Finite Group Databasebeta

← D61×#3392 #30560908

Subgroups — D61×#3392 #30560908

Order 15616.

2238 conjugacy classes of subgroups.

The subgroup lattice has 662 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#557×D61C211yesmaximal, characteristic, complemented
2#550×D61C211yesmaximal, characteristic
3#21458801C211yesmaximal, characteristic, complemented
4#21458800C211yesmaximal, characteristic, complemented
5#21458799C213yesmaximal, complemented
6#21458798C213yesmaximal, complemented
7#21458797C213yesmaximal, complemented
8#21458796C211yesmaximal, characteristic, complemented
9#21458795C213yesmaximal, complemented
10#492×D61C213yesmaximal, complemented
11#460×D61C213yesmaximal, complemented
12#461×D61C213yesmaximal, complemented
13C8×Q8×D61C211yesmaximal, characteristic, complemented
14C4×SD16×D61C213yesmaximal, complemented
15C61×#3392C211yesmaximal, characteristic, complemented
16C2×#41×D61—21no—
17C2×SD16×D61C2213yes—
18C2×SD16×D61C2216yes—
19#16840491C2211yescharacteristic, complemented
20#16840490C2213yescomplemented
21#16840489—23no—
22#16840488—21no—
23#16840487C2213yescomplemented
24#16840434C2211yescharacteristic
25#16840433C2213yes—
26#16840432C2213yes—
27#16840431—21no—
28#16840430—23no—
29C4⋊Q8×D61C2213yes—
30#127×D61C2213yes—
31#125×D61—23no—
32#121×D61—23no—
33C4×Q8×D61—21no—
34C4×Q8×D61C2211yescharacteristic
35C4×Q8×D61C2213yes—
36C4×D4×D61C2213yes—
37#16840136C2213yescomplemented
38#16840135C2213yescomplemented
39#16840134C2213yescomplemented
40#16840133C2216yescomplemented
41#16840133C2216yescomplemented
42#16839968C2213yescomplemented
43#16839967C2213yescomplemented
44#16839966C2216yescomplemented
45#16839965C2216yescomplemented
46#16839964C2213yescomplemented
47#16839963C2213yescomplemented
48#16839962C2213yescomplemented
49#16839961C2216yescomplemented
50#16839960C2216yescomplemented
51#16839959C2213yescomplemented
52Q8×#92418C2211yescharacteristic, complemented
53C8×Dic122C2213yescomplemented
54#16839886C2213yescomplemented
55#16839853C2213yescomplemented
56C4×#11758190C2213yescomplemented
57SD16×Dic61C2213yescomplemented
58C4×#11758201C2213yescomplemented
59#16839849C2213yescomplemented
60C4×#11758200C2213yescomplemented
61#16839847C2213yescomplemented
62#106×D61C2211yescharacteristic
63C4⋊C8×D61C2213yes—
64#103×D61C2213yes—
65#102×D61C2213yes—
66C4×C8×D61C2213yes—
67C61×#557C2211yescharacteristic, complemented
68C61×#550C2211yescharacteristic
69C61×#492C2213yescomplemented
70C61×#461C2213yescomplemented
71C61×#460C2213yescomplemented
72C8×Q8×C61C2211yescharacteristic, complemented
73C4×SD16×C61C2213yescomplemented
74C2×#11758226—21no—
75C2×#11758222—23no—
76C2×#11758224—23no—
77#41×D61—48no—
78C2×Q8×D61D411yescharacteristic
79C2×Q8×D61C2313yes—
80C2×D4×D61—23no—
81C2×D4×D61—23no—
82C2×D4×D61C2313yes—
83C22×C4×D61—23no—
84C2×#11758190C2313yes—
85C2×#11758190C2316yes—
86C2×#11758200C2313yes—
87C2×#11758200C2316yes—
88C2×#11758201C2313yes—
89C2×#11758201C2316yes—
90SD16×D61—212no—
91SD16×D61—412no—
92C2×C8×D61C2313yes—
93C2×C8×D61C2311yescharacteristic
94#14519777C2313yes—
95#14519776—23no—
96#14519776—23no—
97#14519775C2316yes—
98#14519774C2313yes—
99#14519773—23no—
100#14519773—23no—
101#14519773C2316yes—
102#14519769—23no—
103#14519768—23no—
104#14519768—23no—
105#14519767—26no—
106#14519767—26no—
107#14519752—23no—
108#14519751—23no—
109#14519750—26no—
110#14519749—26no—
111#14519748—23no—
112Q8×Dic61—21no—
113Q8×Dic61D411yescharacteristic
114Q8×Dic61C2313yes—
115C4×Dic122—23no—
116C4×Dic122—23no—
117C4×Dic122C2313yes—
118C4×Dic122C2313yes—
119#14519742—23no—
120#14519742—23no—
121#14519742D413yes—
122#14519742C2313yes—
123C4×D244C2313yes—
124D4×Dic61C2313yes—
125#14519739C2313yes—
126C4×C22⋊D61—23no—
127#14519737—23no—
128C4⋊C4×D61—23no—
129C4⋊C4×D61—23no—
130C4⋊C4×D61—23no—
131C4⋊C4×D61D413yes—
132C4⋊C4×D61C2311yescharacteristic
133C4⋊C4×D61C2311yescharacteristic
134C22⋊C4×D61—23no—
135C42×D61—23no—
136C42×D61C2313yes—
137#14519709C2311yescharacteristic
138#14519708C2311yescharacteristic
139#14519708C2311yescharacteristic
140C4⋊#92418C2313yes—
141#14519706C2316yes—
142#14519702C2313yes—
143#14519701C2313yes—
144#14519700C2313yes—
145#14519699C2313yes—
146#14519698C2313yes—
147#14519697C2313yes—
148C4×#92418C2313yes—
149C8×Dic61C2313yes—
150C8×Dic61C2313yes—
151C2×#41×C61—21no—
152C2×SD16×C61C2313yes—
153C2×SD16×C61C2316yes—
154C4⋊Q8×C61C2313yes—
155#127×C61C2313yes—
156#125×C61—23no—
157#121×C61—23no—
158C4×Q8×C61—21no—
159C4×Q8×C61C2313yes—
160C4×Q8×C61C2311yescharacteristic
161C4×D4×C61C2313yes—
162#106×C61C2311yescharacteristic
163C4⋊C8×C61C2313yes—
164#103×C61C2313yes—
165#102×C61C2313yes—
166C4×C8×C61C2313yes—
167C23×D61—23no—
168#41×C61—44no—
169C2×Q8×C61C2×D411yescharacteristic
170C2×Q8×C61C2413yes—
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—26no—
181Q8×D61—26no—
182Q8×D61SD1618yescomplemented
183#11758224—412no—
184D4×D61—26no—
185D4×D61—26no—
186D4×D61—412no—
187D4×D61—86no—
188#11758222—412no—
189C2×D244—23no—
190C2×D244—23no—
191C2×D244C2413yes—
192C2×C4×D61—21no—
193C2×C4×D61—23no—
194C2×C4×D61—21no—
195C2×C4×D61—23no—
196C2×C4×D61—23no—
197C2×C4×D61—412no—
198C2×C4×D61C2×D413yes—
199C2×C4×D61C2×D413yes—
200C2×C4×D61C2411yescharacteristic
201C2×Dic122—23no—
202C2×Dic122C2×D413yes—
203C2×Dic122C2413yes—
204SD16×C61—26no—
205SD16×C61—46no—
206C2×C8×C61C2413yes—
207C2×C8×C61C2411yescharacteristic
208C4⋊C4×C61—23no—
209C4⋊C4×C61—23no—
210C4⋊C4×C61—23no—
211C4⋊C4×C61C2×D413yes—
212C4⋊C4×C61C2411yescharacteristic
213C4⋊C4×C61C2411yescharacteristic
214C22⋊C4×C61—23no—
215C42×C61—23no—
216C42×C61C2413yes—
217C22⋊Dic61—23no—
218#11758201—26no—
219#11758201—46no—
220#11758200—26no—
221#11758200—46no—
222#11758198—23no—
223#11758198—23no—
224C4⋊Dic61—23no—
225C4⋊Dic61—23no—
226C4⋊Dic61—23no—
227C4⋊Dic61C2×D413yes—
228C4⋊Dic61C2411yescharacteristic
229C4⋊Dic61C2411yescharacteristic
230#11758196—23no—
231#11758196—26no—
232#11758196—23no—
233#11758196—26no—
234#11758196—21no—
235#11758196—21no—
236#11758196C2×D416yes—
237C4×Dic61—23no—
238C4×Dic61—23no—
239C4×Dic61—23no—
240C4×Dic61C2×D413yes—
241C4×Dic61C2×D413yes—
242C4×Dic61C2413yes—
243C2×#92418C2413yes—
244C2×#92418C2411yescharacteristic
245#11758190—26no—
246#11758190—46no—
247C8×D61—26no—
248C8×D61#4114yescomplemented
249C23×C61—23no—
250C22×D61—23no—
251C22×D61—46no—
252C22×D61—46no—
253C22×D61—46no—
254C22×D61C22×D411yescharacteristic
255Q8×C61—23no—
256Q8×C61—23no—
257Q8×C61C2×SD1614yescomplemented
258D4×C61—23no—
259D4×C61—23no—
260D4×C61—46no—
261D4×C61—83no—
262C2×C4×C61—21no—
263C2×C4×C61—23no—
264C2×C4×C61—21no—
265C2×C4×C61—46no—
266C2×C4×C61C22×D413yes—
267C2×C4×C61C22×D413yes—
268C2×C4×C61C2511yescharacteristic
269C22⋊D61—46no—
270C22⋊D61—412no—
271C22⋊D61—86no—
272C2×Dic61—21no—
273C2×Dic61—23no—
274C2×Dic61—21no—
275C2×Dic61—46no—
276C2×Dic61C22×D413yes—
277C2×Dic61C22×D411yescharacteristic
278C2×Dic61C22×D413yes—
279D244—23no—
280D244—23no—
281D244—46no—
282D244—83no—
283C4×D61—26no—
284C4×D61—46no—
285C4×D61—42no—
286C4×D61—42no—
287C4×D61—46no—
288C4×D61—46no—
289C4×D61C2×SD16112yes—
290C4×D61C2×#4112yes—
291C4×D61H(5,𝔽2)12yes—
292Dic122—23no—
293Dic122—23no—
294Dic122—46no—
295Dic122C2×SD16112yescomplemented
296C8×C61—23no—
297C8×C61C2×#4112yescomplemented
298#92418—23no—
299#92418C2×#4112yescomplemented
300C2×#3392—611nomaximal, Sylow
301C22×C61—43no—
302C22×C61—43no—
303C22×C61—43no—
304C22×C61C23×D411yescharacteristic
305C2×D61—43no—
306C2×D61—43no—
307C2×D61—43no—
308C2×D61—812no—
309C2×D61#55012yes—
310C2×D61C22×SD1612yes—
311C2×D61#57712yes—
312C4×C61—23no—
313C4×C61—43no—
314C4×C61—41no—
315C4×C61—41no—
316C4×C61C22×SD1616yes—
317C4×C61C22×#4111yescharacteristic
318C4×C61C2×H(5,𝔽2)11yescharacteristic
319Dic61—23no—
320Dic61—43no—
321Dic61—41no—
322Dic61—41no—
323Dic61#55012yes—
324Dic61C22×SD1616yes—
325C2×#557—611no—
326C2×#550—611no—
327#3392—613no—
328#3392—613no—
329#3392—613no—
330#3392—611no—
331#3392—613no—
332#3392—611no—
333#3392—611no—
334#3392D6111yescharacteristic, direct product factor
335C2×#492—613no—
336C2×#460—613no—
337C2×#461—613no—
338C2×C8×Q8—611no—
339C2×C4×SD16—613no—
340C2×C61—83no—
341C2×C61C2×#55011yescharacteristic
342C2×C61C23×SD1611yescharacteristic
343C2×C61C2×#57711yescharacteristic
344D61—83no—
345D61#339214yesdirect product factor
346C22×#41—1221no—
347C22×SD16—613no—
348C22×SD16—616no—
349#557—613no—
350#557—613no—
351#557—611no—
352#557—1221no—
353#557—1223no—
354#557C2×D6111yescharacteristic, complemented
355#550—613no—
356#550—611no—
357#550—613no—
358#550—1223no—
359#550—1221no—
360#550C2×D6111yescharacteristic
361C2×C4⋊Q8—613no—
362C2×#127—613no—
363C2×#125—1223no—
364C2×#121—1223no—
365C2×C4×Q8—611no—
366C2×C4×Q8—613no—
367C2×C4×Q8—1221no—
368C2×C4×D4—613no—
369#492—616no—
370#492—616no—
371#492—613no—
372#492—613no—
373#492—613no—
374#492C2×D6113yescomplemented
375#461—616no—
376#461—613no—
377#461—613no—
378#461—613no—
379#461—616no—
380#461C2×D6113yescomplemented
381#460—616no—
382#460—613no—
383#460—613no—
384#460—613no—
385#460—616no—
386#460C2×D6113yescomplemented
387C8×Q8—611no—
388C8×Q8—613no—
389C8×Q8—613no—
390C8×Q8C2×D6111yescharacteristic, complemented
391C4×SD16—613no—
392C4×SD16—613no—
393C4×SD16—613no—
394C4×SD16—613no—
395C4×SD16—613no—
396C4×SD16—613no—
397C4×SD16—613no—
398C4×SD16C2×D6113yescomplemented
399C2×#106—611no—
400C2×C4⋊C8—613no—
401C2×#103—613no—
402C2×#102—613no—
403C2×C4×C8—613no—
404C61C2×#339211yescharacteristic, complemented, Sylow
405C2×#41—21no—
406C2×#41—1223no—
407C2×#41—1223no—
408C2×#41—1221no—
409C2×#41—2448no—
410C22×Q8—611no—
411C22×Q8—613no—
412C22×D4—613no—
413C22×D4—1223no—
414C22×D4—1223no—
415C23×C4—1223no—
416C2×SD16—613no—
417C2×SD16—613no—
418C2×SD16—616no—
419C2×SD16—616no—
420C2×SD16—613no—
421C2×SD16—616no—
422C2×SD16—12212no—
423C2×SD16—24412no—
424C2×SD16C22×D6113yes—
425C2×SD16C22×D6116yes—
426C22×C8—611no—
427C22×C8—613no—
428C4⋊Q8—616no—
429C4⋊Q8—613no—
430C4⋊Q8—1223no—
431C4⋊Q8—1223no—
432C4⋊Q8C22×D6113yes—
433#127—616no—
434#127—613no—
435#127—1223no—
436#127—1223no—
437#127C22×D6113yes—
438#125—23no—
439#125—1223no—
440#125—1226no—
441#125—1226no—
442#125—1223no—
443#125—1223no—
444#121—23no—
445#121—1226no—
446#121—1223no—
447#121—1223no—
448#121—1226no—
449#121—1223no—
450C4×Q8—21no—
451C4×Q8—611no—
452C4×Q8—613no—
453C4×Q8—613no—
454C4×Q8—613no—
455C4×Q8—613no—
456C4×Q8—613no—
457C4×Q8—1223no—
458C4×Q8—1223no—
459C4×Q8—1223no—
460C4×Q8—1223no—
461C4×Q8—1221no—
462C4×Q8C22×D6113yes—
463C4×Q8C22×D6111yescharacteristic
464C4×D4—613no—
465C4×D4—613no—
466C4×D4—613no—
467C4×D4—1223no—
468C4×D4—1223no—
469C4×D4C22×D6113yes—
470C2×C4⋊C4—613no—
471C2×C4⋊C4—611no—
472C2×C4⋊C4—611no—
473C2×C4⋊C4—1223no—
474C2×C4⋊C4—1223no—
475C2×C4⋊C4—1223no—
476C2×C22⋊C4—1223no—
477C2×C42—613no—
478C2×C42—1223no—
479#106—611no—
480#106—611no—
481#106—611no—
482#106C22×D6111yescharacteristic
483C4⋊C8—616no—
484C4⋊C8—613no—
485C4⋊C8C22×D6113yes—
486#103—613no—
487#103—613no—
488#103—613no—
489#103C22×D6113yes—
490#102—613no—
491#102—613no—
492#102—613no—
493#102C22×D6113yes—
494C4×C8—613no—
495C4×C8—613no—
496C4×C8—613no—
497C4×C8C22×D6113yes—
498C24—1223no—
499#41—44no—
500#41—24412no—
501#41—24412no—
502#41—2444no—
503C2×Q8—618no—
504C2×Q8—613no—
505C2×Q8—613no—
506C2×Q8—1223no—
507C2×Q8—1226no—
508C2×Q8—1226no—
509C2×Q8D4×D6111yescharacteristic
510C2×Q8C23×D6113yes—
511C2×D4—23no—
512C2×D4—23no—
513C2×D4—613no—
514C2×D4—1226no—
515C2×D4—1226no—
516C2×D4—1226no—
517C2×D4—1223no—
518C2×D4—1226no—
519C2×D4—1223no—
520C2×D4—1223no—
521C2×D4—4886no—
522C2×D4—24412no—
523C2×D4C23×D6113yes—
524C22×C4—23no—
525C22×C4—613no—
526C22×C4—611no—
527C22×C4—613no—
528C22×C4—1223no—
529C22×C4—1221no—
530C22×C4—1223no—
531C22×C4—1221no—
532C22×C4—1223no—
533C22×C4—1223no—
534C22×C4—24412no—
535SD16—26no—
536SD16—46no—
537SD16—1226no—
538SD16—1226no—
539SD16—1226no—
540SD16—2446no—
541SD16—2446no—
542SD16—2446no—
543C2×C8—611no—
544C2×C8—614no—
545C2×C8—613no—
546C2×C8—1226no—
547C2×C8C23×D6113yes—
548C2×C8C23×D6111yescharacteristic
549C4⋊C4—23no—
550C4⋊C4—23no—
551C4⋊C4—23no—
552C4⋊C4—616no—
553C4⋊C4—613no—
554C4⋊C4—611no—
555C4⋊C4—611no—
556C4⋊C4—1221no—
557C4⋊C4—1221no—
558C4⋊C4—1223no—
559C4⋊C4—1226no—
560C4⋊C4—1226no—
561C4⋊C4—1223no—
562C4⋊C4—1223no—
563C4⋊C4—1223no—
564C4⋊C4—1223no—
565C4⋊C4D4×D6113yes—
566C4⋊C4C23×D6111yescharacteristic
567C4⋊C4C23×D6111yescharacteristic
568C22⋊C4—23no—
569C22⋊C4—1223no—
570C22⋊C4—1223no—
571C22⋊C4—1223no—
572C42—23no—
573C42—613no—
574C42—613no—
575C42—613no—
576C42—1223no—
577C42—1223no—
578C42—1223no—
579C42C23×D6113yes—
580C23—23no—
581C23—611no—
582C23—1223no—
583C23—2446no—
584C23—2446no—
585C23—2446no—
586Q8—23no—
587Q8—23no—
588Q8—6112no—
589Q8—1223no—
590Q8—1223no—
591Q8—2446no—
592Q8SD16×D6114yescomplemented
593D4—23no—
594D4—23no—
595D4—46no—
596D4—83no—
597D4—1223no—
598D4—1223no—
599D4—4883no—
600D4—4886no—
601D4—2446no—
602D4—24412no—
603D4—2446no—
604C2×C4—21no—
605C2×C4—23no—
606C2×C4—21no—
607C2×C4—46no—
608C2×C4—611no—
609C2×C4—613no—
610C2×C4—6112no—
611C2×C4—612no—
612C2×C4—612no—
613C2×C4—613no—
614C2×C4—1226no—
615C2×C4—1221no—
616C2×C4—1223no—
617C2×C4—1221no—
618C2×C4—2446no—
619C2×C4—2442no—
620C2×C4—2442no—
621C2×C4—2446no—
622C2×C4—2446no—
623C2×C4—2446no—
624C2×C4C2×D4×D6113yes—
625C2×C4C2×D4×D6113yes—
626C2×C4C24×D6111yescharacteristic
627C8—23no—
628C8—612no—
629C8—1223no—
630C8#41×D6112yescomplemented
631C22—43no—
632C22—43no—
633C22—43no—
634C22—612no—
635C22—612no—
636C22—612no—
637C22—48812no—
638C22—2443no—
639C22—2443no—
640C22—2443no—
641C22C22×D4×D6111yescharacteristic
642C4—23no—
643C4—41no—
644C4—41no—
645C4—43no—
646C4—616no—
647C4—612no—
648C4—1223no—
649C4—2441no—
650C4—2441no—
651C4—2443no—
652C4C2×SD16×D6116yes—
653C4C2×#41×D6111yescharacteristic
654C4H(5,𝔽2)×D6111yescharacteristic
655C2—83no—
656C2—614no—
657C2—4883no—
658C2#550×D6111yescharacteristic
659C2C22×SD16×D6111yescharacteristic
660C2#577×D6111yescharacteristic
subgroupG/H
centerC22C22×D4×D61
commutator, cosocleC2×C4×C61C25
FrattiniC2×C4C24×D61
FittingC61×#3392C2
socleC22×C61C23×D4
prime radical#550×D61C2
hypercenter#3392D61
hypocenterC61C2×#3392
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C4×Q8 ◁ #3392

Complements

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

NH ≅ G/Ntype
1. #557×D61C2semidirect
3. #21458801C2semidirect
4. #21458800C2semidirect
5. #21458799C2semidirect
6. #21458798C2semidirect
7. #21458797C2semidirect
8. #21458796C2semidirect
9. #21458795C2semidirect
10. #492×D61C2semidirect
11. #460×D61C2semidirect
12. #461×D61C2semidirect
13. C8×Q8×D61C2semidirect
14. C4×SD16×D61C2semidirect
15. C61×#3392C2semidirect
19. #16840491C22semidirect
20. #16840490C22semidirect
23. #16840487C22semidirect
37. #16840136C22semidirect
38. #16840135C22semidirect
39. #16840134C22semidirect
40. #16840133C22semidirect
41. #16840133C22semidirect
42. #16839968C22semidirect
43. #16839967C22semidirect
44. #16839966C22semidirect
45. #16839965C22semidirect
46. #16839964C22semidirect
47. #16839963C22semidirect
48. #16839962C22semidirect
49. #16839961C22semidirect
50. #16839960C22semidirect
51. #16839959C22semidirect
52. Q8×#92418C22semidirect
53. C8×Dic122C22semidirect
54. #16839886C22semidirect
55. #16839853C22semidirect
56. C4×#11758190C22semidirect
57. SD16×Dic61C22semidirect
58. C4×#11758201C22semidirect
59. #16839849C22semidirect
60. C4×#11758200C22semidirect
61. #16839847C22semidirect
67. C61×#557C22semidirect
69. C61×#492C22semidirect
70. C61×#461C22semidirect
71. C61×#460C22semidirect
72. C8×Q8×C61C22semidirect
73. C4×SD16×C61C22semidirect
182. Q8×D61SD16semidirect
248. C8×D61#41semidirect
257. Q8×C61C2×SD16semidirect
295. Dic122C2×SD16semidirect
297. C8×C61C2×#41semidirect
299. #92418C2×#41semidirect
334. #3392D61direct (N × H)
345. D61#3392direct (N × H)
354. #557C2×D61semidirect
374. #492C2×D61semidirect
380. #461C2×D61semidirect
386. #460C2×D61semidirect
390. C8×Q8C2×D61semidirect
398. C4×SD16C2×D61semidirect
404. C61C2×#3392semidirect
592. Q8SD16×D61semidirect
630. C8#41×D61semidirect

Maximal subgroups

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

subgroupindex
#557×D612
#550×D612
#214588012
#214588002
#214587992
#214587982
#214587972
#214587962
#214587952
#492×D612
#460×D612
#461×D612
C8×Q8×D612
C4×SD16×D612
C61×#33922
C2×#339261