Finite Group Databasebeta

← #30600129

Subgroups — #30600129

Order 15616.

690 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 690 conjugacy classes of subgroups — too many to draw. Only its 106 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21462349C211yesmaximal, characteristic
2#21461998C211yesmaximal, characteristic, complemented
3#21461985C211yesmaximal, characteristic, complemented
4D61⋊#331C211yesmaximal, characteristic
5#21460654C211yesmaximal, characteristic, complemented
6#21460654C211yesmaximal, characteristic, complemented
7#486×D61C211yesmaximal, characteristic
8C2×D4×#6923—21no—
9C2×D4×#6923—21no—
10D61⋊(C4⋊C8)C2211yescharacteristic
11D61⋊(C4⋊C8)C2211yescharacteristic
12C2×#14519840—21no—
13C2×#14519840—21no—
14#16840714—21no—
15#16840714—21no—
16#16840710—22no—
17C4×C4⋊#6923C2211yescharacteristic
18C4×C4⋊#6923C2211yescharacteristic
19#16840635—21no—
20#16840635—21no—
21#16840620—22no—
22C4⋊Q8×D61C2211yescharacteristic
23#127×D61C2211yescharacteristic
24#16840110C411yescharacteristic, complemented
25#16840109C411yescharacteristic, complemented
26#16840108C411yescharacteristic, complemented
27#16840107—22no—
28#102×D61—22no—
29C4×C8×D61C2211yescharacteristic
30C61×#486C411yescharacteristic, complemented
31C23×#6923—22no—
32C2×D61⋊C8—21no—
33C2×D61⋊C8—21no—
34D4×#6923—42no—
35D4×#6923—42no—
36D4×#6923—42no—
37D4×#6923—42no—
38C2×C4⋊#6923D411yescharacteristic
39C2×C4⋊#6923D411yescharacteristic
40C2×C4⋊#6923D411yescharacteristic
41C2×C4⋊#6923D411yescharacteristic
42C2×C22⋊#6923—22no—
43C2×C22⋊#6923—22no—
44C2×C22⋊#6923—22no—
45C2×C4×#6923—21no—
46C2×C4×#6923—21no—
47#14519844—21no—
48#14519844—21no—
49C4⋊#92420—21no—
50C4⋊#92420—21no—
51#14519840—42no—
52#14519840—42no—
53(C2×#6923)⋊C4—21no—
54(C2×#6923)⋊C4—21no—
55C2×Q8×D61—21no—
56C2×D4×D61—21no—
57C2×D4×D61—21no—
58C2×D4×D61D412yes—
59C2×C8×D61—21no—
60#14519777C2×C411yescharacteristic, complemented
61#14519776—21no—
62#14519776—21no—
63#14519775—21no—
64#14519774C2×C411yescharacteristic
65#14519773—21no—
66#14519773—21no—
67#14519773D412yes—
68C4⋊C4×D61—21no—
69C4⋊C4×D61—21no—
70C42×D61C2311yescharacteristic
71#14519699—22no—
72#14519698—22no—
73#14519697—22no—
74C4×#92418C2×C411yescharacteristic, complemented
75C8×Dic61—21no—
76C4⋊Q8×C61C2×C411yescharacteristic, complemented
77#127×C61C2×C411yescharacteristic
78#102×C61—22no—
79C4×C8×C61C2×C411yescharacteristic, complemented
80C23×D61—22no—
81C22×#6923—21no—
82C22×#6923—21no—
83C22×#6923—21no—
84C22×#6923—21no—
85C22×#6923—44no—
86C22×#6923—44no—
87C22×#6923—82no—
88C2×Q8×C61—21no—
89C2×D4×C61—21no—
90C2×D4×C61—21no—
91C2×D4×C61C22⋊C412yes—
92C2×C22⋊D61—22no—
93C2×C22⋊D61—22no—
94C2×C22⋊D61—22no—
95Q8×D61—41no—
96Q8×D61—41no—
97Q8×D61—41no—
98Q8×D61—41no—
99D4×D61—42no—
100D4×D61—42no—
101D4×D61—41no—
102D4×D61—41no—
103D4×D61—41no—
104D4×D61—41no—
105D4×D61—41no—
106D4×D61—41no—
107D4×D61—41no—
108D4×D61—41no—
109C2×D244—21no—
110C2×D244—21no—
111C2×D244C22⋊C412yes—
112C2×C4×D61—41no—
113C2×C4×D61C2×D411yescharacteristic
114C2×C4×D61C2×D411yescharacteristic
115C2×C4×D61C2×D411yescharacteristic
116C2×Dic122—21no—
117C2×Dic122—41no—
118C22⋊#6923—44no—
119C22⋊#6923—44no—
120C22⋊#6923—82no—
121C2×#92420—41no—
122C2×#92420—41no—
123C4⋊#6923—21no—
124C4⋊#6923—21no—
125C4⋊#6923—21no—
126C4⋊#6923—21no—
127C4⋊#6923—21no—
128C4⋊#6923—21no—
129C4⋊#6923D812yes—
130C4⋊#6923SD1612yes—
131C4×#6923—41no—
132C4×#6923—41no—
133C4×#6923—41no—
134C4×#6923—41no—
135D61⋊C8—41no—
136D61⋊C8—41no—
137D61⋊C8—41no—
138D61⋊C8—41no—
139C2×C8×C61—21no—
140C4⋊C4×C61—21no—
141C4⋊C4×C61—21no—
142C42×C61C22×C411yescharacteristic
143C4⋊Dic61—21no—
144C4⋊Dic61—21no—
145#11758196—41no—
146#11758196—41no—
147C4×Dic61C2×D411yescharacteristic
148C4×Dic61#4111yescharacteristic
149C4×Dic61#4111yescharacteristic
150C2×#92418—21no—
151C8×D61—41no—
152C8×D61—41no—
153C23×C61—22no—
154C22×D61—22no—
155C22×D61—42no—
156C22×D61—42no—
157C22×D61—42no—
158C22×D61—42no—
159C22×D61—42no—
160C22×D61—42no—
161C22×D61#12011yescharacteristic
162C2×#6923—22no—
163C2×#6923—22no—
164C2×#6923—41no—
165C2×#6923—41no—
166C2×#6923—41no—
167C2×#6923—41no—
168C2×#6923—41no—
169C2×#6923—41no—
170C2×#6923—41no—
171C2×#6923—41no—
172C2×#6923—41no—
173C2×#6923—41no—
174C2×#6923—88no—
175Q8×C61—41no—
176Q8×C61—41no—
177D4×C61—42no—
178D4×C61—41no—
179D4×C61—41no—
180D4×C61—41no—
181D4×C61—41no—
182C2×C4×C61—41no—
183C2×C4×C61C2×C22⋊C411yescharacteristic
184C2×C4×C61C4×D411yescharacteristic
185C2×C4×C61C4×D411yescharacteristic
186C22⋊D61—42no—
187C22⋊D61—42no—
188C22⋊D61—42no—
189C22⋊D61—42no—
190C22⋊D61—82no—
191C2×Dic61—41no—
192C2×Dic61#12111yescharacteristic
193C2×Dic61#12111yescharacteristic
194C2×Dic61#12311yescharacteristic
195D244—42no—
196D244—41no—
197D244—41no—
198D244—41no—
199D244—41no—
200C4×D61—21no—
201C4×D61—21no—
202C4×D61—81no—
203C4×D61—81no—
204C4×D61C2×D811yescharacteristic
205C4×D61C2×SD1611yescharacteristic
206C4×D61#13511yescharacteristic
207C4×D61#13511yescharacteristic
208C4×D61AGL(1,ℤ/8)11yescharacteristic
209C4×D61AGL(1,ℤ/8)11yescharacteristic
210C4×D61AGL(1,ℤ/8)11yescharacteristic
211C4×D61#13711yescharacteristic
212Dic122—41no—
213Dic122—41no—
214Dic122—81no—
215Dic122—81no—
216#92420—81no—
217#92420—81no—
218C8×C61—41no—
219#92418—41no—
220#9057—611nomaximal, Sylow
221C22×C61—42no—
222C22×C61—42no—
223C22×C61—42no—
224C22×C61#38611yescharacteristic
225C2×D61—42no—
226C2×D61—42no—
227C2×D61—42no—
228C2×D61—82no—
229C2×D61—82no—
230C2×D61—82no—
231C2×D61—82no—
232C2×D61#44711yescharacteristic
233C2×D61#44911yescharacteristic
234C2×D61#45011yescharacteristic
235C2×D61#45211yescharacteristic
236C2×D61#45511yescharacteristic
237C2×D61#45511yescharacteristic
238#6923—42no—
239#6923—42no—
240#6923—81no—
241#6923—81no—
242C4×C61—21no—
243C4×C61—81no—
244C4×C61C4×D811yescharacteristic
245C4×C61C4×SD1611yescharacteristic
246C4×C61#44011yescharacteristic
247C4×C61#44211yescharacteristic
248Dic61—21no—
249Dic61—81no—
250Dic61#45911yescharacteristic
251Dic61#46111yescharacteristic
252Dic61#46311yescharacteristic
253Dic61#46311yescharacteristic
254C2×#486—611no—
255#2366—611no—
256#1988—611no—
257#1981—611no—
258C2×#331—611no—
259#1279—611no—
260#1279—611no—
261C2×C61—82no—
262C2×C61#187711yescharacteristic
263C2×C61#187911yescharacteristic
264C2×C61#188511yescharacteristic
265D61—82no—
266D61#163911yescharacteristic
267D61#164011yescharacteristic
268D61#164111yescharacteristic
269D61#164211yescharacteristic
270C2×C4⋊Q8—611no—
271C2×#127—611no—
272C2×C4×D4—1221no—
273C2×C4×D4—1221no—
274#486—611no—
275#486—611no—
276#486—611no—
277#486—1222no—
278#486#692311yescharacteristic, complemented
279C2×C4⋊C8—611no—
280C2×C4⋊C8—611no—
281C2×#103—1221no—
282C2×#103—1221no—
283C2×#102—1222no—
284C2×C4×C8—611no—
285#390—1221no—
286#390—1221no—
287#386—1222no—
288C4×C4⋊C4—611no—
289C4×C4⋊C4—611no—
290#340—1221no—
291#340—1221no—
292#331—1222no—
293C61#905711yescharacteristic, complemented, Sylow
294C22×Q8—1221no—
295C22×D4—612no—
296C22×D4—1221no—
297C22×D4—1221no—
298C23×C4—1222no—
299C22×C8—1221no—
300C22×C8—1221no—
301C22×C8—1221no—
302C4⋊Q8—611no—
303C4⋊Q8—1221no—
304C4⋊Q8—1221no—
305C4⋊Q8—1221no—
306C4⋊Q8C2×#692311yescharacteristic, complemented
307#127—612no—
308#127—611no—
309#127—1221no—
310#127—1221no—
311#127C2×#692311yescharacteristic
312C4×D4—2442no—
313C4×D4—2442no—
314C4×D4—2442no—
315C4×D4—2442no—
316C2×C4⋊C4—611no—
317C2×C4⋊C4—611no—
318C2×C4⋊C4—611no—
319C2×C4⋊C4—611no—
320C2×C4⋊C4—1221no—
321C2×C4⋊C4—1221no—
322C2×C22⋊C4—1222no—
323C2×C22⋊C4—1222no—
324C2×C22⋊C4—1222no—
325C2×C42—611no—
326C2×C42—1221no—
327C2×C42—1221no—
328C4⋊C8—1221no—
329C4⋊C8—1221no—
330C4⋊C8—1221no—
331C4⋊C8—1221no—
332#103—2442no—
333#103—2442no—
334#102—22no—
335#102—1222no—
336#102—1222no—
337#102—1222no—
338C4×C8—611no—
339C4×C8—1221no—
340C4×C8C2×#692311yescharacteristic, complemented
341#95—1221no—
342#95—1221no—
343C24—1222no—
344C2×Q8—21no—
345C2×Q8—1221no—
346C2×Q8—2441no—
347C2×Q8—2441no—
348C2×Q8—2441no—
349C2×Q8—2441no—
350C2×Q8—2441no—
351C2×D4—21no—
352C2×D4—21no—
353C2×D4—612no—
354C2×D4—1222no—
355C2×D4—1222no—
356C2×D4—1222no—
357C2×D4—1221no—
358C2×D4—1221no—
359C2×D4—2441no—
360C2×D4—2441no—
361C2×D4—2441no—
362C2×D4—2441no—
363C2×D4—2441no—
364C2×D4—2441no—
365C2×D4—2441no—
366C2×D4—2441no—
367C2×D4—2442no—
368C2×D4—2442no—
369C2×D4C22⋊#692312yes—
370C22×C4—611no—
371C22×C4—611no—
372C22×C4—611no—
373C22×C4—1221no—
374C22×C4—1221no—
375C22×C4—1221no—
376C22×C4—1221no—
377C22×C4—4882no—
378C22×C4—2441no—
379C22×C4—2444no—
380C22×C4—2444no—
381C2×C8—21no—
382C2×C8—1221no—
383C2×C8—2441no—
384C2×C8—2441no—
385C2×C8—2441no—
386C2×C8—2441no—
387C2×C8—2441no—
388C2×C8—2441no—
389C2×C8—2441no—
390C2×C8—2441no—
391C4⋊C4—21no—
392C4⋊C4—21no—
393C4⋊C4—612no—
394C4⋊C4—612no—
395C4⋊C4—1221no—
396C4⋊C4—1221no—
397C4⋊C4—1221no—
398C4⋊C4—1221no—
399C4⋊C4—1221no—
400C4⋊C4—1221no—
401C4⋊C4—1221no—
402C4⋊C4—1221no—
403C4⋊C4—2441no—
404C4⋊C4—2441no—
405C22⋊C4—4882no—
406C22⋊C4—2444no—
407C22⋊C4—2444no—
408C42—611no—
409C42—611no—
410C42—611no—
411C42—2441no—
412C42—2441no—
413C42—2441no—
414C42—2441no—
415C42C22×#692311yescharacteristic
416C23—22no—
417C23—611no—
418C23—1222no—
419C23—2442no—
420C23—2442no—
421C23—2442no—
422C23—2442no—
423C23—2442no—
424C23—2442no—
425Q8—41no—
426Q8—41no—
427Q8—4881no—
428Q8—4881no—
429Q8—2441no—
430Q8—2441no—
431D4—42no—
432D4—41no—
433D4—41no—
434D4—41no—
435D4—41no—
436D4—4882no—
437D4—2442no—
438D4—2442no—
439D4—2442no—
440D4—2442no—
441D4—2442no—
442D4—2441no—
443D4—2441no—
444D4—2441no—
445D4—2441no—
446C2×C4—41no—
447C2×C4—611no—
448C2×C4—611no—
449C2×C4—611no—
450C2×C4—611no—
451C2×C4—611no—
452C2×C4—611no—
453C2×C4—611no—
454C2×C4—611no—
455C2×C4—611no—
456C2×C4—611no—
457C2×C4—611no—
458C2×C4—1221no—
459C2×C4—1221no—
460C2×C4—1222no—
461C2×C4—1222no—
462C2×C4—4881no—
463C2×C4—4881no—
464C2×C4—4888no—
465C2×C4—2441no—
466C2×C4—2441no—
467C2×C4—2441no—
468C2×C4—2441no—
469C2×C4—2441no—
470C2×C4—2441no—
471C2×C4—2441no—
472C2×C4—2441no—
473C2×C4—2441no—
474C2×C4—2441no—
475C2×C4—2441no—
476C2×C4C2×C22⋊#692311yescharacteristic
477C2×C4D4×#692311yescharacteristic
478C2×C4D4×#692311yescharacteristic
479C8—41no—
480C8—4881no—
481C8—4881no—
482C8—2441no—
483C22—42no—
484C22—42no—
485C22—42no—
486C22—611no—
487C22—611no—
488C22—611no—
489C22—611no—
490C22—611no—
491C22—611no—
492C22—4882no—
493C22—4882no—
494C22—4882no—
495C22—4882no—
496C22—2442no—
497C22—2442no—
498C22—2442no—
499C22#1684071011yescharacteristic
500C4—21no—
501C4—81no—
502C4—611no—
503C4—611no—
504C4—611no—
505C4—611no—
506C4—1221no—
507C4—4881no—
508C4—4881no—
509C4—4881no—
510C4—2442no—
511C4—2442no—
512C4D8×#692311yescharacteristic
513C4SD16×#692311yescharacteristic
514C4#1684080111yescharacteristic
515C4#1684080311yescharacteristic
516C2—82no—
517C2—611no—
518C2—611no—
519C2—611no—
520C2—611no—
521C2—4882no—
522C2#2146181211yescharacteristic
523C2#2146181411yescharacteristic
524C2#2146182411yescharacteristic
subgroupG/H
centerC22#16840710
commutatorC42×C61C22×C4
FrattiniC42C22×#6923
FittingC61×#486C4
socleC22×C61#386
prime radical#127×D61C22
cosocleC42×D61C23
hypercenter#486#6923
hypocenterC61#9057
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #486

Complements

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

NH ≅ G/Ntype
2. #21461998C2semidirect
3. #21461985C2semidirect
5. #21460654C2semidirect
6. #21460654C2semidirect
24. #16840110C4semidirect
25. #16840109C4semidirect
26. #16840108C4semidirect
30. C61×#486C4semidirect
60. #14519777C2×C4semidirect
74. C4×#92418C2×C4semidirect
76. C4⋊Q8×C61C2×C4semidirect
79. C4×C8×C61C2×C4semidirect
278. #486#6923semidirect
293. C61#9057semidirect
306. C4⋊Q8C2×#6923semidirect
340. C4×C8C2×#6923semidirect

Maximal subgroups

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

subgroupindex
#214623492
#214619982
#214619852
D61⋊#3312
#214606542
#486×D612
#905761