Finite Group Databasebeta

← C61×#24352 #29527785

Subgroups — C61×#24352 #29527785

Order 15616.

1198 conjugacy classes of subgroups.

The subgroup lattice has 608 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C61×#3501C211yesmaximal, characteristic, complemented
2C61×#2626C212yesmaximal, complemented
3C61×#2321C211yesmaximal, characteristic
4C61×#2062C212yesmaximal, complemented
5C61×#2029C212yesmaximal, complemented
6C61×#1996C212yesmaximal, complemented
7C61×#1889C212yesmaximal
8C61×#1867C212yesmaximal, complemented
9C61×#1799C211yesmaximal, characteristic, complemented
10C61×#554—24no—
11C61×#554—22no—
12D4×Q8×C61—22no—
13C61×#544—21no—
14C61×#543—21no—
15C61×#543C2212yes—
16C61×#533—21no—
17C2×C22⋊Q8×C61—21no—
18C2×C22⋊Q8×C61C2212yes—
19C2×#120×C61C2211yescharacteristic
20C61×#518C2212yes—
21C2×C4×D4×C61C2211yescharacteristic
22C22×C22⋊C4×C61C2211yescharacteristic
23C2×#99×C61C2212yes—
24C61×#399—22no—
25C61×#396—24no—
26C61×#393—22no—
27C61×#390—21no—
28C61×#388—21no—
29C61×#388—21no—
30C61×#386—21no—
31C61×#386C2211yescharacteristic
32C61×#385C2211yescharacteristic
33C61×#385C2214yes—
34C61×#380C2212yes—
35C61×#379C2211yescharacteristic
36C61×#379C2214yes—
37C4×C22⋊C4×C61C2211yescharacteristic
38C61×#342C2212yes—
39C61×#342C2214yes—
40C61×#342C2214yes—
41C25×C61C2311yescharacteristic
42C22×Q8×C61—21no—
43C22×Q8×C61D412yescomplemented
44C22×D4×C61—21no—
45C22×D4×C61C2311yescharacteristic
46C23×C4×C61D411yescharacteristic
47C23×C4×C61C2311yescharacteristic
48C23×C4×C61C2311yescharacteristic
49C4⋊Q8×C61—21no—
50C4⋊Q8×C61—21no—
51C4⋊Q8×C61—24no—
52#125×C61—22no—
53#125×C61—24no—
54C22⋊Q8×C61—24no—
55C22⋊Q8×C61—24no—
56C22⋊Q8×C61—22no—
57C22⋊Q8×C61—22no—
58C22⋊Q8×C61—22no—
59C22⋊Q8×C61—22no—
60C22⋊Q8×C61—22no—
61C22⋊Q8×C61—22no—
62C22⋊Q8×C61—24no—
63C22⋊Q8×C61—24no—
64C22⋊Q8×C61—22no—
65C22⋊Q8×C61—22no—
66#120×C61—22no—
67#120×C61Q814yescomplemented
68C4×Q8×C61—22no—
69C4×Q8×C61—22no—
70C4×D4×C61—22no—
71C4×D4×C61—24no—
72C4×D4×C61—22no—
73C4×D4×C61—24no—
74#117×C61—22no—
75C2×C4⋊C4×C61—21no—
76C2×C4⋊C4×C61—22no—
77C2×C4⋊C4×C61—22no—
78C2×C4⋊C4×C61—21no—
79C2×C4⋊C4×C61—24no—
80C2×C4⋊C4×C61D411yescharacteristic
81C2×C4⋊C4×C61D414yescomplemented
82C2×C4⋊C4×C61D412yescomplemented
83C2×C22⋊C4×C61—22no—
84C2×C22⋊C4×C61—21no—
85C2×C22⋊C4×C61—21no—
86C2×C22⋊C4×C61—21no—
87C2×C22⋊C4×C61—22no—
88C2×C22⋊C4×C61—22no—
89C2×C22⋊C4×C61—21no—
90C2×C22⋊C4×C61—21no—
91C2×C22⋊C4×C61—21no—
92C2×C22⋊C4×C61—24no—
93C2×C22⋊C4×C61—24no—
94C2×C22⋊C4×C61D411yescharacteristic
95C2×C22⋊C4×C61C2311yescharacteristic
96C2×C22⋊C4×C61C2314yes—
97C2×C22⋊C4×C61C2311yescharacteristic
98C2×C22⋊C4×C61C2314yes—
99C2×C22⋊C4×C61C2311yescharacteristic
100C2×C42×C61—21no—
101C2×C42×C61D411yescharacteristic
102#99×C61—24no—
103#95×C61—21no—
104#95×C61—22no—
105#95×C61—21no—
106#95×C61—24no—
107#95×C61—21no—
108C24×C61—22no—
109C24×C61—21no—
110C24×C61—21no—
111C24×C61—22no—
112C24×C61—81no—
113C24×C61—81no—
114C24×C61C2×D411yescharacteristic
115C24×C61C2×Q812yescomplemented
116C24×C61C2411yescharacteristic
117C2×Q8×C61—22no—
118C2×Q8×C61—22no—
119C2×Q8×C61—22no—
120C2×Q8×C61—22no—
121C2×Q8×C61—82no—
122C2×Q8×C61—82no—
123C2×D4×C61—22no—
124C2×D4×C61—22no—
125C2×D4×C61—24no—
126C2×D4×C61—81no—
127C2×D4×C61—81no—
128C2×D4×C61C2×Q812yes—
129C2×D4×C61#4112yes—
130C22×C4×C61—21no—
131C22×C4×C61—22no—
132C22×C4×C61—22no—
133C22×C4×C61—21no—
134C22×C4×C61—21no—
135C22×C4×C61—21no—
136C22×C4×C61—21no—
137C22×C4×C61—24no—
138C22×C4×C61—22no—
139C22×C4×C61—81no—
140C22×C4×C61—81no—
141C22×C4×C61—81no—
142C22×C4×C61C2×D411yescharacteristic
143C22×C4×C61C2×D411yescharacteristic
144C22×C4×C61C2×D414yes—
145C22×C4×C61C2×D411yescharacteristic
146C22×C4×C61C2×D414yes—
147C22×C4×C61C2×D411yescharacteristic
148C22×C4×C61C2×D412yes—
149C22×C4×C61C2×D411yescharacteristic
150C22×C4×C61C2×D412yes—
151C4⋊C4×C61—22no—
152C4⋊C4×C61—24no—
153C4⋊C4×C61—22no—
154C4⋊C4×C61—22no—
155C4⋊C4×C61—24no—
156C4⋊C4×C61—22no—
157C4⋊C4×C61—24no—
158C4⋊C4×C61—22no—
159C4⋊C4×C61—22no—
160C4⋊C4×C61—22no—
161C4⋊C4×C61—24no—
162C4⋊C4×C61—82no—
163C4⋊C4×C61—81no—
164C4⋊C4×C61—84no—
165C22⋊C4×C61—24no—
166C22⋊C4×C61—24no—
167C22⋊C4×C61—22no—
168C22⋊C4×C61—24no—
169C22⋊C4×C61—21no—
170C22⋊C4×C61—22no—
171C22⋊C4×C61—24no—
172C22⋊C4×C61—24no—
173C22⋊C4×C61—22no—
174C22⋊C4×C61—24no—
175C22⋊C4×C61—41no—
176C22⋊C4×C61—41no—
177C22⋊C4×C61—42no—
178C22⋊C4×C61—81no—
179C22⋊C4×C61—84no—
180C22⋊C4×C61—81no—
181C22⋊C4×C61—84no—
182C22⋊C4×C61—81no—
183C22⋊C4×C61C2×Q812yescomplemented
184C42×C61—24no—
185C42×C61—21no—
186C42×C61—21no—
187C42×C61—81no—
188C23×C61—21no—
189C23×C61—21no—
190C23×C61—22no—
191C23×C61—24no—
192C23×C61—41no—
193C23×C61—41no—
194C23×C61—41no—
195C23×C61—41no—
196C23×C61—44no—
197C23×C61—41no—
198C23×C61—44no—
199C23×C61—41no—
200C23×C61—41no—
201C23×C61—41no—
202C23×C61—41no—
203C23×C61—41no—
204C23×C61—42no—
205C23×C61—82no—
206C23×C61—82no—
207C23×C61—84no—
208C23×C61C22⋊Q812yes—
209C23×C61C22×D411yescharacteristic
210C23×C61C22×D412yes—
211C23×C61C22×Q811yescharacteristic
212C23×C61C2×#4111yescharacteristic
213C23×C612−1+412yes—
214Q8×C61—44no—
215Q8×C61—82no—
216Q8×C61—82no—
217Q8×C61—82no—
218D4×C61—24no—
219D4×C61—24no—
220D4×C61—84no—
221C2×C4×C61—21no—
222C2×C4×C61—21no—
223C2×C4×C61—21no—
224C2×C4×C61—24no—
225C2×C4×C61—24no—
226C2×C4×C61—21no—
227C2×C4×C61—22no—
228C2×C4×C61—41no—
229C2×C4×C61—42no—
230C2×C4×C61—41no—
231C2×C4×C61—42no—
232C2×C4×C61—42no—
233C2×C4×C61—41no—
234C2×C4×C61—41no—
235C2×C4×C61—44no—
236C2×C4×C61—41no—
237C2×C4×C61—41no—
238C2×C4×C61—44no—
239C2×C4×C61—41no—
240C2×C4×C61—41no—
241C2×C4×C61—44no—
242C2×C4×C61—44no—
243C2×C4×C61—41no—
244C2×C4×C61—44no—
245C2×C4×C61—41no—
246C2×C4×C61—41no—
247C2×C4×C61—41no—
248C2×C4×C61—44no—
249C2×C4×C61—81no—
250C2×C4×C61—82no—
251C2×C4×C61—82no—
252C2×C4×C61#12011yescharacteristic
253C2×C4×C61#12011yescharacteristic
254C2×C4×C61#12012yes—
255C2×C4×C61C22⋊Q812yes—
256#24352C6111yesmaximal, characteristic, direct product factor, Sylow
257C22×C61—22no—
258C22×C61—21no—
259C22×C61—22no—
260C22×C61—21no—
261C22×C61—22no—
262C22×C61—41no—
263C22×C61—41no—
264C22×C61—42no—
265C22×C61—42no—
266C22×C61—42no—
267C22×C61—42no—
268C22×C61—48no—
269C22×C61—41no—
270C22×C61—41no—
271C22×C61—42no—
272C22×C61—81no—
273C22×C61—84no—
274C22×C61—81no—
275C22×C61—81no—
276C22×C61—81no—
277C22×C61—81no—
278C22×C61C2×#12011yescharacteristic
279C22×C61C2×C22⋊Q811yescharacteristic
280C22×C61#53611yescharacteristic
281C22×C61#54812yes—
282C22×C61D4×Q812yes—
283C4×C61—22no—
284C4×C61—21no—
285C4×C61—21no—
286C4×C61—22no—
287C4×C61—42no—
288C4×C61—41no—
289C4×C61—44no—
290C4×C61—44no—
291C4×C61—41no—
292C4×C61—84no—
293C4×C61—81no—
294C4×C61—81no—
295#3501C2×C6111yescharacteristic, complemented
296#2626C2×C6112yescomplemented
297#2321C2×C6111yescharacteristic
298#2062C2×C6112yescomplemented
299#2029C2×C6112yescomplemented
300#1996C2×C6112yescomplemented
301#1889C2×C6112yes—
302#1867C2×C6112yescomplemented
303#1799C2×C6111yescharacteristic, complemented
304C2×C61—21no—
305C2×C61—22no—
306C2×C61—21no—
307C2×C61—22no—
308C2×C61—42no—
309C2×C61—82no—
310C2×C61#240911yescharacteristic
311C2×C61#302611yescharacteristic
312C2×C61#302711yescharacteristic
313#554—24no—
314#554—22no—
315D4×Q8—22no—
316#544—21no—
317#543—21no—
318#543C22×C6112yes—
319#533—21no—
320C2×C22⋊Q8—21no—
321C2×C22⋊Q8C22×C6112yes—
322C2×#120C22×C6111yescharacteristic
323#518C22×C6112yes—
324C2×C4×D4C22×C6111yescharacteristic
325C22×C22⋊C4C22×C6111yescharacteristic
326C2×#99C22×C6112yes—
327#399—22no—
328#396—24no—
329#393—22no—
330#390—21no—
331#388—21no—
332#388—21no—
333#386—21no—
334#386C22×C6111yescharacteristic
335#385C22×C6111yescharacteristic
336#385C22×C6114yes—
337#380C22×C6112yes—
338#379C22×C6111yescharacteristic
339#379C22×C6114yes—
340C4×C22⋊C4C22×C6111yescharacteristic
341#342C22×C6112yes—
342#342C22×C6114yes—
343#342C22×C6114yes—
344C61#2435211yescharacteristic, direct product factor, Sylow
345C25C23×C6111yescharacteristic
346C22×Q8—21no—
347C22×Q8D4×C6112yescomplemented
348C22×D4—21no—
349C22×D4C23×C6111yescharacteristic
350C23×C4D4×C6111yescharacteristic
351C23×C4C23×C6111yescharacteristic
352C23×C4C23×C6111yescharacteristic
353C4⋊Q8—21no—
354C4⋊Q8—21no—
355C4⋊Q8—24no—
356#125—22no—
357#125—24no—
358C22⋊Q8—22no—
359C22⋊Q8—22no—
360C22⋊Q8—24no—
361C22⋊Q8—24no—
362C22⋊Q8—22no—
363C22⋊Q8—22no—
364C22⋊Q8—22no—
365C22⋊Q8—22no—
366C22⋊Q8—22no—
367C22⋊Q8—22no—
368C22⋊Q8—24no—
369C22⋊Q8—24no—
370#120—22no—
371#120Q8×C6114yescomplemented
372C4×Q8—22no—
373C4×Q8—22no—
374C4×D4—22no—
375C4×D4—22no—
376C4×D4—24no—
377C4×D4—24no—
378#117—22no—
379C2×C4⋊C4—21no—
380C2×C4⋊C4—22no—
381C2×C4⋊C4—22no—
382C2×C4⋊C4—21no—
383C2×C4⋊C4—24no—
384C2×C4⋊C4D4×C6111yescharacteristic
385C2×C4⋊C4D4×C6114yescomplemented
386C2×C4⋊C4D4×C6112yescomplemented
387C2×C22⋊C4—22no—
388C2×C22⋊C4—21no—
389C2×C22⋊C4—21no—
390C2×C22⋊C4—21no—
391C2×C22⋊C4—22no—
392C2×C22⋊C4—22no—
393C2×C22⋊C4—21no—
394C2×C22⋊C4—21no—
395C2×C22⋊C4—21no—
396C2×C22⋊C4—24no—
397C2×C22⋊C4—24no—
398C2×C22⋊C4D4×C6111yescharacteristic
399C2×C22⋊C4C23×C6111yescharacteristic
400C2×C22⋊C4C23×C6114yes—
401C2×C22⋊C4C23×C6111yescharacteristic
402C2×C22⋊C4C23×C6114yes—
403C2×C22⋊C4C23×C6111yescharacteristic
404C2×C42—21no—
405C2×C42D4×C6111yescharacteristic
406#99—24no—
407#95—21no—
408#95—22no—
409#95—21no—
410#95—24no—
411#95—21no—
412C24—22no—
413C24—21no—
414C24—21no—
415C24—22no—
416C24—81no—
417C24—81no—
418C24C2×D4×C6111yescharacteristic
419C24C2×Q8×C6112yescomplemented
420C24C24×C6111yescharacteristic
421C2×Q8—22no—
422C2×Q8—22no—
423C2×Q8—22no—
424C2×Q8—22no—
425C2×Q8—82no—
426C2×Q8—82no—
427C2×D4—22no—
428C2×D4—22no—
429C2×D4—24no—
430C2×D4—81no—
431C2×D4—81no—
432C2×D4C2×Q8×C6112yes—
433C2×D4#41×C6112yes—
434C22×C4—21no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—24no—
442C22×C4—22no—
443C22×C4—81no—
444C22×C4—81no—
445C22×C4—81no—
446C22×C4C2×D4×C6111yescharacteristic
447C22×C4C2×D4×C6111yescharacteristic
448C22×C4C2×D4×C6114yes—
449C22×C4C2×D4×C6111yescharacteristic
450C22×C4C2×D4×C6114yes—
451C22×C4C2×D4×C6111yescharacteristic
452C22×C4C2×D4×C6112yes—
453C22×C4C2×D4×C6111yescharacteristic
454C22×C4C2×D4×C6112yes—
455C4⋊C4—22no—
456C4⋊C4—24no—
457C4⋊C4—22no—
458C4⋊C4—22no—
459C4⋊C4—24no—
460C4⋊C4—22no—
461C4⋊C4—24no—
462C4⋊C4—22no—
463C4⋊C4—22no—
464C4⋊C4—22no—
465C4⋊C4—24no—
466C4⋊C4—82no—
467C4⋊C4—81no—
468C4⋊C4—84no—
469C22⋊C4—24no—
470C22⋊C4—24no—
471C22⋊C4—22no—
472C22⋊C4—24no—
473C22⋊C4—21no—
474C22⋊C4—22no—
475C22⋊C4—24no—
476C22⋊C4—24no—
477C22⋊C4—22no—
478C22⋊C4—24no—
479C22⋊C4—41no—
480C22⋊C4—41no—
481C22⋊C4—42no—
482C22⋊C4—81no—
483C22⋊C4—84no—
484C22⋊C4—81no—
485C22⋊C4—84no—
486C22⋊C4—81no—
487C22⋊C4C2×Q8×C6112yescomplemented
488C42—24no—
489C42—21no—
490C42—21no—
491C42—81no—
492C23—21no—
493C23—21no—
494C23—24no—
495C23—22no—
496C23—41no—
497C23—41no—
498C23—41no—
499C23—41no—
500C23—44no—
501C23—41no—
502C23—44no—
503C23—41no—
504C23—41no—
505C23—41no—
506C23—41no—
507C23—42no—
508C23—41no—
509C23—82no—
510C23—82no—
511C23—84no—
512C23C22⋊Q8×C6112yes—
513C23C22×D4×C6111yescharacteristic
514C23C22×D4×C6112yes—
515C23C22×Q8×C6111yescharacteristic
516C23C2×#41×C6111yescharacteristic
517C232−1+4×C6112yes—
518Q8—44no—
519Q8—82no—
520Q8—82no—
521Q8—82no—
522D4—24no—
523D4—24no—
524D4—84no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—24no—
529C2×C4—24no—
530C2×C4—21no—
531C2×C4—22no—
532C2×C4—41no—
533C2×C4—42no—
534C2×C4—41no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—44no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—44no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—44no—
546C2×C4—44no—
547C2×C4—41no—
548C2×C4—44no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—44no—
552C2×C4—41no—
553C2×C4—81no—
554C2×C4—82no—
555C2×C4—82no—
556C2×C4#120×C6111yescharacteristic
557C2×C4#120×C6111yescharacteristic
558C2×C4#120×C6112yes—
559C2×C4C22⋊Q8×C6112yes—
560C22—22no—
561C22—21no—
562C22—22no—
563C22—21no—
564C22—22no—
565C22—41no—
566C22—41no—
567C22—42no—
568C22—42no—
569C22—42no—
570C22—42no—
571C22—48no—
572C22—41no—
573C22—41no—
574C22—42no—
575C22—81no—
576C22—84no—
577C22—81no—
578C22—81no—
579C22—81no—
580C22—81no—
581C22C2×#120×C6111yescharacteristic
582C22C2×C22⋊Q8×C6111yescharacteristic
583C22C61×#53611yescharacteristic
584C22C61×#54812yes—
585C22D4×Q8×C6112yes—
586C4—22no—
587C4—21no—
588C4—21no—
589C4—22no—
590C4—42no—
591C4—41no—
592C4—44no—
593C4—44no—
594C4—41no—
595C4—84no—
596C4—81no—
597C4—81no—
598C2—21no—
599C2—22no—
600C2—21no—
601C2—22no—
602C2—42no—
603C2—82no—
604C2C61×#240911yescharacteristic
605C2C61×#302611yescharacteristic
606C2C61×#302711yescharacteristic
subgroupG/H
center, socleC22×C61C2×#120
commutator, Frattini, cosocleC24C24×C61
prime radicalC2×#120×C61C22
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C22×C61 ◁ C23×C4×C61 ◁ G

Complements

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

NH ≅ G/Ntype
1. C61×#3501C2semidirect
2. C61×#2626C2semidirect
4. C61×#2062C2semidirect
5. C61×#2029C2semidirect
6. C61×#1996C2semidirect
8. C61×#1867C2semidirect
9. C61×#1799C2semidirect
43. C22×Q8×C61D4semidirect
67. #120×C61Q8semidirect
81. C2×C4⋊C4×C61D4semidirect
82. C2×C4⋊C4×C61D4semidirect
115. C24×C61C2×Q8semidirect
183. C22⋊C4×C61C2×Q8semidirect
256. #24352C61direct (N × H)
295. #3501C2×C61semidirect
296. #2626C2×C61semidirect
298. #2062C2×C61semidirect
299. #2029C2×C61semidirect
300. #1996C2×C61semidirect
302. #1867C2×C61semidirect
303. #1799C2×C61semidirect
344. C61#24352direct (N × H)
347. C22×Q8D4×C61semidirect
371. #120Q8×C61semidirect
385. C2×C4⋊C4D4×C61semidirect
386. C2×C4⋊C4D4×C61semidirect
419. C24C2×Q8×C61semidirect
487. C22⋊C4C2×Q8×C61semidirect

Maximal subgroups

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

subgroupindex
C61×#35012
C61×#26262
C61×#23212
C61×#20622
C61×#20292
C61×#19962
C61×#18892
C61×#18672
C61×#17992
#2435261