Finite Group Databasebeta

← #29608413

Subgroups — #29608413

Order 15616.

2018 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×#16839642C212yesmaximal
2C2×#16839637C212yesmaximal, complemented
3#21444072C212yesmaximal, complemented
4#21444064C212yesmaximal, complemented
5#21444064C212yesmaximal, complemented
6#21444049C211yesmaximal, characteristic, complemented
7#21444046C211yesmaximal, characteristic, complemented
8#21444041C212yesmaximal, complemented
9C2×C61×#390C211yesmaximal, characteristic
10C24×Dic61C2212yes—
11C22×C4⋊Dic61C2212yes—
12C22×C4⋊Dic61C2212yes—
13C22×C22⋊Dic61C2212yes—
14C22×C22⋊Dic61C2212yes—
15C22×C22⋊Dic61C2212yes—
16#16839642—28no—
17#16839637—28no—
18C2×#14519684C2212yes—
19C2×#14519684C2212yes—
20C2×#14519684C2212yes—
21C2×#14519684C2211yescharacteristic
22C2×#14519684C2211yescharacteristic
23C2×#14519684C2212yes—
24C2×#14519684C2212yes—
25C2×#14519684C2211yescharacteristic
26C2×#14519684C2212yes—
27C2×#14519684C2211yescharacteristic
28C23×D4×C61C2211yescharacteristic
29C22×C4⋊C4×C61C2211yescharacteristic
30C22×C22⋊C4×C61C2212yes—
31C22×C22⋊C4×C61C2212yes—
32C22×C42×C61C2211yescharacteristic
33C61×#390C418yescomplemented
34C23×Dic61—24no—
35C23×Dic61—24no—
36C23×Dic61—24no—
37C23×Dic61—42no—
38C23×Dic61—42no—
39C23×Dic61—42no—
40C23×Dic61—42no—
41C23×Dic61C2312yes—
42C23×Dic61C2312yes—
43C23×Dic61C2312yes—
44C23×Dic61C2312yes—
45C2×C4⋊Dic61—24no—
46C2×C4⋊Dic61—24no—
47C2×C4⋊Dic61—24no—
48C2×C4⋊Dic61—42no—
49C2×C4⋊Dic61—42no—
50C2×C4⋊Dic61—42no—
51C2×C22⋊Dic61—24no—
52C2×C22⋊Dic61—24no—
53C2×C22⋊Dic61—24no—
54C2×C22⋊Dic61—24no—
55C2×C22⋊Dic61—24no—
56C2×C22⋊Dic61—42no—
57C2×C22⋊Dic61—42no—
58C2×C22⋊Dic61—42no—
59C2×C22⋊Dic61—42no—
60#14519684—24no—
61#14519684—24no—
62#14519684—22no—
63#14519684—24no—
64#14519684—41no—
65#14519684—42no—
66#14519684—42no—
67#14519684—41no—
68#14519684—42no—
69#14519684D414yescomplemented
70C25×C61C2312yes—
71C22×D4×C61—42no—
72C22×D4×C61—42no—
73C22×D4×C61C2×C414yes—
74C22×D4×C61D414yes—
75C22×D4×C61Q814yes—
76C23×C4×C61C2312yes—
77C23×C4×C61C2311yescharacteristic
78C23×C4×C61C2311yescharacteristic
79C23×C4×C61C2311yescharacteristic
80C2×C4⋊C4×C61—42no—
81C2×C4⋊C4×C61C2×C414yescomplemented
82C2×C22⋊C4×C61—24no—
83C2×C22⋊C4×C61—24no—
84C2×C22⋊C4×C61—42no—
85C2×C22⋊C4×C61—42no—
86C2×C22⋊C4×C61C2×C418yescomplemented
87C2×C22⋊C4×C61C2×C418yescomplemented
88C2×C42×C61—42no—
89C2×C42×C61C2×C414yescomplemented
90C24×C61—22no—
91C24×C61—22no—
92C24×C61—22no—
93C24×C61—22no—
94C24×C61—22no—
95C24×C61—22no—
96C24×C61—22no—
97C24×C61—22no—
98C24×C61—22no—
99C24×C61—22no—
100C24×C61—22no—
101C24×C61—22no—
102C24×C61C22×C414yes—
103C24×C61C2×D414yes—
104C24×C61C2×Q814yes—
105C24×C61C2411yescharacteristic
106C2×D4×C61—44no—
107C2×D4×C61—44no—
108C2×D4×C61—48no—
109C2×D4×C61—48no—
110C2×D4×C61C4⋊C4116yescomplemented
111C22×C4×C61—22no—
112C22×C4×C61—22no—
113C22×C4×C61—22no—
114C22×C4×C61—22no—
115C22×C4×C61—22no—
116C22×C4×C61—22no—
117C22×C4×C61—22no—
118C22×C4×C61—22no—
119C22×C4×C61—22no—
120C22×C4×C61—22no—
121C22×C4×C61—22no—
122C22×C4×C61—22no—
123C22×C4×C61—21no—
124C22×C4×C61—21no—
125C22×C4×C61C22×C412yes—
126C22×C4×C61C22×C414yes—
127C22×C4×C61C22×C412yes—
128C22×C4×C61C22×C412yes—
129C22×C4×C61C2×D412yes—
130C22×C4×C61C2×D412yes—
131C22×C4×C61C2×Q812yes—
132C22×C4×C61#4112yes—
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×Dic61—48no—
160C22×Dic61—48no—
161C22×Dic61—48no—
162C22×Dic61—48no—
163C22×Dic61—48no—
164C22×Dic61—48no—
165C22×Dic61—48no—
166C22×Dic61C2×D414yes—
167C22×Dic61#4114yes—
168C4⋊C4×C61—44no—
169C22⋊C4×C61—216no—
170C22⋊C4×C61—48no—
171C42×C61—44no—
172C22⋊Dic61—48no—
173C22⋊Dic61—48no—
174C22⋊Dic61—48no—
175C4⋊Dic61—48no—
176C4⋊Dic61—48no—
177C23×C61—24no—
178C23×C61—24no—
179C23×C61—24no—
180C23×C61—24no—
181C23×C61—24no—
182C23×C61—24no—
183C23×C61—24no—
184C23×C61—24no—
185C23×C61—24no—
186C23×C61—24no—
187C23×C61—24no—
188C23×C61—24no—
189C23×C61—24no—
190C23×C61—24no—
191C23×C61—24no—
192C23×C61—24no—
193C23×C61—24no—
194C23×C61—24no—
195C23×C61—42no—
196C23×C61—42no—
197C23×C61—42no—
198C23×C61—42no—
199C23×C61—42no—
200C23×C61—42no—
201C23×C61—42no—
202C23×C61—42no—
203C23×C61—42no—
204C23×C61—42no—
205C23×C61—42no—
206C23×C61—42no—
207C23×C61—42no—
208C23×C61—42no—
209C23×C61—42no—
210C23×C61—42no—
211C23×C61C2×C4⋊C418yes—
212C23×C61C23×C411yescharacteristic
213C23×C61C22×D411yescharacteristic
214C23×C61C22×D411yescharacteristic
215C23×C61C22×Q811yescharacteristic
216C23×C61C2×#4111yescharacteristic
217C23×C61C2×#4112yes—
218C23×C61H(5,𝔽2)12yes—
219C23×C61H(5,𝔽2)12yes—
220C23×C61H(5,𝔽2)12yes—
221C23×C61H(5,𝔽2)11yescharacteristic
222C23×C612−1+411yescharacteristic
223D4×C61—416no—
224C2×C4×C61—22no—
225C2×C4×C61—22no—
226C2×C4×C61—22no—
227C2×C4×C61—22no—
228C2×C4×C61—24no—
229C2×C4×C61—24no—
230C2×C4×C61—24no—
231C2×C4×C61—24no—
232C2×C4×C61—24no—
233C2×C4×C61—22no—
234C2×C4×C61—22no—
235C2×C4×C61—24no—
236C2×C4×C61—24no—
237C2×C4×C61—22no—
238C2×C4×C61—42no—
239C2×C4×C61—42no—
240C2×C4×C61—42no—
241C2×C4×C61—42no—
242C2×C4×C61—42no—
243C2×C4×C61—42no—
244C2×C4×C61C2×C4⋊C414yes—
245C2×C4×C61C4×D414yes—
246C2×Dic61—24no—
247C2×Dic61—24no—
248C2×Dic61—24no—
249C2×Dic61—24no—
250C2×Dic61—24no—
251C2×Dic61—24no—
252C2×Dic61—432no—
253C2×Dic61—42no—
254C2×Dic61—42no—
255C2×Dic61—42no—
256C2×Dic61—42no—
257C2×Dic61—42no—
258C2×Dic61—42no—
259C2×Dic61—42no—
260C2×Dic61—42no—
261C2×Dic61—42no—
262C2×Dic61—42no—
263C2×Dic61—42no—
264C2×Dic61—42no—
265C2×Dic61—42no—
266C2×Dic61—42no—
267C2×Dic61—42no—
268C2×Dic61—42no—
269C2×Dic61—42no—
270C2×Dic61—42no—
271C2×Dic61—42no—
272C2×Dic61—42no—
273C2×Dic61—42no—
274C2×Dic61—42no—
275#14438—611nomaximal, Sylow
276C22×C61—28no—
277C22×C61—28no—
278C22×C61—28no—
279C22×C61—44no—
280C22×C61—44no—
281C22×C61—44no—
282C22×C61—44no—
283C22×C61—44no—
284C22×C61—44no—
285C22×C61—44no—
286C22×C61—44no—
287C22×C61—44no—
288C22×C61—44no—
289C22×C61—44no—
290C22×C61—44no—
291C22×C61C22×C4⋊C411yescharacteristic
292C22×C61C2×C4×D411yescharacteristic
293C22×C61C4×#4111yescharacteristic
294C22×C61#51811yescharacteristic
295C22×C61#51811yescharacteristic
296C22×C61#51811yescharacteristic
297C22×C61#52011yescharacteristic
298C22×C61#53412yes—
299C22×C61#53512yes—
300C22×C61#53812yes—
301C22×C61#54012yes—
302C22×C61#54212yes—
303C22×C61#54312yes—
304C22×C61D4211yescharacteristic
305C22×C61#54612yes—
306C22×C61#54612yes—
307C22×C61#54711yescharacteristic
308C22×C61D4×Q811yescharacteristic
309C22×C61#55312yes—
310C22×C61#55411yescharacteristic
311C22×C61#55412yes—
312C22×C61#56012yes—
313C22×C61#56212yes—
314C4×C61—24no—
315C4×C61—24no—
316C4×C61—44no—
317C4×C61—42no—
318Dic61—44no—
319Dic61—44no—
320Dic61—82no—
321Dic61—82no—
322C2×#390D6111yescharacteristic
323C2×#386—612no—
324C2×#385—612no—
325#1441—612no—
326#1439—612no—
327#1439—612no—
328C4⋊#95—611no—
329#1434—611no—
330C22⋊#95—612no—
331C2×C61—48no—
332C2×C61#231511yescharacteristic
333C2×C61#232811yescharacteristic
334C2×C61#233711yescharacteristic
335C2×C61D4×C4⋊C411yescharacteristic
336C2×C61#234911yescharacteristic
337C2×C61#236411yescharacteristic
338C2×C61#237711yescharacteristic
339C2×C61#267812yes—
340C2×C61#269212yes—
341C2×C61#270112yes—
342C2×C61#278012yes—
343C23×D4C2×D6111yescharacteristic
344C24×C4—612no—
345C22×C4⋊C4—612no—
346C22×C4⋊C4—612no—
347C22×C4⋊C4C2×D6111yescharacteristic
348C22×C22⋊C4—612no—
349C22×C22⋊C4—612no—
350C22×C22⋊C4—612no—
351C22×C22⋊C4C2×D6112yes—
352C22×C22⋊C4C2×D6112yes—
353C22×C42C2×D6111yescharacteristic
354#390Dic6118yescomplemented
355#386—1228no—
356#385—1228no—
357C2×#95—611no—
358C2×#95—612no—
359C2×#95—611no—
360C2×#95—612no—
361C2×#95—612no—
362C2×#95—611no—
363C2×#95—612no—
364C2×#95—612no—
365C2×#95—611no—
366C2×#95—612no—
367C61#1443811yescharacteristic, complemented, Sylow
368C25C22×D6112yes—
369C22×D4—42no—
370C22×D4—42no—
371C22×D4Dic12214yes—
372C22×D4D24414yes—
373C22×D4C2×Dic6114yes—
374C23×C4—612no—
375C23×C4—612no—
376C23×C4—612no—
377C23×C4—612no—
378C23×C4—1224no—
379C23×C4—1224no—
380C23×C4—1224no—
381C23×C4—2442no—
382C23×C4—2442no—
383C23×C4—2442no—
384C23×C4—2442no—
385C23×C4C22×D6112yes—
386C23×C4C22×D6111yescharacteristic
387C23×C4C22×D6111yescharacteristic
388C23×C4C22×D6111yescharacteristic
389C2×C4⋊C4—42no—
390C2×C4⋊C4—1224no—
391C2×C4⋊C4—1224no—
392C2×C4⋊C4—1224no—
393C2×C4⋊C4—2442no—
394C2×C4⋊C4—2442no—
395C2×C4⋊C4—2442no—
396C2×C4⋊C4C2×Dic6114yescomplemented
397C2×C22⋊C4—24no—
398C2×C22⋊C4—24no—
399C2×C22⋊C4—42no—
400C2×C22⋊C4—42no—
401C2×C22⋊C4—1224no—
402C2×C22⋊C4—1224no—
403C2×C22⋊C4—1224no—
404C2×C22⋊C4—1224no—
405C2×C22⋊C4—1224no—
406C2×C22⋊C4—2442no—
407C2×C22⋊C4—2442no—
408C2×C22⋊C4—2442no—
409C2×C22⋊C4—2442no—
410C2×C22⋊C4C2×Dic6118yescomplemented
411C2×C22⋊C4C2×Dic6118yescomplemented
412C2×C42—42no—
413C2×C42C2×Dic6114yescomplemented
414#95—614no—
415#95—1224no—
416#95—1224no—
417#95—1224no—
418#95—1222no—
419#95—2442no—
420#95—2442no—
421#95—2442no—
422#95—2441no—
423#95—2441no—
424C24—22no—
425C24—22no—
426C24—22no—
427C24—22no—
428C24—22no—
429C24—22no—
430C24—22no—
431C24—22no—
432C24—22no—
433C24—22no—
434C24—22no—
435C24—22no—
436C24C2×Dic12214yes—
437C24C2×D24414yes—
438C24C22×Dic6114yes—
439C24C23×D6111yescharacteristic
440C2×D4—48no—
441C2×D4—48no—
442C2×D4—44no—
443C2×D4—44no—
444C2×D4C4⋊Dic61116yescomplemented
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—21no—
458C22×C4—21no—
459C22×C4—614no—
460C22×C4—614no—
461C22×C4—1222no—
462C22×C4—1222no—
463C22×C4—1222no—
464C22×C4—1222no—
465C22×C4—1222no—
466C22×C4—1222no—
467C22×C4—1222no—
468C22×C4—1222no—
469C22×C4—1222no—
470C22×C4—1222no—
471C22×C4—1222no—
472C22×C4—1222no—
473C22×C4—1222no—
474C22×C4—1222no—
475C22×C4—1222no—
476C22×C4—1222no—
477C22×C4—1222no—
478C22×C4—1222no—
479C22×C4—1222no—
480C22×C4—1222no—
481C22×C4—1222no—
482C22×C4—1222no—
483C22×C4—1222no—
484C22×C4—1222no—
485C22×C4—1222no—
486C22×C4—1222no—
487C22×C4—2448no—
488C22×C4—2448no—
489C22×C4—2448no—
490C22×C4—2448no—
491C22×C4—2448no—
492C22×C4—2448no—
493C22×C4—2448no—
494C22×C4C2×Dic12212yes—
495C22×C4C2×D24412yes—
496C22×C4D4×D6112yes—
497C22×C4#1175822412yes—
498C22×C4C22×Dic6112yes—
499C22×C4C22×Dic6114yes—
500C22×C4C22×Dic6112yes—
501C22×C4C22×Dic6112yes—
502C4⋊C4—44no—
503C4⋊C4—2448no—
504C4⋊C4—2448no—
505C22⋊C4—216no—
506C22⋊C4—48no—
507C22⋊C4—2448no—
508C22⋊C4—2448no—
509C22⋊C4—2448no—
510C42—44no—
511C23—24no—
512C23—24no—
513C23—24no—
514C23—24no—
515C23—24no—
516C23—24no—
517C23—24no—
518C23—24no—
519C23—24no—
520C23—24no—
521C23—24no—
522C23—24no—
523C23—24no—
524C23—24no—
525C23—24no—
526C23—24no—
527C23—24no—
528C23—24no—
529C23—42no—
530C23—42no—
531C23—42no—
532C23—42no—
533C23—42no—
534C23—42no—
535C23—42no—
536C23—42no—
537C23—42no—
538C23—42no—
539C23—42no—
540C23—42no—
541C23—42no—
542C23—42no—
543C23—42no—
544C23—42no—
545C23C2×C4⋊Dic6118yes—
546C23C23×Dic6111yescharacteristic
547C23C2×D4×D6111yescharacteristic
548C23C22×D24411yescharacteristic
549C23C22×Dic12211yescharacteristic
550C23#41×D6112yes—
551C23C2×#1175822411yescharacteristic
552C23#1451982212yes—
553C23#1451982212yes—
554C23#1451982212yes—
555C23#1451982311yescharacteristic
556C23#1451982411yescharacteristic
557D4—416no—
558C2×C4—24no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—24no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—24no—
570C2×C4—24no—
571C2×C4—22no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—1224no—
579C2×C4—1224no—
580C2×C4—1224no—
581C2×C4—1224no—
582C2×C4—1224no—
583C2×C4—1224no—
584C2×C4—2442no—
585C2×C4—2442no—
586C2×C4—2442no—
587C2×C4—2442no—
588C2×C4—2442no—
589C2×C4—2442no—
590C2×C4—2442no—
591C2×C4—2442no—
592C2×C4—2442no—
593C2×C4—2442no—
594C2×C4—2442no—
595C2×C4—2442no—
596C2×C4—2442no—
597C2×C4—2442no—
598C2×C4—24432no—
599C2×C4—2442no—
600C2×C4—2442no—
601C2×C4—2442no—
602C2×C4—2442no—
603C2×C4—2442no—
604C2×C4—2442no—
605C2×C4—2442no—
606C2×C4—2442no—
607C2×C4C2×C4⋊Dic6114yes—
608C2×C4D4×Dic6114yes—
609C22—28no—
610C22—28no—
611C22—28no—
612C22—44no—
613C22—44no—
614C22—44no—
615C22—44no—
616C22—44no—
617C22—44no—
618C22—44no—
619C22—44no—
620C22—44no—
621C22—44no—
622C22—44no—
623C22—44no—
624C22C22×C4⋊Dic6111yescharacteristic
625C22C2×D4×Dic6111yescharacteristic
626C22#41×Dic6111yescharacteristic
627C22#1684023711yescharacteristic
628C22#1684023711yescharacteristic
629C22#1684023711yescharacteristic
630C22#1684024511yescharacteristic
631C22#1684032412yes—
632C22#1684032512yes—
633C22#1684034612yes—
634C22#1684036612yes—
635C22#1684037612yes—
636C22#1684038812yes—
637C22D4×D24411yescharacteristic
638C22#1684040112yes—
639C22#1684040712yes—
640C22#1684041611yescharacteristic
641C22D4×Dic12211yescharacteristic
642C22#1684045512yes—
643C22#1684046911yescharacteristic
644C22#1684047112yes—
645C22#1684050012yes—
646C22#1684050712yes—
647C4—24no—
648C4—24no—
649C4—42no—
650C4—44no—
651C4—4882no—
652C4—4882no—
653C4—2444no—
654C4—2444no—
655C2—48no—
656C2#2144843311yescharacteristic
657C2#2144854511yescharacteristic
658C2#2144863411yescharacteristic
659C2D4×C4⋊Dic6111yescharacteristic
660C2#2144872911yescharacteristic
661C2#2144887111yescharacteristic
662C2#2144897611yescharacteristic
663C2#2145213912yes—
664C2#2145231212yes—
665C2#2145243012yes—
666C2#2145346812yes—
subgroupG/H
center, FrattiniC24C23×D61
commutatorC23×C61C23×C4
FittingC2×C61×#390C2
socle, cosocleC24×C61C24
prime radicalC23×D4×C61C22
hypercenterC2×#390D61
hypocenterC61#14438
LCSG ▷ C23×C61 ▷ C61
UCSC1 ◁ C24 ◁ C2×#390

Complements

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

NH ≅ G/Ntype
2. C2×#16839637C2semidirect
3. #21444072C2semidirect
4. #21444064C2semidirect
5. #21444064C2semidirect
6. #21444049C2semidirect
7. #21444046C2semidirect
8. #21444041C2semidirect
33. C61×#390C4semidirect
69. #14519684D4semidirect
81. C2×C4⋊C4×C61C2×C4semidirect
86. C2×C22⋊C4×C61C2×C4semidirect
87. C2×C22⋊C4×C61C2×C4semidirect
89. C2×C42×C61C2×C4semidirect
110. C2×D4×C61C4⋊C4semidirect
354. #390Dic61semidirect
367. C61#14438semidirect
396. C2×C4⋊C4C2×Dic61semidirect
410. C2×C22⋊C4C2×Dic61semidirect
411. C2×C22⋊C4C2×Dic61semidirect
413. C2×C42C2×Dic61semidirect
444. C2×D4C4⋊Dic61semidirect

Maximal subgroups

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

subgroupindex
C2×#168396422
C2×#168396372
#214440722
#214440642
#214440492
#214440462
#214440412
C2×C61×#3902
#1443861