Finite Group Databasebeta

← C61×#33568 #29537001

Subgroups — C61×#33568 #29537001

Order 15616.

1110 conjugacy classes of subgroups.

The subgroup lattice has 626 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C61×#3545C211yesmaximal, characteristic, complemented
2C61×#3020C212yesmaximal, complemented
3C2×C61×#454C211yesmaximal, characteristic, complemented
4C61×#2204C214yesmaximal, complemented
5C61×#2203C212yesmaximal, complemented
6C61×#2202C212yesmaximal, complemented
7C61×#2133C212yesmaximal, complemented
8C2×C61×#355C211yesmaximal, characteristic, complemented
9C2×2−1+4×C61C2211yescharacteristic, complemented
10C61×#575—22no—
11C2×AGL(1,ℤ/8)×C61—21no—
12C61×#563C2211yescharacteristic, complemented
13C61×#562—21no—
14C61×H(3,𝔽4)C2211yescharacteristic, complemented
15C61×#555C2212yescomplemented
16C61×#551—21no—
17C61×#551—22no—
18C61×#546—22no—
19C61×#542—22no—
20C61×#538—22no—
21C61×#536C2212yes—
22C61×#534—21no—
23C61×#529—21no—
24C2×#124×C61C2211yescharacteristic
25C61×#456—22no—
26C61×#456C2214yescomplemented
27C61×#454—21no—
28C61×#454—21no—
29C61×#454—22no—
30C61×#454C2212yes—
31C61×#454C2214yescomplemented
32C61×#454C2212yescomplemented
33C61×#421—22no—
34C2×#104×C61—21no—
35C2×#101×C61C2212yescomplemented
36C2×#101×C61C2211yescharacteristic, complemented
37C61×#356C2214yescomplemented
38C61×#355C2212yescomplemented
39C61×#355C2212yescomplemented
40C61×#355C2214yescomplemented
412−1+4×C61—21no—
422−1+4×C61—22no—
432−1+4×C61—42no—
442−1+4×C61C2312yes—
45C2×#41×C61—21no—
46C2×#41×C61—22no—
47C2×#41×C61—41no—
48C2×#41×C61D412yes—
49C22×Q8×C61—41no—
50C22×Q8×C61C2311yescharacteristic
51C22×D4×C61—21no—
52C22×D4×C61D411yescharacteristic
53#137×C61—24no—
54#137×C61—42no—
55AGL(1,ℤ/8)×C61—22no—
56AGL(1,ℤ/8)×C61—22no—
57AGL(1,ℤ/8)×C61—24no—
58AGL(1,ℤ/8)×C61—42no—
59AGL(1,ℤ/8)×C61—42no—
60#135×C61—44no—
61C2×SD16×C61—41no—
62C2×D8×C61—41no—
63C2×M16×C61—21no—
64C2×M16×C61—22no—
65C4⋊Q8×C61D412yes—
66C4⋊Q8×C61C2312yes—
67#126×C61—22no—
68#126×C61—22no—
69#126×C61—22no—
70#125×C61—21no—
71#124×C61—22no—
72#124×C61—22no—
73#124×C61—22no—
74#124×C61—21no—
75#124×C61D412yes—
76#124×C61D411yescharacteristic
77#124×C61D411yescharacteristic
78#124×C61C2312yes—
79#124×C61C2311yescharacteristic
80#124×C61C2311yescharacteristic
81#123×C61—21no—
82#123×C61—22no—
83#123×C61—22no—
84#123×C61—22no—
85#123×C61—21no—
86#123×C61—22no—
87#123×C61—22no—
88C22⋊Q8×C61—22no—
89C22⋊Q8×C61—21no—
90C22⋊Q8×C61—22no—
91C22⋊Q8×C61—22no—
92C22⋊Q8×C61—22no—
93#121×C61—22no—
94#121×C61—22no—
95#121×C61—21no—
96#121×C61—22no—
97#121×C61—22no—
98#120×C61—21no—
99#120×C61—22no—
100#120×C61—21no—
101#120×C61—22no—
102C4×Q8×C61—21no—
103C4×D4×C61—21no—
104C4×D4×C61—22no—
105C4×D4×C61—22no—
106C4×D4×C61—22no—
107#117×C61—22no—
108#117×C61—22no—
109#117×C61D412yes—
110C2×C22⋊C4×C61—21no—
111C2×C22⋊C4×C61—22no—
112C2×C22⋊C4×C61—22no—
113C2×C42×C61D411yescharacteristic
114#104×C61—22no—
115#104×C61—22no—
116#104×C61—24no—
117#104×C61—24no—
118#101×C61—21no—
119#101×C61—22no—
120#101×C61C2314yes—
121#101×C61C2312yes—
122C24×C61—21no—
123C24×C61—21no—
124#41×C61—24no—
125#41×C61—22no—
126#41×C61—22no—
127#41×C61—24no—
128#41×C61—42no—
129#41×C61—41no—
130#41×C61—42no—
131#41×C61—42no—
132#41×C61—42no—
133#41×C61—42no—
134#41×C61—44no—
135#41×C61—41no—
136#41×C61—42no—
137C2×Q8×C61—21no—
138C2×Q8×C61—21no—
139C2×Q8×C61—22no—
140C2×Q8×C61—41no—
141C2×Q8×C61—41no—
142C2×Q8×C61—42no—
143C2×Q8×C61—42no—
144C2×Q8×C61—44no—
145C2×Q8×C61C2×D412yes—
146C2×Q8×C61C2×D411yescharacteristic
147C2×Q8×C61C2×D412yes—
148C2×Q8×C61C2411yescharacteristic
149C2×D4×C61—21no—
150C2×D4×C61—22no—
151C2×D4×C61—22no—
152C2×D4×C61—22no—
153C2×D4×C61—22no—
154C2×D4×C61—21no—
155C2×D4×C61—22no—
156C2×D4×C61—21no—
157C2×D4×C61—41no—
158C2×D4×C61—41no—
159C2×D4×C61—41no—
160C2×D4×C61—81no—
161C2×D4×C61—81no—
162C2×D4×C61C2×D412yes—
163C2×D4×C61C2×D411yescharacteristic
164C2×D4×C61C2×D411yescharacteristic
165C22×C4×C61—22no—
166C22×C4×C61—22no—
167C22×C4×C61—21no—
168C22×C4×C61—21no—
169C22×C4×C61—22no—
170C22×C4×C61—41no—
171C22×C4×C61—42no—
172C22×C4×C61C2×D411yescharacteristic
173C22×C4×C61C2×D412yes—
174Q16×C61—44no—
175SD16×C61—44no—
176SD16×C61—42no—
177SD16×C61—42no—
178SD16×C61—44no—
179D8×C61—42no—
180D8×C61—42no—
181D8×C61—44no—
182M16×C61—22no—
183M16×C61—24no—
184M16×C61—41no—
185M16×C61—42no—
186C2×C8×C61—41no—
187C2×C8×C61—42no—
188C4⋊C4×C61—22no—
189C4⋊C4×C61—22no—
190C4⋊C4×C61—21no—
191C4⋊C4×C61—21no—
192C4⋊C4×C61—22no—
193C4⋊C4×C61—22no—
194C4⋊C4×C61—22no—
195C4⋊C4×C61—22no—
196C4⋊C4×C61—22no—
197C22⋊C4×C61—22no—
198C22⋊C4×C61—22no—
199C22⋊C4×C61—21no—
200C22⋊C4×C61—21no—
201C22⋊C4×C61—22no—
202C22⋊C4×C61—22no—
203C22⋊C4×C61—22no—
204C22⋊C4×C61—22no—
205C22⋊C4×C61—22no—
206C22⋊C4×C61—22no—
207C22⋊C4×C61—22no—
208C22⋊C4×C61—22no—
209C22⋊C4×C61—22no—
210C22⋊C4×C61—22no—
211C22⋊C4×C61—22no—
212C42×C61—22no—
213C42×C61—21no—
214C42×C61C2×D411yescharacteristic
215C42×C61C2×D411yescharacteristic
216C42×C61C2×D412yes—
217C42×C61C2×D412yes—
218C23×C61—21no—
219C23×C61—21no—
220C23×C61—22no—
221C23×C61—41no—
222C23×C61—41no—
223C23×C61—41no—
224C23×C61—81no—
225C23×C61—82no—
226C23×C61—82no—
227C23×C61#12011yescharacteristic
228Q8×C61—22no—
229Q8×C61—24no—
230Q8×C61—41no—
231Q8×C61—42no—
232Q8×C61—41no—
233Q8×C61—42no—
234Q8×C61—41no—
235Q8×C61—42no—
236Q8×C61—81no—
237D4×C61—22no—
238D4×C61—24no—
239D4×C61—41no—
240D4×C61—41no—
241D4×C61—42no—
242D4×C61—42no—
243D4×C61—41no—
244D4×C61—42no—
245D4×C61—42no—
246D4×C61—42no—
247D4×C61—42no—
248D4×C61—81no—
249D4×C61—82no—
250D4×C61—82no—
251D4×C61—81no—
252D4×C61—82no—
253C2×C4×C61—21no—
254C2×C4×C61—22no—
255C2×C4×C61—22no—
256C2×C4×C61—21no—
257C2×C4×C61—21no—
258C2×C4×C61—22no—
259C2×C4×C61—21no—
260C2×C4×C61—22no—
261C2×C4×C61—22no—
262C2×C4×C61—42no—
263C2×C4×C61—44no—
264C2×C4×C61—44no—
265C2×C4×C61—42no—
266C2×C4×C61—41no—
267C2×C4×C61—82no—
268C2×C4×C61—82no—
269C2×C4×C61—81no—
270C2×C4×C61—82no—
271C2×C4×C61—81no—
272C2×C4×C61#12011yescharacteristic
273C2×C4×C61#12012yes—
274C2×C4×C61C22×D412yes—
275C2×C4×C61C22×D411yescharacteristic
276C8×C61—42no—
277C8×C61—44no—
278#33568C6111yesmaximal, characteristic, direct product factor, Sylow
279C22×C61—41no—
280C22×C61—42no—
281C22×C61—42no—
282C22×C61—42no—
283C22×C61—42no—
284C22×C61—42no—
285C22×C61—42no—
286C22×C61—41no—
287C22×C61—41no—
288C22×C61—82no—
289C22×C61—161no—
290C22×C61—161no—
291C22×C61—162no—
292C22×C61B(4,𝔽2)11yescharacteristic
293C22×C61B(4,𝔽2)11yescharacteristic
294C22×C61C2×#12011yescharacteristic
295C4×C61—21no—
296C4×C61—22no—
297C4×C61—22no—
298C4×C61—21no—
299C4×C61—41no—
300C4×C61—42no—
301C4×C61—42no—
302C4×C61—41no—
303C4×C61—42no—
304C4×C61—42no—
305#3545C2×C6111yescharacteristic, complemented
306#3020C2×C6112yescomplemented
307C2×#454C2×C6111yescharacteristic, complemented
308#2204C2×C6114yescomplemented
309#2203C2×C6112yescomplemented
310#2202C2×C6112yescomplemented
311#2133C2×C6112yescomplemented
312C2×#355C2×C6111yescharacteristic, complemented
313C2×C61—21no—
314C2×C61—21no—
315C2×C61—21no—
316C2×C61—81no—
317C2×C61—82no—
318C2×C61—81no—
319C2×C61—82no—
320C2×C61C2×B(4,𝔽2)11yescharacteristic
321C2×2−1+4C22×C6111yescharacteristic, complemented
322#575—22no—
323C2×AGL(1,ℤ/8)—21no—
324#563C22×C6111yescharacteristic, complemented
325#562—21no—
326H(3,𝔽4)C22×C6111yescharacteristic, complemented
327#555C22×C6112yescomplemented
328#551—21no—
329#551—22no—
330#546—22no—
331#542—22no—
332#538—22no—
333#536C22×C6112yes—
334#534—21no—
335#529—21no—
336C2×#124C22×C6111yescharacteristic
337#456—22no—
338#456C22×C6114yescomplemented
339#454—21no—
340#454—21no—
341#454—22no—
342#454C22×C6112yes—
343#454C22×C6114yescomplemented
344#454C22×C6112yescomplemented
345#421—22no—
346C2×#104—21no—
347C2×#101C22×C6112yescomplemented
348C2×#101C22×C6111yescharacteristic, complemented
349#356C22×C6114yescomplemented
350#355C22×C6112yescomplemented
351#355C22×C6112yescomplemented
352#355C22×C6114yescomplemented
353C61#3356811yescharacteristic, direct product factor, Sylow
3542−1+4—21no—
3552−1+4—22no—
3562−1+4—42no—
3572−1+4C23×C6112yes—
358C2×#41—21no—
359C2×#41—22no—
360C2×#41—41no—
361C2×#41D4×C6112yes—
362C22×Q8—41no—
363C22×Q8C23×C6111yescharacteristic
364C22×D4—21no—
365C22×D4D4×C6111yescharacteristic
366#137—24no—
367#137—42no—
368AGL(1,ℤ/8)—22no—
369AGL(1,ℤ/8)—22no—
370AGL(1,ℤ/8)—24no—
371AGL(1,ℤ/8)—42no—
372AGL(1,ℤ/8)—42no—
373#135—44no—
374C2×SD16—41no—
375C2×D8—41no—
376C2×M16—21no—
377C2×M16—22no—
378C4⋊Q8D4×C6112yes—
379C4⋊Q8C23×C6112yes—
380#126—22no—
381#126—22no—
382#126—22no—
383#125—21no—
384#124—22no—
385#124—22no—
386#124—21no—
387#124—22no—
388#124D4×C6111yescharacteristic
389#124D4×C6111yescharacteristic
390#124D4×C6112yes—
391#124C23×C6111yescharacteristic
392#124C23×C6111yescharacteristic
393#124C23×C6112yes—
394#123—21no—
395#123—22no—
396#123—22no—
397#123—22no—
398#123—21no—
399#123—22no—
400#123—22no—
401C22⋊Q8—21no—
402C22⋊Q8—22no—
403C22⋊Q8—22no—
404C22⋊Q8—22no—
405C22⋊Q8—22no—
406#121—21no—
407#121—22no—
408#121—22no—
409#121—22no—
410#121—22no—
411#120—21no—
412#120—22no—
413#120—21no—
414#120—22no—
415C4×Q8—21no—
416C4×D4—21no—
417C4×D4—22no—
418C4×D4—22no—
419C4×D4—22no—
420#117—22no—
421#117—22no—
422#117D4×C6112yes—
423C2×C22⋊C4—21no—
424C2×C22⋊C4—22no—
425C2×C22⋊C4—22no—
426C2×C42D4×C6111yescharacteristic
427#104—22no—
428#104—22no—
429#104—24no—
430#104—24no—
431#101—21no—
432#101—22no—
433#101C23×C6114yes—
434#101C23×C6112yes—
435C24—21no—
436C24—21no—
437#41—24no—
438#41—22no—
439#41—22no—
440#41—24no—
441#41—42no—
442#41—41no—
443#41—42no—
444#41—42no—
445#41—42no—
446#41—42no—
447#41—44no—
448#41—41no—
449#41—42no—
450C2×Q8—21no—
451C2×Q8—21no—
452C2×Q8—22no—
453C2×Q8—41no—
454C2×Q8—41no—
455C2×Q8—42no—
456C2×Q8—42no—
457C2×Q8—44no—
458C2×Q8C2×D4×C6112yes—
459C2×Q8C2×D4×C6111yescharacteristic
460C2×Q8C2×D4×C6112yes—
461C2×Q8C24×C6111yescharacteristic
462C2×D4—21no—
463C2×D4—22no—
464C2×D4—22no—
465C2×D4—22no—
466C2×D4—22no—
467C2×D4—21no—
468C2×D4—22no—
469C2×D4—21no—
470C2×D4—41no—
471C2×D4—41no—
472C2×D4—41no—
473C2×D4—81no—
474C2×D4—81no—
475C2×D4C2×D4×C6112yes—
476C2×D4C2×D4×C6111yescharacteristic
477C2×D4C2×D4×C6111yescharacteristic
478C22×C4—22no—
479C22×C4—22no—
480C22×C4—21no—
481C22×C4—21no—
482C22×C4—22no—
483C22×C4—41no—
484C22×C4—42no—
485C22×C4C2×D4×C6111yescharacteristic
486C22×C4C2×D4×C6112yes—
487Q16—44no—
488SD16—44no—
489SD16—42no—
490SD16—42no—
491SD16—44no—
492D8—42no—
493D8—42no—
494D8—44no—
495M16—22no—
496M16—24no—
497M16—41no—
498M16—42no—
499C2×C8—41no—
500C2×C8—42no—
501C4⋊C4—22no—
502C4⋊C4—22no—
503C4⋊C4—21no—
504C4⋊C4—21no—
505C4⋊C4—22no—
506C4⋊C4—22no—
507C4⋊C4—22no—
508C4⋊C4—22no—
509C4⋊C4—22no—
510C22⋊C4—22no—
511C22⋊C4—22no—
512C22⋊C4—21no—
513C22⋊C4—21no—
514C22⋊C4—22no—
515C22⋊C4—22no—
516C22⋊C4—22no—
517C22⋊C4—22no—
518C22⋊C4—22no—
519C22⋊C4—22no—
520C22⋊C4—22no—
521C22⋊C4—22no—
522C22⋊C4—22no—
523C22⋊C4—22no—
524C22⋊C4—22no—
525C42—22no—
526C42—21no—
527C42C2×D4×C6111yescharacteristic
528C42C2×D4×C6111yescharacteristic
529C42C2×D4×C6112yes—
530C42C2×D4×C6112yes—
531C23—21no—
532C23—21no—
533C23—22no—
534C23—41no—
535C23—41no—
536C23—41no—
537C23—81no—
538C23—82no—
539C23—82no—
540C23#120×C6111yescharacteristic
541Q8—22no—
542Q8—24no—
543Q8—41no—
544Q8—42no—
545Q8—41no—
546Q8—42no—
547Q8—42no—
548Q8—41no—
549Q8—81no—
550D4—22no—
551D4—24no—
552D4—41no—
553D4—41no—
554D4—42no—
555D4—42no—
556D4—41no—
557D4—42no—
558D4—42no—
559D4—42no—
560D4—42no—
561D4—81no—
562D4—82no—
563D4—82no—
564D4—81no—
565D4—82no—
566C2×C4—21no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—22no—
572C2×C4—21no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—42no—
576C2×C4—44no—
577C2×C4—44no—
578C2×C4—42no—
579C2×C4—41no—
580C2×C4—82no—
581C2×C4—82no—
582C2×C4—81no—
583C2×C4—82no—
584C2×C4—81no—
585C2×C4#120×C6111yescharacteristic
586C2×C4#120×C6112yes—
587C2×C4C22×D4×C6112yes—
588C2×C4C22×D4×C6111yescharacteristic
589C8—42no—
590C8—44no—
591C22—41no—
592C22—42no—
593C22—42no—
594C22—42no—
595C22—42no—
596C22—42no—
597C22—42no—
598C22—41no—
599C22—41no—
600C22—82no—
601C22—161no—
602C22—161no—
603C22—162no—
604C22C61×B(4,𝔽2)11yescharacteristic
605C22C61×B(4,𝔽2)11yescharacteristic
606C22C2×#120×C6111yescharacteristic
607C4—21no—
608C4—22no—
609C4—22no—
610C4—21no—
611C4—41no—
612C4—42no—
613C4—42no—
614C4—41no—
615C4—42no—
616C4—42no—
617C2—21no—
618C2—21no—
619C2—21no—
620C2—81no—
621C2—81no—
622C2—82no—
623C2—82no—
624C2C2×C61×B(4,𝔽2)11yescharacteristic
subgroupG/H
center, socleC2×C61C2×B(4,𝔽2)
commutator, Frattini, cosocleC2×Q8C24×C61
LCSG ▷ C2×Q8 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C61 ◁ C23×C61 ◁ C22×Q8×C61 ◁ G

Complements

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

NH ≅ G/Ntype
1. C61×#3545C2semidirect
2. C61×#3020C2semidirect
3. C2×C61×#454C2semidirect
4. C61×#2204C2semidirect
5. C61×#2203C2semidirect
6. C61×#2202C2semidirect
7. C61×#2133C2semidirect
8. C2×C61×#355C2semidirect
9. C2×2−1+4×C61C22semidirect
12. C61×#563C22semidirect
14. C61×H(3,𝔽4)C22semidirect
15. C61×#555C22semidirect
26. C61×#456C22semidirect
31. C61×#454C22semidirect
32. C61×#454C22semidirect
35. C2×#101×C61C22semidirect
36. C2×#101×C61C22semidirect
37. C61×#356C22semidirect
38. C61×#355C22semidirect
39. C61×#355C22semidirect
40. C61×#355C22semidirect
278. #33568C61direct (N × H)
305. #3545C2×C61semidirect
306. #3020C2×C61semidirect
307. C2×#454C2×C61semidirect
308. #2204C2×C61semidirect
309. #2203C2×C61semidirect
310. #2202C2×C61semidirect
311. #2133C2×C61semidirect
312. C2×#355C2×C61semidirect
321. C2×2−1+4C22×C61semidirect
324. #563C22×C61semidirect
326. H(3,𝔽4)C22×C61semidirect
327. #555C22×C61semidirect
338. #456C22×C61semidirect
343. #454C22×C61semidirect
344. #454C22×C61semidirect
347. C2×#101C22×C61semidirect
348. C2×#101C22×C61semidirect
349. #356C22×C61semidirect
350. #355C22×C61semidirect
351. #355C22×C61semidirect
352. #355C22×C61semidirect
353. C61#33568direct (N × H)

Maximal subgroups

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

subgroupindex
C61×#35452
C61×#30202
C2×C61×#4542
C61×#22042
C61×#22032
C61×#22022
C61×#21332
C2×C61×#3552
#3356861