Finite Group Databasebeta

← #29699450

Subgroups — #29699450

Order 15616.

866 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×#16839862C211yesmaximal, characteristic
2#21448739C211yesmaximal, characteristic, complemented
3#21446519C211yesmaximal, characteristic
4#21446505C211yesmaximal, characteristic
5#21446411C211yesmaximal, characteristic, complemented
6#21446390C211yesmaximal, characteristic, complemented
7#21446177C211yesmaximal, characteristic
8#21446177C211yesmaximal, characteristic
9#21445954C211yesmaximal, characteristic, complemented
10#21445829C211yesmaximal, characteristic
11#21445821C211yesmaximal, characteristic, complemented
12#21445694C211yesmaximal, characteristic, complemented
13#21445652C211yesmaximal, characteristic, complemented
14C4×#14519700C211yesmaximal, characteristic, complemented
15C61×#2354C211yesmaximal, characteristic
16C22×#11758200C2211yescharacteristic
17C2×C4×Dic122C2211yescharacteristic
18C2×C4×Dic122C2211yescharacteristic
19C2×C4×Dic122C2211yescharacteristic
20#16839862C418yescomplemented
21C2×#14519710C2211yescharacteristic
22C2×C4⋊#92418C2211yescharacteristic
23C2×#14519700C2211yescharacteristic
24C2×#14519700C2211yescharacteristic
25C2×#14519700C2211yescharacteristic
26C2×#14519698C2211yescharacteristic
27C2×#14519698C2211yescharacteristic
28C2×#14519698C2211yescharacteristic
29C2×#14519688C2211yescharacteristic
30C2×#14519688C2211yescharacteristic
31C2×C4×#92418C2211yescharacteristic
32#16839660C2211yescharacteristic
33#16839660C2211yescharacteristic
34#16839657—21no—
35#16839657—21no—
36#16839628C2211yescharacteristic
37#16839626—21no—
38#16839613—21no—
39#16839524C2211yescharacteristic
40#16839524C2211yescharacteristic
41#16839524C2211yescharacteristic
42#16839524C2211yescharacteristic
43#16839524C2211yescharacteristic
44#16839524C2211yescharacteristic
45#16839524C2211yescharacteristic
46#16839524C2211yescharacteristic
47#16839517C2211yescharacteristic
48#16839517C2211yescharacteristic
49C2×C4×D4×C61C2211yescharacteristic
50C2×#117×C61—21no—
51C61×#390C2211yescharacteristic
52C61×#390C2211yescharacteristic
53C61×#388—21no—
54C61×#388—21no—
55C61×#384C2211yescharacteristic
56C61×#380—21no—
57C4×C4⋊C4×C61C2211yescharacteristic
58C4×C4⋊C4×C61C2211yescharacteristic
59C4×C4⋊C4×C61C2211yescharacteristic
60C22×Dic122C2311yescharacteristic
61C2×#11758200—22no—
62C2×#11758200—22no—
63C2×#11758200C2×C414yes—
64C22×#92418C2311yescharacteristic
65C22×#92418C2311yescharacteristic
66C22×#92418C2311yescharacteristic
67C22×#92418C2311yescharacteristic
68C4×Dic122—22no—
69C4×Dic122—22no—
70C4×Dic122—22no—
71C4×Dic122—41no—
72C4×Dic122—41no—
73C4×Dic122C2×C414yes—
74C2×C4⋊Dic61C2311yescharacteristic
75C2×C4⋊Dic61C2311yescharacteristic
76C2×C4⋊Dic61C2311yescharacteristic
77C2×#11758196—21no—
78C2×#11758196—21no—
79C2×#11758196—21no—
80C2×#11758196—21no—
81C2×#11758196—21no—
82C2×C4×Dic61—21no—
83C2×C4×Dic61—21no—
84C2×C4×Dic61—21no—
85#14519710C2×C414yes—
86C4⋊#92418—22no—
87#14519700—22no—
88#14519700—22no—
89#14519700C2×C414yes—
90#14519698—41no—
91#14519698—41no—
92#14519698C2×C414yes—
93#14519688—41no—
94#14519688C2×C414yes—
95C4×#92418—41no—
96#14519684—21no—
97#14519684—21no—
98#14519684—21no—
99#14519684—21no—
100C22×D4×C61C2311yescharacteristic
101C23×C4×C61—21no—
102C4×D4×C61—22no—
103C4×D4×C61C2×C414yes—
104#117×C61—44no—
105C2×C4⋊C4×C61—21no—
106C2×C4⋊C4×C61—21no—
107C2×C4⋊C4×C61D411yescharacteristic
108C2×C4⋊C4×C61D411yescharacteristic
109C2×C4⋊C4×C61C2311yescharacteristic
110C2×C4⋊C4×C61C2311yescharacteristic
111C2×C4⋊C4×C61C2311yescharacteristic
112C2×C22⋊C4×C61—21no—
113C2×C22⋊C4×C61—21no—
114C2×C22⋊C4×C61—21no—
115C2×C22⋊C4×C61—21no—
116C2×C22⋊C4×C61—21no—
117C2×C42×C61—21no—
118C2×C42×C61—21no—
119C2×C42×C61C2311yescharacteristic
120C2×C42×C61C2311yescharacteristic
121C2×C42×C61C2311yescharacteristic
122#95×C61—21no—
123#95×C61—21no—
124#95×C61D411yescharacteristic
125#95×C61D411yescharacteristic
126C24×C61—21no—
127C2×D4×C61—21no—
128C2×D4×C61—21no—
129C2×D4×C61—21no—
130C2×D4×C61—21no—
131C2×D4×C61—21no—
132C2×D4×C61C22×C412yes—
133C22×C4×C61—21no—
134C22×C4×C61—24no—
135C22×C4×C61—21no—
136C22×C4×C61—21no—
137C22×C4×C61—42no—
138C22×C4×C61—42no—
139C22×C4×C61C2×D411yescharacteristic
140C22×C4×C61C2×D411yescharacteristic
141C22×C4×C61C2×D411yescharacteristic
142C22×C4×C61C2×D411yescharacteristic
143C22×C4×C61C2×D411yescharacteristic
144C22×C4×C61C2×D411yescharacteristic
145C22×C4×C61C2411yescharacteristic
146C22×Dic61—21no—
147C22×Dic61—21no—
148C22×Dic61—21no—
149C22×Dic61—21no—
150C2×Dic122—21no—
151C2×Dic122—21no—
152C2×Dic122—21no—
153C2×Dic122C22×C412yes—
154C2×Dic122#4112yes—
155C2×Dic122#4112yes—
156C4⋊C4×C61—22no—
157C4⋊C4×C61—21no—
158C4⋊C4×C61—21no—
159C4⋊C4×C61—21no—
160C4⋊C4×C61—21no—
161C4⋊C4×C61—21no—
162C4⋊C4×C61—41no—
163C4⋊C4×C61—42no—
164C4⋊C4×C61—81no—
165C4⋊C4×C61C22×C412yes—
166C22⋊C4×C61—24no—
167C22⋊C4×C61—42no—
168C22⋊C4×C61—42no—
169C22⋊C4×C61—81no—
170C22⋊C4×C61—81no—
171C42×C61—21no—
172C42×C61—21no—
173C42×C61—21no—
174C42×C61—21no—
175C42×C61—42no—
176C42×C61—42no—
177C42×C61C22×C412yes—
178C42×C61#4112yes—
179#11758200—28no—
180C4⋊Dic61—21no—
181C4⋊Dic61—21no—
182C4⋊Dic61—21no—
183C4⋊Dic61—21no—
184C4⋊Dic61—21no—
185C4⋊Dic61C22×C412yes—
186#11758196—24no—
187#11758196—42no—
188#11758196—42no—
189#11758196—42no—
190#11758196—42no—
191C4×Dic61—24no—
192C4×Dic61—42no—
193C4×Dic61—42no—
194C2×#92418—21no—
195C2×#92418—21no—
196C2×#92418—21no—
197C2×#92418—21no—
198C2×#92418—21no—
199C2×#92418—21no—
200C2×#92418—21no—
201C2×#92418—21no—
202C2×#92418—21no—
203C2×#92418—21no—
204C2×#92418C22×C412yes—
205C2×#92418C22×C412yes—
206C23×C61—22no—
207C23×C61—41no—
208C23×C61—41no—
209C23×C61—41no—
210C23×C61—41no—
211C23×C61—41no—
212C23×C61—41no—
213C23×C61C22×D411yescharacteristic
214D4×C61—22no—
215D4×C61—22no—
216D4×C61—41no—
217D4×C61—41no—
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—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—21no—
234C2×C4×C61—48no—
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×C61C4×D412yes—
244C2×C4×C61C4×D412yes—
245C2×C4×C61AGL(1,ℤ/8)12yes—
246C2×C4×C61C23×C411yescharacteristic
247C2×C4×C61C2×#4111yescharacteristic
248C2×C4×C61C2×#4111yescharacteristic
249C2×C4×C61C2×#4111yescharacteristic
250C2×C4×C61H(5,𝔽2)11yescharacteristic
251C2×C4×C612−1+411yescharacteristic
252C2×Dic61—22no—
253C2×Dic61—22no—
254C2×Dic61—22no—
255C2×Dic61—22no—
256C2×Dic61—41no—
257C2×Dic61—41no—
258C2×Dic61—41no—
259C2×Dic61—41no—
260C2×Dic61—41no—
261C2×Dic61—41no—
262C2×Dic61—41no—
263C2×Dic61—41no—
264Dic122—22no—
265Dic122—22no—
266Dic122—22no—
267Dic122#11714yescomplemented
268#92418—22no—
269#92418—22no—
270#92418—41no—
271#92418—41no—
272#22781—611nomaximal, Sylow
273C22×C61—42no—
274C22×C61—42no—
275C22×C61—42no—
276C22×C61—81no—
277C22×C61—81no—
278C22×C61—81no—
279C22×C61—81no—
280C22×C61C2×C4×D411yescharacteristic
281C22×C61#54611yescharacteristic
282C22×C61#54811yescharacteristic
283C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
284C22×C61#57511yescharacteristic
285C22×C61#57711yescharacteristic
286C22×C61#57811yescharacteristic
287C4×C61—22no—
288C4×C61—22no—
289C4×C61—22no—
290C4×C61—42no—
291C4×C61—41no—
292C4×C61—41no—
293C4×C61—41no—
294C4×C61—81no—
295C4×C61—81no—
296C4×C61C2×#11711yescharacteristic
297C4×C61#52011yescharacteristic
298C4×C61#55211yescharacteristic
299C4×C61#55311yescharacteristic
300Dic61—24no—
301Dic61—42no—
302Dic61—81no—
303Dic61—81no—
304C2×#440—611no—
305#2354D6111yescharacteristic
306#2351—611no—
307#1936—611no—
308#1934—611no—
309#1921—611no—
310#1918—611no—
311#1880—611no—
312#1880—611no—
313#1836—611no—
314#1804—611no—
315#1803—611no—
316#1771—611no—
317#1763—611no—
318C4×#103—611no—
319C2×C61—82no—
320C2×C61#235211yescharacteristic
321C2×C61C4×AGL(1,ℤ/8)11yescharacteristic
322C2×C61#294711yescharacteristic
323C2×C61#330811yescharacteristic
324C2×C61#332111yescharacteristic
325C2×C61#336811yescharacteristic
326C2×C61#337711yescharacteristic
327C22×SD16—611no—
328C2×C4×Q8—611no—
329C2×C4×Q8—611no—
330C2×C4×Q8—611no—
331C2×C4×D4C2×D6111yescharacteristic
332C2×#117—21no—
333#440—618no—
334C2×#107—611no—
335C2×C4⋊C8—611no—
336C2×#103—611no—
337C2×#103—611no—
338C2×#103—611no—
339C2×#102—611no—
340C2×#102—611no—
341C2×#102—611no—
342C2×#97—611no—
343C2×#97—611no—
344C2×C4×C8—611no—
345#390C2×D6111yescharacteristic
346#390C2×D6111yescharacteristic
347#389—611no—
348#389—611no—
349#389—1221no—
350#389—1221no—
351#388—21no—
352#388—21no—
353#384C2×D6111yescharacteristic
354#382—611no—
355#382—1221no—
356#380—21no—
357C4×C4⋊C4—1221no—
358C4×C4⋊C4C2×D6111yescharacteristic
359C4×C4⋊C4C2×D6111yescharacteristic
360C4×C4⋊C4C2×D6111yescharacteristic
361#340—611no—
362#340—611no—
363#340—611no—
364#340—611no—
365#340—611no—
366#340—611no—
367#340—611no—
368#340—611no—
369#336—611no—
370#336—611no—
371C61#2278111yescharacteristic, complemented, Sylow
372C22×Q8—611no—
373C22×D4C22×D6111yescharacteristic
374C23×C4—21no—
375C2×SD16—614no—
376C2×SD16—1222no—
377C2×SD16—1222no—
378C22×C8—611no—
379C22×C8—611no—
380C22×C8—611no—
381C22×C8—611no—
382C4×Q8—614no—
383C4×Q8—1222no—
384C4×Q8—1222no—
385C4×Q8—1222no—
386C4×Q8—2441no—
387C4×Q8—2441no—
388C4×D4—22no—
389C4×D4C4×D6114yes—
390#117—44no—
391C2×C4⋊C4—21no—
392C2×C4⋊C4—21no—
393C2×C4⋊C4—611no—
394C2×C4⋊C4—611no—
395C2×C4⋊C4—611no—
396C2×C4⋊C4—1221no—
397C2×C4⋊C4—1221no—
398C2×C4⋊C4—1221no—
399C2×C4⋊C4—1221no—
400C2×C4⋊C4—1221no—
401C2×C4⋊C4C22⋊D6111yescharacteristic
402C2×C4⋊C4C22⋊D6111yescharacteristic
403C2×C4⋊C4C22×D6111yescharacteristic
404C2×C4⋊C4C22×D6111yescharacteristic
405C2×C4⋊C4C22×D6111yescharacteristic
406C2×C22⋊C4—21no—
407C2×C22⋊C4—21no—
408C2×C22⋊C4—21no—
409C2×C22⋊C4—21no—
410C2×C22⋊C4—21no—
411C2×C42—21no—
412C2×C42—21no—
413C2×C42—1221no—
414C2×C42—1221no—
415C2×C42—1221no—
416C2×C42C22×D6111yescharacteristic
417C2×C42C22×D6111yescharacteristic
418C2×C42C22×D6111yescharacteristic
419#107—614no—
420C4⋊C8—1222no—
421#103—614no—
422#103—1222no—
423#103—1222no—
424#102—614no—
425#102—2441no—
426#102—2441no—
427#97—614no—
428#97—2441no—
429C4×C8—2441no—
430#95—21no—
431#95—21no—
432#95—1221no—
433#95—1221no—
434#95—1221no—
435#95—1221no—
436#95C22⋊D6111yescharacteristic
437#95C22⋊D6111yescharacteristic
438C24—21no—
439C2×Q8—612no—
440C2×Q8—612no—
441C2×Q8—612no—
442C2×Q8—1221no—
443C2×Q8—1221no—
444C2×Q8—1221no—
445C2×D4—21no—
446C2×D4—21no—
447C2×D4—21no—
448C2×D4—21no—
449C2×D4—21no—
450C2×D4C2×C4×D6112yes—
451C22×C4—21no—
452C22×C4—24no—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—42no—
456C22×C4—42no—
457C22×C4—1221no—
458C22×C4—1221no—
459C22×C4—1221no—
460C22×C4—1221no—
461C22×C4C2×C22⋊D6111yescharacteristic
462C22×C4C2×C22⋊D6111yescharacteristic
463C22×C4C2×C22⋊D6111yescharacteristic
464C22×C4C2×C22⋊D6111yescharacteristic
465C22×C4C2×C22⋊D6111yescharacteristic
466C22×C4C2×C22⋊D6111yescharacteristic
467C22×C4C23×D6111yescharacteristic
468SD16—1228no—
469C2×C8—612no—
470C2×C8—612no—
471C2×C8—1221no—
472C2×C8—1221no—
473C2×C8—1221no—
474C2×C8—1221no—
475C2×C8—1221no—
476C2×C8—1221no—
477C2×C8—1221no—
478C2×C8—1221no—
479C2×C8—1221no—
480C2×C8—1221no—
481C4⋊C4—21no—
482C4⋊C4—22no—
483C4⋊C4—21no—
484C4⋊C4—21no—
485C4⋊C4—21no—
486C4⋊C4—21no—
487C4⋊C4—41no—
488C4⋊C4—42no—
489C4⋊C4—81no—
490C4⋊C4—612no—
491C4⋊C4—1221no—
492C4⋊C4—1224no—
493C4⋊C4—1221no—
494C4⋊C4—1221no—
495C4⋊C4—1221no—
496C4⋊C4—1221no—
497C4⋊C4—2442no—
498C4⋊C4—2442no—
499C4⋊C4—2442no—
500C4⋊C4—2442no—
501C4⋊C4C2×C4×D6112yes—
502C22⋊C4—24no—
503C22⋊C4—42no—
504C22⋊C4—42no—
505C22⋊C4—81no—
506C22⋊C4—81no—
507C42—21no—
508C42—21no—
509C42—21no—
510C42—21no—
511C42—42no—
512C42—42no—
513C42—1224no—
514C42—2442no—
515C42—2442no—
516C42C2×C4×D6112yes—
517C42#1175822212yes—
518C23—22no—
519C23—41no—
520C23—41no—
521C23—41no—
522C23—41no—
523C23—41no—
524C23—41no—
525C23C22×C22⋊D6111yescharacteristic
526Q8—614no—
527Q8—1222no—
528Q8—1222no—
529Q8—1222no—
530D4—22no—
531D4—22no—
532D4—41no—
533D4—41no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
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—48no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—1222no—
560C2×C4—1222no—
561C2×C4—1222no—
562C2×C4—1222no—
563C2×C4—2441no—
564C2×C4—2441no—
565C2×C4—2441no—
566C2×C4—2441no—
567C2×C4—2441no—
568C2×C4—2441no—
569C2×C4—2441no—
570C2×C4—2441no—
571C2×C4C4×C22⋊D6112yes—
572C2×C4C4×C22⋊D6112yes—
573C2×C4#1451980312yes—
574C2×C4C22×C4×D6111yescharacteristic
575C2×C4#41×D6111yescharacteristic
576C2×C4#41×D6111yescharacteristic
577C2×C4C2×#1175822211yescharacteristic
578C2×C4#1451982311yescharacteristic
579C2×C4#1451982411yescharacteristic
580C8—1222no—
581C8—1222no—
582C8—2441no—
583C8—2441no—
584C22—42no—
585C22—42no—
586C22—42no—
587C22—81no—
588C22—81no—
589C22—81no—
590C22—81no—
591C22C2×C4×C22⋊D6111yescharacteristic
592C22#1684040011yescharacteristic
593C22#1684041911yescharacteristic
594C22C2×#1451980311yescharacteristic
595C22#1684056011yescharacteristic
596C22#1684057311yescharacteristic
597C22#1684057711yescharacteristic
598C4—22no—
599C4—22no—
600C4—22no—
601C4—42no—
602C4—41no—
603C4—41no—
604C4—41no—
605C4—81no—
606C4—81no—
607C4—1224no—
608C4—4881no—
609C4—4881no—
610C4—2442no—
611C4#117×D6111yescharacteristic
612C4#1684024211yescharacteristic
613C4#1684045111yescharacteristic
614C4#1684045611yescharacteristic
615C2—82no—
616C2#2144874911yescharacteristic
617C2C4×#1451980311yescharacteristic
618C2#2145465811yescharacteristic
619C2#2145780411yescharacteristic
620C2#2145799911yescharacteristic
621C2#2145854411yescharacteristic
622C2#2145865511yescharacteristic
subgroupG/H
centerC23C22×C22⋊D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#2354C2
socleC23×C61C22×D4
prime radicalC22×D4×C61C23
cosocleC22×C4×C61C24
hypercenter#2354D61
hypocenterC61#22781
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #2354

Complements

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

NH ≅ G/Ntype
2. #21448739C2semidirect
5. #21446411C2semidirect
6. #21446390C2semidirect
9. #21445954C2semidirect
11. #21445821C2semidirect
12. #21445694C2semidirect
13. #21445652C2semidirect
14. C4×#14519700C2semidirect
20. #16839862C4semidirect
267. Dic122#117semidirect
371. C61#22781semidirect

Maximal subgroups

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

subgroupindex
C2×#168398622
#214487392
#214465192
#214465052
#214464112
#214463902
#214461772
#214459542
#214458292
#214458212
#214456942
#214456522
C4×#145197002
C61×#23542
#2278161