Finite Group Databasebeta

← C2×#20246057 #26375006

Subgroups — C2×#20246057 #26375006

Order 12032.

3718 conjugacy classes of subgroups.

The subgroup lattice has 580 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#13094007C211yesmaximal, characteristic
2C22×C4×C22⋊D47C211yesmaximal, characteristic
3#20246057C2116yesmaximal, direct product factor
4C2×#15689018C212yesmaximal, complemented
5C2×#15689013C212yesmaximal, complemented
6C2×#15689004C212yesmaximal, complemented
7C2×#15689002C211yesmaximal, characteristic
8C2×#15688996C211yesmaximal, characteristic, complemented
9C2×#15688994C212yesmaximal, complemented
10C2×#15688986C211yesmaximal, characteristic, complemented
11C2×C4×C22⋊Dic47C211yesmaximal, characteristic, complemented
12C23×C4⋊C4×C47C211yesmaximal, characteristic, complemented
13C23×C22⋊D47C2211yescharacteristic
14C23×C4×D47C2211yescharacteristic
15C2×#13094007C4116yescomplemented
16C2×#13094007C2218yes—
17C2×C4×C22⋊D47—28no—
18C2×C4×C22⋊D47C2218yes—
19C22×Dic47⋊C4C2211yescharacteristic
20C22×Dic47⋊C4C2212yes—
21C22×Dic47⋊C4C2212yes—
22C22×C22⋊Dic47C2211yescharacteristic
23C22×C22⋊Dic47C2211yescharacteristic
24C22×C22⋊Dic47C2211yescharacteristic
25C22×(C2×D47)⋊C4C2211yescharacteristic
26C22×(C2×D47)⋊C4C2212yes—
27C22×(C2×D47)⋊C4C2212yes—
28C22×C4×Dic47C2211yescharacteristic
29C22×C4×Dic47C2212yes—
30#15689018C22116yescomplemented
31#15689013C22116yescomplemented
32#15689004C22116yescomplemented
33#15689002C2218yes—
34#15688996C2218yescomplemented
35#15688994C22116yescomplemented
36#15688986C2218yescomplemented
37C4×C22⋊Dic47C2218yescomplemented
38C2×C47⋊#95C2212yes—
39C2×C47⋊#95C2212yes—
40C2×C47⋊#95C2212yes—
41C2×C47⋊#95C2212yes—
42C2×C47⋊#95C2212yes—
43C24×C4×C47C2211yescharacteristic
44C24×C4×C47C2211yescharacteristic
45C24×C4×C47C2211yescharacteristic
46C22×C4⋊C4×C47—28no—
47C22×C4⋊C4×C47C2212yes—
48C22×C4⋊C4×C47C2212yes—
49C22×C4⋊C4×C47C2218yescomplemented
50C24×D47C2311yescharacteristic
51C22×C22⋊D47—24no—
52C22×C22⋊D47—24no—
53C22×C22⋊D47C2×C418yes—
54C22×C22⋊D47C2314yes—
55C23×Dic47C2311yescharacteristic
56C23×Dic47C2312yes—
57C23×Dic47C2311yescharacteristic
58C23×Dic47C2312yes—
59C23×Dic47C2311yescharacteristic
60C22×C4×D47—24no—
61C22×C4×D47—44no—
62C22×C4×D47C2314yes—
63#13094007C2×C4164yescomplemented
64C4×C22⋊D47—232no—
65C2×Dic47⋊C4—24no—
66C2×Dic47⋊C4—44no—
67C2×Dic47⋊C4C2×C4116yescomplemented
68C2×Dic47⋊C4C2314yes—
69C2×Dic47⋊C4C2318yes—
70C2×Dic47⋊C4C2318yes—
71C2×C22⋊Dic47—24no—
72C2×C22⋊Dic47—24no—
73C2×C22⋊Dic47C2×C418yescomplemented
74C2×C22⋊Dic47C2314yes—
75C2×C22⋊Dic47C2314yes—
76C2×C22⋊Dic47C2314yes—
77C2×(C2×D47)⋊C4—24no—
78C2×(C2×D47)⋊C4—44no—
79C2×(C2×D47)⋊C4C2×C4116yescomplemented
80C2×(C2×D47)⋊C4C2314yes—
81C2×(C2×D47)⋊C4C2318yes—
82C2×(C2×D47)⋊C4C2318yes—
83C2×C4×Dic47—24no—
84C2×C4×Dic47—44no—
85C2×C4×Dic47C2314yes—
86C2×C4×Dic47C2318yes—
87C47⋊#95C2318yes—
88C47⋊#95C2318yes—
89C47⋊#95C2318yes—
90C47⋊#95C2318yes—
91C47⋊#95C2318yes—
92C25×C47C2311yescharacteristic
93C23×C4×C47—24no—
94C23×C4×C47—24no—
95C23×C4×C47—24no—
96C23×C4×C47—44no—
97C23×C4×C47—44no—
98C23×C4×C47C2×C418yescomplemented
99C23×C4×C47D418yes—
100C23×C4×C47C2312yes—
101C23×C4×C47C2312yes—
102C23×C4×C47C2314yes—
103C23×C4×C47C2311yescharacteristic
104C23×C4×C47C2311yescharacteristic
105C23×C4×C47C2314yes—
106C23×C4×C47C2314yes—
107C2×C4⋊C4×C47—28no—
108C2×C4⋊C4×C47—28no—
109C2×C4⋊C4×C47—232no—
110C2×C4⋊C4×C47C2318yes—
111C2×C4⋊C4×C47C2318yes—
112C24×C47—24no—
113C24×C47—24no—
114C24×C47C22×C414yes—
115C24×C47C2×D414yes—
116C24×C47#4114yes—
117C24×C47C2412yes—
118C24×C47C2411yescharacteristic
119C23×D47—22no—
120C23×D47—22no—
121C23×D47—24no—
122C23×D47—24no—
123C23×D47C22×C414yes—
124C23×D47C2412yes—
125C22×C4×C47—232no—
126C22×C4×C47—24no—
127C22×C4×C47—28no—
128C22×C4×C47—24no—
129C22×C4×C47—28no—
130C22×C4×C47—24no—
131C22×C4×C47—24no—
132C22×C4×C47—28no—
133C22×C4×C47—24no—
134C22×C4×C47—24no—
135C22×C4×C47—24no—
136C22×C4×C47—416no—
137C22×C4×C47—48no—
138C22×C4×C47—48no—
139C22×C4×C47—416no—
140C22×C4×C47C22×C418yes—
141C22×C4×C47C22×C4116yescomplemented
142C22×C4×C47C2×D414yes—
143C22×C4×C47C2×D414yes—
144C22×C4×C47C2×D4116yes—
145C22×C4×C47C2414yes—
146C22×C4×C47C2414yes—
147C22×C4×C47C2412yes—
148C22×C4×C47C2412yes—
149C2×C22⋊D47—232no—
150C2×C22⋊D47—28no—
151C2×C22⋊D47—28no—
152C2×C22⋊D47C22×C4116yes—
153C22×Dic47—22no—
154C22×Dic47—22no—
155C22×Dic47—24no—
156C22×Dic47—24no—
157C22×Dic47—22no—
158C22×Dic47—22no—
159C22×Dic47—24no—
160C22×Dic47—24no—
161C22×Dic47—24no—
162C22×Dic47—24no—
163C22×Dic47—24no—
164C22×Dic47C22×C418yes—
165C22×Dic47C22×C414yes—
166C22×Dic47C2412yes—
167C22×Dic47C2414yes—
168C22×Dic47C2412yes—
169C22×Dic47C2414yes—
170C22×Dic47C2412yes—
171C2×C4×D47—28no—
172C2×C4×D47—48no—
173C2×C4×D47—416no—
174C4⋊C4×C47—216no—
175C4⋊C4×C47—216no—
176C22⋊Dic47—28no—
177C22⋊Dic47—28no—
178C22⋊Dic47C22×C4116yescomplemented
179(C2×D47)⋊C4—28no—
180(C2×D47)⋊C4—48no—
181(C2×D47)⋊C4C22×C4132yescomplemented
182Dic47⋊C4—28no—
183Dic47⋊C4—48no—
184Dic47⋊C4C22×C4132yescomplemented
185C4×Dic47—28no—
186C4×Dic47—48no—
187C23×C47—216no—
188C23×C47—28no—
189C23×C47—28no—
190C23×C47—216no—
191C23×C47—24no—
192C23×C47—24no—
193C23×C47C4×D4116yes—
194C23×C47C23×C414yes—
195C23×C47C23×C412yes—
196C23×C47C22×D414yes—
197C23×C47C22×D412yes—
198C23×C47C2×#4114yes—
199C23×C47C2×#4112yes—
200C23×C47H(5,𝔽2)14yes—
201C23×C472−1+414yes—
202C23×C47C2511yescharacteristic
203C22×D47—28no—
204C22×D47—216no—
205C22×D47—22no—
206C22×D47—22no—
207C22×D47—24no—
208C22×D47—24no—
209C22×D47—44no—
210C22×D47—44no—
211C22×D47—48no—
212C22×D47C23×C414yes—
213C2×C4×C47—216no—
214C2×C4×C47—232no—
215C2×C4×C47—28no—
216C2×C4×C47—28no—
217C2×C4×C47—24no—
218C2×C4×C47—24no—
219C2×C4×C47—24no—
220C2×C4×C47—24no—
221C2×C4×C47—24no—
222C2×C4×C47—24no—
223C2×C4×C47—24no—
224C2×C4×C47—416no—
225C2×C4×C47—416no—
226C2×C4×C47—48no—
227C2×C4×C47C23×C418yes—
228C2×C4×C47C22×D414yes—
229C2×C4×C47C22×D414yes—
230C22⋊D47—232no—
231C2×Dic47—28no—
232C2×Dic47—216no—
233C2×Dic47—22no—
234C2×Dic47—22no—
235C2×Dic47—24no—
236C2×Dic47—24no—
237C2×Dic47—22no—
238C2×Dic47—22no—
239C2×Dic47—24no—
240C2×Dic47—24no—
241C2×Dic47—24no—
242C2×Dic47—24no—
243C2×Dic47—24no—
244C2×Dic47—44no—
245C2×Dic47—44no—
246C2×Dic47—48no—
247C2×Dic47C23×C414yes—
248C2×Dic47C23×C418yes—
249C4×D47—416no—
250C2×#2322—471nomaximal, Sylow
251C22×C47—232no—
252C22×C47—28no—
253C22×C47—24no—
254C22×C47—24no—
255C22×C47—28no—
256C22×C47C2×C4×D418yes—
257C22×C47C2×C4×D414yes—
258C22×C47#52018yes—
259C22×C47#53414yes—
260C22×C47#53614yes—
261C22×C47#53918yes—
262C22×C47C24×C411yescharacteristic
263C22×C47C23×D411yescharacteristic
264C22×C47C22×#4111yescharacteristic
265C22×C47C2×H(5,𝔽2)12yes—
266C22×C47C2×2−1+412yes—
267C2×D47—24no—
268C2×D47—28no—
269C2×D47—416no—
270C2×D47—42no—
271C2×D47—42no—
272C2×D47—44no—
273C4×C47—28no—
274C4×C47—24no—
275C4×C47—24no—
276C4×C47—44no—
277Dic47—24no—
278Dic47—28no—
279Dic47—42no—
280Dic47—42no—
281Dic47—44no—
282C22×#123—471no—
283C22×C4×D4—471no—
284C23×C4⋊C4D4711yescharacteristic, complemented
285#2322—4716no—
286C2×#389—472no—
287C2×#388—472no—
288C2×#387—472no—
289C2×#386—471no—
290C2×#385—471no—
291C2×#385—472no—
292C2×#381—471no—
293C2×C4×C22⋊C4—471no—
294C2×C47—28no—
295C2×C47#232218yescomplemented
296C2×C47C22×C4×D411yescharacteristic
297C2×C47C2×#52012yes—
298C2×C47C2×#53411yescharacteristic
299C2×C47C2×#53611yescharacteristic
300C2×C47C2×#53912yes—
301D47—44no—
302C23×D4—471no—
303C24×C4—471no—
304C24×C4C2×D4711yescharacteristic
305C24×C4C2×D4711yescharacteristic
306C24×C4C2×D4711yescharacteristic
307C2×#123—4716no—
308C2×#123—478no—
309C2×C4×D4—478no—
310C2×C4×D4—948no—
311C22×C4⋊C4—28no—
312C22×C4⋊C4—472no—
313C22×C4⋊C4—471no—
314C22×C4⋊C4—472no—
315C22×C4⋊C4C2×D4712yes—
316C22×C4⋊C4C2×D4712yes—
317C22×C4⋊C4C2×D4718yescomplemented
318C22×C22⋊C4—471no—
319C22×C22⋊C4—472no—
320C22×C22⋊C4—471no—
321C22×C22⋊C4—471no—
322C22×C22⋊C4—471no—
323C22×C22⋊C4—472no—
324C22×C42—471no—
325C22×C42—472no—
326#389—4716no—
327#388—4716no—
328#387—4716no—
329#386—478no—
330#385—478no—
331#385—4716no—
332#381—478no—
333C4×C22⋊C4—478no—
334C2×#95—472no—
335C2×#95—472no—
336C2×#95—472no—
337C2×#95—472no—
338C2×#95—472no—
339C47C2×#232211yescharacteristic, complemented, Sylow
340C25—471no—
341C25C22×D4711yescharacteristic
342C22×D4—478no—
343C22×D4—474no—
344C22×D4—944no—
345C22×D4—944no—
346C23×C4—24no—
347C23×C4—24no—
348C23×C4—24no—
349C23×C4—44no—
350C23×C4—44no—
351C23×C4—472no—
352C23×C4—471no—
353C23×C4—471no—
354C23×C4—472no—
355C23×C4—471no—
356C23×C4—474no—
357C23×C4—944no—
358C23×C4—1884no—
359C23×C4C4×D4718yescomplemented
360C23×C4C22⋊D4718yes—
361C23×C4C22×D4712yes—
362C23×C4C22×D4712yes—
363C23×C4C22×D4714yes—
364C23×C4C22×D4711yescharacteristic
365C23×C4C22×D4711yescharacteristic
366C23×C4C22×D4714yes—
367C23×C4C22×D4714yes—
368#123—4764no—
369C4×D4—9432no—
370C2×C4⋊C4—28no—
371C2×C4⋊C4—28no—
372C2×C4⋊C4—232no—
373C2×C4⋊C4—4716no—
374C2×C4⋊C4—478no—
375C2×C4⋊C4—474no—
376C2×C4⋊C4—478no—
377C2×C4⋊C4—944no—
378C2×C4⋊C4—1884no—
379C2×C4⋊C4C22×D4718yes—
380C2×C4⋊C4C22×D4718yes—
381C2×C22⋊C4—478no—
382C2×C22⋊C4—474no—
383C2×C22⋊C4—4716no—
384C2×C22⋊C4—478no—
385C2×C22⋊C4—474no—
386C2×C22⋊C4—474no—
387C2×C22⋊C4—474no—
388C2×C22⋊C4—478no—
389C2×C22⋊C4—944no—
390C2×C22⋊C4—944no—
391C2×C22⋊C4—944no—
392C2×C22⋊C4—1884no—
393C2×C42—474no—
394C2×C42—478no—
395C2×C42—944no—
396C2×C42—1884no—
397#95—478no—
398#95—478no—
399#95—478no—
400#95—478no—
401#95—478no—
402C24—24no—
403C24—24no—
404C24—474no—
405C24—472no—
406C24—942no—
407C24—942no—
408C24—944no—
409C24—944no—
410C24C2×C4×D4714yes—
411C24#1062535014yes—
412C24C2×C22⋊D4714yes—
413C24C23×D4712yes—
414C24C23×D4711yescharacteristic
415C2×D4—4716no—
416C2×D4—9432no—
417C2×D4—948no—
418C2×D4—948no—
419C22×C4—232no—
420C22×C4—24no—
421C22×C4—28no—
422C22×C4—24no—
423C22×C4—28no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—28no—
427C22×C4—24no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—416no—
431C22×C4—48no—
432C22×C4—48no—
433C22×C4—416no—
434C22×C4—478no—
435C22×C4—474no—
436C22×C4—474no—
437C22×C4—472no—
438C22×C4—472no—
439C22×C4—474no—
440C22×C4—472no—
441C22×C4—942no—
442C22×C4—942no—
443C22×C4—944no—
444C22×C4—944no—
445C22×C4—942no—
446C22×C4—942no—
447C22×C4—944no—
448C22×C4—944no—
449C22×C4—944no—
450C22×C4—944no—
451C22×C4—944no—
452C22×C4—948no—
453C22×C4—1888no—
454C22×C4—18816no—
455C22×C4C2×C4×D4718yes—
456C22×C4C2×C4×D47116yescomplemented
457C22×C4C2×C22⋊D4714yes—
458C22×C4C2×C22⋊D4714yes—
459C22×C4C2×C22⋊D47116yes—
460C22×C4C23×D4714yes—
461C22×C4C23×D4714yes—
462C22×C4C23×D4712yes—
463C22×C4C23×D4712yes—
464C4⋊C4—216no—
465C4⋊C4—216no—
466C4⋊C4—4732no—
467C4⋊C4—948no—
468C4⋊C4—1888no—
469C22⋊C4—4716no—
470C22⋊C4—4732no—
471C22⋊C4—948no—
472C22⋊C4—948no—
473C22⋊C4—948no—
474C22⋊C4—1888no—
475C42—948no—
476C42—1888no—
477C23—216no—
478C23—28no—
479C23—28no—
480C23—24no—
481C23—24no—
482C23—216no—
483C23—474no—
484C23—948no—
485C23—9416no—
486C23—942no—
487C23—942no—
488C23—944no—
489C23—944no—
490C23—1884no—
491C23—1884no—
492C23—1888no—
493C23C4×C22⋊D47116yes—
494C23C22×C4×D4714yes—
495C23C22×C4×D4712yes—
496C23C22×C22⋊D4714yes—
497C23C22×C22⋊D4712yes—
498C23C2×#1062535014yes—
499C23C2×#1062535012yes—
500C23#1309406914yes—
501C23#1309407214yes—
502C23C24×D4711yescharacteristic
503D4—9432no—
504C2×C4—216no—
505C2×C4—232no—
506C2×C4—28no—
507C2×C4—28no—
508C2×C4—24no—
509C2×C4—24no—
510C2×C4—24no—
511C2×C4—24no—
512C2×C4—24no—
513C2×C4—24no—
514C2×C4—24no—
515C2×C4—416no—
516C2×C4—416no—
517C2×C4—48no—
518C2×C4—478no—
519C2×C4—474no—
520C2×C4—948no—
521C2×C4—9416no—
522C2×C4—942no—
523C2×C4—942no—
524C2×C4—944no—
525C2×C4—944no—
526C2×C4—942no—
527C2×C4—942no—
528C2×C4—944no—
529C2×C4—944no—
530C2×C4—944no—
531C2×C4—944no—
532C2×C4—944no—
533C2×C4—18816no—
534C2×C4—1884no—
535C2×C4—1884no—
536C2×C4—1888no—
537C2×C4C22×C4×D4718yes—
538C2×C4C22×C22⋊D4714yes—
539C2×C4C22×C22⋊D4714yes—
540C22—232no—
541C22—28no—
542C22—24no—
543C22—24no—
544C22—28no—
545C22—944no—
546C22—948no—
547C22—18816no—
548C22—1882no—
549C22—1882no—
550C22—1884no—
551C22C2×C4×C22⋊D4718yes—
552C22C2×C4×C22⋊D4714yes—
553C22#1568960418yes—
554C22#1568968214yes—
555C22#1568969714yes—
556C22#1568972018yes—
557C22C23×C4×D4711yescharacteristic
558C22C23×C22⋊D4711yescharacteristic
559C22C22×#1062535011yescharacteristic
560C22C2×#1309406912yes—
561C22C2×#1309407212yes—
562C4—28no—
563C4—24no—
564C4—24no—
565C4—44no—
566C4—944no—
567C4—948no—
568C4—1882no—
569C4—1882no—
570C4—1884no—
571C2—28no—
572C2—1884no—
573C2#2024605718yesdirect product factor
574C2C22×C4×C22⋊D4711yescharacteristic
575C2C2×#1568960412yes—
576C2C2×#1568968211yescharacteristic
577C2C2×#1568969711yescharacteristic
578C2C2×#1568972012yes—
subgroupG/H
centerC24C23×D47
commutatorC22×C47C24×C4
FrattiniC23C24×D47
FittingC23×C4⋊C4×C47C2
socleC24×C47C24
prime radicalC23×C22⋊D47C22
cosocleC23×C47C25
hypercenterC23×C4⋊C4D47
hypocenterC47C2×#2322
LCSG ▷ C22×C47 ▷ C47
UCSC1 ◁ C24 ◁ C23×C4⋊C4

Complements

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

NH ≅ G/Ntype
3. #20246057C2direct (N × H)
4. C2×#15689018C2semidirect
5. C2×#15689013C2semidirect
6. C2×#15689004C2semidirect
8. C2×#15688996C2semidirect
9. C2×#15688994C2semidirect
10. C2×#15688986C2semidirect
11. C2×C4×C22⋊Dic47C2semidirect
12. C23×C4⋊C4×C47C2semidirect
15. C2×#13094007C4semidirect
30. #15689018C22semidirect
31. #15689013C22semidirect
32. #15689004C22semidirect
34. #15688996C22semidirect
35. #15688994C22semidirect
36. #15688986C22semidirect
37. C4×C22⋊Dic47C22semidirect
49. C22×C4⋊C4×C47C22semidirect
63. #13094007C2×C4semidirect
67. C2×Dic47⋊C4C2×C4semidirect
73. C2×C22⋊Dic47C2×C4semidirect
79. C2×(C2×D47)⋊C4C2×C4semidirect
98. C23×C4×C47C2×C4semidirect
141. C22×C4×C47C22×C4semidirect
178. C22⋊Dic47C22×C4semidirect
181. (C2×D47)⋊C4C22×C4semidirect
184. Dic47⋊C4C22×C4semidirect
284. C23×C4⋊C4D47semidirect
295. C2×C47#2322semidirect
317. C22×C4⋊C4C2×D47semidirect
339. C47C2×#2322semidirect
359. C23×C4C4×D47semidirect
456. C22×C4C2×C4×D47semidirect
573. C2#20246057direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#130940072
C22×C4×C22⋊D472
#202460572
C2×#156890182
C2×#156890132
C2×#156890042
C2×#156890022
C2×#156889962
C2×#156889942
C2×#156889862
C2×C4×C22⋊Dic472
C23×C4⋊C4×C472
C2×#232247