Finite Group Databasebeta

← C61×#24351 #29527784

Subgroups — C61×#24351 #29527784

Order 15616.

1122 conjugacy classes of subgroups.

The subgroup lattice has 556 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C61×#3501C211yesmaximal, characteristic
2C61×#2855C212yesmaximal, complemented
3C4×#123×C61C211yesmaximal, characteristic
4C61×#2062C212yesmaximal, complemented
5C61×#2030C212yesmaximal, complemented
6C61×#1996C212yesmaximal, complemented
7C61×#1890C212yesmaximal
8C61×#1867C212yesmaximal
9C61×#1799C211yesmaximal, characteristic
10C61×#554—24no—
11C61×#554—22no—
12D4×Q8×C61—22no—
13C61×#544—21no—
14C61×#543—21no—
15C61×#543C2212yes—
16C61×#533—21no—
17C2×#123×C61C2211yescharacteristic
18C2×C22⋊Q8×C61—21no—
19C2×C22⋊Q8×C61C2212yes—
20C61×#518C2212yes—
21C2×C4×D4×C61C2211yescharacteristic
22C22×C42×C61C2211yescharacteristic
23C2×#99×C61C2212yes—
24C61×#391—22no—
25C61×#389—22no—
26C61×#388—21no—
27C61×#387—21no—
28C61×#387—24no—
29C61×#386C2211yescharacteristic
30C61×#385C2211yescharacteristic
31C61×#385C2214yes—
32C61×#381C2211yescharacteristic
33C61×#381C2214yes—
34C61×#380C2212yes—
35C4×C4⋊C4×C61—21no—
36C4×C22⋊C4×C61—21no—
37C4×C22⋊C4×C61C2211yescharacteristic
38C61×#342C2212yes—
39C61×#342C2214yes—
40C61×#342C2214yes—
41C22×Q8×C61—21no—
42C22×Q8×C61D412yes—
43C22×D4×C61C2311yescharacteristic
44C23×C4×C61D411yescharacteristic
45C23×C4×C61C2311yescharacteristic
46C23×C4×C61C2311yescharacteristic
47C23×C4×C61C2311yescharacteristic
48C4⋊Q8×C61—21no—
49C4⋊Q8×C61—21no—
50C4⋊Q8×C61—24no—
51#125×C61—22no—
52#125×C61—24no—
53#123×C61—22no—
54#123×C61Q814yescomplemented
55C22⋊Q8×C61—24no—
56C22⋊Q8×C61—24no—
57C22⋊Q8×C61—22no—
58C22⋊Q8×C61—22no—
59C22⋊Q8×C61—22no—
60C22⋊Q8×C61—22no—
61C22⋊Q8×C61—22no—
62C22⋊Q8×C61—22no—
63C22⋊Q8×C61—24no—
64C22⋊Q8×C61—24no—
65C22⋊Q8×C61—22no—
66C22⋊Q8×C61—22no—
67C4×Q8×C61—22no—
68C4×Q8×C61—22no—
69C4×D4×C61—22no—
70C4×D4×C61—24no—
71C4×D4×C61—22no—
72C4×D4×C61—24no—
73#117×C61—22no—
74C2×C4⋊C4×C61—21no—
75C2×C4⋊C4×C61—22no—
76C2×C4⋊C4×C61—21no—
77C2×C4⋊C4×C61—21no—
78C2×C4⋊C4×C61—22no—
79C2×C4⋊C4×C61—24no—
80C2×C4⋊C4×C61D411yescharacteristic
81C2×C4⋊C4×C61D414yes—
82C2×C4⋊C4×C61D412yes—
83C2×C22⋊C4×C61—22no—
84C2×C22⋊C4×C61—21no—
85C2×C22⋊C4×C61—21no—
86C2×C22⋊C4×C61D411yescharacteristic
87C2×C22⋊C4×C61C2311yescharacteristic
88C2×C22⋊C4×C61C2314yes—
89C2×C22⋊C4×C61C2311yescharacteristic
90C2×C22⋊C4×C61C2314yes—
91C2×C22⋊C4×C61C2311yescharacteristic
92C2×C42×C61—21no—
93C2×C42×C61—21no—
94C2×C42×C61—22no—
95C2×C42×C61—21no—
96C2×C42×C61—22no—
97C2×C42×C61D411yescharacteristic
98#99×C61—24no—
99#95×C61—21no—
100#95×C61—21no—
101#95×C61—21no—
102#95×C61—21no—
103#95×C61—21no—
104#95×C61—22no—
105#95×C61—24no—
106#95×C61—24no—
107#95×C61—24no—
108C24×C61C2×D411yescharacteristic
109C24×C61C2411yescharacteristic
110C2×Q8×C61—22no—
111C2×Q8×C61—22no—
112C2×Q8×C61—22no—
113C2×Q8×C61—22no—
114C2×Q8×C61—44no—
115C2×Q8×C61—82no—
116C2×D4×C61—22no—
117C2×D4×C61—22no—
118C2×D4×C61C2×Q812yes—
119C2×D4×C61#4112yes—
120C22×C4×C61—22no—
121C22×C4×C61—21no—
122C22×C4×C61—21no—
123C22×C4×C61—22no—
124C22×C4×C61—21no—
125C22×C4×C61—21no—
126C22×C4×C61—22no—
127C22×C4×C61—22no—
128C22×C4×C61—22no—
129C22×C4×C61—21no—
130C22×C4×C61—21no—
131C22×C4×C61—21no—
132C22×C4×C61—24no—
133C22×C4×C61—42no—
134C22×C4×C61—42no—
135C22×C4×C61—42no—
136C22×C4×C61C2×D411yescharacteristic
137C22×C4×C61C2×D412yes—
138C22×C4×C61C2×D411yescharacteristic
139C22×C4×C61C2×D414yes—
140C22×C4×C61C2×D411yescharacteristic
141C22×C4×C61C2×D411yescharacteristic
142C22×C4×C61C2×D414yes—
143C22×C4×C61C2×D411yescharacteristic
144C22×C4×C61C2×D412yes—
145C22×C4×C61C2×Q812yescomplemented
146C4⋊C4×C61—22no—
147C4⋊C4×C61—24no—
148C4⋊C4×C61—22no—
149C4⋊C4×C61—22no—
150C4⋊C4×C61—24no—
151C4⋊C4×C61—22no—
152C4⋊C4×C61—24no—
153C4⋊C4×C61—22no—
154C4⋊C4×C61—22no—
155C4⋊C4×C61—22no—
156C4⋊C4×C61—24no—
157C4⋊C4×C61—24no—
158C4⋊C4×C61—44no—
159C4⋊C4×C61—42no—
160C4⋊C4×C61—48no—
161C22⋊C4×C61—24no—
162C22⋊C4×C61—24no—
163C22⋊C4×C61—24no—
164C22⋊C4×C61—22no—
165C22⋊C4×C61—24no—
166C22⋊C4×C61—21no—
167C22⋊C4×C61—22no—
168C22⋊C4×C61—24no—
169C22⋊C4×C61—22no—
170C22⋊C4×C61—24no—
171C22⋊C4×C61—42no—
172C22⋊C4×C61C2×Q812yescomplemented
173C42×C61—24no—
174C42×C61—21no—
175C42×C61—21no—
176C42×C61—42no—
177C42×C61—41no—
178C42×C61—42no—
179C42×C61—41no—
180C23×C61—21no—
181C23×C61—21no—
182C23×C61—41no—
183C23×C61—41no—
184C23×C61—41no—
185C23×C61—41no—
186C23×C61C22⋊Q812yes—
187C23×C61C22×D411yescharacteristic
188C23×C61C22×D412yes—
189C23×C61C22×Q811yescharacteristic
190C23×C61C2×#4111yescharacteristic
191C23×C612−1+412yes—
192Q8×C61—44no—
193Q8×C61—82no—
194Q8×C61—82no—
195Q8×C61—82no—
196D4×C61—24no—
197D4×C61—24no—
198C2×C4×C61—21no—
199C2×C4×C61—21no—
200C2×C4×C61—22no—
201C2×C4×C61—24no—
202C2×C4×C61—22no—
203C2×C4×C61—24no—
204C2×C4×C61—21no—
205C2×C4×C61—24no—
206C2×C4×C61—24no—
207C2×C4×C61—21no—
208C2×C4×C61—22no—
209C2×C4×C61—22no—
210C2×C4×C61—24no—
211C2×C4×C61—42no—
212C2×C4×C61—41no—
213C2×C4×C61—41no—
214C2×C4×C61—41no—
215C2×C4×C61—42no—
216C2×C4×C61—41no—
217C2×C4×C61—41no—
218C2×C4×C61—41no—
219C2×C4×C61—41no—
220C2×C4×C61—42no—
221C2×C4×C61—42no—
222C2×C4×C61—44no—
223C2×C4×C61—41no—
224C2×C4×C61—41no—
225C2×C4×C61—44no—
226C2×C4×C61—41no—
227C2×C4×C61—41no—
228C2×C4×C61—44no—
229C2×C4×C61—44no—
230C2×C4×C61—41no—
231C2×C4×C61—44no—
232C2×C4×C61—81no—
233C2×C4×C61—82no—
234C2×C4×C61—82no—
235C2×C4×C61#12011yescharacteristic
236C2×C4×C61#12012yes—
237C2×C4×C61#12011yescharacteristic
238C2×C4×C61C22⋊Q812yes—
239#24351C6111yesmaximal, characteristic, direct product factor, Sylow
240C22×C61—22no—
241C22×C61—21no—
242C22×C61—22no—
243C22×C61—21no—
244C22×C61—22no—
245C22×C61—41no—
246C22×C61—41no—
247C22×C61—42no—
248C22×C61—42no—
249C22×C61—41no—
250C22×C61—41no—
251C22×C61—81no—
252C22×C61C2×#12011yescharacteristic
253C22×C61C2×C22⋊Q811yescharacteristic
254C22×C61#53611yescharacteristic
255C22×C61#54812yes—
256C22×C61D4×Q812yes—
257C4×C61—22no—
258C4×C61—21no—
259C4×C61—22no—
260C4×C61—42no—
261C4×C61—42no—
262C4×C61—41no—
263C4×C61—44no—
264C4×C61—42no—
265C4×C61—44no—
266C4×C61—41no—
267C4×C61—84no—
268C4×C61—81no—
269C4×C61#45812yes—
270#3501C2×C6111yescharacteristic
271#2855C2×C6112yescomplemented
272C4×#123C2×C6111yescharacteristic
273#2062C2×C6112yescomplemented
274#2030C2×C6112yescomplemented
275#1996C2×C6112yescomplemented
276#1890C2×C6112yes—
277#1867C2×C6112yes—
278#1799C2×C6111yescharacteristic
279C2×C61—21no—
280C2×C61—22no—
281C2×C61—21no—
282C2×C61—22no—
283C2×C61—42no—
284C2×C61#240911yescharacteristic
285C2×C61C2×#45811yescharacteristic
286C2×C61#302511yescharacteristic
287#554—24no—
288#554—22no—
289D4×Q8—22no—
290#544—21no—
291#543—21no—
292#543C22×C6112yes—
293#533—21no—
294C2×#123C22×C6111yescharacteristic
295C2×C22⋊Q8—21no—
296C2×C22⋊Q8C22×C6112yes—
297#518C22×C6112yes—
298C2×C4×D4C22×C6111yescharacteristic
299C22×C42C22×C6111yescharacteristic
300C2×#99C22×C6112yes—
301#391—22no—
302#389—22no—
303#388—21no—
304#387—24no—
305#387—21no—
306#386C22×C6111yescharacteristic
307#385C22×C6111yescharacteristic
308#385C22×C6114yes—
309#381C22×C6114yes—
310#381C22×C6111yescharacteristic
311#380C22×C6112yes—
312C4×C4⋊C4—21no—
313C4×C22⋊C4—21no—
314C4×C22⋊C4C22×C6111yescharacteristic
315#342C22×C6112yes—
316#342C22×C6114yes—
317#342C22×C6114yes—
318C61#2435111yescharacteristic, direct product factor, Sylow
319C22×Q8—21no—
320C22×Q8D4×C6112yes—
321C22×D4C23×C6111yescharacteristic
322C23×C4D4×C6111yescharacteristic
323C23×C4C23×C6111yescharacteristic
324C23×C4C23×C6111yescharacteristic
325C23×C4C23×C6111yescharacteristic
326C4⋊Q8—21no—
327C4⋊Q8—21no—
328C4⋊Q8—24no—
329#125—22no—
330#125—24no—
331#123—22no—
332#123Q8×C6114yescomplemented
333C22⋊Q8—22no—
334C22⋊Q8—22no—
335C22⋊Q8—24no—
336C22⋊Q8—24no—
337C22⋊Q8—22no—
338C22⋊Q8—22no—
339C22⋊Q8—22no—
340C22⋊Q8—22no—
341C22⋊Q8—22no—
342C22⋊Q8—22no—
343C22⋊Q8—24no—
344C22⋊Q8—24no—
345C4×Q8—22no—
346C4×Q8—22no—
347C4×D4—22no—
348C4×D4—22no—
349C4×D4—24no—
350C4×D4—24no—
351#117—22no—
352C2×C4⋊C4—21no—
353C2×C4⋊C4—22no—
354C2×C4⋊C4—21no—
355C2×C4⋊C4—21no—
356C2×C4⋊C4—22no—
357C2×C4⋊C4—24no—
358C2×C4⋊C4D4×C6111yescharacteristic
359C2×C4⋊C4D4×C6114yes—
360C2×C4⋊C4D4×C6112yes—
361C2×C22⋊C4—22no—
362C2×C22⋊C4—21no—
363C2×C22⋊C4—21no—
364C2×C22⋊C4D4×C6111yescharacteristic
365C2×C22⋊C4C23×C6111yescharacteristic
366C2×C22⋊C4C23×C6114yes—
367C2×C22⋊C4C23×C6111yescharacteristic
368C2×C22⋊C4C23×C6114yes—
369C2×C22⋊C4C23×C6111yescharacteristic
370C2×C42—21no—
371C2×C42—21no—
372C2×C42—22no—
373C2×C42—21no—
374C2×C42—22no—
375C2×C42D4×C6111yescharacteristic
376#99—24no—
377#95—21no—
378#95—21no—
379#95—21no—
380#95—21no—
381#95—21no—
382#95—22no—
383#95—24no—
384#95—24no—
385#95—24no—
386C24C2×D4×C6111yescharacteristic
387C24C24×C6111yescharacteristic
388C2×Q8—22no—
389C2×Q8—22no—
390C2×Q8—22no—
391C2×Q8—22no—
392C2×Q8—44no—
393C2×Q8—82no—
394C2×D4—22no—
395C2×D4—22no—
396C2×D4C2×Q8×C6112yes—
397C2×D4#41×C6112yes—
398C22×C4—22no—
399C22×C4—21no—
400C22×C4—21no—
401C22×C4—22no—
402C22×C4—21no—
403C22×C4—21no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—21no—
408C22×C4—21no—
409C22×C4—21no—
410C22×C4—24no—
411C22×C4—42no—
412C22×C4—42no—
413C22×C4—42no—
414C22×C4C2×D4×C6111yescharacteristic
415C22×C4C2×D4×C6112yes—
416C22×C4C2×D4×C6111yescharacteristic
417C22×C4C2×D4×C6114yes—
418C22×C4C2×D4×C6111yescharacteristic
419C22×C4C2×D4×C6111yescharacteristic
420C22×C4C2×D4×C6114yes—
421C22×C4C2×D4×C6111yescharacteristic
422C22×C4C2×D4×C6112yes—
423C22×C4C2×Q8×C6112yescomplemented
424C4⋊C4—22no—
425C4⋊C4—24no—
426C4⋊C4—22no—
427C4⋊C4—22no—
428C4⋊C4—24no—
429C4⋊C4—22no—
430C4⋊C4—24no—
431C4⋊C4—22no—
432C4⋊C4—22no—
433C4⋊C4—22no—
434C4⋊C4—24no—
435C4⋊C4—24no—
436C4⋊C4—44no—
437C4⋊C4—42no—
438C4⋊C4—48no—
439C22⋊C4—24no—
440C22⋊C4—24no—
441C22⋊C4—24no—
442C22⋊C4—22no—
443C22⋊C4—24no—
444C22⋊C4—21no—
445C22⋊C4—22no—
446C22⋊C4—24no—
447C22⋊C4—22no—
448C22⋊C4—24no—
449C22⋊C4—42no—
450C22⋊C4C2×Q8×C6112yescomplemented
451C42—24no—
452C42—21no—
453C42—21no—
454C42—42no—
455C42—41no—
456C42—42no—
457C42—41no—
458C23—21no—
459C23—21no—
460C23—41no—
461C23—41no—
462C23—41no—
463C23—41no—
464C23C22⋊Q8×C6112yes—
465C23C22×D4×C6111yescharacteristic
466C23C22×D4×C6112yes—
467C23C22×Q8×C6111yescharacteristic
468C23C2×#41×C6111yescharacteristic
469C232−1+4×C6112yes—
470Q8—44no—
471Q8—82no—
472Q8—82no—
473Q8—82no—
474D4—24no—
475D4—24no—
476C2×C4—21no—
477C2×C4—21no—
478C2×C4—22no—
479C2×C4—24no—
480C2×C4—22no—
481C2×C4—24no—
482C2×C4—24no—
483C2×C4—21no—
484C2×C4—24no—
485C2×C4—24no—
486C2×C4—21no—
487C2×C4—22no—
488C2×C4—22no—
489C2×C4—42no—
490C2×C4—41no—
491C2×C4—41no—
492C2×C4—41no—
493C2×C4—42no—
494C2×C4—41no—
495C2×C4—41no—
496C2×C4—42no—
497C2×C4—41no—
498C2×C4—41no—
499C2×C4—42no—
500C2×C4—44no—
501C2×C4—41no—
502C2×C4—41no—
503C2×C4—44no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—44no—
507C2×C4—44no—
508C2×C4—41no—
509C2×C4—44no—
510C2×C4—81no—
511C2×C4—82no—
512C2×C4—82no—
513C2×C4#120×C6111yescharacteristic
514C2×C4#120×C6112yes—
515C2×C4#120×C6111yescharacteristic
516C2×C4C22⋊Q8×C6112yes—
517C22—22no—
518C22—21no—
519C22—22no—
520C22—21no—
521C22—22no—
522C22—41no—
523C22—41no—
524C22—42no—
525C22—42no—
526C22—41no—
527C22—41no—
528C22—81no—
529C22C2×#120×C6111yescharacteristic
530C22C2×C22⋊Q8×C6111yescharacteristic
531C22C61×#53611yescharacteristic
532C22C61×#54812yes—
533C22D4×Q8×C6112yes—
534C4—22no—
535C4—21no—
536C4—22no—
537C4—42no—
538C4—42no—
539C4—41no—
540C4—44no—
541C4—42no—
542C4—44no—
543C4—41no—
544C4—84no—
545C4—81no—
546C4C61×#45812yes—
547C2—21no—
548C2—22no—
549C2—21no—
550C2—22no—
551C2—42no—
552C2C61×#240911yescharacteristic
553C2C2×C61×#45811yescharacteristic
554C2C61×#302511yescharacteristic
subgroupG/H
center, socleC22×C61C2×#120
commutator, Frattini, cosocleC24C24×C61
prime radicalC22×D4×C61C23
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
2. C61×#2855C2semidirect
4. C61×#2062C2semidirect
5. C61×#2030C2semidirect
6. C61×#1996C2semidirect
54. #123×C61Q8semidirect
145. C22×C4×C61C2×Q8semidirect
172. C22⋊C4×C61C2×Q8semidirect
239. #24351C61direct (N × H)
271. #2855C2×C61semidirect
273. #2062C2×C61semidirect
274. #2030C2×C61semidirect
275. #1996C2×C61semidirect
318. C61#24351direct (N × H)
332. #123Q8×C61semidirect
423. C22×C4C2×Q8×C61semidirect
450. 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×#28552
C4×#123×C612
C61×#20622
C61×#20302
C61×#19962
C61×#18902
C61×#18672
C61×#17992
#2435161