Finite Group Databasebeta

← #29646559

Subgroups — #29646559

Order 15616.

988 conjugacy classes of subgroups.

The subgroup lattice has 618 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#533×D61C211yesmaximal, characteristic
2#21454450C211yesmaximal, characteristic
3#431×D61C211yesmaximal, characteristic
4#21444241C211yesmaximal, characteristic, complemented
5#21444240C211yesmaximal, characteristic, complemented
6#21444239C211yesmaximal, characteristic, complemented
7#21444220C214yesmaximal, complemented
8#21444212C214yesmaximal, complemented
9C61×#1485C211yesmaximal, characteristic, complemented
10#16840319C2211yescharacteristic
11#16840318—21no—
12#16840317C411yescharacteristic
13#16840317C411yescharacteristic
14#16840316—21no—
15#16840316C2212yes—
16#16840315—21no—
17#16840314—21no—
18#16840314C412yes—
19C4⋊Q8×D61C411yescharacteristic
20C4⋊Q8×D61C411yescharacteristic
21#125×D61—21no—
22C22⋊Q8×D61—21no—
23C22⋊Q8×D61C412yes—
24C4×Q8×D61—21no—
25C4×Q8×D61C2212yes—
26#117×D61—21no—
27C2×C42×D61C2211yescharacteristic
28#16839817—21no—
29#16839817—21no—
30#16839815C2211yescharacteristic
31#16839814C2212yes—
32#16839813—21no—
33#16839813C2212yes—
34#16839813C2212yes—
35#16839812—21no—
36#16839785—21no—
37C4⋊C8×D61C2212yes—
38#16839725—21no—
39C22⋊C8×D61—21no—
40C4×#11758194C2211yescharacteristic
41C4×C8×D61—21no—
42#16839494C2214yescomplemented
43Q8⋊#92418C2214yescomplemented
44#16839492C2214yescomplemented
45C61×#533C2211yescharacteristic
46C61×#431C2211yescharacteristic
47C4×M16×C61C2211yescharacteristic
48C61×Q8⋊C8C2214yescomplemented
49C2×Q8×D61—21no—
50C2×Q8×D61C2×C412yes—
51C22×C4×D61C2×C411yescharacteristic
52C22×C4×D61D411yescharacteristic
53C22×C4×D61D411yescharacteristic
54C2×#11758194—21no—
55C2×C8×D61—22no—
56#14519777C2×C411yescharacteristic
57#14519777C2×C411yescharacteristic
58#14519776—21no—
59#14519776—21no—
60#14519776—21no—
61#14519776C2×C412yes—
62#14519775—21no—
63#14519775C2×C412yes—
64#14519769—21no—
65#14519768—21no—
66#14519768—21no—
67#14519767—22no—
68#14519767—22no—
69#14519757—21no—
70#14519757C2×C412yes—
71#14519756—21no—
72#14519756C2×C412yes—
73#14519755—22no—
74#14519755—22no—
75#14519754—22no—
76#14519754—22no—
77#14519753—21no—
78#14519753C2×C412yes—
79Q8×Dic61—21no—
80Q8×Dic61C2×C412yes—
81C4×Dic122—21no—
82C4×Dic122—22no—
83C4×Dic122—22no—
84C4×Dic122C2312yes—
85#14519742—21no—
86#14519742—22no—
87#14519742—22no—
88#14519742C2×C412yes—
89#14519736—21no—
90#14519735—21no—
91#14519735—21no—
92#14519734—22no—
93#14519733—22no—
94C4⋊C4×D61—21no—
95C4⋊C4×D61—21no—
96C4⋊C4×D61—21no—
97C4⋊C4×D61—22no—
98C4⋊C4×D61—22no—
99C4⋊C4×D61C2×C412yes—
100C22⋊C4×D61—22no—
101C2×C4×Dic61C2×C411yescharacteristic
102C2×C4×Dic61D411yescharacteristic
103C2×C4×Dic61D411yescharacteristic
104C42×D61—22no—
105C42×D61C2×C411yescharacteristic
106C42×D61C2×C411yescharacteristic
107C42×D61D411yescharacteristic
108C42×D61D411yescharacteristic
109C42×D61D411yescharacteristic
110C42×D61D411yescharacteristic
111C42×D61C2311yescharacteristic
112C42×D61C2311yescharacteristic
113C4⋊#92418C2312yes—
114#14519706—22no—
115#14519706—22no—
116#14519706—22no—
117C22⋊#92418—21no—
118#14519689—22no—
119#14519689—22no—
120#14519689—22no—
121#14519688—21no—
122C4×#92418—21no—
123C4×#92418C2312yes—
124C8×Dic61—22no—
125C2×M16×C61—21no—
126C4⋊Q8×C61C2×C411yescharacteristic
127C4⋊Q8×C61C2×C411yescharacteristic
128#125×C61—21no—
129C22⋊Q8×C61—21no—
130C22⋊Q8×C61C2×C412yes—
131C4×Q8×C61—21no—
132C4×Q8×C61C2312yes—
133#117×C61—21no—
134C2×C42×C61C2311yescharacteristic
135C4⋊C8×C61C2312yes—
136C22⋊C8×C61—21no—
137#97×C61—21no—
138C4×C8×C61—21no—
139C4×C8×C61C2312yes—
140C23×D61C22⋊C411yescharacteristic
141C2×Q8×C61—21no—
142C2×Q8×C61C22×C412yes—
143C22×C4×C61C22×C411yescharacteristic
144C22×C4×C61C2×D411yescharacteristic
145C22×C4×C61C2×D411yescharacteristic
146C22×Dic61C22⋊C411yescharacteristic
147C22×Dic61C22⋊C411yescharacteristic
148C22×Dic61C22⋊C411yescharacteristic
149Q8×D61—28no—
150Q8×D61—44no—
151C2×C4×D61—24no—
152C2×C4×D61—24no—
153C2×C4×D61—22no—
154C2×C4×D61—22no—
155C2×C4×D61—42no—
156C2×C4×D61C22⋊C411yescharacteristic
157C2×C4×D61C22⋊C411yescharacteristic
158C2×C4×D61C22⋊C411yescharacteristic
159C2×C4×D61C22⋊C411yescharacteristic
160C2×C4×D61C22⋊C411yescharacteristic
161C2×C4×D61C22⋊C411yescharacteristic
162C2×C4×D61C22×C411yescharacteristic
163C2×C4×D61C22×C411yescharacteristic
164C2×C4×D61C2×D411yescharacteristic
165C2×C4×D61C2×D411yescharacteristic
166C2×C4×D61C2×D411yescharacteristic
167C2×C4×D61C2×D411yescharacteristic
168C2×Dic122—21no—
169C2×Dic122—22no—
170C2×Dic122—22no—
171C2×Dic122C22×C412yes—
172M16×C61—42no—
173C2×C8×C61—22no—
174C2×C8×C61—22no—
175C4⋊C4×C61—21no—
176C4⋊C4×C61—21no—
177C4⋊C4×C61—21no—
178C4⋊C4×C61—22no—
179C4⋊C4×C61—22no—
180C4⋊C4×C61C22×C412yes—
181C22⋊C4×C61—22no—
182C42×C61—22no—
183C42×C61C22×C411yescharacteristic
184C42×C61C2×D411yescharacteristic
185C42×C61C2×D411yescharacteristic
186C42×C61C2411yescharacteristic
187C22⋊Dic61—22no—
188#11758198—22no—
189#11758198—22no—
190C4⋊Dic61—21no—
191C4⋊Dic61—21no—
192C4⋊Dic61—21no—
193C4⋊Dic61—22no—
194C4⋊Dic61—22no—
195C4⋊Dic61C22×C412yes—
196#11758196—22no—
197#11758196—22no—
198#11758196—22no—
199#11758196—22no—
200#11758196—22no—
201#11758196—22no—
202#11758196—22no—
203#11758196—22no—
204C4×Dic61—22no—
205C4×Dic61—22no—
206C4×Dic61—22no—
207C4×Dic61C22⋊C411yescharacteristic
208C4×Dic61C22⋊C411yescharacteristic
209C4×Dic61C22⋊C411yescharacteristic
210C4×Dic61C22⋊C411yescharacteristic
211C4×Dic61C22⋊C411yescharacteristic
212C4×Dic61C22⋊C411yescharacteristic
213C4×Dic61C22×C411yescharacteristic
214C4×Dic61C22×C411yescharacteristic
215C4×Dic61C2×D411yescharacteristic
216C4×Dic61C2×D411yescharacteristic
217C4×Dic61C2×D411yescharacteristic
218C4×Dic61C2×D411yescharacteristic
219#11758194—42no—
220C2×#92418—22no—
221C2×#92418—22no—
222C8×D61—44no—
223C23×C61C2×C22⋊C411yescharacteristic
224C22×D61—22no—
225C22×D61—22no—
226C22×D61—41no—
227C22×D61C2×C22⋊C411yescharacteristic
228C22×D61C2×C22⋊C411yescharacteristic
229Q8×C61—24no—
230Q8×C61—42no—
231C2×C4×C61—22no—
232C2×C4×C61—22no—
233C2×C4×C61—22no—
234C2×C4×C61—22no—
235C2×C4×C61—22no—
236C2×C4×C61C2×C22⋊C411yescharacteristic
237C2×C4×C61C2×C22⋊C411yescharacteristic
238C2×C4×C61C2×C22⋊C411yescharacteristic
239C2×C4×C61C23×C411yescharacteristic
240C2×C4×C61C22×D411yescharacteristic
241C2×C4×C61C22×D411yescharacteristic
242C2×Dic61—22no—
243C2×Dic61—22no—
244C2×Dic61—22no—
245C2×Dic61—22no—
246C2×Dic61—41no—
247C2×Dic61C2×C22⋊C411yescharacteristic
248C2×Dic61C2×C22⋊C411yescharacteristic
249C2×Dic61C2×C22⋊C411yescharacteristic
250C2×Dic61C2×C22⋊C411yescharacteristic
251C2×Dic61C2×C22⋊C411yescharacteristic
252C2×Dic61C2×C22⋊C411yescharacteristic
253C4×D61—22no—
254C4×D61—22no—
255C4×D61—22no—
256C4×D61—22no—
257C4×D61—22no—
258C4×D61—22no—
259C4×D61—22no—
260C4×D61—22no—
261C4×D61—41no—
262C4×D61—41no—
263C4×D61—44no—
264C4×D61—44no—
265C4×D61#13114yes—
266C4×D61#13714yes—
267Dic122—24no—
268Dic122—44no—
269Dic122—42no—
270Dic122—44no—
271C8×C61—42no—
272C8×C61—42no—
273#92418—42no—
274#92418—42no—
275#17464—611nomaximal, Sylow
276C22×C61—21no—
277C22×C61—21no—
278C22×C61—21no—
279C22×C61C22×C22⋊C411yescharacteristic
280C2×D61—21no—
281C2×D61—21no—
282C2×D61—21no—
283C2×D61—21no—
284C2×D61—44no—
285C2×D61#40812yes—
286C2×D61#41912yes—
287C4×C61—21no—
288C4×C61—21no—
289C4×C61—21no—
290C4×C61—42no—
291C4×C61—42no—
292C4×C61#42112yes—
293C4×C61C2×#13112yes—
294C4×C61C2×#13712yes—
295Dic61—21no—
296Dic61—21no—
297Dic61—21no—
298Dic61—21no—
299Dic61—42no—
300Dic61—42no—
301Dic61#40812yes—
302Dic61#41912yes—
303C2×#533—611no—
304#2920—611no—
305C2×#431—611no—
306#1485—611no—
307#1485—611no—
308#1485—611no—
309#1485D6111yescharacteristic, complemented
310#1481—614no—
311#1479—614no—
312C2×C61—41no—
313C2×C61C2×#40811yescharacteristic
314C2×C61C2×#41911yescharacteristic
315C2×C61C2×#42111yescharacteristic
316D61—22no—
317D61—22no—
318#533—611no—
319#533—611no—
320#533—612no—
321#533—612no—
322#533—611no—
323#533—1221no—
324#533—1221no—
325#533—1221no—
326#533—1221no—
327#533C2×D6111yescharacteristic
328C2×C4⋊Q8—611no—
329C2×C4⋊Q8—611no—
330C2×#125—1221no—
331C2×C22⋊Q8—612no—
332C2×C22⋊Q8—1221no—
333C2×C4×Q8—612no—
334C2×C4×Q8—1221no—
335C2×#117—1221no—
336C22×C42—611no—
337#432—1221no—
338#432—1221no—
339#431—612no—
340#431—612no—
341#431—611no—
342#431—612no—
343#431—1221no—
344#431—1221no—
345#431C2×D6111yescharacteristic
346#423—1221no—
347C2×C4⋊C8—612no—
348#407—1221no—
349C2×C22⋊C8—1221no—
350C4×M16—611no—
351C4×M16C2×D6111yescharacteristic
352C2×C4×C8—1221no—
353Q8⋊C8—614no—
354Q8⋊C8—614no—
355Q8⋊C8—614no—
356Q8⋊C8C2×D6114yescomplemented
357C61#1746411yescharacteristic, complemented, Sylow
358C22×Q8—612no—
359C22×Q8—1221no—
360C23×C4—611no—
361C23×C4—611no—
362C23×C4—611no—
363C2×M16—21no—
364C2×M16—1221no—
365C22×C8—1222no—
366C4⋊Q8—612no—
367C4⋊Q8—612no—
368C4⋊Q8—611no—
369C4⋊Q8—611no—
370C4⋊Q8—1221no—
371C4⋊Q8—1221no—
372C4⋊Q8—1221no—
373C4⋊Q8—1221no—
374C4⋊Q8C4×D6111yescharacteristic
375C4⋊Q8C4×D6111yescharacteristic
376#125—21no—
377#125—1222no—
378#125—1222no—
379#125—1221no—
380#125—1221no—
381#125—1221no—
382C22⋊Q8—21no—
383C22⋊Q8—612no—
384C22⋊Q8—612no—
385C22⋊Q8—612no—
386C22⋊Q8—1222no—
387C22⋊Q8—1222no—
388C22⋊Q8—1222no—
389C22⋊Q8—1222no—
390C22⋊Q8—1221no—
391C22⋊Q8—1221no—
392C22⋊Q8—1221no—
393C22⋊Q8C4×D6112yes—
394C4×Q8—21no—
395C4×Q8—612no—
396C4×Q8—612no—
397C4×Q8—612no—
398C4×Q8—1222no—
399C4×Q8—1222no—
400C4×Q8—1222no—
401C4×Q8—1221no—
402C4×Q8—1222no—
403C4×Q8—1221no—
404C4×Q8—1221no—
405C4×Q8C22×D6112yes—
406#117—21no—
407#117—1222no—
408#117—1222no—
409#117—1221no—
410#117—1221no—
411#117—1221no—
412C2×C4⋊C4—612no—
413C2×C4⋊C4—1221no—
414C2×C4⋊C4—1221no—
415C2×C4⋊C4—1222no—
416C2×C4⋊C4—1222no—
417C2×C4⋊C4—1221no—
418C2×C22⋊C4—1222no—
419C2×C42—611no—
420C2×C42—611no—
421C2×C42—611no—
422C2×C42—611no—
423C2×C42—611no—
424C2×C42—611no—
425C2×C42—611no—
426C2×C42—611no—
427C2×C42—611no—
428C2×C42—611no—
429C2×C42—611no—
430C2×C42—1222no—
431C2×C42C22×D6111yescharacteristic
432C4⋊C8—612no—
433C4⋊C8—1222no—
434C4⋊C8—1222no—
435C4⋊C8—1222no—
436C4⋊C8C22×D6112yes—
437C22⋊C8—21no—
438C22⋊C8—1222no—
439C22⋊C8—1222no—
440C22⋊C8—1222no—
441C22⋊C8—1221no—
442#97—21no—
443#97—1221no—
444C4×C8—21no—
445C4×C8—612no—
446C4×C8—1222no—
447C4×C8—1221no—
448C4×C8C22×D6112yes—
449C24—611no—
450C2×Q8—21no—
451C2×Q8—612no—
452C2×Q8—1222no—
453C2×Q8—1222no—
454C2×Q8—1228no—
455C2×Q8—1221no—
456C2×Q8—2444no—
457C2×Q8C2×C4×D6112yes—
458C22×C4—611no—
459C22×C4—611no—
460C22×C4—611no—
461C22×C4—611no—
462C22×C4—611no—
463C22×C4—611no—
464C22×C4—611no—
465C22×C4—611no—
466C22×C4—611no—
467C22×C4—611no—
468C22×C4—611no—
469C22×C4—611no—
470C22×C4—611no—
471C22×C4—611no—
472C22×C4—611no—
473C22×C4—1224no—
474C22×C4—1224no—
475C22×C4—1222no—
476C22×C4—1222no—
477C22×C4—2442no—
478C22×C4C2×C4×D6111yescharacteristic
479C22×C4D4×D6111yescharacteristic
480C22×C4D4×D6111yescharacteristic
481M16—42no—
482M16—2442no—
483C2×C8—22no—
484C2×C8—22no—
485C2×C8—1222no—
486C2×C8—1222no—
487C2×C8—2444no—
488C4⋊C4—22no—
489C4⋊C4—21no—
490C4⋊C4—21no—
491C4⋊C4—21no—
492C4⋊C4—22no—
493C4⋊C4—612no—
494C4⋊C4—1222no—
495C4⋊C4—1222no—
496C4⋊C4—1222no—
497C4⋊C4—1222no—
498C4⋊C4—1222no—
499C4⋊C4—1222no—
500C4⋊C4—1222no—
501C4⋊C4—1222no—
502C4⋊C4—1222no—
503C4⋊C4—1222no—
504C4⋊C4—1221no—
505C4⋊C4—1221no—
506C4⋊C4—1221no—
507C4⋊C4C2×C4×D6112yes—
508C22⋊C4—22no—
509C22⋊C4—1222no—
510C22⋊C4—1222no—
511C22⋊C4—1222no—
512C42—22no—
513C42—611no—
514C42—611no—
515C42—611no—
516C42—611no—
517C42—611no—
518C42—611no—
519C42—611no—
520C42—611no—
521C42—611no—
522C42—611no—
523C42—611no—
524C42—611no—
525C42—1222no—
526C42—1222no—
527C42—1222no—
528C42C2×C4×D6111yescharacteristic
529C42D4×D6111yescharacteristic
530C42D4×D6111yescharacteristic
531C42C23×D6111yescharacteristic
532C23—611no—
533C23—611no—
534C23—1222no—
535C23—1222no—
536C23—2441no—
537C23C22⋊C4×D6111yescharacteristic
538Q8—24no—
539Q8—42no—
540Q8—1224no—
541Q8—2444no—
542Q8—2444no—
543Q8—2442no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—611no—
550C2×C4—611no—
551C2×C4—614no—
552C2×C4—614no—
553C2×C4—611no—
554C2×C4—611no—
555C2×C4—611no—
556C2×C4—611no—
557C2×C4—1222no—
558C2×C4—1222no—
559C2×C4—1222no—
560C2×C4—1222no—
561C2×C4—1222no—
562C2×C4—1222no—
563C2×C4—1222no—
564C2×C4—1222no—
565C2×C4—1222no—
566C2×C4—1222no—
567C2×C4—1222no—
568C2×C4—1222no—
569C2×C4—2441no—
570C2×C4—2441no—
571C2×C4—2444no—
572C2×C4—2444no—
573C2×C4—2441no—
574C2×C4C22⋊C4×D6111yescharacteristic
575C2×C4C22⋊C4×D6111yescharacteristic
576C2×C4C22⋊C4×D6111yescharacteristic
577C2×C4C22×C4×D6111yescharacteristic
578C2×C4C2×D4×D6111yescharacteristic
579C2×C4C2×D4×D6111yescharacteristic
580C8—42no—
581C8—42no—
582C8—2442no—
583C8—2442no—
584C22—21no—
585C22—21no—
586C22—21no—
587C22—612no—
588C22—612no—
589C22—1221no—
590C22—1221no—
591C22—1221no—
592C22—1221no—
593C22—2444no—
594C22C2×C22⋊C4×D6111yescharacteristic
595C4—21no—
596C4—21no—
597C4—21no—
598C4—42no—
599C4—42no—
600C4—612no—
601C4—612no—
602C4—1221no—
603C4—1221no—
604C4—1221no—
605C4—1221no—
606C4—2442no—
607C4—2442no—
608C4#1683977712yes—
609C4#131×D6112yes—
610C4#137×D6112yes—
611C2—41no—
612C2—1222no—
613C2—1222no—
614C2#408×D6111yescharacteristic
615C2#419×D6111yescharacteristic
616C2C2×#1683977711yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC42C23×D61
FittingC61×#1485C2
socleC22×C61C22×C22⋊C4
prime radicalC23×D61C22⋊C4
cosocleC42×C61C24
hypercenter#1485D61
hypocenterC61#17464
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1485

Complements

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

NH ≅ G/Ntype
4. #21444241C2semidirect
5. #21444240C2semidirect
6. #21444239C2semidirect
7. #21444220C2semidirect
8. #21444212C2semidirect
9. C61×#1485C2semidirect
42. #16839494C22semidirect
43. Q8⋊#92418C22semidirect
44. #16839492C22semidirect
48. C61×Q8⋊C8C22semidirect
309. #1485D61semidirect
356. Q8⋊C8C2×D61semidirect
357. C61#17464semidirect

Maximal subgroups

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

subgroupindex
#533×D612
#214544502
#431×D612
#214442412
#214442402
#214442392
#214442202
#214442122
C61×#14852
#1746461