Finite Group Databasebeta

← #29693922

Subgroups — #29693922

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#21454736C211yesmaximal, characteristic, complemented
2#21454679C211yesmaximal, characteristic, complemented
3#21446724C211yesmaximal, characteristic, complemented
4#21446718C211yesmaximal, characteristic, complemented
5#21446357C211yesmaximal, characteristic, complemented
6#21446352C211yesmaximal, characteristic, complemented
7#21446347C211yesmaximal, characteristic, complemented
8#21446342C211yesmaximal, characteristic, complemented
9#21446315C211yesmaximal, characteristic, complemented
10#21446308C211yesmaximal, characteristic, complemented
11#21446301C211yesmaximal, characteristic, complemented
12#21446294C211yesmaximal, characteristic, complemented
13#21445943C211yesmaximal, characteristic, complemented
14#21445574C211yesmaximal, characteristic, complemented
15C61×#3154C211yesmaximal, characteristic
16#16840529C2211yescharacteristic
17#16840529C2211yescharacteristic
18#16840240C2211yescharacteristic
19#16840237C2211yescharacteristic
20#16839884—22no—
21#16839879—22no—
22#16839863—22no—
23SD16×Dic61—22no—
24C2×#14519713C2211yescharacteristic
25#16839800C2211yescharacteristic
26#16839779C2211yescharacteristic
27#16839779C2211yescharacteristic
28#16839779C2211yescharacteristic
29#16839779C2211yescharacteristic
30#16839771C2211yescharacteristic
31#16839771C2211yescharacteristic
32#16839767C2211yescharacteristic
33#16839767C2211yescharacteristic
34C2×#14519696C2211yescharacteristic
35C2×#14519694C2211yescharacteristic
36#16839736C2211yescharacteristic
37#16839736C2211yescharacteristic
38#16839717C2211yescharacteristic
39#16839717C2211yescharacteristic
40#16839533C2211yescharacteristic
41#16839533C2211yescharacteristic
42#16839520C2211yescharacteristic
43#16839519C2211yescharacteristic
44#16839518C2211yescharacteristic
45#16839518C2211yescharacteristic
46#16839518C2211yescharacteristic
47#16839518C2211yescharacteristic
48C2×#137×C61C2211yescharacteristic
49C2×AGL(1,ℤ/8)×C61C2211yescharacteristic
50C2×#135×C61C2211yescharacteristic
51C22×SD16×C61C2211yescharacteristic
52C61×#536C2211yescharacteristic
53C61×#535C2211yescharacteristic
54C61×#497C412yescomplemented
55C61×#496C412yescomplemented
56C61×#495C412yescomplemented
57C61×#492C412yescomplemented
58C61×#405C2211yescharacteristic
59#14519784—22no—
60#14519784—22no—
61#14519784—41no—
62#14519784—41no—
63C2×#11758194—41no—
64C2×#11758194—41no—
65C2×#11758194C2311yescharacteristic
66C2×#11758194C2311yescharacteristic
67C2×#11758194C2311yescharacteristic
68C2×#11758194C2311yescharacteristic
69Q8×Dic61—22no—
70Q8×Dic61—41no—
71D4×Dic61—22no—
72D4×Dic61—41no—
73#14519736—41no—
74#14519736C2311yescharacteristic
75#14519736C2311yescharacteristic
76#14519736C2311yescharacteristic
77#14519736C2311yescharacteristic
78C2×C22⋊Dic61—41no—
79#14519713—21no—
80#14519713—21no—
81#14519711—21no—
82#14519709—21no—
83#14519704—22no—
84#14519704—22no—
85#14519704—41no—
86#14519704—41no—
87#14519701—22no—
88#14519701—41no—
89#14519698—22no—
90#14519698—41no—
91#14519696—41no—
92#14519694—41no—
93#14519692—41no—
94#14519692—41no—
95#14519687—21no—
96#14519687D412yescomplemented
97C8×Dic61—21no—
98C8×Dic61D412yescomplemented
99C2×#41×C61C2311yescharacteristic
100C2×#41×C61C2311yescharacteristic
101C22×Q8×C61C2311yescharacteristic
102C22×D4×C61C2311yescharacteristic
103#137×C61—41no—
104#137×C61—41no—
105AGL(1,ℤ/8)×C61—41no—
106AGL(1,ℤ/8)×C61—41no—
107#135×C61—21no—
108#135×C61—21no—
109#135×C61—21no—
110#135×C61—21no—
111C2×Q16×C61C2×C411yescharacteristic, complemented
112C2×Q16×C61C2×C412yescomplemented
113C2×SD16×C61—21no—
114C2×SD16×C61—21no—
115C2×SD16×C61—21no—
116C2×SD16×C61—21no—
117C2×SD16×C61C2×C412yes—
118C2×SD16×C61C2×C412yes—
119C2×SD16×C61C2×C412yes—
120C2×SD16×C61C2×C411yescharacteristic, complemented
121C2×SD16×C61C2×C411yescharacteristic, complemented
122C2×SD16×C61C2×C412yescomplemented
123C2×D8×C61C2×C412yes—
124C2×D8×C61C2×C411yescharacteristic, complemented
125C2×M16×C61C2311yescharacteristic
126C22×C8×C61C2311yescharacteristic
127C4⋊Q8×C61C2×C412yescomplemented
128#127×C61C2×C412yes—
129#124×C61C2×C412yes—
130#124×C61C2×C412yescomplemented
131#123×C61—41no—
132C22⋊Q8×C61—41no—
133#121×C61—41no—
134#120×C61—41no—
135#117×C61C2311yescharacteristic
136#97×C61C2×C412yescomplemented
137C4×C8×C61C2×C412yescomplemented
138C24×C61—21no—
139#41×C61—21no—
140#41×C61—21no—
141#41×C61—21no—
142#41×C61—21no—
143#41×C61—21no—
144#41×C61—21no—
145#41×C61—21no—
146#41×C61—21no—
147C2×Q8×C61—21no—
148C2×Q8×C61—21no—
149C2×Q8×C61—21no—
150C2×Q8×C61—21no—
151C2×Q8×C61—21no—
152C2×Q8×C61C22×C411yescharacteristic
153C2×Q8×C61C22×C412yes—
154C2×Q8×C61C22×C411yescharacteristic
155C2×D4×C61—21no—
156C2×D4×C61—21no—
157C2×D4×C61—21no—
158C2×D4×C61—21no—
159C2×D4×C61—21no—
160C2×D4×C61—21no—
161C2×D4×C61—21no—
162C2×D4×C61—42no—
163C2×D4×C61C22×C412yes—
164C2×D4×C61C22×C411yescharacteristic
165C2×D4×C61C22×C411yescharacteristic
166C22×C4×C61—21no—
167C22×C4×C61—21no—
168C22×C4×C61—21no—
169C22×C4×C61C2411yescharacteristic
170C22×Dic61—42no—
171Q16×C61—21no—
172Q16×C61—21no—
173Q16×C61—42no—
174SD16×C61—22no—
175SD16×C61—22no—
176SD16×C61—22no—
177SD16×C61—22no—
178SD16×C61—21no—
179SD16×C61—21no—
180SD16×C61—21no—
181SD16×C61—21no—
182SD16×C61—42no—
183SD16×C61—42no—
184D8×C61—21no—
185D8×C61—21no—
186D8×C61—42no—
187M16×C61—21no—
188M16×C61—21no—
189C2×C8×C61C22×C412yes—
190C2×C8×C61C22×C411yescharacteristic
191C2×C8×C61C22×C411yescharacteristic
192C2×C8×C61C2×D411yescharacteristic
193C2×C8×C61C2×D411yescharacteristic
194C2×C8×C61#4111yescharacteristic
195C2×C8×C61#4111yescharacteristic
196C4⋊C4×C61—21no—
197C4⋊C4×C61—42no—
198C22⋊C4×C61—21no—
199C22⋊C4×C61—42no—
200C22⋊C4×C61—42no—
201C42×C61C22×C412yes—
202C22⋊Dic61—21no—
203C22⋊Dic61—21no—
204C22⋊Dic61—21no—
205C22⋊Dic61—21no—
206C22⋊Dic61—42no—
207C4⋊Dic61—21no—
208C4⋊Dic61—21no—
209C4⋊Dic61—21no—
210C4⋊Dic61—21no—
211C4⋊Dic61—42no—
212C4×Dic61—21no—
213C4×Dic61—21no—
214C4×Dic61—21no—
215C4×Dic61—42no—
216C4×Dic61C2×D412yes—
217#11758194—21no—
218#11758194—21no—
219#11758194—21no—
220#11758194—21no—
221#11758194—21no—
222#11758194—21no—
223#11758194—21no—
224#11758194—21no—
225#11758194—42no—
226#11758194—42no—
227C2×#92418—21no—
228C2×#92418—21no—
229C2×#92418—21no—
230C2×#92418—42no—
231C2×#92418—42no—
232C2×#92418C2×D412yes—
233C23×C61—21no—
234C23×C61—22no—
235C23×C61—21no—
236C23×C61—41no—
237C23×C61—41no—
238C23×C61—161no—
239C23×C61H(5,𝔽2)11yescharacteristic
240Q8×C61—22no—
241Q8×C61—21no—
242Q8×C61—21no—
243Q8×C61—21no—
244Q8×C61—22no—
245Q8×C61—21no—
246Q8×C61—41no—
247Q8×C61—41no—
248D4×C61—21no—
249D4×C61—21no—
250D4×C61—22no—
251D4×C61—21no—
252D4×C61—22no—
253D4×C61—21no—
254D4×C61—41no—
255D4×C61—41no—
256D4×C61—41no—
257D4×C61—41no—
258D4×C61—41no—
259D4×C61—41no—
260D4×C61—162no—
261C2×C4×C61—22no—
262C2×C4×C61—21no—
263C2×C4×C61—22no—
264C2×C4×C61—21no—
265C2×C4×C61—41no—
266C2×C4×C61—41no—
267C2×C4×C61—41no—
268C2×C4×C61—41no—
269C2×C4×C61—41no—
270C2×C4×C61—41no—
271C2×C4×C61C23×C411yescharacteristic
272C2×C4×C61C22×D411yescharacteristic
273C2×C4×C61C2×#4111yescharacteristic
274C2×C4×C61H(5,𝔽2)11yescharacteristic
275C2×C4×C61H(5,𝔽2)11yescharacteristic
276C2×C4×C61H(5,𝔽2)11yescharacteristic
277C2×Dic61—22no—
278C2×Dic61—41no—
279C2×Dic61—41no—
280C2×Dic61—41no—
281C2×Dic61—162no—
282C8×C61—22no—
283C8×C61C4×D411yescharacteristic
284C8×C61C4×D411yescharacteristic
285C8×C61C4×D411yescharacteristic
286C8×C61C4×D411yescharacteristic
287#92418—22no—
288#92418—41no—
289#92418—41no—
290#92418—41no—
291#22284—611nomaximal, Sylow
292C22×C61—41no—
293C22×C61—41no—
294C22×C61—41no—
295C22×C61—42no—
296C22×C61—81no—
297C22×C61—82no—
298C22×C61—81no—
299C22×C61—82no—
300C22×C61—161no—
301C22×C61#51811yescharacteristic
302C22×C61#53511yescharacteristic
303C22×C61#54011yescharacteristic
304C4×C61—42no—
305C4×C61—41no—
306C4×C61—41no—
307C4×C61—42no—
308C4×C61C2×C4×D411yescharacteristic
309C4×C61#51811yescharacteristic
310C4×C61#53511yescharacteristic
311C4×C61#54011yescharacteristic
312Dic61—42no—
313Dic61—81no—
314Dic61—81no—
315Dic61—81no—
316#3154D6111yescharacteristic
317#2957—611no—
318#2949—611no—
319#1977—611no—
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—81no—
335C2×C61—81no—
336C2×C61—82no—
337C2×C61#232811yescharacteristic
338C2×#137C2×D6111yescharacteristic
339C2×AGL(1,ℤ/8)C2×D6111yescharacteristic
340C2×#135C2×D6111yescharacteristic
341C22×SD16C2×D6111yescharacteristic
342#568—611no—
343#568—611no—
344#536C2×D6111yescharacteristic
345#535C2×D6111yescharacteristic
346#519—611no—
347#518—611no—
348#497Dic6112yescomplemented
349#496Dic6112yescomplemented
350#495Dic6112yescomplemented
351#492Dic6112yescomplemented
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#418—611no—
365#418—611no—
366C2×#101—611no—
367C2×#100—611no—
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×#41C22×D6111yescharacteristic
383C2×#41C22×D6111yescharacteristic
384C22×Q8C22×D6111yescharacteristic
385C22×D4C22×D6111yescharacteristic
386#137—41no—
387#137—41no—
388AGL(1,ℤ/8)—41no—
389AGL(1,ℤ/8)—41no—
390#135—21no—
391#135—21no—
392#135—21no—
393#135—21no—
394C2×Q16C2×Dic6111yescharacteristic, complemented
395C2×Q16C2×Dic6112yescomplemented
396C2×SD16—21no—
397C2×SD16—21no—
398C2×SD16—21no—
399C2×SD16—21no—
400C2×SD16C2×Dic6112yes—
401C2×SD16C2×Dic6112yes—
402C2×SD16C2×Dic6112yes—
403C2×SD16C2×Dic6111yescharacteristic, complemented
404C2×SD16C2×Dic6111yescharacteristic, complemented
405C2×SD16C2×Dic6112yescomplemented
406C2×D8C2×Dic6112yes—
407C2×D8C2×Dic6111yescharacteristic, complemented
408#131—1222no—
409#131—1222no—
410#131—2441no—
411#131—2441no—
412C2×M16—611no—
413C2×M16—611no—
414C2×M16—611no—
415C2×M16—611no—
416C2×M16—2441no—
417C2×M16—2441no—
418C2×M16C22×D6111yescharacteristic
419C22×C8C22×D6111yescharacteristic
420C4⋊Q8C2×Dic6112yescomplemented
421#127C2×Dic6112yes—
422#124C2×Dic6112yes—
423#124C2×Dic6112yescomplemented
424#123—41no—
425C22⋊Q8—41no—
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—611no—
435#117—611no—
436#117—2441no—
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—1222no—
450#102—2441no—
451#101—2441no—
452#100—2441no—
453#99—2441no—
454#99—2441no—
455#97—612no—
456#97—1221no—
457#97C2×Dic6112yescomplemented
458C4×C8—612no—
459C4×C8—1221no—
460C4×C8C2×Dic6112yescomplemented
461C24—21no—
462#41—21no—
463#41—21no—
464#41—21no—
465#41—21no—
466#41—21no—
467#41—21no—
468#41—21no—
469#41—21no—
470C2×Q8—21no—
471C2×Q8—21no—
472C2×Q8—21no—
473C2×Q8—21no—
474C2×Q8—21no—
475C2×Q8C22×Dic6111yescharacteristic
476C2×Q8C22×Dic6112yes—
477C2×Q8C22×Dic6111yescharacteristic
478C2×D4—21no—
479C2×D4—21no—
480C2×D4—21no—
481C2×D4—21no—
482C2×D4—21no—
483C2×D4—21no—
484C2×D4—21no—
485C2×D4—42no—
486C2×D4C22×Dic6112yes—
487C2×D4C22×Dic6111yescharacteristic
488C2×D4C22×Dic6111yescharacteristic
489C22×C4—21no—
490C22×C4—21no—
491C22×C4—21no—
492C22×C4—2442no—
493C22×C4C23×D6111yescharacteristic
494Q16—21no—
495Q16—21no—
496Q16—42no—
497SD16—22no—
498SD16—22no—
499SD16—22no—
500SD16—22no—
501SD16—21no—
502SD16—21no—
503SD16—21no—
504SD16—21no—
505SD16—42no—
506SD16—42no—
507D8—21no—
508D8—21no—
509D8—42no—
510M16—21no—
511M16—21no—
512M16—1221no—
513M16—1221no—
514M16—1221no—
515M16—1221no—
516M16—1221no—
517M16—1221no—
518M16—1221no—
519M16—1221no—
520M16—2442no—
521M16—2442no—
522C2×C8—612no—
523C2×C8—1221no—
524C2×C8—1221no—
525C2×C8—1221no—
526C2×C8—2442no—
527C2×C8—2442no—
528C2×C8D4×D6111yescharacteristic
529C2×C8D4×D6111yescharacteristic
530C2×C8#1175822411yescharacteristic
531C2×C8#1175822411yescharacteristic
532C2×C8C22×Dic6112yes—
533C2×C8C22×Dic6111yescharacteristic
534C2×C8C22×Dic6111yescharacteristic
535C4⋊C4—21no—
536C4⋊C4—42no—
537C4⋊C4—1221no—
538C4⋊C4—1221no—
539C4⋊C4—1221no—
540C4⋊C4—1221no—
541C4⋊C4—2442no—
542C22⋊C4—21no—
543C22⋊C4—42no—
544C22⋊C4—42no—
545C22⋊C4—1221no—
546C22⋊C4—1221no—
547C22⋊C4—1221no—
548C22⋊C4—1221no—
549C22⋊C4—2442no—
550C42—612no—
551C42—1221no—
552C42—1221no—
553C42—1221no—
554C42—2442no—
555C42C22×Dic6112yes—
556C23—21no—
557C23—22no—
558C23—21no—
559C23—41no—
560C23—41no—
561C23—161no—
562C23#1451982211yescharacteristic
563Q8—22no—
564Q8—21no—
565Q8—21no—
566Q8—21no—
567Q8—22no—
568Q8—21no—
569Q8—41no—
570Q8—41no—
571D4—21no—
572D4—21no—
573D4—22no—
574D4—21no—
575D4—22no—
576D4—21no—
577D4—41no—
578D4—41no—
579D4—41no—
580D4—41no—
581D4—41no—
582D4—41no—
583D4—162no—
584C2×C4—22no—
585C2×C4—21no—
586C2×C4—22no—
587C2×C4—21no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—1222no—
595C2×C4—9762no—
596C2×C4—2441no—
597C2×C4—2441no—
598C2×C4—2441no—
599C2×C4C23×Dic6111yescharacteristic
600C2×C4C2×D4×D6111yescharacteristic
601C2×C4C2×#1175822411yescharacteristic
602C2×C4#1451982211yescharacteristic
603C2×C4#1451982211yescharacteristic
604C2×C4#1451982211yescharacteristic
605C8—22no—
606C8—1222no—
607C8—2441no—
608C8—2441no—
609C8—2441no—
610C8D4×Dic6111yescharacteristic
611C8D4×Dic6111yescharacteristic
612C8D4×Dic6111yescharacteristic
613C8D4×Dic6111yescharacteristic
614C22—41no—
615C22—41no—
616C22—41no—
617C22—42no—
618C22—81no—
619C22—82no—
620C22—81no—
621C22—82no—
622C22—161no—
623C22#1684023711yescharacteristic
624C22#1684032511yescharacteristic
625C22#1684036611yescharacteristic
626C4—42no—
627C4—41no—
628C4—41no—
629C4—42no—
630C4—4881no—
631C4—4881no—
632C4—4881no—
633C4—2442no—
634C4C2×D4×Dic6111yescharacteristic
635C4#1684023711yescharacteristic
636C4#1684032511yescharacteristic
637C4#1684036611yescharacteristic
638C2—21no—
639C2—21no—
640C2—21no—
641C2—82no—
642C2—81no—
643C2—81no—
644C2#2144854511yescharacteristic
subgroupG/H
centerC2#21448545
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
Fitting, prime radicalC61×#3154C2
socleC2×C61#2328
cosocleC22×C4×C61C24
hypercenter#3154D61
hypocenterC61#22284
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2 ◁ C22×C4 ◁ #3154

Complements

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

NH ≅ G/Ntype
1. #21454736C2semidirect
2. #21454679C2semidirect
3. #21446724C2semidirect
4. #21446718C2semidirect
5. #21446357C2semidirect
6. #21446352C2semidirect
7. #21446347C2semidirect
8. #21446342C2semidirect
9. #21446315C2semidirect
10. #21446308C2semidirect
11. #21446301C2semidirect
12. #21446294C2semidirect
13. #21445943C2semidirect
14. #21445574C2semidirect
54. C61×#497C4semidirect
55. C61×#496C4semidirect
56. C61×#495C4semidirect
57. C61×#492C4semidirect
96. #14519687D4semidirect
98. C8×Dic61D4semidirect
111. C2×Q16×C61C2×C4semidirect
112. C2×Q16×C61C2×C4semidirect
120. C2×SD16×C61C2×C4semidirect
121. C2×SD16×C61C2×C4semidirect
122. C2×SD16×C61C2×C4semidirect
124. C2×D8×C61C2×C4semidirect
127. C4⋊Q8×C61C2×C4semidirect
130. #124×C61C2×C4semidirect
136. #97×C61C2×C4semidirect
137. C4×C8×C61C2×C4semidirect
348. #497Dic61semidirect
349. #496Dic61semidirect
350. #495Dic61semidirect
351. #492Dic61semidirect
381. C61#22284semidirect
394. C2×Q16C2×Dic61semidirect
395. C2×Q16C2×Dic61semidirect
403. C2×SD16C2×Dic61semidirect
404. C2×SD16C2×Dic61semidirect
405. C2×SD16C2×Dic61semidirect
407. C2×D8C2×Dic61semidirect
420. C4⋊Q8C2×Dic61semidirect
423. #124C2×Dic61semidirect
457. #97C2×Dic61semidirect
460. C4×C8C2×Dic61semidirect

Maximal subgroups

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

subgroupindex
#214547362
#214546792
#214467242
#214467182
#214463572
#214463522
#214463472
#214463422
#214463152
#214463082
#214463012
#214462942
#214459432
#214455742
C61×#31542
#2228461