Finite Group Databasebeta

← #29647169

Subgroups — #29647169

Order 15616.

1150 conjugacy classes of subgroups.

The subgroup lattice has 658 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21459116C211yesmaximal, characteristic
2#21454449C211yesmaximal, characteristic, complemented
3#376×D61C211yesmaximal, characteristic, complemented
4#21444331C211yesmaximal, characteristic, complemented
5#21444328C211yesmaximal, characteristic, complemented
6#21444326C211yesmaximal, characteristic, complemented
7#21444322C214yesmaximal, complemented
8#21444304C214yesmaximal, complemented
9C61×#1506C211yesmaximal, characteristic, complemented
10#16840317C411yescharacteristic, complemented
11#16840317C411yescharacteristic, complemented
12#16840313C411yescharacteristic, complemented
13#16840313C411yescharacteristic, complemented
14#16840312C2211yescharacteristic
15#16840299—21no—
16#16840299C412yescomplemented
17#16840298—21no—
18#16840298C2212yes—
19#16840297—21no—
20#16840297—21no—
21#16840275—21no—
22#16840275C2212yes—
23#16840274—21no—
24C4×#11758226—21no—
25C4×#11758226C412yescomplemented
26#16840230—21no—
27C2×C42×D61—21no—
28C2×C42×D61C2211yescharacteristic
29#16839819C2211yescharacteristic
30#16839817—21no—
31#16839814—21no—
32#16839812C2212yes—
33#16839784—21no—
34#16839724C2212yes—
35#16839714—21no—
36#16839608—21no—
37#16839608C2211yescharacteristic
38#16839607—21no—
39#16839607C2212yes—
40#95×D61—21no—
41#95×D61C2212yes—
42#16839499C2214yes—
43#16839497C2214yes—
44#16839495C2214yes—
45C61×#533C2211yescharacteristic
46C61×#432C2211yescharacteristic
47C61×#376C2211yescharacteristic
48C61×#328C2214yes—
49C2×#11758226—21no—
50C2×#11758226C2×C412yes—
51C2×#11758222—21no—
52C22×C4×D61—22no—
53C22×C4×D61D411yescharacteristic
54C22×C4×D61D411yescharacteristic
55C22×C4×D61C2311yescharacteristic
56C2×#11758189—22no—
57#14519777C2×C411yescharacteristic, complemented
58#14519774C2×C411yescharacteristic, complemented
59#14519772—22no—
60#14519772—22no—
61#14519768—21no—
62#14519768C2×C412yescomplemented
63#14519766—21no—
64#14519766C2×C412yescomplemented
65#14519765—21no—
66#14519762—22no—
67#14519762—22no—
68#14519761—22no—
69#14519757—21no—
70#14519757—21no—
71#14519753—21no—
72#14519753C2×C412yes—
73#14519752—21no—
74#14519752—21no—
75#14519752—21no—
76#14519752C2×C412yes—
77#14519751—21no—
78#14519751C2312yes—
79#14519747—22no—
80Q8×Dic61—21no—
81Q8×Dic61C2×C412yescomplemented
82#14519742—21no—
83#14519742—21no—
84C4×D244—22no—
85C4×D244—22no—
86C4×D244—21no—
87C4×D244C2×C412yescomplemented
88#14519739—22no—
89#14519739—22no—
90#14519739—21no—
91#14519739—21no—
92#14519739—21no—
93#14519739C2×C412yescomplemented
94#14519737—22no—
95#14519737—22no—
96#14519735—21no—
97#14519735—22no—
98#14519735—22no—
99#14519735C2×C412yes—
100#14519734—22no—
101C2×C4×Dic61—21no—
102C2×C4×Dic61—22no—
103C2×C4×Dic61D411yescharacteristic
104C2×C4×Dic61D411yescharacteristic
105C2×C4×Dic61C2311yescharacteristic
106C42×D61—28no—
107C42×D61—22no—
108C42×D61C2×C411yescharacteristic
109C42×D61C2×C411yescharacteristic
110C42×D61C2×C411yescharacteristic
111C42×D61C2×C411yescharacteristic
112C42×D61D411yescharacteristic
113C42×D61D411yescharacteristic
114C42×D61D411yescharacteristic
115C42×D61D411yescharacteristic
116C4⋊#92418—21no—
117#14519706—22no—
118C22⋊#92418C2312yes—
119#14519689—22no—
120#14519688—21no—
121C8×Dic61—22no—
122#14519684—21no—
123#14519684—22no—
124#14519684—22no—
125#14519684C2312yes—
126C4⋊Q8×C61C2×C411yescharacteristic, complemented
127C4⋊Q8×C61C2×C411yescharacteristic, complemented
128#125×C61—21no—
129C22⋊Q8×C61—21no—
130C22⋊Q8×C61C2312yes—
131C4×Q8×C61—21no—
132C4×Q8×C61C2×C412yescomplemented
133#117×C61—21no—
134C2×C42×C61—21no—
135C2×C42×C61C2311yescharacteristic
136C4⋊C8×C61—21no—
137C22⋊C8×C61C2312yes—
138#97×C61—21no—
139#95×C61—21no—
140#95×C61C2312yes—
141C23×D61C2×D411yescharacteristic
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—22no—
149C22×Dic61—22no—
150C22×Dic61C2×D411yescharacteristic
151C22×Dic61C2×D411yescharacteristic
152C22×Dic61C2×D411yescharacteristic
153#11758226—44no—
154#11758226—44no—
155#11758222—44no—
156C2×D244—22no—
157C2×D244—22no—
158C2×D244—21no—
159C2×D244—21no—
160C2×D244C22×C412yes—
161C2×C4×D61—24no—
162C2×C4×D61—28no—
163C2×C4×D61—22no—
164C2×C4×D61—22no—
165C2×C4×D61—22no—
166C2×C4×D61—42no—
167C2×C4×D61—42no—
168C2×C4×D61—44no—
169C2×C4×D61—44no—
170C2×C4×D61C22⋊C411yescharacteristic
171C2×C4×D61C22⋊C411yescharacteristic
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×D61C2×D411yescharacteristic
181C2×C4×D61C2×D411yescharacteristic
182C2×Dic122—21no—
183C2×C8×C61—22no—
184C4⋊C4×C61—21no—
185C4⋊C4×C61—21no—
186C4⋊C4×C61—21no—
187C4⋊C4×C61—22no—
188C4⋊C4×C61—22no—
189C4⋊C4×C61C22×C412yes—
190C22⋊C4×C61—22no—
191C42×C61—24no—
192C42×C61—22no—
193C42×C61C22×C411yescharacteristic
194C42×C61C22×C411yescharacteristic
195C42×C61C2×D411yescharacteristic
196C42×C61C2×D411yescharacteristic
197#11758198—22no—
198#11758198—22no—
199#11758198—22no—
200#11758198—22no—
201#11758198—22no—
202#11758198—22no—
203#11758198—22no—
204#11758198—22no—
205C4⋊Dic61—21no—
206C4⋊Dic61—22no—
207C4⋊Dic61—22no—
208C4⋊Dic61C22×C412yes—
209#11758196—22no—
210#11758196—22no—
211C4×Dic61—28no—
212C4×Dic61—24no—
213C4×Dic61—22no—
214C4×Dic61—22no—
215C4×Dic61C22⋊C411yescharacteristic
216C4×Dic61C22⋊C411yescharacteristic
217C4×Dic61C22⋊C411yescharacteristic
218C4×Dic61C22⋊C411yescharacteristic
219C4×Dic61C22⋊C411yescharacteristic
220C4×Dic61C22⋊C411yescharacteristic
221C4×Dic61C22⋊C411yescharacteristic
222C4×Dic61C22⋊C411yescharacteristic
223C4×Dic61C22×C411yescharacteristic
224C4×Dic61C22×C411yescharacteristic
225C4×Dic61C2×D411yescharacteristic
226C4×Dic61C2×D411yescharacteristic
227C2×#92418—22no—
228#11758189—44no—
229C23×C61C22×D411yescharacteristic
230C22×D61—22no—
231C22×D61—22no—
232C22×D61—22no—
233C22×D61—22no—
234C22×D61C2×C22⋊C411yescharacteristic
235C22×D61C2×C22⋊C411yescharacteristic
236Q8×C61—42no—
237Q8×C61—42no—
238C2×C4×C61—22no—
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—44no—
253C2×Dic61—22no—
254C2×Dic61—22no—
255C2×Dic61—24no—
256C2×Dic61—22no—
257C2×Dic61—41no—
258C2×Dic61—42no—
259C2×Dic61—42no—
260C2×Dic61C2×C22⋊C411yescharacteristic
261C2×Dic61C2×C22⋊C411yescharacteristic
262C2×Dic61C2×C22⋊C411yescharacteristic
263C2×Dic61C2×C22⋊C411yescharacteristic
264C2×Dic61C2×C22⋊C411yescharacteristic
265C2×Dic61C2×C22⋊C411yescharacteristic
266D244—42no—
267D244—42no—
268D244—44no—
269D244—44no—
270D244—42no—
271C4×D61—22no—
272C4×D61—22no—
273C4×D61—22no—
274C4×D61—22no—
275C4×D61—22no—
276C4×D61—41no—
277C4×D61—41no—
278C4×D61—41no—
279C4×D61—41no—
280C4×D61—41no—
281C4×D61—41no—
282C4×D61—44no—
283C4×D61—44no—
284C4×D61—416no—
285C4×D61—44no—
286C4×D61#10414yes—
287Dic122—42no—
288C8×C61—42no—
289#92418—42no—
290#17541—611nomaximal, Sylow
291C22×C61—21no—
292C22×C61—21no—
293C22×C61C22×C22⋊C411yescharacteristic
294C22×C61#57512yes—
295C2×D61—28no—
296C2×D61—21no—
297C2×D61—21no—
298C2×D61—21no—
299C2×D61—21no—
300C2×D61—42no—
301C2×D61—42no—
302C2×D61—42no—
303C2×D61#41012yes—
304C2×D61C2×#10412yes—
305C4×C61—21no—
306C4×C61—21no—
307C4×C61—21no—
308C4×C61—21no—
309C4×C61—21no—
310C4×C61—44no—
311C4×C61—42no—
312C4×C61—42no—
313C4×C61C2×#10412yes—
314Dic61—21no—
315Dic61—21no—
316Dic61—21no—
317Dic61—21no—
318Dic61—21no—
319Dic61—44no—
320Dic61—42no—
321Dic61C2×#10412yes—
322#3444—611no—
323#2920—611no—
324C2×#376—611no—
325#1506—611no—
326#1506D6111yescharacteristic, complemented
327#1505—611no—
328#1505—611no—
329#1504—614no—
330#1500—614no—
331C2×C61—22no—
332C2×C61C2×#41011yescharacteristic
333C2×C61#289611yescharacteristic
334C2×C61C22×#10411yescharacteristic
335D61—22no—
336D61—22no—
337D61—82no—
338#533—611no—
339#533—611no—
340#533C2×D6111yescharacteristic
341#532—611no—
342#532—611no—
343#532—611no—
344#529—612no—
345#529—612no—
346#529—1221no—
347#529—1221no—
348#529—1221no—
349#529—1221no—
350#525—612no—
351#525—1221no—
352#525—1221no—
353C4×#41—612no—
354C4×#41—1221no—
355C4×#41—1221no—
356C22×C42—611no—
357C22×C42—1221no—
358#432—611no—
359#432—1221no—
360#432C2×D6111yescharacteristic
361#431—612no—
362#431—1221no—
363#423—1221no—
364#407—612no—
365C4×M16—1221no—
366#376—611no—
367#376—612no—
368#376—1221no—
369#376—1221no—
370#376C2×D6111yescharacteristic
371C2×#95—612no—
372C2×#95—1221no—
373#328—614no—
374#328C2×D6114yes—
375#327—614no—
376#327—614no—
377C61#1754111yescharacteristic, complemented, Sylow
378C2×#41—612no—
379C2×#41—1221no—
380C2×#41—1221no—
381C23×C4—611no—
382C23×C4—611no—
383C23×C4—611no—
384C23×C4—1222no—
385C2×M16—1222no—
386C4⋊Q8—611no—
387C4⋊Q8C4×D6111yescharacteristic, complemented
388C4⋊Q8C4×D6111yescharacteristic, complemented
389#127—611no—
390#126—1222no—
391#126—1222no—
392#125—21no—
393#125—612no—
394#125—1221no—
395#124—612no—
396#124—1221no—
397#124—1221no—
398#123—1222no—
399#123—1222no—
400#123—1222no—
401C22⋊Q8—21no—
402C22⋊Q8—612no—
403C22⋊Q8—1221no—
404C22⋊Q8—1221no—
405C22⋊Q8—1221no—
406C22⋊Q8C22×D6112yes—
407#121—612no—
408#121—612no—
409#121—1221no—
410#121—1221no—
411#121—1221no—
412#121—1221no—
413#120—1222no—
414C4×Q8—21no—
415C4×Q8—612no—
416C4×Q8—1221no—
417C4×Q8—1221no—
418C4×Q8—1221no—
419C4×Q8C4×D6112yescomplemented
420C4×D4—612no—
421C4×D4—612no—
422C4×D4—1222no—
423C4×D4—1222no—
424C4×D4—1221no—
425C4×D4—1221no—
426C4×D4—1221no—
427C4×D4—1222no—
428C4×D4—1222no—
429C4×D4—1222no—
430C4×D4—1222no—
431C4×D4—1221no—
432#117—21no—
433#117—612no—
434#117—1222no—
435#117—1222no—
436#117—1222no—
437#117—1221no—
438C2×C42—21no—
439C2×C42—611no—
440C2×C42—611no—
441C2×C42—611no—
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—1222no—
451C2×C42—1228no—
452C2×C42—1222no—
453C2×C42—1221no—
454C2×C42C22×D6111yescharacteristic
455C4⋊C8—21no—
456C4⋊C8—1222no—
457C4⋊C8—1221no—
458C22⋊C8—612no—
459C22⋊C8—1222no—
460C22⋊C8C22×D6112yes—
461#97—21no—
462#97—1221no—
463C4×C8—1222no—
464#95—21no—
465#95—612no—
466#95—1222no—
467#95—1222no—
468#95—1221no—
469#95C22×D6112yes—
470C24—611no—
471#41—2444no—
472#41—2444no—
473#41—2444no—
474C2×Q8—21no—
475C2×Q8—1221no—
476C2×Q8C2×C4×D6112yes—
477C2×D4—612no—
478C2×D4—1222no—
479C2×D4—1222no—
480C2×D4—1222no—
481C2×D4—1221no—
482C2×D4—1221no—
483C22×C4—22no—
484C22×C4—611no—
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—611no—
498C22×C4—611no—
499C22×C4—1224no—
500C22×C4—1222no—
501C22×C4—1222no—
502C22×C4—1228no—
503C22×C4—1222no—
504C22×C4—1222no—
505C22×C4—2442no—
506C22×C4—2442no—
507C22×C4—2444no—
508C22×C4—2444no—
509C22×C4D4×D6111yescharacteristic
510C22×C4D4×D6111yescharacteristic
511C22×C4C23×D6111yescharacteristic
512M16—2444no—
513C2×C8—22no—
514C2×C8—1222no—
515C4⋊C4—22no—
516C4⋊C4—21no—
517C4⋊C4—21no—
518C4⋊C4—21no—
519C4⋊C4—22no—
520C4⋊C4—612no—
521C4⋊C4—1222no—
522C4⋊C4—1222no—
523C4⋊C4—1222no—
524C4⋊C4—1222no—
525C4⋊C4—1221no—
526C4⋊C4C2×C4×D6112yes—
527C22⋊C4—22no—
528C22⋊C4—1222no—
529C22⋊C4—1222no—
530C22⋊C4—1222no—
531C22⋊C4—1222no—
532C22⋊C4—1222no—
533C22⋊C4—1222no—
534C22⋊C4—1222no—
535C22⋊C4—1222no—
536C42—24no—
537C42—22no—
538C42—611no—
539C42—611no—
540C42—611no—
541C42—611no—
542C42—611no—
543C42—611no—
544C42—611no—
545C42—611no—
546C42—611no—
547C42—611no—
548C42—611no—
549C42—611no—
550C42—1222no—
551C42—1222no—
552C42—1224no—
553C42—1228no—
554C42C2×C4×D6111yescharacteristic
555C42C2×C4×D6111yescharacteristic
556C42D4×D6111yescharacteristic
557C42D4×D6111yescharacteristic
558C23—611no—
559C23—611no—
560C23—1222no—
561C23—1222no—
562C23—1222no—
563C23—1222no—
564C23C2×D4×D6111yescharacteristic
565Q8—42no—
566Q8—42no—
567Q8—2442no—
568D4—2442no—
569D4—2442no—
570D4—2444no—
571D4—2444no—
572D4—2444no—
573D4—2442no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—24no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—41no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—611no—
583C2×C4—614no—
584C2×C4—611no—
585C2×C4—611no—
586C2×C4—611no—
587C2×C4—611no—
588C2×C4—611no—
589C2×C4—1222no—
590C2×C4—1222no—
591C2×C4—1222no—
592C2×C4—1222no—
593C2×C4—1222no—
594C2×C4—1222no—
595C2×C4—1222no—
596C2×C4—1224no—
597C2×C4—1222no—
598C2×C4—2441no—
599C2×C4—2441no—
600C2×C4—2441no—
601C2×C4—2441no—
602C2×C4—2441no—
603C2×C4—2441no—
604C2×C4—2444no—
605C2×C4—2444no—
606C2×C4—2444no—
607C2×C4—24416no—
608C2×C4—2441no—
609C2×C4—2442no—
610C2×C4—2442no—
611C2×C4C22⋊C4×D6111yescharacteristic
612C2×C4C22⋊C4×D6111yescharacteristic
613C2×C4C22⋊C4×D6111yescharacteristic
614C2×C4C22⋊C4×D6111yescharacteristic
615C2×C4C22×C4×D6111yescharacteristic
616C2×C4C2×D4×D6111yescharacteristic
617C8—42no—
618C8—2442no—
619C22—21no—
620C22—21no—
621C22—612no—
622C22—612no—
623C22—1228no—
624C22—1221no—
625C22—1221no—
626C22—1221no—
627C22—1221no—
628C22—2442no—
629C22—2442no—
630C22—2442no—
631C22C2×C22⋊C4×D6111yescharacteristic
632C22#1684056112yes—
633C4—21no—
634C4—21no—
635C4—21no—
636C4—21no—
637C4—21no—
638C4—44no—
639C4—42no—
640C4—42no—
641C4—612no—
642C4—1221no—
643C4—1221no—
644C4—1221no—
645C4—1221no—
646C4—1221no—
647C4—2444no—
648C4—2442no—
649C4#104×D6112yes—
650C2—22no—
651C2—1222no—
652C2—1222no—
653C2—4882no—
654C2#410×D6111yescharacteristic
655C2#2145433711yescharacteristic
656C2C2×#104×D6111yescharacteristic
subgroupG/H
centerC22C2×C22⋊C4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1506C2
socleC22×C61C22×C22⋊C4
prime radical#21459116C2
cosocleC22×C4×C61C24
hypercenter#1506D61
hypocenterC61#17541
LCSG ▷ C2×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1506

Complements

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

NH ≅ G/Ntype
2. #21454449C2semidirect
3. #376×D61C2semidirect
4. #21444331C2semidirect
5. #21444328C2semidirect
6. #21444326C2semidirect
7. #21444322C2semidirect
8. #21444304C2semidirect
9. C61×#1506C2semidirect
10. #16840317C4semidirect
11. #16840317C4semidirect
12. #16840313C4semidirect
13. #16840313C4semidirect
16. #16840299C4semidirect
25. C4×#11758226C4semidirect
57. #14519777C2×C4semidirect
58. #14519774C2×C4semidirect
62. #14519768C2×C4semidirect
64. #14519766C2×C4semidirect
81. Q8×Dic61C2×C4semidirect
87. C4×D244C2×C4semidirect
93. #14519739C2×C4semidirect
126. C4⋊Q8×C61C2×C4semidirect
127. C4⋊Q8×C61C2×C4semidirect
132. C4×Q8×C61C2×C4semidirect
326. #1506D61semidirect
377. C61#17541semidirect
387. C4⋊Q8C4×D61semidirect
388. C4⋊Q8C4×D61semidirect
419. C4×Q8C4×D61semidirect

Maximal subgroups

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

subgroupindex
#214591162
#214544492
#376×D612
#214443312
#214443282
#214443262
#214443222
#214443042
C61×#15062
#1754161