Finite Group Databasebeta

← #29577074

Subgroups — #29577074

Order 15616.

704 conjugacy classes of subgroups.

The subgroup lattice has 562 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21455071C211yesmaximal, characteristic
2#21454268C211yesmaximal, characteristic, complemented
3#21444386C211yesmaximal, characteristic, complemented
4#21444178C211yesmaximal, characteristic, complemented
5#21443725C211yesmaximal, characteristic, complemented
6#21443707C211yesmaximal, characteristic, complemented
7C61×#1788C211yesmaximal, characteristic, complemented
8C2×#14519802C411yescharacteristic, complemented
9C2×#14519802C411yescharacteristic, complemented
10C2×#14519784—21no—
11C2×#14519751C2211yescharacteristic
12#16839907—21no—
13#16839907—21no—
14#16839901—21no—
15#16839896—21no—
16#16839763C411yescharacteristic, complemented
17#16839763C411yescharacteristic, complemented
18#16839729—22no—
19#16839729—21no—
20#16839726—21no—
21#16839726C2211yescharacteristic
22#16839726C2211yescharacteristic
23C2×C22⋊#92418—21no—
24C2×C22⋊#92418C2211yescharacteristic
25#16839495—21no—
26#16839495—21no—
27#16839485—22no—
28C22×#41×C61C2211yescharacteristic
29C2×#103×C61—21no—
30C2×#102×C61—21no—
31C2×C22⋊C8×C61C2211yescharacteristic
32C61×#380C2211yescharacteristic
33C22×D244C2×C411yescharacteristic
34C22×C22⋊D61—21no—
35#14519802—42no—
36#14519802—42no—
37C2×#11758201—21no—
38C2×#11758201—21no—
39C2×#11758199—21no—
40C2×#11758199—21no—
41#14519784—42no—
42#14519784—42no—
43C2×#11758194—22no—
44C2×#11758194—21no—
45C2×#11758194C2×C411yescharacteristic
46C2×#11758194C2×C411yescharacteristic
47C22×#92418—22no—
48C22×#92418—21no—
49#14519751—21no—
50#14519751—21no—
51#14519751—21no—
52#14519751—21no—
53C2×C4⋊Dic61C2×C411yescharacteristic
54C2×#11758198—21no—
55#14519701—21no—
56#14519701—21no—
57#14519698—21no—
58#14519698—21no—
59C22⋊#92418—22no—
60C22⋊#92418—22no—
61C22⋊#92418—21no—
62C22⋊#92418—21no—
63C22⋊#92418—21no—
64C22⋊#92418—21no—
65C22⋊#92418D411yescharacteristic
66C22⋊#92418D411yescharacteristic
67C22⋊#92418D412yes—
68C2×#41×C61—21no—
69C2×#41×C61—21no—
70C2×#41×C61—42no—
71C2×#41×C61—42no—
72C2×#41×C61C2×C411yescharacteristic
73C2×#41×C61C2×C411yescharacteristic
74C2×#41×C61D411yescharacteristic
75C2×#41×C61D411yescharacteristic
76C22×Q8×C61D411yescharacteristic
77C22×D4×C61—21no—
78C22×D4×C61D411yescharacteristic
79C23×C4×C61—21no—
80C23×C4×C61C2311yescharacteristic
81C22×C8×C61—21no—
82C2×C4⋊C4×C61—21no—
83C2×C22⋊C4×C61—21no—
84#103×C61—42no—
85#102×C61—42no—
86C22⋊C8×C61—21no—
87C22⋊C8×C61—21no—
88#95×C61—21no—
89C24×C61—21no—
90C24×C61#4111yescharacteristic
91C23×D61—21no—
92#41×C61—22no—
93#41×C61—22no—
94#41×C61—22no—
95#41×C61—22no—
96#41×C61—41no—
97#41×C61—41no—
98#41×C61—41no—
99#41×C61—41no—
100#41×C61—44no—
101#41×C61—44no—
102C2×Q8×C61—21no—
103C2×Q8×C61—22no—
104C2×Q8×C61—41no—
105C2×Q8×C61C22⋊C411yescharacteristic
106C2×Q8×C61C22⋊C411yescharacteristic
107C2×D4×C61—21no—
108C2×D4×C61—22no—
109C2×D4×C61—21no—
110C2×D4×C61—21no—
111C2×D4×C61—21no—
112C2×D4×C61—21no—
113C2×D4×C61—41no—
114C2×D4×C61—44no—
115C2×D4×C61C22⋊C411yescharacteristic
116C2×D4×C61C22⋊C411yescharacteristic
117C22×C4×C61—21no—
118C22×C4×C61—21no—
119C22×C4×C61—21no—
120C22×C4×C61—21no—
121C22×C4×C61—21no—
122C22×C4×C61—41no—
123C22×C4×C61—41no—
124C22×C4×C61—44no—
125C22×C4×C61—41no—
126C22×C4×C61C22×C411yescharacteristic
127C22×C4×C61C2×D411yescharacteristic
128C22×C4×C61C2×D411yescharacteristic
129C22×C4×C61C2×D411yescharacteristic
130C22×C4×C61C2×D411yescharacteristic
131C22×C4×C61#4111yescharacteristic
132C2×C22⋊D61—41no—
133C2×C22⋊D61—41no—
134C2×C22⋊D61—81no—
135C2×C22⋊D61—81no—
136C22×Dic61—21no—
137C2×D244—21no—
138C2×D244—21no—
139C2×D244—41no—
140C2×D244—41no—
141C2×C8×C61—41no—
142C2×C8×C61—41no—
143C2×C8×C61—41no—
144C4⋊C4×C61—41no—
145C4⋊C4×C61—41no—
146C22⋊C4×C61—81no—
147#11758201—42no—
148#11758201—42no—
149#11758199—42no—
150#11758199—42no—
151#11758198—41no—
152#11758198—41no—
153C4⋊Dic61—21no—
154C4⋊Dic61—21no—
155#11758194—44no—
156#11758194—41no—
157#11758194—41no—
158#11758194—41no—
159#11758194—41no—
160C2×#92418—21no—
161C2×#92418—21no—
162C2×#92418—21no—
163C2×#92418—22no—
164C2×#92418—21no—
165C2×#92418—44no—
166C2×#92418—41no—
167C2×#92418—41no—
168C23×C61—21no—
169C23×C61—21no—
170C23×C61—21no—
171C23×C61—21no—
172C23×C61—41no—
173C23×C61—41no—
174C23×C61—42no—
175C23×C61—41no—
176C23×C61—41no—
177C23×C61—81no—
178C23×C61C4×D411yescharacteristic
179C23×C61#12111yescharacteristic
180C23×C61#12311yescharacteristic
181C22×D61—41no—
182C22×D61—42no—
183C22×D61—81no—
184C22×D61—81no—
185Q8×C61—21no—
186Q8×C61—21no—
187Q8×C61—21no—
188Q8×C61—42no—
189Q8×C61#9912yes—
190D4×C61—24no—
191D4×C61—21no—
192D4×C61—21no—
193D4×C61—21no—
194D4×C61—42no—
195D4×C61—42no—
196D4×C61—42no—
197D4×C61—42no—
198D4×C61#9912yes—
199C2×C4×C61—24no—
200C2×C4×C61—21no—
201C2×C4×C61—21no—
202C2×C4×C61—21no—
203C2×C4×C61—21no—
204C2×C4×C61—21no—
205C2×C4×C61—21no—
206C2×C4×C61—21no—
207C2×C4×C61—21no—
208C2×C4×C61—41no—
209C2×C4×C61—41no—
210C2×C4×C61—41no—
211C2×C4×C61—42no—
212C2×C4×C61—42no—
213C2×C4×C61—42no—
214C2×C4×C61—42no—
215C2×C4×C61—81no—
216C2×C4×C61—81no—
217C2×C4×C61—81no—
218C2×C4×C61C2×C22⋊C411yescharacteristic
219C2×C4×C61C4×D411yescharacteristic
220C2×C4×C61#12011yescharacteristic
221C2×C4×C61#12111yescharacteristic
222C22⋊D61—82no—
223C22⋊D61—81no—
224C22⋊D61—81no—
225C2×Dic61—41no—
226C2×Dic61—81no—
227D244—41no—
228D244—41no—
229D244—81no—
230C8×C61—81no—
231#92418—41no—
232#92418—42no—
233#92418—41no—
234#92418—41no—
235#92418—41no—
236#9347—611nomaximal, Sylow
237C22×C61—21no—
238C22×C61—21no—
239C22×C61—21no—
240C22×C61—21no—
241C22×C61—21no—
242C22×C61—22no—
243C22×C61—21no—
244C22×C61—21no—
245C22×C61—21no—
246C22×C61—41no—
247C22×C61—41no—
248C22×C61—41no—
249C22×C61—41no—
250C22×C61—41no—
251C22×C61—81no—
252C22×C61—81no—
253C22×C61—82no—
254C22×C61—82no—
255C22×C61#38611yescharacteristic
256C22×C61#47111yescharacteristic
257C22×C61#47211yescharacteristic
258C2×D61—81no—
259C2×D61—81no—
260C2×D61—81no—
261C2×D61—162no—
262C4×C61—21no—
263C4×C61—21no—
264C4×C61—21no—
265C4×C61—22no—
266C4×C61—41no—
267C4×C61—161no—
268C4×C61C2×#9911yescharacteristic
269C4×C61C2×#9911yescharacteristic
270Dic61—81no—
271#3006—611no—
272#2880—611no—
273#1788D6111yescharacteristic, complemented
274#1519—611no—
275#1471—611no—
276#1305—611no—
277#1298—611no—
278C2×C61—21no—
279C2×C61—21no—
280C2×C61—41no—
281C2×C61—41no—
282C2×C61—42no—
283C2×C61—81no—
284C2×C61#188111yescharacteristic
285C2×C61#189811yescharacteristic
286C2×C61#190711yescharacteristic
287D61—161no—
288C22×#41C2×D6111yescharacteristic
289C2×AGL(1,ℤ/8)—611no—
290C2×AGL(1,ℤ/8)—611no—
291C2×#131—1221no—
292C2×#121—611no—
293#449—1221no—
294#449—1221no—
295#448—1221no—
296#447—1221no—
297#417—611no—
298#417—611no—
299C2×#103—21no—
300C2×#102—21no—
301#408—1222no—
302#408—1221no—
303#407—611no—
304#407—611no—
305#407—1221no—
306C2×C22⋊C8—611no—
307C2×C22⋊C8—1221no—
308C2×C22⋊C8C2×D6111yescharacteristic
309#380C2×D6111yescharacteristic
310#327—1221no—
311#327—1221no—
312#323—1222no—
313C61#934711yescharacteristic, complemented, Sylow
314C2×#41—21no—
315C2×#41—21no—
316C2×#41—42no—
317C2×#41—42no—
318C2×#41C4×D6111yescharacteristic
319C2×#41C4×D6111yescharacteristic
320C2×#41D24411yescharacteristic
321C2×#41D24411yescharacteristic
322C22×Q8C22⋊D6111yescharacteristic
323C22×D4—21no—
324C22×D4—611no—
325C22×D4—1221no—
326C22×D4C22⋊D6111yescharacteristic
327C23×C4—21no—
328C23×C4C22×D6111yescharacteristic
329AGL(1,ℤ/8)—2442no—
330AGL(1,ℤ/8)—2442no—
331C2×SD16—1221no—
332C2×SD16—1221no—
333C2×D8—1221no—
334C2×D8—1221no—
335#131—2442no—
336#131—2442no—
337C2×M16—611no—
338C2×M16—611no—
339C2×M16—1222no—
340C2×M16—1221no—
341C22×C8—21no—
342C22×C8—1222no—
343C22×C8—1221no—
344#121—1221no—
345#121—1221no—
346#121—1221no—
347#121—1221no—
348C2×C4⋊C4—21no—
349C2×C4⋊C4—611no—
350C2×C22⋊C4—21no—
351C2×C22⋊C4—1221no—
352#103—42no—
353#103—1221no—
354#103—1221no—
355#102—42no—
356#102—1221no—
357#102—1221no—
358C22⋊C8—21no—
359C22⋊C8—21no—
360C22⋊C8—612no—
361C22⋊C8—611no—
362C22⋊C8—611no—
363C22⋊C8—1222no—
364C22⋊C8—1222no—
365C22⋊C8—1221no—
366C22⋊C8—1221no—
367C22⋊C8—1221no—
368C22⋊C8—1221no—
369#95—21no—
370C24—21no—
371C24—1221no—
372C24#1175822411yescharacteristic
373#41—22no—
374#41—22no—
375#41—22no—
376#41—22no—
377#41—41no—
378#41—41no—
379#41—41no—
380#41—41no—
381#41—44no—
382#41—44no—
383C2×Q8—21no—
384C2×Q8—22no—
385C2×Q8—41no—
386C2×Q8#1175819811yescharacteristic
387C2×Q8#1175819811yescharacteristic
388C2×D4—21no—
389C2×D4—22no—
390C2×D4—21no—
391C2×D4—21no—
392C2×D4—21no—
393C2×D4—21no—
394C2×D4—41no—
395C2×D4—44no—
396C2×D4—1221no—
397C2×D4—1221no—
398C2×D4—4881no—
399C2×D4—4881no—
400C2×D4—2441no—
401C2×D4—2441no—
402C2×D4—2441no—
403C2×D4—2441no—
404C2×D4#1175819811yescharacteristic
405C2×D4#1175819811yescharacteristic
406C22×C4—21no—
407C22×C4—21no—
408C22×C4—21no—
409C22×C4—21no—
410C22×C4—21no—
411C22×C4—41no—
412C22×C4—41no—
413C22×C4—44no—
414C22×C4—41no—
415C22×C4—1221no—
416C22×C4C2×C4×D6111yescharacteristic
417C22×C4C2×D24411yescharacteristic
418C22×C4D4×D6111yescharacteristic
419C22×C4D4×D6111yescharacteristic
420C22×C4#1175822411yescharacteristic
421C22×C4C2×C22⋊D6111yescharacteristic
422SD16—2442no—
423SD16—2442no—
424D8—2442no—
425D8—2442no—
426M16—2444no—
427M16—2441no—
428M16—2441no—
429M16—2441no—
430M16—2441no—
431C2×C8—41no—
432C2×C8—41no—
433C2×C8—41no—
434C2×C8—1221no—
435C2×C8—1221no—
436C2×C8—1221no—
437C2×C8—1222no—
438C2×C8—1221no—
439C2×C8—2444no—
440C2×C8—2441no—
441C2×C8—2441no—
442C4⋊C4—41no—
443C4⋊C4—41no—
444C4⋊C4—1221no—
445C4⋊C4—1221no—
446C22⋊C4—81no—
447C22⋊C4—2441no—
448C22⋊C4—2441no—
449C23—21no—
450C23—21no—
451C23—21no—
452C23—21no—
453C23—41no—
454C23—41no—
455C23—42no—
456C23—41no—
457C23—41no—
458C23—81no—
459C23—4881no—
460C23—4881no—
461C23—2442no—
462C23—2441no—
463C23#1451973711yescharacteristic
464C23#1451974911yescharacteristic
465C23#1451976111yescharacteristic
466Q8—21no—
467Q8—21no—
468Q8—21no—
469Q8—42no—
470Q8#1451969112yes—
471D4—24no—
472D4—21no—
473D4—21no—
474D4—21no—
475D4—42no—
476D4—42no—
477D4—42no—
478D4—42no—
479D4—4881no—
480D4—4882no—
481D4—4881no—
482D4—4881no—
483D4—2441no—
484D4—2441no—
485D4#1451969112yes—
486C2×C4—24no—
487C2×C4—21no—
488C2×C4—21no—
489C2×C4—21no—
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—21no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—41no—
496C2×C4—41no—
497C2×C4—41no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—81no—
503C2×C4—81no—
504C2×C4—81no—
505C2×C4—4881no—
506C2×C4—2441no—
507C2×C4C2×#1175819811yescharacteristic
508C2×C4#1451973711yescharacteristic
509C2×C4#1451974711yescharacteristic
510C2×C4#1451974911yescharacteristic
511C8—81no—
512C8—2441no—
513C8—2442no—
514C8—2441no—
515C8—2441no—
516C8—2441no—
517C22—21no—
518C22—21no—
519C22—21no—
520C22—21no—
521C22—21no—
522C22—22no—
523C22—21no—
524C22—21no—
525C22—21no—
526C22—41no—
527C22—41no—
528C22—41no—
529C22—41no—
530C22—41no—
531C22—81no—
532C22—81no—
533C22—82no—
534C22—82no—
535C22—9762no—
536C22—4881no—
537C22—4881no—
538C22—4881no—
539C22#1683963911yescharacteristic
540C22#1684002711yescharacteristic
541C22#1684003011yescharacteristic
542C4—21no—
543C4—21no—
544C4—22no—
545C4—21no—
546C4—41no—
547C4—161no—
548C4—4881no—
549C4C2×#1451969111yescharacteristic
550C4C2×#1451969111yescharacteristic
551C2—21no—
552C2—21no—
553C2—41no—
554C2—41no—
555C2—42no—
556C2—81no—
557C2—9761no—
558C2#2144617911yescharacteristic
559C2#2144627011yescharacteristic
560C2#2144632511yescharacteristic
subgroupG/H
centerC22#16839639
commutatorC22×C4×C61C22×C4
FrattiniC23×C4C22×D61
FittingC61×#1788C2
socleC22×C61#386
prime radical#21455071C2
cosocleC23×C4×C61C23
hypercenter#1788D61
hypocenterC61#9347
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1788

Complements

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

NH ≅ G/Ntype
2. #21454268C2semidirect
3. #21444386C2semidirect
4. #21444178C2semidirect
5. #21443725C2semidirect
6. #21443707C2semidirect
7. C61×#1788C2semidirect
8. C2×#14519802C4semidirect
9. C2×#14519802C4semidirect
16. #16839763C4semidirect
17. #16839763C4semidirect
273. #1788D61semidirect
313. C61#9347semidirect

Maximal subgroups

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

subgroupindex
#214550712
#214542682
#214443862
#214441782
#214437252
#214437072
C61×#17882
#934761