Finite Group Databasebeta

← #29651055

Subgroups — #29651055

Order 15616.

670 conjugacy classes of subgroups.

The subgroup lattice has 612 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21455862C211yesmaximal, characteristic
2#21454824C211yesmaximal, characteristic, complemented
3#21454554C211yesmaximal, characteristic, complemented
4#21444730C211yesmaximal, characteristic, complemented
5#21444706C211yesmaximal, characteristic, complemented
6#21444694C211yesmaximal, characteristic, complemented
7#21444686C211yesmaximal, characteristic, complemented
8#21444658C211yesmaximal, characteristic, complemented
9#21444645C211yesmaximal, characteristic, complemented
10#21444634C211yesmaximal, characteristic, complemented
11#21444611C211yesmaximal, characteristic, complemented
12#21444610C211yesmaximal, characteristic, complemented
13#21444211C211yesmaximal, characteristic, complemented
14#21444114C211yesmaximal, characteristic, complemented
15C61×#1495C211yesmaximal, characteristic, complemented
16C2×M16×D61—21no—
17C2×C4×D244C2211yescharacteristic
18#16840095C411yescharacteristic
19#16840078C411yescharacteristic
20#16840059C411yescharacteristic
21#16840059C411yescharacteristic
22#16839974C411yescharacteristic
23#16839974C411yescharacteristic
24#16839912C411yescharacteristic
25#16839895C411yescharacteristic
26#16839893C2211yescharacteristic, complemented
27Q8×#92418C2211yescharacteristic, complemented
28C4×#11758201C2211yescharacteristic
29C4×#11758201C2211yescharacteristic
30C4×#11758199C2211yescharacteristic
31C4×#11758199C2211yescharacteristic
32#16839840—21no—
33#16839840C2211yescharacteristic, complemented
34#16839839C2211yescharacteristic
35#16839839C2211yescharacteristic
36#16839834—21no—
37#16839832—21no—
38#16839832—21no—
39D4×#92418C2211yescharacteristic, complemented
40#16839827C2211yescharacteristic
41#16839827C2211yescharacteristic
42#16839823—21no—
43#16839815C2211yescharacteristic
44#16839815C2211yescharacteristic
45#16839725—21no—
46#16839718—21no—
47C4×#11758194C2211yescharacteristic
48#16839514C2211yescharacteristic, complemented
49#16839514C2211yescharacteristic, complemented
50#16839512C2211yescharacteristic, complemented
51#16839512C2211yescharacteristic, complemented
52#16839491C2211yescharacteristic
53#16839491C2211yescharacteristic
54#16839491C2211yescharacteristic
55#16839491C2211yescharacteristic
56#16839484C2211yescharacteristic, complemented
57#16839484C2211yescharacteristic, complemented
58#16839483C2211yescharacteristic, complemented
59C8×#92418C2211yescharacteristic, complemented
60C61×#529C2211yescharacteristic
61C61×#431C2211yescharacteristic
62C61×#423C2211yescharacteristic
63C61×Q8⋊C8C2211yescharacteristic, complemented
64C61×Q8⋊C8C2211yescharacteristic, complemented
65C61×D4⋊C8C2211yescharacteristic, complemented
66C61×D4⋊C8C2211yescharacteristic, complemented
67C22×D244C2×C411yescharacteristic
68C22×C4×D61—21no—
69C2×#11758201C2×C411yescharacteristic
70C2×#11758201C2×C411yescharacteristic
71C2×#11758199C2×C411yescharacteristic
72C2×#11758199C2×C411yescharacteristic
73C2×#11758194—21no—
74C2×#11758194D411yescharacteristic
75C2×#11758194D411yescharacteristic
76C2×#11758189—21no—
77C2×#11758189—21no—
78M16×D61—44no—
79C22×#92418—21no—
80C2×C8×D61—21no—
81C2×C8×D61—21no—
82C4×D244—21no—
83C4×D244—21no—
84C4×D244C2×C411yescharacteristic
85C4×D244C2×C411yescharacteristic
86C4×D244C2311yescharacteristic
87C4×D244C2311yescharacteristic
88C2×C4⋊Dic61C2×C411yescharacteristic
89C2×#11758198—21no—
90#14519710C2×C411yescharacteristic
91#14519710C2×C411yescharacteristic
92#14519708C2×C411yescharacteristic
93#14519708C2×C411yescharacteristic
94C4⋊#92418—21no—
95C4⋊#92418—21no—
96C4⋊#92418C2×C411yescharacteristic
97C4⋊#92418C2×C411yescharacteristic
98C4⋊#92418C2311yescharacteristic
99C4⋊#92418C2311yescharacteristic
100#14519701C2×C411yescharacteristic
101#14519701C2×C411yescharacteristic
102#14519698C2×C411yescharacteristic
103#14519698C2×C411yescharacteristic
104#14519697C2×C411yescharacteristic
105#14519697C2×C411yescharacteristic
106#14519697C2×C411yescharacteristic
107#14519697C2×C411yescharacteristic
108C22⋊#92418—21no—
109C22⋊#92418—21no—
110C22⋊#92418—21no—
111C22⋊#92418C2×C411yescharacteristic
112C22⋊#92418C2×C411yescharacteristic
113#14519689—21no—
114#14519689—21no—
115#14519689—21no—
116#14519689—21no—
117#14519688—21no—
118#14519688D411yescharacteristic
119#14519688D411yescharacteristic
120C4×#92418—21no—
121C4×#92418—21no—
122C4×#92418C2311yescharacteristic
123C4×#92418C2311yescharacteristic
124C4×#92418C2311yescharacteristic
125C4×#92418C2311yescharacteristic
126C2×M16×C61—21no—
127#126×C61—21no—
128#125×C61C2×C411yescharacteristic
129#124×C61C2×C411yescharacteristic
130#123×C61—21no—
131C22⋊Q8×C61C2×C411yescharacteristic
132#121×C61C2×C411yescharacteristic
133C4×Q8×C61C2311yescharacteristic
134C4×D4×C61—21no—
135C4×D4×C61C2311yescharacteristic
136#117×C61—21no—
137C2×C42×C61C2311yescharacteristic
138C4⋊C8×C61—21no—
139C4⋊C8×C61—21no—
140C4⋊C8×C61C2311yescharacteristic
141C4⋊C8×C61C2311yescharacteristic
142C22⋊C8×C61—21no—
143C4×C8×C61C2311yescharacteristic
144C4×C8×C61C2311yescharacteristic
145C23×D61—21no—
146C2×Q8×C61C22×C411yescharacteristic
147C2×D4×C61—21no—
148C2×D4×C61C22×C411yescharacteristic
149C22×C4×C61—21no—
150C22×C4×C61C22×C411yescharacteristic
151C22×C4×C61C2×D411yescharacteristic
152C22×C4×C61#4111yescharacteristic
153C22×Dic61—21no—
154C2×D244—22no—
155C2×D244—21no—
156C2×D244—41no—
157C2×D244C22×C411yescharacteristic
158C2×D244C22×C411yescharacteristic
159C2×C4×D61—21no—
160C2×C4×D61—21no—
161C2×C4×D61—21no—
162C2×C4×D61—21no—
163C2×C4×D61—42no—
164C2×C4×D61—42no—
165M16×C61—42no—
166C2×C8×C61—21no—
167C2×C8×C61—21no—
168C2×C8×C61—21no—
169C2×C8×C61—21no—
170C4⋊C4×C61—21no—
171C4⋊C4×C61—21no—
172C4⋊C4×C61—21no—
173C4⋊C4×C61—21no—
174C4⋊C4×C61C22×C411yescharacteristic
175C4⋊C4×C61C22×C411yescharacteristic
176C22⋊C4×C61—21no—
177C22⋊C4×C61—21no—
178C22⋊C4×C61—21no—
179C22⋊C4×C61—21no—
180C22⋊C4×C61—21no—
181C42×C61—21no—
182C42×C61C22×C411yescharacteristic
183C42×C61C2×D411yescharacteristic
184C42×C61#4111yescharacteristic
185C42×C61C2411yescharacteristic
186#11758201—22no—
187#11758201—41no—
188#11758199—22no—
189#11758199—41no—
190#11758198—21no—
191#11758198—21no—
192#11758198—21no—
193#11758198—21no—
194C4⋊Dic61—21no—
195C4⋊Dic61C22×C411yescharacteristic
196C4⋊Dic61C22×C411yescharacteristic
197#11758194—22no—
198#11758194—41no—
199#11758194—42no—
200C2×#92418—21no—
201C2×#92418—21no—
202C2×#92418—42no—
203C2×#92418C22×C411yescharacteristic
204C2×#92418C22×C411yescharacteristic
205C2×#92418C22×C411yescharacteristic
206C2×#92418C22×C411yescharacteristic
207C2×#92418C2×D411yescharacteristic
208C2×#92418C2×D411yescharacteristic
209C2×#92418C2×D411yescharacteristic
210C2×#92418C2×D411yescharacteristic
211#11758189—42no—
212#11758189—42no—
213C8×D61—42no—
214C8×D61—42no—
215C23×C61—21no—
216C23×C61C4×D411yescharacteristic
217C22×D61—21no—
218C22×D61—21no—
219C22×D61—42no—
220C22×D61—42no—
221C22×D61—81no—
222Q8×C61—21no—
223Q8×C61—21no—
224D4×C61—21no—
225D4×C61—21no—
226D4×C61—81no—
227C2×C4×C61—21no—
228C2×C4×C61—21no—
229C2×C4×C61—21no—
230C2×C4×C61—21no—
231C2×C4×C61—21no—
232C2×C4×C61—21no—
233C2×C4×C61—21no—
234C2×C4×C61—41no—
235C2×C4×C61—41no—
236C2×C4×C61—41no—
237C2×C4×C61C4×D411yescharacteristic
238C2×C4×C61C4×D411yescharacteristic
239C2×C4×C61C4×D411yescharacteristic
240C2×C4×C61C23×C411yescharacteristic
241C2×C4×C61C22×D411yescharacteristic
242C2×C4×C61C2×#4111yescharacteristic
243C2×Dic61—21no—
244C2×Dic61—21no—
245C2×Dic61—42no—
246D244—21no—
247D244—21no—
248D244—21no—
249D244—42no—
250D244#13112yes—
251C4×D61—42no—
252C4×D61—42no—
253C4×D61—41no—
254C4×D61—41no—
255C4×D61—41no—
256C4×D61—41no—
257C8×C61—41no—
258C8×C61—41no—
259C8×C61—41no—
260C8×C61—41no—
261#92418—21no—
262#92418—21no—
263#92418—21no—
264#92418—21no—
265#92418—21no—
266#92418—21no—
267#92418—21no—
268#92418—41no—
269#92418—41no—
270#92418AGL(1,ℤ/8)12yes—
271#18006—611nomaximal, Sylow
272C22×C61—21no—
273C22×C61—21no—
274C22×C61—21no—
275C22×C61—41no—
276C22×C61—41no—
277C22×C61—41no—
278C22×C61C2×C4×D411yescharacteristic
279C2×D61—22no—
280C2×D61—41no—
281C2×D61—41no—
282C2×D61—41no—
283C2×D61—41no—
284C2×D61—41no—
285C2×D61—84no—
286C4×C61—21no—
287C4×C61—21no—
288C4×C61—21no—
289C4×C61—21no—
290C4×C61—41no—
291C4×C61—41no—
292C4×C61—41no—
293C4×C61C2×#13111yescharacteristic
294C4×C61#56811yescharacteristic
295C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
296C4×C61#57511yescharacteristic
297Dic61—41no—
298Dic61—41no—
299#3105—611no—
300#2969—611no—
301D4×M16—611no—
302#1594—611no—
303#1590—611no—
304#1588—611no—
305#1586—611no—
306#1582—611no—
307#1581—611no—
308SD16⋊C8—611no—
309C8×D8—611no—
310#1574—611no—
311#1495D6111yescharacteristic, complemented
312#1478—611no—
313#1453—611no—
314C2×C61—41no—
315C2×C61—81no—
316C2×C61#293311yescharacteristic
317C2×C61C4×AGL(1,ℤ/8)11yescharacteristic
318C2×C61#295711yescharacteristic
319D61—42no—
320D61—81no—
321C22×M16—1221no—
322#529C2×D6111yescharacteristic
323C2×C4×D4—611no—
324#484—611no—
325#481—611no—
326#478—611no—
327#478—611no—
328#463—611no—
329#463—611no—
330#450—611no—
331#447—611no—
332#446—611no—
333C8×Q8—611no—
334C4×SD16—611no—
335C4×SD16—611no—
336C4×D8—611no—
337C4×D8—611no—
338#436—611no—
339#436—611no—
340#436—611no—
341#436—1221no—
342#435—1221no—
343#435—1221no—
344#435—1221no—
345C8×D4—611no—
346C8×D4—611no—
347C8×D4—611no—
348#433—1221no—
349#431—611no—
350#431—611no—
351#431C2×D6111yescharacteristic
352#423C2×D6111yescharacteristic
353#407—1221no—
354#405—1221no—
355C4×M16—611no—
356#335—611no—
357#335—611no—
358#334—611no—
359#334—611no—
360Q8⋊C8C2×D6111yescharacteristic, complemented
361Q8⋊C8C2×D6111yescharacteristic, complemented
362D4⋊C8—611no—
363D4⋊C8—611no—
364D4⋊C8—611no—
365D4⋊C8—611no—
366D4⋊C8C2×D6111yescharacteristic, complemented
367D4⋊C8C2×D6111yescharacteristic, complemented
368#322—611no—
369#322—611no—
370#322—611no—
371C82—611no—
372C61#1800611yescharacteristic, complemented, Sylow
373C22×D4—611no—
374C23×C4—1221no—
375C2×SD16—611no—
376C2×SD16—611no—
377C2×D8—611no—
378C2×D8—611no—
379C2×M16—21no—
380C2×M16—611no—
381C2×M16—611no—
382C2×M16—1221no—
383C2×M16—1221no—
384C2×M16—1221no—
385C2×M16—2444no—
386C22×C8—1221no—
387C22×C8—1221no—
388C22×C8—1221no—
389#126—21no—
390#125C4×D6111yescharacteristic
391#124C4×D6111yescharacteristic
392#123—21no—
393C22⋊Q8C4×D6111yescharacteristic
394#121C4×D6111yescharacteristic
395C4×Q8C22×D6111yescharacteristic
396C4×D4—21no—
397C4×D4—611no—
398C4×D4—611no—
399C4×D4—611no—
400C4×D4—611no—
401C4×D4—1221no—
402C4×D4—1221no—
403C4×D4C22×D6111yescharacteristic
404#117—21no—
405C2×C4⋊C4—611no—
406C2×C22⋊C4—1221no—
407C2×C42C22×D6111yescharacteristic
408#107—611no—
409#107—611no—
410#106—611no—
411#106—611no—
412C4⋊C8—21no—
413C4⋊C8—21no—
414C4⋊C8—611no—
415C4⋊C8—611no—
416C4⋊C8—611no—
417C4⋊C8—611no—
418C4⋊C8—1221no—
419C4⋊C8—1221no—
420C4⋊C8C22×D6111yescharacteristic
421C4⋊C8C22×D6111yescharacteristic
422#103—611no—
423#103—611no—
424#102—611no—
425#102—611no—
426#102—611no—
427#102—611no—
428#102—611no—
429#102—611no—
430C22⋊C8—21no—
431C22⋊C8—611no—
432C22⋊C8—611no—
433C22⋊C8—1221no—
434C22⋊C8—1221no—
435C22⋊C8—1221no—
436C22⋊C8—1221no—
437C22⋊C8—1221no—
438C22⋊C8—1221no—
439C22⋊C8—1221no—
440#97—611no—
441#97—611no—
442#97—1221no—
443C4×C8—611no—
444C4×C8—611no—
445C4×C8—611no—
446C4×C8—611no—
447C4×C8—1221no—
448C4×C8—1221no—
449C4×C8C22×D6111yescharacteristic
450C4×C8C22×D6111yescharacteristic
451C24—1221no—
452C2×Q8C2×C4×D6111yescharacteristic
453C2×D4—21no—
454C2×D4—611no—
455C2×D4—611no—
456C2×D4—1222no—
457C2×D4—1221no—
458C2×D4—2441no—
459C2×D4C2×C4×D6111yescharacteristic
460C22×C4—21no—
461C22×C4—1221no—
462C22×C4—1221no—
463C22×C4—1221no—
464C22×C4—1221no—
465C22×C4—1221no—
466C22×C4—2442no—
467C22×C4—2442no—
468C22×C4C2×C4×D6111yescharacteristic
469C22×C4D4×D6111yescharacteristic
470C22×C4#1175822411yescharacteristic
471SD16—1222no—
472SD16—2441no—
473D8—1222no—
474D8—2441no—
475M16—42no—
476M16—1222no—
477M16—2442no—
478M16—2442no—
479M16—2441no—
480M16—2442no—
481C2×C8—21no—
482C2×C8—21no—
483C2×C8—21no—
484C2×C8—21no—
485C2×C8—611no—
486C2×C8—611no—
487C2×C8—611no—
488C2×C8—611no—
489C2×C8—611no—
490C2×C8—611no—
491C2×C8—611no—
492C2×C8—611no—
493C2×C8—1221no—
494C2×C8—1221no—
495C2×C8—2442no—
496C2×C8—2442no—
497C2×C8—2442no—
498C4⋊C4—21no—
499C4⋊C4—21no—
500C4⋊C4—21no—
501C4⋊C4—21no—
502C4⋊C4—611no—
503C4⋊C4—611no—
504C4⋊C4—1221no—
505C4⋊C4C2×C4×D6111yescharacteristic
506C4⋊C4C2×C4×D6111yescharacteristic
507C22⋊C4—21no—
508C22⋊C4—21no—
509C22⋊C4—21no—
510C22⋊C4—21no—
511C22⋊C4—21no—
512C22⋊C4—1221no—
513C22⋊C4—1221no—
514C22⋊C4—1221no—
515C22⋊C4—1221no—
516C42—21no—
517C42C2×C4×D6111yescharacteristic
518C42D4×D6111yescharacteristic
519C42#1175822411yescharacteristic
520C42C23×D6111yescharacteristic
521C23—21no—
522C23—1221no—
523C23—1221no—
524C23—4881no—
525C23—2442no—
526C23—2442no—
527C23#1451973711yescharacteristic
528Q8—21no—
529Q8—21no—
530D4—21no—
531D4—21no—
532D4—81no—
533D4—612no—
534D4—1221no—
535D4—1221no—
536D4—1221no—
537D4—2442no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—1221no—
549C2×C4—1221no—
550C2×C4—2442no—
551C2×C4—2442no—
552C2×C4—2442no—
553C2×C4—2441no—
554C2×C4—2441no—
555C2×C4—2441no—
556C2×C4—2441no—
557C2×C4#1451973711yescharacteristic
558C2×C4#1451973711yescharacteristic
559C2×C4#1451973711yescharacteristic
560C2×C4C22×C4×D6111yescharacteristic
561C2×C4C2×D4×D6111yescharacteristic
562C2×C4C2×#1175822411yescharacteristic
563C8—41no—
564C8—41no—
565C8—41no—
566C8—41no—
567C8—612no—
568C8—1221no—
569C8—1221no—
570C8—1221no—
571C8—1221no—
572C8—1221no—
573C8—1221no—
574C8—1221no—
575C8—2441no—
576C8—2441no—
577C22—21no—
578C22—21no—
579C22—21no—
580C22—41no—
581C22—41no—
582C22—41no—
583C22—1222no—
584C22—4884no—
585C22—2441no—
586C22—2441no—
587C22—2441no—
588C22—2441no—
589C22—2441no—
590C22C2×#1451973711yescharacteristic
591C4—21no—
592C4—21no—
593C4—21no—
594C4—21no—
595C4—41no—
596C4—41no—
597C4—41no—
598C4—2441no—
599C4—2441no—
600C4#131×D6111yescharacteristic
601C4#1684052711yescharacteristic
602C4AGL(1,ℤ/8)×D6111yescharacteristic
603C4#1684056111yescharacteristic
604C2—41no—
605C2—81no—
606C2—4881no—
607C2—2442no—
608C2#2145454911yescharacteristic
609C2#2145463211yescharacteristic
610C2#2145473211yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC42C23×D61
FittingC61×#1495C2
socleC22×C61C2×C4×D4
prime radical#16839895C4
cosocleC42×C61C24
hypercenter#1495D61
hypocenterC61#18006
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1495

Complements

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

NH ≅ G/Ntype
2. #21454824C2semidirect
3. #21454554C2semidirect
4. #21444730C2semidirect
5. #21444706C2semidirect
6. #21444694C2semidirect
7. #21444686C2semidirect
8. #21444658C2semidirect
9. #21444645C2semidirect
10. #21444634C2semidirect
11. #21444611C2semidirect
12. #21444610C2semidirect
13. #21444211C2semidirect
14. #21444114C2semidirect
15. C61×#1495C2semidirect
26. #16839893C22semidirect
27. Q8×#92418C22semidirect
33. #16839840C22semidirect
39. D4×#92418C22semidirect
48. #16839514C22semidirect
49. #16839514C22semidirect
50. #16839512C22semidirect
51. #16839512C22semidirect
56. #16839484C22semidirect
57. #16839484C22semidirect
58. #16839483C22semidirect
59. C8×#92418C22semidirect
63. C61×Q8⋊C8C22semidirect
64. C61×Q8⋊C8C22semidirect
65. C61×D4⋊C8C22semidirect
66. C61×D4⋊C8C22semidirect
311. #1495D61semidirect
360. Q8⋊C8C2×D61semidirect
361. Q8⋊C8C2×D61semidirect
366. D4⋊C8C2×D61semidirect
367. D4⋊C8C2×D61semidirect
372. C61#18006semidirect

Maximal subgroups

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

subgroupindex
#214558622
#214548242
#214545542
#214447302
#214447062
#214446942
#214446862
#214446582
#214446452
#214446342
#214446112
#214446102
#214442112
#214441142
C61×#14952
#1800661