Finite Group Databasebeta

← C2×#20241891 #26183417

Subgroups — C2×#20241891 #26183417

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×#15689636C211yesmaximal, characteristic
2C2×#15689085C211yesmaximal, characteristic, complemented
3C2×#95×D47C211yesmaximal, characteristic, complemented
4#20241891C218yesmaximal, direct product factor
5C2×#15688860C211yesmaximal, characteristic, complemented
6C2×#15688858C211yesmaximal, characteristic, complemented
7C2×#15688856C211yesmaximal, characteristic, complemented
8C2×C47×#328C211yesmaximal, characteristic, complemented
9C22×#10625354C411yescharacteristic, complemented
10C23×C4×D47—21no—
11C23×C4×D47C2211yescharacteristic
12C22×#10625324—21no—
13#15689636—22no—
14#15689636—22no—
15#15689636C414yescomplemented
16#15689636C2214yes—
17C2×#13094009—21no—
18C2×#13094000C411yescharacteristic, complemented
19C2×#13093999C411yescharacteristic, complemented
20C2×#13093998C2211yescharacteristic
21C2×#13093994—21no—
22C2×#13093986—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#15688860C2214yescomplemented
34#15688858C2214yescomplemented
35#15688856C2214yescomplemented
36C2×C22⋊Q8×C47C2211yescharacteristic
37C2×C22⋊C8×C47C2211yescharacteristic
38C2×#95×C47C2211yescharacteristic
39C47×#328C2214yescomplemented
40C24×D47D411yescharacteristic
41C2×#10625354—22no—
42C2×#10625354—44no—
43C2×#10625354C2×C414yescomplemented
44C22×D188—21no—
45C22×D188—21no—
46C22×D188C2×C411yescharacteristic, complemented
47C22×C22⋊D47—21no—
48C23×Dic47—21no—
49C23×Dic47D411yescharacteristic
50C22×C4×D47—28no—
51C22×C4×D47—24no—
52C22×C4×D47—21no—
53C22×C4×D47—44no—
54C22×C4×D47—44no—
55C22×C4×D47—44no—
56C22×C4×D47C2×C411yescharacteristic
57C22×C4×D47C2×C411yescharacteristic
58C22×C4×D47C2×C412yes—
59C22×C4×D47D411yescharacteristic
60C22×C4×D47D412yes—
61C22×C4×D47D411yescharacteristic
62C22×C4×D47D412yes—
63C22×C4×D47C2312yes—
64C2×#10625324—24no—
65C2×#10625324—44no—
66C22×C47⋊C8—21no—
67#13094009—24no—
68#13094009—24no—
69#13094000—22no—
70#13094000C2×C414yescomplemented
71#13093999—22no—
72#13093999C2×C414yescomplemented
73#13093998—21no—
74#13093998—21no—
75#13093998C2×C412yescomplemented
76#13093998C2312yes—
77#13093994—24no—
78#13093994—24no—
79#13093986—24no—
80#13093986—24no—
81(C22⋊D47)⋊C4—24no—
82(C22⋊D47)⋊C4—24no—
83#13093982—22no—
84#13093982C2×C414yescomplemented
85C2×C4⋊Dic47C2×C411yescharacteristic, complemented
86C2×Dic47⋊C4—21no—
87C2×(C2×D47)⋊C4—21no—
88C2×(C2×D47)⋊C4—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×Q8×C47C2×C411yescharacteristic, complemented
99C23×C4×C47—21no—
100C23×C4×C47C2311yescharacteristic
101C22×C8×C47—21no—
102C22⋊Q8×C47—21no—
103C22⋊Q8×C47—21no—
104C22⋊Q8×C47C2×C412yescomplemented
105C22⋊Q8×C47C2312yes—
106C2×C4⋊C4×C47—21no—
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×C47C2×D411yescharacteristic
114C23×D47—22no—
115C23×D47—22no—
116C23×D47—24no—
117C23×D47—24no—
118C23×D47—21no—
119C23×D47C22⋊C412yes—
120C23×D47C22⋊C411yescharacteristic
121C23×D47C22⋊C411yescharacteristic
122C23×D47C2×D412yes—
123C2×Q8×C47—21no—
124C2×Q8×C47—42no—
125C2×Q8×C47C22×C412yescomplemented
126C22×C4×C47—24no—
127C22×C4×C47—21no—
128C22×C4×C47—21no—
129C22×C4×C47—22no—
130C22×C4×C47—42no—
131C22×C4×C47—42no—
132C22×C4×C47—42no—
133C22×C4×C47C22×C411yescharacteristic
134C22×C4×C47C22×C411yescharacteristic
135C22×C4×C47C2×D411yescharacteristic
136C22×C4×C47C2×D411yescharacteristic
137C22×C4×C47C2×D411yescharacteristic
138C22×C4×C47C2411yescharacteristic
139C2×C22⋊D47—22no—
140C2×C22⋊D47—22no—
141C2×C22⋊D47—42no—
142C2×C22⋊D47—42no—
143C22×Dic47—24no—
144C22×Dic47—21no—
145C22×Dic47—22no—
146C22×Dic47—42no—
147C22×Dic47—42no—
148C22×Dic47—42no—
149C22×Dic47C22⋊C411yescharacteristic
150C22×Dic47C22⋊C411yescharacteristic
151C22×Dic47C22⋊C412yes—
152C22×Dic47C2×D412yes—
153#10625354—48no—
154#10625354—48no—
155C2×D188—24no—
156C2×D188—24no—
157C2×D188—21no—
158C2×D188—42no—
159C2×D188—44no—
160C2×D188—44no—
161C2×D188C22×C412yescomplemented
162C2×C4×D47—24no—
163C2×C4×D47—24no—
164C2×C4×D47—216no—
165C2×C4×D47—24no—
166C2×C4×D47—44no—
167C2×C4×D47—44no—
168C2×C4×D47—44no—
169C2×C4×D47—44no—
170C2×C4×D47—42no—
171C2×C4×D47—42no—
172C2×C4×D47—44no—
173C2×C4×D47—432no—
174C2×C4×D47—48no—
175C2×C4×D47—48no—
176C2×C4×D47C22⋊C412yes—
177C2×C4×D47C22⋊C412yes—
178C2×C4×D47C22⋊C412yes—
179C2×C4×D47C22⋊C412yes—
180C2×C4×D47C22×C412yes—
181C2×C4×D47C22×C412yes—
182C2×C4×D47C2×D412yes—
183C2×C4×D47C2×D412yes—
184C2×C8×C47—22no—
185C2×C8×C47—42no—
186C4⋊C4×C47—22no—
187C4⋊C4×C47—21no—
188C4⋊C4×C47—22no—
189C4⋊C4×C47C22×C412yescomplemented
190C22⋊C4×C47—22no—
191C22⋊C4×C47—22no—
192(C2×D47)⋊C4—24no—
193(C2×D47)⋊C4—24no—
194(C2×D47)⋊C4—22no—
195(C2×D47)⋊C4—22no—
196(C2×D47)⋊C4—24no—
197C4⋊Dic47—21no—
198C4⋊Dic47C22×C412yescomplemented
199Dic47⋊C4—24no—
200C4×Dic47—24no—
201C2×C47⋊C8—22no—
202C2×C47⋊C8—42no—
203#10625324—48no—
204C23×C47—22no—
205C23×C47—22no—
206C23×C47C2×C22⋊C411yescharacteristic
207C23×C47C2×C22⋊C411yescharacteristic
208C23×C47AGL(1,ℤ/8)12yes—
209C23×C47#13712yes—
210C23×C47C22×D411yescharacteristic
211C22×D47—216no—
212C22×D47—216no—
213C22×D47—22no—
214C22×D47—22no—
215C22×D47—22no—
216C22×D47—24no—
217C22×D47—22no—
218C22×D47—22no—
219C22×D47—22no—
220C22×D47—24no—
221C22×D47—24no—
222C22×D47—24no—
223C22×D47—24no—
224C22×D47—22no—
225C22×D47—42no—
226C22×D47—42no—
227C22×D47—42no—
228C22×D47#9914yes—
229C22×D47#10414yes—
230C22×D47C2×C22⋊C412yes—
231C22×D47C2×C22⋊C412yes—
232Q8×C47—42no—
233Q8×C47—42no—
234C2×C4×C47—22no—
235C2×C4×C47—22no—
236C2×C4×C47—24no—
237C2×C4×C47—22no—
238C2×C4×C47—22no—
239C2×C4×C47—41no—
240C2×C4×C47—41no—
241C2×C4×C47—41no—
242C2×C4×C47—41no—
243C2×C4×C47—42no—
244C2×C4×C47—48no—
245C2×C4×C47—42no—
246C2×C4×C47—42no—
247C2×C4×C47—42no—
248C2×C4×C47C2×C22⋊C411yescharacteristic
249C2×C4×C47C2×C22⋊C411yescharacteristic
250C2×C4×C47C23×C411yescharacteristic
251C2×C4×C47C22×D411yescharacteristic
252C22⋊D47—42no—
253C22⋊D47—42no—
254C22⋊D47—44no—
255C2×Dic47—22no—
256C2×Dic47—22no—
257C2×Dic47—24no—
258C2×Dic47—22no—
259C2×Dic47—42no—
260C2×Dic47—42no—
261C2×Dic47—42no—
262C2×Dic47—48no—
263C2×Dic47—42no—
264C2×Dic47—42no—
265C2×Dic47C2×C22⋊C412yes—
266C2×Dic47C2×C22⋊C412yes—
267D188—42no—
268D188—42no—
269D188—48no—
270D188—48no—
271C4×D47—24no—
272C4×D47—24no—
273C4×D47—24no—
274C4×D47—24no—
275C4×D47—42no—
276C4×D47—42no—
277C4×D47—42no—
278C4×D47—42no—
279C4×D47—48no—
280C4×D47—432no—
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×C47C2×#9912yes—
291C22×C47#41712yes—
292C22×C47#41712yes—
293C22×C47C2×#10412yes—
294C22×C47C22×C22⋊C411yescharacteristic
295C22×C47C2×AGL(1,ℤ/8)11yescharacteristic
296C22×C47C2×#13711yescharacteristic
297C2×D47—28no—
298C2×D47—28no—
299C2×D47—216no—
300C2×D47—22no—
301C2×D47—22no—
302C2×D47—22no—
303C2×D47—22no—
304C2×D47—28no—
305C2×D47—28no—
306C2×D47—42no—
307C2×D47—42no—
308C2×D47—42no—
309C2×D47—84no—
310C2×D47C2×#9914yes—
311C2×D47C2×#10414yes—
312C4×C47—21no—
313C4×C47—21no—
314C4×C47—21no—
315C4×C47—21no—
316C4×C47—44no—
317C4×C47—42no—
318C4×C47—42no—
319Dic47—22no—
320Dic47—22no—
321Dic47—44no—
322Dic47—42no—
323C2×#525—471no—
324C2×#407—471no—
325C22×#95—471no—
326#1504—478no—
327C2×#328—471no—
328C2×#328D4711yescharacteristic, complemented
329C2×#327—471no—
330C2×#327—471no—
331C2×C47—21no—
332C2×C47—21no—
333C2×C47—22no—
334C2×C47#150414yescomplemented
335C2×C47C22×#9911yescharacteristic
336C2×C47C2×#41711yescharacteristic
337C2×C47C22×#10411yescharacteristic
338D47—24no—
339D47—24no—
340D47—82no—
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⋊Q8C2×D4711yescharacteristic
352C2×#121—471no—
353C2×#121—471no—
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#328C2×D4714yescomplemented
370#327—474no—
371#327—474no—
372C47C2×#150411yescharacteristic, complemented, Sylow
373C25—471no—
374C2×#41—474no—
375C2×#41—942no—
376C2×#41—1884no—
377C22×Q8C4×D4711yescharacteristic, complemented
378C22×D4—471no—
379C22×D4—941no—
380C22×D4—941no—
381C22×D4—941no—
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—941no—
393C23×C4—944no—
394C23×C4—948no—
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—21no—
407C22⋊Q8—21no—
408C22⋊Q8—474no—
409C22⋊Q8—942no—
410C22⋊Q8C4×D4712yescomplemented
411C22⋊Q8C22×D4712yes—
412#121—472no—
413#121—472no—
414#121—474no—
415#121—942no—
416#121—941no—
417#121—941no—
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—21no—
427C2×C4⋊C4—471no—
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—471no—
447C24—471no—
448C24—472no—
449C24—472no—
450C24—944no—
451C24—944no—
452C24—942no—
453C24—942no—
454C24—941no—
455C24D4×D4711yescharacteristic
456#41—1888no—
457#41—1888no—
458C2×Q8—21no—
459C2×Q8—42no—
460C2×Q8C2×C4×D4712yescomplemented
461C2×D4—472no—
462C2×D4—944no—
463C2×D4—944no—
464C2×D4—942no—
465C2×D4—942no—
466C2×D4—941no—
467C2×D4—1882no—
468C2×D4—1884no—
469C2×D4—1884no—
470C2×D4—1882no—
471C2×D4—1882no—
472C22×C4—24no—
473C22×C4—21no—
474C22×C4—21no—
475C22×C4—22no—
476C22×C4—42no—
477C22×C4—42no—
478C22×C4—42no—
479C22×C4—471no—
480C22×C4—471no—
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—472no—
491C22×C4—944no—
492C22×C4—944no—
493C22×C4—944no—
494C22×C4—9416no—
495C22×C4—944no—
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—1884no—
505C22×C4—1888no—
506C22×C4—18832no—
507C22×C4—1888no—
508C22×C4—1882no—
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—22no—
523C4⋊C4—21no—
524C4⋊C4—22no—
525C4⋊C4—472no—
526C4⋊C4—944no—
527C4⋊C4—941no—
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—472no—
540C23—472no—
541C23—474no—
542C23—474no—
543C23—944no—
544C23—944no—
545C23—944no—
546C23—944no—
547C23—9416no—
548C23—9416no—
549C23—942no—
550C23—944no—
551C23—942no—
552C23—942no—
553C23—942no—
554C23—942no—
555C23—942no—
556C23—942no—
557C23—1882no—
558C23—1882no—
559C23—1882no—
560C23C22⋊C4×D4711yescharacteristic
561C23C22⋊C4×D4711yescharacteristic
562C23#1309405112yes—
563C23#1309405612yes—
564C23C2×D4×D4711yescharacteristic
565Q8—42no—
566Q8—42no—
567D4—1882no—
568D4—1882no—
569D4—1888no—
570D4—1888no—
571D4—1884no—
572D4—1882no—
573D4—1882no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—24no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—42no—
584C2×C4—48no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—472no—
589C2×C4—472no—
590C2×C4—944no—
591C2×C4—944no—
592C2×C4—944no—
593C2×C4—944no—
594C2×C4—942no—
595C2×C4—942no—
596C2×C4—944no—
597C2×C4—942no—
598C2×C4—1882no—
599C2×C4—1882no—
600C2×C4—1882no—
601C2×C4—1882no—
602C2×C4—1888no—
603C2×C4—18832no—
604C2×C4—1882no—
605C2×C4—1882no—
606C2×C4—1882no—
607C2×C4—1888no—
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—474no—
623C22—474no—
624C22—948no—
625C22—9416no—
626C22—948no—
627C22—942no—
628C22—942no—
629C22—948no—
630C22—948no—
631C22—942no—
632C22—942no—
633C22—3764no—
634C22—1882no—
635C22—1882no—
636C22—1882no—
637C22#99×D4712yes—
638C22#1568912312yes—
639C22#1568912312yes—
640C22#104×D4712yes—
641C22C2×C22⋊C4×D4711yescharacteristic
642C22C2×#1309405111yescharacteristic
643C22C2×#1309405611yescharacteristic
644C4—21no—
645C4—21no—
646C4—21no—
647C4—21no—
648C4—44no—
649C4—42no—
650C4—42no—
651C4—942no—
652C4—942no—
653C4—1884no—
654C4—1882no—
655C2—21no—
656C2—22no—
657C2—21no—
658C2—944no—
659C2—944no—
660C2—3762no—
661C2#2024189114yesdirect product factor
662C2C2×#99×D4711yescharacteristic
663C2C2×#1568912311yescharacteristic
664C2C2×#104×D4711yescharacteristic
subgroupG/H
centerC23C22⋊C4×D47
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC2×C47×#328C2
socleC23×C47C2×C22⋊C4
prime radicalC2×#15689636C2
cosocleC22×C4×C47C24
hypercenterC2×#328D47
hypocenterC47C2×#1504
LCSG ▷ C2×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#328

Complements

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

NH ≅ G/Ntype
2. C2×#15689085C2semidirect
3. C2×#95×D47C2semidirect
4. #20241891C2direct (N × H)
5. C2×#15688860C2semidirect
6. C2×#15688858C2semidirect
7. C2×#15688856C2semidirect
8. C2×C47×#328C2semidirect
9. C22×#10625354C4semidirect
15. #15689636C4semidirect
18. C2×#13094000C4semidirect
19. C2×#13093999C4semidirect
24. C2×#13093982C4semidirect
26. #15689085C22semidirect
32. #95×D47C22semidirect
33. #15688860C22semidirect
34. #15688858C22semidirect
35. #15688856C22semidirect
39. C47×#328C22semidirect
43. C2×#10625354C2×C4semidirect
46. C22×D188C2×C4semidirect
70. #13094000C2×C4semidirect
72. #13093999C2×C4semidirect
75. #13093998C2×C4semidirect
84. #13093982C2×C4semidirect
85. C2×C4⋊Dic47C2×C4semidirect
98. C22×Q8×C47C2×C4semidirect
104. C22⋊Q8×C47C2×C4semidirect
107. C2×C4⋊C4×C47C2×C4semidirect
125. C2×Q8×C47C22×C4semidirect
161. C2×D188C22×C4semidirect
189. C4⋊C4×C47C22×C4semidirect
198. C4⋊Dic47C22×C4semidirect
328. C2×#328D47semidirect
334. C2×C47#1504semidirect
369. #328C2×D47semidirect
372. C47C2×#1504semidirect
377. C22×Q8C4×D47semidirect
410. C22⋊Q8C4×D47semidirect
429. C2×C4⋊C4C4×D47semidirect
460. C2×Q8C2×C4×D47semidirect
528. C4⋊C4C2×C4×D47semidirect
661. C2#20241891direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#156896362
C2×#156890852
C2×#95×D472
#202418912
C2×#156888602
C2×#156888582
C2×#156888562
C2×C47×#3282
C2×#150447