Finite Group Databasebeta

← #30570456

Subgroups — #30570456

Order 15616.

1648 conjugacy classes of subgroups.

The subgroup lattice has 702 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21460230C211yesmaximal, characteristic
2#21458751C212yesmaximal, complemented
3#21458736C214yesmaximal, complemented
4Q8×#11758202C212yesmaximal, complemented
5#21458661C214yesmaximal, complemented
6#21458633C212yesmaximal, complemented
7#21458515C212yesmaximal, complemented
8#21457102C214yesmaximal, complemented
9#21457033C211yesmaximal, characteristic, complemented
10#21456955C211yesmaximal, characteristic
11#21456048C214yesmaximal, complemented
12#21455941C211yesmaximal, characteristic
13#21455858C211yesmaximal, characteristic, complemented
14#21454914C211yesmaximal, characteristic, complemented
15C61×#3502C211yesmaximal, characteristic, complemented
16#16840516—22no—
17#16840497C2212yes—
18Q8×Dic122C2212yes—
19#16840486—21no—
20#16840485C2212yes—
21Q8×D244C2212yes—
22#16840427—21no—
23#16840385—22no—
24#16840383C2214yes—
25#16840335—22no—
26#16840319C2211yescharacteristic
27#16840312C2211yescharacteristic
28#16840172C2212yes—
29#16840172C2214yes—
30#16840165C2211yescharacteristic
31#16840165C2214yes—
32#16840158C2211yescharacteristic
33#16840132C2212yes—
34#16840124C2214yes—
35#16840119C2214yes—
36#16840110C2211yescharacteristic
37#16840110C2211yescharacteristic
38#16840103C2214yes—
39#16840097C2212yes—
40#16840086C2214yes—
41#16840075C2212yes—
42#16840066C2214yes—
43#16840059C2214yes—
44#16840059C2214yes—
45#16840057C2214yes—
46#16840057C2212yes—
47#16840057C2212yes—
48#16840054C2212yes—
49#16840054C2214yes—
50#16840047C2214yes—
51#16840044C2212yes—
52#16840039C2214yes—
53#16839984C2214yes—
54#16839984C2214yes—
55#16839972C2212yes—
56#16839962C2212yes—
57#16839918C2212yes—
58#16839901C2212yes—
59#16839893C2214yes—
60#16839893C2212yes—
61Q8×#92418C2212yes—
62#16839867C2214yes—
63#16839860C2214yes—
64#16839860C2214yes—
65C4×#11758202C2212yes—
66C4×#11758202C2214yes—
67C4×#11758202C2212yes—
68C4×#11758201C2214yes—
69#16839823C2214yes—
70#16839819C2211yescharacteristic
71#16839815C2211yescharacteristic
72Q82×C61—21no—
73Q82×C61C2212yes—
74C61×#557—21no—
75C61×#557—22no—
76C61×#557C2214yes—
77C61×#557C2212yes—
78C61×#544—22no—
79C61×#544C2212yes—
80C61×#533—22no—
81C61×#533C2211yescharacteristic
82C61×#533C2211yescharacteristic
83C61×#519C2212yes—
84C2×C4×Q8×C61C2211yescharacteristic
85C2×#11758222—21no—
86C2×Q8×D61—22no—
87C2×#11758202C2312yes—
88C2×#11758202C2312yes—
89C2×#11758201C2314yes—
90#14519777C2311yescharacteristic
91#14519777C2311yescharacteristic
92#14519777C2314yes—
93#14519777C2311yescharacteristic
94#14519776—24no—
95#14519776—22no—
96#14519774C2311yescharacteristic
97#14519772—24no—
98#14519772—24no—
99#14519769—21no—
100#14519766—22no—
101#14519765—21no—
102#14519765C2314yes—
103#14519763—24no—
104#14519762—24no—
105#14519759—24no—
106#14519757—22no—
107#14519757—24no—
108#14519756—21no—
109#14519756C2312yes—
110#14519755—24no—
111#14519753—22no—
112#14519751—21no—
113#14519751C2312yes—
114Q8×Dic61—22no—
115C4×Dic122—21no—
116C4×Dic122C2314yes—
117C4×Dic122C2312yes—
118C4×D244—21no—
119C4×D244C2312yes—
120C4×D244C2314yes—
121#14519739—22no—
122#14519736—21no—
123#14519735—22no—
124#14519710C2312yes—
125#14519710C2314yes—
126#14519710C2314yes—
127#14519710C2314yes—
128#14519710C2312yes—
129#14519708C2314yes—
130#14519708C2314yes—
131C4⋊#92418C2312yes—
132C4⋊#92418C2312yes—
133C4⋊#92418C2314yes—
134C4⋊#92418C2314yes—
135#14519701C2312yes—
136#14519701C2314yes—
137#14519701C2312yes—
138#14519700C2312yes—
139#14519700C2312yes—
140#14519700C2314yes—
141#14519700C2314yes—
142#14519697C2314yes—
143#14519697C2314yes—
144#14519697C2314yes—
145C22⋊#92418C2312yes—
146C22⋊#92418C2312yes—
147#14519688C2312yes—
148#14519688C2314yes—
149C4×#92418C2311yescharacteristic
150C4×#92418C2311yescharacteristic
151C4×#92418C2314yes—
152C22×Q8×C61C2311yescharacteristic
153C4⋊Q8×C61—22no—
154C4⋊Q8×C61—24no—
155C4⋊Q8×C61—22no—
156C4⋊Q8×C61—24no—
157C4⋊Q8×C61—22no—
158C4⋊Q8×C61D411yescharacteristic
159C4⋊Q8×C61D411yescharacteristic
160C4⋊Q8×C61C2311yescharacteristic
161C4⋊Q8×C61C2311yescharacteristic
162C4⋊Q8×C61C2314yes—
163#125×C61—24no—
164#125×C61—24no—
165#125×C61—24no—
166#125×C61—22no—
167#125×C61D412yes—
168#125×C61C2312yes—
169#125×C61C2314yes—
170C22⋊Q8×C61—21no—
171C22⋊Q8×C61—21no—
172C22⋊Q8×C61—24no—
173C22⋊Q8×C61D412yes—
174C22⋊Q8×C61D412yes—
175C4×Q8×C61—22no—
176C4×Q8×C61—24no—
177C4×Q8×C61—22no—
178C4×Q8×C61—22no—
179C4×Q8×C61—22no—
180C4×Q8×C61—21no—
181C4×Q8×C61—21no—
182C4×Q8×C61—21no—
183C4×Q8×C61—21no—
184C4×Q8×C61C2312yes—
185C4×Q8×C61C2312yes—
186C4×Q8×C61C2314yes—
187C4×Q8×C61C2314yes—
188C4×Q8×C61C2312yes—
189C4×Q8×C61C2312yes—
190#117×C61—24no—
191#117×C61C2312yes—
192#117×C61C2312yes—
193C2×C4⋊C4×C61—21no—
194C2×C4⋊C4×C61C2311yescharacteristic
195C2×C42×C61—21no—
196C2×C42×C61C2311yescharacteristic
197C2×Q8×C61—24no—
198C2×Q8×C61—21no—
199C2×Q8×C61—21no—
200C2×Q8×C61—21no—
201C2×Q8×C61—24no—
202C2×Q8×C61C2×D412yes—
203C2×Q8×C61C2×D412yes—
204C2×Q8×C61C2412yes—
205C22×C4×C61—21no—
206C22×C4×C61—21no—
207C22×C4×C61C2×D411yescharacteristic
208C22×C4×C61C2×D411yescharacteristic
209C22×C4×C61C2411yescharacteristic
210C2×C22⋊D61—22no—
211Q8×D61—84no—
212#11758222—82no—
213C2×D244—21no—
214C2×D244—21no—
215C2×D244C2412yes—
216C2×C4×D61—22no—
217C2×C4×D61—24no—
218C2×Dic122—21no—
219C2×Dic122—24no—
220C2×Dic122—21no—
221C2×Dic122C2412yes—
222C4⋊C4×C61—22no—
223C4⋊C4×C61—22no—
224C4⋊C4×C61—21no—
225C4⋊C4×C61—24no—
226C4⋊C4×C61—24no—
227C4⋊C4×C61—22no—
228C4⋊C4×C61—21no—
229C4⋊C4×C61—21no—
230C4⋊C4×C61—24no—
231C4⋊C4×C61—24no—
232C4⋊C4×C61—22no—
233C4⋊C4×C61—24no—
234C4⋊C4×C61C2×D412yes—
235C4⋊C4×C61C2×D412yes—
236C4⋊C4×C61C2×D414yes—
237C4⋊C4×C61C2×D412yes—
238C4⋊C4×C61C2×D412yes—
239C4⋊C4×C61C2×D414yes—
240C4⋊C4×C61C2412yes—
241C4⋊C4×C61C2412yes—
242C4⋊C4×C61C2412yes—
243C22⋊C4×C61—22no—
244C22⋊C4×C61C2×D414yes—
245C42×C61—22no—
246C42×C61—22no—
247C42×C61—24no—
248C42×C61—24no—
249C42×C61—22no—
250C42×C61C2×D411yescharacteristic
251C42×C61C2×D411yescharacteristic
252C42×C61C2411yescharacteristic
253C42×C61C2411yescharacteristic
254C42×C61C2414yes—
255C22⋊Dic61—22no—
256#11758202—24no—
257#11758202—24no—
258#11758201—28no—
259#11758198—24no—
260#11758198—24no—
261#11758198—22no—
262#11758198—22no—
263#11758198—24no—
264C4⋊Dic61—21no—
265C4⋊Dic61—21no—
266C4⋊Dic61—21no—
267C4⋊Dic61—21no—
268C4⋊Dic61—24no—
269C4⋊Dic61C2412yes—
270C4⋊Dic61C2412yes—
271#11758196—22no—
272#11758196—24no—
273#11758196—22no—
274C4×Dic61—24no—
275C4×Dic61—22no—
276C2×#92418C2412yes—
277C2×#92418C2412yes—
278C2×#92418C2414yes—
279C23×C61C22×D411yescharacteristic
280C22×D61—22no—
281Q8×C61—24no—
282Q8×C61—24no—
283Q8×C61—22no—
284Q8×C61—42no—
285Q8×C61—42no—
286Q8×C61—84no—
287Q8×C61—81no—
288Q8×C612−1+414yes—
289C2×C4×C61—24no—
290C2×C4×C61—22no—
291C2×C4×C61—22no—
292C2×C4×C61—22no—
293C2×C4×C61—22no—
294C2×C4×C61—22no—
295C2×C4×C61—22no—
296C2×C4×C61—41no—
297C2×C4×C61—81no—
298C2×C4×C61C22×D414yes—
299C2×C4×C61C22×D411yescharacteristic
300C2×C4×C61C22×D411yescharacteristic
301C2×C4×C61C22×D414yes—
302C2×C4×C61C22×D411yescharacteristic
303C2×C4×C61C22×D411yescharacteristic
304C2×C4×C61C22×D411yescharacteristic
305C2×C4×C61C2511yescharacteristic
306C22⋊D61—82no—
307C2×Dic61—22no—
308C2×Dic61—22no—
309C2×Dic61—22no—
310D244—24no—
311D244—42no—
312D244—81no—
313C4×D61—44no—
314C4×D61—84no—
315Dic122—22no—
316Dic122—22no—
317Dic122—42no—
318Dic122—81no—
319Dic122—84no—
320#92418—24no—
321#92418—22no—
322#92418—22no—
323#61854—611nomaximal, Sylow
324C22×C61—21no—
325C22×C61—21no—
326C22×C61—21no—
327C22×C61C23×D411yescharacteristic
328C2×D61—42no—
329C2×D61—42no—
330C2×D61—42no—
331C4×C61—21no—
332C4×C61—24no—
333C4×C61—21no—
334C4×C61—24no—
335C4×C61—21no—
336C4×C61—21no—
337C4×C61—24no—
338C4×C61—42no—
339C4×C61—42no—
340C4×C61—42no—
341C4×C61#57812yes—
342C4×C61C2×2−1+412yes—
343Dic61—42no—
344Dic61—42no—
345Dic61—42no—
346#3550—611no—
347#3502D6111yescharacteristic, complemented
348#3387—612no—
349#3386—614no—
350Q8×Q16—612no—
351#3377—614no—
352#3375—612no—
353#3365—612no—
354#3243—614no—
355#3237—611no—
356#3228—611no—
357#3121—614no—
358#3112—611no—
359#3104—611no—
360#2985—611no—
361C2×C61—41no—
362C2×C61#348611yescharacteristic
363C2×C61C2×#57811yescharacteristic
364C2×C61#358511yescharacteristic
365D61—82no—
366#563—1222no—
367#559—612no—
368Q82—21no—
369Q82—612no—
370Q82C2×D6112yes—
371#557—21no—
372#557—22no—
373#557C2×D6112yes—
374#557C2×D6114yes—
375#556—612no—
376#556—1221no—
377D4×Q8—612no—
378D4×Q8—1221no—
379#544—22no—
380#544C2×D6112yes—
381#542—614no—
382#542—1222no—
383#536—1222no—
384#533—22no—
385#533—611no—
386#533C2×D6111yescharacteristic
387#533C2×D6111yescharacteristic
388#532—611no—
389#519C2×D6112yes—
390C2×C4×Q8C2×D6111yescharacteristic
391#501—612no—
392#501—614no—
393#500—611no—
394#500—614no—
395#498—611no—
396#491—612no—
397#489—614no—
398#488—614no—
399#486—611no—
400#486—611no—
401#485—614no—
402#484—612no—
403#482—614no—
404#480—612no—
405#479—614no—
406#478—614no—
407#478—614no—
408#477—612no—
409#477—614no—
410#477—612no—
411#477—612no—
412#477—614no—
413#475—612no—
414#475—614no—
415#474—614no—
416#464—614no—
417#464—614no—
418#462—612no—
419#460—612no—
420#451—612no—
421#448—612no—
422#446—612no—
423#446—614no—
424C8×Q8—612no—
425#441—614no—
426#440—614no—
427#440—614no—
428C4×Q16—612no—
429C4×Q16—612no—
430C4×Q16—614no—
431C4×SD16—614no—
432#433—614no—
433#432—611no—
434#431—611no—
435C61#6185411yescharacteristic, complemented, Sylow
436C2×#41—1221no—
437C22×Q8—1222no—
438C22×Q8C22×D6111yescharacteristic
439C2×Q16—612no—
440C2×Q16—612no—
441C2×SD16—614no—
442C4⋊Q8—22no—
443C4⋊Q8—24no—
444C4⋊Q8—22no—
445C4⋊Q8—24no—
446C4⋊Q8—22no—
447C4⋊Q8—611no—
448C4⋊Q8—611no—
449C4⋊Q8—614no—
450C4⋊Q8—611no—
451C4⋊Q8—1224no—
452C4⋊Q8—1222no—
453C4⋊Q8C22⋊D6111yescharacteristic
454C4⋊Q8C22⋊D6111yescharacteristic
455C4⋊Q8C22×D6111yescharacteristic
456C4⋊Q8C22×D6111yescharacteristic
457C4⋊Q8C22×D6114yes—
458#127—611no—
459#126—1224no—
460#126—1224no—
461#125—24no—
462#125—24no—
463#125—24no—
464#125—22no—
465#125—1221no—
466#125C22⋊D6112yes—
467#125C22×D6112yes—
468#125C22×D6114yes—
469#124—614no—
470#124—1222no—
471#124—1224no—
472#124—1221no—
473#123—1224no—
474#123—1224no—
475C22⋊Q8—21no—
476C22⋊Q8—21no—
477C22⋊Q8—24no—
478C22⋊Q8—612no—
479C22⋊Q8—1222no—
480C22⋊Q8—1221no—
481C22⋊Q8—1222no—
482C22⋊Q8—1224no—
483C22⋊Q8—1224no—
484C22⋊Q8C22⋊D6112yes—
485C22⋊Q8C22⋊D6112yes—
486#121—612no—
487#121—1221no—
488C4×Q8—22no—
489C4×Q8—22no—
490C4×Q8—22no—
491C4×Q8—24no—
492C4×Q8—22no—
493C4×Q8—21no—
494C4×Q8—21no—
495C4×Q8—21no—
496C4×Q8—21no—
497C4×Q8—612no—
498C4×Q8—614no—
499C4×Q8—1222no—
500C4×Q8—1221no—
501C4×Q8C22×D6112yes—
502C4×Q8C22×D6112yes—
503C4×Q8C22×D6114yes—
504C4×Q8C22×D6114yes—
505C4×Q8C22×D6112yes—
506C4×Q8C22×D6112yes—
507C4×D4—612no—
508C4×D4—614no—
509C4×D4—1222no—
510C4×D4—1221no—
511#117—24no—
512#117—1221no—
513#117—1222no—
514#117C22×D6112yes—
515#117C22×D6112yes—
516C2×C4⋊C4—21no—
517C2×C4⋊C4C22×D6111yescharacteristic
518C2×C42—21no—
519C2×C42C22×D6111yescharacteristic
520#107—612no—
521#107—614no—
522#107—612no—
523#107—614no—
524#107—614no—
525#106—614no—
526#106—614no—
527C4⋊C8—612no—
528C4⋊C8—612no—
529C4⋊C8—614no—
530C4⋊C8—614no—
531#103—612no—
532#103—612no—
533#103—614no—
534#103—612no—
535#103—612no—
536#103—614no—
537#103—614no—
538#102—614no—
539#102—614no—
540#102—614no—
541C22⋊C8—612no—
542C22⋊C8—612no—
543#97—612no—
544#97—614no—
545C4×C8—611no—
546C4×C8—611no—
547C4×C8—614no—
548#41—4882no—
549C2×Q8—24no—
550C2×Q8—21no—
551C2×Q8—21no—
552C2×Q8—21no—
553C2×Q8—24no—
554C2×Q8—612no—
555C2×Q8—1224no—
556C2×Q8—1221no—
557C2×Q8—1221no—
558C2×Q8—4884no—
559C2×Q8C2×C22⋊D6112yes—
560C2×Q8C2×C22⋊D6112yes—
561C2×Q8C23×D6112yes—
562C2×D4—612no—
563C2×D4—1222no—
564C2×D4—1221no—
565C2×D4—1221no—
566C22×C4—21no—
567C22×C4—21no—
568C22×C4—1224no—
569C22×C4—1222no—
570C22×C4C2×C22⋊D6111yescharacteristic
571C22×C4C2×C22⋊D6111yescharacteristic
572C22×C4C23×D6111yescharacteristic
573Q16—1224no—
574Q16—1224no—
575SD16—1228no—
576C2×C8—612no—
577C2×C8—612no—
578C2×C8—614no—
579C4⋊C4—22no—
580C4⋊C4—22no—
581C4⋊C4—21no—
582C4⋊C4—24no—
583C4⋊C4—24no—
584C4⋊C4—22no—
585C4⋊C4—21no—
586C4⋊C4—21no—
587C4⋊C4—24no—
588C4⋊C4—24no—
589C4⋊C4—22no—
590C4⋊C4—24no—
591C4⋊C4—612no—
592C4⋊C4—612no—
593C4⋊C4—1224no—
594C4⋊C4—1222no—
595C4⋊C4—1222no—
596C4⋊C4—1221no—
597C4⋊C4—1221no—
598C4⋊C4—1221no—
599C4⋊C4—1221no—
600C4⋊C4—1224no—
601C4⋊C4C2×C22⋊D6112yes—
602C4⋊C4C2×C22⋊D6112yes—
603C4⋊C4C2×C22⋊D6112yes—
604C4⋊C4C2×C22⋊D6112yes—
605C4⋊C4C2×C22⋊D6114yes—
606C4⋊C4C2×C22⋊D6114yes—
607C4⋊C4C23×D6112yes—
608C4⋊C4C23×D6112yes—
609C4⋊C4C23×D6112yes—
610C22⋊C4—22no—
611C22⋊C4—1222no—
612C22⋊C4—1224no—
613C22⋊C4—1222no—
614C22⋊C4—1224no—
615C22⋊C4—1222no—
616C22⋊C4—1224no—
617C22⋊C4C2×C22⋊D6114yes—
618C42—22no—
619C42—22no—
620C42—24no—
621C42—24no—
622C42—22no—
623C42—1222no—
624C42—1224no—
625C42C2×C22⋊D6111yescharacteristic
626C42C2×C22⋊D6111yescharacteristic
627C42C23×D6111yescharacteristic
628C42C23×D6114yes—
629C42C23×D6111yescharacteristic
630C23—1222no—
631C23C22×C22⋊D6111yescharacteristic
632Q8—24no—
633Q8—24no—
634Q8—22no—
635Q8—42no—
636Q8—42no—
637Q8—81no—
638Q8—84no—
639Q8—1222no—
640Q8—1222no—
641Q8—4884no—
642Q8—4881no—
643Q8—2442no—
644Q8#1451982514yes—
645D4—1224no—
646D4—4881no—
647D4—4882no—
648D4—2442no—
649C2×C4—24no—
650C2×C4—22no—
651C2×C4—22no—
652C2×C4—22no—
653C2×C4—22no—
654C2×C4—22no—
655C2×C4—22no—
656C2×C4—41no—
657C2×C4—81no—
658C2×C4—1222no—
659C2×C4—1222no—
660C2×C4—1222no—
661C2×C4—4884no—
662C2×C4—2444no—
663C2×C4C22×C22⋊D6111yescharacteristic
664C2×C4C22×C22⋊D6111yescharacteristic
665C2×C4C22×C22⋊D6114yes—
666C2×C4C22×C22⋊D6114yes—
667C2×C4C22×C22⋊D6111yescharacteristic
668C2×C4C22×C22⋊D6111yescharacteristic
669C2×C4C22×C22⋊D6111yescharacteristic
670C2×C4C24×D6111yescharacteristic
671C8—1224no—
672C8—1222no—
673C8—1222no—
674C22—21no—
675C22—21no—
676C22—21no—
677C22—2442no—
678C22—2442no—
679C22—2442no—
680C22C23×C22⋊D6111yescharacteristic
681C4—24no—
682C4—21no—
683C4—21no—
684C4—24no—
685C4—21no—
686C4—21no—
687C4—24no—
688C4—42no—
689C4—42no—
690C4—42no—
691C4—2442no—
692C4—2442no—
693C4—2442no—
694C4#1684057712yes—
695C4C2×#1451982512yes—
696C2—41no—
697C2—4882no—
698C2#2145947011yescharacteristic
699C2C2×#1684057711yescharacteristic
700C2#2146059811yescharacteristic
subgroupG/H
centerC22C23×C22⋊D61
commutator, cosocleC2×C4×C61C25
FrattiniC2×C4C24×D61
FittingC61×#3502C2
socleC22×C61C23×D4
prime radical#16840312C22
hypercenter#3502D61
hypocenterC61#61854
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ #533 ◁ #3502

Complements

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

NH ≅ G/Ntype
2. #21458751C2semidirect
3. #21458736C2semidirect
4. Q8×#11758202C2semidirect
5. #21458661C2semidirect
6. #21458633C2semidirect
7. #21458515C2semidirect
8. #21457102C2semidirect
9. #21457033C2semidirect
11. #21456048C2semidirect
13. #21455858C2semidirect
14. #21454914C2semidirect
15. C61×#3502C2semidirect
347. #3502D61semidirect
435. C61#61854semidirect

Maximal subgroups

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

subgroupindex
#214602302
#214587512
#214587362
Q8×#117582022
#214586612
#214586332
#214585152
#214571022
#214570332
#214569552
#214560482
#214559412
#214558582
#214549142
C61×#35022
#6185461