Finite Group Databasebeta

← C2×#20243757 #26217768

Subgroups — C2×#20243757 #26217768

Order 12032.

1802 conjugacy classes of subgroups.

The subgroup lattice has 664 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#13094054C211yesmaximal, characteristic
2C22×#13093951C211yesmaximal, characteristic
3C2×#15689131C211yesmaximal, characteristic, complemented
4C2×#15689028C211yesmaximal, characteristic, complemented
5#20243757C218yesmaximal, direct product factor
6C2×#15688892C211yesmaximal, characteristic, complemented
7C2×#15688882C211yesmaximal, characteristic, complemented
8C2×C47×#520C211yesmaximal, characteristic
9C23×Dic94C2211yescharacteristic
10C2×#13094054—24no—
11C2×#13094054—22no—
12C2×#13094054C418yescomplemented
13C2×#13094054C2214yes—
14C22×#10625337—21no—
15C22×#10625335—21no—
16C22×#10625329C2211yescharacteristic
17C22×#10625329C2211yescharacteristic
18C22×C4×Dic47C2211yescharacteristic
19C2×#13093951—24no—
20C2×#13093951C2214yes—
21#15689131—22no—
22#15689131C2214yescomplemented
23C2×#13093947—21no—
24C2×#13093947—21no—
25#15689028—22no—
26#15689028C2214yescomplemented
27C2×C47⋊#95—22no—
28#15688892C2214yescomplemented
29#15688882C2214yescomplemented
30C22×#41×C47C2211yescharacteristic
31C47×#520—22no—
32C47×#520—22no—
33C47×#520—22no—
34C47×#520C2214yes—
35C2×C4×Q8×C47—21no—
36C2×C4×D4×C47—21no—
37C2×C4×D4×C47—22no—
38C2×#117×C47—21no—
39C2×#117×C47C2211yescharacteristic
40C22×C4⋊C4×C47—21no—
41C22×C4⋊C4×C47C2211yescharacteristic
42C22×Dic94—21no—
43C22×Dic94—21no—
44C22×Dic94—21no—
45C22×Dic94—22no—
46C22×Dic94—21no—
47C22×Dic94—22no—
48C22×Dic94—21no—
49C22×Dic94—22no—
50C22×Dic94C2×C414yescomplemented
51C22×Dic94C2312yes—
52C23×Dic47—21no—
53C23×Dic47—21no—
54#13094054—216no—
55#13094054—28no—
56#13094054C2×C4116yescomplemented
57C2×#10625337—28no—
58C2×#10625337—24no—
59C2×#10625335—28no—
60C2×#10625335—24no—
61C2×#10625329—22no—
62C2×#10625329—21no—
63C2×#10625329—21no—
64C2×#10625329—22no—
65C2×#10625329—22no—
66C2×#10625329C2×C414yescomplemented
67C2×#10625329C2312yes—
68C2×#10625329C2312yes—
69C22×C47⋊C8—21no—
70C22×C47⋊C8—21no—
71C2×C4×Dic47—22no—
72C2×C4×Dic47—22no—
73C2×C4×Dic47—21no—
74C2×C4×Dic47C2312yes—
75#13093951—28no—
76#13093947—24no—
77#13093947—24no—
78C47⋊#95—28no—
79C2×#41×C47—21no—
80C2×#41×C47—21no—
81C2×#41×C47—22no—
82C2×#41×C47—22no—
83C2×#41×C47C2×C414yes—
84C2×#41×C47D414yes—
85C2×#41×C47D412yes—
86C2×#41×C47C2312yes—
87C22×Q8×C47D411yescharacteristic
88C22×D4×C47—21no—
89C22×D4×C47D411yescharacteristic
90C23×C4×C47—21no—
91C23×C4×C47—22no—
92C23×C4×C47D412yes—
93C23×C4×C47C2311yescharacteristic
94C4×Q8×C47—24no—
95C4×Q8×C47—24no—
96C4×D4×C47—28no—
97C4×D4×C47—24no—
98C4×D4×C47—24no—
99C4×D4×C47—28no—
100#117×C47—24no—
101#117×C47—21no—
102#117×C47—21no—
103#117×C47—24no—
104#117×C47D412yes—
105#117×C47C2312yes—
106C2×C4⋊C4×C47—22no—
107C2×C4⋊C4×C47—24no—
108C2×C4⋊C4×C47—21no—
109C2×C4⋊C4×C47—21no—
110C2×C4⋊C4×C47—21no—
111C2×C4⋊C4×C47—21no—
112C2×C4⋊C4×C47—24no—
113C2×C4⋊C4×C47D412yes—
114C2×C4⋊C4×C47D411yescharacteristic
115C2×C4⋊C4×C47D411yescharacteristic
116C2×C4⋊C4×C47C2312yes—
117C2×C22⋊C4×C47—22no—
118C2×C22⋊C4×C47—21no—
119C2×C42×C47—22no—
120C2×C42×C47—21no—
121C24×C47—21no—
122C24×C47C2×D411yescharacteristic
123#41×C47—24no—
124#41×C47—24no—
125#41×C47—24no—
126#41×C47—24no—
127#41×C47—24no—
128#41×C47—22no—
129#41×C47—22no—
130#41×C47C22⋊C418yes—
131#41×C47C22×C414yes—
132#41×C47C2×D414yes—
133C2×Q8×C47—21no—
134C2×Q8×C47—22no—
135C2×Q8×C47C22⋊C414yes—
136C2×Q8×C47C2×D412yes—
137C2×D4×C47—24no—
138C2×D4×C47—22no—
139C2×D4×C47—24no—
140C2×D4×C47—22no—
141C2×D4×C47—21no—
142C2×D4×C47—22no—
143C2×D4×C47C22⋊C414yes—
144C2×D4×C47C2×D412yes—
145C22×C4×C47—21no—
146C22×C4×C47—24no—
147C22×C4×C47—22no—
148C22×C4×C47—24no—
149C22×C4×C47—22no—
150C22×C4×C47—21no—
151C22×C4×C47—22no—
152C22×C4×C47—21no—
153C22×C4×C47—28no—
154C22×C4×C47—22no—
155C22×C4×C47—44no—
156C22×C4×C47—84no—
157C22×C4×C47C22×C412yes—
158C22×C4×C47C2×D414yes—
159C22×C4×C47C2×D411yescharacteristic
160C22×C4×C47C2×D411yescharacteristic
161C22×C4×C47C2×D412yes—
162C22×C4×C47C2×D411yescharacteristic
163C22×C4×C47C2×D411yescharacteristic
164C22×C4×C47C2×D411yescharacteristic
165C22×C4×C47#4112yes—
166C22×C4×C47#4112yes—
167C22×C4×C47C2411yescharacteristic
168C22×Dic47—24no—
169C22×Dic47—24no—
170C22×Dic47—22no—
171C22×Dic47—22no—
172C22×Dic47—42no—
173C22×Dic47—41no—
174C22×Dic47—41no—
175C22×Dic47—42no—
176C22×Dic47—42no—
177C2×Dic94—24no—
178C2×Dic94—24no—
179C2×Dic94—24no—
180C2×Dic94—24no—
181C2×Dic94—22no—
182C2×Dic94—22no—
183C2×Dic94—22no—
184C2×Dic94—22no—
185C2×Dic94—24no—
186C2×Dic94—22no—
187C2×Dic94—41no—
188C2×Dic94—41no—
189C2×Dic94—42no—
190C2×Dic94—42no—
191C2×Dic94—42no—
192C2×Dic94—42no—
193C2×Dic94—42no—
194C2×Dic94C22×C414yescomplemented
195C4⋊C4×C47—28no—
196C4⋊C4×C47—22no—
197C4⋊C4×C47—21no—
198C4⋊C4×C47—22no—
199C4⋊C4×C47—21no—
200C4⋊C4×C47—22no—
201C4⋊C4×C47—22no—
202C4⋊C4×C47C2×D412yes—
203C4⋊C4×C47C2×D412yes—
204C22⋊C4×C47—28no—
205C22⋊C4×C47—22no—
206C22⋊C4×C47—22no—
207C42×C47—28no—
208C42×C47—22no—
209C42×C47—22no—
210#10625337—216no—
211#10625335—216no—
212C4×Dic47—22no—
213C4×Dic47—22no—
214C4×Dic47—42no—
215#10625329—24no—
216#10625329—22no—
217#10625329—22no—
218#10625329—22no—
219#10625329—42no—
220#10625329C22×C414yescomplemented
221C2×C47⋊C8—24no—
222C2×C47⋊C8—22no—
223C2×C47⋊C8—22no—
224C2×C47⋊C8—42no—
225C23×C47—22no—
226C23×C47—22no—
227C23×C47—22no—
228C23×C47—24no—
229C23×C47—42no—
230C23×C47—42no—
231C23×C47C4×D412yes—
232C23×C47#12011yescharacteristic
233C23×C47#12011yescharacteristic
234C23×C47#12112yes—
235C23×C47C22×D411yescharacteristic
236Q8×C47—22no—
237Q8×C47—24no—
238Q8×C47C2×C22⋊C414yes—
239D4×C47—24no—
240D4×C47—24no—
241D4×C47—28no—
242D4×C47—22no—
243D4×C47—24no—
244D4×C47C2×C22⋊C414yes—
245C2×C4×C47—22no—
246C2×C4×C47—24no—
247C2×C4×C47—24no—
248C2×C4×C47—24no—
249C2×C4×C47—28no—
250C2×C4×C47—22no—
251C2×C4×C47—24no—
252C2×C4×C47—22no—
253C2×C4×C47—44no—
254C2×C4×C47—42no—
255C2×C4×C47—42no—
256C2×C4×C47—44no—
257C2×C4×C47—82no—
258C2×C4×C47—88no—
259C2×C4×C47C2×C22⋊C412yes—
260C2×C4×C47C2×C22⋊C412yes—
261C2×C4×C47C4×D412yes—
262C2×C4×C47C4×D412yes—
263C2×C4×C47C4×D412yes—
264C2×C4×C47#12011yescharacteristic
265C2×C4×C47#12011yescharacteristic
266C2×C4×C47#12112yes—
267C2×C4×C47#12112yes—
268C2×C4×C47#12112yes—
269C2×C4×C47#12312yes—
270C2×C4×C47#12312yes—
271C2×C4×C47C23×C411yescharacteristic
272C2×C4×C47C22×D411yescharacteristic
273C2×C4×C47C22×D411yescharacteristic
274C2×C4×C47C22×D411yescharacteristic
275C2×C4×C47C2×#4111yescharacteristic
276C2×C4×C47C2×#4111yescharacteristic
277C2×Dic47—24no—
278C2×Dic47—24no—
279C2×Dic47—48no—
280C2×Dic47—42no—
281C2×Dic47—42no—
282C2×Dic47—42no—
283C2×Dic47—42no—
284C2×Dic47—44no—
285C2×Dic47—44no—
286Dic94—24no—
287Dic94—24no—
288Dic94—42no—
289Dic94—42no—
290Dic94—44no—
291Dic94—44no—
292C47⋊C8—24no—
293C47⋊C8—42no—
294C2×#1882—471nomaximal, Sylow
295C22×C47—24no—
296C22×C47—24no—
297C22×C47—21no—
298C22×C47—21no—
299C22×C47—21no—
300C22×C47—21no—
301C22×C47—24no—
302C22×C47—48no—
303C22×C47—42no—
304C22×C47—42no—
305C22×C47#38612yes—
306C22×C47#38612yes—
307C22×C47#45512yes—
308C22×C47#45612yes—
309C22×C47C2×C4×D411yescharacteristic
310C22×C47C2×#12011yescharacteristic
311C22×C47C2×#12111yescharacteristic
312C4×C47—24no—
313C4×C47—44no—
314C4×C47—42no—
315C4×C47—42no—
316C4×C47#38612yes—
317C4×C47#38612yes—
318C4×C47C22×C22⋊C411yescharacteristic
319C4×C47C2×C4×D411yescharacteristic
320C4×C47C2×#12111yescharacteristic
321C4×C47C2×#12311yescharacteristic
322Dic47—44no—
323Dic47—44no—
324C22×#137—471no—
325C2×#520D4711yescharacteristic
326C22×#104—471no—
327C2×#419—471no—
328C2×#391—471no—
329#1882—478no—
330C2×#343—471no—
331C2×#339—471no—
332C2×C47—21no—
333C2×C47—21no—
334C2×C47—22no—
335C2×C47—44no—
336C2×C47#188214yescomplemented
337C2×C47C2×#38611yescharacteristic
338C2×C47C2×#45511yescharacteristic
339C2×C47C2×#45611yescharacteristic
340C22×#41C2×D4711yescharacteristic
341C23×Q8—471no—
342C2×#137—478no—
343C2×#137—474no—
344C2×#137—944no—
345C2×#137—942no—
346C22×Q16—941no—
347C22×SD16—941no—
348C22×M16—471no—
349C22×M16—471no—
350#520—22no—
351#520—22no—
352#520—22no—
353#520C2×D4714yes—
354C2×C4×Q8—21no—
355C2×C4×D4—21no—
356C2×C4×D4—22no—
357C2×#117—21no—
358C2×#117C2×D4711yescharacteristic
359C22×C4⋊C4—21no—
360C22×C4⋊C4C2×D4711yescharacteristic
361C22×C42—471no—
362C2×#104—474no—
363C2×#104—944no—
364#419—474no—
365#419—942no—
366C2×#103—941no—
367C2×#103—941no—
368#391—474no—
369#391—942no—
370C2×#95—942no—
371#343—474no—
372#339—474no—
373C47C2×#188211yescharacteristic, complemented, Sylow
374C2×#41—22no—
375C2×#41—22no—
376C2×#41—21no—
377C2×#41—21no—
378C2×#41C4×D4714yes—
379C2×#41D18814yes—
380C2×#41C22⋊D4712yes—
381C2×#41C22×D4712yes—
382C22×Q8—474no—
383C22×Q8—472no—
384C22×Q8—942no—
385C22×Q8—941no—
386C22×Q8—942no—
387C22×Q8—942no—
388C22×Q8—941no—
389C22×Q8—941no—
390C22×Q8—941no—
391C22×Q8—941no—
392C22×Q8C22⋊D4711yescharacteristic
393C22×D4—21no—
394C22×D4C22⋊D4711yescharacteristic
395C23×C4—21no—
396C23×C4—22no—
397C23×C4—941no—
398C23×C4—941no—
399C23×C4C22⋊D4712yes—
400C23×C4C22×D4711yescharacteristic
401#137—4716no—
402#137—9416no—
403#137—948no—
404C2×Q16—948no—
405C2×Q16—944no—
406C2×SD16—948no—
407C2×SD16—944no—
408C2×M16—474no—
409C2×M16—472no—
410C2×M16—472no—
411C2×M16—942no—
412C2×M16—941no—
413C2×M16—941no—
414C2×M16—942no—
415C2×M16—942no—
416C22×C8—941no—
417C22×C8—941no—
418C4×Q8—24no—
419C4×Q8—24no—
420C4×D4—28no—
421C4×D4—24no—
422C4×D4—24no—
423C4×D4—28no—
424#117—24no—
425#117—21no—
426#117—21no—
427#117—24no—
428#117C22⋊D4712yes—
429#117C22×D4712yes—
430C2×C4⋊C4—22no—
431C2×C4⋊C4—24no—
432C2×C4⋊C4—21no—
433C2×C4⋊C4—21no—
434C2×C4⋊C4—21no—
435C2×C4⋊C4—21no—
436C2×C4⋊C4—24no—
437C2×C4⋊C4C22⋊D4712yes—
438C2×C4⋊C4C22⋊D4711yescharacteristic
439C2×C4⋊C4C22⋊D4711yescharacteristic
440C2×C4⋊C4C22×D4712yes—
441C2×C22⋊C4—22no—
442C2×C22⋊C4—21no—
443C2×C42—22no—
444C2×C42—21no—
445C2×C42—472no—
446C2×C42—942no—
447C2×C42—942no—
448C2×C42—941no—
449#104—948no—
450#103—944no—
451#103—944no—
452#95—948no—
453C24—21no—
454C24C2×C22⋊D4711yescharacteristic
455#41—24no—
456#41—24no—
457#41—24no—
458#41—24no—
459#41—24no—
460#41—22no—
461#41—22no—
462#41(C2×D47)⋊C418yes—
463#41C2×C4×D4714yes—
464#41C2×D18814yes—
465C2×Q8—21no—
466C2×Q8—22no—
467C2×Q8—474no—
468C2×Q8—944no—
469C2×Q8—942no—
470C2×Q8—944no—
471C2×Q8—944no—
472C2×Q8—944no—
473C2×Q8—944no—
474C2×Q8—942no—
475C2×Q8—942no—
476C2×Q8—942no—
477C2×Q8—942no—
478C2×Q8—1881no—
479C2×Q8—1881no—
480C2×Q8—1882no—
481C2×Q8—1882no—
482C2×Q8—1882no—
483C2×Q8—1882no—
484C2×Q8—1882no—
485C2×Q8(C2×D47)⋊C414yes—
486C2×Q8C2×C22⋊D4712yes—
487C2×D4—24no—
488C2×D4—22no—
489C2×D4—24no—
490C2×D4—22no—
491C2×D4—21no—
492C2×D4—22no—
493C2×D4(C2×D47)⋊C414yes—
494C2×D4C2×C22⋊D4712yes—
495C22×C4—21no—
496C22×C4—24no—
497C22×C4—22no—
498C22×C4—24no—
499C22×C4—22no—
500C22×C4—21no—
501C22×C4—22no—
502C22×C4—21no—
503C22×C4—28no—
504C22×C4—22no—
505C22×C4—44no—
506C22×C4—84no—
507C22×C4—944no—
508C22×C4—944no—
509C22×C4—942no—
510C22×C4—942no—
511C22×C4—1882no—
512C22×C4—1881no—
513C22×C4—1881no—
514C22×C4—1882no—
515C22×C4—1882no—
516C22×C4C2×C4×D4712yes—
517C22×C4C2×D18812yes—
518C22×C4#1062535012yes—
519C22×C4#1062535012yes—
520C22×C4C2×C22⋊D4714yes—
521C22×C4C2×C22⋊D4711yescharacteristic
522C22×C4C2×C22⋊D4711yescharacteristic
523C22×C4C2×C22⋊D4711yescharacteristic
524C22×C4C2×C22⋊D4711yescharacteristic
525C22×C4C2×C22⋊D4711yescharacteristic
526C22×C4C23×D4711yescharacteristic
527Q16—9416no—
528SD16—9416no—
529M16—474no—
530M16—944no—
531M16—942no—
532M16—942no—
533M16—942no—
534M16—1882no—
535C2×C8—944no—
536C2×C8—942no—
537C2×C8—942no—
538C2×C8—1882no—
539C4⋊C4—28no—
540C4⋊C4—22no—
541C4⋊C4—21no—
542C4⋊C4—22no—
543C4⋊C4—21no—
544C4⋊C4—22no—
545C4⋊C4—22no—
546C4⋊C4C2×C22⋊D4712yes—
547C4⋊C4C2×C22⋊D4712yes—
548C22⋊C4—28no—
549C22⋊C4—22no—
550C22⋊C4—22no—
551C42—28no—
552C42—22no—
553C42—22no—
554C42—942no—
555C42—942no—
556C42—1882no—
557C23—22no—
558C23—22no—
559C23—24no—
560C23—22no—
561C23—42no—
562C23—42no—
563C23C4×C22⋊D4712yes—
564C23#1309399311yescharacteristic
565C23#1309399311yescharacteristic
566C23#1309399812yes—
567C23C22×C22⋊D4711yescharacteristic
568Q8—24no—
569Q8—22no—
570Q8—944no—
571Q8—944no—
572Q8—1884no—
573Q8—1884no—
574Q8—1882no—
575Q8—1882no—
576Q8C2×(C2×D47)⋊C414yes—
577D4—28no—
578D4—24no—
579D4—24no—
580D4—24no—
581D4—22no—
582D4C2×(C2×D47)⋊C414yes—
583C2×C4—24no—
584C2×C4—22no—
585C2×C4—28no—
586C2×C4—24no—
587C2×C4—24no—
588C2×C4—22no—
589C2×C4—24no—
590C2×C4—22no—
591C2×C4—44no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—44no—
595C2×C4—82no—
596C2×C4—88no—
597C2×C4—944no—
598C2×C4—944no—
599C2×C4—1888no—
600C2×C4—1882no—
601C2×C4—1884no—
602C2×C4—1884no—
603C2×C4—1882no—
604C2×C4—1882no—
605C2×C4—1882no—
606C2×C4C2×(C2×D47)⋊C412yes—
607C2×C4C2×(C2×D47)⋊C412yes—
608C2×C4C4×C22⋊D4712yes—
609C2×C4C4×C22⋊D4712yes—
610C2×C4C4×C22⋊D4712yes—
611C2×C4#1309399311yescharacteristic
612C2×C4#1309399311yescharacteristic
613C2×C4#1309399812yes—
614C2×C4#1309399812yes—
615C2×C4#1309399812yes—
616C2×C4#1309400712yes—
617C2×C4#1309400712yes—
618C2×C4C22×C4×D4711yescharacteristic
619C2×C4C22×C22⋊D4711yescharacteristic
620C2×C4C22×C22⋊D4711yescharacteristic
621C2×C4C22×D18811yescharacteristic
622C2×C4C2×#1062535011yescharacteristic
623C2×C4C2×#1062535011yescharacteristic
624C8—944no—
625C8—1882no—
626C22—24no—
627C22—24no—
628C22—21no—
629C22—21no—
630C22—21no—
631C22—21no—
632C22—24no—
633C22—48no—
634C22—42no—
635C22—42no—
636C22#1568900212yes—
637C22#1568900212yes—
638C22#1568930112yes—
639C22#1568930612yes—
640C22C2×C4×C22⋊D4711yescharacteristic
641C22C2×#1309399311yescharacteristic
642C22C2×#1309399811yescharacteristic
643C4—24no—
644C4—44no—
645C4—42no—
646C4—42no—
647C4—1884no—
648C4—1884no—
649C4#1568900212yes—
650C4#1568900212yes—
651C4C22×(C2×D47)⋊C411yescharacteristic
652C4C2×C4×C22⋊D4711yescharacteristic
653C4C2×#1309399811yescharacteristic
654C4C2×#1309400711yescharacteristic
655C2—21no—
656C2—22no—
657C2—21no—
658C2—44no—
659C2#2024375714yesdirect product factor
660C2C2×#1568900211yescharacteristic
661C2C2×#1568930111yescharacteristic
662C2C2×#1568930611yescharacteristic
subgroupG/H
centerC23#13093993
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC2×C47×#520C2
socleC23×C47#120
prime radicalC22×#41×C47C22
cosocleC22×C4×C47C24
hypercenterC2×#520D47
hypocenterC47C2×#1882
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#520

Complements

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

NH ≅ G/Ntype
3. C2×#15689131C2semidirect
4. C2×#15689028C2semidirect
5. #20243757C2direct (N × H)
6. C2×#15688892C2semidirect
7. C2×#15688882C2semidirect
12. C2×#13094054C4semidirect
22. #15689131C22semidirect
26. #15689028C22semidirect
28. #15688892C22semidirect
29. #15688882C22semidirect
50. C22×Dic94C2×C4semidirect
56. #13094054C2×C4semidirect
66. C2×#10625329C2×C4semidirect
194. C2×Dic94C22×C4semidirect
220. #10625329C22×C4semidirect
336. C2×C47#1882semidirect
373. C47C2×#1882semidirect
659. C2#20243757direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#130940542
C22×#130939512
C2×#156891312
C2×#156890282
#202437572
C2×#156888922
C2×#156888822
C2×C47×#5202
C2×#188247