Finite Group Databasebeta

← #29689666

Subgroups — #29689666

Order 15616.

1200 conjugacy classes of subgroups.

The subgroup lattice has 690 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21459116C211yesmaximal, characteristic
2#423×D61C211yesmaximal, characteristic, complemented
3C4×#14519734C211yesmaximal, characteristic, complemented
4#21446797C211yesmaximal, characteristic, complemented
5#21446647C211yesmaximal, characteristic, complemented
6#21446646C211yesmaximal, characteristic, complemented
7#21446120C214yesmaximal, complemented
8#21445838C214yesmaximal, complemented
9C61×#1994C211yesmaximal, characteristic, complemented
10C2×M16×D61C2212yes—
11#16840317C411yescharacteristic, complemented
12#16840317C411yescharacteristic, complemented
13#16840313C411yescharacteristic, complemented
14#16840313C411yescharacteristic, complemented
15#16840312C2211yescharacteristic
16#16840299—21no—
17#16840299—21no—
18#16840298—21no—
19#16840298—21no—
20#16840297C412yescomplemented
21#16840297C412yescomplemented
22#16840275—21no—
23#16840275—21no—
24#16840274C2212yes—
25C4×#11758226—21no—
26C4×#11758226—21no—
27#16840230C2212yes—
28C2×#14519734—21no—
29C2×C42×D61C2211yescharacteristic
30#16839786C2211yescharacteristic
31#16839785C2212yes—
32#16839784—21no—
33#16839784—21no—
34C4⋊C8×D61—21no—
35C4⋊C8×D61—21no—
36C2×#14519705C2214yes—
37C2×#14519703C2214yes—
38C4×#11758196—21no—
39C4×#11758196C2212yes—
40C4×#11758198—21no—
41C4×#11758198C2212yes—
42#16839608—21no—
43C42×Dic61C2211yescharacteristic
44#16839521C2214yes—
45C61×#533C2211yescharacteristic
46C61×#423C2211yescharacteristic
47C43×C61C2211yescharacteristic
48C61×#339C2214yes—
49C2×#11758226—21no—
50C2×#11758226—21no—
51C2×#11758222C2312yes—
52C22×C4×D61D411yescharacteristic
53C22×C4×D61D411yescharacteristic
54C22×C4×D61C2311yescharacteristic
55C2×#11758194C2312yes—
56C2×#11758189—22no—
57M16×D61—44no—
58C2×C8×D61—22no—
59#14519777C2×C411yescharacteristic
60#14519774C2×C411yescharacteristic
61#14519772—22no—
62#14519772—22no—
63#14519768—21no—
64#14519768—21no—
65#14519766—21no—
66#14519766—21no—
67#14519765C2×C412yes—
68#14519762—22no—
69#14519762—22no—
70#14519761—22no—
71#14519757C2×C412yes—
72#14519757C2×C412yes—
73#14519753—21no—
74#14519753—21no—
75#14519752—21no—
76#14519752—21no—
77#14519752C2×C412yes—
78#14519752C2×C412yes—
79#14519751—21no—
80#14519751—21no—
81#14519747—22no—
82Q8×Dic61—21no—
83Q8×Dic61—21no—
84#14519742C2×C412yes—
85#14519742C2×C412yes—
86C4×D244—22no—
87C4×D244—22no—
88C4×D244—21no—
89C4×D244—21no—
90#14519739—22no—
91#14519739—22no—
92#14519739—21no—
93#14519739—21no—
94#14519739C2×C412yes—
95#14519739C2×C412yes—
96#14519737—22no—
97#14519737—22no—
98#14519735—21no—
99#14519735—21no—
100#14519735—22no—
101#14519735—22no—
102#14519734—28no—
103#14519734—22no—
104C2×#11758196—22no—
105C2×#11758198—22no—
106C2×C4×Dic61—21no—
107C2×C4×Dic61—21no—
108C2×C4×Dic61D411yescharacteristic
109C2×C4×Dic61D411yescharacteristic
110C2×C4×Dic61C2312yes—
111C2×C4×Dic61C2311yescharacteristic
112C42×D61—22no—
113C42×D61C2×C411yescharacteristic
114C42×D61C2×C411yescharacteristic
115C42×D61C2×C411yescharacteristic
116C42×D61C2×C411yescharacteristic
117C42×D61D411yescharacteristic
118C42×D61D411yescharacteristic
119C42×D61D411yescharacteristic
120C42×D61D411yescharacteristic
121C4⋊#92418—21no—
122C4⋊#92418—21no—
123#14519706—22no—
124#14519706—22no—
125#14519705—44no—
126#14519703—44no—
127#14519684—22no—
128#14519684—22no—
129C2×M16×C61C2312yes—
130C4⋊Q8×C61C2×C411yescharacteristic
131C4⋊Q8×C61C2×C411yescharacteristic
132#125×C61C2×C412yes—
133C22⋊Q8×C61—21no—
134C22⋊Q8×C61—21no—
135C4×Q8×C61—21no—
136C4×Q8×C61—21no—
137#117×C61C2312yes—
138C2×C42×C61—21no—
139C2×C42×C61—21no—
140C2×C42×C61C2312yes—
141C2×C42×C61C2311yescharacteristic
142C4⋊C8×C61—21no—
143C4⋊C8×C61—21no—
144C23×D61C2×D411yescharacteristic
145C2×Q8×C61—21no—
146C2×Q8×C61—21no—
147C22×C4×C61—22no—
148C22×C4×C61C2×D411yescharacteristic
149C22×C4×C61C2×D411yescharacteristic
150C22×C4×C61C2411yescharacteristic
151C2×C22⋊D61—22no—
152C22×Dic61—22no—
153C22×Dic61C2×D411yescharacteristic
154C22×Dic61C2×D411yescharacteristic
155C22×Dic61C2×D411yescharacteristic
156#11758226—44no—
157#11758226—44no—
158#11758222—24no—
159#11758222—24no—
160C2×D244—22no—
161C2×D244—22no—
162C2×D244—21no—
163C2×D244—21no—
164C2×D244C22×C412yes—
165C2×C4×D61—22no—
166C2×C4×D61—22no—
167C2×C4×D61—24no—
168C2×C4×D61—22no—
169C2×C4×D61—22no—
170C2×C4×D61—22no—
171C2×C4×D61—42no—
172C2×C4×D61C22⋊C411yescharacteristic
173C2×C4×D61C22⋊C411yescharacteristic
174C2×C4×D61C22⋊C411yescharacteristic
175C2×C4×D61C22⋊C411yescharacteristic
176C2×C4×D61C22⋊C411yescharacteristic
177C2×C4×D61C22⋊C411yescharacteristic
178C2×C4×D61C22⋊C411yescharacteristic
179C2×C4×D61C22⋊C411yescharacteristic
180C2×C4×D61C22×C411yescharacteristic
181C2×C4×D61C22×C411yescharacteristic
182C2×C4×D61C2×D411yescharacteristic
183C2×C4×D61C2×D411yescharacteristic
184C2×Dic122C22×C412yes—
185M16×C61—24no—
186C2×C8×C61—22no—
187C4⋊C4×C61—22no—
188C4⋊C4×C61—22no—
189C4⋊C4×C61—21no—
190C4⋊C4×C61—21no—
191C4⋊C4×C61C22×C412yes—
192C4⋊C4×C61C22×C412yes—
193C22⋊C4×C61—22no—
194C42×C61—24no—
195C42×C61—22no—
196C42×C61—22no—
197C42×C61—22no—
198C42×C61—42no—
199C42×C61C22×C411yescharacteristic
200C42×C61C22×C411yescharacteristic
201C42×C61C2×D411yescharacteristic
202C42×C61C2×D411yescharacteristic
203#11758198—28no—
204#11758198—22no—
205#11758198—22no—
206#11758198—22no—
207#11758198—22no—
208#11758198—22no—
209#11758198—22no—
210#11758198—22no—
211#11758198—22no—
212C4⋊Dic61—21no—
213C4⋊Dic61—21no—
214C4⋊Dic61—22no—
215C4⋊Dic61—22no—
216#11758196—28no—
217#11758196—22no—
218#11758196—22no—
219C4×Dic61—24no—
220C4×Dic61—22no—
221C4×Dic61—22no—
222C4×Dic61—22no—
223C4×Dic61—22no—
224C4×Dic61—42no—
225C4×Dic61C22⋊C411yescharacteristic
226C4×Dic61C22⋊C411yescharacteristic
227C4×Dic61C22⋊C411yescharacteristic
228C4×Dic61C22⋊C411yescharacteristic
229C4×Dic61C22⋊C411yescharacteristic
230C4×Dic61C22⋊C411yescharacteristic
231C4×Dic61C22⋊C411yescharacteristic
232C4×Dic61C22⋊C411yescharacteristic
233C4×Dic61C22×C411yescharacteristic
234C4×Dic61C22×C411yescharacteristic
235C4×Dic61C2×D411yescharacteristic
236C4×Dic61C2×D411yescharacteristic
237#11758194—24no—
238C2×#92418—22no—
239#11758189—44no—
240C8×D61—44no—
241C23×C61C22×D411yescharacteristic
242C22×D61—22no—
243C22×D61—22no—
244C22×D61—22no—
245C22×D61—41no—
246C22×D61C2×C22⋊C411yescharacteristic
247C22×D61C2×C22⋊C411yescharacteristic
248Q8×C61—42no—
249Q8×C61—42no—
250C2×C4×C61—22no—
251C2×C4×C61—22no—
252C2×C4×C61—24no—
253C2×C4×C61—22no—
254C2×C4×C61—22no—
255C2×C4×C61—42no—
256C2×C4×C61—42no—
257C2×C4×C61C2×C22⋊C411yescharacteristic
258C2×C4×C61C2×C22⋊C411yescharacteristic
259C2×C4×C61C2×C22⋊C411yescharacteristic
260C2×C4×C61C2×C22⋊C411yescharacteristic
261C2×C4×C61C23×C411yescharacteristic
262C2×C4×C61C22×D411yescharacteristic
263C2×C4×C61H(5,𝔽2)12yes—
264C2×C4×C612−1+412yes—
265C22⋊D61—44no—
266C2×Dic61—22no—
267C2×Dic61—22no—
268C2×Dic61—24no—
269C2×Dic61—22no—
270C2×Dic61—41no—
271C2×Dic61—42no—
272C2×Dic61—42no—
273C2×Dic61C2×C22⋊C411yescharacteristic
274C2×Dic61C2×C22⋊C411yescharacteristic
275C2×Dic61C2×C22⋊C411yescharacteristic
276C2×Dic61C2×C22⋊C411yescharacteristic
277C2×Dic61C2×C22⋊C411yescharacteristic
278C2×Dic61C2×C22⋊C411yescharacteristic
279D244—24no—
280D244—44no—
281D244—44no—
282D244—42no—
283D244—42no—
284C4×D61—22no—
285C4×D61—22no—
286C4×D61—22no—
287C4×D61—21no—
288C4×D61—21no—
289C4×D61—21no—
290C4×D61—21no—
291C4×D61—22no—
292C4×D61—22no—
293C4×D61—22no—
294C4×D61—22no—
295C4×D61—41no—
296C4×D61—41no—
297C4×D61—44no—
298C4×D61—42no—
299C4×D61—42no—
300C4×D61—44no—
301C4×D61#10414yescomplemented
302Dic122—24no—
303C8×C61—42no—
304#92418—42no—
305#21856—611nomaximal, Sylow
306C22×C61—21no—
307C22×C61—21no—
308C22×C61C22×C22⋊C411yescharacteristic
309C22×C61#53712yes—
310C2×D61—21no—
311C2×D61—21no—
312C2×D61—21no—
313C2×D61—21no—
314C2×D61—21no—
315C2×D61—44no—
316C2×D61—42no—
317C2×D61—42no—
318C2×D61—42no—
319C2×D61C2×#10412yes—
320C4×C61—21no—
321C4×C61—21no—
322C4×C61—44no—
323C4×C61—42no—
324C4×C61—42no—
325C4×C61C2×#10412yes—
326C4×C61#42112yes—
327C4×C61#52012yes—
328C4×C61#53511yescharacteristic
329C4×C61#53611yescharacteristic
330Dic61—21no—
331Dic61—21no—
332Dic61—21no—
333Dic61—21no—
334Dic61—21no—
335Dic61—44no—
336Dic61—42no—
337Dic61C2×#10412yes—
338#3444—611no—
339C2×#423—611no—
340C4×#117—611no—
341#1994—611no—
342#1994D6111yescharacteristic, complemented
343#1962—611no—
344#1962—611no—
345#1871—614no—
346#1806—614no—
347C2×C61—22no—
348C2×C61#231111yescharacteristic
349C2×C61C22×#10411yescharacteristic
350C2×C61C2×#42111yescharacteristic
351D61—22no—
352D61—41no—
353D61—82no—
354C22×M16—612no—
355#533—611no—
356#533—611no—
357#533C2×D6111yescharacteristic
358#532—611no—
359#532—611no—
360#532—611no—
361#529—612no—
362#529—612no—
363#529—1221no—
364#529—1221no—
365#529—1221no—
366#529—1221no—
367#525—612no—
368#525—1221no—
369#525—1221no—
370C4×#41—612no—
371C4×#41—1221no—
372C4×#41—1221no—
373C2×#117—1221no—
374C22×C42—611no—
375#423—612no—
376#423—611no—
377#423—1221no—
378#423—1221no—
379#423C2×D6111yescharacteristic
380C2×C4⋊C8—1221no—
381C2×C4⋊C8—1221no—
382C2×#104—614no—
383C2×#104—614no—
384C4×C4⋊C4—612no—
385C4×C4⋊C4—1221no—
386C4×C22⋊C4—612no—
387C4×C22⋊C4—1221no—
388#376—1221no—
389C43—611no—
390C43C2×D6111yescharacteristic
391#339—614no—
392#339C2×D6114yes—
393C61#2185611yescharacteristic, complemented, Sylow
394C2×#41—612no—
395C2×#41—1221no—
396C2×#41—1221no—
397C23×C4—611no—
398C23×C4—611no—
399C23×C4—611no—
400C2×M16—612no—
401C2×M16—1222no—
402C2×M16—2444no—
403C2×M16C22×D6112yes—
404C22×C8—1222no—
405C4⋊Q8—611no—
406C4⋊Q8C4×D6111yescharacteristic
407C4⋊Q8C4×D6111yescharacteristic
408#127—611no—
409#126—1222no—
410#126—1222no—
411#125—1221no—
412#125—1221no—
413#125C4×D6112yes—
414#124—612no—
415#124—1221no—
416#124—1221no—
417#123—1222no—
418#123—1222no—
419#123—1222no—
420C22⋊Q8—21no—
421C22⋊Q8—21no—
422C22⋊Q8—612no—
423C22⋊Q8—612no—
424C22⋊Q8—1221no—
425C22⋊Q8—1221no—
426#121—612no—
427#121—612no—
428#121—1221no—
429#121—1221no—
430#121—1221no—
431#121—1221no—
432#120—1222no—
433C4×Q8—21no—
434C4×Q8—21no—
435C4×Q8—612no—
436C4×Q8—612no—
437C4×Q8—1221no—
438C4×Q8—1221no—
439C4×D4—612no—
440C4×D4—612no—
441C4×D4—1222no—
442C4×D4—1222no—
443C4×D4—1221no—
444C4×D4—1221no—
445C4×D4—1222no—
446C4×D4—1222no—
447C4×D4—1222no—
448C4×D4—1222no—
449C4×D4—1221no—
450C4×D4—1221no—
451#117—1222no—
452#117—1228no—
453#117—1222no—
454#117—1222no—
455#117—1221no—
456#117—1221no—
457#117C22×D6112yes—
458C2×C4⋊C4—1222no—
459C2×C22⋊C4—1222no—
460C2×C42—21no—
461C2×C42—21no—
462C2×C42—611no—
463C2×C42—611no—
464C2×C42—611no—
465C2×C42—611no—
466C2×C42—611no—
467C2×C42—611no—
468C2×C42—611no—
469C2×C42—611no—
470C2×C42—611no—
471C2×C42—611no—
472C2×C42—611no—
473C2×C42—612no—
474C2×C42—1222no—
475C2×C42—1221no—
476C2×C42—1221no—
477C2×C42C22×D6112yes—
478C2×C42C22×D6111yescharacteristic
479C4⋊C8—21no—
480C4⋊C8—21no—
481C4⋊C8—1222no—
482C4⋊C8—1222no—
483C4⋊C8—1221no—
484C4⋊C8—1221no—
485#104—2444no—
486#104—2444no—
487#95—1222no—
488#95—1222no—
489C24—611no—
490#41—1224no—
491#41—1224no—
492#41—2444no—
493#41—2444no—
494C2×Q8—21no—
495C2×Q8—21no—
496C2×Q8—612no—
497C2×D4—612no—
498C2×D4—1222no—
499C2×D4—1222no—
500C2×D4—1222no—
501C2×D4—1221no—
502C2×D4—1221no—
503C22×C4—22no—
504C22×C4—611no—
505C22×C4—611no—
506C22×C4—611no—
507C22×C4—611no—
508C22×C4—611no—
509C22×C4—611no—
510C22×C4—611no—
511C22×C4—611no—
512C22×C4—611no—
513C22×C4—611no—
514C22×C4—611no—
515C22×C4—611no—
516C22×C4—611no—
517C22×C4—611no—
518C22×C4—611no—
519C22×C4—1224no—
520C22×C4—1222no—
521C22×C4—1222no—
522C22×C4—1222no—
523C22×C4—1222no—
524C22×C4—1222no—
525C22×C4—1222no—
526C22×C4—2442no—
527C22×C4D4×D6111yescharacteristic
528C22×C4D4×D6111yescharacteristic
529C22×C4C23×D6111yescharacteristic
530M16—24no—
531M16—1224no—
532M16—2444no—
533C2×C8—22no—
534C2×C8—1222no—
535C2×C8—2444no—
536C4⋊C4—21no—
537C4⋊C4—21no—
538C4⋊C4—22no—
539C4⋊C4—22no—
540C4⋊C4—1222no—
541C4⋊C4—1222no—
542C4⋊C4—1222no—
543C4⋊C4—1222no—
544C4⋊C4—1228no—
545C4⋊C4—1221no—
546C4⋊C4—1221no—
547C4⋊C4C2×C4×D6112yes—
548C4⋊C4C2×C4×D6112yes—
549C22⋊C4—22no—
550C22⋊C4—1222no—
551C22⋊C4—1222no—
552C22⋊C4—1222no—
553C22⋊C4—1222no—
554C22⋊C4—1222no—
555C22⋊C4—1222no—
556C22⋊C4—1228no—
557C22⋊C4—1222no—
558C22⋊C4—1222no—
559C42—24no—
560C42—22no—
561C42—22no—
562C42—22no—
563C42—42no—
564C42—611no—
565C42—611no—
566C42—611no—
567C42—611no—
568C42—611no—
569C42—611no—
570C42—611no—
571C42—611no—
572C42—611no—
573C42—611no—
574C42—611no—
575C42—611no—
576C42—1222no—
577C42—1222no—
578C42—1224no—
579C42—1222no—
580C42—1222no—
581C42—2442no—
582C42C2×C4×D6111yescharacteristic
583C42C2×C4×D6111yescharacteristic
584C42D4×D6111yescharacteristic
585C42D4×D6111yescharacteristic
586C23—611no—
587C23—611no—
588C23—1222no—
589C23—1222no—
590C23—1222no—
591C23—2441no—
592C23C2×D4×D6111yescharacteristic
593Q8—42no—
594Q8—42no—
595Q8—1224no—
596D4—1224no—
597D4—2444no—
598D4—2444no—
599D4—2444no—
600D4—2442no—
601D4—2442no—
602C2×C4—22no—
603C2×C4—22no—
604C2×C4—24no—
605C2×C4—22no—
606C2×C4—22no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—611no—
610C2×C4—614no—
611C2×C4—611no—
612C2×C4—611no—
613C2×C4—611no—
614C2×C4—611no—
615C2×C4—611no—
616C2×C4—1222no—
617C2×C4—1222no—
618C2×C4—1222no—
619C2×C4—1222no—
620C2×C4—1222no—
621C2×C4—1222no—
622C2×C4—1222no—
623C2×C4—1222no—
624C2×C4—1222no—
625C2×C4—1221no—
626C2×C4—1221no—
627C2×C4—1221no—
628C2×C4—1221no—
629C2×C4—1224no—
630C2×C4—1222no—
631C2×C4—2441no—
632C2×C4—2441no—
633C2×C4—2444no—
634C2×C4—2444no—
635C2×C4—2441no—
636C2×C4—2442no—
637C2×C4—2442no—
638C2×C4—2442no—
639C2×C4—2442no—
640C2×C4C22⋊C4×D6111yescharacteristic
641C2×C4C22⋊C4×D6111yescharacteristic
642C2×C4C22⋊C4×D6111yescharacteristic
643C2×C4C22⋊C4×D6111yescharacteristic
644C2×C4C22×C4×D6111yescharacteristic
645C2×C4C2×D4×D6111yescharacteristic
646C2×C4#1451982312yes—
647C2×C4#1451982512yes—
648C8—42no—
649C8—2442no—
650C22—21no—
651C22—21no—
652C22—612no—
653C22—1221no—
654C22—1221no—
655C22—1221no—
656C22—1221no—
657C22—1221no—
658C22—2444no—
659C22—2442no—
660C22—2442no—
661C22—2442no—
662C22C2×C22⋊C4×D6111yescharacteristic
663C22#1684034212yes—
664C4—21no—
665C4—21no—
666C4—44no—
667C4—42no—
668C4—42no—
669C4—612no—
670C4—1221no—
671C4—1221no—
672C4—1221no—
673C4—1221no—
674C4—1221no—
675C4—2444no—
676C4—2442no—
677C4#104×D6112yes—
678C4#1683977712yes—
679C4#1684024412yes—
680C4#1684033011yescharacteristic
681C4#1684033711yescharacteristic
682C2—22no—
683C2—1222no—
684C2—4882no—
685C2—2441no—
686C2#2144841011yescharacteristic
687C2C2×#104×D6111yescharacteristic
688C2C2×#1683977711yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1994C2
socleC22×C61C22×C22⋊C4
prime radical#21459116C2
cosocleC22×C4×C61C24
hypercenter#1994D61
hypocenterC61#21856
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1994

Complements

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

NH ≅ G/Ntype
2. #423×D61C2semidirect
3. C4×#14519734C2semidirect
4. #21446797C2semidirect
5. #21446647C2semidirect
6. #21446646C2semidirect
7. #21446120C2semidirect
8. #21445838C2semidirect
9. C61×#1994C2semidirect
11. #16840317C4semidirect
12. #16840317C4semidirect
13. #16840313C4semidirect
14. #16840313C4semidirect
20. #16840297C4semidirect
21. #16840297C4semidirect
301. C4×D61#104semidirect
342. #1994D61semidirect
393. C61#21856semidirect

Maximal subgroups

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

subgroupindex
#214591162
#423×D612
C4×#145197342
#214467972
#214466472
#214466462
#214461202
#214458382
C61×#19942
#2185661