Finite Group Databasebeta

← C2×#20243692 #26217469

Subgroups — C2×#20243692 #26217469

Order 12032.

2270 conjugacy classes of subgroups.

The subgroup lattice has 590 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#13094000C211yesmaximal, characteristic
2C22×C4⋊C4×D47C211yesmaximal, characteristic
3C22×#13093948C211yesmaximal, characteristic, complemented
4C22×(C2×D47)⋊C8C211yesmaximal, characteristic
5#20243692C218yesmaximal, direct product factor
6C2×#15688886C211yesmaximal, characteristic, complemented
7C2×#15688885C211yesmaximal, characteristic, complemented
8C22×#103×C47C211yesmaximal, characteristic, complemented
9C23×C4×D47—21no—
10C23×C4×D47C2211yescharacteristic
11C23×C47⋊C8C2211yescharacteristic
12C2×#13094000—24no—
13C2×#13094000—22no—
14C2×#13094000C418yescomplemented
15C2×#13094000C2214yes—
16C22×C4⋊Dic47C2211yescharacteristic
17C22×C4⋊Dic47C2211yescharacteristic
18C22×Dic47⋊C4—21no—
19C22×Dic47⋊C4—21no—
20C2×C4⋊C4×D47—22no—
21C2×C4⋊C4×D47—24no—
22C2×C4⋊C4×D47—24no—
23C2×C4⋊C4×D47C2214yes—
24C22×(C2×D47)⋊C4—21no—
25C2×#13093948—24no—
26C2×#13093948C2214yescomplemented
27C2×(C2×D47)⋊C8—24no—
28C2×(C2×D47)⋊C8C2214yes—
29#15688886C2214yescomplemented
30#15688885C2214yescomplemented
31C23×Q8×C47C2211yescharacteristic
32C23×C8×C47C2211yescharacteristic
33C22×C4⋊C4×C47C2211yescharacteristic
34C2×#103×C47—24no—
35C2×#103×C47C2214yescomplemented
36C24×D47D411yescharacteristic
37C23×Dic47—21no—
38C23×Dic47—21no—
39C23×Dic47D411yescharacteristic
40C22×C4×D47—22no—
41C22×C4×D47—22no—
42C22×C4×D47—22no—
43C22×C4×D47—22no—
44C22×C4×D47—28no—
45C22×C4×D47—24no—
46C22×C4×D47—44no—
47C22×C4×D47—44no—
48C22×C4×D47C2×C414yes—
49C22×C4×D47D412yes—
50C22×C4×D47D414yes—
51C22×C4×D47C2312yes—
52C22×C47⋊C8—22no—
53C22×C47⋊C8—22no—
54C22×C47⋊C8—22no—
55C22×C47⋊C8C2312yes—
56#13094000—216no—
57#13094000—28no—
58#13094000C2×C4116yescomplemented
59C2×C4⋊Dic47—21no—
60C2×C4⋊Dic47—22no—
61C2×C4⋊Dic47—21no—
62C2×C4⋊Dic47—22no—
63C2×C4⋊Dic47C2×C414yescomplemented
64C2×C4⋊Dic47D414yes—
65C2×C4⋊Dic47C2312yes—
66C2×C4⋊Dic47C2312yes—
67C2×Dic47⋊C4—28no—
68C2×Dic47⋊C4—28no—
69C2×Dic47⋊C4—24no—
70C2×Dic47⋊C4—24no—
71C4⋊C4×D47—216no—
72C4⋊C4×D47—28no—
73C4⋊C4×D47—28no—
74C2×(C2×D47)⋊C4—28no—
75C2×(C2×D47)⋊C4—24no—
76#13093948—28no—
77(C2×D47)⋊C8—28no—
78C22×Q8×C47—22no—
79C22×Q8×C47—21no—
80C22×Q8×C47—22no—
81C22×Q8×C47—22no—
82C22×Q8×C47—22no—
83C22×Q8×C47—22no—
84C22×Q8×C47C2×C414yescomplemented
85C22×Q8×C47C2312yes—
86C23×C4×C47—21no—
87C23×C4×C47—21no—
88C23×C4×C47C2311yescharacteristic
89C22×C8×C47—22no—
90C22×C8×C47—22no—
91C22×C8×C47—22no—
92C22×C8×C47C2312yes—
93C2×C4⋊C4×C47—21no—
94C2×C4⋊C4×C47—22no—
95C2×C4⋊C4×C47D414yes—
96C2×C4⋊C4×C47C2312yes—
97#103×C47—28no—
98C24×C47C2×D411yescharacteristic
99C23×D47—22no—
100C23×D47—22no—
101C23×D47—22no—
102C23×D47—22no—
103C23×D47C22⋊C414yes—
104C23×D47SD1614yes—
105C23×D47Q1614yes—
106C23×D47C2×D412yes—
107C2×Q8×C47—24no—
108C2×Q8×C47—22no—
109C2×Q8×C47—28no—
110C2×Q8×C47—28no—
111C2×Q8×C47—22no—
112C2×Q8×C47—22no—
113C2×Q8×C47—22no—
114C2×Q8×C47—22no—
115C2×Q8×C47—42no—
116C2×Q8×C47—44no—
117C2×Q8×C47—44no—
118C2×Q8×C47—42no—
119C2×Q8×C47C22×C414yescomplemented
120C22×C4×C47—24no—
121C22×C4×C47—24no—
122C22×C4×C47—22no—
123C22×C4×C47—22no—
124C22×C4×C47—42no—
125C22×C4×C47—42no—
126C22×C4×C47—42no—
127C22×C4×C47—42no—
128C22×C4×C47C22×C412yes—
129C22×C4×C47C2×D411yescharacteristic
130C22×C4×C47C2×D412yes—
131C22×C4×C47C2×D412yes—
132C22×C4×C47C2×D412yes—
133C22×C4×C47#4112yes—
134C22×C4×C47#4112yes—
135C22×C4×C47C2411yescharacteristic
136C22×Dic47—22no—
137C22×Dic47—22no—
138C22×Dic47—24no—
139C22×Dic47—24no—
140C22×Dic47—22no—
141C22×Dic47—22no—
142C22×Dic47—42no—
143C22×Dic47—42no—
144C22×Dic47—42no—
145C22×Dic47—42no—
146C22×Dic47C22⋊C414yes—
147C22×Dic47C2×D412yes—
148C2×C4×D47—28no—
149C2×C4×D47—28no—
150C2×C4×D47—24no—
151C2×C4×D47—28no—
152C2×C4×D47—28no—
153C2×C4×D47—24no—
154C2×C4×D47—22no—
155C2×C4×D47—22no—
156C2×C4×D47—22no—
157C2×C4×D47—22no—
158C2×C4×D47—216no—
159C2×C4×D47—48no—
160C2×C4×D47—432no—
161C2×C4×D47—48no—
162C2×C4×D47C22⋊C418yes—
163C2×C4×D47C22×C414yes—
164C2×C4×D47C2×D414yes—
165C2×C8×C47—22no—
166C2×C8×C47—22no—
167C2×C8×C47—22no—
168C2×C8×C47—44no—
169C4⋊C4×C47—24no—
170C4⋊C4×C47—22no—
171C4⋊C4×C47C2×D414yes—
172(C2×D47)⋊C4—216no—
173C4⋊Dic47—24no—
174C4⋊Dic47—24no—
175C4⋊Dic47—22no—
176C4⋊Dic47—22no—
177C4⋊Dic47C22×C414yescomplemented
178C4⋊Dic47C2×D414yes—
179Dic47⋊C4—216no—
180Dic47⋊C4—216no—
181C2×C47⋊C8—22no—
182C2×C47⋊C8—22no—
183C2×C47⋊C8—22no—
184C2×C47⋊C8—44no—
185C23×C47C2×C22⋊C412yes—
186C23×C47#12012yes—
187C23×C47#12312yes—
188C23×C47C2×SD1612yes—
189C23×C47C2×Q1612yes—
190C23×C47AGL(1,ℤ/8)12yes—
191C23×C47#13712yes—
192C23×C47C22×D411yescharacteristic
193C22×D47—22no—
194C22×D47—22no—
195C22×D47—22no—
196C22×D47—22no—
197C22×D47—28no—
198C22×D47—28no—
199C22×D47—28no—
200C22×D47—28no—
201C22×D47—28no—
202C22×D47—28no—
203C22×D47#103116yescomplemented
204C22×D47C2×C22⋊C414yes—
205C22×D47C2×SD1614yes—
206C22×D47C2×Q1614yes—
207Q8×C47—24no—
208Q8×C47—24no—
209Q8×C47—42no—
210Q8×C47—42no—
211Q8×C47—48no—
212C2×C4×C47—24no—
213C2×C4×C47—24no—
214C2×C4×C47—42no—
215C2×C4×C47—42no—
216C2×C4×C47—42no—
217C2×C4×C47—42no—
218C2×C4×C47—48no—
219C2×C4×C47—48no—
220C2×C4×C47C2×C22⋊C412yes—
221C2×C4×C47C4×D414yes—
222C2×C4×C47C4×D414yes—
223C2×C4×C47#12012yes—
224C2×C4×C47#12114yes—
225C2×C4×C47#12114yes—
226C2×C4×C47#12312yes—
227C2×C4×C47C23×C411yescharacteristic
228C2×C4×C47C22×D411yescharacteristic
229C2×C4×C47C22×D411yescharacteristic
230C2×C4×C47C22×D411yescharacteristic
231C2×C4×C47C2×#4111yescharacteristic
232C2×C4×C47C2×#4111yescharacteristic
233C2×Dic47—22no—
234C2×Dic47—22no—
235C2×Dic47—28no—
236C2×Dic47—24no—
237C2×Dic47—24no—
238C2×Dic47—42no—
239C2×Dic47—42no—
240C2×Dic47—42no—
241C2×Dic47—42no—
242C2×Dic47—48no—
243C2×Dic47—48no—
244C2×Dic47C2×C22⋊C414yes—
245C4×D47—24no—
246C4×D47—24no—
247C4×D47—24no—
248C4×D47—24no—
249C4×D47—216no—
250C4×D47—432no—
251C8×C47—42no—
252C47⋊C8—42no—
253C2×#1872—471nomaximal, Sylow
254C22×C47#38614yes—
255C22×C47C2×#10314yes—
256C22×C47#41714yes—
257C22×C47#45014yes—
258C22×C47#45114yes—
259C22×C47#48314yes—
260C22×C47#48414yes—
261C22×C47C22×C22⋊C411yescharacteristic
262C22×C47C2×#12011yescharacteristic
263C22×C47C2×#12311yescharacteristic
264C22×C47C22×SD1611yescharacteristic
265C22×C47C22×Q1611yescharacteristic
266C22×C47C2×AGL(1,ℤ/8)11yescharacteristic
267C22×C47C2×#13711yescharacteristic
268C2×D47—24no—
269C2×D47—24no—
270C2×D47—24no—
271C2×D47—232no—
272C2×D47—24no—
273C2×D47—24no—
274C2×D47—24no—
275C2×D47C2×#10318yescomplemented
276C4×C47—44no—
277C4×C47—44no—
278C4×C47#38614yes—
279C4×C47C2×C4×D411yescharacteristic
280C4×C47C2×C4×D411yescharacteristic
281C4×C47C2×#12111yescharacteristic
282C4×C47C2×#12111yescharacteristic
283Dic47—24no—
284Dic47—44no—
285Dic47—44no—
286C22×C22⋊Q8—471no—
287C23×C4⋊C4—471no—
288C22×#103—471no—
289C22×#103D4711yescharacteristic, complemented
290C22×C22⋊C8—471no—
291#1872—478no—
292C2×#340—471no—
293C2×#340—471no—
294C2×C47#187218yescomplemented
295C2×C47C2×#38611yescharacteristic
296C2×C47C22×#10311yescharacteristic
297C2×C47C2×#41711yescharacteristic
298C2×C47C2×#45011yescharacteristic
299C2×C47C2×#45111yescharacteristic
300C2×C47C2×#48311yescharacteristic
301C2×C47C2×#48411yescharacteristic
302D47—28no—
303C23×Q8C2×D4711yescharacteristic
304C24×C4—471no—
305C24×C4—941no—
306C23×C8—471no—
307C23×C8C2×D4711yescharacteristic
308C2×C22⋊Q8—478no—
309C2×C22⋊Q8—474no—
310C2×C22⋊Q8—942no—
311C2×C22⋊Q8—944no—
312C22×C4⋊C4—471no—
313C22×C4⋊C4—471no—
314C22×C4⋊C4—474no—
315C22×C4⋊C4—944no—
316C22×C4⋊C4—942no—
317C22×C4⋊C4—941no—
318C22×C4⋊C4—941no—
319C22×C4⋊C4—944no—
320C22×C4⋊C4C2×D4711yescharacteristic
321C22×C22⋊C4—941no—
322C2×#103—24no—
323C2×#103—474no—
324C2×#103—944no—
325C2×#103C2×D4714yescomplemented
326C2×C22⋊C8—474no—
327C2×C22⋊C8—944no—
328#340—474no—
329#340—474no—
330C47C2×#187211yescharacteristic, complemented, Sylow
331C25—471no—
332C22×Q8—22no—
333C22×Q8—21no—
334C22×Q8—22no—
335C22×Q8—22no—
336C22×Q8—22no—
337C22×Q8—22no—
338C22×Q8C4×D4714yescomplemented
339C22×Q8C22×D4712yes—
340C23×C4—21no—
341C23×C4—21no—
342C23×C4—471no—
343C23×C4—474no—
344C23×C4—472no—
345C23×C4—474no—
346C23×C4—472no—
347C23×C4—944no—
348C23×C4—948no—
349C23×C4—942no—
350C23×C4—942no—
351C23×C4—942no—
352C23×C4—942no—
353C23×C4—941no—
354C23×C4—941no—
355C23×C4—1884no—
356C23×C4—1884no—
357C23×C4C22×D4711yescharacteristic
358C22×C8—22no—
359C22×C8—22no—
360C22×C8—22no—
361C22×C8—472no—
362C22×C8—942no—
363C22×C8—942no—
364C22×C8—942no—
365C22×C8C22×D4712yes—
366C22⋊Q8—4716no—
367C22⋊Q8—9416no—
368C22⋊Q8—948no—
369C2×C4⋊C4—21no—
370C2×C4⋊C4—22no—
371C2×C4⋊C4—474no—
372C2×C4⋊C4—474no—
373C2×C4⋊C4—472no—
374C2×C4⋊C4—472no—
375C2×C4⋊C4—948no—
376C2×C4⋊C4—9416no—
377C2×C4⋊C4—944no—
378C2×C4⋊C4—948no—
379C2×C4⋊C4—944no—
380C2×C4⋊C4—948no—
381C2×C4⋊C4—941no—
382C2×C4⋊C4—941no—
383C2×C4⋊C4—942no—
384C2×C4⋊C4—942no—
385C2×C4⋊C4—948no—
386C2×C4⋊C4C22⋊D4714yes—
387C2×C4⋊C4C22×D4712yes—
388C2×C22⋊C4—948no—
389C2×C22⋊C4—944no—
390#103—28no—
391#103—948no—
392C22⋊C8—948no—
393C24—474no—
394C24—474no—
395C24—474no—
396C24—472no—
397C24—942no—
398C24—942no—
399C24—942no—
400C24—942no—
401C24D4×D4711yescharacteristic
402C2×Q8—24no—
403C2×Q8—22no—
404C2×Q8—28no—
405C2×Q8—28no—
406C2×Q8—22no—
407C2×Q8—22no—
408C2×Q8—22no—
409C2×Q8—22no—
410C2×Q8—42no—
411C2×Q8—44no—
412C2×Q8—44no—
413C2×Q8—42no—
414C2×Q8C2×C4×D4714yescomplemented
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—42no—
420C22×C4—42no—
421C22×C4—42no—
422C22×C4—42no—
423C22×C4—474no—
424C22×C4—472no—
425C22×C4—478no—
426C22×C4—474no—
427C22×C4—474no—
428C22×C4—9416no—
429C22×C4—942no—
430C22×C4—942no—
431C22×C4—948no—
432C22×C4—948no—
433C22×C4—944no—
434C22×C4—948no—
435C22×C4—948no—
436C22×C4—944no—
437C22×C4—942no—
438C22×C4—942no—
439C22×C4—942no—
440C22×C4—942no—
441C22×C4—944no—
442C22×C4—944no—
443C22×C4—942no—
444C22×C4—942no—
445C22×C4—1888no—
446C22×C4—18832no—
447C22×C4—1888no—
448C22×C4—1882no—
449C22×C4—1882no—
450C22×C4—1882no—
451C22×C4—1882no—
452C22×C4C2×C4×D4712yes—
453C22×C4#1062535012yes—
454C22×C4D4×D4711yescharacteristic
455C22×C4D4×D4712yes—
456C22×C4D4×D4712yes—
457C22×C4#1062535212yes—
458C22×C4C2×C22⋊D4712yes—
459C22×C4C23×D4711yescharacteristic
460C2×C8—22no—
461C2×C8—22no—
462C2×C8—22no—
463C2×C8—44no—
464C2×C8—942no—
465C2×C8—942no—
466C2×C8—942no—
467C2×C8—1884no—
468C4⋊C4—24no—
469C4⋊C4—22no—
470C4⋊C4—474no—
471C4⋊C4—474no—
472C4⋊C4—9416no—
473C4⋊C4—9416no—
474C4⋊C4—944no—
475C4⋊C4—944no—
476C4⋊C4—942no—
477C4⋊C4—942no—
478C4⋊C4C2×C22⋊D4714yes—
479C22⋊C4—9416no—
480C23—474no—
481C23—4716no—
482C23—474no—
483C23—474no—
484C23—948no—
485C23—948no—
486C23—948no—
487C23—942no—
488C23—942no—
489C23—948no—
490C23—942no—
491C23—942no—
492C23—948no—
493C23—948no—
494C23C22⋊C4×D4712yes—
495C23#1309399212yes—
496C23#1309400612yes—
497C23SD16×D4712yes—
498C23Q16×D4712yes—
499C23#1309405112yes—
500C23#1309405612yes—
501C23C2×D4×D4711yescharacteristic
502Q8—24no—
503Q8—24no—
504Q8—48no—
505Q8—42no—
506Q8—42no—
507C2×C4—24no—
508C2×C4—24no—
509C2×C4—48no—
510C2×C4—48no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—474no—
516C2×C4—948no—
517C2×C4—942no—
518C2×C4—942no—
519C2×C4—9416no—
520C2×C4—944no—
521C2×C4—944no—
522C2×C4—944no—
523C2×C4—944no—
524C2×C4—944no—
525C2×C4—944no—
526C2×C4—18832no—
527C2×C4—1888no—
528C2×C4—1888no—
529C2×C4—1882no—
530C2×C4—1882no—
531C2×C4—1882no—
532C2×C4—1882no—
533C2×C4C22⋊C4×D4712yes—
534C2×C4(C22⋊D47)⋊C414yes—
535C2×C4C4×C22⋊D4714yes—
536C2×C4#1309399212yes—
537C2×C4#1309399514yes—
538C2×C4#1309399714yes—
539C2×C4#1309400612yes—
540C2×C4C22×C4×D4711yescharacteristic
541C2×C4C2×D4×D4711yescharacteristic
542C2×C4C2×D4×D4711yescharacteristic
543C2×C4C22×C22⋊D4711yescharacteristic
544C2×C4C2×#1062535211yescharacteristic
545C2×C4C2×#1062535011yescharacteristic
546C8—42no—
547C8—1882no—
548C22—478no—
549C22—944no—
550C22—944no—
551C22—944no—
552C22—944no—
553C22—944no—
554C22—9432no—
555C22—944no—
556C22#1568899914yes—
557C22#103×D4714yes—
558C22#1568912314yes—
559C22#1568927214yes—
560C22#1568927714yes—
561C22#1568945214yes—
562C22#1568945714yes—
563C22C2×C22⋊C4×D4711yescharacteristic
564C22C2×#1309399211yescharacteristic
565C22C2×#1309400611yescharacteristic
566C22C2×SD16×D4711yescharacteristic
567C22C2×Q16×D4711yescharacteristic
568C22C2×#1309405111yescharacteristic
569C22C2×#1309405611yescharacteristic
570C4—44no—
571C4—44no—
572C4—944no—
573C4—1884no—
574C4—1884no—
575C4#1568899914yes—
576C4C2×(C22⋊D47)⋊C411yescharacteristic
577C4C2×C4×C22⋊D4711yescharacteristic
578C4C2×#1309399511yescharacteristic
579C4C2×#1309399711yescharacteristic
580C2—948no—
581C2#2024369218yesdirect product factor
582C2C2×#1568899911yescharacteristic
583C2C2×#103×D4711yescharacteristic
584C2C2×#1568912311yescharacteristic
585C2C2×#1568927211yescharacteristic
586C2C2×#1568927711yescharacteristic
587C2C2×#1568945211yescharacteristic
588C2C2×#1568945711yescharacteristic
subgroupG/H
centerC24D4×D47
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC22×#103×C47C2
socleC24×C47C2×D4
prime radicalC24×D47D4
cosocleC22×C4×C47C24
hypercenterC22×#103D47
hypocenterC47C2×#1872
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C24 ◁ C23×C4 ◁ C22×#103

Complements

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

NH ≅ G/Ntype
3. C22×#13093948C2semidirect
5. #20243692C2direct (N × H)
6. C2×#15688886C2semidirect
7. C2×#15688885C2semidirect
8. C22×#103×C47C2semidirect
14. C2×#13094000C4semidirect
26. C2×#13093948C22semidirect
29. #15688886C22semidirect
30. #15688885C22semidirect
35. C2×#103×C47C22semidirect
58. #13094000C2×C4semidirect
63. C2×C4⋊Dic47C2×C4semidirect
84. C22×Q8×C47C2×C4semidirect
119. C2×Q8×C47C22×C4semidirect
177. C4⋊Dic47C22×C4semidirect
203. C22×D47#103semidirect
275. C2×D47C2×#103semidirect
289. C22×#103D47semidirect
294. C2×C47#1872semidirect
325. C2×#103C2×D47semidirect
330. C47C2×#1872semidirect
338. C22×Q8C4×D47semidirect
414. C2×Q8C2×C4×D47semidirect
581. C2#20243692direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#130940002
C22×C4⋊C4×D472
C22×#130939482
C22×(C2×D47)⋊C82
#202436922
C2×#156888862
C2×#156888852
C22×#103×C472
C2×#187247