Finite Group Databasebeta

← #29647226

Subgroups — #29647226

Order 15616.

1102 conjugacy classes of subgroups.

The subgroup lattice has 654 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21459863C211yesmaximal, characteristic
2#21454853C211yesmaximal, characteristic, complemented
3#21448179C211yesmaximal, characteristic, complemented
4#21444342C211yesmaximal, characteristic, complemented
5#21444337C212yesmaximal, complemented
6#21444331C211yesmaximal, characteristic, complemented
7#21444322C212yesmaximal, complemented
8#21444309C212yesmaximal, complemented
9#21444306C211yesmaximal, characteristic, complemented
10#328×D61C212yesmaximal, complemented
11C61×#1503C211yesmaximal, characteristic, complemented
12#16840486C411yescharacteristic, complemented
13#16840470—21no—
14#16840453C412yescomplemented
15#16840436—22no—
16#16840427C411yescharacteristic, complemented
17#16840424C412yescomplemented
18#16840415C411yescharacteristic, complemented
19#16840415C411yescharacteristic, complemented
20#16840403—21no—
21#16840384—21no—
22#16840360—22no—
23#16840296C2211yescharacteristic
24#16840275C2212yes—
25C22⋊Q8×D61C2212yes—
26#16840244—21no—
27C2×C4×D244C2211yescharacteristic
28C2×#14519739—21no—
29#16839841—21no—
30#16839832—22no—
31C8×D244—21no—
32#16839819C2211yescharacteristic
33#16839724C2212yes—
34C22⋊C8×D61C2212yes—
35C4⋊C4×Dic61—21no—
36C4×#11758198—22no—
37#16839609C2212yes—
38#16839608C2211yescharacteristic
39#95×D61C2212yes—
40#16839500C2211yescharacteristic
41#16839500C2211yescharacteristic
42#16839499C2212yes—
43#16839499C2212yes—
44#16839498C2212yes—
45#16839497C2211yescharacteristic
46#16839497C2211yescharacteristic
47#16839495C2212yes—
48C61×#533C2211yescharacteristic
49C61×#431C2211yescharacteristic
50C4×C4⋊C4×C61C2211yescharacteristic
51C61×#328C2212yes—
52C61×#328C2212yes—
53C2×#11758226—21no—
54C2×#11758226C2×C412yescomplemented
55C2×Q8×D61C2×C412yescomplemented
56C22×D244D411yescharacteristic
57C22×C4×D61—22no—
58C22×C4×D61C2312yes—
59C2×#11758189—22no—
60C2×C8×D61—22no—
61#14519776—21no—
62#14519772—22no—
63#14519771—21no—
64#14519771—22no—
65#14519769C2×C411yescharacteristic, complemented
66#14519767—22no—
67#14519766—21no—
68#14519765—22no—
69#14519765C2×C411yescharacteristic, complemented
70#14519762—22no—
71#14519762—22no—
72#14519762—22no—
73#14519762—22no—
74#14519760—21no—
75#14519757—22no—
76#14519757C2×C412yescomplemented
77#14519756C2311yescharacteristic
78#14519755—22no—
79#14519755—22no—
80#14519754—22no—
81#14519753—21no—
82#14519753C2×C412yescomplemented
83#14519753C2×C412yescomplemented
84#14519752—22no—
85#14519752—21no—
86#14519752C2×C412yescomplemented
87#14519751C2311yescharacteristic
88#14519747—22no—
89Q8×Dic61—21no—
90C4×Dic122C2×C411yescharacteristic, complemented
91C4×D244—22no—
92C4×D244C2×C412yes—
93C4×D244C2×C411yescharacteristic
94C4×D244C2×C411yescharacteristic
95C4×D244C2×C411yescharacteristic, complemented
96C4×D244D411yescharacteristic
97C4×D244D412yes—
98C4×D244D411yescharacteristic
99#14519739—28no—
100#14519739—21no—
101#14519739—22no—
102C4×C22⋊D61—21no—
103#14519737—22no—
104#14519735C2×C412yescomplemented
105#14519734—21no—
106#14519734—22no—
107C2×C4⋊Dic61D411yescharacteristic
108C4⋊C4×D61—22no—
109C4⋊C4×D61—21no—
110C4⋊C4×D61C2×C412yescomplemented
111C2×#11758198—22no—
112C2×#11758198D412yes—
113C22⋊C4×D61—22no—
114C2×C4×Dic61—21no—
115C2×C4×Dic61—22no—
116C4⋊#92418—21no—
117C22⋊#92418C2312yes—
118#14519689—22no—
119#14519689—22no—
120#14519688—21no—
121#14519684—21no—
122#14519684—22no—
123#14519684C2311yescharacteristic
124#14519684C2311yescharacteristic
125C4⋊Q8×C61C2×C411yescharacteristic, complemented
126C4⋊Q8×C61C2×C411yescharacteristic, complemented
127#125×C61—21no—
128C22⋊Q8×C61—21no—
129C22⋊Q8×C61C2312yes—
130C4×Q8×C61—21no—
131C4×Q8×C61C2×C412yescomplemented
132#117×C61—21no—
133C2×C4⋊C4×C61—21no—
134C2×C42×C61—21no—
135C2×C42×C61C2311yescharacteristic
136C4⋊C8×C61—21no—
137C22⋊C8×C61C2311yescharacteristic
138C22⋊C8×C61C2311yescharacteristic
139C4×C8×C61—21no—
140#95×C61C2312yes—
141C23×D61C2×D412yes—
142C2×Q8×C61—21no—
143C2×Q8×C61C22×C412yes—
144C22×C4×C61—22no—
145C22×C4×C61C2×D411yescharacteristic
146C22×C4×C61C2×D411yescharacteristic
147C22×C4×C61C2411yescharacteristic
148C2×C22⋊D61—21no—
149C22×Dic61—21no—
150C22×Dic61—21no—
151C22×Dic61C2×D412yes—
152#11758226—44no—
153#11758226—82no—
154Q8×D61—82no—
155C2×D244—24no—
156C2×D244—22no—
157C2×D244—22no—
158C2×D244C22⋊C412yes—
159C2×D244C22⋊C411yescharacteristic
160C2×D244C22⋊C411yescharacteristic
161C2×D244C22×C411yescharacteristic
162C2×C4×D61—22no—
163C2×C4×D61—21no—
164C2×C4×D61—28no—
165C2×C4×D61—22no—
166C2×C4×D61—22no—
167C2×C4×D61—82no—
168C2×C4×D61—82no—
169C2×C4×D61—82no—
170C2×C4×D61C22×C412yes—
171C2×C4×D61C22×C412yes—
172C2×C4×D61C2×D412yes—
173C2×C4×D61C2×D412yes—
174C2×Dic122—22no—
175C2×Dic122C22×C411yescharacteristic
176C2×C8×C61—21no—
177C2×C8×C61—21no—
178C4⋊C4×C61—24no—
179C4⋊C4×C61—21no—
180C4⋊C4×C61—21no—
181C4⋊C4×C61—21no—
182C4⋊C4×C61—22no—
183C4⋊C4×C61—22no—
184C4⋊C4×C61C22×C412yes—
185C22⋊C4×C61—22no—
186C42×C61—22no—
187C42×C61—81no—
188C42×C61C22×C411yescharacteristic
189C42×C61C22×C411yescharacteristic
190C42×C61C2×D411yescharacteristic
191C42×C61C2×D411yescharacteristic
192C22⋊Dic61—21no—
193#11758198—28no—
194#11758198—22no—
195#11758198—22no—
196#11758198—22no—
197#11758198—22no—
198#11758198—21no—
199#11758198—21no—
200#11758198—21no—
201#11758198—21no—
202#11758198—22no—
203#11758198C22⋊C412yes—
204#11758198C22⋊C412yes—
205#11758198C22⋊C412yes—
206#11758198C22⋊C412yes—
207C4⋊Dic61—22no—
208C4⋊Dic61C22⋊C411yescharacteristic
209C4⋊Dic61C22⋊C411yescharacteristic
210C4⋊Dic61C22⋊C412yes—
211C4⋊Dic61C22×C411yescharacteristic
212C4⋊Dic61C22×C411yescharacteristic
213#11758196—21no—
214#11758196—21no—
215#11758196—21no—
216#11758196—21no—
217#11758196—22no—
218#11758196—22no—
219#11758196—22no—
220C4×Dic61—24no—
221C4×Dic61—21no—
222C4×Dic61—22no—
223C4×Dic61—82no—
224C2×#92418—22no—
225#11758189—82no—
226C8×D61—82no—
227C23×C61C22×D411yescharacteristic
228C22×D61—22no—
229C22×D61—22no—
230C22×D61—24no—
231C22×D61—21no—
232C22×D61—42no—
233C22×D61C2×C22⋊C412yes—
234C22×D61C2×C22⋊C412yes—
235Q8×C61—42no—
236Q8×C61—42no—
237C2×C4×C61—21no—
238C2×C4×C61—21no—
239C2×C4×C61—22no—
240C2×C4×C61—24no—
241C2×C4×C61—22no—
242C2×C4×C61—22no—
243C2×C4×C61—41no—
244C2×C4×C61—42no—
245C2×C4×C61—42no—
246C2×C4×C61C2×C22⋊C411yescharacteristic
247C2×C4×C61C2×C22⋊C411yescharacteristic
248C2×C4×C61C2×C22⋊C411yescharacteristic
249C2×C4×C61C2×C22⋊C411yescharacteristic
250C2×C4×C61C23×C411yescharacteristic
251C2×C4×C61C22×D411yescharacteristic
252C22⋊D61—41no—
253C22⋊D61—41no—
254C2×Dic61—22no—
255C2×Dic61—22no—
256C2×Dic61—21no—
257C2×Dic61—21no—
258C2×Dic61—21no—
259C2×Dic61—42no—
260C2×Dic61—41no—
261C2×Dic61—41no—
262C2×Dic61—41no—
263C2×Dic61—41no—
264C2×Dic61C2×C22⋊C412yes—
265C2×Dic61C2×C22⋊C412yes—
266D244—24no—
267D244—22no—
268D244—41no—
269D244—44no—
270D244—82no—
271D244#10414yes—
272C4×D61—42no—
273C4×D61—42no—
274C4×D61—42no—
275C4×D61—42no—
276C4×D61—44no—
277C4×D61—42no—
278C4×D61—82no—
279C4×D61—82no—
280C4×D61—88no—
281Dic122—41no—
282Dic122—82no—
283C8×C61—41no—
284C8×C61—41no—
285#92418—42no—
286#17548—611nomaximal, Sylow
287C22×C61—21no—
288C22×C61—21no—
289C22×C61C22×C22⋊C411yescharacteristic
290C22×C61#57711yescharacteristic
291C22×C61#57811yescharacteristic
292C2×D61—22no—
293C2×D61—22no—
294C2×D61—22no—
295C2×D61—22no—
296C2×D61—22no—
297C2×D61—44no—
298C2×D61—44no—
299C2×D61—41no—
300C2×D61—41no—
301C2×D61—41no—
302C2×D61C2×#10414yes—
303C4×C61—21no—
304C4×C61—21no—
305C4×C61—21no—
306C4×C61—21no—
307C4×C61—21no—
308C4×C61—44no—
309C4×C61—42no—
310C4×C61—42no—
311C4×C61C2×#10412yes—
312Dic61—22no—
313Dic61—22no—
314Dic61—42no—
315Dic61—42no—
316Dic61—41no—
317Dic61—41no—
318Dic61—41no—
319#3520—611no—
320#2974—611no—
321#2275—611no—
322#1509—611no—
323#1508—612no—
324#1506—611no—
325#1504—612no—
326#1503D6111yescharacteristic, complemented
327#1501—612no—
328#1500—611no—
329C2×#328—612no—
330C2×C61—21no—
331C2×C61—21no—
332C2×C61#288311yescharacteristic
333C2×C61#289811yescharacteristic
334C2×C61C22×#10411yescharacteristic
335D61—24no—
336D61—42no—
337D61—81no—
338#556—611no—
339#554—1221no—
340#552—612no—
341#551—1222no—
342D4×Q8—611no—
343#548—612no—
344#547—611no—
345#547—611no—
346#546—1221no—
347#542—1221no—
348#539—1222no—
349#533C2×D6111yescharacteristic
350#529—611no—
351#525—612no—
352C2×C22⋊Q8—612no—
353#520—1221no—
354C2×C4×D4—611no—
355C2×C4×D4—1221no—
356#436—1221no—
357#435—1222no—
358C8×D4—1221no—
359#432—611no—
360#431C2×D6111yescharacteristic
361#407—612no—
362C2×C22⋊C8—612no—
363C4×C4⋊C4—1221no—
364C4×C4⋊C4C2×D6111yescharacteristic
365C4×C22⋊C4—612no—
366C4×C22⋊C4—1222no—
367#376—611no—
368C2×#95—612no—
369#328—612no—
370#328—612no—
371#328—611no—
372#328—611no—
373#328—612no—
374#328C2×D6112yes—
375#328C2×D6112yes—
376#327—611no—
377#327—611no—
378#327—612no—
379C61#1754811yescharacteristic, complemented, Sylow
380C2×#41—612no—
381C2×#41—1221no—
382C22×Q8—612no—
383C22×D4—611no—
384C23×C4—612no—
385C23×C4—1222no—
386C2×M16—1222no—
387C22×C8—1222no—
388C4⋊Q8—1221no—
389C4⋊Q8C4×D6111yescharacteristic, complemented
390C4⋊Q8C4×D6111yescharacteristic, complemented
391#126—1222no—
392#126—1222no—
393#126—1221no—
394#125—21no—
395#125—611no—
396#125—1222no—
397#124—611no—
398#124—1222no—
399#124—1221no—
400#123—1222no—
401#123—1222no—
402#123—1222no—
403#123—1222no—
404#123—1221no—
405C22⋊Q8—21no—
406C22⋊Q8—612no—
407C22⋊Q8—612no—
408C22⋊Q8—611no—
409C22⋊Q8—612no—
410C22⋊Q8—1222no—
411C22⋊Q8—1222no—
412C22⋊Q8—1222no—
413C22⋊Q8—1222no—
414C22⋊Q8—1221no—
415C22⋊Q8C22×D6112yes—
416#121—611no—
417#121—612no—
418#121—1222no—
419#121—1221no—
420#120—1222no—
421C4×Q8—21no—
422C4×Q8—611no—
423C4×Q8—1221no—
424C4×Q8C4×D6112yescomplemented
425C4×D4—611no—
426C4×D4—611no—
427C4×D4—612no—
428C4×D4—611no—
429C4×D4—611no—
430C4×D4—612no—
431C4×D4—611no—
432C4×D4—1222no—
433C4×D4—1222no—
434C4×D4—1222no—
435C4×D4—1221no—
436C4×D4—1228no—
437C4×D4—1221no—
438#117—21no—
439#117—612no—
440#117—1222no—
441#117—1221no—
442C2×C4⋊C4—21no—
443C2×C4⋊C4—611no—
444C2×C4⋊C4—612no—
445C2×C4⋊C4—1221no—
446C2×C4⋊C4—1222no—
447C2×C22⋊C4—612no—
448C2×C22⋊C4—1222no—
449C2×C22⋊C4—1222no—
450C2×C42—21no—
451C2×C42—1221no—
452C2×C42—1222no—
453C2×C42C22×D6111yescharacteristic
454C4⋊C8—21no—
455C4⋊C8—1221no—
456C22⋊C8—612no—
457C22⋊C8—1222no—
458C22⋊C8—1222no—
459C22⋊C8C22×D6111yescharacteristic
460C22⋊C8C22×D6111yescharacteristic
461#97—1221no—
462C4×C8—21no—
463#95—611no—
464#95—611no—
465#95—1222no—
466#95—1221no—
467#95C22×D6112yes—
468C24—612no—
469#41—4882no—
470#41—2444no—
471C2×Q8—21no—
472C2×Q8—611no—
473C2×Q8—1222no—
474C2×Q8—4882no—
475C2×Q8C2×C4×D6112yes—
476C2×D4—611no—
477C2×D4—611no—
478C2×D4—612no—
479C2×D4—611no—
480C2×D4—1224no—
481C2×D4—1222no—
482C2×D4—1222no—
483C2×D4—1221no—
484C22×C4—22no—
485C22×C4—612no—
486C22×C4—612no—
487C22×C4—612no—
488C22×C4—612no—
489C22×C4—612no—
490C22×C4—1222no—
491C22×C4—1222no—
492C22×C4—1228no—
493C22×C4—1222no—
494C22×C4—1221no—
495C22×C4—1221no—
496C22×C4—1221no—
497C22×C4—4882no—
498C22×C4—4882no—
499C22×C4—4882no—
500C22×C4D4×D6111yescharacteristic
501C22×C4D4×D6111yescharacteristic
502C22×C4C23×D6111yescharacteristic
503M16—4882no—
504C2×C8—21no—
505C2×C8—21no—
506C2×C8—1222no—
507C2×C8—4882no—
508C4⋊C4—24no—
509C4⋊C4—22no—
510C4⋊C4—21no—
511C4⋊C4—21no—
512C4⋊C4—21no—
513C4⋊C4—22no—
514C4⋊C4—611no—
515C4⋊C4—612no—
516C4⋊C4—611no—
517C4⋊C4—611no—
518C4⋊C4—611no—
519C4⋊C4—1222no—
520C4⋊C4—1222no—
521C4⋊C4—1222no—
522C4⋊C4—1222no—
523C4⋊C4—1221no—
524C4⋊C4—1221no—
525C4⋊C4—1221no—
526C4⋊C4—1221no—
527C4⋊C4C2×C4×D6112yes—
528C22⋊C4—22no—
529C22⋊C4—612no—
530C22⋊C4—612no—
531C22⋊C4—612no—
532C22⋊C4—612no—
533C22⋊C4—1222no—
534C22⋊C4—1222no—
535C22⋊C4—1222no—
536C22⋊C4—1222no—
537C22⋊C4—1222no—
538C22⋊C4—1228no—
539C22⋊C4—1221no—
540C22⋊C4—1221no—
541C22⋊C4—1221no—
542C22⋊C4—1221no—
543C22⋊C4—1221no—
544C42—22no—
545C42—81no—
546C42—1222no—
547C42—1224no—
548C42—1221no—
549C42—4882no—
550C42C2×C4×D6111yescharacteristic
551C42C2×C4×D6111yescharacteristic
552C42D4×D6111yescharacteristic
553C42D4×D6111yescharacteristic
554C23—612no—
555C23—612no—
556C23—1224no—
557C23—1222no—
558C23—1222no—
559C23—1221no—
560C23—2442no—
561C23C2×D4×D6111yescharacteristic
562Q8—42no—
563Q8—42no—
564Q8—4882no—
565Q8—2441no—
566D4—614no—
567D4—1224no—
568D4—1222no—
569D4—4882no—
570D4—2441no—
571D4—2444no—
572D4—2441no—
573D4—2441no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—22no—
577C2×C4—24no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—41no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—612no—
584C2×C4—612no—
585C2×C4—1222no—
586C2×C4—1222no—
587C2×C4—1221no—
588C2×C4—1221no—
589C2×C4—1221no—
590C2×C4—4882no—
591C2×C4—4882no—
592C2×C4—4888no—
593C2×C4—2442no—
594C2×C4—2442no—
595C2×C4—2442no—
596C2×C4—2442no—
597C2×C4—2444no—
598C2×C4—2442no—
599C2×C4—2441no—
600C2×C4—2441no—
601C2×C4—2441no—
602C2×C4—2441no—
603C2×C4—2442no—
604C2×C4C22⋊C4×D6111yescharacteristic
605C2×C4C22⋊C4×D6111yescharacteristic
606C2×C4C22⋊C4×D6111yescharacteristic
607C2×C4C22⋊C4×D6111yescharacteristic
608C2×C4C22×C4×D6111yescharacteristic
609C2×C4C2×D4×D6111yescharacteristic
610C8—41no—
611C8—41no—
612C8—2442no—
613C22—21no—
614C22—21no—
615C22—614no—
616C22—1222no—
617C22—1222no—
618C22—1222no—
619C22—1222no—
620C22—1222no—
621C22—2444no—
622C22—2444no—
623C22—2441no—
624C22—2441no—
625C22—2441no—
626C22C2×C22⋊C4×D6111yescharacteristic
627C22#1684056811yescharacteristic
628C22#1684057511yescharacteristic
629C4—21no—
630C4—21no—
631C4—21no—
632C4—21no—
633C4—21no—
634C4—44no—
635C4—42no—
636C4—42no—
637C4—1222no—
638C4—1222no—
639C4—2442no—
640C4—2442no—
641C4—2441no—
642C4—2441no—
643C4—2441no—
644C4#104×D6112yes—
645C2—21no—
646C2—21no—
647C2—1224no—
648C2—4881no—
649C2—2442no—
650C2#2145427811yescharacteristic
651C2#2145434711yescharacteristic
652C2C2×#104×D6111yescharacteristic
subgroupG/H
centerC22C2×C22⋊C4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1503C2
socleC22×C61C22×C22⋊C4
prime radical#21459863C2
cosocleC22×C4×C61C24
hypercenter#1503D61
hypocenterC61#17548
LCSG ▷ C2×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1503

Complements

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

NH ≅ G/Ntype
2. #21454853C2semidirect
3. #21448179C2semidirect
4. #21444342C2semidirect
5. #21444337C2semidirect
6. #21444331C2semidirect
7. #21444322C2semidirect
8. #21444309C2semidirect
9. #21444306C2semidirect
10. #328×D61C2semidirect
11. C61×#1503C2semidirect
12. #16840486C4semidirect
14. #16840453C4semidirect
16. #16840427C4semidirect
17. #16840424C4semidirect
18. #16840415C4semidirect
19. #16840415C4semidirect
54. C2×#11758226C2×C4semidirect
55. C2×Q8×D61C2×C4semidirect
65. #14519769C2×C4semidirect
69. #14519765C2×C4semidirect
76. #14519757C2×C4semidirect
82. #14519753C2×C4semidirect
83. #14519753C2×C4semidirect
86. #14519752C2×C4semidirect
90. C4×Dic122C2×C4semidirect
95. C4×D244C2×C4semidirect
104. #14519735C2×C4semidirect
110. C4⋊C4×D61C2×C4semidirect
125. C4⋊Q8×C61C2×C4semidirect
126. C4⋊Q8×C61C2×C4semidirect
131. C4×Q8×C61C2×C4semidirect
326. #1503D61semidirect
379. C61#17548semidirect
389. C4⋊Q8C4×D61semidirect
390. C4⋊Q8C4×D61semidirect
424. C4×Q8C4×D61semidirect

Maximal subgroups

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

subgroupindex
#214598632
#214548532
#214481792
#214443422
#214443372
#214443312
#214443222
#214443092
#214443062
#328×D612
C61×#15032
#1754861