Finite Group Databasebeta

← #29686945

Subgroups — #29686945

Order 15616.

876 conjugacy classes of subgroups.

The subgroup lattice has 534 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21455539C211yesmaximal, characteristic
2#21454718C212yesmaximal, complemented
3C2×#16839878C211yesmaximal, characteristic, complemented
4#131×Dic61C211yesmaximal, characteristic, complemented
5#21446715C211yesmaximal, characteristic, complemented
6#21446607C211yesmaximal, characteristic, complemented
7#21446486C212yesmaximal, complemented
8#21446318C212yesmaximal, complemented
9#21445700C211yesmaximal, characteristic, complemented
10#21445618C212yesmaximal, complemented
11C61×#1814C211yesmaximal, characteristic, complemented
12#16840560C2212yes—
13C2×#14519799C2211yescharacteristic
14C2×#14519785C2212yes—
15C2×#14519784C2211yescharacteristic
16#41×Dic61C2211yescharacteristic
17#16840037C412yescomplemented
18#16840032C412yescomplemented
19#16840025C414yescomplemented
20#16839881—24no—
21#16839878—21no—
22#16839878—21no—
23#16839878—21no—
24#16839878—21no—
25#16839876—24no—
26#16839810C2212yes—
27C2×#14519713C2211yescharacteristic
28#16839780C2212yes—
29C2×#14519703C2212yes—
30#16839744C2212yes—
31#16839735C2212yes—
32#16839720—21no—
33#16839720C2211yescharacteristic
34#16839718C2212yes—
35#16839717—21no—
36#16839717C2211yescharacteristic
37M16×Dic61—21no—
38M16×Dic61C2211yescharacteristic
39C2×C8×Dic61—21no—
40C2×C8×Dic61C2211yescharacteristic
41#16839533C2212yes—
42#16839521C2212yes—
43C2×#131×C61C2211yescharacteristic
44C2×#108×C61C2211yescharacteristic
45C61×#427C2211yescharacteristic
46C61×#421C2212yes—
47C61×#344C2212yes—
48C2×#11758222C2312yes—
49#14519808—22no—
50#14519807C2×C414yes—
51#14519803—22no—
52#14519802C2×C414yes—
53#14519799—21no—
54#14519799—21no—
55#14519799—21no—
56#14519799—21no—
57#14519797—24no—
58C2×Dic244C2×C411yescharacteristic, complemented
59C2×#11758190C2×C412yescomplemented
60C2×D488C2×C411yescharacteristic, complemented
61#14519785—22no—
62#14519785—22no—
63#14519785—22no—
64#14519785—22no—
65#14519784—21no—
66#14519784—21no—
67#14519784D412yes—
68#14519784D412yes—
69C2×#11758194—21no—
70C2×#11758194C2312yes—
71C2×#11758194C2311yescharacteristic
72C2×#11758189—22no—
73C22×#92418—21no—
74C22×#92418C2311yescharacteristic
75C2×C8×D61—22no—
76Q8×Dic61—21no—
77D4×Dic61—21no—
78D4×Dic61—22no—
79#14519736—21no—
80#14519736C2311yescharacteristic
81C2×C4×Dic61—21no—
82C2×C4×Dic61C2311yescharacteristic
83#14519713C2×C412yescomplemented
84#14519713C2×C412yescomplemented
85#14519712—24no—
86#14519705—24no—
87#14519703—22no—
88#14519703—22no—
89#14519695C2×C414yescomplemented
90#14519693C2×C414yescomplemented
91#14519691—24no—
92#14519688—22no—
93#14519687—21no—
94#14519687—21no—
95#14519687—21no—
96C4×#92418—22no—
97C8×Dic61—21no—
98C8×Dic61—21no—
99C8×Dic61—21no—
100C8×Dic61D411yescharacteristic
101C8×Dic61D411yescharacteristic
102C8×Dic61D411yescharacteristic
103C8×Dic61D411yescharacteristic
104C2×#41×C61C2311yescharacteristic
105#131×C61—21no—
106#131×C61—21no—
107#131×C61C2×C412yes—
108#131×C61C2×C412yes—
109C2×M16×C61—21no—
110C2×M16×C61C2312yes—
111C2×M16×C61C2311yescharacteristic
112C22×C8×C61—21no—
113C22×C8×C61C2311yescharacteristic
114#117×C61C2312yes—
115#108×C61—21no—
116#108×C61—21no—
117#107×C61—21no—
118#106×C61—21no—
119#104×C61—24no—
120#41×C61—21no—
121#41×C61—21no—
122#41×C61C22×C412yes—
123#41×C61C2×D412yes—
124C2×Q8×C61#4111yescharacteristic
125C2×D4×C61—21no—
126C2×D4×C61#4111yescharacteristic
127C22×C4×C61—21no—
128C22×C4×C61C2411yescharacteristic
129C2×C22⋊D61—22no—
130C22×Dic61—22no—
131C22×Dic61—21no—
132#11758222—22no—
133#11758222—22no—
134#11758222—22no—
135#11758222—22no—
136C2×D244C22×C412yes—
137C2×C4×D61—22no—
138C2×Dic122C22×C412yes—
139M16×C61—21no—
140M16×C61—22no—
141M16×C61—22no—
142M16×C61—22no—
143M16×C61—22no—
144M16×C61C22×C412yes—
145C2×C8×C61—21no—
146C2×C8×C61—22no—
147C2×C8×C61—22no—
148C2×C8×C61—22no—
149C2×C8×C61C22×C411yescharacteristic
150C2×C8×C61C22×C411yescharacteristic
151C2×C8×C61C2×D411yescharacteristic
152C2×C8×C61C2×D411yescharacteristic
153C2×C8×C61#4111yescharacteristic
154C2×C8×C61#4111yescharacteristic
155C4⋊C4×C61—22no—
156C22⋊C4×C61—22no—
157C42×C61—22no—
158C22⋊Dic61—22no—
159C22⋊Dic61#4112yes—
160#11758202—24no—
161#11758201—24no—
162#11758200—24no—
163#11758199—24no—
164C4⋊Dic61—22no—
165C4⋊Dic61—21no—
166C4×Dic61—22no—
167C4×Dic61—21no—
168C4×Dic61—21no—
169C4×Dic61C2×D411yescharacteristic
170C4×Dic61C2×D411yescharacteristic
171#11758194—22no—
172#11758194—21no—
173#11758194—22no—
174#11758194—22no—
175#11758194C22×C414yes—
176#11758194C2×D412yes—
177C2×#92418—21no—
178C2×#92418—22no—
179C2×#92418—22no—
180C2×#92418—22no—
181C2×#92418C2×D411yescharacteristic
182C2×#92418C2×D411yescharacteristic
183C2×#92418C2×D411yescharacteristic
184C2×#92418C2×D411yescharacteristic
185C2×#92418#4111yescharacteristic
186C2×#92418#4111yescharacteristic
187Dic244—21no—
188Dic244—21no—
189D488—21no—
190D488—21no—
191#11758190—22no—
192#11758190—22no—
193#11758189—42no—
194#11758189—42no—
195C8×D61—42no—
196C8×D61—42no—
197C23×C61—21no—
198C23×C61C2×#4111yescharacteristic
199C22×D61—22no—
200Q8×C61—21no—
201Q8×C61C4×D412yes—
202D4×C61—22no—
203D4×C61—22no—
204D4×C61—21no—
205D4×C61C4×D412yes—
206C2×C4×C61—21no—
207C2×C4×C61—22no—
208C2×C4×C61—22no—
209C2×C4×C61—42no—
210C2×C4×C61C23×C411yescharacteristic
211C2×C4×C61C22×D411yescharacteristic
212C2×C4×C61C22×D411yescharacteristic
213C2×C4×C61C2×#4111yescharacteristic
214C2×C4×C61C2×#4111yescharacteristic
215C2×C4×C61C2×#4111yescharacteristic
216C22⋊D61—42no—
217C22⋊D61—42no—
218C2×Dic61—21no—
219C2×Dic61—22no—
220C2×Dic61—22no—
221C2×Dic61—21no—
222C2×Dic61—42no—
223C2×Dic61—44no—
224D244—22no—
225D244—22no—
226C4×D61—42no—
227C4×D61—42no—
228Dic122—22no—
229Dic122—22no—
230C8×C61—22no—
231C8×C61—42no—
232C8×C61C4×D411yescharacteristic
233C8×C61C4×D411yescharacteristic
234C8×C61C4×D411yescharacteristic
235C8×C61C4×D411yescharacteristic
236#92418—24no—
237#92418—22no—
238#92418#12111yescharacteristic
239#92418#12111yescharacteristic
240#92418#12111yescharacteristic
241#92418#12111yescharacteristic
242#21558—611nomaximal, Sylow
243C22×C61—22no—
244C22×C61—41no—
245C22×C61—42no—
246C22×C61C4×#4111yescharacteristic
247C22×C61#52511yescharacteristic
248C22×C61#53211yescharacteristic
249C2×D61—42no—
250C2×D61—82no—
251C4×C61—22no—
252C4×C61—82no—
253C4×C61C2×C4×D411yescharacteristic
254C4×C61C2×C4×D411yescharacteristic
255C4×C61C2×#12111yescharacteristic
256C4×C61#52911yescharacteristic
257Dic61—22no—
258Dic61—42no—
259Dic61—41no—
260Dic61—42no—
261#3067—611no—
262#2955—612no—
263C2×#443—611no—
264C4×#131—611no—
265#1976—611no—
266#1953—611no—
267#1932—612no—
268#1906—612no—
269#1814D6111yescharacteristic, complemented
270#1772—611no—
271#1757—612no—
272C2×C61—21no—
273C2×C61—21no—
274C2×C61—21no—
275C2×C61—42no—
276C2×C61C4×#12111yescharacteristic
277D61—82no—
278#575—612no—
279C2×#135—611no—
280C2×#131—611no—
281C2×#131—612no—
282C2×#131C2×D6111yescharacteristic
283C4×#41—611no—
284#473—612no—
285#472—612no—
286#471—614no—
287#444—1224no—
288#443—1221no—
289#443—1221no—
290#443—1221no—
291#443—1221no—
292#443—1224no—
293#430—612no—
294C2×#108—611no—
295C2×#108C2×D6111yescharacteristic
296#427C2×D6111yescharacteristic
297#421—612no—
298#421C2×D6112yes—
299C2×#104—612no—
300#413—612no—
301#410—612no—
302#405—611no—
303#405—611no—
304#405—612no—
305#405—1221no—
306#405—1221no—
307C4×M16—611no—
308C4×M16—1221no—
309C2×C4×C8—611no—
310C2×C4×C8—1221no—
311#344—612no—
312#344C2×D6112yes—
313#339—612no—
314C61#2155811yescharacteristic, complemented, Sylow
315C2×#41—612no—
316C2×#41C22×D6111yescharacteristic
317#137—614no—
318#137—1222no—
319AGL(1,ℤ/8)—614no—
320AGL(1,ℤ/8)—1222no—
321#135—1224no—
322#135—1221no—
323#135—1221no—
324#135—1221no—
325#135—1221no—
326C2×Q16—611no—
327C2×SD16—612no—
328C2×D8—611no—
329#131—21no—
330#131—21no—
331#131—612no—
332#131—612no—
333#131—1221no—
334#131—1221no—
335#131—1222no—
336#131—1222no—
337#131—1222no—
338#131—1222no—
339#131C4×D6112yes—
340#131C4×D6112yes—
341C2×M16—21no—
342C2×M16—611no—
343C2×M16—612no—
344C2×M16—1222no—
345C2×M16—1221no—
346C2×M16C22×D6112yes—
347C2×M16C22×D6111yescharacteristic
348C22×C8—21no—
349C22×C8—611no—
350C22×C8—1222no—
351C22×C8—1221no—
352C22×C8C22×D6111yescharacteristic
353C4×Q8—1221no—
354C4×D4—1222no—
355C4×D4—1221no—
356#117—611no—
357#117—1221no—
358#117C22×D6112yes—
359C2×C42—611no—
360C2×C42—1221no—
361#108—21no—
362#108—21no—
363#108—612no—
364#108—612no—
365#108—1224no—
366#107—21no—
367#106—21no—
368#104—24no—
369#104—1224no—
370#104—1222no—
371#104—1222no—
372#101—614no—
373#100—614no—
374#99—1224no—
375#97—1222no—
376#97—1221no—
377#97—1221no—
378#97—1221no—
379C4×C8—611no—
380C4×C8—611no—
381C4×C8—611no—
382C4×C8—611no—
383C4×C8—1222no—
384C4×C8—1221no—
385C4×C8—1221no—
386C4×C8—1221no—
387#41—21no—
388#41—21no—
389#41—1222no—
390#41—1222no—
391#41—1222no—
392#41—1222no—
393#41C2×C4×D6112yes—
394#41D4×D6112yes—
395C2×Q8—612no—
396C2×Q8#1175822611yescharacteristic
397C2×D4—21no—
398C2×D4—612no—
399C2×D4—1222no—
400C2×D4#1175822611yescharacteristic
401C22×C4—21no—
402C22×C4—1222no—
403C22×C4—1221no—
404C22×C4—1222no—
405C22×C4C23×D6111yescharacteristic
406Q16—1224no—
407Q16—1221no—
408Q16—1221no—
409SD16—1224no—
410SD16—1224no—
411SD16—1222no—
412SD16—1222no—
413D8—1224no—
414D8—1221no—
415D8—1221no—
416M16—21no—
417M16—22no—
418M16—22no—
419M16—22no—
420M16—22no—
421M16—614no—
422M16—612no—
423M16—1222no—
424M16—1222no—
425M16—1221no—
426M16—1222no—
427M16—2442no—
428M16—2442no—
429M16C2×C4×D6112yes—
430C2×C8—21no—
431C2×C8—22no—
432C2×C8—22no—
433C2×C8—22no—
434C2×C8—611no—
435C2×C8—611no—
436C2×C8—611no—
437C2×C8—611no—
438C2×C8—611no—
439C2×C8—611no—
440C2×C8—1222no—
441C2×C8—1222no—
442C2×C8—1221no—
443C2×C8—1222no—
444C2×C8—2442no—
445C2×C8—2442no—
446C2×C8C2×C4×D6111yescharacteristic
447C2×C8C2×C4×D6111yescharacteristic
448C2×C8D4×D6111yescharacteristic
449C2×C8D4×D6111yescharacteristic
450C2×C8#1175822411yescharacteristic
451C2×C8#1175822411yescharacteristic
452C4⋊C4—22no—
453C4⋊C4—1222no—
454C4⋊C4—1221no—
455C22⋊C4—22no—
456C22⋊C4—612no—
457C22⋊C4—1222no—
458C42—22no—
459C42—611no—
460C42—611no—
461C42—1222no—
462C42—1221no—
463C42—1221no—
464C23—21no—
465C23—1222no—
466C23#41×D6111yescharacteristic
467Q8—21no—
468Q8—1222no—
469Q8—1222no—
470Q8#1451973912yes—
471D4—22no—
472D4—22no—
473D4—21no—
474D4—1222no—
475D4—1222no—
476D4—2442no—
477D4—2442no—
478D4#1451973912yes—
479C2×C4—21no—
480C2×C4—22no—
481C2×C4—22no—
482C2×C4—42no—
483C2×C4—1221no—
484C2×C4—1222no—
485C2×C4—1221no—
486C2×C4—1222no—
487C2×C4—2442no—
488C2×C4—2444no—
489C2×C4—2442no—
490C2×C4—2442no—
491C2×C4C22×C4×D6111yescharacteristic
492C2×C4C2×D4×D6111yescharacteristic
493C2×C4C2×D4×D6111yescharacteristic
494C2×C4#41×D6111yescharacteristic
495C2×C4C2×#1175822411yescharacteristic
496C2×C4C2×#1175822611yescharacteristic
497C8—22no—
498C8—42no—
499C8—611no—
500C8—611no—
501C8—611no—
502C8—611no—
503C8—1224no—
504C8—1222no—
505C8#1451973711yescharacteristic
506C8#1451973711yescharacteristic
507C8#1451973711yescharacteristic
508C8#1451973711yescharacteristic
509C22—22no—
510C22—41no—
511C22—42no—
512C22—4882no—
513C22—2442no—
514C22#1684023011yescharacteristic
515C22#1684027511yescharacteristic
516C22#1684030711yescharacteristic
517C4—22no—
518C4—82no—
519C4—1222no—
520C4—2441no—
521C4—2442no—
522C4—2442no—
523C4C2×#1451973711yescharacteristic
524C4C2×#1451973911yescharacteristic
525C4#121×D6111yescharacteristic
526C4#1684029811yescharacteristic
527C2—21no—
528C2—21no—
529C2—21no—
530C2—42no—
531C2—4882no—
532C2#2144832511yescharacteristic
subgroupG/H
centerC4#121×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1814C2
socleC2×C61C4×#121
prime radical#21455539C2
cosocleC22×C4×C61C24
hypercenter#1814D61
hypocenterC61#21558
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C4 ◁ C22×C4 ◁ #1814

Complements

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

NH ≅ G/Ntype
2. #21454718C2semidirect
3. C2×#16839878C2semidirect
4. #131×Dic61C2semidirect
5. #21446715C2semidirect
6. #21446607C2semidirect
7. #21446486C2semidirect
8. #21446318C2semidirect
9. #21445700C2semidirect
10. #21445618C2semidirect
11. C61×#1814C2semidirect
17. #16840037C4semidirect
18. #16840032C4semidirect
19. #16840025C4semidirect
58. C2×Dic244C2×C4semidirect
59. C2×#11758190C2×C4semidirect
60. C2×D488C2×C4semidirect
83. #14519713C2×C4semidirect
84. #14519713C2×C4semidirect
89. #14519695C2×C4semidirect
90. #14519693C2×C4semidirect
269. #1814D61semidirect
314. C61#21558semidirect

Maximal subgroups

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

subgroupindex
#214555392
#214547182
C2×#168398782
#131×Dic612
#214467152
#214466072
#214464862
#214463182
#214457002
#214456182
C61×#18142
#2155861