Finite Group Databasebeta

← C2×#20249116 #26383324

Subgroups — C2×#20249116 #26383324

Order 12032.

5390 conjugacy classes of subgroups.

The subgroup lattice has 616 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#10625352C211yesmaximal, characteristic
2C22×#13094002C211yesmaximal, characteristic, complemented
3C22×#13094001C212yesmaximal, complemented
4#20249116C2116yesmaximal, direct product factor
5C2×#15689044C211yesmaximal, characteristic, complemented
6C2×#15689043C211yesmaximal, characteristic, complemented
7C2×#15689039C212yesmaximal, complemented
8C2×#15689038C212yesmaximal, complemented
9C2×#15689037C211yesmaximal, characteristic, complemented
10C2×#15688982C211yesmaximal, characteristic, complemented
11C2×#15688981C212yesmaximal, complemented
12C2×C47×#394C211yesmaximal, characteristic, complemented
13C22×#10625352—216no—
14C22×#10625352—28no—
15C22×#10625352C2218yes—
16C23×Dic94C2211yescharacteristic
17C23×Dic94C2211yescharacteristic
18C23×C22⋊D47C2212yes—
19C24×Dic47C2212yes—
20C23×C4×D47C2211yescharacteristic
21C2×#13094002—28no—
22C2×#13094002C2218yescomplemented
23C2×#13094001—216no—
24C2×#13094001C22116yescomplemented
25C22×C4⋊Dic47C2212yes—
26C22×Dic47⋊C4C2212yes—
27C22×Dic47⋊C4C2211yescharacteristic
28C22×Dic47⋊C4C2212yes—
29C22×Dic47⋊C4C2211yescharacteristic
30C22×C22⋊Dic47C2212yes—
31C22×C22⋊Dic47C2212yes—
32C22×(C2×D47)⋊C4C2211yescharacteristic
33C22×(C2×D47)⋊C4C2211yescharacteristic
34C22×(C2×D47)⋊C4C2212yes—
35#15689044C2218yescomplemented
36#15689043C2218yescomplemented
37#15689039C22116yescomplemented
38#15689038C22116yescomplemented
39#15689037C2218yescomplemented
40#15688982C2218yescomplemented
41#15688981C22116yescomplemented
42C2×C47⋊#95C2211yescharacteristic
43C2×C47⋊#95C2212yes—
44C2×C47⋊#95C2212yes—
45C23×D4×C47C2211yescharacteristic
46C22×C4⋊C4×C47C2211yescharacteristic
47C22×C22⋊C4×C47C2212yes—
48C22×C22⋊C4×C47C2212yes—
49C47×#394C2218yescomplemented
50C2×#95×C47C2211yescharacteristic
51C24×D47C2311yescharacteristic
52C2×#10625352—2128no—
53C2×#10625352—264no—
54C2×#10625352—232no—
55C22×Dic94—24no—
56C22×Dic94—28no—
57C22×Dic94—28no—
58C22×Dic94—24no—
59C22×Dic94C2314yes—
60C22×Dic94C2314yes—
61C22×C22⋊D47—28no—
62C22×C22⋊D47—28no—
63C22×C22⋊D47—28no—
64C22×C22⋊D47C2318yes—
65C23×Dic47—28no—
66C23×Dic47—28no—
67C23×Dic47D4116yes—
68C23×Dic47C2311yescharacteristic
69C23×Dic47C2312yes—
70C23×Dic47C2311yescharacteristic
71C23×Dic47C2312yes—
72C23×Dic47C2318yes—
73C23×Dic47C2311yescharacteristic
74C22×C4×D47—28no—
75C22×C4×D47D418yes—
76C22×C4×D47C2314yes—
77#13094002—232no—
78#13094001—264no—
79C2×C4⋊Dic47—28no—
80C2×C4⋊Dic47C2318yes—
81C2×Dic47⋊C4—24no—
82C2×Dic47⋊C4—28no—
83C2×Dic47⋊C4—28no—
84C2×Dic47⋊C4—24no—
85C2×Dic47⋊C4C2314yes—
86C2×Dic47⋊C4C2318yes—
87C2×Dic47⋊C4C2314yes—
88C2×Dic47⋊C4C2318yes—
89C2×C22⋊Dic47—28no—
90C2×C22⋊Dic47—44no—
91C2×C22⋊Dic47C2318yes—
92C2×C22⋊Dic47C2318yes—
93C2×(C2×D47)⋊C4—24no—
94C2×(C2×D47)⋊C4—24no—
95C2×(C2×D47)⋊C4—44no—
96C2×(C2×D47)⋊C4C2314yes—
97C2×(C2×D47)⋊C4C2314yes—
98C2×(C2×D47)⋊C4C2318yes—
99C47⋊#95C2318yes—
100C47⋊#95C2318yes—
101C47⋊#95C2314yes—
102C25×C47C2312yes—
103C22×D4×C47—28no—
104C22×D4×C47—24no—
105C22×D4×C47C2314yes—
106C23×C4×C47C2311yescharacteristic
107C23×C4×C47C2312yes—
108C23×C4×C47C2311yescharacteristic
109C23×C4×C47C2311yescharacteristic
110C2×C4⋊C4×C47—24no—
111C2×C4⋊C4×C47C2314yes—
112C2×C22⋊C4×C47—28no—
113C2×C22⋊C4×C47—44no—
114C2×C22⋊C4×C47C2318yes—
115C2×C22⋊C4×C47C2318yes—
116#95×C47C2314yes—
117C24×C47—24no—
118C24×C47—24no—
119C24×C47—24no—
120C24×C47—24no—
121C24×C47—24no—
122C24×C47—24no—
123C24×C47C2×D418yes—
124C24×C47C2414yes—
125C24×C47C2411yescharacteristic
126C23×D47—24no—
127C23×D47—22no—
128C23×D47—24no—
129C23×D47—22no—
130C23×D47C2×D414yes—
131C23×D47C2412yes—
132C2×D4×C47—232no—
133C2×D4×C47—216no—
134C2×D4×C47—28no—
135C22×C4×C47—24no—
136C22×C4×C47—24no—
137C22×C4×C47—24no—
138C22×C4×C47—22no—
139C22×C4×C47—22no—
140C22×C4×C47—22no—
141C22×C4×C47—24no—
142C22×C4×C47—22no—
143C22×C4×C47—24no—
144C22×C4×C47C2×D414yes—
145C22×C4×C47C2412yes—
146C22×C4×C47C2412yes—
147C22×C4×C47C2414yes—
148C22×C4×C47C2412yes—
149C2×C22⋊D47—264no—
150C2×C22⋊D47—216no—
151C2×C22⋊D47—216no—
152C2×C22⋊D47—216no—
153C22×Dic47—24no—
154C22×Dic47—24no—
155C22×Dic47—264no—
156C22×Dic47—216no—
157C22×Dic47—216no—
158C22×Dic47—22no—
159C22×Dic47—22no—
160C22×Dic47—24no—
161C22×Dic47—24no—
162C22×Dic47—24no—
163C22×Dic47—22no—
164C22×Dic47—24no—
165C22×Dic47—22no—
166C22×Dic47C2×D418yes—
167C22×Dic47C2×D4132yes—
168C22×Dic47C2×D418yes—
169C22×Dic47C2×D414yes—
170C22×Dic47C2412yes—
171C22×Dic47C2414yes—
172C22×Dic47C2412yes—
173C22×Dic47C2414yes—
174C22×Dic47C2412yes—
175#10625352—2256no—
176C2×C4×D47—232no—
177C2×C4×D47—216no—
178C2×C4×D47C2×D4116yes—
179C2×Dic94—232no—
180C2×Dic94—28no—
181C2×Dic94—216no—
182C2×Dic94—216no—
183C2×Dic94—28no—
184C2×Dic94—416no—
185C4⋊C4×C47—28no—
186C22⋊C4×C47—216no—
187C22⋊C4×C47—48no—
188C22⋊Dic47—216no—
189C22⋊Dic47—48no—
190(C2×D47)⋊C4—28no—
191(C2×D47)⋊C4—28no—
192(C2×D47)⋊C4—48no—
193C4⋊Dic47—216no—
194Dic47⋊C4—28no—
195Dic47⋊C4—216no—
196Dic47⋊C4—216no—
197Dic47⋊C4—28no—
198C23×C47—216no—
199C23×C47—216no—
200C23×C47—216no—
201C23×C47—24no—
202C23×C47—24no—
203C23×C47—24no—
204C23×C47—24no—
205C23×C47—24no—
206C23×C47—24no—
207C23×C47—48no—
208C23×C47—48no—
209C23×C47—48no—
210C23×C47—48no—
211C23×C47C22×D418yes—
212C23×C47C22×D412yes—
213C23×C47C22×D414yes—
214C23×C47H(5,𝔽2)12yes—
215C23×C472−1+414yes—
216C23×C472−1+412yes—
217C23×C47C2511yescharacteristic
218C22×D47—28no—
219C22×D47—28no—
220C22×D47—28no—
221C22×D47—24no—
222C22×D47—22no—
223C22×D47—24no—
224C22×D47—22no—
225C22×D47—48no—
226C22×D47—48no—
227C22×D47C22×D414yes—
228D4×C47—232no—
229C2×C4×C47—28no—
230C2×C4×C47—24no—
231C2×C4×C47—24no—
232C2×C4×C47—22no—
233C2×C4×C47—24no—
234C2×C4×C47—22no—
235C2×C4×C47—24no—
236C2×C4×C47—22no—
237C2×C4×C47—22no—
238C2×C4×C47—24no—
239C2×C4×C47—44no—
240C2×C4×C47—48no—
241C2×C4×C47—44no—
242C2×C4×C47C22×D414yes—
243C22⋊D47—264no—
244C2×Dic47—216no—
245C2×Dic47—24no—
246C2×Dic47—24no—
247C2×Dic47—264no—
248C2×Dic47—22no—
249C2×Dic47—24no—
250C2×Dic47—22no—
251C2×Dic47—24no—
252C2×Dic47—24no—
253C2×Dic47—22no—
254C2×Dic47—24no—
255C2×Dic47—22no—
256C2×Dic47—44no—
257C2×Dic47—48no—
258C2×Dic47—44no—
259C2×Dic47#120116yes—
260C2×Dic47C22×D418yes—
261C2×Dic47C22×D418yes—
262C2×Dic47C22×D414yes—
263C4×D47—232no—
264Dic94—232no—
265Dic94—416no—
266C2×#2616—471nomaximal, Sylow
267C22×C47—28no—
268C22×C47—28no—
269C22×C47—28no—
270C22×C47—432no—
271C22×C47—44no—
272C22×C47—44no—
273C22×C47—44no—
274C22×C47—44no—
275C22×C47C2×#12014yes—
276C22×C47#53614yes—
277C22×C47#53618yes—
278C22×C47#53718yes—
279C22×C47#53714yes—
280C22×C47C23×D411yescharacteristic
281C22×C47C23×D412yes—
282C22×C47C2×H(5,𝔽2)11yescharacteristic
283C22×C47C2×2−1+412yes—
284C22×C47C2×2−1+411yescharacteristic
285C2×D47—24no—
286C2×D47—24no—
287C2×D47—24no—
288C2×D47—416no—
289C2×D47—44no—
290C2×D47—44no—
291C4×C47—24no—
292C4×C47—42no—
293C4×C47—44no—
294C4×C47—42no—
295Dic47—28no—
296Dic47—42no—
297Dic47—44no—
298Dic47—42no—
299Dic47C2×#12018yes—
300C23×#41—471no—
301C22×C22⋊Q8—471no—
302C22×C22⋊Q8—472no—
303#2616—4716no—
304C2×#395—471no—
305C2×#395—471no—
306C2×#394—471no—
307C2×#394—472no—
308C2×#394—472no—
309C2×#394D4711yescharacteristic, complemented
310C2×#380—471no—
311C2×#380—472no—
312C2×C47—48no—
313C2×C47#261618yescomplemented
314C2×C47C22×#12011yescharacteristic
315C2×C47C2×#53611yescharacteristic
316C2×C47C2×#53612yes—
317C2×C47C2×#53712yes—
318C2×C47C2×#53711yescharacteristic
319D47—44no—
320C22×#41—478no—
321C22×#41—9416no—
322C22×#41—948no—
323C23×Q8—471no—
324C23×Q8—471no—
325C23×D4—472no—
326C23×D4C2×D4711yescharacteristic
327C24×C4—472no—
328C24×C4—471no—
329C2×C22⋊Q8—478no—
330C2×C22⋊Q8—4716no—
331C2×C22⋊Q8—9416no—
332C2×C22⋊Q8—948no—
333C22×C4⋊C4—472no—
334C22×C4⋊C4—471no—
335C22×C4⋊C4—472no—
336C22×C4⋊C4—472no—
337C22×C4⋊C4—471no—
338C22×C4⋊C4C2×D4711yescharacteristic
339C22×C22⋊C4—472no—
340C22×C22⋊C4—471no—
341C22×C22⋊C4—471no—
342C22×C22⋊C4—472no—
343C22×C22⋊C4—472no—
344C22×C22⋊C4C2×D4712yes—
345C22×C22⋊C4C2×D4712yes—
346#395—478no—
347#395—478no—
348#394—478no—
349#394—4716no—
350#394—4716no—
351#394C2×D4718yescomplemented
352#380—4716no—
353#380—478no—
354C2×#95—472no—
355C2×#95—472no—
356C2×#95—471no—
357C2×#95C2×D4711yescharacteristic
358C47C2×#261611yescharacteristic, complemented, Sylow
359C25—471no—
360C25C22×D4712yes—
361C2×#41—9464no—
362C2×#41—9432no—
363C2×#41—94128no—
364C22×Q8—474no—
365C22×Q8—474no—
366C22×Q8—948no—
367C22×Q8—948no—
368C22×Q8—944no—
369C22×Q8—944no—
370C22×D4—28no—
371C22×D4—24no—
372C22×D4—478no—
373C22×D4—948no—
374C22×D4—948no—
375C22×D4—948no—
376C22×D4C22×D4714yes—
377C23×C4—471no—
378C23×C4—4716no—
379C23×C4—478no—
380C23×C4—472no—
381C23×C4—478no—
382C23×C4—474no—
383C23×C4—471no—
384C23×C4—472no—
385C23×C4—471no—
386C23×C4—948no—
387C23×C4—948no—
388C23×C4—948no—
389C23×C4C22×D4711yescharacteristic
390C23×C4C22×D4712yes—
391C23×C4C22×D4711yescharacteristic
392C23×C4C22×D4711yescharacteristic
393C22⋊Q8—9464no—
394C22⋊Q8—9432no—
395C2×C4⋊C4—24no—
396C2×C4⋊C4—474no—
397C2×C4⋊C4—478no—
398C2×C4⋊C4—478no—
399C2×C4⋊C4—474no—
400C2×C4⋊C4—478no—
401C2×C4⋊C4—948no—
402C2×C4⋊C4—948no—
403C2×C4⋊C4—944no—
404C2×C4⋊C4—948no—
405C2×C4⋊C4—944no—
406C2×C4⋊C4C22×D4714yes—
407C2×C22⋊C4—28no—
408C2×C22⋊C4—44no—
409C2×C22⋊C4—478no—
410C2×C22⋊C4—474no—
411C2×C22⋊C4—474no—
412C2×C22⋊C4—478no—
413C2×C22⋊C4—478no—
414C2×C22⋊C4—944no—
415C2×C22⋊C4—944no—
416C2×C22⋊C4—948no—
417C2×C22⋊C4—1884no—
418C2×C22⋊C4—1884no—
419C2×C22⋊C4C22×D4718yes—
420C2×C22⋊C4C22×D4718yes—
421#95—478no—
422#95—478no—
423#95—474no—
424#95C22×D4714yes—
425C24—24no—
426C24—24no—
427C24—24no—
428C24—24no—
429C24—24no—
430C24—24no—
431C24—474no—
432C24—472no—
433C24—944no—
434C24—942no—
435C24—944no—
436C24—942no—
437C24D4×D4718yes—
438C24C23×D4714yes—
439C24C23×D4711yescharacteristic
440#41—94256no—
441C2×Q8—9416no—
442C2×Q8—9416no—
443C2×Q8—948no—
444C2×Q8—9432no—
445C2×Q8—948no—
446C2×Q8—18816no—
447C2×D4—232no—
448C2×D4—216no—
449C2×D4—28no—
450C2×D4—9416no—
451C2×D4—9416no—
452C2×D4—9464no—
453C2×D4—9416no—
454C22×C4—24no—
455C22×C4—24no—
456C22×C4—24no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—24no—
461C22×C4—24no—
462C22×C4—22no—
463C22×C4—474no—
464C22×C4—478no—
465C22×C4—472no—
466C22×C4—478no—
467C22×C4—4732no—
468C22×C4—474no—
469C22×C4—4716no—
470C22×C4—474no—
471C22×C4—472no—
472C22×C4—472no—
473C22×C4—9432no—
474C22×C4—9416no—
475C22×C4—9464no—
476C22×C4—9416no—
477C22×C4—9416no—
478C22×C4—944no—
479C22×C4—944no—
480C22×C4—944no—
481C22×C4—942no—
482C22×C4—942no—
483C22×C4—944no—
484C22×C4—944no—
485C22×C4—942no—
486C22×C4—944no—
487C22×C4—942no—
488C22×C4D4×D4714yes—
489C22×C4C23×D4712yes—
490C22×C4C23×D4714yes—
491C22×C4C23×D4712yes—
492C22×C4C23×D4712yes—
493C4⋊C4—28no—
494C4⋊C4—9416no—
495C4⋊C4—9416no—
496C4⋊C4—948no—
497C4⋊C4—9416no—
498C4⋊C4—948no—
499C22⋊C4—216no—
500C22⋊C4—48no—
501C22⋊C4—948no—
502C22⋊C4—948no—
503C22⋊C4—9416no—
504C22⋊C4—1888no—
505C22⋊C4—1888no—
506C23—216no—
507C23—24no—
508C23—24no—
509C23—216no—
510C23—24no—
511C23—216no—
512C23—24no—
513C23—24no—
514C23—24no—
515C23—48no—
516C23—48no—
517C23—48no—
518C23—48no—
519C23—474no—
520C23—948no—
521C23—944no—
522C23—948no—
523C23—942no—
524C23—944no—
525C23—942no—
526C23—948no—
527C23—1888no—
528C23—1888no—
529C23C2×D4×D4718yes—
530C23C2×D4×D4712yes—
531C23C2×D4×D4714yes—
532C23#1309406912yes—
533C23#1309407114yes—
534C23#1309407112yes—
535C23C24×D4711yescharacteristic
536Q8—9432no—
537Q8—18816no—
538D4—232no—
539D4—9464no—
540C2×C4—28no—
541C2×C4—24no—
542C2×C4—24no—
543C2×C4—24no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—24no—
548C2×C4—24no—
549C2×C4—22no—
550C2×C4—44no—
551C2×C4—48no—
552C2×C4—44no—
553C2×C4—4716no—
554C2×C4—474no—
555C2×C4—478no—
556C2×C4—478no—
557C2×C4—9432no—
558C2×C4—9464no—
559C2×C4—9416no—
560C2×C4—944no—
561C2×C4—944no—
562C2×C4—944no—
563C2×C4—942no—
564C2×C4—942no—
565C2×C4—944no—
566C2×C4—944no—
567C2×C4—942no—
568C2×C4—944no—
569C2×C4—942no—
570C2×C4—1884no—
571C2×C4—1888no—
572C2×C4—1884no—
573C2×C4C2×D4×D4714yes—
574C22—28no—
575C22—28no—
576C22—28no—
577C22—432no—
578C22—44no—
579C22—44no—
580C22—44no—
581C22—44no—
582C22—944no—
583C22—944no—
584C22—944no—
585C22—18816no—
586C22—1884no—
587C22—1884no—
588C22#120×D4714yes—
589C22#1568969314yes—
590C22#1568969518yes—
591C22#1568970018yes—
592C22#1568970214yes—
593C22C22×D4×D4711yescharacteristic
594C22C22×D4×D4712yes—
595C22C2×#1309406911yescharacteristic
596C22C2×#1309407112yes—
597C22C2×#1309407111yescharacteristic
598C4—24no—
599C4—42no—
600C4—44no—
601C4—42no—
602C4—478no—
603C4—948no—
604C4—1882no—
605C4—1884no—
606C4—1882no—
607C2—48no—
608C2—1884no—
609C2#2024911618yesdirect product factor
610C2C2×#120×D4711yescharacteristic
611C2C2×#1568969311yescharacteristic
612C2C2×#1568969512yes—
613C2C2×#1568970012yes—
614C2C2×#1568970211yescharacteristic
subgroupG/H
centerC24C23×D47
commutator, cosocleC23×C47C25
FrattiniC23C24×D47
FittingC2×C47×#394C2
socleC24×C47C24
prime radicalC23×#10625352C2
hypercenterC2×#394D47
hypocenterC47C2×#2616
LCSG ▷ C23×C47 ▷ C47
UCSC1 ◁ C24 ◁ C2×#394

Complements

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

NH ≅ G/Ntype
2. C22×#13094002C2semidirect
3. C22×#13094001C2semidirect
4. #20249116C2direct (N × H)
5. C2×#15689044C2semidirect
6. C2×#15689043C2semidirect
7. C2×#15689039C2semidirect
8. C2×#15689038C2semidirect
9. C2×#15689037C2semidirect
10. C2×#15688982C2semidirect
11. C2×#15688981C2semidirect
12. C2×C47×#394C2semidirect
22. C2×#13094002C22semidirect
24. C2×#13094001C22semidirect
35. #15689044C22semidirect
36. #15689043C22semidirect
37. #15689039C22semidirect
38. #15689038C22semidirect
39. #15689037C22semidirect
40. #15688982C22semidirect
41. #15688981C22semidirect
49. C47×#394C22semidirect
309. C2×#394D47semidirect
313. C2×C47#2616semidirect
351. #394C2×D47semidirect
358. C47C2×#2616semidirect
609. C2#20249116direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#106253522
C22×#130940022
C22×#130940012
#202491162
C2×#156890442
C2×#156890432
C2×#156890392
C2×#156890382
C2×#156890372
C2×#156889822
C2×#156889812
C2×C47×#3942
C2×#261647