Finite Group Databasebeta

← #29646807

Subgroups — #29646807

Order 15616.

1324 conjugacy classes of subgroups.

The subgroup lattice has 700 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840309C211yesmaximal, characteristic
2#21454449C211yesmaximal, characteristic, complemented
3C4×M16×D61C211yesmaximal, characteristic, complemented
4#21444268C214yesmaximal, complemented
5#21444263C211yesmaximal, characteristic, complemented
6#21444262C211yesmaximal, characteristic, complemented
7#21444261C211yesmaximal, characteristic, complemented
8#21444234C214yesmaximal, complemented
9C61×#1490C211yesmaximal, characteristic, complemented
10C22×#11758222—21no—
11C2×M16×D61—21no—
12#16840309—21no—
13#16840309—21no—
14#16840309—21no—
15#16840309—21no—
16#16840309C412yes—
17#16840309C412yes—
18#16840309C2212yes—
19#16840309C2212yes—
20C2×#14519776C411yescharacteristic
21C2×#14519773C411yescharacteristic
22C2×#14519763—21no—
23C2×#14519752C411yescharacteristic
24C2×#14519750—21no—
25C2×#14519748C411yescharacteristic
26C2×C4×D244C2211yescharacteristic
27C2×C4×C22⋊D61—21no—
28C2×C42×D61C2211yescharacteristic
29#16839819C2211yescharacteristic
30#16839817—21no—
31#16839814C2212yescomplemented
32#16839812—21no—
33#16839784C2212yescomplemented
34#16839724—21no—
35C4×#11758194C2211yescharacteristic
36#16839714—21no—
37#16839714—21no—
38M16×Dic61—21no—
39C4×#11758189C2212yescomplemented
40#97×D61—21no—
41C4×C8×D61C2212yescomplemented
42#16839491C2214yescomplemented
43D4⋊#92418C2214yescomplemented
44#16839489C2214yescomplemented
45C2×C4×D4×C61C2211yescharacteristic
46C61×#432C2211yescharacteristic
47C4×M16×C61C2211yescharacteristic
48C61×D4⋊C8C2214yescomplemented
49C2×#11758222—22no—
50C2×#11758222—22no—
51C2×#11758222—22no—
52C2×#11758222—22no—
53C2×#11758222—48no—
54C22×Dic122—21no—
55C22×D244—21no—
56C22×D244C2×C411yescharacteristic
57C22×C22⋊D61—21no—
58C22×C22⋊D61—21no—
59C22×C22⋊D61—21no—
60C22×C4×D61—21no—
61C22×C4×D61C2×C411yescharacteristic
62C22×C4×D61D411yescharacteristic
63C22×C4×D61D411yescharacteristic
64C2×#11758194—21no—
65C2×#11758189—22no—
66C2×#11758189—22no—
67M16×D61—28no—
68C2×C8×D61—22no—
69#14519776—21no—
70#14519776—21no—
71#14519776C2×C412yes—
72#14519776C2×C412yes—
73#14519773—21no—
74#14519773—21no—
75#14519773C2×C412yes—
76#14519773C2×C412yes—
77#14519763—22no—
78#14519763—22no—
79#14519763—22no—
80#14519763—22no—
81#14519752—21no—
82#14519752—21no—
83#14519752C2×C412yes—
84#14519752C2×C412yes—
85#14519750—22no—
86#14519750—22no—
87#14519750—22no—
88#14519750—22no—
89#14519748—21no—
90#14519748—21no—
91#14519748C2×C412yes—
92#14519748C2×C412yes—
93C4×D244—21no—
94C4×D244—21no—
95C4×D244C2×C412yes—
96C4×D244C2312yes—
97C4×C22⋊D61—22no—
98C4×C22⋊D61—22no—
99C4×C22⋊D61—22no—
100C4×C22⋊D61—22no—
101C2×C4⋊Dic61C2×C411yescharacteristic
102C2×#11758196—21no—
103C2×C22⋊Dic61—21no—
104C2×#11758198—21no—
105C2×#11758198—21no—
106C2×C4×Dic61C2×C411yescharacteristic
107C2×C4×Dic61D411yescharacteristic
108C2×C4×Dic61D411yescharacteristic
109C42×D61C2×C411yescharacteristic
110C42×D61C2×C411yescharacteristic
111C42×D61D411yescharacteristic
112C42×D61D411yescharacteristic
113C42×D61D411yescharacteristic
114C42×D61D411yescharacteristic
115C42×D61C2311yescharacteristic
116C42×D61C2311yescharacteristic
117C4⋊#92418C2312yes—
118#14519706—22no—
119C22⋊#92418—21no—
120#14519689—22no—
121#14519688—21no—
122#14519688—21no—
123#14519687—22no—
124C4×#92418C2312yes—
125C8×Dic61—22no—
126C8×Dic61—22no—
127C22×D4×C61C2×C411yescharacteristic
128C23×C4×C61—21no—
129C2×M16×C61—21no—
130C4×D4×C61—21no—
131C4×D4×C61—21no—
132C4×D4×C61C2×C412yes—
133C4×D4×C61C2312yes—
134C2×C4⋊C4×C61C2×C411yescharacteristic
135C2×C22⋊C4×C61—21no—
136C2×C42×C61C2311yescharacteristic
137C4⋊C8×C61C2312yes—
138C22⋊C8×C61—21no—
139#97×C61—21no—
140#97×C61—21no—
141C4×C8×C61C2312yes—
142C24×C61—21no—
143C23×D61—21no—
144C23×D61C22⋊C411yescharacteristic
145C2×D4×C61—21no—
146C2×D4×C61—42no—
147C2×D4×C61C22×C412yes—
148C22×C4×C61—21no—
149C22×C4×C61—22no—
150C22×C4×C61—22no—
151C22×C4×C61—44no—
152C22×C4×C61C22×C411yescharacteristic
153C22×C4×C61C2×D411yescharacteristic
154C22×C4×C61C2×D411yescharacteristic
155C2×C22⋊D61—22no—
156C2×C22⋊D61—22no—
157C2×C22⋊D61—22no—
158C2×C22⋊D61—22no—
159C2×C22⋊D61—22no—
160C2×C22⋊D61—22no—
161C2×C22⋊D61—44no—
162C2×C22⋊D61—44no—
163C2×C22⋊D61—44no—
164C22×Dic61—21no—
165C22×Dic61C22⋊C411yescharacteristic
166C22×Dic61C22⋊C411yescharacteristic
167C22×Dic61C22⋊C411yescharacteristic
168#11758222—48no—
169#11758222—48no—
170#11758222—48no—
171#11758222—48no—
172C2×D244—22no—
173C2×D244—22no—
174C2×D244—21no—
175C2×D244—42no—
176C2×D244—44no—
177C2×D244C22×C412yes—
178C2×C4×D61—24no—
179C2×C4×D61—24no—
180C2×C4×D61—22no—
181C2×C4×D61—22no—
182C2×C4×D61—42no—
183C2×C4×D61—44no—
184C2×C4×D61C22⋊C411yescharacteristic
185C2×C4×D61C22⋊C411yescharacteristic
186C2×C4×D61C22⋊C411yescharacteristic
187C2×C4×D61C22⋊C411yescharacteristic
188C2×C4×D61C22⋊C411yescharacteristic
189C2×C4×D61C22⋊C411yescharacteristic
190C2×C4×D61C22×C411yescharacteristic
191C2×C4×D61C22×C411yescharacteristic
192C2×C4×D61C2×D411yescharacteristic
193C2×C4×D61C2×D411yescharacteristic
194C2×C4×D61C2×D411yescharacteristic
195C2×C4×D61C2×D411yescharacteristic
196C2×Dic122—22no—
197C2×Dic122—22no—
198C2×Dic122—44no—
199M16×C61—24no—
200C2×C8×C61—22no—
201C2×C8×C61—22no—
202C4⋊C4×C61—21no—
203C4⋊C4×C61C22×C412yes—
204C22⋊C4×C61—22no—
205C22⋊C4×C61—22no—
206C42×C61C22×C411yescharacteristic
207C42×C61C2×D411yescharacteristic
208C42×C61C2×D411yescharacteristic
209C42×C61C2411yescharacteristic
210C22⋊Dic61—22no—
211C22⋊Dic61—22no—
212#11758198—22no—
213#11758198—22no—
214#11758198—22no—
215#11758198—22no—
216C4⋊Dic61—21no—
217C4⋊Dic61C22×C412yes—
218#11758196—22no—
219#11758196—22no—
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×Dic61C2×D411yescharacteristic
229C4×Dic61C2×D411yescharacteristic
230C4×Dic61C2×D411yescharacteristic
231C4×Dic61C2×D411yescharacteristic
232#11758194—24no—
233C2×#92418—22no—
234C2×#92418—22no—
235#11758189—28no—
236#11758189—44no—
237C8×D61—28no—
238C23×C61—22no—
239C23×C61—42no—
240C23×C61—42no—
241C23×C61—42no—
242C23×C61C2×C22⋊C411yescharacteristic
243C22×D61—22no—
244C22×D61—22no—
245C22×D61—22no—
246C22×D61—41no—
247C22×D61—42no—
248C22×D61—42no—
249C22×D61—42no—
250C22×D61C2×C22⋊C411yescharacteristic
251C22×D61C2×C22⋊C411yescharacteristic
252D4×C61—24no—
253D4×C61—42no—
254C2×C4×C61—22no—
255C2×C4×C61—22no—
256C2×C4×C61—22no—
257C2×C4×C61—22no—
258C2×C4×C61—44no—
259C2×C4×C61—42no—
260C2×C4×C61—44no—
261C2×C4×C61C2×C22⋊C411yescharacteristic
262C2×C4×C61C2×C22⋊C411yescharacteristic
263C2×C4×C61C2×C22⋊C411yescharacteristic
264C2×C4×C61C23×C411yescharacteristic
265C2×C4×C61C22×D411yescharacteristic
266C2×C4×C61C22×D411yescharacteristic
267C22⋊D61—44no—
268C22⋊D61—44no—
269C22⋊D61—44no—
270C22⋊D61—44no—
271C22⋊D61—44no—
272C22⋊D61—44no—
273C2×Dic61—22no—
274C2×Dic61—22no—
275C2×Dic61—41no—
276C2×Dic61—42no—
277C2×Dic61#10414yes—
278C2×Dic61C2×C22⋊C411yescharacteristic
279C2×Dic61C2×C22⋊C411yescharacteristic
280C2×Dic61C2×C22⋊C411yescharacteristic
281C2×Dic61C2×C22⋊C411yescharacteristic
282C2×Dic61C2×C22⋊C411yescharacteristic
283C2×Dic61C2×C22⋊C411yescharacteristic
284D244—24no—
285D244—42no—
286D244—44no—
287D244—44no—
288C4×D61—22no—
289C4×D61—22no—
290C4×D61—22no—
291C4×D61—22no—
292C4×D61—22no—
293C4×D61—22no—
294C4×D61—22no—
295C4×D61—22no—
296C4×D61—22no—
297C4×D61—41no—
298C4×D61—41no—
299C4×D61—44no—
300C4×D61—44no—
301C4×D61AGL(1,ℤ/8)14yes—
302Dic122—44no—
303Dic122—44no—
304C8×C61—24no—
305C8×C61—42no—
306#92418—24no—
307#92418—42no—
308#17500—611nomaximal, Sylow
309C22×C61—21no—
310C22×C61—21no—
311C22×C61—42no—
312C22×C61—42no—
313C22×C61—42no—
314C22×C61—84no—
315C22×C61C2×#10412yes—
316C22×C61C22×C22⋊C411yescharacteristic
317C2×D61—21no—
318C2×D61—21no—
319C2×D61—21no—
320C2×D61—21no—
321C2×D61—21no—
322C2×D61—44no—
323C2×D61—42no—
324C2×D61—42no—
325C2×D61—42no—
326C2×D61—84no—
327C2×D61#41812yes—
328C4×C61—21no—
329C4×C61—21no—
330C4×C61—21no—
331C4×C61—21no—
332C4×C61—42no—
333C4×C61#56812yes—
334C4×C61C2×AGL(1,ℤ/8)12yes—
335Dic61—21no—
336Dic61—21no—
337Dic61—21no—
338Dic61—42no—
339Dic61#41812yes—
340Dic61C2×#10412yes—
341Dic61C2×#10412yes—
342C2×#532—611no—
343#2920—611no—
344C2×C4×M16—611no—
345#1492—614no—
346#1490—611no—
347#1490—611no—
348#1490—611no—
349#1490D6111yescharacteristic, complemented
350#1484—614no—
351C2×C61—22no—
352C2×C61—82no—
353C2×C61#287911yescharacteristic
354C2×C61C2×#41811yescharacteristic
355C2×C61C22×#10411yescharacteristic
356D61—22no—
357D61—41no—
358D61—82no—
359C22×#41—1221no—
360C22×M16—1221no—
361#532—612no—
362#532—612no—
363#532—612no—
364#532—612no—
365#532—1221no—
366#532—1221no—
367#532—1221no—
368#532—1221no—
369C2×C4⋊Q8—611no—
370C2×#127—611no—
371C2×#124—1221no—
372C2×#121—611no—
373C2×#121—611no—
374C2×#121—1221no—
375C2×C4×D4—611no—
376C2×C4×D4—1221no—
377C2×C4×D4C2×D6111yescharacteristic
378C22×C42—611no—
379#432—611no—
380#432—1221no—
381#432C2×D6111yescharacteristic
382#431—612no—
383#431—1221no—
384#423—612no—
385#407—1221no—
386C4×M16—611no—
387C4×M16—612no—
388C4×M16—1221no—
389C4×M16—1221no—
390C4×M16—1221no—
391C4×M16C2×D6111yescharacteristic
392C2×#97—1221no—
393C2×C4×C8—612no—
394D4⋊C8—614no—
395D4⋊C8—614no—
396D4⋊C8—614no—
397D4⋊C8C2×D6114yescomplemented
398C61#1750011yescharacteristic, complemented, Sylow
399C2×#41—1222no—
400C2×#41—1222no—
401C2×#41—1222no—
402C2×#41—1222no—
403C2×#41—2448no—
404C22×Q8—1221no—
405C22×D4—611no—
406C22×D4—1221no—
407C22×D4—1221no—
408C22×D4—1221no—
409C22×D4—1221no—
410C22×D4C4×D6111yescharacteristic
411C23×C4—21no—
412C23×C4—611no—
413C23×C4—611no—
414C23×C4—611no—
415C23×C4—1221no—
416C2×M16—21no—
417C2×M16—1222no—
418C2×M16—1222no—
419C2×M16—1228no—
420C2×M16—1221no—
421C22×C8—1222no—
422C4⋊Q8—612no—
423C4⋊Q8—612no—
424C4⋊Q8—1221no—
425C4⋊Q8—1221no—
426#127—612no—
427#127—612no—
428#127—1221no—
429#127—1221no—
430#124—1222no—
431#124—1222no—
432#124—1222no—
433#124—1222no—
434#121—612no—
435#121—612no—
436#121—612no—
437#121—612no—
438#121—1222no—
439#121—1222no—
440#121—1222no—
441#121—1222no—
442#121—1221no—
443#121—1221no—
444#121—1221no—
445#121—1221no—
446C4×D4—21no—
447C4×D4—21no—
448C4×D4—612no—
449C4×D4—612no—
450C4×D4—1222no—
451C4×D4—1222no—
452C4×D4—1222no—
453C4×D4—1222no—
454C4×D4—1221no—
455C4×D4—1221no—
456C4×D4C4×D6112yes—
457C4×D4C22×D6112yes—
458C2×C4⋊C4—611no—
459C2×C4⋊C4—1221no—
460C2×C4⋊C4C4×D6111yescharacteristic
461C2×C22⋊C4—21no—
462C2×C22⋊C4—1221no—
463C2×C22⋊C4—1221no—
464C2×C22⋊C4—1221no—
465C2×C42—611no—
466C2×C42—611no—
467C2×C42—611no—
468C2×C42—611no—
469C2×C42—611no—
470C2×C42—611no—
471C2×C42—611no—
472C2×C42—611no—
473C2×C42—611no—
474C2×C42—611no—
475C2×C42—611no—
476C2×C42C22×D6111yescharacteristic
477C4⋊C8—612no—
478C4⋊C8—1222no—
479C4⋊C8C22×D6112yes—
480C22⋊C8—21no—
481C22⋊C8—1222no—
482C22⋊C8—1221no—
483#97—21no—
484#97—21no—
485#97—1222no—
486#97—1221no—
487#97—1221no—
488C4×C8—612no—
489C4×C8—1222no—
490C4×C8—1222no—
491C4×C8C22×D6112yes—
492C24—21no—
493C24—611no—
494C24—1221no—
495#41—2448no—
496#41—2448no—
497#41—2448no—
498#41—2448no—
499C2×Q8—1222no—
500C2×Q8—1222no—
501C2×Q8—2444no—
502C2×D4—21no—
503C2×D4—42no—
504C2×D4—612no—
505C2×D4—1222no—
506C2×D4—1222no—
507C2×D4—1222no—
508C2×D4—1222no—
509C2×D4—1222no—
510C2×D4—1222no—
511C2×D4—1222no—
512C2×D4—1222no—
513C2×D4—1221no—
514C2×D4—2442no—
515C2×D4—2444no—
516C2×D4—2444no—
517C2×D4—2444no—
518C2×D4—2444no—
519C2×D4C2×C4×D6112yes—
520C22×C4—21no—
521C22×C4—22no—
522C22×C4—22no—
523C22×C4—44no—
524C22×C4—611no—
525C22×C4—611no—
526C22×C4—611no—
527C22×C4—611no—
528C22×C4—611no—
529C22×C4—611no—
530C22×C4—611no—
531C22×C4—611no—
532C22×C4—611no—
533C22×C4—611no—
534C22×C4—611no—
535C22×C4—611no—
536C22×C4—611no—
537C22×C4—611no—
538C22×C4—611no—
539C22×C4—1224no—
540C22×C4—1224no—
541C22×C4—1221no—
542C22×C4—1222no—
543C22×C4—1222no—
544C22×C4—2442no—
545C22×C4—2444no—
546C22×C4C2×C4×D6111yescharacteristic
547C22×C4D4×D6111yescharacteristic
548C22×C4D4×D6111yescharacteristic
549M16—24no—
550M16—1228no—
551M16—1224no—
552M16—2444no—
553C2×C8—22no—
554C2×C8—22no—
555C2×C8—1228no—
556C2×C8—1222no—
557C2×C8—1222no—
558C4⋊C4—21no—
559C4⋊C4—612no—
560C4⋊C4—1222no—
561C4⋊C4—1222no—
562C4⋊C4—1221no—
563C4⋊C4C2×C4×D6112yes—
564C22⋊C4—22no—
565C22⋊C4—22no—
566C22⋊C4—1222no—
567C22⋊C4—1222no—
568C22⋊C4—1222no—
569C22⋊C4—1222no—
570C22⋊C4—1222no—
571C22⋊C4—1222no—
572C42—611no—
573C42—611no—
574C42—611no—
575C42—611no—
576C42—611no—
577C42—611no—
578C42—611no—
579C42—611no—
580C42—611no—
581C42—611no—
582C42—611no—
583C42—611no—
584C42C2×C4×D6111yescharacteristic
585C42D4×D6111yescharacteristic
586C42D4×D6111yescharacteristic
587C42C23×D6111yescharacteristic
588C23—22no—
589C23—42no—
590C23—42no—
591C23—42no—
592C23—611no—
593C23—611no—
594C23—1222no—
595C23—1222no—
596C23—1222no—
597C23—2441no—
598C23—2442no—
599C23—2442no—
600C23—2442no—
601C23C22⋊C4×D6111yescharacteristic
602Q8—2444no—
603Q8—2444no—
604D4—24no—
605D4—42no—
606D4—1224no—
607D4—2442no—
608D4—2444no—
609D4—2444no—
610D4—2444no—
611D4—2444no—
612D4—2444no—
613D4—2444no—
614D4—2444no—
615D4—2444no—
616C2×C4—22no—
617C2×C4—22no—
618C2×C4—22no—
619C2×C4—22no—
620C2×C4—44no—
621C2×C4—42no—
622C2×C4—44no—
623C2×C4—611no—
624C2×C4—611no—
625C2×C4—611no—
626C2×C4—614no—
627C2×C4—614no—
628C2×C4—611no—
629C2×C4—611no—
630C2×C4—611no—
631C2×C4—1222no—
632C2×C4—1222no—
633C2×C4—1222no—
634C2×C4—1222no—
635C2×C4—1222no—
636C2×C4—1222no—
637C2×C4—1222no—
638C2×C4—1222no—
639C2×C4—1222no—
640C2×C4—1222no—
641C2×C4—1222no—
642C2×C4—2441no—
643C2×C4—2441no—
644C2×C4—2444no—
645C2×C4—2441no—
646C2×C4—2442no—
647C2×C4—2444no—
648C2×C4C22⋊C4×D6111yescharacteristic
649C2×C4C22⋊C4×D6111yescharacteristic
650C2×C4C22⋊C4×D6111yescharacteristic
651C2×C4C22×C4×D6111yescharacteristic
652C2×C4C2×D4×D6111yescharacteristic
653C2×C4C2×D4×D6111yescharacteristic
654C8—24no—
655C8—42no—
656C8—1224no—
657C8—2442no—
658C22—21no—
659C22—21no—
660C22—42no—
661C22—42no—
662C22—42no—
663C22—84no—
664C22—612no—
665C22—1221no—
666C22—1221no—
667C22—1221no—
668C22—1221no—
669C22—1221no—
670C22—4884no—
671C22—2444no—
672C22—2442no—
673C22—2442no—
674C22—2442no—
675C22#104×D6112yes—
676C22C2×C22⋊C4×D6111yescharacteristic
677C4—21no—
678C4—21no—
679C4—21no—
680C4—21no—
681C4—42no—
682C4—612no—
683C4—612no—
684C4—612no—
685C4—1221no—
686C4—1221no—
687C4—1221no—
688C4—2442no—
689C4#1684052812yes—
690C4AGL(1,ℤ/8)×D6112yes—
691C2—22no—
692C2—82no—
693C2—1222no—
694C2—4882no—
695C2—2441no—
696C2#2145426211yescharacteristic
697C2#418×D6111yescharacteristic
698C2C2×#104×D6111yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC42C23×D61
FittingC61×#1490C2
socleC22×C61C22×C22⋊C4
prime radicalC2×#16840309C2
cosocleC42×C61C24
hypercenter#1490D61
hypocenterC61#17500
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1490

Complements

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

NH ≅ G/Ntype
2. #21454449C2semidirect
3. C4×M16×D61C2semidirect
4. #21444268C2semidirect
5. #21444263C2semidirect
6. #21444262C2semidirect
7. #21444261C2semidirect
8. #21444234C2semidirect
9. C61×#1490C2semidirect
31. #16839814C22semidirect
33. #16839784C22semidirect
39. C4×#11758189C22semidirect
41. C4×C8×D61C22semidirect
42. #16839491C22semidirect
43. D4⋊#92418C22semidirect
44. #16839489C22semidirect
48. C61×D4⋊C8C22semidirect
349. #1490D61semidirect
397. D4⋊C8C2×D61semidirect
398. C61#17500semidirect

Maximal subgroups

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

subgroupindex
C2×#168403092
#214544492
C4×M16×D612
#214442682
#214442632
#214442622
#214442612
#214442342
C61×#14902
#1750061