Finite Group Databasebeta

← #29602162

Subgroups — #29602162

Order 15616.

1962 conjugacy classes of subgroups.

The subgroup lattice has 678 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839635C212yesmaximal
2C2×#16839626C211yesmaximal, characteristic, complemented
3C2×C4×C22⋊Dic61C211yesmaximal, characteristic
4#21444077C212yesmaximal, complemented
5#21444074C212yesmaximal, complemented
6#21444047C211yesmaximal, characteristic, complemented
7#21444044C211yesmaximal, characteristic, complemented
8#21444043C211yesmaximal, characteristic, complemented
9#21444042C211yesmaximal, characteristic
10#21444042C212yesmaximal
11C22×#95×C61C211yesmaximal, characteristic
12C22×#11758196C2212yes—
13C22×#11758196C2212yes—
14C22×C22⋊Dic61C2212yes—
15C22×C22⋊Dic61C2211yescharacteristic
16C22×C22⋊Dic61C2211yescharacteristic
17C22×C4×Dic61C2211yescharacteristic
18C22×C4×Dic61C2211yescharacteristic
19#16839635—28no—
20#16839626—24no—
21C4×C22⋊Dic61C418yescomplemented
22C2×#14519684C2211yescharacteristic
23C2×#14519684C2212yes—
24C2×#14519684C2212yes—
25C2×#14519684C2212yes—
26C2×#14519684C2211yescharacteristic
27C2×#14519684C2212yes—
28C2×#14519684C2212yes—
29C2×#14519684C2212yes—
30C2×#14519684C2212yes—
31C2×#14519684C2211yescharacteristic
32C2×#14519684C2211yescharacteristic
33C24×C4×C61C2212yes—
34C24×C4×C61C2211yescharacteristic
35C2×#95×C61—24no—
36C2×#95×C61C2211yescharacteristic
37C2×#95×C61C2212yes—
38C2×#95×C61C2211yescharacteristic
39C23×Dic61C2312yes—
40C23×Dic61C2312yes—
41C23×Dic61C2311yescharacteristic
42C23×Dic61C2311yescharacteristic
43C23×Dic61C2311yescharacteristic
44C23×Dic61C2311yescharacteristic
45C2×#11758196—24no—
46C2×#11758196—24no—
47C2×#11758196—24no—
48C2×#11758196—24no—
49C2×#11758196—42no—
50C2×#11758196—42no—
51C2×C22⋊Dic61—24no—
52C2×C22⋊Dic61—24no—
53C2×C22⋊Dic61—24no—
54C2×C22⋊Dic61—24no—
55C2×C22⋊Dic61C2×C414yes—
56C2×C22⋊Dic61C2×C414yes—
57C2×C22⋊Dic61D414yes—
58C2×C22⋊Dic61Q814yes—
59C2×C4×Dic61—22no—
60C2×C4×Dic61—22no—
61C2×C4×Dic61—42no—
62C2×C4×Dic61C2×C414yescomplemented
63C2×C4×Dic61C2×C414yescomplemented
64#14519684—24no—
65#14519684—22no—
66#14519684—24no—
67#14519684—22no—
68#14519684—24no—
69#14519684—24no—
70#14519684—42no—
71#14519684—42no—
72#14519684—42no—
73#14519684C2×C414yescomplemented
74#14519684C2×C414yescomplemented
75C25×C61C2311yescharacteristic
76C23×C4×C61—24no—
77C23×C4×C61—24no—
78C23×C4×C61—24no—
79C23×C4×C61—22no—
80C23×C4×C61—24no—
81C23×C4×C61—22no—
82C23×C4×C61—24no—
83C23×C4×C61—24no—
84C23×C4×C61—42no—
85C23×C4×C61—42no—
86C23×C4×C61—42no—
87C23×C4×C61C2×C414yes—
88C23×C4×C61C2311yescharacteristic
89C23×C4×C61C2312yes—
90C23×C4×C61C2312yes—
91C23×C4×C61C2311yescharacteristic
92#95×C61—24no—
93#95×C61—22no—
94#95×C61D414yes—
95C24×C61—21no—
96C24×C61—21no—
97C24×C61—22no—
98C24×C61—22no—
99C24×C61—21no—
100C24×C61—21no—
101C24×C61C22×C412yes—
102C24×C61C2×D412yes—
103C24×C61C2×Q812yes—
104C24×C61#4114yes—
105C24×C61#4114yes—
106C24×C61C2411yescharacteristic
107C22×C4×C61—22no—
108C22×C4×C61—22no—
109C22×C4×C61—216no—
110C22×C4×C61—28no—
111C22×C4×C61—21no—
112C22×C4×C61—22no—
113C22×C4×C61—22no—
114C22×C4×C61—22no—
115C22×C4×C61—22no—
116C22×C4×C61—21no—
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—22no—
124C22×C4×C61—22no—
125C22×C4×C61—22no—
126C22×C4×C61—22no—
127C22×C4×C61—28no—
128C22×C4×C61—44no—
129C22×C4×C61—48no—
130C22×C4×C61—48no—
131C22×C4×C61—48no—
132C22×C4×C61—48no—
133C22×C4×C61—48no—
134C22×C4×C61—48no—
135C22×C4×C61—48no—
136C22×C4×C61—48no—
137C22×C4×C61C22×C412yes—
138C22×C4×C61C22×C412yes—
139C22×C4×C61C2×D414yes—
140C22×C4×C61C2×D412yes—
141C22×C4×C61#4114yes—
142C22×C4×C61#4112yes—
143C22×Dic61—21no—
144C22×Dic61—21no—
145C22×Dic61—22no—
146C22×Dic61—22no—
147C22×Dic61—21no—
148C22×Dic61—21no—
149C22×Dic61—21no—
150C22×Dic61—22no—
151C22×Dic61—21no—
152C22×Dic61—22no—
153C22×Dic61—22no—
154C22×Dic61—21no—
155C22×Dic61—22no—
156C22×Dic61—21no—
157C22×Dic61—22no—
158C22×Dic61—22no—
159C22×Dic61—22no—
160C22×Dic61—22no—
161C22×Dic61—22no—
162C22×Dic61—22no—
163C22×Dic61—22no—
164C22×Dic61—22no—
165C22×Dic61—22no—
166C22×Dic61—22no—
167C22×Dic61—22no—
168C22×Dic61—22no—
169C22×Dic61—22no—
170C22×Dic61—22no—
171C22×Dic61C22×C412yes—
172C22×Dic61C22×C412yes—
173C22×Dic61C22×C412yes—
174C22×Dic61C22×C412yes—
175C22×Dic61C2×D412yes—
176C22×Dic61C2×D412yes—
177C22×Dic61C2×Q812yes—
178C22×Dic61C2×Q812yes—
179C22⋊Dic61—28no—
180C22⋊Dic61—48no—
181C22⋊Dic61C4⋊C4116yes—
182#11758196—48no—
183#11758196—48no—
184C4×Dic61—28no—
185C4×Dic61—44no—
186C23×C61—22no—
187C23×C61—22no—
188C23×C61—22no—
189C23×C61—22no—
190C23×C61—24no—
191C23×C61—24no—
192C23×C61—24no—
193C23×C61—24no—
194C23×C61—22no—
195C23×C61—24no—
196C23×C61—24no—
197C23×C61—22no—
198C23×C61—22no—
199C23×C61—24no—
200C23×C61—24no—
201C23×C61—24no—
202C23×C61—22no—
203C23×C61—24no—
204C23×C61C2×C4⋊C414yes—
205C23×C61#11714yes—
206C23×C61#11714yes—
207C23×C61#12418yes—
208C23×C61#12518yes—
209C23×C61C23×C411yescharacteristic
210C23×C61C22×D411yescharacteristic
211C23×C61C22×D411yescharacteristic
212C23×C61C22×Q811yescharacteristic
213C23×C61C2×#4112yes—
214C23×C61C2×#4112yes—
215C23×C61C2×#4111yescharacteristic
216C23×C61C2×#4112yes—
217C23×C61H(5,𝔽2)12yes—
218C23×C61H(5,𝔽2)11yescharacteristic
219C23×C612−1+411yescharacteristic
220C2×C4×C61—22no—
221C2×C4×C61—22no—
222C2×C4×C61—24no—
223C2×C4×C61—24no—
224C2×C4×C61—22no—
225C2×C4×C61—22no—
226C2×C4×C61—22no—
227C2×C4×C61—24no—
228C2×C4×C61—24no—
229C2×C4×C61—24no—
230C2×C4×C61—24no—
231C2×C4×C61—24no—
232C2×C4×C61—24no—
233C2×C4×C61—416no—
234C2×C4×C61—432no—
235C2×C4×C61—42no—
236C2×C4×C61—42no—
237C2×C4×C61—42no—
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×C61—42no—
245C2×C4×C61C4×D414yes—
246C2×Dic61—22no—
247C2×Dic61—22no—
248C2×Dic61—22no—
249C2×Dic61—22no—
250C2×Dic61—22no—
251C2×Dic61—24no—
252C2×Dic61—22no—
253C2×Dic61—22no—
254C2×Dic61—22no—
255C2×Dic61—24no—
256C2×Dic61—22no—
257C2×Dic61—22no—
258C2×Dic61—22no—
259C2×Dic61—22no—
260C2×Dic61—22no—
261C2×Dic61—22no—
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—
275C2×Dic61—42no—
276C2×Dic61—42no—
277C2×Dic61—42no—
278C2×Dic61—42no—
279C2×Dic61—42no—
280C2×Dic61C2×C4⋊C414yes—
281C2×Dic61C2×C4⋊C414yes—
282#13896—611nomaximal, Sylow
283C22×C61—24no—
284C22×C61—24no—
285C22×C61—24no—
286C22×C61—24no—
287C22×C61—24no—
288C22×C61—28no—
289C22×C61—28no—
290C22×C61—28no—
291C22×C61—28no—
292C22×C61—24no—
293C22×C61#38218yescomplemented
294C22×C61C22×C4⋊C411yescharacteristic
295C22×C61C2×#11711yescharacteristic
296C22×C61C2×#11711yescharacteristic
297C22×C61C2×C4×D411yescharacteristic
298C22×C61C4×#4111yescharacteristic
299C22×C61#51811yescharacteristic
300C22×C61#52011yescharacteristic
301C22×C61#52512yes—
302C22×C61C2×#12412yes—
303C22×C61C2×#12512yes—
304C22×C61#52912yes—
305C22×C61#53812yes—
306C22×C61#53912yes—
307C22×C61D4211yescharacteristic
308C22×C61#54612yes—
309C22×C61#54612yes—
310C22×C61#54711yescharacteristic
311C22×C61D4×Q811yescharacteristic
312C22×C61#55112yes—
313C22×C61#55112yes—
314C22×C61#55212yes—
315C22×C61#55411yescharacteristic
316C22×C61#55412yes—
317C4×C61—24no—
318C4×C61—42no—
319C4×C61—44no—
320C4×C61—44no—
321Dic61—24no—
322Dic61—24no—
323Dic61—42no—
324Dic61—42no—
325Dic61—82no—
326Dic61—82no—
327C2×#385—612no—
328C2×#382—611no—
329C2×C4×C22⋊C4—611no—
330C22×#95D6111yescharacteristic
331#1442—612no—
332#1442—612no—
333C4⋊#95—611no—
334#1434—611no—
335#1434—611no—
336C22⋊#95—612no—
337C22⋊#95—611no—
338C2×C61—28no—
339C2×C61C2×#38211yescharacteristic
340C2×C61#229011yescharacteristic
341C2×C61#231811yescharacteristic
342C2×C61#234211yescharacteristic
343C2×C61#234311yescharacteristic
344C2×C61D4×C4⋊C411yescharacteristic
345C2×C61#234911yescharacteristic
346C2×C61#247212yes—
347C2×C61#247312yes—
348C2×C61#248812yes—
349C2×C61#250512yes—
350C24×C4C2×D6112yes—
351C24×C4C2×D6111yescharacteristic
352C22×C4⋊C4—612no—
353C22×C4⋊C4—612no—
354C22×C22⋊C4—611no—
355C22×C22⋊C4—611no—
356C22×C22⋊C4—612no—
357C22×C42—611no—
358C22×C42—611no—
359#385—1228no—
360#382—1224no—
361C4×C22⋊C4—618no—
362C2×#95—24no—
363C2×#95—611no—
364C2×#95—611no—
365C2×#95—611no—
366C2×#95—612no—
367C2×#95—611no—
368C2×#95—612no—
369C2×#95—612no—
370C2×#95—612no—
371C2×#95—612no—
372C2×#95—612no—
373C2×#95—612no—
374C2×#95C2×D6111yescharacteristic
375C2×#95C2×D6112yes—
376C2×#95C2×D6111yescharacteristic
377C61#1389611yescharacteristic, complemented, Sylow
378C25C22×D6111yescharacteristic
379C23×C4—24no—
380C23×C4—24no—
381C23×C4—24no—
382C23×C4—22no—
383C23×C4—24no—
384C23×C4—22no—
385C23×C4—24no—
386C23×C4—24no—
387C23×C4—42no—
388C23×C4—42no—
389C23×C4—42no—
390C23×C4—611no—
391C23×C4—611no—
392C23×C4—611no—
393C23×C4—611no—
394C23×C4—612no—
395C23×C4—612no—
396C23×C4C4×D6114yes—
397C23×C4C22×D6111yescharacteristic
398C23×C4C22×D6112yes—
399C23×C4C22×D6112yes—
400C23×C4C22×D6111yescharacteristic
401C2×C4⋊C4—1224no—
402C2×C4⋊C4—1224no—
403C2×C4⋊C4—1224no—
404C2×C4⋊C4—1224no—
405C2×C4⋊C4—2442no—
406C2×C4⋊C4—2442no—
407C2×C22⋊C4—614no—
408C2×C22⋊C4—614no—
409C2×C22⋊C4—614no—
410C2×C22⋊C4—614no—
411C2×C22⋊C4—1224no—
412C2×C22⋊C4—1224no—
413C2×C22⋊C4—1224no—
414C2×C22⋊C4—1224no—
415C2×C42—614no—
416C2×C42—614no—
417C2×C42—1222no—
418C2×C42—1222no—
419C2×C42—2442no—
420#95—24no—
421#95—22no—
422#95—614no—
423#95—614no—
424#95—1224no—
425#95—1224no—
426#95—1224no—
427#95—1222no—
428#95—1224no—
429#95—1222no—
430#95—2442no—
431#95—2442no—
432#95—2442no—
433#95C22⋊D6114yes—
434C24—21no—
435C24—21no—
436C24—22no—
437C24—22no—
438C24—21no—
439C24—21no—
440C24C2×C4×D6112yes—
441C24#1175822214yes—
442C24D4×D6112yes—
443C24#1175822414yes—
444C24Q8×D6112yes—
445C24C23×D6111yescharacteristic
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—216no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—21no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—21no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—28no—
466C22×C4—28no—
467C22×C4—44no—
468C22×C4—48no—
469C22×C4—48no—
470C22×C4—48no—
471C22×C4—48no—
472C22×C4—48no—
473C22×C4—48no—
474C22×C4—48no—
475C22×C4—48no—
476C22×C4—612no—
477C22×C4—612no—
478C22×C4—612no—
479C22×C4—612no—
480C22×C4—612no—
481C22×C4—612no—
482C22×C4—612no—
483C22×C4—612no—
484C22×C4—1221no—
485C22×C4—1221no—
486C22×C4—1222no—
487C22×C4—1222no—
488C22×C4—1221no—
489C22×C4—1221no—
490C22×C4—1221no—
491C22×C4—1222no—
492C22×C4—1221no—
493C22×C4—1222no—
494C22×C4—1222no—
495C22×C4—1221no—
496C22×C4—1222no—
497C22×C4—1221no—
498C22×C4—1222no—
499C22×C4—1222no—
500C22×C4—1222no—
501C22×C4—1222no—
502C22×C4—1222no—
503C22×C4—1222no—
504C22×C4—1222no—
505C22×C4—1222no—
506C22×C4—1222no—
507C22×C4—1222no—
508C22×C4—1222no—
509C22×C4—1222no—
510C22×C4—1222no—
511C22×C4—1222no—
512C22×C4C2×C4×D6112yes—
513C22×C4C2×C4×D6112yes—
514C22×C4#1175822214yes—
515C22×C4#1175822212yes—
516C22×C4C2×C22⋊D6114yes—
517C22×C4C2×C22⋊D6112yes—
518C4⋊C4—2448no—
519C4⋊C4—2448no—
520C22⋊C4—6116no—
521C22⋊C4—1228no—
522C22⋊C4—2448no—
523C42—1228no—
524C42—2444no—
525C23—22no—
526C23—24no—
527C23—22no—
528C23—22no—
529C23—22no—
530C23—22no—
531C23—24no—
532C23—24no—
533C23—24no—
534C23—24no—
535C23—22no—
536C23—24no—
537C23—24no—
538C23—22no—
539C23—22no—
540C23—24no—
541C23—24no—
542C23—24no—
543C23C4⋊C4×D6114yes—
544C23#1451973314yes—
545C23#1451973414yes—
546C23#1451976318yes—
547C23#1451976718yes—
548C23C22×C4×D6111yescharacteristic
549C23C2×D4×D6111yescharacteristic
550C23C22×C22⋊D6111yescharacteristic
551C23C2×Q8×D6111yescharacteristic
552C23C2×#1175822412yes—
553C23C2×#1175822211yescharacteristic
554C23C2×#1175822212yes—
555C23C2×#1175822212yes—
556C23#1451982212yes—
557C23#1451982211yescharacteristic
558C23#1451982511yescharacteristic
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—24no—
562C2×C4—24no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—24no—
570C2×C4—24no—
571C2×C4—24no—
572C2×C4—416no—
573C2×C4—432no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—614no—
585C2×C4—614no—
586C2×C4—1222no—
587C2×C4—1224no—
588C2×C4—1222no—
589C2×C4—1222no—
590C2×C4—1222no—
591C2×C4—1222no—
592C2×C4—1222no—
593C2×C4—1224no—
594C2×C4—1222no—
595C2×C4—1222no—
596C2×C4—1222no—
597C2×C4—1222no—
598C2×C4—1222no—
599C2×C4—1222no—
600C2×C4—1222no—
601C2×C4—1222no—
602C2×C4—2442no—
603C2×C4—2442no—
604C2×C4—2442no—
605C2×C4—2442no—
606C2×C4—2442no—
607C2×C4—2442no—
608C2×C4—2442no—
609C2×C4—2442no—
610C2×C4—2442no—
611C2×C4—2442no—
612C2×C4—2442no—
613C2×C4—2442no—
614C2×C4—2442no—
615C2×C4—2442no—
616C2×C4—2442no—
617C2×C4—2442no—
618C2×C4—2442no—
619C2×C4—2442no—
620C2×C4C4×C22⋊D6114yes—
621C22—24no—
622C22—24no—
623C22—24no—
624C22—24no—
625C22—28no—
626C22—28no—
627C22—28no—
628C22—24no—
629C22—28no—
630C22—24no—
631C22#1683962618yescomplemented
632C22C2×C4⋊C4×D6111yescharacteristic
633C22C2×#1451973311yescharacteristic
634C22C2×#1451973411yescharacteristic
635C22C2×C4×C22⋊D6111yescharacteristic
636C22C4×#1175822211yescharacteristic
637C22#1684023511yescharacteristic
638C22#1684024311yescharacteristic
639C22#1684027312yes—
640C22C2×#1451976312yes—
641C22C2×#1451976712yes—
642C22#1684029612yes—
643C22#1684035012yes—
644C22#1684035912yes—
645C22D4×C22⋊D6111yescharacteristic
646C22#1684040212yes—
647C22#1684040612yes—
648C22#1684041311yescharacteristic
649C22Q8×C22⋊D6111yescharacteristic
650C22#1684043812yes—
651C22#1684043812yes—
652C22#1684044712yes—
653C22#1684046711yescharacteristic
654C22#1684046712yes—
655C4—24no—
656C4—44no—
657C4—44no—
658C4—42no—
659C4—1224no—
660C4—1224no—
661C4—4882no—
662C4—4882no—
663C4—2442no—
664C4—2442no—
665C2—28no—
666C2C2×#1683962611yescharacteristic
667C2#2144826211yescharacteristic
668C2#2144845311yescharacteristic
669C2#2144866411yescharacteristic
670C2#2144867511yescharacteristic
671C2C4⋊C4×C22⋊D6111yescharacteristic
672C2#2144872611yescharacteristic
673C2#2144993912yes—
674C2#2144995412yes—
675C2#2145014512yes—
676C2#2145032012yes—
subgroupG/H
center, FrattiniC24C23×D61
commutatorC23×C61C23×C4
FittingC22×#95×C61C2
socle, cosocleC24×C61C24
prime radicalC25×C61C23
hypercenterC22×#95D61
hypocenterC61#13896
LCSG ▷ C23×C61 ▷ C61
UCSC1 ◁ C24 ◁ C22×#95

Complements

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

NH ≅ G/Ntype
2. C2×#16839626C2semidirect
4. #21444077C2semidirect
5. #21444074C2semidirect
6. #21444047C2semidirect
7. #21444044C2semidirect
8. #21444043C2semidirect
21. C4×C22⋊Dic61C4semidirect
62. C2×C4×Dic61C2×C4semidirect
63. C2×C4×Dic61C2×C4semidirect
73. #14519684C2×C4semidirect
74. #14519684C2×C4semidirect
293. C22×C61#382semidirect
377. C61#13896semidirect
631. C22#16839626semidirect

Maximal subgroups

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

subgroupindex
C2×#168396352
C2×#168396262
C2×C4×C22⋊Dic612
#214440772
#214440742
#214440472
#214440442
#214440432
#214440422
C22×#95×C612
#1389661