
247C8×C61—21no—
248C8×C61—21no—
249#92418—21no—
250#92418—21no—
251#23215—611nomaximal, Sylow
252C22×C61—42no—
253C22×C61—42no—
254C22×C61—42no—
255C22×C61C2×C4×D411yescharacteristic
256C2×D61—42no—
257C2×D61—42no—
258C2×D61—42no—
259C2×D61—82no—
260C2×D61—82no—
261C2×D61—82no—
262C2×D61—82no—
263C2×D61#54611yescharacteristic
264C2×D61#54811yescharacteristic
265C2×D61C22×SD1611yescharacteristic
266C2×D61C2×#13511yescharacteristic
267C2×D61#57611yescharacteristic
268C2×D61#57611yescharacteristic
269#6923—24no—
270#6923—42no—
271#6923—42no—
272#6923—42no—
273#6923—81no—
274#6923—81no—
275C4×C61—21no—
276C4×C61—42no—
277C4×C61C4×SD1612yes—
278C4×C61C4×#4111yescharacteristic
279C4×C61#52011yescharacteristic
280Dic61—21no—
281Dic61—42no—
282Dic61C4×SD1612yes—
283Dic61#55611yescharacteristic
284Dic61#55911yescharacteristic
285C2×#486—611no—
286#2369—611no—
287#2366—611no—
288#1987—611no—
289#1982—611no—
290#1934—612no—
291#1926—612no—
292#1864—612no—
293C4×#106—611no—
294C8×C4⋊C4—611no—
295C4×#102—612no—
296C2×C61—82no—
297C2×C61#236711yescharacteristic
298C2×C61C2×C4×SD1611yescharacteristic
299C2×C61#294811yescharacteristic
300D61—82no—
301D61#329511yescharacteristic
302D61#330511yescharacteristic
303D61#336111yescharacteristic
304D61#336211yescharacteristic
305C2×C4⋊Q8—611no—
306C2×#127—611no—
307C2×C4×D4—612no—
308C2×C4×D4—1221no—
309#486—611no—
310#486—611no—
311#486—612no—
312#486—612no—
313#486—611no—
314#486#692311yescharacteristic, complemented
315C2×#106—611no—
316C2×#106—611no—
317C2×C4⋊C8—611no—
318C2×#102—612no—
319C2×#102—612no—
320C2×#102—612no—
321C2×#102—612no—
322C2×C4×C8—611no—
323C2×C4×C8—612no—
324#391—1222no—
325#390—612no—
326#390—1221no—
327#389—612no—
328#389—1221no—
329#386—1222no—
330C4×C4⋊C4—611no—
331C4×C4⋊C4—611no—
332C4×C4⋊C4—612no—
333C4×C4⋊C4—611no—
334C4×C4⋊C4—1221no—
335#340—612no—
336#340—612no—
337#340—611no—
338#340—611no—
339#336—612no—
340C61#2321511yescharacteristic, complemented, Sylow
341C22×Q8—1221no—
342C22×D4—612no—
343C22×D4—1221no—
344C22×D4—1221no—
345C23×C4—1222no—
346C22×C8—611no—
347C22×C8—611no—
348C22×C8—612no—
349C4⋊Q8—612no—
350C4⋊Q8—611no—
351C4⋊Q8—1221no—
352C4⋊Q8—1221no—
353C4⋊Q8C2×#692311yescharacteristic, complemented
354#127—612no—
355#127—611no—
356#127—1221no—
357#127—1221no—
358#127C2×#692311yescharacteristic
359C4×D4—1224no—
360C4×D4—2442no—
361C4×D4—2444no—
362C2×C4⋊C4—611no—
363C2×C4⋊C4—611no—
364C2×C4⋊C4—611no—
365C2×C4⋊C4—612no—
366C2×C4⋊C4—611no—
367C2×C4⋊C4—611no—
368C2×C4⋊C4—611no—
369C2×C4⋊C4—1221no—
370C2×C4⋊C4—1222no—
371C2×C22⋊C4—1222no—
372C2×C22⋊C4—1222no—
373C2×C22⋊C4—1222no—
374C2×C42—611no—
375C2×C42—612no—
376C2×C42—1222no—
377C2×C42—1221no—
378C2×C42—1221no—
379#106—1222no—
380#106—1222no—
381C4⋊C8—1221no—
382C4⋊C8—1221no—
383#102—612no—
384#102—612no—
385#102—612no—
386#102—612no—
387#102—612no—
388#102—612no—
389#102—1224no—
390#102—2442no—
391#102C2×#692312yescomplemented
392#102C2×#692312yescomplemented
393C4×C8—611no—
394C4×C8—611no—
395C4×C8—611no—
396C4×C8—2442no—
397C4×C8C2×#692311yescharacteristic, complemented
398#95—612no—
399#95—1221no—
400#95—1222no—
401#95—1221no—
402#95—1222no—
403C24—1222no—
404C2×Q8—21no—
405C2×Q8—1222no—
406C2×Q8—1221no—
407C2×Q8—4881no—
408C2×Q8—4881no—
409C2×D4—21no—
410C2×D4—21no—
411C2×D4—612no—
412C2×D4—1222no—
413C2×D4—1222no—
414C2×D4—1222no—
415C2×D4—1222no—
416C2×D4—1222no—
417C2×D4—1222no—
418C2×D4—1222no—
419C2×D4—1221no—
420C2×D4—1221no—
421C2×D4—4881no—
422C2×D4—4881no—
423C2×D4—2442no—
424C2×D4—2442no—
425C2×D4C22×#692312yes—
426C22×C4—611no—
427C22×C4—611no—
428C22×C4—612no—
429C22×C4—611no—
430C22×C4—612no—
431C22×C4—1222no—
432C22×C4—1221no—
433C22×C4—1221no—
434C22×C4—1221no—
435C22×C4—1221no—
436C22×C4—4882no—
437C22×C4—2444no—
438C22×C4—2444no—
439C2×C8—611no—
440C2×C8—611no—
441C2×C8—1221no—
442C2×C8—1221no—
443C2×C8—1221no—
444C2×C8—1221no—
445C2×C8—1222no—
446C2×C8—1222no—
447C2×C8—1222no—
448C2×C8C22×#692311yescharacteristic
449C2×C8C22×#692311yescharacteristic
450C4⋊C4—21no—
451C4⋊C4—614no—
452C4⋊C4—612no—
453C4⋊C4—612no—
454C4⋊C4—612no—
455C4⋊C4—1222no—
456C4⋊C4—1222no—
457C4⋊C4—1221no—
458C4⋊C4—1222no—
459C4⋊C4—1221no—
460C4⋊C4—1221no—
461C4⋊C4—1221no—
462C4⋊C4—1221no—
463C4⋊C4—1221no—
464C4⋊C4—1221no—
465C4⋊C4—4882no—
466C4⋊C4C22×#692312yes—
467C22⋊C4—4882no—
468C22⋊C4—2444no—
469C22⋊C4—2444no—
470C42—611no—
471C42—611no—
472C42—611no—
473C42—1222no—
474C42—1222no—
475C42—4882no—
476C42—2442no—
477C42—2442no—
478C42C22×#692311yescharacteristic
479C23—22no—
480C23—611no—
481C23—1222no—
482C23—2442no—
483C23—2442no—
484C23—2442no—
485C23—2442no—
486C23—2442no—
487C23—2442no—
488Q8—81no—
489Q8—4881no—
490Q8—4882no—
491D4—22no—
492D4—22no—
493D4—42no—
494D4—81no—
495D4—1222no—
496D4—1222no—
497D4—4881no—
498D4—4882no—
499D4—2442no—
500D4—2442no—
501D4—2444no—
502D4—2442no—
503C2×C4—22no—
504C2×C4—611no—
505C2×C4—611no—
506C2×C4—612no—
507C2×C4—612no—
508C2×C4—611no—
509C2×C4—611no—
510C2×C4—611no—
511C2×C4—611no—
512C2×C4—614no—
513C2×C4—611no—
514C2×C4—1221no—
515C2×C4—1221no—
516C2×C4—1222no—
517C2×C4—1222no—
518C2×C4—1222no—
519C2×C4—1222no—
520C2×C4—1222no—
521C2×C4—1222no—
522C2×C4—1222no—
523C2×C4—1222no—
524C2×C4—4888no—
525C2×C4—2442no—
526C2×C4—2442no—
527C2×C4—2441no—
528C2×C4—2441no—
529C2×C4—2441no—
530C2×C4—2441no—
531C2×C4—2441no—
532C2×C4—2441no—
533C2×C4—2441no—
534C2×C4—2441no—
535C2×C4—2441no—
536C2×C4—2441no—
537C2×C4D4×#692311yescharacteristic
538C2×C4D4×#692311yescharacteristic
539C2×C4C23×#692311yescharacteristic
540C8—21no—
541C8—21no—
542C8—1221no—
543C8—1221no—
544C8—2442no—
545C22—42no—
546C22—42no—
547C22—42no—
548C22—611no—
549C22—611no—
550C22—611no—
551C22—611no—
552C22—611no—
553C22—611no—
554C22—4882no—
555C22—4882no—
556C22—4882no—
557C22—4882no—
558C22—2442no—
559C22—2442no—
560C22—2442no—
561C22C2×D4×#692311yescharacteristic
562C4—21no—
563C4—42no—
564C4—612no—
565C4—611no—
566C4—611no—
567C4—1224no—
568C4—1221no—
569C4—4881no—
570C4—4881no—
571C4—2442no—
572C4—2442no—
573C4—2442no—
574C4—2442no—
575C4SD16×#692312yes—
576C4#41×#692311yescharacteristic
577C4#1684083111yescharacteristic
578C2—82no—
579C2—611no—
580C2—611no—
581C2—611no—
582C2—611no—
583C2—4882no—
584C2#2146235011yescharacteristic
585C2C2×SD16×#692311yescharacteristic
586C2#2146257811yescharacteristic
subgroupG/H
centerC22C2×D4×#6923
commutatorC2×C4×C61C23×C4
FrattiniC2×C4C23×#6923
FittingC61×#486C4
socleC22×C61C2×C4×D4
prime radical#127×D61C22
cosocleC2×C4×D61C24
hypercenter#486#6923
hypocenterC61#23215
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #486

Complements

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

NH ≅ G/Ntype
1. #21462352C2semidirect
3. #21461997C2semidirect
5. #21461909C2semidirect
6. #21461897C2semidirect
7. #21461784C2semidirect
8. C4×#14519845C2semidirect
9. C8×C4⋊#6923C2semidirect
10. #102×#6923C2semidirect
38. #16840110C4semidirect
39. #16840109C4semidirect
40. #16840108C4semidirect
41. #16840107C4semidirect
42. #16840107C4semidirect
46. C61×#486C4semidirect
84. #14519777C2×C4semidirect
86. #14519776C2×C4semidirect
95. #14519699C2×C4semidirect
96. #14519699C2×C4semidirect
97. #14519698C2×C4semidirect
98. #14519698C2×C4semidirect
99. #14519697C2×C4semidirect
100. #14519697C2×C4semidirect
101. C4×#92418C2×C4semidirect
102. C8×Dic61C2×C4semidirect
103. C8×Dic61C2×C4semidirect
104. C4⋊Q8×C61C2×C4semidirect
106. #102×C61C2×C4semidirect
107. #102×C61C2×C4semidirect
108. C4×C8×C61C2×C4semidirect
314. #486#6923semidirect
340. C61#23215semidirect
353. C4⋊Q8C2×#6923semidirect
391. #102C2×#6923semidirect
392. #102C2×#6923semidirect
397. C4×C8C2×#6923semidirect

Maximal subgroups

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

subgroupindex
#214623522
#214623492
#214619972
#214619862
#214619092
#214618972
#214617842
C4×#145198452
C8×C4⋊#69232
#102×#69232
#486×D612
#2321561