Finite Group Databasebeta

← #29610773

Subgroups — #29610773

Order 15616.

1586 conjugacy classes of subgroups.

The subgroup lattice has 660 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839655C212yesmaximal, complemented
2#21444076C212yesmaximal, complemented
3#21444072C211yesmaximal, characteristic, complemented
4#21444064C212yesmaximal, complemented
5#21444064C212yesmaximal, complemented
6#21444064C212yesmaximal, complemented
7#21444060C212yesmaximal
8#21444048C211yesmaximal, characteristic, complemented
9C2×C61×#394C211yesmaximal, characteristic
10C22×C4⋊Dic61C2212yes—
11C22×C4⋊Dic61C2211yescharacteristic
12C22×C4⋊Dic61C2211yescharacteristic
13C22×C22⋊Dic61C2212yes—
14C22×C22⋊Dic61C2212yes—
15C22×C22⋊Dic61C2212yes—
16C22×C22⋊Dic61C2212yes—
17C22×C4×Dic61C2211yescharacteristic
18#16839655—28no—
19C2×#14519684C2212yes—
20C2×#14519684C2212yes—
21C2×#14519684C2212yes—
22C2×#14519684C2212yes—
23C2×#14519684C2212yes—
24C2×#14519684C2211yescharacteristic
25C2×#14519684C2212yes—
26C2×#14519684C2212yes—
27C23×D4×C61C2211yescharacteristic
28C22×C4⋊C4×C61C2211yescharacteristic
29C22×C22⋊C4×C61C2212yes—
30C22×C22⋊C4×C61C2212yes—
31C61×#394C418yescomplemented
32C2×#95×C61C2211yescharacteristic
33C23×Dic61C2312yes—
34C23×Dic61C2312yes—
35C23×Dic61C2312yes—
36C23×Dic61C2312yes—
37C2×C4⋊Dic61—22no—
38C2×C4⋊Dic61—24no—
39C2×C4⋊Dic61—42no—
40C2×C4⋊Dic61—42no—
41C2×C4⋊Dic61—41no—
42C2×C4⋊Dic61—41no—
43C2×C4⋊Dic61—42no—
44C2×C4⋊Dic61—41no—
45C2×C22⋊Dic61—24no—
46C2×C22⋊Dic61—24no—
47C2×C22⋊Dic61—42no—
48C2×C22⋊Dic61—42no—
49C2×C22⋊Dic61—42no—
50C2×C22⋊Dic61—42no—
51C2×C22⋊Dic61—42no—
52C2×C22⋊Dic61—42no—
53C2×C22⋊Dic61—42no—
54C2×C22⋊Dic61—42no—
55C2×C22⋊Dic61—42no—
56C2×C22⋊Dic61—42no—
57C2×C4×Dic61—41no—
58C2×C4×Dic61—41no—
59C2×C4×Dic61D414yescomplemented
60#14519684—24no—
61#14519684—24no—
62#14519684—42no—
63#14519684—42no—
64#14519684—42no—
65#14519684—41no—
66#14519684—42no—
67#14519684—42no—
68C25×C61C2312yes—
69C22×D4×C61—41no—
70C22×D4×C61—41no—
71C22×D4×C61—42no—
72C22×D4×C61—42no—
73C22×D4×C61C2×C414yes—
74C23×C4×C61C2312yes—
75C23×C4×C61C2311yescharacteristic
76C23×C4×C61C2311yescharacteristic
77C23×C4×C61C2311yescharacteristic
78C2×C4⋊C4×C61—41no—
79C2×C4⋊C4×C61—41no—
80C2×C4⋊C4×C61C2×C414yes—
81C2×C22⋊C4×C61—24no—
82C2×C22⋊C4×C61—24no—
83C2×C22⋊C4×C61—42no—
84C2×C22⋊C4×C61—42no—
85C2×C22⋊C4×C61C2×C418yes—
86C2×C22⋊C4×C61C2×C418yes—
87#95×C61C2×C414yes—
88C24×C61—22no—
89C24×C61—22no—
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×C61C2411yescharacteristic
104C2×D4×C61—44no—
105C2×D4×C61—48no—
106C2×D4×C61—44no—
107C2×D4×C61—48no—
108C2×D4×C61—44no—
109C22×C4×C61—21no—
110C22×C4×C61—21no—
111C22×C4×C61—21no—
112C22×C4×C61—21no—
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—21no—
120C22×C4×C61—21no—
121C22×C4×C61—21no—
122C22×C4×C61—21no—
123C22×C4×C61—22no—
124C22×C4×C61—22no—
125C22×C4×C61—22no—
126C22×C4×C61—22no—
127C22×C4×C61C22×C414yes—
128C22×C4×C61C22×C412yes—
129C22×C4×C61C22×C412yes—
130C22×C4×C61C22×C412yes—
131C22×C4×C61C2×D412yes—
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—22no—
160C22×Dic61C2×D414yes—
161C4⋊C4×C61—44no—
162C22⋊C4×C61—216no—
163C22⋊C4×C61—48no—
164C22⋊Dic61—48no—
165C22⋊Dic61—48no—
166C22⋊Dic61—48no—
167C22⋊Dic61—48no—
168C4⋊Dic61—48no—
169C4⋊Dic61—44no—
170C4⋊Dic61—44no—
171C4×Dic61—44no—
172C23×C61—24no—
173C23×C61—24no—
174C23×C61—24no—
175C23×C61—24no—
176C23×C61—24no—
177C23×C61—24no—
178C23×C61—24no—
179C23×C61—42no—
180C23×C61—42no—
181C23×C61—42no—
182C23×C61—42no—
183C23×C61—42no—
184C23×C61—42no—
185C23×C61—42no—
186C23×C61—42no—
187C23×C61—42no—
188C23×C61—42no—
189C23×C61—42no—
190C23×C61—42no—
191C23×C61—42no—
192C23×C61—42no—
193C23×C61—42no—
194C23×C61—42no—
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×C61C23×C411yescharacteristic
208C23×C61C22×D411yescharacteristic
209C23×C61C2×#4111yescharacteristic
210C23×C61H(5,𝔽2)11yescharacteristic
211C23×C61H(5,𝔽2)12yes—
212C23×C61H(5,𝔽2)11yescharacteristic
213C23×C61H(5,𝔽2)12yes—
214C23×C61H(5,𝔽2)11yescharacteristic
215C23×C61H(5,𝔽2)12yes—
216C23×C612−1+411yescharacteristic
217C23×C612−1+412yes—
218D4×C61—416no—
219C2×C4×C61—22no—
220C2×C4×C61—22no—
221C2×C4×C61—22no—
222C2×C4×C61—22no—
223C2×C4×C61—22no—
224C2×C4×C61—22no—
225C2×C4×C61—24no—
226C2×C4×C61—24no—
227C2×C4×C61—24no—
228C2×C4×C61—22no—
229C2×C4×C61—22no—
230C2×C4×C61—24no—
231C2×C4×C61—24no—
232C2×C4×C61—42no—
233C2×C4×C61—42no—
234C2×C4×C61—42no—
235C2×C4×C61—42no—
236C2×C4×C61—41no—
237C2×C4×C61—42no—
238C2×C4×C61—41no—
239C2×C4×C61—42no—
240C2×C4×C61—41no—
241C2×C4×C61—41no—
242C2×C4×C61C4×D414yes—
243C2×Dic61—24no—
244C2×Dic61—24no—
245C2×Dic61—24no—
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—
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—
271#14632—611nomaximal, Sylow
272C22×C61—44no—
273C22×C61—44no—
274C22×C61—44no—
275C22×C61—44no—
276C22×C61—44no—
277C22×C61—44no—
278C22×C61—44no—
279C22×C61—82no—
280C22×C61—82no—
281C22×C61—82no—
282C22×C61—82no—
283C22×C61—82no—
284C22×C61—82no—
285C22×C61—82no—
286C22×C61—82no—
287C22×C61C2×C4×D411yescharacteristic
288C22×C61#51812yes—
289C22×C61#51811yescharacteristic
290C22×C61#52012yes—
291C22×C61#52011yescharacteristic
292C22×C61#53411yescharacteristic
293C22×C61#53512yes—
294C22×C61#53711yescharacteristic
295C22×C61#53712yes—
296C22×C61#53811yescharacteristic
297C22×C61#53911yescharacteristic
298C22×C61#54012yes—
299C22×C61#54212yes—
300C22×C61#56011yescharacteristic
301C22×C61#56012yes—
302C22×C61#56012yes—
303C22×C61H(3,𝔽4)11yescharacteristic
304C22×C61H(3,𝔽4)11yescharacteristic
305C22×C61#56212yes—
306C22×C61#56212yes—
307C22×C61#56212yes—
308C22×C61#56311yescharacteristic
309C22×C61#56312yes—
310C4×C61—24no—
311C4×C61—44no—
312C4×C61—42no—
313C4×C61—42no—
314Dic61—44no—
315Dic61—82no—
316Dic61—82no—
317Dic61—82no—
318C2×#394D6111yescharacteristic
319C2×#388—612no—
320#1442—612no—
321#1441—611no—
322#1439—612no—
323#1439—612no—
324#1439—612no—
325#1438—612no—
326C4⋊#95—611no—
327C2×C61—84no—
328C2×C61#232412yes—
329C2×C61#232511yescharacteristic
330C2×C61#237511yescharacteristic
331C2×C61#237612yes—
332C2×C61#237911yescharacteristic
333C2×C61#281312yes—
334C2×C61#281911yescharacteristic
335C2×C61#282311yescharacteristic
336C2×C61#282811yescharacteristic
337C2×C61#283211yescharacteristic
338C2×C61#283712yes—
339C23×D4C2×D6111yescharacteristic
340C22×C4⋊C4—611no—
341C22×C4⋊C4—612no—
342C22×C4⋊C4—611no—
343C22×C4⋊C4C2×D6111yescharacteristic
344C22×C22⋊C4—612no—
345C22×C22⋊C4—612no—
346C22×C22⋊C4—612no—
347C22×C22⋊C4—612no—
348C22×C22⋊C4C2×D6112yes—
349C22×C22⋊C4C2×D6112yes—
350C22×C42—611no—
351#394Dic6118yescomplemented
352#388—1228no—
353C2×#95—612no—
354C2×#95—612no—
355C2×#95—612no—
356C2×#95—612no—
357C2×#95—612no—
358C2×#95—611no—
359C2×#95—612no—
360C2×#95—612no—
361C2×#95C2×D6111yescharacteristic
362C61#1463211yescharacteristic, complemented, Sylow
363C25C22×D6112yes—
364C22×D4—42no—
365C22×D4—42no—
366C22×D4—41no—
367C22×D4—41no—
368C22×D4C2×Dic6114yes—
369C23×C4—612no—
370C23×C4—612no—
371C23×C4—612no—
372C23×C4—612no—
373C23×C4C22×D6111yescharacteristic
374C23×C4C22×D6112yes—
375C23×C4C22×D6111yescharacteristic
376C23×C4C22×D6111yescharacteristic
377C2×C4⋊C4—41no—
378C2×C4⋊C4—41no—
379C2×C4⋊C4—1224no—
380C2×C4⋊C4—1222no—
381C2×C4⋊C4—2441no—
382C2×C4⋊C4—2441no—
383C2×C4⋊C4—2442no—
384C2×C4⋊C4—2442no—
385C2×C4⋊C4—2442no—
386C2×C4⋊C4—2441no—
387C2×C4⋊C4C2×Dic6114yes—
388C2×C22⋊C4—24no—
389C2×C22⋊C4—24no—
390C2×C22⋊C4—42no—
391C2×C22⋊C4—42no—
392C2×C22⋊C4—1224no—
393C2×C22⋊C4—1224no—
394C2×C22⋊C4—2442no—
395C2×C22⋊C4—2442no—
396C2×C22⋊C4—2442no—
397C2×C22⋊C4—2442no—
398C2×C22⋊C4—2442no—
399C2×C22⋊C4—2442no—
400C2×C22⋊C4—2442no—
401C2×C22⋊C4—2442no—
402C2×C22⋊C4—2442no—
403C2×C22⋊C4—2442no—
404C2×C22⋊C4C2×Dic6118yes—
405C2×C22⋊C4C2×Dic6118yes—
406C2×C42—614no—
407C2×C42—2441no—
408C2×C42—2441no—
409#95—1224no—
410#95—1224no—
411#95—2442no—
412#95—2442no—
413#95—2441no—
414#95—2442no—
415#95—2442no—
416#95—2442no—
417#95C2×Dic6114yes—
418C24—22no—
419C24—22no—
420C24—22no—
421C24—22no—
422C24—22no—
423C24—22no—
424C24—22no—
425C24—22no—
426C24—22no—
427C24—22no—
428C24—22no—
429C24—22no—
430C24—22no—
431C24—22no—
432C24C22×Dic6114yes—
433C24C23×D6111yescharacteristic
434C2×D4—44no—
435C2×D4—48no—
436C2×D4—48no—
437C2×D4—44no—
438C2×D4—44no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—21no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—21no—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—614no—
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—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×C4D4×D6112yes—
486C22×C4#1175822412yes—
487C22×C4C22×Dic6114yes—
488C22×C4C22×Dic6112yes—
489C22×C4C22×Dic6112yes—
490C22×C4C22×Dic6112yes—
491C4⋊C4—44no—
492C4⋊C4—2444no—
493C4⋊C4—2448no—
494C4⋊C4—2444no—
495C22⋊C4—216no—
496C22⋊C4—48no—
497C22⋊C4—2448no—
498C22⋊C4—2448no—
499C22⋊C4—2448no—
500C22⋊C4—2448no—
501C42—2444no—
502C23—24no—
503C23—24no—
504C23—24no—
505C23—24no—
506C23—24no—
507C23—24no—
508C23—24no—
509C23—42no—
510C23—42no—
511C23—42no—
512C23—42no—
513C23—42no—
514C23—42no—
515C23—42no—
516C23—42no—
517C23—42no—
518C23—42no—
519C23—42no—
520C23—42no—
521C23—42no—
522C23—42no—
523C23—42no—
524C23—42no—
525C23—42no—
526C23—42no—
527C23—42no—
528C23—42no—
529C23—42no—
530C23—42no—
531C23—42no—
532C23—42no—
533C23—42no—
534C23—42no—
535C23—42no—
536C23—42no—
537C23C23×Dic6111yescharacteristic
538C23C2×D4×D6111yescharacteristic
539C23C2×#1175822411yescharacteristic
540C23#1451982212yes—
541C23#1451982211yescharacteristic
542C23#1451982211yescharacteristic
543C23#1451982212yes—
544C23#1451982311yescharacteristic
545C23#1451982312yes—
546C23#1451982412yes—
547C23#1451982411yescharacteristic
548D4—416no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—24no—
556C2×C4—24no—
557C2×C4—24no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—42no—
563C2×C4—41no—
564C2×C4—42no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—1224no—
573C2×C4—1224no—
574C2×C4—1224no—
575C2×C4—2442no—
576C2×C4—2442no—
577C2×C4—2442no—
578C2×C4—2442no—
579C2×C4—2442no—
580C2×C4—2442no—
581C2×C4—2442no—
582C2×C4—2442no—
583C2×C4—2442no—
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—2442no—
599C2×C4—2442no—
600C2×C4D4×Dic6114yes—
601C22—44no—
602C22—44no—
603C22—44no—
604C22—44no—
605C22—44no—
606C22—44no—
607C22—44no—
608C22—82no—
609C22—82no—
610C22—82no—
611C22—82no—
612C22—82no—
613C22—82no—
614C22—82no—
615C22—82no—
616C22C2×D4×Dic6111yescharacteristic
617C22#1684023712yes—
618C22#1684023711yescharacteristic
619C22#1684024511yescharacteristic
620C22#1684024512yes—
621C22#1684032011yescharacteristic
622C22#1684032812yes—
623C22#1684033911yescharacteristic
624C22#1684034112yes—
625C22#1684034911yescharacteristic
626C22#1684035711yescharacteristic
627C22#1684036512yes—
628C22#1684037912yes—
629C22#1684050112yes—
630C22#1684050111yescharacteristic
631C22#1684050212yes—
632C22#1684050411yescharacteristic
633C22#1684050511yescharacteristic
634C22#1684050612yes—
635C22#1684050912yes—
636C22#1684050912yes—
637C22#1684051211yescharacteristic
638C22#1684051512yes—
639C4—24no—
640C4—42no—
641C4—42no—
642C4—44no—
643C4—4882no—
644C4—4882no—
645C4—4882no—
646C4—2444no—
647C2—84no—
648C2#2144851412yes—
649C2#2144852411yescharacteristic
650C2#2144896011yescharacteristic
651C2#2144896712yes—
652C2#2144899311yescharacteristic
653C2#2145382612yes—
654C2#2145386311yescharacteristic
655C2#2145389011yescharacteristic
656C2#2145394211yescharacteristic
657C2#2145399011yescharacteristic
658C2#2145403512yes—
subgroupG/H
center, FrattiniC24C23×D61
commutatorC23×C61C23×C4
FittingC2×C61×#394C2
socle, cosocleC24×C61C24
prime radicalC23×D4×C61C22
hypercenterC2×#394D61
hypocenterC61#14632
LCSG ▷ C23×C61 ▷ C61
UCSC1 ◁ C24 ◁ C2×#394

Complements

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

NH ≅ G/Ntype
1. C2×#16839655C2semidirect
2. #21444076C2semidirect
3. #21444072C2semidirect
4. #21444064C2semidirect
5. #21444064C2semidirect
6. #21444064C2semidirect
8. #21444048C2semidirect
31. C61×#394C4semidirect
59. C2×C4×Dic61D4semidirect
351. #394Dic61semidirect
362. C61#14632semidirect

Maximal subgroups

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

subgroupindex
C2×#168396552
#214440762
#214440722
#214440642
#214440602
#214440482
C2×C61×#3942
#1463261