Finite Group Databasebeta

← #29646218

Subgroups — #29646218

Order 15616.

944 conjugacy classes of subgroups.

The subgroup lattice has 646 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840445C211yesmaximal, characteristic
2#21454478C212yesmaximal, complemented
3#21444181C212yesmaximal, complemented
4#21444176C212yesmaximal, complemented
5#21444173C212yesmaximal, complemented
6#21444154C211yesmaximal, characteristic
7#21444150C211yesmaximal, characteristic, complemented
8#21444134C211yesmaximal, characteristic
9#21444133C211yesmaximal, characteristic, complemented
10#21444130C211yesmaximal, characteristic, complemented
11C61×#1464C211yesmaximal, characteristic, complemented
12#16840445—22no—
13#16840445—22no—
14#16840445—22no—
15#16840445—22no—
16C2×#14519772C411yescharacteristic
17C2×#14519771C411yescharacteristic
18C2×#14519769C2211yescharacteristic
19C2×#14519767C411yescharacteristic
20C2×#14519767C2211yescharacteristic
21C2×#14519761C411yescharacteristic
22C2×#14519760C411yescharacteristic
23C2×#14519755C2211yescharacteristic
24C2×#14519754C411yescharacteristic
25C2×C4×Dic122C411yescharacteristic
26C2×#14519737C2211yescharacteristic
27C2×#14519734C2211yescharacteristic
28C2×#14519733C2211yescharacteristic
29C22×#11758196C411yescharacteristic
30#16839824—22no—
31#16839822—22no—
32#16839816C2212yes—
33#16839815C2212yes—
34#16839812C2212yes—
35#16839787—22no—
36#16839724C2212yes—
37C22⋊C8×D61C2212yes—
38#16839719—22no—
39#16839488C2212yes—
40#16839487C2212yes—
41#16839487C2212yes—
42#16839487C2212yes—
43#16839487C2212yes—
44#16839486C2211yescharacteristic
45#16839486C2211yescharacteristic
46C2×#117×C61C2211yescharacteristic
47C61×#432C2212yes—
48C61×#324C2212yes—
49C61×#323C2211yescharacteristic
50C61×#323C2211yescharacteristic
51C22×Dic122C2×C411yescharacteristic
52C22×C22⋊D61C2×C411yescharacteristic
53C23×Dic61C2×C411yescharacteristic
54C23×Dic61C2×C411yescharacteristic
55C22×C4×D61C2311yescharacteristic
56C2×#11758189—22no—
57C2×C8×D61—22no—
58#14519772—22no—
59#14519772—22no—
60#14519771—22no—
61#14519771—22no—
62#14519769—22no—
63#14519769—22no—
64#14519767—22no—
65#14519767—22no—
66#14519767—22no—
67#14519767—22no—
68#14519761—22no—
69#14519761—22no—
70#14519760—22no—
71#14519760—22no—
72#14519755—22no—
73#14519755—22no—
74#14519754—22no—
75#14519754—22no—
76C4×Dic122—21no—
77C4×Dic122—21no—
78C4×Dic122—21no—
79C4×Dic122—21no—
80#14519737—22no—
81#14519737—22no—
82#14519734D412yes—
83#14519734D412yes—
84#14519734D412yes—
85#14519734D412yes—
86#14519733—22no—
87#14519733—22no—
88C2×C4⋊Dic61C2×C411yescharacteristic
89C2×C4⋊Dic61C2×C411yescharacteristic
90C2×C4⋊Dic61C2×C411yescharacteristic
91C2×#11758196—21no—
92C2×#11758196—21no—
93C2×#11758196—21no—
94C2×#11758196—21no—
95C2×#11758196C2×C411yescharacteristic
96C2×#11758196C2×C411yescharacteristic
97C2×#11758196C2×C411yescharacteristic
98C2×#11758196C2×C411yescharacteristic
99C2×#11758196C2×C411yescharacteristic
100C2×#11758196C2×C411yescharacteristic
101C2×#11758196C2×C411yescharacteristic
102C2×#11758196C2×C411yescharacteristic
103C2×#11758196C2×C411yescharacteristic
104C2×#11758196C2311yescharacteristic
105C2×#11758196C2311yescharacteristic
106C2×C22⋊Dic61C2×C411yescharacteristic
107C2×#11758198C2×C411yescharacteristic
108C2×#11758198C2×C411yescharacteristic
109C2×#11758198C2×C411yescharacteristic
110C2×#11758198C2×C411yescharacteristic
111C2×#11758198C2×C411yescharacteristic
112C2×C4×Dic61C2×C411yescharacteristic
113C2×C4×Dic61C2×C411yescharacteristic
114C2×C4×Dic61C2311yescharacteristic
115C4⋊#92418—22no—
116#14519706—22no—
117#14519706—22no—
118C22⋊#92418C2312yes—
119C22⋊#92418C2312yes—
120#14519689—22no—
121#14519689—22no—
122#14519687—22no—
123C4×#92418—22no—
124C8×Dic61—22no—
125C23×C4×C61C2×C411yescharacteristic
126#117×C61—22no—
127#117×C61—22no—
128C2×C4⋊C4×C61C2×C411yescharacteristic
129C2×C4⋊C4×C61C2311yescharacteristic
130C2×C22⋊C4×C61C2×C411yescharacteristic
131C2×C22⋊C4×C61C2311yescharacteristic
132C2×C42×C61C2×C411yescharacteristic
133C2×C42×C61C2311yescharacteristic
134C4⋊C8×C61—22no—
135C22⋊C8×C61C2312yes—
136C22⋊C8×C61C2312yes—
137#97×C61—22no—
138C24×C61C22×C411yescharacteristic
139C23×D61C22×C411yescharacteristic
140C22×C4×C61—22no—
141C22×C4×C61—21no—
142C22×C4×C61—21no—
143C22×C4×C61—41no—
144C22×C4×C61C22×C411yescharacteristic
145C22×C4×C61C22×C411yescharacteristic
146C22×C4×C61C22×C411yescharacteristic
147C22×C4×C61C22×C411yescharacteristic
148C22×C4×C61C22×C411yescharacteristic
149C22×C4×C61C2411yescharacteristic
150C2×C22⋊D61—22no—
151C2×C22⋊D61—22no—
152C2×C22⋊D61—41no—
153C22×Dic61—22no—
154C22×Dic61—21no—
155C22×Dic61—21no—
156C22×Dic61—21no—
157C22×Dic61—21no—
158C22×Dic61—41no—
159C22×Dic61—42no—
160C22×Dic61C22×C411yescharacteristic
161C22×Dic61C22×C411yescharacteristic
162C22×Dic61C22×C411yescharacteristic
163C22×Dic61C22×C411yescharacteristic
164C22×Dic61C22×C411yescharacteristic
165C22×Dic61C22×C411yescharacteristic
166C22×Dic61C22×C411yescharacteristic
167C2×C4×D61—42no—
168C2×C4×D61C2×D412yes—
169C2×C4×D61C2×D412yes—
170C2×Dic122—21no—
171C2×Dic122—21no—
172C2×Dic122—42no—
173C2×C8×C61—22no—
174C2×C8×C61—22no—
175C4⋊C4×C61—22no—
176C4⋊C4×C61—22no—
177C22⋊C4×C61—22no—
178C22⋊C4×C61—22no—
179C42×C61—21no—
180C42×C61—21no—
181C42×C61C2×D412yes—
182C42×C61C2×D412yes—
183C22⋊Dic61—22no—
184#11758198—22no—
185#11758198—22no—
186#11758198—22no—
187#11758198C22⋊C412yes—
188#11758198C22⋊C412yes—
189#11758198C22⋊C412yes—
190#11758198C22⋊C412yes—
191C4⋊Dic61—22no—
192C4⋊Dic61—22no—
193C4⋊Dic61—21no—
194C4⋊Dic61—21no—
195#11758196—21no—
196#11758196—21no—
197#11758196—21no—
198#11758196—21no—
199#11758196—22no—
200#11758196—22no—
201#11758196—21no—
202#11758196—21no—
203#11758196—21no—
204#11758196—21no—
205#11758196—22no—
206#11758196—22no—
207#11758196—22no—
208#11758196C22⋊C411yescharacteristic
209#11758196C22⋊C411yescharacteristic
210#11758196C22⋊C411yescharacteristic
211#11758196C22⋊C411yescharacteristic
212#11758196C22⋊C412yes—
213#11758196C22⋊C412yes—
214C4×Dic61—21no—
215C4×Dic61—21no—
216C4×Dic61—21no—
217C4×Dic61—21no—
218C4×Dic61C2×D412yes—
219C4×Dic61C2×D412yes—
220C2×#92418—22no—
221C2×#92418—22no—
222#11758189—44no—
223C8×D61—44no—
224C23×C61—21no—
225C23×C61—21no—
226C23×C61—41no—
227C23×C61—41no—
228C23×C61#13112yes—
229C23×C61C23×C411yescharacteristic
230C22×D61—21no—
231C22×D61—21no—
232C22×D61—41no—
233C22×D61—41no—
234C22×D61C2×C22⋊C412yes—
235C2×C4×C61—21no—
236C2×C4×C61—21no—
237C2×C4×C61—21no—
238C2×C4×C61—21no—
239C2×C4×C61—21no—
240C2×C4×C61—21no—
241C2×C4×C61—21no—
242C2×C4×C61—41no—
243C2×C4×C61—41no—
244C2×C4×C61—42no—
245C2×C4×C61—42no—
246C2×C4×C61—41no—
247C2×C4×C61—41no—
248C2×C4×C61C2×C22⋊C411yescharacteristic
249C2×C4×C61C2×C22⋊C411yescharacteristic
250C2×C4×C61C2×C22⋊C412yes—
251C2×C4×C61C22×D412yes—
252C22⋊D61—44no—
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—21no—
263C2×Dic61—21no—
264C2×Dic61—41no—
265C2×Dic61—42no—
266C2×Dic61—42no—
267C2×Dic61—42no—
268C2×Dic61—42no—
269C2×Dic61—41no—
270C2×Dic61—41no—
271C2×Dic61C2×C22⋊C411yescharacteristic
272C2×Dic61C2×C22⋊C411yescharacteristic
273C2×Dic61C2×C22⋊C411yescharacteristic
274C2×Dic61C2×C22⋊C412yes—
275C2×Dic61C2×C22⋊C411yescharacteristic
276C4×D61—24no—
277C4×D61—42no—
278Dic122—42no—
279Dic122—42no—
280C8×C61—42no—
281C8×C61—42no—
282#92418—42no—
283#92418—42no—
284#17413—611nomaximal, Sylow
285C22×C61—21no—
286C22×C61—21no—
287C22×C61—22no—
288C22×C61—42no—
289C22×C61—42no—
290C22×C61—41no—
291C22×C61—41no—
292C22×C61C22×C22⋊C411yescharacteristic
293C22×C61C2×#13111yescharacteristic
294C22×C61#56811yescharacteristic
295C2×D61—22no—
296C2×D61—22no—
297C2×D61—22no—
298C2×D61—81no—
299C2×D61—81no—
300C4×C61—21no—
301C4×C61—21no—
302C4×C61—22no—
303C4×C61—22no—
304C4×C61—41no—
305C4×C61—41no—
306C4×C61—41no—
307Dic61—21no—
308Dic61—21no—
309Dic61—21no—
310Dic61—22no—
311Dic61—41no—
312Dic61—41no—
313Dic61—41no—
314Dic61—41no—
315Dic61#41012yes—
316C2×#552—611no—
317#2924—612no—
318#1472—612no—
319#1470—612no—
320#1470—612no—
321#1465—611no—
322#1464—611no—
323#1464D6111yescharacteristic, complemented
324#1459—611no—
325#1458—611no—
326#1458—611no—
327C2×C61—21no—
328C2×C61—21no—
329C2×C61—42no—
330C2×C61#288011yescharacteristic
331C2×C61C2×#41011yescharacteristic
332C2×C61#288911yescharacteristic
333D61—42no—
334#552—1222no—
335#552—1222no—
336#552—1222no—
337#552—1222no—
338C2×#126—611no—
339C2×#126—611no—
340C2×#125—611no—
341C2×#125—611no—
342C2×#125—611no—
343C2×#123—611no—
344C2×#123—611no—
345C2×C22⋊Q8—611no—
346C2×C22⋊Q8—611no—
347C2×C4×Q8—611no—
348C2×C4×D4—611no—
349C2×#117—611no—
350C2×#117—611no—
351C2×#117C2×D6111yescharacteristic
352C22×C4⋊C4—611no—
353#433—1222no—
354#433—1222no—
355#432—612no—
356#432C2×D6112yes—
357#431—612no—
358#431—612no—
359#424—1222no—
360#407—612no—
361C2×C22⋊C8—612no—
362#405—1222no—
363#324—612no—
364#324—612no—
365#324—612no—
366#324—612no—
367#324—612no—
368#324C2×D6112yes—
369#323—611no—
370#323—611no—
371#323C2×D6111yescharacteristic
372#323C2×D6111yescharacteristic
373C61#1741311yescharacteristic, complemented, Sylow
374C22×Q8—611no—
375C22×D4—611no—
376C23×C4—611no—
377C23×C4—611no—
378C23×C4—611no—
379C23×C4C4×D6111yescharacteristic
380C2×M16—1222no—
381C22×C8—1222no—
382#126—1222no—
383#126—1222no—
384#126—1222no—
385#126—1222no—
386#125—1222no—
387#125—1222no—
388#125—1222no—
389#125—1222no—
390#125—1222no—
391#125—1222no—
392#123—1222no—
393#123—1222no—
394#123—1222no—
395#123—1222no—
396C22⋊Q8—1222no—
397C22⋊Q8—1222no—
398C22⋊Q8—1222no—
399C22⋊Q8—1222no—
400C4×Q8—1221no—
401C4×Q8—1221no—
402C4×Q8—1221no—
403C4×Q8—1221no—
404C4×D4—1222no—
405C4×D4—1222no—
406#117—22no—
407#117—22no—
408#117—612no—
409#117—612no—
410#117—612no—
411#117—612no—
412#117—1222no—
413#117—1222no—
414C2×C4⋊C4—611no—
415C2×C4⋊C4—611no—
416C2×C4⋊C4—611no—
417C2×C4⋊C4—611no—
418C2×C4⋊C4—611no—
419C2×C4⋊C4—611no—
420C2×C4⋊C4—611no—
421C2×C4⋊C4—611no—
422C2×C4⋊C4—611no—
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⋊C4C4×D6111yescharacteristic
433C2×C4⋊C4C22×D6111yescharacteristic
434C2×C22⋊C4—611no—
435C2×C22⋊C4—611no—
436C2×C22⋊C4—611no—
437C2×C22⋊C4—611no—
438C2×C22⋊C4—611no—
439C2×C22⋊C4—611no—
440C2×C22⋊C4C4×D6111yescharacteristic
441C2×C22⋊C4C22×D6111yescharacteristic
442C2×C42—611no—
443C2×C42—611no—
444C2×C42—611no—
445C2×C42C4×D6111yescharacteristic
446C2×C42C22×D6111yescharacteristic
447C4⋊C8—22no—
448C4⋊C8—1222no—
449C4⋊C8—1222no—
450C4⋊C8—1222no—
451C22⋊C8—612no—
452C22⋊C8—612no—
453C22⋊C8—1222no—
454C22⋊C8—1222no—
455C22⋊C8C22×D6112yes—
456C22⋊C8C22×D6112yes—
457#97—22no—
458#97—1222no—
459C4×C8—1222no—
460C4×C8—1222no—
461C24—611no—
462C24C2×C4×D6111yescharacteristic
463C2×Q8—1221no—
464C2×Q8—1221no—
465C2×Q8—2442no—
466C2×D4—1222no—
467C2×D4—1222no—
468C2×D4—2441no—
469C22×C4—22no—
470C22×C4—21no—
471C22×C4—21no—
472C22×C4—41no—
473C22×C4—611no—
474C22×C4—611no—
475C22×C4—611no—
476C22×C4—612no—
477C22×C4—612no—
478C22×C4—611no—
479C22×C4—611no—
480C22×C4—611no—
481C22×C4—611no—
482C22×C4—1221no—
483C22×C4—1221no—
484C22×C4—1221no—
485C22×C4—1221no—
486C22×C4—1222no—
487C22×C4—2442no—
488C22×C4—2441no—
489C22×C4—2442no—
490C22×C4C2×C4×D6111yescharacteristic
491C22×C4C2×C4×D6111yescharacteristic
492C22×C4C2×C4×D6111yescharacteristic
493C22×C4C2×C4×D6111yescharacteristic
494C22×C4C2×C4×D6111yescharacteristic
495C22×C4C23×D6111yescharacteristic
496M16—2444no—
497C2×C8—22no—
498C2×C8—22no—
499C2×C8—1222no—
500C2×C8—1222no—
501C2×C8—2444no—
502C4⋊C4—22no—
503C4⋊C4—22no—
504C4⋊C4—611no—
505C4⋊C4—611no—
506C4⋊C4—611no—
507C4⋊C4—611no—
508C4⋊C4—612no—
509C4⋊C4—612no—
510C4⋊C4—1221no—
511C4⋊C4—1221no—
512C4⋊C4—1221no—
513C4⋊C4—1221no—
514C4⋊C4—1222no—
515C4⋊C4—1222no—
516C4⋊C4—1222no—
517C4⋊C4—1221no—
518C4⋊C4—1221no—
519C4⋊C4—1222no—
520C4⋊C4—1222no—
521C4⋊C4—1221no—
522C4⋊C4—1221no—
523C4⋊C4—1221no—
524C4⋊C4—1221no—
525C4⋊C4—1222no—
526C4⋊C4—1222no—
527C22⋊C4—22no—
528C22⋊C4—22no—
529C22⋊C4—612no—
530C22⋊C4—612no—
531C22⋊C4—612no—
532C22⋊C4—612no—
533C22⋊C4—1222no—
534C22⋊C4—1222no—
535C22⋊C4—1222no—
536C22⋊C4—1222no—
537C42—21no—
538C42—21no—
539C42—612no—
540C42—612no—
541C42—1221no—
542C42—1221no—
543C42—1221no—
544C42—1221no—
545C42D4×D6112yes—
546C42D4×D6112yes—
547C23—21no—
548C23—21no—
549C23—41no—
550C23—41no—
551C23—612no—
552C23—1221no—
553C23—1221no—
554C23—2441no—
555C23—2441no—
556C23#1451978512yes—
557C23C22×C4×D6111yescharacteristic
558Q8—2442no—
559Q8—2442no—
560D4—2444no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—611no—
575C2×C4—611no—
576C2×C4—611no—
577C2×C4—612no—
578C2×C4—611no—
579C2×C4—1224no—
580C2×C4—1222no—
581C2×C4—1221no—
582C2×C4—1221no—
583C2×C4—1221no—
584C2×C4—1221no—
585C2×C4—1221no—
586C2×C4—1221no—
587C2×C4—1221no—
588C2×C4—1221no—
589C2×C4—1221no—
590C2×C4—1221no—
591C2×C4—2442no—
592C2×C4—2442no—
593C2×C4—2441no—
594C2×C4—2442no—
595C2×C4—2442no—
596C2×C4—2442no—
597C2×C4—2441no—
598C2×C4—2441no—
599C2×C4C22⋊C4×D6111yescharacteristic
600C2×C4C22⋊C4×D6111yescharacteristic
601C2×C4C22⋊C4×D6112yes—
602C2×C4C2×D4×D6112yes—
603C8—42no—
604C8—42no—
605C8—2442no—
606C8—2442no—
607C22—21no—
608C22—21no—
609C22—22no—
610C22—42no—
611C22—42no—
612C22—41no—
613C22—41no—
614C22—1222no—
615C22—1222no—
616C22—1222no—
617C22—4881no—
618C22—4881no—
619C22C2×C22⋊C4×D6111yescharacteristic
620C22C2×#1451978511yescharacteristic
621C22#1684052811yescharacteristic
622C4—21no—
623C4—21no—
624C4—22no—
625C4—22no—
626C4—41no—
627C4—41no—
628C4—41no—
629C4—612no—
630C4—1221no—
631C4—1221no—
632C4—1221no—
633C4—1222no—
634C4—2441no—
635C4—2441no—
636C4—2441no—
637C4—2441no—
638C2—21no—
639C2—21no—
640C2—42no—
641C2—2442no—
642C2#2145426911yescharacteristic
643C2#410×D6111yescharacteristic
644C2#2145430011yescharacteristic
subgroupG/H
centerC22C2×C22⋊C4×D61
commutatorC23×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1464C2
socleC22×C61C22×C22⋊C4
prime radicalC22×C22⋊D61C2×C4
cosocleC22×C4×C61C24
hypercenter#1464D61
hypocenterC61#17413
LCSG ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1464

Complements

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

NH ≅ G/Ntype
2. #21454478C2semidirect
3. #21444181C2semidirect
4. #21444176C2semidirect
5. #21444173C2semidirect
7. #21444150C2semidirect
9. #21444133C2semidirect
10. #21444130C2semidirect
11. C61×#1464C2semidirect
323. #1464D61semidirect
373. C61#17413semidirect

Maximal subgroups

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

subgroupindex
C2×#168404452
#214544782
#214441812
#214441762
#214441732
#214441542
#214441502
#214441342
#214441332
#214441302
C61×#14642
#1741361