Finite Group Databasebeta

← #29835126

Subgroups — #29835126

Order 15616.

568 conjugacy classes of subgroups.

The subgroup lattice has 506 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456461C211yesmaximal, characteristic, complemented
2#21454666C211yesmaximal, characteristic, complemented
3#21448030C211yesmaximal, characteristic, complemented
4#21448012C211yesmaximal, characteristic, complemented
5#21447953C211yesmaximal, characteristic, complemented
6#21447875C211yesmaximal, characteristic, complemented
7#21447874C211yesmaximal, characteristic, complemented
8#21447749C211yesmaximal, characteristic, complemented
9#21447732C211yesmaximal, characteristic
10#21447694C211yesmaximal, characteristic, complemented
11#21447680C211yesmaximal, characteristic, complemented
12#21447670C211yesmaximal, characteristic, complemented
13#21447663C211yesmaximal, characteristic, complemented
14#21447625C211yesmaximal, characteristic, complemented
15C61×#2226C211yesmaximal, characteristic, complemented
16C2×D8×D61C2211yescharacteristic
17#16840386—21no—
18#16840234—21no—
19C2×#14519722C2211yescharacteristic
20C2×#14519721C2211yescharacteristic
21C2×D976C2211yescharacteristic
22C2×#14519718C2211yescharacteristic
23#16840146C2211yescharacteristic, complemented
24#16840124—21no—
25#16840009—21no—
26#16840004C2211yescharacteristic
27#16839999C2211yescharacteristic
28#16839872—21no—
29#16839871—21no—
30#16839861—21no—
31#16839856C2211yescharacteristic, complemented
32C4×D488C2211yescharacteristic
33D8×Dic61C2211yescharacteristic
34#16839844C2211yescharacteristic
35#16839844C2211yescharacteristic, complemented
36#16839799C2211yescharacteristic
37#16839765—21no—
38#16839716C2211yescharacteristic
39#16839587C2211yescharacteristic, complemented
40#16839587C2211yescharacteristic, complemented
41C4⋊#11758185C2211yescharacteristic, complemented
42#16839567C2211yescharacteristic, complemented
43#16839565C2211yescharacteristic, complemented
44#16839564C2211yescharacteristic, complemented
45#16839562C2211yescharacteristic
46#16839562C2211yescharacteristic
47#16839562C2211yescharacteristic, complemented
48#16839541C2211yescharacteristic
49#16839541C2211yescharacteristic
50#16839537C2211yescharacteristic, complemented
51C16×Dic61C2211yescharacteristic, complemented
52C61×#500C2211yescharacteristic, complemented
53C4×D8×C61C2211yescharacteristic
54C61×#367C2211yescharacteristic, complemented
55C61×#367C2211yescharacteristic, complemented
56C61×C4⋊C16C2211yescharacteristic, complemented
57C61×#357C2211yescharacteristic
58C61×#357C2211yescharacteristic
59C2×D4×D61—21no—
60C2×Dic244C2311yescharacteristic
61C2×#11758190—21no—
62C2×D488C2311yescharacteristic
63C2×D488C2311yescharacteristic
64C2×#11758199—21no—
65D8×D61—22no—
66D8×D61—22no—
67C2×#11758189D411yescharacteristic
68C2×C8×D61C2311yescharacteristic
69#14519772—41no—
70#14519766—41no—
71#14519765—21no—
72#14519762—41no—
73#14519757—21no—
74#14519753—41no—
75#14519752—21no—
76#14519742—21no—
77C4×D244—21no—
78D4×Dic61—21no—
79#14519739—21no—
80#14519739—21no—
81C4×C22⋊D61—41no—
82#14519737—41no—
83#14519735—21no—
84#14519734D411yescharacteristic
85C2×#11758198—41no—
86C22⋊C4×D61—41no—
87#14519722—22no—
88#14519721—22no—
89D976—22no—
90#14519718—22no—
91C2×#11758185C2311yescharacteristic
92C2×#11758185C2311yescharacteristic
93#14519711C2311yescharacteristic
94#14519708—21no—
95#14519700—21no—
96#14519700—21no—
97#14519699—21no—
98#14519698—21no—
99#14519697—21no—
100#14519697—21no—
101#14519697—21no—
102#14519688D411yescharacteristic
103#14519687D411yescharacteristic
104C8×Dic61C2311yescharacteristic
105C2×D8×C61C2311yescharacteristic
106C4⋊Q8×C61—21no—
107C4×D4×C61—21no—
108C2×C16×C61C2311yescharacteristic
109C2×C16×C61C2311yescharacteristic
110#107×C61—21no—
111#107×C61C2311yescharacteristic
112#107×C61C2311yescharacteristic
113#107×C61C2311yescharacteristic
114#102×C61—21no—
115C4×C8×C61C2311yescharacteristic
116C23×D61—41no—
117C2×Q8×C61—41no—
118C2×D4×C61—21no—
119C22×C4×C61—41no—
120C2×C22⋊D61—41no—
121C22×Dic61—41no—
122D4×D61—24no—
123D4×D61—42no—
124C2×D244—21no—
125C2×D244—21no—
126C2×D244—41no—
127C2×C4×D61—41no—
128C2×C4×D61—41no—
129C2×C4×D61—41no—
130C2×C4×D61C2×D411yescharacteristic
131C2×Dic122—21no—
132D8×C61—21no—
133D8×C61#4112yes—
134C2×C8×C61C2×D411yescharacteristic
135C2×C8×C61C2411yescharacteristic
136C4⋊C4×C61—21no—
137C4⋊C4×C61—21no—
138C4⋊C4×C61—21no—
139C4⋊C4×C61—41no—
140C22⋊C4×C61—41no—
141C42×C61C2×D411yescharacteristic
142C22⋊Dic61—41no—
143#11758199—24no—
144#11758198—41no—
145#11758198—41no—
146#11758198—41no—
147#11758198—41no—
148#11758198—41no—
149#11758198—41no—
150#11758198—41no—
151#11758198—41no—
152#11758198D811yescharacteristic
153#11758198D811yescharacteristic
154C4⋊Dic61—21no—
155C4⋊Dic61—41no—
156#11758196—41no—
157#11758196D811yescharacteristic
158#11758196D811yescharacteristic
159C4×Dic61—41no—
160C4×Dic61—41no—
161C4×Dic61C2×D411yescharacteristic
162C2×#92418C2×D411yescharacteristic
163C2×#92418C2×D411yescharacteristic
164Dic244—21no—
165Dic244—21no—
166D488—21no—
167D488—21no—
168D488—21no—
169D488#4112yes—
170#11758190—42no—
171#11758189—41no—
172C8×D61—21no—
173C8×D61—21no—
174C16×C61—21no—
175C16×C61—21no—
176#11758185—21no—
177#11758185—21no—
178C23×C61—41no—
179C22×D61—44no—
180C22×D61—41no—
181C22×D61—41no—
182C22×D61—161no—
183C22×D61—161no—
184C22×D61C2×D811yescharacteristic
185Q8×C61—161no—
186D4×C61—22no—
187D4×C61—41no—
188C2×C4×C61—41no—
189C2×C4×C61—41no—
190C2×C4×C61—41no—
191C2×C4×C61—161no—
192C2×C4×C61C2×D811yescharacteristic
193C2×C4×C61C2×D811yescharacteristic
194C2×C4×C61C22×D411yescharacteristic
195C22⋊D61—44no—
196C2×Dic61—41no—
197C2×Dic61—41no—
198C2×Dic61—161no—
199C2×Dic61C2×D811yescharacteristic
200C2×Dic61C2×D811yescharacteristic
201C2×Dic61C2×D811yescharacteristic
202D244—22no—
203D244—41no—
204D244—41no—
205D244—41no—
206D244—161no—
207C4×D61—21no—
208C4×D61—21no—
209C4×D61—161no—
210C4×D61—161no—
211C4×D61—161no—
212Dic122—41no—
213Dic122—41no—
214C8×C61—21no—
215C8×C61C2×#4111yescharacteristic
216C8×C61C2×#4111yescharacteristic
217#92418—21no—
218#92418—21no—
219#33678—611nomaximal, Sylow
220C22×C61—42no—
221C22×C61—81no—
222C22×C61—81no—
223C22×C61C22×D811yescharacteristic
224C2×D61—21no—
225C2×D61—21no—
226C2×D61—21no—
227C2×D61—42no—
228C2×D61—81no—
229C2×D61—81no—
230C2×D61—81no—
231C2×D61—81no—
232C2×D61—81no—
233C2×D61—164no—
234C4×C61—21no—
235C4×C61—21no—
236C4×C61—81no—
237C4×C61—81no—
238C4×C61—81no—
239C4×C61#52511yescharacteristic
240C4×C61#57511yescharacteristic
241Dic61—21no—
242Dic61—21no—
243Dic61—21no—
244Dic61—81no—
245Dic61—81no—
246#3175—611no—
247#2948—611no—
248#2237—611no—
249#2234—611no—
250#2226D6111yescharacteristic, complemented
251#2224—611no—
252#2210—611no—
253#2210—611no—
254#2187—611no—
255#2184—611no—
256#2177—611no—
257#2175—611no—
258C4×SD32—611no—
259C4×D16—611no—
260#2164—611no—
261C2×C61—82no—
262C2×C61#310011yescharacteristic
263C2×C61#341511yescharacteristic
264C2×C61#341611yescharacteristic
265D61—41no—
266D61—82no—
267D61—161no—
268C22×D8—611no—
269#542—1221no—
270#518—1221no—
271C2×SD32—611no—
272C2×SD32—611no—
273C2×D16—611no—
274C2×D16—611no—
275#500C2×D6111yescharacteristic, complemented
276#495—611no—
277#489—1221no—
278#468—1221no—
279#467—611no—
280#466—611no—
281#442—1221no—
282#442—1221no—
283#440—1221no—
284C4×Q16—611no—
285C4×D8—611no—
286C4×D8—611no—
287C4×D8—611no—
288C4×D8—611no—
289C4×D8C2×D6111yescharacteristic
290#427—611no—
291#418—1221no—
292#405—611no—
293#367C2×D6111yescharacteristic, complemented
294#367C2×D6111yescharacteristic, complemented
295#366—611no—
296#366—611no—
297C4⋊C16—611no—
298C4⋊C16C2×D6111yescharacteristic, complemented
299#358—611no—
300#358—611no—
301#357—611no—
302#357—611no—
303#357—611no—
304#357—611no—
305#357C2×D6111yescharacteristic
306#357C2×D6111yescharacteristic
307C22⋊C16—611no—
308C22⋊C16—611no—
309#346—611no—
310C4×C16—611no—
311C61#3367811yescharacteristic, complemented, Sylow
312C22×D4—1221no—
313C2×Q16—611no—
314C2×SD16—1221no—
315C2×D8—611no—
316C2×D8—611no—
317C2×D8—1221no—
318C2×D8—1222no—
319C2×D8—1222no—
320C2×D8C22×D6111yescharacteristic
321C2×M16—611no—
322C22×C8—611no—
323C4⋊Q8—21no—
324#126—2441no—
325#124—1221no—
326#124—2441no—
327#123—2441no—
328C22⋊Q8—1221no—
329C22⋊Q8—2441no—
330#121—1221no—
331C4×Q8—1221no—
332C4×D4—21no—
333C4×D4—1221no—
334C4×D4—1221no—
335C4×D4—1221no—
336C4×D4—1221no—
337C4×D4—2441no—
338C4×D4—2441no—
339#117—611no—
340#117—1221no—
341C2×C22⋊C4—2441no—
342C2×C22⋊C4—2441no—
343SD32—1222no—
344SD32—1222no—
345D16—1222no—
346D16—1222no—
347C2×C16—611no—
348C2×C16—611no—
349C2×C16C22×D6111yescharacteristic
350C2×C16C22×D6111yescharacteristic
351#107—21no—
352#107—611no—
353#107C22×D6111yescharacteristic
354#107C22×D6111yescharacteristic
355#107C22×D6111yescharacteristic
356#106—1221no—
357#103—1221no—
358#103—1221no—
359#102—21no—
360#102—1221no—
361#102—1221no—
362#102—1221no—
363#102—1221no—
364#102—1221no—
365#97—611no—
366#97—611no—
367C4×C8—611no—
368C4×C8C22×D6111yescharacteristic
369C24—2441no—
370C2×Q8—41no—
371C2×Q8—1221no—
372C2×D4—21no—
373C2×D4—1224no—
374C2×D4—1221no—
375C2×D4—1221no—
376C2×D4—2441no—
377C2×D4—2441no—
378C2×D4—2442no—
379C22×C4—41no—
380C22×C4—611no—
381C22×C4—2441no—
382C22×C4—2441no—
383C22×C4—2441no—
384C22×C4—2441no—
385Q16—1221no—
386Q16—1221no—
387SD16—2442no—
388D8—21no—
389D8—612no—
390D8—1224no—
391D8—1221no—
392D8—1221no—
393D8—1221no—
394D8#1175822612yes—
395M16—2441no—
396C2×C8—611no—
397C2×C8—611no—
398C2×C8—1221no—
399C2×C8—1221no—
400C2×C8D4×D6111yescharacteristic
401C2×C8C23×D6111yescharacteristic
402C4⋊C4—21no—
403C4⋊C4—21no—
404C4⋊C4—21no—
405C4⋊C4—41no—
406C4⋊C4—611no—
407C4⋊C4—611no—
408C4⋊C4—1221no—
409C4⋊C4—2441no—
410C4⋊C4—2441no—
411C22⋊C4—41no—
412C22⋊C4—611no—
413C22⋊C4—611no—
414C22⋊C4—2441no—
415C22⋊C4—2441no—
416C22⋊C4—2441no—
417C22⋊C4—2441no—
418C22⋊C4—2441no—
419C22⋊C4—2441no—
420C22⋊C4—2441no—
421C22⋊C4—2441no—
422C22⋊C4—2441no—
423C42—611no—
424C42—2441no—
425C42—2441no—
426C42D4×D6111yescharacteristic
427C16—21no—
428C16—21no—
429C16—1221no—
430C16—1221no—
431C23—41no—
432C23—611no—
433C23—9761no—
434C23—9761no—
435C23—2444no—
436C23—2441no—
437C23—2441no—
438Q8—161no—
439Q8—2441no—
440Q8—2441no—
441D4—22no—
442D4—41no—
443D4—1222no—
444D4—9761no—
445D4—2444no—
446D4—2441no—
447D4—2441no—
448D4—2441no—
449C2×C4—41no—
450C2×C4—41no—
451C2×C4—41no—
452C2×C4—161no—
453C2×C4—611no—
454C2×C4—611no—
455C2×C4—611no—
456C2×C4—1221no—
457C2×C4—1221no—
458C2×C4—9761no—
459C2×C4—9761no—
460C2×C4—9761no—
461C2×C4—9761no—
462C2×C4—2441no—
463C2×C4—2441no—
464C2×C4D8×D6111yescharacteristic
465C2×C4D8×D6111yescharacteristic
466C2×C4C2×D4×D6111yescharacteristic
467C8—21no—
468C8—1221no—
469C8—1221no—
470C8#41×D6111yescharacteristic
471C8C2×#1175822611yescharacteristic
472C22—42no—
473C22—81no—
474C22—81no—
475C22—1221no—
476C22—1221no—
477C22—1221no—
478C22—9764no—
479C22—4881no—
480C22—4881no—
481C22—4881no—
482C22—4881no—
483C22—4881no—
484C22—2442no—
485C22C2×D8×D6111yescharacteristic
486C4—21no—
487C4—21no—
488C4—81no—
489C4—81no—
490C4—81no—
491C4—1221no—
492C4—1221no—
493C4—1221no—
494C4—4881no—
495C4—4881no—
496C4#1684027511yescharacteristic
497C4#1684056111yescharacteristic
498C2—82no—
499C2—9761no—
500C2—4882no—
501C2—2441no—
502C2#2145582111yescharacteristic
503C2#2145895911yescharacteristic
504C2#2145896111yescharacteristic
subgroupG/H
centerC22C2×D8×D61
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC61×#2226C2
socleC22×C61C22×D8
prime radical#21447732C2
hypercenter#2226D61
hypocenterC61#33678
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #2226

Complements

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

NH ≅ G/Ntype
1. #21456461C2semidirect
2. #21454666C2semidirect
3. #21448030C2semidirect
4. #21448012C2semidirect
5. #21447953C2semidirect
6. #21447875C2semidirect
7. #21447874C2semidirect
8. #21447749C2semidirect
10. #21447694C2semidirect
11. #21447680C2semidirect
12. #21447670C2semidirect
13. #21447663C2semidirect
14. #21447625C2semidirect
15. C61×#2226C2semidirect
23. #16840146C22semidirect
31. #16839856C22semidirect
35. #16839844C22semidirect
39. #16839587C22semidirect
40. #16839587C22semidirect
41. C4⋊#11758185C22semidirect
42. #16839567C22semidirect
43. #16839565C22semidirect
44. #16839564C22semidirect
47. #16839562C22semidirect
50. #16839537C22semidirect
51. C16×Dic61C22semidirect
52. C61×#500C22semidirect
54. C61×#367C22semidirect
55. C61×#367C22semidirect
56. C61×C4⋊C16C22semidirect
250. #2226D61semidirect
275. #500C2×D61semidirect
293. #367C2×D61semidirect
294. #367C2×D61semidirect
298. C4⋊C16C2×D61semidirect
311. C61#33678semidirect

Maximal subgroups

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

subgroupindex
#214564612
#214546662
#214480302
#214480122
#214479532
#214478752
#214478742
#214477492
#214477322
#214476942
#214476802
#214476702
#214476632
#214476252
C61×#22262
#3367861