Finite Group Databasebeta

← #30617829

Subgroups — #30617829

Order 15616.

1450 conjugacy classes of subgroups.

The subgroup lattice has 696 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1D61⋊#431C211yesmaximal, characteristic, complemented
2#21462266C211yesmaximal, characteristic, complemented
3#21461987C214yesmaximal, complemented
4#21461948C214yesmaximal, complemented
5#21461658C212yesmaximal, complemented
6#21461657C212yesmaximal
7C2×#124×D61C211yesmaximal, characteristic
8#16840781—21no—
9#16840780—21no—
10#16840779—21no—
11#16840778—21no—
12D61⋊(C4⋊C8)C2212yescomplemented
13C2×#14519840C2218yescomplemented
14C2×#14519839C2218yes—
15D61⋊(C22⋊C8)C2212yes—
16C4×D61⋊C8C2212yescomplemented
17#16840707—21no—
18#16840706C2212yescomplemented
19#16840705—21no—
20#16840704C2212yes—
21C4×C4⋊#6923C2212yescomplemented
22C4×C22⋊#6923—21no—
23#16840698—21no—
24C22×Q8×D61C2211yescharacteristic
25C22×D4×D61C2211yescharacteristic
26C2×#14519766C411yescharacteristic, complemented
27C2×#14519765C411yescharacteristic, complemented
28C2×#14519764C411yescharacteristic, complemented
29C2×#14519763—21no—
30C2×#14519763—21no—
31#124×D61—21no—
32#124×D61—21no—
33#124×D61—21no—
34#124×D61—21no—
35#124×D61C412yescomplemented
36#124×D61C412yescomplemented
37#124×D61C2212yes—
38#124×D61C2212yes—
39C2×C22⋊C4×D61—21no—
40C2×C22⋊C4×D61—21no—
41C2×C42×D61C2211yescharacteristic
42C2×#124×C61C411yescharacteristic, complemented
43C2×D61⋊C8C2314yes—
44C2×C4⋊#6923C2314yes—
45C2×C22⋊#6923—22no—
46C2×C4×#6923—22no—
47#14519844—22no—
48C4⋊#92420—22no—
49#14519840—48no—
50#14519839—48no—
51#14519834—22no—
52C22⋊#92420—22no—
53C4×#92420—22no—
54C4×#92420—22no—
55(C2×#6923)⋊C4—22no—
56(C2×#6923)⋊C4—22no—
57(C2×#6923)⋊C4—22no—
58(C2×#6923)⋊C4—22no—
59C24×D61—21no—
60C22×Dic122—21no—
61C22×Dic122C2×C411yescharacteristic
62C2×Q8×D61—21no—
63C2×Q8×D61—21no—
64C2×Q8×D61—44no—
65C2×Q8×D61C2×C412yes—
66C2×Q8×D61C2312yes—
67C22×D244C2×C411yescharacteristic
68C22×C22⋊D61—21no—
69C2×D4×D61—21no—
70C2×D4×D61—21no—
71C2×D4×D61—44no—
72C2×D4×D61C2×C412yes—
73C2×D4×D61C2312yes—
74C22×C4×D61—21no—
75C22×C4×D61D411yescharacteristic
76C22×C4×D61D411yescharacteristic
77C22×C4×D61C2311yescharacteristic
78#14519766—21no—
79#14519766—21no—
80#14519766C2×C412yes—
81#14519766C2×C412yes—
82#14519765—21no—
83#14519765—21no—
84#14519765C2×C412yes—
85#14519765C2×C412yes—
86#14519764—21no—
87#14519764—21no—
88#14519764C2×C412yes—
89#14519764C2×C412yes—
90#14519763—22no—
91#14519763—22no—
92#14519763—22no—
93#14519763—22no—
94#14519763—22no—
95#14519763—22no—
96#14519763—22no—
97#14519763—22no—
98C2×C22⋊Dic61—21no—
99C2×C22⋊Dic61—21no—
100C2×#11758198—21no—
101C2×#11758198—21no—
102C2×#11758198—21no—
103C2×#11758198—21no—
104C22⋊C4×D61—22no—
105C22⋊C4×D61—22no—
106C22⋊C4×D61—22no—
107C22⋊C4×D61—22no—
108C22⋊C4×D61—22no—
109C22⋊C4×D61—22no—
110C22⋊C4×D61—22no—
111C22⋊C4×D61—22no—
112C2×C4×Dic61C2×C411yescharacteristic
113C2×C4×Dic61D411yescharacteristic
114C2×C4×Dic61D411yescharacteristic
115C42×D61C2×C411yescharacteristic
116C42×D61C2×C411yescharacteristic
117C42×D61D411yescharacteristic
118C42×D61D411yescharacteristic
119C42×D61D411yescharacteristic
120C42×D61D411yescharacteristic
121C42×D61C2311yescharacteristic
122C42×D61C2311yescharacteristic
123C22×Q8×C61C2×C411yescharacteristic
124C22×D4×C61C2×C411yescharacteristic
125#124×C61—21no—
126#124×C61—21no—
127#124×C61C2×C412yes—
128#124×C61C2×C412yes—
129C2×C22⋊C4×C61—21no—
130C2×C22⋊C4×C61—21no—
131C2×C42×C61C2×C411yescharacteristic
132C24×C61—21no—
133C23×D61—21no—
134C23×D61—22no—
135C23×D61—22no—
136C23×D61—44no—
137C23×D61—44no—
138C23×D61—44no—
139C23×D61C2×D411yescharacteristic
140C22×#6923—24no—
141C2×Q8×C61—21no—
142C2×Q8×C61—42no—
143C2×Q8×C61C22×C412yes—
144C2×D4×C61—21no—
145C2×D4×C61—42no—
146C2×D4×C61C22×C412yes—
147C22×C4×C61—21no—
148C22×C4×C61C22⋊C411yescharacteristic
149C22×C4×C61C22⋊C411yescharacteristic
150C22×C4×C61C22×C411yescharacteristic
151C2×C22⋊D61—22no—
152C2×C22⋊D61—22no—
153C2×C22⋊D61—44no—
154C22×Dic61—21no—
155C22×Dic61C22⋊C411yescharacteristic
156C22×Dic61C22⋊C411yescharacteristic
157C22×Dic61C2×D411yescharacteristic
158Q8×D61—28no—
159Q8×D61—44no—
160Q8×D61—44no—
161Q8×D61—44no—
162D4×D61—28no—
163D4×D61—44no—
164D4×D61—44no—
165D4×D61—44no—
166C2×D244—21no—
167C2×D244—42no—
168C2×D244C22×C412yes—
169C2×C4×D61—24no—
170C2×C4×D61—24no—
171C2×C4×D61—22no—
172C2×C4×D61—22no—
173C2×C4×D61—42no—
174C2×C4×D61—44no—
175C2×C4×D61C22⋊C411yescharacteristic
176C2×C4×D61C22⋊C411yescharacteristic
177C2×C4×D61C22⋊C411yescharacteristic
178C2×C4×D61C22⋊C411yescharacteristic
179C2×C4×D61C22×C411yescharacteristic
180C2×C4×D61C2×D411yescharacteristic
181C2×C4×D61C2×D411yescharacteristic
182C2×C4×D61C2×D411yescharacteristic
183C2×C4×D61C2×D411yescharacteristic
184C2×C4×D61C2×D411yescharacteristic
185C2×C4×D61C2×D411yescharacteristic
186C2×C4×D61C2411yescharacteristic
187C2×Dic122—21no—
188C2×Dic122—22no—
189C2×Dic122—22no—
190C2×Dic122—42no—
191C2×Dic122—44no—
192C2×Dic122C22×C412yes—
193C22⋊#6923—44no—
194C2×#92420—24no—
195C4⋊#6923—28no—
196C4×#6923—44no—
197D61⋊C8—28no—
198C22⋊C4×C61—22no—
199C22⋊C4×C61—22no—
200C22⋊C4×C61—22no—
201C22⋊C4×C61—22no—
202C42×C61C22⋊C411yescharacteristic
203C42×C61C22⋊C411yescharacteristic
204C42×C61C22×C411yescharacteristic
205C42×C61C22×C411yescharacteristic
206C22⋊Dic61—22no—
207C22⋊Dic61—22no—
208C22⋊Dic61—22no—
209C22⋊Dic61—22no—
210#11758198—22no—
211#11758198—22no—
212#11758198—22no—
213#11758198—22no—
214#11758198—22no—
215#11758198—22no—
216#11758198—22no—
217#11758198—22no—
218C4×Dic61C22⋊C411yescharacteristic
219C4×Dic61C22⋊C411yescharacteristic
220C4×Dic61C22⋊C411yescharacteristic
221C4×Dic61C22⋊C411yescharacteristic
222C4×Dic61C22⋊C411yescharacteristic
223C4×Dic61C22⋊C411yescharacteristic
224C4×Dic61C22×C411yescharacteristic
225C4×Dic61C22×C411yescharacteristic
226C4×Dic61C2×D411yescharacteristic
227C4×Dic61C2×D411yescharacteristic
228C4×Dic61C2×D411yescharacteristic
229C4×Dic61C2×D411yescharacteristic
230C23×C61—22no—
231C23×C61—42no—
232C23×C61—42no—
233C23×C61—42no—
234C23×C61C2×C22⋊C411yescharacteristic
235C22×D61—22no—
236C22×D61—22no—
237C22×D61—22no—
238C22×D61—44no—
239C22×D61—44no—
240C22×D61—42no—
241C22×D61—42no—
242C22×D61—42no—
243C22×D61—44no—
244C22×D61—44no—
245C22×D61—44no—
246C22×D61—44no—
247C22×D61—816no—
248C22×D61C2×C22⋊C411yescharacteristic
249C22×D61#13514yes—
250C22×D61C22×D411yescharacteristic
251C2×#6923—44no—
252C2×#6923—44no—
253C2×#6923—44no—
254Q8×C61—24no—
255Q8×C61—42no—
256D4×C61—24no—
257D4×C61—42no—
258C2×C4×C61—22no—
259C2×C4×C61—22no—
260C2×C4×C61—22no—
261C2×C4×C61—41no—
262C2×C4×C61—42no—
263C2×C4×C61C2×C22⋊C411yescharacteristic
264C2×C4×C61C2×C22⋊C411yescharacteristic
265C2×C4×C61C2×C22⋊C411yescharacteristic
266C2×C4×C61C2×C22⋊C411yescharacteristic
267C2×C4×C61C2×C22⋊C411yescharacteristic
268C2×C4×C61C23×C411yescharacteristic
269C22⋊D61—44no—
270C22⋊D61—44no—
271C2×Dic61—22no—
272C2×Dic61—22no—
273C2×Dic61—22no—
274C2×Dic61—41no—
275C2×Dic61—42no—
276C2×Dic61C2×C22⋊C411yescharacteristic
277C2×Dic61C2×C22⋊C411yescharacteristic
278C2×Dic61C2×C22⋊C411yescharacteristic
279C2×Dic61C2×C22⋊C411yescharacteristic
280C2×Dic61C2×C22⋊C411yescharacteristic
281C2×Dic61C22×D411yescharacteristic
282D244—24no—
283D244—42no—
284C4×D61—22no—
285C4×D61—22no—
286C4×D61—22no—
287C4×D61—22no—
288C4×D61—22no—
289C4×D61—22no—
290C4×D61—22no—
291C4×D61—22no—
292C4×D61—22no—
293C4×D61—41no—
294C4×D61—41no—
295C4×D61—44no—
296C4×D61—44no—
297C4×D61H(5,𝔽2)14yes—
298Dic122—24no—
299Dic122—44no—
300Dic122—42no—
301Dic122—44no—
302#92420—44no—
303#21806—611nomaximal, Sylow
304C22×C61—21no—
305C22×C61—21no—
306C22×C61—42no—
307C22×C61—42no—
308C22×C61—42no—
309C22×C61—84no—
310C22×C61#41612yes—
311C22×C61C22×C22⋊C411yescharacteristic
312C2×D61—21no—
313C2×D61—21no—
314C2×D61—28no—
315C2×D61—42no—
316C2×D61—42no—
317C2×D61—42no—
318C2×D61—88no—
319C2×D61—84no—
320C2×D61—88no—
321C2×D61#41612yes—
322C2×D61#53512yes—
323C2×D61C2×#13512yes—
324C2×D61#57512yes—
325#6923—84no—
326C4×C61—21no—
327C4×C61—21no—
328C4×C61—21no—
329C4×C61—21no—
330C4×C61—21no—
331C4×C61—42no—
332C4×C61#51812yes—
333Dic61—21no—
334Dic61—21no—
335Dic61—21no—
336Dic61—21no—
337Dic61—21no—
338Dic61—42no—
339Dic61#53512yes—
340C22×#124—611no—
341C2×#431—611no—
342#2283—611no—
343#1983—614no—
344#1958—614no—
345#1787—612no—
346#1786—612no—
347C2×C61—22no—
348C2×C61—82no—
349C2×C61#230911yescharacteristic
350C2×C61C2×#41611yescharacteristic
351C2×C61#289611yescharacteristic
352D61—22no—
353D61—82no—
354D61#321114yes—
355C23×Q8—611no—
356C23×D4—611no—
357C2×#124—611no—
358C2×#124—611no—
359C2×#124—612no—
360C2×#124—612no—
361C2×#124—612no—
362C2×#124—612no—
363C2×#124—611no—
364C2×#124—1221no—
365C2×#124—1221no—
366C2×#124—1221no—
367C2×#124—1221no—
368C2×#124—1221no—
369C2×#124—1221no—
370C2×#124#692311yescharacteristic, complemented
371C22×C22⋊C4—1221no—
372C22×C22⋊C4—1221no—
373C22×C42—611no—
374#431—1221no—
375#431—1221no—
376#431—1221no—
377#431—1221no—
378C2×C4⋊C8—612no—
379C2×#103—618no—
380C2×#102—618no—
381C2×C22⋊C8—612no—
382C2×C4×C8—612no—
383#383—1221no—
384#382—612no—
385#381—1221no—
386#380—612no—
387C4×C4⋊C4—612no—
388C4×C22⋊C4—1221no—
389#376—1221no—
390C61#2180611yescharacteristic, complemented, Sylow
391C25—1221no—
392C22×Q8—612no—
393C22×Q8—612no—
394C22×Q8—611no—
395C22×Q8—1221no—
396C22×Q8—1221no—
397C22×Q8—1221no—
398C22×Q8—2444no—
399C22×Q8C2×#692311yescharacteristic
400C22×D4—612no—
401C22×D4—612no—
402C22×D4—611no—
403C22×D4—1221no—
404C22×D4—1221no—
405C22×D4—1221no—
406C22×D4—2444no—
407C22×D4C2×#692311yescharacteristic
408C23×C4—611no—
409C23×C4—611no—
410C23×C4—611no—
411C23×C4—1221no—
412C22×C8—614no—
413#124—21no—
414#124—21no—
415#124—612no—
416#124—612no—
417#124—612no—
418#124—612no—
419#124—612no—
420#124—612no—
421#124—1222no—
422#124—1222no—
423#124—1222no—
424#124—1222no—
425#124—1222no—
426#124—1222no—
427#124—1222no—
428#124—1222no—
429#124—1221no—
430#124—1221no—
431#124—1221no—
432#124—1221no—
433#124—1221no—
434#124—1221no—
435#124C2×#692312yes—
436#124C2×#692312yes—
437C2×C4⋊C4—614no—
438C2×C22⋊C4—21no—
439C2×C22⋊C4—21no—
440C2×C22⋊C4—1222no—
441C2×C22⋊C4—1222no—
442C2×C22⋊C4—1222no—
443C2×C22⋊C4—1222no—
444C2×C22⋊C4—1222no—
445C2×C22⋊C4—1222no—
446C2×C22⋊C4—1222no—
447C2×C22⋊C4—1222no—
448C2×C22⋊C4—1221no—
449C2×C22⋊C4—1221no—
450C2×C22⋊C4—1221no—
451C2×C22⋊C4—1221no—
452C2×C22⋊C4—1221no—
453C2×C22⋊C4—1221no—
454C2×C22⋊C4—1222no—
455C2×C42—611no—
456C2×C42—611no—
457C2×C42—611no—
458C2×C42—611no—
459C2×C42—611no—
460C2×C42—611no—
461C2×C42—611no—
462C2×C42—611no—
463C2×C42—611no—
464C2×C42—611no—
465C2×C42—611no—
466C2×C42—1222no—
467C2×C42C2×#692311yescharacteristic
468C4⋊C8—1222no—
469C4⋊C8—1222no—
470#103—2448no—
471#102—2448no—
472C22⋊C8—1222no—
473C22⋊C8—1222no—
474C4×C8—1222no—
475C4×C8—1222no—
476#95—1222no—
477#95—1222no—
478#95—1222no—
479#95—1222no—
480C24—21no—
481C24—611no—
482C24—1222no—
483C24—1222no—
484C24—1221no—
485C24—2444no—
486C24—2444no—
487C24—2444no—
488C2×Q8—21no—
489C2×Q8—42no—
490C2×Q8—612no—
491C2×Q8—1222no—
492C2×Q8—1222no—
493C2×Q8—1228no—
494C2×Q8—1221no—
495C2×Q8—2442no—
496C2×Q8—2444no—
497C2×Q8—2444no—
498C2×Q8—2444no—
499C2×Q8—2444no—
500C2×Q8C22×#692312yes—
501C2×D4—21no—
502C2×D4—42no—
503C2×D4—612no—
504C2×D4—1228no—
505C2×D4—1222no—
506C2×D4—1222no—
507C2×D4—1221no—
508C2×D4—2442no—
509C2×D4—2444no—
510C2×D4—2444no—
511C2×D4—2444no—
512C2×D4—2444no—
513C2×D4C22×#692312yes—
514C22×C4—21no—
515C22×C4—611no—
516C22×C4—611no—
517C22×C4—611no—
518C22×C4—611no—
519C22×C4—611no—
520C22×C4—611no—
521C22×C4—611no—
522C22×C4—611no—
523C22×C4—611no—
524C22×C4—611no—
525C22×C4—611no—
526C22×C4—611no—
527C22×C4—611no—
528C22×C4—611no—
529C22×C4—611no—
530C22×C4—1224no—
531C22×C4—1224no—
532C22×C4—1222no—
533C22×C4—1222no—
534C22×C4—1221no—
535C22×C4—1224no—
536C22×C4—2442no—
537C22×C4—2444no—
538C22×C4C22⋊#692311yescharacteristic
539C22×C4C22⋊#692311yescharacteristic
540C22×C4C22×#692311yescharacteristic
541C2×C8—1224no—
542C2×C8—1228no—
543C4⋊C4—1228no—
544C22⋊C4—22no—
545C22⋊C4—22no—
546C22⋊C4—22no—
547C22⋊C4—22no—
548C22⋊C4—1222no—
549C22⋊C4—1222no—
550C22⋊C4—1222no—
551C22⋊C4—1222no—
552C22⋊C4—1222no—
553C22⋊C4—1222no—
554C22⋊C4—1222no—
555C22⋊C4—1222no—
556C22⋊C4—1222no—
557C22⋊C4—1222no—
558C22⋊C4—1222no—
559C22⋊C4—1222no—
560C22⋊C4—2444no—
561C42—611no—
562C42—611no—
563C42—611no—
564C42—611no—
565C42—611no—
566C42—611no—
567C42—611no—
568C42—611no—
569C42—611no—
570C42—611no—
571C42—611no—
572C42—611no—
573C42—2444no—
574C42C22⋊#692311yescharacteristic
575C42C22⋊#692311yescharacteristic
576C42C22×#692311yescharacteristic
577C42C22×#692311yescharacteristic
578C23—22no—
579C23—42no—
580C23—42no—
581C23—42no—
582C23—611no—
583C23—614no—
584C23—611no—
585C23—1222no—
586C23—1222no—
587C23—1222no—
588C23—48816no—
589C23—2442no—
590C23—2442no—
591C23—2444no—
592C23—2444no—
593C23—2444no—
594C23—2444no—
595C23—2444no—
596C23—2444no—
597C23—2442no—
598C23C2×C22⋊#692311yescharacteristic
599Q8—24no—
600Q8—42no—
601Q8—1224no—
602Q8—2442no—
603Q8—2444no—
604Q8—2444no—
605D4—24no—
606D4—42no—
607D4—1224no—
608D4—2442no—
609D4—2444no—
610D4—2444no—
611C2×C4—22no—
612C2×C4—22no—
613C2×C4—22no—
614C2×C4—41no—
615C2×C4—42no—
616C2×C4—611no—
617C2×C4—614no—
618C2×C4—611no—
619C2×C4—611no—
620C2×C4—611no—
621C2×C4—611no—
622C2×C4—611no—
623C2×C4—1222no—
624C2×C4—1222no—
625C2×C4—1222no—
626C2×C4—1222no—
627C2×C4—1222no—
628C2×C4—1222no—
629C2×C4—1222no—
630C2×C4—1222no—
631C2×C4—1222no—
632C2×C4—1222no—
633C2×C4—1222no—
634C2×C4—1222no—
635C2×C4—2441no—
636C2×C4—2441no—
637C2×C4—2444no—
638C2×C4—2444no—
639C2×C4—2441no—
640C2×C4—2444no—
641C2×C4—2444no—
642C2×C4—2444no—
643C2×C4—2442no—
644C2×C4C2×C22⋊#692311yescharacteristic
645C2×C4C2×C22⋊#692311yescharacteristic
646C2×C4C2×C22⋊#692311yescharacteristic
647C2×C4C2×C22⋊#692311yescharacteristic
648C2×C4C2×C22⋊#692311yescharacteristic
649C2×C4C23×#692311yescharacteristic
650C8—2444no—
651C22—21no—
652C22—21no—
653C22—42no—
654C22—42no—
655C22—42no—
656C22—84no—
657C22—612no—
658C22—612no—
659C22—612no—
660C22—612no—
661C22—1228no—
662C22—1221no—
663C22—1221no—
664C22—4884no—
665C22—4888no—
666C22—4888no—
667C22—2442no—
668C22—2442no—
669C22—2442no—
670C22#1684075212yes—
671C22C22×C22⋊#692311yescharacteristic
672C4—21no—
673C4—21no—
674C4—21no—
675C4—21no—
676C4—21no—
677C4—42no—
678C4—612no—
679C4—1221no—
680C4—1221no—
681C4—1221no—
682C4—1221no—
683C4—1221no—
684C4—4884no—
685C4—2442no—
686C4#1684082912yes—
687C2—22no—
688C2—82no—
689C2—614no—
690C2—1222no—
691C2—4882no—
692C2#2146229211yescharacteristic
693C2C2×#1684075211yescharacteristic
694C2#2146243911yescharacteristic
subgroupG/H
centerC22C22×C22⋊#6923
commutatorC2×C4×C61C23×C4
FrattiniC2×C4C23×#6923
FittingC2×#124×C61C4
socleC22×C61C22×C22⋊C4
prime radicalC22×D4×D61C22
cosocleC2×C4×D61C24
hypercenterC2×#124#6923
hypocenterC61#21806
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#124

Complements

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

NH ≅ G/Ntype
1. D61⋊#431C2semidirect
2. #21462266C2semidirect
3. #21461987C2semidirect
4. #21461948C2semidirect
5. #21461658C2semidirect
12. D61⋊(C4⋊C8)C22semidirect
13. C2×#14519840C22semidirect
16. C4×D61⋊C8C22semidirect
18. #16840706C22semidirect
21. C4×C4⋊#6923C22semidirect
26. C2×#14519766C4semidirect
27. C2×#14519765C4semidirect
28. C2×#14519764C4semidirect
35. #124×D61C4semidirect
36. #124×D61C4semidirect
42. C2×#124×C61C4semidirect
370. C2×#124#6923semidirect
390. C61#21806semidirect

Maximal subgroups

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

subgroupindex
D61⋊#4312
#214622662
#214619872
#214619482
#214616582
#214616572
C2×#124×D612
#2180661