Finite Group Databasebeta

← #29647966

Subgroups — #29647966

Order 15616.

1036 conjugacy classes of subgroups.

The subgroup lattice has 624 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#533×D61C211yesmaximal, characteristic
2#21454449C211yesmaximal, characteristic
3C2×#16839714C211yesmaximal, characteristic
4#21444435C211yesmaximal, characteristic, complemented
5#21444434C211yesmaximal, characteristic, complemented
6#21444433C211yesmaximal, characteristic, complemented
7#21444419C214yesmaximal, complemented
8#21444412C214yesmaximal, complemented
9C61×#1531C211yesmaximal, characteristic, complemented
10C22×#11758189—21no—
11#16840319C2211yescharacteristic
12#16840318C2212yes—
13#16840317C411yescharacteristic
14#16840317C411yescharacteristic
15#16840316—21no—
16#16840316—21no—
17#16840315C412yes—
18#16840314—21no—
19#16840314—21no—
20C4⋊Q8×D61C411yescharacteristic
21C4⋊Q8×D61C411yescharacteristic
22#125×D61C2212yes—
23C22⋊Q8×D61—21no—
24C22⋊Q8×D61—21no—
25C4×Q8×D61—21no—
26C4×Q8×D61—21no—
27#117×D61C412yes—
28C2×C42×D61C2211yescharacteristic
29#16839819C2211yescharacteristic
30#16839817C2212yes—
31#16839814—21no—
32#16839812—21no—
33#16839784—21no—
34#16839724—21no—
35#16839714—21no—
36#16839714—21no—
37#16839714C2212yes—
38#16839714C2212yes—
39#16839714C2212yes—
40C2×#14519688C2211yescharacteristic
41C2×C8×Dic61—21no—
42#16839504C2214yescomplemented
43#16839503C2214yescomplemented
44#16839503C2214yescomplemented
45C61×#533C2211yescharacteristic
46C61×#432C2211yescharacteristic
47C2×#97×C61C2211yescharacteristic
48C61×#330C2214yescomplemented
49C2×Q8×D61—21no—
50C2×Q8×D61—21no—
51C22×C4×D61C2×C411yescharacteristic
52C22×C4×D61D411yescharacteristic
53C22×C4×D61D411yescharacteristic
54C2×#11758189—28no—
55C2×#11758189—22no—
56C2×#11758189—22no—
57C2×#11758189—22no—
58C22×#92418—21no—
59#14519777C2×C411yescharacteristic
60#14519777C2×C411yescharacteristic
61#14519776—21no—
62#14519776—21no—
63#14519776C2×C412yes—
64#14519776C2×C412yes—
65#14519775—21no—
66#14519775—21no—
67#14519769C2312yes—
68#14519768C2×C412yes—
69#14519768C2×C412yes—
70#14519767—22no—
71#14519767—22no—
72#14519757—21no—
73#14519757—21no—
74#14519756—21no—
75#14519756—21no—
76#14519755—22no—
77#14519755—22no—
78#14519754—22no—
79#14519754—22no—
80#14519753—21no—
81#14519753—21no—
82Q8×Dic61—21no—
83Q8×Dic61—21no—
84C4×Dic122—21no—
85C4×Dic122—21no—
86C4×Dic122—22no—
87C4×Dic122—22no—
88#14519742—21no—
89#14519742—21no—
90#14519742—22no—
91#14519742—22no—
92#14519736C2×C412yes—
93#14519735C2×C412yes—
94#14519735C2×C412yes—
95#14519734—22no—
96#14519733—22no—
97C4⋊C4×D61—21no—
98C4⋊C4×D61—21no—
99C4⋊C4×D61—22no—
100C4⋊C4×D61—22no—
101C4⋊C4×D61C2×C412yes—
102C4⋊C4×D61C2×C412yes—
103C22⋊C4×D61—22no—
104C2×C4×Dic61C2×C411yescharacteristic
105C2×C4×Dic61D411yescharacteristic
106C2×C4×Dic61D411yescharacteristic
107C42×D61—22no—
108C42×D61C2×C411yescharacteristic
109C42×D61C2×C411yescharacteristic
110C42×D61D411yescharacteristic
111C42×D61D411yescharacteristic
112C42×D61D411yescharacteristic
113C42×D61D411yescharacteristic
114C42×D61C2311yescharacteristic
115C42×D61C2311yescharacteristic
116C4⋊#92418—21no—
117#14519706—22no—
118C22⋊#92418—21no—
119#14519689—22no—
120#14519688—21no—
121#14519688C2312yes—
122#14519688C2312yes—
123C8×Dic61—22no—
124C8×Dic61—22no—
125C8×Dic61—22no—
126C22×C8×C61—21no—
127C4⋊Q8×C61C2×C411yescharacteristic
128C4⋊Q8×C61C2×C411yescharacteristic
129#125×C61C2312yes—
130C22⋊Q8×C61—21no—
131C22⋊Q8×C61—21no—
132C4×Q8×C61—21no—
133C4×Q8×C61—21no—
134#117×C61C2×C412yes—
135C2×C42×C61C2311yescharacteristic
136C4⋊C8×C61—21no—
137C22⋊C8×C61—21no—
138#97×C61—21no—
139#97×C61C2312yes—
140#97×C61C2312yes—
141C23×D61C22⋊C411yescharacteristic
142C2×Q8×C61—21no—
143C2×Q8×C61—21no—
144C22×C4×C61C22×C411yescharacteristic
145C22×C4×C61C2×D411yescharacteristic
146C22×C4×C61C2×D411yescharacteristic
147C22×Dic61C22⋊C411yescharacteristic
148C22×Dic61C22⋊C411yescharacteristic
149C22×Dic61C22⋊C411yescharacteristic
150Q8×D61—44no—
151Q8×D61—44no—
152C2×C4×D61—24no—
153C2×C4×D61—24no—
154C2×C4×D61—22no—
155C2×C4×D61—22no—
156C2×C4×D61—42no—
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×D61C22×C411yescharacteristic
165C2×C4×D61C2×D411yescharacteristic
166C2×C4×D61C2×D411yescharacteristic
167C2×C4×D61C2×D411yescharacteristic
168C2×C4×D61C2×D411yescharacteristic
169C2×Dic122—21no—
170C2×Dic122—21no—
171C2×Dic122—22no—
172C2×Dic122—22no—
173C2×C8×C61—24no—
174C2×C8×C61—22no—
175C2×C8×C61—22no—
176C4⋊C4×C61—21no—
177C4⋊C4×C61—21no—
178C4⋊C4×C61—22no—
179C4⋊C4×C61—22no—
180C4⋊C4×C61C22×C412yes—
181C4⋊C4×C61C22×C412yes—
182C22⋊C4×C61—22no—
183C42×C61—22no—
184C42×C61C22×C411yescharacteristic
185C42×C61C2×D411yescharacteristic
186C42×C61C2×D411yescharacteristic
187C42×C61C2411yescharacteristic
188C22⋊Dic61—22no—
189#11758198—22no—
190#11758198—22no—
191C4⋊Dic61—21no—
192C4⋊Dic61—21no—
193C4⋊Dic61—22no—
194C4⋊Dic61—22no—
195C4⋊Dic61C22×C412yes—
196C4⋊Dic61C22×C412yes—
197#11758196—22no—
198#11758196—22no—
199#11758196—22no—
200#11758196—22no—
201#11758196—22no—
202#11758196—22no—
203#11758196—22no—
204#11758196—22no—
205C4×Dic61—22no—
206C4×Dic61—22no—
207C4×Dic61—22no—
208C4×Dic61C22⋊C411yescharacteristic
209C4×Dic61C22⋊C411yescharacteristic
210C4×Dic61C22⋊C411yescharacteristic
211C4×Dic61C22⋊C411yescharacteristic
212C4×Dic61C22⋊C411yescharacteristic
213C4×Dic61C22⋊C411yescharacteristic
214C4×Dic61C22×C411yescharacteristic
215C4×Dic61C22×C411yescharacteristic
216C4×Dic61C2×D411yescharacteristic
217C4×Dic61C2×D411yescharacteristic
218C4×Dic61C2×D411yescharacteristic
219C4×Dic61C2×D411yescharacteristic
220C2×#92418—24no—
221C2×#92418—22no—
222C2×#92418—22no—
223#11758189—28no—
224#11758189—28no—
225#11758189—44no—
226C23×C61C2×C22⋊C411yescharacteristic
227C22×D61—22no—
228C22×D61—22no—
229C22×D61—41no—
230C22×D61C2×C22⋊C411yescharacteristic
231C22×D61C2×C22⋊C411yescharacteristic
232Q8×C61—42no—
233Q8×C61—42no—
234C2×C4×C61—22no—
235C2×C4×C61—22no—
236C2×C4×C61—22no—
237C2×C4×C61—22no—
238C2×C4×C61—41no—
239C2×C4×C61C2×C22⋊C411yescharacteristic
240C2×C4×C61C2×C22⋊C411yescharacteristic
241C2×C4×C61C2×C22⋊C411yescharacteristic
242C2×C4×C61C23×C411yescharacteristic
243C2×C4×C61C22×D411yescharacteristic
244C2×C4×C61C22×D411yescharacteristic
245C2×Dic61—22no—
246C2×Dic61—22no—
247C2×Dic61—22no—
248C2×Dic61—22no—
249C2×Dic61#10114yes—
250C2×Dic61C2×C22⋊C411yescharacteristic
251C2×Dic61C2×C22⋊C411yescharacteristic
252C2×Dic61C2×C22⋊C411yescharacteristic
253C2×Dic61C2×C22⋊C411yescharacteristic
254C2×Dic61C2×C22⋊C411yescharacteristic
255C2×Dic61C2×C22⋊C411yescharacteristic
256C4×D61—22no—
257C4×D61—22no—
258C4×D61—22no—
259C4×D61—22no—
260C4×D61—22no—
261C4×D61—22no—
262C4×D61—22no—
263C4×D61—22no—
264C4×D61—22no—
265C4×D61—41no—
266C4×D61—41no—
267C4×D61—44no—
268C4×D61—44no—
269C4×D61#10414yes—
270Dic122—42no—
271Dic122—42no—
272Dic122—44no—
273Dic122—44no—
274C8×C61—24no—
275C8×C61—42no—
276#92418—24no—
277#92418—42no—
278#17648—611nomaximal, Sylow
279C22×C61—21no—
280C22×C61—21no—
281C22×C61C2×#10112yes—
282C22×C61C22×C22⋊C411yescharacteristic
283C2×D61—21no—
284C2×D61—21no—
285C2×D61—21no—
286C2×D61—21no—
287C2×D61—21no—
288C2×D61—44no—
289C2×D61C2×#10412yes—
290C4×C61—21no—
291C4×C61—21no—
292C4×C61—21no—
293C4×C61—21no—
294C4×C61—21no—
295C4×C61—42no—
296C4×C61—42no—
297C4×C61C2×#10412yes—
298Dic61—21no—
299Dic61—21no—
300Dic61—42no—
301Dic61—42no—
302Dic61C2×#10112yes—
303Dic61C2×#10112yes—
304Dic61C2×#10412yes—
305Dic61#42112yes—
306C2×#533—611no—
307#2920—611no—
308C2×C4×M16—611no—
309#1531—611no—
310#1531—611no—
311#1531—611no—
312#1531D6111yescharacteristic, complemented
313#1528—614no—
314#1526—614no—
315C2×C61—22no—
316C2×C61C22×#10111yescharacteristic
317C2×C61C22×#10411yescharacteristic
318C2×C61C2×#42111yescharacteristic
319D61—22no—
320D61—41no—
321C22×M16—1221no—
322#533—611no—
323#533—611no—
324#533—612no—
325#533—612no—
326#533—611no—
327#533—1221no—
328#533—1221no—
329#533—1221no—
330#533—1221no—
331#533C2×D6111yescharacteristic
332C2×C4⋊Q8—611no—
333C2×C4⋊Q8—611no—
334C2×#125—612no—
335C2×C22⋊Q8—1221no—
336C2×C22⋊Q8—1221no—
337C2×C4×Q8—1221no—
338C2×C4×Q8—1221no—
339C2×#117—612no—
340C22×C42—611no—
341#432—611no—
342#432—612no—
343#432C2×D6111yescharacteristic
344#431—1221no—
345#431—1221no—
346#423—1221no—
347#407—1221no—
348C4×M16—612no—
349C4×M16—612no—
350C4×M16—612no—
351C4×M16—1221no—
352C4×M16—1221no—
353C2×#97—611no—
354C2×#97C2×D6111yescharacteristic
355C2×C4×C8—1221no—
356#330—614no—
357#330—614no—
358#330—614no—
359#330C2×D6114yescomplemented
360C61#1764811yescharacteristic, complemented, Sylow
361C22×Q8—1221no—
362C22×Q8—1221no—
363C23×C4—611no—
364C23×C4—611no—
365C23×C4—611no—
366C2×M16—1222no—
367C2×M16—1222no—
368C2×M16—1228no—
369C2×M16—1222no—
370C22×C8—21no—
371C22×C8—1221no—
372C4⋊Q8—612no—
373C4⋊Q8—612no—
374C4⋊Q8—611no—
375C4⋊Q8—611no—
376C4⋊Q8—1221no—
377C4⋊Q8—1221no—
378C4⋊Q8—1221no—
379C4⋊Q8—1221no—
380C4⋊Q8C4×D6111yescharacteristic
381C4⋊Q8C4×D6111yescharacteristic
382#125—612no—
383#125—612no—
384#125—612no—
385#125—1222no—
386#125—1222no—
387#125C22×D6112yes—
388C22⋊Q8—21no—
389C22⋊Q8—21no—
390C22⋊Q8—1222no—
391C22⋊Q8—1222no—
392C22⋊Q8—1222no—
393C22⋊Q8—1222no—
394C22⋊Q8—1221no—
395C22⋊Q8—1221no—
396C22⋊Q8—1221no—
397C22⋊Q8—1221no—
398C22⋊Q8—1221no—
399C22⋊Q8—1221no—
400C4×Q8—21no—
401C4×Q8—21no—
402C4×Q8—1222no—
403C4×Q8—1222no—
404C4×Q8—1222no—
405C4×Q8—1222no—
406C4×Q8—1221no—
407C4×Q8—1221no—
408C4×Q8—1221no—
409C4×Q8—1221no—
410C4×Q8—1221no—
411C4×Q8—1221no—
412#117—612no—
413#117—612no—
414#117—612no—
415#117—1222no—
416#117—1222no—
417#117C4×D6112yes—
418C2×C4⋊C4—612no—
419C2×C4⋊C4—612no—
420C2×C4⋊C4—1222no—
421C2×C4⋊C4—1222no—
422C2×C4⋊C4—1221no—
423C2×C4⋊C4—1221no—
424C2×C22⋊C4—1222no—
425C2×C42—611no—
426C2×C42—611no—
427C2×C42—611no—
428C2×C42—611no—
429C2×C42—611no—
430C2×C42—611no—
431C2×C42—611no—
432C2×C42—611no—
433C2×C42—611no—
434C2×C42—611no—
435C2×C42—611no—
436C2×C42—1222no—
437C2×C42C22×D6111yescharacteristic
438C4⋊C8—21no—
439C4⋊C8—1222no—
440C4⋊C8—1221no—
441C22⋊C8—21no—
442C22⋊C8—1222no—
443C22⋊C8—1221no—
444#97—21no—
445#97—612no—
446#97—612no—
447#97—1221no—
448#97C22×D6112yes—
449#97C22×D6112yes—
450C4×C8—1222no—
451C4×C8—1222no—
452C4×C8—1222no—
453C24—611no—
454C2×Q8—21no—
455C2×Q8—21no—
456C2×Q8—1222no—
457C2×Q8—1222no—
458C2×Q8—1221no—
459C2×Q8—1221no—
460C2×Q8—2444no—
461C2×Q8—2444no—
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—611no—
474C22×C4—611no—
475C22×C4—611no—
476C22×C4—611no—
477C22×C4—1224no—
478C22×C4—1224no—
479C22×C4—1222no—
480C22×C4—1222no—
481C22×C4—2442no—
482C22×C4C2×C4×D6111yescharacteristic
483C22×C4D4×D6111yescharacteristic
484C22×C4D4×D6111yescharacteristic
485M16—1228no—
486M16—1228no—
487M16—2444no—
488C2×C8—24no—
489C2×C8—22no—
490C2×C8—22no—
491C2×C8—1224no—
492C2×C8—1222no—
493C2×C8—1222no—
494C4⋊C4—22no—
495C4⋊C4—22no—
496C4⋊C4—21no—
497C4⋊C4—21no—
498C4⋊C4—612no—
499C4⋊C4—612no—
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—1222no—
510C4⋊C4—1221no—
511C4⋊C4—1221no—
512C4⋊C4C2×C4×D6112yes—
513C4⋊C4C2×C4×D6112yes—
514C22⋊C4—22no—
515C22⋊C4—1222no—
516C22⋊C4—1222no—
517C22⋊C4—1222no—
518C42—22no—
519C42—611no—
520C42—611no—
521C42—611no—
522C42—611no—
523C42—611no—
524C42—611no—
525C42—611no—
526C42—611no—
527C42—611no—
528C42—611no—
529C42—611no—
530C42—611no—
531C42—1222no—
532C42—1222no—
533C42—1222no—
534C42C2×C4×D6111yescharacteristic
535C42D4×D6111yescharacteristic
536C42D4×D6111yescharacteristic
537C42C23×D6111yescharacteristic
538C23—611no—
539C23—611no—
540C23—1222no—
541C23—1222no—
542C23—2441no—
543C23C22⋊C4×D6111yescharacteristic
544Q8—42no—
545Q8—42no—
546Q8—2442no—
547Q8—2444no—
548Q8—2444no—
549Q8—2442no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—41no—
555C2×C4—611no—
556C2×C4—611no—
557C2×C4—611no—
558C2×C4—611no—
559C2×C4—614no—
560C2×C4—614no—
561C2×C4—611no—
562C2×C4—611no—
563C2×C4—1222no—
564C2×C4—1222no—
565C2×C4—1222no—
566C2×C4—1222no—
567C2×C4—1222no—
568C2×C4—1222no—
569C2×C4—1222no—
570C2×C4—1222no—
571C2×C4—1222no—
572C2×C4—1222no—
573C2×C4—1222no—
574C2×C4—1222no—
575C2×C4—1222no—
576C2×C4—2441no—
577C2×C4—2441no—
578C2×C4—2444no—
579C2×C4—2444no—
580C2×C4C22⋊C4×D6111yescharacteristic
581C2×C4C22⋊C4×D6111yescharacteristic
582C2×C4C22⋊C4×D6111yescharacteristic
583C2×C4C22×C4×D6111yescharacteristic
584C2×C4C2×D4×D6111yescharacteristic
585C2×C4C2×D4×D6111yescharacteristic
586C8—24no—
587C8—42no—
588C8—1224no—
589C8—2442no—
590C22—21no—
591C22—21no—
592C22—612no—
593C22—1221no—
594C22—1221no—
595C22—1221no—
596C22—1221no—
597C22—1221no—
598C22—2444no—
599C22#101×D6112yes—
600C22C2×C22⋊C4×D6111yescharacteristic
601C4—21no—
602C4—21no—
603C4—21no—
604C4—21no—
605C4—21no—
606C4—42no—
607C4—42no—
608C4—612no—
609C4—612no—
610C4—612no—
611C4—612no—
612C4—1221no—
613C4—1221no—
614C4—2442no—
615C4—2442no—
616C4#104×D6112yes—
617C2—22no—
618C2—1222no—
619C2—2441no—
620C2C2×#101×D6111yescharacteristic
621C2C2×#104×D6111yescharacteristic
622C2#421×D6111yescharacteristic
subgroupG/H
centerC22C2×C22⋊C4×D61
commutatorC2×C4×C61C23×C4
FrattiniC42C23×D61
FittingC61×#1531C2
socleC22×C61C22×C22⋊C4
prime radicalC23×D61C22⋊C4
cosocleC42×C61C24
hypercenter#1531D61
hypocenterC61#17648
LCSG ▷ C2×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1531

Complements

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

NH ≅ G/Ntype
4. #21444435C2semidirect
5. #21444434C2semidirect
6. #21444433C2semidirect
7. #21444419C2semidirect
8. #21444412C2semidirect
9. C61×#1531C2semidirect
42. #16839504C22semidirect
43. #16839503C22semidirect
44. #16839503C22semidirect
48. C61×#330C22semidirect
312. #1531D61semidirect
359. #330C2×D61semidirect
360. C61#17648semidirect

Maximal subgroups

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

subgroupindex
#533×D612
#214544492
C2×#168397142
#214444352
#214444342
#214444332
#214444192
#214444122
C61×#15312
#1764861