Finite Group Databasebeta

← #29651045

Subgroups — #29651045

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
2M16×D244C211yesmaximal, characteristic, complemented
3#21444731C211yesmaximal, characteristic, complemented
4#21444709C211yesmaximal, characteristic, complemented
5#21444697C211yesmaximal, characteristic, complemented
6#21444681C211yesmaximal, characteristic, complemented
7#21444653C211yesmaximal, characteristic, complemented
8#21444648C211yesmaximal, characteristic, complemented
9#21444629C211yesmaximal, characteristic, complemented
10C8×#11758199C211yesmaximal, characteristic, complemented
11#21444607C211yesmaximal, characteristic, complemented
12#21444284C211yesmaximal, characteristic, complemented
13#21444209C211yesmaximal, characteristic, complemented
14#21444116C211yesmaximal, characteristic, complemented
15C61×#2969C211yesmaximal, 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
26C4×#11758201C2211yescharacteristic
27C4×#11758201C2211yescharacteristic
28C4×#11758199C2211yescharacteristic
29C4×#11758199C2211yescharacteristic
30#16839841C2211yescharacteristic
31#16839841C2211yescharacteristic
32#16839835—21no—
33#16839835—21no—
34C8×D244C2211yescharacteristic
35C8×D244C2211yescharacteristic
36#16839815C2211yescharacteristic
37#16839815C2211yescharacteristic
38#16839786C2211yescharacteristic
39#16839725—21no—
40#16839514C2211yescharacteristic, complemented
41#16839514C2211yescharacteristic, complemented
42#16839512C2211yescharacteristic, complemented
43#16839512C2211yescharacteristic, complemented
44Q8⋊#92418C2211yescharacteristic, complemented
45Q8⋊#92418C2211yescharacteristic, complemented
46D4⋊#92418C2211yescharacteristic, complemented
47D4⋊#92418C2211yescharacteristic, complemented
48#16839489C2211yescharacteristic
49#16839489C2211yescharacteristic
50#16839489C2211yescharacteristic
51#16839489C2211yescharacteristic
52#16839484C2211yescharacteristic, complemented
53#16839484C2211yescharacteristic, complemented
54#16839483C2211yescharacteristic, complemented
55C8×#92418C2211yescharacteristic, complemented
56C61×#529C2211yescharacteristic
57C61×#446C2211yescharacteristic, complemented
58C8×Q8×C61C2211yescharacteristic, complemented
59C61×#436—21no—
60C61×#436C2211yescharacteristic, complemented
61C61×#435—21no—
62C8×D4×C61C2211yescharacteristic, complemented
63C61×#433—21no—
64C61×#431C2211yescharacteristic
65C61×#405—21no—
66C4×M16×C61C2211yescharacteristic
67C22×D244C2×C411yescharacteristic
68C22×C4×D61—21no—
69C2×#11758201C2×C411yescharacteristic
70C2×#11758201C2×C411yescharacteristic
71C2×#11758199C2×C411yescharacteristic
72C2×#11758199C2×C411yescharacteristic
73C2×#11758194—21no—
74C2×#11758189—21no—
75C2×#11758189—21no—
76M16×D61—44no—
77C2×C8×D61—21no—
78C2×C8×D61—21no—
79C4×D244—21no—
80C4×D244—21no—
81C4×D244C2×C411yescharacteristic
82C4×D244C2×C411yescharacteristic
83C4×D244C2311yescharacteristic
84C4×D244C2311yescharacteristic
85C2×C4⋊Dic61C2×C411yescharacteristic
86C2×#11758198—21no—
87#14519710C2×C411yescharacteristic
88#14519710C2×C411yescharacteristic
89#14519708C2×C411yescharacteristic
90#14519708C2×C411yescharacteristic
91C4⋊#92418—21no—
92C4⋊#92418—21no—
93C4⋊#92418C2×C411yescharacteristic
94C4⋊#92418C2×C411yescharacteristic
95C4⋊#92418C2311yescharacteristic
96C4⋊#92418C2311yescharacteristic
97#14519701C2×C411yescharacteristic
98#14519701C2×C411yescharacteristic
99#14519698C2×C411yescharacteristic
100#14519698C2×C411yescharacteristic
101#14519697C2×C411yescharacteristic
102#14519697C2×C411yescharacteristic
103#14519697C2×C411yescharacteristic
104#14519697C2×C411yescharacteristic
105C22⋊#92418—21no—
106C22⋊#92418C2×C411yescharacteristic
107C22⋊#92418C2×C411yescharacteristic
108#14519689—21no—
109#14519689—21no—
110#14519689—21no—
111#14519689—21no—
112C4×#92418C2311yescharacteristic
113C4×#92418C2311yescharacteristic
114C4×#92418C2311yescharacteristic
115C4×#92418C2311yescharacteristic
116C2×M16×C61—21no—
117C2×M16×C61D411yescharacteristic
118C2×M16×C61D411yescharacteristic
119C22×C8×C61—21no—
120#126×C61—21no—
121#125×C61C2×C411yescharacteristic
122#124×C61C2×C411yescharacteristic
123#123×C61—21no—
124C22⋊Q8×C61C2×C411yescharacteristic
125#121×C61C2×C411yescharacteristic
126C4×Q8×C61C2311yescharacteristic
127C4×D4×C61—21no—
128C4×D4×C61C2311yescharacteristic
129#117×C61—21no—
130C2×C42×C61C2311yescharacteristic
131C4⋊C8×C61—21no—
132C4⋊C8×C61—21no—
133C4⋊C8×C61C2311yescharacteristic
134C4⋊C8×C61C2311yescharacteristic
135C22⋊C8×C61—21no—
136C22⋊C8×C61—21no—
137C22⋊C8×C61—21no—
138#97×C61—21no—
139#97×C61D411yescharacteristic
140#97×C61D411yescharacteristic
141C4×C8×C61—21no—
142C4×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—22no—
166M16×C61—41no—
167M16×C61—42no—
168C2×C8×C61—21no—
169C2×C8×C61—21no—
170C2×C8×C61—42no—
171C2×C8×C61C2×D411yescharacteristic
172C2×C8×C61C2×D411yescharacteristic
173C2×C8×C61C2×D411yescharacteristic
174C2×C8×C61C2×D411yescharacteristic
175C4⋊C4×C61—21no—
176C4⋊C4×C61—21no—
177C4⋊C4×C61—21no—
178C4⋊C4×C61—21no—
179C4⋊C4×C61C22×C411yescharacteristic
180C4⋊C4×C61C22×C411yescharacteristic
181C22⋊C4×C61—21no—
182C22⋊C4×C61—21no—
183C22⋊C4×C61—21no—
184C22⋊C4×C61—21no—
185C22⋊C4×C61—21no—
186C42×C61—21no—
187C42×C61C22×C411yescharacteristic
188C42×C61C2×D411yescharacteristic
189C42×C61#4111yescharacteristic
190C42×C61C2411yescharacteristic
191#11758201—22no—
192#11758201—41no—
193#11758199—22no—
194#11758199—41no—
195#11758198—21no—
196#11758198—21no—
197#11758198—21no—
198#11758198—21no—
199C4⋊Dic61—21no—
200C4⋊Dic61C22×C411yescharacteristic
201C4⋊Dic61C22×C411yescharacteristic
202#11758194—42no—
203C2×#92418—21no—
204C2×#92418—21no—
205C2×#92418—21no—
206C2×#92418—21no—
207C2×#92418C22×C411yescharacteristic
208C2×#92418C22×C411yescharacteristic
209C2×#92418C22×C411yescharacteristic
210C2×#92418C22×C411yescharacteristic
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—21no—
258C8×C61—21no—
259C8×C61—21no—
260C8×C61—41no—
261C8×C61—41no—
262C8×C61AGL(1,ℤ/8)12yes—
263#92418—21no—
264#92418—21no—
265#92418—21no—
266#92418—21no—
267#92418—41no—
268#92418—41no—
269#92418—41no—
270#92418—41no—
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#2969D6111yescharacteristic, complemented
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#1495—611no—
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#446C2×D6111yescharacteristic, complemented
333C8×Q8C2×D6111yescharacteristic, complemented
334C4×SD16—611no—
335C4×SD16—611no—
336C4×D8—611no—
337C4×D8—611no—
338#436—21no—
339#436—611no—
340#436—611no—
341#436C2×D6111yescharacteristic, complemented
342#435—21no—
343#435—1221no—
344#435—1221no—
345C8×D4—611no—
346C8×D4—611no—
347C8×D4C2×D6111yescharacteristic, complemented
348#433—21no—
349#431—611no—
350#431—611no—
351#431C2×D6111yescharacteristic
352#423—611no—
353#407—1221no—
354#405—21no—
355C4×M16C2×D6111yescharacteristic
356#335—611no—
357#335—611no—
358#334—611no—
359#334—611no—
360Q8⋊C8—611no—
361Q8⋊C8—611no—
362D4⋊C8—611no—
363D4⋊C8—611no—
364D4⋊C8—611no—
365D4⋊C8—611no—
366D4⋊C8—611no—
367D4⋊C8—611no—
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—1221no—
381C2×M16—1221no—
382C2×M16—1221no—
383C2×M16—2444no—
384C2×M16C22⋊D6111yescharacteristic
385C2×M16C22⋊D6111yescharacteristic
386C22×C8—21no—
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—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—21no—
441#97C22⋊D6111yescharacteristic
442#97C22⋊D6111yescharacteristic
443C4×C8—21no—
444C4×C8—21no—
445C4×C8—611no—
446C4×C8—611no—
447C4×C8—611no—
448C4×C8—611no—
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×C4#1175822211yescharacteristic
470C22×C4C2×C22⋊D6111yescharacteristic
471SD16—1222no—
472SD16—2441no—
473D8—1222no—
474D8—2441no—
475M16—22no—
476M16—41no—
477M16—42no—
478M16—2442no—
479M16—2442no—
480M16—2442no—
481C2×C8—21no—
482C2×C8—21no—
483C2×C8—42no—
484C2×C8—611no—
485C2×C8—611no—
486C2×C8—611no—
487C2×C8—611no—
488C2×C8—1221no—
489C2×C8—1221no—
490C2×C8—1221no—
491C2×C8—1221no—
492C2×C8—2442no—
493C2×C8—2442no—
494C2×C8C2×C22⋊D6111yescharacteristic
495C2×C8C2×C22⋊D6111yescharacteristic
496C2×C8C2×C22⋊D6111yescharacteristic
497C2×C8C2×C22⋊D6111yescharacteristic
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
518C42#1175822211yescharacteristic
519C42C2×C22⋊D6111yescharacteristic
520C42C23×D6111yescharacteristic
521C23—21no—
522C23—1221no—
523C23—1221no—
524C23—4881no—
525C23—2442no—
526C23—2442no—
527C23C4×C22⋊D6111yescharacteristic
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×C4C4×C22⋊D6111yescharacteristic
558C2×C4C4×C22⋊D6111yescharacteristic
559C2×C4C4×C22⋊D6111yescharacteristic
560C2×C4C22×C4×D6111yescharacteristic
561C2×C4C22×C22⋊D6111yescharacteristic
562C2×C4C2×#1175822211yescharacteristic
563C8—21no—
564C8—21no—
565C8—21no—
566C8—41no—
567C8—41no—
568C8—1221no—
569C8—1221no—
570C8—1221no—
571C8—1221no—
572C8—2441no—
573C8—2441no—
574C8—2441no—
575C8—2441no—
576C8#1451980212yes—
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×C4×C22⋊D6111yescharacteristic
591C4—21no—
592C4—21no—
593C4—21no—
594C4—21no—
595C4—41no—
596C4—41no—
597C4—41no—
598C4—2441no—
599C4—2441no—
600C4#131×D6111yescharacteristic
601C4#1684052711yescharacteristic
602C4C2×#1451980211yescharacteristic
603C4#1684056011yescharacteristic
604C2—41no—
605C2—81no—
606C2—4881no—
607C2—2442no—
608C2#2145454111yescharacteristic
609C2C4×#1451980211yescharacteristic
610C2#2145473411yescharacteristic
subgroupG/H
centerC2×C4C22×C22⋊D61
commutatorC2×C4×C61C23×C4
FrattiniC42C23×D61
FittingC61×#2969C2
socleC22×C61C2×C4×D4
prime radical#16839895C4
cosocleC42×C61C24
hypercenter#2969D61
hypocenterC61#18006
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C4×M16 ◁ #2969

Complements

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

NH ≅ G/Ntype
2. M16×D244C2semidirect
3. #21444731C2semidirect
4. #21444709C2semidirect
5. #21444697C2semidirect
6. #21444681C2semidirect
7. #21444653C2semidirect
8. #21444648C2semidirect
9. #21444629C2semidirect
10. C8×#11758199C2semidirect
11. #21444607C2semidirect
12. #21444284C2semidirect
13. #21444209C2semidirect
14. #21444116C2semidirect
15. C61×#2969C2semidirect
40. #16839514C22semidirect
41. #16839514C22semidirect
42. #16839512C22semidirect
43. #16839512C22semidirect
44. Q8⋊#92418C22semidirect
45. Q8⋊#92418C22semidirect
46. D4⋊#92418C22semidirect
47. D4⋊#92418C22semidirect
52. #16839484C22semidirect
53. #16839484C22semidirect
54. #16839483C22semidirect
55. C8×#92418C22semidirect
57. C61×#446C22semidirect
58. C8×Q8×C61C22semidirect
60. C61×#436C22semidirect
62. C8×D4×C61C22semidirect
300. #2969D61semidirect
332. #446C2×D61semidirect
333. C8×Q8C2×D61semidirect
341. #436C2×D61semidirect
347. C8×D4C2×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
M16×D2442
#214447312
#214447092
#214446972
#214446812
#214446532
#214446482
#214446292
C8×#117581992
#214446072
#214442842
#214442092
#214441162
C61×#29692
#1800661