Finite Group Databasebeta

← C2×#20246987 #26384298

Subgroups — C2×#20246987 #26384298

Order 12032.

3246 conjugacy classes of subgroups.

The subgroup lattice has 592 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#13094024C211yesmaximal, characteristic, complemented
2C22×#13094003C211yesmaximal, characteristic
3C22×#13094003C211yesmaximal, characteristic
4C22×C4×Dic94C211yesmaximal, characteristic, complemented
5#20246987C2116yesmaximal, direct product factor
6C2×#15689057C212yesmaximal, complemented
7C2×#15689046C212yesmaximal
8C2×#15689029C211yesmaximal, characteristic, complemented
9C2×#15689021C211yesmaximal, characteristic, complemented
10C2×#15689020C212yesmaximal, complemented
11C2×#15689016C212yesmaximal
12C2×C4×C22⋊C4×C47C211yesmaximal, characteristic
13C23×Dic94C2211yescharacteristic
14C23×Dic94C2211yescharacteristic
15C2×#13094024—28no—
16C2×#13094024C2218yescomplemented
17C2×#13094003—28no—
18C2×#13094003—28no—
19C2×#13094003C2218yes—
20C2×#13094003C2218yes—
21C2×C4×Dic94—28no—
22C2×C4×Dic94C2218yescomplemented
23C22×C4⋊Dic47C2212yes—
24C22×C4⋊Dic47C2212yes—
25C22×C4⋊Dic47C2211yescharacteristic
26C22×C4⋊Dic47C2211yescharacteristic
27C22×Dic47⋊C4C2212yes—
28C22×Dic47⋊C4C2212yes—
29C22×Dic47⋊C4C2212yes—
30C22×Dic47⋊C4C2212yes—
31C22×C22⋊Dic47C2212yes—
32C22×C22⋊Dic47C2212yes—
33C22×C4×Dic47C2212yes—
34#15689057C22116yescomplemented
35#15689046C22116yes—
36#15689029C2218yescomplemented
37#15689021C2218yescomplemented
38#15689020C22116yescomplemented
39#15689016C22116yes—
40C2×C47⋊#95C2212yes—
41C2×C47⋊#95C2212yes—
42C2×C47⋊#95C2212yes—
43C24×C4×C47C2211yescharacteristic
44C22×C22⋊C4×C47C2211yescharacteristic
45C22×C22⋊C4×C47C2211yescharacteristic
46C22×C42×C47C2211yescharacteristic
47C22×C42×C47C2211yescharacteristic
48C4×C22⋊C4×C47C2218yes—
49C2×#95×C47C2211yescharacteristic
50C2×#95×C47C2211yescharacteristic
51C22×Dic94—24no—
52C22×Dic94—24no—
53C22×Dic94—24no—
54C22×Dic94—24no—
55C22×Dic94—24no—
56C22×Dic94D418yes—
57C22×Dic94C2314yes—
58C22×Dic94C2314yes—
59C23×Dic47C2312yes—
60C23×Dic47C2312yes—
61C23×Dic47C2312yes—
62C23×Dic47C2312yes—
63#13094024—232no—
64#13094003—232no—
65#13094003—232no—
66C4×Dic94—232no—
67C2×C4⋊Dic47—24no—
68C2×C4⋊Dic47—28no—
69C2×C4⋊Dic47—28no—
70C2×C4⋊Dic47D418yes—
71C2×C4⋊Dic47C2318yes—
72C2×C4⋊Dic47C2318yes—
73C2×C4⋊Dic47C2314yes—
74C2×C4⋊Dic47C2314yes—
75C2×Dic47⋊C4—28no—
76C2×Dic47⋊C4—28no—
77C2×Dic47⋊C4—28no—
78C2×Dic47⋊C4—28no—
79C2×Dic47⋊C4C2318yes—
80C2×Dic47⋊C4C2318yes—
81C2×Dic47⋊C4C2318yes—
82C2×Dic47⋊C4C2318yes—
83C2×C22⋊Dic47—28no—
84C2×C22⋊Dic47—28no—
85C2×C22⋊Dic47C2318yes—
86C2×C22⋊Dic47C2318yes—
87C2×C4×Dic47—28no—
88C2×C4×Dic47C2318yes—
89C47⋊#95C2318yes—
90C47⋊#95C2318yes—
91C47⋊#95C2318yes—
92C25×C47C2311yescharacteristic
93C23×C4×C47—24no—
94C23×C4×C47—24no—
95C23×C4×C47—24no—
96C23×C4×C47C2311yescharacteristic
97C23×C4×C47C2311yescharacteristic
98C23×C4×C47C2314yes—
99C23×C4×C47C2311yescharacteristic
100C23×C4×C47C2311yescharacteristic
101C23×C4×C47C2311yescharacteristic
102C23×C4×C47C2311yescharacteristic
103C2×C22⋊C4×C47—24no—
104C2×C22⋊C4×C47Q818yescomplemented
105C2×C22⋊C4×C47C2314yes—
106C2×C22⋊C4×C47C2314yes—
107C2×C42×C47—24no—
108C2×C42×C47—24no—
109C2×C42×C47C2314yes—
110C2×C42×C47C2314yes—
111#95×C47C2314yes—
112#95×C47C2314yes—
113C24×C47—22no—
114C24×C47—22no—
115C24×C47—22no—
116C24×C47—22no—
117C24×C47—22no—
118C24×C47—22no—
119C24×C47C2×Q814yes—
120C24×C47C2412yes—
121C24×C47C2411yescharacteristic
122C22×C4×C47—22no—
123C22×C4×C47—22no—
124C22×C4×C47—22no—
125C22×C4×C47—22no—
126C22×C4×C47—22no—
127C22×C4×C47—22no—
128C22×C4×C47—22no—
129C22×C4×C47—22no—
130C22×C4×C47—22no—
131C22×C4×C47—22no—
132C22×C4×C47—22no—
133C22×C4×C47—22no—
134C22×C4×C47—28no—
135C22×C4×C47—28no—
136C22×C4×C47—28no—
137C22×C4×C47—416no—
138C22×C4×C47C2×D414yes—
139C22×C4×C47C2×D414yes—
140C22×C4×C47C2×Q814yes—
141C22×C4×C47C2×Q814yes—
142C22×C4×C47#4114yes—
143C22×C4×C47#4114yes—
144C22×C4×C47C2412yes—
145C22×C4×C47C2412yes—
146C22×C4×C47C2412yes—
147C22×C4×C47C2412yes—
148C22×C4×C47C2412yes—
149C22×C4×C47C2412yes—
150C22×Dic47—24no—
151C22×Dic47—24no—
152C22×Dic47—24no—
153C22×Dic47—24no—
154C22×Dic47—24no—
155C22×Dic47—24no—
156C22×Dic47—24no—
157C22×Dic47—24no—
158C22×Dic47—24no—
159C22×Dic47—24no—
160C22×Dic47C2×D418yes—
161C22×Dic47C2×D418yes—
162C22×Dic47C2414yes—
163C22×Dic47C2414yes—
164C22×Dic47C2414yes—
165C22×Dic47C2414yes—
166C2×Dic94—232no—
167C2×Dic94—28no—
168C2×Dic94—28no—
169C2×Dic94—28no—
170C2×Dic94—28no—
171C2×Dic94—28no—
172C2×Dic94—416no—
173C2×Dic94C2×D4116yes—
174C22⋊C4×C47—28no—
175C22⋊C4×C47C2×Q8116yescomplemented
176C42×C47—28no—
177C42×C47—28no—
178C22⋊Dic47—216no—
179C22⋊Dic47—216no—
180C4⋊Dic47—28no—
181C4⋊Dic47—216no—
182C4⋊Dic47—216no—
183C4⋊Dic47C2×D4116yes—
184Dic47⋊C4—216no—
185Dic47⋊C4—216no—
186Dic47⋊C4—216no—
187Dic47⋊C4—216no—
188C4×Dic47—216no—
189C23×C47—28no—
190C23×C47—28no—
191C23×C47—28no—
192C23×C47—22no—
193C23×C47—22no—
194C23×C47—22no—
195C23×C47—22no—
196C23×C47—22no—
197C23×C47—22no—
198C23×C47—44no—
199C23×C47—44no—
200C23×C47—44no—
201C23×C47—44no—
202C23×C47C22×D412yes—
203C23×C47C22×D412yes—
204C23×C47C22×Q814yes—
205C23×C47C22×Q812yes—
206C23×C47C2×#4112yes—
207C23×C47C2×#4112yes—
208C23×C472−1+412yes—
209C23×C472−1+412yes—
210C23×C47C2511yescharacteristic
211C2×C4×C47—28no—
212C2×C4×C47—28no—
213C2×C4×C47—28no—
214C2×C4×C47—22no—
215C2×C4×C47—22no—
216C2×C4×C47—22no—
217C2×C4×C47—22no—
218C2×C4×C47—22no—
219C2×C4×C47—22no—
220C2×C4×C47—22no—
221C2×C4×C47—22no—
222C2×C4×C47—22no—
223C2×C4×C47—22no—
224C2×C4×C47—22no—
225C2×C4×C47—22no—
226C2×C4×C47—44no—
227C2×C4×C47—44no—
228C2×C4×C47—416no—
229C2×C4×C47#121116yes—
230C2×C4×C47C22×D414yes—
231C2×C4×C47C22×D414yes—
232C2×C4×C47C22×Q814yes—
233C2×C4×C47C22×Q814yes—
234C2×C4×C47C2×#4114yes—
235C2×C4×C47C2×#4114yes—
236C2×Dic47—216no—
237C2×Dic47—216no—
238C2×Dic47—24no—
239C2×Dic47—24no—
240C2×Dic47—24no—
241C2×Dic47—24no—
242C2×Dic47—24no—
243C2×Dic47—24no—
244C2×Dic47—24no—
245C2×Dic47—24no—
246C2×Dic47—24no—
247C2×Dic47—24no—
248C2×Dic47—48no—
249C2×Dic47—48no—
250C2×Dic47C22×D418yes—
251C2×Dic47C22×D418yes—
252Dic94—232no—
253Dic94—416no—
254C2×#2429—471nomaximal, Sylow
255C22×C47—24no—
256C22×C47—24no—
257C22×C47—24no—
258C22×C47—42no—
259C22×C47—416no—
260C22×C47—42no—
261C22×C47—42no—
262C22×C47—42no—
263C22×C47C2×#12114yes—
264C22×C47#53314yes—
265C22×C47#54114yes—
266C22×C47#54714yes—
267C22×C47#54814yes—
268C22×C47D4×Q814yes—
269C22×C47D4×Q814yes—
270C22×C47C23×D411yescharacteristic
271C22×C47C23×D411yescharacteristic
272C22×C47C23×Q811yescharacteristic
273C22×C47C22×#4111yescharacteristic
274C22×C47C22×#4111yescharacteristic
275C22×C47C2×2−1+411yescharacteristic
276C22×C47C2×2−1+411yescharacteristic
277C4×C47—24no—
278C4×C47—24no—
279C4×C47—24no—
280C4×C47—42no—
281C4×C47—42no—
282C4×C47C2×#12118yes—
283Dic47—28no—
284Dic47—28no—
285Dic47—44no—
286Dic47—44no—
287C22×C4⋊Q8—471no—
288C22×C22⋊Q8—471no—
289C22×C22⋊Q8—471no—
290C22×C4×Q8—471no—
291#2429—4716no—
292C2×#398—472no—
293C2×#396—472no—
294C2×#391—471no—
295C2×#389—471no—
296C2×#389—472no—
297C2×#388—472no—
298C2×C4×C22⋊C4D4711yescharacteristic
299C2×C47—44no—
300C2×C47#242918yescomplemented
301C2×C47C22×#12111yescharacteristic
302C2×C47C2×#53311yescharacteristic
303C2×C47C2×#54111yescharacteristic
304C2×C47C2×#54711yescharacteristic
305C2×C47C2×#54811yescharacteristic
306C2×C47C2×D4×Q811yescharacteristic
307C2×C47C2×D4×Q811yescharacteristic
308C23×Q8—471no—
309C23×Q8—471no—
310C24×C4C2×D4711yescharacteristic
311C2×C4⋊Q8—478no—
312C2×C4⋊Q8—948no—
313C2×C22⋊Q8—478no—
314C2×C22⋊Q8—478no—
315C2×C22⋊Q8—948no—
316C2×C22⋊Q8—948no—
317C2×C4×Q8—478no—
318C2×C4×Q8—948no—
319C22×C4⋊C4—471no—
320C22×C4⋊C4—472no—
321C22×C4⋊C4—472no—
322C22×C4⋊C4—472no—
323C22×C4⋊C4—472no—
324C22×C4⋊C4—472no—
325C22×C4⋊C4—472no—
326C22×C4⋊C4—471no—
327C22×C22⋊C4—472no—
328C22×C22⋊C4—472no—
329C22×C22⋊C4C2×D4711yescharacteristic
330C22×C22⋊C4C2×D4711yescharacteristic
331C22×C42—472no—
332C22×C42C2×D4711yescharacteristic
333C22×C42C2×D4711yescharacteristic
334#398—4716no—
335#396—4716no—
336#391—478no—
337#389—478no—
338#389—4716no—
339#388—4716no—
340C4×C22⋊C4C2×D4718yes—
341C2×#95—472no—
342C2×#95—472no—
343C2×#95—472no—
344C2×#95C2×D4711yescharacteristic
345C2×#95C2×D4711yescharacteristic
346C47C2×#242911yescharacteristic, complemented, Sylow
347C25C22×D4711yescharacteristic
348C22×Q8—478no—
349C22×Q8—474no—
350C22×Q8—474no—
351C22×Q8—944no—
352C22×Q8—944no—
353C22×Q8—944no—
354C22×Q8—944no—
355C22×Q8—944no—
356C23×C4—24no—
357C23×C4—24no—
358C23×C4—24no—
359C23×C4—472no—
360C23×C4—472no—
361C23×C4—472no—
362C23×C4—472no—
363C23×C4C22×D4714yes—
364C23×C4C22×D4711yescharacteristic
365C23×C4C22×D4711yescharacteristic
366C23×C4C22×D4711yescharacteristic
367C23×C4C22×D4711yescharacteristic
368C23×C4C22×D4711yescharacteristic
369C23×C4C22×D4711yescharacteristic
370C4⋊Q8—9432no—
371C22⋊Q8—9432no—
372C22⋊Q8—9432no—
373C4×Q8—9432no—
374C2×C4⋊C4—478no—
375C2×C4⋊C4—474no—
376C2×C4⋊C4—478no—
377C2×C4⋊C4—478no—
378C2×C4⋊C4—478no—
379C2×C4⋊C4—474no—
380C2×C4⋊C4—478no—
381C2×C4⋊C4—478no—
382C2×C4⋊C4—478no—
383C2×C4⋊C4—948no—
384C2×C4⋊C4—948no—
385C2×C4⋊C4—948no—
386C2×C4⋊C4—948no—
387C2×C4⋊C4—948no—
388C2×C4⋊C4—948no—
389C2×C4⋊C4—944no—
390C2×C22⋊C4—24no—
391C2×C22⋊C4—478no—
392C2×C22⋊C4—478no—
393C2×C22⋊C4—948no—
394C2×C22⋊C4—948no—
395C2×C22⋊C4Dic9418yescomplemented
396C2×C22⋊C4C22×D4714yes—
397C2×C22⋊C4C22×D4714yes—
398C2×C42—24no—
399C2×C42—24no—
400C2×C42—478no—
401C2×C42—948no—
402C2×C42C22×D4714yes—
403C2×C42C22×D4714yes—
404#95—478no—
405#95—478no—
406#95—478no—
407#95C22×D4714yes—
408#95C22×D4714yes—
409C24—22no—
410C24—22no—
411C24—22no—
412C24—22no—
413C24—22no—
414C24—22no—
415C24C2×Dic9414yes—
416C24C23×D4712yes—
417C24C23×D4711yescharacteristic
418C2×Q8—4716no—
419C2×Q8—9432no—
420C2×Q8—948no—
421C2×Q8—948no—
422C2×Q8—948no—
423C2×Q8—948no—
424C2×Q8—948no—
425C2×Q8—18816no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—28no—
439C22×C4—28no—
440C22×C4—28no—
441C22×C4—416no—
442C22×C4—478no—
443C22×C4—478no—
444C22×C4—474no—
445C22×C4—474no—
446C22×C4—474no—
447C22×C4—474no—
448C22×C4—944no—
449C22×C4—944no—
450C22×C4—944no—
451C22×C4—944no—
452C22×C4—944no—
453C22×C4—944no—
454C22×C4—944no—
455C22×C4—944no—
456C22×C4—944no—
457C22×C4—944no—
458C22×C4C2×Dic9414yes—
459C22×C4C2×Dic9414yes—
460C22×C4#1062535014yes—
461C22×C4#1062535014yes—
462C22×C4D4×D4714yes—
463C22×C4D4×D4714yes—
464C22×C4C23×D4712yes—
465C22×C4C23×D4712yes—
466C22×C4C23×D4712yes—
467C22×C4C23×D4712yes—
468C22×C4C23×D4712yes—
469C22×C4C23×D4712yes—
470C4⋊C4—4716no—
471C4⋊C4—9416no—
472C4⋊C4—9416no—
473C4⋊C4—9416no—
474C4⋊C4—9416no—
475C4⋊C4—9416no—
476C4⋊C4—948no—
477C4⋊C4—9416no—
478C22⋊C4—28no—
479C22⋊C4—9416no—
480C22⋊C4—9416no—
481C22⋊C4C2×Dic94116yescomplemented
482C42—28no—
483C42—28no—
484C42—9416no—
485C23—28no—
486C23—28no—
487C23—28no—
488C23—22no—
489C23—22no—
490C23—22no—
491C23—22no—
492C23—22no—
493C23—22no—
494C23—44no—
495C23—44no—
496C23—44no—
497C23—44no—
498C23C2×D4×D4712yes—
499C23C2×D4×D4712yes—
500C23C22×Dic9414yes—
501C23C22×Dic9412yes—
502C23C2×#1062535012yes—
503C23C2×#1062535012yes—
504C23#1309407112yes—
505C23#1309407112yes—
506C23C24×D4711yescharacteristic
507Q8—9432no—
508Q8—18816no—
509C2×C4—28no—
510C2×C4—28no—
511C2×C4—28no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—44no—
525C2×C4—44no—
526C2×C4—416no—
527C2×C4—478no—
528C2×C4—478no—
529C2×C4—9416no—
530C2×C4—9416no—
531C2×C4—944no—
532C2×C4—944no—
533C2×C4—944no—
534C2×C4—944no—
535C2×C4—944no—
536C2×C4—944no—
537C2×C4—944no—
538C2×C4—944no—
539C2×C4—944no—
540C2×C4—944no—
541C2×C4—1888no—
542C2×C4—1888no—
543C2×C4#13093997116yes—
544C2×C4C2×D4×D4714yes—
545C2×C4C2×D4×D4714yes—
546C2×C4C22×Dic9414yes—
547C2×C4C22×Dic9414yes—
548C2×C4C2×#1062535014yes—
549C2×C4C2×#1062535014yes—
550C22—24no—
551C22—24no—
552C22—24no—
553C22—42no—
554C22—42no—
555C22—42no—
556C22—416no—
557C22—42no—
558C22C2×#1309399714yes—
559C22#1568968014yes—
560C22#1568973514yes—
561C22#1568977814yes—
562C22#1568978214yes—
563C22D4×Dic9414yes—
564C22D4×Dic9414yes—
565C22C22×D4×D4711yescharacteristic
566C22C22×D4×D4711yescharacteristic
567C22C23×Dic9411yescharacteristic
568C22C22×#1062535011yescharacteristic
569C22C22×#1062535011yescharacteristic
570C22C2×#1309407111yescharacteristic
571C22C2×#1309407111yescharacteristic
572C4—24no—
573C4—24no—
574C4—24no—
575C4—42no—
576C4—42no—
577C4—948no—
578C4—948no—
579C4—1884no—
580C4—1884no—
581C4C2×#1309399718yes—
582C2—44no—
583C2#2024698718yesdirect product factor
584C2C22×#1309399711yescharacteristic
585C2C2×#1568968011yescharacteristic
586C2C2×#1568973511yescharacteristic
587C2C2×#1568977811yescharacteristic
588C2C2×#1568978211yescharacteristic
589C2C2×D4×Dic9411yescharacteristic
590C2C2×D4×Dic9411yescharacteristic
subgroupG/H
centerC24C23×D47
commutator, cosocleC23×C47C25
FrattiniC23C24×D47
FittingC2×C4×C22⋊C4×C47C2
socleC24×C47C24
prime radicalC25×C47C23
hypercenterC2×C4×C22⋊C4D47
hypocenterC47C2×#2429
LCSG ▷ C23×C47 ▷ C47
UCSC1 ◁ C24 ◁ C2×C4×C22⋊C4

Complements

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

NH ≅ G/Ntype
1. C22×#13094024C2semidirect
4. C22×C4×Dic94C2semidirect
5. #20246987C2direct (N × H)
6. C2×#15689057C2semidirect
8. C2×#15689029C2semidirect
9. C2×#15689021C2semidirect
10. C2×#15689020C2semidirect
16. C2×#13094024C22semidirect
22. C2×C4×Dic94C22semidirect
34. #15689057C22semidirect
36. #15689029C22semidirect
37. #15689021C22semidirect
38. #15689020C22semidirect
104. C2×C22⋊C4×C47Q8semidirect
175. C22⋊C4×C47C2×Q8semidirect
300. C2×C47#2429semidirect
346. C47C2×#2429semidirect
395. C2×C22⋊C4Dic94semidirect
481. C22⋊C4C2×Dic94semidirect
583. C2#20246987direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#130940242
C22×#130940032
C22×C4×Dic942
#202469872
C2×#156890572
C2×#156890462
C2×#156890292
C2×#156890212
C2×#156890202
C2×#156890162
C2×C4×C22⋊C4×C472
C2×#242947