Finite Group Databasebeta

← #29698726

Subgroups — #29698726

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×#16839830C211yesmaximal, characteristic
2#21448763C211yesmaximal, characteristic
3#21448749C211yesmaximal, characteristic
4#21446588C211yesmaximal, characteristic, complemented
5#21446576C211yesmaximal, characteristic, complemented
6#21446020C211yesmaximal, characteristic, complemented
7#21446019C211yesmaximal, characteristic, complemented
8#21446005C211yesmaximal, characteristic, complemented
9#21445951C211yesmaximal, characteristic, complemented
10#21445849C211yesmaximal, characteristic, complemented
11#21445846C211yesmaximal, characteristic, complemented
12#21445718C211yesmaximal, characteristic, complemented
13#21445537C211yesmaximal, characteristic, complemented
14#21445535C211yesmaximal, characteristic, complemented
15C61×#1854C211yesmaximal, characteristic, complemented
16C22×#11758189C2211yescharacteristic
17C23×#92418C2211yescharacteristic
18C2×C4×C22⋊D61C2211yescharacteristic
19C2×#14519736C2211yescharacteristic
20C2×#14519735C2211yescharacteristic
21C2×C4⋊C4×D61C2211yescharacteristic
22#16839830C418yescomplemented
23C2×#14519711C2211yescharacteristic
24C2×#14519710C2211yescharacteristic
25C2×#14519710C2211yescharacteristic
26C2×#14519709C2211yescharacteristic
27C2×#14519706C2211yescharacteristic
28C2×#14519689C2211yescharacteristic
29C2×#14519687C2211yescharacteristic
30#16839662—21no—
31#16839656—21no—
32#16839656—21no—
33#16839651—21no—
34#16839646—21no—
35#16839638—21no—
36#16839631C2211yescharacteristic
37#16839627C2211yescharacteristic
38#16839622C2211yescharacteristic
39#16839619C2211yescharacteristic
40#16839619C2211yescharacteristic
41C4⋊C4×Dic61C2211yescharacteristic
42C4⋊C4×Dic61C2211yescharacteristic
43#16839613—21no—
44C4×#11758198—21no—
45#16839525C2211yescharacteristic
46#16839525C2211yescharacteristic
47#16839524C2211yescharacteristic
48#16839524C2211yescharacteristic
49#16839524C2211yescharacteristic
50#16839524C2211yescharacteristic
51#16839524C2211yescharacteristic
52#16839524C2211yescharacteristic
53C2×#117×C61C2211yescharacteristic
54C2×#107×C61C2211yescharacteristic
55C2×#106×C61C2211yescharacteristic
56C2×C22⋊C8×C61C2211yescharacteristic
57C61×#380C2211yescharacteristic
58C61×#340C2211yescharacteristic
59C61×#340C2211yescharacteristic
60C22×C22⋊D61D411yescharacteristic
61C22×C4×D61—21no—
62C22×C4×D61C2311yescharacteristic
63C2×#11758189—41no—
64C2×#11758189—41no—
65C2×#11758189C2×C414yes—
66C22×#92418—41no—
67C22×#92418—41no—
68C22×#92418C2×C414yescomplemented
69C22×#92418C2311yescharacteristic
70C22×#92418C2311yescharacteristic
71C4×C22⋊D61C2×C414yes—
72C4×C22⋊D61Q814yes—
73#14519736—22no—
74#14519736—41no—
75#14519735—22no—
76#14519735—41no—
77C2×C4⋊Dic61C2311yescharacteristic
78C2×C4⋊Dic61C2311yescharacteristic
79C2×C4⋊Dic61C2311yescharacteristic
80C2×C4⋊Dic61C2311yescharacteristic
81C2×#11758196—21no—
82C2×#11758196—21no—
83C2×#11758196—21no—
84C2×#11758196D411yescharacteristic
85C4⋊C4×D61—22no—
86C4⋊C4×D61—41no—
87C2×C22⋊Dic61—21no—
88C2×C22⋊Dic61—21no—
89C2×C22⋊Dic61D411yescharacteristic
90C2×#11758198—21no—
91C2×#11758198—21no—
92C2×#11758198—21no—
93C2×#11758198D411yescharacteristic
94C2×C4×Dic61—21no—
95C2×C4×Dic61—21no—
96C2×C4×Dic61—21no—
97C2×C4×Dic61—21no—
98C2×C4×Dic61C2311yescharacteristic
99#14519711—41no—
100#14519710—41no—
101#14519710—41no—
102#14519709—22no—
103#14519706C2×C414yescomplemented
104#14519689C2×C414yes—
105#14519687C2×C414yescomplemented
106#14519684—21no—
107#14519684—21no—
108#14519684—21no—
109#14519684—21no—
110#14519684—21no—
111#14519684—21no—
112#14519684—21no—
113C23×C4×C61C2311yescharacteristic
114C22×C8×C61C2311yescharacteristic
115C22×C8×C61C2311yescharacteristic
116#117×C61—22no—
117#117×C61—41no—
118C2×C4⋊C4×C61C2311yescharacteristic
119C2×C4⋊C4×C61C2311yescharacteristic
120C2×C4⋊C4×C61C2311yescharacteristic
121C2×C4⋊C4×C61C2311yescharacteristic
122C2×C22⋊C4×C61—21no—
123C2×C22⋊C4×C61—21no—
124C2×C42×C61—21no—
125#107×C61—22no—
126#106×C61—41no—
127C22⋊C8×C61C2×C414yescomplemented
128#95×C61—21no—
129C24×C61C2×D411yescharacteristic
130C23×D61C2×D411yescharacteristic
131C22×C4×C61—21no—
132C22×C4×C61—21no—
133C22×C4×C61—21no—
134C22×C4×C61—21no—
135C22×C4×C61—21no—
136C22×C4×C61—21no—
137C22×C4×C61—21no—
138C22×C4×C61—21no—
139C22×C4×C61C22×C412yes—
140C22×C4×C61C2×D411yescharacteristic
141C22×C4×C61C2×Q812yes—
142C22×C4×C61C2411yescharacteristic
143C2×C22⋊D61—41no—
144C2×C22⋊D61—41no—
145C2×C22⋊D61C4⋊C414yes—
146C22×Dic61—21no—
147C22×Dic61—21no—
148C22×Dic61—21no—
149C22×Dic61—21no—
150C22×Dic61C2×D411yescharacteristic
151C22×Dic61C2×D411yescharacteristic
152C22×Dic61C2×D411yescharacteristic
153C2×C4×D61—21no—
154C2×C4×D61—21no—
155C2×C4×D61—21no—
156C2×C4×D61—21no—
157C2×C4×D61—42no—
158C2×C4×D61—42no—
159C2×C4×D61—81no—
160C2×C4×D61C22×C412yes—
161C2×C4×D61C2×Q812yes—
162C2×C8×C61—21no—
163C2×C8×C61—21no—
164C2×C8×C61—21no—
165C2×C8×C61—21no—
166C2×C8×C61C22×C412yes—
167C2×C8×C61C22×C412yes—
168C4⋊C4×C61—21no—
169C4⋊C4×C61—21no—
170C4⋊C4×C61—21no—
171C4⋊C4×C61—21no—
172C4⋊C4×C61—21no—
173C4⋊C4×C61—21no—
174C4⋊C4×C61—21no—
175C4⋊C4×C61#4112yes—
176C22⋊C4×C61—42no—
177C22⋊C4×C61—81no—
178C42×C61—42no—
179C22⋊Dic61—42no—
180C22⋊Dic61—81no—
181C22⋊Dic61C4⋊C414yes—
182#11758198—42no—
183#11758198—42no—
184#11758198—81no—
185#11758198C4⋊C414yes—
186C4⋊Dic61—21no—
187C4⋊Dic61—21no—
188C4⋊Dic61—21no—
189C4⋊Dic61—21no—
190C4⋊Dic61—21no—
191C4⋊Dic61—21no—
192C4⋊Dic61—21no—
193C4⋊Dic61#4112yes—
194#11758196—42no—
195#11758196—42no—
196#11758196—81no—
197#11758196C4⋊C414yes—
198C4×Dic61—42no—
199C4×Dic61—42no—
200C4×Dic61—42no—
201C4×Dic61—81no—
202C4×Dic61C22×C412yes—
203C4×Dic61C2×Q812yes—
204C2×#92418—21no—
205C2×#92418—21no—
206C2×#92418—21no—
207C2×#92418—21no—
208C2×#92418—44no—
209C2×#92418C22×C412yes—
210C2×#92418C22×C412yes—
211#11758189—44no—
212C23×C61—21no—
213C23×C61—21no—
214C23×C61—21no—
215C23×C61—21no—
216C23×C61—21no—
217C23×C61—21no—
218C23×C61C2×C4⋊C412yes—
219C23×C61C22×D411yescharacteristic
220C22×D61—21no—
221C22×D61—21no—
222C22×D61—21no—
223C22×D61—21no—
224C22×D61—21no—
225C22×D61—21no—
226C22×D61C2×C4⋊C412yes—
227C2×C4×C61—21no—
228C2×C4×C61—21no—
229C2×C4×C61—22no—
230C2×C4×C61—22no—
231C2×C4×C61—22no—
232C2×C4×C61—22no—
233C2×C4×C61—22no—
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×C61—41no—
243C2×C4×C61—41no—
244C2×C4×C61—41no—
245C2×C4×C61C2×C4⋊C412yes—
246C2×C4×C61C23×C411yescharacteristic
247C2×C4×C61C22×Q811yescharacteristic
248C2×C4×C61C2×#4111yescharacteristic
249C2×C4×C61C2×#4111yescharacteristic
250C2×C4×C61H(5,𝔽2)11yescharacteristic
251C2×C4×C61H(5,𝔽2)11yescharacteristic
252C22⋊D61—44no—
253C2×Dic61—21no—
254C2×Dic61—21no—
255C2×Dic61—21no—
256C2×Dic61—21no—
257C2×Dic61—21no—
258C2×Dic61—21no—
259C2×Dic61—22no—
260C2×Dic61—41no—
261C2×Dic61—41no—
262C2×Dic61—41no—
263C2×Dic61—41no—
264C2×Dic61—41no—
265C2×Dic61—41no—
266C2×Dic61—41no—
267C2×Dic61—41no—
268C2×Dic61—41no—
269C2×Dic61—41no—
270C2×Dic61—41no—
271C2×Dic61C2×C4⋊C412yes—
272C2×Dic61C2×C4⋊C412yes—
273C2×Dic61C2×C4⋊C412yes—
274C4×D61—22no—
275C4×D61—22no—
276C4×D61—22no—
277C4×D61—22no—
278C4×D61—84no—
279C8×C61—22no—
280C8×C61—22no—
281#92418—22no—
282#92418—22no—
283#22712—611nomaximal, Sylow
284C22×C61—22no—
285C22×C61—22no—
286C22×C61—22no—
287C22×C61—41no—
288C22×C61—41no—
289C22×C61—41no—
290C22×C61—41no—
291C22×C61C22×C4⋊C411yescharacteristic
292C22×C61#54611yescharacteristic
293C22×C61#54611yescharacteristic
294C22×C61#57511yescharacteristic
295C22×C61#57511yescharacteristic
296C22×C61#57611yescharacteristic
297C22×C61#57811yescharacteristic
298C2×D61—22no—
299C2×D61—22no—
300C2×D61—22no—
301C2×D61—41no—
302C2×D61—41no—
303C2×D61—41no—
304C2×D61—41no—
305C4×C61—22no—
306C4×C61—42no—
307C4×C61—81no—
308C4×C61—81no—
309C4×C61—81no—
310C4×C61C4×#4111yescharacteristic
311C4×C61#51811yescharacteristic
312C4×C61#55411yescharacteristic
313C4×C61#55411yescharacteristic
314Dic61—22no—
315Dic61—22no—
316Dic61—22no—
317Dic61—42no—
318Dic61—81no—
319Dic61—81no—
320Dic61—81no—
321C2×#435—611no—
322#2354—611no—
323#2352—611no—
324#1949—611no—
325#1947—611no—
326#1854—611no—
327#1854—611no—
328#1854D6111yescharacteristic, complemented
329#1852—611no—
330#1836—611no—
331#1809—611no—
332#1808—611no—
333#1776—611no—
334#1736—611no—
335#1735—611no—
336C2×C61—42no—
337C2×C61#234911yescharacteristic
338C2×C61#291111yescharacteristic
339C2×C61#291311yescharacteristic
340C2×C61#330911yescharacteristic
341C2×C61#331311yescharacteristic
342C2×C61#331511yescharacteristic
343C2×C61#331911yescharacteristic
344D61—42no—
345C22×M16—611no—
346C23×C8—611no—
347C2×C4×D4—611no—
348C2×#117—611no—
349C2×#117—611no—
350C2×#117C2×D6111yescharacteristic
351C22×C4⋊C4—611no—
352#435—618no—
353C2×#107—611no—
354C2×#107—611no—
355C2×#107—611no—
356C2×#107C2×D6111yescharacteristic
357C2×#106—611no—
358C2×#106C2×D6111yescharacteristic
359C2×C4⋊C8—611no—
360C2×C22⋊C8—611no—
361C2×C22⋊C8C2×D6111yescharacteristic
362C2×#97—611no—
363#390—1221no—
364#388—1221no—
365#388—1221no—
366#388—1221no—
367#387—1221no—
368#386—1221no—
369#384—611no—
370#382—611no—
371#380—611no—
372#380—611no—
373#380—611no—
374#380C2×D6111yescharacteristic
375C4×C4⋊C4—611no—
376C4×C4⋊C4—611no—
377C4×C4⋊C4—1221no—
378C4×C22⋊C4—1221no—
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#340C2×D6111yescharacteristic
388#340C2×D6111yescharacteristic
389C61#2271211yescharacteristic, complemented, Sylow
390C22×D4—611no—
391C23×C4—611no—
392C23×C4—1221no—
393C23×C4C22×D6111yescharacteristic
394C2×M16—614no—
395C2×M16—2441no—
396C2×M16—2441no—
397C22×C8—611no—
398C22×C8—611no—
399C22×C8—614no—
400C22×C8—2441no—
401C22×C8—2441no—
402C22×C8C22×D6111yescharacteristic
403C22×C8C22×D6111yescharacteristic
404C4×D4—614no—
405C4×D4—614no—
406#117—22no—
407#117—41no—
408#117—1222no—
409#117—1222no—
410#117—2441no—
411#117—2441no—
412C2×C4⋊C4—611no—
413C2×C4⋊C4—611no—
414C2×C4⋊C4—611no—
415C2×C4⋊C4—611no—
416C2×C4⋊C4—611no—
417C2×C4⋊C4—1222no—
418C2×C4⋊C4—1221no—
419C2×C4⋊C4—1221no—
420C2×C4⋊C4—1221no—
421C2×C4⋊C4—2441no—
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—611no—
429C2×C22⋊C4—611no—
430C2×C22⋊C4—1221no—
431C2×C22⋊C4—1221no—
432C2×C22⋊C4—1221no—
433C2×C22⋊C4—1221no—
434C2×C22⋊C4—1221no—
435C2×C42—21no—
436C2×C42—611no—
437C2×C42—1221no—
438C2×C42—1221no—
439C2×C42—1221no—
440C2×C42—1221no—
441#107—22no—
442#107—2441no—
443#107—2441no—
444#107—2441no—
445#106—41no—
446#106—1222no—
447C4⋊C8—614no—
448C22⋊C8—614no—
449C22⋊C8C4×D6114yescomplemented
450#97—614no—
451#95—21no—
452#95—1221no—
453#95—1221no—
454#95—1221no—
455#95—1221no—
456#95—1221no—
457#95—1221no—
458#95—1221no—
459C24—611no—
460C24D4×D6111yescharacteristic
461C2×D4—614no—
462C2×D4—2441no—
463C2×D4—2441no—
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—611no—
473C22×C4—611no—
474C22×C4—611no—
475C22×C4—612no—
476C22×C4—612no—
477C22×C4—1221no—
478C22×C4—1221no—
479C22×C4—1221no—
480C22×C4—1221no—
481C22×C4—1221no—
482C22×C4—1221no—
483C22×C4—1221no—
484C22×C4—1221no—
485C22×C4—4881no—
486C22×C4—2442no—
487C22×C4—2442no—
488C22×C4C2×C4×D6112yes—
489C22×C4D4×D6111yescharacteristic
490C22×C4Q8×D6112yes—
491C22×C4C23×D6111yescharacteristic
492M16—2444no—
493C2×C8—21no—
494C2×C8—21no—
495C2×C8—21no—
496C2×C8—21no—
497C2×C8—612no—
498C2×C8—612no—
499C2×C8—1221no—
500C2×C8—1221no—
501C2×C8—1221no—
502C2×C8—1221no—
503C2×C8—2444no—
504C2×C8C2×C4×D6112yes—
505C2×C8C2×C4×D6112yes—
506C4⋊C4—21no—
507C4⋊C4—21no—
508C4⋊C4—21no—
509C4⋊C4—21no—
510C4⋊C4—21no—
511C4⋊C4—21no—
512C4⋊C4—21no—
513C4⋊C4—614no—
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—2442no—
525C4⋊C4#1175822412yes—
526C22⋊C4—42no—
527C22⋊C4—81no—
528C22⋊C4—614no—
529C22⋊C4—614no—
530C22⋊C4—4881no—
531C22⋊C4—4881no—
532C22⋊C4—2442no—
533C22⋊C4—2442no—
534C22⋊C4—2442no—
535C42—42no—
536C42—612no—
537C42—612no—
538C42—4881no—
539C42—2442no—
540C42—2442no—
541C42—2442no—
542C23—21no—
543C23—21no—
544C23—21no—
545C23—21no—
546C23—21no—
547C23—21no—
548C23—612no—
549C23—1221no—
550C23—1221no—
551C23—1221no—
552C23—1221no—
553C23—1221no—
554C23—1221no—
555C23C4⋊C4×D6112yes—
556C23C2×D4×D6111yescharacteristic
557D4—2444no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—612no—
577C2×C4—612no—
578C2×C4—612no—
579C2×C4—1221no—
580C2×C4—1221no—
581C2×C4—1221no—
582C2×C4—1221no—
583C2×C4—1221no—
584C2×C4—1221no—
585C2×C4—1222no—
586C2×C4—1222no—
587C2×C4—1222no—
588C2×C4—1222no—
589C2×C4—1222no—
590C2×C4—4884no—
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×C4—2441no—
600C2×C4—2441no—
601C2×C4—2441no—
602C2×C4C4⋊C4×D6112yes—
603C2×C4C22×C4×D6111yescharacteristic
604C2×C4C2×Q8×D6111yescharacteristic
605C2×C4#41×D6111yescharacteristic
606C2×C4C2×#1175822411yescharacteristic
607C2×C4#1451982211yescharacteristic
608C2×C4#1451982311yescharacteristic
609C8—22no—
610C8—22no—
611C8—1222no—
612C8—1222no—
613C22—22no—
614C22—22no—
615C22—22no—
616C22—41no—
617C22—41no—
618C22—41no—
619C22—41no—
620C22—1222no—
621C22—1222no—
622C22—1222no—
623C22—2441no—
624C22—2441no—
625C22—2441no—
626C22—2441no—
627C22C2×C4⋊C4×D6111yescharacteristic
628C22#1684039911yescharacteristic
629C22#1684040411yescharacteristic
630C22#1684055911yescharacteristic
631C22#1684055911yescharacteristic
632C22#1684056611yescharacteristic
633C22#1684057811yescharacteristic
634C4—22no—
635C4—42no—
636C4—81no—
637C4—81no—
638C4—81no—
639C4—1222no—
640C4—1222no—
641C4—1222no—
642C4—4881no—
643C4—4881no—
644C4—4881no—
645C4—2442no—
646C4C4×#1175822411yescharacteristic
647C4#1684023411yescharacteristic
648C4#1684046611yescharacteristic
649C4#1684047011yescharacteristic
650C2—42no—
651C2—2442no—
652C2#2144872511yescharacteristic
653C2#2145440711yescharacteristic
654C2#2145441811yescharacteristic
655C2#2145782311yescharacteristic
656C2#2145788911yescharacteristic
657C2#2145791311yescharacteristic
658C2#2145797911yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1854C2
socleC23×C61C22×D4
prime radicalC22×C22⋊D61D4
cosocleC22×C4×C61C24
hypercenter#1854D61
hypocenterC61#22712
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1854

Complements

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

NH ≅ G/Ntype
4. #21446588C2semidirect
5. #21446576C2semidirect
6. #21446020C2semidirect
7. #21446019C2semidirect
8. #21446005C2semidirect
9. #21445951C2semidirect
10. #21445849C2semidirect
11. #21445846C2semidirect
12. #21445718C2semidirect
13. #21445537C2semidirect
14. #21445535C2semidirect
15. C61×#1854C2semidirect
22. #16839830C4semidirect
68. C22×#92418C2×C4semidirect
103. #14519706C2×C4semidirect
105. #14519687C2×C4semidirect
127. C22⋊C8×C61C2×C4semidirect
328. #1854D61semidirect
389. C61#22712semidirect
449. C22⋊C8C4×D61semidirect

Maximal subgroups

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

subgroupindex
C2×#168398302
#214487632
#214487492
#214465882
#214465762
#214460202
#214460192
#214460052
#214459512
#214458492
#214458462
#214457182
#214455372
#214455352
C61×#18542
#2271261