Finite Group Databasebeta

← #29708105

Subgroups — #29708105

Order 15616.

1110 conjugacy classes of subgroups.

The subgroup lattice has 668 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840392C211yesmaximal, characteristic
2C2×#16839801C211yesmaximal, characteristic
3C2×#16839799C211yesmaximal, characteristic
4#21448434C211yesmaximal, characteristic
5#21447160C211yesmaximal, characteristic, complemented
6#21447155C211yesmaximal, characteristic, complemented
7#21447154C211yesmaximal, characteristic, complemented
8#21447152C211yesmaximal, characteristic, complemented
9#21447150C211yesmaximal, characteristic, complemented
10#21447148C211yesmaximal, characteristic, complemented
11#21447146C211yesmaximal, characteristic, complemented
12#21445537C211yesmaximal, characteristic
13#340×D61C211yesmaximal, characteristic
14#21445514C211yesmaximal, characteristic
15C61×#2059C211yesmaximal, characteristic, complemented
16C22×#11758189C2211yescharacteristic
17C22×C8×D61C2211yescharacteristic
18#16840392—24no—
19#16840392—24no—
20C2×#14519777C2211yescharacteristic
21C2×#14519776—21no—
22C2×#14519776C2211yescharacteristic
23C2×#14519768C2211yescharacteristic
24C2×#14519767—21no—
25C2×#14519757—21no—
26C2×#14519755—21no—
27C2×#14519735C2211yescharacteristic
28C2×#14519735C2211yescharacteristic
29C2×#14519734C2211yescharacteristic
30C2×C4⋊C4×D61C2211yescharacteristic
31C2×C4⋊C4×D61C2211yescharacteristic
32#16839801—24no—
33#16839799—24no—
34C2×#14519711C2211yescharacteristic
35C2×#14519711C2211yescharacteristic
36C2×#14519710C2211yescharacteristic
37C2×#14519708C2211yescharacteristic
38C2×#14519708C2211yescharacteristic
39C2×#14519708C2211yescharacteristic
40#16839661C2211yescharacteristic
41#16839660C2211yescharacteristic
42#16839659—21no—
43#16839658C2211yescharacteristic
44#16839657—21no—
45#16839621—21no—
46#16839619—21no—
47#16839525C2211yescharacteristic
48#16839525C2211yescharacteristic
49#16839524C2211yescharacteristic
50#16839524C2211yescharacteristic
51#16839524C2211yescharacteristic
52#16839524C2211yescharacteristic
53#16839517C2211yescharacteristic
54#16839516C2211yescharacteristic
55#16839516C2211yescharacteristic
56C2×#125×C61C2211yescharacteristic
57C2×#107×C61C2211yescharacteristic
58C2×#106×C61C2211yescharacteristic
59C61×#389C2211yescharacteristic
60C61×#340C2211yescharacteristic
61C61×#340C2211yescharacteristic
62C61×#336C2211yescharacteristic
63C22×Dic122—21no—
64C22×Dic122—21no—
65C22×C4×D61—21no—
66C22×C4×D61—21no—
67C22×C4×D61C2311yescharacteristic
68C2×#11758189—22no—
69C2×#11758189—22no—
70C2×#11758189Q814yescomplemented
71C22×#92418C2311yescharacteristic
72C22×#92418C2311yescharacteristic
73C2×C8×D61—22no—
74C2×C8×D61—22no—
75C2×C8×D61Q814yescomplemented
76#14519777—22no—
77#14519777—22no—
78#14519776—28no—
79#14519776—22no—
80#14519776—22no—
81#14519768—22no—
82#14519768—22no—
83#14519767—28no—
84#14519757—28no—
85#14519755—28no—
86#14519735—22no—
87#14519735—22no—
88#14519735—22no—
89#14519735D414yes—
90#14519734—22no—
91#14519734—22no—
92C2×C4⋊Dic61—21no—
93C2×C4⋊Dic61—21no—
94C2×C4⋊Dic61—21no—
95C2×C4⋊Dic61C2311yescharacteristic
96C2×C4⋊Dic61C2311yescharacteristic
97C2×C4⋊Dic61C2311yescharacteristic
98C2×C4⋊Dic61C2311yescharacteristic
99C2×#11758196—21no—
100C2×#11758196—21no—
101C2×#11758196—21no—
102C2×#11758196—21no—
103C2×#11758196—21no—
104C2×#11758196—21no—
105C2×#11758196D411yescharacteristic
106C2×#11758196D411yescharacteristic
107C4⋊C4×D61—22no—
108C4⋊C4×D61—22no—
109C4⋊C4×D61D414yes—
110C4⋊C4×D61Q814yescomplemented
111C2×#11758198—21no—
112C2×#11758198—21no—
113C2×#11758198D411yescharacteristic
114C2×#11758198D411yescharacteristic
115C2×C4×Dic61—21no—
116C2×C4×Dic61—21no—
117C2×C4×Dic61C2311yescharacteristic
118#14519711—22no—
119#14519711—22no—
120#14519710—22no—
121#14519708—22no—
122#14519708—22no—
123#14519708—22no—
124#14519684—21no—
125#14519684—21no—
126#14519684—21no—
127C22×C8×C61C2311yescharacteristic
128C22×C8×C61C2311yescharacteristic
129#125×C61—22no—
130#125×C61—22no—
131C2×C4⋊C4×C61—21no—
132C2×C4⋊C4×C61—21no—
133C2×C4⋊C4×C61—21no—
134C2×C4⋊C4×C61C2311yescharacteristic
135C2×C4⋊C4×C61C2311yescharacteristic
136C2×C4⋊C4×C61C2311yescharacteristic
137C2×C4⋊C4×C61C2311yescharacteristic
138C2×C42×C61C2311yescharacteristic
139#107×C61—22no—
140#106×C61—22no—
141#95×C61—21no—
142C23×D61C2×D411yescharacteristic
143C22×C4×C61—21no—
144C22×C4×C61—21no—
145C22×C4×C61—21no—
146C22×C4×C61—21no—
147C22×C4×C61C2×D411yescharacteristic
148C22×C4×C61C2×D411yescharacteristic
149C22×C4×C61C2411yescharacteristic
150C22×Dic61—21no—
151C22×Dic61—21no—
152C22×Dic61—21no—
153C22×Dic61—21no—
154C22×Dic61C2×D411yescharacteristic
155C22×Dic61C2×D411yescharacteristic
156C22×Dic61C2×D411yescharacteristic
157C2×C4×D61—24no—
158C2×C4×D61—24no—
159C2×C4×D61—81no—
160C2×C4×D61—81no—
161C2×C4×D61—81no—
162C2×C4×D61—81no—
163C2×C4×D61C2×D412yes—
164C2×C4×D61C2×D412yes—
165C2×C4×D61C2×Q812yes—
166C2×C4×D61C2×Q812yes—
167C2×C4×D61C2×Q812yes—
168C2×C4×D61#4112yes—
169C2×Dic122—24no—
170C2×Dic122—24no—
171C2×Dic122—81no—
172C2×Dic122—81no—
173C2×Dic122—81no—
174C2×Dic122—81no—
175C2×C8×C61—21no—
176C2×C8×C61—21no—
177C2×C8×C61—21no—
178C2×C8×C61—21no—
179C2×C8×C61C2×Q812yes—
180C2×C8×C61C2×Q812yes—
181C4⋊C4×C61—24no—
182C4⋊C4×C61—24no—
183C4⋊C4×C61—21no—
184C4⋊C4×C61—21no—
185C4⋊C4×C61—21no—
186C4⋊C4×C61—21no—
187C4⋊C4×C61—21no—
188C4⋊C4×C61—81no—
189C4⋊C4×C61C2×D412yes—
190C4⋊C4×C61C2×D412yes—
191C4⋊C4×C61C2×Q812yes—
192C42×C61—21no—
193C42×C61—21no—
194#11758198—24no—
195#11758198—22no—
196#11758198—22no—
197#11758198—42no—
198C4⋊Dic61—24no—
199C4⋊Dic61—24no—
200C4⋊Dic61—21no—
201C4⋊Dic61—21no—
202C4⋊Dic61—21no—
203C4⋊Dic61—21no—
204C4⋊Dic61—21no—
205C4⋊Dic61—81no—
206C4⋊Dic61C2×D412yes—
207C4⋊Dic61C2×D412yes—
208C4⋊Dic61C2×Q812yes—
209#11758196—24no—
210#11758196—24no—
211#11758196—24no—
212#11758196—22no—
213#11758196—22no—
214#11758196—24no—
215#11758196—81no—
216#11758196—81no—
217C4×Dic61—21no—
218C4×Dic61—21no—
219C4×Dic61—24no—
220C4×Dic61—42no—
221C2×#92418—21no—
222C2×#92418—21no—
223C2×#92418—21no—
224C2×#92418—21no—
225C2×#92418C2×Q812yes—
226C2×#92418C2×Q812yes—
227#11758189—28no—
228C8×D61—28no—
229C23×C61C22×D411yescharacteristic
230C22×D61—21no—
231C22×D61—21no—
232C22×D61—21no—
233C22×D61—21no—
234C22×D61#12012yes—
235C22×D61C22⋊Q812yes—
236C22×D61C4⋊Q812yes—
237C2×C4×C61—22no—
238C2×C4×C61—22no—
239C2×C4×C61—21no—
240C2×C4×C61—21no—
241C2×C4×C61—22no—
242C2×C4×C61—21no—
243C2×C4×C61—21no—
244C2×C4×C61—21no—
245C2×C4×C61—21no—
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×C61C22×D411yescharacteristic
256C2×C4×C61C22×D411yescharacteristic
257C2×C4×C61C22×Q811yescharacteristic
258C2×C4×C61C22×Q811yescharacteristic
259C2×C4×C61C22×Q811yescharacteristic
260C2×C4×C61C2×#4111yescharacteristic
261C2×Dic61—21no—
262C2×Dic61—21no—
263C2×Dic61—22no—
264C2×Dic61—22no—
265C2×Dic61—22no—
266C2×Dic61—21no—
267C2×Dic61—21no—
268C2×Dic61—21no—
269C2×Dic61—21no—
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#12012yes—
280C2×Dic61C22⋊Q812yes—
281C2×Dic61C4⋊Q812yes—
282C4×D61—84no—
283C4×D61—84no—
284C4×D61C22⋊Q814yes—
285C4×D61C22⋊Q814yes—
286C4×D61C4⋊Q814yes—
287C4×D61C4⋊Q814yes—
288Dic122—84no—
289Dic122—84no—
290C8×C61—22no—
291C8×C61—22no—
292#92418—22no—
293#92418—22no—
294#23557—611nomaximal, Sylow
295C22×C61C2×#12011yescharacteristic
296C22×C61C2×C22⋊Q811yescharacteristic
297C22×C61C2×C4⋊Q811yescharacteristic
298C22×C61#57611yescharacteristic
299C22×C61#57711yescharacteristic
300C22×C61#57711yescharacteristic
301C22×C61#57811yescharacteristic
302C2×D61—22no—
303C2×D61—22no—
304C2×D61—22no—
305C2×D61—22no—
306C2×D61—22no—
307C2×D61—22no—
308C2×D61#39514yes—
309C4×C61—42no—
310C4×C61—41no—
311C4×C61—42no—
312C4×C61—41no—
313C4×C61—42no—
314C4×C61—81no—
315C4×C61C2×C22⋊Q811yescharacteristic
316C4×C61C2×C22⋊Q811yescharacteristic
317C4×C61C2×C4⋊Q811yescharacteristic
318C4×C61C2×C4⋊Q811yescharacteristic
319Dic61—42no—
320Dic61—41no—
321Dic61—42no—
322Dic61—42no—
323Dic61—41no—
324Dic61—81no—
325Dic61#39514yes—
326C2×#544—611no—
327C2×#428—611no—
328C2×#427—611no—
329#2316—611no—
330#2059—611no—
331#2059—611no—
332#2059—611no—
333#2059D6111yescharacteristic, complemented
334#2058—611no—
335#2058—611no—
336#2057—611no—
337#2057—611no—
338#1736—611no—
339C2×#340—611no—
340#1729—611no—
341C2×C61C2×#39511yescharacteristic
342C2×C61#301211yescharacteristic
343C2×C61#301311yescharacteristic
344C2×C61#308111yescharacteristic
345C2×C61#308211yescharacteristic
346C2×C61#316311yescharacteristic
347C2×C61#316411yescharacteristic
348D61—24no—
349C22×M16—611no—
350C23×C8—611no—
351#544—1224no—
352#544—1224no—
353C2×C4⋊Q8—611no—
354C2×C4⋊Q8—611no—
355C2×C4⋊Q8—1221no—
356C2×#125—611no—
357C2×#125—1221no—
358C2×#125C2×D6111yescharacteristic
359C2×C22⋊Q8—1221no—
360C2×C22⋊Q8—1221no—
361C2×#117—611no—
362C2×#117—611no—
363C2×#117—611no—
364C22×C4⋊C4—611no—
365C22×C4⋊C4—611no—
366#428—1224no—
367#427—1224no—
368C2×#107—611no—
369C2×#107—611no—
370C2×#107—611no—
371C2×#107C2×D6111yescharacteristic
372C2×#106—611no—
373C2×#106—611no—
374C2×#106—611no—
375C2×#106C2×D6111yescharacteristic
376#389—611no—
377#389—611no—
378#389—611no—
379#389—1221no—
380#389—1221no—
381#389C2×D6111yescharacteristic
382#380—1221no—
383#380—1221no—
384#340—611no—
385#340—611no—
386#340—611no—
387#340—611no—
388#340—611no—
389#340—611no—
390#340C2×D6111yescharacteristic
391#340C2×D6111yescharacteristic
392#336—611no—
393#336—611no—
394#336—611no—
395#336C2×D6111yescharacteristic
396C61#2355711yescharacteristic, complemented, Sylow
397C22×Q8—1221no—
398C22×Q8—1221no—
399C23×C4—611no—
400C23×C4—1221no—
401C23×C4—1221no—
402C2×M16—614no—
403C2×M16—1222no—
404C2×M16—1222no—
405C22×C8—611no—
406C22×C8—611no—
407C22×C8—614no—
408C22×C8—1222no—
409C22×C8—1222no—
410C22×C8C22×D6111yescharacteristic
411C22×C8C22×D6111yescharacteristic
412C4⋊Q8—1222no—
413C4⋊Q8—1222no—
414C4⋊Q8—1222no—
415C4⋊Q8—1228no—
416C4⋊Q8—1222no—
417#125—22no—
418#125—22no—
419#125—1222no—
420#125—1228no—
421#125—1222no—
422C22⋊Q8—1228no—
423C22⋊Q8—1228no—
424#117—614no—
425#117—1222no—
426#117—1222no—
427#117—1222no—
428#117—1222no—
429#117—1222no—
430C2×C4⋊C4—21no—
431C2×C4⋊C4—21no—
432C2×C4⋊C4—21no—
433C2×C4⋊C4—611no—
434C2×C4⋊C4—611no—
435C2×C4⋊C4—611no—
436C2×C4⋊C4—614no—
437C2×C4⋊C4—614no—
438C2×C4⋊C4—611no—
439C2×C4⋊C4—611no—
440C2×C4⋊C4—611no—
441C2×C4⋊C4—1222no—
442C2×C4⋊C4—1222no—
443C2×C4⋊C4—1221no—
444C2×C4⋊C4—1221no—
445C2×C4⋊C4—1221no—
446C2×C4⋊C4—1221no—
447C2×C4⋊C4—1221no—
448C2×C4⋊C4—1221no—
449C2×C4⋊C4—1221no—
450C2×C4⋊C4—1221no—
451C2×C4⋊C4—1221no—
452C2×C4⋊C4C22×D6111yescharacteristic
453C2×C4⋊C4C22×D6111yescharacteristic
454C2×C4⋊C4C22×D6111yescharacteristic
455C2×C4⋊C4C22×D6111yescharacteristic
456C2×C22⋊C4—611no—
457C2×C22⋊C4—611no—
458C2×C22⋊C4—1221no—
459C2×C22⋊C4—1221no—
460C2×C42—611no—
461C2×C42—1221no—
462C2×C42—1221no—
463C2×C42C22×D6111yescharacteristic
464#107—22no—
465#107—1222no—
466#107—1222no—
467#107—1222no—
468#106—22no—
469#106—1222no—
470#106—1222no—
471#106—1222no—
472#95—21no—
473#95—1221no—
474#95—1221no—
475#95—1221no—
476C24—611no—
477C2×Q8—1224no—
478C2×Q8—1224no—
479C2×Q8—4881no—
480C2×Q8—4881no—
481C2×Q8—4881no—
482C2×Q8—4881no—
483C22×C4—21no—
484C22×C4—21no—
485C22×C4—21no—
486C22×C4—21no—
487C22×C4—611no—
488C22×C4—612no—
489C22×C4—612no—
490C22×C4—612no—
491C22×C4—612no—
492C22×C4—612no—
493C22×C4—612no—
494C22×C4—611no—
495C22×C4—611no—
496C22×C4—1224no—
497C22×C4—1224no—
498C22×C4—1221no—
499C22×C4—1221no—
500C22×C4—1221no—
501C22×C4—1221no—
502C22×C4—4881no—
503C22×C4—4881no—
504C22×C4—4881no—
505C22×C4—4881no—
506C22×C4D4×D6111yescharacteristic
507C22×C4D4×D6111yescharacteristic
508C22×C4C23×D6111yescharacteristic
509M16—1228no—
510C2×C8—21no—
511C2×C8—21no—
512C2×C8—21no—
513C2×C8—21no—
514C2×C8—612no—
515C2×C8—612no—
516C2×C8—1228no—
517C2×C8—1221no—
518C2×C8—1221no—
519C2×C8—1221no—
520C2×C8—1221no—
521C2×C8Q8×D6112yes—
522C2×C8Q8×D6112yes—
523C4⋊C4—24no—
524C4⋊C4—24no—
525C4⋊C4—21no—
526C4⋊C4—21no—
527C4⋊C4—21no—
528C4⋊C4—21no—
529C4⋊C4—21no—
530C4⋊C4—81no—
531C4⋊C4—612no—
532C4⋊C4—612no—
533C4⋊C4—612no—
534C4⋊C4—1224no—
535C4⋊C4—1224no—
536C4⋊C4—1222no—
537C4⋊C4—1222no—
538C4⋊C4—1224no—
539C4⋊C4—1224no—
540C4⋊C4—1224no—
541C4⋊C4—1224no—
542C4⋊C4—1221no—
543C4⋊C4—1221no—
544C4⋊C4—1221no—
545C4⋊C4—1221no—
546C4⋊C4—1221no—
547C4⋊C4—4881no—
548C4⋊C4—4881no—
549C4⋊C4—4881no—
550C4⋊C4D4×D6112yes—
551C4⋊C4D4×D6112yes—
552C4⋊C4Q8×D6112yes—
553C22⋊C4—1222no—
554C22⋊C4—1222no—
555C22⋊C4—1224no—
556C22⋊C4—2442no—
557C42—21no—
558C42—21no—
559C42—1224no—
560C42—1221no—
561C42—1221no—
562C42—2442no—
563C23—612no—
564C23—612no—
565C23—612no—
566C23—1221no—
567C23—1221no—
568C23—1221no—
569C23—1221no—
570C23C2×D4×D6111yescharacteristic
571Q8—4884no—
572Q8—4884no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—22no—
576C2×C4—22no—
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—41no—
590C2×C4—41no—
591C2×C4—612no—
592C2×C4—612no—
593C2×C4—612no—
594C2×C4—614no—
595C2×C4—614no—
596C2×C4—614no—
597C2×C4—614no—
598C2×C4—1222no—
599C2×C4—1221no—
600C2×C4—1221no—
601C2×C4—1222no—
602C2×C4—1222no—
603C2×C4—1221no—
604C2×C4—1221no—
605C2×C4—1221no—
606C2×C4—1221no—
607C2×C4—4884no—
608C2×C4—4884no—
609C2×C4—2441no—
610C2×C4—2441no—
611C2×C4—2441no—
612C2×C4—2441no—
613C2×C4—2441no—
614C2×C4—2441no—
615C2×C4—2441no—
616C2×C4—2441no—
617C2×C4—2441no—
618C2×C4C2×D4×D6111yescharacteristic
619C2×C4C2×D4×D6111yescharacteristic
620C2×C4C2×Q8×D6111yescharacteristic
621C2×C4C2×Q8×D6111yescharacteristic
622C2×C4C2×Q8×D6111yescharacteristic
623C2×C4#41×D6111yescharacteristic
624C8—22no—
625C8—22no—
626C8—1222no—
627C8—1222no—
628C22—614no—
629C22—1222no—
630C22—1222no—
631C22—1222no—
632C22—1222no—
633C22—1222no—
634C22—1222no—
635C22#120×D6111yescharacteristic
636C22C22⋊Q8×D6111yescharacteristic
637C22C4⋊Q8×D6111yescharacteristic
638C22#1684056411yescharacteristic
639C22#1684057111yescharacteristic
640C22#1684057211yescharacteristic
641C22#1684057611yescharacteristic
642C4—41no—
643C4—42no—
644C4—41no—
645C4—42no—
646C4—42no—
647C4—81no—
648C4—614no—
649C4—4881no—
650C4—2441no—
651C4—2442no—
652C4—2442no—
653C4—2441no—
654C4—2442no—
655C4C22⋊Q8×D6111yescharacteristic
656C4C22⋊Q8×D6111yescharacteristic
657C4C4⋊Q8×D6111yescharacteristic
658C4C4⋊Q8×D6111yescharacteristic
659C2—1224no—
660C2#395×D6111yescharacteristic
661C2#2145512711yescharacteristic
662C2#2145513211yescharacteristic
663C2#2145564711yescharacteristic
664C2#2145566011yescharacteristic
665C2#2145634611yescharacteristic
666C2#2145635611yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2059C2
socleC23×C61C22×D4
prime radicalC23×D61C2×D4
hypercenter#2059D61
hypocenterC61#23557
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2059

Complements

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

NH ≅ G/Ntype
5. #21447160C2semidirect
6. #21447155C2semidirect
7. #21447154C2semidirect
8. #21447152C2semidirect
9. #21447150C2semidirect
10. #21447148C2semidirect
11. #21447146C2semidirect
15. C61×#2059C2semidirect
70. C2×#11758189Q8semidirect
75. C2×C8×D61Q8semidirect
110. C4⋊C4×D61Q8semidirect
333. #2059D61semidirect
396. C61#23557semidirect

Maximal subgroups

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

subgroupindex
C2×#168403922
C2×#168398012
C2×#168397992
#214484342
#214471602
#214471552
#214471542
#214471522
#214471502
#214471482
#214471462
#214455372
#340×D612
#214455142
C61×#20592
#2355761