Finite Group Databasebeta

← #29651323

Subgroups — #29651323

Order 15616.

656 conjugacy classes of subgroups.

The subgroup lattice has 630 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21455815C211yesmaximal, characteristic
2#21454886C211yesmaximal, characteristic, complemented
3#21454856C211yesmaximal, characteristic, complemented
4#21444730C211yesmaximal, characteristic, complemented
5#21444701C211yesmaximal, characteristic, complemented
6#21444686C211yesmaximal, characteristic, complemented
7#21444680C211yesmaximal, characteristic, complemented
8#21444651C211yesmaximal, characteristic, complemented
9#21444645C211yesmaximal, characteristic, complemented
10#21444640C211yesmaximal, characteristic, complemented
11#21444611C211yesmaximal, characteristic, complemented
12#21444600C211yesmaximal, characteristic, complemented
13#21444211C211yesmaximal, characteristic, complemented
14#21444119C211yesmaximal, characteristic, complemented
15C61×#1496C211yesmaximal, characteristic, complemented
16C2×C4×D244C2211yescharacteristic
17#16840073C411yescharacteristic
18#16840073C411yescharacteristic
19#16840039C411yescharacteristic
20#16840039C411yescharacteristic
21#16839954C411yescharacteristic
22#16839954C411yescharacteristic
23#16839895C411yescharacteristic
24#16839895C411yescharacteristic
25C4×#11758199C2211yescharacteristic
26C4×#11758199C2211yescharacteristic
27C4×#11758199C2211yescharacteristic
28C4×#11758199C2211yescharacteristic
29#16839840—21no—
30#16839840C2211yescharacteristic, complemented
31#16839840C2211yescharacteristic, complemented
32#16839839C2211yescharacteristic
33#16839839C2211yescharacteristic
34#16839835—21no—
35#16839835—21no—
36D4×#92418—21no—
37D4×#92418C2211yescharacteristic, complemented
38D4×#92418C2211yescharacteristic, complemented
39#16839827C2211yescharacteristic
40#16839827C2211yescharacteristic
41#16839815C2211yescharacteristic
42#16839815C2211yescharacteristic
43#16839815C2211yescharacteristic
44#16839729—21no—
45#16839729—21no—
46#16839724—21no—
47C22⋊C8×D61—21no—
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×#532C2211yescharacteristic
61C61×#432C2211yescharacteristic
62C61×#431C2211yescharacteristic
63C61×D4⋊C8C2211yescharacteristic, complemented
64C61×D4⋊C8C2211yescharacteristic, complemented
65C61×D4⋊C8C2211yescharacteristic, complemented
66C61×D4⋊C8C2211yescharacteristic, complemented
67C22×D244C2×C411yescharacteristic
68C22×C4×D61—21no—
69C2×#11758199C2×C411yescharacteristic
70C2×#11758199C2×C411yescharacteristic
71C2×#11758199C2×C411yescharacteristic
72C2×#11758199C2×C411yescharacteristic
73C2×#11758194—21no—
74C2×#11758194—21no—
75C2×#11758189—21no—
76C2×#11758189—21no—
77C22×#92418—21no—
78C22×#92418—21no—
79C2×C8×D61—21no—
80C2×C8×D61—21no—
81C4×D244—21no—
82C4×D244—21no—
83C4×D244C2×C411yescharacteristic
84C4×D244C2×C411yescharacteristic
85C4×D244C2311yescharacteristic
86C4×D244C2311yescharacteristic
87C2×C4⋊Dic61C2×C411yescharacteristic
88C2×#11758198—21no—
89#14519710C2×C411yescharacteristic
90#14519710C2×C411yescharacteristic
91#14519710C2×C411yescharacteristic
92#14519710C2×C411yescharacteristic
93C4⋊#92418C2×C411yescharacteristic
94C4⋊#92418C2×C411yescharacteristic
95C4⋊#92418D411yescharacteristic
96C4⋊#92418D411yescharacteristic
97C4⋊#92418C2311yescharacteristic
98C4⋊#92418C2311yescharacteristic
99#14519698C2×C411yescharacteristic
100#14519698C2×C411yescharacteristic
101#14519698C2×C411yescharacteristic
102#14519698C2×C411yescharacteristic
103#14519697C2×C411yescharacteristic
104#14519697C2×C411yescharacteristic
105#14519697C2×C411yescharacteristic
106#14519697C2×C411yescharacteristic
107C22⋊#92418—21no—
108C22⋊#92418—21no—
109C22⋊#92418—21no—
110C22⋊#92418—21no—
111C22⋊#92418—21no—
112C22⋊#92418C2×C411yescharacteristic
113C22⋊#92418C2×C411yescharacteristic
114C22⋊#92418D411yescharacteristic
115C22⋊#92418D411yescharacteristic
116#14519689—21no—
117#14519689—21no—
118#14519689—21no—
119#14519689—21no—
120C4×#92418—21no—
121C4×#92418C2311yescharacteristic
122C4×#92418C2311yescharacteristic
123C4×#92418C2311yescharacteristic
124C4×#92418C2311yescharacteristic
125C2×#41×C61—21no—
126C4⋊Q8×C61C2×C411yescharacteristic
127#127×C61C2×C411yescharacteristic
128#124×C61—21no—
129#121×C61—21no—
130#121×C61C2×C411yescharacteristic
131#121×C61C2×C411yescharacteristic
132C4×D4×C61—21no—
133C4×D4×C61C2311yescharacteristic
134C4×D4×C61C2311yescharacteristic
135C2×C42×C61C2311yescharacteristic
136C4⋊C8×C61—21no—
137C4⋊C8×C61C2311yescharacteristic
138C4⋊C8×C61C2311yescharacteristic
139C22⋊C8×C61—21no—
140C22⋊C8×C61—21no—
141#97×C61—21no—
142C4×C8×C61C2311yescharacteristic
143C4×C8×C61C2311yescharacteristic
144C23×D61—21no—
145#41×C61—81no—
146#41×C61—81no—
147C2×Q8×C61—21no—
148C2×D4×C61—21no—
149C2×D4×C61—21no—
150C2×D4×C61—21no—
151C2×D4×C61—21no—
152C2×D4×C61C22×C411yescharacteristic
153C2×D4×C61C22×C411yescharacteristic
154C22×C4×C61—21no—
155C22×C4×C61—21no—
156C22×C4×C61C22×C411yescharacteristic
157C22×C4×C61C2×D411yescharacteristic
158C22×C4×C61#4111yescharacteristic
159C22×Dic61—21no—
160C2×D244—22no—
161C2×D244—21no—
162C2×D244—41no—
163C2×D244C22×C411yescharacteristic
164C2×D244C22×C411yescharacteristic
165C2×C4×D61—21no—
166C2×C4×D61—21no—
167C2×C4×D61—21no—
168C2×C4×D61—21no—
169C2×C4×D61—81no—
170C2×C4×D61—81no—
171C2×C8×C61—21no—
172C2×C8×C61—21no—
173C2×C8×C61—21no—
174C2×C8×C61—21no—
175C4⋊C4×C61—21no—
176C4⋊C4×C61C22×C411yescharacteristic
177C4⋊C4×C61C22×C411yescharacteristic
178C22⋊C4×C61—21no—
179C22⋊C4×C61—21no—
180C22⋊C4×C61—21no—
181C22⋊C4×C61—21no—
182C42×C61C22×C411yescharacteristic
183C42×C61C2×D411yescharacteristic
184C42×C61#4111yescharacteristic
185C42×C61C2411yescharacteristic
186#11758199—22no—
187#11758199—22no—
188#11758199—41no—
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—81no—
198#11758194—81no—
199C2×#92418—21no—
200C2×#92418—21no—
201C2×#92418—81no—
202C2×#92418—81no—
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—81no—
212#11758189—81no—
213C8×D61—81no—
214C8×D61—81no—
215C23×C61—21no—
216C23×C61—21no—
217C23×C61C4×D411yescharacteristic
218C22×D61—21no—
219C22×D61—21no—
220C22×D61—42no—
221C22×D61—42no—
222C22×D61—81no—
223Q8×C61—81no—
224D4×C61—21no—
225D4×C61—21no—
226D4×C61—21no—
227D4×C61—21no—
228D4×C61—42no—
229D4×C61—81no—
230D4×C61—81no—
231D4×C61—81no—
232C2×C4×C61—21no—
233C2×C4×C61—21no—
234C2×C4×C61—21no—
235C2×C4×C61—21no—
236C2×C4×C61—21no—
237C2×C4×C61—21no—
238C2×C4×C61—41no—
239C2×C4×C61—41no—
240C2×C4×C61—41no—
241C2×C4×C61—41no—
242C2×C4×C61—41no—
243C2×C4×C61C4×D411yescharacteristic
244C2×C4×C61C4×D411yescharacteristic
245C2×C4×C61C4×D411yescharacteristic
246C2×C4×C61C23×C411yescharacteristic
247C2×C4×C61C22×D411yescharacteristic
248C2×C4×C61C2×#4111yescharacteristic
249C2×Dic61—21no—
250C2×Dic61—21no—
251C2×Dic61—81no—
252D244—21no—
253D244—21no—
254D244—21no—
255D244—42no—
256D244#13112yes—
257C4×D61—41no—
258C4×D61—41no—
259C4×D61—41no—
260C4×D61—41no—
261C4×D61—81no—
262C4×D61—81no—
263C8×C61—41no—
264C8×C61—41no—
265C8×C61—41no—
266C8×C61—41no—
267#92418—21no—
268#92418—21no—
269#92418—21no—
270#92418—21no—
271#92418—21no—
272#92418—21no—
273#92418—21no—
274#92418—21no—
275#92418—41no—
276#92418—41no—
277#18026—611nomaximal, Sylow
278C22×C61—21no—
279C22×C61—21no—
280C22×C61—21no—
281C22×C61—41no—
282C22×C61—41no—
283C22×C61—41no—
284C22×C61—41no—
285C22×C61—41no—
286C22×C61—41no—
287C22×C61C2×C4×D411yescharacteristic
288C2×D61—22no—
289C2×D61—41no—
290C2×D61—41no—
291C2×D61—41no—
292C2×D61—41no—
293C2×D61—41no—
294C2×D61—84no—
295C4×C61—21no—
296C4×C61—21no—
297C4×C61—21no—
298C4×C61—21no—
299C4×C61—41no—
300C4×C61—41no—
301C4×C61C2×#13111yescharacteristic
302C4×C61#56811yescharacteristic
303C4×C61#57611yescharacteristic
304C4×C61#57611yescharacteristic
305Dic61—41no—
306Dic61—41no—
307#3100—611no—
308#2979—611no—
309#2974—611no—
310#1594—611no—
311#1589—611no—
312#1586—611no—
313#1586—611no—
314#1581—611no—
315#1581—611no—
316#1580—611no—
317C8×D8—611no—
318#1572—611no—
319#1496D6111yescharacteristic, complemented
320#1478—611no—
321#1454—611no—
322C2×C61—41no—
323C2×C61—81no—
324C2×C61—81no—
325C2×C61#293311yescharacteristic
326C2×C61#294811yescharacteristic
327C2×C61#295511yescharacteristic
328D61—42no—
329D61—81no—
330#532C2×D6111yescharacteristic
331C2×C4×D4—611no—
332#480—611no—
333#480—611no—
334#474—611no—
335#474—611no—
336#459—611no—
337#459—611no—
338#447—611no—
339#447—611no—
340C4×D8—611no—
341C4×D8—611no—
342C4×D8—611no—
343C4×D8—611no—
344#436—611no—
345#436—611no—
346#436—611no—
347#436—611no—
348#436—1221no—
349#435—1221no—
350#435—1221no—
351C8×D4—611no—
352C8×D4—611no—
353C8×D4—611no—
354C8×D4—611no—
355C8×D4—1221no—
356#432C2×D6111yescharacteristic
357#431—611no—
358#431—611no—
359#431—611no—
360#431C2×D6111yescharacteristic
361#408—1221no—
362#408—1221no—
363#407—1221no—
364C2×C22⋊C8—1221no—
365#335—611no—
366#335—611no—
367#334—611no—
368#334—611no—
369D4⋊C8—611no—
370D4⋊C8—611no—
371D4⋊C8—611no—
372D4⋊C8—611no—
373D4⋊C8C2×D6111yescharacteristic, complemented
374D4⋊C8C2×D6111yescharacteristic, complemented
375D4⋊C8C2×D6111yescharacteristic, complemented
376D4⋊C8C2×D6111yescharacteristic, complemented
377#322—611no—
378#322—611no—
379#322—611no—
380C82—611no—
381C61#1802611yescharacteristic, complemented, Sylow
382C2×#41—21no—
383C22×D4—611no—
384C23×C4—1221no—
385C2×D8—611no—
386C2×D8—611no—
387C2×D8—611no—
388C2×D8—611no—
389C2×M16—1221no—
390C2×M16—1221no—
391C2×M16—1221no—
392C2×M16—1221no—
393C22×C8—1221no—
394C22×C8—1221no—
395C22×C8—1221no—
396C22×C8—1221no—
397C4⋊Q8C4×D6111yescharacteristic
398#127C4×D6111yescharacteristic
399#124—21no—
400#121—21no—
401#121C4×D6111yescharacteristic
402#121C4×D6111yescharacteristic
403C4×D4—21no—
404C4×D4—611no—
405C4×D4—611no—
406C4×D4—611no—
407C4×D4—611no—
408C4×D4—1221no—
409C4×D4—1221no—
410C4×D4C22×D6111yescharacteristic
411C4×D4C22×D6111yescharacteristic
412C2×C4⋊C4—611no—
413C2×C22⋊C4—1221no—
414C2×C42C22×D6111yescharacteristic
415#107—611no—
416#107—611no—
417#107—611no—
418#107—611no—
419C4⋊C8—21no—
420C4⋊C8—611no—
421C4⋊C8—611no—
422C4⋊C8—611no—
423C4⋊C8—611no—
424C4⋊C8—611no—
425C4⋊C8—611no—
426C4⋊C8C22×D6111yescharacteristic
427C4⋊C8C22×D6111yescharacteristic
428#102—611no—
429#102—611no—
430#102—611no—
431#102—611no—
432#102—611no—
433#102—611no—
434#102—611no—
435#102—611no—
436C22⋊C8—21no—
437C22⋊C8—21no—
438C22⋊C8—611no—
439C22⋊C8—611no—
440C22⋊C8—611no—
441C22⋊C8—611no—
442C22⋊C8—1221no—
443C22⋊C8—1221no—
444C22⋊C8—1221no—
445C22⋊C8—1221no—
446C22⋊C8—1221no—
447C22⋊C8—1221no—
448C22⋊C8—1221no—
449C22⋊C8—1221no—
450C22⋊C8—1221no—
451#97—21no—
452C4×C8—611no—
453C4×C8—611no—
454C4×C8—611no—
455C4×C8—611no—
456C4×C8—1221no—
457C4×C8C22×D6111yescharacteristic
458C4×C8C22×D6111yescharacteristic
459C24—1221no—
460#41—81no—
461#41—81no—
462C2×Q8—21no—
463C2×D4—21no—
464C2×D4—21no—
465C2×D4—21no—
466C2×D4—21no—
467C2×D4—611no—
468C2×D4—611no—
469C2×D4—1222no—
470C2×D4—1221no—
471C2×D4—2441no—
472C2×D4C2×C4×D6111yescharacteristic
473C2×D4C2×C4×D6111yescharacteristic
474C22×C4—21no—
475C22×C4—21no—
476C22×C4—1221no—
477C22×C4—1221no—
478C22×C4—1221no—
479C22×C4—1221no—
480C22×C4—1221no—
481C22×C4—4881no—
482C22×C4—4881no—
483C22×C4C2×C4×D6111yescharacteristic
484C22×C4D4×D6111yescharacteristic
485C22×C4#1175822411yescharacteristic
486D8—1222no—
487D8—1222no—
488D8—2441no—
489D8—2441no—
490M16—4881no—
491M16—4881no—
492M16—4881no—
493M16—4881no—
494C2×C8—21no—
495C2×C8—21no—
496C2×C8—21no—
497C2×C8—21no—
498C2×C8—611no—
499C2×C8—611no—
500C2×C8—611no—
501C2×C8—611no—
502C2×C8—611no—
503C2×C8—611no—
504C2×C8—611no—
505C2×C8—611no—
506C2×C8—1221no—
507C2×C8—1221no—
508C2×C8—4881no—
509C2×C8—4881no—
510C2×C8—4881no—
511C2×C8—4881no—
512C4⋊C4—21no—
513C4⋊C4—611no—
514C4⋊C4—611no—
515C4⋊C4—1221no—
516C4⋊C4C2×C4×D6111yescharacteristic
517C4⋊C4C2×C4×D6111yescharacteristic
518C22⋊C4—21no—
519C22⋊C4—21no—
520C22⋊C4—21no—
521C22⋊C4—21no—
522C22⋊C4—1221no—
523C22⋊C4—1221no—
524C22⋊C4—1221no—
525C22⋊C4—1221no—
526C42C2×C4×D6111yescharacteristic
527C42D4×D6111yescharacteristic
528C42#1175822411yescharacteristic
529C42C23×D6111yescharacteristic
530C23—21no—
531C23—21no—
532C23—1221no—
533C23—1221no—
534C23—4881no—
535C23—2442no—
536C23—2442no—
537C23#1451973711yescharacteristic
538Q8—81no—
539D4—21no—
540D4—21no—
541D4—21no—
542D4—21no—
543D4—42no—
544D4—81no—
545D4—81no—
546D4—81no—
547D4—612no—
548D4—1221no—
549D4—1221no—
550D4—1221no—
551D4—2442no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—1221no—
564C2×C4—1221no—
565C2×C4—4881no—
566C2×C4—4881no—
567C2×C4—4881no—
568C2×C4—2441no—
569C2×C4—2441no—
570C2×C4—2441no—
571C2×C4—2441no—
572C2×C4#1451973711yescharacteristic
573C2×C4#1451973711yescharacteristic
574C2×C4#1451973711yescharacteristic
575C2×C4C22×C4×D6111yescharacteristic
576C2×C4C2×D4×D6111yescharacteristic
577C2×C4C2×#1175822411yescharacteristic
578C8—41no—
579C8—41no—
580C8—41no—
581C8—41no—
582C8—1221no—
583C8—1221no—
584C8—1221no—
585C8—1221no—
586C8—1221no—
587C8—1221no—
588C8—1221no—
589C8—1221no—
590C8—2441no—
591C8—2441no—
592C22—21no—
593C22—21no—
594C22—21no—
595C22—41no—
596C22—41no—
597C22—41no—
598C22—41no—
599C22—41no—
600C22—41no—
601C22—1222no—
602C22—4884no—
603C22—2441no—
604C22—2441no—
605C22—2441no—
606C22—2441no—
607C22—2441no—
608C22C2×#1451973711yescharacteristic
609C4—21no—
610C4—21no—
611C4—21no—
612C4—21no—
613C4—41no—
614C4—41no—
615C4—2441no—
616C4—2441no—
617C4#131×D6111yescharacteristic
618C4#1684052711yescharacteristic
619C4#1684056611yescharacteristic
620C4#1684056611yescharacteristic
621C2—41no—
622C2—81no—
623C2—81no—
624C2—4881no—
625C2—2442no—
626C2#2145454911yescharacteristic
627C2#2145466711yescharacteristic
628C2#2145471811yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC42C23×D61
FittingC61×#1496C2
socleC22×C61C2×C4×D4
prime radical#21455815C2
cosocleC42×C61C24
hypercenter#1496D61
hypocenterC61#18026
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1496

Complements

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

NH ≅ G/Ntype
2. #21454886C2semidirect
3. #21454856C2semidirect
4. #21444730C2semidirect
5. #21444701C2semidirect
6. #21444686C2semidirect
7. #21444680C2semidirect
8. #21444651C2semidirect
9. #21444645C2semidirect
10. #21444640C2semidirect
11. #21444611C2semidirect
12. #21444600C2semidirect
13. #21444211C2semidirect
14. #21444119C2semidirect
15. C61×#1496C2semidirect
30. #16839840C22semidirect
31. #16839840C22semidirect
37. D4×#92418C22semidirect
38. D4×#92418C22semidirect
48. #16839514C22semidirect
49. #16839514C22semidirect
50. #16839512C22semidirect
51. #16839512C22semidirect
56. #16839484C22semidirect
57. #16839484C22semidirect
58. #16839483C22semidirect
59. C8×#92418C22semidirect
63. C61×D4⋊C8C22semidirect
64. C61×D4⋊C8C22semidirect
65. C61×D4⋊C8C22semidirect
66. C61×D4⋊C8C22semidirect
319. #1496D61semidirect
373. D4⋊C8C2×D61semidirect
374. D4⋊C8C2×D61semidirect
375. D4⋊C8C2×D61semidirect
376. D4⋊C8C2×D61semidirect
381. C61#18026semidirect

Maximal subgroups

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

subgroupindex
#214558152
#214548862
#214548562
#214447302
#214447012
#214446862
#214446802
#214446512
#214446452
#214446402
#214446112
#214446002
#214442112
#214441192
C61×#14962
#1802661