Finite Group Databasebeta

← C2×#20243833 #26218034

Subgroups — C2×#20243833 #26218034

Order 12032.

1954 conjugacy classes of subgroups.

The subgroup lattice has 668 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689630C211yesmaximal, characteristic
2C22×#13093981C211yesmaximal, characteristic, complemented
3C22×#13093951C211yesmaximal, characteristic, complemented
4C2×#15689086C211yesmaximal, characteristic, complemented
5#20243833C218yesmaximal, direct product factor
6C2×#15688884C211yesmaximal, characteristic, complemented
7C2×#15688883C211yesmaximal, characteristic, complemented
8C22×#104×C47C211yesmaximal, characteristic, complemented
9C23×C4×D47C2211yescharacteristic
10C22×#10625329C2211yescharacteristic
11#15689630—22no—
12#15689630—22no—
13#15689630C414yescomplemented
14#15689630C2214yes—
15C2×#13094005—21no—
16C2×#13094000C412yescomplemented
17C2×#13093995C412yescomplemented
18C2×#13093992—21no—
19C2×C4×C22⋊D47—22no—
20C2×#13093983C2211yescharacteristic
21C2×#13093981—22no—
22C2×#13093981—24no—
23C2×#13093981—24no—
24C2×#13093981C2214yescomplemented
25C22×Dic47⋊C4—21no—
26C22×(C2×D47)⋊C4—21no—
27C22×C4×Dic47C2211yescharacteristic
28C2×#13093951—24no—
29C2×#13093951C2214yescomplemented
30#15689086—22no—
31#15689086C2214yescomplemented
32C2×(C2×D47)⋊C8—22no—
33#15688884C2214yescomplemented
34#15688883C2214yescomplemented
35C22×#41×C47C2211yescharacteristic
36C22×M16×C47C2211yescharacteristic
37C22×C42×C47C2211yescharacteristic
38C2×#104×C47—24no—
39C2×#104×C47C2214yescomplemented
40C24×D47D411yescharacteristic
41C22×C22⋊D47—22no—
42C23×Dic47—21no—
43C23×Dic47D411yescharacteristic
44C22×C4×D47—22no—
45C22×C4×D47—22no—
46C22×C4×D47—22no—
47C22×C4×D47—22no—
48C22×C4×D47C2×C412yes—
49C22×C4×D47C2×C412yes—
50C22×C4×D47D412yes—
51C22×C4×D47D412yes—
52C22×C4×D47D412yes—
53C22×C4×D47C2312yes—
54C2×#10625329—21no—
55C2×#10625329—22no—
56C2×#10625329—22no—
57C2×#10625329C2312yes—
58C22×C47⋊C8—21no—
59#13094005—24no—
60#13094005—24no—
61#13094000—24no—
62#13094000C2×C418yescomplemented
63#13093995—24no—
64#13093995C2×C418yescomplemented
65#13093992—24no—
66#13093992—24no—
67C4×C22⋊D47—28no—
68C4×C22⋊D47—28no—
69#13093983—21no—
70#13093983—21no—
71#13093983C2×C412yes—
72#13093983C2312yes—
73#13093981—216no—
74#13093981—28no—
75#13093981—28no—
76C2×C4⋊Dic47C2×C412yes—
77C2×Dic47⋊C4—28no—
78C2×Dic47⋊C4—22no—
79C2×Dic47⋊C4—24no—
80C2×C22⋊Dic47—21no—
81C2×C22⋊Dic47—22no—
82C2×(C2×D47)⋊C4—28no—
83C2×(C2×D47)⋊C4—22no—
84C2×(C2×D47)⋊C4—24no—
85C2×C4×Dic47—21no—
86C2×C4×Dic47—22no—
87C2×C4×Dic47—21no—
88C2×C4×Dic47D414yescomplemented
89C2×C4×Dic47C2312yes—
90#13093951—28no—
91(C2×D47)⋊C8—28no—
92C2×#41×C47—21no—
93C2×#41×C47—21no—
94C2×#41×C47—22no—
95C2×#41×C47—22no—
96C2×#41×C47—22no—
97C2×#41×C47—22no—
98C2×#41×C47C2×C412yes—
99C2×#41×C47C2312yes—
100C22×Q8×C47C2×C411yescharacteristic
101C22×D4×C47—21no—
102C22×D4×C47C2×C411yescharacteristic
103C23×C4×C47—21no—
104C23×C4×C47—21no—
105C23×C4×C47C2311yescharacteristic
106C2×M16×C47—21no—
107C2×M16×C47—22no—
108C2×M16×C47—22no—
109C2×M16×C47C2312yes—
110C22×C8×C47—21no—
111C2×C42×C47—21no—
112C2×C42×C47—22no—
113C2×C42×C47D414yescomplemented
114C2×C42×C47C2312yes—
115#104×C47—28no—
116C24×C47—21no—
117C24×C47C2×D411yescharacteristic
118C23×D47—22no—
119C23×D47—22no—
120C23×D47—44no—
121C23×D47C22⋊C412yes—
122C23×D47C22⋊C412yes—
123C23×D47C2×D412yes—
124#41×C47—24no—
125#41×C47—24no—
126#41×C47—22no—
127#41×C47—22no—
128#41×C47—22no—
129#41×C47—22no—
130#41×C47—42no—
131#41×C47—42no—
132#41×C47—42no—
133#41×C47—42no—
134C2×Q8×C47—21no—
135C2×Q8×C47—22no—
136C2×Q8×C47—22no—
137C2×Q8×C47C22×C412yes—
138C2×D4×C47—22no—
139C2×D4×C47—22no—
140C2×D4×C47—21no—
141C2×D4×C47—22no—
142C2×D4×C47—22no—
143C2×D4×C47—42no—
144C2×D4×C47—42no—
145C2×D4×C47C22×C412yes—
146C22×C4×C47—21no—
147C22×C4×C47—24no—
148C22×C4×C47—22no—
149C22×C4×C47—22no—
150C22×C4×C47—22no—
151C22×C4×C47—44no—
152C22×C4×C47—42no—
153C22×C4×C47—42no—
154C22×C4×C47C22×C411yescharacteristic
155C22×C4×C47C22×C411yescharacteristic
156C22×C4×C47C2×D411yescharacteristic
157C22×C4×C47C2×D411yescharacteristic
158C22×C4×C47C2×D411yescharacteristic
159C22×C4×C47C2×D412yes—
160C22×C4×C47C2×D412yes—
161C22×C4×C47#4112yes—
162C22×C4×C47#4112yes—
163C22×C4×C47C2411yescharacteristic
164C2×C22⋊D47—28no—
165C2×C22⋊D47—48no—
166C22×Dic47—22no—
167C22×Dic47—22no—
168C22×Dic47—22no—
169C22×Dic47—24no—
170C22×Dic47—22no—
171C22×Dic47—44no—
172C22×Dic47C22⋊C412yes—
173C22×Dic47C22⋊C412yes—
174C22×Dic47C2×D412yes—
175C2×C4×D47—24no—
176C2×C4×D47—24no—
177C2×C4×D47—24no—
178C2×C4×D47—24no—
179C2×C4×D47—24no—
180C2×C4×D47—24no—
181C2×C4×D47—24no—
182C2×C4×D47—24no—
183C2×C4×D47—24no—
184C2×C4×D47—24no—
185C2×C4×D47—22no—
186C2×C4×D47—22no—
187C2×C4×D47—22no—
188C2×C4×D47—22no—
189C2×C4×D47C22⋊C414yes—
190C2×C4×D47C22⋊C414yes—
191C2×C4×D47C22×C414yes—
192C2×C4×D47C2×D414yes—
193M16×C47—22no—
194M16×C47—22no—
195M16×C47—42no—
196C2×C8×C47—22no—
197C2×C8×C47—42no—
198C42×C47—24no—
199C42×C47—22no—
200C42×C47C2×D414yescomplemented
201C22⋊Dic47—22no—
202C22⋊Dic47—22no—
203C22⋊Dic47—28no—
204(C2×D47)⋊C4—216no—
205(C2×D47)⋊C4—28no—
206C4⋊Dic47—22no—
207C4⋊Dic47C22×C414yes—
208Dic47⋊C4—216no—
209Dic47⋊C4—28no—
210C4×Dic47—22no—
211C4×Dic47—24no—
212C4×Dic47—22no—
213C4×Dic47—22no—
214C4×Dic47C2×D414yescomplemented
215#10625329—22no—
216#10625329—22no—
217#10625329—42no—
218C2×C47⋊C8—22no—
219C2×C47⋊C8—42no—
220C23×C47—24no—
221C23×C47—22no—
222C23×C47—42no—
223C23×C47—44no—
224C23×C47C2×C22⋊C411yescharacteristic
225C23×C47C2×C22⋊C411yescharacteristic
226C23×C47#12012yes—
227C23×C47#12312yes—
228C23×C47C22×D411yescharacteristic
229C22×D47—22no—
230C22×D47—22no—
231C22×D47—24no—
232C22×D47—24no—
233C22×D47—24no—
234C22×D47—24no—
235C22×D47—44no—
236C22×D47—44no—
237C22×D47—48no—
238C22×D47—48no—
239C22×D47C2×C22⋊C414yes—
240Q8×C47—22no—
241Q8×C47—22no—
242Q8×C47—42no—
243D4×C47—22no—
244D4×C47—22no—
245D4×C47—42no—
246D4×C47—42no—
247D4×C47—44no—
248D4×C47—42no—
249C2×C4×C47—22no—
250C2×C4×C47—24no—
251C2×C4×C47—44no—
252C2×C4×C47—48no—
253C2×C4×C47—42no—
254C2×C4×C47—42no—
255C2×C4×C47—42no—
256C2×C4×C47—44no—
257C2×C4×C47C2×C22⋊C411yescharacteristic
258C2×C4×C47C2×C22⋊C411yescharacteristic
259C2×C4×C47C4×D412yes—
260C2×C4×C47C4×D412yes—
261C2×C4×C47C4×D412yes—
262C2×C4×C47C4×D412yes—
263C2×C4×C47#12012yes—
264C2×C4×C47#12112yes—
265C2×C4×C47#12112yes—
266C2×C4×C47#12112yes—
267C2×C4×C47#12112yes—
268C2×C4×C47#12312yes—
269C2×C4×C47C23×C411yescharacteristic
270C2×C4×C47C22×D411yescharacteristic
271C2×C4×C47C22×D411yescharacteristic
272C2×C4×C47C22×D411yescharacteristic
273C2×C4×C47C2×#4111yescharacteristic
274C2×C4×C47C2×#4111yescharacteristic
275C22⋊D47—416no—
276C2×Dic47—22no—
277C2×Dic47—22no—
278C2×Dic47—24no—
279C2×Dic47—24no—
280C2×Dic47—24no—
281C2×Dic47—24no—
282C2×Dic47—44no—
283C2×Dic47—48no—
284C2×Dic47—44no—
285C2×Dic47C2×C22⋊C414yes—
286C4×D47—24no—
287C4×D47—24no—
288C4×D47—24no—
289C4×D47—24no—
290C4×D47—28no—
291C4×D47—28no—
292C8×C47—42no—
293C47⋊C8—42no—
294C2×#1897—471nomaximal, Sylow
295C22×C47—22no—
296C22×C47—22no—
297C22×C47—24no—
298C22×C47—24no—
299C22×C47—44no—
300C22×C47—42no—
301C22×C47—84no—
302C22×C47#38612yes—
303C22×C47#38612yes—
304C22×C47#42114yes—
305C22×C47C22×C22⋊C411yescharacteristic
306C22×C47C2×#12011yescharacteristic
307C22×C47C2×#12311yescharacteristic
308C2×D47—24no—
309C2×D47—24no—
310C2×D47—24no—
311C2×D47—44no—
312C2×D47—44no—
313C2×D47—48no—
314C2×D47—48no—
315C4×C47—44no—
316C4×C47—42no—
317C4×C47#38612yes—
318C4×C47#38612yes—
319C4×C47C2×C4×D411yescharacteristic
320C4×C47C2×C4×D411yescharacteristic
321C4×C47C2×#12111yescharacteristic
322C4×C47C2×#12111yescharacteristic
323Dic47—24no—
324Dic47—44no—
325Dic47—44no—
326C2×#525—471no—
327C22×#117—471no—
328C22×#104—471no—
329C22×#104D4711yescharacteristic, complemented
330C2×#407—471no—
331#1897—478no—
332C2×#339—471no—
333C2×#339—471no—
334C2×C47—21no—
335C2×C47—21no—
336C2×C47—22no—
337C2×C47—82no—
338C2×C47#189714yescomplemented
339C2×C47C2×#38611yescharacteristic
340C2×C47C2×#42112yes—
341D47—44no—
342C22×#41C2×D4711yescharacteristic
343C24×C4—471no—
344C22×M16—471no—
345C22×M16C2×D4711yescharacteristic
346#525—474no—
347#525—474no—
348#525—942no—
349#525—942no—
350C2×#123—941no—
351C2×C22⋊Q8—472no—
352C2×#121—472no—
353C2×#120—941no—
354C2×C4×D4—942no—
355C2×#117—474no—
356C2×#117—471no—
357C2×#117—944no—
358C2×#117—942no—
359C2×#117—944no—
360C22×C4⋊C4—941no—
361C22×C22⋊C4—941no—
362C22×C42—471no—
363C22×C42C2×D4711yescharacteristic
364C2×#104—24no—
365C2×#104—474no—
366C2×#104—944no—
367C2×#104C2×D4714yescomplemented
368#407—474no—
369#407—942no—
370C2×C22⋊C8—942no—
371#339—474no—
372#339—474no—
373C47C2×#189711yescharacteristic, complemented, Sylow
374C25—471no—
375C2×#41—21no—
376C2×#41—21no—
377C2×#41—22no—
378C2×#41—22no—
379C2×#41—22no—
380C2×#41—22no—
381C2×#41C4×D4712yes—
382C2×#41C22×D4712yes—
383C22×Q8C4×D4711yescharacteristic
384C22×D4—21no—
385C22×D4—942no—
386C22×D4C4×D4711yescharacteristic
387C23×C4—21no—
388C23×C4—21no—
389C23×C4—471no—
390C23×C4—472no—
391C23×C4—472no—
392C23×C4—472no—
393C23×C4—472no—
394C23×C4—472no—
395C23×C4—472no—
396C23×C4—942no—
397C23×C4—942no—
398C23×C4—942no—
399C23×C4—942no—
400C23×C4—941no—
401C23×C4C22×D4711yescharacteristic
402C2×M16—21no—
403C2×M16—22no—
404C2×M16—22no—
405C2×M16—472no—
406C2×M16—941no—
407C2×M16—942no—
408C2×M16—942no—
409C2×M16C22×D4712yes—
410C22×C8—21no—
411C22×C8—941no—
412#123—944no—
413#123—944no—
414C22⋊Q8—478no—
415C22⋊Q8—944no—
416#121—478no—
417#121—944no—
418#120—944no—
419#120—944no—
420C4×D4—948no—
421C4×D4—948no—
422#117—472no—
423#117—472no—
424#117—948no—
425#117—9416no—
426#117—941no—
427#117—941no—
428#117—948no—
429C2×C4⋊C4—472no—
430C2×C4⋊C4—944no—
431C2×C4⋊C4—942no—
432C2×C4⋊C4—948no—
433C2×C22⋊C4—944no—
434C2×C22⋊C4—942no—
435C2×C22⋊C4—948no—
436C2×C22⋊C4—942no—
437C2×C22⋊C4—941no—
438C2×C42—21no—
439C2×C42—22no—
440C2×C42—474no—
441C2×C42—472no—
442C2×C42—941no—
443C2×C42—941no—
444C2×C42—942no—
445C2×C42C22⋊D4714yescomplemented
446C2×C42C22×D4712yes—
447#104—28no—
448#104—948no—
449C22⋊C8—948no—
450C24—21no—
451C24—472no—
452C24—472no—
453C24—472no—
454C24—942no—
455C24—942no—
456C24—1884no—
457C24D4×D4711yescharacteristic
458#41—24no—
459#41—24no—
460#41—22no—
461#41—22no—
462#41—22no—
463#41—22no—
464#41—42no—
465#41—42no—
466#41—42no—
467#41—42no—
468C2×Q8—21no—
469C2×Q8—22no—
470C2×Q8—22no—
471C2×Q8C2×C4×D4712yes—
472C2×D4—22no—
473C2×D4—22no—
474C2×D4—21no—
475C2×D4—22no—
476C2×D4—22no—
477C2×D4—42no—
478C2×D4—42no—
479C2×D4—948no—
480C2×D4—1888no—
481C2×D4C2×C4×D4712yes—
482C22×C4—21no—
483C22×C4—24no—
484C22×C4—22no—
485C22×C4—22no—
486C22×C4—22no—
487C22×C4—44no—
488C22×C4—42no—
489C22×C4—42no—
490C22×C4—472no—
491C22×C4—472no—
492C22×C4—472no—
493C22×C4—474no—
494C22×C4—474no—
495C22×C4—474no—
496C22×C4—474no—
497C22×C4—942no—
498C22×C4—942no—
499C22×C4—944no—
500C22×C4—944no—
501C22×C4—944no—
502C22×C4—944no—
503C22×C4—944no—
504C22×C4—944no—
505C22×C4—944no—
506C22×C4—944no—
507C22×C4—944no—
508C22×C4—944no—
509C22×C4—942no—
510C22×C4—942no—
511C22×C4—942no—
512C22×C4—942no—
513C22×C4—942no—
514C22×C4—944no—
515C22×C4—942no—
516C22×C4—1884no—
517C22×C4C2×C4×D4711yescharacteristic
518C22×C4C2×C4×D4711yescharacteristic
519C22×C4#1062535012yes—
520C22×C4D4×D4711yescharacteristic
521C22×C4D4×D4711yescharacteristic
522C22×C4D4×D4712yes—
523C22×C4D4×D4711yescharacteristic
524C22×C4#1062535212yes—
525C22×C4C2×C22⋊D4712yes—
526C22×C4C23×D4711yescharacteristic
527M16—22no—
528M16—22no—
529M16—42no—
530M16—942no—
531M16—942no—
532M16—1882no—
533C2×C8—22no—
534C2×C8—42no—
535C2×C8—942no—
536C2×C8—1882no—
537C4⋊C4—474no—
538C4⋊C4—948no—
539C4⋊C4—9416no—
540C4⋊C4—942no—
541C22⋊C4—948no—
542C22⋊C4—9416no—
543C22⋊C4—948no—
544C22⋊C4—942no—
545C22⋊C4—942no—
546C42—24no—
547C42—22no—
548C42—474no—
549C42—942no—
550C42—944no—
551C42—942no—
552C42—942no—
553C42C2×C22⋊D4714yescomplemented
554C23—24no—
555C23—22no—
556C23—42no—
557C23—44no—
558C23—474no—
559C23—944no—
560C23—944no—
561C23—942no—
562C23—942no—
563C23—944no—
564C23—944no—
565C23—1884no—
566C23—1884no—
567C23—1888no—
568C23—1888no—
569C23C22⋊C4×D4711yescharacteristic
570C23C22⋊C4×D4711yescharacteristic
571C23#1309399212yes—
572C23#1309400612yes—
573C23C2×D4×D4711yescharacteristic
574Q8—22no—
575Q8—22no—
576Q8—42no—
577D4—22no—
578D4—22no—
579D4—44no—
580D4—42no—
581D4—42no—
582D4—42no—
583D4—18816no—
584C2×C4—22no—
585C2×C4—24no—
586C2×C4—44no—
587C2×C4—48no—
588C2×C4—42no—
589C2×C4—44no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—474no—
593C2×C4—944no—
594C2×C4—944no—
595C2×C4—942no—
596C2×C4—942no—
597C2×C4—948no—
598C2×C4—948no—
599C2×C4—944no—
600C2×C4—944no—
601C2×C4—944no—
602C2×C4—944no—
603C2×C4—944no—
604C2×C4—944no—
605C2×C4—1884no—
606C2×C4—1888no—
607C2×C4—1884no—
608C2×C4C22⋊C4×D4711yescharacteristic
609C2×C4C22⋊C4×D4711yescharacteristic
610C2×C4(C22⋊D47)⋊C412yes—
611C2×C4(C22⋊D47)⋊C412yes—
612C2×C4C4×C22⋊D4712yes—
613C2×C4C4×C22⋊D4712yes—
614C2×C4#1309399212yes—
615C2×C4#1309399512yes—
616C2×C4#1309399512yes—
617C2×C4#1309399712yes—
618C2×C4#1309399712yes—
619C2×C4#1309400612yes—
620C2×C4C22×C4×D4711yescharacteristic
621C2×C4C2×D4×D4711yescharacteristic
622C2×C4C2×D4×D4711yescharacteristic
623C2×C4C22×C22⋊D4711yescharacteristic
624C2×C4C2×#1062535211yescharacteristic
625C2×C4C2×#1062535011yescharacteristic
626C8—42no—
627C8—1882no—
628C22—22no—
629C22—22no—
630C22—24no—
631C22—24no—
632C22—44no—
633C22—42no—
634C22—84no—
635C22—944no—
636C22—944no—
637C22—944no—
638C22—1884no—
639C22—1888no—
640C22—1884no—
641C22—1888no—
642C22#1568899912yes—
643C22#1568899912yes—
644C22#1568913814yes—
645C22C2×C22⋊C4×D4711yescharacteristic
646C22C2×#1309399211yescharacteristic
647C22C2×#1309400611yescharacteristic
648C4—44no—
649C4—42no—
650C4—944no—
651C4—1884no—
652C4—1884no—
653C4#1568899912yes—
654C4#1568899912yes—
655C4C2×(C22⋊D47)⋊C411yescharacteristic
656C4C2×C4×C22⋊D4711yescharacteristic
657C4C2×#1309399511yescharacteristic
658C4C2×#1309399711yescharacteristic
659C2—21no—
660C2—22no—
661C2—21no—
662C2—82no—
663C2—1884no—
664C2#2024383314yesdirect product factor
665C2C2×#1568899911yescharacteristic
666C2C2×#1568913812yes—
subgroupG/H
centerC22×C4D4×D47
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC22×#104×C47C2
socleC23×C47C2×C22⋊C4
prime radicalC2×#15689630C2
cosocleC22×C4×C47C24
hypercenterC22×#104D47
hypocenterC47C2×#1897
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22×C4 ◁ C23×C4 ◁ C22×#104

Complements

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

NH ≅ G/Ntype
2. C22×#13093981C2semidirect
3. C22×#13093951C2semidirect
4. C2×#15689086C2semidirect
5. #20243833C2direct (N × H)
6. C2×#15688884C2semidirect
7. C2×#15688883C2semidirect
8. C22×#104×C47C2semidirect
13. #15689630C4semidirect
16. C2×#13094000C4semidirect
17. C2×#13093995C4semidirect
24. C2×#13093981C22semidirect
29. C2×#13093951C22semidirect
31. #15689086C22semidirect
33. #15688884C22semidirect
34. #15688883C22semidirect
39. C2×#104×C47C22semidirect
62. #13094000C2×C4semidirect
64. #13093995C2×C4semidirect
88. C2×C4×Dic47D4semidirect
113. C2×C42×C47D4semidirect
200. C42×C47C2×D4semidirect
214. C4×Dic47C2×D4semidirect
329. C22×#104D47semidirect
338. C2×C47#1897semidirect
367. C2×#104C2×D47semidirect
373. C47C2×#1897semidirect
445. C2×C42C22⋊D47semidirect
553. C42C2×C22⋊D47semidirect
664. C2#20243833direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#156896302
C22×#130939812
C22×#130939512
C2×#156890862
#202438332
C2×#156888842
C2×#156888832
C22×#104×C472
C2×#189747