Finite Group Databasebeta

← #29647193

Subgroups — #29647193

Order 15616.

1084 conjugacy classes of subgroups.

The subgroup lattice has 656 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#533×D61C211yesmaximal, characteristic
2#21454447C211yesmaximal, characteristic
3C4×#14519735C211yesmaximal, characteristic
4#21444337C214yesmaximal
5#21444329C214yesmaximal, complemented
6#21444319C211yesmaximal, characteristic, complemented
7#21444318C211yesmaximal, characteristic, complemented
8#21444317C211yesmaximal, characteristic, complemented
9C61×#1503C211yesmaximal, characteristic, complemented
10#16840319C2211yescharacteristic
11#16840318—21no—
12#16840317C411yescharacteristic, complemented
13#16840317C411yescharacteristic, complemented
14#16840316—21no—
15#16840316C412yescomplemented
16#16840315—21no—
17#16840314—21no—
18#16840314C2212yes—
19C4⋊Q8×D61C411yescharacteristic, complemented
20C4⋊Q8×D61C411yescharacteristic, complemented
21#125×D61—21no—
22C22⋊Q8×D61—21no—
23C22⋊Q8×D61C2212yes—
24C4×Q8×D61—21no—
25C4×Q8×D61C412yescomplemented
26C2×#14519735—21no—
27#117×D61—21no—
28C2×C42×D61C2211yescharacteristic
29#16839818—21no—
30#16839816C2212yes—
31#16839815C2211yescharacteristic
32#16839813—21no—
33#16839784—21no—
34#16839724C2212yes—
35C4×#11758189—21no—
36C4⋊C4×Dic61—21no—
37C4×C4⋊Dic61C2211yescharacteristic
38C4×#11758198—21no—
39#16839609C2212yes—
40#16839607C2212yes—
41C42×Dic61—21no—
42#16839500C2214yes—
43#16839499C2214yes—
44#16839498C2214yes—
45C61×#533C2211yescharacteristic
46C61×#431C2211yescharacteristic
47C4×C4⋊C4×C61C2211yescharacteristic
48C61×#328C2214yes—
49C2×Q8×D61—21no—
50C2×Q8×D61C2×C412yes—
51C22×C4×D61D411yescharacteristic
52C22×C4×D61D411yescharacteristic
53C22×C4×D61C2311yescharacteristic
54C2×#11758189—22no—
55#14519777C2×C411yescharacteristic
56#14519777C2×C411yescharacteristic
57#14519776—21no—
58#14519776—21no—
59#14519776—21no—
60#14519776C2×C412yes—
61#14519775—21no—
62#14519775C2×C412yes—
63#14519769—21no—
64#14519768—21no—
65#14519768—21no—
66#14519767—22no—
67#14519767—22no—
68#14519757—21no—
69#14519757C2×C412yes—
70#14519756—21no—
71#14519756C2312yes—
72#14519755—22no—
73#14519755—22no—
74#14519754—22no—
75#14519754—22no—
76#14519753—21no—
77#14519753C2×C412yes—
78Q8×Dic61—21no—
79Q8×Dic61C2×C412yes—
80C4×Dic122—21no—
81C4×Dic122—22no—
82C4×Dic122—22no—
83C4×Dic122C2×C412yes—
84#14519742—21no—
85#14519742—22no—
86#14519742—22no—
87#14519742C2×C412yes—
88#14519736—21no—
89#14519735—28no—
90#14519735—21no—
91#14519735—21no—
92#14519734—22no—
93#14519733—22no—
94C2×C4⋊Dic61—21no—
95C4⋊C4×D61—21no—
96C4⋊C4×D61—21no—
97C4⋊C4×D61—21no—
98C4⋊C4×D61—22no—
99C4⋊C4×D61—22no—
100C4⋊C4×D61C2×C412yes—
101C2×#11758198—22no—
102C22⋊C4×D61—22no—
103C2×C4×Dic61—22no—
104C2×C4×Dic61—21no—
105C2×C4×Dic61—22no—
106C2×C4×Dic61D411yescharacteristic
107C2×C4×Dic61D411yescharacteristic
108C2×C4×Dic61C2311yescharacteristic
109C42×D61—22no—
110C42×D61C2×C411yescharacteristic
111C42×D61C2×C411yescharacteristic
112C42×D61C2×C411yescharacteristic
113C42×D61C2×C411yescharacteristic
114C42×D61D411yescharacteristic
115C42×D61D411yescharacteristic
116C42×D61D411yescharacteristic
117C42×D61D411yescharacteristic
118C4⋊#92418—21no—
119#14519706—22no—
120C22⋊#92418C2312yes—
121#14519689—22no—
122#14519687—22no—
123C4×#92418—21no—
124#14519684—22no—
125#14519684C2312yes—
126C4⋊Q8×C61C2×C411yescharacteristic
127C4⋊Q8×C61C2×C411yescharacteristic
128#125×C61—21no—
129C22⋊Q8×C61—21no—
130C22⋊Q8×C61C2312yes—
131C4×Q8×C61—21no—
132C4×Q8×C61C2×C412yes—
133#117×C61—21no—
134C2×C4⋊C4×C61—21no—
135C2×C42×C61—21no—
136C2×C42×C61C2311yescharacteristic
137C4⋊C8×C61—21no—
138C22⋊C8×C61C2312yes—
139C4×C8×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
148C22×Dic61—22no—
149C22×Dic61C2×D411yescharacteristic
150C22×Dic61C2×D411yescharacteristic
151C22×Dic61C2×D411yescharacteristic
152Q8×D61—44no—
153Q8×D61—44no—
154C2×C4×D61—22no—
155C2×C4×D61—22no—
156C2×C4×D61—24no—
157C2×C4×D61—22no—
158C2×C4×D61—22no—
159C2×C4×D61—42no—
160C2×C4×D61C22⋊C411yescharacteristic
161C2×C4×D61C22⋊C411yescharacteristic
162C2×C4×D61C22⋊C411yescharacteristic
163C2×C4×D61C22⋊C411yescharacteristic
164C2×C4×D61C22⋊C411yescharacteristic
165C2×C4×D61C22⋊C411yescharacteristic
166C2×C4×D61C22⋊C411yescharacteristic
167C2×C4×D61C22⋊C411yescharacteristic
168C2×C4×D61C22×C411yescharacteristic
169C2×C4×D61C22×C411yescharacteristic
170C2×C4×D61C2×D411yescharacteristic
171C2×C4×D61C2×D411yescharacteristic
172C2×Dic122—21no—
173C2×Dic122—22no—
174C2×Dic122—22no—
175C2×Dic122C22×C412yes—
176C2×C8×C61—22no—
177C4⋊C4×C61—24no—
178C4⋊C4×C61—21no—
179C4⋊C4×C61—21no—
180C4⋊C4×C61—21no—
181C4⋊C4×C61—22no—
182C4⋊C4×C61—22no—
183C4⋊C4×C61C22×C412yes—
184C22⋊C4×C61—22no—
185C42×C61—22no—
186C42×C61—42no—
187C42×C61C22×C411yescharacteristic
188C42×C61C22×C411yescharacteristic
189C42×C61C2×D411yescharacteristic
190C42×C61C2×D411yescharacteristic
191C22⋊Dic61—22no—
192#11758198—28no—
193#11758198—22no—
194#11758198—22no—
195C4⋊Dic61—24no—
196C4⋊Dic61—21no—
197C4⋊Dic61—21no—
198C4⋊Dic61—21no—
199C4⋊Dic61—22no—
200C4⋊Dic61—22no—
201C4⋊Dic61C22×C412yes—
202#11758196—22no—
203#11758196—22no—
204#11758196—22no—
205#11758196—22no—
206#11758196—22no—
207#11758196—22no—
208#11758196—22no—
209#11758196—22no—
210C4×Dic61—28no—
211C4×Dic61—22no—
212C4×Dic61—22no—
213C4×Dic61—22no—
214C4×Dic61—44no—
215C4×Dic61—42no—
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×Dic61C22×C411yescharacteristic
226C4×Dic61C2×D411yescharacteristic
227C4×Dic61C2×D411yescharacteristic
228C2×#92418—22no—
229#11758189—44no—
230C23×C61C22×D411yescharacteristic
231C22×D61—21no—
232C22×D61—21no—
233C22×D61—22no—
234C22×D61—41no—
235C22×D61C2×C22⋊C411yescharacteristic
236C22×D61C2×C22⋊C411yescharacteristic
237Q8×C61—42no—
238Q8×C61—42no—
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×C61#13712yes—
251C2×C4×C61#13712yes—
252C2×C4×C61C23×C411yescharacteristic
253C2×C4×C61C22×D411yescharacteristic
254C2×Dic61—21no—
255C2×Dic61—21no—
256C2×Dic61—22no—
257C2×Dic61—22no—
258C2×Dic61—24no—
259C2×Dic61—22no—
260C2×Dic61—22no—
261C2×Dic61—42no—
262C2×Dic61—42no—
263C2×Dic61C2×C22⋊C411yescharacteristic
264C2×Dic61C2×C22⋊C411yescharacteristic
265C2×Dic61C2×C22⋊C411yescharacteristic
266C2×Dic61C2×C22⋊C411yescharacteristic
267C2×Dic61C2×C22⋊C411yescharacteristic
268C2×Dic61C2×C22⋊C411yescharacteristic
269C4×D61—22no—
270C4×D61—22no—
271C4×D61—22no—
272C4×D61—22no—
273C4×D61—22no—
274C4×D61—22no—
275C4×D61—22no—
276C4×D61—22no—
277C4×D61—22no—
278C4×D61—41no—
279C4×D61—41no—
280C4×D61—44no—
281C4×D61—44no—
282C4×D61#10414yes—
283Dic122—42no—
284Dic122—44no—
285Dic122—44no—
286Dic122—42no—
287C8×C61—42no—
288#92418—42no—
289#17545—611nomaximal, Sylow
290C22×C61—21no—
291C22×C61—21no—
292C22×C61C22×C22⋊C411yescharacteristic
293C22×C61C2×#13711yescharacteristic
294C22×C61C2×#13711yescharacteristic
295C2×D61—21no—
296C2×D61—21no—
297C2×D61—21no—
298C2×D61—21no—
299C2×D61—21no—
300C2×D61—42no—
301C2×D61—42no—
302C2×D61C2×#10412yes—
303C4×C61—21no—
304C4×C61—21no—
305C4×C61—21no—
306C4×C61—44no—
307C4×C61—42no—
308C4×C61—42no—
309C4×C61#41912yes—
310C4×C61#41912yes—
311C4×C61C2×#10412yes—
312Dic61—21no—
313Dic61—21no—
314Dic61—21no—
315Dic61—21no—
316Dic61—44no—
317Dic61—42no—
318Dic61—42no—
319Dic61#41012yes—
320Dic61C2×#10412yes—
321C2×#533—611no—
322#2920—611no—
323C4×#117—611no—
324#1508—614no—
325#1506—614no—
326#1503—611no—
327#1503—611no—
328#1503—611no—
329#1503D6111yescharacteristic, complemented
330C2×C61—21no—
331C2×C61—21no—
332C2×C61C2×#41011yescharacteristic
333C2×C61C2×#41911yescharacteristic
334C2×C61C22×#10411yescharacteristic
335D61—22no—
336D61—41no—
337#533—611no—
338#533—611no—
339#533—612no—
340#533—612no—
341#533—611no—
342#533—1221no—
343#533—1221no—
344#533—1221no—
345#533—1221no—
346#533C2×D6111yescharacteristic
347C2×C4⋊Q8—611no—
348C2×C4⋊Q8—611no—
349C2×#125—1221no—
350C2×C22⋊Q8—612no—
351C2×C22⋊Q8—1221no—
352C2×C4×Q8—612no—
353C2×C4×Q8—1221no—
354C2×#117—1221no—
355C2×#117—1221no—
356C22×C42—611no—
357#432—612no—
358#432—1221no—
359#431—611no—
360#431—1221no—
361#431C2×D6111yescharacteristic
362#423—1221no—
363#407—612no—
364C4×M16—1221no—
365C4×C4⋊C4—611no—
366C4×C4⋊C4—1221no—
367C4×C4⋊C4C2×D6111yescharacteristic
368C4×C22⋊C4—612no—
369C4×C22⋊C4—1221no—
370#376—612no—
371C43—1221no—
372#328—614no—
373#328—614no—
374#328—614no—
375#328C2×D6114yes—
376C61#1754511yescharacteristic, complemented, Sylow
377C22×Q8—612no—
378C22×Q8—1221no—
379C23×C4—611no—
380C23×C4—611no—
381C23×C4—611no—
382C2×M16—1222no—
383C4⋊Q8—612no—
384C4⋊Q8—612no—
385C4⋊Q8—611no—
386C4⋊Q8—611no—
387C4⋊Q8—1221no—
388C4⋊Q8—1221no—
389C4⋊Q8—1221no—
390C4⋊Q8—1221no—
391C4⋊Q8C4×D6111yescharacteristic
392C4⋊Q8C4×D6111yescharacteristic
393#125—21no—
394#125—1222no—
395#125—1222no—
396#125—1221no—
397#125—1221no—
398#125—1221no—
399C22⋊Q8—21no—
400C22⋊Q8—612no—
401C22⋊Q8—612no—
402C22⋊Q8—612no—
403C22⋊Q8—1222no—
404C22⋊Q8—1222no—
405C22⋊Q8—1222no—
406C22⋊Q8—1222no—
407C22⋊Q8—1221no—
408C22⋊Q8—1221no—
409C22⋊Q8—1221no—
410C22⋊Q8C22×D6112yes—
411C4×Q8—21no—
412C4×Q8—612no—
413C4×Q8—612no—
414C4×Q8—612no—
415C4×Q8—1222no—
416C4×Q8—1222no—
417C4×Q8—1222no—
418C4×Q8—1222no—
419C4×Q8—1221no—
420C4×Q8—1221no—
421C4×Q8—1221no—
422C4×Q8C4×D6112yes—
423#117—21no—
424#117—1222no—
425#117—1222no—
426#117—1221no—
427#117—1221no—
428#117—1228no—
429#117—1221no—
430C2×C4⋊C4—21no—
431C2×C4⋊C4—612no—
432C2×C4⋊C4—1221no—
433C2×C4⋊C4—1221no—
434C2×C4⋊C4—1222no—
435C2×C4⋊C4—1222no—
436C2×C4⋊C4—1221no—
437C2×C4⋊C4—1221no—
438C2×C22⋊C4—1222no—
439C2×C22⋊C4—1222no—
440C2×C42—21no—
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—611no—
451C2×C42—611no—
452C2×C42—1222no—
453C2×C42—1222no—
454C2×C42—1222no—
455C2×C42—1221no—
456C2×C42C22×D6111yescharacteristic
457C4⋊C8—21no—
458C4⋊C8—1222no—
459C4⋊C8—1221no—
460C22⋊C8—612no—
461C22⋊C8—1222no—
462C22⋊C8C22×D6112yes—
463#97—1222no—
464C4×C8—21no—
465C4×C8—1221no—
466#95—612no—
467#95—1222no—
468#95C22×D6112yes—
469C24—611no—
470C2×Q8—21no—
471C2×Q8—612no—
472C2×Q8—1222no—
473C2×Q8—1222no—
474C2×Q8—1221no—
475C2×Q8—2444no—
476C2×Q8—2444no—
477C2×Q8C2×C4×D6112yes—
478C22×C4—22no—
479C22×C4—611no—
480C22×C4—611no—
481C22×C4—611no—
482C22×C4—611no—
483C22×C4—611no—
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—1224no—
495C22×C4—1222no—
496C22×C4—1222no—
497C22×C4—1222no—
498C22×C4—1222no—
499C22×C4—1222no—
500C22×C4—2442no—
501C22×C4D4×D6111yescharacteristic
502C22×C4D4×D6111yescharacteristic
503C22×C4C23×D6111yescharacteristic
504M16—2444no—
505C2×C8—22no—
506C2×C8—1222no—
507C4⋊C4—24no—
508C4⋊C4—22no—
509C4⋊C4—21no—
510C4⋊C4—21no—
511C4⋊C4—21no—
512C4⋊C4—22no—
513C4⋊C4—612no—
514C4⋊C4—1222no—
515C4⋊C4—1222no—
516C4⋊C4—1222no—
517C4⋊C4—1222no—
518C4⋊C4—1222no—
519C4⋊C4—1222no—
520C4⋊C4—1222no—
521C4⋊C4—1222no—
522C4⋊C4—1224no—
523C4⋊C4—1222no—
524C4⋊C4—1222no—
525C4⋊C4—1221no—
526C4⋊C4—1221no—
527C4⋊C4—1221no—
528C4⋊C4C2×C4×D6112yes—
529C22⋊C4—22no—
530C22⋊C4—1222no—
531C22⋊C4—1222no—
532C22⋊C4—1228no—
533C22⋊C4—1222no—
534C42—22no—
535C42—42no—
536C42—611no—
537C42—611no—
538C42—611no—
539C42—611no—
540C42—611no—
541C42—611no—
542C42—611no—
543C42—611no—
544C42—611no—
545C42—611no—
546C42—611no—
547C42—611no—
548C42—1222no—
549C42—1222no—
550C42—1228no—
551C42—1222no—
552C42—2444no—
553C42—2442no—
554C42C2×C4×D6111yescharacteristic
555C42C2×C4×D6111yescharacteristic
556C42D4×D6111yescharacteristic
557C42D4×D6111yescharacteristic
558C23—611no—
559C23—611no—
560C23—1222no—
561C23—1221no—
562C23—1221no—
563C23—2441no—
564C23C2×D4×D6111yescharacteristic
565Q8—42no—
566Q8—42no—
567Q8—2442no—
568Q8—2444no—
569Q8—2444no—
570Q8—2442no—
571C2×C4—22no—
572C2×C4—24no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—41no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—611no—
579C2×C4—611no—
580C2×C4—614no—
581C2×C4—611no—
582C2×C4—611no—
583C2×C4—611no—
584C2×C4—611no—
585C2×C4—1222no—
586C2×C4—1222no—
587C2×C4—1222no—
588C2×C4—1222no—
589C2×C4—1222no—
590C2×C4—1222no—
591C2×C4—1221no—
592C2×C4—1221no—
593C2×C4—1222no—
594C2×C4—1222no—
595C2×C4—1222no—
596C2×C4—1222no—
597C2×C4—1222no—
598C2×C4—1224no—
599C2×C4—1222no—
600C2×C4—1222no—
601C2×C4—2441no—
602C2×C4—2441no—
603C2×C4—2444no—
604C2×C4—2444no—
605C2×C4—2442no—
606C2×C4—2442no—
607C2×C4C22⋊C4×D6111yescharacteristic
608C2×C4C22⋊C4×D6111yescharacteristic
609C2×C4C22⋊C4×D6111yescharacteristic
610C2×C4C22⋊C4×D6111yescharacteristic
611C2×C4#1451980612yes—
612C2×C4#1451980912yes—
613C2×C4C22×C4×D6111yescharacteristic
614C2×C4C2×D4×D6111yescharacteristic
615C8—42no—
616C8—2442no—
617C22—21no—
618C22—21no—
619C22—612no—
620C22—1221no—
621C22—1221no—
622C22—1221no—
623C22—1221no—
624C22—1221no—
625C22—2442no—
626C22—2442no—
627C22C2×C22⋊C4×D6111yescharacteristic
628C22C2×#1451980611yescharacteristic
629C22C2×#1451980911yescharacteristic
630C4—21no—
631C4—21no—
632C4—21no—
633C4—44no—
634C4—42no—
635C4—42no—
636C4—612no—
637C4—612no—
638C4—1221no—
639C4—1221no—
640C4—1221no—
641C4—1221no—
642C4—2444no—
643C4—2442no—
644C4—2442no—
645C4#1683976912yes—
646C4#1683976912yes—
647C4#104×D6112yes—
648C2—21no—
649C2—21no—
650C2—1222no—
651C2—2441no—
652C2#410×D6111yescharacteristic
653C2C2×#1683976911yescharacteristic
654C2C2×#104×D6111yescharacteristic
subgroupG/H
centerC2×C4C22⋊C4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1503C2
socleC22×C61C22×C22⋊C4
prime radicalC23×D61C2×D4
cosocleC22×C4×C61C24
hypercenter#1503D61
hypocenterC61#17545
LCSG ▷ C2×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1503

Complements

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

NH ≅ G/Ntype
5. #21444329C2semidirect
6. #21444319C2semidirect
7. #21444318C2semidirect
8. #21444317C2semidirect
9. C61×#1503C2semidirect
12. #16840317C4semidirect
13. #16840317C4semidirect
15. #16840316C4semidirect
19. C4⋊Q8×D61C4semidirect
20. C4⋊Q8×D61C4semidirect
25. C4×Q8×D61C4semidirect
329. #1503D61semidirect
376. C61#17545semidirect

Maximal subgroups

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

subgroupindex
#533×D612
#214544472
C4×#145197352
#214443372
#214443292
#214443192
#214443182
#214443172
C61×#15032
#1754561