Finite Group Databasebeta

← #29686417

Subgroups — #29686417

Order 15616.

1008 conjugacy classes of subgroups.

The subgroup lattice has 666 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839965C211yesmaximal, characteristic
2C2×#16839859C211yesmaximal, characteristic
3C2×SD16×Dic61C211yesmaximal, characteristic
4#21448537C211yesmaximal, characteristic, complemented
5#21448180C211yesmaximal, characteristic
6#21446740C211yesmaximal, characteristic, complemented
7#21446586C211yesmaximal, characteristic, complemented
8#21446457C211yesmaximal, characteristic, complemented
9#21446454C211yesmaximal, characteristic, complemented
10#21446153C211yesmaximal, characteristic
11#21445827C211yesmaximal, characteristic
12#21445675C211yesmaximal, characteristic, complemented
13#21445651C211yesmaximal, characteristic, complemented
14#21445638C211yesmaximal, characteristic, complemented
15C61×#1877C211yesmaximal, characteristic
16C22×#11758200C2211yescharacteristic
17C2×#14519775C2211yescharacteristic
18C2×Q8×Dic61C2211yescharacteristic
19C2×#14519742C2211yescharacteristic
20C2×D4×Dic61C2211yescharacteristic
21C2×D4×Dic61C2211yescharacteristic
22#16839965C418yescomplemented
23#16839859—24no—
24SD16×Dic61—24no—
25C2×#14519710C2211yescharacteristic
26C2×#14519709C2211yescharacteristic
27C2×#14519706C2211yescharacteristic
28C2×#14519706C2211yescharacteristic
29C2×#14519702C2211yescharacteristic
30C2×#14519702C2211yescharacteristic
31C2×#14519701C2211yescharacteristic
32C2×#14519701C2211yescharacteristic
33C2×#14519700C2211yescharacteristic
34C2×#14519698C2211yescharacteristic
35C2×#14519687C2211yescharacteristic
36C2×C8×Dic61C2211yescharacteristic
37#16839668C2211yescharacteristic
38#16839649—21no—
39#16839649—21no—
40#16839644—21no—
41#16839644—21no—
42#16839627C2211yescharacteristic
43C4⋊C4×Dic61C2211yescharacteristic
44C4⋊C4×Dic61C2211yescharacteristic
45C4×C4⋊Dic61C2211yescharacteristic
46C22⋊C4×Dic61—21no—
47C22⋊C4×Dic61—21no—
48C4×C22⋊Dic61—21no—
49#16839608C2211yescharacteristic
50C2×#14519684—21no—
51#16839525C2211yescharacteristic
52#16839524C2211yescharacteristic
53#16839524C2211yescharacteristic
54#16839517C2211yescharacteristic
55C22×SD16×C61C2211yescharacteristic
56C2×C4×D4×C61C2211yescharacteristic
57C2×#103×C61C2211yescharacteristic
58C2×#102×C61C2211yescharacteristic
59C61×#391C2211yescharacteristic
60C61×#340C2211yescharacteristic
61C61×#336C2211yescharacteristic
62C22×Dic122C2311yescharacteristic
63C23×Dic61—21no—
64C23×Dic61—21no—
65C2×#11758200—22no—
66C2×#11758200—22no—
67C2×#11758200C2×C414yes—
68C22×#92418C2311yescharacteristic
69C22×#92418C2311yescharacteristic
70#14519775—22no—
71#14519775C2×C414yescomplemented
72Q8×Dic61—22no—
73Q8×Dic61—22no—
74#14519742—22no—
75#14519742—22no—
76D4×Dic61—22no—
77D4×Dic61—22no—
78D4×Dic61C2×C414yes—
79D4×Dic61D414yes—
80C2×C4⋊Dic61—21no—
81C2×C4⋊Dic61C2311yescharacteristic
82C2×C4⋊Dic61C2311yescharacteristic
83C2×C4⋊Dic61C2311yescharacteristic
84C2×#11758196—21no—
85C2×#11758196—21no—
86C2×#11758196—21no—
87C2×C22⋊Dic61—21no—
88C2×C22⋊Dic61—21no—
89C2×C4×Dic61—21no—
90C2×C4×Dic61—21no—
91C2×C4×Dic61—21no—
92C2×C4×Dic61—21no—
93C2×C4×Dic61—21no—
94C2×C4×Dic61C2311yescharacteristic
95C2×C4×Dic61C2311yescharacteristic
96#14519710—22no—
97#14519709—22no—
98#14519706—22no—
99#14519706C2×C414yescomplemented
100#14519702—22no—
101#14519702C2×C414yescomplemented
102#14519701—22no—
103#14519701C2×C414yescomplemented
104#14519700—22no—
105#14519698—22no—
106#14519687—22no—
107C8×Dic61—22no—
108#14519684—21no—
109#14519684—21no—
110#14519684—21no—
111#14519684—21no—
112#14519684—21no—
113#14519684—21no—
114#14519684—21no—
115#14519684D411yescharacteristic
116#14519684D411yescharacteristic
117#14519684D411yescharacteristic
118#14519684D411yescharacteristic
119C22×Q8×C61C2311yescharacteristic
120C22×D4×C61C2311yescharacteristic
121C23×C4×C61—21no—
122C2×SD16×C61—22no—
123C2×SD16×C61—22no—
124C2×SD16×C61C2×C414yes—
125C22×C8×C61C2311yescharacteristic
126C22×C8×C61C2311yescharacteristic
127C4×D4×C61—22no—
128C4×D4×C61—22no—
129C2×C4⋊C4×C61C2311yescharacteristic
130C2×C4⋊C4×C61C2311yescharacteristic
131C2×C22⋊C4×C61—21no—
132C2×C42×C61C2311yescharacteristic
133#103×C61—22no—
134#102×C61—22no—
135#95×C61—21no—
136#95×C61—21no—
137C24×C61—21no—
138C2×Q8×C61—21no—
139C2×Q8×C61—21no—
140C2×Q8×C61—21no—
141C2×Q8×C61—21no—
142C2×Q8×C61—21no—
143C2×Q8×C61C22×C412yes—
144C2×D4×C61—21no—
145C2×D4×C61—21no—
146C2×D4×C61—21no—
147C2×D4×C61C22×C412yes—
148C2×D4×C61C2×D412yes—
149C2×D4×C61#4112yes—
150C22×C4×C61—21no—
151C22×C4×C61—21no—
152C22×C4×C61—24no—
153C22×C4×C61—21no—
154C22×C4×C61—81no—
155C22×C4×C61—81no—
156C22×C4×C61C2×D411yescharacteristic
157C22×C4×C61C2×D411yescharacteristic
158C22×C4×C61C2411yescharacteristic
159C22×Dic61—21no—
160C22×Dic61—21no—
161C22×Dic61—21no—
162C22×Dic61—24no—
163C22×Dic61—24no—
164C22×Dic61—21no—
165C22×Dic61—81no—
166C22×Dic61—81no—
167C22×Dic61—81no—
168C22×Dic61—81no—
169C22×Dic61C2×D411yescharacteristic
170C22×Dic61C2×D411yescharacteristic
171C22×Dic61C2×D411yescharacteristic
172C22×Dic61C2×D411yescharacteristic
173C2×Dic122—21no—
174C2×Dic122—21no—
175C2×Dic122—21no—
176C2×Dic122—21no—
177C2×Dic122—21no—
178C2×Dic122C22×C412yes—
179SD16×C61—28no—
180C2×C8×C61—21no—
181C2×C8×C61—21no—
182C2×C8×C61—21no—
183C2×C8×C61—21no—
184C2×C8×C61—21no—
185C2×C8×C61C22×C412yes—
186C4⋊C4×C61—21no—
187C4⋊C4×C61—21no—
188C4⋊C4×C61—21no—
189C4⋊C4×C61#4112yes—
190C22⋊C4×C61—24no—
191C42×C61—21no—
192C42×C61—21no—
193C22⋊Dic61—24no—
194C22⋊Dic61—24no—
195#11758200—28no—
196C4⋊Dic61—21no—
197C4⋊Dic61—21no—
198C4⋊Dic61—21no—
199C4⋊Dic61—21no—
200C4⋊Dic61—42no—
201C4⋊Dic61C22×C412yes—
202C4⋊Dic61C2×D412yes—
203#11758196—24no—
204#11758196—24no—
205#11758196—42no—
206C4×Dic61—42no—
207C4×Dic61—42no—
208C4×Dic61—81no—
209C4×Dic61—81no—
210C4×Dic61—81no—
211C4×Dic61C22×C412yes—
212C4×Dic61C2×D412yes—
213C4×Dic61#4112yes—
214C4×Dic61#4112yes—
215C2×#92418—21no—
216C2×#92418—21no—
217C2×#92418—21no—
218C2×#92418—21no—
219C2×#92418—21no—
220C2×#92418C22×C412yes—
221C23×C61—22no—
222C23×C61—22no—
223C23×C61—22no—
224C23×C61—41no—
225C23×C61—41no—
226C23×C61—41no—
227C23×C61—41no—
228C23×C61C22×D411yescharacteristic
229Q8×C61—22no—
230Q8×C61—22no—
231Q8×C61—41no—
232Q8×C61—41no—
233D4×C61—22no—
234D4×C61—22no—
235D4×C61—22no—
236D4×C61C4×D414yes—
237C2×C4×C61—22no—
238C2×C4×C61—21no—
239C2×C4×C61—21no—
240C2×C4×C61—22no—
241C2×C4×C61—21no—
242C2×C4×C61—21no—
243C2×C4×C61—21no—
244C2×C4×C61—21no—
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—84no—
253C2×C4×C61C23×C411yescharacteristic
254C2×C4×C61C22×D411yescharacteristic
255C2×C4×C61C2×#4111yescharacteristic
256C2×C4×C61C2×#4111yescharacteristic
257C2×C4×C61C2×#4111yescharacteristic
258C2×C4×C61C2×#4111yescharacteristic
259C2×Dic61—21no—
260C2×Dic61—21no—
261C2×Dic61—21no—
262C2×Dic61—21no—
263C2×Dic61—21no—
264C2×Dic61—21no—
265C2×Dic61—21no—
266C2×Dic61—21no—
267C2×Dic61—22no—
268C2×Dic61—22no—
269C2×Dic61—22no—
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—
279C2×Dic61—84no—
280C2×Dic61—84no—
281C2×Dic61C4×D412yes—
282C2×Dic61C4×D412yes—
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#21514—611nomaximal, Sylow
294C22×C61—24no—
295C22×C61—42no—
296C22×C61—42no—
297C22×C61—42no—
298C22×C61—42no—
299C22×C61—42no—
300C22×C61—42no—
301C22×C61C2×C4×D411yescharacteristic
302C22×C61C2×#12111yescharacteristic
303C22×C61#52511yescharacteristic
304C22×C61#57711yescharacteristic
305C22×C61#57711yescharacteristic
306C22×C61#57711yescharacteristic
307C22×C61#57811yescharacteristic
308C4×C61—41no—
309C4×C61—41no—
310C4×C61—42no—
311C4×C61—42no—
312C4×C61—81no—
313C4×C61C2×C4×D411yescharacteristic
314C4×C61C4×#4111yescharacteristic
315C4×C61#52911yescharacteristic
316C4×C61#53211yescharacteristic
317Dic61—22no—
318Dic61—22no—
319Dic61—22no—
320Dic61—22no—
321Dic61—42no—
322Dic61—42no—
323Dic61—42no—
324Dic61—81no—
325C2×#461—611no—
326C2×#440—611no—
327C2×C4×SD16—611no—
328#2327—611no—
329#2275—611no—
330#1981—611no—
331#1949—611no—
332#1928—611no—
333#1927—611no—
334#1877—611no—
335#1877D6111yescharacteristic
336#1804—611no—
337#1768—611no—
338#1763—611no—
339C4×#103—611no—
340C2×C61—44no—
341C2×C61C4×#12111yescharacteristic
342C2×C61#294711yescharacteristic
343C2×C61#294911yescharacteristic
344C2×C61#304211yescharacteristic
345C2×C61#304611yescharacteristic
346C2×C61#312011yescharacteristic
347C2×C61#312311yescharacteristic
348C22×SD16—611no—
349C22×SD16C2×D6111yescharacteristic
350C2×C4⋊Q8—611no—
351C2×C4×Q8—611no—
352C2×C4×Q8—611no—
353C2×C4×D4—611no—
354C2×C4×D4—611no—
355C2×C4×D4C2×D6111yescharacteristic
356#461—618no—
357#440—1224no—
358C4×SD16—1224no—
359C2×#107—611no—
360C2×#106—611no—
361C2×C4⋊C8—611no—
362C2×C4⋊C8—611no—
363C2×#103—611no—
364C2×#103—611no—
365C2×#103—611no—
366C2×#103—611no—
367C2×#103—611no—
368C2×#103C2×D6111yescharacteristic
369C2×#102—611no—
370C2×#102C2×D6111yescharacteristic
371C2×#97—611no—
372C2×C4×C8—611no—
373#391—611no—
374#391C2×D6111yescharacteristic
375#387—1221no—
376#387—1221no—
377#387—1221no—
378#387—1221no—
379#382—611no—
380C4×C4⋊C4—611no—
381C4×C4⋊C4—611no—
382C4×C4⋊C4—611no—
383C4×C22⋊C4—1221no—
384C4×C22⋊C4—1221no—
385C4×C22⋊C4—1221no—
386#376—611no—
387C2×#95—1221no—
388#340—611no—
389#340—611no—
390#340—611no—
391#340C2×D6111yescharacteristic
392#336—611no—
393#336C2×D6111yescharacteristic
394C61#2151411yescharacteristic, complemented, Sylow
395C22×Q8—611no—
396C22×Q8C22×D6111yescharacteristic
397C22×D4C22×D6111yescharacteristic
398C23×C4—21no—
399C23×C4—1221no—
400C23×C4—1221no—
401C2×SD16—22no—
402C2×SD16—22no—
403C2×SD16—614no—
404C2×SD16—1222no—
405C2×SD16—1222no—
406C2×SD16C4×D6114yes—
407C22×C8—611no—
408C22×C8—611no—
409C22×C8C22×D6111yescharacteristic
410C22×C8C22×D6111yescharacteristic
411C4⋊Q8—614no—
412C4⋊Q8—1222no—
413C4×Q8—1222no—
414C4×Q8—1222no—
415C4×Q8—1222no—
416C4×Q8—1222no—
417C4×D4—22no—
418C4×D4—22no—
419C4×D4—614no—
420C4×D4—614no—
421C4×D4—1222no—
422C4×D4—1222no—
423C2×C4⋊C4—611no—
424C2×C4⋊C4—611no—
425C2×C4⋊C4—611no—
426C2×C4⋊C4—1221no—
427C2×C4⋊C4—1221no—
428C2×C4⋊C4—1221no—
429C2×C4⋊C4—1221no—
430C2×C4⋊C4C22×D6111yescharacteristic
431C2×C4⋊C4C22×D6111yescharacteristic
432C2×C22⋊C4—21no—
433C2×C22⋊C4—1221no—
434C2×C22⋊C4—1221no—
435C2×C42—611no—
436C2×C42—611no—
437C2×C42—1221no—
438C2×C42—1221no—
439C2×C42—1221no—
440C2×C42—1221no—
441C2×C42—1221no—
442C2×C42C22×D6111yescharacteristic
443#107—1222no—
444#106—1222no—
445C4⋊C8—614no—
446C4⋊C8—1222no—
447#103—22no—
448#103—614no—
449#103—614no—
450#103—1222no—
451#103—1222no—
452#103—1222no—
453#102—22no—
454#102—1222no—
455#97—1222no—
456C4×C8—1222no—
457#95—21no—
458#95—21no—
459#95—611no—
460#95—611no—
461#95—611no—
462#95—611no—
463#95—1221no—
464#95—1221no—
465#95—1221no—
466#95—1221no—
467#95—1221no—
468#95—1221no—
469#95—1221no—
470C24—21no—
471C2×Q8—21no—
472C2×Q8—21no—
473C2×Q8—21no—
474C2×Q8—21no—
475C2×Q8—21no—
476C2×Q8—612no—
477C2×Q8—1221no—
478C2×Q8—1221no—
479C2×Q8—1221no—
480C2×Q8—1221no—
481C2×Q8—1221no—
482C2×Q8C2×C4×D6112yes—
483C2×D4—21no—
484C2×D4—21no—
485C2×D4—21no—
486C2×D4C2×C4×D6112yes—
487C2×D4D4×D6112yes—
488C2×D4#1175822412yes—
489C22×C4—21no—
490C22×C4—21no—
491C22×C4—24no—
492C22×C4—21no—
493C22×C4—81no—
494C22×C4—81no—
495C22×C4—611no—
496C22×C4—611no—
497C22×C4—611no—
498C22×C4—611no—
499C22×C4—1224no—
500C22×C4—1224no—
501C22×C4—1221no—
502C22×C4—1221no—
503C22×C4—1221no—
504C22×C4—1221no—
505C22×C4—4881no—
506C22×C4—4881no—
507C22×C4—4881no—
508C22×C4—4881no—
509C22×C4D4×D6111yescharacteristic
510C22×C4D4×D6111yescharacteristic
511C22×C4C23×D6111yescharacteristic
512SD16—28no—
513SD16—1228no—
514C2×C8—21no—
515C2×C8—21no—
516C2×C8—21no—
517C2×C8—21no—
518C2×C8—21no—
519C2×C8—612no—
520C2×C8—1221no—
521C2×C8—1221no—
522C2×C8—1221no—
523C2×C8—1221no—
524C2×C8—1221no—
525C2×C8C2×C4×D6112yes—
526C4⋊C4—21no—
527C4⋊C4—21no—
528C4⋊C4—21no—
529C4⋊C4—612no—
530C4⋊C4—612no—
531C4⋊C4—1224no—
532C4⋊C4—1224no—
533C4⋊C4—1221no—
534C4⋊C4—1221no—
535C4⋊C4—1221no—
536C4⋊C4—1221no—
537C4⋊C4—2442no—
538C4⋊C4—2442no—
539C4⋊C4#1175822412yes—
540C22⋊C4—24no—
541C22⋊C4—1224no—
542C22⋊C4—1224no—
543C42—21no—
544C42—21no—
545C42—612no—
546C42—612no—
547C42—612no—
548C42—612no—
549C42—4881no—
550C42—4881no—
551C42—4881no—
552C42—2442no—
553C42—2442no—
554C23—22no—
555C23—22no—
556C23—22no—
557C23—41no—
558C23—41no—
559C23—41no—
560C23—41no—
561C23C2×D4×D6111yescharacteristic
562Q8—22no—
563Q8—22no—
564Q8—41no—
565Q8—41no—
566Q8—1222no—
567Q8—1222no—
568Q8—2441no—
569Q8—2441no—
570D4—22no—
571D4—22no—
572D4—22no—
573D4#1451973714yes—
574C2×C4—22no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—22no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—84no—
590C2×C4—612no—
591C2×C4—612no—
592C2×C4—612no—
593C2×C4—612no—
594C2×C4—1221no—
595C2×C4—1221no—
596C2×C4—1221no—
597C2×C4—1221no—
598C2×C4—1221no—
599C2×C4—1221no—
600C2×C4—1221no—
601C2×C4—1221no—
602C2×C4—1222no—
603C2×C4—1222no—
604C2×C4—1222no—
605C2×C4—4884no—
606C2×C4—4884no—
607C2×C4—2441no—
608C2×C4—2441no—
609C2×C4—2441no—
610C2×C4—2441no—
611C2×C4—2441no—
612C2×C4—2441no—
613C2×C4—2441no—
614C2×C4—2441no—
615C2×C4—2441no—
616C2×C4C22×C4×D6111yescharacteristic
617C2×C4C2×D4×D6111yescharacteristic
618C2×C4#41×D6111yescharacteristic
619C2×C4#41×D6111yescharacteristic
620C2×C4C2×#1175822411yescharacteristic
621C2×C4C2×#1175822411yescharacteristic
622C8—22no—
623C8—41no—
624C8—1222no—
625C8—2441no—
626C22—24no—
627C22—42no—
628C22—42no—
629C22—42no—
630C22—42no—
631C22—42no—
632C22—42no—
633C22C4×D4×D6111yescharacteristic
634C22#121×D6111yescharacteristic
635C22#1684026811yescharacteristic
636C22#1684056811yescharacteristic
637C22#1684056911yescharacteristic
638C22#1684057211yescharacteristic
639C22#1684057811yescharacteristic
640C4—42no—
641C4—41no—
642C4—41no—
643C4—42no—
644C4—81no—
645C4—1222no—
646C4—1222no—
647C4—1222no—
648C4—1222no—
649C4—4881no—
650C4—2442no—
651C4—2442no—
652C4—2442no—
653C4C2×#1451973711yescharacteristic
654C4C4×#1175822411yescharacteristic
655C4#1684028911yescharacteristic
656C4#1684030711yescharacteristic
657C2—44no—
658C2#2144832911yescharacteristic
659C2#2145465911yescharacteristic
660C2#2145467311yescharacteristic
661C2#2145532911yescharacteristic
662C2#2145536411yescharacteristic
663C2#2145602611yescharacteristic
664C2#2145606511yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1877C2
socleC23×C61C22×D4
prime radicalC22×D4×C61C23
cosocleC22×C4×C61C24
hypercenter#1877D61
hypocenterC61#21514
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1877

Complements

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

NH ≅ G/Ntype
4. #21448537C2semidirect
6. #21446740C2semidirect
7. #21446586C2semidirect
8. #21446457C2semidirect
9. #21446454C2semidirect
12. #21445675C2semidirect
13. #21445651C2semidirect
14. #21445638C2semidirect
22. #16839965C4semidirect
71. #14519775C2×C4semidirect
99. #14519706C2×C4semidirect
101. #14519702C2×C4semidirect
103. #14519701C2×C4semidirect
394. C61#21514semidirect

Maximal subgroups

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

subgroupindex
C2×#168399652
C2×#168398592
C2×SD16×Dic612
#214485372
#214481802
#214467402
#214465862
#214464572
#214464542
#214461532
#214458272
#214456752
#214456512
#214456382
C61×#18772
#2151461