Finite Group Databasebeta

← C61×#24002 #29527435

Subgroups — C61×#24002 #29527435

Order 15616.

1844 conjugacy classes of subgroups.

The subgroup lattice has 612 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#121×C61C211yesmaximal, characteristic
2C2×C61×#480C212yesmaximal, complemented
3C2×C61×#380C211yesmaximal, characteristic, complemented
4C61×#2033C214yesmaximal, complemented
5C61×#1874C212yesmaximal, complemented
6C61×#1852C212yesmaximal, complemented
7C61×#1786C212yesmaximal, complemented
8C61×#1779C211yesmaximal, characteristic, complemented
9C23×D4×C61—21no—
10C23×D4×C61C2211yescharacteristic
11C24×C4×C61C2211yescharacteristic
12C2×#121×C61—21no—
13C2×#121×C61—21no—
14C2×#121×C61—28no—
15C2×#121×C61C2212yes—
16C2×#121×C61C2212yes—
17C22×C4⋊C4×C61C2211yescharacteristic
18C22×C4⋊C4×C61C2211yescharacteristic, complemented
19C22×C4⋊C4×C61C2211yescharacteristic, complemented
20C22×C22⋊C4×C61—21no—
21C22×C22⋊C4×C61—21no—
22C61×#480—28no—
23C2×#107×C61C2214yescomplemented
24C2×#102×C61C2214yescomplemented
25C2×#102×C61C2214yescomplemented
26C2×C22⋊C8×C61C2212yes—
27C2×C22⋊C8×C61C2212yescomplemented
28C2×C22⋊C8×C61C2212yescomplemented
29C61×#380—21no—
30C61×#380—21no—
31C61×#380C2212yes—
32C61×#380C2212yescomplemented
33C2×#95×C61—21no—
34C61×#340C2214yescomplemented
35C25×C61—21no—
36C25×C61D411yescharacteristic
37C22×D4×C61—28no—
38C22×D4×C61—22no—
39C22×D4×C61—22no—
40C22×D4×C61—24no—
41C22×D4×C61—21no—
42C22×D4×C61—24no—
43C22×D4×C61—42no—
44C22×D4×C61—44no—
45C22×D4×C61—44no—
46C22×D4×C61C2312yes—
47C23×C4×C61—21no—
48C23×C4×C61—24no—
49C23×C4×C61—24no—
50C23×C4×C61—21no—
51C23×C4×C61—21no—
52C23×C4×C61—42no—
53C23×C4×C61D411yescharacteristic
54C23×C4×C61D411yescharacteristic
55C23×C4×C61D411yescharacteristic
56C23×C4×C61C2311yescharacteristic
57C23×C4×C61C2311yescharacteristic
58C23×C4×C61C2311yescharacteristic
59C22×C8×C61C2314yes—
60#121×C61—28no—
61#121×C61—28no—
62#121×C61—24no—
63#121×C61—24no—
64#121×C61—24no—
65#121×C61D418yes—
66C2×C4⋊C4×C61—21no—
67C2×C4⋊C4×C61—24no—
68C2×C4⋊C4×C61—24no—
69C2×C4⋊C4×C61—21no—
70C2×C4⋊C4×C61—21no—
71C2×C4⋊C4×C61—42no—
72C2×C4⋊C4×C61C2312yes—
73C2×C4⋊C4×C61C2312yes—
74C2×C4⋊C4×C61C2312yes—
75C2×C22⋊C4×C61—28no—
76C2×C22⋊C4×C61—22no—
77C2×C22⋊C4×C61—22no—
78C2×C22⋊C4×C61—22no—
79C2×C22⋊C4×C61—22no—
80C2×C22⋊C4×C61—44no—
81#107×C61—28no—
82#102×C61—28no—
83#102×C61—44no—
84C22⋊C8×C61—24no—
85C22⋊C8×C61—24no—
86C22⋊C8×C61—42no—
87#95×C61—22no—
88#95×C61—22no—
89C24×C61—22no—
90C24×C61—22no—
91C24×C61—22no—
92C24×C61—22no—
93C24×C61—24no—
94C24×C61—22no—
95C24×C61—41no—
96C24×C61—41no—
97C24×C61—42no—
98C24×C61—42no—
99C24×C61—42no—
100C24×C61—42no—
101C24×C61—42no—
102C24×C61—42no—
103C24×C61D814yes—
104C24×C61C2×D411yescharacteristic
105C24×C61C2×D411yescharacteristic
106C24×C61C2×D411yescharacteristic
107C2×D4×C61—24no—
108C2×D4×C61—28no—
109C2×D4×C61—28no—
110C2×D4×C61—22no—
111C2×D4×C61—24no—
112C2×D4×C61—24no—
113C2×D4×C61—416no—
114C2×D4×C61—42no—
115C2×D4×C61—432no—
116C2×D4×C61—44no—
117C2×D4×C61—44no—
118C2×D4×C61—44no—
119C2×D4×C61—44no—
120C2×D4×C61—42no—
121C2×D4×C61C2×D414yes—
122C22×C4×C61—22no—
123C22×C4×C61—22no—
124C22×C4×C61—22no—
125C22×C4×C61—22no—
126C22×C4×C61—24no—
127C22×C4×C61—22no—
128C22×C4×C61—22no—
129C22×C4×C61—22no—
130C22×C4×C61—22no—
131C22×C4×C61—22no—
132C22×C4×C61—22no—
133C22×C4×C61—22no—
134C22×C4×C61—22no—
135C22×C4×C61—22no—
136C22×C4×C61—22no—
137C22×C4×C61—22no—
138C22×C4×C61—22no—
139C22×C4×C61—41no—
140C22×C4×C61—41no—
141C22×C4×C61—41no—
142C22×C4×C61—416no—
143C22×C4×C61—42no—
144C22×C4×C61—42no—
145C22×C4×C61—42no—
146C22×C4×C61—42no—
147C22×C4×C61—42no—
148C22×C4×C61—42no—
149C22×C4×C61—42no—
150C22×C4×C61—42no—
151C22×C4×C61C2×D411yescharacteristic
152C22×C4×C61C2×D411yescharacteristic
153C22×C4×C61C2×D414yes—
154C22×C4×C61C2×D411yescharacteristic
155C22×C4×C61#4114yes—
156C22×C4×C61C2411yescharacteristic
157C2×C8×C61—24no—
158C2×C8×C61—24no—
159C2×C8×C61—24no—
160C4⋊C4×C61—24no—
161C4⋊C4×C61—22no—
162C4⋊C4×C61—24no—
163C4⋊C4×C61—24no—
164C4⋊C4×C61—42no—
165C4⋊C4×C61—42no—
166C4⋊C4×C61C2×D414yes—
167C22⋊C4×C61—28no—
168C22⋊C4×C61—28no—
169C22⋊C4×C61—44no—
170C22⋊C4×C61—44no—
171C23×C61—21no—
172C23×C61—21no—
173C23×C61—21no—
174C23×C61—21no—
175C23×C61—21no—
176C23×C61—21no—
177C23×C61—21no—
178C23×C61—21no—
179C23×C61—21no—
180C23×C61—21no—
181C23×C61—21no—
182C23×C61—21no—
183C23×C61—21no—
184C23×C61—21no—
185C23×C61—28no—
186C23×C61—28no—
187C23×C61—24no—
188C23×C61—44no—
189C23×C61—44no—
190C23×C61—44no—
191C23×C61—44no—
192C23×C61—44no—
193C23×C61—48no—
194C23×C61—48no—
195C23×C61—48no—
196C23×C61—42no—
197C23×C61—42no—
198C23×C61—42no—
199C23×C61—42no—
200C23×C61—42no—
201C23×C61—42no—
202C23×C61—84no—
203C23×C61—84no—
204C23×C61—84no—
205C23×C61—84no—
206C23×C61#12012yes—
207C23×C61#12312yes—
208C23×C61C2×D812yes—
209C23×C61C2×D812yes—
210C23×C61C2×D812yes—
211C23×C61AGL(1,ℤ/8)12yes—
212C23×C61#13712yes—
213C23×C61C22×D411yescharacteristic
214D4×C61—28no—
215D4×C61—48no—
216D4×C61—416no—
217D4×C61—416no—
218D4×C61—44no—
219C2×C4×C61—21no—
220C2×C4×C61—21no—
221C2×C4×C61—21no—
222C2×C4×C61—21no—
223C2×C4×C61—28no—
224C2×C4×C61—28no—
225C2×C4×C61—21no—
226C2×C4×C61—24no—
227C2×C4×C61—21no—
228C2×C4×C61—21no—
229C2×C4×C61—44no—
230C2×C4×C61—44no—
231C2×C4×C61—44no—
232C2×C4×C61—48no—
233C2×C4×C61—42no—
234C2×C4×C61—42no—
235C2×C4×C61—44no—
236C2×C4×C61—42no—
237C2×C4×C61—42no—
238C2×C4×C61—42no—
239C2×C4×C61—42no—
240C2×C4×C61—42no—
241C2×C4×C61—42no—
242C2×C4×C61—84no—
243C2×C4×C61—84no—
244C2×C4×C61#12012yes—
245C2×C4×C61#12312yes—
246C2×C4×C61C22×D412yes—
247C2×C4×C61C2×#4112yes—
248C2×C4×C61H(5,𝔽2)12yes—
249C8×C61—44no—
250#24002C6111yesmaximal, characteristic, direct product factor, Sylow
251C22×C61—22no—
252C22×C61—22no—
253C22×C61—22no—
254C22×C61—22no—
255C22×C61—22no—
256C22×C61—22no—
257C22×C61—22no—
258C22×C61—22no—
259C22×C61—22no—
260C22×C61—22no—
261C22×C61—22no—
262C22×C61—22no—
263C22×C61—22no—
264C22×C61—22no—
265C22×C61—22no—
266C22×C61—416no—
267C22×C61—41no—
268C22×C61—41no—
269C22×C61—41no—
270C22×C61—41no—
271C22×C61—44no—
272C22×C61—44no—
273C22×C61—44no—
274C22×C61—816no—
275C22×C61—82no—
276C22×C61—82no—
277C22×C61—82no—
278C22×C61—82no—
279C22×C61#44714yes—
280C22×C61#48014yescomplemented
281C22×C61C2×#12011yescharacteristic
282C22×C61C2×#12311yescharacteristic
283C22×C61#53411yescharacteristic
284C22×C61C22×D811yescharacteristic
285C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
286C22×C61C2×#13711yescharacteristic
287C22×C61#57511yescharacteristic
288C4×C61—22no—
289C4×C61—22no—
290C4×C61—41no—
291C4×C61—44no—
292C4×C61—82no—
293C4×C61—82no—
294C4×C61#54614yes—
295C22×#121C2×C6111yescharacteristic
296C2×#480C2×C6112yescomplemented
297C2×#380C2×C6111yescharacteristic, complemented
298#2033C2×C6114yescomplemented
299#1874C2×C6112yescomplemented
300#1852C2×C6112yescomplemented
301#1786C2×C6112yescomplemented
302#1779C2×C6111yescharacteristic, complemented
303C2×C61—24no—
304C2×C61—24no—
305C2×C61—42no—
306C2×C61—84no—
307C2×C61#240511yescharacteristic
308C2×C61C2×#44711yescharacteristic
309C2×C61#300711yescharacteristic
310C2×C61C2×#48011yescharacteristic
311C2×C61#309411yescharacteristic
312C2×C61#318611yescharacteristic
313C2×C61#319311yescharacteristic
314C23×D4—21no—
315C23×D4C22×C6111yescharacteristic
316C24×C4C22×C6111yescharacteristic
317C2×#121—21no—
318C2×#121—21no—
319C2×#121—28no—
320C2×#121C22×C6112yes—
321C2×#121C22×C6112yes—
322C22×C4⋊C4C22×C6111yescharacteristic
323C22×C4⋊C4C22×C6111yescharacteristic, complemented
324C22×C4⋊C4C22×C6111yescharacteristic, complemented
325C22×C22⋊C4—21no—
326C22×C22⋊C4—21no—
327#480—28no—
328C2×#107C22×C6114yescomplemented
329C2×#102C22×C6114yescomplemented
330C2×#102C22×C6114yescomplemented
331C2×C22⋊C8C22×C6112yes—
332C2×C22⋊C8C22×C6112yescomplemented
333C2×C22⋊C8C22×C6112yescomplemented
334#380—21no—
335#380—21no—
336#380C22×C6112yes—
337#380C22×C6112yescomplemented
338C2×#95—21no—
339#340C22×C6114yescomplemented
340C61#2400211yescharacteristic, direct product factor, Sylow
341C25—21no—
342C25D4×C6111yescharacteristic
343C22×D4—28no—
344C22×D4—22no—
345C22×D4—22no—
346C22×D4—24no—
347C22×D4—21no—
348C22×D4—24no—
349C22×D4—42no—
350C22×D4—44no—
351C22×D4—44no—
352C22×D4C23×C6112yes—
353C23×C4—21no—
354C23×C4—24no—
355C23×C4—24no—
356C23×C4—21no—
357C23×C4—21no—
358C23×C4—42no—
359C23×C4D4×C6111yescharacteristic
360C23×C4D4×C6111yescharacteristic
361C23×C4D4×C6111yescharacteristic
362C23×C4C23×C6111yescharacteristic
363C23×C4C23×C6111yescharacteristic
364C23×C4C23×C6111yescharacteristic
365C22×C8C23×C6114yes—
366#121—28no—
367#121—28no—
368#121—24no—
369#121—24no—
370#121—24no—
371#121D4×C6118yes—
372C2×C4⋊C4—21no—
373C2×C4⋊C4—24no—
374C2×C4⋊C4—24no—
375C2×C4⋊C4—21no—
376C2×C4⋊C4—21no—
377C2×C4⋊C4—42no—
378C2×C4⋊C4C23×C6112yes—
379C2×C4⋊C4C23×C6112yes—
380C2×C4⋊C4C23×C6112yes—
381C2×C22⋊C4—28no—
382C2×C22⋊C4—22no—
383C2×C22⋊C4—22no—
384C2×C22⋊C4—22no—
385C2×C22⋊C4—22no—
386C2×C22⋊C4—44no—
387#107—28no—
388#102—28no—
389#102—44no—
390C22⋊C8—24no—
391C22⋊C8—24no—
392C22⋊C8—42no—
393#95—22no—
394#95—22no—
395C24—22no—
396C24—22no—
397C24—22no—
398C24—22no—
399C24—24no—
400C24—22no—
401C24—41no—
402C24—41no—
403C24—42no—
404C24—42no—
405C24—42no—
406C24—42no—
407C24—42no—
408C24—42no—
409C24D8×C6114yes—
410C24C2×D4×C6111yescharacteristic
411C24C2×D4×C6111yescharacteristic
412C24C2×D4×C6111yescharacteristic
413C2×D4—24no—
414C2×D4—28no—
415C2×D4—28no—
416C2×D4—22no—
417C2×D4—24no—
418C2×D4—24no—
419C2×D4—416no—
420C2×D4—42no—
421C2×D4—432no—
422C2×D4—44no—
423C2×D4—44no—
424C2×D4—44no—
425C2×D4—44no—
426C2×D4—42no—
427C2×D4C2×D4×C6114yes—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—24no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—41no—
446C22×C4—41no—
447C22×C4—41no—
448C22×C4—416no—
449C22×C4—42no—
450C22×C4—42no—
451C22×C4—42no—
452C22×C4—42no—
453C22×C4—42no—
454C22×C4—42no—
455C22×C4—42no—
456C22×C4—42no—
457C22×C4C2×D4×C6111yescharacteristic
458C22×C4C2×D4×C6111yescharacteristic
459C22×C4C2×D4×C6114yes—
460C22×C4C2×D4×C6111yescharacteristic
461C22×C4#41×C6114yes—
462C22×C4C24×C6111yescharacteristic
463C2×C8—24no—
464C2×C8—24no—
465C2×C8—24no—
466C4⋊C4—24no—
467C4⋊C4—22no—
468C4⋊C4—24no—
469C4⋊C4—24no—
470C4⋊C4—42no—
471C4⋊C4—42no—
472C4⋊C4C2×D4×C6114yes—
473C22⋊C4—28no—
474C22⋊C4—28no—
475C22⋊C4—44no—
476C22⋊C4—44no—
477C23—21no—
478C23—21no—
479C23—21no—
480C23—21no—
481C23—21no—
482C23—21no—
483C23—21no—
484C23—21no—
485C23—21no—
486C23—21no—
487C23—21no—
488C23—21no—
489C23—28no—
490C23—28no—
491C23—21no—
492C23—21no—
493C23—24no—
494C23—44no—
495C23—44no—
496C23—44no—
497C23—44no—
498C23—44no—
499C23—48no—
500C23—48no—
501C23—48no—
502C23—42no—
503C23—42no—
504C23—42no—
505C23—42no—
506C23—42no—
507C23—42no—
508C23—84no—
509C23—84no—
510C23—84no—
511C23—84no—
512C23#120×C6112yes—
513C23#123×C6112yes—
514C23C2×D8×C6112yes—
515C23C2×D8×C6112yes—
516C23C2×D8×C6112yes—
517C23AGL(1,ℤ/8)×C6112yes—
518C23#137×C6112yes—
519C23C22×D4×C6111yescharacteristic
520D4—28no—
521D4—48no—
522D4—416no—
523D4—416no—
524D4—44no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—28no—
530C2×C4—28no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—24no—
534C2×C4—21no—
535C2×C4—44no—
536C2×C4—44no—
537C2×C4—44no—
538C2×C4—48no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—44no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—84no—
549C2×C4—84no—
550C2×C4#120×C6112yes—
551C2×C4#123×C6112yes—
552C2×C4C22×D4×C6112yes—
553C2×C4C2×#41×C6112yes—
554C2×C4H(5,𝔽2)×C6112yes—
555C8—44no—
556C22—22no—
557C22—22no—
558C22—22no—
559C22—22no—
560C22—22no—
561C22—22no—
562C22—22no—
563C22—22no—
564C22—22no—
565C22—22no—
566C22—22no—
567C22—22no—
568C22—22no—
569C22—22no—
570C22—22no—
571C22—416no—
572C22—41no—
573C22—41no—
574C22—41no—
575C22—41no—
576C22—44no—
577C22—44no—
578C22—44no—
579C22—816no—
580C22—82no—
581C22—82no—
582C22—82no—
583C22—82no—
584C22C61×#44714yes—
585C22C61×#48014yescomplemented
586C22C2×#120×C6111yescharacteristic
587C22C2×#123×C6111yescharacteristic
588C22C61×#53411yescharacteristic
589C22C22×D8×C6111yescharacteristic
590C22C2×AGL(1,ℤ/8)×C6111yescharacteristic
591C22C2×#137×C6111yescharacteristic
592C22C61×#57511yescharacteristic
593C4—22no—
594C4—22no—
595C4—41no—
596C4—44no—
597C4—82no—
598C4—82no—
599C4C61×#54614yes—
600C2—24no—
601C2—24no—
602C2—42no—
603C2—84no—
604C2C61×#240511yescharacteristic
605C2C2×C61×#44711yescharacteristic
606C2C61×#300711yescharacteristic
607C2C2×C61×#48011yescharacteristic
608C2C61×#309411yescharacteristic
609C2C61×#318611yescharacteristic
610C2C61×#319311yescharacteristic
subgroupG/H
center, socleC23×C61C22×D4
commutator, Frattini, cosocleC22×C4C24×C61
prime radicalC22×#121×C61C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C61 ◁ C24×C4×C61 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C61×#480C2semidirect
3. C2×C61×#380C2semidirect
4. C61×#2033C2semidirect
5. C61×#1874C2semidirect
6. C61×#1852C2semidirect
7. C61×#1786C2semidirect
8. C61×#1779C2semidirect
18. C22×C4⋊C4×C61C22semidirect
19. C22×C4⋊C4×C61C22semidirect
23. C2×#107×C61C22semidirect
24. C2×#102×C61C22semidirect
25. C2×#102×C61C22semidirect
27. C2×C22⋊C8×C61C22semidirect
28. C2×C22⋊C8×C61C22semidirect
32. C61×#380C22semidirect
34. C61×#340C22semidirect
250. #24002C61direct (N × H)
280. C22×C61#480semidirect
296. C2×#480C2×C61semidirect
297. C2×#380C2×C61semidirect
298. #2033C2×C61semidirect
299. #1874C2×C61semidirect
300. #1852C2×C61semidirect
301. #1786C2×C61semidirect
302. #1779C2×C61semidirect
323. C22×C4⋊C4C22×C61semidirect
324. C22×C4⋊C4C22×C61semidirect
328. C2×#107C22×C61semidirect
329. C2×#102C22×C61semidirect
330. C2×#102C22×C61semidirect
332. C2×C22⋊C8C22×C61semidirect
333. C2×C22⋊C8C22×C61semidirect
337. #380C22×C61semidirect
339. #340C22×C61semidirect
340. C61#24002direct (N × H)
585. C22C61×#480semidirect

Maximal subgroups

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

subgroupindex
C22×#121×C612
C2×C61×#4802
C2×C61×#3802
C61×#20332
C61×#18742
C61×#18522
C61×#17862
C61×#17792
#2400261