Finite Group Databasebeta

← #29684497

Subgroups — #29684497

Order 15616.

1154 conjugacy classes of subgroups.

The subgroup lattice has 670 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#533×D61C211yesmaximal, characteristic
2C4×M16×D61C211yesmaximal, characteristic
3#21448223C211yesmaximal, characteristic
4#21446764C211yesmaximal, characteristic, complemented
5#21446763C211yesmaximal, characteristic, complemented
6#21446762C211yesmaximal, characteristic, complemented
7#21446761C214yesmaximal, complemented
8#21445527C214yesmaximal
9C61×#1986C211yesmaximal, characteristic, complemented
10C2×M16×D61C2212yes—
11#16840319C2211yescharacteristic
12#16840318C412yescomplemented
13#16840317C411yescharacteristic, complemented
14#16840317C411yescharacteristic, complemented
15#16840316—21no—
16#16840316—21no—
17#16840315C2212yes—
18#16840314—21no—
19#16840314—21no—
20C4⋊Q8×D61C411yescharacteristic, complemented
21C4⋊Q8×D61C411yescharacteristic, complemented
22#125×D61C412yescomplemented
23C22⋊Q8×D61—21no—
24C22⋊Q8×D61—21no—
25C4×Q8×D61—21no—
26C4×Q8×D61—21no—
27C2×#14519734C2212yes—
28#117×D61C2212yes—
29C2×C42×D61C2211yescharacteristic
30C4×#11758194C2211yescharacteristic
31#16839714—21no—
32M16×Dic61C2212yes—
33C4×#11758189—21no—
34#97×D61—21no—
35C4×C8×D61—21no—
36#16839631—21no—
37#16839626—21no—
38#16839621—21no—
39#16839619—21no—
40C4×#11758196C2212yes—
41C4×C4⋊Dic61C2211yescharacteristic
42#16839523C2214yes—
43#16839522C2214yes—
44#16839521C2214yes—
45C61×#533C2211yescharacteristic
46C4×M16×C61C2211yescharacteristic
47C4×C4⋊C4×C61C2211yescharacteristic
48C61×#339C2214yes—
49C2×Q8×D61—21no—
50C2×Q8×D61—21no—
51C22×C4×D61D411yescharacteristic
52C22×C4×D61D411yescharacteristic
53C22×C4×D61C2311yescharacteristic
54C2×#11758194C2312yes—
55C2×#11758189—22no—
56M16×D61—24no—
57M16×D61Q818yescomplemented
58C2×C8×D61—22no—
59#14519777C2×C411yescharacteristic
60#14519777C2×C411yescharacteristic
61#14519776—21no—
62#14519776—21no—
63#14519776C2×C412yes—
64#14519776C2×C412yes—
65#14519775—21no—
66#14519775—21no—
67#14519769C2×C412yes—
68#14519768C2×C412yes—
69#14519768C2×C412yes—
70#14519767—22no—
71#14519767—22no—
72#14519757—21no—
73#14519757—21no—
74#14519756—21no—
75#14519756—21no—
76#14519755—22no—
77#14519755—22no—
78#14519754—22no—
79#14519754—22no—
80#14519753—21no—
81#14519753—21no—
82Q8×Dic61—21no—
83Q8×Dic61—21no—
84C4×Dic122—21no—
85C4×Dic122—21no—
86C4×Dic122—22no—
87C4×Dic122—22no—
88#14519742—21no—
89#14519742—21no—
90#14519742—22no—
91#14519742—22no—
92#14519736C2312yes—
93#14519735C2×C412yes—
94#14519735C2×C412yes—
95#14519734—22no—
96#14519734—28no—
97#14519733—22no—
98C2×C4⋊Dic61—21no—
99C2×#11758196—22no—
100C2×#11758196—22no—
101C4⋊C4×D61—22no—
102C4⋊C4×D61—22no—
103C4⋊C4×D61—21no—
104C4⋊C4×D61—21no—
105C4⋊C4×D61C2×C412yes—
106C4⋊C4×D61C2×C412yes—
107C2×#11758198—22no—
108C22⋊C4×D61—22no—
109C2×C4×Dic61D411yescharacteristic
110C2×C4×Dic61D411yescharacteristic
111C2×C4×Dic61C2312yes—
112C2×C4×Dic61C2311yescharacteristic
113C42×D61—22no—
114C42×D61C2×C411yescharacteristic
115C42×D61C2×C411yescharacteristic
116C42×D61C2×C411yescharacteristic
117C42×D61C2×C411yescharacteristic
118C42×D61D411yescharacteristic
119C42×D61D411yescharacteristic
120C42×D61D411yescharacteristic
121C42×D61D411yescharacteristic
122#14519688—21no—
123#14519687—22no—
124C4×#92418—21no—
125C8×Dic61—22no—
126#14519684—21no—
127#14519684—22no—
128C2×M16×C61C2312yes—
129C4⋊Q8×C61C2×C411yescharacteristic
130C4⋊Q8×C61C2×C411yescharacteristic
131#125×C61C2×C412yes—
132C22⋊Q8×C61—21no—
133C22⋊Q8×C61—21no—
134C4×Q8×C61—21no—
135C4×Q8×C61—21no—
136#117×C61C2312yes—
137C2×C4⋊C4×C61—21no—
138C2×C42×C61C2312yes—
139C2×C42×C61C2311yescharacteristic
140#97×C61—21no—
141C4×C8×C61—21no—
142#95×C61—21no—
143C23×D61C2×D411yescharacteristic
144C2×Q8×C61—21no—
145C2×Q8×C61—21no—
146C22×C4×C61—22no—
147C22×C4×C61C2×D411yescharacteristic
148C22×C4×C61C2×D411yescharacteristic
149C22×C4×C61C2411yescharacteristic
150C22×Dic61—22no—
151C22×Dic61C2×D411yescharacteristic
152C22×Dic61C2×D411yescharacteristic
153C22×Dic61C2×D411yescharacteristic
154Q8×D61—44no—
155Q8×D61—44no—
156C2×C4×D61—22no—
157C2×C4×D61—22no—
158C2×C4×D61—42no—
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×C4×D61C2×Q814yes—
173C2×C4×D61#4114yes—
174C2×Dic122—21no—
175C2×Dic122—21no—
176C2×Dic122—22no—
177C2×Dic122—22no—
178M16×C61—22no—
179M16×C61C2×Q814yescomplemented
180C2×C8×C61—22no—
181C4⋊C4×C61—22no—
182C4⋊C4×C61—22no—
183C4⋊C4×C61—21no—
184C4⋊C4×C61—21no—
185C4⋊C4×C61—42no—
186C4⋊C4×C61C22×C412yes—
187C4⋊C4×C61C22×C412yes—
188C22⋊C4×C61—22no—
189C42×C61—22no—
190C42×C61—24no—
191C42×C61C22×C411yescharacteristic
192C42×C61C22×C411yescharacteristic
193C42×C61C2×D411yescharacteristic
194C42×C61C2×D411yescharacteristic
195C22⋊Dic61—22no—
196#11758198—22no—
197#11758198—22no—
198#11758198—44no—
199C4⋊Dic61—21no—
200C4⋊Dic61—21no—
201C4⋊Dic61—22no—
202C4⋊Dic61—22no—
203C4⋊Dic61—42no—
204C4⋊Dic61C22×C412yes—
205C4⋊Dic61C22×C412yes—
206#11758196—22no—
207#11758196—22no—
208#11758196—22no—
209#11758196—22no—
210#11758196—22no—
211#11758196—22no—
212#11758196—22no—
213#11758196—22no—
214#11758196—44no—
215#11758196—44no—
216C4×Dic61—22no—
217C4×Dic61—22no—
218C4×Dic61—22no—
219C4×Dic61—24no—
220C4×Dic61C22⋊C411yescharacteristic
221C4×Dic61C22⋊C411yescharacteristic
222C4×Dic61C22⋊C411yescharacteristic
223C4×Dic61C22⋊C411yescharacteristic
224C4×Dic61C22⋊C411yescharacteristic
225C4×Dic61C22⋊C411yescharacteristic
226C4×Dic61C22⋊C411yescharacteristic
227C4×Dic61C22⋊C411yescharacteristic
228C4×Dic61C22×C411yescharacteristic
229C4×Dic61C22×C411yescharacteristic
230C4×Dic61C2×D411yescharacteristic
231C4×Dic61C2×D411yescharacteristic
232#11758194—22no—
233#11758194C2×Q814yescomplemented
234C2×#92418—22no—
235#11758189—28no—
236C8×D61—28no—
237C23×C61C22×D411yescharacteristic
238C22×D61—41no—
239C22×D61—41no—
240C22×D61C2×C22⋊C411yescharacteristic
241C22×D61C2×C22⋊C411yescharacteristic
242C22×D61#12412yes—
243C22×D61C4⋊Q812yes—
244Q8×C61—42no—
245Q8×C61—42no—
246C2×C4×C61—22no—
247C2×C4×C61—22no—
248C2×C4×C61—22no—
249C2×C4×C61—41no—
250C2×C4×C61—42no—
251C2×C4×C61—42no—
252C2×C4×C61—42no—
253C2×C4×C61C2×C22⋊C411yescharacteristic
254C2×C4×C61C2×C22⋊C411yescharacteristic
255C2×C4×C61C2×C22⋊C411yescharacteristic
256C2×C4×C61C2×C22⋊C411yescharacteristic
257C2×C4×C61C23×C411yescharacteristic
258C2×C4×C61C22×D411yescharacteristic
259C2×C4×C61C22×Q812yes—
260C2×C4×C61C2×#4112yes—
261C2×Dic61—22no—
262C2×Dic61—22no—
263C2×Dic61—22no—
264C2×Dic61—41no—
265C2×Dic61—42no—
266C2×Dic61—42no—
267C2×Dic61—42no—
268C2×Dic61C2×C22⋊C411yescharacteristic
269C2×Dic61C2×C22⋊C411yescharacteristic
270C2×Dic61C2×C22⋊C411yescharacteristic
271C2×Dic61C2×C22⋊C411yescharacteristic
272C2×Dic61C2×C22⋊C411yescharacteristic
273C2×Dic61C2×C22⋊C411yescharacteristic
274C2×Dic61#12412yes—
275C2×Dic61C4⋊Q812yes—
276C4×D61—22no—
277C4×D61—22no—
278C4×D61—22no—
279C4×D61—22no—
280C4×D61—41no—
281C4×D61—41no—
282C4×D61—41no—
283C4×D61—41no—
284C4×D61—44no—
285C4×D61—44no—
286C4×D61C4×Q814yes—
287C4×D61C4×Q814yes—
288C4×D61C22⋊Q814yes—
289C4×D61C22⋊Q814yes—
290Dic122—44no—
291Dic122—42no—
292Dic122—42no—
293Dic122—44no—
294C8×C61—24no—
295#92418—24no—
296#21306—611nomaximal, Sylow
297C22×C61—21no—
298C22×C61—21no—
299C22×C61C22×C22⋊C411yescharacteristic
300C22×C61C2×#12411yescharacteristic
301C22×C61C2×C4⋊Q811yescharacteristic
302C2×D61—21no—
303C2×D61—21no—
304C2×D61—21no—
305C2×D61—21no—
306C2×D61—42no—
307C2×D61—42no—
308C2×D61#39112yes—
309C2×D61#39112yes—
310C4×C61—21no—
311C4×C61—21no—
312C4×C61—21no—
313C4×C61—42no—
314C4×C61—42no—
315C4×C61—82no—
316C4×C61#42112yes—
317C4×C61C2×C4×Q812yes—
318C4×C61C2×C22⋊Q812yes—
319Dic61—21no—
320Dic61—21no—
321Dic61—21no—
322Dic61—42no—
323Dic61—42no—
324Dic61—82no—
325Dic61#39112yes—
326Dic61#39112yes—
327Dic61#42112yes—
328C2×#533—611no—
329C2×C4×M16—611no—
330#2285—611no—
331#1986—611no—
332#1986—611no—
333#1986—614no—
334#1986—611no—
335#1986D6111yescharacteristic, complemented
336#1733—614no—
337C2×C61—21no—
338C2×C61—21no—
339C2×C61C2×#39111yescharacteristic
340C2×C61C2×#42111yescharacteristic
341C2×C61C2×#42111yescharacteristic
342D61—22no—
343D61—22no—
344C22×M16—612no—
345#533—611no—
346#533—611no—
347#533—612no—
348#533—612no—
349#533—611no—
350#533—1221no—
351#533—1221no—
352#533—1221no—
353#533—1221no—
354#533C2×D6111yescharacteristic
355C2×C4⋊Q8—611no—
356C2×C4⋊Q8—611no—
357C2×#125—612no—
358C2×C22⋊Q8—1221no—
359C2×C22⋊Q8—1221no—
360C2×C4×Q8—1221no—
361C2×C4×Q8—1221no—
362C2×#117—612no—
363C2×#117—612no—
364C22×C42—611no—
365C4×M16—612no—
366C4×M16—611no—
367C4×M16—1221no—
368C4×M16—1221no—
369C4×M16C2×D6111yescharacteristic
370C2×#97—1221no—
371C2×C4×C8—1221no—
372#384—1221no—
373#382—1221no—
374#380—1221no—
375#380—1221no—
376C4×C4⋊C4—612no—
377C4×C4⋊C4—611no—
378C4×C4⋊C4C2×D6111yescharacteristic
379#339—614no—
380#339—614no—
381#339—614no—
382#339C2×D6114yes—
383C61#2130611yescharacteristic, complemented, Sylow
384C22×Q8—1221no—
385C22×Q8—1221no—
386C23×C4—611no—
387C23×C4—611no—
388C23×C4—611no—
389C2×M16—612no—
390C2×M16—618no—
391C2×M16—1222no—
392C2×M16—1224no—
393C2×M16C22×D6112yes—
394C22×C8—1222no—
395C4⋊Q8—612no—
396C4⋊Q8—612no—
397C4⋊Q8—611no—
398C4⋊Q8—611no—
399C4⋊Q8—1221no—
400C4⋊Q8—1221no—
401C4⋊Q8—1221no—
402C4⋊Q8—1221no—
403C4⋊Q8C4×D6111yescharacteristic
404C4⋊Q8C4×D6111yescharacteristic
405#125—612no—
406#125—612no—
407#125—612no—
408#125—1222no—
409#125—1222no—
410#125C4×D6112yes—
411C22⋊Q8—21no—
412C22⋊Q8—21no—
413C22⋊Q8—1222no—
414C22⋊Q8—1222no—
415C22⋊Q8—1222no—
416C22⋊Q8—1222no—
417C22⋊Q8—1221no—
418C22⋊Q8—1221no—
419C22⋊Q8—1221no—
420C22⋊Q8—1221no—
421C22⋊Q8—1221no—
422C22⋊Q8—1221no—
423C4×Q8—21no—
424C4×Q8—21no—
425C4×Q8—1222no—
426C4×Q8—1222no—
427C4×Q8—1222no—
428C4×Q8—1222no—
429C4×Q8—1221no—
430C4×Q8—1221no—
431C4×Q8—1221no—
432C4×Q8—1221no—
433C4×Q8—1221no—
434C4×Q8—1221no—
435#117—612no—
436#117—612no—
437#117—612no—
438#117—1222no—
439#117—1228no—
440#117—1222no—
441#117C22×D6112yes—
442C2×C4⋊C4—21no—
443C2×C4⋊C4—612no—
444C2×C4⋊C4—612no—
445C2×C4⋊C4—1222no—
446C2×C4⋊C4—1222no—
447C2×C4⋊C4—1221no—
448C2×C4⋊C4—1221no—
449C2×C4⋊C4—1222no—
450C2×C4⋊C4—1222no—
451C2×C4⋊C4—1221no—
452C2×C22⋊C4—1222no—
453C2×C22⋊C4—1222no—
454C2×C42—611no—
455C2×C42—611no—
456C2×C42—611no—
457C2×C42—611no—
458C2×C42—611no—
459C2×C42—611no—
460C2×C42—611no—
461C2×C42—611no—
462C2×C42—611no—
463C2×C42—611no—
464C2×C42—611no—
465C2×C42—612no—
466C2×C42—1222no—
467C2×C42C22×D6112yes—
468C2×C42C22×D6111yescharacteristic
469#97—21no—
470#97—1222no—
471#97—1221no—
472C4×C8—21no—
473C4×C8—1222no—
474C4×C8—1221no—
475#95—21no—
476#95—1222no—
477#95—1221no—
478C24—611no—
479C2×Q8—21no—
480C2×Q8—21no—
481C2×Q8—1222no—
482C2×Q8—1222no—
483C2×Q8—1221no—
484C2×Q8—1221no—
485C2×Q8—2444no—
486C2×Q8—2444no—
487C22×C4—22no—
488C22×C4—611no—
489C22×C4—611no—
490C22×C4—611no—
491C22×C4—611no—
492C22×C4—611no—
493C22×C4—614no—
494C22×C4—614no—
495C22×C4—611no—
496C22×C4—611no—
497C22×C4—611no—
498C22×C4—611no—
499C22×C4—611no—
500C22×C4—611no—
501C22×C4—611no—
502C22×C4—611no—
503C22×C4—611no—
504C22×C4—611no—
505C22×C4—1222no—
506C22×C4—1222no—
507C22×C4—1222no—
508C22×C4—2442no—
509C22×C4—2442no—
510C22×C4D4×D6111yescharacteristic
511C22×C4D4×D6111yescharacteristic
512C22×C4C23×D6111yescharacteristic
513M16—22no—
514M16—614no—
515M16—1228no—
516M16—1222no—
517M16Q8×D6114yescomplemented
518C2×C8—22no—
519C2×C8—1228no—
520C2×C8—1222no—
521C4⋊C4—22no—
522C4⋊C4—22no—
523C4⋊C4—21no—
524C4⋊C4—21no—
525C4⋊C4—42no—
526C4⋊C4—612no—
527C4⋊C4—612no—
528C4⋊C4—1222no—
529C4⋊C4—1222no—
530C4⋊C4—1222no—
531C4⋊C4—1222no—
532C4⋊C4—1222no—
533C4⋊C4—1222no—
534C4⋊C4—1222no—
535C4⋊C4—1222no—
536C4⋊C4—1222no—
537C4⋊C4—1222no—
538C4⋊C4—1221no—
539C4⋊C4—1221no—
540C4⋊C4—2442no—
541C4⋊C4—2444no—
542C4⋊C4—2444no—
543C4⋊C4C2×C4×D6112yes—
544C4⋊C4C2×C4×D6112yes—
545C22⋊C4—22no—
546C22⋊C4—1222no—
547C22⋊C4—1222no—
548C22⋊C4—1222no—
549C22⋊C4—2444no—
550C42—22no—
551C42—24no—
552C42—611no—
553C42—611no—
554C42—611no—
555C42—611no—
556C42—611no—
557C42—611no—
558C42—611no—
559C42—611no—
560C42—611no—
561C42—611no—
562C42—611no—
563C42—611no—
564C42—1222no—
565C42—1222no—
566C42—1222no—
567C42—1224no—
568C42C2×C4×D6111yescharacteristic
569C42C2×C4×D6111yescharacteristic
570C42D4×D6111yescharacteristic
571C42D4×D6111yescharacteristic
572C23—611no—
573C23—611no—
574C23—612no—
575C23—612no—
576C23—2441no—
577C23—2441no—
578C23C2×D4×D6111yescharacteristic
579Q8—42no—
580Q8—42no—
581Q8—2442no—
582Q8—2444no—
583Q8—2444no—
584Q8—2442no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—41no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—611no—
593C2×C4—611no—
594C2×C4—611no—
595C2×C4—612no—
596C2×C4—612no—
597C2×C4—614no—
598C2×C4—614no—
599C2×C4—614no—
600C2×C4—614no—
601C2×C4—611no—
602C2×C4—611no—
603C2×C4—611no—
604C2×C4—1222no—
605C2×C4—1222no—
606C2×C4—1222no—
607C2×C4—1222no—
608C2×C4—1222no—
609C2×C4—1222no—
610C2×C4—1222no—
611C2×C4—2441no—
612C2×C4—2441no—
613C2×C4—2441no—
614C2×C4—2441no—
615C2×C4—2444no—
616C2×C4—2444no—
617C2×C4—2441no—
618C2×C4—2442no—
619C2×C4—2442no—
620C2×C4—2442no—
621C2×C4C22⋊C4×D6111yescharacteristic
622C2×C4C22⋊C4×D6111yescharacteristic
623C2×C4C22⋊C4×D6111yescharacteristic
624C2×C4C22⋊C4×D6111yescharacteristic
625C2×C4C22×C4×D6111yescharacteristic
626C2×C4C2×D4×D6111yescharacteristic
627C2×C4C2×Q8×D6112yes—
628C2×C4#41×D6112yes—
629C8—24no—
630C8—1224no—
631C22—21no—
632C22—21no—
633C22—612no—
634C22—612no—
635C22—1221no—
636C22—1221no—
637C22—1221no—
638C22—1221no—
639C22—2442no—
640C22—2442no—
641C22C2×C22⋊C4×D6111yescharacteristic
642C22#124×D6111yescharacteristic
643C22C4⋊Q8×D6111yescharacteristic
644C4—21no—
645C4—21no—
646C4—21no—
647C4—42no—
648C4—42no—
649C4—82no—
650C4—612no—
651C4—612no—
652C4—612no—
653C4—1221no—
654C4—1221no—
655C4—1221no—
656C4—4882no—
657C4—2442no—
658C4—2442no—
659C4#1683977712yes—
660C4C4×Q8×D6112yes—
661C4C22⋊Q8×D6112yes—
662C2—21no—
663C2—21no—
664C2—1222no—
665C2—1222no—
666C2#391×D6111yescharacteristic
667C2#421×D6111yescharacteristic
668C2C2×#1683977711yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1986C2
socleC22×C61C22×C22⋊C4
prime radicalC23×D61C2×D4
cosocleC22×C4×C61C24
hypercenter#1986D61
hypocenterC61#21306
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1986

Complements

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

NH ≅ G/Ntype
4. #21446764C2semidirect
5. #21446763C2semidirect
6. #21446762C2semidirect
7. #21446761C2semidirect
9. C61×#1986C2semidirect
12. #16840318C4semidirect
13. #16840317C4semidirect
14. #16840317C4semidirect
20. C4⋊Q8×D61C4semidirect
21. C4⋊Q8×D61C4semidirect
22. #125×D61C4semidirect
57. M16×D61Q8semidirect
179. M16×C61C2×Q8semidirect
233. #11758194C2×Q8semidirect
335. #1986D61semidirect
383. C61#21306semidirect
517. M16Q8×D61semidirect

Maximal subgroups

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

subgroupindex
#533×D612
C4×M16×D612
#214482232
#214467642
#214467632
#214467622
#214467612
#214455272
C61×#19862
#2130661