Finite Group Databasebeta

← #29836860

Subgroups — #29836860

Order 15616.

568 conjugacy classes of subgroups.

The subgroup lattice has 522 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21458188C211yesmaximal, characteristic, complemented
2#21457506C211yesmaximal, characteristic
3#21448051C211yesmaximal, characteristic, complemented
4#21448017C211yesmaximal, characteristic, complemented
5#21448011C211yesmaximal, characteristic, complemented
6#21447977C211yesmaximal, characteristic, complemented
7#21447915C211yesmaximal, characteristic, complemented
8#21447910C211yesmaximal, characteristic, complemented
9#21447882C211yesmaximal, characteristic, complemented
10#21447761C211yesmaximal, characteristic, complemented
11#21447750C211yesmaximal, characteristic, complemented
12C4×#14519722C211yesmaximal, characteristic, complemented
13#21447527C211yesmaximal, characteristic, complemented
14C2×#16839564C211yesmaximal, characteristic, complemented
15D4×C16×C61C211yesmaximal, characteristic, complemented
16C2×#14519799C2211yescharacteristic
17C22×D488C2211yescharacteristic
18#16840452—21no—
19#16840408—21no—
20C2×Dic488C2211yescharacteristic, complemented
21C2×#14519722C2211yescharacteristic
22C2×#14519722C2211yescharacteristic
23#16840164C2211yescharacteristic, complemented
24#16840146C2211yescharacteristic
25#16840104—21no—
26#16840093—21no—
27#16840070—21no—
28#16840003C2211yescharacteristic
29#16840003C2211yescharacteristic
30#16840003C2211yescharacteristic, complemented
31#16839998C2211yescharacteristic
32#16839978—21no—
33#16839915—21no—
34#16839909—21no—
35C4×Dic244C2211yescharacteristic, complemented
36C4×D488C2211yescharacteristic
37#16839800C2211yescharacteristic, complemented
38C2×#14519711C2211yescharacteristic
39#16839590C2211yescharacteristic, complemented
40#16839590C2211yescharacteristic, complemented
41#16839588C2211yescharacteristic, complemented
42#16839567C2211yescharacteristic, complemented
43#16839567C2211yescharacteristic, complemented
44#16839567C2211yescharacteristic, complemented
45#16839567C2211yescharacteristic, complemented
46#16839564C2211yescharacteristic
47#16839564C2211yescharacteristic
48#16839564C2211yescharacteristic
49#16839564C2211yescharacteristic
50#16839564C2211yescharacteristic
51#16839564C2211yescharacteristic
52C22×C16×C61C2211yescharacteristic
53C22×C16×C61C2211yescharacteristic, complemented
54C8×D4×C61C2211yescharacteristic
55C61×C4⋊C16C2211yescharacteristic, complemented
56C61×C22⋊C16C2211yescharacteristic
57C61×C22⋊C16C2211yescharacteristic, complemented
58C4×C16×C61C2211yescharacteristic, complemented
59C2×#11758222—21no—
60C22×D244—21no—
61#14519799—22no—
62#14519799—41no—
63C2×Dic244C2311yescharacteristic
64C2×Dic244C2311yescharacteristic
65C2×#11758190—21no—
66C2×#11758190—21no—
67C2×D488—22no—
68C2×D488—21no—
69C2×D488—41no—
70C2×D488C2311yescharacteristic
71C2×D488C2311yescharacteristic
72#14519770—41no—
73#14519769—21no—
74#14519767—41no—
75#14519765—21no—
76#14519762—41no—
77#14519758—41no—
78#14519756—21no—
79#14519756—21no—
80#14519756—21no—
81#14519751—21no—
82#14519750—41no—
83#14519745—41no—
84C4×Dic122—21no—
85C4×D244—21no—
86#14519736—21no—
87#14519733—41no—
88C2×C4⋊Dic61—21no—
89C22⋊C4×D61—41no—
90Dic488—41no—
91#14519722—22no—
92#14519722—41no—
93#14519711—21no—
94#14519711C2311yescharacteristic
95#14519711C2311yescharacteristic
96#14519711C2311yescharacteristic
97#14519711C2311yescharacteristic
98#14519709—21no—
99#14519709—21no—
100#14519702—21no—
101#14519702—21no—
102#14519702—21no—
103#14519702—21no—
104#14519699—21no—
105#14519699—21no—
106C22×C8×C61C2311yescharacteristic
107C22×C8×C61C2311yescharacteristic
108C4×D4×C61D411yescharacteristic
109C2×C16×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—
125C22×Dic61—41no—
126#11758222—42no—
127#11758222—81no—
128C2×D244—21no—
129C2×D244—21no—
130C2×D244—42no—
131C2×D244—41no—
132C2×D244—81no—
133C2×C4×D61—41no—
134C2×C4×D61—41no—
135C2×Dic122—21no—
136C2×Dic122—21no—
137C2×C8×C61—21no—
138C2×C8×C61—21no—
139C2×C8×C61—21no—
140C2×C8×C61—21no—
141C2×C8×C61C2×D411yescharacteristic
142C2×C8×C61C2×D411yescharacteristic
143C2×C8×C61C2×D411yescharacteristic
144C2×C8×C61C2411yescharacteristic
145C4⋊C4×C61D811yescharacteristic
146C22⋊C4×C61D811yescharacteristic
147C22⋊C4×C61D811yescharacteristic
148C42×C61C2×D411yescharacteristic
149C22⋊Dic61—41no—
150C22⋊Dic61—41no—
151C22⋊Dic61—41no—
152C22⋊Dic61—41no—
153#11758198—41no—
154#11758198—41no—
155#11758198—41no—
156#11758198—41no—
157C4⋊Dic61—21no—
158C4⋊Dic61—21no—
159C4⋊Dic61—21no—
160C4⋊Dic61—21no—
161C4⋊Dic61—41no—
162#11758196—41no—
163#11758196—41no—
164#11758196—41no—
165#11758196—41no—
166#11758196—41no—
167#11758196—41no—
168C4×Dic61—41no—
169C4×Dic61—41no—
170Dic244—21no—
171Dic244—21no—
172Dic244—21no—
173Dic244—21no—
174D488—21no—
175D488—21no—
176D488—21no—
177D488—42no—
178D488#4112yes—
179#11758190—42no—
180#11758190—81no—
181C16×C61—21no—
182C16×C61—21no—
183C16×C61—21no—
184C16×C61—21no—
185C23×C61C2×D811yescharacteristic
186C23×C61C2×D811yescharacteristic
187C22×D61—41no—
188C22×D61—41no—
189C22×D61—82no—
190C22×D61—82no—
191C22×D61—161no—
192D4×C61—22no—
193C2×C4×C61—21no—
194C2×C4×C61—21no—
195C2×C4×C61—21no—
196C2×C4×C61—21no—
197C2×C4×C61C2×D811yescharacteristic
198C2×C4×C61C2×D811yescharacteristic
199C2×C4×C61C2×D811yescharacteristic
200C2×C4×C61C2×D811yescharacteristic
201C2×C4×C61C22×D411yescharacteristic
202C22⋊D61—82no—
203C22⋊D61—161no—
204C2×Dic61—41no—
205C2×Dic61—41no—
206C2×Dic61—41no—
207C2×Dic61—41no—
208C2×Dic61—41no—
209C2×Dic61—41no—
210C2×Dic61—82no—
211D244—22no—
212D244—41no—
213D244—41no—
214D244—41no—
215D244—82no—
216C4×D61—82no—
217C4×D61—161no—
218Dic122—41no—
219Dic122—41no—
220Dic122—41no—
221Dic122—41no—
222C8×C61—21no—
223C8×C61—21no—
224C8×C61—21no—
225C8×C61C2×#4111yescharacteristic
226C8×C61H(5,𝔽2)11yescharacteristic
227#33863—611nomaximal, Sylow
228C22×C61—21no—
229C22×C61—21no—
230C22×C61—21no—
231C22×C61—21no—
232C22×C61—21no—
233C22×C61#50812yes—
234C22×C61C22×D811yescharacteristic
235C2×D61—42no—
236C2×D61—81no—
237C2×D61—81no—
238C2×D61—81no—
239C2×D61—81no—
240C2×D61—81no—
241C2×D61—164no—
242C4×C61—21no—
243C4×C61—21no—
244C4×C61—21no—
245C4×C61—21no—
246C4×C61#54611yescharacteristic
247C4×C61#57711yescharacteristic
248Dic61—81no—
249Dic61—81no—
250Dic61—81no—
251Dic61—81no—
252Dic61—81no—
253Dic61—81no—
254#3336—611no—
255#3280—611no—
256#2242—611no—
257#2235—611no—
258#2234—611no—
259#2228—611no—
260#2217—611no—
261#2216—611no—
262#2211—611no—
263#2189—611no—
264#2187—611no—
265C4×SD32—611no—
266D4×C16D6111yescharacteristic, complemented
267#2138—611no—
268C2×#357—611no—
269C2×C61—22no—
270C2×C61—41no—
271C2×C61#329411yescharacteristic
272C2×C61C2×#50811yescharacteristic
273C2×C61#341611yescharacteristic
274D61—82no—
275D61—161no—
276C2×#135—611no—
277C22×D8—611no—
278#552—1221no—
279#546—1221no—
280C2×Q32—611no—
281C2×SD32—611no—
282C2×SD32—611no—
283C22×C16C2×D6111yescharacteristic
284C22×C16C2×D6111yescharacteristic, complemented
285#499—611no—
286#495—611no—
287#485—1221no—
288#483—1221no—
289#479—1221no—
290#467—611no—
291#467—611no—
292#467—611no—
293#466—611no—
294#463—1221no—
295#450—1221no—
296#449—1221no—
297C4×Q16—611no—
298C4×D8—611no—
299C8×D4C2×D6111yescharacteristic
300#427—611no—
301C2×#107—611no—
302#367—611no—
303#367—611no—
304#366—611no—
305C4⋊C16C2×D6111yescharacteristic, complemented
306#358—611no—
307#358—611no—
308#358—611no—
309#358—611no—
310#357—611no—
311#357—611no—
312#357—611no—
313#357—611no—
314#357—611no—
315#357—611no—
316C22⋊C16C2×D6111yescharacteristic
317C22⋊C16C2×D6111yescharacteristic, complemented
318C4×C16C2×D6111yescharacteristic, complemented
319C61#3386311yescharacteristic, complemented, Sylow
320C2×#41—1221no—
321C22×D4—1221no—
322#135—1222no—
323#135—2441no—
324C2×Q16—611no—
325C2×Q16—611no—
326C2×SD16—1221no—
327C2×SD16—1221no—
328C2×D8—611no—
329C2×D8—611no—
330C2×D8—1222no—
331C2×D8—1221no—
332C2×D8—2441no—
333C22×C8C22×D6111yescharacteristic
334C22×C8C22×D6111yescharacteristic
335#126—2441no—
336#125—1221no—
337#125—2441no—
338#124—1221no—
339#123—2441no—
340#123—2441no—
341C22⋊Q8—1221no—
342C22⋊Q8—1221no—
343C22⋊Q8—1221no—
344#121—1221no—
345#121—2441no—
346#120—2441no—
347C4×Q8—1221no—
348C4×D4—1221no—
349C4×D4D24411yescharacteristic
350#117—1221no—
351#117—2441no—
352C2×C4⋊C4—1221no—
353C2×C22⋊C4—2441no—
354Q32—2441no—
355SD32—1222no—
356SD32—2441no—
357C2×C16—41no—
358C2×C16—41no—
359C2×C16C22×D6111yescharacteristic
360C2×C16C22×D6111yescharacteristic
361C2×C16C22×D6111yescharacteristic
362C2×C16C22×D6111yescharacteristic
363#107—611no—
364#107—611no—
365#107—611no—
366#107—611no—
367#107—1221no—
368#106—1221no—
369#106—1221no—
370C4⋊C8D24411yescharacteristic
371#103—1221no—
372#103—1221no—
373#103—1221no—
374#103—1221no—
375#102—1221no—
376#102—1221no—
377C22⋊C8D24411yescharacteristic
378C22⋊C8D24411yescharacteristic
379C4×C8C22×D6111yescharacteristic
380C24—2441no—
381#41—4881no—
382#41—2442no—
383C2×Q8—1221no—
384C2×Q8—1221no—
385C2×D4—1221no—
386C2×D4—1221no—
387C2×D4—4881no—
388C2×D4—2442no—
389C2×D4—2441no—
390C2×D4—2441no—
391C2×D4—2441no—
392C2×D4D48811yescharacteristic
393C22×C4—2441no—
394C22×C4—2441no—
395C22×C4—2441no—
396C22×C4C2×D24411yescharacteristic
397C22×C4C2×D24411yescharacteristic
398Q16—1221no—
399Q16—1221no—
400Q16—1221no—
401Q16—1221no—
402SD16—4881no—
403SD16—2442no—
404D8—612no—
405D8—1221no—
406D8—1221no—
407D8—1221no—
408D8—2442no—
409C2×C8—21no—
410C2×C8—21no—
411C2×C8—21no—
412C2×C8—21no—
413C2×C8C2×D24411yescharacteristic
414C2×C8C2×D24411yescharacteristic
415C2×C8C2×D24411yescharacteristic
416C2×C8C23×D6111yescharacteristic
417C4⋊C4—1221no—
418C4⋊C4—1221no—
419C4⋊C4—1221no—
420C4⋊C4—1221no—
421C4⋊C4—2441no—
422C4⋊C4—2441no—
423C4⋊C4—2441no—
424C4⋊C4—2441no—
425C4⋊C4—2441no—
426C4⋊C4—2441no—
427C4⋊C4—2441no—
428C4⋊C4D48811yescharacteristic
429C22⋊C4—2441no—
430C22⋊C4—2441no—
431C22⋊C4—2441no—
432C22⋊C4—2441no—
433C22⋊C4—2441no—
434C22⋊C4—2441no—
435C22⋊C4—2441no—
436C22⋊C4—2441no—
437C22⋊C4D48811yescharacteristic
438C22⋊C4D48811yescharacteristic
439C42—2441no—
440C42—2441no—
441C42C2×D24411yescharacteristic
442C16—21no—
443C16—21no—
444C16—21no—
445C16—21no—
446C23—9761no—
447C23—4882no—
448C23—4882no—
449C23—2441no—
450C23—2441no—
451C23C2×D48811yescharacteristic
452C23C2×D48811yescharacteristic
453Q8—2441no—
454Q8—2441no—
455Q8—2441no—
456Q8—2441no—
457D4—22no—
458D4—1222no—
459D4—9761no—
460D4—4882no—
461D4—4882no—
462D4—2441no—
463D4—2441no—
464D4—2441no—
465C2×C4—21no—
466C2×C4—21no—
467C2×C4—21no—
468C2×C4—21no—
469C2×C4—9761no—
470C2×C4—4882no—
471C2×C4—4882no—
472C2×C4—2441no—
473C2×C4—2441no—
474C2×C4—2441no—
475C2×C4—2441no—
476C2×C4—2441no—
477C2×C4—2441no—
478C2×C4C2×D48811yescharacteristic
479C2×C4C2×D48811yescharacteristic
480C2×C4C2×D48811yescharacteristic
481C2×C4C2×D48811yescharacteristic
482C2×C4C22×D24411yescharacteristic
483C8—21no—
484C8—21no—
485C8—21no—
486C8#41×D6111yescharacteristic
487C8#1451982211yescharacteristic
488C22—21no—
489C22—21no—
490C22—21no—
491C22—21no—
492C22—21no—
493C22—9764no—
494C22—4881no—
495C22—4881no—
496C22—4881no—
497C22—4881no—
498C22—4881no—
499C22—2442no—
500C22#1684019412yes—
501C22C22×D48811yescharacteristic
502C4—21no—
503C4—21no—
504C4—21no—
505C4—21no—
506C4—4881no—
507C4—4881no—
508C4—4881no—
509C4—4881no—
510C4—4881no—
511C4—4881no—
512C4#1684040711yescharacteristic
513C4#1684057411yescharacteristic
514C2—22no—
515C2—41no—
516C2—9761no—
517C2—4882no—
518C2#2145763311yescharacteristic
519C2C2×#1684019411yescharacteristic
520C2#2145896711yescharacteristic
subgroupG/H
centerC22C22×D488
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingD4×C16×C61C2
socleC22×C61C22×D8
prime radical#21457506C2
hypercenterD4×C16D61
hypocenterC61#33863
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C4×D4 ◁ C8×D4 ◁ D4×C16

Complements

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

NH ≅ G/Ntype
1. #21458188C2semidirect
3. #21448051C2semidirect
4. #21448017C2semidirect
5. #21448011C2semidirect
6. #21447977C2semidirect
7. #21447915C2semidirect
8. #21447910C2semidirect
9. #21447882C2semidirect
10. #21447761C2semidirect
11. #21447750C2semidirect
12. C4×#14519722C2semidirect
13. #21447527C2semidirect
14. C2×#16839564C2semidirect
15. D4×C16×C61C2semidirect
20. C2×Dic488C22semidirect
23. #16840164C22semidirect
30. #16840003C22semidirect
35. C4×Dic244C22semidirect
37. #16839800C22semidirect
39. #16839590C22semidirect
40. #16839590C22semidirect
41. #16839588C22semidirect
42. #16839567C22semidirect
43. #16839567C22semidirect
44. #16839567C22semidirect
45. #16839567C22semidirect
53. C22×C16×C61C22semidirect
55. C61×C4⋊C16C22semidirect
57. C61×C22⋊C16C22semidirect
58. C4×C16×C61C22semidirect
266. D4×C16D61semidirect
284. C22×C16C2×D61semidirect
305. C4⋊C16C2×D61semidirect
317. C22⋊C16C2×D61semidirect
318. C4×C16C2×D61semidirect
319. C61#33863semidirect

Maximal subgroups

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

subgroupindex
#214581882
#214575062
#214480512
#214480172
#214480112
#214479772
#214479152
#214479102
#214478822
#214477612
#214477502
C4×#145197222
#214475272
C2×#168395642
D4×C16×C612
#3386361