Finite Group Databasebeta

← #29733340

Subgroups — #29733340

Order 15616.

1326 conjugacy classes of subgroups.

The subgroup lattice has 642 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840458C211yesmaximal, characteristic, complemented
2C2×#16840126C211yesmaximal, characteristic, complemented
3C2×#16839954C211yesmaximal, characteristic, complemented
4#21446815C212yesmaximal, complemented
5#21446815C212yesmaximal, complemented
6#21446776C211yesmaximal, characteristic, complemented
7#21446618C211yesmaximal, characteristic, complemented
8#21446142C212yesmaximal, complemented
9#21446137C212yesmaximal, complemented
10#21445647C211yesmaximal, characteristic, complemented
11C61×#2541C211yesmaximal, characteristic, complemented
12C22×#11758199C2212yes—
13#16840458—24no—
14#16840458—24no—
15C2×#14519772—21no—
16C2×#14519769C2211yescharacteristic
17C2×#14519759—21no—
18C2×#14519751C2211yescharacteristic
19C2×#14519751C2211yescharacteristic
20C2×#14519749—21no—
21C2×C4×D244C2211yescharacteristic
22C2×#14519737—21no—
23C2×#14519736C2211yescharacteristic
24C22×C4⋊Dic61C2211yescharacteristic
25#16840126—24no—
26#16839954—24no—
27C2×C4⋊#92418C2211yescharacteristic
28C2×#14519698C2212yes—
29C2×#14519698C2212yes—
30C2×#14519698C2212yes—
31C2×#14519697C2212yes—
32C2×C22⋊#92418C2212yes—
33C2×C22⋊#92418C2212yes—
34C2×#14519688C2211yescharacteristic
35#16839524C2212yes—
36#16839524C2212yes—
37#16839524C2212yes—
38C2×#127×C61—21no—
39C2×#127×C61C2211yescharacteristic
40C2×#121×C61C2212yes—
41C2×#121×C61C2212yes—
42C2×C4×D4×C61C2211yescharacteristic
43C61×#392—21no—
44C61×#392—21no—
45C61×#390—21no—
46C4×C4⋊C4×C61C2211yescharacteristic
47C22×D244C2311yescharacteristic
48C22×C22⋊D61—21no—
49C22×C22⋊D61—21no—
50C23×Dic61—21no—
51C22×C4×D61—21no—
52C2×#11758199—24no—
53C2×#11758199—42no—
54C2×#11758199—42no—
55C22×#92418C2312yes—
56C22×#92418C2312yes—
57#14519772—28no—
58#14519769—22no—
59#14519769D414yescomplemented
60#14519759—28no—
61#14519751—22no—
62#14519751—22no—
63#14519751—22no—
64#14519751—22no—
65#14519749—28no—
66C4×D244—22no—
67C4×D244D414yescomplemented
68#14519737—28no—
69#14519736—22no—
70#14519736—22no—
71C2×C4⋊Dic61—21no—
72C2×C4⋊Dic61—22no—
73C2×C4⋊Dic61—22no—
74C2×C4⋊Dic61C2311yescharacteristic
75C2×C4⋊Dic61C2311yescharacteristic
76C2×C4⋊Dic61C2311yescharacteristic
77C2×#11758196—21no—
78C2×#11758196—21no—
79C2×C22⋊Dic61—21no—
80C2×#11758198—21no—
81C2×#11758198—21no—
82C2×#11758198—21no—
83C2×C4×Dic61—21no—
84C4⋊#92418—22no—
85#14519698—24no—
86#14519698—24no—
87#14519698—24no—
88#14519697—24no—
89C22⋊#92418—42no—
90C22⋊#92418—42no—
91#14519688—22no—
92C22×D4×C61—22no—
93C22×D4×C61—22no—
94C22×D4×C61—21no—
95C22×D4×C61—21no—
96C22×D4×C61D411yescharacteristic
97C22×D4×C61C2312yes—
98C23×C4×C61C2311yescharacteristic
99C23×C4×C61C2311yescharacteristic
100#127×C61—22no—
101#127×C61—22no—
102#127×C61—44no—
103#121×C61—24no—
104#121×C61—24no—
105#121×C61—42no—
106#121×C61—42no—
107C4×D4×C61—22no—
108C4×D4×C61—22no—
109C2×C4⋊C4×C61D411yescharacteristic
110C2×C4⋊C4×C61C2312yes—
111C2×C22⋊C4×C61—22no—
112C2×C22⋊C4×C61—22no—
113C2×C22⋊C4×C61D411yescharacteristic
114C2×C22⋊C4×C61D411yescharacteristic
115C2×C42×C61—21no—
116C2×C42×C61C2311yescharacteristic
117#95×C61—21no—
118C24×C61—22no—
119C24×C61C2×D411yescharacteristic
120C24×C61C2×D411yescharacteristic
121C23×D61—21no—
122C2×D4×C61—22no—
123C2×D4×C61—22no—
124C2×D4×C61—22no—
125C2×D4×C61—22no—
126C2×D4×C61—22no—
127C2×D4×C61—22no—
128C2×D4×C61—22no—
129C2×D4×C61—22no—
130C2×D4×C61—41no—
131C2×D4×C61—44no—
132C2×D4×C61—44no—
133C2×D4×C61—42no—
134C2×D4×C61—42no—
135C2×D4×C61—44no—
136C2×D4×C61—44no—
137C2×D4×C61—42no—
138C2×D4×C61—42no—
139C2×D4×C61—82no—
140C2×D4×C61—82no—
141C2×D4×C61—82no—
142C22×C4×C61—22no—
143C22×C4×C61—21no—
144C22×C4×C61—21no—
145C22×C4×C61—21no—
146C22×C4×C61—21no—
147C22×C4×C61—22no—
148C22×C4×C61—22no—
149C22×C4×C61—22no—
150C22×C4×C61—22no—
151C22×C4×C61C2×D411yescharacteristic
152C22×C4×C61C2×D411yescharacteristic
153C22×C4×C61C2×D411yescharacteristic
154C22×C4×C61C2×D411yescharacteristic
155C22×C4×C61C2411yescharacteristic
156C2×C22⋊D61—24no—
157C2×C22⋊D61—24no—
158C2×C22⋊D61—42no—
159C2×C22⋊D61—42no—
160C2×C22⋊D61—82no—
161C22×Dic61—21no—
162C22×Dic61—21no—
163C22×Dic61—21no—
164C22×Dic61—24no—
165C22×Dic61—82no—
166C2×D244—22no—
167C2×D244—22no—
168C2×D244—21no—
169C2×D244C2×D412yes—
170C2×C4×D61—24no—
171C2×C4×D61—82no—
172C4⋊C4×C61—22no—
173C4⋊C4×C61—22no—
174C4⋊C4×C61#4114yes—
175C22⋊C4×C61—22no—
176C22⋊C4×C61—22no—
177C22⋊C4×C61—44no—
178C22⋊C4×C61—44no—
179C42×C61—42no—
180C42×C61C2×D412yes—
181C42×C61C2×D412yes—
182C22⋊Dic61—24no—
183#11758199—48no—
184#11758198—24no—
185#11758198—24no—
186#11758198—24no—
187C4⋊Dic61—24no—
188C4⋊Dic61—21no—
189C4⋊Dic61—21no—
190C4⋊Dic61—21no—
191C4⋊Dic61C2×D412yes—
192C4⋊Dic61C2×D412yes—
193C4⋊Dic61C2×D412yes—
194#11758196—24no—
195#11758196—24no—
196C4×Dic61—24no—
197C2×#92418—22no—
198C2×#92418—22no—
199C2×#92418—22no—
200C2×#92418—22no—
201C2×#92418—22no—
202C2×#92418—22no—
203C23×C61—21no—
204C23×C61—22no—
205C23×C61—21no—
206C23×C61—22no—
207C23×C61—21no—
208C23×C61—22no—
209C23×C61—21no—
210C23×C61—21no—
211C23×C61—22no—
212C23×C61—42no—
213C23×C61—42no—
214C23×C61—42no—
215C23×C61—42no—
216C23×C61—42no—
217C23×C61—42no—
218C23×C61—42no—
219C23×C61AGL(1,ℤ/8)12yes—
220C23×C61C22×D411yescharacteristic
221C22×D61—22no—
222C22×D61—42no—
223C22×D61—41no—
224C22×D61—41no—
225C22×D61—42no—
226D4×C61—44no—
227D4×C61—42no—
228D4×C61—42no—
229D4×C61—42no—
230D4×C61—42no—
231D4×C61—84no—
232D4×C61—84no—
233D4×C61—88no—
234D4×C61—88no—
235C2×C4×C61—21no—
236C2×C4×C61—21no—
237C2×C4×C61—22no—
238C2×C4×C61—21no—
239C2×C4×C61—21no—
240C2×C4×C61—24no—
241C2×C4×C61—22no—
242C2×C4×C61—22no—
243C2×C4×C61—42no—
244C2×C4×C61—42no—
245C2×C4×C61—41no—
246C2×C4×C61—41no—
247C2×C4×C61—41no—
248C2×C4×C61—41no—
249C2×C4×C61—42no—
250C2×C4×C61—42no—
251C2×C4×C61#12012yes—
252C2×C4×C61#12012yes—
253C2×C4×C61C22×D411yescharacteristic
254C2×C4×C61C22×D411yescharacteristic
255C2×C4×C61C2×#4112yes—
256C2×C4×C61H(5,𝔽2)12yes—
257C22⋊D61—48no—
258C22⋊D61—84no—
259C2×Dic61—22no—
260C2×Dic61—22no—
261C2×Dic61—22no—
262C2×Dic61—41no—
263C2×Dic61—41no—
264C2×Dic61—42no—
265C2×Dic61—42no—
266C2×Dic61—84no—
267D244—22no—
268D244—22no—
269D244—42no—
270C4×D61—84no—
271#92418—42no—
272#92418—42no—
273#25581—611nomaximal, Sylow
274C22×C61—22no—
275C22×C61—24no—
276C22×C61—41no—
277C22×C61—41no—
278C22×C61—42no—
279C22×C61—41no—
280C22×C61—41no—
281C22×C61—41no—
282C22×C61—42no—
283C22×C61—42no—
284C22×C61—82no—
285C22×C61—82no—
286C22×C61—82no—
287C22×C61—82no—
288C22×C61—82no—
289C22×C61—82no—
290C22×C61—82no—
291C22×C61C2×#12011yescharacteristic
292C22×C61#54612yes—
293C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
294C22×C61#57511yescharacteristic
295C22×C61#57612yes—
296C2×D61—42no—
297C2×D61—42no—
298C2×D61—42no—
299C2×D61—82no—
300C2×D61—82no—
301C4×C61—22no—
302C4×C61—22no—
303C4×C61—41no—
304C4×C61—41no—
305C4×C61—44no—
306C4×C61#53211yescharacteristic
307C4×C61#53511yescharacteristic
308C4×C61#55012yes—
309Dic61—42no—
310Dic61—42no—
311Dic61—42no—
312C2×#553—611no—
313C2×#490—611no—
314C2×#459—611no—
315#2541D6111yescharacteristic, complemented
316#1999—612no—
317#1999—612no—
318#1989—611no—
319#1955—611no—
320#1875—612no—
321#1874—612no—
322#1762—611no—
323C2×C61—42no—
324C2×C61—81no—
325C2×C61—162no—
326C2×C61#243111yescharacteristic
327C2×C61#300611yescharacteristic
328C2×C61#300811yescharacteristic
329C2×C61#330912yes—
330C2×C61#331812yes—
331D61—82no—
332C22×D8—612no—
333#553—1224no—
334#553—1224no—
335C2×#127—21no—
336C2×#127C2×D6111yescharacteristic
337C2×#126—1221no—
338C2×#125—611no—
339C2×#123—1221no—
340C2×#121—611no—
341C2×#121—611no—
342C2×#121—1221no—
343C2×#121C2×D6112yes—
344C2×#121C2×D6112yes—
345C2×C4×D4—611no—
346C2×C4×D4—1221no—
347C2×C4×D4C2×D6111yescharacteristic
348C2×#117—611no—
349C22×C4⋊C4—611no—
350#490—1224no—
351#459—1224no—
352C2×C4⋊C8—611no—
353C2×#102—612no—
354C2×#102—612no—
355C2×#102—612no—
356C2×#102—612no—
357C2×C22⋊C8—612no—
358C2×C22⋊C8—612no—
359C2×#97—611no—
360#392—21no—
361#392—21no—
362#390—21no—
363C4×C4⋊C4C2×D6111yescharacteristic
364#340—612no—
365#340—612no—
366#340—612no—
367C61#2558111yescharacteristic, complemented, Sylow
368C22×D4—22no—
369C22×D4—22no—
370C22×D4—21no—
371C22×D4—21no—
372C22×D4—611no—
373C22×D4—1221no—
374C22×D4—1221no—
375C22×D4C22⋊D6111yescharacteristic
376C22×D4C22×D6112yes—
377C23×C4—1221no—
378C23×C4—1221no—
379C23×C4C22×D6111yescharacteristic
380C23×C4C22×D6111yescharacteristic
381C2×D8—1224no—
382C2×D8—2442no—
383C2×D8—2442no—
384C22×C8—612no—
385C22×C8—612no—
386#127—22no—
387#127—22no—
388#127—44no—
389#126—1228no—
390#125—614no—
391#125—1222no—
392#123—1228no—
393#121—24no—
394#121—24no—
395#121—42no—
396#121—42no—
397#121—1222no—
398#121—1222no—
399#121—1222no—
400#121—1222no—
401#121—1228no—
402C4×D4—22no—
403C4×D4—22no—
404C4×D4—614no—
405C4×D4—1228no—
406C4×D4—1222no—
407#117—1222no—
408#117—1222no—
409C2×C4⋊C4—611no—
410C2×C4⋊C4—611no—
411C2×C4⋊C4—611no—
412C2×C4⋊C4—1222no—
413C2×C4⋊C4—1222no—
414C2×C4⋊C4—1221no—
415C2×C4⋊C4—1221no—
416C2×C4⋊C4—1221no—
417C2×C4⋊C4C22⋊D6111yescharacteristic
418C2×C4⋊C4C22×D6112yes—
419C2×C22⋊C4—22no—
420C2×C22⋊C4—22no—
421C2×C22⋊C4—1221no—
422C2×C22⋊C4—1221no—
423C2×C22⋊C4—1221no—
424C2×C22⋊C4—1221no—
425C2×C22⋊C4C22⋊D6111yescharacteristic
426C2×C22⋊C4C22⋊D6111yescharacteristic
427C2×C42—21no—
428C2×C42—1221no—
429C2×C42C22×D6111yescharacteristic
430C4⋊C8—1222no—
431#102—1224no—
432#102—1224no—
433#102—1224no—
434#102—1224no—
435C22⋊C8—2442no—
436C22⋊C8—2442no—
437#97—1222no—
438#95—21no—
439C24—22no—
440C24—1221no—
441C24C2×C22⋊D6111yescharacteristic
442C24C2×C22⋊D6111yescharacteristic
443C2×D4—22no—
444C2×D4—22no—
445C2×D4—22no—
446C2×D4—22no—
447C2×D4—22no—
448C2×D4—22no—
449C2×D4—22no—
450C2×D4—22no—
451C2×D4—41no—
452C2×D4—44no—
453C2×D4—44no—
454C2×D4—42no—
455C2×D4—42no—
456C2×D4—44no—
457C2×D4—44no—
458C2×D4—42no—
459C2×D4—42no—
460C2×D4—82no—
461C2×D4—82no—
462C2×D4—82no—
463C2×D4—612no—
464C2×D4—1224no—
465C2×D4—1224no—
466C2×D4—1221no—
467C2×D4—1222no—
468C2×D4—1222no—
469C2×D4—4882no—
470C2×D4—2442no—
471C2×D4—2442no—
472C22×C4—22no—
473C22×C4—21no—
474C22×C4—21no—
475C22×C4—21no—
476C22×C4—21no—
477C22×C4—22no—
478C22×C4—22no—
479C22×C4—22no—
480C22×C4—22no—
481C22×C4—1224no—
482C22×C4—1221no—
483C22×C4—1221no—
484C22×C4—1221no—
485C22×C4—1224no—
486C22×C4—4882no—
487C22×C4—4882no—
488C22×C4C2×C22⋊D6111yescharacteristic
489C22×C4C2×C22⋊D6111yescharacteristic
490C22×C4C2×C22⋊D6111yescharacteristic
491C22×C4C2×C22⋊D6111yescharacteristic
492C22×C4C23×D6111yescharacteristic
493D8—2448no—
494C2×C8—1222no—
495C2×C8—1222no—
496C2×C8—1222no—
497C2×C8—1222no—
498C2×C8—1222no—
499C2×C8—1222no—
500C4⋊C4—22no—
501C4⋊C4—22no—
502C4⋊C4—612no—
503C4⋊C4—612no—
504C4⋊C4—612no—
505C4⋊C4—1224no—
506C4⋊C4—1221no—
507C4⋊C4—1221no—
508C4⋊C4—1221no—
509C4⋊C4—1224no—
510C4⋊C4—1224no—
511C4⋊C4#1175822414yes—
512C22⋊C4—22no—
513C22⋊C4—22no—
514C22⋊C4—44no—
515C22⋊C4—44no—
516C22⋊C4—1224no—
517C22⋊C4—1224no—
518C22⋊C4—1224no—
519C22⋊C4—1224no—
520C42—42no—
521C42—1224no—
522C42D4×D6112yes—
523C42D4×D6112yes—
524C23—21no—
525C23—21no—
526C23—22no—
527C23—22no—
528C23—21no—
529C23—21no—
530C23—21no—
531C23—22no—
532C23—22no—
533C23—42no—
534C23—42no—
535C23—42no—
536C23—42no—
537C23—42no—
538C23—42no—
539C23—42no—
540C23—1222no—
541C23—2441no—
542C23—2441no—
543C23—2442no—
544C23—2442no—
545C23#1451980312yes—
546C23C22×C22⋊D6111yescharacteristic
547D4—44no—
548D4—42no—
549D4—42no—
550D4—42no—
551D4—42no—
552D4—84no—
553D4—84no—
554D4—88no—
555D4—88no—
556D4—1222no—
557D4—1222no—
558D4—4884no—
559D4—2448no—
560D4—2442no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—22no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—24no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—1222no—
578C2×C4—1222no—
579C2×C4—1222no—
580C2×C4—4884no—
581C2×C4—4884no—
582C2×C4—2441no—
583C2×C4—2441no—
584C2×C4—2442no—
585C2×C4—2442no—
586C2×C4#1451974512yes—
587C2×C4#1451974512yes—
588C2×C4C2×D4×D6111yescharacteristic
589C2×C4C2×D4×D6111yescharacteristic
590C2×C4C2×#1175822412yes—
591C2×C4#1451982212yes—
592C8—2442no—
593C8—2442no—
594C22—24no—
595C22—22no—
596C22—42no—
597C22—41no—
598C22—41no—
599C22—42no—
600C22—42no—
601C22—41no—
602C22—41no—
603C22—41no—
604C22—82no—
605C22—82no—
606C22—82no—
607C22—82no—
608C22—82no—
609C22—82no—
610C22—82no—
611C22—4882no—
612C22—4882no—
613C22—2442no—
614C22—2442no—
615C22—2442no—
616C22C2×#1451974511yescharacteristic
617C22#1684040612yes—
618C22C2×#1451980311yescharacteristic
619C22#1684056011yescharacteristic
620C22#1684056512yes—
621C4—22no—
622C4—22no—
623C4—44no—
624C4—41no—
625C4—41no—
626C4—2442no—
627C4—2442no—
628C4—2442no—
629C4#1684031011yescharacteristic
630C4#1684032511yescharacteristic
631C4#1684043212yes—
632C2—42no—
633C2—81no—
634C2—162no—
635C2—4882no—
636C2#2144943511yescharacteristic
637C2#2145506911yescharacteristic
638C2#2145508511yescharacteristic
639C2#2145782512yes—
640C2#2145796012yes—
subgroupG/H
centerC23C22×C22⋊D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2541C2
socleC23×C61C22×D4
hypercenter#2541D61
hypocenterC61#25581
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ #2541

Complements

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

NH ≅ G/Ntype
1. C2×#16840458C2semidirect
2. C2×#16840126C2semidirect
3. C2×#16839954C2semidirect
4. #21446815C2semidirect
5. #21446815C2semidirect
6. #21446776C2semidirect
7. #21446618C2semidirect
8. #21446142C2semidirect
9. #21446137C2semidirect
10. #21445647C2semidirect
11. C61×#2541C2semidirect
59. #14519769D4semidirect
67. C4×D244D4semidirect
315. #2541D61semidirect
367. C61#25581semidirect

Maximal subgroups

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

subgroupindex
C2×#168404582
C2×#168401262
C2×#168399542
#214468152
#214467762
#214466182
#214461422
#214461372
#214456472
C61×#25412
#2558161