Finite Group Databasebeta

← #30616077

Subgroups — #30616077

Order 15616.

814 conjugacy classes of subgroups.

The subgroup lattice has 548 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21462303C211yesmaximal, characteristic
2#21462259C211yesmaximal, characteristic
3#21461898C212yesmaximal
4#21461883C212yesmaximal
5#21461813C212yesmaximal
6#21461785C212yesmaximal
7C4×#14519845C211yesmaximal, characteristic
8#21461591C211yesmaximal, characteristic
9#21461591C211yesmaximal, characteristic
10#21461537C211yesmaximal, characteristic
11C4×Q16×D61C211yesmaximal, characteristic
12C2×Q8×#6923C2212yes—
13C2×#14519845C2212yes—
14C2×#14519840C2212yes—
15C2×#14519840C2212yes—
16#16840721C2211yescharacteristic
17(C2×#6923)⋊Q8C2212yes—
18#16840713C2212yes—
19#16840711—22no—
20#16840707—21no—
21#16840706—21no—
22C4⋊C4×#6923—22no—
23C4⋊C4×#6923C2212yes—
24C4×C4⋊#6923—21no—
25C4×C4⋊#6923C2211yescharacteristic
26C4×C4⋊#6923C2211yescharacteristic
27#16840698—21no—
28#16840698C2211yescharacteristic
29#16840636C2212yes—
30#16840635C2212yes—
31#16840635C2212yes—
32(C2×#6923)⋊C8C2212yes—
33(C2×#6923)⋊C8C2212yes—
34C2×Q16×D61C2211yescharacteristic
35C4×Q8×D61C2211yescharacteristic
36C4×Q8×D61C2211yescharacteristic
37C4×Dic244C411yescharacteristic, complemented
38Q16×Dic61C411yescharacteristic, complemented
39#16839856C411yescharacteristic, complemented
40C4×#11758202C411yescharacteristic, complemented
41C4×#11758202C411yescharacteristic, complemented
42#16839854C411yescharacteristic, complemented
43#16839854C411yescharacteristic, complemented
44#107×D61C2211yescharacteristic
45#103×D61C2211yescharacteristic
46#103×D61C2211yescharacteristic
47C4×C8×D61C2211yescharacteristic
48C4×Q16×C61C411yescharacteristic, complemented
49C2×D61⋊C8C2312yes—
50Q8×#6923—24no—
51Q8×#6923—24no—
52C2×C4⋊#6923—22no—
53C2×C4⋊#6923—22no—
54C2×C4⋊#6923C2312yes—
55C2×C4⋊#6923C2312yes—
56C2×C4×#6923—22no—
57C2×C4×#6923—22no—
58C2×C4×#6923—21no—
59C2×C4×#6923—21no—
60C2×C4×#6923C2312yes—
61#14519845—24no—
62#14519840—24no—
63#14519840—24no—
64#14519832—21no—
65#14519832—21no—
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×Q8×D61C2311yescharacteristic
77C2×Dic244C2×C411yescharacteristic
78C2×#11758202C2×C411yescharacteristic
79C2×#11758202C2×C411yescharacteristic
80Q16×D61—24no—
81C2×C8×D61C2311yescharacteristic
82C2×C8×D61C2311yescharacteristic
83Q8×Dic61C2×C411yescharacteristic
84Q8×Dic61C2×C411yescharacteristic
85C4×Dic122—21no—
86C4×Dic122—21no—
87C4×Dic122C2×C411yescharacteristic
88C4×Dic122C2×C411yescharacteristic
89#14519742—21no—
90#14519742—21no—
91#14519742C2×C411yescharacteristic
92#14519742C2×C411yescharacteristic
93C4⋊C4×D61—21no—
94C4⋊C4×D61—21no—
95C4⋊C4×D61C2311yescharacteristic
96C4⋊C4×D61C2311yescharacteristic
97C42×D61—21no—
98C42×D61—21no—
99C42×D61C2311yescharacteristic
100#14519711C2×C411yescharacteristic
101#14519710C2×C411yescharacteristic
102#14519710C2×C411yescharacteristic
103#14519702C2×C411yescharacteristic
104#14519702C2×C411yescharacteristic
105#14519701C2×C411yescharacteristic
106#14519701C2×C411yescharacteristic
107#14519700C2×C411yescharacteristic
108#14519700C2×C411yescharacteristic
109C4×#92418C2×C411yescharacteristic
110C8×Dic61C2×C411yescharacteristic
111C8×Dic61C2×C411yescharacteristic
112C2×Q16×C61C2×C411yescharacteristic
113C4×Q8×C61C2×C411yescharacteristic
114C4×Q8×C61C2×C411yescharacteristic
115#107×C61C2×C411yescharacteristic
116#103×C61C2×C411yescharacteristic
117#103×C61C2×C411yescharacteristic
118C4×C8×C61C2×C411yescharacteristic
119C22×#6923—22no—
120C22×#6923—22no—
121C22×#6923C2×D412yes—
122C22×#6923C2×D412yes—
123C2×Q8×C61C22×C411yescharacteristic
124C2×Q8×C61C22×C411yescharacteristic
125Q8×D61—22no—
126Q8×D61—22no—
127Q8×D61—22no—
128Q8×D61#4114yes—
129C2×C4×D61—21no—
130C2×C4×D61—21no—
131C2×C4×D61—21no—
132C2×C4×D61—21no—
133C2×C4×D61C2×D411yescharacteristic
134C2×C4×D61C2×D411yescharacteristic
135C2×C4×D61C2411yescharacteristic
136C2×Dic122—21no—
137C2×Dic122—21no—
138C2×Dic122C22×C411yescharacteristic
139C2×Dic122C22×C411yescharacteristic
140C2×#92420—22no—
141C4⋊#6923—28no—
142C4⋊#6923—22no—
143C4⋊#6923—22no—
144C4⋊#6923—22no—
145C4⋊#6923—82no—
146C4⋊#6923#4114yes—
147C4×#6923—28no—
148C4×#6923—22no—
149C4×#6923—22no—
150C4×#6923—81no—
151C4×#6923—81no—
152C4×#6923—82no—
153D61⋊C8—22no—
154D61⋊C8—22no—
155Q16×C61—22no—
156C2×C8×C61C22×C411yescharacteristic
157C2×C8×C61C22×C411yescharacteristic
158C4⋊C4×C61—21no—
159C4⋊C4×C61—21no—
160C4⋊C4×C61C22×C411yescharacteristic
161C4⋊C4×C61C22×C411yescharacteristic
162C42×C61—21no—
163C42×C61—21no—
164C42×C61C22×C411yescharacteristic
165#11758202—22no—
166#11758202—22no—
167C4⋊Dic61—21no—
168C4⋊Dic61—21no—
169C4⋊Dic61C22×C411yescharacteristic
170C4⋊Dic61C22×C411yescharacteristic
171#11758196—21no—
172#11758196—21no—
173#11758196—21no—
174#11758196—21no—
175#11758196—21no—
176#11758196—21no—
177C4×Dic61—21no—
178C4×Dic61—21no—
179C4×Dic61—21no—
180C4×Dic61—21no—
181C4×Dic61—21no—
182C4×Dic61—21no—
183C4×Dic61C22×C411yescharacteristic
184C4×Dic61#4111yescharacteristic
185C4×Dic61#4111yescharacteristic
186C2×#92418C22×C411yescharacteristic
187C2×#92418C22×C411yescharacteristic
188Dic244—22no—
189C8×D61—22no—
190C8×D61—22no—
191C22×D61C22×D411yescharacteristic
192C2×#6923—22no—
193C2×#6923—22no—
194C2×#6923—24no—
195C2×#6923—22no—
196C2×#6923—22no—
197C2×#6923—22no—
198C2×#6923—22no—
199C2×#6923—42no—
200C2×#6923—42no—
201C2×#6923—42no—
202C2×#6923—42no—
203C2×#6923—42no—
204Q8×C61—21no—
205Q8×C61—21no—
206Q8×C61—21no—
207Q8×C61#11712yes—
208C2×C4×C61—21no—
209C2×C4×C61—21no—
210C2×C4×C61—21no—
211C2×C4×C61—21no—
212C2×C4×C61C4×D411yescharacteristic
213C2×C4×C61C4×D411yescharacteristic
214C2×C4×C61C23×C411yescharacteristic
215C2×Dic61—21no—
216C2×Dic61—21no—
217C2×Dic61—21no—
218C2×Dic61—21no—
219C2×Dic61C4×D411yescharacteristic
220C2×Dic61C4×D411yescharacteristic
221C2×Dic61C2×#4111yescharacteristic
222C4×D61—24no—
223C4×D61—21no—
224C4×D61—21no—
225C4×D61—22no—
226C4×D61—42no—
227C4×D61—41no—
228C4×D61—41no—
229C4×D61—41no—
230C4×D61—41no—
231C4×D61C2×#4112yes—
232C4×D61C2×#4112yes—
233Dic122—21no—
234Dic122—24no—
235Dic122—21no—
236Dic122—21no—
237Dic122—42no—
238Dic122#11712yes—
239#92420—42no—
240C8×C61—21no—
241C8×C61—21no—
242#92418—21no—
243#92418—21no—
244#21132—611nomaximal, Sylow
245C22×C61C2×C4×D411yescharacteristic
246C2×D61#52512yes—
247C2×D61#57711yescharacteristic
248C2×D61#57711yescharacteristic
249C2×D61#57812yes—
250#6923—42no—
251#6923—44no—
252#6923—42no—
253#6923—82no—
254C4×C61—21no—
255C4×C61—21no—
256C4×C61—22no—
257C4×C61—41no—
258C4×C61—41no—
259C4×C61—41no—
260C4×C61C2×#11711yescharacteristic
261C4×C61C4×#4111yescharacteristic
262Dic61—21no—
263Dic61—21no—
264Dic61—22no—
265Dic61—41no—
266Dic61—41no—
267Dic61—41no—
268Dic61#52912yes—
269C2×C4×Q16—611no—
270#2320—611no—
271#2276—611no—
272#1927—612no—
273#1921—612no—
274#1878—612no—
275#1865—612no—
276C4×#106—611no—
277#1763—611no—
278#1763—611no—
279#1746—611no—
280C2×C61#229011yescharacteristic
281C2×C61#295011yescharacteristic
282C2×C61#295211yescharacteristic
283D61#312012yes—
284D61#312112yes—
285C22×Q16—611no—
286C2×C4×Q8—611no—
287C2×C4×Q8—611no—
288C2×C4×Q8—612no—
289C4×Q16—611no—
290C4×Q16—611no—
291C4×Q16—611no—
292C4×Q16—611no—
293C4×Q16—611no—
294C4×Q16—611no—
295C4×Q16—611no—
296C4×Q16#692311yescharacteristic, complemented
297C2×#107—611no—
298C2×#106—612no—
299C2×#103—611no—
300C2×#103—611no—
301C2×#103—612no—
302C2×#103—612no—
303C2×#97—611no—
304C2×C4×C8—611no—
305#391—612no—
306#389—612no—
307#387—1222no—
308#383—1221no—
309#382—1221no—
310C4×C4⋊C4—611no—
311C4×C4⋊C4—612no—
312C4×C4⋊C4—611no—
313C4×C4⋊C4—1221no—
314C4×C4⋊C4—1222no—
315#376—611no—
316#376—1221no—
317#340—612no—
318#340—612no—
319#340—612no—
320#336—612no—
321#336—612no—
322C61#2113211yescharacteristic, complemented, Sylow
323C22×Q8—611no—
324C22×Q8—611no—
325C2×Q16—611no—
326C2×Q16—611no—
327C2×Q16—611no—
328C2×Q16—1224no—
329C2×Q16C2×#692311yescharacteristic
330C22×C8—611no—
331C22×C8—611no—
332C22×C8—612no—
333C4×Q8—611no—
334C4×Q8—611no—
335C4×Q8—611no—
336C4×Q8—611no—
337C4×Q8—611no—
338C4×Q8—611no—
339C4×Q8—1221no—
340C4×Q8—1221no—
341C4×Q8—1221no—
342C4×Q8—1221no—
343C4×Q8—1224no—
344C4×Q8—1224no—
345C4×Q8C2×#692311yescharacteristic
346C4×Q8C2×#692311yescharacteristic
347C2×C4⋊C4—612no—
348C2×C4⋊C4—611no—
349C2×C4⋊C4—611no—
350C2×C4⋊C4—612no—
351C2×C4⋊C4—1221no—
352C2×C4⋊C4—1221no—
353C2×C4⋊C4—1222no—
354C2×C4⋊C4—1222no—
355C2×C42—611no—
356C2×C42—612no—
357C2×C42—1221no—
358C2×C42—1221no—
359C2×C42—1222no—
360C2×C42—1221no—
361C2×C42—1222no—
362C2×C42—1221no—
363#107—611no—
364#107—611no—
365#107—611no—
366#107C2×#692311yescharacteristic
367#106—1224no—
368#103—611no—
369#103—611no—
370#103—611no—
371#103—611no—
372#103—611no—
373#103—611no—
374#103—1224no—
375#103—1224no—
376#103C2×#692311yescharacteristic
377#103C2×#692311yescharacteristic
378#97—1221no—
379#97—1221no—
380C4×C8—611no—
381C4×C8—611no—
382C4×C8—611no—
383C4×C8C2×#692311yescharacteristic
384#95—611no—
385#95—611no—
386#95—612no—
387#95—1221no—
388#95—1222no—
389#95—1221no—
390#95—1222no—
391#95—1222no—
392#95—1222no—
393C2×Q8—614no—
394C2×Q8—611no—
395C2×Q8—611no—
396C2×Q8—1221no—
397C2×Q8—1221no—
398C2×Q8—1222no—
399C2×Q8—1222no—
400C2×Q8—1222no—
401C2×Q8C22×#692311yescharacteristic
402C2×Q8C22×#692311yescharacteristic
403C22×C4—611no—
404C22×C4—611no—
405C22×C4—611no—
406C22×C4—612no—
407C22×C4—612no—
408C22×C4—1221no—
409C22×C4—1221no—
410C22×C4—1221no—
411C22×C4—1221no—
412C22×C4—1222no—
413C22×C4—1222no—
414Q16—22no—
415Q16—1222no—
416Q16—1222no—
417Q16—1222no—
418C2×C8—611no—
419C2×C8—611no—
420C2×C8—1222no—
421C2×C8—1222no—
422C2×C8—1222no—
423C2×C8—1222no—
424C2×C8—1222no—
425C2×C8C22×#692311yescharacteristic
426C2×C8C22×#692311yescharacteristic
427C4⋊C4—21no—
428C4⋊C4—21no—
429C4⋊C4—614no—
430C4⋊C4—611no—
431C4⋊C4—611no—
432C4⋊C4—1221no—
433C4⋊C4—1221no—
434C4⋊C4—1221no—
435C4⋊C4—1221no—
436C4⋊C4—1221no—
437C4⋊C4—1221no—
438C4⋊C4—1222no—
439C4⋊C4—1222no—
440C4⋊C4—1222no—
441C4⋊C4—1228no—
442C4⋊C4—1221no—
443C4⋊C4—1221no—
444C4⋊C4—4882no—
445C4⋊C4C22×#692311yescharacteristic
446C4⋊C4C22×#692311yescharacteristic
447C42—21no—
448C42—21no—
449C42—611no—
450C42—611no—
451C42—611no—
452C42—1221no—
453C42—1221no—
454C42—1221no—
455C42—1221no—
456C42—1222no—
457C42—1222no—
458C42—1228no—
459C42—1221no—
460C42—1221no—
461C42—4881no—
462C42—4881no—
463C42—4882no—
464C42C22×#692311yescharacteristic
465C23—611no—
466Q8—21no—
467Q8—21no—
468Q8—21no—
469Q8—612no—
470Q8—1224no—
471Q8—1221no—
472Q8—1221no—
473Q8—1221no—
474Q8—2442no—
475Q8#1451985612yes—
476C2×C4—21no—
477C2×C4—21no—
478C2×C4—21no—
479C2×C4—21no—
480C2×C4—611no—
481C2×C4—612no—
482C2×C4—612no—
483C2×C4—611no—
484C2×C4—611no—
485C2×C4—1224no—
486C2×C4—1221no—
487C2×C4—1221no—
488C2×C4—1222no—
489C2×C4—1221no—
490C2×C4—1222no—
491C2×C4—1224no—
492C2×C4—1222no—
493C2×C4—1222no—
494C2×C4—1222no—
495C2×C4—1222no—
496C2×C4—1222no—
497C2×C4—1221no—
498C2×C4—1221no—
499C2×C4—1221no—
500C2×C4—2441no—
501C2×C4—2441no—
502C2×C4—2441no—
503C2×C4—2441no—
504C2×C4—2442no—
505C2×C4—2442no—
506C2×C4—2442no—
507C2×C4—2442no—
508C2×C4—2442no—
509C2×C4—2442no—
510C2×C4D4×#692311yescharacteristic
511C2×C4D4×#692311yescharacteristic
512C2×C4C23×#692311yescharacteristic
513C8—21no—
514C8—21no—
515C8—1221no—
516C8—1221no—
517C8—2442no—
518C22—612no—
519C22—612no—
520C22—611no—
521C22—611no—
522C22C2×D4×#692311yescharacteristic
523C4—22no—
524C4—21no—
525C4—21no—
526C4—41no—
527C4—41no—
528C4—41no—
529C4—612no—
530C4—1222no—
531C4—1221no—
532C4—1221no—
533C4—4882no—
534C4—2442no—
535C4—2442no—
536C4—2444no—
537C4—2441no—
538C4—2441no—
539C4—2441no—
540C4C2×#1451985611yescharacteristic
541C4#41×#692311yescharacteristic
542C2—612no—
543C2—612no—
544C2#2146227311yescharacteristic
545C2#2146258011yescharacteristic
546C2#2146258211yescharacteristic
subgroupG/H
centerC22C2×D4×#6923
commutatorC2×C4×C61C23×C4
FrattiniC2×C4C23×#6923
FittingC4×Q16×C61C4
socleC22×C61C2×C4×D4
prime radicalC22×D61C22×D4
cosocleC2×C4×D61C24
hypercenterC4×Q16#6923
hypocenterC61#21132
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ C4×Q16

Complements

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

NH ≅ G/Ntype
37. C4×Dic244C4semidirect
38. Q16×Dic61C4semidirect
39. #16839856C4semidirect
40. C4×#11758202C4semidirect
41. C4×#11758202C4semidirect
42. #16839854C4semidirect
43. #16839854C4semidirect
48. C4×Q16×C61C4semidirect
296. C4×Q16#6923semidirect
322. C61#21132semidirect

Maximal subgroups

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

subgroupindex
#214623032
#214622592
#214618982
#214618832
#214618132
#214617852
C4×#145198452
#214615912
#214615372
C4×Q16×D612
#2113261