Finite Group Databasebeta

← #29646714

Subgroups — #29646714

Order 15616.

1026 conjugacy classes of subgroups.

The subgroup lattice has 640 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840316C211yesmaximal, characteristic
2#431×D61C211yesmaximal, characteristic
3#423×D61C211yesmaximal, characteristic
4#21444272C214yesmaximal, complemented
5#21444249C211yesmaximal, characteristic, complemented
6#21444248C211yesmaximal, characteristic, complemented
7#21444247C211yesmaximal, characteristic, complemented
8#21444220C214yesmaximal, complemented
9C61×#1487C211yesmaximal, characteristic, complemented
10C2×M16×D61—21no—
11#16840316—21no—
12#16840316—21no—
13#16840316—21no—
14#16840316—21no—
15#16840316C412yes—
16#16840316C412yes—
17#16840316C2212yes—
18#16840316C2212yes—
19C2×#14519776C411yescharacteristic
20C2×#14519775C411yescharacteristic
21C2×#14519767—21no—
22C2×#14519757C411yescharacteristic
23C2×#14519755—21no—
24C2×#14519753C411yescharacteristic
25C2×C4×Dic122—21no—
26C2×C4×Dic122C2211yescharacteristic
27C2×#14519734—21no—
28C2×C42×D61C2211yescharacteristic
29#16839815C2211yescharacteristic
30#16839814—21no—
31#16839813C2212yes—
32#16839812—21no—
33#16839786C2211yescharacteristic
34#16839785—21no—
35#16839784—21no—
36#16839784C2212yes—
37C4⋊C8×D61—21no—
38C4⋊C8×D61—21no—
39C4⋊C8×D61C2212yes—
40C22⋊C8×D61—21no—
41C4×C8×D61C2212yes—
42#16839494C2214yescomplemented
43Q8⋊#92418C2214yescomplemented
44#16839492C2214yescomplemented
45C2×C4×Q8×C61C2211yescharacteristic
46C61×#431C2211yescharacteristic
47C61×#423C2211yescharacteristic
48C61×Q8⋊C8C2214yescomplemented
49C22×Dic122—21no—
50C22×Dic122C2×C411yescharacteristic
51C22×C4×D61C2×C411yescharacteristic
52C22×C4×D61D411yescharacteristic
53C22×C4×D61D411yescharacteristic
54C2×#11758194—21no—
55C2×#11758189—22no—
56M16×D61—44no—
57C2×C8×D61—22no—
58C2×C8×D61—22no—
59#14519776—21no—
60#14519776—21no—
61#14519776C2×C412yes—
62#14519776C2×C412yes—
63#14519775—21no—
64#14519775—21no—
65#14519775C2×C412yes—
66#14519775C2×C412yes—
67#14519767—22no—
68#14519767—22no—
69#14519767—22no—
70#14519767—22no—
71#14519757—21no—
72#14519757—21no—
73#14519757C2×C412yes—
74#14519757C2×C412yes—
75#14519755—22no—
76#14519755—22no—
77#14519755—22no—
78#14519755—22no—
79#14519753—21no—
80#14519753—21no—
81#14519753C2×C412yes—
82#14519753C2×C412yes—
83C4×Dic122—21no—
84C4×Dic122—21no—
85C4×Dic122—22no—
86C4×Dic122—22no—
87C4×Dic122—22no—
88C4×Dic122—22no—
89C4×Dic122C2×C412yes—
90C4×Dic122C2312yes—
91#14519734—22no—
92#14519734—22no—
93#14519734—22no—
94#14519734—22no—
95C2×C4⋊Dic61—21no—
96C2×C4⋊Dic61C2×C411yescharacteristic
97C2×#11758196—21no—
98C2×#11758196—21no—
99C2×#11758196—21no—
100C2×#11758196—21no—
101C2×#11758196—21no—
102C2×#11758198—21no—
103C2×#11758198—21no—
104C2×C4×Dic61—21no—
105C2×C4×Dic61C2×C411yescharacteristic
106C2×C4×Dic61D411yescharacteristic
107C2×C4×Dic61D411yescharacteristic
108C42×D61C2×C411yescharacteristic
109C42×D61C2×C411yescharacteristic
110C42×D61D411yescharacteristic
111C42×D61D411yescharacteristic
112C42×D61D411yescharacteristic
113C42×D61D411yescharacteristic
114C42×D61C2311yescharacteristic
115C42×D61C2311yescharacteristic
116C4⋊#92418—21no—
117C4⋊#92418—21no—
118C4⋊#92418C2312yes—
119#14519706—22no—
120#14519706—22no—
121#14519706—22no—
122C22⋊#92418—21no—
123#14519689—22no—
124C4×#92418C2312yes—
125C8×Dic61—22no—
126C22×Q8×C61C2×C411yescharacteristic
127C2×M16×C61—21no—
128C4×Q8×C61—21no—
129C4×Q8×C61—21no—
130C4×Q8×C61C2×C412yes—
131C4×Q8×C61C2312yes—
132C2×C4⋊C4×C61—21no—
133C2×C4⋊C4×C61C2×C411yescharacteristic
134C2×C42×C61—21no—
135C2×C42×C61C2311yescharacteristic
136C4⋊C8×C61—21no—
137C4⋊C8×C61—21no—
138C4⋊C8×C61C2312yes—
139C22⋊C8×C61—21no—
140C4×C8×C61C2312yes—
141C23×D61C22⋊C411yescharacteristic
142C2×Q8×C61—21no—
143C2×Q8×C61—42no—
144C2×Q8×C61C22×C412yes—
145C22×C4×C61—21no—
146C22×C4×C61—21no—
147C22×C4×C61C22×C411yescharacteristic
148C22×C4×C61C2×D411yescharacteristic
149C22×C4×C61C2×D411yescharacteristic
150C22×Dic61—21no—
151C22×Dic61—21no—
152C22×Dic61C22⋊C411yescharacteristic
153C22×Dic61C22⋊C411yescharacteristic
154C22×Dic61C22⋊C411yescharacteristic
155C2×C4×D61—42no—
156C2×C4×D61—42no—
157C2×C4×D61C22⋊C411yescharacteristic
158C2×C4×D61C22⋊C411yescharacteristic
159C2×C4×D61C22⋊C411yescharacteristic
160C2×C4×D61C22⋊C411yescharacteristic
161C2×C4×D61C22⋊C411yescharacteristic
162C2×C4×D61C22⋊C411yescharacteristic
163C2×C4×D61SD1614yes—
164C2×C4×D61Q1614yes—
165C2×C4×D61C22×C411yescharacteristic
166C2×C4×D61C22×C411yescharacteristic
167C2×C4×D61C2×D411yescharacteristic
168C2×C4×D61C2×D411yescharacteristic
169C2×C4×D61C2×D411yescharacteristic
170C2×C4×D61C2×D411yescharacteristic
171C2×Dic122—21no—
172C2×Dic122—22no—
173C2×Dic122—22no—
174C2×Dic122—42no—
175C2×Dic122—44no—
176C2×Dic122C22×C412yes—
177M16×C61—42no—
178C2×C8×C61—22no—
179C2×C8×C61—22no—
180C4⋊C4×C61—22no—
181C4⋊C4×C61—21no—
182C4⋊C4×C61—22no—
183C4⋊C4×C61C22×C412yes—
184C42×C61—22no—
185C42×C61—22no—
186C42×C61C22×C411yescharacteristic
187C42×C61C2×D411yescharacteristic
188C42×C61C2×D411yescharacteristic
189C42×C61C2411yescharacteristic
190#11758198—22no—
191#11758198—22no—
192#11758198—22no—
193#11758198—22no—
194C4⋊Dic61—21no—
195C4⋊Dic61—22no—
196C4⋊Dic61—22no—
197C4⋊Dic61C22×C412yes—
198#11758196—22no—
199#11758196—22no—
200#11758196—22no—
201#11758196—22no—
202#11758196—22no—
203#11758196—22no—
204#11758196—22no—
205#11758196—22no—
206#11758196—22no—
207#11758196—22no—
208C4×Dic61—22no—
209C4×Dic61—22no—
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×Dic61C2×D411yescharacteristic
219C4×Dic61C2×D411yescharacteristic
220C4×Dic61C2×D411yescharacteristic
221C4×Dic61C2×D411yescharacteristic
222#11758194—42no—
223C2×#92418—22no—
224C2×#92418—22no—
225#11758189—44no—
226C8×D61—44no—
227C8×D61—44no—
228C23×C61C2×C22⋊C411yescharacteristic
229C22×D61—41no—
230C22×D61—41no—
231C22×D61#10314yes—
232C22×D61C2×C22⋊C411yescharacteristic
233C22×D61C2×C22⋊C411yescharacteristic
234Q8×C61—24no—
235Q8×C61—42no—
236C2×C4×C61—22no—
237C2×C4×C61—22no—
238C2×C4×C61—22no—
239C2×C4×C61—41no—
240C2×C4×C61—42no—
241C2×C4×C61—42no—
242C2×C4×C61C2×C22⋊C411yescharacteristic
243C2×C4×C61C2×C22⋊C411yescharacteristic
244C2×C4×C61C2×C22⋊C411yescharacteristic
245C2×C4×C61C2×SD1612yes—
246C2×C4×C61C2×Q1612yes—
247C2×C4×C61C23×C411yescharacteristic
248C2×C4×C61C22×D411yescharacteristic
249C2×C4×C61C22×D411yescharacteristic
250C2×Dic61—22no—
251C2×Dic61—22no—
252C2×Dic61—22no—
253C2×Dic61—41no—
254C2×Dic61—42no—
255C2×Dic61—42no—
256C2×Dic61#10314yes—
257C2×Dic61C2×C22⋊C411yescharacteristic
258C2×Dic61C2×C22⋊C411yescharacteristic
259C2×Dic61C2×C22⋊C411yescharacteristic
260C2×Dic61C2×C22⋊C411yescharacteristic
261C2×Dic61C2×C22⋊C411yescharacteristic
262C2×Dic61C2×C22⋊C411yescharacteristic
263C4×D61—22no—
264C4×D61—22no—
265C4×D61—22no—
266C4×D61—22no—
267C4×D61—41no—
268C4×D61—41no—
269C4×D61—41no—
270C4×D61—41no—
271C4×D61#10314yes—
272C4×D61#10314yes—
273C4×D61C2×SD1612yes—
274C4×D61C2×SD1612yes—
275C4×D61C2×Q1612yes—
276C4×D61C2×Q1612yes—
277Dic122—24no—
278Dic122—44no—
279Dic122—42no—
280Dic122—44no—
281C8×C61—42no—
282C8×C61—42no—
283#92418—42no—
284#92418—42no—
285#17486—611nomaximal, Sylow
286C22×C61—21no—
287C22×C61—21no—
288C22×C61C2×#10312yes—
289C22×C61C22×C22⋊C411yescharacteristic
290C2×D61—21no—
291C2×D61—21no—
292C2×D61—21no—
293C2×D61—21no—
294C2×D61—44no—
295C2×D61C2×#10312yes—
296C2×D61C2×#10312yes—
297C4×C61—21no—
298C4×C61—21no—
299C4×C61—21no—
300C4×C61—42no—
301C4×C61—42no—
302C4×C61C2×#10312yes—
303C4×C61#56812yes—
304C4×C61C22×SD1611yescharacteristic
305C4×C61C22×Q1611yescharacteristic
306Dic61—21no—
307Dic61—21no—
308Dic61—21no—
309Dic61—42no—
310Dic61—42no—
311Dic61C2×#10312yes—
312Dic61C2×#10312yes—
313Dic61#42112yes—
314C2×#533—611no—
315C2×#431—611no—
316C2×#423—611no—
317#1493—614no—
318#1487—611no—
319#1487—611no—
320#1487—611no—
321#1487D6111yescharacteristic, complemented
322#1481—614no—
323C2×C61—22no—
324C2×C61#287911yescharacteristic
325C2×C61C22×#10311yescharacteristic
326C2×C61C2×#42111yescharacteristic
327D61—22no—
328D61—22no—
329C22×M16—1221no—
330#533—612no—
331#533—612no—
332#533—612no—
333#533—612no—
334#533—1221no—
335#533—1221no—
336#533—1221no—
337#533—1221no—
338C2×C4⋊Q8—611no—
339C2×C4⋊Q8—611no—
340C2×#125—1221no—
341C2×C22⋊Q8—611no—
342C2×C22⋊Q8—611no—
343C2×C22⋊Q8—1221no—
344C2×C4×Q8—611no—
345C2×C4×Q8—1221no—
346C2×C4×Q8C2×D6111yescharacteristic
347C2×#117—1221no—
348C22×C42—611no—
349#431—611no—
350#431—612no—
351#431—1221no—
352#431—1221no—
353#431C2×D6111yescharacteristic
354#423—611no—
355#423—612no—
356#423—1221no—
357#423—1221no—
358#423C2×D6111yescharacteristic
359C2×C4⋊C8—612no—
360C2×C4⋊C8—1221no—
361C2×C4⋊C8—1221no—
362C2×C22⋊C8—1221no—
363C2×C4×C8—612no—
364Q8⋊C8—614no—
365Q8⋊C8—614no—
366Q8⋊C8—614no—
367Q8⋊C8C2×D6114yescomplemented
368C61#1748611yescharacteristic, complemented, Sylow
369C22×Q8—611no—
370C22×Q8—1221no—
371C22×Q8C4×D6111yescharacteristic
372C23×C4—611no—
373C23×C4—611no—
374C23×C4—611no—
375C2×M16—21no—
376C2×M16—1222no—
377C2×M16—1221no—
378C2×M16—2444no—
379C22×C8—1222no—
380C22×C8—1222no—
381C4⋊Q8—612no—
382C4⋊Q8—612no—
383C4⋊Q8—612no—
384C4⋊Q8—612no—
385C4⋊Q8—1221no—
386C4⋊Q8—1221no—
387C4⋊Q8—1221no—
388C4⋊Q8—1221no—
389#125—1222no—
390#125—1222no—
391#125—1222no—
392#125—1222no—
393C22⋊Q8—612no—
394C22⋊Q8—612no—
395C22⋊Q8—612no—
396C22⋊Q8—612no—
397C22⋊Q8—1221no—
398C22⋊Q8—1221no—
399C22⋊Q8—1222no—
400C22⋊Q8—1222no—
401C22⋊Q8—1222no—
402C22⋊Q8—1222no—
403C22⋊Q8—1221no—
404C22⋊Q8—1221no—
405C4×Q8—21no—
406C4×Q8—21no—
407C4×Q8—612no—
408C4×Q8—612no—
409C4×Q8—1222no—
410C4×Q8—1222no—
411C4×Q8—1222no—
412C4×Q8—1222no—
413C4×Q8—1221no—
414C4×Q8—1221no—
415C4×Q8C4×D6112yes—
416C4×Q8C22×D6112yes—
417#117—1222no—
418#117—1222no—
419#117—1222no—
420#117—1222no—
421C2×C4⋊C4—21no—
422C2×C4⋊C4—611no—
423C2×C4⋊C4—1221no—
424C2×C4⋊C4—1221no—
425C2×C4⋊C4—1221no—
426C2×C4⋊C4—1221no—
427C2×C4⋊C4—1221no—
428C2×C4⋊C4—1221no—
429C2×C4⋊C4C4×D6111yescharacteristic
430C2×C22⋊C4—1221no—
431C2×C22⋊C4—1221no—
432C2×C42—21no—
433C2×C42—611no—
434C2×C42—611no—
435C2×C42—611no—
436C2×C42—611no—
437C2×C42—611no—
438C2×C42—611no—
439C2×C42—611no—
440C2×C42—611no—
441C2×C42—611no—
442C2×C42—611no—
443C2×C42—611no—
444C2×C42—1221no—
445C2×C42C22×D6111yescharacteristic
446C4⋊C8—21no—
447C4⋊C8—21no—
448C4⋊C8—612no—
449C4⋊C8—1222no—
450C4⋊C8—1222no—
451C4⋊C8—1222no—
452C4⋊C8—1221no—
453C4⋊C8—1221no—
454C4⋊C8C22×D6112yes—
455C22⋊C8—21no—
456C22⋊C8—1222no—
457C22⋊C8—1221no—
458C4×C8—612no—
459C4×C8—1222no—
460C4×C8C22×D6112yes—
461C24—611no—
462C2×Q8—21no—
463C2×Q8—42no—
464C2×Q8—612no—
465C2×Q8—1222no—
466C2×Q8—1222no—
467C2×Q8—1221no—
468C2×Q8—2442no—
469C2×Q8—2444no—
470C2×Q8C2×C4×D6112yes—
471C22×C4—21no—
472C22×C4—21no—
473C22×C4—611no—
474C22×C4—611no—
475C22×C4—611no—
476C22×C4—611no—
477C22×C4—611no—
478C22×C4—614no—
479C22×C4—614no—
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—1221no—
491C22×C4—1221no—
492C22×C4—2442no—
493C22×C4—2442no—
494C22×C4C2×C4×D6111yescharacteristic
495C22×C4D4×D6111yescharacteristic
496C22×C4D4×D6111yescharacteristic
497M16—42no—
498M16—2444no—
499M16—2442no—
500C2×C8—22no—
501C2×C8—22no—
502C2×C8—1222no—
503C2×C8—1222no—
504C2×C8—2444no—
505C2×C8—2444no—
506C4⋊C4—22no—
507C4⋊C4—22no—
508C4⋊C4—21no—
509C4⋊C4—612no—
510C4⋊C4—1222no—
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—1221no—
522C4⋊C4—1222no—
523C4⋊C4C2×C4×D6112yes—
524C22⋊C4—1222no—
525C22⋊C4—1222no—
526C22⋊C4—1222no—
527C22⋊C4—1222no—
528C42—22no—
529C42—22no—
530C42—611no—
531C42—611no—
532C42—611no—
533C42—611no—
534C42—611no—
535C42—611no—
536C42—611no—
537C42—611no—
538C42—611no—
539C42—611no—
540C42—611no—
541C42—611no—
542C42—1222no—
543C42—1222no—
544C42C2×C4×D6111yescharacteristic
545C42D4×D6111yescharacteristic
546C42D4×D6111yescharacteristic
547C42C23×D6111yescharacteristic
548C23—611no—
549C23—611no—
550C23—614no—
551C23—2441no—
552C23—2441no—
553C23C22⋊C4×D6111yescharacteristic
554Q8—24no—
555Q8—42no—
556Q8—1224no—
557Q8—2442no—
558Q8—2444no—
559Q8—2444no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—41no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—611no—
567C2×C4—611no—
568C2×C4—611no—
569C2×C4—614no—
570C2×C4—614no—
571C2×C4—614no—
572C2×C4—612no—
573C2×C4—612no—
574C2×C4—612no—
575C2×C4—612no—
576C2×C4—611no—
577C2×C4—611no—
578C2×C4—611no—
579C2×C4—1222no—
580C2×C4—1222no—
581C2×C4—1222no—
582C2×C4—1222no—
583C2×C4—1222no—
584C2×C4—1222no—
585C2×C4—1222no—
586C2×C4—2441no—
587C2×C4—2441no—
588C2×C4—2441no—
589C2×C4—2441no—
590C2×C4—2441no—
591C2×C4—2442no—
592C2×C4—2442no—
593C2×C4C22⋊C4×D6111yescharacteristic
594C2×C4C22⋊C4×D6111yescharacteristic
595C2×C4C22⋊C4×D6111yescharacteristic
596C2×C4SD16×D6112yes—
597C2×C4Q16×D6112yes—
598C2×C4C22×C4×D6111yescharacteristic
599C2×C4C2×D4×D6111yescharacteristic
600C2×C4C2×D4×D6111yescharacteristic
601C8—42no—
602C8—42no—
603C8—2442no—
604C8—2442no—
605C22—21no—
606C22—21no—
607C22—612no—
608C22—612no—
609C22—1221no—
610C22—1221no—
611C22—1221no—
612C22—1221no—
613C22—2444no—
614C22#103×D6112yes—
615C22C2×C22⋊C4×D6111yescharacteristic
616C4—21no—
617C4—21no—
618C4—21no—
619C4—42no—
620C4—42no—
621C4—612no—
622C4—612no—
623C4—612no—
624C4—1221no—
625C4—1221no—
626C4—1221no—
627C4—2442no—
628C4—2442no—
629C4#103×D6112yes—
630C4#1684052812yes—
631C4C2×SD16×D6111yescharacteristic
632C4C2×Q16×D6111yescharacteristic
633C2—22no—
634C2—1222no—
635C2—1222no—
636C2#2145426211yescharacteristic
637C2C2×#103×D6111yescharacteristic
638C2#421×D6111yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC42C23×D61
FittingC61×#1487C2
socleC22×C61C22×C22⋊C4
prime radicalC23×D61C22⋊C4
cosocleC42×C61C24
hypercenter#1487D61
hypocenterC61#17486
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1487

Complements

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

NH ≅ G/Ntype
4. #21444272C2semidirect
5. #21444249C2semidirect
6. #21444248C2semidirect
7. #21444247C2semidirect
8. #21444220C2semidirect
9. C61×#1487C2semidirect
42. #16839494C22semidirect
43. Q8⋊#92418C22semidirect
44. #16839492C22semidirect
48. C61×Q8⋊C8C22semidirect
321. #1487D61semidirect
367. Q8⋊C8C2×D61semidirect
368. C61#17486semidirect

Maximal subgroups

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

subgroupindex
C2×#168403162
#431×D612
#423×D612
#214442722
#214442492
#214442482
#214442472
#214442202
C61×#14872
#1748661