Finite Group Databasebeta

← #30616397

Subgroups — #30616397

Order 15616.

814 conjugacy classes of subgroups.

The subgroup lattice has 578 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21462303C211yesmaximal, characteristic, complemented
2#21462302C211yesmaximal, characteristic
3#21461969C212yesmaximal
4#21461925C212yesmaximal, complemented
5#21461909C212yesmaximal
6#21461908C212yesmaximal, complemented
7C4×#14519846C211yesmaximal, characteristic, complemented
8#21461591C211yesmaximal, characteristic, complemented
9#21461589C211yesmaximal, characteristic
10#21461475C211yesmaximal, characteristic, complemented
11C4×SD16×D61C211yesmaximal, characteristic
12C2×#14519846C2212yes—
13C2×#14519840C2212yes—
14C2×#14519839C2212yes—
15#16840721C2211yescharacteristic
16C2×#14519831C2212yes—
17C2×C8×#6923C2212yes—
18(C2×#6923)⋊Q8C2212yes—
19#16840714C2212yes—
20#16840713C2212yes—
21#16840713C2212yes—
22#16840712—22no—
23#16840711—22no—
24#16840707—21no—
25#16840706—21no—
26#16840705—21no—
27#16840704—21no—
28C4×C4⋊#6923C2211yescharacteristic
29C4×C4⋊#6923C2211yescharacteristic
30#16840698C2211yescharacteristic
31#16840636C2212yes—
32#16840635C2212yes—
33#16840635C2212yes—
34C2×SD16×D61C2211yescharacteristic
35C4×Q8×D61C2211yescharacteristic
36C4×D4×D61C2211yescharacteristic
37#16839853C411yescharacteristic, complemented
38C4×#11758190C411yescharacteristic, complemented
39SD16×Dic61C411yescharacteristic, complemented
40C4×#11758201C411yescharacteristic, complemented
41#16839849C411yescharacteristic, complemented
42C4×#11758200C411yescharacteristic, complemented
43#16839847C411yescharacteristic, complemented
44#106×D61C2211yescharacteristic
45#103×D61C2211yescharacteristic
46#102×D61C2211yescharacteristic
47C4×C8×D61C2211yescharacteristic
48C4×SD16×C61C411yescharacteristic, complemented
49C2×D61⋊C8C2312yes—
50C2×C4⋊#6923—22no—
51C2×C4⋊#6923—22no—
52C2×C4⋊#6923C2312yes—
53C2×C4⋊#6923C2312yes—
54C2×C22⋊#6923—22no—
55C2×C22⋊#6923—22no—
56C2×C4×#6923—21no—
57C2×C4×#6923—21no—
58C2×C4×#6923C2312yes—
59#14519846—24no—
60#14519840—24no—
61#14519839—24no—
62#14519832—21no—
63#14519832—21no—
64#14519831—24no—
65C8×#6923—24no—
66(C2×#6923)⋊C4—22no—
67(C2×#6923)⋊C4—22no—
68(C2×#6923)⋊C4—22no—
69(C2×#6923)⋊C4—22no—
70(C2×#6923)⋊C4—21no—
71(C2×#6923)⋊C4—21no—
72(C2×#6923)⋊C4D411yescharacteristic
73(C2×#6923)⋊C4D411yescharacteristic
74(C2×#6923)⋊C4D412yes—
75C2×Q8×D61C2311yescharacteristic
76C2×D4×D61C2311yescharacteristic
77C22×C4×D61—21no—
78C2×#11758190C2×C411yescharacteristic
79C2×#11758200C2×C411yescharacteristic
80C2×#11758201C2×C411yescharacteristic
81SD16×D61—24no—
82C2×C8×D61C2311yescharacteristic
83C2×C8×D61C2311yescharacteristic
84Q8×Dic61C2×C411yescharacteristic
85C4×Dic122—21no—
86C4×Dic122C2×C411yescharacteristic
87#14519742—21no—
88#14519742C2×C411yescharacteristic
89C4×D244C2×C411yescharacteristic
90D4×Dic61C2×C411yescharacteristic
91#14519739C2×C411yescharacteristic
92C4×C22⋊D61—21no—
93#14519737—21no—
94C4⋊C4×D61—21no—
95C4⋊C4×D61C2311yescharacteristic
96C4⋊C4×D61C2311yescharacteristic
97C22⋊C4×D61—21no—
98C42×D61—21no—
99C42×D61C2311yescharacteristic
100#14519709C2×C411yescharacteristic
101#14519708C2×C411yescharacteristic
102#14519708C2×C411yescharacteristic
103#14519702C2×C411yescharacteristic
104#14519701C2×C411yescharacteristic
105#14519700C2×C411yescharacteristic
106#14519699C2×C411yescharacteristic
107#14519698C2×C411yescharacteristic
108#14519697C2×C411yescharacteristic
109C4×#92418C2×C411yescharacteristic
110C8×Dic61C2×C411yescharacteristic
111C8×Dic61C2×C411yescharacteristic
112C2×SD16×C61C2×C411yescharacteristic
113C4×Q8×C61C2×C411yescharacteristic
114C4×D4×C61C2×C411yescharacteristic
115#106×C61C2×C411yescharacteristic
116#103×C61C2×C411yescharacteristic
117#102×C61C2×C411yescharacteristic
118C4×C8×C61C2×C411yescharacteristic
119C23×D61—21no—
120C22×#6923—22no—
121C22×#6923—22no—
122C22×#6923C2×D412yes—
123C22×#6923C2×D412yes—
124C2×Q8×C61C22×C411yescharacteristic
125C2×D4×C61C22×C411yescharacteristic
126C22×C4×C61—21no—
127C2×C22⋊D61—21no—
128C22×Dic61—21no—
129Q8×D61—22no—
130Q8×D61—22no—
131D4×D61—22no—
132D4×D61—22no—
133C2×D244C22×C411yescharacteristic
134C2×C4×D61—21no—
135C2×C4×D61—21no—
136C2×C4×D61—21no—
137C2×C4×D61—21no—
138C2×C4×D61—21no—
139C2×C4×D61—82no—
140C2×C4×D61C2×D411yescharacteristic
141C2×C4×D61C2×D411yescharacteristic
142C2×C4×D61C2411yescharacteristic
143C2×Dic122—21no—
144C2×Dic122C22×C411yescharacteristic
145C22⋊#6923—82no—
146C22⋊#6923—82no—
147C2×#92420—22no—
148C4⋊#6923—22no—
149C4⋊#6923—22no—
150C4⋊#6923—22no—
151C4⋊#6923—22no—
152C4⋊#6923—82no—
153C4⋊#6923—82no—
154C4×#6923—22no—
155C4×#6923—22no—
156C4×#6923—81no—
157C4×#6923—81no—
158D61⋊C8—22no—
159D61⋊C8#4114yes—
160SD16×C61—22no—
161C2×C8×C61C22×C411yescharacteristic
162C2×C8×C61C22×C411yescharacteristic
163C4⋊C4×C61—21no—
164C4⋊C4×C61C22×C411yescharacteristic
165C4⋊C4×C61C22×C411yescharacteristic
166C22⋊C4×C61—21no—
167C42×C61—21no—
168C42×C61C22×C411yescharacteristic
169C22⋊Dic61—21no—
170#11758201—22no—
171#11758200—22no—
172#11758198—21no—
173#11758198—21no—
174C4⋊Dic61—21no—
175C4⋊Dic61C22×C411yescharacteristic
176C4⋊Dic61C22×C411yescharacteristic
177#11758196—21no—
178#11758196—21no—
179#11758196—21no—
180#11758196—21no—
181C4×Dic61—21no—
182C4×Dic61—21no—
183C4×Dic61—21no—
184C4×Dic61C22×C411yescharacteristic
185C4×Dic61#4111yescharacteristic
186C4×Dic61#4111yescharacteristic
187C2×#92418C22×C411yescharacteristic
188C2×#92418C22×C411yescharacteristic
189#11758190—22no—
190C8×D61—22no—
191C8×D61#4114yes—
192C23×C61—21no—
193C22×D61—21no—
194C22×D61—42no—
195C22×D61—41no—
196C22×D61—42no—
197C22×D61—41no—
198C22×D61C22×D411yescharacteristic
199C2×#6923—22no—
200C2×#6923—22no—
201C2×#6923—22no—
202C2×#6923—22no—
203C2×#6923—22no—
204C2×#6923—22no—
205C2×#6923—42no—
206C2×#6923—42no—
207C2×#6923—42no—
208C2×#6923—42no—
209C2×#6923—42no—
210C2×#6923—42no—
211Q8×C61—21no—
212Q8×C61—21no—
213D4×C61—21no—
214D4×C61—21no—
215C2×C4×C61—21no—
216C2×C4×C61—21no—
217C2×C4×C61—21no—
218C2×C4×C61—81no—
219C2×C4×C61C4×D411yescharacteristic
220C2×C4×C61C4×D411yescharacteristic
221C2×C4×C61C23×C411yescharacteristic
222C22⋊D61—42no—
223C2×Dic61—21no—
224C2×Dic61—21no—
225C2×Dic61—21no—
226C2×Dic61—81no—
227C2×Dic61C4×D411yescharacteristic
228C2×Dic61C4×D411yescharacteristic
229C2×Dic61C2×#4111yescharacteristic
230D244—21no—
231D244—21no—
232C4×D61—21no—
233C4×D61—22no—
234C4×D61—21no—
235C4×D61—42no—
236C4×D61—41no—
237C4×D61—42no—
238C4×D61—41no—
239C4×D61—81no—
240C4×D61—81no—
241C4×D61C2×#4112yes—
242C4×D61C2×#4112yes—
243Dic122—21no—
244Dic122—21no—
245Dic122—42no—
246#92420—24no—
247C8×C61—21no—
248C8×C61#11712yes—
249#92418—21no—
250#92418#11712yes—
251#21285—611nomaximal, Sylow
252C22×C61—41no—
253C22×C61—41no—
254C22×C61—41no—
255C22×C61C2×C4×D411yescharacteristic
256C2×D61—41no—
257C2×D61—41no—
258C2×D61—41no—
259C2×D61—82no—
260C2×D61—82no—
261C2×D61#53212yes—
262C2×D61#57611yescharacteristic
263C2×D61#57712yes—
264C2×D61#57811yescharacteristic
265#6923—42no—
266#6923—42no—
267#6923—82no—
268#6923—82no—
269C4×C61—21no—
270C4×C61—21no—
271C4×C61—41no—
272C4×C61—41no—
273C4×C61—41no—
274C4×C61C2×#11711yescharacteristic
275C4×C61C4×#4111yescharacteristic
276Dic61—21no—
277Dic61—21no—
278Dic61—41no—
279Dic61—41no—
280Dic61—41no—
281Dic61#52912yes—
282C2×C4×SD16—611no—
283#2320—611no—
284#2319—611no—
285#1973—612no—
286#1944—612no—
287#1934—612no—
288#1933—612no—
289C4×#107—611no—
290#1763—611no—
291#1762—611no—
292#1726—611no—
293C2×C61—81no—
294C2×C61#229311yescharacteristic
295C2×C61#294911yescharacteristic
296C2×C61#295111yescharacteristic
297D61—81no—
298D61#317812yes—
299D61#317912yes—
300C22×SD16—611no—
301C2×C4×Q8—611no—
302C2×C4×D4—611no—
303C4×SD16—611no—
304C4×SD16—611no—
305C4×SD16—611no—
306C4×SD16—611no—
307C4×SD16—611no—
308C4×SD16—611no—
309C4×SD16—611no—
310C4×SD16#692311yescharacteristic, complemented
311C2×#107—612no—
312C2×#106—611no—
313C2×#103—611no—
314C2×#103—612no—
315C2×#102—611no—
316C2×#102—612no—
317C2×#97—612no—
318C2×#97—611no—
319C2×C4×C8—611no—
320C2×C4×C8—612no—
321#391—612no—
322#390—612no—
323#389—612no—
324#389—612no—
325#388—1222no—
326#387—1222no—
327#383—1221no—
328#382—1221no—
329#381—1221no—
330#380—1221no—
331C4×C4⋊C4—611no—
332C4×C4⋊C4—611no—
333#376—611no—
334#340—612no—
335#340—612no—
336#340—612no—
337C61#2128511yescharacteristic, complemented, Sylow
338C22×Q8—611no—
339C22×D4—611no—
340C23×C4—1221no—
341C2×SD16—611no—
342C2×SD16—611no—
343C2×SD16—611no—
344C2×SD16—1224no—
345C2×SD16C2×#692311yescharacteristic
346C22×C8—611no—
347C22×C8—612no—
348C22×C8—611no—
349C4×Q8—611no—
350C4×Q8—611no—
351C4×Q8—611no—
352C4×Q8—1221no—
353C4×Q8—1221no—
354C4×Q8C2×#692311yescharacteristic
355C4×D4—611no—
356C4×D4—611no—
357C4×D4—611no—
358C4×D4—1221no—
359C4×D4—1221no—
360C4×D4C2×#692311yescharacteristic
361C2×C4⋊C4—611no—
362C2×C4⋊C4—611no—
363C2×C4⋊C4—612no—
364C2×C4⋊C4—612no—
365C2×C4⋊C4—1221no—
366C2×C4⋊C4—1222no—
367C2×C4⋊C4—1222no—
368C2×C22⋊C4—1221no—
369C2×C22⋊C4—1222no—
370C2×C22⋊C4—1222no—
371C2×C42—611no—
372C2×C42—612no—
373C2×C42—1221no—
374C2×C42—1221no—
375C2×C42—1221no—
376#107—1224no—
377#106—611no—
378#106—611no—
379#106—611no—
380#106C2×#692311yescharacteristic
381#103—611no—
382#103—611no—
383#103—611no—
384#103—1224no—
385#103C2×#692311yescharacteristic
386#102—611no—
387#102—611no—
388#102—611no—
389#102—1224no—
390#102C2×#692311yescharacteristic
391#97—1221no—
392#97—1221no—
393#97—1224no—
394C4×C8—611no—
395C4×C8—611no—
396C4×C8—611no—
397C4×C8—1224no—
398C4×C8C2×#692311yescharacteristic
399#95—611no—
400#95—611no—
401#95—612no—
402#95—1221no—
403#95—1221no—
404#95—1222no—
405#95—1222no—
406#95—1222no—
407#95—1222no—
408C24—1221no—
409C2×Q8—611no—
410C2×Q8—1221no—
411C2×Q8—1222no—
412C2×Q8—1222no—
413C2×Q8C22×#692311yescharacteristic
414C2×D4—611no—
415C2×D4—1222no—
416C2×D4—1222no—
417C2×D4—1221no—
418C2×D4C22×#692311yescharacteristic
419C22×C4—21no—
420C22×C4—611no—
421C22×C4—611no—
422C22×C4—611no—
423C22×C4—612no—
424C22×C4—612no—
425C22×C4—1221no—
426C22×C4—1221no—
427C22×C4—1221no—
428C22×C4—1221no—
429C22×C4—1221no—
430C22×C4—1221no—
431C22×C4—1222no—
432C22×C4—1222no—
433C22×C4—4882no—
434SD16—22no—
435SD16—1222no—
436SD16—1222no—
437SD16—1222no—
438C2×C8—611no—
439C2×C8—614no—
440C2×C8—614no—
441C2×C8—611no—
442C2×C8—1222no—
443C2×C8—1222no—
444C2×C8—1222no—
445C2×C8C22×#692311yescharacteristic
446C2×C8C22×#692311yescharacteristic
447C4⋊C4—21no—
448C4⋊C4—611no—
449C4⋊C4—611no—
450C4⋊C4—1221no—
451C4⋊C4—1221no—
452C4⋊C4—1221no—
453C4⋊C4—1221no—
454C4⋊C4—1222no—
455C4⋊C4—1222no—
456C4⋊C4—1222no—
457C4⋊C4—1222no—
458C4⋊C4—1221no—
459C4⋊C4—4882no—
460C4⋊C4—4882no—
461C4⋊C4C22×#692311yescharacteristic
462C4⋊C4C22×#692311yescharacteristic
463C22⋊C4—21no—
464C22⋊C4—1221no—
465C22⋊C4—1221no—
466C22⋊C4—1221no—
467C22⋊C4—4882no—
468C22⋊C4—4882no—
469C42—21no—
470C42—611no—
471C42—611no—
472C42—611no—
473C42—1221no—
474C42—1221no—
475C42—1222no—
476C42—1222no—
477C42—1221no—
478C42—4881no—
479C42—4881no—
480C42C22×#692311yescharacteristic
481C23—21no—
482C23—611no—
483C23—1221no—
484C23—2441no—
485C23—2442no—
486C23—2442no—
487C23—2441no—
488Q8—21no—
489Q8—21no—
490Q8—1221no—
491Q8—1221no—
492Q8—2442no—
493D4—21no—
494D4—21no—
495D4—1221no—
496D4—1221no—
497D4—2442no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—81no—
502C2×C4—611no—
503C2×C4—612no—
504C2×C4—612no—
505C2×C4—611no—
506C2×C4—611no—
507C2×C4—1221no—
508C2×C4—1221no—
509C2×C4—1222no—
510C2×C4—1222no—
511C2×C4—1222no—
512C2×C4—1222no—
513C2×C4—1222no—
514C2×C4—1222no—
515C2×C4—1222no—
516C2×C4—1221no—
517C2×C4—1221no—
518C2×C4—1221no—
519C2×C4—4881no—
520C2×C4—4881no—
521C2×C4—4881no—
522C2×C4—2441no—
523C2×C4—2442no—
524C2×C4—2441no—
525C2×C4—2442no—
526C2×C4—2442no—
527C2×C4—2442no—
528C2×C4—2442no—
529C2×C4—2442no—
530C2×C4—2442no—
531C2×C4—2442no—
532C2×C4D4×#692311yescharacteristic
533C2×C4D4×#692311yescharacteristic
534C2×C4C23×#692311yescharacteristic
535C8—21no—
536C8—612no—
537C8—1224no—
538C8—1221no—
539C8#1451985612yes—
540C22—41no—
541C22—41no—
542C22—41no—
543C22—612no—
544C22—612no—
545C22—611no—
546C22—611no—
547C22—4882no—
548C22—4882no—
549C22—2441no—
550C22—2441no—
551C22—2441no—
552C22C2×D4×#692311yescharacteristic
553C4—21no—
554C4—21no—
555C4—41no—
556C4—41no—
557C4—41no—
558C4—612no—
559C4—1221no—
560C4—1221no—
561C4—4882no—
562C4—4882no—
563C4—2442no—
564C4—2442no—
565C4—2441no—
566C4—2441no—
567C4—2441no—
568C4C2×#1451985611yescharacteristic
569C4#41×#692311yescharacteristic
570C2—81no—
571C2—612no—
572C2—612no—
573C2—4881no—
574C2#2146227611yescharacteristic
575C2#2146257911yescharacteristic
576C2#2146258111yescharacteristic
subgroupG/H
centerC22C2×D4×#6923
commutatorC2×C4×C61C23×C4
FrattiniC2×C4C23×#6923
FittingC4×SD16×C61C4
socleC22×C61C2×C4×D4
prime radicalC2×D4×D61C23
cosocleC2×C4×D61C24
hypercenterC4×SD16#6923
hypocenterC61#21285
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ C4×SD16

Complements

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

NH ≅ G/Ntype
1. #21462303C2semidirect
4. #21461925C2semidirect
6. #21461908C2semidirect
7. C4×#14519846C2semidirect
8. #21461591C2semidirect
10. #21461475C2semidirect
37. #16839853C4semidirect
38. C4×#11758190C4semidirect
39. SD16×Dic61C4semidirect
40. C4×#11758201C4semidirect
41. #16839849C4semidirect
42. C4×#11758200C4semidirect
43. #16839847C4semidirect
48. C4×SD16×C61C4semidirect
310. C4×SD16#6923semidirect
337. C61#21285semidirect

Maximal subgroups

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

subgroupindex
#214623032
#214623022
#214619692
#214619252
#214619092
#214619082
C4×#145198462
#214615912
#214615892
#214614752
C4×SD16×D612
#2128561