Finite Group Databasebeta

← #29733339

Subgroups — #29733339

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×#16840128C211yesmaximal, characteristic, complemented
2C2×#16839957C211yesmaximal, characteristic, complemented
3#21450765C211yesmaximal, characteristic, complemented
4#21446815C212yesmaximal, complemented
5#21446815C212yesmaximal, complemented
6#21446777C211yesmaximal, characteristic, complemented
7#21446618C211yesmaximal, characteristic, complemented
8#21446141C212yesmaximal, complemented
9#21446136C212yesmaximal, complemented
10#21445644C211yesmaximal, characteristic, complemented
11C2×C61×#553C211yesmaximal, characteristic, complemented
12C22×#11758199C2212yes—
13C2×#14519774C2211yescharacteristic
14C2×#14519773—21no—
15C2×#14519751C2212yes—
16C2×#14519751C2212yes—
17#16840128—24no—
18#16839957—24no—
19C2×C4⋊#92418C2211yescharacteristic
20C2×#14519698C2212yes—
21C2×#14519697C2212yes—
22C2×#14519697C2212yes—
23C2×#14519697C2212yes—
24C2×C22⋊#92418C2212yes—
25C2×C22⋊#92418C2212yes—
26C2×#14519688C2211yescharacteristic
27#16839671—21no—
28#16839671—21no—
29#16839663—21no—
30C4×C4⋊Dic61C2211yescharacteristic
31#16839524C2212yes—
32#16839524C2212yes—
33#16839524C2212yes—
34C61×#553—24no—
35C61×#553—24no—
36C2×#126×C61—21no—
37C2×#125×C61C2211yescharacteristic
38C2×#123×C61—21no—
39C2×#121×C61—21no—
40C2×#121×C61C2211yescharacteristic
41C2×#121×C61C2211yescharacteristic
42C2×C4×D4×C61—21no—
43C2×C4×D4×C61C2211yescharacteristic
44C2×C4×D4×C61C2211yescharacteristic
45C2×#117×C61C2211yescharacteristic
46C22×C4⋊C4×C61C2211yescharacteristic
47C22×D244—21no—
48C22×D244—21no—
49C22×D244C2312yes—
50C22×C22⋊D61—22no—
51C22×C22⋊D61—22no—
52C2×#11758199—24no—
53C2×#11758199—42no—
54C2×#11758199—42no—
55C22×#92418C2312yes—
56C22×#92418C2312yes—
57#14519774—22no—
58#14519774—22no—
59#14519773—44no—
60#14519751—24no—
61#14519751—24no—
62#14519751—42no—
63#14519751—42no—
64C2×C4⋊Dic61C2312yes—
65C2×#11758198—22no—
66C2×#11758198—22no—
67C2×C4×Dic61—21no—
68C4⋊#92418—22no—
69#14519698—24no—
70#14519697—24no—
71#14519697—24no—
72#14519697—24no—
73C22⋊#92418—42no—
74C22⋊#92418—42no—
75#14519688—22no—
76#14519684—21no—
77C22×D4×C61—21no—
78C22×D4×C61—21no—
79C22×D4×C61D411yescharacteristic
80C22×D4×C61C2311yescharacteristic
81C23×C4×C61—21no—
82C23×C4×C61—21no—
83C23×C4×C61C2311yescharacteristic
84C23×C4×C61C2311yescharacteristic
85#126×C61—28no—
86#125×C61—22no—
87#125×C61D414yescomplemented
88#123×C61—28no—
89#121×C61—28no—
90#121×C61—22no—
91#121×C61—22no—
92#121×C61—22no—
93#121×C61—22no—
94C4×D4×C61—28no—
95C4×D4×C61—22no—
96C4×D4×C61—22no—
97C4×D4×C61—22no—
98C4×D4×C61D414yescomplemented
99#117×C61—22no—
100#117×C61—22no—
101C2×C4⋊C4×C61—21no—
102C2×C4⋊C4×C61—21no—
103C2×C4⋊C4×C61—21no—
104C2×C4⋊C4×C61—22no—
105C2×C4⋊C4×C61—22no—
106C2×C4⋊C4×C61D411yescharacteristic
107C2×C4⋊C4×C61C2311yescharacteristic
108C2×C4⋊C4×C61C2311yescharacteristic
109C2×C4⋊C4×C61C2311yescharacteristic
110C2×C22⋊C4×C61—21no—
111C2×C22⋊C4×C61—21no—
112C2×C22⋊C4×C61—21no—
113C2×C22⋊C4×C61—21no—
114C2×C22⋊C4×C61D411yescharacteristic
115C2×C22⋊C4×C61D411yescharacteristic
116C2×C42×C61—21no—
117C2×C42×C61C2311yescharacteristic
118C24×C61—21no—
119C24×C61C2×D411yescharacteristic
120C24×C61C2×D411yescharacteristic
121C23×D61—22no—
122C2×D4×C61—24no—
123C2×D4×C61—22no—
124C2×D4×C61—24no—
125C2×D4×C61—22no—
126C2×D4×C61—21no—
127C2×D4×C61—22no—
128C2×D4×C61—22no—
129C2×D4×C61—41no—
130C2×D4×C61—42no—
131C2×D4×C61—42no—
132C2×D4×C61—82no—
133C2×D4×C61C2×D412yes—
134C22×C4×C61—21no—
135C22×C4×C61—21no—
136C22×C4×C61—21no—
137C22×C4×C61—24no—
138C22×C4×C61—21no—
139C22×C4×C61—21no—
140C22×C4×C61—21no—
141C22×C4×C61—21no—
142C22×C4×C61—22no—
143C22×C4×C61—22no—
144C22×C4×C61—22no—
145C22×C4×C61—22no—
146C22×C4×C61—24no—
147C22×C4×C61—82no—
148C22×C4×C61—82no—
149C22×C4×C61C2×D411yescharacteristic
150C22×C4×C61C2×D411yescharacteristic
151C22×C4×C61C2×D411yescharacteristic
152C22×C4×C61C2×D411yescharacteristic
153C22×C4×C61C2411yescharacteristic
154C2×C22⋊D61—44no—
155C2×C22⋊D61—44no—
156C2×C22⋊D61—44no—
157C2×C22⋊D61—44no—
158C2×C22⋊D61—82no—
159C2×C22⋊D61—82no—
160C22×Dic61—22no—
161C2×D244—22no—
162C2×D244—22no—
163C2×D244—22no—
164C2×D244—22no—
165C2×D244—22no—
166C2×D244—22no—
167C2×D244—42no—
168C2×D244—42no—
169C2×D244—42no—
170C2×D244—42no—
171C2×D244—82no—
172C4⋊C4×C61—24no—
173C4⋊C4×C61—24no—
174C4⋊C4×C61—24no—
175C4⋊C4×C61—22no—
176C4⋊C4×C61—21no—
177C4⋊C4×C61—21no—
178C4⋊C4×C61—21no—
179C4⋊C4×C61C2×D412yes—
180C4⋊C4×C61C2×D412yes—
181C4⋊C4×C61C2×D412yes—
182C22⋊C4×C61—24no—
183C22⋊C4×C61—24no—
184C22⋊C4×C61—24no—
185C22⋊C4×C61—24no—
186C22⋊C4×C61—22no—
187C22⋊C4×C61—22no—
188C42×C61—24no—
189C42×C61C2×D412yes—
190C42×C61C2×D412yes—
191#11758199—48no—
192#11758198—44no—
193#11758198—44no—
194C4⋊Dic61—22no—
195C4⋊Dic61#4114yes—
196C4×Dic61—42no—
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—22no—
210C23×C61—21no—
211C23×C61—21no—
212C23×C61—22no—
213C23×C61—42no—
214C23×C61—41no—
215C23×C61—41no—
216C23×C61—42no—
217C23×C61AGL(1,ℤ/8)12yes—
218C23×C61C22×D411yescharacteristic
219C22×D61—42no—
220C22×D61—42no—
221C22×D61—42no—
222C22×D61—42no—
223C22×D61—42no—
224C22×D61—42no—
225C22×D61—42no—
226D4×C61—22no—
227D4×C61—22no—
228D4×C61—44no—
229D4×C61—48no—
230D4×C61—42no—
231D4×C61—84no—
232C2×C4×C61—21no—
233C2×C4×C61—21no—
234C2×C4×C61—22no—
235C2×C4×C61—21no—
236C2×C4×C61—22no—
237C2×C4×C61—22no—
238C2×C4×C61—21no—
239C2×C4×C61—22no—
240C2×C4×C61—22no—
241C2×C4×C61—22no—
242C2×C4×C61—41no—
243C2×C4×C61—41no—
244C2×C4×C61—42no—
245C2×C4×C61—42no—
246C2×C4×C61—41no—
247C2×C4×C61—41no—
248C2×C4×C61—41no—
249C2×C4×C61—41no—
250C2×C4×C61—42no—
251C2×C4×C61—42no—
252C2×C4×C61—84no—
253C2×C4×C61—84no—
254C2×C4×C61#12012yes—
255C2×C4×C61#12012yes—
256C2×C4×C61C22×D411yescharacteristic
257C2×C4×C61C22×D411yescharacteristic
258C2×C4×C61C2×#4112yes—
259C2×C4×C61H(5,𝔽2)12yes—
260C22⋊D61—88no—
261C22⋊D61—88no—
262C2×Dic61—24no—
263C2×Dic61—42no—
264C2×Dic61—42no—
265D244—42no—
266D244—42no—
267D244—42no—
268D244—42no—
269D244—84no—
270D244—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—42no—
281C22×C61—41no—
282C22×C61—42no—
283C22×C61—42no—
284C22×C61—41no—
285C22×C61—42no—
286C22×C61—42no—
287C22×C61—82no—
288C22×C61—82no—
289C22×C61C2×#12011yescharacteristic
290C22×C61#54612yes—
291C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
292C22×C61#57511yescharacteristic
293C22×C61#57612yes—
294C2×D61—82no—
295C2×D61—82no—
296C2×D61—82no—
297C2×D61—82no—
298C2×D61—82no—
299C2×D61—82no—
300C2×D61—82no—
301C4×C61—22no—
302C4×C61—22no—
303C4×C61—41no—
304C4×C61—41no—
305C4×C61—42no—
306C4×C61—42no—
307C4×C61—42no—
308C4×C61#53211yescharacteristic
309C4×C61#53511yescharacteristic
310C4×C61#55012yes—
311Dic61—44no—
312C2×#553D6111yescharacteristic, complemented
313C2×#490—611no—
314C2×#459—611no—
315#2541—611no—
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—82no—
326C2×C61#243111yescharacteristic
327C2×C61#300611yescharacteristic
328C2×C61#300811yescharacteristic
329C2×C61#330912yes—
330C2×C61#331812yes—
331D61—162no—
332C22×D8—612no—
333#553—24no—
334#553—24no—
335C2×#127—611no—
336C2×#127—1221no—
337C2×#126—21no—
338C2×#125C2×D6111yescharacteristic
339C2×#123—21no—
340C2×#121—21no—
341C2×#121—612no—
342C2×#121—612no—
343C2×#121C2×D6111yescharacteristic
344C2×#121C2×D6111yescharacteristic
345C2×C4×D4—21no—
346C2×C4×D4C2×D6111yescharacteristic
347C2×C4×D4C2×D6111yescharacteristic
348C2×#117C2×D6111yescharacteristic
349C22×C4⋊C4C2×D6111yescharacteristic
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—1221no—
361#392—1221no—
362#390—1221no—
363C4×C4⋊C4—611no—
364#340—612no—
365#340—612no—
366#340—612no—
367C61#2558111yescharacteristic, complemented, Sylow
368C22×D4—21no—
369C22×D4—21no—
370C22×D4—612no—
371C22×D4—1222no—
372C22×D4—1222no—
373C22×D4—1221no—
374C22×D4—1221no—
375C22×D4C22⋊D6111yescharacteristic
376C22×D4C22×D6111yescharacteristic
377C23×C4—21no—
378C23×C4—21no—
379C23×C4C22×D6111yescharacteristic
380C23×C4C22×D6111yescharacteristic
381C2×D8—1224no—
382C2×D8—2442no—
383C2×D8—2442no—
384C22×C8—612no—
385C22×C8—612no—
386#127—1222no—
387#127—1222no—
388#127—2444no—
389#126—28no—
390#125—22no—
391#125C22⋊D6114yescomplemented
392#123—28no—
393#121—28no—
394#121—22no—
395#121—22no—
396#121—22no—
397#121—22no—
398#121—1224no—
399#121—1224no—
400#121—2442no—
401#121—2442no—
402C4×D4—28no—
403C4×D4—22no—
404C4×D4—22no—
405C4×D4—22no—
406C4×D4C22⋊D6114yescomplemented
407#117—22no—
408#117—22no—
409C2×C4⋊C4—21no—
410C2×C4⋊C4—21no—
411C2×C4⋊C4—21no—
412C2×C4⋊C4—22no—
413C2×C4⋊C4—22no—
414C2×C4⋊C4—612no—
415C2×C4⋊C4C22⋊D6111yescharacteristic
416C2×C4⋊C4C22×D6111yescharacteristic
417C2×C4⋊C4C22×D6111yescharacteristic
418C2×C4⋊C4C22×D6111yescharacteristic
419C2×C22⋊C4—21no—
420C2×C22⋊C4—21no—
421C2×C22⋊C4—21no—
422C2×C22⋊C4—21no—
423C2×C22⋊C4—1222no—
424C2×C22⋊C4—1222no—
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—1221no—
439C24—21no—
440C24—1222no—
441C24C2×C22⋊D6111yescharacteristic
442C24C2×C22⋊D6111yescharacteristic
443C2×D4—24no—
444C2×D4—22no—
445C2×D4—24no—
446C2×D4—21no—
447C2×D4—22no—
448C2×D4—22no—
449C2×D4—22no—
450C2×D4—41no—
451C2×D4—42no—
452C2×D4—42no—
453C2×D4—82no—
454C2×D4—1222no—
455C2×D4—1222no—
456C2×D4—1222no—
457C2×D4—1222no—
458C2×D4—1222no—
459C2×D4—1222no—
460C2×D4—4882no—
461C2×D4—4882no—
462C2×D4—4882no—
463C2×D4—2442no—
464C2×D4—2442no—
465C2×D4—2444no—
466C2×D4—2444no—
467C2×D4—2444no—
468C2×D4—2444no—
469C2×D4—2442no—
470C2×D4—2442no—
471C2×D4C2×C22⋊D6112yes—
472C22×C4—21no—
473C22×C4—21no—
474C22×C4—21no—
475C22×C4—24no—
476C22×C4—21no—
477C22×C4—21no—
478C22×C4—21no—
479C22×C4—21no—
480C22×C4—22no—
481C22×C4—22no—
482C22×C4—22no—
483C22×C4—22no—
484C22×C4—24no—
485C22×C4—82no—
486C22×C4—82no—
487C22×C4—1222no—
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—24no—
501C4⋊C4—21no—
502C4⋊C4—21no—
503C4⋊C4—21no—
504C4⋊C4—24no—
505C4⋊C4—24no—
506C4⋊C4—22no—
507C4⋊C4—614no—
508C4⋊C4—1222no—
509C4⋊C4C2×C22⋊D6112yes—
510C4⋊C4C2×C22⋊D6112yes—
511C4⋊C4C2×C22⋊D6112yes—
512C22⋊C4—24no—
513C22⋊C4—24no—
514C22⋊C4—24no—
515C22⋊C4—24no—
516C22⋊C4—22no—
517C22⋊C4—22no—
518C22⋊C4—2444no—
519C22⋊C4—2444no—
520C42—24no—
521C42—2442no—
522C42C2×C22⋊D6112yes—
523C42C2×C22⋊D6112yes—
524C23—21no—
525C23—21no—
526C23—22no—
527C23—22no—
528C23—21no—
529C23—22no—
530C23—21no—
531C23—21no—
532C23—22no—
533C23—22no—
534C23—41no—
535C23—41no—
536C23—42no—
537C23—42no—
538C23—2442no—
539C23—2442no—
540C23—2442no—
541C23—2442no—
542C23—2442no—
543C23—2442no—
544C23—2442no—
545C23#1451980212yes—
546C23C22×C22⋊D6111yescharacteristic
547D4—22no—
548D4—22no—
549D4—44no—
550D4—48no—
551D4—42no—
552D4—84no—
553D4—4884no—
554D4—4884no—
555D4—4888no—
556D4—4888no—
557D4—2442no—
558D4—2442no—
559D4—2442no—
560D4—2442no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—22no—
564C2×C4—21no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—21no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—84no—
582C2×C4—84no—
583C2×C4—1224no—
584C2×C4—2442no—
585C2×C4—2442no—
586C2×C4#1451974612yes—
587C2×C4#1451974612yes—
588C2×C4C22×C22⋊D6111yescharacteristic
589C2×C4C22×C22⋊D6111yescharacteristic
590C2×C4#41×D6112yes—
591C2×C4#1451982312yes—
592C8—2442no—
593C8—2442no—
594C22—24no—
595C22—22no—
596C22—42no—
597C22—41no—
598C22—41no—
599C22—42no—
600C22—42no—
601C22—41no—
602C22—42no—
603C22—41no—
604C22—42no—
605C22—42no—
606C22—41no—
607C22—82no—
608C22—82no—
609C22—4882no—
610C22—4882no—
611C22—4882no—
612C22—4882no—
613C22—4882no—
614C22—4882no—
615C22—4882no—
616C22C2×#1451974611yescharacteristic
617C22#1684040012yes—
618C22C2×#1451980211yescharacteristic
619C22#1684056011yescharacteristic
620C22#1684056512yes—
621C4—22no—
622C4—22no—
623C4—41no—
624C4—41no—
625C4—42no—
626C4—42no—
627C4—42no—
628C4—2444no—
629C4#1684030811yescharacteristic
630C4#1684032611yescharacteristic
631C4#1684043112yes—
632C2—42no—
633C2—81no—
634C2—82no—
635C2—9762no—
636C2#2144943411yescharacteristic
637C2#2145507111yescharacteristic
638C2#2145508711yescharacteristic
639C2#2145781912yes—
640C2#2145795412yes—
subgroupG/H
centerC23C22×C22⋊D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C61×#553C2
socleC23×C61C22×D4
hypercenterC2×#553D61
hypocenterC61#25581
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ C2×#553

Complements

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

NH ≅ G/Ntype
1. C2×#16840128C2semidirect
2. C2×#16839957C2semidirect
3. #21450765C2semidirect
4. #21446815C2semidirect
5. #21446815C2semidirect
6. #21446777C2semidirect
7. #21446618C2semidirect
8. #21446141C2semidirect
9. #21446136C2semidirect
10. #21445644C2semidirect
11. C2×C61×#553C2semidirect
87. #125×C61D4semidirect
98. C4×D4×C61D4semidirect
312. C2×#553D61semidirect
367. C61#25581semidirect
391. #125C22⋊D61semidirect
406. C4×D4C22⋊D61semidirect

Maximal subgroups

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

subgroupindex
C2×#168401282
C2×#168399572
#214507652
#214468152
#214467772
#214466182
#214461412
#214461362
#214456442
C2×C61×#5532
#2558161