Finite Group Databasebeta

← #41×#11758200 #30561433

Subgroups — #41×#11758200 #30561433

Order 15616.

2116 conjugacy classes of subgroups.

The subgroup lattice has 650 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#41×Dic122C211yesmaximal, characteristic, complemented
2#21458800C211yesmaximal, characteristic, complemented
3Q8×#11758200C211yesmaximal, characteristic, complemented
4#21458486C211yesmaximal, characteristic, complemented
5#21458461C211yesmaximal, characteristic, complemented
6#21458176C213yesmaximal, complemented
7#21457678C213yesmaximal, complemented
8#21457643C213yesmaximal, complemented
9D4×#11758200C213yesmaximal, complemented
10#21456374C213yesmaximal, complemented
11#21455846C213yesmaximal, complemented
12#21455827C213yesmaximal, complemented
13#41×#92418C211yesmaximal, characteristic, complemented
14C2×C4×#11758200C213yesmaximal, complemented
15D4×#41×C61C211yesmaximal, characteristic
16C22×#11758200C2213yes—
17Q8×Dic122C2211yescharacteristic, complemented
18#16840491C2211yescharacteristic, complemented
19#16840488—21no—
20#16840471—23no—
21#16840469C2213yescomplemented
22D4×Dic122C2213yescomplemented
23#16840319C2213yescomplemented
24#16840315—23no—
25#41×Dic61—21no—
26C2×C4×Dic122C2213yescomplemented
27#16840158C2213yescomplemented
28#16840134C2213yescomplemented
29#16840114C2213yescomplemented
30#16840108C2213yescomplemented
31#16840092C2216yescomplemented
32#16840078C2216yescomplemented
33#16840052C2213yes—
34#16840052C2213yescomplemented
35#16840044C2213yescomplemented
36#16840044C2213yescomplemented
37#16839989C2216yescomplemented
38#16839989C2216yescomplemented
39#16839967C2213yes—
40#16839967C2213yescomplemented
41#16839959C2213yescomplemented
42#16839959C2213yescomplemented
43#16839913C2216yes—
44#16839900C2216yescomplemented
45Q8×#92418C2211yescharacteristic, complemented
46Q8×#92418C2211yescharacteristic, complemented
47C4×#11758200C2213yes—
48C4×#11758200C2213yes—
49C4×#11758200C2213yes—
50C4×#11758200C2211yescharacteristic
51C4×#11758200C2213yescomplemented
52C4×#11758200C2213yescomplemented
53C4×#11758200C2216yescomplemented
54C4×#11758200C2213yescomplemented
55C4×#11758200C2213yescomplemented
56D4×#92418C2213yescomplemented
57D4×#92418C2213yescomplemented
58#16839815C2213yescomplemented
59C2×#14519708C2213yescomplemented
60C2×#14519700C2213yescomplemented
61C2×#14519698C2213yes—
62C2×C4×#92418C2213yescomplemented
63C22×#41×C61—21no—
64C61×#550C2211yescharacteristic
65D4×Q8×C61C2211yescharacteristic
66C61×#548—21no—
67C61×#547C2213yes—
68C61×#546—23no—
69D42×C61C2213yes—
70C61×#532C2213yes—
71C61×#525—23no—
72C4×#41×C61C2211yescharacteristic
73C2×C4×D4×C61C2213yes—
74C22×Dic122C2313yes—
75C2×#11758200—212no—
76C2×#11758200C2313yes—
77C2×#11758200C2313yes—
78C2×#11758200C2313yes—
79C2×#11758200C2316yes—
80C2×#11758200C2311yescharacteristic
81C22×#92418C2313yes—
82C22×#92418C2313yes—
83#14519777C2313yes—
84#14519777C2313yes—
85#14519776—23no—
86#14519776—23no—
87#14519769—23no—
88#14519768—23no—
89#14519768—23no—
90#14519756—23no—
91#14519756C2316yes—
92#14519754—26no—
93#14519754—26no—
94Q8×Dic61—21no—
95Q8×Dic61—21no—
96C4×Dic122—23no—
97C4×Dic122—23no—
98C4×Dic122—23no—
99C4×Dic122C2313yes—
100C4×Dic122C2313yes—
101C4×Dic122C2311yescharacteristic
102C4×Dic122C2313yes—
103D4×Dic61—23no—
104D4×Dic61—23no—
105#14519736—23no—
106C2×C4⋊Dic61C2313yes—
107C2×#11758196—23no—
108C2×C4×Dic61—23no—
109#14519708C2313yes—
110#14519708C2316yes—
111#14519708C2313yes—
112#14519708C2311yescharacteristic
113#14519708C2313yes—
114C4⋊#92418C2313yes—
115C4⋊#92418C2313yes—
116#14519700C2316yes—
117#14519700C2313yes—
118#14519700C2313yes—
119#14519700C2311yescharacteristic
120#14519700C2313yes—
121#14519698C2313yes—
122#14519698C2316yes—
123#14519698C2313yes—
124#14519698C2311yescharacteristic
125#14519698C2313yes—
126C22⋊#92418C2316yes—
127C4×#92418C2311yescharacteristic
128C4×#92418C2313yes—
129C4×#92418C2313yes—
130C4×#92418C2313yes—
131C2×#41×C61—21no—
132C2×#41×C61—23no—
133C2×#41×C61—23no—
134C2×#41×C61—48no—
135C2×#41×C61D411yescharacteristic
136C22×Q8×C61—21no—
137C22×D4×C61—23no—
138C22×D4×C61C2313yes—
139C23×C4×C61—23no—
140C4⋊Q8×C61C2313yes—
141#127×C61C2313yes—
142#124×C61—23no—
143#123×C61—26no—
144C22⋊Q8×C61—23no—
145C22⋊Q8×C61—23no—
146#121×C61—23no—
147#121×C61—23no—
148#121×C61—23no—
149#121×C61C2316yes—
150#120×C61—26no—
151C4×Q8×C61D411yescharacteristic
152C4×Q8×C61C2311yescharacteristic
153C4×D4×C61—23no—
154C4×D4×C61—23no—
155C4×D4×C61—23no—
156C4×D4×C61D413yes—
157C4×D4×C61C2313yes—
158C4×D4×C61C2313yes—
159C4×D4×C61C2313yes—
160C4×D4×C61C2311yescharacteristic
161C4×D4×C61C2313yes—
162#117×C61D413yes—
163C2×C4⋊C4×C61C2313yes—
164C2×C22⋊C4×C61—23no—
165C2×C42×C61C2313yes—
166C24×C61—23no—
167#41×C61—44no—
168#41×C61—412no—
169#41×C61—412no—
170#41×C61SD1618yescomplemented
171C2×Q8×C61—23no—
172C2×Q8×C61—44no—
173C2×Q8×C61C2×D411yescharacteristic
174C2×D4×C61—26no—
175C2×D4×C61—26no—
176C2×D4×C61—23no—
177C2×D4×C61—23no—
178C2×D4×C61—23no—
179C2×D4×C61—43no—
180C2×D4×C61—412no—
181C2×D4×C61C2×D413yes—
182C2×D4×C61C2413yes—
183C2×D4×C61C2411yescharacteristic
184C22×C4×C61—23no—
185C22×C4×C61—21no—
186C22×C4×C61—23no—
187C22×C4×C61—23no—
188C22×C4×C61—23no—
189C22×C4×C61—412no—
190C22×C4×C61C2×D413yes—
191C22×C4×C61C2×D413yes—
192C22×C4×C61C2413yes—
193C22×Dic61—23no—
194C22×Dic61—23no—
195C2×Dic122—26no—
196C2×Dic122—23no—
197C2×Dic122—23no—
198C2×Dic122—43no—
199C2×Dic122C2413yes—
200C2×Dic122C2411yescharacteristic
201C4⋊C4×C61—23no—
202C4⋊C4×C61—23no—
203C4⋊C4×C61C2×D413yes—
204C4⋊C4×C61C2×D413yes—
205C4⋊C4×C61C2411yescharacteristic
206C4⋊C4×C61C2413yes—
207C22⋊C4×C61—21no—
208C22⋊C4×C61—23no—
209C22⋊C4×C61—23no—
210C22⋊C4×C61—23no—
211C22⋊C4×C61—26no—
212C22⋊C4×C61C2×D416yes—
213C42×C61C2×D413yes—
214C42×C61C2×D413yes—
215C42×C61C2411yescharacteristic
216C42×C61C2413yes—
217C22⋊Dic61—26no—
218#11758200—26no—
219#11758200—26no—
220#11758200—26no—
221#11758200—46no—
222#11758200#4114yesdirect product factor
223C4⋊Dic61—23no—
224C4⋊Dic61—23no—
225C4⋊Dic61—23no—
226C4⋊Dic61—23no—
227C4⋊Dic61C2411yescharacteristic
228C4⋊Dic61C2413yes—
229#11758196—21no—
230#11758196—23no—
231#11758196—23no—
232#11758196—23no—
233#11758196—26no—
234C4×Dic61—21no—
235C4×Dic61—23no—
236C4×Dic61—23no—
237C4×Dic61—23no—
238C2×#92418—26no—
239C2×#92418—43no—
240C2×#92418C2411yescharacteristic
241C2×#92418C2413yes—
242C2×#92418C2413yes—
243C2×#92418C2411yescharacteristic
244C23×C61—21no—
245C23×C61—23no—
246C23×C61—46no—
247C23×C61—46no—
248C23×C61—83no—
249C23×C61C22×D413yes—
250Q8×C61—46no—
251Q8×C61C2×SD1614yescomplemented
252D4×C61—23no—
253D4×C61—21no—
254D4×C61—23no—
255D4×C61—46no—
256D4×C61—412no—
257D4×C61—46no—
258D4×C61—83no—
259D4×C61C2×SD16112yescomplemented
260D4×C61C2×#4112yes—
261C2×C4×C61—23no—
262C2×C4×C61—23no—
263C2×C4×C61—21no—
264C2×C4×C61—23no—
265C2×C4×C61—43no—
266C2×C4×C61—46no—
267C2×C4×C61—42no—
268C2×C4×C61—46no—
269C2×C4×C61—46no—
270C2×C4×C61—46no—
271C2×C4×C61C2×SD16112yescomplemented
272C2×C4×C61C22×D411yescharacteristic
273C2×C4×C61C22×D413yes—
274C2×C4×C61C22×D413yes—
275C2×C4×C61C22×D413yes—
276C2×C4×C61C22×D411yescharacteristic
277C2×C4×C61C2511yescharacteristic
278C2×Dic61—21no—
279C2×Dic61—21no—
280C2×Dic61—23no—
281C2×Dic61—23no—
282C2×Dic61—46no—
283C2×Dic61—83no—
284Dic122—21no—
285Dic122—23no—
286Dic122—23no—
287Dic122—46no—
288Dic122—83no—
289Dic122C2×#4112yescomplemented
290#92418—23no—
291#92418—21no—
292#92418—23no—
293#92418C2×#4112yescomplemented
294SD16×#41—611nomaximal, Sylow
295C22×C61—23no—
296C22×C61—23no—
297C22×C61—22no—
298C22×C61—41no—
299C22×C61—41no—
300C22×C61—43no—
301C22×C61—43no—
302C22×C61—43no—
303C22×C61—812no—
304C22×C61C22×SD1616yes—
305C22×C61C23×D411yescharacteristic
306C4×C61—23no—
307C4×C61—21no—
308C4×C61—23no—
309C4×C61—41no—
310C4×C61—43no—
311C4×C61C22×SD1612yes—
312C4×C61C22×SD1616yes—
313C4×C61C22×#4111yescharacteristic
314C4×C61#58511yescharacteristic
315Dic61—22no—
316Dic61—43no—
317Dic61—41no—
318Dic61—43no—
319Q8×#41—611no—
320D4×#41D6111yescharacteristic
321#3392—611no—
322Q8×SD16—611no—
323#3362—611no—
324#3359—611no—
325#3335—613no—
326#3298—613no—
327#3295—613no—
328D4×SD16—613no—
329#3167—613no—
330#3103—613no—
331#3101—613no—
332C8×#41—611no—
333C2×C4×SD16—613no—
334C2×C61—26no—
335C2×C61—42no—
336C2×C61—83no—
337C2×C61D4×#4111yescharacteristic
338C2×C61C23×SD1611yescharacteristic
339C2×C61#358411yescharacteristic
340C22×#41—21no—
341C22×SD16—613no—
342Q82—611no—
343#557—611no—
344#557—1221no—
345#554—613no—
346#554—1223no—
347#550C2×D6111yescharacteristic
348D4×Q8—613no—
349D4×Q8C2×D6111yescharacteristic
350#548—21no—
351#547C2×D6113yes—
352#546—23no—
353D42C2×D6113yes—
354#533—613no—
355#533—1223no—
356#532C2×D6113yes—
357#525—23no—
358C4×#41—1221no—
359C4×#41C2×D6111yescharacteristic
360C2×C4×Q8—613no—
361C2×C4×D4C2×D6113yes—
362#498—613no—
363#492—613no—
364#487—613no—
365#486—613no—
366#483—616no—
367#481—616no—
368#476—613no—
369#476—613no—
370#475—613no—
371#475—613no—
372#465—616no—
373#465—616no—
374#461—613no—
375#461—613no—
376#460—613no—
377#460—613no—
378#450—616no—
379#448—616no—
380C8×Q8—611no—
381C8×Q8—611no—
382C4×SD16—613no—
383C4×SD16—613no—
384C4×SD16—616no—
385C4×SD16—613no—
386C4×SD16—613no—
387C4×SD16—613no—
388C4×SD16—613no—
389C4×SD16—613no—
390C4×SD16—611no—
391C8×D4—613no—
392C8×D4—613no—
393#431—613no—
394C2×#106—613no—
395C2×#103—613no—
396C2×#102—613no—
397C2×C4×C8—613no—
398C61SD16×#4111yescharacteristic, complemented, Sylow
399C2×#41—21no—
400C2×#41—23no—
401C2×#41—23no—
402C2×#41—48no—
403C2×#41C22⋊D6111yescharacteristic
404C22×Q8—21no—
405C22×Q8—613no—
406C22×D4—23no—
407C22×D4C22×D6113yes—
408C23×C4—23no—
409C2×SD16—611no—
410C2×SD16—613no—
411C2×SD16—613no—
412C2×SD16—613no—
413C2×SD16—616no—
414C2×SD16—12212no—
415C22×C8—613no—
416C22×C8—613no—
417C4⋊Q8—613no—
418C4⋊Q8—613no—
419C4⋊Q8—1223no—
420C4⋊Q8—1223no—
421C4⋊Q8C22×D6113yes—
422#127C22×D6113yes—
423#125—1223no—
424#125—1223no—
425#125—1223no—
426#124—23no—
427#123—26no—
428C22⋊Q8—23no—
429C22⋊Q8—23no—
430C22⋊Q8—616no—
431C22⋊Q8—1223no—
432C22⋊Q8—1226no—
433C22⋊Q8—1226no—
434#121—23no—
435#121—23no—
436#121—23no—
437#121C22×D6116yes—
438#120—26no—
439C4×Q8—611no—
440C4×Q8—613no—
441C4×Q8—613no—
442C4×Q8—613no—
443C4×Q8—1221no—
444C4×Q8—1221no—
445C4×Q8—1223no—
446C4×Q8—1223no—
447C4×Q8—1223no—
448C4×Q8C22⋊D6111yescharacteristic
449C4×Q8C22×D6111yescharacteristic
450C4×D4—23no—
451C4×D4—23no—
452C4×D4—23no—
453C4×D4—1223no—
454C4×D4—1223no—
455C4×D4C22⋊D6113yes—
456C4×D4C22×D6111yescharacteristic
457C4×D4C22×D6113yes—
458C4×D4C22×D6113yes—
459C4×D4C22×D6113yes—
460C4×D4C22×D6113yes—
461#117—1223no—
462#117C22⋊D6113yes—
463C2×C4⋊C4—613no—
464C2×C4⋊C4—1223no—
465C2×C4⋊C4C22×D6113yes—
466C2×C22⋊C4—23no—
467C2×C42—1223no—
468C2×C42C22×D6113yes—
469#106—611no—
470#106—613no—
471#106—613no—
472#106—613no—
473#106—616no—
474C4⋊C8—613no—
475C4⋊C8—613no—
476#103—611no—
477#103—613no—
478#103—613no—
479#103—613no—
480#103—616no—
481#102—611no—
482#102—613no—
483#102—613no—
484#102—613no—
485#102—616no—
486C22⋊C8—616no—
487C4×C8—611no—
488C4×C8—613no—
489C4×C8—613no—
490C4×C8—613no—
491C24—23no—
492#41—44no—
493#41—412no—
494#41—412no—
495#41#1175820018yesdirect product factor
496C2×Q8—23no—
497C2×Q8—44no—
498C2×Q8—611no—
499C2×Q8—613no—
500C2×Q8—1226no—
501C2×Q8—1223no—
502C2×Q8—1223no—
503C2×Q8—2443no—
504C2×Q8C2×C22⋊D6111yescharacteristic
505C2×D4—26no—
506C2×D4—26no—
507C2×D4—23no—
508C2×D4—23no—
509C2×D4—23no—
510C2×D4—43no—
511C2×D4—412no—
512C2×D4C2×C22⋊D6113yes—
513C2×D4C23×D6113yes—
514C2×D4C23×D6111yescharacteristic
515C22×C4—23no—
516C22×C4—23no—
517C22×C4—23no—
518C22×C4—21no—
519C22×C4—23no—
520C22×C4—412no—
521C22×C4—1223no—
522C22×C4—1223no—
523C22×C4C2×C22⋊D6113yes—
524C22×C4C2×C22⋊D6113yes—
525C22×C4C23×D6113yes—
526SD16—614no—
527SD16—1226no—
528SD16—1226no—
529SD16—1226no—
530SD16—2446no—
531C2×C8—611no—
532C2×C8—611no—
533C2×C8—613no—
534C2×C8—613no—
535C2×C8—1226no—
536C2×C8—2443no—
537C4⋊C4—23no—
538C4⋊C4—23no—
539C4⋊C4—611no—
540C4⋊C4—613no—
541C4⋊C4—1223no—
542C4⋊C4—1221no—
543C4⋊C4—1223no—
544C4⋊C4—1223no—
545C4⋊C4—1223no—
546C4⋊C4—1226no—
547C4⋊C4—1223no—
548C4⋊C4—1223no—
549C4⋊C4—1223no—
550C4⋊C4C2×C22⋊D6113yes—
551C4⋊C4C2×C22⋊D6113yes—
552C4⋊C4C23×D6111yescharacteristic
553C4⋊C4C23×D6113yes—
554C22⋊C4—21no—
555C22⋊C4—23no—
556C22⋊C4—23no—
557C22⋊C4—23no—
558C22⋊C4—26no—
559C22⋊C4—1226no—
560C22⋊C4C2×C22⋊D6116yes—
561C42—1223no—
562C42—1223no—
563C42—1221no—
564C42—1223no—
565C42C2×C22⋊D6113yes—
566C42C2×C22⋊D6113yes—
567C42C23×D6113yes—
568C42C23×D6111yescharacteristic
569C23—21no—
570C23—23no—
571C23—46no—
572C23—46no—
573C23—83no—
574C23C22×C22⋊D6113yes—
575Q8—46no—
576Q8—612no—
577Q8—1221no—
578Q8—1223no—
579Q8—1223no—
580Q8—4883no—
581Q8—2446no—
582Q8C2×#1175820014yescomplemented
583D4—23no—
584D4—21no—
585D4—23no—
586D4—46no—
587D4—412no—
588D4—46no—
589D4—83no—
590D4C2×#11758200112yescomplemented
591D4#41×D6112yes—
592C2×C4—23no—
593C2×C4—23no—
594C2×C4—21no—
595C2×C4—23no—
596C2×C4—43no—
597C2×C4—46no—
598C2×C4—42no—
599C2×C4—46no—
600C2×C4—46no—
601C2×C4—46no—
602C2×C4—1221no—
603C2×C4—1221no—
604C2×C4—1223no—
605C2×C4—1223no—
606C2×C4—4883no—
607C2×C4—2446no—
608C2×C4C2×#11758200112yescomplemented
609C2×C4C22×C22⋊D6111yescharacteristic
610C2×C4C22×C22⋊D6113yes—
611C2×C4C22×C22⋊D6113yes—
612C2×C4C22×C22⋊D6113yes—
613C2×C4C22×C22⋊D6111yescharacteristic
614C2×C4C24×D6111yescharacteristic
615C8—612no—
616C8—1223no—
617C8—1221no—
618C8—1223no—
619C22—23no—
620C22—23no—
621C22—22no—
622C22—41no—
623C22—41no—
624C22—43no—
625C22—43no—
626C22—43no—
627C22—812no—
628C22C22×#1175820016yes—
629C22C23×C22⋊D6111yescharacteristic
630C4—23no—
631C4—21no—
632C4—23no—
633C4—41no—
634C4—43no—
635C4—1222no—
636C4—2443no—
637C4—2441no—
638C4—2443no—
639C4C22×#1175820012yes—
640C4C22×#1175820016yes—
641C4C2×#41×D6111yescharacteristic
642C4#1684059811yescharacteristic
643C2—26no—
644C2—42no—
645C2—83no—
646C2#41×C22⋊D6111yescharacteristic
647C2C23×#1175820011yescharacteristic
648C2#2146059011yescharacteristic
subgroupG/H
centerC2×C4C22×C22⋊D61
commutator, cosocleC2×C4×C61C25
FrattiniC2×C4C24×D61
Fitting, prime radicalD4×#41×C61C2
socleC22×C61C23×D4
hypercenterD4×#41D61
hypocenterC61SD16×#41
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C4×#41 ◁ D4×#41

Complements

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

NH ≅ G/Ntype
1. #41×Dic122C2semidirect
2. #21458800C2semidirect
3. Q8×#11758200C2semidirect
4. #21458486C2semidirect
5. #21458461C2semidirect
6. #21458176C2semidirect
7. #21457678C2semidirect
8. #21457643C2semidirect
9. D4×#11758200C2semidirect
10. #21456374C2semidirect
11. #21455846C2semidirect
12. #21455827C2semidirect
13. #41×#92418C2semidirect
14. C2×C4×#11758200C2semidirect
17. Q8×Dic122C22semidirect
18. #16840491C22semidirect
21. #16840469C22semidirect
22. D4×Dic122C22semidirect
23. #16840319C22semidirect
26. C2×C4×Dic122C22semidirect
27. #16840158C22semidirect
28. #16840134C22semidirect
29. #16840114C22semidirect
30. #16840108C22semidirect
31. #16840092C22semidirect
32. #16840078C22semidirect
34. #16840052C22semidirect
35. #16840044C22semidirect
36. #16840044C22semidirect
37. #16839989C22semidirect
38. #16839989C22semidirect
40. #16839967C22semidirect
41. #16839959C22semidirect
42. #16839959C22semidirect
44. #16839900C22semidirect
45. Q8×#92418C22semidirect
46. Q8×#92418C22semidirect
51. C4×#11758200C22semidirect
52. C4×#11758200C22semidirect
53. C4×#11758200C22semidirect
54. C4×#11758200C22semidirect
55. C4×#11758200C22semidirect
56. D4×#92418C22semidirect
57. D4×#92418C22semidirect
58. #16839815C22semidirect
59. C2×#14519708C22semidirect
60. C2×#14519700C22semidirect
62. C2×C4×#92418C22semidirect
170. #41×C61SD16semidirect
222. #11758200#41direct (N × H)
251. Q8×C61C2×SD16semidirect
259. D4×C61C2×SD16semidirect
271. C2×C4×C61C2×SD16semidirect
289. Dic122C2×#41semidirect
293. #92418C2×#41semidirect
398. C61SD16×#41semidirect
495. #41#11758200direct (N × H)
582. Q8C2×#11758200semidirect
590. D4C2×#11758200semidirect
608. C2×C4C2×#11758200semidirect

Maximal subgroups

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

subgroupindex
#41×Dic1222
#214588002
Q8×#117582002
#214584862
#214584612
#214581762
#214576782
#214576432
D4×#117582002
#214563742
#214558462
#214558272
#41×#924182
C2×C4×#117582002
D4×#41×C612
SD16×#4161