Finite Group Databasebeta

← #29606783

Subgroups — #29606783

Order 15616.

1830 conjugacy classes of subgroups.

The subgroup lattice has 664 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4×Dic122C211yesmaximal, characteristic, complemented
2C2×#16839674C211yesmaximal, characteristic
3C2×#16839650C212yesmaximal
4#21444089C211yesmaximal, characteristic, complemented
5#21444083C211yesmaximal, characteristic, complemented
6#21444082C212yesmaximal, complemented
7#21444064C211yesmaximal, characteristic
8#21444062C212yesmaximal
9#21444046C211yesmaximal, characteristic, complemented
10#21444043C212yesmaximal, complemented
11C2×C61×#388C211yesmaximal, characteristic
12C23×Dic122C2211yescharacteristic
13C2×C4×Dic122—28no—
14C2×C4×Dic122—24no—
15C22×C4⋊Dic61C2211yescharacteristic
16C22×#11758196C2212yes—
17C22×#11758196C2212yes—
18C22×C22⋊Dic61C2212yes—
19C22×C22⋊Dic61C2212yes—
20C22×C4×Dic61C2212yes—
21#16839674C418yescomplemented
22#16839650—28no—
23C2×#14519684C2212yes—
24C2×#14519684C2212yes—
25C2×#14519684C2212yes—
26C2×#14519684C2211yescharacteristic
27C2×#14519684C2212yes—
28C2×#14519684C2212yes—
29C2×#14519684C2212yes—
30C2×#14519684C2212yes—
31C2×#14519684C2211yescharacteristic
32C22×C4⋊C4×C61C2211yescharacteristic
33C22×C22⋊C4×C61C2211yescharacteristic
34C22×C22⋊C4×C61C2211yescharacteristic
35C22×C22⋊C4×C61C2211yescharacteristic
36C22×C42×C61C2211yescharacteristic
37C61×#388—24no—
38C2×#95×C61C2211yescharacteristic
39C2×#95×C61C2211yescharacteristic
40C22×Dic122—22no—
41C22×Dic122—22no—
42C22×Dic122—24no—
43C22×Dic122—24no—
44C22×Dic122C2×C414yes—
45C23×Dic61C2312yes—
46C23×Dic61C2312yes—
47C23×Dic61C2312yes—
48C23×Dic61C2312yes—
49C4×Dic122—232no—
50C2×C4⋊Dic61—22no—
51C2×C4⋊Dic61—24no—
52C2×#11758196—24no—
53C2×#11758196—24no—
54C2×#11758196—24no—
55C2×#11758196—24no—
56C2×#11758196—42no—
57C2×#11758196—42no—
58C2×C22⋊Dic61—24no—
59C2×C22⋊Dic61—24no—
60C2×C22⋊Dic61—24no—
61C2×C22⋊Dic61—42no—
62C2×C22⋊Dic61—42no—
63C2×C22⋊Dic61C2×C418yes—
64C2×C4×Dic61—24no—
65C2×C4×Dic61—24no—
66C2×C4×Dic61D418yes—
67#14519684—24no—
68#14519684—22no—
69#14519684—24no—
70#14519684—24no—
71#14519684—42no—
72#14519684—42no—
73#14519684—42no—
74#14519684C2×C414yes—
75#14519684C2×C418yes—
76C25×C61C2311yescharacteristic
77C23×C4×C61C2311yescharacteristic
78C23×C4×C61C2311yescharacteristic
79C23×C4×C61C2311yescharacteristic
80C23×C4×C61C2311yescharacteristic
81C23×C4×C61C2311yescharacteristic
82C23×C4×C61C2311yescharacteristic
83C2×C4⋊C4×C61—22no—
84C2×C4⋊C4×C61—42no—
85C2×C22⋊C4×C61—22no—
86C2×C22⋊C4×C61—22no—
87C2×C22⋊C4×C61—22no—
88C2×C22⋊C4×C61—22no—
89C2×C22⋊C4×C61—24no—
90C2×C22⋊C4×C61—42no—
91C2×C22⋊C4×C61C2×C414yes—
92C2×C42×C61—24no—
93C2×C42×C61D414yes—
94#95×C61—22no—
95#95×C61—22no—
96C24×C61—21no—
97C24×C61—22no—
98C24×C61—21no—
99C24×C61—22no—
100C24×C61—22no—
101C24×C61—21no—
102C24×C61—21no—
103C24×C61—21no—
104C24×C61—22no—
105C24×C61—21no—
106C24×C61C22×C412yes—
107C24×C61C2411yescharacteristic
108C22×C4×C61—22no—
109C22×C4×C61—22no—
110C22×C4×C61—21no—
111C22×C4×C61—21no—
112C22×C4×C61—22no—
113C22×C4×C61—21no—
114C22×C4×C61—22no—
115C22×C4×C61—21no—
116C22×C4×C61—22no—
117C22×C4×C61—22no—
118C22×C4×C61—21no—
119C22×C4×C61—22no—
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×C61—21no—
128C22×C4×C61—21no—
129C22×C4×C61—21no—
130C22×C4×C61—21no—
131C22×C4×C61C22×C412yes—
132C22×C4×C61C22×C412yes—
133C22×C4×C61C2×D412yes—
134C22×C4×C61C2×D412yes—
135C22×C4×C61C2×D414yes—
136C22×C4×C61C2×D412yes—
137C22×C4×C61#4112yes—
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×Dic61—22no—
161C22×Dic61C22×C414yes—
162C22×Dic61C22×C414yes—
163C22×Dic61C2×D414yes—
164C22×Dic61C2×D414yes—
165C22×Dic61#4114yes—
166C2×Dic122—216no—
167C2×Dic122—28no—
168C2×Dic122—28no—
169C2×Dic122—28no—
170C2×Dic122—216no—
171C4⋊C4×C61—44no—
172C22⋊C4×C61—28no—
173C22⋊C4×C61—44no—
174C22⋊C4×C61—44no—
175C42×C61—28no—
176C22⋊Dic61—216no—
177C22⋊Dic61—48no—
178C4⋊Dic61—28no—
179#11758196—216no—
180#11758196—48no—
181C4×Dic61—216no—
182C23×C61—24no—
183C23×C61—22no—
184C23×C61—22no—
185C23×C61—24no—
186C23×C61—22no—
187C23×C61—41no—
188C23×C61—41no—
189C23×C61—42no—
190C23×C61—42no—
191C23×C61—41no—
192C23×C61—41no—
193C23×C61—42no—
194C23×C61—41no—
195C23×C61—42no—
196C23×C61—41no—
197C23×C61—42no—
198C23×C61—42no—
199C23×C61—41no—
200C23×C61—42no—
201C23×C61—41no—
202C23×C61—42no—
203C23×C61—42no—
204C23×C61—42no—
205C23×C61C23×C411yescharacteristic
206C23×C61C22×D412yes—
207C23×C61C22×D411yescharacteristic
208C23×C61C2×#4112yes—
209C23×C61C2×#4111yescharacteristic
210C23×C61H(5,𝔽2)12yes—
211C23×C61H(5,𝔽2)11yescharacteristic
212C23×C612−1+411yescharacteristic
213C23×C612−1+411yescharacteristic
214C23×C612−1+412yes—
215C23×C612−1+411yescharacteristic
216C2×C4×C61—24no—
217C2×C4×C61—22no—
218C2×C4×C61—22no—
219C2×C4×C61—24no—
220C2×C4×C61—22no—
221C2×C4×C61—22no—
222C2×C4×C61—24no—
223C2×C4×C61—24no—
224C2×C4×C61—22no—
225C2×C4×C61—24no—
226C2×C4×C61—22no—
227C2×C4×C61—22no—
228C2×C4×C61—22no—
229C2×C4×C61—41no—
230C2×C4×C61—42no—
231C2×C4×C61—41no—
232C2×C4×C61—42no—
233C2×C4×C61—42no—
234C2×C4×C61—41no—
235C2×C4×C61—41no—
236C2×C4×C61—42no—
237C2×C4×C61—41no—
238C2×C4×C61—41no—
239C2×C4×C61—41no—
240C2×C4×C61—41no—
241C2×C4×C61—42no—
242C2×C4×C61—42no—
243C2×C4×C61—42no—
244C2×C4×C61C4×D414yes—
245C2×C4×C61#12014yes—
246C2×Dic61—24no—
247C2×Dic61—24no—
248C2×Dic61—24no—
249C2×Dic61—24no—
250C2×Dic61—24no—
251C2×Dic61—24no—
252C2×Dic61—24no—
253C2×Dic61—24no—
254C2×Dic61—24no—
255C2×Dic61—24no—
256C2×Dic61—24no—
257C2×Dic61—24no—
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×Dic61C4×D418yes—
274Dic122—232no—
275#14301—611nomaximal, Sylow
276C22×C61—44no—
277C22×C61—42no—
278C22×C61—44no—
279C22×C61—42no—
280C22×C61—42no—
281C22×C61—81no—
282C22×C61—81no—
283C22×C61—81no—
284C22×C61—82no—
285C22×C61—81no—
286C22×C61—82no—
287C22×C61C2×C4×D412yes—
288C22×C61C2×C4×D411yescharacteristic
289C22×C61#51911yescharacteristic
290C22×C61#52012yes—
291C22×C61#52011yescharacteristic
292C22×C61C2×#12011yescharacteristic
293C22×C61#52511yescharacteristic
294C22×C61#52512yes—
295C22×C61#53512yes—
296C22×C61#53511yescharacteristic
297C22×C61#53611yescharacteristic
298C22×C61#53611yescharacteristic
299C22×C61#53612yes—
300C22×C61#53612yes—
301C22×C61#53711yescharacteristic
302C22×C61#53712yes—
303C22×C61#54112yes—
304C22×C61#54111yescharacteristic
305C22×C61#54212yes—
306C22×C61#54212yes—
307C22×C61#54212yes—
308C22×C61#54212yes—
309C22×C61#54211yescharacteristic
310C4×C61—24no—
311C4×C61—24no—
312C4×C61—42no—
313C4×C61—42no—
314C4×C61—81no—
315C4×C61—81no—
316Dic61—28no—
317Dic61—44no—
318Dic61—44no—
319Dic61—82no—
320C22×C4×Q8—611no—
321C2×#393—611no—
322C2×#388—612no—
323C2×#388D6111yescharacteristic
324#1446—611no—
325#1444—611no—
326#1444—612no—
327#1439—612no—
328#1439—611no—
329#1434—611no—
330#1434—612no—
331C2×C61—82no—
332C2×C61C4×#12011yescharacteristic
333C2×C61#232312yes—
334C2×C61#232311yescharacteristic
335C2×C61#232612yes—
336C2×C61#232611yescharacteristic
337C2×C61#261411yescharacteristic
338C2×C61#261511yescharacteristic
339C2×C61#263211yescharacteristic
340C2×C61#263212yes—
341C2×C61#263711yescharacteristic
342C2×C61#263712yes—
343C23×Q8—611no—
344C2×C4×Q8—1228no—
345C2×C4×Q8—1224no—
346C22×C4⋊C4—612no—
347C22×C4⋊C4—611no—
348C22×C4⋊C4—612no—
349C22×C4⋊C4C2×D6111yescharacteristic
350C22×C22⋊C4—612no—
351C22×C22⋊C4—612no—
352C22×C22⋊C4C2×D6111yescharacteristic
353C22×C22⋊C4C2×D6111yescharacteristic
354C22×C22⋊C4C2×D6111yescharacteristic
355C22×C42—612no—
356C22×C42C2×D6111yescharacteristic
357#393—618no—
358#388—24no—
359#388—1228no—
360C2×#95—612no—
361C2×#95—611no—
362C2×#95—612no—
363C2×#95—612no—
364C2×#95—611no—
365C2×#95—612no—
366C2×#95—612no—
367C2×#95—612no—
368C2×#95—612no—
369C2×#95C2×D6111yescharacteristic
370C2×#95C2×D6111yescharacteristic
371C61#1430111yescharacteristic, complemented, Sylow
372C25C22×D6111yescharacteristic
373C22×Q8—614no—
374C22×Q8—1224no—
375C22×Q8—1224no—
376C22×Q8—1222no—
377C22×Q8—1222no—
378C23×C4—612no—
379C23×C4—612no—
380C23×C4—612no—
381C23×C4—612no—
382C23×C4C22×D6111yescharacteristic
383C23×C4C22×D6111yescharacteristic
384C23×C4C22×D6111yescharacteristic
385C23×C4C22×D6111yescharacteristic
386C23×C4C22×D6111yescharacteristic
387C23×C4C22×D6111yescharacteristic
388C4×Q8—12232no—
389C2×C4⋊C4—22no—
390C2×C4⋊C4—42no—
391C2×C4⋊C4—1224no—
392C2×C4⋊C4—1224no—
393C2×C4⋊C4—1224no—
394C2×C4⋊C4—1224no—
395C2×C4⋊C4—1222no—
396C2×C4⋊C4—1224no—
397C2×C4⋊C4—2442no—
398C2×C4⋊C4—2442no—
399C2×C22⋊C4—22no—
400C2×C22⋊C4—22no—
401C2×C22⋊C4—24no—
402C2×C22⋊C4—22no—
403C2×C22⋊C4—22no—
404C2×C22⋊C4—42no—
405C2×C22⋊C4—618no—
406C2×C22⋊C4—1224no—
407C2×C22⋊C4—1224no—
408C2×C22⋊C4—1224no—
409C2×C22⋊C4—2442no—
410C2×C22⋊C4—2442no—
411C2×C22⋊C4C4×D6114yes—
412C2×C42—24no—
413C2×C42—618no—
414C2×C42—1224no—
415C2×C42—1224no—
416C2×C42C22⋊D6114yes—
417#95—22no—
418#95—22no—
419#95—618no—
420#95—614no—
421#95—1224no—
422#95—1224no—
423#95—1222no—
424#95—1224no—
425#95—2442no—
426#95—2442no—
427#95—2442no—
428C24—22no—
429C24—21no—
430C24—22no—
431C24—21no—
432C24—22no—
433C24—21no—
434C24—21no—
435C24—21no—
436C24—22no—
437C24—21no—
438C24C2×C4×D6112yes—
439C24C23×D6111yescharacteristic
440C2×Q8—12216no—
441C2×Q8—1228no—
442C2×Q8—1228no—
443C2×Q8—12216no—
444C2×Q8—1228no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—21no—
448C22×C4—22no—
449C22×C4—21no—
450C22×C4—22no—
451C22×C4—21no—
452C22×C4—21no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—21no—
456C22×C4—22no—
457C22×C4—21no—
458C22×C4—22no—
459C22×C4—21no—
460C22×C4—21no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—21no—
465C22×C4—21no—
466C22×C4—21no—
467C22×C4—21no—
468C22×C4—614no—
469C22×C4—614no—
470C22×C4—614no—
471C22×C4—614no—
472C22×C4—614no—
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—1222no—
488C22×C4—1222no—
489C22×C4—1222no—
490C22×C4—1222no—
491C22×C4—1222no—
492C22×C4—1222no—
493C22×C4—1222no—
494C22×C4—1222no—
495C22×C4—1222no—
496C22×C4C2×C4×D6112yes—
497C22×C4C2×C4×D6112yes—
498C22×C4#1175822212yes—
499C22×C4D4×D6114yes—
500C22×C4C2×C22⋊D6112yes—
501C22×C4C2×C22⋊D6112yes—
502C22×C4C2×C22⋊D6112yes—
503C4⋊C4—44no—
504C4⋊C4—12216no—
505C4⋊C4—1228no—
506C4⋊C4—2448no—
507C22⋊C4—28no—
508C22⋊C4—44no—
509C22⋊C4—44no—
510C22⋊C4—12216no—
511C22⋊C4—2448no—
512C42—28no—
513C42—12216no—
514C23—24no—
515C23—22no—
516C23—22no—
517C23—24no—
518C23—22no—
519C23—41no—
520C23—41no—
521C23—42no—
522C23—42no—
523C23—41no—
524C23—41no—
525C23—41no—
526C23—42no—
527C23—42no—
528C23—41no—
529C23—42no—
530C23—42no—
531C23—41no—
532C23—42no—
533C23—42no—
534C23—42no—
535C23—42no—
536C23—41no—
537C23C22×C4×D6111yescharacteristic
538C23C2×D4×D6112yes—
539C23C22×C22⋊D6111yescharacteristic
540C23#41×D6112yes—
541C23C2×#1175822211yescharacteristic
542C23#1451982212yes—
543C23#1451982311yescharacteristic
544C23#1451982411yescharacteristic
545C23#1451982411yescharacteristic
546C23#1451982412yes—
547C23#1451982411yescharacteristic
548Q8—12232no—
549C2×C4—24no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—24no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—24no—
556C2×C4—24no—
557C2×C4—22no—
558C2×C4—24no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—42no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—42no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—618no—
578C2×C4—1224no—
579C2×C4—1224no—
580C2×C4—1224no—
581C2×C4—1224no—
582C2×C4—1224no—
583C2×C4—1224no—
584C2×C4—1224no—
585C2×C4—1224no—
586C2×C4—1224no—
587C2×C4—1224no—
588C2×C4—1224no—
589C2×C4—1224no—
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×C4—2442no—
601C2×C4—2442no—
602C2×C4—2442no—
603C2×C4—2442no—
604C2×C4—2442no—
605C2×C4C4×C22⋊D6114yes—
606C2×C4#1451974514yes—
607C22—44no—
608C22—42no—
609C22—44no—
610C22—42no—
611C22—42no—
612C22—81no—
613C22—81no—
614C22—81no—
615C22—82no—
616C22—81no—
617C22—82no—
618C22C4×D4×D6112yes—
619C22C2×C4×C22⋊D6111yescharacteristic
620C22#1684023911yescharacteristic
621C22#1684024112yes—
622C22#1684024211yescharacteristic
623C22C2×#1451974511yescharacteristic
624C22#1684026911yescharacteristic
625C22#1684027112yes—
626C22#1684032812yes—
627C22#1684032911yescharacteristic
628C22#1684033311yescharacteristic
629C22#1684033311yescharacteristic
630C22#1684033412yes—
631C22#1684033412yes—
632C22#1684034011yescharacteristic
633C22#1684034112yes—
634C22#1684036912yes—
635C22#1684037411yescharacteristic
636C22#1684037612yes—
637C22#1684037612yes—
638C22#1684037712yes—
639C22#1684037812yes—
640C22#1684038311yescharacteristic
641C4—24no—
642C4—24no—
643C4—42no—
644C4—42no—
645C4—81no—
646C4—81no—
647C4—1228no—
648C4—4882no—
649C4—2444no—
650C4—2444no—
651C2—82no—
652C2C4×#1451974511yescharacteristic
653C2#2144849812yes—
654C2#2144850111yescharacteristic
655C2#2144852812yes—
656C2#2144853111yescharacteristic
657C2#2145154011yescharacteristic
658C2#2145154411yescharacteristic
659C2#2145168011yescharacteristic
660C2#2145168312yes—
661C2#2145172711yescharacteristic
662C2#2145173112yes—
subgroupG/H
center, FrattiniC24C23×D61
commutatorC23×C61C23×C4
FittingC2×C61×#388C2
socle, cosocleC24×C61C24
prime radicalC25×C61C23
hypercenterC2×#388D61
hypocenterC61#14301
LCSG ▷ C23×C61 ▷ C61
UCSC1 ◁ C24 ◁ C2×#388

Complements

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

NH ≅ G/Ntype
1. C22×C4×Dic122C2semidirect
4. #21444089C2semidirect
5. #21444083C2semidirect
6. #21444082C2semidirect
9. #21444046C2semidirect
10. #21444043C2semidirect
21. #16839674C4semidirect
371. C61#14301semidirect

Maximal subgroups

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

subgroupindex
C22×C4×Dic1222
C2×#168396742
C2×#168396502
#214440892
#214440832
#214440822
#214440642
#214440622
#214440462
#214440432
C2×C61×#3882
#1430161