Finite Group Databasebeta

← #29647261

Subgroups — #29647261

Order 15616.

1024 conjugacy classes of subgroups.

The subgroup lattice has 660 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840298C211yesmaximal, characteristic
2#21454448C211yesmaximal, characteristic, complemented
3#21448210C211yesmaximal, characteristic, complemented
4#21444356C214yesmaximal, complemented
5#21444352C211yesmaximal, characteristic, complemented
6#21444349C211yesmaximal, characteristic
7#21444347C211yesmaximal, characteristic
8#21444304C214yesmaximal, complemented
9C61×#1511C211yesmaximal, characteristic, complemented
10#16840298—21no—
11#16840298—21no—
12#16840298—21no—
13#16840298—21no—
14#16840298C412yescomplemented
15#16840298C412yescomplemented
16#16840298C2212yes—
17#16840298C2212yes—
18C2×#14519772—21no—
19C2×#14519768C411yescharacteristic, complemented
20C2×#14519766C411yescharacteristic, complemented
21C2×#14519761—21no—
22C2×#14519751C2211yescharacteristic
23C2×Q8×Dic61C411yescharacteristic, complemented
24C2×#14519739C411yescharacteristic, complemented
25C2×#14519737—21no—
26C2×#14519735—21no—
27C22×C4×Dic61C2211yescharacteristic
28#16839816—21no—
29#16839816C2212yescomplemented
30#16839812—21no—
31#16839812C2212yescomplemented
32#16839785—21no—
33#16839726C2211yescharacteristic
34M16×Dic61—21no—
35#16839629—21no—
36#16839626C2212yescomplemented
37#16839625C2211yescharacteristic
38#16839620—21no—
39#16839613—21no—
40C22⋊C4×Dic61—21no—
41#16839607C2212yescomplemented
42#16839499C2214yescomplemented
43#16839497C2214yes—
44#16839495C2214yes—
45C2×C22⋊Q8×C61C2211yescharacteristic
46C61×#407C2211yescharacteristic
47C61×#381C2211yescharacteristic
48C61×#328C2214yescomplemented
49C22×D244C2×C411yescharacteristic
50C22×C22⋊D61—21no—
51C23×Dic61D411yescharacteristic
52C23×Dic61D411yescharacteristic
53C22×C4×D61—21no—
54C2×#11758194—21no—
55#14519772—22no—
56#14519772—22no—
57#14519772—22no—
58#14519772—22no—
59#14519768—21no—
60#14519768—21no—
61#14519768C2×C412yes—
62#14519768C2×C412yes—
63#14519766—21no—
64#14519766—21no—
65#14519766C2×C412yes—
66#14519766C2×C412yes—
67#14519761—22no—
68#14519761—22no—
69#14519761—22no—
70#14519761—22no—
71#14519751—21no—
72#14519751—21no—
73#14519751C2×C412yes—
74#14519751C2312yes—
75Q8×Dic61—21no—
76Q8×Dic61—21no—
77Q8×Dic61C2×C412yes—
78Q8×Dic61C2×C412yes—
79#14519739—21no—
80#14519739—21no—
81#14519739C2×C412yes—
82#14519739C2×C412yes—
83#14519737—22no—
84#14519737—22no—
85#14519737—22no—
86#14519737—22no—
87#14519735—22no—
88#14519735—22no—
89#14519735—22no—
90#14519735—22no—
91C2×C4⋊Dic61—21no—
92C2×C4⋊Dic61—21no—
93C2×C4⋊Dic61C2×C411yescharacteristic
94C2×#11758196—21no—
95C2×#11758196—22no—
96C2×C22⋊Dic61—21no—
97C2×#11758198—21no—
98C2×#11758198—21no—
99C2×#11758198—21no—
100C2×#11758198—21no—
101C2×C4×Dic61—21no—
102C2×C4×Dic61—22no—
103C2×C4×Dic61C2×C411yescharacteristic
104C2×C4×Dic61C2×C411yescharacteristic
105C2×C4×Dic61C2×C411yescharacteristic
106C2×C4×Dic61C2×C411yescharacteristic
107C2×C4×Dic61D411yescharacteristic
108C2×C4×Dic61D411yescharacteristic
109C2×C4×Dic61D411yescharacteristic
110C2×C4×Dic61D411yescharacteristic
111C2×C4×Dic61D411yescharacteristic
112C2×C4×Dic61D411yescharacteristic
113C2×C4×Dic61C2311yescharacteristic
114C2×C4×Dic61C2311yescharacteristic
115#14519706—22no—
116#14519706—22no—
117C22⋊#92418—21no—
118C22⋊#92418C2312yes—
119#14519687—22no—
120C8×Dic61—22no—
121#14519684—21no—
122#14519684—22no—
123#14519684—22no—
124#14519684C2312yes—
125C22×Q8×C61C2×C411yescharacteristic
126C23×C4×C61C2311yescharacteristic
127C2×M16×C61—21no—
128C22⋊Q8×C61—21no—
129C22⋊Q8×C61—21no—
130C22⋊Q8×C61C2×C412yes—
131C22⋊Q8×C61C2312yes—
132C2×C4⋊C4×C61—21no—
133C2×C4⋊C4×C61C2×C411yescharacteristic
134C2×C22⋊C4×C61—21no—
135C2×C22⋊C4×C61—21no—
136C22⋊C8×C61—21no—
137C22⋊C8×C61C2312yes—
138#95×C61—21no—
139#95×C61C2312yes—
140C24×C61C2×D411yescharacteristic
141C23×D61—21no—
142C2×Q8×C61—21no—
143C2×Q8×C61—42no—
144C2×Q8×C61C22×C412yes—
145C22×C4×C61—21no—
146C22×C4×C61—21no—
147C22×C4×C61—22no—
148C22×C4×C61—42no—
149C22×C4×C61C22×C411yescharacteristic
150C22×C4×C61C22×C411yescharacteristic
151C22×C4×C61C2×D411yescharacteristic
152C22×C4×C61C2×D411yescharacteristic
153C22×C4×C61C2×D411yescharacteristic
154C22×C4×C61C2411yescharacteristic
155C2×C22⋊D61—22no—
156C2×C22⋊D61—22no—
157C2×C22⋊D61—44no—
158C22×Dic61—21no—
159C22×Dic61—22no—
160C22×Dic61—42no—
161C22×Dic61—42no—
162C22×Dic61C22⋊C411yescharacteristic
163C22×Dic61C22⋊C411yescharacteristic
164C22×Dic61C22⋊C411yescharacteristic
165C22×Dic61C22⋊C411yescharacteristic
166C22×Dic61C22⋊C411yescharacteristic
167C22×Dic61C22⋊C411yescharacteristic
168C22×Dic61C22⋊C411yescharacteristic
169C22×Dic61C22⋊C411yescharacteristic
170C22×Dic61C2×D411yescharacteristic
171C22×Dic61C2×D411yescharacteristic
172C22×Dic61C2×D411yescharacteristic
173C22×Dic61C2×D411yescharacteristic
174C2×D244—21no—
175C2×D244—42no—
176C2×D244C22×C412yes—
177C2×C4×D61—22no—
178C2×C4×D61—22no—
179C2×C4×D61—44no—
180M16×C61—42no—
181C2×C8×C61—22no—
182C4⋊C4×C61—22no—
183C4⋊C4×C61—21no—
184C4⋊C4×C61—22no—
185C4⋊C4×C61C22×C412yes—
186C22⋊C4×C61—22no—
187C22⋊C4×C61—22no—
188C22⋊C4×C61—42no—
189C22⋊Dic61—42no—
190#11758198—22no—
191#11758198—22no—
192#11758198—22no—
193#11758198—22no—
194#11758198—22no—
195#11758198—22no—
196#11758198—22no—
197#11758198—22no—
198C4⋊Dic61—21no—
199C4⋊Dic61—22no—
200C4⋊Dic61—22no—
201C4⋊Dic61—22no—
202C4⋊Dic61—22no—
203C4⋊Dic61C22×C412yes—
204#11758196—22no—
205#11758196—22no—
206#11758196—44no—
207C4×Dic61—22no—
208C4×Dic61—22no—
209C4×Dic61—44no—
210C4×Dic61C22⋊C411yescharacteristic
211C4×Dic61C22⋊C411yescharacteristic
212C4×Dic61C22⋊C411yescharacteristic
213C4×Dic61C22⋊C411yescharacteristic
214C4×Dic61C22⋊C411yescharacteristic
215C4×Dic61C22⋊C411yescharacteristic
216C4×Dic61C22⋊C411yescharacteristic
217C4×Dic61C22⋊C411yescharacteristic
218C4×Dic61C22×C411yescharacteristic
219C4×Dic61C22×C411yescharacteristic
220C4×Dic61C22×C411yescharacteristic
221C4×Dic61C22×C411yescharacteristic
222C4×Dic61C2×D411yescharacteristic
223C4×Dic61C2×D411yescharacteristic
224C4×Dic61C2×D411yescharacteristic
225C4×Dic61C2×D411yescharacteristic
226#11758194—42no—
227C2×#92418—22no—
228C23×C61—22no—
229C23×C61—22no—
230C23×C61—41no—
231C23×C61C2×C22⋊C411yescharacteristic
232C23×C61C2×C22⋊C411yescharacteristic
233C23×C61C22×D411yescharacteristic
234C22×D61—22no—
235C22×D61—42no—
236C22×D61—42no—
237C22×D61—42no—
238Q8×C61—42no—
239Q8×C61—42no—
240C2×C4×C61—22no—
241C2×C4×C61—22no—
242C2×C4×C61—22no—
243C2×C4×C61—22no—
244C2×C4×C61—41no—
245C2×C4×C61—41no—
246C2×C4×C61—41no—
247C2×C4×C61—41no—
248C2×C4×C61—42no—
249C2×C4×C61—42no—
250C2×C4×C61—42no—
251C2×C4×C61—42no—
252C2×C4×C61—42no—
253C2×C4×C61C2×C22⋊C411yescharacteristic
254C2×C4×C61C2×C22⋊C411yescharacteristic
255C2×C4×C61C23×C411yescharacteristic
256C2×C4×C61C22×D411yescharacteristic
257C22⋊D61—44no—
258C22⋊D61—44no—
259C2×Dic61—22no—
260C2×Dic61—22no—
261C2×Dic61—22no—
262C2×Dic61—22no—
263C2×Dic61—22no—
264C2×Dic61—22no—
265C2×Dic61—41no—
266C2×Dic61—41no—
267C2×Dic61—41no—
268C2×Dic61—41no—
269C2×Dic61—41no—
270C2×Dic61—41no—
271C2×Dic61—41no—
272C2×Dic61—42no—
273C2×Dic61—42no—
274C2×Dic61—42no—
275C2×Dic61—42no—
276C2×Dic61C2×C22⋊C411yescharacteristic
277C2×Dic61C2×C22⋊C411yescharacteristic
278C2×Dic61C2×C22⋊C411yescharacteristic
279C2×Dic61C2×C22⋊C411yescharacteristic
280C2×Dic61C2×C22⋊C411yescharacteristic
281C2×Dic61C2×C22⋊C411yescharacteristic
282C2×Dic61C2×C22⋊C411yescharacteristic
283C2×Dic61C2×C22⋊C411yescharacteristic
284D244—42no—
285D244—42no—
286C4×D61—44no—
287C4×D61—44no—
288C8×C61—42no—
289#92418—42no—
290#17554—611nomaximal, Sylow
291C22×C61—21no—
292C22×C61—21no—
293C22×C61—21no—
294C22×C61—21no—
295C22×C61—21no—
296C22×C61—21no—
297C22×C61—21no—
298C22×C61—44no—
299C22×C61#41012yes—
300C22×C61C22×C22⋊C411yescharacteristic
301C22×C61#57512yes—
302C2×D61—42no—
303C2×D61—42no—
304C2×D61—42no—
305C2×D61—84no—
306C4×C61—21no—
307C4×C61—21no—
308C4×C61—21no—
309C4×C61—21no—
310C4×C61—42no—
311C4×C61—42no—
312C4×C61—82no—
313Dic61—21no—
314Dic61—21no—
315Dic61—21no—
316Dic61—21no—
317Dic61—21no—
318Dic61—21no—
319Dic61—21no—
320Dic61—42no—
321Dic61—82no—
322Dic61#42112yes—
323C2×#529—611no—
324#2920—611no—
325#2283—611no—
326#1512—614no—
327#1511—611no—
328#1511D6111yescharacteristic, complemented
329#1510—611no—
330#1510—611no—
331#1500—614no—
332C2×C61—22no—
333C2×C61—22no—
334C2×C61—41no—
335C2×C61C2×#41011yescharacteristic
336C2×C61#289611yescharacteristic
337C2×C61C2×#42111yescharacteristic
338D61—82no—
339#529—612no—
340#529—612no—
341#529—612no—
342#529—612no—
343#529—1221no—
344#529—1221no—
345#529—1221no—
346#529—1221no—
347C2×#126—1221no—
348C2×#125—611no—
349C2×#124—611no—
350C2×#123—1221no—
351C2×C22⋊Q8C2×D6111yescharacteristic
352C2×#121—611no—
353C2×C4×Q8—611no—
354C2×C4×D4—611no—
355C2×C4×D4—1221no—
356C2×#117—1221no—
357C22×C42—611no—
358#432—612no—
359#432—1221no—
360#431—612no—
361#431—1221no—
362#423—1221no—
363#407—611no—
364#407C2×D6111yescharacteristic
365C4×M16—1221no—
366#383—1221no—
367#382—612no—
368#381—611no—
369#381C2×D6111yescharacteristic
370#380—1221no—
371C4×C4⋊C4—1221no—
372C4×C22⋊C4—1221no—
373#376—612no—
374#328—614no—
375#328C2×D6114yescomplemented
376#327—614no—
377#327—614no—
378C61#1755411yescharacteristic, complemented, Sylow
379C22×Q8C4×D6111yescharacteristic
380C22×D4—611no—
381C22×D4—1221no—
382C23×C4—611no—
383C23×C4—611no—
384C23×C4—1221no—
385C23×C4C22×D6111yescharacteristic
386C2×M16—21no—
387C2×M16—1221no—
388#126—1222no—
389#126—1222no—
390#126—1222no—
391#126—1222no—
392#125—612no—
393#125—612no—
394#125—1221no—
395#125—1221no—
396#124—612no—
397#124—612no—
398#124—1221no—
399#124—1221no—
400#123—1222no—
401#123—1222no—
402#123—1222no—
403#123—1222no—
404C22⋊Q8—21no—
405C22⋊Q8—21no—
406C22⋊Q8C4×D6112yes—
407C22⋊Q8C22×D6112yes—
408#121—612no—
409#121—612no—
410#121—1221no—
411#121—1221no—
412C4×Q8—612no—
413C4×Q8—612no—
414C4×Q8—1221no—
415C4×Q8—1221no—
416C4×D4—612no—
417C4×D4—612no—
418C4×D4—1221no—
419C4×D4—1221no—
420C4×D4—1222no—
421C4×D4—1222no—
422C4×D4—1222no—
423C4×D4—1222no—
424#117—1222no—
425#117—1222no—
426#117—1222no—
427#117—1222no—
428C2×C4⋊C4—21no—
429C2×C4⋊C4—611no—
430C2×C4⋊C4—1221no—
431C2×C4⋊C4—1221no—
432C2×C4⋊C4—1221no—
433C2×C4⋊C4—1222no—
434C2×C4⋊C4C4×D6111yescharacteristic
435C2×C22⋊C4—21no—
436C2×C22⋊C4—21no—
437C2×C22⋊C4—1221no—
438C2×C22⋊C4—1221no—
439C2×C22⋊C4—1221no—
440C2×C22⋊C4—1221no—
441C2×C22⋊C4—1221no—
442C2×C42—611no—
443C2×C42—611no—
444C2×C42—611no—
445C2×C42—611no—
446C2×C42—611no—
447C2×C42—611no—
448C2×C42—611no—
449C2×C42—611no—
450C2×C42—611no—
451C2×C42—611no—
452C2×C42—611no—
453C2×C42—611no—
454C2×C42—1221no—
455C2×C42—1222no—
456C4⋊C8—1222no—
457C4⋊C8—1222no—
458C22⋊C8—21no—
459C22⋊C8—612no—
460C22⋊C8—1221no—
461C22⋊C8C22×D6112yes—
462#97—1222no—
463C4×C8—1222no—
464#95—21no—
465#95—612no—
466#95—1222no—
467#95—1222no—
468#95—1221no—
469#95C22×D6112yes—
470C24—1221no—
471C24D4×D6111yescharacteristic
472C2×Q8—21no—
473C2×Q8—42no—
474C2×Q8C2×C4×D6112yes—
475C2×D4—612no—
476C2×D4—1222no—
477C2×D4—1222no—
478C2×D4—1221no—
479C2×D4—2442no—
480C2×D4—2444no—
481C22×C4—21no—
482C22×C4—21no—
483C22×C4—22no—
484C22×C4—42no—
485C22×C4—611no—
486C22×C4—611no—
487C22×C4—611no—
488C22×C4—611no—
489C22×C4—611no—
490C22×C4—611no—
491C22×C4—611no—
492C22×C4—611no—
493C22×C4—611no—
494C22×C4—611no—
495C22×C4—611no—
496C22×C4—611no—
497C22×C4—1222no—
498C22×C4—1222no—
499C22×C4—1221no—
500C22×C4—1222no—
501C22×C4—2444no—
502C22×C4—2442no—
503C22×C4—2442no—
504C22×C4C2×C4×D6111yescharacteristic
505C22×C4C2×C4×D6111yescharacteristic
506C22×C4D4×D6111yescharacteristic
507C22×C4D4×D6111yescharacteristic
508C22×C4D4×D6111yescharacteristic
509C22×C4C23×D6111yescharacteristic
510M16—42no—
511M16—2442no—
512C2×C8—22no—
513C2×C8—1222no—
514C4⋊C4—22no—
515C4⋊C4—22no—
516C4⋊C4—21no—
517C4⋊C4—612no—
518C4⋊C4—1222no—
519C4⋊C4—1222no—
520C4⋊C4—1222no—
521C4⋊C4—1222no—
522C4⋊C4—1222no—
523C4⋊C4—1222no—
524C4⋊C4—1221no—
525C4⋊C4—2444no—
526C4⋊C4C2×C4×D6112yes—
527C22⋊C4—22no—
528C22⋊C4—22no—
529C22⋊C4—42no—
530C22⋊C4—1222no—
531C22⋊C4—1222no—
532C22⋊C4—1222no—
533C22⋊C4—1222no—
534C22⋊C4—1222no—
535C22⋊C4—1222no—
536C22⋊C4—1222no—
537C22⋊C4—1222no—
538C22⋊C4—2442no—
539C42—611no—
540C42—611no—
541C42—611no—
542C42—611no—
543C42—611no—
544C42—611no—
545C42—611no—
546C42—611no—
547C42—611no—
548C42—611no—
549C42—611no—
550C42—611no—
551C42—611no—
552C42—611no—
553C42—611no—
554C42—611no—
555C42—1222no—
556C42—1222no—
557C42—2444no—
558C23—22no—
559C23—22no—
560C23—41no—
561C23—1222no—
562C23—2442no—
563C23—2442no—
564C23—2442no—
565C23C22⋊C4×D6111yescharacteristic
566C23C22⋊C4×D6111yescharacteristic
567C23C2×D4×D6111yescharacteristic
568Q8—42no—
569Q8—42no—
570D4—2442no—
571D4—2442no—
572D4—2444no—
573D4—2444no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—611no—
588C2×C4—611no—
589C2×C4—611no—
590C2×C4—611no—
591C2×C4—611no—
592C2×C4—611no—
593C2×C4—611no—
594C2×C4—611no—
595C2×C4—1222no—
596C2×C4—1222no—
597C2×C4—1222no—
598C2×C4—1222no—
599C2×C4—1222no—
600C2×C4—1222no—
601C2×C4—2444no—
602C2×C4—2444no—
603C2×C4—2441no—
604C2×C4—2441no—
605C2×C4—2441no—
606C2×C4—2441no—
607C2×C4—2441no—
608C2×C4—2441no—
609C2×C4—2441no—
610C2×C4—2442no—
611C2×C4—2442no—
612C2×C4—2442no—
613C2×C4—2442no—
614C2×C4C22⋊C4×D6111yescharacteristic
615C2×C4C22⋊C4×D6111yescharacteristic
616C2×C4C22×C4×D6111yescharacteristic
617C2×C4C2×D4×D6111yescharacteristic
618C8—42no—
619C8—2442no—
620C22—21no—
621C22—21no—
622C22—21no—
623C22—21no—
624C22—21no—
625C22—21no—
626C22—21no—
627C22—44no—
628C22—4884no—
629C22—2442no—
630C22—2442no—
631C22—2442no—
632C22#1683973412yes—
633C22C2×C22⋊C4×D6111yescharacteristic
634C22#1684056112yes—
635C4—21no—
636C4—21no—
637C4—21no—
638C4—21no—
639C4—42no—
640C4—42no—
641C4—82no—
642C4—612no—
643C4—1221no—
644C4—1221no—
645C4—1221no—
646C4—1221no—
647C4—1221no—
648C4—1221no—
649C4—1221no—
650C4—4882no—
651C4—2442no—
652C2—22no—
653C2—22no—
654C2—41no—
655C2—4882no—
656C2C2×#1683973411yescharacteristic
657C2#2145433711yescharacteristic
658C2#421×D6111yescharacteristic
subgroupG/H
centerC22C2×C22⋊C4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1511C2
socleC22×C61C22×C22⋊C4
prime radicalC2×#14519751C22
cosocleC22×C4×C61C24
hypercenter#1511D61
hypocenterC61#17554
LCSG ▷ C2×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1511

Complements

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

NH ≅ G/Ntype
2. #21454448C2semidirect
3. #21448210C2semidirect
4. #21444356C2semidirect
5. #21444352C2semidirect
8. #21444304C2semidirect
9. C61×#1511C2semidirect
14. #16840298C4semidirect
15. #16840298C4semidirect
19. C2×#14519768C4semidirect
20. C2×#14519766C4semidirect
23. C2×Q8×Dic61C4semidirect
24. C2×#14519739C4semidirect
29. #16839816C22semidirect
31. #16839812C22semidirect
36. #16839626C22semidirect
41. #16839607C22semidirect
42. #16839499C22semidirect
48. C61×#328C22semidirect
328. #1511D61semidirect
375. #328C2×D61semidirect
378. C61#17554semidirect

Maximal subgroups

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

subgroupindex
C2×#168402982
#214544482
#214482102
#214443562
#214443522
#214443492
#214443472
#214443042
C61×#15112
#1755461