Finite Group Databasebeta

← C2×#21446121 #29681187

Subgroups — C2×#21446121 #29681187

Order 15616.

2050 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×#16840273C211yesmaximal, characteristic
2C22×#14519703C212yesmaximal, complemented
3C2×C4×C22⋊Dic61C211yesmaximal, characteristic, complemented
4#21446121C218yesmaximal, direct product factor
5C2×#16839523C212yesmaximal, complemented
6C23×M16×C61C211yesmaximal, characteristic, complemented
7C22×#11758222C2211yescharacteristic
8#16840273—22no—
9#16840273—22no—
10#16840273C414yescomplemented
11#16840273C2214yes—
12C2×#14519760—21no—
13C2×#14519756C411yescharacteristic, complemented
14C2×#14519756C411yescharacteristic, complemented
15C2×#14519751C411yescharacteristic, complemented
16C2×#14519751C411yescharacteristic, complemented
17C2×#14519746—21no—
18C2×C4×C22⋊D61—21no—
19C2×C4×C22⋊D61—21no—
20C2×#14519736C2211yescharacteristic
21C22×C22⋊Dic61—21no—
22C22×C4×Dic61C2212yes—
23C2×#14519703—28no—
24C2×#14519703C2218yescomplemented
25C4×C22⋊Dic61—22no—
26C4×C22⋊Dic61C2214yescomplemented
27C2×#14519684—21no—
28#16839523C2218yescomplemented
29C24×C4×C61C2211yescharacteristic
30C22×M16×C61—21no—
31C22×M16×C61—22no—
32C22×M16×C61—28no—
33C22×M16×C61C2212yes—
34C22×M16×C61C2214yescomplemented
35C23×C8×C61—21no—
36C2×#11758222—21no—
37C2×#11758222—21no—
38C2×#11758222—24no—
39C2×#11758222—24no—
40C2×#11758222C2×C412yes—
41C2×#11758222C2312yes—
42C22×Dic122C2×C411yescharacteristic
43C22×D244C2×C411yescharacteristic
44C22×C22⋊D61—21no—
45C22×C22⋊D61—21no—
46C22×C22⋊D61—21no—
47C23×Dic61—22no—
48C22×C4×D61—21no—
49#14519760—24no—
50#14519760—24no—
51#14519756—22no—
52#14519756—22no—
53#14519756C2×C414yescomplemented
54#14519756C2×C414yescomplemented
55#14519751—22no—
56#14519751—22no—
57#14519751C2×C414yescomplemented
58#14519751C2×C414yescomplemented
59#14519746—24no—
60#14519746—24no—
61C4×C22⋊D61—24no—
62C4×C22⋊D61—24no—
63C4×C22⋊D61—24no—
64C4×C22⋊D61—24no—
65#14519736—21no—
66#14519736—21no—
67#14519736C2×C412yes—
68#14519736C2312yes—
69C2×C4⋊Dic61C2×C411yescharacteristic
70C2×C4⋊Dic61C2×C411yescharacteristic
71C2×#11758196—21no—
72C2×#11758196—21no—
73C2×C22⋊Dic61—28no—
74C2×C22⋊Dic61—21no—
75C2×C22⋊Dic61—21no—
76C2×C22⋊Dic61—24no—
77C2×C22⋊Dic61—21no—
78C2×#11758198—21no—
79C2×#11758198—21no—
80C2×C4×Dic61—21no—
81C2×C4×Dic61—24no—
82C2×C4×Dic61—24no—
83C2×C4×Dic61—22no—
84C2×C4×Dic61C2314yes—
85#14519703—216no—
86#14519684—24no—
87C25×C61D411yescharacteristic
88C23×C4×C61—24no—
89C23×C4×C61—24no—
90C23×C4×C61C2×C411yescharacteristic
91C23×C4×C61C2×C411yescharacteristic
92C23×C4×C61C2×C412yes—
93C23×C4×C61D411yescharacteristic
94C23×C4×C61D412yes—
95C23×C4×C61D411yescharacteristic
96C23×C4×C61D411yescharacteristic
97C23×C4×C61D412yes—
98C23×C4×C61C2312yes—
99C23×C4×C61C2311yescharacteristic
100C2×M16×C61—216no—
101C2×M16×C61—28no—
102C2×M16×C61—24no—
103C2×M16×C61—22no—
104C2×M16×C61—24no—
105C2×M16×C61—216no—
106C2×M16×C61D418yescomplemented
107C2×M16×C61C2314yes—
108C22×C8×C61—28no—
109C22×C8×C61—22no—
110C22×C8×C61—24no—
111C24×C61—24no—
112C24×C61—24no—
113C24×C61—42no—
114C24×C61C22⋊C411yescharacteristic
115C24×C61C22⋊C412yes—
116C24×C61C22⋊C411yescharacteristic
117C24×C61C2×D412yes—
118C24×C61C2×D411yescharacteristic
119C23×D61—21no—
120C22×C4×C61—24no—
121C22×C4×C61—24no—
122C22×C4×C61—24no—
123C22×C4×C61—28no—
124C22×C4×C61—22no—
125C22×C4×C61—24no—
126C22×C4×C61—22no—
127C22×C4×C61—28no—
128C22×C4×C61—22no—
129C22×C4×C61—22no—
130C22×C4×C61—24no—
131C22×C4×C61—24no—
132C22×C4×C61—24no—
133C22×C4×C61C22⋊C411yescharacteristic
134C22×C4×C61C22⋊C411yescharacteristic
135C22×C4×C61C22⋊C412yes—
136C22×C4×C61C22⋊C412yes—
137C22×C4×C61C22⋊C412yes—
138C22×C4×C61C22⋊C412yes—
139C22×C4×C61C22⋊C412yes—
140C22×C4×C61C22×C412yes—
141C22×C4×C61C22×C412yes—
142C22×C4×C61C22×C411yescharacteristic
143C22×C4×C61C22×C411yescharacteristic
144C22×C4×C61C2×D412yes—
145C22×C4×C61C2×D412yes—
146C22×C4×C61C2×D412yes—
147C22×C4×C61C2×D411yescharacteristic
148C22×C4×C61C2×D411yescharacteristic
149C22×C4×C61C2×D411yescharacteristic
150C22×C4×C61C2×D414yes—
151C22×C4×C61#4114yes—
152C22×C4×C61C2411yescharacteristic
153C2×C22⋊D61—22no—
154C2×C22⋊D61—24no—
155C2×C22⋊D61—24no—
156C2×C22⋊D61—22no—
157C2×C22⋊D61—44no—
158C2×C22⋊D61—44no—
159C2×C22⋊D61—44no—
160C22×Dic61—28no—
161C22×Dic61—21no—
162C22×Dic61—21no—
163C22×Dic61—21no—
164C22×Dic61—24no—
165C22×Dic61—44no—
166C22×Dic61—44no—
167#11758222—28no—
168#11758222—24no—
169#11758222—24no—
170#11758222—42no—
171#11758222—44no—
172#11758222—42no—
173C2×D244—24no—
174C2×D244—21no—
175C2×D244C22×C412yes—
176C2×C4×D61—22no—
177C2×C4×D61—22no—
178C2×C4×D61—42no—
179C2×C4×D61—42no—
180C2×Dic122—24no—
181C2×Dic122—21no—
182C2×Dic122C22×C412yes—
183M16×C61—216no—
184M16×C61—28no—
185M16×C61—24no—
186M16×C61C2×D418yescomplemented
187C2×C8×C61—216no—
188C2×C8×C61—28no—
189C2×C8×C61—24no—
190C22⋊Dic61—216no—
191C22⋊Dic61—22no—
192C22⋊Dic61—24no—
193C22⋊Dic61—24no—
194C22⋊Dic61—22no—
195#11758198—24no—
196#11758198—24no—
197C4⋊Dic61—21no—
198C4⋊Dic61—21no—
199C4⋊Dic61C22×C412yes—
200C4⋊Dic61C22×C412yes—
201#11758196—24no—
202#11758196—24no—
203C4×Dic61—22no—
204C4×Dic61—22no—
205C4×Dic61—24no—
206C4×Dic61—24no—
207C4×Dic61—44no—
208C23×C61—22no—
209C23×C61—22no—
210C23×C61—22no—
211C23×C61—22no—
212C23×C61—24no—
213C23×C61—24no—
214C23×C61—22no—
215C23×C61—22no—
216C23×C61—24no—
217C23×C61—24no—
218C23×C61—22no—
219C23×C61—42no—
220C23×C61—42no—
221C23×C61—416no—
222C23×C61#10414yescomplemented
223C23×C61C2×C22⋊C412yes—
224C23×C61C2×C22⋊C411yescharacteristic
225C23×C61C2×C22⋊C412yes—
226C23×C61C2×C22⋊C411yescharacteristic
227C23×C61#12114yes—
228C23×C61C22×D411yescharacteristic
229C22×D61—22no—
230C22×D61—42no—
231C22×D61—42no—
232C22×D61—42no—
233C2×C4×C61—24no—
234C2×C4×C61—24no—
235C2×C4×C61—22no—
236C2×C4×C61—22no—
237C2×C4×C61—24no—
238C2×C4×C61—24no—
239C2×C4×C61—22no—
240C2×C4×C61—22no—
241C2×C4×C61—22no—
242C2×C4×C61—22no—
243C2×C4×C61—24no—
244C2×C4×C61—24no—
245C2×C4×C61—24no—
246C2×C4×C61—24no—
247C2×C4×C61C2×C22⋊C412yes—
248C2×C4×C61C2×C22⋊C412yes—
249C2×C4×C61C2×C22⋊C411yescharacteristic
250C2×C4×C61C2×C22⋊C411yescharacteristic
251C2×C4×C61C4×D414yes—
252C2×C4×C61C4×D414yes—
253C2×C4×C61#12012yes—
254C2×C4×C61#12012yes—
255C2×C4×C61#12114yes—
256C2×C4×C61#12312yes—
257C2×C4×C61#12312yes—
258C2×C4×C61C23×C411yescharacteristic
259C2×C4×C61C22×D412yes—
260C2×C4×C61C22×D411yescharacteristic
261C2×C4×C61C2×#4112yes—
262C22⋊D61—48no—
263C22⋊D61—44no—
264C22⋊D61—44no—
265C22⋊D61—48no—
266C2×Dic61—28no—
267C2×Dic61—22no—
268C2×Dic61—22no—
269C2×Dic61—22no—
270C2×Dic61—416no—
271C2×Dic61—44no—
272C2×Dic61—44no—
273C2×Dic61—42no—
274C2×Dic61—42no—
275C2×Dic61—42no—
276D244—24no—
277D244—42no—
278C4×D61—42no—
279C4×D61—42no—
280C4×D61—44no—
281Dic122—24no—
282Dic122—42no—
283C8×C61—28no—
284C2×#1871—611nomaximal, Sylow
285C22×C61—28no—
286C22×C61—28no—
287C22×C61—21no—
288C22×C61—21no—
289C22×C61—21no—
290C22×C61—21no—
291C22×C61—22no—
292C22×C61—22no—
293C22×C61—22no—
294C22×C61—22no—
295C22×C61—22no—
296C22×C61—48no—
297C22×C61—48no—
298C22×C61—44no—
299C22×C61#38612yes—
300C22×C61#38612yes—
301C22×C61C2×#10412yes—
302C22×C61C2×#10414yescomplemented
303C22×C61#42112yes—
304C22×C61C22×C22⋊C411yescharacteristic
305C22×C61C2×#12112yes—
306C2×D61—42no—
307C2×D61—42no—
308C2×D61—42no—
309C2×D61—84no—
310C4×C61—22no—
311C4×C61—22no—
312C4×C61#38612yes—
313C4×C61#38612yes—
314C4×C61C2×C4×D412yes—
315C4×C61C2×#12011yescharacteristic
316C4×C61C2×#12311yescharacteristic
317Dic61—48no—
318Dic61—42no—
319Dic61—42no—
320Dic61—42no—
321C23×M16D6111yescharacteristic, complemented
322C2×#525—611no—
323C22×#104—612no—
324C2×C4×C22⋊C4—611no—
325#1871—618no—
326C2×#339—612no—
327C2×C61—22no—
328C2×C61—22no—
329C2×C61—24no—
330C2×C61—42no—
331C2×C61#187114yescomplemented
332C2×C61C2×#38611yescharacteristic
333C2×C61C22×#10411yescharacteristic
334C2×C61C2×#42111yescharacteristic
335D61—82no—
336C22×#41—611no—
337C24×C4C2×D6111yescharacteristic
338C22×M16—22no—
339C22×M16—21no—
340C22×M16—28no—
341C22×M16C2×D6112yes—
342C22×M16C2×D6114yescomplemented
343C23×C8—21no—
344#525—614no—
345#525—614no—
346#525—1222no—
347#525—1222no—
348C2×#123—1221no—
349C2×C22⋊Q8—611no—
350C2×C22⋊Q8—611no—
351C2×#121—611no—
352C2×#121—611no—
353C2×#120—1221no—
354C2×C4×D4—1221no—
355C2×C4×D4—1221no—
356C2×#117—611no—
357C22×C22⋊C4—1221no—
358C22×C42—612no—
359C2×#104—618no—
360C2×#104—1228no—
361C4×C22⋊C4—614no—
362C4×C22⋊C4—1222no—
363C2×#95—1221no—
364#339—618no—
365C61C2×#187111yescharacteristic, complemented, Sylow
366C25D24411yescharacteristic
367C2×#41—612no—
368C2×#41—612no—
369C2×#41—1221no—
370C2×#41—1221no—
371C2×#41—1224no—
372C2×#41—1224no—
373C22×Q8—611no—
374C22×D4—611no—
375C22×D4—1221no—
376C22×D4—1221no—
377C22×D4—1221no—
378C23×C4—24no—
379C23×C4—24no—
380C23×C4—1222no—
381C23×C4—1221no—
382C23×C4C4×D6111yescharacteristic
383C23×C4C4×D6111yescharacteristic
384C23×C4C4×D6112yes—
385C23×C4D24411yescharacteristic
386C23×C4D24412yes—
387C23×C4C22⋊D6112yes—
388C23×C4C22⋊D6111yescharacteristic
389C23×C4C22⋊D6111yescharacteristic
390C23×C4C22×D6112yes—
391C23×C4C22×D6111yescharacteristic
392C2×M16—216no—
393C2×M16—28no—
394C2×M16—22no—
395C2×M16—24no—
396C2×M16—24no—
397C2×M16—216no—
398C2×M16C22⋊D6118yescomplemented
399C2×M16C22×D6114yes—
400C22×C8—28no—
401C22×C8—24no—
402C22×C8—22no—
403#123—1224no—
404#123—1224no—
405C22⋊Q8—614no—
406C22⋊Q8—614no—
407C22⋊Q8—1222no—
408C22⋊Q8—1222no—
409#121—614no—
410#121—614no—
411#121—1222no—
412#121—1222no—
413#120—1224no—
414#120—1224no—
415C4×D4—1224no—
416C4×D4—1224no—
417C4×D4—1224no—
418C4×D4—1224no—
419#117—612no—
420#117—612no—
421#117—1221no—
422#117—1221no—
423C2×C4⋊C4—611no—
424C2×C4⋊C4—611no—
425C2×C4⋊C4—1221no—
426C2×C4⋊C4—1221no—
427C2×C22⋊C4—1228no—
428C2×C22⋊C4—1221no—
429C2×C22⋊C4—1221no—
430C2×C22⋊C4—1224no—
431C2×C22⋊C4—1221no—
432C2×C22⋊C4—1221no—
433C2×C22⋊C4—1221no—
434C2×C42—614no—
435C2×C42—1221no—
436C2×C42—1224no—
437C2×C42—1224no—
438C2×C42—1222no—
439#104—12216no—
440#95—1224no—
441C24—24no—
442C24—24no—
443C24—42no—
444C24—1221no—
445C24#1175819811yescharacteristic
446C24#1175819812yes—
447C24#1175819811yescharacteristic
448C24C2×D24412yes—
449C24C2×D24411yescharacteristic
450#41—1224no—
451#41—1224no—
452#41—1228no—
453#41—2442no—
454#41—2444no—
455#41—2442no—
456C2×Q8—612no—
457C2×Q8—1221no—
458C2×Q8—1224no—
459C2×D4—612no—
460C2×D4—1222no—
461C2×D4—1224no—
462C2×D4—1224no—
463C2×D4—1222no—
464C2×D4—1221no—
465C2×D4—1224no—
466C2×D4—2444no—
467C2×D4—2444no—
468C2×D4—2444no—
469C22×C4—24no—
470C22×C4—24no—
471C22×C4—24no—
472C22×C4—24no—
473C22×C4—24no—
474C22×C4—28no—
475C22×C4—22no—
476C22×C4—24no—
477C22×C4—22no—
478C22×C4—28no—
479C22×C4—22no—
480C22×C4—22no—
481C22×C4—24no—
482C22×C4—1228no—
483C22×C4—1224no—
484C22×C4—1221no—
485C22×C4—1221no—
486C22×C4—1221no—
487C22×C4—1222no—
488C22×C4—1222no—
489C22×C4—2444no—
490C22×C4—2444no—
491C22×C4—2442no—
492C22×C4—2442no—
493C22×C4#1175819811yescharacteristic
494C22×C4#1175819811yescharacteristic
495C22×C4#1175819812yes—
496C22×C4#1175819812yes—
497C22×C4#1175819812yes—
498C22×C4#1175819812yes—
499C22×C4#1175819812yes—
500C22×C4C2×C4×D6112yes—
501C22×C4C2×C4×D6112yes—
502C22×C4C2×C4×D6111yescharacteristic
503C22×C4C2×C4×D6111yescharacteristic
504C22×C4C2×D24412yes—
505C22×C4C2×D24411yescharacteristic
506C22×C4#1175822214yes—
507C22×C4C2×C22⋊D6112yes—
508C22×C4C2×C22⋊D6112yes—
509C22×C4C2×C22⋊D6111yescharacteristic
510C22×C4C2×C22⋊D6114yes—
511C22×C4C2×C22⋊D6111yescharacteristic
512C22×C4C23×D6111yescharacteristic
513M16—216no—
514M16—24no—
515M16—28no—
516M16C2×C22⋊D6118yescomplemented
517C2×C8—216no—
518C2×C8—24no—
519C2×C8—28no—
520C4⋊C4—612no—
521C4⋊C4—612no—
522C4⋊C4—1221no—
523C4⋊C4—1221no—
524C4⋊C4—1224no—
525C4⋊C4—1224no—
526C22⋊C4—12216no—
527C22⋊C4—1222no—
528C22⋊C4—1222no—
529C22⋊C4—1224no—
530C22⋊C4—1224no—
531C22⋊C4—1224no—
532C22⋊C4—1224no—
533C42—1222no—
534C42—1222no—
535C42—1224no—
536C42—1224no—
537C42—2444no—
538C23—22no—
539C23—22no—
540C23—22no—
541C23—22no—
542C23—24no—
543C23—22no—
544C23—24no—
545C23—24no—
546C23—22no—
547C23—24no—
548C23—22no—
549C23—42no—
550C23—42no—
551C23—416no—
552C23—1222no—
553C23—2442no—
554C23—2442no—
555C23—2442no—
556C23#1451970314yescomplemented
557C23C2×#1175819812yes—
558C23C2×#1175819812yes—
559C23C2×#1175819811yescharacteristic
560C23C2×#1175819811yescharacteristic
561C23#1451975114yes—
562C23C22×D24411yescharacteristic
563Q8—1224no—
564Q8—2442no—
565D4—1224no—
566D4—2448no—
567D4—2444no—
568D4—2448no—
569D4—2444no—
570D4—2442no—
571C2×C4—24no—
572C2×C4—24no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—24no—
576C2×C4—24no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—24no—
582C2×C4—24no—
583C2×C4—24no—
584C2×C4—24no—
585C2×C4—1228no—
586C2×C4—1222no—
587C2×C4—1222no—
588C2×C4—1222no—
589C2×C4—2444no—
590C2×C4—24416no—
591C2×C4—2444no—
592C2×C4—2442no—
593C2×C4—2442no—
594C2×C4—2442no—
595C2×C4—2442no—
596C2×C4—2442no—
597C2×C4—2444no—
598C2×C4C2×#1175819812yes—
599C2×C4C2×#1175819812yes—
600C2×C4C2×#1175819811yescharacteristic
601C2×C4C2×#1175819811yescharacteristic
602C2×C4C4×C22⋊D6114yes—
603C2×C4C4×C22⋊D6114yes—
604C2×C4#1451974612yes—
605C2×C4#1451974612yes—
606C2×C4#1451975114yes—
607C2×C4#1451976012yes—
608C2×C4#1451976012yes—
609C2×C4C22×C4×D6111yescharacteristic
610C2×C4C22×C22⋊D6112yes—
611C2×C4C22×C22⋊D6111yescharacteristic
612C2×C4C2×#1175822212yes—
613C8—28no—
614C22—21no—
615C22—21no—
616C22—21no—
617C22—22no—
618C22—21no—
619C22—22no—
620C22—28no—
621C22—28no—
622C22—22no—
623C22—22no—
624C22—22no—
625C22—48no—
626C22—48no—
627C22—44no—
628C22—4884no—
629C22—2442no—
630C22—2442no—
631C22—2442no—
632C22#1683964112yes—
633C22#1683964112yes—
634C22C2×#1451970312yes—
635C22C2×#1451970314yescomplemented
636C22#1683977812yes—
637C22C22×#1175819811yescharacteristic
638C22C2×#1451975112yes—
639C4—22no—
640C4—22no—
641C4—2448no—
642C4—2442no—
643C4—2442no—
644C4—2442no—
645C4#1683964112yes—
646C4#1683964112yes—
647C4C2×C4×C22⋊D6112yes—
648C4C2×#1451974611yescharacteristic
649C4C2×#1451976011yescharacteristic
650C2—22no—
651C2—24no—
652C2—22no—
653C2—42no—
654C2—4882no—
655C2#2144612114yesdirect product factor
656C2C2×#1683964111yescharacteristic
657C2C22×#1451970311yescharacteristic
658C2C2×#1683977811yescharacteristic
subgroupG/H
centerC22×C4C2×C22⋊D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC23×M16×C61C2
socleC23×C61C2×C22⋊C4
prime radicalC2×#16840273C2
cosocleC22×C4×C61C24
hypercenterC23×M16D61
hypocenterC61C2×#1871
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22×C4 ◁ C24×C4 ◁ C23×M16

Complements

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

NH ≅ G/Ntype
2. C22×#14519703C2semidirect
3. C2×C4×C22⋊Dic61C2semidirect
4. #21446121C2direct (N × H)
5. C2×#16839523C2semidirect
6. C23×M16×C61C2semidirect
10. #16840273C4semidirect
13. C2×#14519756C4semidirect
14. C2×#14519756C4semidirect
15. C2×#14519751C4semidirect
16. C2×#14519751C4semidirect
24. C2×#14519703C22semidirect
26. C4×C22⋊Dic61C22semidirect
28. #16839523C22semidirect
34. C22×M16×C61C22semidirect
53. #14519756C2×C4semidirect
54. #14519756C2×C4semidirect
57. #14519751C2×C4semidirect
58. #14519751C2×C4semidirect
106. C2×M16×C61D4semidirect
186. M16×C61C2×D4semidirect
222. C23×C61#104semidirect
302. C22×C61C2×#104semidirect
321. C23×M16D61semidirect
331. C2×C61#1871semidirect
342. C22×M16C2×D61semidirect
365. C61C2×#1871semidirect
398. C2×M16C22⋊D61semidirect
516. M16C2×C22⋊D61semidirect
556. C23#14519703semidirect
635. C22C2×#14519703semidirect
655. C2#21446121direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#168402732
C22×#145197032
C2×C4×C22⋊Dic612
#214461212
C2×#168395232
C23×M16×C612
C2×#187161