Finite Group Databasebeta

← #29647192

Subgroups — #29647192

Order 15616.

1084 conjugacy classes of subgroups.

The subgroup lattice has 650 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840314C211yesmaximal, characteristic
2#21454448C211yesmaximal, characteristic
3C4×#14519733C211yesmaximal, characteristic
4#21444340C211yesmaximal, characteristic
5#21444339C211yesmaximal, characteristic
6#21444338C211yesmaximal, characteristic
7#21444329C214yesmaximal, complemented
8#21444316C214yesmaximal, complemented
9C61×#1508C211yesmaximal, characteristic
10#16840314—21no—
11#16840314—21no—
12#16840314—21no—
13#16840314—21no—
14#16840314C412yescomplemented
15#16840314C412yescomplemented
16#16840314C2212yes—
17#16840314C2212yes—
18C2×#14519776C411yescharacteristic, complemented
19C2×#14519775C411yescharacteristic, complemented
20C2×#14519767—21no—
21C2×#14519756C2211yescharacteristic
22C2×#14519754—21no—
23C2×Q8×Dic61C411yescharacteristic, complemented
24C2×#14519742—21no—
25C2×#14519742C411yescharacteristic, complemented
26C2×#14519733—21no—
27C2×#14519733—21no—
28C22×C4×Dic61C2211yescharacteristic
29#16839816—21no—
30#16839816C2212yes—
31#16839812—21no—
32#16839812C2212yes—
33#16839785—21no—
34#16839726C2211yescharacteristic
35M16×Dic61—21no—
36C4×#11758196—21no—
37#16839613C2212yes—
38C22⋊C4×Dic61—21no—
39C4×C22⋊Dic61C2211yescharacteristic
40#16839607C2212yes—
41C42×Dic61—21no—
42#16839500C2214yes—
43#16839499C2214yes—
44#16839498C2214yes—
45C2×C22⋊Q8×C61C2211yescharacteristic
46C61×#407C2211yescharacteristic
47C4×C22⋊C4×C61C2211yescharacteristic
48C61×#328C2214yes—
49C22×Dic122—21no—
50C22×Dic122C2×C411yescharacteristic
51C23×Dic61D411yescharacteristic
52C23×Dic61D411yescharacteristic
53C2×#11758194—21no—
54#14519776—21no—
55#14519776—21no—
56#14519776C2×C412yes—
57#14519776C2×C412yes—
58#14519775—21no—
59#14519775—21no—
60#14519775C2×C412yes—
61#14519775C2×C412yes—
62#14519767—22no—
63#14519767—22no—
64#14519767—22no—
65#14519767—22no—
66#14519756—21no—
67#14519756—21no—
68#14519756C2×C412yes—
69#14519756C2312yes—
70#14519754—22no—
71#14519754—22no—
72#14519754—22no—
73#14519754—22no—
74Q8×Dic61—21no—
75Q8×Dic61—21no—
76Q8×Dic61C2×C412yes—
77Q8×Dic61C2×C412yes—
78#14519742—22no—
79#14519742—22no—
80#14519742—21no—
81#14519742—21no—
82#14519742—22no—
83#14519742—22no—
84#14519742C2×C412yes—
85#14519742C2×C412yes—
86#14519733—28no—
87#14519733—22no—
88#14519733—22no—
89#14519733—22no—
90#14519733—22no—
91C2×C4⋊Dic61—21no—
92C2×C4⋊Dic61C2×C411yescharacteristic
93C2×#11758196—21no—
94C2×#11758196—21no—
95C2×#11758196—21no—
96C2×#11758196—21no—
97C2×#11758196—21no—
98C2×#11758196—22no—
99C2×C22⋊Dic61—21no—
100C2×C22⋊Dic61—21no—
101C2×C4×Dic61—21no—
102C2×C4×Dic61—21no—
103C2×C4×Dic61—22no—
104C2×C4×Dic61—21no—
105C2×C4×Dic61—22no—
106C2×C4×Dic61C2×C411yescharacteristic
107C2×C4×Dic61C2×C411yescharacteristic
108C2×C4×Dic61C2×C411yescharacteristic
109C2×C4×Dic61C2×C411yescharacteristic
110C2×C4×Dic61D411yescharacteristic
111C2×C4×Dic61D411yescharacteristic
112C2×C4×Dic61D411yescharacteristic
113C2×C4×Dic61D411yescharacteristic
114C2×C4×Dic61D411yescharacteristic
115C2×C4×Dic61D411yescharacteristic
116C2×C4×Dic61C2311yescharacteristic
117C2×C4×Dic61C2311yescharacteristic
118#14519706—22no—
119#14519706—22no—
120C22⋊#92418—21no—
121C22⋊#92418C2312yes—
122#14519687—22no—
123C8×Dic61—22no—
124#14519684—22no—
125#14519684C2312yes—
126C22×Q8×C61C2×C411yescharacteristic
127C23×C4×C61C2311yescharacteristic
128C2×M16×C61—21no—
129C22⋊Q8×C61—21no—
130C22⋊Q8×C61—21no—
131C22⋊Q8×C61C2×C412yes—
132C22⋊Q8×C61C2312yes—
133C2×C4⋊C4×C61—21no—
134C2×C4⋊C4×C61C2×C411yescharacteristic
135C2×C22⋊C4×C61—21no—
136C2×C22⋊C4×C61—21no—
137C2×C42×C61—21no—
138C22⋊C8×C61—21no—
139C22⋊C8×C61C2312yes—
140#95×C61C2312yes—
141C24×C61C2×D411yescharacteristic
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
155C22×Dic61—24no—
156C22×Dic61—24no—
157C22×Dic61—21no—
158C22×Dic61—21no—
159C22×Dic61—22no—
160C22×Dic61C22⋊C411yescharacteristic
161C22×Dic61C22⋊C411yescharacteristic
162C22×Dic61C22⋊C411yescharacteristic
163C22×Dic61C22⋊C411yescharacteristic
164C22×Dic61C22⋊C411yescharacteristic
165C22×Dic61C22⋊C411yescharacteristic
166C22×Dic61C22⋊C411yescharacteristic
167C22×Dic61C22⋊C411yescharacteristic
168C22×Dic61C2×D411yescharacteristic
169C22×Dic61C2×D411yescharacteristic
170C22×Dic61C2×D411yescharacteristic
171C22×Dic61C2×D411yescharacteristic
172C2×Dic122—21no—
173C2×Dic122—22no—
174C2×Dic122—22no—
175C2×Dic122—44no—
176C2×Dic122—42no—
177C2×Dic122C22×C412yes—
178M16×C61—42no—
179C2×C8×C61—22no—
180C4⋊C4×C61—22no—
181C4⋊C4×C61—21no—
182C4⋊C4×C61—22no—
183C4⋊C4×C61C22×C412yes—
184C22⋊C4×C61—24no—
185C22⋊C4×C61—22no—
186C22⋊C4×C61—22no—
187C42×C61—42no—
188C22⋊Dic61—24no—
189C22⋊Dic61—22no—
190C22⋊Dic61—22no—
191C4⋊Dic61—21no—
192C4⋊Dic61—22no—
193C4⋊Dic61—22no—
194C4⋊Dic61C22×C412yes—
195#11758196—28no—
196#11758196—22no—
197#11758196—22no—
198#11758196—22no—
199#11758196—22no—
200#11758196—22no—
201#11758196—22no—
202#11758196—22no—
203#11758196—22no—
204#11758196—22no—
205#11758196—22no—
206C4×Dic61—28no—
207C4×Dic61—22no—
208C4×Dic61—22no—
209C4×Dic61—22no—
210C4×Dic61—22no—
211C4×Dic61—42no—
212C4×Dic61—44no—
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×Dic61C22×C411yescharacteristic
223C4×Dic61C22×C411yescharacteristic
224C4×Dic61C22×C411yescharacteristic
225C4×Dic61C2×D411yescharacteristic
226C4×Dic61C2×D411yescharacteristic
227C4×Dic61C2×D411yescharacteristic
228C4×Dic61C2×D411yescharacteristic
229#11758194—42no—
230C2×#92418—22no—
231C23×C61—22no—
232C23×C61—22no—
233C23×C61—41no—
234C23×C61C2×C22⋊C411yescharacteristic
235C23×C61C2×C22⋊C411yescharacteristic
236C23×C61C22×D411yescharacteristic
237Q8×C61—42no—
238Q8×C61—42no—
239C2×C4×C61—22no—
240C2×C4×C61—22no—
241C2×C4×C61—22no—
242C2×C4×C61—24no—
243C2×C4×C61—22no—
244C2×C4×C61—22no—
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×C61C2×C22⋊C411yescharacteristic
253C2×C4×C61C2×C22⋊C411yescharacteristic
254C2×C4×C61C23×C411yescharacteristic
255C2×C4×C61C22×D411yescharacteristic
256C2×Dic61—22no—
257C2×Dic61—22no—
258C2×Dic61—22no—
259C2×Dic61—22no—
260C2×Dic61—22no—
261C2×Dic61—22no—
262C2×Dic61—22no—
263C2×Dic61—22no—
264C2×Dic61—22no—
265C2×Dic61—22no—
266C2×Dic61—24no—
267C2×Dic61—22no—
268C2×Dic61—22no—
269C2×Dic61—42no—
270C2×Dic61—42no—
271C2×Dic61—42no—
272C2×Dic61—42no—
273C2×Dic61#10414yes—
274C2×Dic61C2×C22⋊C411yescharacteristic
275C2×Dic61C2×C22⋊C411yescharacteristic
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×Dic61#13714yes—
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×C61—21no—
293C22×C61—21no—
294C22×C61—21no—
295C22×C61—21no—
296C22×C61—21no—
297C22×C61—44no—
298C22×C61C2×#10412yes—
299C22×C61C22×C22⋊C411yescharacteristic
300C22×C61C2×#13712yes—
301C4×C61—21no—
302C4×C61—21no—
303C4×C61—21no—
304C4×C61—44no—
305C4×C61—42no—
306C4×C61—42no—
307C4×C61#41012yes—
308Dic61—21no—
309Dic61—21no—
310Dic61—21no—
311Dic61—21no—
312Dic61—44no—
313Dic61—42no—
314Dic61—42no—
315Dic61#41912yes—
316Dic61#41912yes—
317Dic61C2×#10412yes—
318Dic61C2×#10412yes—
319C2×#533—611no—
320#2920—611no—
321C4×#117—611no—
322#1508—611no—
323#1508—611no—
324#1508—611no—
325#1508D6111yescharacteristic
326#1506—614no—
327#1503—614no—
328C2×C61—22no—
329C2×C61—22no—
330C2×C61—41no—
331C2×C61C2×#41011yescharacteristic
332C2×C61C2×#41911yescharacteristic
333C2×C61C22×#10411yescharacteristic
334#533—612no—
335#533—612no—
336#533—612no—
337#533—612no—
338#533—1221no—
339#533—1221no—
340#533—1221no—
341#533—1221no—
342C2×C4⋊Q8—611no—
343C2×C4⋊Q8—611no—
344C2×#125—1221no—
345C2×C22⋊Q8—611no—
346C2×C22⋊Q8—1221no—
347C2×C22⋊Q8C2×D6111yescharacteristic
348C2×C4×Q8—611no—
349C2×C4×Q8—611no—
350C2×C4×Q8—1221no—
351C2×#117—1221no—
352C2×#117—1221no—
353C22×C42—611no—
354#432—612no—
355#432—1221no—
356#431—612no—
357#431—1221no—
358#423—1221no—
359#407—611no—
360#407C2×D6111yescharacteristic
361C4×M16—1221no—
362C4×C4⋊C4—612no—
363C4×C4⋊C4—1221no—
364C4×C22⋊C4—611no—
365C4×C22⋊C4—1221no—
366C4×C22⋊C4C2×D6111yescharacteristic
367#376—612no—
368C43—1221no—
369#328—614no—
370#328—614no—
371#328—614no—
372#328C2×D6114yes—
373C61#1754511yescharacteristic, complemented, Sylow
374C22×Q8—611no—
375C22×Q8—1221no—
376C22×Q8C4×D6111yescharacteristic
377C23×C4—611no—
378C23×C4—611no—
379C23×C4C22×D6111yescharacteristic
380C2×M16—21no—
381C2×M16—1221no—
382C4⋊Q8—612no—
383C4⋊Q8—612no—
384C4⋊Q8—612no—
385C4⋊Q8—612no—
386C4⋊Q8—1221no—
387C4⋊Q8—1221no—
388C4⋊Q8—1221no—
389C4⋊Q8—1221no—
390#125—1222no—
391#125—1222no—
392#125—1222no—
393#125—1222no—
394C22⋊Q8—21no—
395C22⋊Q8—21no—
396C22⋊Q8—612no—
397C22⋊Q8—612no—
398C22⋊Q8—1222no—
399C22⋊Q8—1222no—
400C22⋊Q8—1222no—
401C22⋊Q8—1222no—
402C22⋊Q8—1221no—
403C22⋊Q8—1221no—
404C22⋊Q8C4×D6112yes—
405C22⋊Q8C22×D6112yes—
406C4×Q8—612no—
407C4×Q8—612no—
408C4×Q8—612no—
409C4×Q8—612no—
410C4×Q8—1221no—
411C4×Q8—1221no—
412C4×Q8—1222no—
413C4×Q8—1222no—
414C4×Q8—1222no—
415C4×Q8—1222no—
416C4×Q8—1221no—
417C4×Q8—1221no—
418#117—1222no—
419#117—1222no—
420#117—1222no—
421#117—1222no—
422#117—1228no—
423C2×C4⋊C4—21no—
424C2×C4⋊C4—611no—
425C2×C4⋊C4—1221no—
426C2×C4⋊C4—1221no—
427C2×C4⋊C4—1222no—
428C2×C4⋊C4—1221no—
429C2×C4⋊C4—1221no—
430C2×C4⋊C4—1221no—
431C2×C4⋊C4—1221no—
432C2×C4⋊C4C4×D6111yescharacteristic
433C2×C22⋊C4—21no—
434C2×C22⋊C4—21no—
435C2×C22⋊C4—1221no—
436C2×C22⋊C4—1221no—
437C2×C42—21no—
438C2×C42—611no—
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—1221no—
451C2×C42—1221no—
452C2×C42—1222no—
453C2×C42—1222no—
454C2×C42—1221no—
455C4⋊C8—1222no—
456C4⋊C8—1222no—
457C22⋊C8—21no—
458C22⋊C8—612no—
459C22⋊C8—1221no—
460C22⋊C8C22×D6112yes—
461#97—1222no—
462C4×C8—1222no—
463#95—612no—
464#95—1222no—
465#95C22×D6112yes—
466C24D4×D6111yescharacteristic
467C2×Q8—21no—
468C2×Q8—42no—
469C2×Q8—612no—
470C2×Q8—1222no—
471C2×Q8—1222no—
472C2×Q8—1221no—
473C2×Q8—2442no—
474C2×Q8—2444no—
475C2×Q8C2×C4×D6112yes—
476C22×C4—22no—
477C22×C4—21no—
478C22×C4—21no—
479C22×C4—42no—
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—1224no—
493C22×C4—1224no—
494C22×C4—1222no—
495C22×C4—1221no—
496C22×C4—1221no—
497C22×C4C2×C4×D6111yescharacteristic
498C22×C4C2×C4×D6111yescharacteristic
499C22×C4D4×D6111yescharacteristic
500C22×C4D4×D6111yescharacteristic
501C22×C4D4×D6111yescharacteristic
502C22×C4C23×D6111yescharacteristic
503M16—42no—
504M16—2442no—
505C2×C8—22no—
506C2×C8—1222no—
507C4⋊C4—22no—
508C4⋊C4—22no—
509C4⋊C4—21no—
510C4⋊C4—612no—
511C4⋊C4—1222no—
512C4⋊C4—1222no—
513C4⋊C4—1222no—
514C4⋊C4—1222no—
515C4⋊C4—1222no—
516C4⋊C4—1222no—
517C4⋊C4—1222no—
518C4⋊C4—1222no—
519C4⋊C4—1222no—
520C4⋊C4—1222no—
521C4⋊C4—1228no—
522C4⋊C4—1222no—
523C4⋊C4—1221no—
524C4⋊C4—1222no—
525C4⋊C4C2×C4×D6112yes—
526C22⋊C4—24no—
527C22⋊C4—22no—
528C22⋊C4—22no—
529C22⋊C4—1224no—
530C22⋊C4—1222no—
531C22⋊C4—1222no—
532C42—42no—
533C42—611no—
534C42—611no—
535C42—611no—
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—611no—
549C42—1222no—
550C42—1222no—
551C42—1222no—
552C42—1222no—
553C42—1228no—
554C42—2442no—
555C42—2444no—
556C23—22no—
557C23—22no—
558C23—41no—
559C23C22⋊C4×D6111yescharacteristic
560C23C22⋊C4×D6111yescharacteristic
561C23C2×D4×D6111yescharacteristic
562Q8—42no—
563Q8—42no—
564Q8—2442no—
565Q8—2444no—
566Q8—2444no—
567Q8—2442no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—24no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—611no—
582C2×C4—611no—
583C2×C4—611no—
584C2×C4—611no—
585C2×C4—614no—
586C2×C4—614no—
587C2×C4—611no—
588C2×C4—611no—
589C2×C4—611no—
590C2×C4—611no—
591C2×C4—1222no—
592C2×C4—1222no—
593C2×C4—1222no—
594C2×C4—1222no—
595C2×C4—1222no—
596C2×C4—1222no—
597C2×C4—1222no—
598C2×C4—1222no—
599C2×C4—1222no—
600C2×C4—1222no—
601C2×C4—1224no—
602C2×C4—1222no—
603C2×C4—1222no—
604C2×C4—2442no—
605C2×C4—2442no—
606C2×C4—2442no—
607C2×C4—2442no—
608C2×C4C22⋊C4×D6111yescharacteristic
609C2×C4C22⋊C4×D6111yescharacteristic
610C2×C4C22×C4×D6111yescharacteristic
611C2×C4C2×D4×D6111yescharacteristic
612C8—42no—
613C8—2442no—
614C22—21no—
615C22—21no—
616C22—21no—
617C22—21no—
618C22—21no—
619C22—21no—
620C22—21no—
621C22—44no—
622C22#104×D6112yes—
623C22C2×C22⋊C4×D6111yescharacteristic
624C22#137×D6112yes—
625C4—21no—
626C4—21no—
627C4—21no—
628C4—44no—
629C4—42no—
630C4—42no—
631C4—612no—
632C4—612no—
633C4—612no—
634C4—612no—
635C4—1221no—
636C4—1221no—
637C4—1221no—
638C4—1221no—
639C4—2444no—
640C4—2442no—
641C4—2442no—
642C4#1683973412yes—
643C2—22no—
644C2—22no—
645C2—41no—
646C2C2×#1683973411yescharacteristic
647C2#419×D6111yescharacteristic
648C2C2×#104×D6111yescharacteristic
subgroupG/H
centerC22C2×C22⋊C4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1508C2
socleC22×C61C22×C22⋊C4
prime radicalC24×C61C2×D4
cosocleC22×C4×C61C24
hypercenter#1508D61
hypocenterC61#17545
LCSG ▷ C2×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1508

Complements

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

NH ≅ G/Ntype
7. #21444329C2semidirect
8. #21444316C2semidirect
14. #16840314C4semidirect
15. #16840314C4semidirect
18. C2×#14519776C4semidirect
19. C2×#14519775C4semidirect
23. C2×Q8×Dic61C4semidirect
25. C2×#14519742C4semidirect
373. C61#17545semidirect

Maximal subgroups

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

subgroupindex
C2×#168403142
#214544482
C4×#145197332
#214443402
#214443392
#214443382
#214443292
#214443162
C61×#15082
#1754561