Finite Group Databasebeta

← #29836986

Subgroups — #29836986

Order 15616.

602 conjugacy classes of subgroups.

The subgroup lattice has 540 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21458167C211yesmaximal, characteristic, complemented
2#21457506C211yesmaximal, characteristic, complemented
3#21448056C211yesmaximal, characteristic, complemented
4#21448017C211yesmaximal, characteristic, complemented
5#21448005C211yesmaximal, characteristic, complemented
6#21447967C211yesmaximal, characteristic, complemented
7#21447922C211yesmaximal, characteristic, complemented
8#21447915C211yesmaximal, characteristic, complemented
9#21447875C211yesmaximal, characteristic, complemented
10#21447744C211yesmaximal, characteristic, complemented
11#21447733C211yesmaximal, characteristic, complemented
12#21447696C211yesmaximal, characteristic, complemented
13#21447532C211yesmaximal, characteristic, complemented
14C2×#16839564C211yesmaximal, characteristic, complemented
15C61×#2168C211yesmaximal, characteristic, complemented
16C2×#14519799C2211yescharacteristic
17C22×D488C2211yescharacteristic
18#16840416—21no—
19#16840408—21no—
20C2×#14519722C2211yescharacteristic, complemented
21C2×D976C2211yescharacteristic
22C2×D976C2211yescharacteristic
23#16840167C2211yescharacteristic, complemented
24#16840146C2211yescharacteristic
25#16840076—21no—
26#16840076—21no—
27#16840043—21no—
28#16840003C2211yescharacteristic
29#16839998C2211yescharacteristic
30#16839998C2211yescharacteristic
31#16839998C2211yescharacteristic, complemented
32#16839978—21no—
33#16839915—21no—
34#16839909—21no—
35C4×D488C2211yescharacteristic
36C4×D488C2211yescharacteristic, complemented
37C2×#14519711C2211yescharacteristic
38C2×#14519711C2211yescharacteristic, complemented
39#16839590C2211yescharacteristic, complemented
40#16839590C2211yescharacteristic, complemented
41#16839590C2211yescharacteristic, complemented
42#16839567C2211yescharacteristic, complemented
43#16839567C2211yescharacteristic, complemented
44#16839567C2211yescharacteristic, complemented
45#16839564C2211yescharacteristic
46#16839564C2211yescharacteristic
47#16839564C2211yescharacteristic
48#16839564C2211yescharacteristic
49#16839564C2211yescharacteristic
50#16839564C2211yescharacteristic
51#16839564C2211yescharacteristic, complemented
52C2×M32×C61C2211yescharacteristic, complemented
53C22×C16×C61C2211yescharacteristic
54C8×D4×C61C2211yescharacteristic
55C61×C4⋊C16C2211yescharacteristic, complemented
56C61×C22⋊C16C2211yescharacteristic
57C61×C22⋊C16C2211yescharacteristic, complemented
58C61×#346C2211yescharacteristic, complemented
59C2×#11758222—21no—
60C2×#11758224—41no—
61C22×D244—21no—
62#14519799—22no—
63#14519799—41no—
64C2×Dic244C2311yescharacteristic
65C2×#11758190—21no—
66C2×#11758190—21no—
67C2×D488—22no—
68C2×D488—21no—
69C2×D488—41no—
70C2×D488C2311yescharacteristic
71C2×D488C2311yescharacteristic
72C2×D488C2311yescharacteristic
73#14519777—21no—
74#14519765—21no—
75#14519762—41no—
76#14519761—41no—
77#14519761—41no—
78#14519757—41no—
79#14519756—21no—
80#14519751—21no—
81#14519751—21no—
82#14519751—21no—
83#14519750—41no—
84#14519745—41no—
85C4×D244—21no—
86C4×D244—21no—
87C2×C4⋊Dic61—21no—
88C2×C4⋊Dic61—21no—
89C22⋊C4×D61—41no—
90#14519722—41no—
91D976—22no—
92D976—41no—
93#14519711—21no—
94#14519711—21no—
95#14519711—21no—
96#14519711C2311yescharacteristic
97#14519711C2311yescharacteristic
98#14519711C2311yescharacteristic
99#14519711C2311yescharacteristic
100#14519702—21no—
101#14519699—21no—
102#14519699—21no—
103#14519699—21no—
104#14519699—21no—
105#14519699—21no—
106C22×C8×C61C2311yescharacteristic
107C22×C8×C61C2311yescharacteristic
108C4×D4×C61D411yescharacteristic
109M32×C61—41no—
110C2×C16×C61—41no—
111C2×C16×C61C2311yescharacteristic
112C2×C16×C61C2311yescharacteristic
113C2×C16×C61C2311yescharacteristic
114C2×C16×C61C2311yescharacteristic
115C4⋊C8×C61D411yescharacteristic
116C22⋊C8×C61D411yescharacteristic
117C22⋊C8×C61D411yescharacteristic
118C4×C8×C61C2311yescharacteristic
119C23×D61—41no—
120C2×D4×C61D811yescharacteristic
121C22×C4×C61C2×D411yescharacteristic
122C22×C4×C61C2×D411yescharacteristic
123C2×C22⋊D61—41no—
124C2×C22⋊D61—41no—
125C2×C22⋊D61—41no—
126C2×C22⋊D61—41no—
127C22×Dic61—41no—
128C22×Dic61—41no—
129#11758224—84no—
130#11758222—42no—
131#11758222—81no—
132C2×D244—21no—
133C2×D244—21no—
134C2×D244—21no—
135C2×D244—42no—
136C2×D244—41no—
137C2×D244—81no—
138C2×C4×D61—41no—
139C2×C4×D61—41no—
140C2×C4×D61—41no—
141C2×Dic122—21no—
142C2×Dic122—41no—
143C2×C8×C61—21no—
144C2×C8×C61—21no—
145C2×C8×C61—21no—
146C2×C8×C61—21no—
147C2×C8×C61C2×D411yescharacteristic
148C2×C8×C61C2×D411yescharacteristic
149C2×C8×C61C2×D411yescharacteristic
150C2×C8×C61C2411yescharacteristic
151C4⋊C4×C61D811yescharacteristic
152C22⋊C4×C61D811yescharacteristic
153C22⋊C4×C61D811yescharacteristic
154C42×C61C2×D411yescharacteristic
155C22⋊Dic61—41no—
156#11758198—41no—
157#11758198—41no—
158#11758198—41no—
159#11758198—41no—
160#11758198—41no—
161#11758198—41no—
162#11758198—41no—
163C4⋊Dic61—21no—
164C4⋊Dic61—21no—
165C4⋊Dic61—21no—
166C4⋊Dic61—21no—
167C4⋊Dic61—41no—
168C4⋊Dic61—41no—
169C4⋊Dic61—41no—
170#11758196—41no—
171Dic244—21no—
172Dic244—21no—
173D488—21no—
174D488—21no—
175D488—21no—
176D488—21no—
177D488—21no—
178D488—42no—
179D488#4112yescomplemented
180#11758190—42no—
181#11758190—81no—
182C16×C61—21no—
183C16×C61—21no—
184C16×C61—21no—
185C16×C61—21no—
186C23×C61C2×D811yescharacteristic
187C23×C61C2×D811yescharacteristic
188C22×D61—41no—
189C22×D61—41no—
190C22×D61—41no—
191C22×D61—82no—
192C22×D61—82no—
193C22×D61—161no—
194D4×C61—22no—
195C2×C4×C61—21no—
196C2×C4×C61—21no—
197C2×C4×C61—21no—
198C2×C4×C61—21no—
199C2×C4×C61C2×D811yescharacteristic
200C2×C4×C61C2×D811yescharacteristic
201C2×C4×C61C2×D811yescharacteristic
202C2×C4×C61C2×D811yescharacteristic
203C2×C4×C61C22×D411yescharacteristic
204C22⋊D61—82no—
205C22⋊D61—82no—
206C22⋊D61—82no—
207C22⋊D61—161no—
208C2×Dic61—41no—
209C2×Dic61—41no—
210C2×Dic61—41no—
211C2×Dic61—41no—
212C2×Dic61—41no—
213C2×Dic61—82no—
214C2×Dic61—82no—
215D244—22no—
216D244—41no—
217D244—41no—
218D244—41no—
219D244—41no—
220D244—41no—
221D244—82no—
222C4×D61—82no—
223C4×D61—82no—
224C4×D61—161no—
225Dic122—41no—
226Dic122—41no—
227Dic122—82no—
228C8×C61—21no—
229C8×C61—21no—
230C8×C61—21no—
231C8×C61C2×#4111yescharacteristic
232C8×C61H(5,𝔽2)11yescharacteristic
233#33873—611nomaximal, Sylow
234C22×C61—21no—
235C22×C61—21no—
236C22×C61—21no—
237C22×C61—21no—
238C22×C61—21no—
239C22×C61#50912yes—
240C22×C61C22×D811yescharacteristic
241C2×D61—42no—
242C2×D61—81no—
243C2×D61—81no—
244C2×D61—81no—
245C2×D61—81no—
246C2×D61—81no—
247C2×D61—81no—
248C2×D61—81no—
249C2×D61—81no—
250C2×D61—164no—
251C4×C61—21no—
252C4×C61—21no—
253C4×C61—21no—
254C4×C61—21no—
255C4×C61#54611yescharacteristic
256C4×C61#57711yescharacteristic
257Dic61—81no—
258Dic61—81no—
259Dic61—81no—
260Dic61—81no—
261Dic61—81no—
262#3334—611no—
263#3280—611no—
264#2243—611no—
265#2235—611no—
266#2233—611no—
267#2226—611no—
268#2218—611no—
269#2217—611no—
270#2210—611no—
271#2186—611no—
272#2184—611no—
273#2177—611no—
274#2168D6111yescharacteristic, complemented
275#2139—611no—
276C2×#357—611no—
277C2×C61—22no—
278C2×C61—41no—
279C2×C61#329411yescharacteristic
280C2×C61C2×#50911yescharacteristic
281C2×C61#341611yescharacteristic
282D61—82no—
283D61—161no—
284D61—161no—
285C2×#135—611no—
286C22×D8—611no—
287#547—1221no—
288#546—1221no—
289C2×SD32—611no—
290C2×D16—611no—
291C2×D16—611no—
292C2×M32C2×D6111yescharacteristic, complemented
293C22×C16C2×D6111yescharacteristic
294#500—611no—
295#495—611no—
296#480—1221no—
297#480—1221no—
298#474—1221no—
299#467—611no—
300#466—611no—
301#466—611no—
302#466—611no—
303#463—1221no—
304#450—1221no—
305#449—1221no—
306C4×D8—611no—
307C4×D8—611no—
308C8×D4C2×D6111yescharacteristic
309C2×#107—611no—
310C2×#107—611no—
311#367—611no—
312#367—611no—
313#367—611no—
314C4⋊C16C2×D6111yescharacteristic, complemented
315#358—611no—
316#358—611no—
317#358—611no—
318#357—611no—
319#357—611no—
320#357—611no—
321#357—611no—
322#357—611no—
323#357—611no—
324#357—611no—
325C22⋊C16C2×D6111yescharacteristic
326C22⋊C16C2×D6111yescharacteristic, complemented
327#346C2×D6111yescharacteristic, complemented
328C61#3387311yescharacteristic, complemented, Sylow
329C2×#41—1221no—
330C2×#41—2441no—
331C22×D4—1221no—
332#135—1222no—
333#135—2441no—
334C2×Q16—611no—
335C2×SD16—1221no—
336C2×SD16—1221no—
337C2×D8—611no—
338C2×D8—611no—
339C2×D8—611no—
340C2×D8—1222no—
341C2×D8—1221no—
342C2×D8—2441no—
343C22×C8C22×D6111yescharacteristic
344C22×C8C22×D6111yescharacteristic
345C4⋊Q8—1221no—
346#124—1221no—
347#123—2441no—
348#123—2441no—
349#123—2441no—
350C22⋊Q8—1221no—
351C22⋊Q8—2441no—
352#121—1221no—
353#121—1221no—
354#121—1221no—
355#121—2441no—
356#120—2441no—
357C4×D4—1221no—
358C4×D4—1221no—
359C4×D4D24411yescharacteristic
360C2×C4⋊C4—1221no—
361C2×C4⋊C4—1221no—
362C2×C22⋊C4—2441no—
363SD32—2441no—
364D16—1222no—
365D16—2441no—
366M32—41no—
367C2×C16—41no—
368C2×C16C22×D6111yescharacteristic
369C2×C16C22×D6111yescharacteristic
370C2×C16C22×D6111yescharacteristic
371C2×C16C22×D6111yescharacteristic
372#107—611no—
373#107—611no—
374#107—611no—
375#107—611no—
376#107—1221no—
377#107—1221no—
378#107—1221no—
379C4⋊C8D24411yescharacteristic
380#103—1221no—
381#102—1221no—
382#102—1221no—
383#102—1221no—
384#102—1221no—
385#102—1221no—
386C22⋊C8D24411yescharacteristic
387C22⋊C8D24411yescharacteristic
388C4×C8C22×D6111yescharacteristic
389C24—2441no—
390#41—4881no—
391#41—4884no—
392#41—2442no—
393C2×Q8—1221no—
394C2×Q8—2441no—
395C2×D4—1221no—
396C2×D4—1221no—
397C2×D4—1221no—
398C2×D4—4881no—
399C2×D4—2442no—
400C2×D4—2441no—
401C2×D4—2441no—
402C2×D4—2441no—
403C2×D4—2441no—
404C2×D4—2441no—
405C2×D4D48811yescharacteristic
406C22×C4—2441no—
407C22×C4—2441no—
408C22×C4—2441no—
409C22×C4—2441no—
410C22×C4—2441no—
411C22×C4C2×D24411yescharacteristic
412C22×C4C2×D24411yescharacteristic
413Q16—1221no—
414Q16—1221no—
415SD16—4881no—
416SD16—2442no—
417D8—612no—
418D8—1221no—
419D8—1221no—
420D8—1221no—
421D8—1221no—
422D8—1221no—
423D8—2442no—
424C2×C8—21no—
425C2×C8—21no—
426C2×C8—21no—
427C2×C8—21no—
428C2×C8C2×D24411yescharacteristic
429C2×C8C2×D24411yescharacteristic
430C2×C8C2×D24411yescharacteristic
431C2×C8C23×D6111yescharacteristic
432C4⋊C4—1221no—
433C4⋊C4—1221no—
434C4⋊C4—1221no—
435C4⋊C4—1221no—
436C4⋊C4—2441no—
437C4⋊C4—2441no—
438C4⋊C4—2441no—
439C4⋊C4—2441no—
440C4⋊C4D48811yescharacteristic
441C22⋊C4—2441no—
442C22⋊C4—2441no—
443C22⋊C4—2441no—
444C22⋊C4—2441no—
445C22⋊C4—2441no—
446C22⋊C4—2441no—
447C22⋊C4—2441no—
448C22⋊C4—2441no—
449C22⋊C4D48811yescharacteristic
450C22⋊C4D48811yescharacteristic
451C42C2×D24411yescharacteristic
452C16—21no—
453C16—21no—
454C16—21no—
455C16—21no—
456C23—9761no—
457C23—4882no—
458C23—4882no—
459C23—2441no—
460C23—2441no—
461C23—2441no—
462C23C2×D48811yescharacteristic
463C23C2×D48811yescharacteristic
464Q8—4882no—
465Q8—2441no—
466Q8—2441no—
467D4—22no—
468D4—1222no—
469D4—9761no—
470D4—4882no—
471D4—4882no—
472D4—4882no—
473D4—4882no—
474D4—2441no—
475D4—2441no—
476D4—2441no—
477D4—2441no—
478D4—2441no—
479C2×C4—21no—
480C2×C4—21no—
481C2×C4—21no—
482C2×C4—21no—
483C2×C4—9761no—
484C2×C4—4882no—
485C2×C4—4882no—
486C2×C4—4882no—
487C2×C4—4882no—
488C2×C4—2441no—
489C2×C4—2441no—
490C2×C4—2441no—
491C2×C4—2441no—
492C2×C4—2441no—
493C2×C4C2×D48811yescharacteristic
494C2×C4C2×D48811yescharacteristic
495C2×C4C2×D48811yescharacteristic
496C2×C4C2×D48811yescharacteristic
497C2×C4C22×D24411yescharacteristic
498C8—21no—
499C8—21no—
500C8—21no—
501C8#41×D6111yescharacteristic
502C8#1451982211yescharacteristic
503C22—21no—
504C22—21no—
505C22—21no—
506C22—21no—
507C22—21no—
508C22—9764no—
509C22—4881no—
510C22—4881no—
511C22—4881no—
512C22—4881no—
513C22—4881no—
514C22—4881no—
515C22—4881no—
516C22—4881no—
517C22—2442no—
518C22#1684020012yes—
519C22C22×D48811yescharacteristic
520C4—21no—
521C4—21no—
522C4—21no—
523C4—21no—
524C4—4881no—
525C4—4881no—
526C4—4881no—
527C4—4881no—
528C4—4881no—
529C4#1684040711yescharacteristic
530C4#1684057411yescharacteristic
531C2—22no—
532C2—41no—
533C2—9761no—
534C2—9761no—
535C2—4882no—
536C2#2145763311yescharacteristic
537C2C2×#1684020011yescharacteristic
538C2#2145896711yescharacteristic
subgroupG/H
centerC22C22×D488
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC61×#2168C2
socleC22×C61C22×D8
hypercenter#2168D61
hypocenterC61#33873
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C4×D4 ◁ C8×D4 ◁ #2168

Complements

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

NH ≅ G/Ntype
1. #21458167C2semidirect
2. #21457506C2semidirect
3. #21448056C2semidirect
4. #21448017C2semidirect
5. #21448005C2semidirect
6. #21447967C2semidirect
7. #21447922C2semidirect
8. #21447915C2semidirect
9. #21447875C2semidirect
10. #21447744C2semidirect
11. #21447733C2semidirect
12. #21447696C2semidirect
13. #21447532C2semidirect
14. C2×#16839564C2semidirect
15. C61×#2168C2semidirect
20. C2×#14519722C22semidirect
23. #16840167C22semidirect
31. #16839998C22semidirect
36. C4×D488C22semidirect
38. C2×#14519711C22semidirect
39. #16839590C22semidirect
40. #16839590C22semidirect
41. #16839590C22semidirect
42. #16839567C22semidirect
43. #16839567C22semidirect
44. #16839567C22semidirect
51. #16839564C22semidirect
52. C2×M32×C61C22semidirect
55. C61×C4⋊C16C22semidirect
57. C61×C22⋊C16C22semidirect
58. C61×#346C22semidirect
179. D488#41semidirect
274. #2168D61semidirect
292. C2×M32C2×D61semidirect
314. C4⋊C16C2×D61semidirect
326. C22⋊C16C2×D61semidirect
327. #346C2×D61semidirect
328. C61#33873semidirect

Maximal subgroups

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

subgroupindex
#214581672
#214575062
#214480562
#214480172
#214480052
#214479672
#214479222
#214479152
#214478752
#214477442
#214477332
#214476962
#214475322
C2×#168395642
C61×#21682
#3387361