Finite Group Databasebeta

← #29648368

Subgroups — #29648368

Order 15616.

1018 conjugacy classes of subgroups.

The subgroup lattice has 646 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#533×D61C211yesmaximal, characteristic
2#431×D61C211yesmaximal, characteristic
3C2×#16839784C211yesmaximal, characteristic
4#21444508C212yesmaximal, complemented
5#21444506C212yesmaximal, complemented
6#21444502C212yesmaximal, complemented
7#21444498C212yesmaximal, complemented
8#21444477C211yesmaximal, characteristic, complemented
9#21444476C211yesmaximal, characteristic, complemented
10#21444474C211yesmaximal, characteristic, complemented
11C61×#1542C211yesmaximal, characteristic, complemented
12C22×#11758189—21no—
13#16840319C2211yescharacteristic
14#16840318—21no—
15#16840317C2211yescharacteristic
16#16840317C2211yescharacteristic
17#16840316—21no—
18#16840316C412yes—
19#16840315—21no—
20#16840314—21no—
21#16840314C412yes—
22C4⋊Q8×D61C2211yescharacteristic
23C4⋊Q8×D61C2211yescharacteristic
24#125×D61—21no—
25C22⋊Q8×D61—21no—
26C22⋊Q8×D61C412yes—
27C4×Q8×D61—21no—
28C4×Q8×D61C412yes—
29#117×D61—21no—
30C2×C42×D61C2211yescharacteristic
31#16839815C2211yescharacteristic
32#16839814C2212yes—
33#16839813—21no—
34#16839812—21no—
35#16839784—21no—
36#16839784—21no—
37#16839784C2212yes—
38#16839784C2212yes—
39C2×C4⋊#92418C2211yescharacteristic
40C2×#14519706—21no—
41C4⋊C8×D61C2212yes—
42C22⋊C8×D61—21no—
43C4×C8×D61—21no—
44#16839511C2212yescomplemented
45#16839510C2212yescomplemented
46#16839510C2212yescomplemented
47#16839509C2212yescomplemented
48#16839508C2212yescomplemented
49#16839507C2212yescomplemented
50C61×#533C2211yescharacteristic
51C61×#431C2211yescharacteristic
52C2×C4⋊C8×C61C2211yescharacteristic
53C61×#333C2212yescomplemented
54C61×#332C2212yescomplemented
55C2×Q8×D61—21no—
56C2×Q8×D61C2×C412yes—
57C22×C4×D61C2×C411yescharacteristic
58C22×C4×D61D411yescharacteristic
59C22×C4×D61D411yescharacteristic
60C2×#11758189—22no—
61C2×#11758189—22no—
62C2×#11758189—44no—
63C22×#92418—21no—
64C2×C8×D61—22no—
65#14519777C2311yescharacteristic
66#14519777C2311yescharacteristic
67#14519776—21no—
68#14519776—21no—
69#14519776—21no—
70#14519776C2×C412yes—
71#14519775—21no—
72#14519775C2×C412yes—
73#14519769—21no—
74#14519768—21no—
75#14519768—21no—
76#14519767—22no—
77#14519767—22no—
78#14519757—21no—
79#14519757C2×C412yes—
80#14519756—21no—
81#14519756C2×C412yes—
82#14519755—22no—
83#14519755—22no—
84#14519754—22no—
85#14519754—22no—
86#14519753—21no—
87#14519753C2×C412yes—
88Q8×Dic61—21no—
89Q8×Dic61C2×C412yes—
90C4×Dic122—21no—
91C4×Dic122—22no—
92C4×Dic122—22no—
93C4×Dic122C2×C412yes—
94#14519742—22no—
95#14519742—22no—
96#14519742—21no—
97#14519742C2×C412yes—
98#14519736—21no—
99#14519735—21no—
100#14519735—21no—
101#14519734—22no—
102#14519733—22no—
103C4⋊C4×D61—22no—
104C4⋊C4×D61—22no—
105C4⋊C4×D61—21no—
106C4⋊C4×D61—21no—
107C4⋊C4×D61—21no—
108C4⋊C4×D61C2×C412yes—
109C22⋊C4×D61—22no—
110C2×C4×Dic61C2×C411yescharacteristic
111C2×C4×Dic61D411yescharacteristic
112C2×C4×Dic61D411yescharacteristic
113C42×D61—22no—
114C42×D61C2×C411yescharacteristic
115C42×D61C2×C411yescharacteristic
116C42×D61D411yescharacteristic
117C42×D61D411yescharacteristic
118C42×D61D411yescharacteristic
119C42×D61D411yescharacteristic
120C42×D61C2311yescharacteristic
121C42×D61C2311yescharacteristic
122C4⋊#92418—21no—
123C4⋊#92418C2312yes—
124C4⋊#92418C2312yes—
125#14519706—22no—
126#14519706—22no—
127#14519706—22no—
128C22⋊#92418—21no—
129#14519689—22no—
130C4×#92418—21no—
131C8×Dic61—22no—
132C22×C8×C61—21no—
133C4⋊Q8×C61C2311yescharacteristic
134C4⋊Q8×C61C2311yescharacteristic
135#125×C61—21no—
136C22⋊Q8×C61—21no—
137C22⋊Q8×C61C2×C412yes—
138C4×Q8×C61—21no—
139C4×Q8×C61C2×C412yes—
140#117×C61—21no—
141C2×C42×C61C2311yescharacteristic
142C4⋊C8×C61—21no—
143C4⋊C8×C61C2312yes—
144C4⋊C8×C61C2312yes—
145C22⋊C8×C61—21no—
146C4×C8×C61—21no—
147C23×D61C22⋊C411yescharacteristic
148C2×Q8×C61—21no—
149C2×Q8×C61C22×C412yes—
150C22×C4×C61C22×C411yescharacteristic
151C22×C4×C61C2×D411yescharacteristic
152C22×C4×C61C2×D411yescharacteristic
153C22×Dic61C22⋊C411yescharacteristic
154C22×Dic61C22⋊C411yescharacteristic
155C22×Dic61C22⋊C411yescharacteristic
156Q8×D61—44no—
157Q8×D61—44no—
158C2×C4×D61—22no—
159C2×C4×D61—22no—
160C2×C4×D61—42no—
161C2×C4×D61—42no—
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×D61SD1614yes—
169C2×C4×D61Q1614yes—
170C2×C4×D61C22×C411yescharacteristic
171C2×C4×D61C22×C411yescharacteristic
172C2×C4×D61C2×D411yescharacteristic
173C2×C4×D61C2×D411yescharacteristic
174C2×C4×D61C2×D411yescharacteristic
175C2×C4×D61C2×D411yescharacteristic
176C2×Dic122—21no—
177C2×Dic122—22no—
178C2×Dic122—22no—
179C2×Dic122C22×C412yes—
180C2×C8×C61—22no—
181C2×C8×C61—22no—
182C2×C8×C61—42no—
183C4⋊C4×C61—22no—
184C4⋊C4×C61—22no—
185C4⋊C4×C61—21no—
186C4⋊C4×C61—21no—
187C4⋊C4×C61—21no—
188C4⋊C4×C61C22×C412yes—
189C22⋊C4×C61—22no—
190C42×C61—22no—
191C42×C61C22×C411yescharacteristic
192C42×C61C2×D411yescharacteristic
193C42×C61C2×D411yescharacteristic
194C42×C61C2411yescharacteristic
195C22⋊Dic61—22no—
196#11758198—22no—
197#11758198—22no—
198C4⋊Dic61—21no—
199C4⋊Dic61—21no—
200C4⋊Dic61—21no—
201C4⋊Dic61—22no—
202C4⋊Dic61—22no—
203C4⋊Dic61C22×C412yes—
204#11758196—22no—
205#11758196—22no—
206#11758196—22no—
207#11758196—22no—
208#11758196—22no—
209#11758196—22no—
210#11758196—22no—
211#11758196—22no—
212C4×Dic61—22no—
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×Dic61C2×D411yescharacteristic
224C4×Dic61C2×D411yescharacteristic
225C4×Dic61C2×D411yescharacteristic
226C4×Dic61C2×D411yescharacteristic
227C2×#92418—22no—
228C2×#92418—22no—
229C2×#92418—42no—
230#11758189—44no—
231#11758189—44no—
232C8×D61—44no—
233C23×C61C2×C22⋊C411yescharacteristic
234C22×D61—41no—
235C22×D61—41no—
236C22×D61#10314yes—
237C22×D61C2×C22⋊C411yescharacteristic
238C22×D61C2×C22⋊C411yescharacteristic
239Q8×C61—42no—
240Q8×C61—42no—
241C2×C4×C61—22no—
242C2×C4×C61—22no—
243C2×C4×C61—22no—
244C2×C4×C61—41no—
245C2×C4×C61C2×C22⋊C411yescharacteristic
246C2×C4×C61C2×C22⋊C411yescharacteristic
247C2×C4×C61C2×C22⋊C411yescharacteristic
248C2×C4×C61C2×SD1612yes—
249C2×C4×C61C2×Q1612yes—
250C2×C4×C61C23×C411yescharacteristic
251C2×C4×C61C22×D411yescharacteristic
252C2×C4×C61C22×D411yescharacteristic
253C2×Dic61—22no—
254C2×Dic61—22no—
255C2×Dic61—22no—
256C2×Dic61—41no—
257C2×Dic61#10314yes—
258C2×Dic61C2×C22⋊C411yescharacteristic
259C2×Dic61C2×C22⋊C411yescharacteristic
260C2×Dic61C2×C22⋊C411yescharacteristic
261C2×Dic61C2×C22⋊C411yescharacteristic
262C2×Dic61C2×C22⋊C411yescharacteristic
263C2×Dic61C2×C22⋊C411yescharacteristic
264C4×D61—22no—
265C4×D61—22no—
266C4×D61—22no—
267C4×D61—22no—
268C4×D61—41no—
269C4×D61—41no—
270C4×D61—41no—
271C4×D61—41no—
272C4×D61—44no—
273C4×D61—44no—
274C4×D61#10314yes—
275C4×D61#10314yes—
276C4×D61C2×SD1612yes—
277C4×D61C2×SD1612yes—
278C4×D61C2×Q1612yes—
279C4×D61C2×Q1612yes—
280Dic122—44no—
281Dic122—42no—
282Dic122—44no—
283Dic122—42no—
284C8×C61—42no—
285C8×C61—42no—
286#92418—42no—
287#92418—42no—
288#17708—611nomaximal, Sylow
289C22×C61—21no—
290C22×C61—21no—
291C22×C61C2×#10312yes—
292C22×C61C22×C22⋊C411yescharacteristic
293C2×D61—21no—
294C2×D61—21no—
295C2×D61—21no—
296C2×D61—21no—
297C2×D61—44no—
298C2×D61C2×#10312yes—
299C2×D61C2×#10312yes—
300C4×C61—21no—
301C4×C61—21no—
302C4×C61—21no—
303C4×C61—42no—
304C4×C61—42no—
305C4×C61C2×#10312yes—
306C4×C61C22×SD1611yescharacteristic
307C4×C61C22×Q1611yescharacteristic
308C4×C61#57512yes—
309Dic61—21no—
310Dic61—21no—
311Dic61—21no—
312Dic61—42no—
313Dic61—42no—
314Dic61#41312yes—
315Dic61C2×#10312yes—
316Dic61C2×#10312yes—
317C2×#533—611no—
318C2×#431—611no—
319C2×#423—611no—
320#1550—612no—
321#1550—612no—
322#1549—612no—
323#1548—612no—
324#1542—611no—
325#1542—611no—
326#1542—611no—
327#1542D6111yescharacteristic, complemented
328C2×C61—22no—
329C2×C61C2×#41311yescharacteristic
330C2×C61C22×#10311yescharacteristic
331C2×C61#289611yescharacteristic
332D61—22no—
333D61—22no—
334C22×M16—1221no—
335#533—611no—
336#533—611no—
337#533—612no—
338#533—612no—
339#533—611no—
340#533—1221no—
341#533—1221no—
342#533—1221no—
343#533—1221no—
344#533C2×D6111yescharacteristic
345C2×C4⋊Q8—611no—
346C2×C4⋊Q8—611no—
347C2×#125—1221no—
348C2×C22⋊Q8—612no—
349C2×C22⋊Q8—1221no—
350C2×C4×Q8—612no—
351C2×C4×Q8—1221no—
352C2×#117—1221no—
353C22×C42—611no—
354#431—611no—
355#431—612no—
356#431—1221no—
357#431—1221no—
358#431C2×D6111yescharacteristic
359#423—612no—
360#423—612no—
361#423—1221no—
362#423—1221no—
363C2×C4⋊C8—611no—
364C2×C4⋊C8—612no—
365C2×C4⋊C8—1221no—
366C2×C4⋊C8C2×D6111yescharacteristic
367C2×C22⋊C8—1221no—
368C2×C4×C8—1221no—
369#333—612no—
370#333—612no—
371#333—612no—
372#333C2×D6112yescomplemented
373#332—612no—
374#332—612no—
375#332—612no—
376#332C2×D6112yescomplemented
377C61#1770811yescharacteristic, complemented, Sylow
378C22×Q8—612no—
379C22×Q8—1221no—
380C23×C4—611no—
381C23×C4—611no—
382C23×C4—611no—
383C2×M16—1222no—
384C2×M16—1222no—
385C2×M16—2444no—
386C22×C8—21no—
387C22×C8—1222no—
388C22×C8—1221no—
389C4⋊Q8—612no—
390C4⋊Q8—612no—
391C4⋊Q8—611no—
392C4⋊Q8—611no—
393C4⋊Q8—1221no—
394C4⋊Q8—1221no—
395C4⋊Q8—1221no—
396C4⋊Q8—1221no—
397C4⋊Q8C22×D6111yescharacteristic
398C4⋊Q8C22×D6111yescharacteristic
399#125—21no—
400#125—1222no—
401#125—1222no—
402#125—1221no—
403#125—1221no—
404#125—1221no—
405C22⋊Q8—21no—
406C22⋊Q8—612no—
407C22⋊Q8—612no—
408C22⋊Q8—612no—
409C22⋊Q8—1222no—
410C22⋊Q8—1222no—
411C22⋊Q8—1222no—
412C22⋊Q8—1222no—
413C22⋊Q8—1221no—
414C22⋊Q8—1221no—
415C22⋊Q8—1221no—
416C22⋊Q8C4×D6112yes—
417C4×Q8—21no—
418C4×Q8—612no—
419C4×Q8—612no—
420C4×Q8—612no—
421C4×Q8—1222no—
422C4×Q8—1222no—
423C4×Q8—1222no—
424C4×Q8—1221no—
425C4×Q8—1222no—
426C4×Q8—1221no—
427C4×Q8—1221no—
428C4×Q8C4×D6112yes—
429#117—21no—
430#117—1222no—
431#117—1222no—
432#117—1221no—
433#117—1221no—
434#117—1221no—
435C2×C4⋊C4—612no—
436C2×C4⋊C4—1221no—
437C2×C4⋊C4—1221no—
438C2×C4⋊C4—1222no—
439C2×C4⋊C4—1222no—
440C2×C4⋊C4—1221no—
441C2×C22⋊C4—1222no—
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—611no—
453C2×C42—1222no—
454C2×C42C22×D6111yescharacteristic
455C4⋊C8—21no—
456C4⋊C8—612no—
457C4⋊C8—612no—
458C4⋊C8—1222no—
459C4⋊C8—1222no—
460C4⋊C8—1222no—
461C4⋊C8—1221no—
462C4⋊C8C22×D6112yes—
463C4⋊C8C22×D6112yes—
464C22⋊C8—21no—
465C22⋊C8—1222no—
466C22⋊C8—1221no—
467C4×C8—21no—
468C4×C8—1222no—
469C4×C8—1221no—
470C24—611no—
471C2×Q8—21no—
472C2×Q8—612no—
473C2×Q8—1222no—
474C2×Q8—1222no—
475C2×Q8—1221no—
476C2×Q8—2444no—
477C2×Q8—2444no—
478C2×Q8C2×C4×D6112yes—
479C22×C4—611no—
480C22×C4—611no—
481C22×C4—611no—
482C22×C4—611no—
483C22×C4—611no—
484C22×C4—614no—
485C22×C4—614no—
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—1222no—
497C22×C4—1222no—
498C22×C4—2442no—
499C22×C4—2442no—
500C22×C4C2×C4×D6111yescharacteristic
501C22×C4D4×D6111yescharacteristic
502C22×C4D4×D6111yescharacteristic
503M16—2444no—
504M16—2444no—
505C2×C8—22no—
506C2×C8—22no—
507C2×C8—42no—
508C2×C8—1222no—
509C2×C8—1222no—
510C2×C8—2444no—
511C2×C8—2442no—
512C4⋊C4—21no—
513C4⋊C4—21no—
514C4⋊C4—21no—
515C4⋊C4—22no—
516C4⋊C4—22no—
517C4⋊C4—612no—
518C4⋊C4—1222no—
519C4⋊C4—1222no—
520C4⋊C4—1222no—
521C4⋊C4—1222no—
522C4⋊C4—1222no—
523C4⋊C4—1222no—
524C4⋊C4—1222no—
525C4⋊C4—1222no—
526C4⋊C4—1221no—
527C4⋊C4—1221no—
528C4⋊C4—1221no—
529C4⋊C4—1222no—
530C4⋊C4—1222no—
531C4⋊C4C2×C4×D6112yes—
532C22⋊C4—22no—
533C22⋊C4—1222no—
534C22⋊C4—1222no—
535C22⋊C4—1222no—
536C42—22no—
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—
552C42C2×C4×D6111yescharacteristic
553C42D4×D6111yescharacteristic
554C42D4×D6111yescharacteristic
555C42C23×D6111yescharacteristic
556C23—611no—
557C23—611no—
558C23—614no—
559C23—2441no—
560C23—2441no—
561C23C22⋊C4×D6111yescharacteristic
562Q8—42no—
563Q8—42no—
564Q8—2442no—
565Q8—2444no—
566Q8—2444no—
567Q8—2442no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—41no—
572C2×C4—611no—
573C2×C4—611no—
574C2×C4—611no—
575C2×C4—614no—
576C2×C4—614no—
577C2×C4—614no—
578C2×C4—612no—
579C2×C4—612no—
580C2×C4—612no—
581C2×C4—612no—
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—1222no—
592C2×C4—2441no—
593C2×C4—2441no—
594C2×C4—2441no—
595C2×C4—2441no—
596C2×C4—2444no—
597C2×C4—2444no—
598C2×C4—2441no—
599C2×C4C22⋊C4×D6111yescharacteristic
600C2×C4C22⋊C4×D6111yescharacteristic
601C2×C4C22⋊C4×D6111yescharacteristic
602C2×C4SD16×D6112yes—
603C2×C4Q16×D6112yes—
604C2×C4C22×C4×D6111yescharacteristic
605C2×C4C2×D4×D6111yescharacteristic
606C2×C4C2×D4×D6111yescharacteristic
607C8—42no—
608C8—42no—
609C8—2442no—
610C8—2442no—
611C22—21no—
612C22—21no—
613C22—612no—
614C22—612no—
615C22—1221no—
616C22—1221no—
617C22—1221no—
618C22—1221no—
619C22—2444no—
620C22#103×D6112yes—
621C22C2×C22⋊C4×D6111yescharacteristic
622C4—21no—
623C4—21no—
624C4—21no—
625C4—42no—
626C4—42no—
627C4—612no—
628C4—612no—
629C4—612no—
630C4—1221no—
631C4—1221no—
632C4—1221no—
633C4—2442no—
634C4—2442no—
635C4#103×D6112yes—
636C4C2×SD16×D6111yescharacteristic
637C4C2×Q16×D6111yescharacteristic
638C4#1684055912yes—
639C2—22no—
640C2—1222no—
641C2—1222no—
642C2#413×D6111yescharacteristic
643C2C2×#103×D6111yescharacteristic
644C2#2145433411yescharacteristic
subgroupG/H
centerC2×C4C22⋊C4×D61
commutatorC2×C4×C61C23×C4
FrattiniC42C23×D61
FittingC61×#1542C2
socleC22×C61C22×C22⋊C4
prime radicalC23×D61C22⋊C4
cosocleC42×C61C24
hypercenter#1542D61
hypocenterC61#17708
LCSG ▷ C2×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1542

Complements

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

NH ≅ G/Ntype
4. #21444508C2semidirect
5. #21444506C2semidirect
6. #21444502C2semidirect
7. #21444498C2semidirect
8. #21444477C2semidirect
9. #21444476C2semidirect
10. #21444474C2semidirect
11. C61×#1542C2semidirect
44. #16839511C22semidirect
45. #16839510C22semidirect
46. #16839510C22semidirect
47. #16839509C22semidirect
48. #16839508C22semidirect
49. #16839507C22semidirect
53. C61×#333C22semidirect
54. C61×#332C22semidirect
327. #1542D61semidirect
372. #333C2×D61semidirect
376. #332C2×D61semidirect
377. C61#17708semidirect

Maximal subgroups

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

subgroupindex
#533×D612
#431×D612
C2×#168397842
#214445082
#214445062
#214445022
#214444982
#214444772
#214444762
#214444742
C61×#15422
#1770861