Finite Group Databasebeta

← D61×#3362 #30559603

Subgroups — D61×#3362 #30559603

Order 15616.

2174 conjugacy classes of subgroups.

The subgroup lattice has 660 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1Q82×D61C211yesmaximal, characteristic, complemented
2#550×D61C211yesmaximal, characteristic
3#21458491C211yesmaximal, characteristic, complemented
4#21458490C211yesmaximal, characteristic, complemented
5#21458489C213yesmaximal, complemented
6#21458488C213yesmaximal, complemented
7#21458487C213yesmaximal, complemented
8#21458486C211yesmaximal, characteristic, complemented
9#21458485C213yesmaximal, complemented
10#486×D61C213yesmaximal, complemented
11#476×D61C213yesmaximal, complemented
12#460×D61C213yesmaximal, complemented
13C8×Q8×D61C211yesmaximal, characteristic, complemented
14C4×SD16×D61C213yesmaximal, complemented
15C61×#3362C211yesmaximal, characteristic, complemented
16C2×#41×D61—21no—
17C2×SD16×D61C2213yes—
18Q8×Dic122—21no—
19Q8×Dic122C2213yescomplemented
20Q8×Dic122C2211yescharacteristic, complemented
21#16840492—23no—
22#16840492C2213yescomplemented
23#16840434C2211yescharacteristic
24#16840433C2213yes—
25#16840432C2213yes—
26#16840431—21no—
27#16840430—23no—
28C4⋊Q8×D61—23no—
29C4⋊Q8×D61C2213yes—
30#127×D61C2213yes—
31#121×D61—23no—
32C4×Q8×D61—21no—
33C4×Q8×D61C2211yescharacteristic
34C4×Q8×D61C2213yes—
35C4×D4×D61C2213yes—
36#16840110C2213yescomplemented
37#16840109C2213yescomplemented
38#16840108C2213yescomplemented
39#16840107C2216yescomplemented
40#16840107C2216yescomplemented
41#16840053C2213yescomplemented
42#16840052C2213yescomplemented
43#16840051C2216yescomplemented
44#16840050C2216yescomplemented
45#16840049C2213yescomplemented
46#16839963C2213yescomplemented
47#16839962C2213yescomplemented
48#16839961C2216yescomplemented
49#16839960C2216yescomplemented
50#16839959C2213yescomplemented
51Q8×#92418C2211yescharacteristic, complemented
52C8×Dic122C2213yescomplemented
53#16839886C2213yescomplemented
54#16839853C2213yescomplemented
55C4×#11758190C2213yescomplemented
56SD16×Dic61C2213yescomplemented
57C4×#11758201C2213yescomplemented
58#16839849C2213yescomplemented
59C4×#11758200C2213yescomplemented
60#16839847C2213yescomplemented
61#106×D61C2213yes—
62C4⋊C8×D61C2213yes—
63#103×D61C2211yescharacteristic
64#102×D61C2216yes—
65#102×D61C2213yes—
66C4×C8×D61C2213yes—
67Q82×C61C2211yescharacteristic, complemented
68C61×#550C2211yescharacteristic
69C61×#486C2213yescomplemented
70C61×#476C2213yescomplemented
71C61×#460C2213yescomplemented
72C8×Q8×C61C2211yescharacteristic, complemented
73C4×SD16×C61C2213yescomplemented
74C2×#11758226—21no—
75C2×#11758222—23no—
76C2×#11758224—23no—
77#41×D61—48no—
78C2×Q8×D61—23no—
79C2×Q8×D61D411yescharacteristic
80C2×Q8×D61C2311yescharacteristic
81C2×D4×D61—23no—
82C2×D4×D61—23no—
83C2×D4×D61C2313yes—
84C22×C4×D61—23no—
85C2×#11758190C2313yes—
86C2×#11758200C2313yes—
87C2×#11758201C2313yes—
88SD16×D61—212no—
89C2×C8×D61C2311yescharacteristic
90C2×C8×D61C2313yes—
91#14519777—23no—
92#14519777C2313yes—
93#14519776—23no—
94#14519776—23no—
95#14519776—26no—
96#14519776—23no—
97#14519776C2316yes—
98#14519775—26no—
99#14519775—23no—
100#14519774C2313yes—
101#14519773—23no—
102#14519773—23no—
103#14519773C2316yes—
104#14519752—23no—
105#14519751—23no—
106#14519750—26no—
107#14519749—26no—
108#14519748—23no—
109Q8×Dic61—21no—
110Q8×Dic61D411yescharacteristic
111Q8×Dic61C2313yes—
112C4×Dic122—23no—
113C4×Dic122—23no—
114C4×Dic122C2313yes—
115C4×Dic122C2313yes—
116#14519742—23no—
117#14519742—23no—
118#14519742D413yes—
119#14519742C2313yes—
120C4×D244C2313yes—
121D4×Dic61C2313yes—
122#14519739C2313yes—
123C4×C22⋊D61—23no—
124#14519737—23no—
125C4⋊C4×D61—23no—
126C4⋊C4×D61—23no—
127C4⋊C4×D61D413yes—
128C4⋊C4×D61C2311yescharacteristic
129C4⋊C4×D61C2313yes—
130C22⋊C4×D61—23no—
131C42×D61—23no—
132C42×D61C2313yes—
133#14519709C2313yes—
134#14519708C2313yes—
135#14519708C2313yes—
136C4⋊#92418C2313yes—
137#14519706C2316yes—
138#14519702C2311yescharacteristic
139#14519701C2311yescharacteristic
140#14519700C2311yescharacteristic
141#14519699C2316yes—
142#14519699C2313yes—
143#14519698C2316yes—
144#14519698C2313yes—
145#14519697C2316yes—
146#14519697C2313yes—
147C4×#92418C2313yes—
148C8×Dic61C2313yes—
149C8×Dic61C2313yes—
150C2×#41×C61—21no—
151C2×SD16×C61C2313yes—
152C4⋊Q8×C61—23no—
153C4⋊Q8×C61C2313yes—
154#127×C61C2313yes—
155#121×C61—23no—
156C4×Q8×C61—21no—
157C4×Q8×C61C2313yes—
158C4×Q8×C61C2311yescharacteristic
159C4×D4×C61C2313yes—
160#106×C61C2313yes—
161C4⋊C8×C61C2313yes—
162#103×C61C2311yescharacteristic
163#102×C61C2316yes—
164#102×C61C2313yes—
165C4×C8×C61C2313yes—
166C23×D61—23no—
167#41×C61—44no—
168C2×Q8×C61—23no—
169C2×Q8×C61C2×D411yescharacteristic
170C2×Q8×C61C2411yescharacteristic
171C2×D4×C61—23no—
172C2×D4×C61—23no—
173C2×D4×C61C2413yes—
174C22×C4×C61—23no—
175C2×C22⋊D61—26no—
176C2×C22⋊D61—23no—
177C2×C22⋊D61—26no—
178C22×Dic61—23no—
179#11758226—44no—
180Q8×D61—22no—
181Q8×D61—86no—
182Q8×D61SD1618yescomplemented
183Q8×D61#4114yescomplemented
184#11758224—412no—
185D4×D61—26no—
186D4×D61—26no—
187D4×D61—412no—
188D4×D61—86no—
189#11758222—412no—
190C2×D244—23no—
191C2×D244—23no—
192C2×D244C2413yes—
193C2×C4×D61—21no—
194C2×C4×D61—21no—
195C2×C4×D61—23no—
196C2×C4×D61—23no—
197C2×C4×D61—23no—
198C2×C4×D61—412no—
199C2×C4×D61C2×D413yes—
200C2×C4×D61C2×D413yes—
201C2×C4×D61C2411yescharacteristic
202C2×Dic122—21no—
203C2×Dic122—23no—
204C2×Dic122—26no—
205C2×Dic122C2×D413yes—
206C2×Dic122C2411yescharacteristic
207SD16×C61—26no—
208C2×C8×C61C2411yescharacteristic
209C2×C8×C61C2413yes—
210C4⋊C4×C61—23no—
211C4⋊C4×C61—23no—
212C4⋊C4×C61C2×D413yes—
213C4⋊C4×C61C2413yes—
214C4⋊C4×C61C2411yescharacteristic
215C22⋊C4×C61—23no—
216C42×C61—23no—
217C42×C61C2413yes—
218C22⋊Dic61—23no—
219#11758201—26no—
220#11758200—26no—
221#11758198—23no—
222#11758198—23no—
223C4⋊Dic61—23no—
224C4⋊Dic61—23no—
225C4⋊Dic61C2×D413yes—
226C4⋊Dic61C2413yes—
227C4⋊Dic61C2411yescharacteristic
228#11758196—23no—
229#11758196—26no—
230#11758196—23no—
231#11758196—21no—
232#11758196—23no—
233#11758196C2×D416yes—
234C4×Dic61—23no—
235C4×Dic61—23no—
236C4×Dic61—23no—
237C4×Dic61C2×D413yes—
238C4×Dic61C2×D413yes—
239C4×Dic61C2413yes—
240C2×#92418C2411yescharacteristic
241C2×#92418C2413yes—
242#11758190—26no—
243C8×D61—26no—
244C8×D61—22no—
245C23×C61—23no—
246C22×D61—23no—
247C22×D61—46no—
248C22×D61—46no—
249C22×D61—46no—
250C22×D61C22×D411yescharacteristic
251Q8×C61—21no—
252Q8×C61—83no—
253Q8×C61C2×SD1614yescomplemented
254Q8×C61C2×#4112yescomplemented
255D4×C61—23no—
256D4×C61—23no—
257D4×C61—46no—
258D4×C61—83no—
259C2×C4×C61—21no—
260C2×C4×C61—23no—
261C2×C4×C61—21no—
262C2×C4×C61—46no—
263C2×C4×C61C22×D413yes—
264C2×C4×C61C22×D413yes—
265C2×C4×C61C2511yescharacteristic
266C22⋊D61—46no—
267C22⋊D61—412no—
268C22⋊D61—86no—
269C2×Dic61—21no—
270C2×Dic61—23no—
271C2×Dic61—21no—
272C2×Dic61—46no—
273C2×Dic61C22×D413yes—
274C2×Dic61C22×D411yescharacteristic
275C2×Dic61C22×D413yes—
276D244—23no—
277D244—23no—
278D244—46no—
279D244—83no—
280C4×D61—24no—
281C4×D61—26no—
282C4×D61—42no—
283C4×D61—46no—
284C4×D61—46no—
285C4×D61—46no—
286C4×D61C2×SD16112yes—
287C4×D61C2×#4112yes—
288C4×D612−1+412yes—
289Dic122—21no—
290Dic122—24no—
291Dic122—86no—
292Dic122—83no—
293Dic122C2×SD16112yescomplemented
294Dic122C2×#4112yescomplemented
295C8×C61—23no—
296C8×C61—21no—
297#92418—23no—
298#92418—21no—
299C2×#3362—611nomaximal, Sylow
300C22×C61—43no—
301C22×C61—43no—
302C22×C61—43no—
303C22×C61C23×D411yescharacteristic
304C2×D61—43no—
305C2×D61—43no—
306C2×D61—43no—
307C2×D61—812no—
308C2×D61#54812yes—
309C2×D61C22×SD1612yes—
310C2×D61#57612yes—
311C4×C61—22no—
312C4×C61—23no—
313C4×C61—41no—
314C4×C61—43no—
315C4×C61C22×SD1616yes—
316C4×C61C22×#4111yescharacteristic
317C4×C61C2×2−1+411yescharacteristic
318Dic61—22no—
319Dic61—23no—
320Dic61—41no—
321Dic61—43no—
322Dic61#54812yes—
323Dic61C22×SD1616yes—
324C2×Q82—611no—
325C2×#550—611no—
326#3362—613no—
327#3362—613no—
328#3362—613no—
329#3362—611no—
330#3362—613no—
331#3362—611no—
332#3362—611no—
333#3362D6111yescharacteristic, direct product factor
334C2×#486—613no—
335C2×#476—613no—
336C2×#460—613no—
337C2×C8×Q8—611no—
338C2×C4×SD16—613no—
339C2×C61—83no—
340C2×C61C2×#54811yescharacteristic
341C2×C61C23×SD1611yescharacteristic
342C2×C61C2×#57611yescharacteristic
343D61—83no—
344D61#336214yesdirect product factor
345C22×#41—1221no—
346C22×SD16—613no—
347Q82—613no—
348Q82—613no—
349Q82—611no—
350Q82—1221no—
351Q82—1223no—
352Q82C2×D6111yescharacteristic, complemented
353#550—613no—
354#550—611no—
355#550—613no—
356#550—1223no—
357#550—1221no—
358#550C2×D6111yescharacteristic
359C2×C4⋊Q8—613no—
360C2×C4⋊Q8—1223no—
361C2×#127—613no—
362C2×#121—1223no—
363C2×C4×Q8—611no—
364C2×C4×Q8—613no—
365C2×C4×Q8—1221no—
366C2×C4×D4—613no—
367#486—613no—
368#486—613no—
369#486—616no—
370#486—616no—
371#486—613no—
372#486C2×D6113yescomplemented
373#476—613no—
374#476—613no—
375#476—616no—
376#476—613no—
377#476—616no—
378#476C2×D6113yescomplemented
379#460—616no—
380#460—613no—
381#460—613no—
382#460—613no—
383#460—616no—
384#460C2×D6113yescomplemented
385C8×Q8—611no—
386C8×Q8—613no—
387C8×Q8—613no—
388C8×Q8C2×D6111yescharacteristic, complemented
389C4×SD16—613no—
390C4×SD16—613no—
391C4×SD16—613no—
392C4×SD16—613no—
393C4×SD16—613no—
394C4×SD16—613no—
395C4×SD16—613no—
396C4×SD16C2×D6113yescomplemented
397C2×#106—613no—
398C2×C4⋊C8—613no—
399C2×#103—611no—
400C2×#102—613no—
401C2×#102—616no—
402C2×C4×C8—613no—
403C61C2×#336211yescharacteristic, complemented, Sylow
404C2×#41—21no—
405C2×#41—1223no—
406C2×#41—1223no—
407C2×#41—1221no—
408C2×#41—2448no—
409C22×Q8—611no—
410C22×Q8—611no—
411C22×Q8—1223no—
412C22×D4—613no—
413C22×D4—1223no—
414C22×D4—1223no—
415C23×C4—1223no—
416C2×SD16—613no—
417C2×SD16—613no—
418C2×SD16—613no—
419C2×SD16—12212no—
420C2×SD16C22×D6113yes—
421C22×C8—613no—
422C22×C8—611no—
423C4⋊Q8—23no—
424C4⋊Q8—616no—
425C4⋊Q8—613no—
426C4⋊Q8—1223no—
427C4⋊Q8—1223no—
428C4⋊Q8—1223no—
429C4⋊Q8—1226no—
430C4⋊Q8—1223no—
431C4⋊Q8—1226no—
432C4⋊Q8—1223no—
433C4⋊Q8C22×D6113yes—
434#127—616no—
435#127—613no—
436#127—1223no—
437#127—1223no—
438#127C22×D6113yes—
439#121—23no—
440#121—1226no—
441#121—1223no—
442#121—1223no—
443#121—1226no—
444#121—1223no—
445C4×Q8—21no—
446C4×Q8—611no—
447C4×Q8—613no—
448C4×Q8—613no—
449C4×Q8—613no—
450C4×Q8—613no—
451C4×Q8—613no—
452C4×Q8—1223no—
453C4×Q8—1223no—
454C4×Q8—1223no—
455C4×Q8—1223no—
456C4×Q8—1221no—
457C4×Q8C22×D6113yes—
458C4×Q8C22×D6111yescharacteristic
459C4×D4—613no—
460C4×D4—613no—
461C4×D4—613no—
462C4×D4—1223no—
463C4×D4—1223no—
464C4×D4C22×D6113yes—
465C2×C4⋊C4—613no—
466C2×C4⋊C4—611no—
467C2×C4⋊C4—613no—
468C2×C4⋊C4—1223no—
469C2×C4⋊C4—1223no—
470C2×C22⋊C4—1223no—
471C2×C42—613no—
472C2×C42—1223no—
473#106—613no—
474#106—613no—
475#106—613no—
476#106C22×D6113yes—
477C4⋊C8—616no—
478C4⋊C8—613no—
479C4⋊C8C22×D6113yes—
480#103—611no—
481#103—611no—
482#103—611no—
483#103C22×D6111yescharacteristic
484#102—613no—
485#102—616no—
486#102—613no—
487#102—616no—
488#102—613no—
489#102—616no—
490#102C22×D6113yes—
491#102C22×D6116yes—
492C4×C8—613no—
493C4×C8—613no—
494C4×C8—613no—
495C4×C8C22×D6113yes—
496C24—1223no—
497#41—44no—
498#41—24412no—
499#41—24412no—
500#41—2444no—
501C2×Q8—23no—
502C2×Q8—618no—
503C2×Q8—613no—
504C2×Q8—614no—
505C2×Q8—611no—
506C2×Q8—1221no—
507C2×Q8—1226no—
508C2×Q8—1222no—
509C2×Q8—1223no—
510C2×Q8—4886no—
511C2×Q8D4×D6111yescharacteristic
512C2×Q8C23×D6111yescharacteristic
513C2×D4—23no—
514C2×D4—23no—
515C2×D4—613no—
516C2×D4—1226no—
517C2×D4—1226no—
518C2×D4—1226no—
519C2×D4—1223no—
520C2×D4—1226no—
521C2×D4—1223no—
522C2×D4—1223no—
523C2×D4—4886no—
524C2×D4—24412no—
525C2×D4C23×D6113yes—
526C22×C4—23no—
527C22×C4—613no—
528C22×C4—611no—
529C22×C4—613no—
530C22×C4—1223no—
531C22×C4—1221no—
532C22×C4—1223no—
533C22×C4—1221no—
534C22×C4—1223no—
535C22×C4—1223no—
536C22×C4—24412no—
537SD16—26no—
538SD16—1226no—
539SD16—1226no—
540SD16—1226no—
541C2×C8—611no—
542C2×C8—613no—
543C2×C8—1222no—
544C2×C8—1226no—
545C2×C8C23×D6111yescharacteristic
546C2×C8C23×D6113yes—
547C4⋊C4—23no—
548C4⋊C4—23no—
549C4⋊C4—616no—
550C4⋊C4—613no—
551C4⋊C4—613no—
552C4⋊C4—611no—
553C4⋊C4—1221no—
554C4⋊C4—1223no—
555C4⋊C4—1223no—
556C4⋊C4—1226no—
557C4⋊C4—1223no—
558C4⋊C4—1223no—
559C4⋊C4—1223no—
560C4⋊C4D4×D6113yes—
561C4⋊C4C23×D6113yes—
562C4⋊C4C23×D6111yescharacteristic
563C22⋊C4—23no—
564C22⋊C4—1223no—
565C22⋊C4—1223no—
566C22⋊C4—1223no—
567C42—23no—
568C42—613no—
569C42—613no—
570C42—613no—
571C42—1223no—
572C42—1223no—
573C42—1223no—
574C42C23×D6113yes—
575C23—23no—
576C23—611no—
577C23—1223no—
578C23—2446no—
579C23—2446no—
580C23—2446no—
581Q8—21no—
582Q8—83no—
583Q8—6112no—
584Q8—612no—
585Q8—1224no—
586Q8—1221no—
587Q8—4883no—
588Q8—4886no—
589Q8SD16×D6114yescomplemented
590Q8#41×D6112yescomplemented
591D4—23no—
592D4—23no—
593D4—46no—
594D4—83no—
595D4—1223no—
596D4—1223no—
597D4—4883no—
598D4—4886no—
599D4—2446no—
600D4—24412no—
601D4—2446no—
602C2×C4—21no—
603C2×C4—23no—
604C2×C4—21no—
605C2×C4—46no—
606C2×C4—611no—
607C2×C4—613no—
608C2×C4—6112no—
609C2×C4—612no—
610C2×C4—612no—
611C2×C4—613no—
612C2×C4—1226no—
613C2×C4—1224no—
614C2×C4—1221no—
615C2×C4—1223no—
616C2×C4—1221no—
617C2×C4—2446no—
618C2×C4—2442no—
619C2×C4—2446no—
620C2×C4—2446no—
621C2×C4—2446no—
622C2×C4C2×D4×D6113yes—
623C2×C4C2×D4×D6113yes—
624C2×C4C24×D6111yescharacteristic
625C8—23no—
626C8—21no—
627C8—1223no—
628C8—1221no—
629C22—43no—
630C22—43no—
631C22—43no—
632C22—612no—
633C22—612no—
634C22—612no—
635C22—48812no—
636C22—2443no—
637C22—2443no—
638C22—2443no—
639C22C22×D4×D6111yescharacteristic
640C4—22no—
641C4—23no—
642C4—41no—
643C4—43no—
644C4—616no—
645C4—612no—
646C4—1222no—
647C4—1223no—
648C4—2441no—
649C4—2443no—
650C4C2×SD16×D6116yes—
651C4C2×#41×D6111yescharacteristic
652C42−1+4×D6111yescharacteristic
653C2—83no—
654C2—614no—
655C2—4883no—
656C2#548×D6111yescharacteristic
657C2C22×SD16×D6111yescharacteristic
658C2#576×D6111yescharacteristic
subgroupG/H
centerC22C22×D4×D61
commutator, cosocleC2×C4×C61C25
FrattiniC2×C4C24×D61
FittingC61×#3362C2
socleC22×C61C23×D4
prime radical#550×D61C2
hypercenter#3362D61
hypocenterC61C2×#3362
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C4×Q8 ◁ #3362

Complements

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

NH ≅ G/Ntype
1. Q82×D61C2semidirect
3. #21458491C2semidirect
4. #21458490C2semidirect
5. #21458489C2semidirect
6. #21458488C2semidirect
7. #21458487C2semidirect
8. #21458486C2semidirect
9. #21458485C2semidirect
10. #486×D61C2semidirect
11. #476×D61C2semidirect
12. #460×D61C2semidirect
13. C8×Q8×D61C2semidirect
14. C4×SD16×D61C2semidirect
15. C61×#3362C2semidirect
19. Q8×Dic122C22semidirect
20. Q8×Dic122C22semidirect
22. #16840492C22semidirect
36. #16840110C22semidirect
37. #16840109C22semidirect
38. #16840108C22semidirect
39. #16840107C22semidirect
40. #16840107C22semidirect
41. #16840053C22semidirect
42. #16840052C22semidirect
43. #16840051C22semidirect
44. #16840050C22semidirect
45. #16840049C22semidirect
46. #16839963C22semidirect
47. #16839962C22semidirect
48. #16839961C22semidirect
49. #16839960C22semidirect
50. #16839959C22semidirect
51. Q8×#92418C22semidirect
52. C8×Dic122C22semidirect
53. #16839886C22semidirect
54. #16839853C22semidirect
55. C4×#11758190C22semidirect
56. SD16×Dic61C22semidirect
57. C4×#11758201C22semidirect
58. #16839849C22semidirect
59. C4×#11758200C22semidirect
60. #16839847C22semidirect
67. Q82×C61C22semidirect
69. C61×#486C22semidirect
70. C61×#476C22semidirect
71. C61×#460C22semidirect
72. C8×Q8×C61C22semidirect
73. C4×SD16×C61C22semidirect
182. Q8×D61SD16semidirect
183. Q8×D61#41semidirect
253. Q8×C61C2×SD16semidirect
254. Q8×C61C2×#41semidirect
293. Dic122C2×SD16semidirect
294. Dic122C2×#41semidirect
333. #3362D61direct (N × H)
344. D61#3362direct (N × H)
352. Q82C2×D61semidirect
372. #486C2×D61semidirect
378. #476C2×D61semidirect
384. #460C2×D61semidirect
388. C8×Q8C2×D61semidirect
396. C4×SD16C2×D61semidirect
403. C61C2×#3362semidirect
589. Q8SD16×D61semidirect
590. Q8#41×D61semidirect

Maximal subgroups

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

subgroupindex
Q82×D612
#550×D612
#214584912
#214584902
#214584892
#214584882
#214584872
#214584862
#214584852
#486×D612
#476×D612
#460×D612
C8×Q8×D612
C4×SD16×D612
C61×#33622
C2×#336261