Finite Group Databasebeta

← #30622350

Subgroups — #30622350

Order 15616.

826 conjugacy classes of subgroups.

The subgroup lattice has 574 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21462352C211yesmaximal, characteristic, complemented
2#21462337C211yesmaximal, characteristic
3#21462000C211yesmaximal, characteristic, complemented
4#21461931C211yesmaximal, characteristic, complemented
5#21461909C212yesmaximal
6#21461897C212yesmaximal
7#21461882C212yesmaximal, complemented
8#106×#6923C212yesmaximal, complemented
9#21461612C211yesmaximal, characteristic, complemented
10C4×#14519839C211yesmaximal, characteristic
11#476×D61C211yesmaximal, characteristic
12C2×#14519856—21no—
13C2×#14519845C2212yes—
14C2×#14519839C2211yescharacteristic
15C2×#14519839C2211yescharacteristic
16C4×D61⋊C8C2211yescharacteristic
17C2×C8×#6923C2212yes—
18(C2×#6923)⋊Q8—22no—
19#16840714C2212yes—
20#16840713—21no—
21#16840713C2212yes—
22#16840712—22no—
23#16840708C2211yescharacteristic
24#16840704—21no—
25C4⋊C4×#6923—21no—
26C4⋊C4×#6923C2212yes—
27C4⋊C4×#6923C2212yes—
28C4×C4⋊#6923C2211yescharacteristic
29C4×C4⋊#6923C2211yescharacteristic
30#16840636C2212yes—
31#16840635C2211yescharacteristic
32#16840635C2211yescharacteristic
33#16840635C2212yes—
34#16840635C2212yes—
35(C2×#6923)⋊C8C2212yes—
36C4⋊Q8×D61C2211yescharacteristic
37C4×D4×D61C2211yescharacteristic
38#16840053C411yescharacteristic, complemented
39#16840052C411yescharacteristic, complemented
40#16840051C412yescomplemented
41#16840050C412yescomplemented
42#16840049C411yescharacteristic, complemented
43#106×D61C2212yes—
44C4⋊C8×D61C2211yescharacteristic
45#102×D61C2212yes—
46C61×#476C411yescharacteristic, complemented
47C2×D61⋊C8C2311yescharacteristic
48C2×D61⋊C8C2311yescharacteristic
49#14519856—44no—
50C2×C4⋊#6923—21no—
51C2×C4⋊#6923—22no—
52C2×C4⋊#6923—21no—
53C2×C4⋊#6923D411yescharacteristic
54C2×C4⋊#6923D411yescharacteristic
55C2×C4⋊#6923C2312yes—
56C2×C4⋊#6923C2311yescharacteristic
57C2×C4⋊#6923C2311yescharacteristic
58C2×C22⋊#6923—22no—
59C2×C22⋊#6923—22no—
60C2×C4×#6923—22no—
61C2×C4×#6923—21no—
62C2×C4×#6923—22no—
63C2×C4×#6923C2312yes—
64#14519845—24no—
65#14519839—22no—
66#14519839—22no—
67C4×#92420—21no—
68C4×#92420—21no—
69C8×#6923—24no—
70(C2×#6923)⋊C4—22no—
71(C2×#6923)⋊C4—22no—
72(C2×#6923)⋊C4—22no—
73(C2×#6923)⋊C4—21no—
74(C2×#6923)⋊C4D412yes—
75C2×Q8×D61—21no—
76C2×D4×D61C2311yescharacteristic
77C22×C4×D61—21no—
78C2×C8×D61C2312yes—
79#14519777C2×C411yescharacteristic
80#14519776—21no—
81#14519776C2×C412yes—
82#14519775—21no—
83C4×D244C2×C411yescharacteristic
84D4×Dic61C2×C411yescharacteristic
85#14519739C2×C411yescharacteristic
86C4×C22⋊D61—21no—
87#14519737—21no—
88C4⋊C4×D61—21no—
89C4⋊C4×D61C2312yes—
90C4⋊C4×D61C2311yescharacteristic
91C22⋊C4×D61—21no—
92C42×D61C2311yescharacteristic
93#14519709C2×C412yes—
94#14519708C2×C412yes—
95#14519708C2×C412yes—
96C4⋊#92418C2×C411yescharacteristic
97#14519706C2×C412yes—
98#14519699C2×C412yes—
99#14519698C2×C412yes—
100#14519697C2×C412yes—
101C4⋊Q8×C61C2×C411yescharacteristic
102C4×D4×C61C2×C411yescharacteristic
103#106×C61C2×C412yes—
104C4⋊C8×C61C2×C411yescharacteristic
105#102×C61C2×C412yes—
106C23×D61—21no—
107C22×#6923—21no—
108C22×#6923—21no—
109C22×#6923—22no—
110C22×#6923C2×D412yes—
111C22×#6923C2×D412yes—
112C2×Q8×C61—21no—
113C2×D4×C61C22×C411yescharacteristic
114C22×C4×C61—21no—
115C2×C22⋊D61—21no—
116C22×Dic61—21no—
117Q8×D61—81no—
118Q8×D61—81no—
119D4×D61—21no—
120D4×D61—21no—
121D4×D61—21no—
122D4×D61—21no—
123C2×D244C22×C411yescharacteristic
124C2×C4×D61—21no—
125C2×C4×D61—22no—
126C2×C4×D61—21no—
127C2×C4×D61—21no—
128C2×C4×D61—42no—
129C2×C4×D61—42no—
130C2×C4×D61C2×D411yescharacteristic
131C2×C4×D61C2×D411yescharacteristic
132C2×C4×D61C2411yescharacteristic
133C2×Dic122—21no—
134C2×Dic122—22no—
135C22⋊#6923—44no—
136C22⋊#6923—82no—
137C2×#92420—21no—
138C2×#92420—21no—
139C4⋊#6923—22no—
140C4⋊#6923—21no—
141C4⋊#6923—22no—
142C4⋊#6923—21no—
143C4⋊#6923—22no—
144C4⋊#6923—21no—
145C4⋊#6923—42no—
146C4⋊#6923—81no—
147C4⋊#6923—82no—
148C4⋊#6923SD1614yes—
149C4⋊#6923#4112yes—
150C4×#6923—22no—
151C4×#6923—22no—
152C4×#6923—44no—
153C4×#6923—42no—
154C4×#6923—82no—
155D61⋊C8—21no—
156D61⋊C8—21no—
157D61⋊C8—21no—
158D61⋊C8#4112yes—
159C2×C8×C61C22×C412yes—
160C4⋊C4×C61—21no—
161C4⋊C4×C61C22×C412yes—
162C4⋊C4×C61C22×C411yescharacteristic
163C22⋊C4×C61—21no—
164C42×C61C22×C411yescharacteristic
165C22⋊Dic61—21no—
166#11758198—21no—
167#11758198—21no—
168C4⋊Dic61—21no—
169C4⋊Dic61C22×C412yes—
170C4⋊Dic61C22×C411yescharacteristic
171#11758196—22no—
172#11758196—21no—
173#11758196—22no—
174C4×Dic61C22×C411yescharacteristic
175C4×Dic61#4111yescharacteristic
176C4×Dic61#4111yescharacteristic
177C2×#92418C22×C412yes—
178C8×D61—22no—
179C8×D61—22no—
180C23×C61—21no—
181C22×D61—21no—
182C22×D61—41no—
183C22×D61—41no—
184C22×D61—41no—
185C22×D61—41no—
186C22×D61—41no—
187C22×D61—41no—
188C22×D61C22×D411yescharacteristic
189C2×#6923—22no—
190C2×#6923—22no—
191C2×#6923—22no—
192C2×#6923—22no—
193C2×#6923—22no—
194C2×#6923—22no—
195C2×#6923—41no—
196C2×#6923—41no—
197C2×#6923—41no—
198C2×#6923—42no—
199C2×#6923—42no—
200C2×#6923—41no—
201C2×#6923—41no—
202C2×#6923—42no—
203C2×#6923C2×SD1614yes—
204Q8×C61—81no—
205D4×C61—21no—
206D4×C61—21no—
207C2×C4×C61—22no—
208C2×C4×C61—21no—
209C2×C4×C61—42no—
210C2×C4×C61C4×D411yescharacteristic
211C2×C4×C61C4×D411yescharacteristic
212C2×C4×C61C23×C411yescharacteristic
213C22⋊D61—41no—
214C22⋊D61—41no—
215C2×Dic61—22no—
216C2×Dic61—21no—
217C2×Dic61—42no—
218C2×Dic61C4×D411yescharacteristic
219C2×Dic61C4×D411yescharacteristic
220C2×Dic61C2×#4111yescharacteristic
221D244—21no—
222D244—21no—
223C4×D61—21no—
224C4×D61—21no—
225C4×D61—41no—
226C4×D61—41no—
227C4×D61—42no—
228C4×D61—42no—
229C4×D61—42no—
230C4×D61—42no—
231C4×D61C2×SD1612yes—
232C4×D61#13512yes—
233C4×D61C2×#4111yescharacteristic
234C4×D61C2×#4111yescharacteristic
235C4×D612−1+411yescharacteristic
236C4×D612−1+411yescharacteristic
237Dic122—82no—
238Dic122—81no—
239#92420—22no—
240#92420—41no—
241C8×C61—22no—
242#92418—22no—
243#23061—611nomaximal, Sylow
244C22×C61—41no—
245C22×C61—41no—
246C22×C61—41no—
247C22×C61C2×C4×D411yescharacteristic
248C2×D61—41no—
249C2×D61—41no—
250C2×D61—41no—
251C2×D61—81no—
252C2×D61—81no—
253C2×D61—81no—
254C2×D61—81no—
255C2×D61#54711yescharacteristic
256C2×D61#54811yescharacteristic
257C2×D61C22×SD1611yescharacteristic
258C2×D61C2×#13511yescharacteristic
259C2×D61#57611yescharacteristic
260C2×D61#57711yescharacteristic
261#6923—24no—
262#6923—42no—
263#6923—42no—
264#6923—81no—
265#6923—82no—
266C4×C61—21no—
267C4×C61—41no—
268C4×C61—42no—
269C4×C61C4×SD1612yes—
270C4×C61C4×#4111yescharacteristic
271C4×C61#51911yescharacteristic
272Dic61—21no—
273Dic61—41no—
274Dic61—42no—
275Dic61C4×SD1612yes—
276Dic61#55211yescharacteristic
277Dic61#55611yescharacteristic
278C2×#476—611no—
279#2369—611no—
280#2354—611no—
281#1990—611no—
282#1948—611no—
283#1934—612no—
284#1926—612no—
285#1920—612no—
286C4×#106—612no—
287C8×C4⋊C4—611no—
288C4×#102—611no—
289C2×C61—81no—
290C2×C61#236211yescharacteristic
291C2×C61C2×C4×SD1611yescharacteristic
292C2×C61#294611yescharacteristic
293D61—81no—
294D61#333511yescharacteristic
295D61#334111yescharacteristic
296D61#336011yescharacteristic
297D61#336211yescharacteristic
298C2×C4⋊Q8—611no—
299C2×C4×D4—611no—
300C2×#117—1221no—
301#476—611no—
302#476—611no—
303#476—612no—
304#476—611no—
305#476—612no—
306#476#692311yescharacteristic, complemented
307C2×#106—612no—
308C2×#106—612no—
309C2×C4⋊C8—611no—
310C2×#102—612no—
311C2×#102—611no—
312C2×#102—611no—
313C2×C4×C8—612no—
314C2×C4×C8—611no—
315#391—1222no—
316#390—612no—
317#389—612no—
318#389—1221no—
319#388—1222no—
320#384—611no—
321#380—1221no—
322C4×C4⋊C4—611no—
323C4×C4⋊C4—612no—
324C4×C4⋊C4—612no—
325C4×C4⋊C4—611no—
326C4×C4⋊C4—1221no—
327#340—611no—
328#340—612no—
329#340—612no—
330#340—612no—
331#340—611no—
332#336—612no—
333C61#2306111yescharacteristic, complemented, Sylow
334C22×Q8—1221no—
335C22×D4—611no—
336C23×C4—1221no—
337C22×C8—611no—
338C22×C8—612no—
339C22×C8—611no—
340C4⋊Q8—612no—
341C4⋊Q8—611no—
342C4⋊Q8—1221no—
343C4⋊Q8—1221no—
344C4⋊Q8C2×#692311yescharacteristic
345C4×D4—611no—
346C4×D4—611no—
347C4×D4—611no—
348C4×D4—1221no—
349C4×D4—1221no—
350C4×D4C2×#692311yescharacteristic
351#117—2444no—
352C2×C4⋊C4—611no—
353C2×C4⋊C4—611no—
354C2×C4⋊C4—611no—
355C2×C4⋊C4—612no—
356C2×C4⋊C4—611no—
357C2×C4⋊C4—612no—
358C2×C4⋊C4—611no—
359C2×C4⋊C4—1221no—
360C2×C4⋊C4—1221no—
361C2×C4⋊C4—1222no—
362C2×C4⋊C4—1221no—
363C2×C22⋊C4—1221no—
364C2×C22⋊C4—1222no—
365C2×C22⋊C4—1222no—
366C2×C42—611no—
367C2×C42—612no—
368C2×C42—1222no—
369C2×C42—1222no—
370C2×C42—1221no—
371#106—612no—
372#106—612no—
373#106—612no—
374#106—1224no—
375#106C2×#692312yes—
376C4⋊C8—612no—
377C4⋊C8—611no—
378C4⋊C8C2×#692311yescharacteristic
379#102—612no—
380#102—612no—
381#102—612no—
382#102—1222no—
383#102—1222no—
384#102C2×#692312yes—
385C4×C8—1221no—
386C4×C8—1221no—
387C4×C8—1224no—
388#95—612no—
389#95—1222no—
390#95—1222no—
391#95—1221no—
392#95—1222no—
393C24—1221no—
394C2×Q8—21no—
395C2×Q8—1222no—
396C2×Q8—1221no—
397C2×Q8—4881no—
398C2×Q8—4881no—
399C2×D4—611no—
400C2×D4—1221no—
401C2×D4—1221no—
402C2×D4—1221no—
403C2×D4—1221no—
404C2×D4—1221no—
405C2×D4C22×#692311yescharacteristic
406C22×C4—21no—
407C22×C4—611no—
408C22×C4—611no—
409C22×C4—612no—
410C22×C4—611no—
411C22×C4—612no—
412C22×C4—1221no—
413C22×C4—1221no—
414C22×C4—1222no—
415C22×C4—1221no—
416C22×C4—1221no—
417C22×C4—1221no—
418C22×C4—1221no—
419C22×C4—1222no—
420C22×C4—2442no—
421C22×C4—2442no—
422C2×C8—612no—
423C2×C8—612no—
424C2×C8—1222no—
425C2×C8—1222no—
426C2×C8—1221no—
427C2×C8—1221no—
428C2×C8—1221no—
429C2×C8—1221no—
430C2×C8—1221no—
431C2×C8C22×#692312yes—
432C4⋊C4—21no—
433C4⋊C4—614no—
434C4⋊C4—612no—
435C4⋊C4—612no—
436C4⋊C4—611no—
437C4⋊C4—1221no—
438C4⋊C4—1222no—
439C4⋊C4—1222no—
440C4⋊C4—1222no—
441C4⋊C4—1221no—
442C4⋊C4—1221no—
443C4⋊C4—1222no—
444C4⋊C4—1221no—
445C4⋊C4—1222no—
446C4⋊C4—1221no—
447C4⋊C4—4881no—
448C4⋊C4—4882no—
449C4⋊C4—2442no—
450C4⋊C4C22×#692312yes—
451C4⋊C4C22×#692311yescharacteristic
452C22⋊C4—21no—
453C22⋊C4—1221no—
454C22⋊C4—1221no—
455C22⋊C4—1221no—
456C22⋊C4—4882no—
457C22⋊C4—2444no—
458C42—611no—
459C42—611no—
460C42—611no—
461C42—1222no—
462C42—1222no—
463C42—4882no—
464C42—2444no—
465C42—2442no—
466C42C22×#692311yescharacteristic
467C23—21no—
468C23—611no—
469C23—1221no—
470C23—2441no—
471C23—2441no—
472C23—2441no—
473C23—2441no—
474C23—2441no—
475C23—2441no—
476Q8—81no—
477Q8—4881no—
478Q8—4882no—
479D4—21no—
480D4—21no—
481D4—1221no—
482D4—1221no—
483D4—2441no—
484D4—2441no—
485C2×C4—21no—
486C2×C4—22no—
487C2×C4—42no—
488C2×C4—611no—
489C2×C4—611no—
490C2×C4—612no—
491C2×C4—612no—
492C2×C4—611no—
493C2×C4—611no—
494C2×C4—611no—
495C2×C4—611no—
496C2×C4—614no—
497C2×C4—611no—
498C2×C4—1221no—
499C2×C4—1221no—
500C2×C4—1222no—
501C2×C4—1222no—
502C2×C4—1222no—
503C2×C4—1222no—
504C2×C4—1222no—
505C2×C4—1222no—
506C2×C4—1221no—
507C2×C4—1222no—
508C2×C4—2442no—
509C2×C4—2442no—
510C2×C4—2441no—
511C2×C4—2442no—
512C2×C4—2441no—
513C2×C4—2442no—
514C2×C4—2442no—
515C2×C4—2441no—
516C2×C4—2442no—
517C2×C4—2441no—
518C2×C4—2442no—
519C2×C4—2441no—
520C2×C4—2441no—
521C2×C4—2441no—
522C2×C4—2442no—
523C2×C4D4×#692311yescharacteristic
524C2×C4D4×#692311yescharacteristic
525C2×C4C23×#692311yescharacteristic
526C8—22no—
527C8—1222no—
528C8—1222no—
529C8—2441no—
530C22—41no—
531C22—41no—
532C22—41no—
533C22—611no—
534C22—611no—
535C22—611no—
536C22—611no—
537C22—611no—
538C22—611no—
539C22—4881no—
540C22—4881no—
541C22—4881no—
542C22—4881no—
543C22—2441no—
544C22—2441no—
545C22—2441no—
546C22C2×D4×#692311yescharacteristic
547C4—21no—
548C4—41no—
549C4—42no—
550C4—612no—
551C4—611no—
552C4—611no—
553C4—1224no—
554C4—1221no—
555C4—4882no—
556C4—4881no—
557C4—2442no—
558C4—2442no—
559C4—2441no—
560C4—2442no—
561C4SD16×#692312yes—
562C4#41×#692311yescharacteristic
563C4#1684083011yescharacteristic
564C2—81no—
565C2—611no—
566C2—611no—
567C2—611no—
568C2—611no—
569C2—4881no—
570C2#2146234511yescharacteristic
571C2C2×SD16×#692311yescharacteristic
572C2#2146257611yescharacteristic
subgroupG/H
centerC22C2×D4×#6923
commutatorC2×C4×C61C23×C4
FrattiniC2×C4C23×#6923
FittingC61×#476C4
socleC22×C61C2×C4×D4
prime radicalC2×D4×D61C23
cosocleC2×C4×D61C24
hypercenter#476#6923
hypocenterC61#23061
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #476

Complements

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

NH ≅ G/Ntype
1. #21462352C2semidirect
3. #21462000C2semidirect
4. #21461931C2semidirect
7. #21461882C2semidirect
8. #106×#6923C2semidirect
9. #21461612C2semidirect
38. #16840053C4semidirect
39. #16840052C4semidirect
40. #16840051C4semidirect
41. #16840050C4semidirect
42. #16840049C4semidirect
46. C61×#476C4semidirect
306. #476#6923semidirect
333. C61#23061semidirect

Maximal subgroups

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

subgroupindex
#214623522
#214623372
#214620002
#214619312
#214619092
#214618972
#214618822
#106×#69232
#214616122
C4×#145198392
#476×D612
#2306161