Finite Group Databasebeta

← C2×#20253331 #27084079

Subgroups — C2×#20253331 #27084079

Order 12032.

2326 conjugacy classes of subgroups.

The subgroup lattice has 622 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C22⋊Q8×D47C211yesmaximal, characteristic
2C22×#13093982C211yesmaximal, characteristic
3#20253331C2116yesmaximal, direct product factor
4C2×#15689459C211yesmaximal, characteristic, complemented
5C2×#15689456C211yesmaximal, characteristic, complemented
6C2×#15689467C212yesmaximal, complemented
7C2×#15689463C212yesmaximal, complemented
8C2×#15689453C211yesmaximal, characteristic, complemented
9C2×#15689162C212yesmaximal, complemented
10C2×#15689130C212yesmaximal, complemented
11C2×#15689085C211yesmaximal, characteristic
12C2×C47×#483C211yesmaximal, characteristic, complemented
13C22×Q8×D47C2211yescharacteristic
14C23×C4×D47C2211yescharacteristic
15C22×#10625324C2212yes—
16C2×#13094004C2211yescharacteristic
17C2×#13094003C2211yescharacteristic
18C2×#13094002—21no—
19C2×#13094001—21no—
20C2×#13094000C2211yescharacteristic
21C22⋊Q8×D47—24no—
22C22⋊Q8×D47C2218yes—
23C2×#13093982—21no—
24C2×#13093982—24no—
25C2×#13093982—21no—
26C2×#13093982C2212yes—
27C2×#13093982C2212yes—
28C2×#13093982C2218yes—
29C22×C4⋊Dic47C2211yescharacteristic
30C2×C4⋊C4×D47—21no—
31C2×C4⋊C4×D47C2211yescharacteristic
32C22×(C2×D47)⋊C4—21no—
33C2×C22⋊C4×D47—21no—
34C22×C4×Dic47—21no—
35#15689467C22116yescomplemented
36#15689463C22116yescomplemented
37#15689459C2218yescomplemented
38#15689456C2218yescomplemented
39#15689453C2218yescomplemented
40#15689162C22116yescomplemented
41C2×#13093958C2212yescomplemented
42C2×#13093957C2212yescomplemented
43C2×#13093955C2212yescomplemented
44#15689130C22116yescomplemented
45C2×#13093949C2212yescomplemented
46C2×#13093948C2212yescomplemented
47C2×#13093947C2212yescomplemented
48#15689085C2218yes—
49C2×C22⋊(C47⋊C8)C2211yescharacteristic
50C2×(C2×D47)⋊C8C2212yes—
51C2×C22⋊Q8×C47C2211yescharacteristic
52C22×C4⋊C4×C47C2211yescharacteristic
53C47×#483C2218yescomplemented
54C2×#106×C47C2212yescomplemented
55C2×#103×C47C2212yescomplemented
56C2×C22⋊C8×C47C2211yescharacteristic
57C24×D47D411yescharacteristic
58C22×Dic94—21no—
59C22×Dic94C2311yescharacteristic
60C2×Q8×D47—22no—
61C2×Q8×D47—28no—
62C2×Q8×D47C2314yes—
63C23×Dic47—21no—
64C23×Dic47D411yescharacteristic
65C22×C4×D47—28no—
66C22×C4×D47—21no—
67C22×C4×D47—21no—
68C22×C4×D47D411yescharacteristic
69C22×C4×D47D414yes—
70C22×C4×D47D411yescharacteristic
71C22×C4×D47C2311yescharacteristic
72C22×C4×D47C2314yes—
73C22×C4×D47C2311yescharacteristic
74C2×#10625324—28no—
75C2×#10625324C2318yes—
76C22×C47⋊C8C2312yes—
77#13094004—22no—
78#13094004C2314yes—
79#13094003—22no—
80#13094003C2314yes—
81#13094002—28no—
82#13094001—28no—
83#13094000—22no—
84#13094000C2314yes—
85#13093982—28no—
86#13093982—24no—
87#13093982—28no—
88#13093982—24no—
89#13093982C2318yes—
90#13093982C2318yes—
91C2×C4⋊Dic47—21no—
92C2×C4⋊Dic47—22no—
93C2×C4⋊Dic47—21no—
94C2×C4⋊Dic47C2311yescharacteristic
95C2×C4⋊Dic47C2312yes—
96C2×C4⋊Dic47C2314yes—
97C2×Dic47⋊C4—21no—
98C2×Dic47⋊C4—21no—
99C2×Dic47⋊C4—21no—
100C4⋊C4×D47—28no—
101C4⋊C4×D47—22no—
102C4⋊C4×D47C2314yes—
103C2×C22⋊Dic47—21no—
104C2×(C2×D47)⋊C4—21no—
105C2×(C2×D47)⋊C4—21no—
106C2×(C2×D47)⋊C4—22no—
107C2×(C2×D47)⋊C4—28no—
108C2×(C2×D47)⋊C4—22no—
109C22⋊C4×D47—28no—
110C2×C4×Dic47—22no—
111C2×C4×Dic47—28no—
112C2×C4×Dic47—22no—
113#13093958C2318yescomplemented
114#13093957C2318yescomplemented
115#13093955C2318yescomplemented
116#13093949C2318yescomplemented
117#13093948C2318yescomplemented
118#13093947C2318yescomplemented
119C22⋊(C47⋊C8)C2314yes—
120(C2×D47)⋊C8C2318yes—
121C22×Q8×C47C2311yescharacteristic
122C23×C4×C47—21no—
123C23×C4×C47C2311yescharacteristic
124C22×C8×C47C2312yes—
125C22⋊Q8×C47—22no—
126C22⋊Q8×C47C2314yes—
127C2×C4⋊C4×C47—21no—
128C2×C4⋊C4×C47—22no—
129C2×C4⋊C4×C47—21no—
130C2×C4⋊C4×C47C2314yes—
131C2×C4⋊C4×C47C2312yes—
132C2×C4⋊C4×C47C2311yescharacteristic
133C2×C22⋊C4×C47—21no—
134#106×C47C2318yescomplemented
135#103×C47C2318yescomplemented
136C22⋊C8×C47C2314yes—
137C24×C47C2×D411yescharacteristic
138C23×D47—24no—
139C23×D47—24no—
140C23×D47—42no—
141C23×D47C2×D411yescharacteristic
142C23×D47C2×D414yes—
143C23×D47C2×D411yescharacteristic
144C2×Q8×C47—21no—
145C2×Q8×C47—24no—
146C2×Q8×C47C2412yes—
147C22×C4×C47—24no—
148C22×C4×C47—21no—
149C22×C4×C47—21no—
150C22×C4×C47—24no—
151C22×C4×C47—22no—
152C22×C4×C47—44no—
153C22×C4×C47C2×D411yescharacteristic
154C22×C4×C47C2×D412yes—
155C22×C4×C47C2412yes—
156C22×C4×C47C2411yescharacteristic
157C22×Dic47—24no—
158C22×Dic47—24no—
159C22×Dic47—22no—
160C22×Dic47—21no—
161C22×Dic47—21no—
162C22×Dic47—44no—
163C22×Dic47C2×D411yescharacteristic
164C22×Dic47C2×D414yes—
165C22×Dic47C2×D411yescharacteristic
166Q8×D47—216no—
167Q8×D47—48no—
168C2×C4×D47—24no—
169C2×C4×D47—24no—
170C2×C4×D47—24no—
171C2×C4×D47—216no—
172C2×C4×D47—216no—
173C2×C4×D47—24no—
174C2×C4×D47—24no—
175C2×C4×D47—44no—
176C2×C4×D47—44no—
177C2×C4×D47C2×D414yes—
178C2×C4×D47C2×D412yes—
179C2×C4×D47C2×D414yes—
180C2×C4×D47C2×D412yes—
181C2×C4×D47#4118yes—
182C2×C4×D47C2412yes—
183C2×C4×D47C2412yes—
184C2×Dic94—24no—
185C2×Dic94—21no—
186C2×Dic94—24no—
187C2×Dic94—44no—
188C2×Dic94C2412yes—
189C2×C8×C47—24no—
190C2×C8×C47C2414yes—
191C4⋊C4×C47—24no—
192C4⋊C4×C47—22no—
193C4⋊C4×C47—24no—
194C4⋊C4×C47—21no—
195C4⋊C4×C47C2414yes—
196C4⋊C4×C47C2412yes—
197C22⋊C4×C47—24no—
198C22⋊Dic47—24no—
199(C2×D47)⋊C4—24no—
200(C2×D47)⋊C4—24no—
201(C2×D47)⋊C4—28no—
202(C2×D47)⋊C4—28no—
203C4⋊Dic47—21no—
204C4⋊Dic47—24no—
205C4⋊Dic47—22no—
206C4⋊Dic47—24no—
207C4⋊Dic47C2414yes—
208C4⋊Dic47C2412yes—
209Dic47⋊C4—24no—
210Dic47⋊C4—24no—
211Dic47⋊C4—24no—
212C4×Dic47—28no—
213C4×Dic47—28no—
214C2×C47⋊C8—24no—
215C2×C47⋊C8C2414yes—
216#10625324—216no—
217C23×C47—22no—
218C23×C47—22no—
219C23×C47#13714yes—
220C23×C47C22×D412yes—
221C23×C47C22×D411yescharacteristic
222C22×D47—22no—
223C22×D47—22no—
224C22×D47—22no—
225C22×D47—28no—
226C22×D47—22no—
227C22×D47—22no—
228C22×D47—28no—
229C22×D47—44no—
230C22×D47—416no—
231C22×D47#12314yes—
232C22×D47C22×D412yes—
233C22×D47C22×D412yes—
234Q8×C47—24no—
235Q8×C47—42no—
236C2×C4×C47—22no—
237C2×C4×C47—24no—
238C2×C4×C47—24no—
239C2×C4×C47—24no—
240C2×C4×C47—22no—
241C2×C4×C47—22no—
242C2×C4×C47—42no—
243C2×C4×C47—48no—
244C2×C4×C47—44no—
245C2×C4×C47—42no—
246C2×C4×C47C22×D412yes—
247C2×C4×C47C22×D411yescharacteristic
248C2×C4×C47C2×#4114yes—
249C2×C4×C47C2511yescharacteristic
250C2×Dic47—22no—
251C2×Dic47—28no—
252C2×Dic47—24no—
253C2×Dic47—22no—
254C2×Dic47—22no—
255C2×Dic47—48no—
256C2×Dic47—42no—
257C2×Dic47—42no—
258C2×Dic47—44no—
259C2×Dic47#12314yes—
260C2×Dic47C22×D412yes—
261C2×Dic47C22×D412yes—
262C4×D47—28no—
263C4×D47—24no—
264C4×D47—24no—
265C4×D47—28no—
266C4×D47—48no—
267C4×D47—48no—
268C4×D47#12318yes—
269C4×D47C2×#4118yes—
270Dic94—24no—
271Dic94—42no—
272Dic94—48no—
273C8×C47—24no—
274C47⋊C8—24no—
275C2×#3095—471nomaximal, Sylow
276C22×C47—28no—
277C22×C47—22no—
278C22×C47—22no—
279C22×C47C2×#12312yes—
280C22×C47C2×#13714yes—
281C22×C47C2×#13712yes—
282C22×C47#57512yes—
283C22×C47C23×D411yescharacteristic
284C2×D47—28no—
285C2×D47—22no—
286C2×D47—22no—
287C2×D47—22no—
288C2×D47—22no—
289C2×D47—22no—
290C2×D47—416no—
291C2×D47—44no—
292C2×D47C2×#12314yes—
293C4×C47—22no—
294C4×C47—42no—
295C4×C47—44no—
296C4×C47—42no—
297C4×C47C2×#12312yes—
298C4×C47C22×#4112yes—
299Dic47—22no—
300Dic47—42no—
301Dic47—42no—
302Dic47—44no—
303Dic47C2×#12314yes—
304C22×C22⋊Q8—471no—
305C22×#117—471no—
306#3095—4716no—
307C2×#484—471no—
308C2×#484—471no—
309C2×#485—472no—
310C2×#485—472no—
311C2×#483—471no—
312C2×#483D4711yescharacteristic, complemented
313C2×#428—472no—
314C2×#419—472no—
315C2×#407—471no—
316C2×C47—24no—
317C2×C47#309514yescomplemented
318C2×C47C22×#12311yescharacteristic
319C2×C47C22×#13711yescharacteristic
320C2×C47C2×#57511yescharacteristic
321D47—24no—
322D47—42no—
323C23×Q8—471no—
324C24×C4—471no—
325C22×M16—472no—
326C2×C22⋊Q8—471no—
327C2×C22⋊Q8—471no—
328C2×C22⋊Q8—478no—
329C2×C22⋊Q8—471no—
330C2×C22⋊Q8—941no—
331C2×C22⋊Q8—941no—
332C2×C22⋊Q8—944no—
333C2×C22⋊Q8C2×D4711yescharacteristic
334C2×#117—472no—
335C2×#117—478no—
336C2×#117—472no—
337C2×#117—941no—
338C2×#117—944no—
339C2×#117—941no—
340C22×C4⋊C4—471no—
341C22×C4⋊C4—471no—
342C22×C4⋊C4—941no—
343C22×C4⋊C4C2×D4711yescharacteristic
344C22×C22⋊C4—941no—
345C22×C22⋊C4—941no—
346C22×C42—941no—
347#485—4716no—
348#485—4716no—
349#484—478no—
350#484—478no—
351#483—478no—
352#483C2×D4718yescomplemented
353#428—4716no—
354C2×#107—472no—
355C2×#107—472no—
356C2×#106—472no—
357C2×#106C2×D4712yescomplemented
358#419—4716no—
359C2×#103—472no—
360C2×#103—472no—
361C2×#103—472no—
362C2×#103C2×D4712yescomplemented
363#407—478no—
364C2×C22⋊C8—471no—
365C2×C22⋊C8—472no—
366C2×C22⋊C8C2×D4711yescharacteristic
367C47C2×#309511yescharacteristic, complemented, Sylow
368C25—471no—
369C22×Q8—474no—
370C22×Q8—471no—
371C22×Q8—941no—
372C22×Q8—942no—
373C22×Q8—948no—
374C22×Q8C22×D4711yescharacteristic
375C23×C4—21no—
376C23×C4—471no—
377C23×C4—471no—
378C23×C4—471no—
379C23×C4—471no—
380C23×C4—474no—
381C23×C4—471no—
382C23×C4—474no—
383C23×C4—948no—
384C23×C4—941no—
385C23×C4—941no—
386C23×C4—941no—
387C23×C4C22×D4711yescharacteristic
388C2×M16—478no—
389C2×M16—948no—
390C22×C8—472no—
391C22×C8C22×D4712yes—
392C22⋊Q8—22no—
393C22⋊Q8—474no—
394C22⋊Q8—474no—
395C22⋊Q8—474no—
396C22⋊Q8—948no—
397C22⋊Q8—948no—
398C22⋊Q8—942no—
399C22⋊Q8—942no—
400C22⋊Q8—942no—
401C22⋊Q8C22×D4714yes—
402#117—478no—
403#117—478no—
404#117—944no—
405#117—948no—
406#117—944no—
407#117—948no—
408C2×C4⋊C4—21no—
409C2×C4⋊C4—22no—
410C2×C4⋊C4—21no—
411C2×C4⋊C4—472no—
412C2×C4⋊C4—474no—
413C2×C4⋊C4—471no—
414C2×C4⋊C4—474no—
415C2×C4⋊C4—942no—
416C2×C4⋊C4—948no—
417C2×C4⋊C4—941no—
418C2×C4⋊C4—941no—
419C2×C4⋊C4—941no—
420C2×C4⋊C4—942no—
421C2×C4⋊C4—941no—
422C2×C4⋊C4—941no—
423C2×C4⋊C4C22×D4711yescharacteristic
424C2×C4⋊C4C22×D4712yes—
425C2×C4⋊C4C22×D4714yes—
426C2×C22⋊C4—21no—
427C2×C22⋊C4—948no—
428C2×C22⋊C4—948no—
429C2×C22⋊C4—942no—
430C2×C22⋊C4—942no—
431C2×C22⋊C4—941no—
432C2×C22⋊C4—941no—
433C2×C22⋊C4—941no—
434C2×C42—948no—
435C2×C42—942no—
436C2×C42—942no—
437#107—478no—
438#107—478no—
439#106—478no—
440#106C22×D4718yescomplemented
441#103—478no—
442#103—478no—
443#103—478no—
444#103C22×D4718yescomplemented
445C22⋊C8—474no—
446C22⋊C8—478no—
447C22⋊C8C22×D4714yes—
448C24—471no—
449C24—471no—
450C24—474no—
451C24—944no—
452C24—944no—
453C24—1882no—
454C24D4×D4711yescharacteristic
455C2×Q8—21no—
456C2×Q8—24no—
457C2×Q8—472no—
458C2×Q8—944no—
459C2×Q8—9416no—
460C2×Q8—941no—
461C2×Q8—944no—
462C2×Q8—1884no—
463C2×Q8—1888no—
464C2×Q8C23×D4712yes—
465C22×C4—24no—
466C22×C4—24no—
467C22×C4—22no—
468C22×C4—21no—
469C22×C4—21no—
470C22×C4—44no—
471C22×C4—471no—
472C22×C4—472no—
473C22×C4—478no—
474C22×C4—472no—
475C22×C4—474no—
476C22×C4—471no—
477C22×C4—474no—
478C22×C4—474no—
479C22×C4—472no—
480C22×C4—472no—
481C22×C4—944no—
482C22×C4—9416no—
483C22×C4—9416no—
484C22×C4—944no—
485C22×C4—944no—
486C22×C4—944no—
487C22×C4—944no—
488C22×C4—944no—
489C22×C4—944no—
490C22×C4—942no—
491C22×C4—941no—
492C22×C4—941no—
493C22×C4—1884no—
494C22×C4—1884no—
495C22×C4—1884no—
496C22×C4D4×D4712yes—
497C22×C4D4×D4711yescharacteristic
498C22×C4C23×D4712yes—
499C22×C4C23×D4711yescharacteristic
500M16—9416no—
501C2×C8—24no—
502C2×C8—474no—
503C2×C8—944no—
504C2×C8C23×D4714yes—
505C4⋊C4—24no—
506C4⋊C4—22no—
507C4⋊C4—24no—
508C4⋊C4—21no—
509C4⋊C4—474no—
510C4⋊C4—472no—
511C4⋊C4—944no—
512C4⋊C4—944no—
513C4⋊C4—944no—
514C4⋊C4—944no—
515C4⋊C4—944no—
516C4⋊C4—942no—
517C4⋊C4—941no—
518C4⋊C4C23×D4714yes—
519C4⋊C4C23×D4712yes—
520C22⋊C4—24no—
521C22⋊C4—948no—
522C22⋊C4—948no—
523C22⋊C4—944no—
524C22⋊C4—944no—
525C22⋊C4—944no—
526C42—948no—
527C42—948no—
528C23—22no—
529C23—22no—
530C23—474no—
531C23—472no—
532C23—472no—
533C23—948no—
534C23—942no—
535C23—942no—
536C23—942no—
537C23—948no—
538C23—942no—
539C23—942no—
540C23—1884no—
541C23—18816no—
542C23#1309405314yes—
543C23C2×D4×D4712yes—
544C23C2×D4×D4711yescharacteristic
545Q8—24no—
546Q8—42no—
547Q8—944no—
548Q8—1888no—
549Q8—1882no—
550C2×C4—22no—
551C2×C4—24no—
552C2×C4—24no—
553C2×C4—24no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—42no—
557C2×C4—48no—
558C2×C4—44no—
559C2×C4—42no—
560C2×C4—474no—
561C2×C4—472no—
562C2×C4—478no—
563C2×C4—478no—
564C2×C4—472no—
565C2×C4—948no—
566C2×C4—948no—
567C2×C4—948no—
568C2×C4—942no—
569C2×C4—944no—
570C2×C4—944no—
571C2×C4—944no—
572C2×C4—942no—
573C2×C4—942no—
574C2×C4—1888no—
575C2×C4—1888no—
576C2×C4—1888no—
577C2×C4—1882no—
578C2×C4—1882no—
579C2×C4—1884no—
580C2×C4C2×D4×D4712yes—
581C2×C4C2×D4×D4711yescharacteristic
582C2×C4#41×D4714yes—
583C2×C4C24×D4711yescharacteristic
584C8—24no—
585C8—944no—
586C22—22no—
587C22—22no—
588C22—28no—
589C22—474no—
590C22—942no—
591C22—942no—
592C22—948no—
593C22—942no—
594C22—942no—
595C22—942no—
596C22—18816no—
597C22—1884no—
598C22#123×D4712yes—
599C22C2×#1309405314yes—
600C22C2×#1309405312yes—
601C22#1568992212yes—
602C22C22×D4×D4711yescharacteristic
603C4—22no—
604C4—44no—
605C4—42no—
606C4—42no—
607C4—474no—
608C4—942no—
609C4—1884no—
610C4—1882no—
611C4—1882no—
612C4#123×D4712yes—
613C4C2×#41×D4712yes—
614C2—24no—
615C2—944no—
616C2—1882no—
617C2#2025333114yesdirect product factor
618C2C2×#123×D4711yescharacteristic
619C2C22×#1309405311yescharacteristic
620C2C2×#1568992211yescharacteristic
subgroupG/H
centerC23C2×D4×D47
commutator, cosocleC2×C4×C47C25
FrattiniC2×C4C24×D47
FittingC2×C47×#483C2
socleC23×C47C22×D4
prime radicalC24×D47D4
hypercenterC2×#483D47
hypocenterC47C2×#3095
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#483

Complements

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

NH ≅ G/Ntype
3. #20253331C2direct (N × H)
4. C2×#15689459C2semidirect
5. C2×#15689456C2semidirect
6. C2×#15689467C2semidirect
7. C2×#15689463C2semidirect
8. C2×#15689453C2semidirect
9. C2×#15689162C2semidirect
10. C2×#15689130C2semidirect
12. C2×C47×#483C2semidirect
35. #15689467C22semidirect
36. #15689463C22semidirect
37. #15689459C22semidirect
38. #15689456C22semidirect
39. #15689453C22semidirect
40. #15689162C22semidirect
41. C2×#13093958C22semidirect
42. C2×#13093957C22semidirect
43. C2×#13093955C22semidirect
44. #15689130C22semidirect
45. C2×#13093949C22semidirect
46. C2×#13093948C22semidirect
47. C2×#13093947C22semidirect
53. C47×#483C22semidirect
54. C2×#106×C47C22semidirect
55. C2×#103×C47C22semidirect
113. #13093958C23semidirect
114. #13093957C23semidirect
115. #13093955C23semidirect
116. #13093949C23semidirect
117. #13093948C23semidirect
118. #13093947C23semidirect
134. #106×C47C23semidirect
135. #103×C47C23semidirect
312. C2×#483D47semidirect
317. C2×C47#3095semidirect
352. #483C2×D47semidirect
357. C2×#106C2×D47semidirect
362. C2×#103C2×D47semidirect
367. C47C2×#3095semidirect
440. #106C22×D47semidirect
444. #103C22×D47semidirect
617. C2#20253331direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C22⋊Q8×D472
C22×#130939822
#202533312
C2×#156894592
C2×#156894562
C2×#156894672
C2×#156894632
C2×#156894532
C2×#156891622
C2×#156891302
C2×#156890852
C2×C47×#4832
C2×#309547