Finite Group Databasebeta

← #29691214

Subgroups — #29691214

Order 15616.

1296 conjugacy classes of subgroups.

The subgroup lattice has 624 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839895C211yesmaximal, characteristic
2C2×#16839664C211yesmaximal, characteristic, complemented
3#21446694C212yesmaximal, complemented
4#21446515C212yesmaximal, complemented
5#21446446C212yesmaximal, complemented
6#21446167C212yesmaximal, complemented
7#21446137C211yesmaximal, characteristic, complemented
8#21446011C211yesmaximal, characteristic, complemented
9#21445749C211yesmaximal, characteristic, complemented
10#21445581C211yesmaximal, characteristic, complemented
11C4×#121×C61C211yesmaximal, characteristic, complemented
12C23×D244C2211yescharacteristic
13C22×#11758199C2212yes—
14C22×C4⋊Dic61C2211yescharacteristic
15C22×#11758198—21no—
16C22×#11758198—21no—
17#16839895C418yescomplemented
18C2×#14519708C2212yescomplemented
19C2×#14519698C2212yescomplemented
20C2×#14519697C2212yes—
21C2×#14519697C2212yes—
22C2×#14519697C2212yes—
23C2×C22⋊#92418C2211yescharacteristic
24C2×C22⋊#92418C2211yescharacteristic
25C2×#14519688C2212yescomplemented
26#16839664—21no—
27#16839664—21no—
28#16839664C2212yes—
29#16839664C2212yes—
30#16839524C2212yescomplemented
31#16839524C2212yescomplemented
32#16839517C2212yescomplemented
33C2×#121×C61C2211yescharacteristic
34C2×C4×D4×C61—21no—
35C2×C4×D4×C61C2211yescharacteristic, complemented
36C22×C42×C61C2211yescharacteristic
37C61×#390C2211yescharacteristic, complemented
38C61×#389C2211yescharacteristic, complemented
39C61×#388—21no—
40C61×#386—21no—
41C61×#380C2211yescharacteristic
42C4×C4⋊C4×C61C2211yescharacteristic, complemented
43C4×C22⋊C4×C61—21no—
44C24×D61—21no—
45C22×D244—21no—
46C22×D244—22no—
47C22×D244—42no—
48C22×D244—42no—
49C22×D244C2×C414yescomplemented
50C22×D244C2312yes—
51C23×Dic61—21no—
52C2×#11758199—42no—
53C2×#11758199—42no—
54C2×#11758199C2×C418yescomplemented
55C22×#92418C2312yes—
56C22×#92418C2312yes—
57C2×C4⋊Dic61—21no—
58C2×C4⋊Dic61—42no—
59C2×C4⋊Dic61C2312yes—
60C2×#11758198—22no—
61C2×#11758198—22no—
62C2×#11758198—22no—
63C2×#11758198—22no—
64C2×#11758198—44no—
65C2×#11758198—44no—
66#14519708—42no—
67#14519698—42no—
68#14519697—42no—
69#14519697—42no—
70#14519697C2×C418yescomplemented
71C22⋊#92418—22no—
72C22⋊#92418C2×C414yescomplemented
73#14519688—42no—
74C22×D4×C61—21no—
75C22×D4×C61C2311yescharacteristic
76C23×C4×C61—21no—
77C23×C4×C61D411yescharacteristic
78C23×C4×C61D411yescharacteristic
79C23×C4×C61C2311yescharacteristic
80#121×C61—22no—
81#121×C61C2×C414yes—
82C4×D4×C61—42no—
83C4×D4×C61—44no—
84C2×C4⋊C4×C61—21no—
85C2×C4⋊C4×C61C2311yescharacteristic
86C2×C4⋊C4×C61C2311yescharacteristic
87C2×C4⋊C4×C61C2311yescharacteristic
88C2×C22⋊C4×C61—21no—
89C2×C22⋊C4×C61—21no—
90C2×C22⋊C4×C61—21no—
91C2×C22⋊C4×C61—21no—
92C2×C22⋊C4×C61—21no—
93C2×C42×C61—21no—
94C2×C42×C61—42no—
95C2×C42×C61D411yescharacteristic
96C2×C42×C61D411yescharacteristic
97C2×C42×C61C2311yescharacteristic
98C2×C42×C61C2311yescharacteristic
99#95×C61—21no—
100#95×C61—21no—
101#95×C61—21no—
102C24×C61—21no—
103C24×C61C2×D411yescharacteristic
104C23×D61—24no—
105C23×D61—22no—
106C23×D61—42no—
107C23×D61—42no—
108C23×D61—42no—
109C23×D61—42no—
110C23×D61—42no—
111C23×D61—42no—
112C2×D4×C61—24no—
113C2×D4×C61—21no—
114C2×D4×C61—22no—
115C2×D4×C61—22no—
116C2×D4×C61—42no—
117C2×D4×C61—42no—
118C2×D4×C61C22×C412yes—
119C22×C4×C61—22no—
120C22×C4×C61—22no—
121C22×C4×C61—22no—
122C22×C4×C61—22no—
123C22×C4×C61—21no—
124C22×C4×C61—22no—
125C22×C4×C61—21no—
126C22×C4×C61—21no—
127C22×C4×C61—41no—
128C22×C4×C61—41no—
129C22×C4×C61—41no—
130C22×C4×C61—42no—
131C22×C4×C61—42no—
132C22×C4×C61—42no—
133C22×C4×C61C22×C412yes—
134C22×C4×C61C2×D411yescharacteristic
135C22×C4×C61C2×D411yescharacteristic
136C22×C4×C61C2×D411yescharacteristic
137C22×C4×C61C2×D411yescharacteristic
138C22×C4×C61C2×D411yescharacteristic
139C22×C4×C61#4112yes—
140C22×C4×C61C2411yescharacteristic
141C22×Dic61—22no—
142C22×Dic61—42no—
143C22×Dic61—42no—
144C22×Dic61—42no—
145C2×D244—22no—
146C2×D244—22no—
147C2×D244—22no—
148C2×D244—22no—
149C2×D244—24no—
150C2×D244—22no—
151C2×D244—48no—
152C2×D244—42no—
153C2×D244—42no—
154C2×D244—48no—
155C2×D244C22×C414yes—
156C4⋊C4×C61—21no—
157C4⋊C4×C61—22no—
158C4⋊C4×C61—22no—
159C4⋊C4×C61—42no—
160C4⋊C4×C61C22×C412yes—
161C22⋊C4×C61—24no—
162C22⋊C4×C61—42no—
163C22⋊C4×C61—42no—
164C22⋊C4×C61—42no—
165C22⋊C4×C61—42no—
166C42×C61—22no—
167C42×C61—22no—
168C42×C61—41no—
169C42×C61—41no—
170C42×C61—42no—
171#11758199—48no—
172#11758198—44no—
173#11758198—44no—
174#11758198—44no—
175#11758198—44no—
176C4⋊Dic61—22no—
177C4⋊Dic61—22no—
178C4⋊Dic61—42no—
179C2×#92418—22no—
180C2×#92418—22no—
181C2×#92418—22no—
182C2×#92418—22no—
183C2×#92418—22no—
184C2×#92418C22×C414yes—
185C23×C61—21no—
186C23×C61—21no—
187C23×C61—21no—
188C23×C61—21no—
189C23×C61—22no—
190C23×C61—41no—
191C23×C61—41no—
192C23×C61—41no—
193C23×C61—41no—
194C23×C61—41no—
195C23×C61—41no—
196C23×C61C4×D412yes—
197C23×C61AGL(1,ℤ/8)12yes—
198C23×C61AGL(1,ℤ/8)12yes—
199C23×C61C22×D411yescharacteristic
200C22×D61—24no—
201C22×D61—48no—
202C22×D61—48no—
203C22×D61—42no—
204C22×D61—42no—
205C22×D61—42no—
206C22×D61—42no—
207C22×D61—42no—
208C22×D61—42no—
209C22×D61—48no—
210C22×D61—84no—
211C22×D61—84no—
212C22×D61—84no—
213C22×D61—84no—
214D4×C61—24no—
215D4×C61—48no—
216D4×C61—42no—
217C2×C4×C61—21no—
218C2×C4×C61—21no—
219C2×C4×C61—21no—
220C2×C4×C61—21no—
221C2×C4×C61—24no—
222C2×C4×C61—24no—
223C2×C4×C61—22no—
224C2×C4×C61—21no—
225C2×C4×C61—21no—
226C2×C4×C61—21no—
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—44no—
234C2×C4×C61—44no—
235C2×C4×C61—41no—
236C2×C4×C61—41no—
237C2×C4×C61—41no—
238C2×C4×C61—41no—
239C2×C4×C61—41no—
240C2×C4×C61—41no—
241C2×C4×C61—41no—
242C2×C4×C61—41no—
243C2×C4×C61—84no—
244C2×C4×C61C4×D412yes—
245C2×C4×C61AGL(1,ℤ/8)12yes—
246C2×C4×C61C23×C411yescharacteristic
247C2×C4×C61C2×#4111yescharacteristic
248C2×C4×C61H(5,𝔽2)12yes—
249C2×C4×C61H(5,𝔽2)12yes—
250C2×Dic61—42no—
251C2×Dic61—42no—
252C2×Dic61—42no—
253C2×Dic61—84no—
254D244—24no—
255D244—24no—
256D244—48no—
257D244—42no—
258D244—42no—
259#92418—24no—
260#92418—42no—
261#22046—611nomaximal, Sylow
262C22×C61—22no—
263C22×C61—22no—
264C22×C61—22no—
265C22×C61—22no—
266C22×C61—22no—
267C22×C61—22no—
268C22×C61—42no—
269C22×C61—42no—
270C22×C61—42no—
271C22×C61—81no—
272C22×C61—81no—
273C22×C61—81no—
274C22×C61—81no—
275C22×C61#44214yes—
276C22×C61C2×C4×D411yescharacteristic
277C22×C61#53411yescharacteristic
278C22×C61#53511yescharacteristic
279C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
280C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
281C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
282C22×C61#57511yescharacteristic
283C2×D61—44no—
284C2×D61—44no—
285C2×D61—44no—
286C2×D61—816no—
287C2×D61—82no—
288C2×D61—82no—
289C2×D61—82no—
290C2×D61—82no—
291C4×C61—22no—
292C4×C61—22no—
293C4×C61—41no—
294C4×C61—41no—
295C4×C61—41no—
296C4×C61—42no—
297C4×C61—81no—
298C4×C61—81no—
299C4×C61#51812yes—
300C4×C61#53812yes—
301Dic61—82no—
302C2×#447—611no—
303C4×#121D6111yescharacteristic, complemented
304C2×#390—611no—
305#1972—612no—
306#1936—612no—
307#1926—612no—
308#1879—612no—
309#1874—611no—
310#1853—611no—
311#1786—611no—
312#1749—611no—
313C2×C61—24no—
314C2×C61—82no—
315C2×C61#232111yescharacteristic
316C2×C61C2×#44211yescharacteristic
317C2×C61C4×AGL(1,ℤ/8)11yescharacteristic
318C2×C61#319211yescharacteristic
319C2×C61#319411yescharacteristic
320C2×C61#321311yescharacteristic
321C2×C61#321711yescharacteristic
322D61—84no—
323C23×D4—611no—
324C22×D8—612no—
325C2×#121C2×D6111yescharacteristic
326C2×C4×D4—21no—
327C2×C4×D4C2×D6111yescharacteristic, complemented
328C22×C4⋊C4—611no—
329C22×C22⋊C4—1221no—
330C22×C22⋊C4—1221no—
331C22×C42C2×D6111yescharacteristic
332#447—618no—
333C2×#106—612no—
334C2×#102—612no—
335C2×#102—612no—
336C2×#102—612no—
337C2×#102—612no—
338C2×C22⋊C8—611no—
339C2×C22⋊C8—611no—
340C2×#97—612no—
341#390—612no—
342#390—612no—
343#390—1221no—
344#390—1221no—
345#390C2×D6111yescharacteristic, complemented
346#389C2×D6111yescharacteristic, complemented
347#388—21no—
348#386—21no—
349#380C2×D6111yescharacteristic
350C4×C4⋊C4C2×D6111yescharacteristic, complemented
351C4×C22⋊C4—21no—
352#340—612no—
353#340—612no—
354#336—612no—
355C61#2204611yescharacteristic, complemented, Sylow
356C25—1221no—
357C22×D4—21no—
358C22×D4—614no—
359C22×D4—612no—
360C22×D4—1222no—
361C22×D4—1221no—
362C22×D4—2442no—
363C22×D4—2442no—
364C22×D4C22×D6111yescharacteristic
365C23×C4—21no—
366C23×C4—1221no—
367C23×C4C22⋊D6111yescharacteristic
368C23×C4C22⋊D6111yescharacteristic
369C23×C4C22×D6111yescharacteristic
370C2×D8—618no—
371C2×D8—2442no—
372C2×D8—2442no—
373C22×C8—612no—
374C22×C8—612no—
375#121—22no—
376#121C4×D6114yes—
377C4×D4—42no—
378C4×D4—44no—
379C2×C4⋊C4—21no—
380C2×C4⋊C4—612no—
381C2×C4⋊C4—1221no—
382C2×C4⋊C4—2442no—
383C2×C4⋊C4C22×D6111yescharacteristic
384C2×C4⋊C4C22×D6111yescharacteristic
385C2×C4⋊C4C22×D6111yescharacteristic
386C2×C22⋊C4—21no—
387C2×C22⋊C4—21no—
388C2×C22⋊C4—21no—
389C2×C22⋊C4—21no—
390C2×C22⋊C4—21no—
391C2×C22⋊C4—1222no—
392C2×C22⋊C4—1222no—
393C2×C22⋊C4—1222no—
394C2×C22⋊C4—1222no—
395C2×C22⋊C4—2444no—
396C2×C22⋊C4—2444no—
397C2×C42—21no—
398C2×C42—42no—
399C2×C42C22⋊D6111yescharacteristic
400C2×C42C22⋊D6111yescharacteristic
401C2×C42C22×D6111yescharacteristic
402C2×C42C22×D6111yescharacteristic
403#106—2442no—
404#102—618no—
405#102—2442no—
406#102—2442no—
407#102—2442no—
408C22⋊C8—614no—
409C22⋊C8—1222no—
410#97—2442no—
411#95—21no—
412#95—21no—
413#95—21no—
414C24—21no—
415C24—1222no—
416C24—1224no—
417C24—2442no—
418C24—2442no—
419C24—2442no—
420C24—2442no—
421C24—2442no—
422C24—2442no—
423C24C2×C22⋊D6111yescharacteristic
424C2×D4—24no—
425C2×D4—21no—
426C2×D4—22no—
427C2×D4—22no—
428C2×D4—42no—
429C2×D4—42no—
430C2×D4—614no—
431C2×D4—1222no—
432C2×D4—1222no—
433C2×D4—1222no—
434C2×D4—1222no—
435C2×D4—1222no—
436C2×D4—1224no—
437C2×D4—2448no—
438C2×D4—2442no—
439C2×D4—2442no—
440C2×D4—2448no—
441C2×D4C2×C4×D6112yes—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—21no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—21no—
449C22×C4—21no—
450C22×C4—41no—
451C22×C4—41no—
452C22×C4—41no—
453C22×C4—42no—
454C22×C4—42no—
455C22×C4—42no—
456C22×C4—1222no—
457C22×C4—2442no—
458C22×C4—2442no—
459C22×C4—2442no—
460C22×C4C2×C4×D6112yes—
461C22×C4#1175822212yes—
462C22×C4C2×C22⋊D6111yescharacteristic
463C22×C4C2×C22⋊D6111yescharacteristic
464C22×C4C2×C22⋊D6111yescharacteristic
465C22×C4C2×C22⋊D6111yescharacteristic
466C22×C4C2×C22⋊D6111yescharacteristic
467C22×C4C23×D6111yescharacteristic
468D8—2448no—
469C2×C8—614no—
470C2×C8—1222no—
471C2×C8—1222no—
472C2×C8—1222no—
473C2×C8—1222no—
474C2×C8—1222no—
475C4⋊C4—21no—
476C4⋊C4—22no—
477C4⋊C4—22no—
478C4⋊C4—42no—
479C4⋊C4—1222no—
480C4⋊C4—1222no—
481C4⋊C4—2442no—
482C4⋊C4C2×C4×D6112yes—
483C22⋊C4—24no—
484C22⋊C4—42no—
485C22⋊C4—42no—
486C22⋊C4—42no—
487C22⋊C4—42no—
488C22⋊C4—2444no—
489C22⋊C4—2444no—
490C22⋊C4—2444no—
491C22⋊C4—2444no—
492C42—22no—
493C42—22no—
494C42—41no—
495C42—41no—
496C42—42no—
497C23—21no—
498C23—21no—
499C23—21no—
500C23—21no—
501C23—22no—
502C23—41no—
503C23—41no—
504C23—41no—
505C23—41no—
506C23—41no—
507C23—41no—
508C23—1224no—
509C23—4884no—
510C23—4884no—
511C23—4884no—
512C23—4884no—
513C23—2448no—
514C23—2448no—
515C23—2442no—
516C23—2442no—
517C23—2442no—
518C23—2442no—
519C23—2442no—
520C23—2442no—
521C23—2448no—
522C23C4×C22⋊D6112yes—
523C23#1451980212yes—
524C23#1451980312yes—
525C23C22×C22⋊D6111yescharacteristic
526D4—24no—
527D4—48no—
528D4—42no—
529D4—1224no—
530D4—1224no—
531D4—2448no—
532D4—2442no—
533D4—2442no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—24no—
539C2×C4—24no—
540C2×C4—22no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—44no—
551C2×C4—44no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—84no—
561C2×C4—4884no—
562C2×C4—2442no—
563C2×C4—2442no—
564C2×C4—2442no—
565C2×C4C4×C22⋊D6112yes—
566C2×C4#1451980212yes—
567C2×C4C22×C4×D6111yescharacteristic
568C2×C4C2×#1175822211yescharacteristic
569C2×C4#1451982212yes—
570C2×C4#1451982312yes—
571C8—1224no—
572C8—2442no—
573C22—22no—
574C22—22no—
575C22—22no—
576C22—22no—
577C22—22no—
578C22—22no—
579C22—42no—
580C22—42no—
581C22—42no—
582C22—81no—
583C22—81no—
584C22—81no—
585C22—81no—
586C22—48816no—
587C22—4882no—
588C22—4882no—
589C22—4882no—
590C22—4882no—
591C22—2444no—
592C22—2444no—
593C22—2444no—
594C22#1683987214yes—
595C22C2×C4×C22⋊D6111yescharacteristic
596C22#1684032111yescharacteristic
597C22#1684032611yescharacteristic
598C22C2×#1451980211yescharacteristic
599C22C2×#1451980211yescharacteristic
600C22C2×#1451980311yescharacteristic
601C22#1684056011yescharacteristic
602C4—22no—
603C4—22no—
604C4—42no—
605C4—41no—
606C4—41no—
607C4—41no—
608C4—81no—
609C4—81no—
610C4—4882no—
611C4#1684023412yes—
612C4#1684034512yes—
613C2—24no—
614C2—82no—
615C2—4884no—
616C2#2144847311yescharacteristic
617C2C2×#1683987211yescharacteristic
618C2C4×#1451980211yescharacteristic
619C2#2145661611yescharacteristic
620C2#2145662911yescharacteristic
621C2#2145680211yescharacteristic
622C2#2145683411yescharacteristic
subgroupG/H
centerC23C22×C22⋊D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC4×#121×C61C2
socleC23×C61C22×D4
prime radicalC2×#16839895C2
cosocleC22×C4×C61C24
hypercenterC4×#121D61
hypocenterC61#22046
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C22×C42 ◁ C4×#121

Complements

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

NH ≅ G/Ntype
2. C2×#16839664C2semidirect
3. #21446694C2semidirect
4. #21446515C2semidirect
5. #21446446C2semidirect
6. #21446167C2semidirect
7. #21446137C2semidirect
8. #21446011C2semidirect
9. #21445749C2semidirect
10. #21445581C2semidirect
11. C4×#121×C61C2semidirect
17. #16839895C4semidirect
18. C2×#14519708C22semidirect
19. C2×#14519698C22semidirect
25. C2×#14519688C22semidirect
30. #16839524C22semidirect
31. #16839524C22semidirect
32. #16839517C22semidirect
35. C2×C4×D4×C61C22semidirect
37. C61×#390C22semidirect
38. C61×#389C22semidirect
42. C4×C4⋊C4×C61C22semidirect
49. C22×D244C2×C4semidirect
54. C2×#11758199C2×C4semidirect
70. #14519697C2×C4semidirect
72. C22⋊#92418C2×C4semidirect
303. C4×#121D61semidirect
327. C2×C4×D4C2×D61semidirect
345. #390C2×D61semidirect
346. #389C2×D61semidirect
350. C4×C4⋊C4C2×D61semidirect
355. C61#22046semidirect

Maximal subgroups

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

subgroupindex
C2×#168398952
C2×#168396642
#214466942
#214465152
#214464462
#214461672
#214461372
#214460112
#214457492
#214455812
C4×#121×C612
#2204661