Finite Group Databasebeta

← #29651759

Subgroups — #29651759

Order 15616.

652 conjugacy classes of subgroups.

The subgroup lattice has 510 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21454945C211yesmaximal, characteristic, complemented
2#21454944C211yesmaximal, characteristic, complemented
3C4×#14519797C211yesmaximal, characteristic
4#21444725C212yesmaximal, complemented
5#21444658C212yesmaximal, complemented
6#21444652C212yesmaximal, complemented
7#21444640C212yesmaximal, complemented
8#21444574C211yesmaximal, characteristic, complemented
9#21444259C211yesmaximal, characteristic, complemented
10#21444106C211yesmaximal, characteristic, complemented
11C61×#1489C211yesmaximal, characteristic, complemented
12C2×#14519797C411yescharacteristic
13C4×#11758222C2211yescharacteristic
14#16839893C2212yes—
15#16839889C2212yes—
16C4×#11758202C411yescharacteristic
17C4×#11758202C2211yescharacteristic
18C4×#11758201C411yescharacteristic
19C4×#11758201C2211yescharacteristic
20C4×#11758200C411yescharacteristic
21C4×#11758200C2211yescharacteristic
22C4×#11758199C411yescharacteristic
23C4×#11758199C2211yescharacteristic
24#16839834—22no—
25#16839831—22no—
26D4×#92418C2212yes—
27#16839827C2212yes—
28#16839819—21no—
29#16839818—21no—
30#16839815C2211yescharacteristic
31#16839798C411yescharacteristic
32#16839786C2211yescharacteristic
33C2×C4⋊#92418—21no—
34C4⋊C8×D61—21no—
35#16839758C411yescharacteristic
36#16839756C411yescharacteristic
37C2×C4×#92418C2211yescharacteristic
38#16839512C2212yes—
39#16839512C2212yes—
40#16839494C2212yes—
41#16839491C2212yes—
42#16839484C2212yes—
43#16839484C2212yes—
44C4×#41×C61C2211yescharacteristic
45C61×#423C2211yescharacteristic
46C4×M16×C61C2211yescharacteristic
47C61×Q8⋊C8C2212yes—
48C61×D4⋊C8C2212yes—
49C2×#11758222C2×C411yescharacteristic
50#14519797—24no—
51C2×#11758202C2×C411yescharacteristic
52C2×#11758200C2×C411yescharacteristic
53C2×#11758201C2×C411yescharacteristic
54C2×#11758199C2×C411yescharacteristic
55C2×#11758194—21no—
56C22×#92418—22no—
57C22×#92418C2×C411yescharacteristic
58C22×#92418C2×C411yescharacteristic
59C2×C8×D61—22no—
60C4×Dic122C2×C411yescharacteristic
61C4×Dic122C2311yescharacteristic
62C4×D244C2×C411yescharacteristic
63C4×D244C2311yescharacteristic
64C4×C22⋊D61—21no—
65C4×C22⋊D61—21no—
66#14519736C2×C411yescharacteristic
67#14519734—21no—
68C42×D61—21no—
69#14519710C2×C411yescharacteristic
70#14519710C2×C411yescharacteristic
71#14519708C2×C411yescharacteristic
72#14519708C2×C411yescharacteristic
73C4⋊#92418—21no—
74C4⋊#92418—22no—
75C4⋊#92418D411yescharacteristic
76C4⋊#92418D411yescharacteristic
77C4⋊#92418C2312yes—
78#14519706—22no—
79#14519701C2×C411yescharacteristic
80#14519701C2×C411yescharacteristic
81#14519700C2×C411yescharacteristic
82#14519700C2×C411yescharacteristic
83#14519698C2×C411yescharacteristic
84#14519698C2×C411yescharacteristic
85#14519697C2×C411yescharacteristic
86#14519697C2×C411yescharacteristic
87C22⋊#92418—22no—
88C22⋊#92418D412yes—
89#14519689—22no—
90#14519688—22no—
91#14519687—22no—
92C4×#92418C2×C411yescharacteristic
93C4×#92418C2×C411yescharacteristic
94C4×#92418C2311yescharacteristic
95C4×#92418C2311yescharacteristic
96C4×#92418C2312yes—
97C2×#41×C61C2×C411yescharacteristic
98C2×M16×C61—21no—
99C2×M16×C61—21no—
100C4×Q8×C61C2×C411yescharacteristic
101C4×Q8×C61C2311yescharacteristic
102C4×D4×C61—21no—
103C4×D4×C61—21no—
104C4×D4×C61C2×C411yescharacteristic
105C4×D4×C61C2311yescharacteristic
106#117×C61—21no—
107#117×C61C2×C411yescharacteristic
108C2×C42×C61—21no—
109C2×C42×C61C2311yescharacteristic
110C4⋊C8×C61—21no—
111C4⋊C8×C61C2312yes—
112#97×C61—21no—
113C4×C8×C61C2312yes—
114#41×C61—22no—
115#41×C61—41no—
116C2×Q8×C61C22×C411yescharacteristic
117C2×D4×C61—21no—
118C2×D4×C61C22×C411yescharacteristic
119C22×C4×C61—21no—
120C22×C4×C61—21no—
121C22×C4×C61—21no—
122C22×C4×C61C22×C411yescharacteristic
123C22×C4×C61C2×D411yescharacteristic
124C22×C4×C61#4111yescharacteristic
125C2×C22⋊D61—21no—
126#11758222—22no—
127#11758222—41no—
128C2×D244C22×C411yescharacteristic
129C2×C4×D61—21no—
130C2×C4×D61—21no—
131C2×C4×D61—21no—
132C2×Dic122C22×C411yescharacteristic
133M16×C61—42no—
134M16×C61—81no—
135C2×C8×C61—22no—
136C2×C8×C61—22no—
137C4⋊C4×C61—21no—
138C4⋊C4×C61—21no—
139C4⋊C4×C61C22×C411yescharacteristic
140C4⋊C4×C61C22×C411yescharacteristic
141C22⋊C4×C61—21no—
142C22⋊C4×C61—21no—
143C22⋊C4×C61—21no—
144C42×C61—21no—
145C42×C61—21no—
146C42×C61—21no—
147C42×C61C22×C411yescharacteristic
148C42×C61C2×D411yescharacteristic
149C42×C61#4111yescharacteristic
150C42×C61C2411yescharacteristic
151C22⋊Dic61—21no—
152#11758202—22no—
153#11758201—22no—
154#11758200—22no—
155#11758199M1614yescomplemented
156#11758198—21no—
157#11758198—21no—
158C4⋊Dic61C22×C411yescharacteristic
159C4⋊Dic61C22×C411yescharacteristic
160#11758196—21no—
161#11758196—21no—
162C4×Dic61—21no—
163C4×Dic61—21no—
164C4×Dic61—21no—
165#11758194—81no—
166C2×#92418—22no—
167C2×#92418—22no—
168C2×#92418—41no—
169C2×#92418—82no—
170C2×#92418C22×C411yescharacteristic
171C2×#92418C22×C411yescharacteristic
172C2×#92418C22×C411yescharacteristic
173C2×#92418C22×C411yescharacteristic
174C2×#92418C2×D412yes—
175C2×#92418C2×D412yes—
176C8×D61—82no—
177C23×C61—21no—
178C23×C61C4×D411yescharacteristic
179C22×D61—21no—
180Q8×C61—21no—
181Q8×C61—21no—
182D4×C61—21no—
183D4×C61—42no—
184D4×C61C2×M1612yes—
185C2×C4×C61—21no—
186C2×C4×C61—21no—
187C2×C4×C61—21no—
188C2×C4×C61—21no—
189C2×C4×C61—22no—
190C2×C4×C61—21no—
191C2×C4×C61—21no—
192C2×C4×C61—21no—
193C2×C4×C61—42no—
194C2×C4×C61—41no—
195C2×C4×C61—41no—
196C2×C4×C61—42no—
197C2×C4×C61C4×D411yescharacteristic
198C2×C4×C61C4×D411yescharacteristic
199C2×C4×C61C4×D411yescharacteristic
200C2×C4×C61C23×C411yescharacteristic
201C2×C4×C61C22×D411yescharacteristic
202C2×C4×C61C2×#4111yescharacteristic
203C22⋊D61—42no—
204C2×Dic61—21no—
205C2×Dic61—21no—
206C2×Dic61—21no—
207D244—21no—
208D244C2×M1612yes—
209C4×D61—42no—
210C4×D61—41no—
211C4×D61—41no—
212C4×D61—42no—
213Dic122—21no—
214Dic122—21no—
215C8×C61—42no—
216C8×C61—42no—
217#92418—21no—
218#92418—21no—
219#92418—21no—
220#92418—22no—
221#92418—22no—
222#92418—42no—
223#92418C2×M1612yes—
224#18065—611nomaximal, Sylow
225C22×C61—21no—
226C22×C61—22no—
227C22×C61—21no—
228C22×C61—21no—
229C22×C61—41no—
230C22×C61—41no—
231C22×C61C2×C4×D411yescharacteristic
232C2×D61—22no—
233C2×D61—41no—
234C2×D61—41no—
235C4×C61—21no—
236C4×C61—21no—
237C4×C61—21no—
238C4×C61—41no—
239C4×C61—41no—
240C4×C61—41no—
241C4×C61#44412yes—
242C4×C61C22×M1611yescharacteristic
243C4×C61#56811yescharacteristic
244C4×C61#57711yescharacteristic
245C4×C61#57711yescharacteristic
246Dic61—41no—
247Dic61—41no—
248Dic61—41no—
249#2990—611no—
250#2990—611no—
251C4×#135—611no—
252#1593—612no—
253#1582—612no—
254#1582—612no—
255#1580—612no—
256#1566—611no—
257#1489—611no—
258#1489D6111yescharacteristic, complemented
259#1451—611no—
260C2×C61—41no—
261C2×C61—42no—
262C2×C61D4×M1611yescharacteristic
263C2×C61#295211yescharacteristic
264C2×C61C2×#44411yescharacteristic
265D61—42no—
266C2×#135—611no—
267C4×#41—611no—
268C4×#41C2×D6111yescharacteristic
269#446—612no—
270#446—612no—
271C4×Q16—611no—
272C4×Q16—611no—
273C4×SD16—611no—
274C4×SD16—611no—
275C4×SD16—611no—
276C4×SD16—611no—
277C4×D8—611no—
278C4×D8—611no—
279#435—1222no—
280#435—1222no—
281C8×D4—612no—
282C8×D4—612no—
283#432—1221no—
284#432—1221no—
285#431—611no—
286#427—611no—
287#423—611no—
288#423C2×D6111yescharacteristic
289C2×C4⋊C8—1221no—
290C2×C4⋊C8—1221no—
291#416—611no—
292#416—611no—
293C4×M16C2×D6111yescharacteristic
294C2×C4×C8—611no—
295#334—612no—
296#334—612no—
297Q8⋊C8—612no—
298Q8⋊C8C2×D6112yes—
299D4⋊C8—612no—
300D4⋊C8C2×D6112yes—
301#322—612no—
302#322—612no—
303C61#1806511yescharacteristic, complemented, Sylow
304C2×#41—611no—
305C2×#41C4×D6111yescharacteristic
306#135—1224no—
307C2×Q16—611no—
308C2×SD16—611no—
309C2×SD16—611no—
310C2×D8—611no—
311C2×M16—21no—
312C2×M16—21no—
313C2×M16—1221no—
314C22×C8—611no—
315C22×C8—611no—
316C22×C8—1222no—
317C22×C8—1222no—
318C4×Q8—611no—
319C4×Q8—611no—
320C4×Q8C4×D6111yescharacteristic
321C4×Q8C22×D6111yescharacteristic
322C4×D4—21no—
323C4×D4—21no—
324C4×D4—611no—
325C4×D4—611no—
326C4×D4—1221no—
327C4×D4—1221no—
328C4×D4C4×D6111yescharacteristic
329C4×D4C22×D6111yescharacteristic
330#117—21no—
331#117—611no—
332#117—1221no—
333#117C4×D6111yescharacteristic
334C2×C42—21no—
335C2×C42—1221no—
336C2×C42C22×D6111yescharacteristic
337#107—611no—
338#107—611no—
339#106—611no—
340#106—611no—
341C4⋊C8—21no—
342C4⋊C8—611no—
343C4⋊C8—611no—
344C4⋊C8—612no—
345C4⋊C8—1222no—
346C4⋊C8—1222no—
347C4⋊C8—1221no—
348C4⋊C8C22×D6112yes—
349#103—611no—
350#103—611no—
351#103—611no—
352#103—611no—
353#102—611no—
354#102—611no—
355#102—611no—
356#102—611no—
357C22⋊C8—612no—
358C22⋊C8—1222no—
359C22⋊C8—1222no—
360#97—21no—
361#97—1222no—
362#97—1222no—
363C4×C8—611no—
364C4×C8—611no—
365C4×C8—611no—
366C4×C8—611no—
367C4×C8—612no—
368C4×C8C22×D6112yes—
369#41—22no—
370#41—41no—
371#41—1222no—
372#41—2441no—
373C2×Q8—611no—
374C2×Q8C2×C4×D6111yescharacteristic
375C2×D4—21no—
376C2×D4—611no—
377C2×D4—1221no—
378C2×D4C2×C4×D6111yescharacteristic
379C22×C4—21no—
380C22×C4—21no—
381C22×C4—21no—
382C22×C4—1221no—
383C22×C4—1221no—
384C22×C4—1221no—
385C22×C4C2×C4×D6111yescharacteristic
386C22×C4D4×D6111yescharacteristic
387C22×C4#1175822411yescharacteristic
388Q16—1222no—
389SD16—1222no—
390SD16—1222no—
391D8—614no—
392M16—42no—
393M16—81no—
394M16—4881no—
395C2×C8—22no—
396C2×C8—22no—
397C2×C8—611no—
398C2×C8—611no—
399C2×C8—611no—
400C2×C8—611no—
401C2×C8—612no—
402C2×C8—612no—
403C2×C8—1222no—
404C2×C8—1222no—
405C2×C8—4882no—
406C2×C8—4882no—
407C2×C8—2441no—
408C4⋊C4—21no—
409C4⋊C4—21no—
410C4⋊C4—611no—
411C4⋊C4—611no—
412C4⋊C4—1221no—
413C4⋊C4—1221no—
414C4⋊C4C2×C4×D6111yescharacteristic
415C4⋊C4C2×C4×D6111yescharacteristic
416C22⋊C4—21no—
417C22⋊C4—21no—
418C22⋊C4—21no—
419C22⋊C4—1221no—
420C22⋊C4—1221no—
421C22⋊C4—1221no—
422C42—21no—
423C42—21no—
424C42—21no—
425C42—1221no—
426C42—1221no—
427C42—1221no—
428C42C2×C4×D6111yescharacteristic
429C42D4×D6111yescharacteristic
430C42#1175822411yescharacteristic
431C42C23×D6111yescharacteristic
432C23—21no—
433C23—1221no—
434C23#1451973711yescharacteristic
435Q8—21no—
436Q8—21no—
437Q8—1221no—
438Q8—1221no—
439D4—21no—
440D4—42no—
441D4—612no—
442D4—1221no—
443D4—2442no—
444D4M16×D6112yes—
445C2×C4—21no—
446C2×C4—21no—
447C2×C4—21no—
448C2×C4—21no—
449C2×C4—22no—
450C2×C4—21no—
451C2×C4—21no—
452C2×C4—21no—
453C2×C4—42no—
454C2×C4—41no—
455C2×C4—41no—
456C2×C4—42no—
457C2×C4—1221no—
458C2×C4—1221no—
459C2×C4—1221no—
460C2×C4—2442no—
461C2×C4—2441no—
462C2×C4—2441no—
463C2×C4—2442no—
464C2×C4#1451973711yescharacteristic
465C2×C4#1451973711yescharacteristic
466C2×C4#1451973711yescharacteristic
467C2×C4C22×C4×D6111yescharacteristic
468C2×C4C2×D4×D6111yescharacteristic
469C2×C4C2×#1175822411yescharacteristic
470C8—42no—
471C8—42no—
472C8—612no—
473C8—1221no—
474C8—1221no—
475C8—1221no—
476C8—1222no—
477C8—1222no—
478C8—2442no—
479C22—21no—
480C22—21no—
481C22—21no—
482C22—22no—
483C22—41no—
484C22—41no—
485C22—1222no—
486C22—2441no—
487C22—2441no—
488C22C2×#1451973711yescharacteristic
489C4—21no—
490C4—21no—
491C4—21no—
492C4—41no—
493C4—41no—
494C4—41no—
495C4—2441no—
496C4—2441no—
497C4—2441no—
498C4#1683988112yes—
499C4C2×M16×D6111yescharacteristic
500C4#1684052811yescharacteristic
501C4#1684057111yescharacteristic
502C4#1684057211yescharacteristic
503C2—41no—
504C2—42no—
505C2—2442no—
506C2#2145455111yescharacteristic
507C2#2145469811yescharacteristic
508C2C2×#1683988111yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC42C23×D61
FittingC61×#1489C2
socleC22×C61C2×C4×D4
prime radicalC2×#14519797C4
cosocleC42×C61C24
hypercenter#1489D61
hypocenterC61#18065
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1489

Complements

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

NH ≅ G/Ntype
1. #21454945C2semidirect
2. #21454944C2semidirect
4. #21444725C2semidirect
5. #21444658C2semidirect
6. #21444652C2semidirect
7. #21444640C2semidirect
8. #21444574C2semidirect
9. #21444259C2semidirect
10. #21444106C2semidirect
11. C61×#1489C2semidirect
155. #11758199M16semidirect
258. #1489D61semidirect
303. C61#18065semidirect

Maximal subgroups

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

subgroupindex
#214549452
#214549442
C4×#145197972
#214447252
#214446582
#214446522
#214446402
#214445742
#214442592
#214441062
C61×#14892
#1806561