Finite Group Databasebeta

← #29716162

Subgroups — #29716162

Order 15616.

1078 conjugacy classes of subgroups.

The subgroup lattice has 674 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#125×D61C211yesmaximal, characteristic
2C2×#16840072C211yesmaximal, characteristic, complemented
3C2×#16840068C211yesmaximal, characteristic, complemented
4#21448349C211yesmaximal, characteristic
5#21447157C211yesmaximal, characteristic, complemented
6#21446971C211yesmaximal, characteristic, complemented
7#21446967C211yesmaximal, characteristic, complemented
8#21446775C211yesmaximal, characteristic, complemented
9#21446773C211yesmaximal, characteristic, complemented
10#21446125C211yesmaximal, characteristic
11#21446125C211yesmaximal, characteristic
12#21446010C211yesmaximal, characteristic
13#21446005C211yesmaximal, characteristic
14#21445798C211yesmaximal, characteristic
15C61×#2059C211yesmaximal, characteristic, complemented
16C2×#14519769C2211yescharacteristic
17C2×#14519768C2211yescharacteristic
18C2×#14519768C2211yescharacteristic
19C2×#14519767—21no—
20C2×#14519767—21no—
21#125×D61—24no—
22#125×D61—24no—
23C2×#14519757C2211yescharacteristic
24C2×#14519742C2211yescharacteristic
25C2×C4⋊C4×D61—21no—
26C2×C4⋊C4×D61—21no—
27C2×C4⋊C4×D61C2211yescharacteristic
28C2×C4⋊C4×D61C2211yescharacteristic
29C2×C42×D61C2211yescharacteristic
30#16840072—24no—
31#16840068—24no—
32C2×#14519710C2211yescharacteristic
33C2×#14519708C2211yescharacteristic
34C2×#14519706C2211yescharacteristic
35C2×#14519706C2211yescharacteristic
36C2×#14519702C2211yescharacteristic
37C2×#14519702C2211yescharacteristic
38C2×#14519700C2211yescharacteristic
39C2×#14519700C2211yescharacteristic
40C2×#14519689C2211yescharacteristic
41C2×#14519689C2211yescharacteristic
42#16839668C2211yescharacteristic
43#16839645—21no—
44#16839633—21no—
45#16839619C2211yescharacteristic
46C4⋊C4×Dic61—21no—
47C4⋊C4×Dic61C2211yescharacteristic
48#16839609—21no—
49#16839525C2211yescharacteristic
50#16839525C2211yescharacteristic
51#16839524C2211yescharacteristic
52#16839524C2211yescharacteristic
53#16839524C2211yescharacteristic
54#16839524C2211yescharacteristic
55#16839517C2211yescharacteristic
56C2×#125×C61C2211yescharacteristic
57C2×#107×C61C2211yescharacteristic
58C2×#106×C61C2211yescharacteristic
59C61×#389C2211yescharacteristic
60C61×#340C2211yescharacteristic
61C61×#340C2211yescharacteristic
62C61×#336C2211yescharacteristic
63C22×Dic122C2311yescharacteristic
64C22×C4×D61—21no—
65C22×C4×D61—21no—
66C22×C4×D61D411yescharacteristic
67C22×C4×D61D411yescharacteristic
68C22×C4×D61C2311yescharacteristic
69C22×#92418C2311yescharacteristic
70C22×#92418C2311yescharacteristic
71#14519769—22no—
72#14519769—22no—
73#14519768—22no—
74#14519768—22no—
75#14519768D414yes—
76#14519768D414yes—
77#14519767—28no—
78#14519767—28no—
79#14519757—22no—
80#14519757Q814yes—
81#14519742—22no—
82#14519742—22no—
83C2×C4⋊Dic61—21no—
84C2×C4⋊Dic61—21no—
85C2×C4⋊Dic61—21no—
86C2×C4⋊Dic61C2311yescharacteristic
87C2×C4⋊Dic61C2311yescharacteristic
88C2×C4⋊Dic61C2311yescharacteristic
89C2×#11758196—21no—
90C2×#11758196—21no—
91C2×#11758196—21no—
92C2×#11758196—21no—
93C2×#11758196—21no—
94C2×#11758196—21no—
95C2×#11758196—21no—
96C4⋊C4×D61—28no—
97C4⋊C4×D61—28no—
98C4⋊C4×D61—22no—
99C4⋊C4×D61—22no—
100C4⋊C4×D61—22no—
101C4⋊C4×D61—22no—
102C2×#11758198—21no—
103C2×#11758198—21no—
104C2×C4×Dic61—21no—
105C2×C4×Dic61—21no—
106C2×C4×Dic61D411yescharacteristic
107C2×C4×Dic61D411yescharacteristic
108C2×C4×Dic61C2311yescharacteristic
109C42×D61—22no—
110C42×D61—22no—
111#14519710—22no—
112#14519708—22no—
113#14519706—22no—
114#14519706—22no—
115#14519702—22no—
116#14519702—22no—
117#14519700—22no—
118#14519700—22no—
119#14519689—22no—
120#14519689—22no—
121#14519684—21no—
122#14519684—21no—
123#14519684—21no—
124#14519684—21no—
125C22×C8×C61C2311yescharacteristic
126C22×C8×C61C2311yescharacteristic
127#125×C61—22no—
128#125×C61—22no—
129C2×C4⋊C4×C61—21no—
130C2×C4⋊C4×C61—21no—
131C2×C4⋊C4×C61—21no—
132C2×C4⋊C4×C61C2311yescharacteristic
133C2×C4⋊C4×C61C2311yescharacteristic
134C2×C4⋊C4×C61C2311yescharacteristic
135C2×C4⋊C4×C61C2311yescharacteristic
136C2×C42×C61C2311yescharacteristic
137#107×C61—22no—
138#106×C61—22no—
139#95×C61—21no—
140C23×D61C2×D411yescharacteristic
141C22×C4×C61—21no—
142C22×C4×C61—21no—
143C22×C4×C61—21no—
144C22×C4×C61—21no—
145C22×C4×C61C2×D411yescharacteristic
146C22×C4×C61C2×D411yescharacteristic
147C22×C4×C61C2411yescharacteristic
148C22×Dic61—21no—
149C22×Dic61—21no—
150C22×Dic61—21no—
151C22×Dic61—21no—
152C22×Dic61C2×D411yescharacteristic
153C22×Dic61C2×D411yescharacteristic
154C22×Dic61C2×D411yescharacteristic
155C2×C4×D61—21no—
156C2×C4×D61—21no—
157C2×C4×D61—21no—
158C2×C4×D61—21no—
159C2×C4×D61—24no—
160C2×C4×D61—24no—
161C2×C4×D61—22no—
162C2×C4×D61—22no—
163C2×C4×D61—22no—
164C2×C4×D61—22no—
165C2×C4×D61—41no—
166C2×C4×D61—41no—
167C2×C4×D61—42no—
168C2×C4×D61—42no—
169C2×C4×D61—42no—
170C2×C4×D61—42no—
171C2×C4×D61C2×Q812yes—
172C2×C4×D61#4112yes—
173C2×Dic122—21no—
174C2×Dic122—21no—
175C2×Dic122—21no—
176C2×Dic122—21no—
177C2×Dic122—21no—
178C2×Dic122C2×Q812yes—
179C2×C8×C61—21no—
180C2×C8×C61—21no—
181C2×C8×C61—21no—
182C2×C8×C61—21no—
183C2×C8×C61—21no—
184C2×C8×C61—21no—
185C4⋊C4×C61—24no—
186C4⋊C4×C61—24no—
187C4⋊C4×C61—21no—
188C4⋊C4×C61—21no—
189C4⋊C4×C61—21no—
190C4⋊C4×C61—21no—
191C4⋊C4×C61—21no—
192C4⋊C4×C61—42no—
193C4⋊C4×C61C2×D412yes—
194C4⋊C4×C61C2×D412yes—
195C4⋊C4×C61C2×Q812yes—
196C42×C61—21no—
197C42×C61—21no—
198#11758198—24no—
199#11758198—42no—
200C4⋊Dic61—21no—
201C4⋊Dic61—21no—
202C4⋊Dic61—24no—
203C4⋊Dic61—24no—
204C4⋊Dic61—24no—
205C4⋊Dic61—21no—
206C4⋊Dic61—21no—
207C4⋊Dic61C2×D412yes—
208C4⋊Dic61C2×D412yes—
209#11758196—24no—
210#11758196—24no—
211#11758196—24no—
212#11758196—24no—
213#11758196—24no—
214#11758196—24no—
215#11758196—42no—
216C4×Dic61—22no—
217C4×Dic61—22no—
218C4×Dic61—42no—
219C4×Dic61—42no—
220C4×Dic61C2×D412yes—
221C4×Dic61C2×D412yes—
222C2×#92418—21no—
223C2×#92418—21no—
224C2×#92418—21no—
225C2×#92418—21no—
226C2×#92418—21no—
227C2×#92418—21no—
228C23×C61C22×D411yescharacteristic
229C22×D61—21no—
230C22×D61—21no—
231C22×D61—21no—
232C22×D61—21no—
233C22×D61C22⋊Q812yes—
234C22×D61#13512yes—
235C22×D61#13712yes—
236C2×C4×C61—21no—
237C2×C4×C61—21no—
238C2×C4×C61—22no—
239C2×C4×C61—22no—
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—41no—
253C2×C4×C61—41no—
254C2×C4×C61C22×D411yescharacteristic
255C2×C4×C61C22×D411yescharacteristic
256C2×C4×C61C22×Q811yescharacteristic
257C2×C4×C61C2×#4111yescharacteristic
258C2×C4×C612−1+411yescharacteristic
259C2×C4×C612−1+411yescharacteristic
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—22no—
269C2×Dic61—41no—
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#12012yes—
279C2×Dic61#12012yes—
280C2×Dic61C22⋊Q812yes—
281C4×D61—22no—
282C4×D61—22no—
283C4×D61—22no—
284C4×D61—22no—
285C4×D61—44no—
286C4×D61—44no—
287C4×D61—48no—
288C4×D61—48no—
289Dic122—22no—
290Dic122—22no—
291Dic122—41no—
292Dic122—41no—
293C8×C61—41no—
294C8×C61—41no—
295#92418—41no—
296#92418—41no—
297#24307—611nomaximal, Sylow
298C22×C61C2×#12011yescharacteristic
299C22×C61C2×C22⋊Q811yescharacteristic
300C22×C61#53611yescharacteristic
301C22×C61C2×#13511yescharacteristic
302C22×C61C2×#13711yescharacteristic
303C22×C61#57511yescharacteristic
304C22×C61#57511yescharacteristic
305C2×D61—22no—
306C2×D61—22no—
307C2×D61—22no—
308C2×D61—22no—
309C2×D61—22no—
310C2×D61—22no—
311C2×D61#47914yescomplemented
312C4×C61—42no—
313C4×C61—41no—
314C4×C61—42no—
315C4×C61—41no—
316C4×C61—42no—
317C4×C61—81no—
318C4×C61#54811yescharacteristic
319C4×C61#54811yescharacteristic
320C4×C61D4×Q811yescharacteristic
321C4×C61D4×Q811yescharacteristic
322Dic61—22no—
323Dic61—22no—
324Dic61—22no—
325Dic61—42no—
326Dic61—42no—
327Dic61—42no—
328Dic61—81no—
329C22×#125—611no—
330C2×#479—611no—
331C2×#479—611no—
332C4×C22⋊Q8—611no—
333#2059—611no—
334#2059D6111yescharacteristic, complemented
335#2026—611no—
336#2026—611no—
337#1989—611no—
338#1989—611no—
339#1872—611no—
340#1872—611no—
341#1853—611no—
342#1852—611no—
343#1797—611no—
344C2×C61#240911yescharacteristic
345C2×C61#300211yescharacteristic
346C2×C61#300711yescharacteristic
347C2×C61C2×#47911yescharacteristic
348C2×C61#308011yescharacteristic
349C2×C61#323211yescharacteristic
350C2×C61#323611yescharacteristic
351D61—24no—
352C2×#125—611no—
353C2×#125—611no—
354C2×#125—611no—
355C2×#125—1221no—
356C2×#125—1221no—
357C2×#125—1224no—
358C2×#125—1224no—
359C2×#125C2×D6111yescharacteristic
360C2×C22⋊Q8—611no—
361C2×C4×Q8—611no—
362C22×C4⋊C4—611no—
363C22×C4⋊C4—611no—
364C22×C4⋊C4—1221no—
365C22×C4⋊C4—1221no—
366C22×C42—611no—
367#479—1224no—
368#479—1224no—
369C2×#107—611no—
370C2×#107C2×D6111yescharacteristic
371C2×#106—611no—
372C2×#106C2×D6111yescharacteristic
373C2×C4⋊C8—611no—
374C2×C4⋊C8—611no—
375C2×#103—611no—
376C2×#103—611no—
377C2×#103—611no—
378C2×#103—611no—
379C2×C22⋊C8—611no—
380C2×C22⋊C8—611no—
381#391—611no—
382#389C2×D6111yescharacteristic
383#387—1221no—
384#385—1221no—
385#380—611no—
386C4×C4⋊C4—611no—
387C4×C4⋊C4—1221no—
388C4×C22⋊C4—1221no—
389#340—611no—
390#340—611no—
391#340—611no—
392#340—611no—
393#340—611no—
394#340—611no—
395#340C2×D6111yescharacteristic
396#340C2×D6111yescharacteristic
397#336—611no—
398#336C2×D6111yescharacteristic
399C61#2430711yescharacteristic, complemented, Sylow
400C22×Q8—611no—
401C23×C4—611no—
402C23×C4—611no—
403C23×C4—611no—
404C23×C4—1221no—
405C23×C4—1221no—
406C22×C8—611no—
407C22×C8—611no—
408C22×C8C22×D6111yescharacteristic
409C22×C8C22×D6111yescharacteristic
410#125—22no—
411#125—22no—
412#125—614no—
413#125—614no—
414#125—1222no—
415#125—1222no—
416#125—1222no—
417#125—1222no—
418#125—1228no—
419#125—1228no—
420C22⋊Q8—614no—
421C22⋊Q8—1222no—
422C4×Q8—1222no—
423C4×Q8—1222no—
424C2×C4⋊C4—21no—
425C2×C4⋊C4—21no—
426C2×C4⋊C4—21no—
427C2×C4⋊C4—611no—
428C2×C4⋊C4—611no—
429C2×C4⋊C4—611no—
430C2×C4⋊C4—1222no—
431C2×C4⋊C4—1222no—
432C2×C4⋊C4—1228no—
433C2×C4⋊C4—1228no—
434C2×C4⋊C4—1221no—
435C2×C4⋊C4—1221no—
436C2×C4⋊C4—1221no—
437C2×C4⋊C4—1221no—
438C2×C4⋊C4—1221no—
439C2×C4⋊C4—1221no—
440C2×C4⋊C4—1221no—
441C2×C4⋊C4—1221no—
442C2×C4⋊C4—1221no—
443C2×C4⋊C4—1222no—
444C2×C4⋊C4—1222no—
445C2×C4⋊C4—1221no—
446C2×C4⋊C4C22×D6111yescharacteristic
447C2×C4⋊C4C22×D6111yescharacteristic
448C2×C4⋊C4C22×D6111yescharacteristic
449C2×C4⋊C4C22×D6111yescharacteristic
450C2×C22⋊C4—1221no—
451C2×C22⋊C4—1221no—
452C2×C42—611no—
453C2×C42—611no—
454C2×C42—611no—
455C2×C42—1222no—
456C2×C42—1222no—
457C2×C42—1221no—
458C2×C42—1221no—
459C2×C42C22×D6111yescharacteristic
460#107—22no—
461#107—1222no—
462#106—22no—
463#106—1222no—
464C4⋊C8—1222no—
465C4⋊C8—1222no—
466#103—1222no—
467#103—1222no—
468#103—1222no—
469#103—1222no—
470C22⋊C8—1222no—
471C22⋊C8—1222no—
472#95—21no—
473#95—1221no—
474#95—1221no—
475#95—1221no—
476#95—1221no—
477C24—611no—
478C2×Q8—612no—
479C2×Q8—1221no—
480C2×Q8—1221no—
481C2×Q8—1221no—
482C2×Q8—1221no—
483C2×Q8—1221no—
484C22×C4—21no—
485C22×C4—21no—
486C22×C4—21no—
487C22×C4—21no—
488C22×C4—611no—
489C22×C4—611no—
490C22×C4—611no—
491C22×C4—612no—
492C22×C4—612no—
493C22×C4—1222no—
494C22×C4—1222no—
495C22×C4—1222no—
496C22×C4—1222no—
497C22×C4—1224no—
498C22×C4—1224no—
499C22×C4—1221no—
500C22×C4—1221no—
501C22×C4—1221no—
502C22×C4—1221no—
503C22×C4—1221no—
504C22×C4—1221no—
505C22×C4—1221no—
506C22×C4—1221no—
507C22×C4—2441no—
508C22×C4—2441no—
509C22×C4—2442no—
510C22×C4—2442no—
511C22×C4—2442no—
512C22×C4—2442no—
513C22×C4D4×D6111yescharacteristic
514C22×C4D4×D6111yescharacteristic
515C22×C4C23×D6111yescharacteristic
516C2×C8—21no—
517C2×C8—21no—
518C2×C8—21no—
519C2×C8—21no—
520C2×C8—21no—
521C2×C8—21no—
522C2×C8—1221no—
523C2×C8—1221no—
524C2×C8—1221no—
525C2×C8—1221no—
526C2×C8—1221no—
527C2×C8—1221no—
528C4⋊C4—24no—
529C4⋊C4—24no—
530C4⋊C4—21no—
531C4⋊C4—21no—
532C4⋊C4—21no—
533C4⋊C4—21no—
534C4⋊C4—21no—
535C4⋊C4—42no—
536C4⋊C4—612no—
537C4⋊C4—612no—
538C4⋊C4—1224no—
539C4⋊C4—1224no—
540C4⋊C4—1224no—
541C4⋊C4—1224no—
542C4⋊C4—1224no—
543C4⋊C4—1224no—
544C4⋊C4—1224no—
545C4⋊C4—1224no—
546C4⋊C4—1224no—
547C4⋊C4—1221no—
548C4⋊C4—1221no—
549C4⋊C4—1221no—
550C4⋊C4—1221no—
551C4⋊C4—2442no—
552C4⋊C4D4×D6112yes—
553C4⋊C4D4×D6112yes—
554C4⋊C4Q8×D6112yes—
555C22⋊C4—1224no—
556C22⋊C4—2442no—
557C42—21no—
558C42—21no—
559C42—612no—
560C42—612no—
561C42—1222no—
562C42—1222no—
563C42—2442no—
564C42—2442no—
565C23—612no—
566C23—612no—
567C23—612no—
568C23—1221no—
569C23—1221no—
570C23—1221no—
571C23—1221no—
572C23C2×D4×D6111yescharacteristic
573Q8—1222no—
574Q8—1222no—
575Q8—2441no—
576Q8—2441no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—21no—
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—612no—
596C2×C4—612no—
597C2×C4—612no—
598C2×C4—1221no—
599C2×C4—1221no—
600C2×C4—1221no—
601C2×C4—1221no—
602C2×C4—1221no—
603C2×C4—1221no—
604C2×C4—1222no—
605C2×C4—1222no—
606C2×C4—1222no—
607C2×C4—1222no—
608C2×C4—1222no—
609C2×C4—1222no—
610C2×C4—1222no—
611C2×C4—2444no—
612C2×C4—2444no—
613C2×C4—2448no—
614C2×C4—2448no—
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×C4—2441no—
624C2×C4C2×D4×D6111yescharacteristic
625C2×C4C2×D4×D6111yescharacteristic
626C2×C4C2×Q8×D6111yescharacteristic
627C2×C4#41×D6111yescharacteristic
628C2×C4#1451982511yescharacteristic
629C2×C4#1451982511yescharacteristic
630C8—41no—
631C8—41no—
632C8—2441no—
633C8—2441no—
634C22—614no—
635C22—1222no—
636C22—1222no—
637C22—1222no—
638C22—1222no—
639C22—1222no—
640C22—1222no—
641C22#120×D6111yescharacteristic
642C22C22⋊Q8×D6111yescharacteristic
643C22#1684033711yescharacteristic
644C22#135×D6111yescharacteristic
645C22#137×D6111yescharacteristic
646C22#1684055811yescharacteristic
647C22#1684055911yescharacteristic
648C4—42no—
649C4—41no—
650C4—42no—
651C4—41no—
652C4—42no—
653C4—81no—
654C4—1222no—
655C4—1222no—
656C4—1222no—
657C4—4881no—
658C4—2442no—
659C4—2442no—
660C4—2442no—
661C4#1684042211yescharacteristic
662C4#1684042211yescharacteristic
663C4#1684042711yescharacteristic
664C4#1684042711yescharacteristic
665C2—1224no—
666C2#2144917911yescharacteristic
667C2#2145503411yescharacteristic
668C2#2145507611yescharacteristic
669C2#479×D6111yescharacteristic
670C2#2145563611yescharacteristic
671C2#2145698811yescharacteristic
672C2#2145702611yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2059C2
socleC23×C61C22×D4
prime radicalC23×D61C2×D4
hypercenter#2059D61
hypocenterC61#24307
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
2. C2×#16840072C2semidirect
3. C2×#16840068C2semidirect
5. #21447157C2semidirect
6. #21446971C2semidirect
7. #21446967C2semidirect
8. #21446775C2semidirect
9. #21446773C2semidirect
15. C61×#2059C2semidirect
311. C2×D61#479semidirect
334. #2059D61semidirect
399. C61#24307semidirect

Maximal subgroups

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

subgroupindex
C2×#125×D612
C2×#168400722
C2×#168400682
#214483492
#214471572
#214469712
#214469672
#214467752
#214467732
#214461252
#214460102
#214460052
#214457982
C61×#20592
#2430761