Finite Group Databasebeta

← #41×#11758190 #30561434

Subgroups — #41×#11758190 #30561434

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#41×D244C211yesmaximal, characteristic
3#21458801C211yesmaximal, characteristic, complemented
4Q8×#11758190C211yesmaximal, characteristic, complemented
5#21458491C211yesmaximal, characteristic, complemented
6#21458462C211yesmaximal, characteristic, complemented
7#21458177C213yesmaximal, complemented
8#21457685C213yesmaximal, complemented
9#21457644C213yesmaximal, complemented
10D4×#11758190C213yesmaximal, complemented
11#21456378C213yesmaximal, complemented
12#21455851C213yesmaximal, complemented
13#21455828C213yesmaximal, complemented
14C2×C4×#11758190C213yesmaximal, complemented
15C8×#41×C61C211yesmaximal, characteristic, complemented
16C2×#41×D61—21no—
17C22×#11758190C2213yes—
18Q8×Dic122C2211yescharacteristic, complemented
19#16840491C2211yescharacteristic, complemented
20#16840488—21no—
21#16840471—23no—
22#16840469C2213yescomplemented
23#16840434C2211yescharacteristic
24Q8×D244C2211yescharacteristic
25D4×Dic122C2213yescomplemented
26#16840424—21no—
27#16840416C2213yes—
28#16840407—23no—
29D4×D244C2213yes—
30#16840319C2213yescomplemented
31#16840315—23no—
32#16840312C2213yes—
33#16840274—23no—
34#41×Dic61—21no—
35C2×C4×Dic122C2213yescomplemented
36C2×C4×D244C2213yes—
37#16840160C2213yescomplemented
38#16840135C2213yescomplemented
39#16840113C2213yescomplemented
40#16840109C2213yescomplemented
41#16840093C2216yescomplemented
42#16840081C2216yescomplemented
43#16840053C2213yes—
44#16840053C2213yescomplemented
45#16840048C2213yescomplemented
46#16840048C2213yescomplemented
47#16839991C2216yescomplemented
48#16839991C2216yescomplemented
49#16839968C2213yes—
50#16839968C2213yescomplemented
51#16839963C2213yescomplemented
52#16839963C2213yescomplemented
53#16839915C2216yes—
54#16839903C2216yescomplemented
55C4×#11758190C2213yes—
56C4×#11758190C2213yes—
57C4×#11758190C2213yes—
58C4×#11758190C2211yescharacteristic
59C4×#11758190C2213yescomplemented
60C4×#11758190C2213yescomplemented
61C4×#11758190C2216yescomplemented
62C4×#11758190C2213yescomplemented
63C4×#11758190C2213yescomplemented
64C2×#14519709C2213yescomplemented
65C2×#14519702C2213yescomplemented
66C2×#14519699C2213yes—
67C4×#41×C61C2211yescharacteristic
68C8×Q8×C61C2211yescharacteristic, complemented
69C8×Q8×C61C2211yescharacteristic, complemented
70C8×D4×C61C2213yescomplemented
71C8×D4×C61C2213yescomplemented
72C61×#431C2213yescomplemented
73C2×C4×C8×C61C2213yescomplemented
74C2×#11758226—21no—
75C2×#11758222—23no—
76C2×#11758224—23no—
77#41×D61—48no—
78C22×Dic122C2313yes—
79C2×Q8×D61—21no—
80C22×D244C2313yes—
81C2×D4×D61—23no—
82C22×C4×D61—23no—
83C2×#11758190—212no—
84C2×#11758190C2313yes—
85C2×#11758190C2313yes—
86C2×#11758190C2313yes—
87C2×#11758190C2316yes—
88C2×#11758190C2311yescharacteristic
89#14519777C2313yes—
90#14519777C2313yes—
91#14519777C2313yes—
92#14519776—23no—
93#14519776—23no—
94#14519774C2313yes—
95#14519769—23no—
96#14519768—23no—
97#14519768—23no—
98#14519765—23no—
99#14519761—26no—
100#14519757—23no—
101#14519757—23no—
102#14519756—23no—
103#14519756C2316yes—
104#14519754—26no—
105#14519754—26no—
106#14519752—23no—
107#14519752—23no—
108#14519751—23no—
109#14519751C2316yes—
110#14519747—26no—
111Q8×Dic61—21no—
112Q8×Dic61—21no—
113C4×Dic122—23no—
114C4×Dic122—23no—
115C4×Dic122—23no—
116C4×Dic122C2313yes—
117C4×Dic122C2313yes—
118C4×Dic122C2311yescharacteristic
119C4×Dic122C2313yes—
120C4×D244—23no—
121C4×D244—23no—
122C4×D244—23no—
123C4×D244C2313yes—
124C4×D244C2313yes—
125C4×D244C2311yescharacteristic
126C4×D244C2313yes—
127D4×Dic61—23no—
128D4×Dic61—23no—
129#14519736—23no—
130C2×C4⋊Dic61C2313yes—
131C2×C4⋊Dic61C2313yes—
132C2×#11758196—23no—
133C2×#11758198—23no—
134C2×C4×Dic61—23no—
135#14519709C2313yes—
136#14519709C2316yes—
137#14519709C2313yes—
138#14519709C2311yescharacteristic
139#14519709C2313yes—
140#14519702C2316yes—
141#14519702C2313yes—
142#14519702C2313yes—
143#14519702C2311yescharacteristic
144#14519702C2313yes—
145#14519699C2313yes—
146#14519699C2316yes—
147#14519699C2313yes—
148#14519699C2311yescharacteristic
149#14519699C2313yes—
150C2×#41×C61D411yescharacteristic
151C22×C8×C61C2313yes—
152C22×C8×C61C2313yes—
153C4×Q8×C61D411yescharacteristic
154C4×Q8×C61C2311yescharacteristic
155C4×D4×C61D413yes—
156C4×D4×C61C2313yes—
157#117×C61D413yes—
158C2×C42×C61C2313yes—
159C4⋊C8×C61C2313yes—
160C4⋊C8×C61C2313yes—
161C22⋊C8×C61C2316yes—
162C4×C8×C61C2311yescharacteristic
163C4×C8×C61C2313yes—
164C4×C8×C61C2313yes—
165C4×C8×C61C2313yes—
166C23×D61—23no—
167#41×C61SD1618yescomplemented
168C2×Q8×C61C2×D411yescharacteristic
169C2×D4×C61C2×D413yes—
170C22×C4×C61C2×D413yes—
171C22×C4×C61C2×D413yes—
172C22×C4×C61C2413yes—
173C2×C22⋊D61—26no—
174C22×Dic61—23no—
175C22×Dic61—23no—
176C22×Dic61—23no—
177#11758226—44no—
178Q8×D61—44no—
179#11758224—412no—
180D4×D61—412no—
181#11758222—412no—
182C2×D244—26no—
183C2×D244—23no—
184C2×D244—23no—
185C2×D244—23no—
186C2×D244—43no—
187C2×D244C2413yes—
188C2×D244C2411yescharacteristic
189C2×C4×D61—21no—
190C2×C4×D61—23no—
191C2×C4×D61—23no—
192C2×C4×D61—23no—
193C2×C4×D61—412no—
194C2×Dic122—26no—
195C2×Dic122—23no—
196C2×Dic122—23no—
197C2×Dic122—23no—
198C2×Dic122—43no—
199C2×Dic122C2413yes—
200C2×Dic122C2411yescharacteristic
201C2×C8×C61—26no—
202C2×C8×C61—43no—
203C2×C8×C61C2411yescharacteristic
204C2×C8×C61C2413yes—
205C2×C8×C61C2413yes—
206C2×C8×C61C2411yescharacteristic
207C4⋊C4×C61C2×D413yes—
208C4⋊C4×C61C2×D413yes—
209C22⋊C4×C61C2×D416yes—
210C42×C61C2×D413yes—
211C42×C61C2×D413yes—
212C42×C61C2411yescharacteristic
213C42×C61C2413yes—
214C22⋊Dic61—26no—
215#11758198—21no—
216#11758198—23no—
217#11758198—23no—
218#11758198—23no—
219#11758198—26no—
220C4⋊Dic61—23no—
221C4⋊Dic61—23no—
222C4⋊Dic61—23no—
223C4⋊Dic61—23no—
224C4⋊Dic61—23no—
225C4⋊Dic61—23no—
226C4⋊Dic61C2411yescharacteristic
227C4⋊Dic61C2413yes—
228C4⋊Dic61C2411yescharacteristic
229C4⋊Dic61C2413yes—
230#11758196—21no—
231#11758196—23no—
232#11758196—23no—
233#11758196—23no—
234#11758196—26no—
235C4×Dic61—21no—
236C4×Dic61—23no—
237C4×Dic61—23no—
238C4×Dic61—23no—
239#11758190—26no—
240#11758190—26no—
241#11758190—26no—
242#11758190—46no—
243#11758190#4114yesdirect product factor
244C23×C61C22×D413yes—
245C22×D61—21no—
246C22×D61—23no—
247C22×D61—46no—
248C22×D61—46no—
249C22×D61—83no—
250Q8×C61C2×SD1614yescomplemented
251D4×C61C2×SD16112yescomplemented
252C2×C4×C61—23no—
253C2×C4×C61—23no—
254C2×C4×C61—43no—
255C2×C4×C61C2×SD16112yescomplemented
256C2×C4×C61C22×D411yescharacteristic
257C2×C4×C61C22×D413yes—
258C2×C4×C61C22×D413yes—
259C2×C4×C61C22×D413yes—
260C2×C4×C61C22×D411yescharacteristic
261C2×C4×C61C2511yescharacteristic
262C22⋊D61—412no—
263C2×Dic61—21no—
264C2×Dic61—21no—
265C2×Dic61—23no—
266C2×Dic61—23no—
267C2×Dic61—23no—
268C2×Dic61—21no—
269C2×Dic61—46no—
270C2×Dic61—46no—
271C2×Dic61—83no—
272D244—23no—
273D244—21no—
274D244—23no—
275D244—46no—
276D244—46no—
277D244—83no—
278D244C2×#4112yes—
279C4×D61—42no—
280C4×D61—46no—
281C4×D61—46no—
282C4×D61—46no—
283Dic122—21no—
284Dic122—23no—
285Dic122—23no—
286Dic122—46no—
287Dic122—46no—
288Dic122—83no—
289Dic122C2×#4112yescomplemented
290C8×C61—23no—
291C8×C61—23no—
292C8×C61—21no—
293C8×C61C2×#4112yescomplemented
294SD16×#41—611nomaximal, Sylow
295C22×C61—23no—
296C22×C61—23no—
297C22×C61C22×SD1616yes—
298C22×C61C23×D411yescharacteristic
299C2×D61—22no—
300C2×D61—41no—
301C2×D61—41no—
302C2×D61—43no—
303C2×D61—43no—
304C2×D61—43no—
305C2×D61—812no—
306C4×C61—23no—
307C4×C61—21no—
308C4×C61—23no—
309C4×C61C22×SD1616yes—
310C4×C61C22×SD1612yes—
311C4×C61C22×#4111yescharacteristic
312C4×C61#58511yescharacteristic
313Dic61—22no—
314Dic61—43no—
315Dic61—41no—
316Dic61—43no—
317Dic61—43no—
318Dic61—41no—
319Q8×#41—611no—
320D4×#41—611no—
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×#41D6111yescharacteristic, complemented
333C2×C4×SD16—613no—
334C2×C61—26no—
335C2×C61D4×#4111yescharacteristic
336C2×C61C23×SD1611yescharacteristic
337C2×C61#358411yescharacteristic
338D61—42no—
339D61—83no—
340C22×#41—1221no—
341C22×SD16—613no—
342Q82—611no—
343#557—611no—
344#557—1221no—
345#554—613no—
346#554—1223no—
347#550—611no—
348D4×Q8—611no—
349D4×Q8—613no—
350#548—1221no—
351#547—613no—
352#546—1223no—
353D42—613no—
354#533—613no—
355#533—1223no—
356#532—613no—
357#525—1223no—
358C4×#41—1221no—
359C4×#41C2×D6111yescharacteristic
360C2×C4×Q8—613no—
361C2×C4×D4—613no—
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×Q8C2×D6111yescharacteristic, complemented
381C8×Q8C2×D6111yescharacteristic, complemented
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×D4C2×D6113yescomplemented
392C8×D4C2×D6113yescomplemented
393#431C2×D6113yescomplemented
394C2×#106—613no—
395C2×#103—613no—
396C2×#102—613no—
397C2×C4×C8C2×D6113yescomplemented
398C61SD16×#4111yescharacteristic, complemented, Sylow
399C2×#41—1221no—
400C2×#41—1223no—
401C2×#41—1223no—
402C2×#41—2448no—
403C2×#41D24411yescharacteristic
404C22×Q8—613no—
405C22×Q8—1221no—
406C22×D4—613no—
407C22×D4—1223no—
408C23×C4—1223no—
409C2×SD16—611no—
410C2×SD16—613no—
411C2×SD16—613no—
412C2×SD16—613no—
413C2×SD16—616no—
414C2×SD16—12212no—
415C22×C8C22×D6113yes—
416C22×C8C22×D6113yes—
417C4⋊Q8—613no—
418C4⋊Q8—613no—
419C4⋊Q8—613no—
420C4⋊Q8—1223no—
421C4⋊Q8—1223no—
422#127—613no—
423#125—1223no—
424#125—1223no—
425#125—1223no—
426#124—1223no—
427#123—1226no—
428C22⋊Q8—616no—
429C22⋊Q8—1223no—
430C22⋊Q8—1223no—
431C22⋊Q8—1223no—
432C22⋊Q8—1226no—
433C22⋊Q8—1226no—
434#121—616no—
435#121—1223no—
436#121—1223no—
437#121—1223no—
438#120—1226no—
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×Q8D24411yescharacteristic
449C4×Q8C22×D6111yescharacteristic
450C4×D4—611no—
451C4×D4—613no—
452C4×D4—613no—
453C4×D4—613no—
454C4×D4—1223no—
455C4×D4—1223no—
456C4×D4—1223no—
457C4×D4—1223no—
458C4×D4—1223no—
459C4×D4D24413yes—
460C4×D4C22×D6113yes—
461#117—1223no—
462#117D24413yes—
463C2×C4⋊C4—613no—
464C2×C4⋊C4—613no—
465C2×C4⋊C4—1223no—
466C2×C22⋊C4—1223no—
467C2×C42—1223no—
468C2×C42C22×D6113yes—
469#106—611no—
470#106—613no—
471#106—613no—
472#106—613no—
473#106—616no—
474C4⋊C8C22×D6113yes—
475C4⋊C8C22×D6113yes—
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⋊C8C22×D6116yes—
487C4×C8C22×D6111yescharacteristic
488C4×C8C22×D6113yes—
489C4×C8C22×D6113yes—
490C4×C8C22×D6113yes—
491C24—1223no—
492#41—2444no—
493#41—24412no—
494#41—24412no—
495#41#1175819018yesdirect product factor
496C2×Q8—611no—
497C2×Q8—613no—
498C2×Q8—1226no—
499C2×Q8—1223no—
500C2×Q8—1223no—
501C2×Q8—1223no—
502C2×Q8—2443no—
503C2×Q8—2444no—
504C2×Q8C2×D24411yescharacteristic
505C2×D4—611no—
506C2×D4—613no—
507C2×D4—1226no—
508C2×D4—1226no—
509C2×D4—1223no—
510C2×D4—1223no—
511C2×D4—1223no—
512C2×D4—2443no—
513C2×D4—24412no—
514C2×D4C2×D24413yes—
515C22×C4—1223no—
516C22×C4—1223no—
517C22×C4—1223no—
518C22×C4—1223no—
519C22×C4—1223no—
520C22×C4—1221no—
521C22×C4—1223no—
522C22×C4—24412no—
523C22×C4C2×D24413yes—
524C22×C4C2×D24413yes—
525C22×C4C23×D6113yes—
526SD16—614no—
527SD16—1226no—
528SD16—1226no—
529SD16—1226no—
530SD16—2446no—
531C2×C8—26no—
532C2×C8—43no—
533C2×C8C23×D6111yescharacteristic
534C2×C8C23×D6113yes—
535C2×C8C23×D6113yes—
536C2×C8C23×D6111yescharacteristic
537C4⋊C4—611no—
538C4⋊C4—613no—
539C4⋊C4—611no—
540C4⋊C4—613no—
541C4⋊C4—1223no—
542C4⋊C4—1223no—
543C4⋊C4—1221no—
544C4⋊C4—1223no—
545C4⋊C4—1223no—
546C4⋊C4—1223no—
547C4⋊C4—1226no—
548C4⋊C4—1223no—
549C4⋊C4—1223no—
550C4⋊C4—1223no—
551C4⋊C4—1223no—
552C4⋊C4C2×D24413yes—
553C4⋊C4C2×D24413yes—
554C22⋊C4—1226no—
555C22⋊C4—1221no—
556C22⋊C4—1223no—
557C22⋊C4—1223no—
558C22⋊C4—1223no—
559C22⋊C4—1226no—
560C22⋊C4C2×D24416yes—
561C42—1223no—
562C42—1223no—
563C42—1221no—
564C42—1223no—
565C42C2×D24413yes—
566C42C2×D24413yes—
567C42C23×D6113yes—
568C42C23×D6111yescharacteristic
569C23—1221no—
570C23—1223no—
571C23—4883no—
572C23—2446no—
573C23—2446no—
574C23C22×D24413yes—
575Q8—612no—
576Q8—1221no—
577Q8—1223no—
578Q8—1223no—
579Q8—4883no—
580Q8—2446no—
581Q8—2446no—
582Q8C2×#1175819014yescomplemented
583D4—612no—
584D4—1223no—
585D4—1221no—
586D4—1223no—
587D4—4883no—
588D4—2446no—
589D4—24412no—
590D4—2446no—
591D4C2×#11758190112yescomplemented
592C2×C4—23no—
593C2×C4—23no—
594C2×C4—43no—
595C2×C4—1221no—
596C2×C4—1221no—
597C2×C4—1223no—
598C2×C4—1223no—
599C2×C4—1221no—
600C2×C4—1223no—
601C2×C4—4883no—
602C2×C4—2446no—
603C2×C4—2446no—
604C2×C4—2442no—
605C2×C4—2446no—
606C2×C4—2446no—
607C2×C4—2446no—
608C2×C4C2×#11758190112yescomplemented
609C2×C4C22×D24411yescharacteristic
610C2×C4C22×D24413yes—
611C2×C4C22×D24413yes—
612C2×C4C22×D24413yes—
613C2×C4C22×D24411yescharacteristic
614C2×C4C24×D6111yescharacteristic
615C8—23no—
616C8—21no—
617C8—23no—
618C8#41×D6112yescomplemented
619C22—23no—
620C22—23no—
621C22—1222no—
622C22—48812no—
623C22—2443no—
624C22—2443no—
625C22—2443no—
626C22—2441no—
627C22—2441no—
628C22C22×#1175819016yes—
629C22C23×D24411yescharacteristic
630C4—23no—
631C4—21no—
632C4—23no—
633C4—1222no—
634C4—2441no—
635C4—2443no—
636C4—2443no—
637C4—2441no—
638C4—2443no—
639C4C22×#1175819012yes—
640C4C22×#1175819016yes—
641C4C2×#41×D6111yescharacteristic
642C4#1684059811yescharacteristic
643C2—26no—
644C2—4883no—
645C2—2442no—
646C2#41×D24411yescharacteristic
647C2C23×#1175819011yescharacteristic
648C2#2146059311yescharacteristic
subgroupG/H
centerC2×C4C22×D244
commutator, cosocleC2×C4×C61C25
FrattiniC2×C4C24×D61
FittingC8×#41×C61C2
socleC22×C61C23×D4
prime radical#41×D244C2
hypercenterC8×#41D61
hypocenterC61SD16×#41
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C4×#41 ◁ C8×#41

Complements

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

NH ≅ G/Ntype
1. #41×Dic122C2semidirect
3. #21458801C2semidirect
4. Q8×#11758190C2semidirect
5. #21458491C2semidirect
6. #21458462C2semidirect
7. #21458177C2semidirect
8. #21457685C2semidirect
9. #21457644C2semidirect
10. D4×#11758190C2semidirect
11. #21456378C2semidirect
12. #21455851C2semidirect
13. #21455828C2semidirect
14. C2×C4×#11758190C2semidirect
15. C8×#41×C61C2semidirect
18. Q8×Dic122C22semidirect
19. #16840491C22semidirect
22. #16840469C22semidirect
25. D4×Dic122C22semidirect
30. #16840319C22semidirect
35. C2×C4×Dic122C22semidirect
37. #16840160C22semidirect
38. #16840135C22semidirect
39. #16840113C22semidirect
40. #16840109C22semidirect
41. #16840093C22semidirect
42. #16840081C22semidirect
44. #16840053C22semidirect
45. #16840048C22semidirect
46. #16840048C22semidirect
47. #16839991C22semidirect
48. #16839991C22semidirect
50. #16839968C22semidirect
51. #16839963C22semidirect
52. #16839963C22semidirect
54. #16839903C22semidirect
59. C4×#11758190C22semidirect
60. C4×#11758190C22semidirect
61. C4×#11758190C22semidirect
62. C4×#11758190C22semidirect
63. C4×#11758190C22semidirect
64. C2×#14519709C22semidirect
65. C2×#14519702C22semidirect
68. C8×Q8×C61C22semidirect
69. C8×Q8×C61C22semidirect
70. C8×D4×C61C22semidirect
71. C8×D4×C61C22semidirect
72. C61×#431C22semidirect
73. C2×C4×C8×C61C22semidirect
167. #41×C61SD16semidirect
243. #11758190#41direct (N × H)
250. Q8×C61C2×SD16semidirect
251. D4×C61C2×SD16semidirect
255. C2×C4×C61C2×SD16semidirect
289. Dic122C2×#41semidirect
293. C8×C61C2×#41semidirect
332. C8×#41D61semidirect
380. C8×Q8C2×D61semidirect
381. C8×Q8C2×D61semidirect
391. C8×D4C2×D61semidirect
392. C8×D4C2×D61semidirect
393. #431C2×D61semidirect
397. C2×C4×C8C2×D61semidirect
398. C61SD16×#41semidirect
495. #41#11758190direct (N × H)
582. Q8C2×#11758190semidirect
591. D4C2×#11758190semidirect
608. C2×C4C2×#11758190semidirect
618. C8#41×D61semidirect

Maximal subgroups

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

subgroupindex
#41×Dic1222
#41×D2442
#214588012
Q8×#117581902
#214584912
#214584622
#214581772
#214576852
#214576442
D4×#117581902
#214563782
#214558512
#214558282
C2×C4×#117581902
C8×#41×C612
SD16×#4161