Finite Group Databasebeta

← #30622830

Subgroups — #30622830

Order 15616.

962 conjugacy classes of subgroups.

The subgroup lattice has 590 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21462349C211yesmaximal, characteristic
2#21462348C211yesmaximal, characteristic, complemented
3#21461998C211yesmaximal, characteristic, complemented
4#21461986C211yesmaximal, characteristic
5#21461909C212yesmaximal, complemented
6#21461896C212yesmaximal, complemented
7#21461782C212yesmaximal, complemented
8C4×#14519846C211yesmaximal, characteristic, complemented
9C8×C4⋊#6923C211yesmaximal, characteristic, complemented
10#102×#6923C212yesmaximal, complemented
11#486×D61C211yesmaximal, characteristic
12C2×Q8×#6923—21no—
13C2×D4×#6923—21no—
14C2×D4×#6923C2212yes—
15C2×#14519846C2211yescharacteristic
16C2×#14519846C2211yescharacteristic
17D61⋊(C4⋊C8)C2211yescharacteristic
18C2×#14519839C2212yes—
19C2×#14519839C2212yes—
20C2×C8×#6923C2212yes—
21(C2×#6923)⋊Q8—21no—
22#16840714—21no—
23#16840714C2212yes—
24#16840713C2212yes—
25#16840711—22no—
26#16840710—22no—
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—
49Q8×#6923—44no—
50D4×#6923—24no—
51D4×#6923—42no—
52D4×#6923—44no—
53C2×C4⋊#6923—22no—
54C2×C4⋊#6923D411yescharacteristic
55C2×C4⋊#6923D411yescharacteristic
56C2×C4⋊#6923C2311yescharacteristic
57C2×C4⋊#6923C2311yescharacteristic
58C2×C4⋊#6923C2311yescharacteristic
59C2×C4⋊#6923C2311yescharacteristic
60C2×C22⋊#6923—22no—
61C2×C22⋊#6923—22no—
62C2×C22⋊#6923—22no—
63C2×C4×#6923—22no—
64C2×C4×#6923—21no—
65C2×C4×#6923—21no—
66C2×C4×#6923C2312yes—
67#14519846—22no—
68#14519846—22no—
69#14519844—21no—
70C4⋊#92420—21no—
71#14519839—24no—
72#14519839—42no—
73C8×#6923—42no—
74(C2×#6923)⋊C4—22no—
75(C2×#6923)⋊C4—22no—
76(C2×#6923)⋊C4—21no—
77(C2×#6923)⋊C4—21no—
78(C2×#6923)⋊C4D412yes—
79C2×Q8×D61—21no—
80C2×D4×D61—21no—
81C2×D4×D61—21no—
82C2×D4×D61C2312yes—
83C2×C8×D61C2311yescharacteristic
84C2×C8×D61C2311yescharacteristic
85#14519777C2×C411yescharacteristic, complemented
86#14519776—21no—
87#14519776C2×C412yescomplemented
88#14519775—21no—
89#14519774C2×C411yescharacteristic
90#14519773—21no—
91#14519773—21no—
92#14519773C2×C412yes—
93C4⋊C4×D61—21no—
94C4⋊C4×D61C2312yes—
95C42×D61C2311yescharacteristic
96#14519699C2×C412yescomplemented
97#14519699C2×C412yescomplemented
98#14519698C2×C412yescomplemented
99#14519698C2×C412yescomplemented
100#14519697C2×C412yescomplemented
101#14519697C2×C412yescomplemented
102C4×#92418C2×C411yescharacteristic, complemented
103C8×Dic61C2×C411yescharacteristic, complemented
104C8×Dic61C2×C411yescharacteristic, complemented
105C4⋊Q8×C61C2×C411yescharacteristic, complemented
106#127×C61C2×C411yescharacteristic
107#102×C61C2×C412yescomplemented
108#102×C61C2×C412yescomplemented
109C4×C8×C61C2×C411yescharacteristic, complemented
110C23×D61—22no—
111C22×#6923—21no—
112C22×#6923—21no—
113C22×#6923—21no—
114C22×#6923—21no—
115C22×#6923—44no—
116C22×#6923—44no—
117C22×#6923—82no—
118C22×#6923C2×D412yes—
119C22×#6923C2×D412yes—
120C2×Q8×C61—21no—
121C2×D4×C61—21no—
122C2×D4×C61—21no—
123C2×D4×C61C22×C412yes—
124C2×C22⋊D61—22no—
125C2×C22⋊D61—22no—
126C2×C22⋊D61—22no—
127Q8×D61—42no—
128Q8×D61—42no—
129D4×D61—22no—
130D4×D61—22no—
131D4×D61—22no—
132D4×D61—22no—
133D4×D61—42no—
134D4×D61—42no—
135D4×D61—81no—
136D4×D61—81no—
137C2×D244—21no—
138C2×D244—21no—
139C2×D244C22×C412yes—
140C2×C4×D61—22no—
141C2×C4×D61C2×D411yescharacteristic
142C2×C4×D61C2×D411yescharacteristic
143C2×C4×D61C2411yescharacteristic
144C2×Dic122—21no—
145C2×Dic122—22no—
146C22⋊#6923—44no—
147C22⋊#6923—44no—
148C22⋊#6923—82no—
149C2×#92420—22no—
150C4⋊#6923—22no—
151C4⋊#6923—21no—
152C4⋊#6923—21no—
153C4⋊#6923—21no—
154C4⋊#6923—21no—
155C4⋊#6923—21no—
156C4⋊#6923—21no—
157C4⋊#6923—44no—
158C4⋊#6923D814yes—
159C4⋊#6923#4112yes—
160C4⋊#6923#4112yes—
161C4×#6923—22no—
162C4×#6923—22no—
163C4×#6923—44no—
164C4×#6923—42no—
165C4×#6923—42no—
166D61⋊C8—22no—
167D61⋊C8—22no—
168C2×C8×C61C22×C411yescharacteristic
169C2×C8×C61C22×C411yescharacteristic
170C4⋊C4×C61—21no—
171C4⋊C4×C61C22×C412yes—
172C42×C61C22×C411yescharacteristic
173C4⋊Dic61—21no—
174C4⋊Dic61C22×C412yes—
175#11758196—22no—
176#11758196—22no—
177C4×Dic61C22×C411yescharacteristic
178C4×Dic61#4111yescharacteristic
179C4×Dic61#4111yescharacteristic
180C2×#92418C22×C411yescharacteristic
181C2×#92418C22×C411yescharacteristic
182C8×D61—21no—
183C8×D61—21no—
184C8×D61—21no—
185C8×D61—21no—
186C23×C61—22no—
187C22×D61—22no—
188C22×D61—42no—
189C22×D61—42no—
190C22×D61—42no—
191C22×D61—42no—
192C22×D61—42no—
193C22×D61—42no—
194C22×D61C22×D411yescharacteristic
195C2×#6923—22no—
196C2×#6923—22no—
197C2×#6923—22no—
198C2×#6923—22no—
199C2×#6923—22no—
200C2×#6923—22no—
201C2×#6923—22no—
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—41no—
212C2×#6923—88no—
213C2×#6923C2×D814yes—
214Q8×C61—42no—
215D4×C61—22no—
216D4×C61—22no—
217D4×C61—42no—
218D4×C61—81no—
219C2×C4×C61—22no—
220C2×C4×C61C4×D411yescharacteristic
221C2×C4×C61C4×D411yescharacteristic
222C2×C4×C61C23×C411yescharacteristic
223C22⋊D61—42no—
224C22⋊D61—42no—
225C22⋊D61—44no—
226C22⋊D61—82no—
227C2×Dic61—22no—
228C2×Dic61C4×D411yescharacteristic
229C2×Dic61C4×D411yescharacteristic
230C2×Dic61C2×#4111yescharacteristic
231D244—22no—
232D244—22no—
233D244—42no—
234D244—81no—
235C4×D61—21no—
236C4×D61—21no—
237C4×D61—42no—
238C4×D61—42no—
239C4×D61C2×D812yes—
240C4×D61#13512yes—
241C4×D61C2×#4111yescharacteristic
242C4×D61C2×#4111yescharacteristic
243C4×D61H(5,𝔽2)11yescharacteristic
244C4×D612−1+411yescharacteristic
245Dic122—42no—
246Dic122—44no—
247#92420—42no—
248C8×C61—21no—
249C8×C61—21no—
250#92418—21no—
251#92418—21no—
252#23216—611nomaximal, Sylow
253C22×C61—42no—
254C22×C61—42no—
255C22×C61—42no—
256C22×C61C2×C4×D411yescharacteristic
257C2×D61—42no—
258C2×D61—42no—
259C2×D61—42no—
260C2×D61—82no—
261C2×D61—82no—
262C2×D61—82no—
263C2×D61—82no—
264C2×D61#54611yescharacteristic
265C2×D61#54811yescharacteristic
266C2×D61C22×D811yescharacteristic
267C2×D61C2×#13511yescharacteristic
268C2×D61#57711yescharacteristic
269C2×D61#57711yescharacteristic
270#6923—24no—
271#6923—42no—
272#6923—42no—
273#6923—42no—
274#6923—81no—
275#6923—81no—
276C4×C61—21no—
277C4×C61—42no—
278C4×C61C4×D812yes—
279C4×C61C4×#4111yescharacteristic
280C4×C61#52011yescharacteristic
281Dic61—21no—
282Dic61—42no—
283Dic61C4×D812yes—
284Dic61#55611yescharacteristic
285Dic61#55911yescharacteristic
286C2×#486—611no—
287#2366—611no—
288#2365—611no—
289#1988—611no—
290#1982—611no—
291#1934—612no—
292#1925—612no—
293#1862—612no—
294C4×#107—611no—
295C8×C4⋊C4—611no—
296C4×#102—612no—
297C2×C61—82no—
298C2×C61#236711yescharacteristic
299C2×C61C2×C4×D811yescharacteristic
300C2×C61#294911yescharacteristic
301D61—82no—
302D61#329411yescharacteristic
303D61#330411yescharacteristic
304D61#335811yescharacteristic
305D61#336511yescharacteristic
306C2×C4⋊Q8—611no—
307C2×#127—611no—
308C2×C4×Q8—1221no—
309C2×C4×D4—612no—
310C2×C4×D4—1221no—
311#486—611no—
312#486—611no—
313#486—612no—
314#486—612no—
315#486—611no—
316#486#692311yescharacteristic, complemented
317C2×#107—611no—
318C2×#107—611no—
319C2×C4⋊C8—611no—
320C2×#102—612no—
321C2×#102—612no—
322C2×#102—612no—
323C2×#102—612no—
324C2×C4×C8—611no—
325C2×C4×C8—612no—
326#391—1221no—
327#390—612no—
328#390—1221no—
329#389—612no—
330#387—1222no—
331#386—1222no—
332C4×C4⋊C4—611no—
333C4×C4⋊C4—612no—
334C4×C4⋊C4—611no—
335C4×C4⋊C4—611no—
336#340—612no—
337#340—612no—
338#340—611no—
339#340—611no—
340#336—612no—
341C61#2321611yescharacteristic, complemented, Sylow
342C22×Q8—1221no—
343C22×D4—612no—
344C22×D4—1221no—
345C22×D4—1221no—
346C23×C4—1222no—
347C22×C8—611no—
348C22×C8—611no—
349C22×C8—612no—
350C4⋊Q8—612no—
351C4⋊Q8—611no—
352C4⋊Q8—1221no—
353C4⋊Q8—1221no—
354C4⋊Q8C2×#692311yescharacteristic, complemented
355#127—612no—
356#127—611no—
357#127—1221no—
358#127—1221no—
359#127C2×#692311yescharacteristic
360C4×Q8—2444no—
361C4×D4—1224no—
362C4×D4—2442no—
363C4×D4—2444no—
364C2×C4⋊C4—611no—
365C2×C4⋊C4—611no—
366C2×C4⋊C4—611no—
367C2×C4⋊C4—612no—
368C2×C4⋊C4—611no—
369C2×C4⋊C4—611no—
370C2×C4⋊C4—611no—
371C2×C4⋊C4—1221no—
372C2×C4⋊C4—1222no—
373C2×C22⋊C4—1222no—
374C2×C22⋊C4—1222no—
375C2×C22⋊C4—1222no—
376C2×C42—611no—
377C2×C42—612no—
378C2×C42—1221no—
379C2×C42—1222no—
380C2×C42—1221no—
381#107—1222no—
382#107—1222no—
383C4⋊C8—1221no—
384C4⋊C8—1221no—
385#102—612no—
386#102—612no—
387#102—612no—
388#102—612no—
389#102—612no—
390#102—612no—
391#102—1224no—
392#102—2442no—
393#102C2×#692312yescomplemented
394#102C2×#692312yescomplemented
395C4×C8—611no—
396C4×C8—611no—
397C4×C8—611no—
398C4×C8—2442no—
399C4×C8C2×#692311yescharacteristic, complemented
400#95—612no—
401#95—1221no—
402#95—1221no—
403#95—1222no—
404#95—1222no—
405C24—1222no—
406C2×Q8—21no—
407C2×Q8—1222no—
408C2×Q8—1221no—
409C2×Q8—2442no—
410C2×Q8—2442no—
411C2×D4—21no—
412C2×D4—21no—
413C2×D4—612no—
414C2×D4—1222no—
415C2×D4—1222no—
416C2×D4—1222no—
417C2×D4—1222no—
418C2×D4—1222no—
419C2×D4—1222no—
420C2×D4—1222no—
421C2×D4—1221no—
422C2×D4—1221no—
423C2×D4—4881no—
424C2×D4—4881no—
425C2×D4—2442no—
426C2×D4—2442no—
427C2×D4C22×#692312yes—
428C22×C4—611no—
429C22×C4—611no—
430C22×C4—612no—
431C22×C4—611no—
432C22×C4—612no—
433C22×C4—1222no—
434C22×C4—1221no—
435C22×C4—1221no—
436C22×C4—1221no—
437C22×C4—1221no—
438C22×C4—4882no—
439C22×C4—2444no—
440C22×C4—2444no—
441C2×C8—611no—
442C2×C8—611no—
443C2×C8—1221no—
444C2×C8—1221no—
445C2×C8—1221no—
446C2×C8—1221no—
447C2×C8—1222no—
448C2×C8—1222no—
449C2×C8—1222no—
450C2×C8C22×#692311yescharacteristic
451C2×C8C22×#692311yescharacteristic
452C4⋊C4—21no—
453C4⋊C4—614no—
454C4⋊C4—612no—
455C4⋊C4—612no—
456C4⋊C4—612no—
457C4⋊C4—1222no—
458C4⋊C4—1222no—
459C4⋊C4—1221no—
460C4⋊C4—1222no—
461C4⋊C4—1221no—
462C4⋊C4—1221no—
463C4⋊C4—1221no—
464C4⋊C4—1221no—
465C4⋊C4—1221no—
466C4⋊C4—1221no—
467C4⋊C4—2444no—
468C4⋊C4C22×#692312yes—
469C22⋊C4—4882no—
470C22⋊C4—2444no—
471C22⋊C4—2444no—
472C42—611no—
473C42—611no—
474C42—611no—
475C42—1222no—
476C42—1222no—
477C42—2442no—
478C42—2442no—
479C42—2444no—
480C42C22×#692311yescharacteristic
481C23—22no—
482C23—611no—
483C23—1222no—
484C23—2442no—
485C23—2442no—
486C23—2442no—
487C23—2442no—
488C23—2442no—
489C23—2442no—
490Q8—42no—
491Q8—2444no—
492Q8—2442no—
493D4—22no—
494D4—22no—
495D4—42no—
496D4—81no—
497D4—1222no—
498D4—1222no—
499D4—4881no—
500D4—4882no—
501D4—2442no—
502D4—2442no—
503D4—2444no—
504D4—2442no—
505C2×C4—22no—
506C2×C4—611no—
507C2×C4—611no—
508C2×C4—612no—
509C2×C4—612no—
510C2×C4—611no—
511C2×C4—611no—
512C2×C4—611no—
513C2×C4—611no—
514C2×C4—614no—
515C2×C4—611no—
516C2×C4—1221no—
517C2×C4—1221no—
518C2×C4—1222no—
519C2×C4—1222no—
520C2×C4—1222no—
521C2×C4—1222no—
522C2×C4—1222no—
523C2×C4—1222no—
524C2×C4—1222no—
525C2×C4—1222no—
526C2×C4—4888no—
527C2×C4—2442no—
528C2×C4—2442no—
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×C4—2441no—
538C2×C4—2441no—
539C2×C4D4×#692311yescharacteristic
540C2×C4D4×#692311yescharacteristic
541C2×C4C23×#692311yescharacteristic
542C8—21no—
543C8—21no—
544C8—1221no—
545C8—1221no—
546C8—2442no—
547C22—42no—
548C22—42no—
549C22—42no—
550C22—611no—
551C22—611no—
552C22—611no—
553C22—611no—
554C22—611no—
555C22—611no—
556C22—4882no—
557C22—4882no—
558C22—4882no—
559C22—4882no—
560C22—2442no—
561C22—2442no—
562C22—2442no—
563C22C2×D4×#692311yescharacteristic
564C4—21no—
565C4—42no—
566C4—612no—
567C4—611no—
568C4—611no—
569C4—1224no—
570C4—1221no—
571C4—4881no—
572C4—4881no—
573C4—2442no—
574C4—2442no—
575C4—2442no—
576C4—2442no—
577C4D8×#692312yes—
578C4#41×#692311yescharacteristic
579C4#1684083111yescharacteristic
580C2—82no—
581C2—611no—
582C2—611no—
583C2—611no—
584C2—611no—
585C2—4882no—
586C2#2146235011yescharacteristic
587C2C2×D8×#692311yescharacteristic
588C2#2146257911yescharacteristic
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#23216
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
2. #21462348C2semidirect
3. #21461998C2semidirect
5. #21461909C2semidirect
6. #21461896C2semidirect
7. #21461782C2semidirect
8. C4×#14519846C2semidirect
9. C8×C4⋊#6923C2semidirect
10. #102×#6923C2semidirect
38. #16840110C4semidirect
39. #16840109C4semidirect
40. #16840108C4semidirect
41. #16840107C4semidirect
42. #16840107C4semidirect
46. C61×#486C4semidirect
85. #14519777C2×C4semidirect
87. #14519776C2×C4semidirect
96. #14519699C2×C4semidirect
97. #14519699C2×C4semidirect
98. #14519698C2×C4semidirect
99. #14519698C2×C4semidirect
100. #14519697C2×C4semidirect
101. #14519697C2×C4semidirect
102. C4×#92418C2×C4semidirect
103. C8×Dic61C2×C4semidirect
104. C8×Dic61C2×C4semidirect
105. C4⋊Q8×C61C2×C4semidirect
107. #102×C61C2×C4semidirect
108. #102×C61C2×C4semidirect
109. C4×C8×C61C2×C4semidirect
316. #486#6923semidirect
341. C61#23216semidirect
354. C4⋊Q8C2×#6923semidirect
393. #102C2×#6923semidirect
394. #102C2×#6923semidirect
399. C4×C8C2×#6923semidirect

Maximal subgroups

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

subgroupindex
#214623492
#214623482
#214619982
#214619862
#214619092
#214618962
#214617822
C4×#145198462
C8×C4⋊#69232
#102×#69232
#486×D612
#2321661