Finite Group Databasebeta

← #29692163

Subgroups — #29692163

Order 15616.

784 conjugacy classes of subgroups.

The subgroup lattice has 592 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840060C211yesmaximal, characteristic
2#21448378C211yesmaximal, characteristic, complemented
3#21448281C211yesmaximal, characteristic
4#21446775C211yesmaximal, characteristic, complemented
5#21446586C211yesmaximal, characteristic, complemented
6#21446559C211yesmaximal, characteristic, complemented
7#21446557C211yesmaximal, characteristic, complemented
8#21446519C211yesmaximal, characteristic
9#21446448C211yesmaximal, characteristic
10#21446442C211yesmaximal, characteristic
11#21446395C211yesmaximal, characteristic, complemented
12#21446389C211yesmaximal, characteristic, complemented
13#21445827C211yesmaximal, characteristic
14#21445673C211yesmaximal, characteristic, complemented
15C61×#1936C211yesmaximal, characteristic
16C2×#14519768C2211yescharacteristic
17C2×D4×Dic61C2211yescharacteristic
18C2×#14519733—21no—
19#16840060C418yescomplemented
20C2×#14519711C2211yescharacteristic
21C2×#14519710C2211yescharacteristic
22C2×#14519709C2211yescharacteristic
23C2×#14519708C2211yescharacteristic
24C2×#14519706C2211yescharacteristic
25C2×#14519706C2211yescharacteristic
26C2×#14519698C2211yescharacteristic
27C2×#14519698C2211yescharacteristic
28C2×#14519688C2211yescharacteristic
29#16839665C2211yescharacteristic
30#16839660C2211yescharacteristic
31#16839660C2211yescharacteristic
32#16839658C2211yescharacteristic
33#16839656—21no—
34#16839650—21no—
35#16839647—21no—
36#16839643—21no—
37#16839627C2211yescharacteristic
38C4⋊C4×Dic61—21no—
39C4⋊C4×Dic61C2211yescharacteristic
40C4⋊C4×Dic61C2211yescharacteristic
41C4⋊C4×Dic61C2211yescharacteristic
42#16839613—21no—
43C22⋊C4×Dic61—21no—
44C42×Dic61C2211yescharacteristic
45#16839525C2211yescharacteristic
46#16839525C2211yescharacteristic
47#16839524C2211yescharacteristic
48#16839524C2211yescharacteristic
49#16839524C2211yescharacteristic
50#16839524C2211yescharacteristic
51#16839516C2211yescharacteristic
52#16839516C2211yescharacteristic
53C2×#102×C61C2211yescharacteristic
54C2×#102×C61C2211yescharacteristic
55C2×#97×C61C2211yescharacteristic
56C61×#390C2211yescharacteristic
57C61×#389C2211yescharacteristic
58C61×#340C2211yescharacteristic
59C61×#340C2211yescharacteristic
60C23×Dic61—21no—
61C22×#92418C2311yescharacteristic
62C22×#92418C2311yescharacteristic
63#14519768—22no—
64#14519768C2×C414yes—
65D4×Dic61—22no—
66D4×Dic61C2×C414yes—
67#14519733—44no—
68C2×C4⋊Dic61—21no—
69C2×C4⋊Dic61C2311yescharacteristic
70C2×C4⋊Dic61C2311yescharacteristic
71C2×C4⋊Dic61C2311yescharacteristic
72C2×C4⋊Dic61C2311yescharacteristic
73C2×#11758196—21no—
74C2×#11758196—21no—
75C2×#11758196—21no—
76C2×#11758196—21no—
77C2×C22⋊Dic61—21no—
78C2×C22⋊Dic61—21no—
79C2×C22⋊Dic61—21no—
80C2×C4×Dic61—21no—
81C2×C4×Dic61—21no—
82C2×C4×Dic61—21no—
83C2×C4×Dic61D411yescharacteristic
84C2×C4×Dic61D411yescharacteristic
85C2×C4×Dic61D411yescharacteristic
86C2×C4×Dic61D411yescharacteristic
87C2×C4×Dic61C2311yescharacteristic
88C2×C4×Dic61C2311yescharacteristic
89#14519711C2×C414yes—
90#14519710—41no—
91#14519709—41no—
92#14519708C2×C414yes—
93#14519706—22no—
94#14519706C2×C414yes—
95#14519698—41no—
96#14519698C2×C414yes—
97#14519688—41no—
98#14519684—21no—
99#14519684—21no—
100#14519684—21no—
101#14519684—21no—
102#14519684—21no—
103#14519684—21no—
104#14519684—21no—
105#14519684—21no—
106C22×D4×C61C2311yescharacteristic
107C22×C8×C61C2311yescharacteristic
108C22×C8×C61C2311yescharacteristic
109C2×C4⋊C4×C61—21no—
110C2×C4⋊C4×C61C2311yescharacteristic
111C2×C4⋊C4×C61C2311yescharacteristic
112C2×C4⋊C4×C61C2311yescharacteristic
113C2×C22⋊C4×C61—21no—
114C2×C22⋊C4×C61—21no—
115C2×C42×C61C2311yescharacteristic
116#102×C61—41no—
117#102×C61C2×C414yes—
118#97×C61—41no—
119#95×C61—21no—
120C24×C61—21no—
121C2×D4×C61—21no—
122C2×D4×C61—21no—
123C2×D4×C61—21no—
124C2×D4×C61—21no—
125C2×D4×C61—21no—
126C2×D4×C61C22×C412yes—
127C22×C4×C61—21no—
128C22×C4×C61—21no—
129C22×C4×C61—21no—
130C22×C4×C61C2×D411yescharacteristic
131C22×C4×C61C2×D411yescharacteristic
132C22×C4×C61C2411yescharacteristic
133C22×Dic61—21no—
134C22×Dic61—21no—
135C22×Dic61—24no—
136C22×Dic61—21no—
137C22×Dic61—21no—
138C22×Dic61—42no—
139C22×Dic61—42no—
140C22×Dic61C2×D411yescharacteristic
141C22×Dic61C2×D411yescharacteristic
142C22×Dic61C2×D411yescharacteristic
143C22×Dic61C2×D411yescharacteristic
144C2×C8×C61—21no—
145C2×C8×C61—21no—
146C2×C8×C61—21no—
147C2×C8×C61—21no—
148C2×C8×C61—21no—
149C2×C8×C61C22×C412yes—
150C4⋊C4×C61—21no—
151C4⋊C4×C61—21no—
152C4⋊C4×C61—21no—
153C4⋊C4×C61—21no—
154C4⋊C4×C61—21no—
155C4⋊C4×C61—42no—
156C4⋊C4×C61C22×C412yes—
157C22⋊C4×C61—42no—
158C22⋊C4×C61—42no—
159C42×C61—21no—
160C42×C61—21no—
161C22⋊Dic61—24no—
162C22⋊Dic61—42no—
163C22⋊Dic61—42no—
164C4⋊Dic61—21no—
165C4⋊Dic61—21no—
166C4⋊Dic61—24no—
167C4⋊Dic61—21no—
168C4⋊Dic61—21no—
169C4⋊Dic61—21no—
170C4⋊Dic61—21no—
171C4⋊Dic61C22×C412yes—
172C4⋊Dic61C22×C412yes—
173#11758196—24no—
174#11758196—42no—
175#11758196—42no—
176#11758196—42no—
177C4×Dic61—21no—
178C4×Dic61—21no—
179C4×Dic61—22no—
180C4×Dic61—41no—
181C4×Dic61—41no—
182C4×Dic61—41no—
183C4×Dic61—42no—
184C4×Dic61—42no—
185C4×Dic61—42no—
186C4×Dic61C22×C412yes—
187C4×Dic61#4112yes—
188C2×#92418—21no—
189C2×#92418—21no—
190C2×#92418—21no—
191C2×#92418—21no—
192C2×#92418—21no—
193C2×#92418C22×C412yes—
194C23×C61—22no—
195C23×C61—41no—
196C23×C61—41no—
197C23×C61—41no—
198C23×C61—41no—
199C23×C61—41no—
200C23×C61—41no—
201C23×C61C22×D411yescharacteristic
202D4×C61—22no—
203D4×C61—22no—
204D4×C61—41no—
205D4×C61—41no—
206C2×C4×C61—22no—
207C2×C4×C61—21no—
208C2×C4×C61—21no—
209C2×C4×C61—21no—
210C2×C4×C61—21no—
211C2×C4×C61—21no—
212C2×C4×C61—21no—
213C2×C4×C61—41no—
214C2×C4×C61—41no—
215C2×C4×C61—41no—
216C2×C4×C61—41no—
217C2×C4×C61—41no—
218C2×C4×C61—41no—
219C2×C4×C61—41no—
220C2×C4×C61—41no—
221C2×C4×C61C23×C411yescharacteristic
222C2×C4×C61C2×#4111yescharacteristic
223C2×C4×C61H(5,𝔽2)11yescharacteristic
224C2×C4×C612−1+411yescharacteristic
225C2×C4×C612−1+411yescharacteristic
226C2×C4×C612−1+411yescharacteristic
227C2×Dic61—21no—
228C2×Dic61—21no—
229C2×Dic61—21no—
230C2×Dic61—21no—
231C2×Dic61—21no—
232C2×Dic61—21no—
233C2×Dic61—22no—
234C2×Dic61—22no—
235C2×Dic61—21no—
236C2×Dic61—21no—
237C2×Dic61—21no—
238C2×Dic61—48no—
239C2×Dic61—41no—
240C2×Dic61—41no—
241C2×Dic61—41no—
242C2×Dic61—41no—
243C2×Dic61—41no—
244C2×Dic61—41no—
245C2×Dic61—41no—
246C2×Dic61—41no—
247C2×Dic61—41no—
248C2×Dic61—41no—
249C2×Dic61C4×D412yes—
250C2×Dic61C4×D412yes—
251C2×Dic61AGL(1,ℤ/8)12yes—
252C8×C61—22no—
253C8×C61—41no—
254#92418—22no—
255#92418—41no—
256#22120—611nomaximal, Sylow
257C22×C61—42no—
258C22×C61—42no—
259C22×C61—42no—
260C22×C61—81no—
261C22×C61—81no—
262C22×C61—81no—
263C22×C61—81no—
264C22×C61C2×C4×D411yescharacteristic
265C22×C61#53611yescharacteristic
266C22×C61#53711yescharacteristic
267C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
268C22×C61#57511yescharacteristic
269C22×C61#57511yescharacteristic
270C22×C61#57511yescharacteristic
271C4×C61—42no—
272C4×C61—41no—
273C4×C61—41no—
274C4×C61—81no—
275C4×C61—81no—
276C4×C61#51911yescharacteristic
277C4×C61#52011yescharacteristic
278C4×C61#54111yescharacteristic
279C4×C61#54211yescharacteristic
280Dic61—22no—
281Dic61—22no—
282Dic61—22no—
283Dic61—42no—
284Dic61—42no—
285Dic61—41no—
286Dic61—81no—
287Dic61—81no—
288C2×#478—611no—
289C4×#125—611no—
290#2293—611no—
291#1989—611no—
292#1949—611no—
293#1944—611no—
294#1943—611no—
295#1936—611no—
296#1936D6111yescharacteristic
297#1926—611no—
298#1925—611no—
299#1919—611no—
300#1918—611no—
301#1804—611no—
302#1767—611no—
303C2×C61—82no—
304C2×C61#232311yescharacteristic
305C2×C61#294311yescharacteristic
306C2×C61C4×AGL(1,ℤ/8)11yescharacteristic
307C2×C61#323411yescharacteristic
308C2×C61#324011yescharacteristic
309C2×C61#325111yescharacteristic
310C2×C61#325611yescharacteristic
311C2×#125—611no—
312C2×C4×D4—611no—
313C2×#117—1221no—
314#478—618no—
315C2×#107—611no—
316C2×#107—611no—
317C2×#106—611no—
318C2×#106—611no—
319C2×C4⋊C8—611no—
320C2×C4⋊C8—611no—
321C2×#102—611no—
322C2×#102—611no—
323C2×#102C2×D6111yescharacteristic
324C2×#102C2×D6111yescharacteristic
325C2×#97—611no—
326C2×#97C2×D6111yescharacteristic
327#390—611no—
328#390C2×D6111yescharacteristic
329#389—611no—
330#389—611no—
331#389—611no—
332#389C2×D6111yescharacteristic
333#388—1221no—
334#388—1221no—
335#387—1221no—
336#387—1221no—
337#382—611no—
338C4×C4⋊C4—611no—
339C4×C4⋊C4—611no—
340C4×C4⋊C4—611no—
341C4×C4⋊C4—1221no—
342C4×C4⋊C4—1221no—
343C4×C22⋊C4—1221no—
344C43—611no—
345#340—611no—
346#340—611no—
347#340—611no—
348#340—611no—
349#340—611no—
350#340—611no—
351#340C2×D6111yescharacteristic
352#340C2×D6111yescharacteristic
353#336—611no—
354#336—611no—
355C61#2212011yescharacteristic, complemented, Sylow
356C22×D4C22×D6111yescharacteristic
357C23×C4—1221no—
358C22×C8—611no—
359C22×C8—611no—
360C22×C8C22×D6111yescharacteristic
361C22×C8C22×D6111yescharacteristic
362#125—614no—
363#125—1222no—
364C4×D4—614no—
365C4×D4—1222no—
366#117—2444no—
367C2×C4⋊C4—21no—
368C2×C4⋊C4—611no—
369C2×C4⋊C4—611no—
370C2×C4⋊C4—611no—
371C2×C4⋊C4—611no—
372C2×C4⋊C4—1221no—
373C2×C4⋊C4—1221no—
374C2×C4⋊C4—1221no—
375C2×C4⋊C4—1221no—
376C2×C4⋊C4—1221no—
377C2×C4⋊C4C22×D6111yescharacteristic
378C2×C4⋊C4C22×D6111yescharacteristic
379C2×C4⋊C4C22×D6111yescharacteristic
380C2×C22⋊C4—21no—
381C2×C22⋊C4—21no—
382C2×C22⋊C4—1221no—
383C2×C22⋊C4—1221no—
384C2×C22⋊C4—1221no—
385C2×C42—611no—
386C2×C42—611no—
387C2×C42—611no—
388C2×C42—611no—
389C2×C42—611no—
390C2×C42—611no—
391C2×C42—1221no—
392C2×C42—1221no—
393C2×C42—1221no—
394C2×C42C22×D6111yescharacteristic
395#107—614no—
396#107—2441no—
397#106—614no—
398#106—2441no—
399C4⋊C8—614no—
400C4⋊C8—1222no—
401#102—41no—
402#102—614no—
403#102—2441no—
404#102C4×D6114yes—
405#97—41no—
406#97—2441no—
407#95—21no—
408#95—1221no—
409#95—1221no—
410#95—1221no—
411#95—1221no—
412#95—1221no—
413#95—1221no—
414#95—1221no—
415#95—1221no—
416C24—21no—
417C2×D4—21no—
418C2×D4—21no—
419C2×D4—21no—
420C2×D4—21no—
421C2×D4—21no—
422C2×D4C2×C4×D6112yes—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—611no—
427C22×C4—611no—
428C22×C4—611no—
429C22×C4—611no—
430C22×C4—1224no—
431C22×C4—1221no—
432C22×C4—1221no—
433C22×C4—1221no—
434C22×C4—1221no—
435C22×C4—2442no—
436C22×C4—2442no—
437C22×C4D4×D6111yescharacteristic
438C22×C4D4×D6111yescharacteristic
439C22×C4C23×D6111yescharacteristic
440C2×C8—21no—
441C2×C8—21no—
442C2×C8—21no—
443C2×C8—21no—
444C2×C8—21no—
445C2×C8—612no—
446C2×C8—1221no—
447C2×C8—1221no—
448C2×C8—1221no—
449C2×C8—1221no—
450C2×C8—1221no—
451C2×C8C2×C4×D6112yes—
452C4⋊C4—21no—
453C4⋊C4—21no—
454C4⋊C4—21no—
455C4⋊C4—21no—
456C4⋊C4—21no—
457C4⋊C4—42no—
458C4⋊C4—612no—
459C4⋊C4—612no—
460C4⋊C4—1224no—
461C4⋊C4—1224no—
462C4⋊C4—1221no—
463C4⋊C4—1221no—
464C4⋊C4—1221no—
465C4⋊C4—1221no—
466C4⋊C4—1221no—
467C4⋊C4—1221no—
468C4⋊C4—2442no—
469C4⋊C4—2442no—
470C4⋊C4—2442no—
471C4⋊C4C2×C4×D6112yes—
472C22⋊C4—42no—
473C22⋊C4—42no—
474C22⋊C4—1224no—
475C22⋊C4—2442no—
476C22⋊C4—2442no—
477C42—21no—
478C42—21no—
479C42—612no—
480C42—612no—
481C42—1222no—
482C42—1221no—
483C42—1221no—
484C42—2441no—
485C42—2441no—
486C42—2441no—
487C42—2442no—
488C42—2442no—
489C42—2442no—
490C23—22no—
491C23—41no—
492C23—41no—
493C23—41no—
494C23—41no—
495C23—41no—
496C23—41no—
497C23C2×D4×D6111yescharacteristic
498D4—22no—
499D4—22no—
500D4—41no—
501D4—41no—
502C2×C4—22no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—612no—
518C2×C4—612no—
519C2×C4—612no—
520C2×C4—1221no—
521C2×C4—1221no—
522C2×C4—1221no—
523C2×C4—1221no—
524C2×C4—1221no—
525C2×C4—1221no—
526C2×C4—1222no—
527C2×C4—1222no—
528C2×C4—1221no—
529C2×C4—1221no—
530C2×C4—1221no—
531C2×C4—2448no—
532C2×C4—2441no—
533C2×C4—2441no—
534C2×C4—2441no—
535C2×C4—2441no—
536C2×C4—2441no—
537C2×C4—2441no—
538C2×C4—2441no—
539C2×C4—2441no—
540C2×C4—2441no—
541C2×C4—2441no—
542C2×C4C22×C4×D6111yescharacteristic
543C2×C4#41×D6111yescharacteristic
544C2×C4#1451982211yescharacteristic
545C2×C4#1451982411yescharacteristic
546C2×C4#1451982411yescharacteristic
547C2×C4#1451982411yescharacteristic
548C8—22no—
549C8—41no—
550C8—1222no—
551C8—2441no—
552C22—42no—
553C22—42no—
554C22—42no—
555C22—81no—
556C22—81no—
557C22—81no—
558C22—81no—
559C22C4×D4×D6111yescharacteristic
560C22#1684033211yescharacteristic
561C22#1684033911yescharacteristic
562C22AGL(1,ℤ/8)×D6111yescharacteristic
563C22#1684055811yescharacteristic
564C22#1684055811yescharacteristic
565C22#1684056111yescharacteristic
566C4—41no—
567C4—42no—
568C4—41no—
569C4—81no—
570C4—81no—
571C4—1222no—
572C4—1222no—
573C4—1222no—
574C4—4881no—
575C4—4881no—
576C4—2442no—
577C4—2442no—
578C4—2441no—
579C4#1684023911yescharacteristic
580C4#1684024111yescharacteristic
581C4#1684036911yescharacteristic
582C4#1684038011yescharacteristic
583C2—82no—
584C2#2144849411yescharacteristic
585C2#2145461611yescharacteristic
586C2#2145462411yescharacteristic
587C2#2145700411yescharacteristic
588C2#2145705611yescharacteristic
589C2#2145717411yescharacteristic
590C2#2145721611yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1936C2
socleC23×C61C22×D4
prime radicalC22×D4×C61C23
cosocleC22×C4×C61C24
hypercenter#1936D61
hypocenterC61#22120
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1936

Complements

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

NH ≅ G/Ntype
2. #21448378C2semidirect
4. #21446775C2semidirect
5. #21446586C2semidirect
6. #21446559C2semidirect
7. #21446557C2semidirect
11. #21446395C2semidirect
12. #21446389C2semidirect
14. #21445673C2semidirect
19. #16840060C4semidirect
355. C61#22120semidirect

Maximal subgroups

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

subgroupindex
C2×#168400602
#214483782
#214482812
#214467752
#214465862
#214465592
#214465572
#214465192
#214464482
#214464422
#214463952
#214463892
#214458272
#214456732
C61×#19362
#2212061