Finite Group Databasebeta

← C2×#20253311 #27084056

Subgroups — C2×#20253311 #27084056

Order 12032.

2650 conjugacy classes of subgroups.

The subgroup lattice has 582 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689634C211yesmaximal, characteristic
2#20253311C2116yesmaximal, direct product factor
3C2×#15689458C212yesmaximal, complemented
4C2×#15689441C212yesmaximal, complemented
5C2×#15689453C212yesmaximal, complemented
6C2×#15689436C212yesmaximal, complemented
7C2×#15689163C212yesmaximal, complemented
8C2×#15689122C212yesmaximal, complemented
9C2×#15689087C211yesmaximal, characteristic, complemented
10C23×C4⋊C4×C47C211yesmaximal, characteristic, complemented
11C22×#10625350C2211yescharacteristic
12C22×#10625329C2212yes—
13#15689634—24no—
14#15689634C2218yes—
15C2×#13094007—21no—
16C2×#13094003C2211yescharacteristic
17C2×#13094003C2211yescharacteristic
18C2×#13093998C2211yescharacteristic
19C2×#13093998C2211yescharacteristic
20C2×#13093993—21no—
21C2×C4×C22⋊D47—21no—
22C2×C4×C22⋊D47—21no—
23C2×#13093983C2211yescharacteristic
24#15689458C22116yescomplemented
25#15689453C22116yescomplemented
26#15689441C22116yescomplemented
27#15689436C22116yescomplemented
28#15689163C22116yescomplemented
29C2×#13093957C2214yescomplemented
30C2×#13093955C2214yescomplemented
31#15689122C22116yescomplemented
32C2×#13093947C2214yescomplemented
33C2×#13093944C2214yescomplemented
34#15689087C2218yescomplemented
35C2×C22⋊(C47⋊C8)C2212yes—
36C2×C22⋊(C47⋊C8)C2212yes—
37C24×C4×C47—21no—
38C24×C4×C47C2211yescharacteristic
39C22×C4⋊C4×C47—24no—
40C22×C4⋊C4×C47—21no—
41C22×C4⋊C4×C47—21no—
42C22×C4⋊C4×C47—21no—
43C22×C4⋊C4×C47C2212yes—
44C22×C4⋊C4×C47C2212yes—
45C22×C4⋊C4×C47C2212yes—
46C22×C4⋊C4×C47C2218yescomplemented
47C2×#10625350—22no—
48C2×#10625350—28no—
49C2×#10625350C2314yes—
50C22×Dic94C2311yescharacteristic
51C22×D188C2311yescharacteristic
52C22×C22⋊D47—21no—
53C22×C22⋊D47—21no—
54C22×C22⋊D47—21no—
55C22×C4×D47—21no—
56C2×#10625329—28no—
57C2×#10625329C2318yes—
58C22×C47⋊C8C2314yes—
59#13094007—28no—
60#13094003—22no—
61#13094003—22no—
62#13094003C2314yes—
63#13094003C2314yes—
64#13093998—22no—
65#13093998—22no—
66#13093998C2314yes—
67#13093998C2314yes—
68#13093993—28no—
69C4×C22⋊D47—28no—
70C4×C22⋊D47—28no—
71#13093983—22no—
72#13093983C2314yes—
73C2×C4⋊Dic47C2311yescharacteristic
74C2×C4⋊Dic47C2311yescharacteristic
75C2×Dic47⋊C4—21no—
76C2×Dic47⋊C4—21no—
77C2×C22⋊Dic47—21no—
78C2×C22⋊Dic47—21no—
79C2×C22⋊Dic47—21no—
80C2×(C2×D47)⋊C4—21no—
81C2×(C2×D47)⋊C4—21no—
82C2×C4×Dic47—21no—
83#13093957C23116yescomplemented
84#13093955C23116yescomplemented
85#13093947C23116yescomplemented
86#13093944C23116yescomplemented
87C22⋊(C47⋊C8)C2318yes—
88C22⋊(C47⋊C8)C2318yes—
89C25×C47D411yescharacteristic
90C23×C4×C47—28no—
91C23×C4×C47—22no—
92C23×C4×C47—22no—
93C23×C4×C47—22no—
94C23×C4×C47—28no—
95C23×C4×C47—48no—
96C23×C4×C47D411yescharacteristic
97C23×C4×C47D411yescharacteristic
98C23×C4×C47D411yescharacteristic
99C23×C4×C47D414yes—
100C23×C4×C47C2311yescharacteristic
101C23×C4×C47C2311yescharacteristic
102C23×C4×C47C2314yes—
103C23×C4×C47C2311yescharacteristic
104C2×C4⋊C4×C47—22no—
105C2×C4⋊C4×C47—22no—
106C2×C4⋊C4×C47—24no—
107C2×C4⋊C4×C47—24no—
108C2×C4⋊C4×C47—22no—
109C2×C4⋊C4×C47—24no—
110C2×C4⋊C4×C47—28no—
111C2×C4⋊C4×C47—28no—
112C2×C4⋊C4×C47—28no—
113C2×C4⋊C4×C47C2314yes—
114C2×C4⋊C4×C47C2318yes—
115C2×C4⋊C4×C47C2318yes—
116C2×C4⋊C4×C47C2318yes—
117C24×C47—24no—
118C24×C47—24no—
119C24×C47—24no—
120C24×C47C2×D414yes—
121C24×C47C2×D411yescharacteristic
122C24×C47C2×D411yescharacteristic
123C24×C47C2×D411yescharacteristic
124C23×D47—21no—
125C22×C4×C47—24no—
126C22×C4×C47—24no—
127C22×C4×C47—24no—
128C22×C4×C47—216no—
129C22×C4×C47—24no—
130C22×C4×C47—24no—
131C22×C4×C47—216no—
132C22×C4×C47—28no—
133C22×C4×C47—28no—
134C22×C4×C47—28no—
135C22×C4×C47—24no—
136C22×C4×C47—48no—
137C22×C4×C47—48no—
138C22×C4×C47—432no—
139C22×C4×C47—48no—
140C22×C4×C47C2×D411yescharacteristic
141C22×C4×C47C2×D411yescharacteristic
142C22×C4×C47C2×D414yes—
143C22×C4×C47C2×D414yes—
144C22×C4×C47C2×D412yes—
145C22×C4×C47C2×D412yes—
146C22×C4×C47C2×D414yes—
147C22×C4×C47C2×D411yescharacteristic
148C22×C4×C47C2×D412yes—
149C22×C4×C47#4118yes—
150C22×C4×C47C2412yes—
151C22×C4×C47C2412yes—
152C22×C4×C47C2411yescharacteristic
153C22×C4×C47C2412yes—
154C2×C22⋊D47—24no—
155C2×C22⋊D47—24no—
156C2×C22⋊D47—24no—
157C2×C22⋊D47—44no—
158C2×C22⋊D47—44no—
159C2×C22⋊D47—44no—
160C22×Dic47—21no—
161C22×Dic47—21no—
162C22×Dic47—21no—
163#10625350—216no—
164#10625350—48no—
165C2×D188—24no—
166C2×D188—21no—
167C2×D188C2412yes—
168C2×C4×D47—24no—
169C2×C4×D47—44no—
170C2×Dic94—24no—
171C2×Dic94—21no—
172C2×Dic94C2412yes—
173C4⋊C4×C47—28no—
174C4⋊C4×C47—28no—
175C4⋊C4×C47—28no—
176C4⋊C4×C47—24no—
177C4⋊C4×C47C2418yes—
178C22⋊Dic47—24no—
179C22⋊Dic47—24no—
180C22⋊Dic47—24no—
181(C2×D47)⋊C4—24no—
182(C2×D47)⋊C4—24no—
183C4⋊Dic47—21no—
184C4⋊Dic47—21no—
185C4⋊Dic47C2412yes—
186C4⋊Dic47C2412yes—
187Dic47⋊C4—24no—
188Dic47⋊C4—24no—
189C4×Dic47—24no—
190#10625329—216no—
191C2×C47⋊C8—28no—
192C2×C47⋊C8C2418yes—
193C23×C47—232no—
194C23×C47—28no—
195C23×C47—22no—
196C23×C47—22no—
197C23×C47—28no—
198C23×C47—22no—
199C23×C47—22no—
200C23×C47—28no—
201C23×C47—22no—
202C23×C47—22no—
203C23×C47#12314yes—
204C23×C47AGL(1,ℤ/8)14yes—
205C23×C47#13714yes—
206C23×C47C22×D412yes—
207C23×C47C22×D412yes—
208C23×C47C22×D412yes—
209C23×C47C22×D411yescharacteristic
210C22×D47—22no—
211C22×D47—42no—
212C22×D47—42no—
213C22×D47—42no—
214C2×C4×C47—28no—
215C2×C4×C47—22no—
216C2×C4×C47—22no—
217C2×C4×C47—24no—
218C2×C4×C47—24no—
219C2×C4×C47—28no—
220C2×C4×C47—28no—
221C2×C4×C47—24no—
222C2×C4×C47—24no—
223C2×C4×C47—28no—
224C2×C4×C47—416no—
225C2×C4×C47—416no—
226C2×C4×C47—416no—
227C2×C4×C47—48no—
228C2×C4×C47#12314yes—
229C2×C4×C47#12318yes—
230C2×C4×C47C22×D412yes—
231C2×C4×C47C22×D412yes—
232C2×C4×C47C22×D412yes—
233C2×C4×C47C22×D411yescharacteristic
234C2×C4×C47C2×#4114yes—
235C2×C4×C47C2×#4118yes—
236C2×C4×C47C2511yescharacteristic
237C22⋊D47—48no—
238C22⋊D47—48no—
239C22⋊D47—48no—
240C2×Dic47—22no—
241C2×Dic47—22no—
242C2×Dic47—22no—
243C2×Dic47—42no—
244C2×Dic47—42no—
245C2×Dic47—42no—
246D188—24no—
247D188—42no—
248C4×D47—48no—
249Dic94—24no—
250Dic94—42no—
251C47⋊C8—28no—
252C2×#3092—471nomaximal, Sylow
253C22×C47—232no—
254C22×C47—28no—
255C22×C47—28no—
256C22×C47—28no—
257C22×C47—22no—
258C22×C47—22no—
259C22×C47—22no—
260C22×C47—22no—
261C22×C47—22no—
262C22×C47—22no—
263C22×C47C2×#12314yes—
264C22×C47C2×#12312yes—
265C22×C47C2×AGL(1,ℤ/8)14yes—
266C22×C47C2×AGL(1,ℤ/8)12yes—
267C22×C47C2×#13714yes—
268C22×C47C2×#13712yes—
269C22×C47C23×D411yescharacteristic
270C2×D47—42no—
271C2×D47—42no—
272C2×D47—42no—
273C2×D47—84no—
274C4×C47—22no—
275C4×C47—22no—
276C4×C47—48no—
277C4×C47C2×#12312yes—
278C4×C47C2×#12314yes—
279C4×C47C22×#4112yes—
280Dic47—42no—
281Dic47—42no—
282Dic47—42no—
283C2×#525—471no—
284C23×C4⋊C4D4711yescharacteristic, complemented
285#3092—4716no—
286C2×#484—472no—
287C2×#481—472no—
288C2×#483—472no—
289C2×#480—472no—
290C2×#428—472no—
291C2×#417—472no—
292C2×#407—471no—
293C2×C47—24no—
294C2×C47—24no—
295C2×C47—24no—
296C2×C47#309214yescomplemented
297C2×C47C22×#12311yescharacteristic
298C2×C47C22×AGL(1,ℤ/8)11yescharacteristic
299C2×C47C22×#13711yescharacteristic
300D47—82no—
301C22×#41—471no—
302C24×C4—21no—
303C24×C4C2×D4711yescharacteristic
304C22×M16—472no—
305#525—478no—
306#525—944no—
307C2×#123—941no—
308C2×C22⋊Q8—471no—
309C2×C22⋊Q8—471no—
310C2×#121—471no—
311C2×#121—471no—
312C2×#120—941no—
313C2×C4×D4—941no—
314C2×C4×D4—941no—
315C2×#117—471no—
316C22×C4⋊C4—24no—
317C22×C4⋊C4—21no—
318C22×C4⋊C4—21no—
319C22×C4⋊C4—21no—
320C22×C4⋊C4C2×D4712yes—
321C22×C4⋊C4C2×D4712yes—
322C22×C4⋊C4C2×D4712yes—
323C22×C4⋊C4C2×D4718yescomplemented
324#484—4716no—
325#483—4716no—
326#481—4716no—
327#480—4716no—
328#428—4716no—
329C2×#107—474no—
330C2×#106—474no—
331#417—4716no—
332C2×#103—474no—
333C2×#102—474no—
334#407—478no—
335C2×C22⋊C8—472no—
336C2×C22⋊C8—472no—
337C47C2×#309211yescharacteristic, complemented, Sylow
338C25C22⋊D4711yescharacteristic
339C2×#41—474no—
340C2×#41—942no—
341C2×#41—948no—
342C22×Q8—471no—
343C22×D4—471no—
344C22×D4—941no—
345C22×D4—941no—
346C22×D4—941no—
347C23×C4—28no—
348C23×C4—22no—
349C23×C4—22no—
350C23×C4—22no—
351C23×C4—28no—
352C23×C4—48no—
353C23×C4—941no—
354C23×C4C22⋊D4711yescharacteristic
355C23×C4C22⋊D4711yescharacteristic
356C23×C4C22⋊D4711yescharacteristic
357C23×C4C22⋊D4714yes—
358C23×C4C22×D4711yescharacteristic
359C23×C4C22×D4711yescharacteristic
360C23×C4C22×D4714yes—
361C23×C4C22×D4711yescharacteristic
362C2×M16—478no—
363C2×M16—948no—
364C22×C8—474no—
365#123—948no—
366C22⋊Q8—474no—
367C22⋊Q8—474no—
368C22⋊Q8—942no—
369C22⋊Q8—942no—
370#121—474no—
371#121—474no—
372#121—942no—
373#121—942no—
374#120—948no—
375C4×D4—948no—
376C4×D4—948no—
377#117—474no—
378#117—942no—
379C2×C4⋊C4—24no—
380C2×C4⋊C4—24no—
381C2×C4⋊C4—24no—
382C2×C4⋊C4—22no—
383C2×C4⋊C4—22no—
384C2×C4⋊C4—22no—
385C2×C4⋊C4—28no—
386C2×C4⋊C4—28no—
387C2×C4⋊C4—28no—
388C2×C4⋊C4—471no—
389C2×C4⋊C4—471no—
390C2×C4⋊C4—941no—
391C2×C4⋊C4—941no—
392C2×C4⋊C4C22×D4714yes—
393C2×C4⋊C4C22×D4718yes—
394C2×C4⋊C4C22×D4718yes—
395C2×C4⋊C4C22×D4718yes—
396C2×C22⋊C4—941no—
397C2×C22⋊C4—941no—
398C2×C22⋊C4—941no—
399C2×C22⋊C4—941no—
400C2×C22⋊C4—941no—
401C2×C42—941no—
402#107—4716no—
403#106—4716no—
404#103—4716no—
405#102—4716no—
406C22⋊C8—478no—
407C22⋊C8—478no—
408C24—24no—
409C24—24no—
410C24—24no—
411C24—941no—
412C24C2×C22⋊D4714yes—
413C24C2×C22⋊D4711yescharacteristic
414C24C2×C22⋊D4711yescharacteristic
415C24C2×C22⋊D4711yescharacteristic
416#41—9416no—
417#41—1888no—
418C2×Q8—472no—
419C2×Q8—941no—
420C2×Q8—944no—
421C2×D4—472no—
422C2×D4—944no—
423C2×D4—944no—
424C2×D4—944no—
425C2×D4—941no—
426C2×D4—944no—
427C2×D4—1884no—
428C2×D4—1884no—
429C2×D4—1884no—
430C22×C4—24no—
431C22×C4—24no—
432C22×C4—24no—
433C22×C4—216no—
434C22×C4—24no—
435C22×C4—24no—
436C22×C4—216no—
437C22×C4—28no—
438C22×C4—28no—
439C22×C4—28no—
440C22×C4—24no—
441C22×C4—48no—
442C22×C4—48no—
443C22×C4—432no—
444C22×C4—48no—
445C22×C4—941no—
446C22×C4—941no—
447C22×C4—941no—
448C22×C4—944no—
449C22×C4—1884no—
450C22×C4#1062535018yes—
451C22×C4C2×C22⋊D4714yes—
452C22×C4C2×C22⋊D4714yes—
453C22×C4C2×C22⋊D4712yes—
454C22×C4C2×C22⋊D4712yes—
455C22×C4C2×C22⋊D4714yes—
456C22×C4C2×C22⋊D4711yescharacteristic
457C22×C4C2×C22⋊D4711yescharacteristic
458C22×C4C2×C22⋊D4712yes—
459C22×C4C2×C22⋊D4711yescharacteristic
460C22×C4C23×D4712yes—
461C22×C4C23×D4712yes—
462C22×C4C23×D4712yes—
463C22×C4C23×D4711yescharacteristic
464M16—9416no—
465C2×C8—478no—
466C2×C8—948no—
467C4⋊C4—28no—
468C4⋊C4—28no—
469C4⋊C4—28no—
470C4⋊C4—24no—
471C4⋊C4—472no—
472C4⋊C4—472no—
473C4⋊C4—944no—
474C4⋊C4—944no—
475C4⋊C4—941no—
476C4⋊C4—941no—
477C4⋊C4C23×D4718yes—
478C22⋊C4—944no—
479C22⋊C4—944no—
480C22⋊C4—944no—
481C22⋊C4—944no—
482C22⋊C4—944no—
483C42—944no—
484C23—232no—
485C23—28no—
486C23—22no—
487C23—22no—
488C23—28no—
489C23—22no—
490C23—22no—
491C23—28no—
492C23—22no—
493C23—22no—
494C23—942no—
495C23—1882no—
496C23—1882no—
497C23—1882no—
498C23#1309400714yes—
499C23#1309405014yes—
500C23#1309405514yes—
501C23C22×C22⋊D4712yes—
502C23C22×C22⋊D4712yes—
503C23C22×C22⋊D4712yes—
504C23C22×C22⋊D4711yescharacteristic
505Q8—944no—
506Q8—1882no—
507D4—944no—
508D4—1888no—
509D4—1888no—
510D4—1888no—
511D4—1882no—
512C2×C4—28no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—28no—
516C2×C4—28no—
517C2×C4—24no—
518C2×C4—24no—
519C2×C4—24no—
520C2×C4—24no—
521C2×C4—28no—
522C2×C4—416no—
523C2×C4—416no—
524C2×C4—416no—
525C2×C4—48no—
526C2×C4—942no—
527C2×C4—942no—
528C2×C4—942no—
529C2×C4—1882no—
530C2×C4—1882no—
531C2×C4—1882no—
532C2×C4—1888no—
533C2×C4#1309400714yes—
534C2×C4#1309400718yes—
535C2×C4C22×C22⋊D4712yes—
536C2×C4C22×C22⋊D4712yes—
537C2×C4C22×C22⋊D4712yes—
538C2×C4C22×C22⋊D4711yescharacteristic
539C2×C4C2×#1062535014yes—
540C2×C4C2×#1062535018yes—
541C2×C4C24×D4711yescharacteristic
542C8—948no—
543C22—232no—
544C22—22no—
545C22—22no—
546C22—22no—
547C22—22no—
548C22—22no—
549C22—22no—
550C22—28no—
551C22—28no—
552C22—28no—
553C22—3764no—
554C22—1882no—
555C22—1882no—
556C22—1882no—
557C22C2×#1309400714yes—
558C22C2×#1309400712yes—
559C22C2×#1309405014yes—
560C22C2×#1309405012yes—
561C22C2×#1309405514yes—
562C22C2×#1309405512yes—
563C22C23×C22⋊D4711yescharacteristic
564C4—22no—
565C4—22no—
566C4—48no—
567C4—1882no—
568C4—1882no—
569C4—1882no—
570C4C2×#1309400714yes—
571C4C2×#1309400712yes—
572C4C22×#1062535012yes—
573C2—24no—
574C2—24no—
575C2—24no—
576C2—3762no—
577C2#2025331114yesdirect product factor
578C2C22×#1309400711yescharacteristic
579C2C22×#1309405011yescharacteristic
580C2C22×#1309405511yescharacteristic
subgroupG/H
centerC23C22×C22⋊D47
commutator, cosocleC2×C4×C47C25
FrattiniC2×C4C24×D47
FittingC23×C4⋊C4×C47C2
socleC23×C47C22×D4
prime radicalC2×#15689634C2
hypercenterC23×C4⋊C4D47
hypocenterC47C2×#3092
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C24×C4 ◁ C23×C4⋊C4

Complements

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

NH ≅ G/Ntype
2. #20253311C2direct (N × H)
3. C2×#15689458C2semidirect
4. C2×#15689441C2semidirect
5. C2×#15689453C2semidirect
6. C2×#15689436C2semidirect
7. C2×#15689163C2semidirect
8. C2×#15689122C2semidirect
9. C2×#15689087C2semidirect
10. C23×C4⋊C4×C47C2semidirect
24. #15689458C22semidirect
25. #15689453C22semidirect
26. #15689441C22semidirect
27. #15689436C22semidirect
28. #15689163C22semidirect
29. C2×#13093957C22semidirect
30. C2×#13093955C22semidirect
31. #15689122C22semidirect
32. C2×#13093947C22semidirect
33. C2×#13093944C22semidirect
34. #15689087C22semidirect
46. C22×C4⋊C4×C47C22semidirect
83. #13093957C23semidirect
84. #13093955C23semidirect
85. #13093947C23semidirect
86. #13093944C23semidirect
284. C23×C4⋊C4D47semidirect
296. C2×C47#3092semidirect
323. C22×C4⋊C4C2×D47semidirect
337. C47C2×#3092semidirect
577. C2#20253311direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#156896342
#202533112
C2×#156894582
C2×#156894412
C2×#156894532
C2×#156894362
C2×#156891632
C2×#156891222
C2×#156890872
C23×C4⋊C4×C472
C2×#309247