Finite Group Databasebeta

← #29607250

Subgroups — #29607250

Order 15616.

1422 conjugacy classes of subgroups.

The subgroup lattice has 632 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21444081C212yesmaximal, complemented
2#21444076C212yesmaximal, complemented
3#21444068C212yesmaximal
4#21444064C212yesmaximal
5#21444049C211yesmaximal, characteristic, complemented
6#21444046C211yesmaximal, characteristic, complemented
7#21444046C211yesmaximal, characteristic, complemented
8#21444046C211yesmaximal, characteristic, complemented
9#21444042C211yesmaximal, characteristic
10#21444042C211yesmaximal, characteristic
11C2×C4×C22⋊C4×C61C211yesmaximal, characteristic
12C22×C4⋊Dic61C2212yes—
13C22×C22⋊Dic61C2212yes—
14C22×C22⋊Dic61C2212yes—
15C2×#14519684C2211yescharacteristic
16C2×#14519684C2211yescharacteristic
17C2×#14519684C2211yescharacteristic
18C2×#14519684C2211yescharacteristic
19C2×#14519684C2212yes—
20C2×#14519684C2212yes—
21C2×#14519684C2212yes—
22C2×#14519684C2212yes—
23C2×#14519684C2212yes—
24C2×#14519684C2212yes—
25C2×#14519684C2212yes—
26C2×#14519684C2212yes—
27C2×#14519684C2212yes—
28C24×C4×C61C2211yescharacteristic
29C22×C22⋊C4×C61C2211yescharacteristic
30C22×C22⋊C4×C61C2211yescharacteristic
31C22×C42×C61C2211yescharacteristic
32C22×C42×C61C2211yescharacteristic
33C4×C22⋊C4×C61C418yescomplemented
34C2×#95×C61C2211yescharacteristic
35C2×#95×C61C2211yescharacteristic
36C23×Dic61C2312yes—
37C23×Dic61C2312yes—
38C23×Dic61C2312yes—
39C23×Dic61C2312yes—
40C2×C4⋊Dic61—24no—
41C2×C4⋊Dic61—42no—
42C2×C4⋊Dic61—42no—
43C2×C22⋊Dic61—24no—
44C2×C22⋊Dic61—24no—
45C2×C22⋊Dic61—42no—
46C2×C22⋊Dic61—42no—
47C2×C22⋊Dic61—42no—
48C2×C22⋊Dic61—42no—
49#14519684—24no—
50#14519684—22no—
51#14519684—22no—
52#14519684—22no—
53#14519684—22no—
54#14519684—42no—
55#14519684—42no—
56#14519684—42no—
57#14519684—42no—
58#14519684—42no—
59#14519684—42no—
60#14519684—42no—
61#14519684—42no—
62C25×C61C2311yescharacteristic
63C23×C4×C61—22no—
64C23×C4×C61—22no—
65C23×C4×C61—42no—
66C23×C4×C61—42no—
67C23×C4×C61C2×C414yes—
68C23×C4×C61C2311yescharacteristic
69C23×C4×C61C2311yescharacteristic
70C23×C4×C61C2311yescharacteristic
71C23×C4×C61C2311yescharacteristic
72C23×C4×C61C2311yescharacteristic
73C23×C4×C61C2311yescharacteristic
74C2×C22⋊C4×C61—24no—
75C2×C22⋊C4×C61—24no—
76C2×C22⋊C4×C61C2×C414yes—
77C2×C22⋊C4×C61C2×C414yes—
78C2×C42×C61—42no—
79C2×C42×C61—42no—
80C2×C42×C61C2×C414yes—
81C2×C42×C61C2×C414yes—
82#95×C61C2×C414yes—
83#95×C61C2×C414yes—
84C24×C61—22no—
85C24×C61—21no—
86C24×C61—21no—
87C24×C61—22no—
88C24×C61—22no—
89C24×C61—21no—
90C24×C61—21no—
91C24×C61—21no—
92C24×C61—21no—
93C24×C61C22×C412yes—
94C24×C61#4114yes—
95C24×C61C2411yescharacteristic
96C22×C4×C61—22no—
97C22×C4×C61—22no—
98C22×C4×C61—22no—
99C22×C4×C61—21no—
100C22×C4×C61—21no—
101C22×C4×C61—22no—
102C22×C4×C61—22no—
103C22×C4×C61—22no—
104C22×C4×C61—21no—
105C22×C4×C61—21no—
106C22×C4×C61—22no—
107C22×C4×C61—22no—
108C22×C4×C61—22no—
109C22×C4×C61—22no—
110C22×C4×C61—22no—
111C22×C4×C61—21no—
112C22×C4×C61—21no—
113C22×C4×C61—21no—
114C22×C4×C61—21no—
115C22×C4×C61—21no—
116C22×C4×C61—21no—
117C22×C4×C61—28no—
118C22×C4×C61—44no—
119C22×C4×C61—44no—
120C22×C4×C61—48no—
121C22×C4×C61—48no—
122C22×C4×C61C22×C412yes—
123C22×C4×C61C22×C412yes—
124C22×C4×C61C22×C412yes—
125C22×C4×C61C22×C412yes—
126C22×C4×C61C22×C412yes—
127C22×C4×C61C22×C412yes—
128C22×C4×C61#4114yes—
129C22×C4×C61#4112yes—
130C22×C4×C61#4112yes—
131C22×Dic61—22no—
132C22×Dic61—22no—
133C22×Dic61—22no—
134C22×Dic61—22no—
135C22×Dic61—22no—
136C22×Dic61—22no—
137C22×Dic61—22no—
138C22×Dic61—22no—
139C22×Dic61—22no—
140C22×Dic61—22no—
141C22×Dic61—22no—
142C22×Dic61—22no—
143C22×Dic61—22no—
144C22×Dic61—22no—
145C22×Dic61—22no—
146C22×Dic61—22no—
147C22×Dic61—22no—
148C22×Dic61—22no—
149C22×Dic61—22no—
150C22×Dic61—22no—
151C22×Dic61—22no—
152C22×Dic61—22no—
153C22×Dic61—22no—
154C22×Dic61—22no—
155C22×Dic61—22no—
156C22×Dic61—22no—
157C22×Dic61—22no—
158C22×Dic61—22no—
159C22⋊C4×C61—28no—
160C22⋊C4×C61—28no—
161C42×C61—44no—
162C42×C61—44no—
163C22⋊Dic61—48no—
164C22⋊Dic61—48no—
165C4⋊Dic61—48no—
166C23×C61—22no—
167C23×C61—22no—
168C23×C61—22no—
169C23×C61—22no—
170C23×C61—22no—
171C23×C61—22no—
172C23×C61—24no—
173C23×C61—24no—
174C23×C61—24no—
175C23×C61—24no—
176C23×C61—24no—
177C23×C61—24no—
178C23×C61—42no—
179C23×C61—42no—
180C23×C61—42no—
181C23×C61—42no—
182C23×C61—41no—
183C23×C61—42no—
184C23×C61—41no—
185C23×C61—41no—
186C23×C61—41no—
187C23×C61—42no—
188C23×C61#11714yes—
189C23×C61C23×C411yescharacteristic
190C23×C61C2×#4112yes—
191C23×C61C2×#4111yescharacteristic
192C23×C61C2×#4111yescharacteristic
193C23×C61C2×#4112yes—
194C23×C61H(5,𝔽2)12yes—
195C23×C61H(5,𝔽2)12yes—
196C23×C61H(5,𝔽2)11yescharacteristic
197C23×C61H(5,𝔽2)11yescharacteristic
198C23×C612−1+411yescharacteristic
199C23×C612−1+411yescharacteristic
200C2×C4×C61—22no—
201C2×C4×C61—22no—
202C2×C4×C61—22no—
203C2×C4×C61—22no—
204C2×C4×C61—24no—
205C2×C4×C61—24no—
206C2×C4×C61—24no—
207C2×C4×C61—24no—
208C2×C4×C61—22no—
209C2×C4×C61—22no—
210C2×C4×C61—22no—
211C2×C4×C61—22no—
212C2×C4×C61—22no—
213C2×C4×C61—22no—
214C2×C4×C61—22no—
215C2×C4×C61—22no—
216C2×C4×C61—22no—
217C2×C4×C61—22no—
218C2×C4×C61—22no—
219C2×C4×C61—22no—
220C2×C4×C61—416no—
221C2×C4×C61—42no—
222C2×C4×C61—42no—
223C2×C4×C61—42no—
224C2×C4×C61—42no—
225C2×C4×C61—42no—
226C2×C4×C61—42no—
227C2×C4×C61—42no—
228C2×C4×C61—42no—
229C2×C4×C61#11714yes—
230C2×C4×C61#11714yes—
231C2×Dic61—42no—
232C2×Dic61—42no—
233C2×Dic61—42no—
234C2×Dic61—42no—
235C2×Dic61—42no—
236C2×Dic61—42no—
237C2×Dic61—42no—
238C2×Dic61—42no—
239C2×Dic61—42no—
240C2×Dic61—42no—
241C2×Dic61—42no—
242C2×Dic61—42no—
243C2×Dic61—42no—
244C2×Dic61—42no—
245C2×Dic61—42no—
246C2×Dic61—42no—
247C2×Dic61—42no—
248C2×Dic61—42no—
249C2×Dic61—42no—
250C2×Dic61—42no—
251C2×Dic61—42no—
252C2×Dic61—42no—
253C2×Dic61—42no—
254C2×Dic61—42no—
255C2×Dic61—42no—
256C2×Dic61—42no—
257C2×Dic61—42no—
258C2×Dic61—42no—
259#14351—611nomaximal, Sylow
260C22×C61—24no—
261C22×C61—24no—
262C22×C61—24no—
263C22×C61—42no—
264C22×C61—42no—
265C22×C61—42no—
266C22×C61—42no—
267C22×C61—44no—
268C22×C61—44no—
269C22×C61—44no—
270C22×C61—44no—
271C22×C61C2×#11711yescharacteristic
272C22×C61C2×#11711yescharacteristic
273C22×C61C2×#11711yescharacteristic
274C22×C61#51811yescharacteristic
275C22×C61#51811yescharacteristic
276C22×C61#52011yescharacteristic
277C22×C61#52011yescharacteristic
278C22×C61#52912yes—
279C22×C61#52912yes—
280C22×C61#53812yes—
281C22×C61#53812yes—
282C22×C61#53812yes—
283C22×C61#53812yes—
284C22×C61#53812yes—
285C22×C61#53912yes—
286C22×C61#53912yes—
287C22×C61#53912yes—
288C22×C61#54012yes—
289C22×C61#54011yescharacteristic
290C22×C61#54111yescharacteristic
291C22×C61#54212yes—
292C22×C61#54211yescharacteristic
293C22×C61#54211yescharacteristic
294C4×C61—24no—
295C4×C61—24no—
296C4×C61—42no—
297C4×C61—42no—
298C4×C61—42no—
299C4×C61—42no—
300Dic61—82no—
301Dic61—82no—
302Dic61—82no—
303Dic61—82no—
304C2×C4×C22⋊C4D6111yescharacteristic
305#1443—612no—
306#1442—612no—
307#1440—612no—
308#1439—612no—
309C4⋊#95—611no—
310#1434—611no—
311#1434—611no—
312#1434—611no—
313C22⋊#95—611no—
314C22⋊#95—611no—
315C2×C61—44no—
316C2×C61#228311yescharacteristic
317C2×C61#231211yescharacteristic
318C2×C61#231411yescharacteristic
319C2×C61#231811yescharacteristic
320C2×C61#231811yescharacteristic
321C2×C61#231811yescharacteristic
322C2×C61#231911yescharacteristic
323C2×C61#264812yes—
324C2×C61#264912yes—
325C2×C61#265212yes—
326C2×C61#265312yes—
327C24×C4C2×D6111yescharacteristic
328C22×C4⋊C4—612no—
329C22×C22⋊C4—612no—
330C22×C22⋊C4—612no—
331C22×C22⋊C4C2×D6111yescharacteristic
332C22×C22⋊C4C2×D6111yescharacteristic
333C22×C42C2×D6111yescharacteristic
334C22×C42C2×D6111yescharacteristic
335C4×C22⋊C4Dic6118yescomplemented
336C2×#95—611no—
337C2×#95—611no—
338C2×#95—611no—
339C2×#95—611no—
340C2×#95—612no—
341C2×#95—612no—
342C2×#95—612no—
343C2×#95—612no—
344C2×#95—612no—
345C2×#95—612no—
346C2×#95—612no—
347C2×#95—612no—
348C2×#95—612no—
349C2×#95C2×D6111yescharacteristic
350C2×#95C2×D6111yescharacteristic
351C61#1435111yescharacteristic, complemented, Sylow
352C25C22×D6111yescharacteristic
353C23×C4—22no—
354C23×C4—22no—
355C23×C4—42no—
356C23×C4—42no—
357C23×C4—612no—
358C23×C4—612no—
359C23×C4—612no—
360C23×C4—612no—
361C23×C4C2×Dic6114yes—
362C23×C4C22×D6111yescharacteristic
363C23×C4C22×D6111yescharacteristic
364C23×C4C22×D6111yescharacteristic
365C23×C4C22×D6111yescharacteristic
366C23×C4C22×D6111yescharacteristic
367C23×C4C22×D6111yescharacteristic
368C2×C4⋊C4—1224no—
369C2×C4⋊C4—2442no—
370C2×C4⋊C4—2442no—
371C2×C22⋊C4—24no—
372C2×C22⋊C4—24no—
373C2×C22⋊C4—1224no—
374C2×C22⋊C4—1224no—
375C2×C22⋊C4—2442no—
376C2×C22⋊C4—2442no—
377C2×C22⋊C4—2442no—
378C2×C22⋊C4—2442no—
379C2×C22⋊C4C2×Dic6114yes—
380C2×C22⋊C4C2×Dic6114yes—
381C2×C42—42no—
382C2×C42—42no—
383C2×C42C2×Dic6114yes—
384C2×C42C2×Dic6114yes—
385#95—1222no—
386#95—1222no—
387#95—1222no—
388#95—1222no—
389#95—1224no—
390#95—2442no—
391#95—2442no—
392#95—2442no—
393#95—2442no—
394#95—2442no—
395#95—2442no—
396#95—2442no—
397#95—2442no—
398#95C2×Dic6114yes—
399#95C2×Dic6114yes—
400C24—22no—
401C24—21no—
402C24—21no—
403C24—22no—
404C24—22no—
405C24—21no—
406C24—21no—
407C24—21no—
408C24—21no—
409C24#1175822214yes—
410C24C22×Dic6112yes—
411C24C23×D6111yescharacteristic
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—21no—
432C22×C4—21no—
433C22×C4—28no—
434C22×C4—44no—
435C22×C4—44no—
436C22×C4—48no—
437C22×C4—48no—
438C22×C4—1222no—
439C22×C4—1222no—
440C22×C4—1222no—
441C22×C4—1222no—
442C22×C4—1222no—
443C22×C4—1222no—
444C22×C4—1222no—
445C22×C4—1222no—
446C22×C4—1222no—
447C22×C4—1222no—
448C22×C4—1222no—
449C22×C4—1222no—
450C22×C4—1222no—
451C22×C4—1222no—
452C22×C4—1222no—
453C22×C4—1222no—
454C22×C4—1222no—
455C22×C4—1222no—
456C22×C4—1222no—
457C22×C4—1222no—
458C22×C4—1222no—
459C22×C4—1222no—
460C22×C4—1222no—
461C22×C4—1222no—
462C22×C4—1222no—
463C22×C4—1222no—
464C22×C4—1222no—
465C22×C4—1222no—
466C22×C4#1175822212yes—
467C22×C4#1175822212yes—
468C22×C4#1175822214yes—
469C22×C4C22×Dic6112yes—
470C22×C4C22×Dic6112yes—
471C22×C4C22×Dic6112yes—
472C22×C4C22×Dic6112yes—
473C22×C4C22×Dic6112yes—
474C22×C4C22×Dic6112yes—
475C4⋊C4—2448no—
476C22⋊C4—28no—
477C22⋊C4—28no—
478C22⋊C4—2448no—
479C22⋊C4—2448no—
480C42—44no—
481C42—44no—
482C23—22no—
483C23—22no—
484C23—22no—
485C23—22no—
486C23—22no—
487C23—22no—
488C23—24no—
489C23—24no—
490C23—24no—
491C23—24no—
492C23—24no—
493C23—24no—
494C23—42no—
495C23—42no—
496C23—42no—
497C23—42no—
498C23—42no—
499C23—41no—
500C23—41no—
501C23—42no—
502C23—41no—
503C23—41no—
504C23#1451973614yes—
505C23C23×Dic6111yescharacteristic
506C23C2×#1175822212yes—
507C23C2×#1175822212yes—
508C23C2×#1175822211yescharacteristic
509C23C2×#1175822211yescharacteristic
510C23#1451982212yes—
511C23#1451982212yes—
512C23#1451982311yescharacteristic
513C23#1451982311yescharacteristic
514C23#1451982411yescharacteristic
515C23#1451982411yescharacteristic
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—24no—
521C2×C4—24no—
522C2×C4—24no—
523C2×C4—24no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—416no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—2442no—
546C2×C4—2442no—
547C2×C4—2442no—
548C2×C4—2442no—
549C2×C4—2442no—
550C2×C4—2442no—
551C2×C4—2442no—
552C2×C4—2442no—
553C2×C4—2442no—
554C2×C4—2442no—
555C2×C4—2442no—
556C2×C4—2442no—
557C2×C4—2442no—
558C2×C4—2442no—
559C2×C4—2442no—
560C2×C4—2442no—
561C2×C4—2442no—
562C2×C4—2442no—
563C2×C4—2442no—
564C2×C4—2442no—
565C2×C4—2442no—
566C2×C4—2442no—
567C2×C4—2442no—
568C2×C4—2442no—
569C2×C4—2442no—
570C2×C4—2442no—
571C2×C4—2442no—
572C2×C4—2442no—
573C2×C4#1451973614yes—
574C2×C4#1451973614yes—
575C22—24no—
576C22—24no—
577C22—24no—
578C22—42no—
579C22—42no—
580C22—44no—
581C22—44no—
582C22—44no—
583C22—44no—
584C22—42no—
585C22—42no—
586C22C2×#1451973611yescharacteristic
587C22C2×#1451973611yescharacteristic
588C22C2×#1451973611yescharacteristic
589C22#1684023711yescharacteristic
590C22#1684023711yescharacteristic
591C22#1684024511yescharacteristic
592C22#1684024511yescharacteristic
593C22#1684029612yes—
594C22#1684029612yes—
595C22#1684034512yes—
596C22#1684034512yes—
597C22#1684034512yes—
598C22#1684035012yes—
599C22#1684035012yes—
600C22#1684035412yes—
601C22#1684035412yes—
602C22#1684035412yes—
603C22#1684036312yes—
604C22#1684036711yescharacteristic
605C22#1684037411yescharacteristic
606C22#1684037712yes—
607C22#1684038311yescharacteristic
608C22#1684038311yescharacteristic
609C4—24no—
610C4—24no—
611C4—42no—
612C4—42no—
613C4—42no—
614C4—42no—
615C4—4882no—
616C4—4882no—
617C4—4882no—
618C4—4882no—
619C2—44no—
620C2#2144821711yescharacteristic
621C2#2144841711yescharacteristic
622C2#2144842811yescharacteristic
623C2#2144845211yescharacteristic
624C2#2144845211yescharacteristic
625C2#2144845211yescharacteristic
626C2#2144846211yescharacteristic
627C2#2145180112yes—
628C2#2145181612yes—
629C2#2145186112yes—
630C2#2145187612yes—
subgroupG/H
center, FrattiniC24C23×D61
commutatorC23×C61C23×C4
FittingC2×C4×C22⋊C4×C61C2
socle, cosocleC24×C61C24
prime radicalC25×C61C23
hypercenterC2×C4×C22⋊C4D61
hypocenterC61#14351
LCSG ▷ C23×C61 ▷ C61
UCSC1 ◁ C24 ◁ C2×C4×C22⋊C4

Complements

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

NH ≅ G/Ntype
1. #21444081C2semidirect
2. #21444076C2semidirect
5. #21444049C2semidirect
6. #21444046C2semidirect
7. #21444046C2semidirect
8. #21444046C2semidirect
33. C4×C22⋊C4×C61C4semidirect
335. C4×C22⋊C4Dic61semidirect
351. C61#14351semidirect

Maximal subgroups

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

subgroupindex
#214440812
#214440762
#214440682
#214440642
#214440492
#214440462
#214440422
C2×C4×C22⋊C4×C612
#1435161