Finite Group Databasebeta

← #29572765

Subgroups — #29572765

Order 15616.

742 conjugacy classes of subgroups.

The subgroup lattice has 574 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21448325C211yesmaximal, characteristic
2#21445797C211yesmaximal, characteristic, complemented
3C2×#16839497C211yesmaximal, characteristic
4C2×#16839495C211yesmaximal, characteristic
5#21444047C211yesmaximal, characteristic, complemented
6#21443697C211yesmaximal, characteristic, complemented
7C61×#1294C211yesmaximal, characteristic, complemented
8C2×#14519751C2211yescharacteristic
9C2×#14519739C411yescharacteristic, complemented
10C2×#14519737—21no—
11C22×#11758196—21no—
12C22×C4×Dic61C2211yescharacteristic
13C2×#14519706—21no—
14C2×C22⋊#92418C2211yescharacteristic
15#16839663C411yescharacteristic, complemented
16#16839658C411yescharacteristic, complemented
17#16839651—21no—
18#16839639—21no—
19C4⋊C4×Dic61C411yescharacteristic, complemented
20#16839609—21no—
21C2×#14519684—21no—
22C2×#14519684C2211yescharacteristic
23#16839516—21no—
24#16839497—22no—
25#16839495—22no—
26C2×C22⋊C8×C61C2211yescharacteristic
27C61×#380C2211yescharacteristic
28C2×#95×C61C2211yescharacteristic
29C22×D244C2×C411yescharacteristic
30C22×C22⋊D61—21no—
31C23×Dic61—21no—
32C23×Dic61D411yescharacteristic
33C23×Dic61D411yescharacteristic
34C22×C4×D61—21no—
35C22×#92418—21no—
36#14519751—21no—
37#14519751—21no—
38#14519751D412yes—
39#14519751Q812yes—
40#14519739—42no—
41#14519737—42no—
42#14519737—42no—
43C2×C4⋊Dic61C2×C411yescharacteristic
44C2×#11758196—24no—
45C2×#11758196—21no—
46C2×#11758196—42no—
47C2×#11758196—42no—
48C2×#11758198—21no—
49C2×#11758198—21no—
50C2×#11758198—21no—
51C2×#11758198—21no—
52C2×C4×Dic61—21no—
53C2×C4×Dic61—42no—
54C2×C4×Dic61C2×C411yescharacteristic
55C2×C4×Dic61C2×C411yescharacteristic
56C2×C4×Dic61D411yescharacteristic
57C2×C4×Dic61D411yescharacteristic
58#14519706—42no—
59C22⋊#92418—21no—
60C22⋊#92418—21no—
61#14519684—21no—
62#14519684—21no—
63#14519684—21no—
64#14519684—21no—
65#14519684—42no—
66C23×C4×C61—21no—
67C23×C4×C61C2311yescharacteristic
68C22×C8×C61—21no—
69C2×C4⋊C4×C61C2×C411yescharacteristic
70C2×C4⋊C4×C61C2×C411yescharacteristic
71C2×C22⋊C4×C61—21no—
72C22⋊C8×C61—21no—
73C22⋊C8×C61—21no—
74#95×C61—21no—
75#95×C61—21no—
76#95×C61—21no—
77C24×C61C2×D411yescharacteristic
78C23×D61—21no—
79C22×C4×C61—22no—
80C22×C4×C61—21no—
81C22×C4×C61—21no—
82C22×C4×C61—41no—
83C22×C4×C61—41no—
84C22×C4×C61—41no—
85C22×C4×C61—41no—
86C22×C4×C61—41no—
87C22×C4×C61—41no—
88C22×C4×C61—41no—
89C22×C4×C61—41no—
90C22×C4×C61C22×C411yescharacteristic
91C22×C4×C61C2×D411yescharacteristic
92C22×C4×C61C2×D411yescharacteristic
93C22×C4×C61C2×Q811yescharacteristic
94C22×C4×C61#4111yescharacteristic
95C22×C4×C61#4111yescharacteristic
96C2×C22⋊D61—22no—
97C2×C22⋊D61—22no—
98C2×C22⋊D61—41no—
99C2×C22⋊D61—41no—
100C2×C22⋊D61—41no—
101C2×C22⋊D61—41no—
102C22×Dic61—21no—
103C22×Dic61—21no—
104C22×Dic61—22no—
105C22×Dic61—22no—
106C22×Dic61—21no—
107C22×Dic61—22no—
108C22×Dic61—41no—
109C22×Dic61—41no—
110C22×Dic61—41no—
111C22×Dic61—41no—
112C22×Dic61—41no—
113C22×Dic61—41no—
114C22×Dic61—41no—
115C22×Dic61—41no—
116C22×Dic61C22⋊C411yescharacteristic
117C22×Dic61C22⋊C411yescharacteristic
118C22×Dic61C22⋊C411yescharacteristic
119C22×Dic61C22⋊C411yescharacteristic
120C22×Dic61D812yes—
121C22×Dic61SD1612yes—
122C2×D244—21no—
123C2×D244—41no—
124C2×D244—41no—
125C2×D244C4⋊C412yescomplemented
126C2×C4×D61—42no—
127C2×C4×D61—42no—
128C2×C4×D61—42no—
129C2×C8×C61—41no—
130C2×C8×C61—41no—
131C2×C8×C61—41no—
132C4⋊C4×C61—41no—
133C4⋊C4×C61—41no—
134C22⋊C4×C61—41no—
135C22⋊C4×C61—41no—
136#11758198—22no—
137#11758198—22no—
138#11758198—42no—
139#11758198—42no—
140#11758198—42no—
141C4⋊Dic61—21no—
142C4⋊Dic61C4⋊C412yescomplemented
143#11758196—48no—
144#11758196—42no—
145C4×Dic61—22no—
146C4×Dic61—22no—
147C4×Dic61—41no—
148C4×Dic61—41no—
149C4×Dic61—42no—
150C2×#92418—41no—
151C2×#92418—41no—
152C2×#92418—41no—
153C23×C61—21no—
154C23×C61—21no—
155C23×C61—21no—
156C23×C61—21no—
157C23×C61C2×C22⋊C411yescharacteristic
158C23×C61#12111yescharacteristic
159C23×C61C22⋊Q811yescharacteristic
160C23×C61C2×D811yescharacteristic
161C23×C61C2×SD1611yescharacteristic
162C23×C61AGL(1,ℤ/8)11yescharacteristic
163C23×C61#13711yescharacteristic
164C22×D61—22no—
165C22×D61—41no—
166C22×D61—41no—
167C22×D61—41no—
168C22×D61—41no—
169C22×D61—41no—
170C22×D61—41no—
171C2×C4×C61—21no—
172C2×C4×C61—21no—
173C2×C4×C61—21no—
174C2×C4×C61—21no—
175C2×C4×C61—41no—
176C2×C4×C61—41no—
177C2×C4×C61—41no—
178C2×C4×C61—41no—
179C2×C4×C61—42no—
180C2×C4×C61—42no—
181C2×C4×C61—42no—
182C2×C4×C61—41no—
183C2×C4×C61—41no—
184C2×C4×C61—41no—
185C2×C4×C61—41no—
186C2×C4×C61—41no—
187C2×C4×C61—41no—
188C2×C4×C61—81no—
189C2×C4×C61—81no—
190C2×C4×C61—81no—
191C2×C4×C61—81no—
192C2×C4×C61C2×C4⋊C411yescharacteristic
193C2×C4×C61#11711yescharacteristic
194C2×C4×C61#12111yescharacteristic
195C2×C4×C61C22⋊Q811yescharacteristic
196C22⋊D61—42no—
197C22⋊D61—42no—
198C22⋊D61—42no—
199C22⋊D61—42no—
200C2×Dic61—24no—
201C2×Dic61—24no—
202C2×Dic61—21no—
203C2×Dic61—21no—
204C2×Dic61—21no—
205C2×Dic61—21no—
206C2×Dic61—21no—
207C2×Dic61—21no—
208C2×Dic61—21no—
209C2×Dic61—21no—
210C2×Dic61—22no—
211C2×Dic61—42no—
212C2×Dic61—42no—
213C2×Dic61—41no—
214C2×Dic61—41no—
215C2×Dic61—42no—
216C2×Dic61—42no—
217C2×Dic61—42no—
218C2×Dic61—81no—
219C2×Dic61—81no—
220C2×Dic61—81no—
221C2×Dic61—81no—
222C2×Dic61#9912yes—
223C2×Dic61#10212yes—
224C2×Dic61#10212yes—
225C2×Dic61#10412yes—
226D244—42no—
227D244—42no—
228C4×D61—84no—
229C8×C61—81no—
230#92418—81no—
231#8577—611nomaximal, Sylow
232C22×C61—21no—
233C22×C61—21no—
234C22×C61—21no—
235C22×C61—21no—
236C22×C61—21no—
237C22×C61—21no—
238C22×C61—21no—
239C22×C61—21no—
240C22×C61—21no—
241C22×C61—21no—
242C22×C61—21no—
243C22×C61—21no—
244C22×C61#38011yescharacteristic
245C22×C61C2×#9911yescharacteristic
246C22×C61#41011yescharacteristic
247C22×C61C2×#10211yescharacteristic
248C22×C61#41711yescharacteristic
249C22×C61C2×#10411yescharacteristic
250C22×C61#42111yescharacteristic
251C22×C61#45911yescharacteristic
252C22×C61#46111yescharacteristic
253C22×C61#47411yescharacteristic
254C22×C61#47611yescharacteristic
255C2×D61—42no—
256C2×D61—42no—
257C2×D61—42no—
258C2×D61—81no—
259C2×D61—81no—
260C2×D61—81no—
261C2×D61—81no—
262C4×C61—21no—
263C4×C61—21no—
264C4×C61—21no—
265C4×C61—21no—
266C4×C61—82no—
267C4×C61—81no—
268C4×C61—81no—
269Dic61—22no—
270Dic61—22no—
271Dic61—22no—
272Dic61—22no—
273Dic61—42no—
274Dic61—82no—
275C4×#121—611no—
276#1797—611no—
277C2×#327—611no—
278C2×#327—611no—
279C4⋊#95—611no—
280#1294—611no—
281#1294D6111yescharacteristic, complemented
282C2×C61—21no—
283C2×C61—21no—
284C2×C61—21no—
285C2×C61—21no—
286C2×C61#150211yescharacteristic
287C2×C61#150411yescharacteristic
288C2×C61#151211yescharacteristic
289C2×C61#151911yescharacteristic
290C2×C61#179911yescharacteristic
291C2×C61#180211yescharacteristic
292C2×C61#180611yescharacteristic
293D61—82no—
294C2×#121—611no—
295C2×C4×D4—611no—
296C2×C4×D4—1221no—
297C22×C4⋊C4—1221no—
298C22×C42—611no—
299C2×C4⋊C8—1221no—
300C2×C22⋊C8—611no—
301C2×C22⋊C8C2×D6111yescharacteristic
302#390—611no—
303#389—611no—
304#388—1221no—
305#386—1221no—
306#380C2×D6111yescharacteristic
307C4×C4⋊C4—611no—
308C4×C22⋊C4—1221no—
309C2×#95—611no—
310C2×#95—1221no—
311C2×#95C2×D6111yescharacteristic
312#336—1221no—
313#327—1222no—
314#327—1222no—
315C61#857711yescharacteristic, complemented, Sylow
316C22×D4—611no—
317C22×D4—1221no—
318C23×C4—21no—
319C23×C4—611no—
320C23×C4—611no—
321C23×C4—1221no—
322C23×C4—1221no—
323C23×C4C22×D6111yescharacteristic
324C22×C8—21no—
325C22×C8—1221no—
326#121—612no—
327#121—612no—
328#121—1221no—
329#121—1221no—
330C4×D4—2442no—
331C4×D4—2442no—
332C4×D4—2442no—
333C2×C4⋊C4—611no—
334C2×C4⋊C4—1221no—
335C2×C4⋊C4—1224no—
336C2×C4⋊C4—2442no—
337C2×C4⋊C4—2442no—
338C2×C4⋊C4C4×D6111yescharacteristic
339C2×C4⋊C4C4×D6111yescharacteristic
340C2×C22⋊C4—21no—
341C2×C22⋊C4—1221no—
342C2×C22⋊C4—1221no—
343C2×C22⋊C4—1221no—
344C2×C22⋊C4—1221no—
345C2×C42—611no—
346C2×C42—611no—
347C2×C42—611no—
348C2×C42—611no—
349C2×C42—1221no—
350C2×C42—2442no—
351C4⋊C8—2442no—
352C22⋊C8—21no—
353C22⋊C8—21no—
354C22⋊C8—1221no—
355C22⋊C8—1221no—
356#95—21no—
357#95—21no—
358#95—21no—
359#95—1221no—
360#95—1221no—
361#95—1221no—
362#95—1221no—
363#95—2442no—
364C24—1221no—
365C24D4×D6111yescharacteristic
366C2×D4—612no—
367C2×D4—1222no—
368C2×D4—1222no—
369C2×D4—1221no—
370C2×D4—2441no—
371C2×D4—2441no—
372C2×D4—2441no—
373C2×D4—2441no—
374C2×D4—2441no—
375C2×D4—2441no—
376C22×C4—22no—
377C22×C4—21no—
378C22×C4—21no—
379C22×C4—41no—
380C22×C4—41no—
381C22×C4—41no—
382C22×C4—41no—
383C22×C4—41no—
384C22×C4—41no—
385C22×C4—41no—
386C22×C4—41no—
387C22×C4—611no—
388C22×C4—611no—
389C22×C4—611no—
390C22×C4—611no—
391C22×C4—612no—
392C22×C4—612no—
393C22×C4—1222no—
394C22×C4—1222no—
395C22×C4—1221no—
396C22×C4—1221no—
397C22×C4—1221no—
398C22×C4—1222no—
399C22×C4—2442no—
400C22×C4—2442no—
401C22×C4—2442no—
402C22×C4—2441no—
403C22×C4—2441no—
404C22×C4—2441no—
405C22×C4—2441no—
406C22×C4—2441no—
407C22×C4—2441no—
408C22×C4—2441no—
409C22×C4—2441no—
410C22×C4C2×C4×D6111yescharacteristic
411C22×C4#1175822211yescharacteristic
412C22×C4#1175822211yescharacteristic
413C22×C4D4×D6111yescharacteristic
414C22×C4D4×D6111yescharacteristic
415C22×C4Q8×D6111yescharacteristic
416C2×C8—41no—
417C2×C8—41no—
418C2×C8—41no—
419C2×C8—2441no—
420C2×C8—2441no—
421C2×C8—2441no—
422C4⋊C4—41no—
423C4⋊C4—41no—
424C4⋊C4—612no—
425C4⋊C4—1221no—
426C4⋊C4—2442no—
427C4⋊C4—2448no—
428C22⋊C4—41no—
429C22⋊C4—41no—
430C22⋊C4—1222no—
431C22⋊C4—1222no—
432C22⋊C4—2442no—
433C22⋊C4—2442no—
434C22⋊C4—2442no—
435C42—1222no—
436C42—1222no—
437C42—2441no—
438C42—2441no—
439C42—2442no—
440C23—21no—
441C23—21no—
442C23—21no—
443C23—21no—
444C23—1222no—
445C23—2441no—
446C23—2441no—
447C23—2441no—
448C23—2441no—
449C23—2441no—
450C23—2441no—
451C23C22⋊C4×D6111yescharacteristic
452C23#1451975011yescharacteristic
453C23#1451975511yescharacteristic
454C23D8×D6111yescharacteristic
455C23SD16×D6111yescharacteristic
456C23#1451980411yescharacteristic
457C23#1451980911yescharacteristic
458D4—2442no—
459D4—2442no—
460D4—2442no—
461D4—2442no—
462D4—2442no—
463D4—2442no—
464C2×C4—21no—
465C2×C4—21no—
466C2×C4—21no—
467C2×C4—21no—
468C2×C4—41no—
469C2×C4—41no—
470C2×C4—41no—
471C2×C4—41no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—41no—
476C2×C4—41no—
477C2×C4—41no—
478C2×C4—41no—
479C2×C4—41no—
480C2×C4—41no—
481C2×C4—81no—
482C2×C4—81no—
483C2×C4—81no—
484C2×C4—81no—
485C2×C4—612no—
486C2×C4—612no—
487C2×C4—612no—
488C2×C4—612no—
489C2×C4—1224no—
490C2×C4—1224no—
491C2×C4—1221no—
492C2×C4—1221no—
493C2×C4—1221no—
494C2×C4—1221no—
495C2×C4—1221no—
496C2×C4—1221no—
497C2×C4—1221no—
498C2×C4—1221no—
499C2×C4—1222no—
500C2×C4—4884no—
501C2×C4—4881no—
502C2×C4—4881no—
503C2×C4—4881no—
504C2×C4—4881no—
505C2×C4—2442no—
506C2×C4—2442no—
507C2×C4—2442no—
508C2×C4—2442no—
509C2×C4—2442no—
510C2×C4—2441no—
511C2×C4—2441no—
512C2×C4C4⋊C4×D6111yescharacteristic
513C2×C4#1451973411yescharacteristic
514C2×C4#1451975011yescharacteristic
515C2×C4#1451975511yescharacteristic
516C8—81no—
517C8—4881no—
518C22—21no—
519C22—21no—
520C22—21no—
521C22—21no—
522C22—21no—
523C22—21no—
524C22—21no—
525C22—21no—
526C22—21no—
527C22—21no—
528C22—21no—
529C22—21no—
530C22—4881no—
531C22—4881no—
532C22—4881no—
533C22—4881no—
534C22—2442no—
535C22—2442no—
536C22—2442no—
537C22#1683962111yescharacteristic
538C22#99×D6111yescharacteristic
539C22#1683973411yescharacteristic
540C22#102×D6111yescharacteristic
541C22#1683976211yescharacteristic
542C22#104×D6111yescharacteristic
543C22#1683977711yescharacteristic
544C22#1683995611yescharacteristic
545C22#1683996611yescharacteristic
546C22#1684004111yescharacteristic
547C22#1684005111yescharacteristic
548C4—21no—
549C4—21no—
550C4—21no—
551C4—21no—
552C4—82no—
553C4—81no—
554C4—81no—
555C4—1222no—
556C4—1222no—
557C4—1222no—
558C4—1222no—
559C4—4882no—
560C4—2442no—
561C2—21no—
562C2—21no—
563C2—21no—
564C2—21no—
565C2—4882no—
566C2#2144431211yescharacteristic
567C2#2144432211yescharacteristic
568C2#2144435611yescharacteristic
569C2#2144438511yescharacteristic
570C2#2144580611yescharacteristic
571C2#2144581811yescharacteristic
572C2#2144583811yescharacteristic
subgroupG/H
centerC23C22⋊C4×D61
commutatorC22×C4×C61C22×C4
FrattiniC23×C4C22×D61
FittingC61×#1294C2
socleC23×C61C2×C22⋊C4
prime radicalC2×#14519751C22
cosocleC23×C4×C61C23
hypercenter#1294D61
hypocenterC61#8577
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1294

Complements

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

NH ≅ G/Ntype
2. #21445797C2semidirect
5. #21444047C2semidirect
6. #21443697C2semidirect
7. C61×#1294C2semidirect
9. C2×#14519739C4semidirect
15. #16839663C4semidirect
16. #16839658C4semidirect
19. C4⋊C4×Dic61C4semidirect
125. C2×D244C4⋊C4semidirect
142. C4⋊Dic61C4⋊C4semidirect
281. #1294D61semidirect
315. C61#8577semidirect

Maximal subgroups

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

subgroupindex
#214483252
#214457972
C2×#168394972
C2×#168394952
#214440472
#214436972
C61×#12942
#857761