Finite Group Databasebeta

← C2×#20241889 #26183415

Subgroups — C2×#20241889 #26183415

Order 12032.

1834 conjugacy classes of subgroups.

The subgroup lattice has 666 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689633C211yesmaximal, characteristic
2C2×#15689085C211yesmaximal, characteristic, complemented
3C2×#95×D47C211yesmaximal, characteristic, complemented
4#20241889C218yesmaximal, direct product factor
5C2×#15688861C211yesmaximal, characteristic, complemented
6C2×#15688859C211yesmaximal, characteristic, complemented
7C2×#15688857C211yesmaximal, characteristic, complemented
8C2×C47×#327C211yesmaximal, characteristic, complemented
9C22×#10625352C411yescharacteristic, complemented
10C23×C4×D47—21no—
11C23×C4×D47C2211yescharacteristic
12C22×#10625324—21no—
13#15689633—22no—
14#15689633—22no—
15#15689633C414yescomplemented
16#15689633C2214yes—
17C2×#13094006—21no—
18C2×#13094004C411yescharacteristic, complemented
19C2×#13094003C2211yescharacteristic
20C2×#13093995C411yescharacteristic, complemented
21C2×#13093992—21no—
22C2×D4×Dic47—21no—
23C2×(C22⋊D47)⋊C4—21no—
24C2×#13093982C411yescharacteristic, complemented
25#15689085—22no—
26#15689085C2214yescomplemented
27C2×C22⋊(C47⋊C8)C2211yescharacteristic
28C2×(C2×D47)⋊C8—21no—
29C2×C47⋊#95—21no—
30C2×C47⋊#95C2211yescharacteristic
31#95×D47—22no—
32#95×D47C2214yescomplemented
33#15688861C2214yescomplemented
34#15688859C2214yescomplemented
35#15688857C2214yescomplemented
36C2×#121×C47C2211yescharacteristic
37C2×C22⋊C8×C47C2211yescharacteristic
38C2×#95×C47C2211yescharacteristic
39C47×#327C2214yescomplemented
40C24×D47D411yescharacteristic
41C2×#10625352—22no—
42C2×#10625352—44no—
43C2×#10625352C2×C414yescomplemented
44C22×Dic94C2×C411yescharacteristic, complemented
45C22×C22⋊D47—21no—
46C22×C22⋊D47—21no—
47C23×Dic47—21no—
48C23×Dic47—21no—
49C23×Dic47D411yescharacteristic
50C22×C4×D47—28no—
51C22×C4×D47—24no—
52C22×C4×D47—44no—
53C22×C4×D47—44no—
54C22×C4×D47—44no—
55C22×C4×D47C2×C411yescharacteristic
56C22×C4×D47C2×C411yescharacteristic
57C22×C4×D47C2×C412yes—
58C22×C4×D47D411yescharacteristic
59C22×C4×D47D412yes—
60C22×C4×D47D411yescharacteristic
61C22×C4×D47D412yes—
62C22×C4×D47C2312yes—
63C2×#10625324—24no—
64C2×#10625324—44no—
65C22×C47⋊C8—21no—
66#13094006—24no—
67#13094006—24no—
68#13094004—22no—
69#13094004C2×C414yescomplemented
70#13094003—21no—
71#13094003—21no—
72#13094003C2×C412yescomplemented
73#13094003C2312yes—
74#13093995—22no—
75#13093995C2×C414yescomplemented
76#13093992—24no—
77#13093992—24no—
78D4×Dic47—24no—
79D4×Dic47—24no—
80(C22⋊D47)⋊C4—24no—
81(C22⋊D47)⋊C4—24no—
82#13093982—22no—
83#13093982C2×C414yescomplemented
84C2×C4⋊Dic47—21no—
85C2×C4⋊Dic47C2×C411yescharacteristic, complemented
86C2×Dic47⋊C4—21no—
87C2×C22⋊Dic47—21no—
88C2×C22⋊Dic47—21no—
89C2×(C2×D47)⋊C4—21no—
90C2×(C2×D47)⋊C4—21no—
91C2×C4×Dic47—21no—
92C22⋊(C47⋊C8)—21no—
93C22⋊(C47⋊C8)C2312yes—
94(C2×D47)⋊C8—24no—
95C47⋊#95—21no—
96C47⋊#95—24no—
97C47⋊#95C2312yes—
98C22×D4×C47—21no—
99C22×D4×C47C2×C411yescharacteristic, complemented
100C23×C4×C47—21no—
101C23×C4×C47C2311yescharacteristic
102C22×C8×C47—21no—
103#121×C47—21no—
104#121×C47—21no—
105#121×C47C2×C412yescomplemented
106#121×C47C2312yes—
107C2×C4⋊C4×C47C2×C411yescharacteristic, complemented
108C2×C22⋊C4×C47—21no—
109C22⋊C8×C47—21no—
110C22⋊C8×C47C2312yes—
111#95×C47—21no—
112#95×C47C2312yes—
113C24×C47—21no—
114C24×C47C2×D411yescharacteristic
115C23×D47—22no—
116C23×D47—22no—
117C23×D47—24no—
118C23×D47—24no—
119C23×D47C22⋊C412yes—
120C23×D47C22⋊C411yescharacteristic
121C23×D47C22⋊C411yescharacteristic
122C23×D47C2×D412yes—
123C2×D4×C47—22no—
124C2×D4×C47—22no—
125C2×D4×C47—21no—
126C2×D4×C47—42no—
127C2×D4×C47—42no—
128C2×D4×C47—42no—
129C2×D4×C47C22×C412yescomplemented
130C22×C4×C47—24no—
131C22×C4×C47—21no—
132C22×C4×C47—22no—
133C22×C4×C47—42no—
134C22×C4×C47—42no—
135C22×C4×C47—42no—
136C22×C4×C47C22×C411yescharacteristic
137C22×C4×C47C22×C411yescharacteristic
138C22×C4×C47C2×D411yescharacteristic
139C22×C4×C47C2×D411yescharacteristic
140C22×C4×C47C2×D411yescharacteristic
141C22×C4×C47C2411yescharacteristic
142C2×C22⋊D47—24no—
143C2×C22⋊D47—24no—
144C2×C22⋊D47—44no—
145C2×C22⋊D47—44no—
146C22×Dic47—24no—
147C22×Dic47—21no—
148C22×Dic47—21no—
149C22×Dic47—22no—
150C22×Dic47—24no—
151C22×Dic47—42no—
152C22×Dic47—44no—
153C22×Dic47—42no—
154C22×Dic47—42no—
155C22×Dic47C22⋊C411yescharacteristic
156C22×Dic47C22⋊C411yescharacteristic
157C22×Dic47C22⋊C412yes—
158C22×Dic47C2×D412yes—
159#10625352—48no—
160#10625352—48no—
161C2×C4×D47—24no—
162C2×C4×D47—24no—
163C2×C4×D47—216no—
164C2×C4×D47—44no—
165C2×C4×D47—44no—
166C2×C4×D47—44no—
167C2×C4×D47—44no—
168C2×C4×D47—42no—
169C2×C4×D47—42no—
170C2×C4×D47—432no—
171C2×C4×D47—48no—
172C2×C4×D47—48no—
173C2×C4×D47C22⋊C412yes—
174C2×C4×D47C22⋊C412yes—
175C2×C4×D47C22⋊C412yes—
176C2×C4×D47C22⋊C412yes—
177C2×C4×D47C22×C412yes—
178C2×C4×D47C22×C412yes—
179C2×C4×D47C2×D412yes—
180C2×C4×D47C2×D412yes—
181C2×Dic94—21no—
182C2×Dic94—42no—
183C2×Dic94C22×C412yescomplemented
184C2×C8×C47—22no—
185C2×C8×C47—42no—
186C4⋊C4×C47—21no—
187C4⋊C4×C47C22×C412yescomplemented
188C22⋊C4×C47—22no—
189C22⋊C4×C47—22no—
190C22⋊Dic47—22no—
191C22⋊Dic47—22no—
192C22⋊Dic47—24no—
193(C2×D47)⋊C4—24no—
194(C2×D47)⋊C4—24no—
195C4⋊Dic47—21no—
196C4⋊Dic47—24no—
197C4⋊Dic47C22×C412yescomplemented
198Dic47⋊C4—22no—
199Dic47⋊C4—22no—
200C4×Dic47—24no—
201C2×C47⋊C8—22no—
202C2×C47⋊C8—42no—
203#10625324—48no—
204C23×C47—22no—
205C23×C47—22no—
206C23×C47—22no—
207C23×C47—42no—
208C23×C47—42no—
209C23×C47—42no—
210C23×C47C2×C22⋊C411yescharacteristic
211C23×C47C2×C22⋊C411yescharacteristic
212C23×C47AGL(1,ℤ/8)12yes—
213C23×C47#13712yes—
214C23×C47C22×D411yescharacteristic
215C22×D47—216no—
216C22×D47—216no—
217C22×D47—22no—
218C22×D47—22no—
219C22×D47—22no—
220C22×D47—24no—
221C22×D47—22no—
222C22×D47—22no—
223C22×D47—22no—
224C22×D47—24no—
225C22×D47—24no—
226C22×D47—24no—
227C22×D47—24no—
228C22×D47#9914yes—
229C22×D47#10414yes—
230C22×D47C2×C22⋊C412yes—
231C22×D47C2×C22⋊C412yes—
232D4×C47—42no—
233D4×C47—42no—
234D4×C47—42no—
235D4×C47—42no—
236D4×C47—44no—
237C2×C4×C47—22no—
238C2×C4×C47—22no—
239C2×C4×C47—24no—
240C2×C4×C47—22no—
241C2×C4×C47—41no—
242C2×C4×C47—41no—
243C2×C4×C47—41no—
244C2×C4×C47—41no—
245C2×C4×C47—42no—
246C2×C4×C47—48no—
247C2×C4×C47—42no—
248C2×C4×C47—42no—
249C2×C4×C47C2×C22⋊C411yescharacteristic
250C2×C4×C47C2×C22⋊C411yescharacteristic
251C2×C4×C47C23×C411yescharacteristic
252C2×C4×C47C22×D411yescharacteristic
253C22⋊D47—48no—
254C22⋊D47—48no—
255C2×Dic47—22no—
256C2×Dic47—22no—
257C2×Dic47—24no—
258C2×Dic47—22no—
259C2×Dic47—22no—
260C2×Dic47—42no—
261C2×Dic47—42no—
262C2×Dic47—48no—
263C2×Dic47—42no—
264C2×Dic47—48no—
265C2×Dic47—42no—
266C2×Dic47—42no—
267C2×Dic47—42no—
268C2×Dic47C2×C22⋊C412yes—
269C2×Dic47C2×C22⋊C412yes—
270C4×D47—24no—
271C4×D47—24no—
272C4×D47—24no—
273C4×D47—24no—
274C4×D47—42no—
275C4×D47—42no—
276C4×D47—42no—
277C4×D47—42no—
278C4×D47—432no—
279Dic94—42no—
280Dic94—42no—
281C8×C47—42no—
282C47⋊C8—42no—
283C2×#1504—471nomaximal, Sylow
284C22×C47—21no—
285C22×C47—21no—
286C22×C47—21no—
287C22×C47—21no—
288C22×C47—24no—
289C22×C47—24no—
290C22×C47—42no—
291C22×C47—42no—
292C22×C47—42no—
293C22×C47—84no—
294C22×C47C2×#9912yes—
295C22×C47#41712yes—
296C22×C47#41712yes—
297C22×C47C2×#10412yes—
298C22×C47C22×C22⋊C411yescharacteristic
299C22×C47C2×AGL(1,ℤ/8)11yescharacteristic
300C22×C47C2×#13711yescharacteristic
301C2×D47—28no—
302C2×D47—28no—
303C2×D47—216no—
304C2×D47—22no—
305C2×D47—22no—
306C2×D47—22no—
307C2×D47—22no—
308C2×D47—28no—
309C2×D47—28no—
310C2×D47C2×#9914yes—
311C2×D47C2×#10414yes—
312C4×C47—21no—
313C4×C47—21no—
314C4×C47—21no—
315C4×C47—21no—
316C4×C47—44no—
317C4×C47—42no—
318Dic47—22no—
319Dic47—22no—
320Dic47—44no—
321Dic47—42no—
322Dic47—42no—
323C2×#525—471no—
324C2×#407—471no—
325C22×#95—471no—
326#1504—478no—
327C2×#328—471no—
328C2×#328—471no—
329C2×#327—471no—
330C2×#327D4711yescharacteristic, complemented
331C2×C47—21no—
332C2×C47—21no—
333C2×C47—22no—
334C2×C47—82no—
335C2×C47#150414yescomplemented
336C2×C47C22×#9911yescharacteristic
337C2×C47C2×#41711yescharacteristic
338C2×C47C22×#10411yescharacteristic
339D47—24no—
340D47—24no—
341C22×#41—471no—
342C24×C4—471no—
343C24×C4—941no—
344C22×M16—941no—
345#525—474no—
346#525—474no—
347#525—942no—
348#525—942no—
349C2×#123—941no—
350C2×C22⋊Q8—471no—
351C2×C22⋊Q8—471no—
352C2×#121—471no—
353C2×#121C2×D4711yescharacteristic
354C2×#120—941no—
355C2×C4×D4—941no—
356C2×C4×D4—941no—
357C2×#117—471no—
358#407—474no—
359#407—942no—
360C2×C22⋊C8—471no—
361C2×C22⋊C8—941no—
362C2×C22⋊C8C2×D4711yescharacteristic
363C2×#95—471no—
364C2×#95—474no—
365C2×#95—942no—
366C2×#95—941no—
367C2×#95C2×D4711yescharacteristic
368#328—474no—
369#328—474no—
370#327—474no—
371#327C2×D4714yescomplemented
372C47C2×#150411yescharacteristic, complemented, Sylow
373C25—471no—
374C2×#41—474no—
375C2×#41—942no—
376C2×#41—1884no—
377C22×Q8—471no—
378C22×D4—21no—
379C22×D4—941no—
380C22×D4—941no—
381C22×D4C4×D4711yescharacteristic, complemented
382C23×C4—21no—
383C23×C4—471no—
384C23×C4—471no—
385C23×C4—471no—
386C23×C4—471no—
387C23×C4—472no—
388C23×C4—471no—
389C23×C4—472no—
390C23×C4—472no—
391C23×C4—472no—
392C23×C4—944no—
393C23×C4—948no—
394C23×C4—941no—
395C23×C4—941no—
396C23×C4—1884no—
397C23×C4—1884no—
398C23×C4—1884no—
399C23×C4C22×D4711yescharacteristic
400C2×M16—944no—
401C2×M16—1884no—
402C22×C8—21no—
403C22×C8—941no—
404#123—944no—
405#123—944no—
406C22⋊Q8—472no—
407C22⋊Q8—472no—
408C22⋊Q8—474no—
409C22⋊Q8—942no—
410C22⋊Q8—941no—
411C22⋊Q8—941no—
412#121—21no—
413#121—21no—
414#121—474no—
415#121—942no—
416#121C4×D4712yescomplemented
417#121C22×D4712yes—
418#120—944no—
419#120—944no—
420C4×D4—944no—
421C4×D4—944no—
422C4×D4—944no—
423C4×D4—944no—
424#117—474no—
425#117—942no—
426C2×C4⋊C4—471no—
427C2×C4⋊C4—941no—
428C2×C4⋊C4—941no—
429C2×C4⋊C4C4×D4711yescharacteristic, complemented
430C2×C22⋊C4—21no—
431C2×C22⋊C4—941no—
432C2×C22⋊C4—941no—
433C2×C22⋊C4—941no—
434C2×C22⋊C4—941no—
435C2×C42—941no—
436C22⋊C8—21no—
437C22⋊C8—472no—
438C22⋊C8—944no—
439C22⋊C8—941no—
440C22⋊C8C22×D4712yes—
441#95—21no—
442#95—472no—
443#95—944no—
444#95—941no—
445#95C22×D4712yes—
446C24—21no—
447C24—471no—
448C24—471no—
449C24—472no—
450C24—472no—
451C24—944no—
452C24—944no—
453C24—942no—
454C24—942no—
455C24D4×D4711yescharacteristic
456#41—1888no—
457#41—1888no—
458C2×Q8—472no—
459C2×Q8—941no—
460C2×Q8—1882no—
461C2×D4—22no—
462C2×D4—22no—
463C2×D4—21no—
464C2×D4—42no—
465C2×D4—42no—
466C2×D4—42no—
467C2×D4—944no—
468C2×D4—944no—
469C2×D4—1884no—
470C2×D4—1884no—
471C2×D4C2×C4×D4712yescomplemented
472C22×C4—24no—
473C22×C4—21no—
474C22×C4—22no—
475C22×C4—42no—
476C22×C4—42no—
477C22×C4—42no—
478C22×C4—471no—
479C22×C4—471no—
480C22×C4—472no—
481C22×C4—472no—
482C22×C4—472no—
483C22×C4—472no—
484C22×C4—472no—
485C22×C4—472no—
486C22×C4—472no—
487C22×C4—472no—
488C22×C4—472no—
489C22×C4—472no—
490C22×C4—944no—
491C22×C4—944no—
492C22×C4—9416no—
493C22×C4—944no—
494C22×C4—944no—
495C22×C4—941no—
496C22×C4—941no—
497C22×C4—942no—
498C22×C4—1882no—
499C22×C4—1882no—
500C22×C4—1884no—
501C22×C4—1884no—
502C22×C4—1884no—
503C22×C4—1884no—
504C22×C4—1888no—
505C22×C4—18832no—
506C22×C4—1888no—
507C22×C4—1882no—
508C22×C4—1884no—
509C22×C4—1882no—
510C22×C4—1882no—
511C22×C4C2×C4×D4711yescharacteristic
512C22×C4C2×C4×D4711yescharacteristic
513C22×C4D4×D4711yescharacteristic
514C22×C4D4×D4711yescharacteristic
515C22×C4D4×D4711yescharacteristic
516C22×C4C23×D4711yescharacteristic
517M16—1888no—
518C2×C8—22no—
519C2×C8—42no—
520C2×C8—942no—
521C2×C8—1882no—
522C4⋊C4—21no—
523C4⋊C4—472no—
524C4⋊C4—944no—
525C4⋊C4—942no—
526C4⋊C4—941no—
527C4⋊C4—942no—
528C4⋊C4C2×C4×D4712yescomplemented
529C22⋊C4—22no—
530C22⋊C4—22no—
531C22⋊C4—944no—
532C22⋊C4—944no—
533C22⋊C4—944no—
534C22⋊C4—942no—
535C22⋊C4—942no—
536C42—944no—
537C23—22no—
538C23—22no—
539C23—22no—
540C23—42no—
541C23—42no—
542C23—42no—
543C23—472no—
544C23—472no—
545C23—474no—
546C23—474no—
547C23—944no—
548C23—944no—
549C23—944no—
550C23—944no—
551C23—9416no—
552C23—9416no—
553C23—942no—
554C23—944no—
555C23—942no—
556C23—942no—
557C23—942no—
558C23—942no—
559C23—942no—
560C23C22⋊C4×D4711yescharacteristic
561C23C22⋊C4×D4711yescharacteristic
562C23#1309404812yes—
563C23#1309405312yes—
564C23C2×D4×D4711yescharacteristic
565Q8—1882no—
566Q8—1882no—
567D4—42no—
568D4—42no—
569D4—44no—
570D4—42no—
571D4—42no—
572D4—1888no—
573D4—1888no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—24no—
577C2×C4—22no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—42no—
583C2×C4—48no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—472no—
587C2×C4—472no—
588C2×C4—944no—
589C2×C4—944no—
590C2×C4—944no—
591C2×C4—944no—
592C2×C4—942no—
593C2×C4—942no—
594C2×C4—944no—
595C2×C4—942no—
596C2×C4—942no—
597C2×C4—1882no—
598C2×C4—1882no—
599C2×C4—1882no—
600C2×C4—1882no—
601C2×C4—18832no—
602C2×C4—1882no—
603C2×C4—1882no—
604C2×C4—1888no—
605C2×C4—1882no—
606C2×C4—1888no—
607C2×C4—1882no—
608C2×C4—1882no—
609C2×C4—1882no—
610C2×C4C22⋊C4×D4711yescharacteristic
611C2×C4C22⋊C4×D4711yescharacteristic
612C2×C4C22×C4×D4711yescharacteristic
613C2×C4C2×D4×D4711yescharacteristic
614C8—42no—
615C8—1882no—
616C22—21no—
617C22—21no—
618C22—21no—
619C22—21no—
620C22—24no—
621C22—24no—
622C22—42no—
623C22—42no—
624C22—42no—
625C22—84no—
626C22—474no—
627C22—474no—
628C22—948no—
629C22—9416no—
630C22—948no—
631C22—942no—
632C22—942no—
633C22—948no—
634C22—948no—
635C22—942no—
636C22—942no—
637C22#99×D4712yes—
638C22#1568912112yes—
639C22#1568912112yes—
640C22#104×D4712yes—
641C22C2×C22⋊C4×D4711yescharacteristic
642C22C2×#1309404811yescharacteristic
643C22C2×#1309405311yescharacteristic
644C4—21no—
645C4—21no—
646C4—21no—
647C4—21no—
648C4—44no—
649C4—42no—
650C4—942no—
651C4—942no—
652C4—1884no—
653C4—1882no—
654C4—1882no—
655C2—21no—
656C2—22no—
657C2—21no—
658C2—82no—
659C2—944no—
660C2—944no—
661C2#2024188914yesdirect product factor
662C2C2×#99×D4711yescharacteristic
663C2C2×#1568912111yescharacteristic
664C2C2×#104×D4711yescharacteristic
subgroupG/H
centerC23C22⋊C4×D47
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC2×C47×#327C2
socleC23×C47C2×C22⋊C4
prime radicalC2×#15689633C2
cosocleC22×C4×C47C24
hypercenterC2×#327D47
hypocenterC47C2×#1504
LCSG ▷ C2×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#327

Complements

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

NH ≅ G/Ntype
2. C2×#15689085C2semidirect
3. C2×#95×D47C2semidirect
4. #20241889C2direct (N × H)
5. C2×#15688861C2semidirect
6. C2×#15688859C2semidirect
7. C2×#15688857C2semidirect
8. C2×C47×#327C2semidirect
9. C22×#10625352C4semidirect
15. #15689633C4semidirect
18. C2×#13094004C4semidirect
20. C2×#13093995C4semidirect
24. C2×#13093982C4semidirect
26. #15689085C22semidirect
32. #95×D47C22semidirect
33. #15688861C22semidirect
34. #15688859C22semidirect
35. #15688857C22semidirect
39. C47×#327C22semidirect
43. C2×#10625352C2×C4semidirect
44. C22×Dic94C2×C4semidirect
69. #13094004C2×C4semidirect
72. #13094003C2×C4semidirect
75. #13093995C2×C4semidirect
83. #13093982C2×C4semidirect
85. C2×C4⋊Dic47C2×C4semidirect
99. C22×D4×C47C2×C4semidirect
105. #121×C47C2×C4semidirect
107. C2×C4⋊C4×C47C2×C4semidirect
129. C2×D4×C47C22×C4semidirect
183. C2×Dic94C22×C4semidirect
187. C4⋊C4×C47C22×C4semidirect
197. C4⋊Dic47C22×C4semidirect
330. C2×#327D47semidirect
335. C2×C47#1504semidirect
371. #327C2×D47semidirect
372. C47C2×#1504semidirect
381. C22×D4C4×D47semidirect
416. #121C4×D47semidirect
429. C2×C4⋊C4C4×D47semidirect
471. C2×D4C2×C4×D47semidirect
528. C4⋊C4C2×C4×D47semidirect
661. C2#20241889direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#156896332
C2×#156890852
C2×#95×D472
#202418892
C2×#156888612
C2×#156888592
C2×#156888572
C2×C47×#3272
C2×#150447