Finite Group Databasebeta

← #29836832

Subgroups — #29836832

Order 15616.

654 conjugacy classes of subgroups.

The subgroup lattice has 560 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21458164C211yesmaximal, characteristic, complemented
2D8×C22⋊D61C211yesmaximal, characteristic, complemented
3#21448041C211yesmaximal, characteristic, complemented
4#21447999C211yesmaximal, characteristic, complemented
5#21447998C211yesmaximal, characteristic, complemented
6#21447954C211yesmaximal, characteristic, complemented
7#21447905C211yesmaximal, characteristic, complemented
8#21447904C211yesmaximal, characteristic, complemented
9#21447853C211yesmaximal, characteristic, complemented
10#21447732C211yesmaximal, characteristic, complemented
11#21447731C211yesmaximal, characteristic, complemented
12#21447663C211yesmaximal, characteristic, complemented
13C16×C22⋊D61C211yesmaximal, characteristic, complemented
14#357×D61C211yesmaximal, characteristic, complemented
15C2×C61×#357C211yesmaximal, characteristic, complemented
16C2×D8×D61C2211yescharacteristic
17#16840413—21no—
18D4×C22⋊D61—21no—
19C2×D976C2211yescharacteristic, complemented
20C2×#14519718C2211yescharacteristic
21C2×#14519718C2211yescharacteristic, complemented
22C2×C16×D61C2211yescharacteristic, complemented
23#16840166C2211yescharacteristic, complemented
24#16840137C2211yescharacteristic
25#16840077—21no—
26#16840075—21no—
27#16840041—21no—
28#16839999C2211yescharacteristic, complemented
29#16839998C2211yescharacteristic
30#16839998C2211yescharacteristic, complemented
31#16839995C2211yescharacteristic
32#16839955—21no—
33#16839896—21no—
34#16839894—21no—
35D8×Dic61C2211yescharacteristic
36#16839844C2211yescharacteristic, complemented
37C8×C22⋊D61C2211yescharacteristic
38#107×D61C2211yescharacteristic, complemented
39#16839588C2211yescharacteristic, complemented
40#16839587C2211yescharacteristic, complemented
41#16839587C2211yescharacteristic, complemented
42#16839580C2211yescharacteristic, complemented
43#16839564C2211yescharacteristic, complemented
44#16839564C2211yescharacteristic, complemented
45#16839563C2211yescharacteristic
46#16839563C2211yescharacteristic, complemented
47#16839562C2211yescharacteristic, complemented
48#16839562C2211yescharacteristic, complemented
49C22⋊#11758185C2211yescharacteristic, complemented
50#16839541C2211yescharacteristic, complemented
51C16×Dic61C2211yescharacteristic, complemented
52C22×D8×C61C2211yescharacteristic
53C22×C16×C61C2211yescharacteristic, complemented
54C2×#107×C61C2211yescharacteristic, complemented
55C61×#357C2211yescharacteristic
56C61×#357C2211yescharacteristic
57C61×#357C2211yescharacteristic, complemented
58C61×#357C2211yescharacteristic, complemented
59C2×#11758222—41no—
60C22×C22⋊D61—41no—
61C2×D4×D61—21no—
62C2×D488C2311yescharacteristic
63C2×D488C2311yescharacteristic
64C2×#11758199—21no—
65C2×#11758199—21no—
66D8×D61—22no—
67D8×D61—41no—
68C2×C8×D61C2311yescharacteristic
69#14519776—21no—
70#14519773—21no—
71#14519762—41no—
72#14519760—41no—
73#14519755—41no—
74#14519752—21no—
75#14519751—21no—
76#14519751—21no—
77#14519749—41no—
78#14519748—21no—
79#14519746—41no—
80#14519745—41no—
81D4×Dic61—21no—
82#14519739—21no—
83C4×C22⋊D61D411yescharacteristic
84C4⋊C4×D61—21no—
85D976—41no—
86#14519718—22no—
87#14519718—41no—
88C2×#11758185C2311yescharacteristic
89C2×#11758185C2311yescharacteristic
90C16×D61—41no—
91#14519711C2311yescharacteristic
92#14519711C2311yescharacteristic
93#14519710—21no—
94#14519710—21no—
95#14519706D411yescharacteristic
96#14519699—21no—
97#14519699—21no—
98#14519698—21no—
99#14519698—21no—
100#14519697—21no—
101#14519697—21no—
102C22⋊#92418D411yescharacteristic
103#14519689D411yescharacteristic
104C8×Dic61C2311yescharacteristic
105C22×D4×C61—21no—
106C2×D8×C61—22no—
107C2×D8×C61—21no—
108C2×D8×C61—41no—
109C2×D8×C61C2311yescharacteristic
110C2×D8×C61C2311yescharacteristic
111C22×C8×C61C2311yescharacteristic
112C2×C4⋊C4×C61—21no—
113C2×C16×C61—41no—
114C2×C16×C61C2311yescharacteristic
115C2×C16×C61C2311yescharacteristic
116#107×C61—21no—
117#107×C61C2311yescharacteristic
118#107×C61C2311yescharacteristic
119C24×C61—41no—
120C23×D61—41no—
121C2×D4×C61—21no—
122C2×D4×C61—21no—
123C2×D4×C61—42no—
124C2×D4×C61—41no—
125C2×D4×C61—81no—
126C22×C4×C61—41no—
127C22×C4×C61C2×D411yescharacteristic
128C2×C22⋊D61—41no—
129C2×C22⋊D61—41no—
130C2×C22⋊D61—41no—
131C2×C22⋊D61—41no—
132C2×C22⋊D61—41no—
133C2×C22⋊D61—41no—
134C2×C22⋊D61—84no—
135C2×C22⋊D61D811yescharacteristic
136C22×Dic61—41no—
137D4×D61—42no—
138D4×D61—81no—
139#11758222—84no—
140C2×D244—21no—
141C2×D244—21no—
142C2×D244—41no—
143C2×C4×D61—41no—
144C2×C4×D61—41no—
145C2×C4×D61C2×D411yescharacteristic
146C2×Dic122—41no—
147D8×C61—21no—
148D8×C61—21no—
149D8×C61—21no—
150D8×C61—42no—
151D8×C61#4112yescomplemented
152C2×C8×C61—21no—
153C2×C8×C61—21no—
154C2×C8×C61C2×D411yescharacteristic
155C2×C8×C61C2411yescharacteristic
156C4⋊C4×C61—21no—
157C4⋊C4×C61—21no—
158C4⋊C4×C61—41no—
159C22⋊Dic61—41no—
160C22⋊Dic61—41no—
161C22⋊Dic61D811yescharacteristic
162#11758199—42no—
163#11758199—81no—
164#11758198—41no—
165#11758198—41no—
166#11758198—41no—
167#11758198—41no—
168#11758198D811yescharacteristic
169C4⋊Dic61—21no—
170C4⋊Dic61—21no—
171#11758196—41no—
172#11758196—41no—
173#11758196D811yescharacteristic
174C4×Dic61C2×D411yescharacteristic
175C2×#92418C2×D411yescharacteristic
176C2×#92418C2×D411yescharacteristic
177D488—21no—
178D488—21no—
179D488—21no—
180D488—21no—
181C8×D61—21no—
182C8×D61—21no—
183C16×C61—21no—
184C16×C61—21no—
185#11758185—21no—
186#11758185—21no—
187C23×C61—41no—
188C23×C61—41no—
189C23×C61—82no—
190C23×C61—82no—
191C23×C61—161no—
192C23×C61C2×D811yescharacteristic
193C22×D61—41no—
194C22×D61—41no—
195C22×D61—82no—
196C22×D61—82no—
197C22×D61—161no—
198C22×D61C2×D811yescharacteristic
199D4×C61—22no—
200D4×C61—41no—
201D4×C61—41no—
202D4×C61—41no—
203D4×C61—82no—
204C2×C4×C61—21no—
205C2×C4×C61—21no—
206C2×C4×C61—41no—
207C2×C4×C61—41no—
208C2×C4×C61—82no—
209C2×C4×C61C2×D811yescharacteristic
210C2×C4×C61C22×D411yescharacteristic
211C22⋊D61—82no—
212C22⋊D61—82no—
213C22⋊D61—82no—
214C22⋊D61—82no—
215C22⋊D61—82no—
216C22⋊D61—161no—
217C22⋊D61D1614yescomplemented
218C2×Dic61—41no—
219C2×Dic61—41no—
220C2×Dic61—82no—
221C2×Dic61C2×D811yescharacteristic
222C2×Dic61C2×D811yescharacteristic
223C2×Dic61C2×D811yescharacteristic
224D244—41no—
225D244—41no—
226D244—41no—
227D244—41no—
228D244—82no—
229C4×D61—21no—
230C4×D61—21no—
231C4×D61—82no—
232C4×D61—82no—
233Dic122—82no—
234C8×C61—21no—
235C8×C61C2×#4111yescharacteristic
236C8×C61H(5,𝔽2)11yescharacteristic
237#92418—21no—
238#92418—21no—
239#33861—611nomaximal, Sylow
240C22×C61—21no—
241C22×C61—21no—
242C22×C61—42no—
243C22×C61—81no—
244C22×C61—81no—
245C22×C61—81no—
246C22×C61—81no—
247C22×C61—81no—
248C22×C61—164no—
249C22×C61C2×D1612yes—
250C22×C61C22×D811yescharacteristic
251C2×D61—21no—
252C2×D61—21no—
253C2×D61—81no—
254C2×D61—81no—
255C2×D61—81no—
256C2×D61—81no—
257C2×D61—81no—
258C2×D61—81no—
259C2×D61—164no—
260C2×D61C2×D1612yes—
261C4×C61—21no—
262C4×C61—81no—
263C4×C61—81no—
264C4×C61#54611yescharacteristic
265C4×C61#57711yescharacteristic
266Dic61—21no—
267Dic61—21no—
268Dic61—81no—
269Dic61—81no—
270Dic61C2×D1612yes—
271#3334—611no—
272D4×D8—611no—
273#2240—611no—
274#2232—611no—
275#2232—611no—
276#2224—611no—
277#2215—611no—
278#2215—611no—
279#2206—611no—
280#2184—611no—
281#2184—611no—
282C4×D16—611no—
283D4×C16—611no—
284C2×#357—611no—
285C2×#357D6111yescharacteristic, complemented
286C2×C61—22no—
287C2×C61—82no—
288C2×C61—161no—
289C2×C61#329411yescharacteristic
290C2×C61C22×D1611yescharacteristic
291C2×C61#341611yescharacteristic
292D61—22no—
293D61—161no—
294D61—161no—
295C22×D8—611no—
296C22×D8C2×D6111yescharacteristic
297#547—1221no—
298D42—1221no—
299C2×D16—611no—
300C2×D16—611no—
301C2×D16—611no—
302C22×C16—611no—
303C22×C16C2×D6111yescharacteristic, complemented
304#500—611no—
305#493—611no—
306#480—1221no—
307#480—1221no—
308#474—1221no—
309#466—611no—
310#466—611no—
311#466—611no—
312#466—611no—
313#459—1221no—
314#447—1221no—
315#447—1221no—
316C4×D8—611no—
317C4×D8—611no—
318C8×D4—611no—
319C2×#107—611no—
320C2×#107C2×D6111yescharacteristic, complemented
321#366—611no—
322#366—611no—
323#366—611no—
324C4⋊C16—611no—
325#357—611no—
326#357—611no—
327#357—611no—
328#357—611no—
329#357—611no—
330#357—611no—
331#357C2×D6111yescharacteristic
332#357C2×D6111yescharacteristic
333#357C2×D6111yescharacteristic, complemented
334#357C2×D6111yescharacteristic, complemented
335C22⋊C16—611no—
336C22⋊C16—611no—
337C4×C16—611no—
338C61#3386111yescharacteristic, complemented, Sylow
339C2×#41—2441no—
340C22×D4—21no—
341C22×D4—1221no—
342C22×D4—2441no—
343C2×D8—22no—
344C2×D8—21no—
345C2×D8—41no—
346C2×D8—611no—
347C2×D8—611no—
348C2×D8—1221no—
349C2×D8—1221no—
350C2×D8—1222no—
351C2×D8—2441no—
352C2×D8C22×D6111yescharacteristic
353C2×D8C22×D6111yescharacteristic
354C22×C8—611no—
355C22×C8C22×D6111yescharacteristic
356C4⋊Q8—1221no—
357#127—1221no—
358#123—2441no—
359#123—2441no—
360C22⋊Q8—2441no—
361#121—1221no—
362#121—1221no—
363#121—1221no—
364#121—1221no—
365#121—2441no—
366#120—2441no—
367#120—2441no—
368C4×D4—611no—
369C4×D4—1221no—
370C4×D4—1221no—
371C2×C4⋊C4—21no—
372C2×C4⋊C4—1221no—
373D16—1222no—
374D16—2441no—
375D16—2441no—
376C2×C16—41no—
377C2×C16—611no—
378C2×C16—611no—
379C2×C16—2441no—
380C2×C16C22×D6111yescharacteristic
381C2×C16C22×D6111yescharacteristic
382#107—21no—
383#107—611no—
384#107—611no—
385#107—1221no—
386#107—1221no—
387#107C22×D6111yescharacteristic
388#107C22×D6111yescharacteristic
389C4⋊C8—611no—
390#102—1221no—
391#102—1221no—
392#102—1221no—
393#102—1221no—
394#102—1221no—
395#102—1221no—
396C22⋊C8—611no—
397C22⋊C8—611no—
398C4×C8—611no—
399C24—41no—
400C24—2441no—
401#41—4884no—
402C2×Q8—2441no—
403C2×D4—21no—
404C2×D4—21no—
405C2×D4—42no—
406C2×D4—41no—
407C2×D4—81no—
408C2×D4—611no—
409C2×D4—1221no—
410C2×D4—1221no—
411C2×D4—4881no—
412C2×D4—4884no—
413C2×D4—2441no—
414C2×D4—2441no—
415C2×D4—2441no—
416C2×D4—2441no—
417C2×D4—2442no—
418C2×D4—2441no—
419C2×D4—2441no—
420C2×D4—2441no—
421C22×C4—41no—
422C22×C4—611no—
423C22×C4—2441no—
424C22×C4—2441no—
425C22×C4—2441no—
426C22×C4D4×D6111yescharacteristic
427D8—21no—
428D8—21no—
429D8—21no—
430D8—42no—
431D8—1221no—
432D8—1221no—
433D8—1221no—
434D8—1221no—
435D8—4881no—
436D8—2442no—
437D8#1175822212yescomplemented
438C2×C8—21no—
439C2×C8—21no—
440C2×C8—611no—
441C2×C8—611no—
442C2×C8—1221no—
443C2×C8—1221no—
444C2×C8D4×D6111yescharacteristic
445C2×C8C23×D6111yescharacteristic
446C4⋊C4—21no—
447C4⋊C4—21no—
448C4⋊C4—41no—
449C4⋊C4—611no—
450C4⋊C4—1221no—
451C4⋊C4—1221no—
452C4⋊C4—2441no—
453C4⋊C4—2441no—
454C22⋊C4—611no—
455C22⋊C4—611no—
456C22⋊C4—2441no—
457C22⋊C4—2441no—
458C22⋊C4—2441no—
459C22⋊C4—2441no—
460C22⋊C4—2441no—
461C22⋊C4—2441no—
462C42—611no—
463C16—21no—
464C16—21no—
465C16—1221no—
466C16—1221no—
467C23—41no—
468C23—41no—
469C23—82no—
470C23—82no—
471C23—161no—
472C23—611no—
473C23—9761no—
474C23—4882no—
475C23—4882no—
476C23—2441no—
477C23—2441no—
478C23D8×D6111yescharacteristic
479Q8—4882no—
480D4—22no—
481D4—41no—
482D4—41no—
483D4—41no—
484D4—82no—
485D4—614no—
486D4—9761no—
487D4—4882no—
488D4—4882no—
489D4—4882no—
490D4—4882no—
491D4—4882no—
492D4—4882no—
493D4—2441no—
494D4—2441no—
495D4—2441no—
496D4—2441no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—41no—
500C2×C4—41no—
501C2×C4—82no—
502C2×C4—611no—
503C2×C4—611no—
504C2×C4—611no—
505C2×C4—1221no—
506C2×C4—1221no—
507C2×C4—4882no—
508C2×C4—4882no—
509C2×C4—4882no—
510C2×C4—2441no—
511C2×C4—2441no—
512C2×C4D8×D6111yescharacteristic
513C2×C4C2×D4×D6111yescharacteristic
514C8—21no—
515C8—1221no—
516C8—1221no—
517C8C2×#1175822211yescharacteristic
518C8#1451982211yescharacteristic
519C22—21no—
520C22—21no—
521C22—42no—
522C22—81no—
523C22—81no—
524C22—81no—
525C22—81no—
526C22—81no—
527C22—164no—
528C22—612no—
529C22—1221no—
530C22—1221no—
531C22—9764no—
532C22—4881no—
533C22—4881no—
534C22—4881no—
535C22—4881no—
536C22—4881no—
537C22—4881no—
538C22D16×D6112yes—
539C22C2×D8×D6111yescharacteristic
540C4—21no—
541C4—81no—
542C4—81no—
543C4—612no—
544C4—1221no—
545C4—1221no—
546C4—4881no—
547C4—4881no—
548C4#1684040211yescharacteristic
549C4#1684056811yescharacteristic
550C2—22no—
551C2—82no—
552C2—161no—
553C2—1222no—
554C2—9761no—
555C2—9761no—
556C2#2145762811yescharacteristic
557C2C2×D16×D6111yescharacteristic
558C2#2145896111yescharacteristic
subgroupG/H
centerC22C2×D8×D61
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC2×C61×#357C2
socleC22×C61C22×D8
hypercenterC2×#357D61
hypocenterC61#33861
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ C2×#357

Complements

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

NH ≅ G/Ntype
1. #21458164C2semidirect
2. D8×C22⋊D61C2semidirect
3. #21448041C2semidirect
4. #21447999C2semidirect
5. #21447998C2semidirect
6. #21447954C2semidirect
7. #21447905C2semidirect
8. #21447904C2semidirect
9. #21447853C2semidirect
10. #21447732C2semidirect
11. #21447731C2semidirect
12. #21447663C2semidirect
13. C16×C22⋊D61C2semidirect
14. #357×D61C2semidirect
15. C2×C61×#357C2semidirect
19. C2×D976C22semidirect
21. C2×#14519718C22semidirect
22. C2×C16×D61C22semidirect
23. #16840166C22semidirect
28. #16839999C22semidirect
30. #16839998C22semidirect
36. #16839844C22semidirect
38. #107×D61C22semidirect
39. #16839588C22semidirect
40. #16839587C22semidirect
41. #16839587C22semidirect
42. #16839580C22semidirect
43. #16839564C22semidirect
44. #16839564C22semidirect
46. #16839563C22semidirect
47. #16839562C22semidirect
48. #16839562C22semidirect
49. C22⋊#11758185C22semidirect
50. #16839541C22semidirect
51. C16×Dic61C22semidirect
53. C22×C16×C61C22semidirect
54. C2×#107×C61C22semidirect
57. C61×#357C22semidirect
58. C61×#357C22semidirect
151. D8×C61#41semidirect
217. C22⋊D61D16semidirect
285. C2×#357D61semidirect
303. C22×C16C2×D61semidirect
320. C2×#107C2×D61semidirect
333. #357C2×D61semidirect
334. #357C2×D61semidirect
338. C61#33861semidirect
437. D8#11758222semidirect

Maximal subgroups

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

subgroupindex
#214581642
D8×C22⋊D612
#214480412
#214479992
#214479982
#214479542
#214479052
#214479042
#214478532
#214477322
#214477312
#214476632
C16×C22⋊D612
#357×D612
C2×C61×#3572
#3386161