Finite Group Databasebeta

← #30544703

Subgroups — #30544703

Order 15616.

1950 conjugacy classes of subgroups.

The subgroup lattice has 626 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21459417C211yesmaximal, characteristic
2#21454839C211yesmaximal, characteristic, complemented
3#21454838C211yesmaximal, characteristic, complemented
4#21454820C213yesmaximal, complemented
5#21454817C213yesmaximal, complemented
6#21454794C213yesmaximal, complemented
7#21454791C213yesmaximal, complemented
8C8×#11758226C211yesmaximal, characteristic, complemented
9Q8×#11758189C211yesmaximal, characteristic, complemented
10C8×Q8×D61C211yesmaximal, characteristic, complemented
11#21454558C213yesmaximal, complemented
12#21454526C213yesmaximal, complemented
13#21454480C213yesmaximal, complemented
14C4×#14519783C213yesmaximal, complemented
15C8×#41×C61C211yesmaximal, characteristic, complemented
16C2×#14519783C2213yes—
17C2×#14519783C2213yes—
18#16840495C413yes—
19#16840488C411yescharacteristic
20#16840480C413yes—
21#16840462C413yes—
22#16840456C413yes—
23#16840431C411yescharacteristic
24#16840424C411yescharacteristic
25#16840419C411yescharacteristic
26#16840309C2213yes—
27#16840296C2213yes—
28#16840294C2213yes—
29C4×#11758226C2211yescharacteristic
30C4×#11758222C2213yes—
31C4×Q8×D61C2211yescharacteristic
32#16839892C2213yes—
33#16839889C2213yes—
34Q8×#92418C2211yescharacteristic
35Q8×#92418C2211yescharacteristic
36C8×Dic122C2213yes—
37#16839886C2213yes—
38#16839841C2213yes—
39#16839840C2213yes—
40#16839840C2213yes—
41#16839839C2213yes—
42#16839837C2216yes—
43#16839836C2216yes—
44#16839833C2216yes—
45C8×D244C2213yes—
46#16839827C2213yes—
47C8×C22⋊D61C2216yes—
48#16839821C2216yes—
49#16839820C2216yes—
50#16839819C2213yes—
51#16839818C2213yes—
52#16839814C2213yes—
53#16839784C2213yes—
54C4⋊C8×D61C2213yes—
55#16839727C2216yes—
56#16839717C2213yes—
57#16839717C2213yes—
58#16839716C2213yes—
59#16839716C2211yescharacteristic
60#16839716C2213yes—
61#16839716C2213yes—
62C4×#11758194C2213yes—
63C4×#11758189C2213yes—
64C4×#11758189C2213yes—
65C4×C8×D61C2213yes—
66C4×C8×D61C2213yes—
67C4×#41×C61C2211yescharacteristic
68C8×Q8×C61C2211yescharacteristic
69C8×Q8×C61C2211yescharacteristic
70C8×D4×C61C2213yes—
71C8×D4×C61C2213yes—
72C61×#431C2213yes—
73C2×C4×C8×C61C2213yes—
74C2×#11758226C2×C411yescharacteristic
75C2×#11758222C2313yes—
76C2×Q8×D61C2×C411yescharacteristic
77#14519783—212no—
78#14519783—212no—
79C2×#11758194C2313yes—
80C2×#11758194C2313yes—
81C2×#11758189C2313yes—
82C2×#11758189C2313yes—
83C2×#11758189C2313yes—
84C2×#11758189C2311yescharacteristic
85C2×C8×D61C2313yes—
86C2×C8×D61C2313yes—
87C2×C8×D61C2313yes—
88C2×C8×D61C2311yescharacteristic
89#14519776C2×C413yes—
90#14519773C2×C413yes—
91#14519771C2×C416yes—
92#14519770C2×C416yes—
93#14519769C2×C413yes—
94#14519768C2×C413yes—
95#14519765C2×C413yes—
96#14519764C2×C413yes—
97#14519763C2×C416yes—
98#14519760C2×C416yes—
99#14519759C2×C416yes—
100#14519757C2×C413yes—
101#14519756C2×C413yes—
102#14519752C2×C413yes—
103#14519751C2×C413yes—
104#14519750C2×C416yes—
105#14519748C2×C413yes—
106#14519748C2×C413yes—
107Q8×Dic61C2×C411yescharacteristic
108Q8×Dic61C2×C411yescharacteristic
109C4×Dic122C2×C413yes—
110C4×Dic122C2×C413yes—
111C4×Dic122C2313yes—
112#14519742C2×C413yes—
113C4×D244C2×C413yes—
114C4×D244C2×C413yes—
115C4×D244C2313yes—
116#14519739C2×C413yes—
117C4×C22⋊D61C2×C413yes—
118C4×C22⋊D61C2×C413yes—
119C4×C22⋊D61C2×C413yes—
120C4×C22⋊D61C2×C413yes—
121C4×C22⋊D61C2316yes—
122#14519736C2313yes—
123#14519735C2×C413yes—
124#14519734C2311yescharacteristic
125#14519734C2313yes—
126C4⋊C4×D61C2×C413yes—
127C42×D61C2313yes—
128C42×D61C2313yes—
129C4⋊#92418C2313yes—
130C4⋊#92418C2313yes—
131#14519706C2316yes—
132#14519706C2316yes—
133C22⋊#92418C2316yes—
134#14519689C2316yes—
135#14519689C2316yes—
136#14519688C2311yescharacteristic
137#14519688C2313yes—
138#14519687C2311yescharacteristic
139#14519687C2313yes—
140#14519687C2313yes—
141#14519687C2313yes—
142C4×#92418C2313yes—
143C4×#92418C2313yes—
144C8×Dic61C2311yescharacteristic
145C8×Dic61C2313yes—
146C8×Dic61C2313yes—
147C8×Dic61C2313yes—
148C2×#41×C61C2×C411yescharacteristic
149C22×C8×C61C2313yes—
150C22×C8×C61C2313yes—
151C4×Q8×C61C2×C411yescharacteristic
152C4×Q8×C61C2311yescharacteristic
153C4×D4×C61C2×C413yes—
154C4×D4×C61C2313yes—
155#117×C61C2×C413yes—
156C2×C42×C61C2313yes—
157C4⋊C8×C61C2313yes—
158C4⋊C8×C61C2313yes—
159C22⋊C8×C61C2316yes—
160C4×C8×C61C2313yes—
161C4×C8×C61C2313yes—
162C4×C8×C61C2311yescharacteristic
163C4×C8×C61C2313yes—
164#41×C61—24no—
165C2×Q8×C61C22×C411yescharacteristic
166C2×D4×C61C22×C413yes—
167C22×C4×C61C22×C413yes—
168C22×C4×C61C22×C413yes—
169C22×C4×C61C2413yes—
170C2×C22⋊D61C22×C413yes—
171C2×C22⋊D61C22×C413yes—
172C2×C22⋊D61C22×C416yes—
173#11758226—24no—
174Q8×D61—24no—
175#11758222—26no—
176#11758222—26no—
177C2×D244C22×C413yes—
178C2×D244C22×C413yes—
179C2×C4×D61C22×C413yes—
180C2×C4×D61C22×C413yes—
181C2×C4×D61C22×C413yes—
182C2×C4×D61C22×C413yes—
183C2×C4×D61C2413yes—
184C2×C4×D61C2411yescharacteristic
185C2×Dic122C22×C413yes—
186C2×Dic122C22×C413yes—
187C2×C8×C61—26no—
188C2×C8×C61—26no—
189C2×C8×C61C2413yes—
190C2×C8×C61C2413yes—
191C2×C8×C61C2411yescharacteristic
192C2×C8×C61C2411yescharacteristic
193C4⋊C4×C61C22×C413yes—
194C4⋊C4×C61C22×C413yes—
195C22⋊C4×C61C22×C416yes—
196C42×C61C22×C413yes—
197C42×C61C22×C413yes—
198C42×C61C2411yescharacteristic
199C42×C61C2413yes—
200C22⋊Dic61C22×C413yes—
201C22⋊Dic61C22×C413yes—
202C22⋊Dic61C22×C416yes—
203#11758198C22×C411yescharacteristic
204#11758198C22×C411yescharacteristic
205#11758198C22×C413yes—
206#11758198C22×C413yes—
207#11758198C22×C416yes—
208#11758198C22×C416yes—
209C4⋊Dic61C22×C413yes—
210C4⋊Dic61C22×C413yes—
211C4⋊Dic61C22×C413yes—
212C4⋊Dic61C22×C413yes—
213#11758196C22×C416yes—
214#11758196C22×C411yescharacteristic
215#11758196C22×C411yescharacteristic
216#11758196C22×C413yes—
217#11758196C22×C413yes—
218#11758196C22×C416yes—
219C4×Dic61C22×C413yes—
220C4×Dic61C22×C413yes—
221C4×Dic61C22×C413yes—
222C4×Dic61C22×C413yes—
223C4×Dic61C2411yescharacteristic
224C4×Dic61C2413yes—
225#11758194—26no—
226#11758194—26no—
227C2×#92418C2413yes—
228C2×#92418C2413yes—
229C2×#92418C2411yescharacteristic
230C2×#92418C2411yescharacteristic
231#11758189—26no—
232#11758189—26no—
233#11758189—26no—
234#11758189#4114yescomplemented
235C8×D61—26no—
236C8×D61—26no—
237C8×D61—26no—
238C8×D61#4114yescomplemented
239C23×C61C23×C413yes—
240C22×D61C23×C413yes—
241C22×D61C23×C411yescharacteristic
242Q8×C61#13114yescomplemented
243D4×C61—26no—
244C2×C4×C61—26no—
245C2×C4×C61—23no—
246C2×C4×C61—23no—
247C2×C4×C61—43no—
248C2×C4×C61C23×C413yes—
249C2×C4×C61C23×C411yescharacteristic
250C2×C4×C61C23×C411yescharacteristic
251C2×C4×C61C23×C413yes—
252C2×C4×C61C23×C413yes—
253C2×C4×C61C2511yescharacteristic
254C22⋊D61—26no—
255C22⋊D61—26no—
256C22⋊D61—46no—
257C2×Dic61C23×C411yescharacteristic
258C2×Dic61C23×C411yescharacteristic
259C2×Dic61C23×C413yes—
260C2×Dic61C23×C413yes—
261C2×Dic61C23×C413yes—
262C2×Dic61C23×C411yescharacteristic
263D244—26no—
264D244—26no—
265C4×D61—26no—
266C4×D61—26no—
267C4×D61—23no—
268C4×D61—23no—
269C4×D61—43no—
270C4×D61—43no—
271C4×D61C2×#4112yes—
272C4×D61C2×#4112yes—
273Dic122—26no—
274Dic122—26no—
275C8×C61—23no—
276C8×C61—23no—
277C8×C61C2×#4112yes—
278C8×C61C2×#4112yes—
279#92418—23no—
280#92418—23no—
281#92418C2×#4112yes—
282#92418C2×#4112yes—
283#60218—611nomaximal, Sylow
284C22×C61—23no—
285C22×C61—23no—
286C22×C61—23no—
287C22×C61C24×C411yescharacteristic
288C2×D61—21no—
289C2×D61—21no—
290C2×D61—23no—
291C2×D61—23no—
292C2×D61—23no—
293C2×D61C4×#4112yes—
294C4×C61—23no—
295C4×C61—21no—
296C4×C61—23no—
297C4×C61—21no—
298C4×C61C2×#13116yes—
299C4×C61C22×#4111yescharacteristic
300C4×C61C22×#4111yescharacteristic
301Dic61—23no—
302Dic61—23no—
303Dic61—21no—
304Dic61—21no—
305Dic61—23no—
306Dic61C4×#4112yes—
307#3480—611no—
308#2971—611no—
309#2971—611no—
310#2969—613no—
311#2969—613no—
312#2966—613no—
313#2966—613no—
314C8×#41—611no—
315C8×#41D6111yescharacteristic, complemented
316Q8×M16—611no—
317C2×C8×Q8—611no—
318#2935—613no—
319#2931—613no—
320#2925—613no—
321C4×#131—613no—
322C2×C61—43no—
323C2×C61C2×C4×#4111yescharacteristic
324C2×C61C22×#13111yescharacteristic
325C2×C61#357311yescharacteristic
326D61—22no—
327D61—43no—
328C2×#131—613no—
329C2×#131—613no—
330#559—613no—
331#557—611no—
332#556—613no—
333#553—613no—
334#553—613no—
335#550—611no—
336#548—611no—
337#548—611no—
338#532—613no—
339#529—613no—
340#529—613no—
341C4×#41—611no—
342C4×#41—613no—
343C4×#41C2×D6111yescharacteristic
344C2×C4×Q8—611no—
345#446—613no—
346#446—613no—
347C8×Q8—611no—
348C8×Q8—611no—
349C8×Q8—613no—
350C8×Q8—613no—
351C8×Q8C2×D6111yescharacteristic
352C8×Q8C2×D6111yescharacteristic
353#436—613no—
354#436—613no—
355#436—613no—
356#436—616no—
357#436—613no—
358#435—616no—
359#435—616no—
360C8×D4—616no—
361C8×D4—613no—
362C8×D4—613no—
363C8×D4C2×D6113yes—
364C8×D4C2×D6113yes—
365#433—616no—
366#433—616no—
367#432—613no—
368#432—613no—
369#431—613no—
370#431C2×D6113yes—
371#423—613no—
372C2×C4⋊C8—613no—
373#408—616no—
374#405—613no—
375#405—613no—
376#405—611no—
377#405—613no—
378#405—613no—
379#405—613no—
380C4×M16—613no—
381C4×M16—613no—
382C4×M16—613no—
383C2×C4×C8—613no—
384C2×C4×C8—613no—
385C2×C4×C8C2×D6113yes—
386C61#6021811yescharacteristic, complemented, Sylow
387C2×#41—611no—
388C2×#41—613no—
389C2×#41C4×D6111yescharacteristic
390C22×Q8—611no—
391#131—12212no—
392#131—12212no—
393C2×M16—613no—
394C2×M16—613no—
395C2×M16—611no—
396C2×M16—613no—
397C2×M16—613no—
398C2×M16—613no—
399C22×C8—611no—
400C22×C8—613no—
401C22×C8—613no—
402C22×C8—613no—
403C22×C8C22×D6113yes—
404C22×C8C22×D6113yes—
405C4⋊Q8—613no—
406#127—613no—
407#126—616no—
408#126—616no—
409#125—613no—
410#125—613no—
411#124—613no—
412#124—616no—
413#124—613no—
414#123—616no—
415#123—616no—
416C22⋊Q8—613no—
417C22⋊Q8—613no—
418#121—613no—
419#121—613no—
420#121—613no—
421#121—613no—
422#121—616no—
423C4×Q8—611no—
424C4×Q8—611no—
425C4×Q8—613no—
426C4×Q8—613no—
427C4×Q8—613no—
428C4×Q8—613no—
429C4×Q8C4×D6111yescharacteristic
430C4×Q8C22×D6111yescharacteristic
431C4×D4—613no—
432C4×D4—613no—
433C4×D4—613no—
434C4×D4—613no—
435C4×D4—616no—
436C4×D4—613no—
437C4×D4—613no—
438C4×D4—613no—
439C4×D4—613no—
440C4×D4C4×D6113yes—
441C4×D4C22×D6113yes—
442#117—613no—
443#117—613no—
444#117—611no—
445#117—613no—
446#117C4×D6113yes—
447C2×C4⋊C4—613no—
448C2×C42—613no—
449C2×C42—613no—
450C2×C42C22×D6113yes—
451C4⋊C8—616no—
452C4⋊C8—616no—
453C4⋊C8—613no—
454C4⋊C8—613no—
455C4⋊C8C22×D6113yes—
456C4⋊C8C22×D6113yes—
457C22⋊C8—616no—
458C22⋊C8—616no—
459C22⋊C8—616no—
460C22⋊C8C22×D6116yes—
461#97—611no—
462#97—613no—
463#97—613no—
464#97—613no—
465#97—611no—
466#97—613no—
467C4×C8—613no—
468C4×C8—613no—
469C4×C8—611no—
470C4×C8—613no—
471C4×C8—613no—
472C4×C8—613no—
473C4×C8C22×D6113yes—
474C4×C8C22×D6113yes—
475C4×C8C22×D6113yes—
476C4×C8C22×D6111yescharacteristic
477#41—24no—
478#41—1226no—
479#41—1226no—
480#41—1224no—
481C2×Q8—613no—
482C2×Q8—613no—
483C2×Q8—1224no—
484C2×Q8C2×C4×D6111yescharacteristic
485C2×D4—613no—
486C2×D4—613no—
487C2×D4—616no—
488C2×D4—613no—
489C2×D4—613no—
490C2×D4C2×C4×D6113yes—
491C22×C4—611no—
492C22×C4—613no—
493C22×C4—613no—
494C22×C4—613no—
495C22×C4—613no—
496C22×C4—613no—
497C22×C4C2×C4×D6113yes—
498C22×C4C2×C4×D6113yes—
499C22×C4C23×D6113yes—
500M16—614no—
501M16—1226no—
502M16—1226no—
503M16—1226no—
504M16—1226no—
505M16—1226no—
506C2×C8—26no—
507C2×C8—26no—
508C2×C8—611no—
509C2×C8—611no—
510C2×C8—614no—
511C2×C8—613no—
512C2×C8—613no—
513C2×C8—1226no—
514C2×C8—1226no—
515C2×C8—1226no—
516C2×C8C23×D6113yes—
517C2×C8C23×D6113yes—
518C2×C8C23×D6111yescharacteristic
519C2×C8C23×D6111yescharacteristic
520C4⋊C4—616no—
521C4⋊C4—613no—
522C4⋊C4—611no—
523C4⋊C4—611no—
524C4⋊C4—613no—
525C4⋊C4—613no—
526C4⋊C4—616no—
527C4⋊C4—613no—
528C4⋊C4—613no—
529C4⋊C4—613no—
530C4⋊C4C2×C4×D6113yes—
531C4⋊C4C2×C4×D6113yes—
532C22⋊C4—613no—
533C22⋊C4—613no—
534C22⋊C4—616no—
535C22⋊C4—611no—
536C22⋊C4—611no—
537C22⋊C4—613no—
538C22⋊C4—616no—
539C22⋊C4—613no—
540C22⋊C4—616no—
541C22⋊C4C2×C4×D6116yes—
542C42—613no—
543C42—613no—
544C42—613no—
545C42—613no—
546C42—611no—
547C42—613no—
548C42C2×C4×D6113yes—
549C42C2×C4×D6113yes—
550C42C23×D6113yes—
551C42C23×D6111yescharacteristic
552C23—611no—
553C23—613no—
554C23C22×C4×D6113yes—
555Q8—1226no—
556Q8—1226no—
557Q8#1451978314yescomplemented
558D4—26no—
559D4—1226no—
560D4—1226no—
561D4—1226no—
562D4—1226no—
563D4—2446no—
564C2×C4—26no—
565C2×C4—23no—
566C2×C4—23no—
567C2×C4—43no—
568C2×C4—611no—
569C2×C4—612no—
570C2×C4—612no—
571C2×C4—611no—
572C2×C4—611no—
573C2×C4—613no—
574C2×C4—613no—
575C2×C4—613no—
576C2×C4—1226no—
577C2×C4—1226no—
578C2×C4—1223no—
579C2×C4—1223no—
580C2×C4—2443no—
581C2×C4—2443no—
582C2×C4C22×C4×D6113yes—
583C2×C4C22×C4×D6111yescharacteristic
584C2×C4C22×C4×D6111yescharacteristic
585C2×C4C22×C4×D6113yes—
586C2×C4C22×C4×D6113yes—
587C2×C4C24×D6111yescharacteristic
588C8—23no—
589C8—23no—
590C8—612no—
591C8—612no—
592C8—1223no—
593C8—1223no—
594C8#41×D6112yes—
595C8#41×D6112yes—
596C22—23no—
597C22—23no—
598C22—23no—
599C22—612no—
600C22—1221no—
601C22—1221no—
602C22—1223no—
603C22—1223no—
604C22—1223no—
605C22C23×C4×D6111yescharacteristic
606C4—21no—
607C4—21no—
608C4—23no—
609C4—23no—
610C4—612no—
611C4—1221no—
612C4—1223no—
613C4—1221no—
614C4—1223no—
615C4—1223no—
616C4C2×#1451978316yes—
617C4C2×#41×D6111yescharacteristic
618C4C2×#41×D6111yescharacteristic
619C2—43no—
620C2—1222no—
621C2—2443no—
622C2C4×#41×D6111yescharacteristic
623C2C22×#1451978311yescharacteristic
624C2#2146053011yescharacteristic
subgroupG/H
centerC2×C8C23×D61
commutator, socleC22×C61C24×C4
FrattiniC2×C4C24×D61
FittingC8×#41×C61C2
prime radical#21459417C2
cosocleC2×C4×C61C25
hypercenterC8×#41D61
hypocenterC61#60218
LCSG ▷ C22×C61 ▷ C61
UCSC1 ◁ C2×C8 ◁ C8×#41

Complements

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

NH ≅ G/Ntype
2. #21454839C2semidirect
3. #21454838C2semidirect
4. #21454820C2semidirect
5. #21454817C2semidirect
6. #21454794C2semidirect
7. #21454791C2semidirect
8. C8×#11758226C2semidirect
9. Q8×#11758189C2semidirect
10. C8×Q8×D61C2semidirect
11. #21454558C2semidirect
12. #21454526C2semidirect
13. #21454480C2semidirect
14. C4×#14519783C2semidirect
15. C8×#41×C61C2semidirect
234. #11758189#41semidirect
238. C8×D61#41semidirect
242. Q8×C61#131semidirect
315. C8×#41D61semidirect
386. C61#60218semidirect
557. Q8#14519783semidirect

Maximal subgroups

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

subgroupindex
#214594172
#214548392
#214548382
#214548202
#214548172
#214547942
#214547912
C8×#117582262
Q8×#117581892
C8×Q8×D612
#214545582
#214545262
#214544802
C4×#145197832
C8×#41×C612
#6021861