Finite Group Databasebeta

← #29650022

Subgroups — #29650022

Order 15616.

712 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#21455870C211yesmaximal, characteristic
2#21454796C211yesmaximal, characteristic, complemented
3C8×D4×D61C211yesmaximal, characteristic, complemented
4#21444658C211yesmaximal, characteristic, complemented
5#21444656C211yesmaximal, characteristic, complemented
6#21444647C211yesmaximal, characteristic, complemented
7#21444645C211yesmaximal, characteristic, complemented
8#21444644C211yesmaximal, characteristic, complemented
9#21444641C211yesmaximal, characteristic, complemented
10#21444629C211yesmaximal, characteristic, complemented
11#21444603C211yesmaximal, characteristic, complemented
12#21444260C211yesmaximal, characteristic, complemented
13#21444231C211yesmaximal, characteristic, complemented
14#21444107C211yesmaximal, characteristic, complemented
15C61×SD16⋊C8C211yesmaximal, characteristic, complemented
16C22×C8×D61—21no—
17#16840299C2211yescharacteristic
18C4×D4×D61C2211yescharacteristic
19#16840096C411yescharacteristic
20#16840082C411yescharacteristic
21#16840061C411yescharacteristic
22#16840060C411yescharacteristic
23#16839980C411yescharacteristic
24#16839979C411yescharacteristic
25#16839911C411yescharacteristic
26#16839897C411yescharacteristic
27#16839893C2211yescharacteristic, complemented
28C4×#11758201C2211yescharacteristic
29C4×D488C2211yescharacteristic
30C4×#11758199C2211yescharacteristic
31#16839839C2211yescharacteristic, complemented
32#16839832—21no—
33C8×D244—21no—
34C8×D244C2211yescharacteristic
35C8×D244C2211yescharacteristic, complemented
36D4×#92418C2211yescharacteristic
37#16839827C2211yescharacteristic
38C8×C22⋊D61—21no—
39#16839825—21no—
40#16839822—21no—
41#16839818C2211yescharacteristic
42#16839813C2211yescharacteristic
43C4⋊C8×D61C2211yescharacteristic
44C22⋊C8×D61—21no—
45#16839716—21no—
46C4×#11758189C2211yescharacteristic
47C4×C8×D61C2211yescharacteristic
48#16839514C2211yescharacteristic, complemented
49#16839513C2211yescharacteristic, complemented
50#16839512C2211yescharacteristic, complemented
51Q8⋊#92418C2211yescharacteristic, complemented
52#16839491C2211yescharacteristic
53#16839491C2211yescharacteristic, complemented
54D4⋊#92418C2211yescharacteristic
55#16839489C2211yescharacteristic
56#16839489C2211yescharacteristic, complemented
57#16839484C2211yescharacteristic, complemented
58#16839484C2211yescharacteristic, complemented
59#16839484C2211yescharacteristic, complemented
60C8×Q8×C61C2211yescharacteristic, complemented
61C4×SD16×C61C2211yescharacteristic
62C8×D4×C61C2211yescharacteristic
63C61×#335C2211yescharacteristic, complemented
64C61×Q8⋊C8C2211yescharacteristic, complemented
65C61×D4⋊C8C2211yescharacteristic
66C61×#322C2211yescharacteristic, complemented
67C2×D4×D61C2×C411yescharacteristic
68C22×C4×D61—21no—
69C2×#11758201C2×C411yescharacteristic
70C2×D488C2×C411yescharacteristic
71C2×#11758199C2×C411yescharacteristic
72C2×#11758189—21no—
73C2×#11758189—21no—
74C22×#92418—21no—
75C2×C8×D61—21no—
76C2×C8×D61—21no—
77C2×C8×D61—21no—
78C2×C8×D61—21no—
79C2×C8×D61—44no—
80C2×C8×D61D411yescharacteristic
81C2×C8×D61D411yescharacteristic
82#14519772—21no—
83#14519768C2×C411yescharacteristic
84#14519766C2×C411yescharacteristic
85#14519762—21no—
86#14519753C2×C411yescharacteristic
87#14519752C2×C411yescharacteristic
88C4×D244—21no—
89C4×D244C2311yescharacteristic
90C4×D244C2311yescharacteristic
91D4×Dic61C2×C411yescharacteristic
92#14519739C2×C411yescharacteristic
93C4×C22⋊D61—21no—
94#14519737—21no—
95#14519734—21no—
96C4⋊C4×D61C2×C411yescharacteristic
97C22⋊C4×D61—21no—
98C42×D61C2311yescharacteristic
99#14519711C2×C411yescharacteristic
100#14519710C2×C411yescharacteristic
101#14519708C2×C411yescharacteristic
102C4⋊#92418—21no—
103C4⋊#92418C2311yescharacteristic
104C4⋊#92418C2311yescharacteristic
105#14519706—21no—
106#14519706C2×C411yescharacteristic
107#14519706C2×C411yescharacteristic
108#14519701C2×C411yescharacteristic
109#14519699C2×C411yescharacteristic
110#14519699C2×C411yescharacteristic
111#14519698C2×C411yescharacteristic
112#14519697C2×C411yescharacteristic
113#14519697C2×C411yescharacteristic
114C22⋊#92418—21no—
115#14519689—21no—
116#14519689—21no—
117#14519689—21no—
118#14519689—21no—
119#14519689—21no—
120#14519689C2×C411yescharacteristic
121#14519689C2×C411yescharacteristic
122#14519688—21no—
123#14519687—21no—
124#14519687—21no—
125C4×#92418C2311yescharacteristic
126C4×#92418C2311yescharacteristic
127C4×#92418C2311yescharacteristic
128C8×Dic61D411yescharacteristic
129C8×Dic61D411yescharacteristic
130C2×SD16×C61C2×C411yescharacteristic
131C22×C8×C61—21no—
132C4×Q8×C61C2311yescharacteristic
133C4×D4×C61C2311yescharacteristic
134#106×C61C2×C411yescharacteristic
135C4⋊C8×C61—21no—
136C4⋊C8×C61C2311yescharacteristic
137C4⋊C8×C61C2311yescharacteristic
138#103×C61C2×C411yescharacteristic
139#102×C61C2×C411yescharacteristic
140C22⋊C8×C61—21no—
141C4×C8×C61—21no—
142C4×C8×C61C2311yescharacteristic
143C4×C8×C61C2311yescharacteristic
144C4×C8×C61C2311yescharacteristic
145C23×D61—21no—
146C2×Q8×C61C22×C411yescharacteristic
147C2×D4×C61C22×C411yescharacteristic
148C22×C4×C61—21no—
149C2×C22⋊D61—21no—
150C22×Dic61—21no—
151D4×D61—22no—
152D4×D61—22no—
153C2×D244—21no—
154C2×D244C22×C411yescharacteristic
155C2×D244C22×C411yescharacteristic
156C2×C4×D61—21no—
157C2×C4×D61—21no—
158C2×C4×D61—21no—
159C2×C4×D61—21no—
160C2×C4×D61—42no—
161C2×C4×D61—42no—
162C2×C4×D61C22×C411yescharacteristic
163C2×C4×D61C2×D411yescharacteristic
164C2×C4×D61#4111yescharacteristic
165SD16×C61—22no—
166C2×C8×C61—21no—
167C2×C8×C61—21no—
168C2×C8×C61—21no—
169C2×C8×C61—42no—
170C2×C8×C61C22×C411yescharacteristic
171C2×C8×C61C22×C411yescharacteristic
172C2×C8×C61C2×D411yescharacteristic
173C2×C8×C61C2×D411yescharacteristic
174C4⋊C4×C61—21no—
175C4⋊C4×C61C22×C411yescharacteristic
176C4⋊C4×C61C22×C411yescharacteristic
177C22⋊C4×C61—21no—
178C42×C61—21no—
179C42×C61C2411yescharacteristic
180C22⋊Dic61—21no—
181#11758201—22no—
182#11758199—22no—
183#11758198—21no—
184#11758198—21no—
185#11758198—21no—
186#11758198—21no—
187#11758198—21no—
188#11758198—21no—
189#11758198—21no—
190C4⋊Dic61—21no—
191C4⋊Dic61C22×C411yescharacteristic
192C4⋊Dic61C22×C411yescharacteristic
193#11758196—21no—
194#11758196—21no—
195#11758196—21no—
196C4×Dic61C22×C411yescharacteristic
197C4×Dic61C2×D411yescharacteristic
198C4×Dic61#4111yescharacteristic
199C2×#92418—21no—
200C2×#92418—21no—
201C2×#92418—21no—
202C2×#92418—42no—
203C2×#92418C22×C411yescharacteristic
204C2×#92418C22×C411yescharacteristic
205C2×#92418C2×D411yescharacteristic
206C2×#92418C2×D411yescharacteristic
207D488—22no—
208#11758189—42no—
209#11758189—42no—
210C8×D61—22no—
211C8×D61—41no—
212C8×D61—42no—
213C8×D61—42no—
214C8×D61—42no—
215C8×D61—42no—
216C23×C61—21no—
217C22×D61—24no—
218C22×D61—21no—
219C22×D61—21no—
220C22×D61—42no—
221C22×D61—42no—
222C22×D61C4×D411yescharacteristic
223Q8×C61—21no—
224Q8×C61—21no—
225D4×C61—21no—
226D4×C61#13112yes—
227C2×C4×C61—21no—
228C2×C4×C61—21no—
229C2×C4×C61—21no—
230C2×C4×C61—41no—
231C2×C4×C61—41no—
232C2×C4×C61C23×C411yescharacteristic
233C2×C4×C61C22×D411yescharacteristic
234C2×C4×C61C2×#4111yescharacteristic
235C22⋊D61—24no—
236C2×Dic61—21no—
237C2×Dic61—21no—
238C2×Dic61—42no—
239C2×Dic61C4×D411yescharacteristic
240C2×Dic61C4×D411yescharacteristic
241C2×Dic61C4×D411yescharacteristic
242D244—21no—
243D244—21no—
244D244—21no—
245D244—42no—
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—
256C4×D61—41no—
257C4×D61—41no—
258C8×C61—21no—
259C8×C61—21no—
260C8×C61—21no—
261C8×C61—21no—
262C8×C61—41no—
263C8×C61—41no—
264C8×C61—41no—
265#92418—21no—
266#92418—21no—
267#92418—21no—
268#92418—41no—
269#92418—41no—
270#92418—41no—
271#92418AGL(1,ℤ/8)12yes—
272#17914—611nomaximal, Sylow
273C22×C61—22no—
274C22×C61—41no—
275C22×C61—41no—
276C22×C61C2×C4×D411yescharacteristic
277C2×D61—21no—
278C2×D61—21no—
279C2×D61—22no—
280C2×D61—48no—
281C2×D61—41no—
282C2×D61—41no—
283C2×D61—41no—
284C2×D61—41no—
285C2×D61—41no—
286C2×D61#44412yes—
287C4×C61—21no—
288C4×C61—41no—
289C4×C61—41no—
290C4×C61—41no—
291C4×C61C2×#13111yescharacteristic
292C4×C61C2×#13111yescharacteristic
293C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
294C4×C61#57511yescharacteristic
295Dic61—21no—
296Dic61—21no—
297Dic61—21no—
298Dic61—41no—
299Dic61—41no—
300#3105—611no—
301#2966—611no—
302C2×C8×D4—611no—
303#1582—611no—
304#1582—611no—
305#1581—611no—
306#1581—611no—
307#1580—611no—
308#1580—611no—
309SD16⋊C8—611no—
310SD16⋊C8D6111yescharacteristic, complemented
311#1573—611no—
312#1490—611no—
313#1483—611no—
314#1451—611no—
315C2×C61—42no—
316C2×C61#292911yescharacteristic
317C2×C61C4×AGL(1,ℤ/8)11yescharacteristic
318C2×C61C2×#44411yescharacteristic
319D61—22no—
320D61—42no—
321D61—81no—
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×Q8C2×D6111yescharacteristic, complemented
335C4×SD16—611no—
336C4×SD16C2×D6111yescharacteristic
337C4×D8—611no—
338C4×D8—611no—
339#436—611no—
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#335C2×D6111yescharacteristic, complemented
359#334—611no—
360#334—611no—
361Q8⋊C8—611no—
362Q8⋊C8C2×D6111yescharacteristic, complemented
363D4⋊C8—611no—
364D4⋊C8—611no—
365D4⋊C8—611no—
366D4⋊C8—611no—
367D4⋊C8—611no—
368D4⋊C8C2×D6111yescharacteristic
369#322—611no—
370#322—611no—
371#322—611no—
372#322C2×D6111yescharacteristic, complemented
373C61#1791411yescharacteristic, complemented, Sylow
374C22×D4—611no—
375C23×C4—1221no—
376C2×SD16—611no—
377C2×SD16C4×D6111yescharacteristic
378C2×D8—611no—
379C2×D8—611no—
380C2×M16—1221no—
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×Q8C22×D6111yescharacteristic
398C4×D4—611no—
399C4×D4—611no—
400C4×D4—611no—
401C4×D4—611no—
402C4×D4—1221no—
403C4×D4—1221no—
404C4×D4—1221no—
405C4×D4C22×D6111yescharacteristic
406#117—1221no—
407C2×C4⋊C4—611no—
408C2×C22⋊C4—1221no—
409C2×C42—611no—
410#107—611no—
411#107—611no—
412#106—611no—
413#106C4×D6111yescharacteristic
414C4⋊C8—21no—
415C4⋊C8—611no—
416C4⋊C8—611no—
417C4⋊C8—611no—
418C4⋊C8—611no—
419C4⋊C8—1221no—
420C4⋊C8—1221no—
421C4⋊C8C22×D6111yescharacteristic
422C4⋊C8C22×D6111yescharacteristic
423#103—611no—
424#103C4×D6111yescharacteristic
425#102—611no—
426#102—611no—
427#102—611no—
428#102—611no—
429#102—611no—
430#102C4×D6111yescharacteristic
431C22⋊C8—21no—
432C22⋊C8—611no—
433C22⋊C8—611no—
434C22⋊C8—1221no—
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—21no—
444C4×C8—611no—
445C4×C8—611no—
446C4×C8—611no—
447C4×C8—611no—
448C4×C8—611no—
449C4×C8C22×D6111yescharacteristic
450C4×C8C22×D6111yescharacteristic
451C4×C8C22×D6111yescharacteristic
452C24—1221no—
453C2×Q8C2×C4×D6111yescharacteristic
454C2×D4—611no—
455C2×D4—611no—
456C2×D4—1221no—
457C2×D4—1221no—
458C2×D4—1222no—
459C2×D4—1222no—
460C2×D4C2×C4×D6111yescharacteristic
461C22×C4—21no—
462C22×C4—611no—
463C22×C4—611no—
464C22×C4—611no—
465C22×C4—1221no—
466C22×C4—1221no—
467C22×C4—1221no—
468C22×C4—1221no—
469C22×C4—1221no—
470C22×C4—2442no—
471C22×C4—2442no—
472SD16—22no—
473SD16—1222no—
474D8—1222no—
475D8—1222no—
476M16—2442no—
477M16—2442no—
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—21no—
501C4⋊C4—611no—
502C4⋊C4—611no—
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—1221no—
511C22⋊C4—1221no—
512C22⋊C4—1221no—
513C22⋊C4—1221no—
514C22⋊C4—1221no—
515C22⋊C4—1221no—
516C22⋊C4—1221no—
517C22⋊C4—1221no—
518C42—21no—
519C42—611no—
520C42—611no—
521C42—611no—
522C42C23×D6111yescharacteristic
523C23—21no—
524C23—611no—
525C23—1224no—
526C23—1221no—
527C23—1221no—
528C23—2442no—
529C23—2442no—
530Q8—21no—
531Q8—21no—
532D4—21no—
533D4—612no—
534D4—1224no—
535D4—1221no—
536D4—1221no—
537D4—1221no—
538D4—2442no—
539D4#1451978312yes—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—611no—
546C2×C4—611no—
547C2×C4—611no—
548C2×C4—1222no—
549C2×C4—1221no—
550C2×C4—1221no—
551C2×C4—1221no—
552C2×C4—1221no—
553C2×C4—1221no—
554C2×C4—1221no—
555C2×C4—2442no—
556C2×C4—2442no—
557C2×C4—2442no—
558C2×C4—2441no—
559C2×C4—2441no—
560C2×C4—2441no—
561C2×C4—2441no—
562C2×C4C22×C4×D6111yescharacteristic
563C2×C4C2×D4×D6111yescharacteristic
564C2×C4#41×D6111yescharacteristic
565C8—21no—
566C8—21no—
567C8—21no—
568C8—21no—
569C8—41no—
570C8—41no—
571C8—41no—
572C8—612no—
573C8—1221no—
574C8—1221no—
575C8—1221no—
576C8—2441no—
577C8—2441no—
578C8—2441no—
579C22—22no—
580C22—41no—
581C22—41no—
582C22—612no—
583C22—1221no—
584C22—1221no—
585C22—1222no—
586C22—2448no—
587C22—2441no—
588C22—2441no—
589C22—2441no—
590C22—2441no—
591C22—2441no—
592C22C4×D4×D6111yescharacteristic
593C4—21no—
594C4—41no—
595C4—41no—
596C4—41no—
597C4—1221no—
598C4—1221no—
599C4—1221no—
600C4—2441no—
601C4—2441no—
602C4#131×D6111yescharacteristic
603C4C2×#1451978311yescharacteristic
604C4AGL(1,ℤ/8)×D6111yescharacteristic
605C4#1684056111yescharacteristic
606C2—42no—
607C2—1222no—
608C2—4881no—
609C2—2442no—
610C2#2145450811yescharacteristic
611C2#2145462411yescharacteristic
612C2#444×D6111yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC42C23×D61
FittingC61×SD16⋊C8C2
socleC22×C61C2×C4×D4
prime radical#16839897C4
cosocleC42×C61C24
hypercenterSD16⋊C8D61
hypocenterC61#17914
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C4×C8 ◁ SD16⋊C8

Complements

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

NH ≅ G/Ntype
2. #21454796C2semidirect
3. C8×D4×D61C2semidirect
4. #21444658C2semidirect
5. #21444656C2semidirect
6. #21444647C2semidirect
7. #21444645C2semidirect
8. #21444644C2semidirect
9. #21444641C2semidirect
10. #21444629C2semidirect
11. #21444603C2semidirect
12. #21444260C2semidirect
13. #21444231C2semidirect
14. #21444107C2semidirect
15. C61×SD16⋊C8C2semidirect
27. #16839893C22semidirect
31. #16839839C22semidirect
35. C8×D244C22semidirect
48. #16839514C22semidirect
49. #16839513C22semidirect
50. #16839512C22semidirect
51. Q8⋊#92418C22semidirect
53. #16839491C22semidirect
56. #16839489C22semidirect
57. #16839484C22semidirect
58. #16839484C22semidirect
59. #16839484C22semidirect
60. C8×Q8×C61C22semidirect
63. C61×#335C22semidirect
64. C61×Q8⋊C8C22semidirect
66. C61×#322C22semidirect
310. SD16⋊C8D61semidirect
334. C8×Q8C2×D61semidirect
358. #335C2×D61semidirect
362. Q8⋊C8C2×D61semidirect
372. #322C2×D61semidirect
373. C61#17914semidirect

Maximal subgroups

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

subgroupindex
#214558702
#214547962
C8×D4×D612
#214446582
#214446562
#214446472
#214446452
#214446442
#214446412
#214446292
#214446032
#214442602
#214442312
#214441072
C61×SD16⋊C82
#1791461