Finite Group Databasebeta

← #29693924

Subgroups — #29693924

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#21456289C211yesmaximal, characteristic
2#21454735C211yesmaximal, characteristic, complemented
3#21454679C211yesmaximal, characteristic, complemented
4#21446717C211yesmaximal, characteristic, complemented
5#21446354C211yesmaximal, characteristic, complemented
6#21446349C211yesmaximal, characteristic, complemented
7#21446343C211yesmaximal, characteristic, complemented
8#21446338C211yesmaximal, characteristic, complemented
9#21446311C211yesmaximal, characteristic, complemented
10#21446304C211yesmaximal, characteristic, complemented
11#21446302C211yesmaximal, characteristic, complemented
12#21446295C211yesmaximal, characteristic, complemented
13#21445944C211yesmaximal, characteristic, complemented
14#21445574C211yesmaximal, characteristic, complemented
15C61×#1977C211yesmaximal, characteristic, complemented
16C2×#14519808C2211yescharacteristic
17C2×#14519803C2211yescharacteristic
18C2×#14519799C2211yescharacteristic
19#16840528C2211yescharacteristic
20#16840528C2211yescharacteristic
21#16840336C2211yescharacteristic
22#16840329C2211yescharacteristic
23#16840240C2211yescharacteristic
24#16840237C2211yescharacteristic
25#16840153C412yescomplemented
26#16840148C412yescomplemented
27#16840143C412yescomplemented
28#16840133C412yescomplemented
29#16839883—22no—
30#16839878—22no—
31#16839863—22no—
32SD16×Dic61—22no—
33#16839800C2211yescharacteristic
34#16839778C2211yescharacteristic
35#16839778C2211yescharacteristic
36#16839778C2211yescharacteristic
37#16839778C2211yescharacteristic
38#16839771C2211yescharacteristic
39#16839767C2211yescharacteristic
40#16839735C2211yescharacteristic
41#16839735C2211yescharacteristic
42#16839717C2211yescharacteristic
43#16839717C2211yescharacteristic
44#16839534C2211yescharacteristic
45#16839534C2211yescharacteristic
46#16839520C2211yescharacteristic
47#16839519C2211yescharacteristic
48#16839519C2211yescharacteristic
49#16839519C2211yescharacteristic
50#16839518C2211yescharacteristic
51#16839518C2211yescharacteristic
52C22×SD16×C61C2211yescharacteristic
53C2×#108×C61C2211yescharacteristic
54C61×#419C2211yescharacteristic
55C61×#418C2211yescharacteristic
56C2×#101×C61C2211yescharacteristic
57C2×#100×C61C2211yescharacteristic
58C61×#405C2211yescharacteristic
59C2×#11758222C2311yescharacteristic
60C2×#11758222C2311yescharacteristic
61#14519808—41no—
62#14519808—41no—
63#14519803—41no—
64#14519803—41no—
65#14519799—21no—
66#14519799—21no—
67#14519799—21no—
68#14519799—21no—
69C2×Dic244C2×C411yescharacteristic, complemented
70C2×#11758202C2×C412yescomplemented
71C2×#11758190C2×C411yescharacteristic, complemented
72C2×#11758190C2×C411yescharacteristic, complemented
73C2×#11758200C2×C412yes—
74C2×#11758201C2×C412yescomplemented
75C2×D488C2×C411yescharacteristic, complemented
76C2×#11758199C2×C412yes—
77#14519785—22no—
78#14519785—22no—
79#14519783—41no—
80#14519783—41no—
81C2×#11758194C2311yescharacteristic
82C2×#11758194C2311yescharacteristic
83M16×D61—41no—
84M16×D61—41no—
85#14519775C2×C412yescomplemented
86#14519773C2×C412yes—
87#14519766C2×C412yescomplemented
88#14519764C2×C412yes—
89#14519760—41no—
90#14519753—41no—
91#14519748—41no—
92#14519746—41no—
93Q8×Dic61—22no—
94Q8×Dic61—41no—
95D4×Dic61—22no—
96D4×Dic61—41no—
97#14519736—41no—
98#14519736C2311yescharacteristic
99#14519736C2311yescharacteristic
100#14519736C2311yescharacteristic
101#14519736C2311yescharacteristic
102C2×C22⋊Dic61—41no—
103#14519711—21no—
104#14519709—21no—
105#14519703—22no—
106#14519703—22no—
107#14519703—41no—
108#14519703—41no—
109#14519701—22no—
110#14519698—22no—
111#14519691—41no—
112#14519691—41no—
113#14519687C2×C412yescomplemented
114#14519687D412yescomplemented
115C8×Dic61C2×C412yescomplemented
116C8×Dic61D412yescomplemented
117C22×Q8×C61C2311yescharacteristic
118C22×D4×C61C2311yescharacteristic
119C2×SD16×C61—21no—
120C2×SD16×C61—21no—
121C2×SD16×C61—21no—
122C2×SD16×C61—21no—
123C2×SD16×C61C2×C412yes—
124C2×SD16×C61C2×C412yes—
125C2×M16×C61C2311yescharacteristic
126C2×M16×C61C2311yescharacteristic
127C2×M16×C61C2311yescharacteristic
128C22×C8×C61C2311yescharacteristic
129#117×C61C2311yescharacteristic
130#108×C61—21no—
131#108×C61—21no—
132#103×C61—41no—
133#102×C61—41no—
134#101×C61—41no—
135#100×C61—41no—
136#97×C61—21no—
137C4×C8×C61—21no—
138C24×C61—21no—
139C2×Q8×C61—21no—
140C2×Q8×C61—21no—
141C2×Q8×C61—21no—
142C2×Q8×C61—21no—
143C2×Q8×C61—21no—
144C2×Q8×C61C22×C412yes—
145C2×D4×C61—21no—
146C2×D4×C61—21no—
147C2×D4×C61—21no—
148C2×D4×C61—21no—
149C2×D4×C61—21no—
150C2×D4×C61C22×C412yes—
151C22×C4×C61—21no—
152C22×C4×C61C2411yescharacteristic
153C2×C22⋊D61—21no—
154C2×C22⋊D61—21no—
155C2×C22⋊D61—42no—
156C22×Dic61—42no—
157#11758222—21no—
158#11758222—21no—
159#11758222—21no—
160#11758222—21no—
161#11758222—21no—
162#11758222—21no—
163#11758222—21no—
164#11758222—21no—
165C2×D244C22×C411yescharacteristic
166C2×D244C22×C411yescharacteristic
167C2×C4×D61—21no—
168C2×C4×D61—21no—
169C2×Dic122C22×C411yescharacteristic
170C2×Dic122C22×C411yescharacteristic
171SD16×C61—22no—
172SD16×C61—22no—
173SD16×C61—22no—
174SD16×C61—22no—
175M16×C61—21no—
176M16×C61—21no—
177M16×C61—21no—
178M16×C61—21no—
179M16×C61—21no—
180M16×C61—21no—
181C2×C8×C61—21no—
182C2×C8×C61—21no—
183C2×C8×C61—21no—
184C2×C8×C61C22×C411yescharacteristic
185C2×C8×C61C22×C411yescharacteristic
186C2×C8×C61C2×D411yescharacteristic
187C2×C8×C61C2×D411yescharacteristic
188C2×C8×C61#4111yescharacteristic
189C2×C8×C61#4111yescharacteristic
190C4⋊C4×C61—21no—
191C22⋊C4×C61—21no—
192C42×C61—21no—
193C22⋊Dic61—21no—
194C22⋊Dic61—21no—
195C22⋊Dic61—21no—
196C22⋊Dic61—21no—
197C22⋊Dic61—42no—
198C22⋊Dic61—42no—
199#11758202—42no—
200#11758201—42no—
201#11758200—42no—
202#11758199—42no—
203#11758198—42no—
204C4⋊Dic61—21no—
205C4⋊Dic61—21no—
206C4⋊Dic61—21no—
207C4⋊Dic61—21no—
208C4⋊Dic61—42no—
209#11758196—42no—
210C4×Dic61—21no—
211C4×Dic61—21no—
212C4×Dic61—42no—
213C4×Dic61C22×C412yes—
214C4×Dic61C2×D412yes—
215#11758194—21no—
216#11758194—21no—
217#11758194—21no—
218#11758194—21no—
219C2×#92418C22×C412yes—
220C2×#92418C2×D412yes—
221Dic244—21no—
222Dic244—21no—
223D488—21no—
224D488—21no—
225#11758190—21no—
226#11758190—21no—
227#11758190—21no—
228#11758190—21no—
229#11758189—42no—
230#11758189—42no—
231C8×D61—42no—
232C8×D61—42no—
233C23×C61—22no—
234C23×C61—41no—
235C23×C61—41no—
236C23×C61—161no—
237C23×C61H(5,𝔽2)11yescharacteristic
238C22×D61—21no—
239C22×D61—21no—
240Q8×C61—22no—
241Q8×C61—22no—
242Q8×C61—41no—
243Q8×C61—41no—
244D4×C61—22no—
245D4×C61—22no—
246D4×C61—41no—
247D4×C61—41no—
248C2×C4×C61—22no—
249C2×C4×C61—41no—
250C2×C4×C61—41no—
251C2×C4×C61—41no—
252C2×C4×C61C23×C411yescharacteristic
253C2×C4×C61C22×D411yescharacteristic
254C2×C4×C61C2×#4111yescharacteristic
255C2×C4×C61H(5,𝔽2)11yescharacteristic
256C2×C4×C61H(5,𝔽2)11yescharacteristic
257C2×C4×C61H(5,𝔽2)11yescharacteristic
258C22⋊D61—41no—
259C22⋊D61—41no—
260C22⋊D61—41no—
261C22⋊D61—41no—
262C22⋊D61—162no—
263C2×Dic61—22no—
264C2×Dic61—21no—
265C2×Dic61—21no—
266C2×Dic61—22no—
267C2×Dic61—41no—
268C2×Dic61—41no—
269C2×Dic61—162no—
270D244—21no—
271D244—21no—
272D244—21no—
273D244—21no—
274C4×D61—41no—
275C4×D61—41no—
276C4×D61—41no—
277C4×D61—41no—
278Dic122—21no—
279Dic122—21no—
280Dic122—21no—
281Dic122—21no—
282C8×C61—41no—
283C8×C61—41no—
284C8×C61—41no—
285C8×C61C4×D411yescharacteristic
286C8×C61C4×D411yescharacteristic
287C8×C61C4×D411yescharacteristic
288C8×C61C4×D411yescharacteristic
289#92418—22no—
290#92418—22no—
291#22284—611nomaximal, Sylow
292C22×C61—41no—
293C22×C61—42no—
294C22×C61—82no—
295C22×C61—82no—
296C22×C61—161no—
297C22×C61#51811yescharacteristic
298C22×C61#53511yescharacteristic
299C22×C61#54011yescharacteristic
300C2×D61—41no—
301C2×D61—41no—
302C2×D61—81no—
303C2×D61—81no—
304C4×C61—42no—
305C4×C61—81no—
306C4×C61C2×C4×D411yescharacteristic
307C4×C61#51811yescharacteristic
308C4×C61#53511yescharacteristic
309C4×C61#54011yescharacteristic
310Dic61—42no—
311Dic61—42no—
312Dic61—41no—
313Dic61—41no—
314Dic61—81no—
315Dic61—81no—
316#3154—611no—
317#2957—611no—
318#2949—611no—
319#1977D6111yescharacteristic, complemented
320#1976—611no—
321#1912—611no—
322#1911—611no—
323#1910—611no—
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—82no—
335C2×C61#232811yescharacteristic
336D61—81no—
337D61—81no—
338C2×#137—611no—
339C2×AGL(1,ℤ/8)—611no—
340C2×#135—611no—
341C22×SD16C2×D6111yescharacteristic
342#568—611no—
343#568—611no—
344#536—611no—
345#535—611no—
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×#108C2×D6111yescharacteristic
357#427—611no—
358#421—611no—
359#421—611no—
360#421—611no—
361#421—611no—
362#419—611no—
363#419C2×D6111yescharacteristic
364#418—611no—
365#418C2×D6111yescharacteristic
366C2×#101C2×D6111yescharacteristic
367C2×#100C2×D6111yescharacteristic
368#410—611no—
369#410—611no—
370#405—611no—
371#405—611no—
372#405C2×D6111yescharacteristic
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)—611no—
380H(3,ℤ/4)—611no—
381C61#2228411yescharacteristic, complemented, Sylow
382C2×#41—611no—
383C2×#41—611no—
384C22×Q8C22×D6111yescharacteristic
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—612no—
395C2×Q16—611no—
396C2×SD16—21no—
397C2×SD16—21no—
398C2×SD16—21no—
399C2×SD16—21no—
400C2×SD16—612no—
401C2×SD16—612no—
402C2×SD16—611no—
403C2×SD16—611no—
404C2×SD16C4×D6112yes—
405C2×SD16C4×D6112yes—
406C2×D8—612no—
407C2×D8—611no—
408#131—1222no—
409#131—1222no—
410#131—2441no—
411#131—2441no—
412C2×M16—611no—
413C2×M16—611no—
414C2×M16—2441no—
415C2×M16—2441no—
416C2×M16C22×D6111yescharacteristic
417C2×M16C22×D6111yescharacteristic
418C2×M16C22×D6111yescharacteristic
419C22×C8C22×D6111yescharacteristic
420C4⋊Q8—612no—
421#127—612no—
422#124—612no—
423#124—612no—
424#123—2441no—
425C22⋊Q8—2441no—
426#121—2441no—
427#120—2441no—
428C4×Q8—1222no—
429C4×Q8—2441no—
430C4×D4—1222no—
431C4×D4—2441no—
432#117—611no—
433#117—611no—
434#117—611no—
435#117—611no—
436#117—2441no—
437#117C22×D6111yescharacteristic
438C2×C22⋊C4—2441no—
439#108—21no—
440#108—21no—
441#107—1221no—
442#106—1221no—
443#104—1222no—
444#104—1222no—
445#104—2441no—
446#104—2441no—
447#103—41no—
448#103—1222no—
449#102—41no—
450#102—1222no—
451#101—41no—
452#100—41no—
453#99—2441no—
454#99—2441no—
455#97—21no—
456#97—612no—
457#97—612no—
458C4×C8—21no—
459C4×C8—612no—
460C4×C8—612no—
461C24—21no—
462#41—1221no—
463#41—1221no—
464#41—1221no—
465#41—1221no—
466#41—1221no—
467#41—1221no—
468#41—1221no—
469#41—1221no—
470C2×Q8—21no—
471C2×Q8—21no—
472C2×Q8—21no—
473C2×Q8—21no—
474C2×Q8—21no—
475C2×Q8—611no—
476C2×Q8—611no—
477C2×Q8C2×C4×D6112yes—
478C2×D4—21no—
479C2×D4—21no—
480C2×D4—21no—
481C2×D4—21no—
482C2×D4—21no—
483C2×D4—611no—
484C2×D4—611no—
485C2×D4—1221no—
486C2×D4—1221no—
487C2×D4—2442no—
488C2×D4C2×C4×D6112yes—
489C22×C4—21no—
490C22×C4—1221no—
491C22×C4—1221no—
492C22×C4—2442no—
493C22×C4C23×D6111yescharacteristic
494Q16—1221no—
495Q16—1221no—
496Q16—2442no—
497SD16—22no—
498SD16—22no—
499SD16—22no—
500SD16—22no—
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—21no—
515M16—21no—
516M16—1221no—
517M16—1221no—
518M16—1221no—
519M16—1221no—
520M16—2442no—
521M16—2442no—
522C2×C8—21no—
523C2×C8—21no—
524C2×C8—21no—
525C2×C8—612no—
526C2×C8—612no—
527C2×C8—2442no—
528C2×C8—2442no—
529C2×C8C2×C4×D6111yescharacteristic
530C2×C8C2×C4×D6111yescharacteristic
531C2×C8D4×D6111yescharacteristic
532C2×C8D4×D6111yescharacteristic
533C2×C8#1175822411yescharacteristic
534C2×C8#1175822411yescharacteristic
535C4⋊C4—21no—
536C4⋊C4—1221no—
537C4⋊C4—1221no—
538C4⋊C4—1221no—
539C4⋊C4—1221no—
540C4⋊C4—2442no—
541C4⋊C4—2442no—
542C22⋊C4—21no—
543C22⋊C4—1221no—
544C22⋊C4—1221no—
545C22⋊C4—1221no—
546C22⋊C4—1221no—
547C22⋊C4—2442no—
548C22⋊C4—2442no—
549C22⋊C4—2442no—
550C42—21no—
551C42—612no—
552C42—612no—
553C42—1221no—
554C42—1221no—
555C42—2442no—
556C23—22no—
557C23—41no—
558C23—41no—
559C23—161no—
560C23—1221no—
561C23—1221no—
562C23#1451982311yescharacteristic
563Q8—22no—
564Q8—22no—
565Q8—41no—
566Q8—41no—
567Q8—1221no—
568Q8—1221no—
569Q8—1221no—
570Q8—1221no—
571D4—22no—
572D4—22no—
573D4—41no—
574D4—41no—
575D4—1221no—
576D4—1221no—
577D4—1221no—
578D4—1221no—
579D4—9762no—
580D4—2441no—
581D4—2441no—
582D4—2441no—
583D4—2441no—
584C2×C4—22no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—1222no—
589C2×C4—1221no—
590C2×C4—1221no—
591C2×C4—1222no—
592C2×C4—9762no—
593C2×C4—2441no—
594C2×C4—2441no—
595C2×C4—2441no—
596C2×C4—2441no—
597C2×C4—2441no—
598C2×C4—2441no—
599C2×C4C22×C4×D6111yescharacteristic
600C2×C4C2×D4×D6111yescharacteristic
601C2×C4C2×#1175822411yescharacteristic
602C2×C4#1451982211yescharacteristic
603C2×C4#1451982211yescharacteristic
604C2×C4#1451982311yescharacteristic
605C8—41no—
606C8—41no—
607C8—41no—
608C8—1222no—
609C8—1222no—
610C8#1451973711yescharacteristic
611C8#1451973711yescharacteristic
612C8#1451973711yescharacteristic
613C8#1451973711yescharacteristic
614C22—41no—
615C22—42no—
616C22—82no—
617C22—82no—
618C22—161no—
619C22—4881no—
620C22—4881no—
621C22—2441no—
622C22—2441no—
623C22#1684023411yescharacteristic
624C22#1684033011yescharacteristic
625C22#1684036511yescharacteristic
626C4—42no—
627C4—81no—
628C4—4881no—
629C4—4881no—
630C4—2442no—
631C4—2442no—
632C4—2441no—
633C4—2441no—
634C4C2×#1451973711yescharacteristic
635C4#1684023411yescharacteristic
636C4#1684032511yescharacteristic
637C4#1684036511yescharacteristic
638C2—21no—
639C2—21no—
640C2—21no—
641C2—82no—
642C2—4881no—
643C2—4881no—
644C2#2144854311yescharacteristic
subgroupG/H
centerC2#21448543
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1977C2
socleC2×C61#2328
prime radical#21456289C2
cosocleC22×C4×C61C24
hypercenter#1977D61
hypocenterC61#22284
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2 ◁ C22×C4 ◁ #1977

Complements

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

NH ≅ G/Ntype
2. #21454735C2semidirect
3. #21454679C2semidirect
4. #21446717C2semidirect
5. #21446354C2semidirect
6. #21446349C2semidirect
7. #21446343C2semidirect
8. #21446338C2semidirect
9. #21446311C2semidirect
10. #21446304C2semidirect
11. #21446302C2semidirect
12. #21446295C2semidirect
13. #21445944C2semidirect
14. #21445574C2semidirect
15. C61×#1977C2semidirect
25. #16840153C4semidirect
26. #16840148C4semidirect
27. #16840143C4semidirect
28. #16840133C4semidirect
69. C2×Dic244C2×C4semidirect
70. C2×#11758202C2×C4semidirect
71. C2×#11758190C2×C4semidirect
72. C2×#11758190C2×C4semidirect
74. C2×#11758201C2×C4semidirect
75. C2×D488C2×C4semidirect
85. #14519775C2×C4semidirect
87. #14519766C2×C4semidirect
113. #14519687C2×C4semidirect
114. #14519687D4semidirect
115. C8×Dic61C2×C4semidirect
116. C8×Dic61D4semidirect
319. #1977D61semidirect
381. C61#22284semidirect

Maximal subgroups

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

subgroupindex
#214562892
#214547352
#214546792
#214467172
#214463542
#214463492
#214463432
#214463382
#214463112
#214463042
#214463022
#214462952
#214459442
#214455742
C61×#19772
#2228461