Finite Group Databasebeta

← #29663268

Subgroups — #29663268

Order 15616.

570 conjugacy classes of subgroups.

The subgroup lattice has 544 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21457163C211yesmaximal, characteristic
2#21457162C211yesmaximal, characteristic
3#21456328C211yesmaximal, characteristic
4#21454748C211yesmaximal, characteristic
5#21445346C211yesmaximal, characteristic, complemented
6#21445345C211yesmaximal, characteristic, complemented
7#21445336C211yesmaximal, characteristic, complemented
8#21445335C211yesmaximal, characteristic, complemented
9#21445202C211yesmaximal, characteristic, complemented
10#21445201C211yesmaximal, characteristic, complemented
11#21445198C211yesmaximal, characteristic, complemented
12#334×D61C211yesmaximal, characteristic
13#21444508C211yesmaximal, characteristic
14#21444506C211yesmaximal, characteristic
15C61×#1672C211yesmaximal, characteristic, complemented
16#16840317C2211yescharacteristic
17#16840317C2211yescharacteristic
18#16840316C2211yescharacteristic
19#16840165C2211yescharacteristic
20#16840165C2211yescharacteristic
21#16840162—21no—
22#16840162—21no—
23#16840160C2211yescharacteristic
24#16840058C2211yescharacteristic
25#16840057C2211yescharacteristic
26#16840056—21no—
27#16840055—21no—
28#16840054C2211yescharacteristic
29#16840004—21no—
30#16840002—21no—
31#16840002—21no—
32#16840000—21no—
33#16839973C2211yescharacteristic
34#16839972C2211yescharacteristic
35#16839971—21no—
36#16839970—21no—
37#16839969C2211yescharacteristic
38#16839893C2211yescharacteristic
39#16839891—21no—
40#16839889C2211yescharacteristic
41C8×Dic122—21no—
42C8×Dic122C2211yescharacteristic
43#16839797—21no—
44#16839797—21no—
45#16839787—21no—
46#16839784C2211yescharacteristic
47C4⋊C8×D61C2211yescharacteristic
48C4⋊C8×D61C2211yescharacteristic
49#16839716—21no—
50C4×C8×D61C2211yescharacteristic
51#16839513C2211yescharacteristic
52#16839512C2211yescharacteristic
53#16839512C2211yescharacteristic
54#16839511C2211yescharacteristic
55#16839510C2211yescharacteristic
56#16839510C2211yescharacteristic
57#16839509C2211yescharacteristic
58#16839508C2211yescharacteristic
59#16839507C2211yescharacteristic
60C61×#498C2211yescharacteristic
61C61×#477C2211yescharacteristic
62C61×#462C2211yescharacteristic
63C8×Q8×C61C2211yescharacteristic
64C61×#334C2211yescharacteristic
65C61×#333C2211yescharacteristic
66C61×#332C2211yescharacteristic
67C2×Dic244—21no—
68C2×#11758202—21no—
69C2×#11758202—21no—
70C2×#11758189—21no—
71C2×C8×D61—21no—
72C2×C8×D61—21no—
73C2×C8×D61D411yescharacteristic
74C2×C8×D61D411yescharacteristic
75#14519777C2311yescharacteristic
76#14519777C2311yescharacteristic
77#14519776D411yescharacteristic
78#14519775D411yescharacteristic
79#14519768—21no—
80#14519768—21no—
81#14519767—21no—
82#14519757—21no—
83#14519757—21no—
84#14519757D411yescharacteristic
85#14519755—21no—
86#14519753—21no—
87#14519753—21no—
88#14519753D411yescharacteristic
89Q8×Dic61—21no—
90Q8×Dic61—21no—
91C4×Dic122—21no—
92C4×Dic122C2311yescharacteristic
93#14519742—21no—
94#14519742—21no—
95#14519735—21no—
96#14519735—21no—
97#14519734—21no—
98C42×D61C2311yescharacteristic
99#14519711—21no—
100#14519710—21no—
101#14519710—21no—
102#14519710—21no—
103#14519710—21no—
104#14519710—21no—
105#14519710—21no—
106#14519709—21no—
107#14519709—21no—
108C4⋊#92418—21no—
109C4⋊#92418C2311yescharacteristic
110C4⋊#92418C2311yescharacteristic
111C4⋊#92418C2311yescharacteristic
112#14519706—21no—
113#14519706—21no—
114#14519706—21no—
115#14519706—21no—
116#14519706—21no—
117#14519702—21no—
118#14519702—21no—
119#14519701—21no—
120#14519701—21no—
121#14519700—21no—
122#14519700—21no—
123#14519688—21no—
124#14519687—21no—
125C4×#92418C2311yescharacteristic
126C8×Dic61—21no—
127C8×Dic61D411yescharacteristic
128C8×Dic61D411yescharacteristic
129C2×Q16×C61—21no—
130C4⋊Q8×C61C2311yescharacteristic
131C4⋊Q8×C61C2311yescharacteristic
132C4×Q8×C61C2311yescharacteristic
133#107×C61—21no—
134#106×C61—21no—
135#106×C61—21no—
136C4⋊C8×C61—21no—
137C4⋊C8×C61C2311yescharacteristic
138C4⋊C8×C61C2311yescharacteristic
139C4⋊C8×C61C2311yescharacteristic
140#103×C61—21no—
141#103×C61—21no—
142C4×C8×C61—21no—
143C4×C8×C61C2311yescharacteristic
144C2×Q8×C61—21no—
145C2×Q8×C61—21no—
146C2×Q8×C61C2×D411yescharacteristic
147C2×C4×D61C2×D411yescharacteristic
148C2×C4×D61C2×D411yescharacteristic
149C2×C4×D61#4111yescharacteristic
150C2×Dic122—21no—
151C2×Dic122—21no—
152C2×Dic122—21no—
153C2×Dic122C2×D411yescharacteristic
154Q16×C61—81no—
155C2×C8×C61—21no—
156C2×C8×C61—21no—
157C2×C8×C61—21no—
158C2×C8×C61C2×D411yescharacteristic
159C2×C8×C61C2×D411yescharacteristic
160C4⋊C4×C61—21no—
161C4⋊C4×C61—21no—
162C4⋊C4×C61—21no—
163C4⋊C4×C61—21no—
164C4⋊C4×C61—21no—
165C4⋊C4×C61C2×D411yescharacteristic
166C42×C61—21no—
167C42×C61C2411yescharacteristic
168#11758202—81no—
169#11758202—81no—
170#11758198—21no—
171#11758198—21no—
172#11758198—41no—
173#11758198—41no—
174C4⋊Dic61—21no—
175C4⋊Dic61—21no—
176C4⋊Dic61—21no—
177C4⋊Dic61—21no—
178C4⋊Dic61—21no—
179C4⋊Dic61—41no—
180C4⋊Dic61—41no—
181C4⋊Dic61C2×D411yescharacteristic
182#11758196—21no—
183#11758196—21no—
184#11758196—21no—
185#11758196—21no—
186#11758196—21no—
187#11758196—41no—
188#11758196—41no—
189C4×Dic61—21no—
190C4×Dic61—41no—
191C4×Dic61—41no—
192C4×Dic61C2×D411yescharacteristic
193C4×Dic61C2×D411yescharacteristic
194C4×Dic61#4111yescharacteristic
195C2×#92418—21no—
196C2×#92418—21no—
197C2×#92418—21no—
198C2×#92418C2×D411yescharacteristic
199C2×#92418C2×D411yescharacteristic
200Dic244—81no—
201#11758189—42no—
202C8×D61—22no—
203C8×D61—81no—
204C8×D61—81no—
205C8×D61Q1614yescomplemented
206C22×D61#12111yescharacteristic
207Q8×C61—21no—
208Q8×C61—21no—
209Q8×C61—41no—
210Q8×C61—41no—
211Q8×C61—81no—
212C2×C4×C61—21no—
213C2×C4×C61—21no—
214C2×C4×C61—41no—
215C2×C4×C61—41no—
216C2×C4×C61C22×D411yescharacteristic
217C2×C4×C61C22×D411yescharacteristic
218C2×C4×C61C2×#4111yescharacteristic
219C2×Dic61—21no—
220C2×Dic61—21no—
221C2×Dic61—41no—
222C2×Dic61—41no—
223C2×Dic61#12111yescharacteristic
224C2×Dic61#12111yescharacteristic
225C2×Dic61#12111yescharacteristic
226C4×D61—21no—
227C4×D61—21no—
228C4×D61—41no—
229C4×D61C2×Q1612yes—
230C4×D61#13712yes—
231Dic122—21no—
232Dic122—21no—
233Dic122—42no—
234Dic122—41no—
235Dic122—41no—
236Dic122—81no—
237C8×C61—21no—
238C8×C61—41no—
239C8×C61—41no—
240C8×C61—41no—
241C8×C61C2×Q1612yes—
242#92418—21no—
243#92418—41no—
244#92418—41no—
245#92418—41no—
246#92418C2×Q1612yes—
247#19210—611nomaximal, Sylow
248C22×C61C2×#12111yescharacteristic
249C2×D61—21no—
250C2×D61—21no—
251C2×D61#46212yes—
252C4×C61—21no—
253C4×C61—41no—
254C4×C61—41no—
255C4×C61—81no—
256C4×C61—81no—
257C4×C61C22×Q1611yescharacteristic
258C4×C61C2×#13711yescharacteristic
259C4×C61#57711yescharacteristic
260C4×C61#57711yescharacteristic
261Dic61—21no—
262Dic61—21no—
263Dic61—41no—
264Dic61—41no—
265Dic61—81no—
266Dic61—81no—
267Dic61#46212yes—
268#3249—611no—
269#3249—611no—
270#3160—611no—
271#2960—611no—
272#1696—611no—
273#1696—611no—
274#1694—611no—
275#1694—611no—
276#1672—611no—
277#1672—611no—
278#1672—611no—
279#1672D6111yescharacteristic, complemented
280C2×#334—611no—
281#1550—611no—
282#1550—611no—
283C2×C61C2×#46211yescharacteristic
284C2×C61#306011yescharacteristic
285C2×C61#306711yescharacteristic
286D61—22no—
287#533—611no—
288#533—611no—
289#533—611no—
290#500—611no—
291#500—611no—
292#499—1221no—
293#499—1221no—
294#498—611no—
295#498C2×D6111yescharacteristic
296#477—611no—
297#477—611no—
298#477—611no—
299#477—1221no—
300#477—1221no—
301#477C2×D6111yescharacteristic
302#467—1221no—
303#467—1221no—
304#467—1221no—
305#467—1221no—
306#462—611no—
307#462—611no—
308#462—611no—
309#462—1221no—
310#462—1221no—
311#462C2×D6111yescharacteristic
312#446—611no—
313#446—611no—
314#446—1221no—
315C8×Q8—611no—
316C8×Q8—1221no—
317C8×Q8C2×D6111yescharacteristic
318#427—1221no—
319#427—1221no—
320#424—1221no—
321#423—611no—
322C2×C4⋊C8—611no—
323C2×C4⋊C8—611no—
324#405—1221no—
325C2×C4×C8—611no—
326#334—611no—
327#334—611no—
328#334—611no—
329#334C2×D6111yescharacteristic
330#333—611no—
331#333—611no—
332#333—611no—
333#333C2×D6111yescharacteristic
334#332—611no—
335#332—611no—
336#332—611no—
337#332C2×D6111yescharacteristic
338C61#1921011yescharacteristic, complemented, Sylow
339C2×Q16—21no—
340C2×Q16—1221no—
341C2×Q16—1221no—
342C2×Q16—1221no—
343C2×M16—1221no—
344C22×C8—611no—
345C22×C8—611no—
346C22×C8—1221no—
347C22×C8—1221no—
348C4⋊Q8—611no—
349C4⋊Q8—611no—
350C4⋊Q8—611no—
351C4⋊Q8—611no—
352C4⋊Q8C22×D6111yescharacteristic
353C4⋊Q8C22×D6111yescharacteristic
354#125—1221no—
355#125—1221no—
356#125—1221no—
357C22⋊Q8—611no—
358C22⋊Q8—611no—
359C22⋊Q8—1221no—
360C22⋊Q8—1221no—
361C22⋊Q8—1221no—
362C22⋊Q8—1221no—
363C22⋊Q8—1221no—
364C4×Q8—611no—
365C4×Q8—1221no—
366C4×Q8—1221no—
367C4×Q8—1221no—
368C4×Q8—1221no—
369C4×Q8—1221no—
370C4×Q8C22×D6111yescharacteristic
371#117—1221no—
372#117—1221no—
373#117—1221no—
374C2×C42—611no—
375#107—21no—
376#107—1221no—
377#107—1221no—
378#107—1221no—
379#107—1221no—
380#107—1221no—
381#107—1221no—
382#107—1221no—
383#106—21no—
384#106—21no—
385#106—1221no—
386#106—1221no—
387C4⋊C8—21no—
388C4⋊C8—611no—
389C4⋊C8—611no—
390C4⋊C8—611no—
391C4⋊C8—1221no—
392C4⋊C8—1221no—
393C4⋊C8—1221no—
394C4⋊C8—1221no—
395C4⋊C8—1221no—
396C4⋊C8—1221no—
397C4⋊C8C22×D6111yescharacteristic
398C4⋊C8C22×D6111yescharacteristic
399C4⋊C8C22×D6111yescharacteristic
400#103—21no—
401#103—21no—
402#103—1221no—
403#103—1221no—
404#103—1221no—
405#103—1221no—
406#103—1221no—
407#103—1221no—
408#97—1221no—
409#97—1221no—
410C4×C8—21no—
411C4×C8—611no—
412C4×C8—611no—
413C4×C8—611no—
414C4×C8—1221no—
415C4×C8C22×D6111yescharacteristic
416C2×Q8—21no—
417C2×Q8—21no—
418C2×Q8—611no—
419C2×Q8—1221no—
420C2×Q8—1221no—
421C2×Q8—1221no—
422C2×Q8D4×D6111yescharacteristic
423C22×C4—611no—
424C22×C4—611no—
425C22×C4—611no—
426Q16—81no—
427Q16—4881no—
428Q16—4881no—
429Q16—4881no—
430M16—2442no—
431C2×C8—21no—
432C2×C8—21no—
433C2×C8—21no—
434C2×C8—611no—
435C2×C8—614no—
436C2×C8—611no—
437C2×C8—1222no—
438C2×C8—1221no—
439C2×C8—1221no—
440C2×C8—1221no—
441C2×C8—4881no—
442C2×C8—4881no—
443C2×C8D4×D6111yescharacteristic
444C2×C8D4×D6111yescharacteristic
445C4⋊C4—21no—
446C4⋊C4—21no—
447C4⋊C4—21no—
448C4⋊C4—21no—
449C4⋊C4—21no—
450C4⋊C4—611no—
451C4⋊C4—1221no—
452C4⋊C4—1221no—
453C4⋊C4—1221no—
454C4⋊C4—1221no—
455C4⋊C4—1221no—
456C4⋊C4—1221no—
457C4⋊C4—1221no—
458C4⋊C4—1221no—
459C4⋊C4—1221no—
460C4⋊C4—1221no—
461C4⋊C4—2441no—
462C4⋊C4—2441no—
463C4⋊C4—2441no—
464C4⋊C4—2441no—
465C4⋊C4D4×D6111yescharacteristic
466C22⋊C4—1221no—
467C22⋊C4—1221no—
468C22⋊C4—2441no—
469C22⋊C4—2441no—
470C42—21no—
471C42—611no—
472C42—611no—
473C42—611no—
474C42—1221no—
475C42—2441no—
476C42—2441no—
477C42C23×D6111yescharacteristic
478C23—611no—
479Q8—21no—
480Q8—21no—
481Q8—41no—
482Q8—41no—
483Q8—81no—
484Q8—1221no—
485Q8—1221no—
486Q8—4881no—
487Q8—2442no—
488Q8—2441no—
489Q8—2441no—
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—41no—
493C2×C4—41no—
494C2×C4—611no—
495C2×C4—612no—
496C2×C4—612no—
497C2×C4—611no—
498C2×C4—611no—
499C2×C4—1221no—
500C2×C4—1221no—
501C2×C4—1221no—
502C2×C4—1221no—
503C2×C4—2441no—
504C2×C4—2441no—
505C2×C4—2441no—
506C2×C4C2×D4×D6111yescharacteristic
507C2×C4C2×D4×D6111yescharacteristic
508C2×C4#41×D6111yescharacteristic
509C8—21no—
510C8—41no—
511C8—41no—
512C8—41no—
513C8—612no—
514C8—1221no—
515C8—2441no—
516C8—2441no—
517C8—2441no—
518C8Q16×D6112yes—
519C22—612no—
520C22—1221no—
521C22—1221no—
522C22#121×D6111yescharacteristic
523C4—21no—
524C4—41no—
525C4—41no—
526C4—81no—
527C4—81no—
528C4—612no—
529C4—1221no—
530C4—1221no—
531C4—4881no—
532C4—4881no—
533C4—2441no—
534C4—2441no—
535C4C2×Q16×D6111yescharacteristic
536C4#137×D6111yescharacteristic
537C4#1684056811yescharacteristic
538C4#1684056911yescharacteristic
539C2—1222no—
540C2#462×D6111yescharacteristic
541C2#2145547611yescharacteristic
542C2#2145553411yescharacteristic
subgroupG/H
centerC22#121×D61
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#1672C2
socleC22×C61C2×#121
prime radicalC22×D61#121
hypercenter#1672D61
hypocenterC61#19210
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #1672

Complements

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

NH ≅ G/Ntype
5. #21445346C2semidirect
6. #21445345C2semidirect
7. #21445336C2semidirect
8. #21445335C2semidirect
9. #21445202C2semidirect
10. #21445201C2semidirect
11. #21445198C2semidirect
15. C61×#1672C2semidirect
205. C8×D61Q16semidirect
279. #1672D61semidirect
338. C61#19210semidirect

Maximal subgroups

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

subgroupindex
#214571632
#214571622
#214563282
#214547482
#214453462
#214453452
#214453362
#214453352
#214452022
#214452012
#214451982
#334×D612
#214445082
#214445062
C61×#16722
#1921061