Finite Group Databasebeta

← #29611342

Subgroups — #29611342

Order 15616.

2842 conjugacy classes of subgroups.

The subgroup lattice has 648 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14519761C211yesmaximal, characteristic
2C23×C4×Dic61C211yesmaximal, characteristic, complemented
3C2×#16839641C211yesmaximal, characteristic
4C2×#16839639C211yesmaximal, characteristic
5C2×#16839623C211yesmaximal, characteristic, complemented
6#21444055C214yesmaximal, complemented
7#21444053C214yesmaximal, complemented
8#21444041C211yesmaximal, characteristic, complemented
9C61×C22⋊#95C211yesmaximal, characteristic, complemented
10C23×C22⋊D61C2211yescharacteristic
11C24×Dic61C2211yescharacteristic
12C24×Dic61C2211yescharacteristic
13C2×#14519761—24no—
14C2×#14519761—24no—
15C2×#14519761C418yescomplemented
16C22×C4⋊Dic61C2212yes—
17C22×C22⋊Dic61C2211yescharacteristic
18C22×#11758198C2212yes—
19C22×#11758198C2212yes—
20C22×#11758198C2212yes—
21C22×C4×Dic61—24no—
22C22×C4×Dic61—24no—
23C22×C4×Dic61—24no—
24C22×C4×Dic61C2214yes—
25#16839641—24no—
26#16839639—24no—
27#16839623—24no—
28C2×#14519684C2214yes—
29C2×#14519684C2212yes—
30C2×#14519684C2212yes—
31C2×#14519684C2214yes—
32C24×C4×C61C2211yescharacteristic
33C22×C22⋊C4×C61C2211yescharacteristic
34C22×C22⋊C4×C61C2211yescharacteristic
35C2×#95×C61C2214yes—
36C24×D61C2311yescharacteristic
37C22×C22⋊D61—22no—
38C22×C22⋊D61—22no—
39C22×C22⋊D61—22no—
40C22×C22⋊D61—22no—
41C22×C22⋊D61—22no—
42C22×C22⋊D61—22no—
43C22×C22⋊D61C2×C414yes—
44C23×Dic61—22no—
45C23×Dic61—22no—
46C23×Dic61—22no—
47C23×Dic61—22no—
48C23×Dic61—22no—
49C23×Dic61—22no—
50C23×Dic61—28no—
51C23×Dic61C2×C414yes—
52C23×Dic61D414yes—
53C23×Dic61D414yes—
54C23×Dic61D414yes—
55C23×Dic61C2312yes—
56C23×Dic61C2311yescharacteristic
57C23×Dic61C2312yes—
58C23×Dic61C2312yes—
59#14519761—232no—
60C2×C4⋊Dic61—24no—
61C2×C4⋊Dic61—24no—
62C2×C4⋊Dic61C2×C418yescomplemented
63C2×C22⋊Dic61—22no—
64C2×C22⋊Dic61—22no—
65C2×C22⋊Dic61—22no—
66C2×#11758198—24no—
67C2×#11758198—24no—
68C2×#11758198—24no—
69C2×#11758198—28no—
70C2×#11758198—24no—
71C2×#11758198—28no—
72C2×#11758198C2×C418yescomplemented
73C2×C4×Dic61—232no—
74C2×C4×Dic61—28no—
75C2×C4×Dic61—28no—
76C2×C4×Dic61—28no—
77#14519684—24no—
78#14519684—28no—
79#14519684—24no—
80#14519684—28no—
81C25×C61C2311yescharacteristic
82C23×C4×C61—22no—
83C23×C4×C61—22no—
84C23×C4×C61—28no—
85C23×C4×C61D414yes—
86C23×C4×C61C2312yes—
87C23×C4×C61C2312yes—
88C23×C4×C61C2312yes—
89C2×C22⋊C4×C61—22no—
90C2×C22⋊C4×C61—22no—
91C2×C22⋊C4×C61—22no—
92C2×C22⋊C4×C61—22no—
93C2×C22⋊C4×C61—22no—
94C2×C22⋊C4×C61C2×C414yes—
95#95×C61—28no—
96C24×C61—24no—
97C24×C61—24no—
98C24×C61C22×C412yes—
99C24×C61C2×D412yes—
100C24×C61C2×D412yes—
101C24×C61C2×D412yes—
102C24×C61C2×D412yes—
103C24×C61#4112yes—
104C24×C61#4112yes—
105C24×C61C2411yescharacteristic
106C23×D61—21no—
107C23×D61—21no—
108C23×D61—21no—
109C23×D61—21no—
110C23×D61—21no—
111C23×D61—21no—
112C23×D61—24no—
113C23×D61—24no—
114C23×D61C22×C412yes—
115C22×C4×C61—216no—
116C22×C4×C61—216no—
117C22×C4×C61—28no—
118C22×C4×C61—22no—
119C22×C4×C61—22no—
120C22×C4×C61—24no—
121C22×C4×C61—216no—
122C22×C4×C61—22no—
123C22×C4×C61—22no—
124C22×C4×C61—24no—
125C22×C4×C61—24no—
126C22×C4×C61—22no—
127C22×C4×C61—22no—
128C22×C4×C61—24no—
129C22×C4×C61—22no—
130C22×C4×C61—24no—
131C22×C4×C61C22×C414yes—
132C22×C4×C61C2×D414yes—
133C22×C4×C61#4118yes—
134C2×C22⋊D61—28no—
135C2×C22⋊D61—28no—
136C2×C22⋊D61—28no—
137C2×C22⋊D61—44no—
138C2×C22⋊D61—44no—
139C2×C22⋊D61—44no—
140C2×C22⋊D61—44no—
141C22×Dic61—216no—
142C22×Dic61—216no—
143C22×Dic61—28no—
144C22×Dic61—28no—
145C22×Dic61—28no—
146C22×Dic61—28no—
147C22×Dic61—28no—
148C22×Dic61—28no—
149C22×Dic61—28no—
150C22×Dic61—22no—
151C22×Dic61—22no—
152C22×Dic61—22no—
153C22×Dic61—22no—
154C22×Dic61—22no—
155C22×Dic61—22no—
156C22×Dic61—24no—
157C22×Dic61—216no—
158C22×Dic61—21no—
159C22×Dic61—21no—
160C22×Dic61—21no—
161C22×Dic61—21no—
162C22×Dic61—21no—
163C22×Dic61—21no—
164C22×Dic61—22no—
165C22×Dic61—22no—
166C22×Dic61—24no—
167C22×Dic61—24no—
168C22×Dic61—22no—
169C22×Dic61C22⋊C4116yescomplemented
170C22×Dic61C22×C412yes—
171C22×Dic61C22×C414yes—
172C22×Dic61C2×D414yes—
173C22×Dic61C2×D414yes—
174C22×Dic61C2×D414yes—
175C22×Dic61#4118yes—
176C22⋊C4×C61—28no—
177C22⋊C4×C61—44no—
178C22⋊Dic61—44no—
179#11758198—216no—
180#11758198—48no—
181#11758198—48no—
182C4⋊Dic61—216no—
183C4×Dic61—232no—
184C23×C61—28no—
185C23×C61—24no—
186C23×C61—24no—
187C23×C61—24no—
188C23×C61—24no—
189C23×C61—24no—
190C23×C61—24no—
191C23×C61—28no—
192C23×C61—28no—
193C23×C61—28no—
194C23×C61C2×C22⋊C414yes—
195C23×C61C4×D414yes—
196C23×C61C4×D414yes—
197C23×C61#12014yes—
198C23×C61#12114yes—
199C23×C61#12114yes—
200C23×C61#12314yes—
201C23×C61C23×C411yescharacteristic
202C23×C61C22×D411yescharacteristic
203C23×C61C22×D411yescharacteristic
204C23×C61C22×D411yescharacteristic
205C23×C61C22×D411yescharacteristic
206C23×C61C2×#4111yescharacteristic
207C23×C61C2×#4111yescharacteristic
208C23×C61C2×#4114yes—
209C23×C61C2×#4114yes—
210C22×D61—22no—
211C22×D61—22no—
212C22×D61—22no—
213C22×D61—28no—
214C22×D61—41no—
215C22×D61—41no—
216C22×D61—41no—
217C22×D61—42no—
218C22×D61—42no—
219C22×D61—41no—
220C22×D61—42no—
221C22×D61—42no—
222C22×D61—42no—
223C22×D61—42no—
224C22×D61—44no—
225C22×D61—44no—
226C22×D61—44no—
227C2×C4×C61—232no—
228C2×C4×C61—24no—
229C2×C4×C61—24no—
230C2×C4×C61—28no—
231C2×C4×C61—28no—
232C2×C4×C61—24no—
233C2×C4×C61—28no—
234C2×C4×C61—42no—
235C2×C4×C61—42no—
236C2×C4×C61—44no—
237C2×C4×C61—42no—
238C2×C4×C61—42no—
239C2×C4×C61—42no—
240C2×C4×C61—44no—
241C2×C4×C61—44no—
242C2×C4×C61#12318yescomplemented
243C22⋊D61—416no—
244C2×Dic61—232no—
245C2×Dic61—24no—
246C2×Dic61—24no—
247C2×Dic61—24no—
248C2×Dic61—24no—
249C2×Dic61—24no—
250C2×Dic61—24no—
251C2×Dic61—24no—
252C2×Dic61—24no—
253C2×Dic61—28no—
254C2×Dic61—28no—
255C2×Dic61—232no—
256C2×Dic61—22no—
257C2×Dic61—22no—
258C2×Dic61—22no—
259C2×Dic61—41no—
260C2×Dic61—41no—
261C2×Dic61—41no—
262C2×Dic61—41no—
263C2×Dic61—42no—
264C2×Dic61—42no—
265C2×Dic61—42no—
266C2×Dic61—44no—
267C2×Dic61—44no—
268C2×Dic61C2×C22⋊C418yes—
269C2×Dic61#12318yescomplemented
270#14706—611nomaximal, Sylow
271C22×C61—24no—
272C22×C61—24no—
273C22×C61—28no—
274C22×C61—28no—
275C22×C61—28no—
276C22×C61—28no—
277C22×C61—28no—
278C22×C61—28no—
279C22×C61#38618yes—
280C22×C61C22×C22⋊C411yescharacteristic
281C22×C61C2×C4×D411yescharacteristic
282C22×C61C2×C4×D411yescharacteristic
283C22×C61C4×#4114yes—
284C22×C61C2×#12011yescharacteristic
285C22×C61C2×#12111yescharacteristic
286C22×C61C2×#12111yescharacteristic
287C22×C61C2×#12312yes—
288C22×C61C2×#12312yes—
289C22×C61C2×#12311yescharacteristic
290C22×C61#52512yes—
291C22×C61#52512yes—
292C22×C61#52512yes—
293C22×C61#52512yes—
294C22×C61#52914yes—
295C22×C61#52914yes—
296C22×C61#53214yes—
297C2×D61—42no—
298C2×D61—44no—
299C2×D61—44no—
300C2×D61—44no—
301C2×D61—81no—
302C2×D61—81no—
303C2×D61—82no—
304C2×D61—82no—
305C2×D61—82no—
306C4×C61—28no—
307C4×C61—44no—
308C4×C61—82no—
309Dic61—28no—
310Dic61—28no—
311Dic61—42no—
312Dic61—82no—
313C22×#123—611no—
314C23×C42—611no—
315C2×#386—611no—
316C2×#386—611no—
317C2×#381—611no—
318#1437—614no—
319#1436—614no—
320C22⋊#95—611no—
321C22⋊#95D6111yescharacteristic, complemented
322C2×C61—28no—
323C2×C61C2×#38611yescharacteristic
324C2×C61#229712yes—
325C2×C61C4×#12312yes—
326C2×C61C4×#12312yes—
327C2×C61#284911yescharacteristic
328C2×C61#284911yescharacteristic
329C2×C61#285312yes—
330C2×C61#285412yes—
331C2×C61#285412yes—
332D61—82no—
333C23×D4—611no—
334C24×C4—611no—
335C24×C4—611no—
336C24×C4C2×D6111yescharacteristic
337C2×#123—618no—
338C2×#123—1224no—
339C2×#123—1224no—
340C22×C4⋊C4—612no—
341C22×C22⋊C4—612no—
342C22×C22⋊C4—611no—
343C22×C22⋊C4—612no—
344C22×C22⋊C4—612no—
345C22×C22⋊C4C2×D6111yescharacteristic
346C22×C22⋊C4C2×D6111yescharacteristic
347C22×C42—614no—
348C22×C42—1224no—
349C22×C42—1224no—
350C22×C42—1224no—
351#386—1224no—
352#386—1224no—
353#381—1224no—
354C2×#95—614no—
355C2×#95—612no—
356C2×#95—612no—
357C2×#95—614no—
358C2×#95C2×D6114yes—
359C61#1470611yescharacteristic, complemented, Sylow
360C25—611no—
361C25C22×D6111yescharacteristic
362C22×D4—614no—
363C22×D4—1222no—
364C22×D4—1222no—
365C22×D4—1222no—
366C22×D4—1222no—
367C22×D4—1222no—
368C22×D4—1222no—
369C23×C4—22no—
370C23×C4—22no—
371C23×C4—28no—
372C23×C4—614no—
373C23×C4—614no—
374C23×C4—614no—
375C23×C4—612no—
376C23×C4—612no—
377C23×C4—614no—
378C23×C4—611no—
379C23×C4—612no—
380C23×C4—1228no—
381C23×C4—1222no—
382C23×C4—1222no—
383C23×C4—1222no—
384C23×C4—1222no—
385C23×C4—1222no—
386C23×C4—1222no—
387C23×C4D24414yes—
388C23×C4C22×D6112yes—
389C23×C4C22×D6112yes—
390C23×C4C22×D6112yes—
391#123—12232no—
392C2×C4⋊C4—618no—
393C2×C4⋊C4—1224no—
394C2×C4⋊C4—1224no—
395C2×C22⋊C4—22no—
396C2×C22⋊C4—22no—
397C2×C22⋊C4—22no—
398C2×C22⋊C4—22no—
399C2×C22⋊C4—22no—
400C2×C22⋊C4—618no—
401C2×C22⋊C4—1224no—
402C2×C22⋊C4—1224no—
403C2×C22⋊C4—1224no—
404C2×C22⋊C4—1228no—
405C2×C22⋊C4—1222no—
406C2×C22⋊C4—1222no—
407C2×C22⋊C4—1222no—
408C2×C22⋊C4—1224no—
409C2×C22⋊C4—1228no—
410C2×C22⋊C4C4×D6114yes—
411C2×C42—1228no—
412C2×C42—1228no—
413C2×C42—12232no—
414C2×C42—1228no—
415#95—28no—
416#95—1228no—
417#95—1224no—
418#95—1228no—
419#95—1224no—
420C24—24no—
421C24—24no—
422C24—612no—
423C24—1221no—
424C24—1221no—
425C24—1221no—
426C24—1221no—
427C24—1224no—
428C24—1224no—
429C24—1221no—
430C24—1221no—
431C24C2×C4×D6112yes—
432C24C2×D24412yes—
433C24#1175822212yes—
434C24D4×D6112yes—
435C24D4×D6112yes—
436C24D4×D6112yes—
437C24#1175822612yes—
438C24C23×D6111yescharacteristic
439C2×D4—1228no—
440C2×D4—1228no—
441C2×D4—1228no—
442C2×D4—2444no—
443C2×D4—2444no—
444C2×D4—2444no—
445C2×D4—2444no—
446C22×C4—216no—
447C22×C4—216no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—24no—
451C22×C4—28no—
452C22×C4—216no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—24no—
456C22×C4—24no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—24no—
460C22×C4—24no—
461C22×C4—22no—
462C22×C4—614no—
463C22×C4—614no—
464C22×C4—614no—
465C22×C4—614no—
466C22×C4—618no—
467C22×C4—6116no—
468C22×C4—612no—
469C22×C4—12216no—
470C22×C4—12216no—
471C22×C4—12216no—
472C22×C4—1222no—
473C22×C4—1222no—
474C22×C4—1222no—
475C22×C4—1222no—
476C22×C4—1224no—
477C22×C4—1222no—
478C22×C4—1222no—
479C22×C4—1228no—
480C22×C4—1228no—
481C22×C4—1228no—
482C22×C4—1228no—
483C22×C4—1228no—
484C22×C4—1228no—
485C22×C4—1228no—
486C22×C4—1221no—
487C22×C4—1221no—
488C22×C4—1221no—
489C22×C4—1221no—
490C22×C4—1221no—
491C22×C4—1221no—
492C22×C4—1222no—
493C22×C4—1222no—
494C22×C4—1224no—
495C22×C4—1224no—
496C22×C4—1222no—
497C22×C4C2×C4×D6114yes—
498C22×C4C2×D24414yes—
499C22×C4#1175822418yes—
500C4⋊C4—12216no—
501C22⋊C4—28no—
502C22⋊C4—44no—
503C22⋊C4—12216no—
504C22⋊C4—2448no—
505C22⋊C4—2444no—
506C22⋊C4—2448no—
507C42—12232no—
508C23—28no—
509C23—24no—
510C23—24no—
511C23—24no—
512C23—24no—
513C23—24no—
514C23—24no—
515C23—28no—
516C23—28no—
517C23—28no—
518C23—1222no—
519C23—1222no—
520C23—1222no—
521C23—1228no—
522C23—2441no—
523C23—2441no—
524C23—2441no—
525C23—2442no—
526C23—2442no—
527C23—2441no—
528C23—2442no—
529C23—2442no—
530C23—2442no—
531C23—2442no—
532C23—2444no—
533C23—2444no—
534C23—2444no—
535C23C22⋊C4×D6114yes—
536C23#1451973914yes—
537C23C4×D24414yes—
538C23#1451974714yes—
539C23#1451975014yes—
540C23#1451975214yes—
541C23#1451976214yes—
542C23C22×C4×D6111yescharacteristic
543C23C2×D4×D6111yescharacteristic
544C23C2×D4×D6111yescharacteristic
545C23C2×D4×D6111yescharacteristic
546C23C22×D24411yescharacteristic
547C23#41×D6114yes—
548C23C2×#1175822414yes—
549C23C2×#1175822211yescharacteristic
550C23C2×#1175822611yescharacteristic
551D4—24416no—
552C2×C4—232no—
553C2×C4—24no—
554C2×C4—24no—
555C2×C4—28no—
556C2×C4—28no—
557C2×C4—24no—
558C2×C4—28no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—44no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—44no—
566C2×C4—44no—
567C2×C4—618no—
568C2×C4—618no—
569C2×C4—12232no—
570C2×C4—1224no—
571C2×C4—1224no—
572C2×C4—1224no—
573C2×C4—1224no—
574C2×C4—1224no—
575C2×C4—1224no—
576C2×C4—1224no—
577C2×C4—1224no—
578C2×C4—1228no—
579C2×C4—1228no—
580C2×C4—12232no—
581C2×C4—1222no—
582C2×C4—1222no—
583C2×C4—1222no—
584C2×C4—2442no—
585C2×C4—2442no—
586C2×C4—2442no—
587C2×C4—2444no—
588C2×C4—2444no—
589C2×C4—2441no—
590C2×C4—2441no—
591C2×C4—2441no—
592C2×C4—2441no—
593C2×C4#1451976118yescomplemented
594C22—24no—
595C22—24no—
596C22—28no—
597C22—28no—
598C22—28no—
599C22—28no—
600C22—28no—
601C22—28no—
602C22—4881no—
603C22—4881no—
604C22—4882no—
605C22—4882no—
606C22—4882no—
607C22—2442no—
608C22—2444no—
609C22—2444no—
610C22—2444no—
611C22#1683964018yes—
612C22C2×C22⋊C4×D6111yescharacteristic
613C22C2×#1451973911yescharacteristic
614C22C2×C4×D24411yescharacteristic
615C22C4×#1175822414yes—
616C22C2×#1451974711yescharacteristic
617C22C2×#1451975011yescharacteristic
618C22C2×#1451975211yescharacteristic
619C22#123×D6112yes—
620C22C2×#1451976112yes—
621C22C2×#1451976211yescharacteristic
622C22#1684026812yes—
623C22#1684026812yes—
624C22#1684027012yes—
625C22#1684027412yes—
626C22#1684029214yes—
627C22#1684029814yes—
628C22#1684030714yes—
629C4—28no—
630C4—44no—
631C4—82no—
632C4—1228no—
633C4—1228no—
634C4—4882no—
635C4—2442no—
636C2—28no—
637C2—4882no—
638C2C2×#1683964011yescharacteristic
639C2#2144830412yes—
640C2#2144833812yes—
641C2C4×#1451976112yes—
642C2#2145411211yescharacteristic
643C2#2145411311yescharacteristic
644C2#2145414412yes—
645C2#2145414712yes—
646C2#2145415112yes—
subgroupG/H
center, FrattiniC24C23×D61
commutatorC23×C61C23×C4
FittingC61×C22⋊#95C2
socle, cosocleC24×C61C24
prime radicalC23×C22⋊D61C22
hypercenterC22⋊#95D61
hypocenterC61#14706
LCSG ▷ C23×C61 ▷ C61
UCSC1 ◁ C24 ◁ C22⋊#95

Complements

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

NH ≅ G/Ntype
2. C23×C4×Dic61C2semidirect
5. C2×#16839623C2semidirect
6. #21444055C2semidirect
7. #21444053C2semidirect
8. #21444041C2semidirect
9. C61×C22⋊#95C2semidirect
15. C2×#14519761C4semidirect
62. C2×C4⋊Dic61C2×C4semidirect
72. C2×#11758198C2×C4semidirect
169. C22×Dic61C22⋊C4semidirect
242. C2×C4×C61#123semidirect
269. C2×Dic61#123semidirect
321. C22⋊#95D61semidirect
359. C61#14706semidirect
593. C2×C4#14519761semidirect

Maximal subgroups

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

subgroupindex
C22×#145197612
C23×C4×Dic612
C2×#168396412
C2×#168396392
C2×#168396232
#214440552
#214440532
#214440412
C61×C22⋊#952
#1470661