Finite Group Databasebeta

← #29681720

Subgroups — #29681720

Order 15616.

1894 conjugacy classes of subgroups.

The subgroup lattice has 632 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×M16×D61C211yesmaximal, characteristic
2C22×#14519753C211yesmaximal, characteristic
3C2×#16839771C211yesmaximal, characteristic, complemented
4#21448198C211yesmaximal, characteristic
5#21446249C214yesmaximal, complemented
6#21446239C214yesmaximal, complemented
7#21445540C211yesmaximal, characteristic, complemented
8#21445539C211yesmaximal, characteristic, complemented
9C2×C61×#419C211yesmaximal, characteristic, complemented
10C23×C4×D61C2211yescharacteristic
11C22×#11758194C2211yescharacteristic
12C22×#11758189C2212yes—
13C2×M16×D61—24no—
14C2×M16×D61—24no—
15C2×M16×D61—24no—
16C22×C8×D61C2212yes—
17C2×#14519753—21no—
18C2×#14519753—21no—
19C2×#14519753—24no—
20C2×#14519753C418yescomplemented
21C2×#14519753C2212yes—
22C2×#14519753C2212yes—
23C2×#14519736C2211yescharacteristic
24C22×C4⋊Dic61C2211yescharacteristic
25C22×#11758196—21no—
26C22×#11758198—21no—
27#16839771—24no—
28C2×#14519701C2214yescomplemented
29#16839623—21no—
30#16839619C2212yes—
31#16839619C2212yes—
32#16839617—21no—
33C4×#11758198—21no—
34C4×#11758198—21no—
35#16839525C2214yescomplemented
36#16839524C2214yescomplemented
37C23×Q8×C61C2211yescharacteristic
38C22×M16×C61C2211yescharacteristic
39C2×#117×C61C2211yescharacteristic
40C61×#419—24no—
41C2×#103×C61C2214yescomplemented
42C24×D61D411yescharacteristic
43C23×Dic61—21no—
44C23×Dic61D411yescharacteristic
45C22×C4×D61—22no—
46C22×C4×D61—22no—
47C22×C4×D61—22no—
48C22×C4×D61—22no—
49C22×C4×D61C2×C414yes—
50C22×C4×D61D411yescharacteristic
51C22×C4×D61D411yescharacteristic
52C22×C4×D61D414yes—
53C22×C4×D61C2311yescharacteristic
54C22×C4×D61C2311yescharacteristic
55C2×#11758194—22no—
56C2×#11758194—22no—
57C2×#11758194D414yes—
58C2×#11758189—24no—
59C2×#11758189—24no—
60C2×#11758189—24no—
61M16×D61—232no—
62C22×#92418C2312yes—
63C2×C8×D61—24no—
64C2×C8×D61—24no—
65C2×C8×D61—24no—
66#14519753—28no—
67#14519753—28no—
68#14519753—24no—
69#14519753—24no—
70#14519753C2×C418yes—
71#14519753C2×C418yes—
72#14519736—22no—
73#14519736—22no—
74C2×C4⋊Dic61—21no—
75C2×C4⋊Dic61—22no—
76C2×C4⋊Dic61C2×C414yes—
77C2×C4⋊Dic61C2312yes—
78C2×C4⋊Dic61C2312yes—
79C2×#11758196—28no—
80C2×#11758196—22no—
81C2×#11758196—22no—
82C2×C22⋊Dic61—21no—
83C2×#11758198—28no—
84C2×#11758198—22no—
85C2×#11758198—22no—
86C2×#11758198—22no—
87C2×#11758198—22no—
88C2×C4×Dic61—21no—
89#14519701—28no—
90#14519684—22no—
91#14519684—22no—
92C22×Q8×C61—22no—
93C22×Q8×C61—21no—
94C22×Q8×C61—22no—
95C22×Q8×C61—22no—
96C22×Q8×C61—22no—
97C22×Q8×C61—22no—
98C22×Q8×C61C2×C414yes—
99C22×Q8×C61C2312yes—
100C23×C4×C61—21no—
101C23×C4×C61C2311yescharacteristic
102C2×M16×C61—22no—
103C2×M16×C61—22no—
104C2×M16×C61D414yes—
105C22×C8×C61C2312yes—
106#117×C61—22no—
107#117×C61—22no—
108C2×C4⋊C4×C61C2312yes—
109C2×C22⋊C4×C61—21no—
110C2×C42×C61—21no—
111#103×C61—28no—
112C24×C61C2×D411yescharacteristic
113C23×D61—22no—
114C23×D61—22no—
115C23×D61—24no—
116C23×D61—42no—
117C23×D61C22⋊C414yes—
118C23×D61C2×D411yescharacteristic
119C23×D61C2×D411yescharacteristic
120C2×Q8×C61—24no—
121C2×Q8×C61—22no—
122C2×Q8×C61—28no—
123C2×Q8×C61—28no—
124C2×Q8×C61—22no—
125C2×Q8×C61—22no—
126C2×Q8×C61—22no—
127C2×Q8×C61—22no—
128C2×Q8×C61—44no—
129C2×Q8×C61—42no—
130C2×Q8×C61—42no—
131C2×Q8×C61—44no—
132C2×Q8×C61C22×C414yes—
133C22×C4×C61—24no—
134C22×C4×C61—22no—
135C22×C4×C61—22no—
136C22×C4×C61—42no—
137C22×C4×C61—42no—
138C22×C4×C61C22×C412yes—
139C22×C4×C61C2×D411yescharacteristic
140C22×C4×C61C2×D412yes—
141C22×C4×C61C2×D412yes—
142C22×C4×C61C2×D412yes—
143C22×C4×C61#4112yes—
144C22×C4×C61#4112yes—
145C22×C4×C61C2411yescharacteristic
146C22×Dic61—22no—
147C22×Dic61—22no—
148C22×Dic61—24no—
149C22×Dic61—22no—
150C22×Dic61—22no—
151C22×Dic61—44no—
152C22×Dic61C22⋊C414yes—
153C22×Dic61C2×D411yescharacteristic
154C22×Dic61C2×D411yescharacteristic
155C2×C4×D61—28no—
156C2×C4×D61—28no—
157C2×C4×D61—28no—
158C2×C4×D61—22no—
159C2×C4×D61—22no—
160C2×C4×D61—22no—
161C2×C4×D61—22no—
162C2×C4×D61—28no—
163C2×C4×D61—21no—
164C2×C4×D61—21no—
165C2×C4×D61—21no—
166C2×C4×D61—21no—
167C2×C4×D61—21no—
168C2×C4×D61—21no—
169C2×C4×D61—21no—
170C2×C4×D61—21no—
171C2×C4×D61C22⋊C414yes—
172C2×C4×D61C22⋊C414yes—
173C2×C4×D61C22×C412yes—
174C2×C4×D61C22×C412yes—
175C2×C4×D61C2×D412yes—
176C2×C4×D61C2×D412yes—
177M16×C61—28no—
178C2×C8×C61—22no—
179C2×C8×C61—22no—
180C2×C8×C61C2×D414yes—
181C4⋊C4×C61—22no—
182C4⋊C4×C61—22no—
183C22⋊C4×C61—42no—
184C42×C61—42no—
185C22⋊Dic61—42no—
186#11758198—28no—
187#11758198—28no—
188#11758198—44no—
189#11758198—44no—
190C4⋊Dic61—24no—
191C4⋊Dic61—22no—
192C4⋊Dic61—22no—
193C4⋊Dic61—22no—
194C4⋊Dic61C22×C414yes—
195#11758196—28no—
196#11758196—28no—
197C4×Dic61—42no—
198#11758194—28no—
199C2×#92418—22no—
200C2×#92418—22no—
201C2×#92418C2×D414yes—
202#11758189—216no—
203C8×D61—216no—
204C23×C61—22no—
205C23×C61—22no—
206C23×C61C2×C22⋊C412yes—
207C23×C61#12112yes—
208C23×C61#12112yes—
209C23×C61C22×D411yescharacteristic
210C22×D61—21no—
211C22×D61—21no—
212C22×D61—22no—
213C22×D61—21no—
214C22×D61—21no—
215C22×D61—22no—
216C22×D61—22no—
217C22×D61—28no—
218C22×D61—28no—
219C22×D61—44no—
220C22×D61—48no—
221C22×D61—48no—
222C22×D61C2×C22⋊C412yes—
223C22×D61C2×C22⋊C412yes—
224C22×D61#13714yes—
225Q8×C61—24no—
226Q8×C61—24no—
227Q8×C61—42no—
228Q8×C61—42no—
229Q8×C61—48no—
230C2×C4×C61—24no—
231C2×C4×C61—42no—
232C2×C4×C61—42no—
233C2×C4×C61—48no—
234C2×C4×C61—42no—
235C2×C4×C61—42no—
236C2×C4×C61—42no—
237C2×C4×C61C2×C22⋊C412yes—
238C2×C4×C61C4×D414yes—
239C2×C4×C61C4×D414yes—
240C2×C4×C61#12014yes—
241C2×C4×C61#12112yes—
242C2×C4×C61#12112yes—
243C2×C4×C61#12314yes—
244C2×C4×C61C23×C411yescharacteristic
245C2×C4×C61C22×D411yescharacteristic
246C2×C4×C61C22×D411yescharacteristic
247C2×C4×C61C22×D411yescharacteristic
248C2×C4×C61C2×#4111yescharacteristic
249C2×C4×C61C2×#4111yescharacteristic
250C2×Dic61—21no—
251C2×Dic61—21no—
252C2×Dic61—21no—
253C2×Dic61—21no—
254C2×Dic61—28no—
255C2×Dic61—24no—
256C2×Dic61—48no—
257C2×Dic61—44no—
258C2×Dic61—42no—
259C2×Dic61—42no—
260C2×Dic61—42no—
261C2×Dic61C2×C22⋊C412yes—
262C2×Dic61C2×C22⋊C412yes—
263C4×D61—22no—
264C4×D61—22no—
265C4×D61—22no—
266C4×D61—22no—
267C4×D61—22no—
268C4×D61—22no—
269C4×D61—22no—
270C4×D61—22no—
271C4×D61—28no—
272C4×D61—28no—
273C8×C61—24no—
274#92418—24no—
275#20972—611nomaximal, Sylow
276C22×C61—22no—
277C22×C61—22no—
278C22×C61—24no—
279C22×C61—24no—
280C22×C61#38614yes—
281C22×C61C22×C22⋊C411yescharacteristic
282C22×C61C2×#12111yescharacteristic
283C22×C61C2×#12111yescharacteristic
284C22×C61C2×#13712yes—
285C22×C61#57512yes—
286C2×D61—22no—
287C2×D61—22no—
288C2×D61—24no—
289C2×D61—24no—
290C2×D61—22no—
291C2×D61—22no—
292C2×D61—22no—
293C2×D61—416no—
294C2×D61—42no—
295C2×D61—42no—
296C2×D61#41914yescomplemented
297C4×C61—44no—
298C4×C61—82no—
299C4×C61#38614yes—
300C4×C61C2×C4×D411yescharacteristic
301C4×C61C2×C4×D411yescharacteristic
302C4×C61C2×#12011yescharacteristic
303C4×C61C2×#12311yescharacteristic
304Dic61—22no—
305Dic61—22no—
306Dic61—44no—
307Dic61—82no—
308C23×M16—611no—
309C22×C22⋊Q8—611no—
310C2×#419—611no—
311C2×#419D6111yescharacteristic, complemented
312#2280—611no—
313#1894—614no—
314#1892—614no—
315#1736—611no—
316#1736—611no—
317C2×C61—24no—
318C2×C61C2×#38611yescharacteristic
319C2×C61C2×#41911yescharacteristic
320C2×C61#289611yescharacteristic
321C2×C61#305612yes—
322C2×C61#305612yes—
323D61—24no—
324D61—42no—
325C23×Q8C2×D6111yescharacteristic
326C24×C4—611no—
327C22×M16—611no—
328C22×M16—612no—
329C22×M16—1224no—
330C22×M16—1224no—
331C22×M16—1224no—
332C22×M16C2×D6111yescharacteristic
333C23×C8—612no—
334C2×C22⋊Q8—618no—
335C2×C22⋊Q8—612no—
336C2×C22⋊Q8—612no—
337C2×C22⋊Q8—1224no—
338C2×C22⋊Q8—1221no—
339C2×C22⋊Q8—1221no—
340C2×#117—611no—
341C2×#117C2×D6111yescharacteristic
342C22×C4⋊C4—611no—
343C22×C4⋊C4—1221no—
344C22×C22⋊C4—1221no—
345#419—24no—
346#419—1224no—
347C2×#103—614no—
348C2×#103C2×D6114yescomplemented
349#381—1221no—
350#380—612no—
351#380—612no—
352#379—1221no—
353C4×C22⋊C4—1221no—
354C4×C22⋊C4—1221no—
355#340—614no—
356#340—614no—
357C61#2097211yescharacteristic, complemented, Sylow
358C25—611no—
359C22×Q8—21no—
360C22×Q8—22no—
361C22×Q8—22no—
362C22×Q8—22no—
363C22×Q8—22no—
364C22×Q8—22no—
365C22×Q8C4×D6114yes—
366C22×Q8C22×D6112yes—
367C23×C4—21no—
368C23×C4—611no—
369C23×C4—614no—
370C23×C4—614no—
371C23×C4—611no—
372C23×C4—611no—
373C23×C4—611no—
374C23×C4—611no—
375C23×C4—1222no—
376C23×C4—1222no—
377C23×C4—1222no—
378C23×C4—1222no—
379C23×C4—1221no—
380C23×C4C22×D6111yescharacteristic
381C2×M16—22no—
382C2×M16—22no—
383C2×M16—614no—
384C2×M16—1224no—
385C2×M16—1224no—
386C2×M16—12232no—
387C2×M16—1222no—
388C2×M16—1222no—
389C2×M16—1224no—
390C2×M16C22⋊D6114yes—
391C22×C8—612no—
392C22×C8—1224no—
393C22×C8—1224no—
394C22×C8—1224no—
395C22×C8C22×D6112yes—
396C22⋊Q8—618no—
397C22⋊Q8—618no—
398C22⋊Q8—1224no—
399C22⋊Q8—1224no—
400C22⋊Q8—1228no—
401C22⋊Q8—1228no—
402#117—22no—
403#117—22no—
404#117—1222no—
405#117—1222no—
406C2×C4⋊C4—614no—
407C2×C4⋊C4—612no—
408C2×C4⋊C4—612no—
409C2×C4⋊C4—1222no—
410C2×C4⋊C4—1222no—
411C2×C4⋊C4—1228no—
412C2×C4⋊C4—1222no—
413C2×C4⋊C4—1221no—
414C2×C4⋊C4C22×D6112yes—
415C2×C22⋊C4—21no—
416C2×C22⋊C4—1222no—
417C2×C22⋊C4—1222no—
418C2×C22⋊C4—1228no—
419C2×C22⋊C4—1222no—
420C2×C22⋊C4—1222no—
421C2×C22⋊C4—1221no—
422C2×C42—21no—
423C2×C42—1221no—
424#103—28no—
425#103—1228no—
426#95—1222no—
427#95—1222no—
428C24—611no—
429C24—611no—
430C24—614no—
431C24—1224no—
432C24—1222no—
433C24—1222no—
434C24—2442no—
435C24D4×D6111yescharacteristic
436C2×Q8—22no—
437C2×Q8—24no—
438C2×Q8—22no—
439C2×Q8—22no—
440C2×Q8—22no—
441C2×Q8—22no—
442C2×Q8—28no—
443C2×Q8—28no—
444C2×Q8—44no—
445C2×Q8—42no—
446C2×Q8—42no—
447C2×Q8—44no—
448C2×Q8C2×C4×D6114yes—
449C22×C4—22no—
450C22×C4—24no—
451C22×C4—22no—
452C22×C4—42no—
453C22×C4—42no—
454C22×C4—611no—
455C22×C4—611no—
456C22×C4—614no—
457C22×C4—612no—
458C22×C4—612no—
459C22×C4—612no—
460C22×C4—612no—
461C22×C4—614no—
462C22×C4—614no—
463C22×C4—1222no—
464C22×C4—1222no—
465C22×C4—1221no—
466C22×C4—1221no—
467C22×C4—1221no—
468C22×C4—1221no—
469C22×C4—1221no—
470C22×C4—1221no—
471C22×C4—1221no—
472C22×C4—1221no—
473C22×C4—1228no—
474C22×C4—1228no—
475C22×C4—1228no—
476C22×C4—1222no—
477C22×C4—1222no—
478C22×C4—1222no—
479C22×C4—1222no—
480C22×C4—1228no—
481C22×C4—1222no—
482C22×C4—1224no—
483C22×C4—1222no—
484C22×C4—2444no—
485C22×C4C2×C4×D6112yes—
486C22×C4#1175822212yes—
487C22×C4D4×D6112yes—
488C22×C4D4×D6112yes—
489C22×C4D4×D6111yescharacteristic
490C22×C4#1175822412yes—
491C22×C4C2×C22⋊D6112yes—
492C22×C4C23×D6111yescharacteristic
493M16—28no—
494M16—12216no—
495M16—1228no—
496C2×C8—22no—
497C2×C8—22no—
498C2×C8—614no—
499C2×C8—12216no—
500C2×C8—1222no—
501C2×C8—1222no—
502C2×C8C2×C22⋊D6114yes—
503C4⋊C4—22no—
504C4⋊C4—22no—
505C4⋊C4—614no—
506C4⋊C4—1228no—
507C4⋊C4—1228no—
508C4⋊C4—1222no—
509C4⋊C4—1224no—
510C4⋊C4—1222no—
511C4⋊C4—1222no—
512C22⋊C4—42no—
513C22⋊C4—1228no—
514C22⋊C4—1228no—
515C22⋊C4—2444no—
516C22⋊C4—2444no—
517C22⋊C4—2442no—
518C42—42no—
519C42—2442no—
520C23—22no—
521C23—22no—
522C23—612no—
523C23—614no—
524C23—612no—
525C23—1228no—
526C23—1221no—
527C23—1221no—
528C23—1222no—
529C23—1221no—
530C23—1221no—
531C23—1222no—
532C23—1222no—
533C23—1228no—
534C23—2444no—
535C23—2448no—
536C23—2448no—
537C23C22⋊C4×D6112yes—
538C23#1451974812yes—
539C23#1451975012yes—
540C23C2×D4×D6111yescharacteristic
541Q8—24no—
542Q8—24no—
543Q8—42no—
544Q8—42no—
545Q8—48no—
546C2×C4—24no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—48no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—612no—
554C2×C4—612no—
555C2×C4—1221no—
556C2×C4—1221no—
557C2×C4—1221no—
558C2×C4—1221no—
559C2×C4—1228no—
560C2×C4—1222no—
561C2×C4—1222no—
562C2×C4—1222no—
563C2×C4—1222no—
564C2×C4—1222no—
565C2×C4—1222no—
566C2×C4—1222no—
567C2×C4—1222no—
568C2×C4—1228no—
569C2×C4—1228no—
570C2×C4—1224no—
571C2×C4—2442no—
572C2×C4—2442no—
573C2×C4—2442no—
574C2×C4—2448no—
575C2×C4—2444no—
576C2×C4C22⋊C4×D6112yes—
577C2×C4#1451973714yes—
578C2×C4C4×C22⋊D6114yes—
579C2×C4#1451974514yes—
580C2×C4#1451974812yes—
581C2×C4#1451975012yes—
582C2×C4#1451975914yes—
583C2×C4C22×C4×D6111yescharacteristic
584C2×C4C2×D4×D6111yescharacteristic
585C2×C4C2×D4×D6111yescharacteristic
586C2×C4C22×C22⋊D6111yescharacteristic
587C2×C4C2×#1175822411yescharacteristic
588C2×C4C2×#1175822211yescharacteristic
589C8—24no—
590C8—1224no—
591C22—22no—
592C22—24no—
593C22—24no—
594C22—22no—
595C22—614no—
596C22—1222no—
597C22—1222no—
598C22—1224no—
599C22—1224no—
600C22—1222no—
601C22—1222no—
602C22—1222no—
603C22—24416no—
604C22—2442no—
605C22—2442no—
606C22#1683963814yes—
607C22C2×C22⋊C4×D6111yescharacteristic
608C22C2×#1451974811yescharacteristic
609C22C2×#1451975011yescharacteristic
610C22#137×D6112yes—
611C22#1684056112yes—
612C4—44no—
613C4—82no—
614C4—1222no—
615C4—1222no—
616C4—4882no—
617C4—2444no—
618C4#1683963814yes—
619C4C2×#1451973711yescharacteristic
620C4C2×C4×C22⋊D6111yescharacteristic
621C4C2×#1451974511yescharacteristic
622C4C2×#1451975911yescharacteristic
623C2—24no—
624C2—1224no—
625C2—2442no—
626C2C2×#1683963811yescharacteristic
627C2#419×D6111yescharacteristic
628C2#2145433711yescharacteristic
629C2#2145544312yes—
630C2#2145544712yes—
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC2×C61×#419C2
socleC23×C61C22×D4
prime radicalC24×D61D4
cosocleC22×C4×C61C24
hypercenterC2×#419D61
hypocenterC61#20972
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#419

Complements

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

NH ≅ G/Ntype
3. C2×#16839771C2semidirect
5. #21446249C2semidirect
6. #21446239C2semidirect
7. #21445540C2semidirect
8. #21445539C2semidirect
9. C2×C61×#419C2semidirect
20. C2×#14519753C4semidirect
28. C2×#14519701C22semidirect
35. #16839525C22semidirect
36. #16839524C22semidirect
41. C2×#103×C61C22semidirect
296. C2×D61#419semidirect
311. C2×#419D61semidirect
348. C2×#103C2×D61semidirect
357. C61#20972semidirect

Maximal subgroups

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

subgroupindex
C22×M16×D612
C22×#145197532
C2×#168397712
#214481982
#214462492
#214462392
#214455402
#214455392
C2×C61×#4192
#2097261