Finite Group Databasebeta

← D61×#3359 #30559571

Subgroups — D61×#3359 #30559571

Order 15616.

2058 conjugacy classes of subgroups.

The subgroup lattice has 644 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#557×D61C211yesmaximal, characteristic, complemented
2D4×Q8×D61C211yesmaximal, characteristic
3#21458462C211yesmaximal, characteristic, complemented
4#21458461C211yesmaximal, characteristic, complemented
5#21458460C213yesmaximal, complemented
6#21458459C213yesmaximal, complemented
7#21458458C213yesmaximal, complemented
8#21458457C211yesmaximal, characteristic, complemented
9#21458456C213yesmaximal, complemented
10#487×D61C213yesmaximal, complemented
11#475×D61C213yesmaximal, complemented
12#461×D61C213yesmaximal
13C8×Q8×D61C211yesmaximal, characteristic, complemented
14C4×SD16×D61C213yesmaximal
15C61×#3359C211yesmaximal, characteristic, complemented
16C22×Q8×D61—21no—
17C2×SD16×D61C2213yes—
18#16840491C2211yescharacteristic, complemented
19#16840490C2213yescomplemented
20#16840489—23no—
21#16840488—21no—
22#16840487C2213yescomplemented
23Q8×D244C2211yescharacteristic
24D4×Dic122C2213yes—
25#16840427C2213yes—
26Q8×C22⋊D61—21no—
27#16840425—23no—
28C4⋊Q8×D61C2213yes—
29C4⋊Q8×D61C2213yes—
30#125×D61—23no—
31C22⋊Q8×D61—23no—
32C4×Q8×D61—21no—
33C4×Q8×D61C2211yescharacteristic
34C4×Q8×D61C2213yes—
35C4×D4×D61C2213yes—
36#16840114C2213yescomplemented
37#16840113C2213yescomplemented
38#16840112C2213yescomplemented
39#16840111C2216yescomplemented
40#16840111C2216yescomplemented
41#16840048C2213yescomplemented
42#16840047C2213yescomplemented
43#16840046C2216yescomplemented
44#16840045C2216yescomplemented
45#16840044C2213yescomplemented
46#16839968C2213yes—
47#16839967C2213yes—
48#16839966C2216yes—
49#16839965C2216yes—
50#16839964C2213yes—
51Q8×#92418C2211yescharacteristic, complemented
52C8×Dic122C2213yescomplemented
53#16839886C2213yescomplemented
54#16839853C2213yes—
55C4×#11758190C2213yes—
56SD16×Dic61C2213yes—
57C4×#11758201C2213yes—
58#16839849C2213yes—
59C4×#11758200C2213yes—
60#16839847C2213yes—
61#106×D61C2213yes—
62C4⋊C8×D61C2213yes—
63#103×D61C2216yes—
64#103×D61C2213yes—
65#102×D61C2211yescharacteristic
66C4×C8×D61C2213yes—
67C61×#557C2211yescharacteristic, complemented
68D4×Q8×C61C2211yescharacteristic
69C61×#487C2213yescomplemented
70C61×#475C2213yescomplemented
71C61×#461C2213yes—
72C8×Q8×C61C2211yescharacteristic, complemented
73C4×SD16×C61C2213yes—
74C22×Dic122—23no—
75C2×Q8×D61—21no—
76C2×Q8×D61—23no—
77C2×Q8×D61—23no—
78C2×Q8×D61—48no—
79C2×Q8×D61D411yescharacteristic
80C2×Q8×D61C2313yes—
81C2×D4×D61C2311yescharacteristic
82C22×C4×D61—23no—
83C2×#11758190C2313yes—
84C2×#11758200C2313yes—
85C2×#11758201C2313yes—
86SD16×D61—212no—
87C2×C8×D61C2311yescharacteristic
88C2×C8×D61C2313yes—
89#14519777C2313yes—
90#14519777C2313yes—
91#14519776—23no—
92#14519776—23no—
93#14519776—23no—
94#14519776C2316yes—
95#14519775—23no—
96#14519775C2316yes—
97#14519769—23no—
98#14519768—23no—
99#14519768—23no—
100#14519767—26no—
101#14519767—26no—
102#14519757—23no—
103#14519756—23no—
104#14519755—26no—
105#14519754—26no—
106#14519753—23no—
107Q8×Dic61—21no—
108Q8×Dic61D411yescharacteristic
109Q8×Dic61C2313yes—
110C4×Dic122—23no—
111C4×Dic122—23no—
112C4×Dic122C2313yes—
113C4×Dic122C2313yes—
114#14519742—23no—
115#14519742—23no—
116#14519742D413yes—
117#14519742C2313yes—
118C4×D244C2313yes—
119D4×Dic61C2313yes—
120#14519739C2313yes—
121C4×C22⋊D61—23no—
122#14519737—23no—
123C4⋊C4×D61—23no—
124C4⋊C4×D61—23no—
125C4⋊C4×D61—23no—
126C4⋊C4×D61—23no—
127C4⋊C4×D61D413yes—
128C4⋊C4×D61C2313yes—
129C4⋊C4×D61C2311yescharacteristic
130C22⋊C4×D61—23no—
131C42×D61—23no—
132C42×D61C2313yes—
133#14519709C2313yes—
134#14519708C2313yes—
135#14519708C2313yes—
136C4⋊#92418C2313yes—
137#14519706C2316yes—
138#14519702C2316yes—
139#14519702C2313yes—
140#14519701C2316yes—
141#14519701C2313yes—
142#14519700C2316yes—
143#14519700C2313yes—
144#14519699C2311yescharacteristic
145#14519698C2311yescharacteristic
146#14519697C2311yescharacteristic
147C4×#92418C2313yes—
148C8×Dic61C2313yes—
149C8×Dic61C2313yes—
150C22×Q8×C61—21no—
151C2×SD16×C61C2313yes—
152C4⋊Q8×C61C2313yes—
153C4⋊Q8×C61C2313yes—
154#125×C61—23no—
155C22⋊Q8×C61—23no—
156C4×Q8×C61—21no—
157C4×Q8×C61C2313yes—
158C4×Q8×C61C2311yescharacteristic
159C4×D4×C61C2313yes—
160#106×C61C2313yes—
161C4⋊C8×C61C2313yes—
162#103×C61C2316yes—
163#103×C61C2313yes—
164#102×C61C2311yescharacteristic
165C4×C8×C61C2313yes—
166C23×D61—21no—
167C2×Q8×C61—23no—
168C2×Q8×C61—44no—
169C2×Q8×C61C2×D411yescharacteristic
170C2×Q8×C61C2413yes—
171C2×D4×C61C2411yescharacteristic
172C22×C4×C61—23no—
173C2×C22⋊D61—21no—
174C22×Dic61—23no—
175Q8×D61—26no—
176Q8×D61—26no—
177Q8×D61—44no—
178Q8×D61—412no—
179Q8×D61—412no—
180Q8×D61SD1618yescomplemented
181D4×D61—22no—
182D4×D61#4114yes—
183C2×D244C2411yescharacteristic
184C2×C4×D61—23no—
185C2×C4×D61—23no—
186C2×C4×D61—21no—
187C2×C4×D61—23no—
188C2×C4×D61—23no—
189C2×C4×D61—412no—
190C2×C4×D61C2×D413yes—
191C2×C4×D61C2×D413yes—
192C2×C4×D61C2411yescharacteristic
193C2×Dic122—23no—
194C2×Dic122—26no—
195C2×Dic122—23no—
196C2×Dic122—412no—
197C2×Dic122C2×D413yes—
198C2×Dic122C2413yes—
199SD16×C61—26no—
200C2×C8×C61C2411yescharacteristic
201C2×C8×C61C2413yes—
202C4⋊C4×C61—23no—
203C4⋊C4×C61—23no—
204C4⋊C4×C61—23no—
205C4⋊C4×C61—23no—
206C4⋊C4×C61C2×D413yes—
207C4⋊C4×C61C2411yescharacteristic
208C4⋊C4×C61C2413yes—
209C22⋊C4×C61—23no—
210C42×C61—23no—
211C42×C61C2413yes—
212C22⋊Dic61—23no—
213#11758201—26no—
214#11758200—26no—
215#11758198—23no—
216#11758198—23no—
217C4⋊Dic61—23no—
218C4⋊Dic61—23no—
219C4⋊Dic61—23no—
220C4⋊Dic61—23no—
221C4⋊Dic61C2×D413yes—
222C4⋊Dic61C2411yescharacteristic
223C4⋊Dic61C2413yes—
224#11758196—23no—
225#11758196—26no—
226#11758196—23no—
227#11758196—26no—
228#11758196—26no—
229#11758196—23no—
230#11758196—21no—
231#11758196C2×D416yes—
232C4×Dic61—23no—
233C4×Dic61—23no—
234C4×Dic61—23no—
235C4×Dic61C2×D413yes—
236C4×Dic61C2×D413yes—
237C4×Dic61C2413yes—
238C2×#92418C2411yescharacteristic
239C2×#92418C2413yes—
240#11758190—26no—
241C8×D61—26no—
242C8×D61—22no—
243C23×C61—21no—
244C22×D61—21no—
245C22×D61—24no—
246C22×D61—42no—
247C22×D61—42no—
248C22×D61C22×D411yescharacteristic
249Q8×C61—23no—
250Q8×C61—23no—
251Q8×C61—46no—
252Q8×C61C2×SD1614yescomplemented
253D4×C61—21no—
254D4×C61C2×#4112yes—
255C2×C4×C61—21no—
256C2×C4×C61—23no—
257C2×C4×C61—23no—
258C2×C4×C61—46no—
259C2×C4×C61C22×D413yes—
260C2×C4×C61C22×D413yes—
261C2×C4×C61C2511yescharacteristic
262C22⋊D61—24no—
263C2×Dic61—21no—
264C2×Dic61—23no—
265C2×Dic61—23no—
266C2×Dic61—46no—
267C2×Dic61C22×D413yes—
268C2×Dic61C22×D411yescharacteristic
269C2×Dic61C22×D413yes—
270D244—21no—
271D244C2×#4112yes—
272C4×D61—26no—
273C4×D61—46no—
274C4×D61—46no—
275C4×D61—42no—
276C4×D61—46no—
277C4×D61—46no—
278C4×D61C2×SD16112yes—
279C4×D61C2×#4112yes—
280C4×D612−1+412yes—
281Dic122—23no—
282Dic122—23no—
283Dic122—46no—
284Dic122—46no—
285Dic122—412no—
286Dic122C2×SD16112yescomplemented
287C8×C61—23no—
288C8×C61—21no—
289#92418—23no—
290#92418—21no—
291C2×#3359—611nomaximal, Sylow
292C22×C61—22no—
293C22×C61—41no—
294C22×C61—41no—
295C22×C61C23×D411yescharacteristic
296C2×D61—22no—
297C2×D61—41no—
298C2×D61—41no—
299C2×D61—48no—
300C2×D61#54812yes—
301C2×D61C22×SD1612yes—
302C2×D61#57812yes—
303C4×C61—23no—
304C4×C61—43no—
305C4×C61—43no—
306C4×C61—41no—
307C4×C61C22×SD1616yes—
308C4×C61C22×#4111yescharacteristic
309C4×C61C2×2−1+411yescharacteristic
310Dic61—23no—
311Dic61—43no—
312Dic61—43no—
313Dic61—41no—
314Dic61#54812yes—
315Dic61C22×SD1616yes—
316C2×#557—611no—
317C2×D4×Q8—611no—
318#3359—613no—
319#3359—613no—
320#3359—613no—
321#3359—611no—
322#3359—613no—
323#3359—611no—
324#3359—611no—
325#3359D6111yescharacteristic, direct product factor
326C2×#487—613no—
327C2×#475—613no—
328C2×#461—613no—
329C2×C8×Q8—611no—
330C2×C4×SD16—613no—
331C2×C61—42no—
332C2×C61C2×#54811yescharacteristic
333C2×C61C23×SD1611yescharacteristic
334C2×C61C2×#57811yescharacteristic
335D61—42no—
336D61#335914yesdirect product factor
337C23×Q8—1221no—
338C22×SD16—613no—
339#557—613no—
340#557—613no—
341#557—611no—
342#557—1221no—
343#557—1223no—
344#557C2×D6111yescharacteristic, complemented
345D4×Q8—613no—
346D4×Q8—611no—
347D4×Q8—613no—
348D4×Q8—1223no—
349D4×Q8—1221no—
350D4×Q8C2×D6111yescharacteristic
351C2×C4⋊Q8—613no—
352C2×C4⋊Q8—613no—
353C2×#125—1223no—
354C2×C22⋊Q8—1223no—
355C2×C4×Q8—611no—
356C2×C4×Q8—613no—
357C2×C4×Q8—1221no—
358C2×C4×D4—613no—
359#487—613no—
360#487—613no—
361#487—616no—
362#487—616no—
363#487—613no—
364#487C2×D6113yescomplemented
365#475—613no—
366#475—613no—
367#475—616no—
368#475—613no—
369#475—616no—
370#475C2×D6113yescomplemented
371#461—616no—
372#461—613no—
373#461—613no—
374#461—613no—
375#461—616no—
376#461C2×D6113yes—
377C8×Q8—611no—
378C8×Q8—613no—
379C8×Q8—613no—
380C8×Q8C2×D6111yescharacteristic, complemented
381C4×SD16—613no—
382C4×SD16—613no—
383C4×SD16—613no—
384C4×SD16—613no—
385C4×SD16—613no—
386C4×SD16—613no—
387C4×SD16—613no—
388C4×SD16C2×D6113yes—
389C2×#106—613no—
390C2×C4⋊C8—613no—
391C2×#103—616no—
392C2×#103—613no—
393C2×#102—611no—
394C2×C4×C8—613no—
395C61C2×#335911yescharacteristic, complemented, Sylow
396C22×Q8—21no—
397C22×Q8—611no—
398C22×Q8—613no—
399C22×Q8—1223no—
400C22×Q8—1223no—
401C22×Q8—1223no—
402C22×Q8—1221no—
403C22×Q8—2448no—
404C22×D4—611no—
405C23×C4—1223no—
406C2×SD16—613no—
407C2×SD16—613no—
408C2×SD16—613no—
409C2×SD16—12212no—
410C2×SD16C22×D6113yes—
411C22×C8—613no—
412C22×C8—611no—
413C4⋊Q8—616no—
414C4⋊Q8—616no—
415C4⋊Q8—613no—
416C4⋊Q8—613no—
417C4⋊Q8—1223no—
418C4⋊Q8—1223no—
419C4⋊Q8—1223no—
420C4⋊Q8—1223no—
421C4⋊Q8C22×D6113yes—
422C4⋊Q8C22×D6113yes—
423#125—23no—
424#125—1223no—
425#125—1226no—
426#125—1226no—
427#125—1223no—
428#125—1223no—
429C22⋊Q8—23no—
430C22⋊Q8—1226no—
431C22⋊Q8—1226no—
432C22⋊Q8—1223no—
433C22⋊Q8—1223no—
434C22⋊Q8—1223no—
435C4×Q8—21no—
436C4×Q8—611no—
437C4×Q8—613no—
438C4×Q8—613no—
439C4×Q8—613no—
440C4×Q8—613no—
441C4×Q8—613no—
442C4×Q8—1223no—
443C4×Q8—1223no—
444C4×Q8—1223no—
445C4×Q8—1223no—
446C4×Q8—1221no—
447C4×Q8C22×D6113yes—
448C4×Q8C22×D6111yescharacteristic
449C4×D4—613no—
450C4×D4—613no—
451C4×D4—613no—
452C4×D4—1223no—
453C4×D4—1223no—
454C4×D4C22×D6113yes—
455C2×C4⋊C4—613no—
456C2×C4⋊C4—613no—
457C2×C4⋊C4—611no—
458C2×C4⋊C4—1223no—
459C2×C4⋊C4—1223no—
460C2×C4⋊C4—1223no—
461C2×C4⋊C4—1223no—
462C2×C22⋊C4—1223no—
463C2×C42—613no—
464C2×C42—1223no—
465#106—613no—
466#106—613no—
467#106—613no—
468#106C22×D6113yes—
469C4⋊C8—616no—
470C4⋊C8—613no—
471C4⋊C8C22×D6113yes—
472#103—613no—
473#103—616no—
474#103—613no—
475#103—616no—
476#103—613no—
477#103—616no—
478#103C22×D6113yes—
479#103C22×D6116yes—
480#102—611no—
481#102—611no—
482#102—611no—
483#102C22×D6111yescharacteristic
484C4×C8—613no—
485C4×C8—613no—
486C4×C8—613no—
487C4×C8C22×D6113yes—
488C24—1221no—
489C2×Q8—23no—
490C2×Q8—44no—
491C2×Q8—618no—
492C2×Q8—613no—
493C2×Q8—613no—
494C2×Q8—1223no—
495C2×Q8—1226no—
496C2×Q8—1226no—
497C2×Q8—1226no—
498C2×Q8—1223no—
499C2×Q8—24412no—
500C2×Q8—24412no—
501C2×Q8—24412no—
502C2×Q8—2444no—
503C2×Q8D4×D6111yescharacteristic
504C2×Q8C23×D6113yes—
505C2×D4—614no—
506C2×D4—611no—
507C2×D4—1222no—
508C2×D4—1221no—
509C2×D4C23×D6111yescharacteristic
510C22×C4—23no—
511C22×C4—613no—
512C22×C4—611no—
513C22×C4—613no—
514C22×C4—1223no—
515C22×C4—1221no—
516C22×C4—1223no—
517C22×C4—1223no—
518C22×C4—1223no—
519C22×C4—1223no—
520C22×C4—24412no—
521SD16—26no—
522SD16—1226no—
523SD16—1226no—
524SD16—1226no—
525C2×C8—611no—
526C2×C8—613no—
527C2×C8—1222no—
528C2×C8—1226no—
529C2×C8C23×D6111yescharacteristic
530C2×C8C23×D6113yes—
531C4⋊C4—23no—
532C4⋊C4—23no—
533C4⋊C4—23no—
534C4⋊C4—23no—
535C4⋊C4—616no—
536C4⋊C4—613no—
537C4⋊C4—611no—
538C4⋊C4—613no—
539C4⋊C4—1223no—
540C4⋊C4—1221no—
541C4⋊C4—1223no—
542C4⋊C4—1226no—
543C4⋊C4—1226no—
544C4⋊C4—1226no—
545C4⋊C4—1223no—
546C4⋊C4—1223no—
547C4⋊C4—1223no—
548C4⋊C4—1223no—
549C4⋊C4—1223no—
550C4⋊C4D4×D6113yes—
551C4⋊C4C23×D6111yescharacteristic
552C4⋊C4C23×D6113yes—
553C22⋊C4—23no—
554C22⋊C4—1223no—
555C22⋊C4—1223no—
556C22⋊C4—1223no—
557C42—23no—
558C42—613no—
559C42—613no—
560C42—613no—
561C42—1223no—
562C42—1223no—
563C42—1223no—
564C42C23×D6113yes—
565C23—21no—
566C23—611no—
567C23—1224no—
568C23—1221no—
569C23—2442no—
570C23—2442no—
571Q8—23no—
572Q8—23no—
573Q8—46no—
574Q8—6112no—
575Q8—1223no—
576Q8—1223no—
577Q8—2446no—
578Q8—24412no—
579Q8—2446no—
580Q8SD16×D6114yescomplemented
581D4—21no—
582D4—612no—
583D4—1224no—
584D4—1221no—
585D4#41×D6112yes—
586C2×C4—21no—
587C2×C4—23no—
588C2×C4—23no—
589C2×C4—46no—
590C2×C4—611no—
591C2×C4—613no—
592C2×C4—6112no—
593C2×C4—612no—
594C2×C4—612no—
595C2×C4—613no—
596C2×C4—1226no—
597C2×C4—1221no—
598C2×C4—1223no—
599C2×C4—1223no—
600C2×C4—2446no—
601C2×C4—2442no—
602C2×C4—2446no—
603C2×C4—2446no—
604C2×C4—2446no—
605C2×C4—2446no—
606C2×C4C2×D4×D6113yes—
607C2×C4C2×D4×D6113yes—
608C2×C4C24×D6111yescharacteristic
609C8—23no—
610C8—21no—
611C8—1223no—
612C8—1221no—
613C22—22no—
614C22—41no—
615C22—41no—
616C22—612no—
617C22—612no—
618C22—612no—
619C22—1222no—
620C22—2448no—
621C22—2441no—
622C22—2441no—
623C22C22×D4×D6111yescharacteristic
624C4—23no—
625C4—43no—
626C4—43no—
627C4—41no—
628C4—616no—
629C4—612no—
630C4—1223no—
631C4—2443no—
632C4—2443no—
633C4—2441no—
634C4C2×SD16×D6116yes—
635C4C2×#41×D6111yescharacteristic
636C42−1+4×D6111yescharacteristic
637C2—42no—
638C2—614no—
639C2—2442no—
640C2#548×D6111yescharacteristic
641C2C22×SD16×D6111yescharacteristic
642C2#578×D6111yescharacteristic
subgroupG/H
centerC22C22×D4×D61
commutator, cosocleC2×C4×C61C25
FrattiniC2×C4C24×D61
FittingC61×#3359C2
socleC22×C61C23×D4
prime radicalC2×D4×D61C23
hypercenter#3359D61
hypocenterC61C2×#3359
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C4×Q8 ◁ #3359

Complements

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

NH ≅ G/Ntype
1. #557×D61C2semidirect
3. #21458462C2semidirect
4. #21458461C2semidirect
5. #21458460C2semidirect
6. #21458459C2semidirect
7. #21458458C2semidirect
8. #21458457C2semidirect
9. #21458456C2semidirect
10. #487×D61C2semidirect
11. #475×D61C2semidirect
13. C8×Q8×D61C2semidirect
15. C61×#3359C2semidirect
18. #16840491C22semidirect
19. #16840490C22semidirect
22. #16840487C22semidirect
36. #16840114C22semidirect
37. #16840113C22semidirect
38. #16840112C22semidirect
39. #16840111C22semidirect
40. #16840111C22semidirect
41. #16840048C22semidirect
42. #16840047C22semidirect
43. #16840046C22semidirect
44. #16840045C22semidirect
45. #16840044C22semidirect
51. Q8×#92418C22semidirect
52. C8×Dic122C22semidirect
53. #16839886C22semidirect
67. C61×#557C22semidirect
69. C61×#487C22semidirect
70. C61×#475C22semidirect
72. C8×Q8×C61C22semidirect
180. Q8×D61SD16semidirect
252. Q8×C61C2×SD16semidirect
286. Dic122C2×SD16semidirect
325. #3359D61direct (N × H)
336. D61#3359direct (N × H)
344. #557C2×D61semidirect
364. #487C2×D61semidirect
370. #475C2×D61semidirect
380. C8×Q8C2×D61semidirect
395. C61C2×#3359semidirect
580. Q8SD16×D61semidirect

Maximal subgroups

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

subgroupindex
#557×D612
D4×Q8×D612
#214584622
#214584612
#214584602
#214584592
#214584582
#214584572
#214584562
#487×D612
#475×D612
#461×D612
C8×Q8×D612
C4×SD16×D612
C61×#33592
C2×#335961