Finite Group Databasebeta

← C2×#20241829 #26182693

Subgroups — C2×#20241829 #26182693

Order 12032.

1734 conjugacy classes of subgroups.

The subgroup lattice has 604 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×D4×D47C211yesmaximal, characteristic
2C2×#15689179C211yesmaximal, characteristic, complemented
3C2×C4×#10625324C211yesmaximal, characteristic, complemented
4#20241829C218yesmaximal, direct product factor
5C2×#15688852C211yesmaximal, characteristic, complemented
6C2×D4⋊(C47⋊C8)C211yesmaximal, characteristic, complemented
7C2×#15688850C211yesmaximal, characteristic, complemented
8C2×C47×D4⋊C8C211yesmaximal, characteristic, complemented
9C22×D4×D47C411yescharacteristic
10C23×C4×D47—21no—
11C22×#10625324—21no—
12C2×C4×D188C2211yescharacteristic
13C2×D4×Dic47C411yescharacteristic
14C2×#13093986C411yescharacteristic
15C2×C4×C22⋊D47—21no—
16C2×(C22⋊D47)⋊C4—21no—
17C4×D4×D47—22no—
18C4×D4×D47—22no—
19C4×D4×D47C414yes—
20C4×D4×D47C2214yes—
21C2×C4⋊C4×D47C411yescharacteristic
22C2×C22⋊C4×D47—21no—
23C2×C42×D47C2211yescharacteristic
24#15689179—22no—
25#15689179C2214yescomplemented
26C2×C4⋊(C47⋊C8)C2211yescharacteristic, complemented
27C2×(C2×D47)⋊C8—21no—
28C4×#10625324—22no—
29C4×#10625324C2214yescomplemented
30C2×#13093934—21no—
31C2×#13093934—21no—
32C2×C4×C47⋊C8C2211yescharacteristic, complemented
33#15688852C2214yescomplemented
34D4⋊(C47⋊C8)C2214yescomplemented
35#15688850C2214yescomplemented
36C2×C4×D4×C47C2211yescharacteristic
37C2×C4⋊C8×C47C2211yescharacteristic, complemented
38C2×C4×C8×C47C2211yescharacteristic, complemented
39C47×D4⋊C8C2214yescomplemented
40C24×D47—21no—
41C22×D188C2×C411yescharacteristic
42C22×C22⋊D47—21no—
43C2×D4×D47—24no—
44C2×D4×D47—24no—
45C2×D4×D47—22no—
46C2×D4×D47C2×C414yes—
47C23×Dic47—21no—
48C22×C4×D47—21no—
49C22×C4×D47—21no—
50C22×C4×D47—21no—
51C22×C4×D47—24no—
52C22×C4×D47—24no—
53C22×C4×D47—44no—
54C22×C4×D47—44no—
55C22×C4×D47C2×C411yescharacteristic
56C22×C4×D47D411yescharacteristic
57C22×C4×D47D411yescharacteristic
58C2×#10625324—28no—
59C2×#10625324—24no—
60C22×C47⋊C8—21no—
61C22×C47⋊C8—21no—
62C4×D188—21no—
63C4×D188—21no—
64C4×D188C2×C412yes—
65C4×D188C2312yes—
66D4×Dic47—22no—
67D4×Dic47C2×C414yes—
68#13093986—22no—
69#13093986C2×C414yes—
70C4×C22⋊D47—24no—
71C4×C22⋊D47—24no—
72(C22⋊D47)⋊C4—24no—
73(C22⋊D47)⋊C4—24no—
74C2×C4⋊Dic47C2×C411yescharacteristic
75C2×Dic47⋊C4—21no—
76C4⋊C4×D47—22no—
77C4⋊C4×D47C2×C414yes—
78C2×C22⋊Dic47—21no—
79C2×(C2×D47)⋊C4—21no—
80C2×(C2×D47)⋊C4—21no—
81C22⋊C4×D47—24no—
82C22⋊C4×D47—24no—
83C2×C4×Dic47C2×C411yescharacteristic
84C2×C4×Dic47D411yescharacteristic
85C2×C4×Dic47D411yescharacteristic
86C42×D47C2×C412yes—
87C42×D47D412yes—
88C42×D47D412yes—
89C42×D47C2312yes—
90C4⋊(C47⋊C8)—21no—
91C4⋊(C47⋊C8)C2312yescomplemented
92(C2×D47)⋊C8—24no—
93#13093934—24no—
94#13093934—24no—
95C4×C47⋊C8—21no—
96C4×C47⋊C8C2312yescomplemented
97C22×D4×C47C2×C411yescharacteristic
98C23×C4×C47—21no—
99C22×C8×C47—21no—
100C22×C8×C47—21no—
101C4×D4×C47—21no—
102C4×D4×C47—21no—
103C4×D4×C47C2×C412yes—
104C4×D4×C47C2312yes—
105C2×C4⋊C4×C47C2×C411yescharacteristic
106C2×C22⋊C4×C47—21no—
107C2×C42×C47C2311yescharacteristic
108C4⋊C8×C47—21no—
109C4⋊C8×C47C2312yescomplemented
110C4×C8×C47—21no—
111C4×C8×C47C2312yescomplemented
112C24×C47—21no—
113C23×D47—28no—
114C23×D47—21no—
115C23×D47—24no—
116C23×D47—44no—
117C23×D47—44no—
118C23×D47C22⋊C411yescharacteristic
119C2×D4×C47—21no—
120C2×D4×C47—22no—
121C2×D4×C47M1614yescomplemented
122C2×D4×C47C22×C412yes—
123C22×C4×C47—21no—
124C22×C4×C47—22no—
125C22×C4×C47—22no—
126C22×C4×C47—42no—
127C22×C4×C47—42no—
128C22×C4×C47C22×C411yescharacteristic
129C22×C4×C47C2×D411yescharacteristic
130C22×C4×C47C2×D411yescharacteristic
131C2×C22⋊D47—28no—
132C2×C22⋊D47—24no—
133C22×Dic47—21no—
134C22×Dic47—24no—
135C22×Dic47—44no—
136C22×Dic47C22⋊C411yescharacteristic
137C22×Dic47C22⋊C411yescharacteristic
138C22×Dic47C22⋊C411yescharacteristic
139D4×D47—216no—
140D4×D47—28no—
141D4×D47—28no—
142C2×D188—21no—
143C2×D188—22no—
144C2×D188M1614yescomplemented
145C2×D188C22×C412yes—
146C2×C4×D47—24no—
147C2×C4×D47—22no—
148C2×C4×D47—22no—
149C2×C4×D47—22no—
150C2×C4×D47—22no—
151C2×C4×D47—24no—
152C2×C4×D47—22no—
153C2×C4×D47—22no—
154C2×C4×D47—24no—
155C2×C4×D47—48no—
156C2×C4×D47—48no—
157C2×C4×D47—416no—
158C2×C4×D47—44no—
159C2×C4×D47—44no—
160C2×C4×D47—42no—
161C2×C4×D47—42no—
162C2×C4×D47C22⋊C412yes—
163C2×C4×D47C22⋊C412yes—
164C2×C4×D47C22⋊C412yes—
165C2×C4×D47C22×C412yes—
166C2×C4×D47C2×D412yes—
167C2×C4×D47C2×D412yes—
168C2×C8×C47—24no—
169C2×C8×C47—22no—
170C2×C8×C47—22no—
171C2×C8×C47—42no—
172C4⋊C4×C47—21no—
173C4⋊C4×C47C22×C412yes—
174C22⋊C4×C47—22no—
175C22⋊C4×C47—22no—
176C42×C47C22×C411yescharacteristic
177C42×C47C2×D411yescharacteristic
178C42×C47C2×D411yescharacteristic
179C42×C47C2411yescharacteristic
180C22⋊Dic47—24no—
181(C2×D47)⋊C4—22no—
182(C2×D47)⋊C4—22no—
183(C2×D47)⋊C4—24no—
184C4⋊Dic47—21no—
185C4⋊Dic47C22×C412yes—
186Dic47⋊C4—24no—
187C4×Dic47C22⋊C412yes—
188C4×Dic47C22⋊C412yes—
189C4×Dic47C22⋊C412yes—
190C4×Dic47C22×C412yes—
191C4×Dic47C2×D412yes—
192C4×Dic47C2×D412yes—
193C2×C47⋊C8—24no—
194C2×C47⋊C8—22no—
195C2×C47⋊C8—22no—
196C2×C47⋊C8—42no—
197#10625324—216no—
198C23×C47—22no—
199C23×C47—24no—
200C23×C47—42no—
201C23×C47—42no—
202C23×C47C2×C22⋊C411yescharacteristic
203C22×D47—22no—
204C22×D47—22no—
205C22×D47—216no—
206C22×D47—22no—
207C22×D47—24no—
208C22×D47—48no—
209C22×D47—432no—
210C22×D47—42no—
211C22×D47—42no—
212C22×D47—48no—
213C22×D47#10414yes—
214C22×D47C2×C22⋊C412yes—
215D4×C47—22no—
216D4×C47—24no—
217D4×C47C2×M1614yescomplemented
218C2×C4×C47—22no—
219C2×C4×C47—22no—
220C2×C4×C47—42no—
221C2×C4×C47—42no—
222C2×C4×C47—42no—
223C2×C4×C47—44no—
224C2×C4×C47C2×C22⋊C411yescharacteristic
225C2×C4×C47C2×C22⋊C411yescharacteristic
226C2×C4×C47C2×C22⋊C411yescharacteristic
227C2×C4×C47C2×M1612yes—
228C2×C4×C47C2×M1612yes—
229C2×C4×C47AGL(1,ℤ/8)12yes—
230C2×C4×C47AGL(1,ℤ/8)12yes—
231C2×C4×C47C23×C411yescharacteristic
232C2×C4×C47C22×D411yescharacteristic
233C2×C4×C47C22×D411yescharacteristic
234C22⋊D47—216no—
235C2×Dic47—22no—
236C2×Dic47—22no—
237C2×Dic47—22no—
238C2×Dic47—22no—
239C2×Dic47—48no—
240C2×Dic47—42no—
241C2×Dic47C2×C22⋊C412yes—
242C2×Dic47C2×C22⋊C412yes—
243C2×Dic47C2×C22⋊C412yes—
244D188—22no—
245D188—24no—
246D188C2×M1614yescomplemented
247C4×D47—24no—
248C4×D47—24no—
249C4×D47—22no—
250C4×D47—22no—
251C4×D47—22no—
252C4×D47—22no—
253C4×D47—24no—
254C4×D47—24no—
255C4×D47—48no—
256C4×D47—48no—
257C4×D47—42no—
258C4×D47—42no—
259C4×D47—44no—
260C8×C47—24no—
261C8×C47—42no—
262C47⋊C8—24no—
263C47⋊C8—42no—
264C2×#1490—471nomaximal, Sylow
265C22×C47—24no—
266C22×C47—42no—
267C22×C47—42no—
268C22×C47—48no—
269C22×C47#40712yes—
270C22×C47#41812yes—
271C22×C47C2×#10412yes—
272C22×C47C22×C22⋊C411yescharacteristic
273C2×D47—22no—
274C2×D47—22no—
275C2×D47—28no—
276C2×D47—24no—
277C2×D47—432no—
278C2×D47—42no—
279C2×D47—42no—
280C2×D47—48no—
281C2×D47C2×#10414yes—
282C4×C47—21no—
283C4×C47—21no—
284C4×C47—42no—
285C4×C47#40712yes—
286C4×C47#41812yes—
287C4×C47C22×M1611yescharacteristic
288C4×C47C22×M1611yescharacteristic
289C4×C47C2×AGL(1,ℤ/8)11yescharacteristic
290C4×C47C2×AGL(1,ℤ/8)11yescharacteristic
291Dic47—22no—
292Dic47—22no—
293Dic47—22no—
294Dic47—42no—
295C22×C4×D4—471no—
296C2×#432—471no—
297C2×C4×M16—471no—
298#1490—478no—
299C2×D4⋊C8—471no—
300C2×D4⋊C8—471no—
301C2×D4⋊C8—471no—
302C2×D4⋊C8D4711yescharacteristic, complemented
303C2×C47—44no—
304C2×C47#149014yescomplemented
305C2×C47C2×#40711yescharacteristic
306C2×C47C2×#41811yescharacteristic
307C2×C47C22×#10411yescharacteristic
308D47—24no—
309D47—44no—
310C23×D4—471no—
311C24×C4—941no—
312C22×M16—941no—
313C2×C4×D4—471no—
314C2×C4×D4—474no—
315C2×C4×D4—474no—
316C2×C4×D4—471no—
317C2×C4×D4—471no—
318C2×C4×D4—941no—
319C2×C4×D4—942no—
320C2×C4×D4—942no—
321C2×C4×D4—941no—
322C2×C4×D4C2×D4711yescharacteristic
323C22×C4⋊C4—471no—
324C22×C22⋊C4—941no—
325C22×C42—471no—
326#432—474no—
327#432—942no—
328C2×C4⋊C8—471no—
329C2×C4⋊C8C2×D4711yescharacteristic, complemented
330C2×C22⋊C8—941no—
331C4×M16—474no—
332C4×M16—942no—
333C2×#97—941no—
334C2×#97—941no—
335C2×C4×C8—471no—
336C2×C4×C8C2×D4711yescharacteristic, complemented
337D4⋊C8—474no—
338D4⋊C8—474no—
339D4⋊C8—474no—
340D4⋊C8C2×D4714yescomplemented
341C47C2×#149011yescharacteristic, complemented, Sylow
342C25—941no—
343C22×D4—474no—
344C22×D4—471no—
345C22×D4—942no—
346C22×D4—941no—
347C22×D4—944no—
348C22×D4—944no—
349C22×D4C4×D4711yescharacteristic
350C23×C4—21no—
351C23×C4—471no—
352C23×C4—471no—
353C23×C4—471no—
354C23×C4—941no—
355C23×C4—944no—
356C23×C4—944no—
357C23×C4—941no—
358C23×C4—941no—
359C23×C4—941no—
360C23×C4—1884no—
361C23×C4—1884no—
362C2×M16—944no—
363C2×M16—948no—
364C22×C8—21no—
365C22×C8—21no—
366C22×C8—941no—
367C22×C8—941no—
368C4×D4—21no—
369C4×D4—21no—
370C4×D4—474no—
371C4×D4—474no—
372C4×D4—472no—
373C4×D4—472no—
374C4×D4—944no—
375C4×D4—944no—
376C4×D4—942no—
377C4×D4—944no—
378C4×D4—944no—
379C4×D4—942no—
380C4×D4—941no—
381C4×D4—941no—
382C4×D4C4×D4712yes—
383C4×D4C22×D4712yes—
384C2×C4⋊C4—471no—
385C2×C4⋊C4—474no—
386C2×C4⋊C4—942no—
387C2×C4⋊C4—941no—
388C2×C4⋊C4C4×D4711yescharacteristic
389C2×C22⋊C4—21no—
390C2×C22⋊C4—944no—
391C2×C22⋊C4—944no—
392C2×C22⋊C4—941no—
393C2×C22⋊C4—941no—
394C2×C22⋊C4—941no—
395C2×C42—471no—
396C2×C42—471no—
397C2×C42—471no—
398C2×C42—472no—
399C2×C42—472no—
400C2×C42—472no—
401C2×C42—472no—
402C2×C42C22×D4711yescharacteristic
403C4⋊C8—21no—
404C4⋊C8—472no—
405C4⋊C8—941no—
406C4⋊C8C22×D4712yescomplemented
407C22⋊C8—944no—
408#97—944no—
409#97—944no—
410C4×C8—21no—
411C4×C8—472no—
412C4×C8—941no—
413C4×C8C22×D4712yescomplemented
414C24—21no—
415C24—471no—
416C24—944no—
417C24—948no—
418C24—941no—
419C24—1884no—
420C24—1884no—
421C2×D4—21no—
422C2×D4—22no—
423C2×D4—474no—
424C2×D4—472no—
425C2×D4—944no—
426C2×D4—948no—
427C2×D4—9416no—
428C2×D4—948no—
429C2×D4—948no—
430C2×D4—941no—
431C2×D4—942no—
432C2×D4#1062532414yescomplemented
433C2×D4C2×C4×D4712yes—
434C22×C4—21no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—42no—
438C22×C4—42no—
439C22×C4—471no—
440C22×C4—471no—
441C22×C4—471no—
442C22×C4—472no—
443C22×C4—472no—
444C22×C4—472no—
445C22×C4—472no—
446C22×C4—472no—
447C22×C4—472no—
448C22×C4—944no—
449C22×C4—944no—
450C22×C4—944no—
451C22×C4—944no—
452C22×C4—942no—
453C22×C4—942no—
454C22×C4—942no—
455C22×C4—942no—
456C22×C4—941no—
457C22×C4—942no—
458C22×C4—942no—
459C22×C4—1884no—
460C22×C4—18816no—
461C22×C4—1888no—
462C22×C4—1888no—
463C22×C4—1884no—
464C22×C4—1884no—
465C22×C4—1882no—
466C22×C4—1882no—
467C22×C4C2×C4×D4711yescharacteristic
468C22×C4D4×D4711yescharacteristic
469C22×C4D4×D4711yescharacteristic
470M16—9416no—
471C2×C8—24no—
472C2×C8—22no—
473C2×C8—22no—
474C2×C8—42no—
475C2×C8—944no—
476C2×C8—942no—
477C2×C8—942no—
478C2×C8—1882no—
479C4⋊C4—21no—
480C4⋊C4—472no—
481C4⋊C4—944no—
482C4⋊C4—941no—
483C4⋊C4C2×C4×D4712yes—
484C22⋊C4—22no—
485C22⋊C4—22no—
486C22⋊C4—944no—
487C22⋊C4—944no—
488C22⋊C4—942no—
489C22⋊C4—942no—
490C42—472no—
491C42—472no—
492C42—472no—
493C42—472no—
494C42—472no—
495C42—472no—
496C42C2×C4×D4711yescharacteristic
497C42D4×D4711yescharacteristic
498C42D4×D4711yescharacteristic
499C42C23×D4711yescharacteristic
500C23—22no—
501C23—24no—
502C23—42no—
503C23—42no—
504C23—472no—
505C23—474no—
506C23—9416no—
507C23—942no—
508C23—942no—
509C23—942no—
510C23—944no—
511C23—1888no—
512C23—1888no—
513C23—18832no—
514C23—1882no—
515C23—1882no—
516C23C22⋊C4×D4711yescharacteristic
517D4—24no—
518D4—22no—
519D4—474no—
520D4—9416no—
521D4—944no—
522D4—942no—
523D4C2×#1062532414yescomplemented
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—42no—
527C2×C4—44no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—472no—
531C2×C4—472no—
532C2×C4—472no—
533C2×C4—944no—
534C2×C4—944no—
535C2×C4—942no—
536C2×C4—942no—
537C2×C4—942no—
538C2×C4—944no—
539C2×C4—942no—
540C2×C4—942no—
541C2×C4—944no—
542C2×C4—942no—
543C2×C4—942no—
544C2×C4—942no—
545C2×C4—1888no—
546C2×C4—1888no—
547C2×C4—1888no—
548C2×C4—1882no—
549C2×C4—1884no—
550C2×C4—1882no—
551C2×C4—1882no—
552C2×C4C22⋊C4×D4711yescharacteristic
553C2×C4C22⋊C4×D4711yescharacteristic
554C2×C4C22⋊C4×D4711yescharacteristic
555C2×C4M16×D4712yes—
556C2×C4C2×#1062532412yes—
557C2×C4#1309404812yes—
558C2×C4#1309405112yes—
559C2×C4C22×C4×D4711yescharacteristic
560C2×C4C2×D4×D4711yescharacteristic
561C2×C4C2×D4×D4711yescharacteristic
562C8—24no—
563C8—42no—
564C8—944no—
565C8—1882no—
566C22—24no—
567C22—42no—
568C22—48no—
569C22—42no—
570C22—474no—
571C22—948no—
572C22—942no—
573C22—942no—
574C22—944no—
575C22—18832no—
576C22—1882no—
577C22—1888no—
578C22—1882no—
579C22#1568908512yes—
580C22#1568912612yes—
581C22#104×D4712yes—
582C22C2×C22⋊C4×D4711yescharacteristic
583C4—21no—
584C4—21no—
585C4—42no—
586C4—942no—
587C4—942no—
588C4—942no—
589C4—1882no—
590C4#1568908512yes—
591C4#1568912612yes—
592C4C2×M16×D4711yescharacteristic
593C4C22×#1062532411yescharacteristic
594C4C2×#1309404811yescharacteristic
595C4C2×#1309405111yescharacteristic
596C2—44no—
597C2—944no—
598C2—1884no—
599C2#2024182914yesdirect product factor
600C2C2×#1568908511yescharacteristic
601C2C2×#1568912611yescharacteristic
602C2C2×#104×D4711yescharacteristic
subgroupG/H
centerC22×C4D4×D47
commutatorC2×C4×C47C23×C4
FrattiniC42C23×D47
FittingC2×C47×D4⋊C8C2
socleC23×C47C2×C22⋊C4
prime radicalC22×D4×D47C4
cosocleC42×C47C24
hypercenterC2×D4⋊C8D47
hypocenterC47C2×#1490
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22×C4 ◁ C2×C42 ◁ C2×D4⋊C8

Complements

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

NH ≅ G/Ntype
2. C2×#15689179C2semidirect
3. C2×C4×#10625324C2semidirect
4. #20241829C2direct (N × H)
5. C2×#15688852C2semidirect
6. C2×D4⋊(C47⋊C8)C2semidirect
7. C2×#15688850C2semidirect
8. C2×C47×D4⋊C8C2semidirect
25. #15689179C22semidirect
26. C2×C4⋊(C47⋊C8)C22semidirect
29. C4×#10625324C22semidirect
32. C2×C4×C47⋊C8C22semidirect
33. #15688852C22semidirect
34. D4⋊(C47⋊C8)C22semidirect
35. #15688850C22semidirect
37. C2×C4⋊C8×C47C22semidirect
38. C2×C4×C8×C47C22semidirect
39. C47×D4⋊C8C22semidirect
91. C4⋊(C47⋊C8)C23semidirect
96. C4×C47⋊C8C23semidirect
109. C4⋊C8×C47C23semidirect
111. C4×C8×C47C23semidirect
121. C2×D4×C47M16semidirect
144. C2×D188M16semidirect
217. D4×C47C2×M16semidirect
246. D188C2×M16semidirect
302. C2×D4⋊C8D47semidirect
304. C2×C47#1490semidirect
329. C2×C4⋊C8C2×D47semidirect
336. C2×C4×C8C2×D47semidirect
340. D4⋊C8C2×D47semidirect
341. C47C2×#1490semidirect
406. C4⋊C8C22×D47semidirect
413. C4×C8C22×D47semidirect
432. C2×D4#10625324semidirect
523. D4C2×#10625324semidirect
599. C2#20241829direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C4×D4×D472
C2×#156891792
C2×C4×#106253242
#202418292
C2×#156888522
C2×D4⋊(C47⋊C8)2
C2×#156888502
C2×C47×D4⋊C82
C2×#149047