Finite Group Databasebeta

← #29729864

Subgroups — #29729864

Order 15616.

838 conjugacy classes of subgroups.

The subgroup lattice has 654 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840111C211yesmaximal, characteristic, complemented
2C2×#16839965C211yesmaximal, characteristic
3#21450990C211yesmaximal, characteristic
4#21448873C211yesmaximal, characteristic, complemented
5#21447240C211yesmaximal, characteristic, complemented
6#21447239C211yesmaximal, characteristic, complemented
7#21447136C211yesmaximal, characteristic
8#21447132C211yesmaximal, characteristic
9#21446765C211yesmaximal, characteristic, complemented
10#21446536C211yesmaximal, characteristic, complemented
11#21446171C211yesmaximal, characteristic
12#21445834C211yesmaximal, characteristic, complemented
13#21445833C211yesmaximal, characteristic, complemented
14#102×Dic61C211yesmaximal, characteristic
15C61×#1880C211yesmaximal, characteristic
16C22×#11758200C2211yescharacteristic
17C2×#14519776C2211yescharacteristic
18C2×#14519775C2211yescharacteristic
19C2×#14519742C2211yescharacteristic
20C2×D4×Dic61C2211yescharacteristic
21#16840111—24no—
22#16839965—24no—
23C2×C4⋊#92418C2211yescharacteristic
24C2×#14519706C2211yescharacteristic
25C2×#14519706C2211yescharacteristic
26C2×#14519702C2211yescharacteristic
27C2×#14519702C2211yescharacteristic
28C2×#14519701C2211yescharacteristic
29C2×#14519701C2211yescharacteristic
30C2×#14519700C2211yescharacteristic
31C2×#14519698C2211yescharacteristic
32C2×C8×Dic61C2211yescharacteristic
33#16839674—21no—
34#16839674—21no—
35#16839669C2211yescharacteristic
36#16839668C2211yescharacteristic
37#16839665C2211yescharacteristic
38#16839657—21no—
39#16839646—21no—
40#16839643—21no—
41#16839628C2211yescharacteristic
42#16839627C2211yescharacteristic
43#16839620—21no—
44#16839620—21no—
45C4⋊C4×Dic61C2211yescharacteristic
46C4×#11758196—21no—
47C4×#11758196C2211yescharacteristic
48#16839525C2211yescharacteristic
49#16839525C2211yescharacteristic
50#16839524C2211yescharacteristic
51#16839524C2211yescharacteristic
52#16839524C2211yescharacteristic
53#16839516C2211yescharacteristic
54C22×SD16×C61C2211yescharacteristic
55C2×C4×Q8×C61C2211yescharacteristic
56C2×#103×C61C2211yescharacteristic
57C2×#102×C61C2211yescharacteristic
58C61×#390C2211yescharacteristic
59C61×#340C2211yescharacteristic
60C61×#336C2211yescharacteristic
61C22×Dic122—21no—
62C22×Dic122C2311yescharacteristic
63C23×Dic61—21no—
64C2×#11758200—22no—
65C2×#11758200—22no—
66C2×#11758200—41no—
67C22×#92418C2311yescharacteristic
68C22×#92418C2311yescharacteristic
69#14519776—22no—
70#14519776—22no—
71#14519775—22no—
72#14519775D414yes—
73#14519742—22no—
74#14519742—41no—
75D4×Dic61—22no—
76D4×Dic61—22no—
77C2×C4⋊Dic61C2311yescharacteristic
78C2×C4⋊Dic61C2311yescharacteristic
79C2×C4⋊Dic61C2311yescharacteristic
80C2×#11758196—21no—
81C2×#11758196—21no—
82C2×#11758196—21no—
83C2×#11758196—21no—
84C2×#11758196D411yescharacteristic
85C2×#11758196D411yescharacteristic
86C2×C22⋊Dic61—21no—
87C2×C22⋊Dic61—21no—
88C2×C22⋊Dic61—21no—
89C2×C4×Dic61—21no—
90C2×C4×Dic61—21no—
91C2×C4×Dic61C2311yescharacteristic
92C2×C4×Dic61C2311yescharacteristic
93C4⋊#92418—41no—
94#14519706—22no—
95#14519706—41no—
96#14519702—22no—
97#14519702—22no—
98#14519701—22no—
99#14519701—22no—
100#14519700—22no—
101#14519698—22no—
102C8×Dic61—22no—
103#14519684—21no—
104#14519684—21no—
105#14519684—21no—
106#14519684—21no—
107#14519684—21no—
108#14519684—21no—
109#14519684—21no—
110#14519684—21no—
111#14519684—21no—
112#14519684D411yescharacteristic
113#14519684D411yescharacteristic
114C22×Q8×C61C2311yescharacteristic
115C22×D4×C61C2311yescharacteristic
116C2×SD16×C61—22no—
117C2×SD16×C61—22no—
118C2×SD16×C61—41no—
119C22×C8×C61C2311yescharacteristic
120C22×C8×C61C2311yescharacteristic
121C4×Q8×C61—22no—
122C4×Q8×C61—41no—
123C2×C4⋊C4×C61—21no—
124C2×C4⋊C4×C61C2311yescharacteristic
125C2×C4⋊C4×C61C2311yescharacteristic
126C2×C22⋊C4×C61—21no—
127C2×C22⋊C4×C61—21no—
128C2×C42×C61—21no—
129C2×C42×C61C2311yescharacteristic
130#103×C61—22no—
131#102×C61—22no—
132C24×C61—21no—
133C2×Q8×C61—21no—
134C2×Q8×C61—21no—
135C2×Q8×C61—21no—
136C2×Q8×C61—21no—
137C2×Q8×C61—21no—
138C2×Q8×C61C2×D412yes—
139C2×D4×C61—21no—
140C2×D4×C61—21no—
141C2×D4×C61—21no—
142C2×D4×C61—21no—
143C2×D4×C61—21no—
144C2×D4×C61#4112yes—
145C22×C4×C61—21no—
146C22×C4×C61—21no—
147C22×C4×C61—21no—
148C22×C4×C61C2×D411yescharacteristic
149C22×C4×C61C2×D411yescharacteristic
150C22×C4×C61C2411yescharacteristic
151C22×Dic61—21no—
152C22×Dic61—21no—
153C22×Dic61—21no—
154C22×Dic61—21no—
155C22×Dic61—42no—
156C22×Dic61—42no—
157C22×Dic61—42no—
158C22×Dic61C2×D411yescharacteristic
159C22×Dic61C2×D411yescharacteristic
160C22×Dic61C2×D411yescharacteristic
161C22×Dic61C2×D411yescharacteristic
162C2×Dic122—21no—
163C2×Dic122—21no—
164C2×Dic122—21no—
165C2×Dic122—21no—
166C2×Dic122—21no—
167C2×Dic122—42no—
168C2×Dic122—81no—
169C2×Dic122—81no—
170C2×Dic122C2×D412yes—
171SD16×C61—44no—
172C2×C8×C61—21no—
173C2×C8×C61—21no—
174C2×C8×C61—21no—
175C2×C8×C61—21no—
176C2×C8×C61—21no—
177C2×C8×C61—21no—
178C4⋊C4×C61—21no—
179C4⋊C4×C61—21no—
180C4⋊C4×C61—21no—
181C4⋊C4×C61—42no—
182C4⋊C4×C61#4112yes—
183C22⋊C4×C61—81no—
184C22⋊C4×C61—81no—
185C42×C61—21no—
186C42×C61—21no—
187C42×C61—42no—
188C22⋊Dic61—42no—
189C22⋊Dic61—81no—
190C22⋊Dic61—81no—
191#11758200—44no—
192C4⋊Dic61—21no—
193C4⋊Dic61—21no—
194C4⋊Dic61—21no—
195C4⋊Dic61—21no—
196C4⋊Dic61—21no—
197C4⋊Dic61—21no—
198#11758196—24no—
199#11758196—24no—
200#11758196—22no—
201#11758196—22no—
202#11758196—42no—
203#11758196—42no—
204C4×Dic61—21no—
205C4×Dic61—21no—
206C4×Dic61—42no—
207C4×Dic61—42no—
208C4×Dic61C2×D412yes—
209C4×Dic61#4112yes—
210C2×#92418—21no—
211C2×#92418—21no—
212C2×#92418—21no—
213C2×#92418—21no—
214C2×#92418—21no—
215C2×#92418—21no—
216C23×C61—22no—
217C23×C61—41no—
218C23×C61—41no—
219C23×C61—41no—
220C23×C61—41no—
221C23×C61—41no—
222C23×C61—41no—
223C23×C61C22×D411yescharacteristic
224Q8×C61—22no—
225Q8×C61—22no—
226Q8×C61—41no—
227Q8×C61—41no—
228D4×C61—22no—
229D4×C61—22no—
230D4×C61—41no—
231D4×C61—41no—
232C2×C4×C61—22no—
233C2×C4×C61—22no—
234C2×C4×C61—21no—
235C2×C4×C61—21no—
236C2×C4×C61—21no—
237C2×C4×C61—21no—
238C2×C4×C61—21no—
239C2×C4×C61—21no—
240C2×C4×C61—41no—
241C2×C4×C61—41no—
242C2×C4×C61—41no—
243C2×C4×C61—41no—
244C2×C4×C61—41no—
245C2×C4×C61—41no—
246C2×C4×C61—41no—
247C2×C4×C61C22×D411yescharacteristic
248C2×C4×C61C2×#4111yescharacteristic
249C2×C4×C61C2×#4111yescharacteristic
250C2×C4×C61C2×#4111yescharacteristic
251C2×C4×C61H(5,𝔽2)11yescharacteristic
252C2×C4×C612−1+411yescharacteristic
253C2×Dic61—21no—
254C2×Dic61—21no—
255C2×Dic61—22no—
256C2×Dic61—21no—
257C2×Dic61—21no—
258C2×Dic61—21no—
259C2×Dic61—21no—
260C2×Dic61—21no—
261C2×Dic61—21no—
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×Dic61—41no—
272C2×Dic61—41no—
273C2×Dic61—84no—
274C2×Dic61#12112yes—
275C2×Dic61#12112yes—
276C2×Dic61#13512yes—
277C2×Dic61#13712yes—
278Dic122—22no—
279Dic122—22no—
280Dic122—41no—
281Dic122—41no—
282Dic122—84no—
283C8×C61—41no—
284C8×C61—41no—
285#92418—41no—
286#92418—41no—
287#25319—611nomaximal, Sylow
288C22×C61—42no—
289C22×C61—42no—
290C22×C61—42no—
291C22×C61—81no—
292C22×C61—81no—
293C22×C61—81no—
294C22×C61—81no—
295C22×C61C2×#12111yescharacteristic
296C22×C61#54611yescharacteristic
297C22×C61#54811yescharacteristic
298C22×C61C2×#13511yescharacteristic
299C22×C61C2×#13711yescharacteristic
300C22×C61#57711yescharacteristic
301C22×C61#57711yescharacteristic
302C4×C61—41no—
303C4×C61—42no—
304C4×C61—41no—
305C4×C61—42no—
306C4×C61—81no—
307C4×C61#52911yescharacteristic
308C4×C61#54211yescharacteristic
309C4×C61#54611yescharacteristic
310C4×C61#54711yescharacteristic
311Dic61—22no—
312Dic61—41no—
313Dic61—42no—
314Dic61—41no—
315Dic61—81no—
316Dic61—81no—
317Dic61—81no—
318Dic61#46414yes—
319C2×#487—611no—
320C2×#461—611no—
321#2563—611no—
322#2365—611no—
323#2074—611no—
324#2074—611no—
325#2055—611no—
326#2055—611no—
327#1987—611no—
328#1939—611no—
329#1880—611no—
330#1880D6111yescharacteristic
331#1805—611no—
332#1805—611no—
333C4×#102—611no—
334C2×C61—82no—
335C2×C61#242711yescharacteristic
336C2×C61C2×#46411yescharacteristic
337C2×C61#304311yescharacteristic
338C2×C61#330311yescharacteristic
339C2×C61#332911yescharacteristic
340C2×C61#336011yescharacteristic
341C2×C61#337211yescharacteristic
342C22×SD16—611no—
343C22×SD16C2×D6111yescharacteristic
344C2×C4⋊Q8—611no—
345C2×C4⋊Q8—611no—
346C2×C4×Q8—611no—
347C2×C4×Q8C2×D6111yescharacteristic
348C2×C4×D4—611no—
349#487—1224no—
350#461—1224no—
351C2×C4⋊C8—611no—
352C2×C4⋊C8—611no—
353C2×C4⋊C8—611no—
354C2×#103—611no—
355C2×#103—611no—
356C2×#103—611no—
357C2×#103—611no—
358C2×#103—611no—
359C2×#103C2×D6111yescharacteristic
360C2×#102—611no—
361C2×#102C2×D6111yescharacteristic
362C2×C4×C8—611no—
363#393—1221no—
364#393—1221no—
365#391—611no—
366#391—611no—
367#390—611no—
368#390C2×D6111yescharacteristic
369#389—1221no—
370#387—1221no—
371#387—1221no—
372#382—611no—
373#382—611no—
374#380—1221no—
375#380—1221no—
376C4×C4⋊C4—611no—
377C4×C4⋊C4—611no—
378C4×C4⋊C4—1221no—
379#340—611no—
380#340—611no—
381#340—611no—
382#340—611no—
383#340—611no—
384#340C2×D6111yescharacteristic
385#336—611no—
386#336C2×D6111yescharacteristic
387C61#2531911yescharacteristic, complemented, Sylow
388C22×Q8—611no—
389C22×Q8—1221no—
390C22×Q8C22×D6111yescharacteristic
391C22×D4C22×D6111yescharacteristic
392C23×C4—1221no—
393C2×SD16—22no—
394C2×SD16—22no—
395C2×SD16—41no—
396C2×SD16—1222no—
397C2×SD16—1222no—
398C2×SD16—2441no—
399C22×C8—611no—
400C22×C8—611no—
401C22×C8C22×D6111yescharacteristic
402C22×C8C22×D6111yescharacteristic
403C4⋊Q8—614no—
404C4⋊Q8—1222no—
405C4⋊Q8—1222no—
406C4⋊Q8—1222no—
407C4×Q8—22no—
408C4×Q8—41no—
409C4×Q8—1222no—
410C4×Q8—2441no—
411C4×D4—1222no—
412C4×D4—1222no—
413C2×C4⋊C4—21no—
414C2×C4⋊C4—611no—
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⋊C4—1221no—
422C2×C4⋊C4—1221no—
423C2×C4⋊C4C22×D6111yescharacteristic
424C2×C4⋊C4C22×D6111yescharacteristic
425C2×C22⋊C4—21no—
426C2×C22⋊C4—21no—
427C2×C22⋊C4—1221no—
428C2×C22⋊C4—1221no—
429C2×C22⋊C4—1221no—
430C2×C42—21no—
431C2×C42—611no—
432C2×C42—611no—
433C2×C42—1221no—
434C2×C42—1221no—
435C2×C42C22×D6111yescharacteristic
436C4⋊C8—1222no—
437C4⋊C8—2441no—
438C4⋊C8—2441no—
439#103—22no—
440#103—1222no—
441#103—1222no—
442#103—1222no—
443#103—1222no—
444#103—1222no—
445#102—22no—
446#102—1222no—
447C4×C8—1222no—
448#95—611no—
449#95—611no—
450#95—1221no—
451#95—1221no—
452#95—1221no—
453#95—1221no—
454#95—1221no—
455#95—1221no—
456#95—1221no—
457#95—1221no—
458#95—1221no—
459C24—21no—
460C2×Q8—21no—
461C2×Q8—21no—
462C2×Q8—21no—
463C2×Q8—21no—
464C2×Q8—21no—
465C2×Q8—612no—
466C2×Q8—1221no—
467C2×Q8—1221no—
468C2×Q8—1221no—
469C2×Q8—1221no—
470C2×Q8—1221no—
471C2×Q8—4881no—
472C2×Q8—4881no—
473C2×Q8—2442no—
474C2×Q8D4×D6112yes—
475C2×D4—21no—
476C2×D4—21no—
477C2×D4—21no—
478C2×D4—21no—
479C2×D4—21no—
480C2×D4#1175822212yes—
481C22×C4—21no—
482C22×C4—21no—
483C22×C4—21no—
484C22×C4—611no—
485C22×C4—611no—
486C22×C4—611no—
487C22×C4—611no—
488C22×C4—1221no—
489C22×C4—1221no—
490C22×C4—1221no—
491C22×C4—1221no—
492C22×C4—2442no—
493C22×C4—2442no—
494C22×C4—2442no—
495C22×C4D4×D6111yescharacteristic
496C22×C4D4×D6111yescharacteristic
497C22×C4C23×D6111yescharacteristic
498SD16—44no—
499SD16—2444no—
500C2×C8—21no—
501C2×C8—21no—
502C2×C8—21no—
503C2×C8—21no—
504C2×C8—21no—
505C2×C8—21no—
506C2×C8—1221no—
507C2×C8—1221no—
508C2×C8—1221no—
509C2×C8—1221no—
510C2×C8—1221no—
511C2×C8—1221no—
512C4⋊C4—21no—
513C4⋊C4—21no—
514C4⋊C4—21no—
515C4⋊C4—42no—
516C4⋊C4—1222no—
517C4⋊C4—1222no—
518C4⋊C4—1224no—
519C4⋊C4—1224no—
520C4⋊C4—1221no—
521C4⋊C4—1221no—
522C4⋊C4—1221no—
523C4⋊C4—1221no—
524C4⋊C4—1221no—
525C4⋊C4—1221no—
526C4⋊C4—2442no—
527C4⋊C4—2442no—
528C4⋊C4#1175822412yes—
529C22⋊C4—81no—
530C22⋊C4—81no—
531C22⋊C4—4881no—
532C22⋊C4—4881no—
533C22⋊C4—2442no—
534C42—21no—
535C42—21no—
536C42—42no—
537C42—612no—
538C42—612no—
539C42—1221no—
540C42—1221no—
541C42—2442no—
542C42—2442no—
543C23—22no—
544C23—41no—
545C23—41no—
546C23—41no—
547C23—41no—
548C23—41no—
549C23—41no—
550C23C2×D4×D6111yescharacteristic
551Q8—22no—
552Q8—22no—
553Q8—41no—
554Q8—41no—
555Q8—1222no—
556Q8—1222no—
557Q8—4884no—
558Q8—2441no—
559Q8—2441no—
560D4—22no—
561D4—22no—
562D4—41no—
563D4—41no—
564C2×C4—22no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—22no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—612no—
580C2×C4—612no—
581C2×C4—612no—
582C2×C4—612no—
583C2×C4—1221no—
584C2×C4—1221no—
585C2×C4—1222no—
586C2×C4—1221no—
587C2×C4—1221no—
588C2×C4—1221no—
589C2×C4—1221no—
590C2×C4—1221no—
591C2×C4—1221no—
592C2×C4—4884no—
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×C4—2441no—
603C2×C4—2441no—
604C2×C4C2×D4×D6111yescharacteristic
605C2×C4#41×D6111yescharacteristic
606C2×C4C2×#1175822411yescharacteristic
607C2×C4C2×#1175822211yescharacteristic
608C2×C4#1451982211yescharacteristic
609C2×C4#1451982411yescharacteristic
610C8—41no—
611C8—41no—
612C8—2441no—
613C8—2441no—
614C22—42no—
615C22—42no—
616C22—42no—
617C22—81no—
618C22—81no—
619C22—81no—
620C22—81no—
621C22#121×D6111yescharacteristic
622C22#1684039911yescharacteristic
623C22#1684042111yescharacteristic
624C22#135×D6111yescharacteristic
625C22#137×D6111yescharacteristic
626C22#1684056811yescharacteristic
627C22#1684057211yescharacteristic
628C4—41no—
629C4—42no—
630C4—42no—
631C4—41no—
632C4—81no—
633C4—614no—
634C4—1222no—
635C4—4881no—
636C4—4881no—
637C4—4881no—
638C4—2441no—
639C4—2442no—
640C4—2441no—
641C4#1684029211yescharacteristic
642C4#1684038011yescharacteristic
643C4#1684040211yescharacteristic
644C4#1684041211yescharacteristic
645C2—82no—
646C2#2144939311yescharacteristic
647C2#464×D6111yescharacteristic
648C2#2145533511yescharacteristic
649C2#2145774011yescharacteristic
650C2#2145810611yescharacteristic
651C2#2145846911yescharacteristic
652C2#2145860111yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1880C2
socleC23×C61C22×D4
prime radicalC22×D4×C61C23
hypercenter#1880D61
hypocenterC61#25319
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1880

Complements

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

NH ≅ G/Ntype
1. C2×#16840111C2semidirect
4. #21448873C2semidirect
5. #21447240C2semidirect
6. #21447239C2semidirect
9. #21446765C2semidirect
10. #21446536C2semidirect
12. #21445834C2semidirect
13. #21445833C2semidirect
387. C61#25319semidirect

Maximal subgroups

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

subgroupindex
C2×#168401112
C2×#168399652
#214509902
#214488732
#214472402
#214472392
#214471362
#214471322
#214467652
#214465362
#214461712
#214458342
#214458332
#102×Dic612
C61×#18802
#2531961