Finite Group Databasebeta

← #29684354

Subgroups — #29684354

Order 15616.

874 conjugacy classes of subgroups.

The subgroup lattice has 620 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839860C211yesmaximal, characteristic
2#21448283C211yesmaximal, characteristic
3#21446703C211yesmaximal, characteristic
4#21446676C211yesmaximal, characteristic
5#21446554C211yesmaximal, characteristic, complemented
6#21446549C211yesmaximal, characteristic, complemented
7#21446515C211yesmaximal, characteristic
8#21446512C211yesmaximal, characteristic, complemented
9#21446501C211yesmaximal, characteristic
10#21446498C211yesmaximal, characteristic, complemented
11#21445717C211yesmaximal, characteristic, complemented
12#21445649C211yesmaximal, characteristic, complemented
13#21445647C211yesmaximal, characteristic
14C4×#14519688C211yesmaximal, characteristic, complemented
15C61×#2295C211yesmaximal, characteristic, complemented
16C22×#11758201C2211yescharacteristic
17C2×C4×D244C2211yescharacteristic
18C2×#14519734—21no—
19#16839860C418yescomplemented
20C2×#14519710C2211yescharacteristic
21C2×#14519710C2211yescharacteristic
22C2×#14519708C2211yescharacteristic
23C2×#14519701C2211yescharacteristic
24C2×#14519701C2211yescharacteristic
25C2×#14519701C2211yescharacteristic
26C2×#14519697C2211yescharacteristic
27C2×#14519697C2211yescharacteristic
28C2×#14519697C2211yescharacteristic
29C2×#14519688C2211yescharacteristic
30C2×#14519688C2211yescharacteristic
31C2×#14519688C2211yescharacteristic
32C2×#14519688C2211yescharacteristic
33C2×C4×#92418C2211yescharacteristic
34C2×C4×#92418C2211yescharacteristic
35#16839664C2211yescharacteristic
36#16839664C2211yescharacteristic
37#16839660C2211yescharacteristic
38#16839660C2211yescharacteristic
39#16839653—21no—
40#16839653—21no—
41C4×C4⋊Dic61C2211yescharacteristic
42C4×#11758198—21no—
43#16839524C2211yescharacteristic
44#16839524C2211yescharacteristic
45#16839524C2211yescharacteristic
46#16839524C2211yescharacteristic
47#16839524C2211yescharacteristic
48#16839524C2211yescharacteristic
49C2×C4×Q8×C61C2211yescharacteristic
50C61×#391C2211yescharacteristic
51C61×#391C2211yescharacteristic
52C61×#389C2211yescharacteristic
53C61×#389C2211yescharacteristic
54C61×#387—21no—
55C61×#387—21no—
56C4×C4⋊C4×C61—21no—
57C4×C4⋊C4×C61C2211yescharacteristic
58C61×#376—21no—
59C43×C61C2211yescharacteristic
60C22×D244C2311yescharacteristic
61C22×C4×D61—21no—
62C2×#11758201—22no—
63C2×#11758201—22no—
64C2×#11758201C2×C414yes—
65C22×#92418C2311yescharacteristic
66C22×#92418C2311yescharacteristic
67C22×#92418C2311yescharacteristic
68C22×#92418C2311yescharacteristic
69C4×D244—22no—
70C4×D244C2×C414yes—
71#14519734—44no—
72C2×C4⋊Dic61C2311yescharacteristic
73C2×C4⋊Dic61C2311yescharacteristic
74C2×C4⋊Dic61C2311yescharacteristic
75C2×#11758196—21no—
76C2×#11758196—21no—
77C2×#11758198—21no—
78C2×#11758198—21no—
79C2×#11758198—21no—
80C2×#11758198—21no—
81C2×#11758198—21no—
82C2×C4×Dic61—21no—
83#14519710—22no—
84#14519710C2×C414yes—
85#14519708—22no—
86#14519701—22no—
87#14519701—22no—
88#14519701C2×C414yes—
89#14519697—22no—
90#14519697—22no—
91#14519697C2×C414yes—
92#14519688—22no—
93#14519688—22no—
94#14519688—22no—
95#14519688C2×C414yes—
96C4×#92418—22no—
97C4×#92418—22no—
98#14519684—21no—
99#14519684—21no—
100#14519684—21no—
101C22×Q8×C61C2311yescharacteristic
102C4×Q8×C61—22no—
103C4×Q8×C61C2×C414yes—
104C2×C4⋊C4×C61—21no—
105C2×C4⋊C4×C61—21no—
106C2×C4⋊C4×C61—21no—
107C2×C4⋊C4×C61C2311yescharacteristic
108C2×C4⋊C4×C61C2311yescharacteristic
109C2×C4⋊C4×C61C2311yescharacteristic
110C2×C42×C61—21no—
111C2×C42×C61—21no—
112C2×C42×C61D411yescharacteristic
113C2×C42×C61D411yescharacteristic
114C2×C42×C61D411yescharacteristic
115C2×C42×C61D411yescharacteristic
116C2×C42×C61C2311yescharacteristic
117C2×C42×C61C2311yescharacteristic
118C2×C42×C61C2311yescharacteristic
119#95×C61—21no—
120#95×C61—21no—
121#95×C61—21no—
122#95×C61—21no—
123#95×C61—21no—
124#95×C61—21no—
125#95×C61—21no—
126C23×D61—21no—
127C2×Q8×C61—21no—
128C2×Q8×C61—21no—
129C2×Q8×C61—21no—
130C2×Q8×C61—21no—
131C2×Q8×C61—21no—
132C2×Q8×C61C22×C412yes—
133C22×C4×C61—21no—
134C22×C4×C61—21no—
135C22×C4×C61—21no—
136C22×C4×C61—21no—
137C22×C4×C61C2×D411yescharacteristic
138C22×C4×C61C2×D411yescharacteristic
139C22×C4×C61C2×D411yescharacteristic
140C22×C4×C61C2×D411yescharacteristic
141C22×C4×C61C2×D411yescharacteristic
142C22×C4×C61C2×D411yescharacteristic
143C22×C4×C61C2411yescharacteristic
144C22×Dic61—21no—
145C22×Dic61—21no—
146C22×Dic61—21no—
147C2×D244—21no—
148C2×D244—21no—
149C2×D244—21no—
150C2×D244—21no—
151C2×D244—21no—
152C2×D244C22×C412yes—
153C2×C4×D61—24no—
154C2×C4×D61—42no—
155C2×C4×D61—42no—
156C4⋊C4×C61—24no—
157C4⋊C4×C61—21no—
158C4⋊C4×C61—21no—
159C4⋊C4×C61—21no—
160C4⋊C4×C61—21no—
161C4⋊C4×C61—21no—
162C4⋊C4×C61—42no—
163C4⋊C4×C61—42no—
164C4⋊C4×C61C22×C412yes—
165C42×C61—24no—
166C42×C61—22no—
167C42×C61—22no—
168C42×C61—21no—
169C42×C61—21no—
170C42×C61—21no—
171C42×C61—21no—
172C42×C61—41no—
173C42×C61—41no—
174C42×C61—42no—
175C42×C61C22×C412yes—
176C42×C61#4112yes—
177#11758201—28no—
178#11758198—24no—
179#11758198—42no—
180#11758198—42no—
181#11758198—42no—
182#11758198—42no—
183C4⋊Dic61—21no—
184C4⋊Dic61—21no—
185C4⋊Dic61—21no—
186C4⋊Dic61—21no—
187C4⋊Dic61—21no—
188C4⋊Dic61C22×C412yes—
189#11758196—42no—
190#11758196—42no—
191C4×Dic61—42no—
192C2×#92418—21no—
193C2×#92418—21no—
194C2×#92418—21no—
195C2×#92418—21no—
196C2×#92418—21no—
197C2×#92418—21no—
198C2×#92418C22×C412yes—
199C2×#92418C22×C412yes—
200C2×#92418#4112yes—
201C2×#92418#4112yes—
202C2×#92418#4112yes—
203C2×#92418#4112yes—
204C23×C61C22×D411yescharacteristic
205C22×D61—22no—
206C22×D61—41no—
207C22×D61—41no—
208C22×D61—41no—
209C22×D61—41no—
210C22×D61—41no—
211C22×D61—41no—
212Q8×C61—22no—
213Q8×C61—22no—
214Q8×C61—41no—
215Q8×C61—41no—
216C2×C4×C61—21no—
217C2×C4×C61—21no—
218C2×C4×C61—21no—
219C2×C4×C61—21no—
220C2×C4×C61—21no—
221C2×C4×C61—21no—
222C2×C4×C61—21no—
223C2×C4×C61—21no—
224C2×C4×C61—22no—
225C2×C4×C61—22no—
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—41no—
233C2×C4×C61—41no—
234C2×C4×C61—41no—
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×C61C4×D412yes—
243C2×C4×C61C4×D412yes—
244C2×C4×C61AGL(1,ℤ/8)12yes—
245C2×C4×C61#13712yes—
246C2×C4×C61C23×C411yescharacteristic
247C2×C4×C61C2×#4111yescharacteristic
248C2×C4×C61C2×#4111yescharacteristic
249C2×C4×C61C2×#4111yescharacteristic
250C2×C4×C61C2×#4111yescharacteristic
251C2×C4×C61C2×#4111yescharacteristic
252C2×Dic61—22no—
253C2×Dic61—41no—
254C2×Dic61—41no—
255C2×Dic61—41no—
256C2×Dic61—41no—
257C2×Dic61—41no—
258C2×Dic61—41no—
259C2×Dic61—41no—
260C2×Dic61—41no—
261D244—22no—
262D244—22no—
263D244—41no—
264D244—41no—
265C4×D61—48no—
266#92418—22no—
267#92418—22no—
268#92418—22no—
269#92418#11714yescomplemented
270#21288—611nomaximal, Sylow
271C22×C61C2×C4×D411yescharacteristic
272C22×C61#53211yescharacteristic
273C22×C61#53211yescharacteristic
274C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
275C22×C61C2×#13711yescharacteristic
276C22×C61#57511yescharacteristic
277C22×C61#57511yescharacteristic
278C2×D61—42no—
279C2×D61—42no—
280C2×D61—42no—
281C2×D61—81no—
282C2×D61—81no—
283C2×D61—81no—
284C2×D61—81no—
285C4×C61—22no—
286C4×C61—22no—
287C4×C61—22no—
288C4×C61—22no—
289C4×C61—41no—
290C4×C61—41no—
291C4×C61—42no—
292C4×C61—42no—
293C4×C61—81no—
294C4×C61—81no—
295C4×C61C2×#11711yescharacteristic
296C4×C61C4×#4111yescharacteristic
297C4×C61#52911yescharacteristic
298C4×C61#52911yescharacteristic
299Dic61—42no—
300Dic61—81no—
301Dic61—81no—
302C2×#440—611no—
303#2295D6111yescharacteristic, complemented
304#2293—611no—
305#1973—611no—
306#1968—611no—
307#1943—611no—
308#1942—611no—
309#1936—611no—
310#1935—611no—
311#1934—611no—
312#1933—611no—
313#1776—611no—
314#1763—611no—
315#1762—611no—
316C4×#97—611no—
317C2×C61#229311yescharacteristic
318C2×C61C4×AGL(1,ℤ/8)11yescharacteristic
319C2×C61C4×#13711yescharacteristic
320C2×C61#318011yescharacteristic
321C2×C61#318111yescharacteristic
322C2×C61#318211yescharacteristic
323C2×C61#318311yescharacteristic
324D61—82no—
325C22×SD16—611no—
326C2×C4×Q8C2×D6111yescharacteristic
327C2×C4×D4—611no—
328C2×#117—1221no—
329#440—618no—
330C2×#107—611no—
331C2×#107—611no—
332C2×#106—611no—
333C2×#103—611no—
334C2×#103—611no—
335C2×#103—611no—
336C2×#102—611no—
337C2×#102—611no—
338C2×#102—611no—
339C2×#97—611no—
340C2×#97—611no—
341C2×#97—611no—
342C2×#97—611no—
343C2×C4×C8—611no—
344C2×C4×C8—611no—
345#391C2×D6111yescharacteristic
346#391C2×D6111yescharacteristic
347#390—611no—
348#390—611no—
349#389—611no—
350#389—611no—
351#389C2×D6111yescharacteristic
352#389C2×D6111yescharacteristic
353#388—1221no—
354#388—1221no—
355#387—21no—
356#387—21no—
357C4×C4⋊C4—21no—
358C4×C4⋊C4—611no—
359C4×C4⋊C4C2×D6111yescharacteristic
360C4×C22⋊C4—1221no—
361#376—21no—
362C43C2×D6111yescharacteristic
363#340—611no—
364#340—611no—
365#340—611no—
366#340—611no—
367#340—611no—
368#340—611no—
369C61#2128811yescharacteristic, complemented, Sylow
370C22×Q8C22×D6111yescharacteristic
371C22×D4—611no—
372C23×C4—1221no—
373C2×SD16—614no—
374C2×SD16—1222no—
375C2×SD16—1222no—
376C22×C8—611no—
377C22×C8—611no—
378C22×C8—611no—
379C22×C8—611no—
380C4×Q8—22no—
381C4×Q8C4×D6114yes—
382C4×D4—614no—
383C4×D4—1222no—
384#117—2444no—
385C2×C4⋊C4—21no—
386C2×C4⋊C4—21no—
387C2×C4⋊C4—21no—
388C2×C4⋊C4—611no—
389C2×C4⋊C4—611no—
390C2×C4⋊C4—611no—
391C2×C4⋊C4—1221no—
392C2×C4⋊C4—1221no—
393C2×C4⋊C4C22×D6111yescharacteristic
394C2×C4⋊C4C22×D6111yescharacteristic
395C2×C4⋊C4C22×D6111yescharacteristic
396C2×C22⋊C4—1221no—
397C2×C22⋊C4—1221no—
398C2×C22⋊C4—1221no—
399C2×C22⋊C4—1221no—
400C2×C22⋊C4—1221no—
401C2×C42—21no—
402C2×C42—21no—
403C2×C42—1221no—
404C2×C42C22⋊D6111yescharacteristic
405C2×C42C22⋊D6111yescharacteristic
406C2×C42C22⋊D6111yescharacteristic
407C2×C42C22⋊D6111yescharacteristic
408C2×C42C22×D6111yescharacteristic
409C2×C42C22×D6111yescharacteristic
410C2×C42C22×D6111yescharacteristic
411#107—614no—
412#107—1222no—
413#106—1222no—
414#103—614no—
415#103—1222no—
416#103—1222no—
417#102—614no—
418#102—1222no—
419#102—1222no—
420#97—614no—
421#97—1222no—
422#97—1222no—
423#97—1222no—
424C4×C8—1222no—
425C4×C8—1222no—
426#95—21no—
427#95—21no—
428#95—21no—
429#95—21no—
430#95—21no—
431#95—21no—
432#95—21no—
433#95—1221no—
434#95—1221no—
435#95—1221no—
436C24—1221no—
437C2×Q8—21no—
438C2×Q8—21no—
439C2×Q8—21no—
440C2×Q8—21no—
441C2×Q8—21no—
442C2×Q8C2×C4×D6112yes—
443C2×D4—612no—
444C2×D4—1221no—
445C2×D4—1221no—
446C2×D4—1221no—
447C2×D4—1221no—
448C2×D4—1221no—
449C22×C4—21no—
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—21no—
453C22×C4—1221no—
454C22×C4—1224no—
455C22×C4—1221no—
456C22×C4—1221no—
457C22×C4—2442no—
458C22×C4—2442no—
459C22×C4C2×C22⋊D6111yescharacteristic
460C22×C4C2×C22⋊D6111yescharacteristic
461C22×C4C2×C22⋊D6111yescharacteristic
462C22×C4C2×C22⋊D6111yescharacteristic
463C22×C4C2×C22⋊D6111yescharacteristic
464C22×C4C2×C22⋊D6111yescharacteristic
465C22×C4C23×D6111yescharacteristic
466SD16—1228no—
467C2×C8—612no—
468C2×C8—612no—
469C2×C8—612no—
470C2×C8—612no—
471C2×C8—612no—
472C2×C8—612no—
473C2×C8—1221no—
474C2×C8—1221no—
475C2×C8—1221no—
476C2×C8—1221no—
477C2×C8—1221no—
478C2×C8—1221no—
479C4⋊C4—21no—
480C4⋊C4—24no—
481C4⋊C4—21no—
482C4⋊C4—21no—
483C4⋊C4—21no—
484C4⋊C4—21no—
485C4⋊C4—42no—
486C4⋊C4—42no—
487C4⋊C4—612no—
488C4⋊C4—1221no—
489C4⋊C4—1221no—
490C4⋊C4—1221no—
491C4⋊C4—1221no—
492C4⋊C4—1221no—
493C4⋊C4—2442no—
494C4⋊C4—2442no—
495C4⋊C4C2×C4×D6112yes—
496C22⋊C4—1224no—
497C22⋊C4—2442no—
498C22⋊C4—2442no—
499C22⋊C4—2442no—
500C22⋊C4—2442no—
501C42—22no—
502C42—21no—
503C42—21no—
504C42—21no—
505C42—21no—
506C42—24no—
507C42—22no—
508C42—41no—
509C42—41no—
510C42—42no—
511C42—2442no—
512C42C2×C4×D6112yes—
513C42#1175822212yes—
514C23—1222no—
515C23—2441no—
516C23—2441no—
517C23—2441no—
518C23—2441no—
519C23—2441no—
520C23—2441no—
521C23C22×C22⋊D6111yescharacteristic
522Q8—22no—
523Q8—22no—
524Q8—41no—
525Q8—41no—
526D4—1222no—
527D4—1222no—
528D4—2441no—
529D4—2441no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—1222no—
557C2×C4—2448no—
558C2×C4—2441no—
559C2×C4—2441no—
560C2×C4—2441no—
561C2×C4—2441no—
562C2×C4—2441no—
563C2×C4—2441no—
564C2×C4—2441no—
565C2×C4—2441no—
566C2×C4C4×C22⋊D6112yes—
567C2×C4C4×C22⋊D6112yes—
568C2×C4#1451980212yes—
569C2×C4#1451980812yes—
570C2×C4C22×C4×D6111yescharacteristic
571C2×C4#41×D6111yescharacteristic
572C2×C4#41×D6111yescharacteristic
573C2×C4#41×D6111yescharacteristic
574C2×C4#41×D6111yescharacteristic
575C2×C4C2×#1175822211yescharacteristic
576C8—614no—
577C8—1222no—
578C8—1222no—
579C8—1222no—
580C22—4881no—
581C22—4881no—
582C22—4881no—
583C22—4881no—
584C22—2442no—
585C22—2442no—
586C22—2442no—
587C22C2×C4×C22⋊D6111yescharacteristic
588C22#1684030811yescharacteristic
589C22#1684030811yescharacteristic
590C22C2×#1451980211yescharacteristic
591C22C2×#1451980811yescharacteristic
592C22#1684056011yescharacteristic
593C22#1684056011yescharacteristic
594C4—22no—
595C4—22no—
596C4—22no—
597C4—22no—
598C4—42no—
599C4—42no—
600C4—41no—
601C4—41no—
602C4—81no—
603C4—81no—
604C4—4881no—
605C4—4881no—
606C4—2442no—
607C4#117×D6111yescharacteristic
608C4#1684023011yescharacteristic
609C4#1684029011yescharacteristic
610C4#1684029011yescharacteristic
611C2—4882no—
612C2#2144827811yescharacteristic
613C2C4×#1451980211yescharacteristic
614C2C4×#1451980811yescharacteristic
615C2#2145651811yescharacteristic
616C2#2145652211yescharacteristic
617C2#2145653811yescharacteristic
618C2#2145654211yescharacteristic
subgroupG/H
centerC23C22×C22⋊D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#2295C2
socleC23×C61C22×D4
prime radicalC22×D244C23
cosocleC22×C4×C61C24
hypercenter#2295D61
hypocenterC61#21288
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C43 ◁ #2295

Complements

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

NH ≅ G/Ntype
5. #21446554C2semidirect
6. #21446549C2semidirect
8. #21446512C2semidirect
10. #21446498C2semidirect
11. #21445717C2semidirect
12. #21445649C2semidirect
14. C4×#14519688C2semidirect
15. C61×#2295C2semidirect
19. #16839860C4semidirect
269. #92418#117semidirect
303. #2295D61semidirect
369. C61#21288semidirect

Maximal subgroups

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

subgroupindex
C2×#168398602
#214482832
#214467032
#214466762
#214465542
#214465492
#214465152
#214465122
#214465012
#214464982
#214457172
#214456492
#214456472
C4×#145196882
C61×#22952
#2128861