Finite Group Databasebeta

← #30600604

Subgroups — #30600604

Order 15616.

590 conjugacy classes of subgroups.

The subgroup lattice has 528 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1Q8×D61⋊C8C211yesmaximal, characteristic, complemented
2#21461947C211yesmaximal, characteristic
3#21461688C211yesmaximal, characteristic, complemented
4D61⋊#331C211yesmaximal, characteristic
5#21460698C211yesmaximal, characteristic, complemented
6#21460648C211yesmaximal, characteristic, complemented
7#460×D61C211yesmaximal, characteristic
8#16840811—21no—
9#16840811—22no—
10Q8×#92420—21no—
11D61⋊(C4⋊C8)—21no—
12D61⋊(C4⋊C8)C2211yescharacteristic
13D61⋊(C4⋊C8)C2211yescharacteristic
14C2×#14519840—21no—
15C2×#14519839—21no—
16C2×#14519839—21no—
17C4×D61⋊C8—21no—
18C4×D61⋊C8C2211yescharacteristic
19#16840708C2211yescharacteristic
20#16840635—21no—
21#16840620—22no—
22C2×SD16×D61—21no—
23#127×D61C2211yescharacteristic
24C4×Q8×D61C2211yescharacteristic
25#16839963C411yescharacteristic, complemented
26#16839962C411yescharacteristic, complemented
27#16839961—21no—
28#16839960—21no—
29#16839959C411yescharacteristic, complemented
30C4⋊C8×D61C2211yescharacteristic
31#102×D61—21no—
32C61×#460C411yescharacteristic, complemented
33C2×D61⋊C8—21no—
34C2×D61⋊C8—21no—
35C2×D61⋊C8D411yescharacteristic
36C2×D61⋊C8D411yescharacteristic
37C2×C4⋊#6923—21no—
38C2×C4⋊#6923—21no—
39C2×C4⋊#6923—21no—
40#14519844—22no—
41#14519844—21no—
42#14519844D412yes—
43C4⋊#92420—22no—
44C4⋊#92420—21no—
45C4⋊#92420—21no—
46#14519840—42no—
47#14519839—42no—
48#14519839—42no—
49C4×#92420—22no—
50C4×#92420—22no—
51C4×#92420—21no—
52C4×#92420—21no—
53C2×Q8×D61D411yescharacteristic
54C2×D4×D61—21no—
55C2×D4×D61—21no—
56C2×D4×D61—21no—
57C2×#11758190—21no—
58C2×#11758200—21no—
59C2×#11758201—21no—
60SD16×D61—42no—
61SD16×D61—42no—
62C2×C8×D61—21no—
63#14519774C2×C411yescharacteristic
64#14519773—21no—
65#14519773—21no—
66#14519773—21no—
67Q8×Dic61D411yescharacteristic
68C4×Dic122—21no—
69C4×Dic122C2×C411yescharacteristic
70#14519742—21no—
71#14519742D411yescharacteristic
72C4⋊C4×D61—21no—
73C4⋊C4×D61D411yescharacteristic
74C42×D61—21no—
75C42×D61C2311yescharacteristic
76C4⋊#92418C2×C411yescharacteristic
77#14519706—21no—
78#14519699—21no—
79#14519698—21no—
80#14519697—21no—
81C2×SD16×C61—21no—
82#127×C61C2×C411yescharacteristic
83C4×Q8×C61C2×C411yescharacteristic
84C4⋊C8×C61C2×C411yescharacteristic
85#102×C61—21no—
86C23×D61—41no—
87C22×#6923—41no—
88C2×Q8×C61C22⋊C411yescharacteristic
89C2×D4×C61—21no—
90C2×D4×C61—21no—
91C2×D4×C61—21no—
92C2×C22⋊D61—41no—
93C2×C22⋊D61—41no—
94C2×C22⋊D61—41no—
95Q8×D61—21no—
96Q8×D61—21no—
97Q8×D61D812yes—
98Q8×D61SD1612yes—
99D4×D61—41no—
100D4×D61—41no—
101D4×D61—41no—
102D4×D61—41no—
103D4×D61—41no—
104D4×D61—41no—
105D4×D61—41no—
106D4×D61—41no—
107D4×D61—81no—
108D4×D61—81no—
109C2×D244—21no—
110C2×D244—21no—
111C2×D244—21no—
112C2×C4×D61—21no—
113C2×C4×D61—21no—
114C2×C4×D61C2×D411yescharacteristic
115C2×C4×D61C2×D411yescharacteristic
116C2×C4×D61#4111yescharacteristic
117C2×Dic122—21no—
118C2×Dic122C22⋊C411yescharacteristic
119C2×#92420—21no—
120C2×#92420—21no—
121C2×#92420—22no—
122C2×#92420—41no—
123C4⋊#6923—41no—
124C4⋊#6923—41no—
125C4⋊#6923—41no—
126C4⋊#6923—41no—
127C4⋊#6923—81no—
128D61⋊C8—21no—
129D61⋊C8—21no—
130D61⋊C8—21no—
131D61⋊C8—41no—
132D61⋊C8—41no—
133D61⋊C8—41no—
134D61⋊C8—41no—
135D61⋊C8SD1612yescomplemented
136SD16×C61—42no—
137C2×C8×C61—21no—
138C4⋊C4×C61—21no—
139C4⋊C4×C61C22⋊C411yescharacteristic
140C42×C61—21no—
141C42×C61C22×C411yescharacteristic
142#11758201—42no—
143#11758200—42no—
144C4⋊Dic61—21no—
145C4⋊Dic61C22⋊C411yescharacteristic
146#11758196—21no—
147#11758196—21no—
148#11758196—21no—
149C4×Dic61—21no—
150C4×Dic61—21no—
151C4×Dic61—21no—
152C4×Dic61C2×D411yescharacteristic
153C4×Dic61C2×D411yescharacteristic
154C4×Dic61#4111yescharacteristic
155C2×#92418—21no—
156#11758190—42no—
157C8×D61—41no—
158C8×D61—41no—
159C23×C61—41no—
160C22×D61—41no—
161C22×D61—81no—
162C22×D61—81no—
163C22×D61—81no—
164C22×D61—81no—
165C22×D61—81no—
166C22×D61—81no—
167C22×D61#12111yescharacteristic
168C2×#6923—81no—
169C2×#6923—81no—
170C2×#6923—81no—
171Q8×C61—21no—
172Q8×C61#10212yescomplemented
173D4×C61—41no—
174D4×C61—41no—
175D4×C61—41no—
176D4×C61—41no—
177D4×C61—81no—
178C2×C4×C61—21no—
179C2×C4×C61—21no—
180C2×C4×C61C2×C22⋊C411yescharacteristic
181C2×C4×C61C4×D411yescharacteristic
182C2×C4×C61C4×D411yescharacteristic
183C22⋊D61—82no—
184C22⋊D61—81no—
185C22⋊D61—81no—
186C22⋊D61—81no—
187C22⋊D61—81no—
188C2×Dic61—21no—
189C2×Dic61—21no—
190C2×Dic61#12011yescharacteristic
191C2×Dic61#12111yescharacteristic
192C2×Dic61#12311yescharacteristic
193D244—41no—
194D244—41no—
195D244—41no—
196D244—41no—
197D244—81no—
198C4×D61—22no—
199C4×D61—22no—
200C4×D61—21no—
201C4×D61—21no—
202C4×D61—41no—
203C4×D61—41no—
204C4×D61C2×D811yescharacteristic
205C4×D61C2×SD1611yescharacteristic
206C4×D61C2×SD1611yescharacteristic
207C4×D61#13511yescharacteristic
208C4×D61#13511yescharacteristic
209C4×D61#13511yescharacteristic
210C4×D61AGL(1,ℤ/8)11yescharacteristic
211C4×D61#13711yescharacteristic
212Dic122—21no—
213Dic122—22no—
214Dic122—22no—
215Dic122#10212yescomplemented
216#92420—22no—
217#92420—41no—
218#92420—42no—
219#92420—81no—
220C8×C61—41no—
221#92418—41no—
222#9222—611nomaximal, Sylow
223C22×C61—81no—
224C22×C61—81no—
225C22×C61—81no—
226C22×C61#38611yescharacteristic
227C2×D61—81no—
228C2×D61—81no—
229C2×D61—81no—
230C2×D61—161no—
231C2×D61—161no—
232C2×D61—161no—
233C2×D61—161no—
234C2×D61#46011yescharacteristic
235C2×D61#46411yescharacteristic
236C2×D61#46511yescharacteristic
237C2×D61#46611yescharacteristic
238C2×D61#47111yescharacteristic
239C2×D61#47211yescharacteristic
240#6923—161no—
241C4×C61—22no—
242C4×C61—21no—
243C4×C61—41no—
244C4×C61C2×#10211yescharacteristic
245C4×C61#41611yescharacteristic
246C4×C61C4×SD1611yescharacteristic
247C4×C61#44011yescharacteristic
248Dic61—22no—
249Dic61—21no—
250Dic61—41no—
251Dic61#44811yescharacteristic
252Dic61#44911yescharacteristic
253Dic61#46511yescharacteristic
254Dic61#46611yescharacteristic
255C2×#460—611no—
256C2×C8×Q8—611no—
257#1957—611no—
258#1803—611no—
259C2×#331—611no—
260#1296—611no—
261#1277—611no—
262C2×C61—161no—
263C2×C61#188011yescharacteristic
264C2×C61#189311yescharacteristic
265C2×C61#190711yescharacteristic
266D61—161no—
267D61#166611yescharacteristic
268D61#166811yescharacteristic
269D61#167811yescharacteristic
270D61#168211yescharacteristic
271C22×SD16—1221no—
272C2×#127—611no—
273C2×C4×Q8—611no—
274#460—611no—
275#460—611no—
276#460—611no—
277#460—1221no—
278#460—1221no—
279#460#692311yescharacteristic, complemented
280C8×Q8—1222no—
281C8×Q8—1221no—
282C8×Q8—1221no—
283C2×C4⋊C8—611no—
284C2×C4⋊C8—611no—
285C2×C4⋊C8—611no—
286C2×C4⋊C8—1221no—
287C2×#103—1221no—
288C2×#102—1221no—
289C2×#102—1221no—
290C2×#102—1221no—
291C2×C4×C8—611no—
292C2×C4×C8—1221no—
293#384—611no—
294#340—1221no—
295#331—1222no—
296C61#922211yescharacteristic, complemented, Sylow
297C22×Q8—611no—
298C22×D4—1221no—
299C22×D4—1221no—
300C22×D4—1221no—
301C2×SD16—21no—
302C2×SD16—1221no—
303C2×SD16—1221no—
304C2×SD16—1221no—
305C2×SD16—2442no—
306C2×SD16—2442no—
307C22×C8—611no—
308C22×C8—611no—
309C22×C8—1221no—
310C22×C8—1221no—
311C22×C8—1221no—
312#127—611no—
313#127—1221no—
314#127—1221no—
315#127—1221no—
316#127C2×#692311yescharacteristic
317C4×Q8—611no—
318C4×Q8—611no—
319C4×Q8—611no—
320C4×Q8—1221no—
321C4×Q8—1221no—
322C4×Q8C2×#692311yescharacteristic
323C2×C4⋊C4—611no—
324C2×C4⋊C4—1221no—
325C2×C4⋊C4—1221no—
326C2×C4⋊C4—1221no—
327C2×C4⋊C4—1221no—
328C2×C42—611no—
329C2×C42—1221no—
330C4⋊C8—612no—
331C4⋊C8—611no—
332C4⋊C8—1221no—
333C4⋊C8—1221no—
334C4⋊C8—1221no—
335C4⋊C8—1221no—
336C4⋊C8—1222no—
337C4⋊C8—1222no—
338C4⋊C8C2×#692311yescharacteristic
339#103—2442no—
340#102—21no—
341#102—1221no—
342#102—1221no—
343#102—1221no—
344#102—2442no—
345#102—2442no—
346C4×C8—1221no—
347C4×C8—1221no—
348C4×C8—1222no—
349C4×C8—1222no—
350C24—2441no—
351C2×Q8—612no—
352C2×Q8—612no—
353C2×Q8—611no—
354C2×Q8—1221no—
355C2×Q8—1221no—
356C2×Q8—1221no—
357C2×Q8C22⋊#692311yescharacteristic
358C2×D4—21no—
359C2×D4—21no—
360C2×D4—21no—
361C2×D4—1221no—
362C2×D4—1221no—
363C2×D4—1221no—
364C2×D4—4881no—
365C2×D4—4881no—
366C2×D4—2441no—
367C2×D4—2441no—
368C2×D4—2441no—
369C2×D4—2441no—
370C2×D4—2441no—
371C2×D4—2441no—
372C2×D4—2441no—
373C2×D4—2441no—
374C2×D4—2441no—
375C2×D4—2441no—
376C2×D4—2441no—
377C22×C4—611no—
378C22×C4—611no—
379C22×C4—611no—
380C22×C4—1221no—
381C22×C4—1221no—
382C22×C4—2441no—
383SD16—42no—
384SD16—2442no—
385SD16—2442no—
386SD16—2442no—
387C2×C8—21no—
388C2×C8—612no—
389C2×C8—1221no—
390C2×C8—1221no—
391C2×C8—1222no—
392C2×C8—1221no—
393C2×C8—1221no—
394C2×C8—1221no—
395C2×C8—1221no—
396C2×C8—2441no—
397C2×C8—2441no—
398C2×C8—2441no—
399C2×C8—2441no—
400C2×C8—2441no—
401C2×C8—2441no—
402C2×C8—2441no—
403C4⋊C4—21no—
404C4⋊C4—611no—
405C4⋊C4—1221no—
406C4⋊C4—1221no—
407C4⋊C4—1221no—
408C4⋊C4—1221no—
409C4⋊C4—4881no—
410C4⋊C4—2441no—
411C4⋊C4—2441no—
412C4⋊C4—2441no—
413C4⋊C4—2441no—
414C4⋊C4C22⋊#692311yescharacteristic
415C42—21no—
416C42—611no—
417C42—611no—
418C42—611no—
419C42—1221no—
420C42—1221no—
421C42—1221no—
422C42C22×#692311yescharacteristic
423C23—41no—
424C23—611no—
425C23—4881no—
426C23—4881no—
427C23—4881no—
428C23—4881no—
429C23—4881no—
430C23—4881no—
431C23—2441no—
432Q8—21no—
433Q8—612no—
434Q8—1222no—
435Q8—1222no—
436Q8—1221no—
437Q8#1451983912yescomplemented
438D4—41no—
439D4—41no—
440D4—41no—
441D4—41no—
442D4—81no—
443D4—4881no—
444D4—4881no—
445D4—4881no—
446D4—4881no—
447D4—4881no—
448D4—4882no—
449D4—2441no—
450D4—2441no—
451D4—2441no—
452D4—2441no—
453C2×C4—21no—
454C2×C4—21no—
455C2×C4—611no—
456C2×C4—611no—
457C2×C4—611no—
458C2×C4—611no—
459C2×C4—611no—
460C2×C4—611no—
461C2×C4—611no—
462C2×C4—611no—
463C2×C4—611no—
464C2×C4—611no—
465C2×C4—611no—
466C2×C4—1221no—
467C2×C4—1221no—
468C2×C4—1222no—
469C2×C4—1222no—
470C2×C4—1221no—
471C2×C4—1221no—
472C2×C4—4881no—
473C2×C4—4881no—
474C2×C4—4881no—
475C2×C4—2441no—
476C2×C4—2441no—
477C2×C4C2×C22⋊#692311yescharacteristic
478C2×C4D4×#692311yescharacteristic
479C2×C4D4×#692311yescharacteristic
480C8—41no—
481C8—1222no—
482C8—4881no—
483C8—2441no—
484C8—2442no—
485C8—2441no—
486C22—81no—
487C22—81no—
488C22—81no—
489C22—611no—
490C22—611no—
491C22—611no—
492C22—611no—
493C22—611no—
494C22—611no—
495C22—9761no—
496C22—9761no—
497C22—9761no—
498C22—9761no—
499C22—4881no—
500C22—4881no—
501C22—4881no—
502C22#1684071011yescharacteristic
503C4—22no—
504C4—21no—
505C4—41no—
506C4—611no—
507C4—611no—
508C4—611no—
509C4—611no—
510C4—1222no—
511C4—1221no—
512C4—9761no—
513C4—2441no—
514C4C2×#1451983911yescharacteristic
515C4#1684075211yescharacteristic
516C4SD16×#692311yescharacteristic
517C4#1684080111yescharacteristic
518C2—161no—
519C2—611no—
520C2—611no—
521C2—611no—
522C2—611no—
523C2—9761no—
524C2#2146181511yescharacteristic
525C2#2146183311yescharacteristic
526C2#2146185811yescharacteristic
subgroupG/H
centerC22#16840710
commutatorC42×C61C22×C4
FrattiniC42C22×#6923
FittingC61×#460C4
socleC22×C61#386
prime radical#127×D61C22
cosocleC42×D61C23
hypercenter#460#6923
hypocenterC61#9222
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #460

Complements

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

NH ≅ G/Ntype
1. Q8×D61⋊C8C2semidirect
3. #21461688C2semidirect
5. #21460698C2semidirect
6. #21460648C2semidirect
25. #16839963C4semidirect
26. #16839962C4semidirect
29. #16839959C4semidirect
32. C61×#460C4semidirect
135. D61⋊C8SD16semidirect
172. Q8×C61#102semidirect
215. Dic122#102semidirect
279. #460#6923semidirect
296. C61#9222semidirect
437. Q8#14519839semidirect

Maximal subgroups

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

subgroupindex
Q8×D61⋊C82
#214619472
#214616882
D61⋊#3312
#214606982
#214606482
#460×D612
#922261