Finite Group Databasebeta

← #29684502

Subgroups — #29684502

Order 15616.

1106 conjugacy classes of subgroups.

The subgroup lattice has 628 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21459222C211yesmaximal, characteristic
2#21454236C211yesmaximal, characteristic
3#21448299C211yesmaximal, characteristic
4#21446764C211yesmaximal, characteristic, complemented
5#21446763C211yesmaximal, characteristic, complemented
6#21446762C211yesmaximal, characteristic, complemented
7#21446761C212yesmaximal, complemented
8#21446761C212yesmaximal, complemented
9#21445527C212yesmaximal
10#339×D61C212yesmaximal
11C61×#1986C211yesmaximal, characteristic, complemented
12C2×M16×D61C2212yes—
13#16840392C412yescomplemented
14#16840392C412yescomplemented
15#16840391—21no—
16#16840391—21no—
17#16840389C412yescomplemented
18#16840389C412yescomplemented
19#16840387—21no—
20#16840387—21no—
21#16840319C2211yescharacteristic
22#16840316—22no—
23#16840316—22no—
24#16840315C2212yes—
25C2×#14519734C2211yescharacteristic
26C2×#14519734C2211yescharacteristic
27#117×D61C2212yes—
28C2×C42×D61C2211yescharacteristic
29#16839719—22no—
30#16839716—22no—
31C4×#11758194C2211yescharacteristic
32M16×Dic61C2212yes—
33#16839659—21no—
34#16839657—21no—
35#16839636—21no—
36#16839633—21no—
37C4×#11758196C2211yescharacteristic
38C4×#11758196C2211yescharacteristic
39C4×C4⋊Dic61C2211yescharacteristic
40#16839523C2212yes—
41#16839523C2212yes—
42#16839522C2212yes—
43#16839522C2212yes—
44#16839521C2212yes—
45#16839521C2212yes—
46C61×#533C2211yescharacteristic
47C4×M16×C61C2211yescharacteristic
48C4×C4⋊C4×C61C2211yescharacteristic
49C61×#339C2212yes—
50C61×#339C2212yes—
51C22×C4×D61—21no—
52C22×C4×D61C2311yescharacteristic
53C2×#11758194C2312yes—
54C2×#11758189—22no—
55M16×D61—24no—
56M16×D61Q818yescomplemented
57C2×C8×D61—22no—
58#14519777C2×C411yescharacteristic, complemented
59#14519777C2×C411yescharacteristic, complemented
60#14519776—22no—
61#14519776—22no—
62#14519776C2×C412yescomplemented
63#14519776C2×C412yescomplemented
64#14519775—22no—
65#14519775—22no—
66#14519769C2×C412yescomplemented
67#14519768C2×C412yescomplemented
68#14519768C2×C412yescomplemented
69#14519767—22no—
70#14519767—22no—
71#14519767—22no—
72#14519767—22no—
73#14519757—22no—
74#14519757—22no—
75#14519756—21no—
76#14519756—21no—
77#14519755—22no—
78#14519755—22no—
79#14519755—22no—
80#14519755—22no—
81#14519754—22no—
82#14519754—22no—
83#14519753—22no—
84#14519753—22no—
85C4×Dic122—21no—
86C4×Dic122—21no—
87C4×Dic122—22no—
88C4×Dic122—22no—
89#14519736C2312yes—
90#14519735C2×C412yes—
91#14519735C2×C412yes—
92#14519734—22no—
93#14519734—24no—
94#14519734C2×C412yes—
95#14519734C2×C412yes—
96#14519734D412yes—
97#14519734D412yes—
98#14519733—22no—
99C2×C4⋊Dic61—21no—
100C2×#11758196—21no—
101C2×#11758196—21no—
102C2×#11758196—21no—
103C2×#11758196D411yescharacteristic
104C2×#11758196D411yescharacteristic
105C4⋊C4×D61C2×C412yes—
106C4⋊C4×D61C2×C412yes—
107C2×#11758198—21no—
108C2×#11758198D411yescharacteristic
109C2×#11758198D411yescharacteristic
110C22⋊C4×D61—22no—
111C2×C4×Dic61—21no—
112C2×C4×Dic61C2311yescharacteristic
113C2×C4×Dic61C2311yescharacteristic
114C2×C4×Dic61C2311yescharacteristic
115C42×D61—22no—
116C42×D61—24no—
117#14519688—21no—
118#14519687—22no—
119C4×#92418—21no—
120C8×Dic61—22no—
121#14519684—21no—
122#14519684—21no—
123#14519684—21no—
124C2×M16×C61C2312yes—
125C4⋊Q8×C61C2×C411yescharacteristic, complemented
126C4⋊Q8×C61C2×C411yescharacteristic, complemented
127#125×C61C2×C412yescomplemented
128C22⋊Q8×C61—21no—
129C22⋊Q8×C61—21no—
130C4×Q8×C61—21no—
131C4×Q8×C61—21no—
132#117×C61C2312yes—
133C2×C4⋊C4×C61—21no—
134C2×C42×C61C2311yescharacteristic
135C2×C42×C61C2311yescharacteristic
136C2×C42×C61C2311yescharacteristic
137#97×C61—21no—
138C4×C8×C61—21no—
139#95×C61—21no—
140C23×D61C2×D411yescharacteristic
141C2×Q8×C61—21no—
142C2×Q8×C61—21no—
143C22×C4×C61—21no—
144C22×C4×C61—21no—
145C22×C4×C61C2×D411yescharacteristic
146C22×C4×C61C2×D411yescharacteristic
147C22×C4×C61C2411yescharacteristic
148C22×Dic61—21no—
149C22×Dic61—21no—
150C22×Dic61C2×D411yescharacteristic
151C22×Dic61C2×D411yescharacteristic
152C22×Dic61C2×D411yescharacteristic
153C2×C4×D61—22no—
154C2×C4×D61—22no—
155C2×C4×D61—42no—
156C2×C4×D61—42no—
157C2×C4×D61—42no—
158C2×C4×D61C22×C412yes—
159C2×C4×D61C2×D412yes—
160C2×C4×D61C2×Q814yes—
161C2×C4×D61#4114yes—
162C2×Dic122—21no—
163C2×Dic122—21no—
164C2×Dic122—22no—
165C2×Dic122—22no—
166M16×C61—22no—
167M16×C61C2×Q814yes—
168C2×C8×C61—22no—
169C4⋊C4×C61—22no—
170C4⋊C4×C61—22no—
171C4⋊C4×C61—21no—
172C4⋊C4×C61—21no—
173C4⋊C4×C61—42no—
174C4⋊C4×C61C22×C412yes—
175C4⋊C4×C61C22×C412yes—
176C22⋊C4×C61—22no—
177C42×C61—22no—
178C42×C61—22no—
179C42×C61—22no—
180C42×C61C22×C411yescharacteristic
181C42×C61C22×C411yescharacteristic
182C42×C61C2×D411yescharacteristic
183C42×C61C2×D411yescharacteristic
184C22⋊Dic61—22no—
185#11758198—22no—
186#11758198—22no—
187#11758198—42no—
188#11758198C22⋊C412yes—
189#11758198C22⋊C412yes—
190#11758198C22⋊C412yes—
191#11758198C22⋊C412yes—
192C4⋊Dic61—21no—
193C4⋊Dic61—21no—
194C4⋊Dic61—22no—
195C4⋊Dic61—22no—
196C4⋊Dic61—42no—
197C4⋊Dic61C22×C412yes—
198C4⋊Dic61C22×C412yes—
199#11758196—22no—
200#11758196—22no—
201#11758196—22no—
202#11758196—22no—
203#11758196—22no—
204#11758196—22no—
205#11758196—22no—
206#11758196—22no—
207#11758196—42no—
208#11758196—42no—
209#11758196—42no—
210#11758196C22⋊C412yes—
211#11758196C22⋊C412yes—
212#11758196C22⋊C412yes—
213#11758196C22⋊C412yes—
214C4×Dic61—22no—
215C4×Dic61—22no—
216C4×Dic61—22no—
217C4×Dic61—22no—
218C4×Dic61—22no—
219C4×Dic61—42no—
220C4×Dic61C22×C412yes—
221C4×Dic61C2×D412yes—
222#11758194—22no—
223#11758194C2×Q814yes—
224C2×#92418—22no—
225#11758189—28no—
226C8×D61—28no—
227C23×C61C22×D411yescharacteristic
228C22×D61—41no—
229C22×D61—41no—
230C22×D61C2×C22⋊C412yes—
231C22×D61#12412yes—
232C22×D61C4⋊Q812yes—
233Q8×C61—81no—
234Q8×C61—81no—
235C2×C4×C61—22no—
236C2×C4×C61—22no—
237C2×C4×C61—22no—
238C2×C4×C61—41no—
239C2×C4×C61—41no—
240C2×C4×C61—41no—
241C2×C4×C61—41no—
242C2×C4×C61—41no—
243C2×C4×C61—41no—
244C2×C4×C61—41no—
245C2×C4×C61C2×C22⋊C411yescharacteristic
246C2×C4×C61C2×C22⋊C411yescharacteristic
247C2×C4×C61C2×C22⋊C411yescharacteristic
248C2×C4×C61C2×C22⋊C411yescharacteristic
249C2×C4×C61C23×C411yescharacteristic
250C2×C4×C61C22×D411yescharacteristic
251C2×C4×C61C22×Q812yes—
252C2×C4×C61C2×#4112yes—
253C2×Dic61—22no—
254C2×Dic61—22no—
255C2×Dic61—22no—
256C2×Dic61—41no—
257C2×Dic61—41no—
258C2×Dic61—41no—
259C2×Dic61—41no—
260C2×Dic61—41no—
261C2×Dic61—41no—
262C2×Dic61—41no—
263C2×Dic61C2×C22⋊C412yes—
264C2×Dic61C2×C22⋊C412yes—
265C2×Dic61C2×C22⋊C412yes—
266C2×Dic61#12412yes—
267C2×Dic61C4⋊Q812yes—
268C4×D61—48no—
269C4×D61—82no—
270C4×D61—82no—
271C4×D61C4×Q818yes—
272C4×D61C22⋊Q814yes—
273C4×D61C22⋊Q814yes—
274Dic122—82no—
275Dic122—82no—
276Dic122—81no—
277Dic122—81no—
278C8×C61—24no—
279#92418—24no—
280#21307—611nomaximal, Sylow
281C22×C61—21no—
282C22×C61—21no—
283C22×C61C22×C22⋊C411yescharacteristic
284C22×C61C2×#12411yescharacteristic
285C22×C61C2×C4⋊Q811yescharacteristic
286C2×D61—22no—
287C2×D61—22no—
288C2×D61—42no—
289C2×D61—42no—
290C2×D61#39114yes—
291C4×C61—21no—
292C4×C61—21no—
293C4×C61—21no—
294C4×C61—21no—
295C4×C61—42no—
296C4×C61—42no—
297C4×C61—42no—
298C4×C61—81no—
299C4×C61C2×C4×Q812yes—
300C4×C61C2×C22⋊Q812yes—
301Dic61—22no—
302Dic61—22no—
303Dic61—42no—
304Dic61—42no—
305Dic61—42no—
306Dic61—81no—
307Dic61#39114yes—
308#3458—611no—
309#2873—611no—
310#2296—611no—
311#1986—611no—
312#1986—611no—
313#1986—612no—
314#1986—612no—
315#1986—611no—
316#1986D6111yescharacteristic, complemented
317#1733—612no—
318C2×#339—612no—
319C2×C61—21no—
320C2×C61—21no—
321C2×C61C2×#39111yescharacteristic
322C2×C61#290111yescharacteristic
323C2×C61#290111yescharacteristic
324D61—24no—
325C22×M16—612no—
326#544—612no—
327#544—612no—
328#544—612no—
329#544—612no—
330#544—1221no—
331#544—1221no—
332#543—1221no—
333#543—1221no—
334#533—612no—
335#533—611no—
336#533—1222no—
337#533—1222no—
338#533C2×D6111yescharacteristic
339C2×#117—611no—
340C2×#117—611no—
341C2×#117—612no—
342C22×C42—611no—
343#405—1222no—
344#405—1222no—
345C4×M16—612no—
346C4×M16—611no—
347C4×M16C2×D6111yescharacteristic
348#389—1221no—
349#389—1221no—
350#385—1221no—
351#385—1221no—
352C4×C4⋊C4—611no—
353C4×C4⋊C4—611no—
354C4×C4⋊C4—611no—
355C4×C4⋊C4C2×D6111yescharacteristic
356#339—612no—
357#339—612no—
358#339—612no—
359#339—612no—
360#339—612no—
361#339—612no—
362#339C2×D6112yes—
363#339C2×D6112yes—
364C61#2130711yescharacteristic, complemented, Sylow
365C23×C4—611no—
366C23×C4—1221no—
367C2×M16—612no—
368C2×M16—618no—
369C2×M16—1222no—
370C2×M16—1224no—
371C2×M16C22×D6112yes—
372C22×C8—1222no—
373C4⋊Q8—612no—
374C4⋊Q8—612no—
375C4⋊Q8—611no—
376C4⋊Q8—611no—
377C4⋊Q8—1222no—
378C4⋊Q8—1222no—
379C4⋊Q8—1222no—
380C4⋊Q8—1222no—
381C4⋊Q8C4×D6111yescharacteristic, complemented
382C4⋊Q8C4×D6111yescharacteristic, complemented
383#125—612no—
384#125—612no—
385#125—612no—
386#125—1222no—
387#125—1222no—
388#125—1222no—
389#125—1222no—
390#125C4×D6112yescomplemented
391C22⋊Q8—21no—
392C22⋊Q8—21no—
393C22⋊Q8—1222no—
394C22⋊Q8—1222no—
395C22⋊Q8—1222no—
396C22⋊Q8—1222no—
397C22⋊Q8—1222no—
398C22⋊Q8—1222no—
399C22⋊Q8—1222no—
400C22⋊Q8—1222no—
401C22⋊Q8—1222no—
402C22⋊Q8—1222no—
403C22⋊Q8—1221no—
404C22⋊Q8—1221no—
405C4×Q8—21no—
406C4×Q8—21no—
407C4×Q8—1222no—
408C4×Q8—1222no—
409C4×Q8—1221no—
410C4×Q8—1221no—
411#117—612no—
412#117—612no—
413#117—612no—
414#117—612no—
415#117—612no—
416#117—612no—
417#117—612no—
418#117—1222no—
419#117—1224no—
420#117—1222no—
421#117C22×D6112yes—
422C2×C4⋊C4—21no—
423C2×C4⋊C4—611no—
424C2×C4⋊C4—611no—
425C2×C4⋊C4—612no—
426C2×C4⋊C4—612no—
427C2×C4⋊C4—1221no—
428C2×C4⋊C4—1221no—
429C2×C4⋊C4—1221no—
430C2×C4⋊C4—1221no—
431C2×C22⋊C4—611no—
432C2×C22⋊C4—611no—
433C2×C22⋊C4—1222no—
434C2×C22⋊C4—1221no—
435C2×C42—611no—
436C2×C42—611no—
437C2×C42—611no—
438C2×C42—1222no—
439C2×C42—1224no—
440C2×C42—1221no—
441C2×C42C22×D6111yescharacteristic
442C2×C42C22×D6111yescharacteristic
443C2×C42C22×D6111yescharacteristic
444#97—21no—
445#97—1222no—
446#97—1221no—
447C4×C8—21no—
448C4×C8—1222no—
449C4×C8—1221no—
450#95—21no—
451#95—1221no—
452#95—1221no—
453#95—1221no—
454C24—611no—
455C2×Q8—21no—
456C2×Q8—21no—
457C2×Q8—1222no—
458C2×Q8—1222no—
459C2×Q8—1221no—
460C2×Q8—1221no—
461C22×C4—21no—
462C22×C4—21no—
463C22×C4—611no—
464C22×C4—612no—
465C22×C4—612no—
466C22×C4—614no—
467C22×C4—614no—
468C22×C4—611no—
469C22×C4—611no—
470C22×C4—1222no—
471C22×C4—1222no—
472C22×C4—1221no—
473C22×C4—1221no—
474C22×C4—2442no—
475C22×C4—2442no—
476C22×C4—2442no—
477C22×C4D4×D6111yescharacteristic
478C22×C4D4×D6111yescharacteristic
479C22×C4C23×D6111yescharacteristic
480M16—22no—
481M16—614no—
482M16—1228no—
483M16—1222no—
484M16Q8×D6114yes—
485C2×C8—22no—
486C2×C8—1228no—
487C2×C8—1222no—
488C4⋊C4—21no—
489C4⋊C4—21no—
490C4⋊C4—22no—
491C4⋊C4—22no—
492C4⋊C4—42no—
493C4⋊C4—612no—
494C4⋊C4—612no—
495C4⋊C4—612no—
496C4⋊C4—612no—
497C4⋊C4—612no—
498C4⋊C4—612no—
499C4⋊C4—1222no—
500C4⋊C4—1222no—
501C4⋊C4—1222no—
502C4⋊C4—1222no—
503C4⋊C4—1222no—
504C4⋊C4—1222no—
505C4⋊C4—1222no—
506C4⋊C4—1222no—
507C4⋊C4—1222no—
508C4⋊C4—1222no—
509C4⋊C4—1221no—
510C4⋊C4—1221no—
511C4⋊C4—2442no—
512C4⋊C4—2442no—
513C4⋊C4—2442no—
514C4⋊C4—2442no—
515C4⋊C4C2×C4×D6112yes—
516C4⋊C4C2×C4×D6112yes—
517C22⋊C4—22no—
518C22⋊C4—612no—
519C22⋊C4—612no—
520C22⋊C4—612no—
521C22⋊C4—612no—
522C22⋊C4—1222no—
523C22⋊C4—1222no—
524C22⋊C4—1222no—
525C22⋊C4—2442no—
526C42—22no—
527C42—22no—
528C42—22no—
529C42—612no—
530C42—612no—
531C42—1222no—
532C42—1222no—
533C42—1222no—
534C42—1222no—
535C42—1222no—
536C42—2442no—
537C42C2×C4×D6111yescharacteristic
538C42C2×C4×D6111yescharacteristic
539C42D4×D6111yescharacteristic
540C42D4×D6111yescharacteristic
541C23—612no—
542C23—612no—
543C23—612no—
544C23—2441no—
545C23—2441no—
546C23C2×D4×D6111yescharacteristic
547Q8—81no—
548Q8—81no—
549Q8—4881no—
550Q8—4881no—
551Q8—4882no—
552Q8—4882no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—612no—
564C2×C4—612no—
565C2×C4—612no—
566C2×C4—618no—
567C2×C4—614no—
568C2×C4—614no—
569C2×C4—612no—
570C2×C4—612no—
571C2×C4—1222no—
572C2×C4—1222no—
573C2×C4—1222no—
574C2×C4—4882no—
575C2×C4—4882no—
576C2×C4—2448no—
577C2×C4—2441no—
578C2×C4—2441no—
579C2×C4—2441no—
580C2×C4—2441no—
581C2×C4—2441no—
582C2×C4—2441no—
583C2×C4—2441no—
584C2×C4C22⋊C4×D6111yescharacteristic
585C2×C4C22⋊C4×D6111yescharacteristic
586C2×C4C22⋊C4×D6111yescharacteristic
587C2×C4C22⋊C4×D6111yescharacteristic
588C2×C4C22×C4×D6111yescharacteristic
589C2×C4C2×D4×D6111yescharacteristic
590C2×C4C2×Q8×D6112yes—
591C2×C4#41×D6112yes—
592C8—24no—
593C8—1224no—
594C22—21no—
595C22—21no—
596C22—614no—
597C22—1222no—
598C22—1222no—
599C22—2442no—
600C22—2442no—
601C22C2×C22⋊C4×D6111yescharacteristic
602C22#124×D6111yescharacteristic
603C22C4⋊Q8×D6111yescharacteristic
604C4—21no—
605C4—21no—
606C4—21no—
607C4—21no—
608C4—42no—
609C4—42no—
610C4—42no—
611C4—81no—
612C4—614no—
613C4—1222no—
614C4—1222no—
615C4—4881no—
616C4—2442no—
617C4—2442no—
618C4—2442no—
619C4C4×Q8×D6112yes—
620C4C22⋊Q8×D6112yes—
621C2—21no—
622C2—21no—
623C2—1224no—
624C2#391×D6111yescharacteristic
625C2#2145436311yescharacteristic
626C2#2145436511yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1986C2
socleC22×C61C22×C22⋊C4
prime radicalC23×D61C2×D4
cosocleC22×C4×C61C24
hypercenter#1986D61
hypocenterC61#21307
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1986

Complements

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

NH ≅ G/Ntype
4. #21446764C2semidirect
5. #21446763C2semidirect
6. #21446762C2semidirect
7. #21446761C2semidirect
8. #21446761C2semidirect
11. C61×#1986C2semidirect
13. #16840392C4semidirect
14. #16840392C4semidirect
17. #16840389C4semidirect
18. #16840389C4semidirect
56. M16×D61Q8semidirect
58. #14519777C2×C4semidirect
59. #14519777C2×C4semidirect
62. #14519776C2×C4semidirect
63. #14519776C2×C4semidirect
66. #14519769C2×C4semidirect
67. #14519768C2×C4semidirect
68. #14519768C2×C4semidirect
125. C4⋊Q8×C61C2×C4semidirect
126. C4⋊Q8×C61C2×C4semidirect
127. #125×C61C2×C4semidirect
316. #1986D61semidirect
364. C61#21307semidirect
381. C4⋊Q8C4×D61semidirect
382. C4⋊Q8C4×D61semidirect
390. #125C4×D61semidirect

Maximal subgroups

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

subgroupindex
#214592222
#214542362
#214482992
#214467642
#214467632
#214467622
#214467612
#214455272
#339×D612
C61×#19862
#2130761