Finite Group Databasebeta

← #29709557

Subgroups — #29709557

Order 15616.

876 conjugacy classes of subgroups.

The subgroup lattice has 680 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839853C211yesmaximal, characteristic, complemented
2C2×#16839820C211yesmaximal, characteristic, complemented
3#21448514C211yesmaximal, characteristic
4#21448497C211yesmaximal, characteristic, complemented
5#21447322C211yesmaximal, characteristic, complemented
6#21447255C211yesmaximal, characteristic, complemented
7#21447195C211yesmaximal, characteristic, complemented
8#21447191C211yesmaximal, characteristic, complemented
9#21447135C211yesmaximal, characteristic, complemented
10#21447016C211yesmaximal, characteristic, complemented
11#21446822C211yesmaximal, characteristic
12#106×Dic61C211yesmaximal, characteristic, complemented
13#21445648C211yesmaximal, characteristic, complemented
14#21445646C211yesmaximal, characteristic, complemented
15C61×#2034C211yesmaximal, characteristic
16C22×#11758200C2211yescharacteristic
17C2×#14519756C2211yescharacteristic
18C2×#14519742C2211yescharacteristic
19C2×D4×Dic61C2211yescharacteristic
20C2×#14519736C2211yescharacteristic
21#16839853—24no—
22#16839820—24no—
23C2×#14519709C2211yescharacteristic
24C2×#14519708C2211yescharacteristic
25C2×#14519706C2211yescharacteristic
26C2×#14519706C2211yescharacteristic
27C2×#14519702C2211yescharacteristic
28C2×#14519702C2211yescharacteristic
29C2×#14519700C2211yescharacteristic
30C2×#14519698C2211yescharacteristic
31C2×C22⋊#92418C2211yescharacteristic
32C2×#14519687C2211yescharacteristic
33C2×C8×Dic61C2211yescharacteristic
34#16839667C2211yescharacteristic
35#16839665—21no—
36#16839665C2211yescharacteristic
37#16839661C2211yescharacteristic
38#16839659—21no—
39#16839658C2211yescharacteristic
40#16839656—21no—
41#16839655—21no—
42#16839650—21no—
43#16839649—21no—
44#16839642—21no—
45#16839637—21no—
46#16839622C2211yescharacteristic
47C4⋊C4×Dic61C2211yescharacteristic
48C4⋊C4×Dic61C2211yescharacteristic
49#16839525C2211yescharacteristic
50#16839524C2211yescharacteristic
51#16839524C2211yescharacteristic
52#16839524C2211yescharacteristic
53#16839524C2211yescharacteristic
54C2×#121×C61C2211yescharacteristic
55C2×#106×C61C2211yescharacteristic
56C2×#102×C61C2211yescharacteristic
57C2×#102×C61C2211yescharacteristic
58C2×C22⋊C8×C61C2211yescharacteristic
59C61×#380C2211yescharacteristic
60C61×#340C2211yescharacteristic
61C22×Dic122C2311yescharacteristic
62C23×Dic61—21no—
63C2×#11758200—22no—
64C2×#11758200—22no—
65C2×#11758200—22no—
66C22×#92418C2311yescharacteristic
67C22×#92418C2311yescharacteristic
68#14519756—22no—
69#14519756—22no—
70#14519742—22no—
71#14519742—22no—
72D4×Dic61—22no—
73D4×Dic61—22no—
74#14519736—22no—
75#14519736—22no—
76C2×C4⋊Dic61—21no—
77C2×C4⋊Dic61D411yescharacteristic
78C2×C4⋊Dic61D411yescharacteristic
79C2×C4⋊Dic61C2311yescharacteristic
80C2×C4⋊Dic61C2311yescharacteristic
81C2×C4⋊Dic61C2311yescharacteristic
82C2×#11758196—21no—
83C2×#11758196—21no—
84C2×#11758196—21no—
85C2×C22⋊Dic61—21no—
86C2×C22⋊Dic61—21no—
87C2×C22⋊Dic61—21no—
88C2×C22⋊Dic61—21no—
89C2×C22⋊Dic61—21no—
90C2×C22⋊Dic61D411yescharacteristic
91C2×C22⋊Dic61D411yescharacteristic
92C2×C4×Dic61—21no—
93C2×C4×Dic61—21no—
94C2×C4×Dic61—21no—
95C2×C4×Dic61C2311yescharacteristic
96C2×C4×Dic61C2311yescharacteristic
97#14519709—22no—
98#14519708—22no—
99#14519706—22no—
100#14519706—22no—
101#14519702—22no—
102#14519702—22no—
103#14519700—22no—
104#14519698—22no—
105C22⋊#92418—22no—
106#14519687—22no—
107C8×Dic61D414yescomplemented
108#14519684—21no—
109#14519684—21no—
110#14519684—21no—
111#14519684—21no—
112#14519684—21no—
113#14519684—21no—
114#14519684—21no—
115C22×D4×C61—21no—
116C22×D4×C61C2311yescharacteristic
117C23×C4×C61C2311yescharacteristic
118C22×C8×C61C2311yescharacteristic
119C22×C8×C61C2311yescharacteristic
120#121×C61—22no—
121#121×C61—22no—
122C2×C4⋊C4×C61C2311yescharacteristic
123C2×C4⋊C4×C61C2311yescharacteristic
124C2×C4⋊C4×C61C2311yescharacteristic
125C2×C22⋊C4×C61—21no—
126C2×C22⋊C4×C61—21no—
127#106×C61—22no—
128#102×C61—22no—
129#102×C61—22no—
130C22⋊C8×C61—22no—
131#95×C61—21no—
132C24×C61—21no—
133C24×C61C2×D411yescharacteristic
134C2×D4×C61—24no—
135C2×D4×C61—21no—
136C2×D4×C61—21no—
137C2×D4×C61—21no—
138C2×D4×C61—21no—
139C2×D4×C61—21no—
140C2×D4×C61—81no—
141C2×D4×C61—81no—
142C2×D4×C61#4112yes—
143C22×C4×C61—21no—
144C22×C4×C61—21no—
145C22×C4×C61—21no—
146C22×C4×C61—21no—
147C22×C4×C61—21no—
148C22×C4×C61—21no—
149C22×C4×C61—21no—
150C22×C4×C61—21no—
151C22×C4×C61—21no—
152C22×C4×C61C2×D411yescharacteristic
153C22×C4×C61C2411yescharacteristic
154C22×Dic61—21no—
155C22×Dic61—21no—
156C22×Dic61—21no—
157C22×Dic61—21no—
158C22×Dic61—42no—
159C22×Dic61—81no—
160C22×Dic61—81no—
161C22×Dic61C2×D411yescharacteristic
162C22×Dic61C2×D411yescharacteristic
163C22×Dic61C2×D411yescharacteristic
164C22×Dic61C2×D411yescharacteristic
165C2×Dic122—21no—
166C2×Dic122—21no—
167C2×Dic122—21no—
168C2×Dic122—21no—
169C2×Dic122—21no—
170C2×Dic122#4112yes—
171C2×C8×C61—21no—
172C2×C8×C61—21no—
173C2×C8×C61—21no—
174C2×C8×C61—21no—
175C2×C8×C61—21no—
176C2×C8×C61C2×D412yes—
177C4⋊C4×C61—21no—
178C4⋊C4×C61—21no—
179C4⋊C4×C61—21no—
180C4⋊C4×C61—21no—
181C4⋊C4×C61—21no—
182C4⋊C4×C61#4112yes—
183C22⋊C4×C61—24no—
184C22⋊C4×C61—81no—
185C22⋊Dic61—24no—
186C22⋊Dic61—22no—
187C22⋊Dic61—22no—
188C22⋊Dic61—42no—
189C22⋊Dic61—81no—
190C22⋊Dic61—81no—
191C22⋊Dic61—81no—
192#11758200—28no—
193C4⋊Dic61—21no—
194C4⋊Dic61—22no—
195C4⋊Dic61—22no—
196C4⋊Dic61—21no—
197C4⋊Dic61—21no—
198C4⋊Dic61—21no—
199C4⋊Dic61—21no—
200C4⋊Dic61—81no—
201C4⋊Dic61#4112yes—
202#11758196—24no—
203#11758196—42no—
204#11758196—81no—
205C4×Dic61—21no—
206C4×Dic61—21no—
207C4×Dic61—42no—
208C4×Dic61—81no—
209C4×Dic61—81no—
210C4×Dic61C2×D412yes—
211C4×Dic61#4112yes—
212C2×#92418—21no—
213C2×#92418—21no—
214C2×#92418—21no—
215C2×#92418—21no—
216C2×#92418—21no—
217C2×#92418C2×D412yes—
218C23×C61—21no—
219C23×C61—21no—
220C23×C61—22no—
221C23×C61—21no—
222C23×C61—21no—
223C23×C61—21no—
224C23×C61—21no—
225C23×C61—21no—
226C23×C61—41no—
227C23×C61—41no—
228C23×C61—41no—
229C23×C61—41no—
230C23×C61—41no—
231C23×C61—41no—
232C23×C61C22×D411yescharacteristic
233D4×C61—22no—
234D4×C61—22no—
235D4×C61—41no—
236D4×C61—41no—
237D4×C61—84no—
238C2×C4×C61—21no—
239C2×C4×C61—21no—
240C2×C4×C61—21no—
241C2×C4×C61—22no—
242C2×C4×C61—41no—
243C2×C4×C61—41no—
244C2×C4×C61—41no—
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×C61C22×D411yescharacteristic
255C2×C4×C61C2×#4111yescharacteristic
256C2×C4×C61C2×#4111yescharacteristic
257C2×C4×C61C2×#4111yescharacteristic
258C2×C4×C61C2×#4111yescharacteristic
259C2×C4×C61C2×#4111yescharacteristic
260C2×Dic61—21no—
261C2×Dic61—21no—
262C2×Dic61—21no—
263C2×Dic61—21no—
264C2×Dic61—21no—
265C2×Dic61—21no—
266C2×Dic61—22no—
267C2×Dic61—22no—
268C2×Dic61—21no—
269C2×Dic61—21no—
270C2×Dic61—21no—
271C2×Dic61—21no—
272C2×Dic61—41no—
273C2×Dic61—41no—
274C2×Dic61—41no—
275C2×Dic61—41no—
276C2×Dic61—41no—
277C2×Dic61—41no—
278C2×Dic61—41no—
279C2×Dic61—41no—
280C2×Dic61—41no—
281C2×Dic61—41no—
282C2×Dic61—84no—
283C2×Dic61#12112yes—
284C2×Dic61#12112yes—
285Dic122—22no—
286Dic122—22no—
287Dic122—41no—
288Dic122—41no—
289C8×C61—22no—
290C8×C61—41no—
291#92418—22no—
292#92418—41no—
293#23707—611nomaximal, Sylow
294C22×C61—41no—
295C22×C61—42no—
296C22×C61—41no—
297C22×C61—41no—
298C22×C61—42no—
299C22×C61—41no—
300C22×C61—41no—
301C22×C61—42no—
302C22×C61—41no—
303C22×C61—41no—
304C22×C61—81no—
305C22×C61—81no—
306C22×C61—81no—
307C22×C61—81no—
308C22×C61C2×#12111yescharacteristic
309C22×C61#52511yescharacteristic
310C22×C61#52511yescharacteristic
311C22×C61#57711yescharacteristic
312C22×C61#57711yescharacteristic
313C22×C61#57711yescharacteristic
314C22×C61#57711yescharacteristic
315C4×C61—42no—
316C4×C61—41no—
317C4×C61—81no—
318C4×C61—81no—
319C4×C61#52511yescharacteristic
320C4×C61#52911yescharacteristic
321C4×C61#52911yescharacteristic
322C4×C61#53211yescharacteristic
323Dic61—22no—
324Dic61—22no—
325Dic61—42no—
326Dic61—42no—
327Dic61—41no—
328Dic61—41no—
329Dic61—81no—
330Dic61—81no—
331C2×C4×SD16—611no—
332C2×#433—611no—
333#2324—611no—
334#2323—611no—
335#2088—611no—
336#2077—611no—
337#2066—611no—
338#2065—611no—
339#2055—611no—
340#2035—611no—
341#2034D6111yescharacteristic
342#2000—611no—
343C4×#106—611no—
344#1763—611no—
345#1762—611no—
346C2×C61—81no—
347C2×C61—82no—
348C2×C61#239211yescharacteristic
349C2×C61#305711yescharacteristic
350C2×C61#306011yescharacteristic
351C2×C61#312311yescharacteristic
352C2×C61#312511yescharacteristic
353C2×C61#312511yescharacteristic
354C2×C61#312611yescharacteristic
355C22×SD16—611no—
356C2×C22⋊Q8—611no—
357C2×#121C2×D6111yescharacteristic
358C2×C4×Q8—611no—
359C2×C4×D4—611no—
360C2×#117—611no—
361C4×SD16—1224no—
362#433—1224no—
363C2×#106—611no—
364C2×#106—611no—
365C2×#106C2×D6111yescharacteristic
366C2×C4⋊C8—611no—
367C2×C4⋊C8—611no—
368C2×#103—611no—
369C2×#103—611no—
370C2×#103—611no—
371C2×#102—611no—
372C2×#102C2×D6111yescharacteristic
373C2×#102C2×D6111yescharacteristic
374C2×C22⋊C8—611no—
375C2×C22⋊C8C2×D6111yescharacteristic
376C2×#97—611no—
377C2×C4×C8—611no—
378#391—611no—
379#390—611no—
380#390—1221no—
381#389—611no—
382#389—611no—
383#389—1221no—
384#388—1221no—
385#388—1221no—
386#388—1221no—
387#387—1221no—
388#386—1221no—
389#385—1221no—
390#380—611no—
391#380C2×D6111yescharacteristic
392C4×C4⋊C4—611no—
393C4×C4⋊C4—611no—
394#340—611no—
395#340—611no—
396#340—611no—
397#340—611no—
398#340—611no—
399#340C2×D6111yescharacteristic
400C61#2370711yescharacteristic, complemented, Sylow
401C22×Q8—611no—
402C22×D4—21no—
403C22×D4C22×D6111yescharacteristic
404C23×C4—1221no—
405C23×C4C22×D6111yescharacteristic
406C2×SD16—1222no—
407C2×SD16—1222no—
408C2×SD16—1222no—
409C22×C8—611no—
410C22×C8—611no—
411C22×C8C22×D6111yescharacteristic
412C22×C8C22×D6111yescharacteristic
413C22⋊Q8—1222no—
414C22⋊Q8—1222no—
415#121—22no—
416#121—22no—
417C4×Q8—1222no—
418C4×Q8—1222no—
419C4×D4—1222no—
420C4×D4—1222no—
421#117—1222no—
422#117—1222no—
423C2×C4⋊C4—611no—
424C2×C4⋊C4—611no—
425C2×C4⋊C4—611no—
426C2×C4⋊C4—611no—
427C2×C4⋊C4—611no—
428C2×C4⋊C4—1221no—
429C2×C4⋊C4—1221no—
430C2×C4⋊C4—1221no—
431C2×C4⋊C4—1221no—
432C2×C4⋊C4C22×D6111yescharacteristic
433C2×C4⋊C4C22×D6111yescharacteristic
434C2×C4⋊C4C22×D6111yescharacteristic
435C2×C22⋊C4—21no—
436C2×C22⋊C4—21no—
437C2×C22⋊C4—611no—
438C2×C22⋊C4—611no—
439C2×C22⋊C4—1221no—
440C2×C22⋊C4—1221no—
441C2×C22⋊C4—1221no—
442C2×C22⋊C4—1221no—
443C2×C22⋊C4—1221no—
444C2×C42—611no—
445C2×C42—611no—
446C2×C42—1221no—
447C2×C42—1221no—
448C2×C42—1221no—
449#106—22no—
450#106—1222no—
451#106—1222no—
452C4⋊C8—1222no—
453C4⋊C8—1222no—
454#103—1222no—
455#103—1222no—
456#103—1222no—
457#102—22no—
458#102—22no—
459#102—1222no—
460C22⋊C8—22no—
461C22⋊C8—1222no—
462#97—1222no—
463C4×C8—614no—
464#95—21no—
465#95—1221no—
466#95—1221no—
467#95—1221no—
468#95—1221no—
469#95—1221no—
470#95—1221no—
471#95—1221no—
472C24—21no—
473C24D4×D6111yescharacteristic
474C2×Q8—612no—
475C2×Q8—1221no—
476C2×Q8—1221no—
477C2×Q8—1221no—
478C2×Q8—1221no—
479C2×Q8—1221no—
480C2×D4—24no—
481C2×D4—21no—
482C2×D4—21no—
483C2×D4—21no—
484C2×D4—21no—
485C2×D4—21no—
486C2×D4—81no—
487C2×D4—81no—
488C2×D4#1175822412yes—
489C22×C4—21no—
490C22×C4—21no—
491C22×C4—21no—
492C22×C4—21no—
493C22×C4—21no—
494C22×C4—21no—
495C22×C4—21no—
496C22×C4—21no—
497C22×C4—21no—
498C22×C4—611no—
499C22×C4—611no—
500C22×C4—611no—
501C22×C4—611no—
502C22×C4—1221no—
503C22×C4—1221no—
504C22×C4—1221no—
505C22×C4—1221no—
506C22×C4—4881no—
507C22×C4—4881no—
508C22×C4—2442no—
509C22×C4D4×D6111yescharacteristic
510C22×C4C23×D6111yescharacteristic
511SD16—1228no—
512C2×C8—21no—
513C2×C8—21no—
514C2×C8—21no—
515C2×C8—21no—
516C2×C8—21no—
517C2×C8—612no—
518C2×C8—1221no—
519C2×C8—1221no—
520C2×C8—1221no—
521C2×C8—1221no—
522C2×C8—1221no—
523C2×C8D4×D6112yes—
524C4⋊C4—21no—
525C4⋊C4—21no—
526C4⋊C4—21no—
527C4⋊C4—21no—
528C4⋊C4—21no—
529C4⋊C4—612no—
530C4⋊C4—1222no—
531C4⋊C4—1222no—
532C4⋊C4—1224no—
533C4⋊C4—1221no—
534C4⋊C4—1221no—
535C4⋊C4—1221no—
536C4⋊C4—1221no—
537C4⋊C4—1221no—
538C4⋊C4—4881no—
539C4⋊C4—4881no—
540C4⋊C4—2442no—
541C4⋊C4#1175822412yes—
542C22⋊C4—24no—
543C22⋊C4—81no—
544C22⋊C4—1222no—
545C22⋊C4—1222no—
546C22⋊C4—1224no—
547C22⋊C4—4881no—
548C22⋊C4—4881no—
549C22⋊C4—4881no—
550C22⋊C4—2442no—
551C42—612no—
552C42—612no—
553C42—1221no—
554C42—1221no—
555C42—4881no—
556C42—4881no—
557C42—2442no—
558C23—21no—
559C23—22no—
560C23—21no—
561C23—21no—
562C23—21no—
563C23—21no—
564C23—21no—
565C23—21no—
566C23—41no—
567C23—41no—
568C23—41no—
569C23—41no—
570C23—41no—
571C23—41no—
572C23C2×D4×D6111yescharacteristic
573Q8—1222no—
574Q8—1222no—
575Q8—2441no—
576Q8—2441no—
577D4—22no—
578D4—22no—
579D4—41no—
580D4—41no—
581D4—84no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—22no—
585C2×C4—21no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—612no—
599C2×C4—612no—
600C2×C4—1221no—
601C2×C4—1221no—
602C2×C4—1221no—
603C2×C4—1221no—
604C2×C4—1221no—
605C2×C4—1221no—
606C2×C4—1222no—
607C2×C4—1222no—
608C2×C4—1221no—
609C2×C4—1221no—
610C2×C4—1221no—
611C2×C4—1221no—
612C2×C4—4884no—
613C2×C4—2441no—
614C2×C4—2441no—
615C2×C4—2441no—
616C2×C4—2441no—
617C2×C4—2441no—
618C2×C4—2441no—
619C2×C4—2441no—
620C2×C4—2441no—
621C2×C4—2441no—
622C2×C4—2441no—
623C2×C4C2×D4×D6111yescharacteristic
624C2×C4#41×D6111yescharacteristic
625C2×C4#41×D6111yescharacteristic
626C2×C4#41×D6111yescharacteristic
627C2×C4C2×#1175822411yescharacteristic
628C2×C4C2×#1175822411yescharacteristic
629C8—22no—
630C8—41no—
631C8—1222no—
632C8—2441no—
633C22—41no—
634C22—41no—
635C22—42no—
636C22—41no—
637C22—41no—
638C22—42no—
639C22—41no—
640C22—42no—
641C22—41no—
642C22—41no—
643C22—81no—
644C22—81no—
645C22—81no—
646C22—81no—
647C22#121×D6111yescharacteristic
648C22#1684026811yescharacteristic
649C22#1684027011yescharacteristic
650C22#1684056811yescharacteristic
651C22#1684056911yescharacteristic
652C22#1684057211yescharacteristic
653C22#1684057211yescharacteristic
654C4—41no—
655C4—42no—
656C4—81no—
657C4—81no—
658C4—1222no—
659C4—1222no—
660C4—4881no—
661C4—4881no—
662C4—2441no—
663C4—2442no—
664C4—2441no—
665C4—2442no—
666C4#1684027211yescharacteristic
667C4#1684028911yescharacteristic
668C4#1684029111yescharacteristic
669C4#1684030711yescharacteristic
670C2—81no—
671C2—82no—
672C2#2144906211yescharacteristic
673C2#2145545311yescharacteristic
674C2#2145547911yescharacteristic
675C2#2145606511yescharacteristic
676C2#2145609711yescharacteristic
677C2#2145609911yescharacteristic
678C2#2145610611yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2034C2
socleC23×C61C22×D4
prime radicalC2×#121×C61C22
hypercenter#2034D61
hypocenterC61#23707
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ #2034

Complements

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

NH ≅ G/Ntype
1. C2×#16839853C2semidirect
2. C2×#16839820C2semidirect
4. #21448497C2semidirect
5. #21447322C2semidirect
6. #21447255C2semidirect
7. #21447195C2semidirect
8. #21447191C2semidirect
9. #21447135C2semidirect
10. #21447016C2semidirect
12. #106×Dic61C2semidirect
13. #21445648C2semidirect
14. #21445646C2semidirect
107. C8×Dic61D4semidirect
400. C61#23707semidirect

Maximal subgroups

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

subgroupindex
C2×#168398532
C2×#168398202
#214485142
#214484972
#214473222
#214472552
#214471952
#214471912
#214471352
#214470162
#214468222
#106×Dic612
#214456482
#214456462
C61×#20342
#2370761