Finite Group Databasebeta

← #29649986

Subgroups — #29649986

Order 15616.

720 conjugacy classes of subgroups.

The subgroup lattice has 614 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21455864C211yesmaximal, characteristic
2#21454797C211yesmaximal, characteristic, complemented
3C8×D4×D61C211yesmaximal, characteristic, complemented
4#21444681C211yesmaximal, characteristic, complemented
5#21444678C211yesmaximal, characteristic, complemented
6#21444676C211yesmaximal, characteristic, complemented
7#21444622C211yesmaximal, characteristic, complemented
8#21444618C211yesmaximal, characteristic, complemented
9D8×#92418C211yesmaximal, characteristic, complemented
10#21444611C211yesmaximal, characteristic, complemented
11#21444601C211yesmaximal, characteristic, complemented
12#21444260C211yesmaximal, characteristic, complemented
13#21444229C211yesmaximal, characteristic, complemented
14#21444098C211yesmaximal, characteristic, complemented
15C61×#1586C211yesmaximal, characteristic, complemented
16C22×C8×D61—21no—
17#16840294C2211yescharacteristic
18C4×D4×D61C2211yescharacteristic
19#16840099C411yescharacteristic
20#16840079C411yescharacteristic
21#16840065C411yescharacteristic
22#16840064C411yescharacteristic
23#16839976C411yescharacteristic
24#16839975C411yescharacteristic
25#16839914C411yescharacteristic
26#16839894C411yescharacteristic
27#16839892C2211yescharacteristic, complemented
28#16839886C2211yescharacteristic, complemented
29C4×#11758190C2211yescharacteristic
30C4×#11758200C2211yescharacteristic
31C4×#11758199C2211yescharacteristic
32#16839831—21no—
33C8×D244C2211yescharacteristic
34D4×#92418C2211yescharacteristic
35D4×#92418C2211yescharacteristic, complemented
36#16839827C2211yescharacteristic
37C8×C22⋊D61—21no—
38#16839825—21no—
39#16839825—21no—
40#16839820—21no—
41#16839818C2211yescharacteristic
42#16839813C2211yescharacteristic
43C4⋊C8×D61C2211yescharacteristic
44C22⋊C8×D61—21no—
45#16839720—21no—
46C4×#11758189C2211yescharacteristic
47C4×C8×D61C2211yescharacteristic
48#16839515C2211yescharacteristic, complemented
49#16839514C2211yescharacteristic, complemented
50#16839512C2211yescharacteristic, complemented
51#16839494C2211yescharacteristic, complemented
52#16839492C2211yescharacteristic, complemented
53#16839491C2211yescharacteristic
54D4⋊#92418C2211yescharacteristic
55D4⋊#92418C2211yescharacteristic, complemented
56#16839489C2211yescharacteristic
57#16839484C2211yescharacteristic, complemented
58C8×#92418C2211yescharacteristic, complemented
59C8×#92418C2211yescharacteristic, complemented
60C4×D8×C61C2211yescharacteristic
61C61×#436C2211yescharacteristic, complemented
62C8×D4×C61C2211yescharacteristic
63C61×#334C2211yescharacteristic, complemented
64C61×D4⋊C8C2211yescharacteristic
65C61×D4⋊C8C2211yescharacteristic, complemented
66C61×#322C2211yescharacteristic, complemented
67C2×D4×D61C2×C411yescharacteristic
68C22×C4×D61—21no—
69C2×#11758190C2×C411yescharacteristic
70C2×#11758200C2×C411yescharacteristic
71C2×#11758199C2×C411yescharacteristic
72C2×#11758189—21no—
73C22×#92418—21no—
74C22×#92418—21no—
75C2×C8×D61—21no—
76C2×C8×D61—21no—
77C2×C8×D61—21no—
78C2×C8×D61—44no—
79C2×C8×D61D411yescharacteristic
80C2×C8×D61D411yescharacteristic
81#14519770—21no—
82#14519768C2×C411yescharacteristic
83#14519764C2×C411yescharacteristic
84#14519759—21no—
85#14519757C2×C411yescharacteristic
86#14519748C2×C411yescharacteristic
87C4×Dic122C2311yescharacteristic
88C4×D244C2311yescharacteristic
89D4×Dic61C2×C411yescharacteristic
90#14519739C2×C411yescharacteristic
91C4×C22⋊D61—21no—
92C4×C22⋊D61—21no—
93#14519737—21no—
94#14519736—21no—
95C4⋊C4×D61C2×C411yescharacteristic
96C22⋊C4×D61—21no—
97C42×D61C2311yescharacteristic
98#14519710C2×C411yescharacteristic
99#14519709C2×C411yescharacteristic
100#14519708C2×C411yescharacteristic
101C4⋊#92418C2311yescharacteristic
102C4⋊#92418C2311yescharacteristic
103#14519706—21no—
104#14519706—21no—
105#14519706—21no—
106#14519706C2×C411yescharacteristic
107#14519706C2×C411yescharacteristic
108#14519702C2×C411yescharacteristic
109#14519700C2×C411yescharacteristic
110#14519699C2×C411yescharacteristic
111#14519698C2×C411yescharacteristic
112#14519698C2×C411yescharacteristic
113#14519697C2×C411yescharacteristic
114C22⋊#92418—21no—
115C22⋊#92418—21no—
116#14519689—21no—
117#14519689—21no—
118#14519689—21no—
119#14519689C2×C411yescharacteristic
120#14519689C2×C411yescharacteristic
121#14519687—21no—
122#14519687—21no—
123#14519687—21no—
124C4×#92418C2311yescharacteristic
125C4×#92418C2311yescharacteristic
126C4×#92418C2311yescharacteristic
127C8×Dic61—21no—
128C8×Dic61D411yescharacteristic
129C8×Dic61D411yescharacteristic
130C2×D8×C61C2×C411yescharacteristic
131C2×M16×C61—21no—
132C22×C8×C61—21no—
133C4×D4×C61C2311yescharacteristic
134C4×D4×C61C2311yescharacteristic
135#107×C61C2×C411yescharacteristic
136C4⋊C8×C61C2311yescharacteristic
137C4⋊C8×C61C2311yescharacteristic
138#102×C61C2×C411yescharacteristic
139#102×C61C2×C411yescharacteristic
140C22⋊C8×C61—21no—
141C22⋊C8×C61—21no—
142C4×C8×C61C2311yescharacteristic
143C4×C8×C61C2311yescharacteristic
144C4×C8×C61C2311yescharacteristic
145C23×D61—21no—
146C2×D4×C61C22×C411yescharacteristic
147C2×D4×C61C22×C411yescharacteristic
148C22×C4×C61—21no—
149C22×C4×C61—21no—
150C2×C22⋊D61—21no—
151C2×C22⋊D61—21no—
152C22×Dic61—21no—
153D4×D61—22no—
154D4×D61—22no—
155C2×D244C22×C411yescharacteristic
156C2×C4×D61—21no—
157C2×C4×D61—21no—
158C2×C4×D61—21no—
159C2×C4×D61—42no—
160C2×C4×D61—42no—
161C2×C4×D61C22×C411yescharacteristic
162C2×C4×D61C2×D411yescharacteristic
163C2×C4×D61#4111yescharacteristic
164C2×Dic122C22×C411yescharacteristic
165D8×C61—22no—
166M16×C61—42no—
167C2×C8×C61—21no—
168C2×C8×C61—21no—
169C2×C8×C61—21no—
170C2×C8×C61—42no—
171C2×C8×C61C22×C411yescharacteristic
172C2×C8×C61C22×C411yescharacteristic
173C2×C8×C61C2×D411yescharacteristic
174C2×C8×C61C2×D411yescharacteristic
175C4⋊C4×C61C22×C411yescharacteristic
176C4⋊C4×C61C22×C411yescharacteristic
177C22⋊C4×C61—21no—
178C22⋊C4×C61—21no—
179C42×C61C2411yescharacteristic
180C22⋊Dic61—21no—
181C22⋊Dic61—21no—
182C22⋊Dic61—21no—
183C22⋊Dic61—21no—
184#11758200—22no—
185#11758199—22no—
186#11758198—21no—
187#11758198—21no—
188#11758198—21no—
189C4⋊Dic61—21no—
190C4⋊Dic61—21no—
191C4⋊Dic61C22×C411yescharacteristic
192C4⋊Dic61C22×C411yescharacteristic
193#11758196—21no—
194#11758196—21no—
195#11758196—21no—
196C4×Dic61—21no—
197C4×Dic61C22×C411yescharacteristic
198C4×Dic61C2×D411yescharacteristic
199C4×Dic61#4111yescharacteristic
200C2×#92418—21no—
201C2×#92418—21no—
202C2×#92418—21no—
203C2×#92418—42no—
204C2×#92418—42no—
205C2×#92418C22×C411yescharacteristic
206C2×#92418C22×C411yescharacteristic
207C2×#92418C2×D411yescharacteristic
208C2×#92418C2×D411yescharacteristic
209#11758190—22no—
210#11758189—24no—
211C8×D61—22no—
212C8×D61—41no—
213C8×D61—42no—
214C8×D61—42no—
215C8×D61—42no—
216C23×C61—21no—
217C23×C61—21no—
218C22×D61—24no—
219C22×D61—21no—
220C22×D61—42no—
221C22×D61—42no—
222C22×D61C4×D411yescharacteristic
223D4×C61—21no—
224D4×C61—21no—
225D4×C61—21no—
226D4×C61#13112yes—
227C2×C4×C61—21no—
228C2×C4×C61—21no—
229C2×C4×C61—41no—
230C2×C4×C61—41no—
231C2×C4×C61—41no—
232C2×C4×C61—41no—
233C2×C4×C61C23×C411yescharacteristic
234C2×C4×C61C22×D411yescharacteristic
235C2×C4×C61C2×#4111yescharacteristic
236C22⋊D61—24no—
237C22⋊D61—42no—
238C2×Dic61—21no—
239C2×Dic61—21no—
240C2×Dic61—21no—
241C2×Dic61—42no—
242C2×Dic61C4×D411yescharacteristic
243C2×Dic61C4×D411yescharacteristic
244C2×Dic61C4×D411yescharacteristic
245D244—21no—
246D244#13112yes—
247C4×D61—22no—
248C4×D61—21no—
249C4×D61—21no—
250C4×D61—21no—
251C4×D61—21no—
252C4×D61—42no—
253C4×D61—42no—
254C4×D61—41no—
255C4×D61—41no—
256Dic122—21no—
257Dic122—21no—
258C8×C61—22no—
259C8×C61—21no—
260C8×C61—21no—
261C8×C61—21no—
262C8×C61—21no—
263C8×C61—41no—
264C8×C61—41no—
265#92418—22no—
266#92418—21no—
267#92418—21no—
268#92418—21no—
269#92418—41no—
270#92418—41no—
271#92418AGL(1,ℤ/8)12yes—
272#17910—611nomaximal, Sylow
273C22×C61—22no—
274C22×C61—41no—
275C22×C61—41no—
276C22×C61—41no—
277C22×C61—41no—
278C22×C61—41no—
279C22×C61C2×C4×D411yescharacteristic
280C2×D61—21no—
281C2×D61—21no—
282C2×D61—22no—
283C2×D61—48no—
284C2×D61—41no—
285C2×D61—41no—
286C2×D61#44312yes—
287C4×C61—21no—
288C4×C61—41no—
289C4×C61—41no—
290C4×C61C2×#13111yescharacteristic
291C4×C61C2×#13111yescharacteristic
292C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
293C4×C61#57511yescharacteristic
294Dic61—21no—
295Dic61—21no—
296Dic61—21no—
297Dic61—41no—
298Dic61—41no—
299Dic61—41no—
300#3105—611no—
301#2966—611no—
302C2×C8×D4—611no—
303#1586—611no—
304#1586D6111yescharacteristic, complemented
305#1585—611no—
306#1585—611no—
307C8×SD16—611no—
308C8×SD16—611no—
309C8×D8—611no—
310C8×D8—611no—
311#1573—611no—
312#1490—611no—
313#1483—611no—
314C8×M16—611no—
315C2×C61—42no—
316C2×C61—81no—
317C2×C61#292911yescharacteristic
318C2×C61C4×AGL(1,ℤ/8)11yescharacteristic
319C2×C61C2×#44311yescharacteristic
320D61—22no—
321D61—42no—
322C23×C8—1221no—
323#529—611no—
324C2×C4×D4—611no—
325#484—611no—
326#481—611no—
327#478—611no—
328#478—611no—
329#463—611no—
330#463—611no—
331#450—611no—
332#447—611no—
333#446—611no—
334C8×Q8—611no—
335C4×SD16—611no—
336C4×SD16—611no—
337C4×D8—611no—
338C4×D8C2×D6111yescharacteristic
339#436C2×D6111yescharacteristic, complemented
340#435—1221no—
341C8×D4—611no—
342C8×D4—611no—
343C8×D4—611no—
344C8×D4—611no—
345C8×D4—1221no—
346C8×D4—1221no—
347C8×D4—1221no—
348C8×D4C2×D6111yescharacteristic
349#433—1221no—
350#432—611no—
351#431—611no—
352C2×C4⋊C8—611no—
353C2×C22⋊C8—1221no—
354#405—1221no—
355C4×M16—611no—
356C2×C4×C8—611no—
357#335—611no—
358#335—611no—
359#334—611no—
360#334C2×D6111yescharacteristic, complemented
361Q8⋊C8—611no—
362Q8⋊C8—611no—
363D4⋊C8—611no—
364D4⋊C8—611no—
365D4⋊C8—611no—
366D4⋊C8—611no—
367D4⋊C8C2×D6111yescharacteristic
368D4⋊C8C2×D6111yescharacteristic, complemented
369#322—611no—
370#322C2×D6111yescharacteristic, complemented
371C82—611no—
372C82—611no—
373C61#1791011yescharacteristic, complemented, Sylow
374C22×D4—611no—
375C23×C4—1221no—
376C2×SD16—611no—
377C2×SD16—611no—
378C2×D8—611no—
379C2×D8C4×D6111yescharacteristic
380C2×M16—21no—
381C2×M16—1221no—
382C22×C8—21no—
383C22×C8—611no—
384C22×C8—611no—
385C22×C8—1221no—
386C22×C8—1221no—
387C22×C8—1221no—
388C22×C8—1221no—
389C22×C8—1221no—
390C22×C8—2444no—
391#126—1221no—
392#125—611no—
393#124—611no—
394#123—1221no—
395C22⋊Q8—611no—
396#121—611no—
397C4×Q8—611no—
398C4×D4—611no—
399C4×D4—611no—
400C4×D4—611no—
401C4×D4—1221no—
402C4×D4—1221no—
403C4×D4—1221no—
404C4×D4C22×D6111yescharacteristic
405C4×D4C22×D6111yescharacteristic
406#117—1221no—
407C2×C4⋊C4—611no—
408C2×C22⋊C4—1221no—
409C2×C42—611no—
410#107—611no—
411#107C4×D6111yescharacteristic
412#106—611no—
413#106—611no—
414C4⋊C8—611no—
415C4⋊C8—611no—
416C4⋊C8—611no—
417C4⋊C8—611no—
418C4⋊C8—1221no—
419C4⋊C8—1221no—
420C4⋊C8—1221no—
421C4⋊C8C22×D6111yescharacteristic
422C4⋊C8C22×D6111yescharacteristic
423#103—611no—
424#103—611no—
425#102—611no—
426#102—611no—
427#102—611no—
428#102—611no—
429#102C4×D6111yescharacteristic
430#102C4×D6111yescharacteristic
431C22⋊C8—21no—
432C22⋊C8—21no—
433C22⋊C8—611no—
434C22⋊C8—611no—
435C22⋊C8—1221no—
436C22⋊C8—1221no—
437C22⋊C8—1221no—
438C22⋊C8—1221no—
439C22⋊C8—1221no—
440#97—1221no—
441#97—1221no—
442#97—1221no—
443C4×C8—611no—
444C4×C8—611no—
445C4×C8—611no—
446C4×C8—611no—
447C4×C8—611no—
448C4×C8—1221no—
449C4×C8C22×D6111yescharacteristic
450C4×C8C22×D6111yescharacteristic
451C4×C8C22×D6111yescharacteristic
452C24—1221no—
453C2×Q8—611no—
454C2×D4—611no—
455C2×D4—1221no—
456C2×D4—1222no—
457C2×D4—1222no—
458C2×D4—1221no—
459C2×D4C2×C4×D6111yescharacteristic
460C2×D4C2×C4×D6111yescharacteristic
461C22×C4—21no—
462C22×C4—21no—
463C22×C4—611no—
464C22×C4—611no—
465C22×C4—611no—
466C22×C4—1221no—
467C22×C4—1221no—
468C22×C4—1221no—
469C22×C4—1221no—
470C22×C4—2442no—
471C22×C4—2442no—
472SD16—1222no—
473SD16—1222no—
474D8—22no—
475D8—1222no—
476M16—42no—
477M16—1224no—
478C2×C8—21no—
479C2×C8—21no—
480C2×C8—21no—
481C2×C8—42no—
482C2×C8—611no—
483C2×C8—611no—
484C2×C8—611no—
485C2×C8—611no—
486C2×C8—1222no—
487C2×C8—1221no—
488C2×C8—1221no—
489C2×C8—1221no—
490C2×C8—2441no—
491C2×C8—2442no—
492C2×C8—2442no—
493C2×C8—2442no—
494C2×C8—2442no—
495C2×C8—2442no—
496C2×C8C2×C4×D6111yescharacteristic
497C2×C8C2×C4×D6111yescharacteristic
498C2×C8D4×D6111yescharacteristic
499C2×C8D4×D6111yescharacteristic
500C4⋊C4—611no—
501C4⋊C4—611no—
502C4⋊C4—1221no—
503C4⋊C4—1221no—
504C4⋊C4—1221no—
505C4⋊C4—1221no—
506C4⋊C4—1221no—
507C4⋊C4C2×C4×D6111yescharacteristic
508C4⋊C4C2×C4×D6111yescharacteristic
509C22⋊C4—21no—
510C22⋊C4—21no—
511C22⋊C4—1221no—
512C22⋊C4—1221no—
513C22⋊C4—1221no—
514C22⋊C4—1221no—
515C22⋊C4—1221no—
516C22⋊C4—1221no—
517C22⋊C4—1221no—
518C42—611no—
519C42—611no—
520C42—611no—
521C42—1221no—
522C42C23×D6111yescharacteristic
523C23—21no—
524C23—21no—
525C23—611no—
526C23—1224no—
527C23—1221no—
528C23—2442no—
529C23—2442no—
530Q8—1221no—
531Q8—1221no—
532D4—21no—
533D4—21no—
534D4—21no—
535D4—612no—
536D4—1224no—
537D4—1221no—
538D4—2442no—
539D4#1451978512yes—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—611no—
547C2×C4—611no—
548C2×C4—611no—
549C2×C4—1222no—
550C2×C4—1221no—
551C2×C4—1221no—
552C2×C4—1221no—
553C2×C4—1221no—
554C2×C4—1221no—
555C2×C4—1221no—
556C2×C4—1221no—
557C2×C4—2442no—
558C2×C4—2442no—
559C2×C4—2442no—
560C2×C4—2441no—
561C2×C4—2441no—
562C2×C4C22×C4×D6111yescharacteristic
563C2×C4C2×D4×D6111yescharacteristic
564C2×C4#41×D6111yescharacteristic
565C8—22no—
566C8—21no—
567C8—21no—
568C8—21no—
569C8—21no—
570C8—41no—
571C8—41no—
572C8—612no—
573C8—1222no—
574C8—1221no—
575C8—1221no—
576C8—1221no—
577C8—2441no—
578C8—2441no—
579C22—22no—
580C22—41no—
581C22—41no—
582C22—41no—
583C22—41no—
584C22—41no—
585C22—612no—
586C22—1221no—
587C22—1221no—
588C22—1222no—
589C22—2448no—
590C22—2441no—
591C22—2441no—
592C22C4×D4×D6111yescharacteristic
593C4—21no—
594C4—41no—
595C4—41no—
596C4—1221no—
597C4—1221no—
598C4—1221no—
599C4—2441no—
600C4—2441no—
601C4—2441no—
602C4#131×D6111yescharacteristic
603C4C2×#1451978511yescharacteristic
604C4AGL(1,ℤ/8)×D6111yescharacteristic
605C4#1684055811yescharacteristic
606C2—42no—
607C2—81no—
608C2—1222no—
609C2—2442no—
610C2#2145451011yescharacteristic
611C2#2145463011yescharacteristic
612C2#443×D6111yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC42C23×D61
FittingC61×#1586C2
socleC22×C61C2×C4×D4
prime radical#16839894C4
cosocleC42×C61C24
hypercenter#1586D61
hypocenterC61#17910
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C4×C8 ◁ #1586

Complements

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

NH ≅ G/Ntype
2. #21454797C2semidirect
3. C8×D4×D61C2semidirect
4. #21444681C2semidirect
5. #21444678C2semidirect
6. #21444676C2semidirect
7. #21444622C2semidirect
8. #21444618C2semidirect
9. D8×#92418C2semidirect
10. #21444611C2semidirect
11. #21444601C2semidirect
12. #21444260C2semidirect
13. #21444229C2semidirect
14. #21444098C2semidirect
15. C61×#1586C2semidirect
27. #16839892C22semidirect
28. #16839886C22semidirect
35. D4×#92418C22semidirect
48. #16839515C22semidirect
49. #16839514C22semidirect
50. #16839512C22semidirect
51. #16839494C22semidirect
52. #16839492C22semidirect
55. D4⋊#92418C22semidirect
57. #16839484C22semidirect
58. C8×#92418C22semidirect
59. C8×#92418C22semidirect
61. C61×#436C22semidirect
63. C61×#334C22semidirect
65. C61×D4⋊C8C22semidirect
66. C61×#322C22semidirect
304. #1586D61semidirect
339. #436C2×D61semidirect
360. #334C2×D61semidirect
368. D4⋊C8C2×D61semidirect
370. #322C2×D61semidirect
373. C61#17910semidirect

Maximal subgroups

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

subgroupindex
#214558642
#214547972
C8×D4×D612
#214446812
#214446782
#214446762
#214446222
#214446182
D8×#924182
#214446112
#214446012
#214442602
#214442292
#214440982
C61×#15862
#1791061