Finite Group Databasebeta

← #29698692

Subgroups — #29698692

Order 15616.

898 conjugacy classes of subgroups.

The subgroup lattice has 660 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839834C211yesmaximal, characteristic
2#21448768C211yesmaximal, characteristic
3#21446585C211yesmaximal, characteristic, complemented
4#21446582C211yesmaximal, characteristic, complemented
5#21446018C211yesmaximal, characteristic, complemented
6#21446016C211yesmaximal, characteristic, complemented
7#21446016C211yesmaximal, characteristic, complemented
8#21446012C211yesmaximal, characteristic, complemented
9#21445952C211yesmaximal, characteristic, complemented
10#21445850C211yesmaximal, characteristic, complemented
11#21445842C211yesmaximal, characteristic, complemented
12#21445717C211yesmaximal, characteristic, complemented
13#21445538C211yesmaximal, characteristic, complemented
14#21445535C211yesmaximal, characteristic, complemented
15C61×#2352C211yesmaximal, characteristic
16C22×#11758194C2211yescharacteristic
17C23×#92418C2211yescharacteristic
18C2×#14519736C2211yescharacteristic
19C2×#14519736C2211yescharacteristic
20#16839834C418yescomplemented
21C2×#14519710C2211yescharacteristic
22C2×#14519710C2211yescharacteristic
23C2×#14519708C2211yescharacteristic
24C2×#14519708C2211yescharacteristic
25C2×#14519708C2211yescharacteristic
26C2×#14519708C2211yescharacteristic
27C2×C4⋊#92418C2211yescharacteristic
28C2×C22⋊#92418C2211yescharacteristic
29C2×C22⋊#92418C2211yescharacteristic
30C2×#14519688C2211yescharacteristic
31#16839665—21no—
32#16839655—21no—
33#16839655—21no—
34#16839632C2211yescharacteristic
35#16839622C2211yescharacteristic
36#16839622C2211yescharacteristic
37C4⋊C4×Dic61—21no—
38C4×C4⋊Dic61C2211yescharacteristic
39#16839524C2211yescharacteristic
40#16839524C2211yescharacteristic
41#16839524C2211yescharacteristic
42#16839524C2211yescharacteristic
43#16839524C2211yescharacteristic
44#16839524C2211yescharacteristic
45#16839524C2211yescharacteristic
46#16839524C2211yescharacteristic
47#16839524C2211yescharacteristic
48#16839524C2211yescharacteristic
49C2×C4×D4×C61C2211yescharacteristic
50C2×#117×C61C2211yescharacteristic
51C22×C4⋊C4×C61C2211yescharacteristic
52C61×#388—21no—
53C61×#387—21no—
54C61×#386—21no—
55C61×#382C2211yescharacteristic
56C61×#380C2211yescharacteristic
57C61×#380C2211yescharacteristic
58C4×C4⋊C4×C61C2211yescharacteristic
59C4×C22⋊C4×C61—21no—
60C2×#11758194—41no—
61C2×#11758194—41no—
62C2×#11758194C2×C414yes—
63C22×#92418—41no—
64C22×#92418—41no—
65C22×#92418C2×C414yescomplemented
66C22×#92418C2311yescharacteristic
67C22×#92418C2311yescharacteristic
68C22×#92418C2311yescharacteristic
69C22×#92418C2311yescharacteristic
70#14519736—22no—
71#14519736—22no—
72#14519736—41no—
73#14519736—41no—
74C2×C4⋊Dic61—21no—
75C2×C4⋊Dic61—21no—
76C2×C4⋊Dic61C2311yescharacteristic
77C2×C4⋊Dic61C2311yescharacteristic
78C2×C4⋊Dic61C2311yescharacteristic
79C2×C4⋊Dic61C2311yescharacteristic
80C2×C22⋊Dic61—21no—
81C2×C22⋊Dic61—21no—
82C2×C22⋊Dic61—21no—
83C2×C22⋊Dic61—21no—
84C2×C4×Dic61—21no—
85C2×C4×Dic61—21no—
86C2×C4×Dic61—21no—
87#14519710—22no—
88#14519710—41no—
89#14519708—22no—
90#14519708—41no—
91#14519708—41no—
92#14519708—41no—
93C4⋊#92418C2×C414yescomplemented
94C22⋊#92418C2×C414yes—
95C22⋊#92418C2×C414yescomplemented
96#14519688C2×C414yescomplemented
97#14519684—21no—
98#14519684—21no—
99#14519684—21no—
100C22×D4×C61D411yescharacteristic
101C23×C4×C61—21no—
102C23×C4×C61C2311yescharacteristic
103C23×C4×C61C2311yescharacteristic
104C4×D4×C61C2×C414yes—
105C4×D4×C61Q814yes—
106#117×C61—22no—
107#117×C61—41no—
108C2×C4⋊C4×C61—22no—
109C2×C4⋊C4×C61—21no—
110C2×C4⋊C4×C61—41no—
111C2×C4⋊C4×C61D411yescharacteristic
112C2×C4⋊C4×C61C2311yescharacteristic
113C2×C4⋊C4×C61C2311yescharacteristic
114C2×C4⋊C4×C61C2311yescharacteristic
115C2×C4⋊C4×C61C2311yescharacteristic
116C2×C22⋊C4×C61—21no—
117C2×C22⋊C4×C61—21no—
118C2×C22⋊C4×C61—21no—
119C2×C22⋊C4×C61D411yescharacteristic
120C2×C22⋊C4×C61D411yescharacteristic
121C2×C42×C61—21no—
122C2×C42×C61—21no—
123C2×C42×C61C2311yescharacteristic
124#95×C61—21no—
125#95×C61—21no—
126#95×C61—21no—
127#95×C61—21no—
128#95×C61—21no—
129C24×C61C2×D411yescharacteristic
130C24×C61C2×D411yescharacteristic
131C2×D4×C61—41no—
132C2×D4×C61—41no—
133C2×D4×C61C4⋊C414yes—
134C22×C4×C61—21no—
135C22×C4×C61—21no—
136C22×C4×C61—21no—
137C22×C4×C61—21no—
138C22×C4×C61—21no—
139C22×C4×C61—21no—
140C22×C4×C61—21no—
141C22×C4×C61—21no—
142C22×C4×C61—21no—
143C22×C4×C61—21no—
144C22×C4×C61—21no—
145C22×C4×C61—21no—
146C22×C4×C61—42no—
147C22×C4×C61—42no—
148C22×C4×C61—81no—
149C22×C4×C61C22×C412yes—
150C22×C4×C61C22×C412yes—
151C22×C4×C61C2×D411yescharacteristic
152C22×C4×C61C2×D411yescharacteristic
153C22×C4×C61C2×D411yescharacteristic
154C22×C4×C61C2×D411yescharacteristic
155C22×C4×C61C2×Q812yes—
156C22×C4×C61C2×Q812yes—
157C22×C4×C61C2411yescharacteristic
158C22×Dic61—21no—
159C22×Dic61—21no—
160C22×Dic61—21no—
161C22×Dic61—21no—
162C4⋊C4×C61—21no—
163C4⋊C4×C61—21no—
164C4⋊C4×C61—21no—
165C4⋊C4×C61—21no—
166C4⋊C4×C61—21no—
167C4⋊C4×C61—21no—
168C4⋊C4×C61—21no—
169C4⋊C4×C61—42no—
170C4⋊C4×C61C4⋊C414yes—
171C4⋊C4×C61#4112yes—
172C22⋊C4×C61—42no—
173C22⋊C4×C61—42no—
174C22⋊C4×C61—81no—
175C22⋊C4×C61C4⋊C414yes—
176C22⋊C4×C61C4⋊C414yes—
177C42×C61—42no—
178C42×C61—81no—
179C42×C61C22×C412yes—
180C42×C61C2×Q812yes—
181C22⋊Dic61—42no—
182C22⋊Dic61—42no—
183C22⋊Dic61—81no—
184C22⋊Dic61—81no—
185C4⋊Dic61—21no—
186C4⋊Dic61—21no—
187C4⋊Dic61—21no—
188C4⋊Dic61—21no—
189C4⋊Dic61—21no—
190C4⋊Dic61—21no—
191C4⋊Dic61—21no—
192C4⋊Dic61—42no—
193C4⋊Dic61—81no—
194C4⋊Dic61#4112yes—
195C4×Dic61—42no—
196C4×Dic61—42no—
197C4×Dic61—42no—
198#11758194—44no—
199C2×#92418—21no—
200C2×#92418—21no—
201C2×#92418—21no—
202C2×#92418—21no—
203C2×#92418—21no—
204C2×#92418—21no—
205C2×#92418—21no—
206C2×#92418—21no—
207C2×#92418—44no—
208C2×#92418C22×C412yes—
209C2×#92418C22×C412yes—
210C2×#92418C22×C412yes—
211C2×#92418C22×C412yes—
212C23×C61—21no—
213C23×C61—21no—
214C23×C61—21no—
215C23×C61—21no—
216C23×C61—21no—
217C23×C61—21no—
218C23×C61—21no—
219C23×C61—21no—
220C23×C61—21no—
221C23×C61—21no—
222C23×C61—21no—
223C23×C61—21no—
224C23×C61C2×C4⋊C412yes—
225C23×C61C2×C4⋊C412yes—
226C23×C61C22×D411yescharacteristic
227D4×C61—44no—
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—21no—
235C2×C4×C61—21no—
236C2×C4×C61—22no—
237C2×C4×C61—22no—
238C2×C4×C61—22no—
239C2×C4×C61—22no—
240C2×C4×C61—22no—
241C2×C4×C61—22no—
242C2×C4×C61—22no—
243C2×C4×C61—22no—
244C2×C4×C61—22no—
245C2×C4×C61—41no—
246C2×C4×C61—41no—
247C2×C4×C61—41no—
248C2×C4×C61—41no—
249C2×C4×C61—41no—
250C2×C4×C61—41no—
251C2×C4×C61—41no—
252C2×C4×C61—41no—
253C2×C4×C61—41no—
254C2×C4×C61—41no—
255C2×C4×C61—41no—
256C2×C4×C61—84no—
257C2×C4×C61C2×C4⋊C412yes—
258C2×C4×C61C2×C4⋊C412yes—
259C2×C4×C61C2×C4⋊C412yes—
260C2×C4×C61C2×C4⋊C412yes—
261C2×C4×C61C23×C411yescharacteristic
262C2×C4×C61C22×Q811yescharacteristic
263C2×C4×C61C2×#4111yescharacteristic
264C2×C4×C61C2×#4111yescharacteristic
265C2×C4×C61H(5,𝔽2)11yescharacteristic
266C2×C4×C61H(5,𝔽2)11yescharacteristic
267C2×Dic61—22no—
268C2×Dic61—41no—
269C2×Dic61—41no—
270C2×Dic61—41no—
271C2×Dic61—41no—
272C2×Dic61—41no—
273C2×Dic61—41no—
274C2×Dic61—41no—
275C2×Dic61—41no—
276C2×Dic61—41no—
277C2×Dic61—41no—
278C2×Dic61—41no—
279#92418—22no—
280#92418—22no—
281#92418—22no—
282#92418—22no—
283#22710—611nomaximal, Sylow
284C22×C61—22no—
285C22×C61—22no—
286C22×C61—22no—
287C22×C61—22no—
288C22×C61—22no—
289C22×C61—22no—
290C22×C61—41no—
291C22×C61—41no—
292C22×C61—41no—
293C22×C61—41no—
294C22×C61—41no—
295C22×C61—41no—
296C22×C61—41no—
297C22×C61—41no—
298C22×C61C22×C4⋊C411yescharacteristic
299C22×C61#54611yescharacteristic
300C22×C61#54611yescharacteristic
301C22×C61#57511yescharacteristic
302C22×C61#57511yescharacteristic
303C22×C61#57711yescharacteristic
304C22×C61#57711yescharacteristic
305C4×C61—22no—
306C4×C61—22no—
307C4×C61—22no—
308C4×C61—22no—
309C4×C61—42no—
310C4×C61—81no—
311C4×C61—81no—
312C4×C61—81no—
313C4×C61C4×#4111yescharacteristic
314C4×C61#51811yescharacteristic
315C4×C61#55411yescharacteristic
316C4×C61#55411yescharacteristic
317Dic61—42no—
318Dic61—81no—
319Dic61—81no—
320Dic61—81no—
321C2×#435—611no—
322#2354—611no—
323#2352D6111yescharacteristic
324#1949—611no—
325#1948—611no—
326#1854—611no—
327#1854—611no—
328#1854—611no—
329#1853—611no—
330#1836—611no—
331#1809—611no—
332#1807—611no—
333#1776—611no—
334#1736—611no—
335#1735—611no—
336C2×C61—42no—
337C2×C61—42no—
338C2×C61#234911yescharacteristic
339C2×C61#291111yescharacteristic
340C2×C61#291211yescharacteristic
341C2×C61#330711yescharacteristic
342C2×C61#331111yescharacteristic
343C2×C61#331711yescharacteristic
344C2×C61#332111yescharacteristic
345C22×M16—611no—
346C23×C8—611no—
347C2×C4×D4C2×D6111yescharacteristic
348C2×#117—611no—
349C2×#117—611no—
350C2×#117C2×D6111yescharacteristic
351C22×C4⋊C4C2×D6111yescharacteristic
352#435—618no—
353C2×#107—611no—
354C2×#107—611no—
355C2×#106—611no—
356C2×#106—611no—
357C2×#106—611no—
358C2×#106—611no—
359C2×C4⋊C8—611no—
360C2×C22⋊C8—611no—
361C2×C22⋊C8—611no—
362C2×#97—611no—
363#390—1221no—
364#388—21no—
365#388—1221no—
366#388—1221no—
367#387—21no—
368#386—21no—
369#384—611no—
370#382C2×D6111yescharacteristic
371#380—611no—
372#380—611no—
373#380C2×D6111yescharacteristic
374#380C2×D6111yescharacteristic
375C4×C4⋊C4—611no—
376C4×C4⋊C4—1221no—
377C4×C4⋊C4C2×D6111yescharacteristic
378C4×C22⋊C4—21no—
379#340—611no—
380#340—611no—
381#340—611no—
382#340—611no—
383#340—611no—
384#340—611no—
385#340—611no—
386#340—611no—
387#340—611no—
388#340—611no—
389C61#2271011yescharacteristic, complemented, Sylow
390C22×D4C22⋊D6111yescharacteristic
391C23×C4—21no—
392C23×C4C22×D6111yescharacteristic
393C23×C4C22×D6111yescharacteristic
394C2×M16—614no—
395C2×M16—2441no—
396C2×M16—2441no—
397C22×C8—611no—
398C22×C8—611no—
399C22×C8—611no—
400C22×C8—611no—
401C22×C8—614no—
402C22×C8—2441no—
403C22×C8—2441no—
404C4×D4Dic12214yes—
405C4×D4C4×D6114yes—
406#117—22no—
407#117—41no—
408#117—1222no—
409#117—1222no—
410#117—2441no—
411#117—2441no—
412C2×C4⋊C4—22no—
413C2×C4⋊C4—21no—
414C2×C4⋊C4—41no—
415C2×C4⋊C4—611no—
416C2×C4⋊C4—611no—
417C2×C4⋊C4—611no—
418C2×C4⋊C4—611no—
419C2×C4⋊C4—1221no—
420C2×C4⋊C4—1221no—
421C2×C4⋊C4C22⋊D6111yescharacteristic
422C2×C4⋊C4C22×D6111yescharacteristic
423C2×C4⋊C4C22×D6111yescharacteristic
424C2×C4⋊C4C22×D6111yescharacteristic
425C2×C4⋊C4C22×D6111yescharacteristic
426C2×C22⋊C4—21no—
427C2×C22⋊C4—21no—
428C2×C22⋊C4—21no—
429C2×C22⋊C4—1221no—
430C2×C22⋊C4—1221no—
431C2×C22⋊C4—1221no—
432C2×C22⋊C4—1221no—
433C2×C22⋊C4C22⋊D6111yescharacteristic
434C2×C22⋊C4C22⋊D6111yescharacteristic
435C2×C42—21no—
436C2×C42—21no—
437C2×C42—1221no—
438C2×C42—1221no—
439C2×C42—1221no—
440C2×C42C22×D6111yescharacteristic
441#107—1222no—
442#107—2441no—
443#106—1222no—
444#106—2441no—
445#106—2441no—
446#106—2441no—
447C4⋊C8—614no—
448C22⋊C8—614no—
449C22⋊C8—614no—
450#97—614no—
451#95—21no—
452#95—21no—
453#95—21no—
454#95—21no—
455#95—21no—
456#95—1221no—
457#95—1221no—
458#95—1221no—
459C24C2×C22⋊D6111yescharacteristic
460C24C2×C22⋊D6111yescharacteristic
461C2×D4—41no—
462C2×D4—41no—
463C2×D4#1175819614yes—
464C22×C4—21no—
465C22×C4—21no—
466C22×C4—21no—
467C22×C4—21no—
468C22×C4—21no—
469C22×C4—21no—
470C22×C4—21no—
471C22×C4—21no—
472C22×C4—21no—
473C22×C4—21no—
474C22×C4—21no—
475C22×C4—21no—
476C22×C4—42no—
477C22×C4—42no—
478C22×C4—81no—
479C22×C4—1221no—
480C22×C4—1221no—
481C22×C4—1221no—
482C22×C4—1221no—
483C22×C4C2×Dic12212yes—
484C22×C4C2×Dic12212yes—
485C22×C4C2×C4×D6112yes—
486C22×C4C2×C4×D6112yes—
487C22×C4C2×C22⋊D6111yescharacteristic
488C22×C4C2×C22⋊D6111yescharacteristic
489C22×C4C2×C22⋊D6111yescharacteristic
490C22×C4C2×C22⋊D6111yescharacteristic
491C22×C4C23×D6111yescharacteristic
492M16—2444no—
493C2×C8—612no—
494C2×C8—612no—
495C2×C8—612no—
496C2×C8—612no—
497C2×C8—1221no—
498C2×C8—1221no—
499C2×C8—1221no—
500C2×C8—1221no—
501C2×C8—1221no—
502C2×C8—1221no—
503C2×C8—1221no—
504C2×C8—1221no—
505C2×C8—2444no—
506C4⋊C4—21no—
507C4⋊C4—21no—
508C4⋊C4—21no—
509C4⋊C4—21no—
510C4⋊C4—21no—
511C4⋊C4—21no—
512C4⋊C4—21no—
513C4⋊C4—42no—
514C4⋊C4—612no—
515C4⋊C4—1221no—
516C4⋊C4—1221no—
517C4⋊C4—1221no—
518C4⋊C4—1221no—
519C4⋊C4—1221no—
520C4⋊C4—1221no—
521C4⋊C4—1221no—
522C4⋊C4—4881no—
523C4⋊C4—2442no—
524C4⋊C4#1175819614yes—
525C4⋊C4#1175822412yes—
526C22⋊C4—42no—
527C22⋊C4—42no—
528C22⋊C4—81no—
529C22⋊C4—4881no—
530C22⋊C4—4881no—
531C22⋊C4—2442no—
532C22⋊C4—2442no—
533C22⋊C4#1175819614yes—
534C22⋊C4#1175819614yes—
535C42—42no—
536C42—81no—
537C42—2442no—
538C42—2442no—
539C42—2442no—
540C42C2×Dic12212yes—
541C42C2×C4×D6112yes—
542C23—21no—
543C23—21no—
544C23—21no—
545C23—21no—
546C23—21no—
547C23—21no—
548C23—21no—
549C23—21no—
550C23—21no—
551C23—21no—
552C23—21no—
553C23—21no—
554C23C2×#1175819612yes—
555C23C2×#1175819612yes—
556C23C22×C22⋊D6111yescharacteristic
557D4—44no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—84no—
587C2×C4—1222no—
588C2×C4—2441no—
589C2×C4—2441no—
590C2×C4—2441no—
591C2×C4—2441no—
592C2×C4—2441no—
593C2×C4—2441no—
594C2×C4—2441no—
595C2×C4—2441no—
596C2×C4—2441no—
597C2×C4—2441no—
598C2×C4—2441no—
599C2×C4C2×#1175819612yes—
600C2×C4C2×#1175819612yes—
601C2×C4C2×#1175819612yes—
602C2×C4C2×#1175819612yes—
603C2×C4C22×C4×D6111yescharacteristic
604C2×C4C22×Dic12211yescharacteristic
605C2×C4#41×D6111yescharacteristic
606C2×C4C2×#1175822411yescharacteristic
607C2×C4#1451982211yescharacteristic
608C2×C4#1451982311yescharacteristic
609C8—1222no—
610C8—1222no—
611C8—1222no—
612C8—1222no—
613C22—22no—
614C22—22no—
615C22—22no—
616C22—22no—
617C22—22no—
618C22—22no—
619C22—41no—
620C22—41no—
621C22—41no—
622C22—41no—
623C22—41no—
624C22—41no—
625C22—41no—
626C22—41no—
627C22C22×#1175819611yescharacteristic
628C22#1684040011yescharacteristic
629C22#1684040611yescharacteristic
630C22#1684056011yescharacteristic
631C22#1684056011yescharacteristic
632C22#1684057011yescharacteristic
633C22#1684057311yescharacteristic
634C4—22no—
635C4—22no—
636C4—22no—
637C4—22no—
638C4—42no—
639C4—81no—
640C4—81no—
641C4—81no—
642C4—4881no—
643C4—4881no—
644C4—4881no—
645C4—2442no—
646C4C4×#1175822411yescharacteristic
647C4#1684023411yescharacteristic
648C4#1684046911yescharacteristic
649C4#1684047111yescharacteristic
650C2—42no—
651C2—42no—
652C2#2144872411yescharacteristic
653C2#2145440811yescharacteristic
654C2#2145441211yescharacteristic
655C2#2145778911yescharacteristic
656C2#2145785511yescharacteristic
657C2#2145793911yescharacteristic
658C2#2145800511yescharacteristic
subgroupG/H
centerC23C22×C22⋊D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#2352C2
socleC23×C61C22×D4
prime radicalC22×D4×C61D4
cosocleC22×C4×C61C24
hypercenter#2352D61
hypocenterC61#22710
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ #2352

Complements

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

NH ≅ G/Ntype
3. #21446585C2semidirect
4. #21446582C2semidirect
5. #21446018C2semidirect
6. #21446016C2semidirect
7. #21446016C2semidirect
8. #21446012C2semidirect
9. #21445952C2semidirect
10. #21445850C2semidirect
11. #21445842C2semidirect
12. #21445717C2semidirect
13. #21445538C2semidirect
14. #21445535C2semidirect
20. #16839834C4semidirect
65. C22×#92418C2×C4semidirect
93. C4⋊#92418C2×C4semidirect
95. C22⋊#92418C2×C4semidirect
96. #14519688C2×C4semidirect
389. C61#22710semidirect

Maximal subgroups

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

subgroupindex
C2×#168398342
#214487682
#214465852
#214465822
#214460182
#214460162
#214460122
#214459522
#214458502
#214458422
#214457172
#214455382
#214455352
C61×#23522
#2271061