Finite Group Databasebeta

← #29693919

Subgroups — #29693919

Order 15616.

796 conjugacy classes of subgroups.

The subgroup lattice has 646 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456291C211yesmaximal, characteristic
2#21454734C211yesmaximal, characteristic, complemented
3#21454681C211yesmaximal, characteristic, complemented
4#21446725C211yesmaximal, characteristic, complemented
5#21446714C211yesmaximal, characteristic, complemented
6#21446355C211yesmaximal, characteristic, complemented
7#21446352C211yesmaximal, characteristic, complemented
8#21446340C211yesmaximal, characteristic, complemented
9#21446316C211yesmaximal, characteristic, complemented
10#21446304C211yesmaximal, characteristic, complemented
11#21446296C211yesmaximal, characteristic, complemented
12#21446292C211yesmaximal, characteristic, complemented
13#21445941C211yesmaximal, characteristic, complemented
14#21445573C211yesmaximal, characteristic, complemented
15C61×#1910C211yesmaximal, characteristic, complemented
16C2×#14519807C2211yescharacteristic
17C2×#14519803C2211yescharacteristic
18C2×#14519797C2211yescharacteristic
19C22×#11758200C2211yescharacteristic
20#16840528C2211yescharacteristic
21#16840338C2211yescharacteristic
22#16840239C2211yescharacteristic
23#16840237C2211yescharacteristic
24#16840157C412yescomplemented
25#16840149C412yescomplemented
26#16840144C412yescomplemented
27#16840134C412yescomplemented
28#16839882—22no—
29#16839877—22no—
30#16839859—22no—
31#16839853—22no—
32C2×#14519712C2211yescharacteristic
33#16839798C2211yescharacteristic
34#16839780C2211yescharacteristic
35#16839779C2211yescharacteristic
36#16839779C2211yescharacteristic
37#16839778C2211yescharacteristic
38#16839772C2211yescharacteristic
39#16839770C2211yescharacteristic
40C2×#14519695C2211yescharacteristic
41C2×#14519694C2211yescharacteristic
42#16839735C2211yescharacteristic
43#16839720C2211yescharacteristic
44#16839718C2211yescharacteristic
45#16839718C2211yescharacteristic
46#16839534C2211yescharacteristic
47#16839532C2211yescharacteristic
48#16839520C2211yescharacteristic
49#16839518C2211yescharacteristic
50#16839518C2211yescharacteristic
51#16839518C2211yescharacteristic
52C61×#568C2211yescharacteristic
53C61×#535C2211yescharacteristic
54C61×#418C2211yescharacteristic
55C61×#418C2211yescharacteristic
56C61×#410C2211yescharacteristic
57C61×H(3,ℤ/4)C2211yescharacteristic
58C61×H(3,ℤ/4)C2211yescharacteristic
59C2×#11758222C2311yescharacteristic
60C22×Dic122C2311yescharacteristic
61#14519807—41no—
62#14519807—41no—
63#14519803—41no—
64#14519803—41no—
65#14519797—21no—
66#14519797—21no—
67#14519797—21no—
68#14519797—21no—
69C2×#11758202C2×C411yescharacteristic, complemented
70C2×#11758202C2×C412yescomplemented
71C2×#11758200—21no—
72C2×#11758200—21no—
73C2×#11758200—21no—
74C2×#11758200—21no—
75C2×#11758200C2×C412yes—
76C2×#11758200C2×C412yes—
77C2×#11758200C2×C412yes—
78C2×#11758200C2×C411yescharacteristic, complemented
79C2×#11758200C2×C412yescomplemented
80C2×#11758201C2×C411yescharacteristic, complemented
81C2×#11758199C2×C412yes—
82C2×#11758199C2×C411yescharacteristic, complemented
83#14519785—41no—
84#14519783—22no—
85C2×#11758194C2311yescharacteristic
86C2×#11758194C2311yescharacteristic
87C2×#11758194C2311yescharacteristic
88M16×D61—41no—
89C22×#92418C2311yescharacteristic
90#14519777C2×C412yescomplemented
91#14519765C2×C412yescomplemented
92#14519761—41no—
93#14519757—41no—
94C4×Dic122—41no—
95#14519742—22no—
96D4×Dic61—22no—
97D4×Dic61—41no—
98#14519736C2311yescharacteristic
99#14519736C2311yescharacteristic
100#14519733—41no—
101C2×C22⋊Dic61—41no—
102#14519712—21no—
103#14519712—21no—
104#14519710—21no—
105#14519708—21no—
106#14519705—41no—
107#14519704—22no—
108#14519704—41no—
109#14519703—22no—
110#14519702—22no—
111#14519700—41no—
112#14519695—41no—
113#14519694—41no—
114#14519691—41no—
115#14519688—21no—
116#14519688C2×C412yescomplemented
117#14519687D412yescomplemented
118C4×#92418—21no—
119C4×#92418C2×C412yescomplemented
120C8×Dic61D412yescomplemented
121C2×#41×C61C2311yescharacteristic
122C22×D4×C61C2311yescharacteristic
123#131×C61—22no—
124#131×C61—41no—
125C2×M16×C61—41no—
126C2×M16×C61C2311yescharacteristic
127C2×M16×C61C2311yescharacteristic
128#127×C61C2×C412yes—
129#124×C61C2×C412yes—
130#121×C61—41no—
131#120×C61—41no—
132#117×C61C2311yescharacteristic
133#117×C61C2311yescharacteristic
134#117×C61C2311yescharacteristic
135#102×C61—22no—
136#102×C61—41no—
137#99×C61—41no—
138C24×C61—21no—
139#41×C61—21no—
140#41×C61—21no—
141#41×C61—21no—
142#41×C61—21no—
143C2×Q8×C61C22×C411yescharacteristic
144C2×D4×C61—21no—
145C2×D4×C61—21no—
146C2×D4×C61—21no—
147C2×D4×C61—21no—
148C2×D4×C61—21no—
149C2×D4×C61—21no—
150C2×D4×C61—42no—
151C2×D4×C61C22×C412yes—
152C2×D4×C61C22×C411yescharacteristic
153C22×C4×C61—21no—
154C22×C4×C61C2411yescharacteristic
155C2×C22⋊D61—21no—
156C22×Dic61—21no—
157C22×Dic61—42no—
158#11758222—21no—
159#11758222—21no—
160#11758222—21no—
161#11758222—21no—
162C2×D244C22×C411yescharacteristic
163C2×C4×D61—21no—
164C2×Dic122—21no—
165C2×Dic122—21no—
166C2×Dic122—21no—
167C2×Dic122—21no—
168C2×Dic122—21no—
169C2×Dic122C22×C412yes—
170C2×Dic122C22×C411yescharacteristic
171M16×C61—21no—
172M16×C61—21no—
173M16×C61—21no—
174M16×C61—21no—
175M16×C61—42no—
176C2×C8×C61—21no—
177C2×C8×C61—42no—
178C2×C8×C61C2×D412yes—
179C4⋊C4×C61—21no—
180C4⋊C4×C61—21no—
181C4⋊C4×C61—21no—
182C22⋊C4×C61—21no—
183C22⋊C4×C61—21no—
184C22⋊C4×C61—21no—
185C22⋊C4×C61—42no—
186C42×C61—21no—
187C42×C61—21no—
188C42×C61C22×C412yes—
189C22⋊Dic61—21no—
190C22⋊Dic61—21no—
191C22⋊Dic61—42no—
192#11758202—21no—
193#11758202—21no—
194#11758202—42no—
195#11758201—21no—
196#11758201—21no—
197#11758200—22no—
198#11758200—22no—
199#11758200—22no—
200#11758200—22no—
201#11758200—21no—
202#11758200—21no—
203#11758200—42no—
204#11758200—42no—
205#11758199—21no—
206#11758199—21no—
207#11758199—42no—
208#11758198—42no—
209C4⋊Dic61—21no—
210C4⋊Dic61—21no—
211C4⋊Dic61—42no—
212#11758196—42no—
213C4×Dic61—21no—
214C4×Dic61—42no—
215C4×Dic61C2×D412yes—
216#11758194—21no—
217#11758194—21no—
218#11758194—21no—
219#11758194—21no—
220#11758194—21no—
221#11758194—21no—
222C2×#92418—21no—
223C2×#92418—21no—
224C2×#92418C22×C412yes—
225C2×#92418C22×C411yescharacteristic
226C2×#92418C22×C411yescharacteristic
227C2×#92418C2×D411yescharacteristic
228C2×#92418C2×D411yescharacteristic
229C2×#92418#4111yescharacteristic
230C2×#92418#4111yescharacteristic
231#11758189—42no—
232C8×D61—42no—
233C23×C61—22no—
234C23×C61—21no—
235C23×C61—41no—
236C23×C61—41no—
237C23×C61—161no—
238C23×C61H(5,𝔽2)11yescharacteristic
239C22×D61—21no—
240Q8×C61—21no—
241Q8×C61—21no—
242D4×C61—21no—
243D4×C61—22no—
244D4×C61—22no—
245D4×C61—21no—
246D4×C61—41no—
247D4×C61—41no—
248D4×C61—41no—
249D4×C61—41no—
250D4×C61—162no—
251C2×C4×C61—22no—
252C2×C4×C61—21no—
253C2×C4×C61—41no—
254C2×C4×C61—41no—
255C2×C4×C61—41no—
256C2×C4×C61—41no—
257C2×C4×C61C23×C411yescharacteristic
258C2×C4×C61C22×D411yescharacteristic
259C2×C4×C61C2×#4111yescharacteristic
260C2×C4×C61H(5,𝔽2)11yescharacteristic
261C2×C4×C61H(5,𝔽2)11yescharacteristic
262C2×C4×C61H(5,𝔽2)11yescharacteristic
263C22⋊D61—41no—
264C22⋊D61—41no—
265C2×Dic61—21no—
266C2×Dic61—22no—
267C2×Dic61—22no—
268C2×Dic61—41no—
269C2×Dic61—41no—
270C2×Dic61—41no—
271C2×Dic61—162no—
272D244—21no—
273D244—21no—
274C4×D61—41no—
275C4×D61—41no—
276Dic122—22no—
277Dic122—21no—
278Dic122—21no—
279Dic122—22no—
280Dic122—41no—
281Dic122—41no—
282C8×C61—22no—
283C8×C61—41no—
284#92418—22no—
285#92418—41no—
286#92418—41no—
287#92418C4×D411yescharacteristic
288#92418C4×D411yescharacteristic
289#92418C4×D411yescharacteristic
290#92418C4×D411yescharacteristic
291#22284—611nomaximal, Sylow
292C22×C61—41no—
293C22×C61—41no—
294C22×C61—42no—
295C22×C61—81no—
296C22×C61—82no—
297C22×C61—82no—
298C22×C61—161no—
299C22×C61#51811yescharacteristic
300C22×C61#53511yescharacteristic
301C22×C61#54011yescharacteristic
302C2×D61—41no—
303C2×D61—81no—
304C4×C61—42no—
305C4×C61—41no—
306C4×C61—81no—
307C4×C61—81no—
308C4×C61C2×C4×D411yescharacteristic
309C4×C61#51811yescharacteristic
310C4×C61#53511yescharacteristic
311C4×C61#54011yescharacteristic
312Dic61—42no—
313Dic61—42no—
314Dic61—41no—
315Dic61—81no—
316#3154—611no—
317#2957—611no—
318#2949—611no—
319#1977—611no—
320#1976—611no—
321#1912—611no—
322#1911—611no—
323#1910D6111yescharacteristic, complemented
324#1909—611no—
325#1905—611no—
326#1904—611no—
327#1903—611no—
328#1902—611no—
329#1833—611no—
330#1747—611no—
331C2×C61—21no—
332C2×C61—21no—
333C2×C61—21no—
334C2×C61—81no—
335C2×C61—82no—
336C2×C61#232811yescharacteristic
337D61—81no—
338C2×#137—611no—
339C2×AGL(1,ℤ/8)—611no—
340C2×#135—611no—
341C22×SD16—611no—
342#568—611no—
343#568C2×D6111yescharacteristic
344#536—611no—
345#535C2×D6111yescharacteristic
346#519—611no—
347#518—611no—
348#497—612no—
349#496—612no—
350#495—612no—
351#492—612no—
352#444—1222no—
353#443—1222no—
354#440—1222no—
355C4×SD16—1222no—
356C2×#108—611no—
357#427—611no—
358#421—611no—
359#421—611no—
360#421—611no—
361#421—611no—
362#419—611no—
363#419—611no—
364#418C2×D6111yescharacteristic
365#418C2×D6111yescharacteristic
366C2×#101—611no—
367C2×#100—611no—
368#410—611no—
369#410C2×D6111yescharacteristic
370#405—611no—
371#405—611no—
372#405—611no—
373#344—611no—
374#344—611no—
375#338—611no—
376H(3,ℤ/4)—611no—
377H(3,ℤ/4)—611no—
378H(3,ℤ/4)—611no—
379H(3,ℤ/4)C2×D6111yescharacteristic
380H(3,ℤ/4)C2×D6111yescharacteristic
381C61#2228411yescharacteristic, complemented, Sylow
382C2×#41—611no—
383C2×#41C22×D6111yescharacteristic
384C22×Q8—611no—
385C22×D4C22×D6111yescharacteristic
386#137—2441no—
387#137—2441no—
388AGL(1,ℤ/8)—2441no—
389AGL(1,ℤ/8)—2441no—
390#135—1221no—
391#135—1221no—
392#135—1221no—
393#135—1221no—
394C2×Q16—611no—
395C2×Q16—612no—
396C2×SD16—611no—
397C2×SD16—612no—
398C2×SD16—612no—
399C2×SD16—611no—
400C2×SD16—612no—
401C2×SD16—612no—
402C2×SD16—1221no—
403C2×SD16—1221no—
404C2×SD16—1221no—
405C2×SD16—1221no—
406C2×D8—611no—
407C2×D8—612no—
408#131—22no—
409#131—41no—
410#131—1222no—
411#131—2441no—
412C2×M16—41no—
413C2×M16—611no—
414C2×M16—611no—
415C2×M16—611no—
416C2×M16—2441no—
417C2×M16C22×D6111yescharacteristic
418C2×M16C22×D6111yescharacteristic
419C22×C8—611no—
420C4⋊Q8—612no—
421#127C4×D6112yes—
422#124—612no—
423#124C4×D6112yes—
424#123—2441no—
425C22⋊Q8—2441no—
426#121—41no—
427#120—41no—
428C4×Q8—1222no—
429C4×Q8—2441no—
430C4×D4—1222no—
431C4×D4—2441no—
432#117—611no—
433#117—611no—
434#117—2441no—
435#117C22×D6111yescharacteristic
436#117C22×D6111yescharacteristic
437#117C22×D6111yescharacteristic
438C2×C22⋊C4—2441no—
439#108—1221no—
440#108—1221no—
441#107—1221no—
442#106—1221no—
443#104—1222no—
444#104—1222no—
445#104—2441no—
446#104—2441no—
447#103—1222no—
448#103—2441no—
449#102—22no—
450#102—41no—
451#101—2441no—
452#100—2441no—
453#99—41no—
454#99—2441no—
455#97—612no—
456#97—612no—
457#97—1221no—
458C4×C8—612no—
459C4×C8—612no—
460C4×C8—1221no—
461C24—21no—
462#41—21no—
463#41—21no—
464#41—21no—
465#41—21no—
466#41—1221no—
467#41—1221no—
468#41—1221no—
469#41—1221no—
470C2×Q8—612no—
471C2×Q8—611no—
472C2×Q8—1221no—
473C2×Q8—1221no—
474C2×Q8—1221no—
475C2×Q8—1221no—
476C2×Q8—1221no—
477C2×Q8C2×C4×D6111yescharacteristic
478C2×D4—21no—
479C2×D4—21no—
480C2×D4—21no—
481C2×D4—21no—
482C2×D4—21no—
483C2×D4—21no—
484C2×D4—42no—
485C2×D4—611no—
486C2×D4—1221no—
487C2×D4C2×C4×D6112yes—
488C2×D4C2×C4×D6111yescharacteristic
489C22×C4—21no—
490C22×C4—1221no—
491C22×C4—1221no—
492C22×C4—2442no—
493C22×C4C23×D6111yescharacteristic
494Q16—1221no—
495Q16—1221no—
496Q16—2442no—
497SD16—1222no—
498SD16—1222no—
499SD16—1222no—
500SD16—1222no—
501SD16—1221no—
502SD16—1221no—
503SD16—1221no—
504SD16—1221no—
505SD16—2442no—
506SD16—2442no—
507D8—1221no—
508D8—1221no—
509D8—2442no—
510M16—21no—
511M16—21no—
512M16—21no—
513M16—21no—
514M16—42no—
515M16—1221no—
516M16—1221no—
517M16—1221no—
518M16—1221no—
519M16—1221no—
520M16—1221no—
521M16—2442no—
522C2×C8—21no—
523C2×C8—42no—
524C2×C8—611no—
525C2×C8—611no—
526C2×C8—611no—
527C2×C8—611no—
528C2×C8—611no—
529C2×C8—611no—
530C2×C8—612no—
531C2×C8—1221no—
532C2×C8—1221no—
533C2×C8—2442no—
534C2×C8D4×D6112yes—
535C4⋊C4—21no—
536C4⋊C4—21no—
537C4⋊C4—21no—
538C4⋊C4—1221no—
539C4⋊C4—1221no—
540C4⋊C4—2442no—
541C4⋊C4—2442no—
542C22⋊C4—21no—
543C22⋊C4—21no—
544C22⋊C4—21no—
545C22⋊C4—42no—
546C22⋊C4—1221no—
547C22⋊C4—1221no—
548C22⋊C4—2442no—
549C22⋊C4—2442no—
550C42—21no—
551C42—21no—
552C42—612no—
553C42—1221no—
554C42—2442no—
555C42C2×C4×D6112yes—
556C23—22no—
557C23—21no—
558C23—41no—
559C23—41no—
560C23—161no—
561C23—1221no—
562C23#1451982211yescharacteristic
563Q8—21no—
564Q8—21no—
565Q8—1222no—
566Q8—1221no—
567Q8—1221no—
568Q8—1222no—
569Q8—2441no—
570Q8—2441no—
571D4—21no—
572D4—22no—
573D4—22no—
574D4—21no—
575D4—41no—
576D4—41no—
577D4—41no—
578D4—41no—
579D4—162no—
580D4—1221no—
581D4—1221no—
582D4—2441no—
583D4—2441no—
584C2×C4—22no—
585C2×C4—21no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—1222no—
591C2×C4—1221no—
592C2×C4—1222no—
593C2×C4—9762no—
594C2×C4—2441no—
595C2×C4—2441no—
596C2×C4—2441no—
597C2×C4—2441no—
598C2×C4—2441no—
599C2×C4C22×C4×D6111yescharacteristic
600C2×C4C2×D4×D6111yescharacteristic
601C2×C4#41×D6111yescharacteristic
602C2×C4#1451982211yescharacteristic
603C2×C4#1451982211yescharacteristic
604C2×C4#1451982311yescharacteristic
605C8—22no—
606C8—41no—
607C8—611no—
608C8—611no—
609C8—611no—
610C8—611no—
611C8—1222no—
612C8—2441no—
613C8—2441no—
614C22—41no—
615C22—41no—
616C22—42no—
617C22—81no—
618C22—82no—
619C22—82no—
620C22—161no—
621C22—4881no—
622C22—2441no—
623C22#1684023511yescharacteristic
624C22#1684032811yescharacteristic
625C22#1684036211yescharacteristic
626C4—42no—
627C4—41no—
628C4—81no—
629C4—81no—
630C4—4881no—
631C4—2442no—
632C4—2441no—
633C4—2442no—
634C4C4×D4×D6111yescharacteristic
635C4#1684023411yescharacteristic
636C4#1684032711yescharacteristic
637C4#1684036411yescharacteristic
638C2—21no—
639C2—21no—
640C2—21no—
641C2—82no—
642C2—81no—
643C2—4881no—
644C2#2144854011yescharacteristic
subgroupG/H
centerC2#21448540
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1910C2
socleC2×C61#2328
prime radical#21456291C2
cosocleC22×C4×C61C24
hypercenter#1910D61
hypocenterC61#22284
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2 ◁ C22×C4 ◁ #1910

Complements

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

NH ≅ G/Ntype
2. #21454734C2semidirect
3. #21454681C2semidirect
4. #21446725C2semidirect
5. #21446714C2semidirect
6. #21446355C2semidirect
7. #21446352C2semidirect
8. #21446340C2semidirect
9. #21446316C2semidirect
10. #21446304C2semidirect
11. #21446296C2semidirect
12. #21446292C2semidirect
13. #21445941C2semidirect
14. #21445573C2semidirect
15. C61×#1910C2semidirect
24. #16840157C4semidirect
25. #16840149C4semidirect
26. #16840144C4semidirect
27. #16840134C4semidirect
69. C2×#11758202C2×C4semidirect
70. C2×#11758202C2×C4semidirect
78. C2×#11758200C2×C4semidirect
79. C2×#11758200C2×C4semidirect
80. C2×#11758201C2×C4semidirect
82. C2×#11758199C2×C4semidirect
90. #14519777C2×C4semidirect
91. #14519765C2×C4semidirect
116. #14519688C2×C4semidirect
117. #14519687D4semidirect
119. C4×#92418C2×C4semidirect
120. C8×Dic61D4semidirect
323. #1910D61semidirect
381. C61#22284semidirect

Maximal subgroups

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

subgroupindex
#214562912
#214547342
#214546812
#214467252
#214467142
#214463552
#214463522
#214463402
#214463162
#214463042
#214462962
#214462922
#214459412
#214455732
C61×#19102
#2228461