Finite Group Databasebeta

← #30600609

Subgroups — #30600609

Order 15616.

562 conjugacy classes of subgroups.

The subgroup lattice has 518 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1D61⋊#436C211yesmaximal, characteristic
2#21461946C211yesmaximal, characteristic, complemented
3#21461687C211yesmaximal, characteristic
4D61⋊#332C211yesmaximal, characteristic, complemented
5#21460698C211yesmaximal, characteristic, complemented
6#21460648C211yesmaximal, characteristic, complemented
7#474×D61C211yesmaximal, characteristic
8C2×D61⋊M16—21no—
9#16840797—21no—
10#16840796—21no—
11#16840795—22no—
12D61⋊(C4⋊C8)C2211yescharacteristic
13D61⋊(C4⋊C8)C2211yescharacteristic
14C2×#14519840—21no—
15C2×#14519840—21no—
16C2×#14519839—21no—
17D61⋊(C22⋊C8)—21no—
18C4×D61⋊C8C2211yescharacteristic
19#16840708C2211yescharacteristic
20#16840635—21no—
21#16840621—22no—
22C4⋊Q8×D61C2211yescharacteristic
23C4×D4×D61C2211yescharacteristic
24#16840043C411yescharacteristic, complemented
25#16840042C411yescharacteristic, complemented
26#16840041—21no—
27#16840040—21no—
28#16840039C411yescharacteristic, complemented
29#107×D61—21no—
30C4⋊C8×D61C2211yescharacteristic
31#102×D61—21no—
32C61×#474C411yescharacteristic, complemented
33C2×#11758213—22no—
34D61⋊M16—42no—
35D61⋊M16—42no—
36C2×#11758217—22no—
37C2×D61⋊C8—21no—
38C2×D61⋊C8—21no—
39C2×D61⋊C8D411yescharacteristic
40C2×D61⋊C8D411yescharacteristic
41C2×C4⋊#6923—21no—
42C2×C4⋊#6923—21no—
43C2×C4⋊#6923—21no—
44#14519844—21no—
45#14519844D412yes—
46C4⋊#92420—21no—
47C4⋊#92420—21no—
48#14519840—42no—
49#14519840—42no—
50#14519839—42no—
51#14519834—22no—
52C22⋊#92420—22no—
53C4×#92420—21no—
54C4×#92420—21no—
55C2×Q8×D61—21no—
56C2×D4×D61D411yescharacteristic
57C22×C4×D61—21no—
58C2×C8×D61—21no—
59#14519777C2×C411yescharacteristic
60#14519776—21no—
61#14519776—21no—
62#14519775—21no—
63C4×D244C2×C411yescharacteristic
64D4×Dic61D411yescharacteristic
65#14519739D411yescharacteristic
66C4×C22⋊D61—21no—
67#14519737—21no—
68C4⋊C4×D61—21no—
69C4⋊C4×D61—21no—
70C4⋊C4×D61D411yescharacteristic
71C22⋊C4×D61—21no—
72C42×D61C2311yescharacteristic
73#14519711—21no—
74#14519710—21no—
75#14519710—21no—
76C4⋊#92418C2×C411yescharacteristic
77#14519706—21no—
78#14519699—21no—
79#14519698—21no—
80#14519697—21no—
81C4⋊Q8×C61C2×C411yescharacteristic
82C4×D4×C61C2×C411yescharacteristic
83#107×C61—21no—
84C4⋊C8×C61C2×C411yescharacteristic
85#102×C61—21no—
86C23×D61—21no—
87C22×#6923—41no—
88C2×Q8×C61—21no—
89C2×D4×C61C22⋊C411yescharacteristic
90C22×C4×C61—21no—
91C2×C22⋊D61—21no—
92C22×Dic61—21no—
93Q8×D61—41no—
94Q8×D61—41no—
95Q8×D61—41no—
96Q8×D61—41no—
97D4×D61—21no—
98D4×D61—21no—
99D4×D61—21no—
100D4×D61—21no—
101C2×D244C22⋊C411yescharacteristic
102C2×C4×D61—21no—
103C2×C4×D61—21no—
104C2×C4×D61—21no—
105C2×C4×D61—41no—
106C2×C4×D61—41no—
107C2×C4×D61—41no—
108C2×C4×D61—41no—
109C2×C4×D61—41no—
110C2×C4×D61C2×D411yescharacteristic
111C2×C4×D61C2×D411yescharacteristic
112C2×C4×D61#4111yescharacteristic
113C2×Dic122—21no—
114C2×Dic122—41no—
115#11758217—44no—
116C2×#92420—21no—
117C2×#92420—21no—
118C2×#92420—22no—
119C2×#92420—41no—
120C4⋊#6923—41no—
121C4⋊#6923—41no—
122C4⋊#6923—41no—
123C4⋊#6923—41no—
124C4⋊#6923—81no—
125#11758213—44no—
126D61⋊C8—21no—
127D61⋊C8—21no—
128D61⋊C8—21no—
129D61⋊C8—41no—
130D61⋊C8—41no—
131D61⋊C8—41no—
132D61⋊C8—41no—
133D61⋊C8D812yes—
134C2×C8×C61—21no—
135C4⋊C4×C61—21no—
136C4⋊C4×C61—21no—
137C4⋊C4×C61C22⋊C411yescharacteristic
138C22⋊C4×C61—21no—
139C42×C61C22×C411yescharacteristic
140C22⋊Dic61—21no—
141#11758198—21no—
142#11758198—21no—
143C4⋊Dic61—21no—
144C4⋊Dic61—21no—
145C4⋊Dic61C22⋊C411yescharacteristic
146#11758196—21no—
147#11758196—41no—
148#11758196—41no—
149C4×Dic61C2×D411yescharacteristic
150C4×Dic61C2×D411yescharacteristic
151C4×Dic61#4111yescharacteristic
152C2×#92418—21no—
153C8×D61—41no—
154C8×D61—41no—
155C23×C61—21no—
156C22×D61—21no—
157C22×D61—41no—
158C22×D61—41no—
159C22×D61—41no—
160C22×D61—41no—
161C22×D61—41no—
162C22×D61—41no—
163C22×D61#12111yescharacteristic
164C2×#6923—81no—
165C2×#6923—81no—
166C2×#6923—81no—
167Q8×C61—41no—
168Q8×C61—41no—
169D4×C61—21no—
170D4×C61—21no—
171C2×C4×C61—21no—
172C2×C4×C61—41no—
173C2×C4×C61—41no—
174C2×C4×C61—41no—
175C2×C4×C61C2×C22⋊C411yescharacteristic
176C2×C4×C61C4×D411yescharacteristic
177C2×C4×C61C4×D411yescharacteristic
178C22⋊D61—41no—
179C22⋊D61—41no—
180C2×Dic61—21no—
181C2×Dic61—41no—
182C2×Dic61—41no—
183C2×Dic61—41no—
184C2×Dic61#12011yescharacteristic
185C2×Dic61#12111yescharacteristic
186C2×Dic61#12311yescharacteristic
187D244—21no—
188D244—21no—
189C4×D61—21no—
190C4×D61—21no—
191C4×D61—41no—
192C4×D61—41no—
193C4×D61—41no—
194C4×D61—41no—
195C4×D61—41no—
196C4×D61—41no—
197C4×D61—81no—
198C4×D61—81no—
199C4×D61C2×D811yescharacteristic
200C4×D61#13511yescharacteristic
201C4×D61AGL(1,ℤ/8)11yescharacteristic
202C4×D61AGL(1,ℤ/8)11yescharacteristic
203C4×D61AGL(1,ℤ/8)11yescharacteristic
204C4×D61#13711yescharacteristic
205C4×D61#13711yescharacteristic
206C4×D61#13711yescharacteristic
207Dic122—41no—
208Dic122—41no—
209Dic122—81no—
210Dic122—81no—
211#92420—22no—
212#92420—42no—
213#92420—41no—
214#92420—81no—
215C8×C61—41no—
216#92418—41no—
217#9227—611nomaximal, Sylow
218C22×C61—41no—
219C22×C61—41no—
220C22×C61—41no—
221C22×C61#38611yescharacteristic
222C2×D61—41no—
223C2×D61—41no—
224C2×D61—41no—
225C2×D61—81no—
226C2×D61—81no—
227C2×D61—81no—
228C2×D61—81no—
229C2×D61#45911yescharacteristic
230C2×D61#46311yescharacteristic
231C2×D61#46811yescharacteristic
232C2×D61#47011yescharacteristic
233C2×D61#47111yescharacteristic
234C2×D61#47311yescharacteristic
235#6923—161no—
236C4×C61—21no—
237C4×C61—41no—
238C4×C61—81no—
239C4×C61#41711yescharacteristic
240C4×C61#41911yescharacteristic
241C4×C61C4×D811yescharacteristic
242C4×C61#44211yescharacteristic
243Dic61—21no—
244Dic61—41no—
245Dic61—81no—
246Dic61#44711yescharacteristic
247Dic61#44911yescharacteristic
248Dic61#46811yescharacteristic
249Dic61#47011yescharacteristic
250C2×#474—611no—
251C2×#436—611no—
252#1956—611no—
253#1802—611no—
254C2×#332—611no—
255#1296—611no—
256#1277—611no—
257C2×C61—81no—
258C2×C61#187911yescharacteristic
259C2×C61#189411yescharacteristic
260C2×C61#190811yescharacteristic
261D61—81no—
262D61#168511yescharacteristic
263D61#168911yescharacteristic
264D61#169911yescharacteristic
265D61#170311yescharacteristic
266C22×M16—1221no—
267C2×C4⋊Q8—611no—
268C2×C4×D4—611no—
269#474—611no—
270#474—611no—
271#474—611no—
272#474—1221no—
273#474—1221no—
274#474#692311yescharacteristic, complemented
275#436—1222no—
276#436—1221no—
277#436—1221no—
278C2×#107—1221no—
279C2×C4⋊C8—611no—
280C2×C4⋊C8—611no—
281C2×C4⋊C8—611no—
282C2×#103—1221no—
283C2×#103—1221no—
284C2×#102—1221no—
285C2×#102—1221no—
286C2×C22⋊C8—1221no—
287C2×C4×C8—611no—
288#384—611no—
289#340—1221no—
290#332—1222no—
291C61#922711yescharacteristic, complemented, Sylow
292C22×Q8—1221no—
293C22×D4—611no—
294C23×C4—1221no—
295C2×M16—1222no—
296C2×M16—1222no—
297C2×M16—2442no—
298C2×M16—2442no—
299C22×C8—611no—
300C22×C8—611no—
301C22×C8—1221no—
302C22×C8—1221no—
303C22×C8—1221no—
304C4⋊Q8—611no—
305C4⋊Q8—1221no—
306C4⋊Q8—1221no—
307C4⋊Q8—1221no—
308C4⋊Q8C2×#692311yescharacteristic
309C4×D4—611no—
310C4×D4—611no—
311C4×D4—611no—
312C4×D4—1221no—
313C4×D4—1221no—
314C4×D4C2×#692311yescharacteristic
315C2×C4⋊C4—611no—
316C2×C4⋊C4—1221no—
317C2×C4⋊C4—1221no—
318C2×C4⋊C4—1221no—
319C2×C4⋊C4—1221no—
320C2×C4⋊C4—1221no—
321C2×C22⋊C4—1221no—
322C2×C42—611no—
323#107—21no—
324#107—1221no—
325#107—1221no—
326#107—1221no—
327C4⋊C8—612no—
328C4⋊C8—611no—
329C4⋊C8—1221no—
330C4⋊C8—1221no—
331C4⋊C8—1221no—
332C4⋊C8—1221no—
333C4⋊C8C2×#692311yescharacteristic
334#103—2442no—
335#103—2442no—
336#102—21no—
337#102—1221no—
338#102—1221no—
339#102—1221no—
340#102—2442no—
341C22⋊C8—1222no—
342C22⋊C8—1222no—
343C4×C8—1221no—
344C4×C8—1221no—
345C24—1221no—
346C2×Q8—21no—
347C2×Q8—1221no—
348C2×Q8—2441no—
349C2×Q8—2441no—
350C2×Q8—2441no—
351C2×Q8—2441no—
352C2×Q8—2441no—
353C2×D4—611no—
354C2×D4—1221no—
355C2×D4—1221no—
356C2×D4—1221no—
357C2×D4—1221no—
358C2×D4—1221no—
359C2×D4C22⋊#692311yescharacteristic
360C22×C4—21no—
361C22×C4—611no—
362C22×C4—611no—
363C22×C4—611no—
364C22×C4—1221no—
365C22×C4—1221no—
366C22×C4—1221no—
367C22×C4—1221no—
368C22×C4—2441no—
369C22×C4—2441no—
370C22×C4—2441no—
371C22×C4—2441no—
372C22×C4—2441no—
373C22×C4—2441no—
374M16—2444no—
375M16—2444no—
376C2×C8—21no—
377C2×C8—612no—
378C2×C8—1221no—
379C2×C8—1221no—
380C2×C8—1221no—
381C2×C8—1222no—
382C2×C8—1221no—
383C2×C8—1221no—
384C2×C8—1221no—
385C2×C8—2441no—
386C2×C8—2441no—
387C2×C8—2441no—
388C2×C8—2441no—
389C2×C8—2441no—
390C2×C8—2441no—
391C2×C8—2441no—
392C4⋊C4—21no—
393C4⋊C4—21no—
394C4⋊C4—611no—
395C4⋊C4—1221no—
396C4⋊C4—1221no—
397C4⋊C4—1221no—
398C4⋊C4—4881no—
399C4⋊C4—2441no—
400C4⋊C4—2441no—
401C4⋊C4—2441no—
402C4⋊C4—2441no—
403C4⋊C4—2441no—
404C4⋊C4—2441no—
405C4⋊C4C22⋊#692311yescharacteristic
406C22⋊C4—21no—
407C22⋊C4—1221no—
408C22⋊C4—1221no—
409C22⋊C4—1221no—
410C42—611no—
411C42—611no—
412C42—611no—
413C42C22×#692311yescharacteristic
414C23—21no—
415C23—611no—
416C23—1221no—
417C23—2441no—
418C23—2441no—
419C23—2441no—
420C23—2441no—
421C23—2441no—
422C23—2441no—
423Q8—41no—
424Q8—41no—
425Q8—4881no—
426Q8—4881no—
427Q8—2441no—
428Q8—2441no—
429D4—21no—
430D4—21no—
431D4—1221no—
432D4—1221no—
433D4—2441no—
434D4—2441no—
435C2×C4—21no—
436C2×C4—41no—
437C2×C4—41no—
438C2×C4—41no—
439C2×C4—611no—
440C2×C4—611no—
441C2×C4—611no—
442C2×C4—611no—
443C2×C4—611no—
444C2×C4—611no—
445C2×C4—611no—
446C2×C4—611no—
447C2×C4—611no—
448C2×C4—611no—
449C2×C4—611no—
450C2×C4—1221no—
451C2×C4—1221no—
452C2×C4—1221no—
453C2×C4—4881no—
454C2×C4—4881no—
455C2×C4—4881no—
456C2×C4—4881no—
457C2×C4—4881no—
458C2×C4—2441no—
459C2×C4—2441no—
460C2×C4—2441no—
461C2×C4—2441no—
462C2×C4—2441no—
463C2×C4—2441no—
464C2×C4—2441no—
465C2×C4—2441no—
466C2×C4—2441no—
467C2×C4C2×C22⋊#692311yescharacteristic
468C2×C4D4×#692311yescharacteristic
469C2×C4D4×#692311yescharacteristic
470C8—41no—
471C8—1222no—
472C8—4881no—
473C8—2441no—
474C8—2442no—
475C8—2441no—
476C22—41no—
477C22—41no—
478C22—41no—
479C22—611no—
480C22—611no—
481C22—611no—
482C22—611no—
483C22—611no—
484C22—611no—
485C22—4881no—
486C22—4881no—
487C22—4881no—
488C22—4881no—
489C22—2441no—
490C22—2441no—
491C22—2441no—
492C22#1684071011yescharacteristic
493C4—21no—
494C4—41no—
495C4—81no—
496C4—611no—
497C4—611no—
498C4—611no—
499C4—611no—
500C4—1221no—
501C4—9761no—
502C4—4881no—
503C4—2441no—
504C4#1684075311yescharacteristic
505C4#1684075511yescharacteristic
506C4D8×#692311yescharacteristic
507C4#1684080311yescharacteristic
508C2—81no—
509C2—611no—
510C2—611no—
511C2—611no—
512C2—611no—
513C2—4881no—
514C2#2146181411yescharacteristic
515C2#2146183411yescharacteristic
516C2#2146185911yescharacteristic
subgroupG/H
centerC22#16840710
commutatorC42×C61C22×C4
FrattiniC42C22×#6923
FittingC61×#474C4
socleC22×C61#386
prime radicalC2×D4×D61D4
cosocleC42×D61C23
hypercenter#474#6923
hypocenterC61#9227
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #474

Complements

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

NH ≅ G/Ntype
2. #21461946C2semidirect
4. D61⋊#332C2semidirect
5. #21460698C2semidirect
6. #21460648C2semidirect
24. #16840043C4semidirect
25. #16840042C4semidirect
28. #16840039C4semidirect
32. C61×#474C4semidirect
274. #474#6923semidirect
291. C61#9227semidirect

Maximal subgroups

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

subgroupindex
D61⋊#4362
#214619462
#214616872
D61⋊#3322
#214606982
#214606482
#474×D612
#922761