Finite Group Databasebeta

← C2×#20243801 #26217897

Subgroups — C2×#20243801 #26217897

Order 12032.

2102 conjugacy classes of subgroups.

The subgroup lattice has 530 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#13094005C211yesmaximal, characteristic
2C22×#13093939C211yesmaximal, characteristic
3C2×#15688984C212yesmaximal, complemented
4#20243801C218yesmaximal, direct product factor
5C2×#15688890C212yesmaximal, complemented
6C2×C47×#518C211yesmaximal, characteristic
7C24×Dic47C2211yescharacteristic
8C2×#13094005—24no—
9C2×#13094005—22no—
10C2×#13094005C418yescomplemented
11C2×#13094005C2214yes—
12C22×Dic47⋊C4—21no—
13C22×C22⋊Dic47—21no—
14C22×C22⋊Dic47C2212yes—
15C22×C22⋊Dic47C2211yescharacteristic
16C2×#13093939—24no—
17C2×#13093939C2214yes—
18#15688984—24no—
19#15688984C2218yescomplemented
20C2×C47⋊#95—24no—
21#15688890C2218yescomplemented
22C23×D4×C47C2211yescharacteristic
23C47×#518—22no—
24C47×#518—24no—
25C47×#518C2214yes—
26C2×C4×D4×C47—24no—
27C2×#117×C47—21no—
28C22×C22⋊C4×C47—21no—
29C22×C22⋊C4×C47C2212yes—
30C23×Dic47—22no—
31C23×Dic47—22no—
32C23×Dic47—21no—
33C23×Dic47—21no—
34C23×Dic47—22no—
35C23×Dic47—21no—
36C23×Dic47—24no—
37C23×Dic47—22no—
38C23×Dic47C2×C414yescomplemented
39C23×Dic47C2312yes—
40#13094005—216no—
41#13094005—28no—
42#13094005C2×C4116yescomplemented
43C2×Dic47⋊C4—28no—
44C2×Dic47⋊C4—24no—
45C2×C22⋊Dic47—28no—
46C2×C22⋊Dic47—24no—
47C2×C22⋊Dic47—21no—
48C2×C22⋊Dic47—22no—
49C2×C22⋊Dic47—24no—
50C2×C22⋊Dic47—24no—
51C2×C22⋊Dic47—22no—
52C2×C22⋊Dic47C2×C414yescomplemented
53C2×C22⋊Dic47C2314yes—
54C2×C22⋊Dic47C2312yes—
55#13093939—28no—
56C47⋊#95—216no—
57C25×C47D411yescharacteristic
58C25×C47C2311yescharacteristic
59C22×D4×C47—22no—
60C22×D4×C47—22no—
61C22×D4×C47—24no—
62C22×D4×C47C2×C414yes—
63C22×D4×C47D414yes—
64C22×D4×C47D412yes—
65C22×D4×C47C2312yes—
66C23×C4×C47—22no—
67C23×C4×C47D411yescharacteristic
68C23×C4×C47D414yes—
69C4×D4×C47—216no—
70C4×D4×C47—216no—
71#117×C47—24no—
72#117×C47—24no—
73C2×C4⋊C4×C47—22no—
74C2×C22⋊C4×C47—22no—
75C2×C22⋊C4×C47—24no—
76C2×C22⋊C4×C47—22no—
77C2×C22⋊C4×C47—24no—
78C2×C22⋊C4×C47—22no—
79C2×C22⋊C4×C47—22no—
80C2×C22⋊C4×C47—22no—
81C2×C22⋊C4×C47D414yes—
82C2×C22⋊C4×C47C2314yes—
83C2×C42×C47—22no—
84C24×C47—21no—
85C24×C47—21no—
86C24×C47—22no—
87C24×C47—44no—
88C24×C47—44no—
89C24×C47C22⋊C414yes—
90C24×C47C22×C412yes—
91C24×C47C2×D411yescharacteristic
92C24×C47C2×D412yes—
93C24×C47C2×D412yes—
94C24×C47C2×D411yescharacteristic
95C24×C47C2×D412yes—
96C24×C47C2×D412yes—
97C24×C47#4114yes—
98C24×C47C2411yescharacteristic
99C2×D4×C47—28no—
100C2×D4×C47—28no—
101C2×D4×C47—24no—
102C2×D4×C47—28no—
103C2×D4×C47—28no—
104C2×D4×C47—24no—
105C2×D4×C47—24no—
106C2×D4×C47—24no—
107C2×D4×C47C22⋊C418yes—
108C2×D4×C47C22×C414yes—
109C2×D4×C47C2×D414yes—
110C22×C4×C47—21no—
111C22×C4×C47—21no—
112C22×C4×C47—22no—
113C22×C4×C47—24no—
114C22×C4×C47—28no—
115C22×C4×C47—24no—
116C22×C4×C47—48no—
117C22×C4×C47—84no—
118C22×C4×C47C22⋊C414yes—
119C22×C4×C47C2×D412yes—
120C22×C4×C47C2×D418yes—
121C22×Dic47—24no—
122C22×Dic47—24no—
123C22×Dic47—24no—
124C22×Dic47—28no—
125C22×Dic47—28no—
126C22×Dic47—24no—
127C22×Dic47—22no—
128C22×Dic47—22no—
129C22×Dic47—22no—
130C22×Dic47—24no—
131C22×Dic47—24no—
132C22×Dic47—44no—
133C22×Dic47—41no—
134C22×Dic47—42no—
135C22×Dic47—41no—
136C22×Dic47—44no—
137C22×Dic47—44no—
138C22×Dic47—42no—
139C22×Dic47—42no—
140C22×Dic47—48no—
141C22×Dic47—44no—
142C22×Dic47C22×C414yescomplemented
143C4⋊C4×C47—28no—
144C22⋊C4×C47—28no—
145C22⋊C4×C47—24no—
146C22⋊C4×C47—24no—
147C22⋊C4×C47—24no—
148C22⋊C4×C47—24no—
149C42×C47—28no—
150C22⋊Dic47—216no—
151C22⋊Dic47—24no—
152C22⋊Dic47—22no—
153C22⋊Dic47—24no—
154C22⋊Dic47—24no—
155C22⋊Dic47—44no—
156C22⋊Dic47C22×C414yescomplemented
157Dic47⋊C4—216no—
158C23×C47—28no—
159C23×C47—24no—
160C23×C47—22no—
161C23×C47—24no—
162C23×C47—22no—
163C23×C47—24no—
164C23×C47—44no—
165C23×C47—44no—
166C23×C47—44no—
167C23×C47—42no—
168C23×C47—48no—
169C23×C47—42no—
170C23×C47—48no—
171C23×C47—44no—
172C23×C47—44no—
173C23×C47—84no—
174C23×C47—82no—
175C23×C47C2×C22⋊C412yes—
176C23×C47C2×C22⋊C414yes—
177C23×C47C2×C22⋊C412yes—
178C23×C47C4×D414yes—
179C23×C47C4×D414yes—
180C23×C47#12011yescharacteristic
181C23×C47#12012yes—
182C23×C47#12011yescharacteristic
183C23×C47#12114yes—
184C23×C47#12114yes—
185C23×C47#12312yes—
186C23×C47#12312yes—
187C23×C47C23×C411yescharacteristic
188C23×C47C22×D411yescharacteristic
189C23×C47C22×D412yes—
190C23×C47C22×D411yescharacteristic
191C23×C47C2×#4112yes—
192D4×C47—28no—
193D4×C47—28no—
194D4×C47—216no—
195C2×C4×C47—22no—
196C2×C4×C47—22no—
197C2×C4×C47—24no—
198C2×C4×C47—24no—
199C2×C4×C47—28no—
200C2×C4×C47—48no—
201C2×C4×C47—48no—
202C2×C4×C47—84no—
203C2×C4×C47—88no—
204C2×C4×C47C2×C22⋊C414yes—
205C2×Dic47—24no—
206C2×Dic47—24no—
207C2×Dic47—28no—
208C2×Dic47—48no—
209C2×Dic47—44no—
210C2×Dic47—42no—
211C2×Dic47—44no—
212C2×Dic47—42no—
213C2×Dic47—44no—
214C2×Dic47—48no—
215C2×Dic47—44no—
216C2×Dic47—48no—
217C2×Dic47—44no—
218C2×Dic47—88no—
219C2×#1890—471nomaximal, Sylow
220C22×C47—24no—
221C22×C47—28no—
222C22×C47—24no—
223C22×C47—28no—
224C22×C47—24no—
225C22×C47—22no—
226C22×C47—24no—
227C22×C47—22no—
228C22×C47—24no—
229C22×C47—42no—
230C22×C47—44no—
231C22×C47—44no—
232C22×C47—42no—
233C22×C47—41no—
234C22×C47—41no—
235C22×C47—42no—
236C22×C47—44no—
237C22×C47—48no—
238C22×C47—44no—
239C22×C47—82no—
240C22×C47—84no—
241C22×C47#38612yes—
242C22×C47#38614yes—
243C22×C47#38612yes—
244C22×C47#45814yes—
245C22×C47C22×C22⋊C411yescharacteristic
246C22×C47C2×C4×D412yes—
247C22×C47C2×#12011yescharacteristic
248C22×C47C2×#12112yes—
249C22×C47C2×#12311yescharacteristic
250C4×C47—24no—
251C4×C47—48no—
252Dic47—44no—
253Dic47—44no—
254Dic47—84no—
255C22×#123—471no—
256C2×#518D4711yescharacteristic
257C22×#99—471no—
258C2×#381—472no—
259#1890—478no—
260C2×#342—472no—
261C2×C47—21no—
262C2×C47—22no—
263C2×C47—21no—
264C2×C47—22no—
265C2×C47—22no—
266C2×C47—24no—
267C2×C47—44no—
268C2×C47#189014yescomplemented
269C2×C47C2×#38611yescharacteristic
270C2×C47C2×#45812yes—
271C23×D4C2×D4711yescharacteristic
272C24×C4—471no—
273C2×#123—478no—
274C2×#123—474no—
275C2×#123—944no—
276C2×#123—942no—
277#518—22no—
278#518—24no—
279#518C2×D4714yes—
280C2×C4×D4—24no—
281C2×#117—21no—
282C22×C4⋊C4—941no—
283C22×C22⋊C4—21no—
284C22×C22⋊C4—471no—
285C22×C22⋊C4—472no—
286C22×C22⋊C4—941no—
287C22×C22⋊C4C2×D4712yes—
288C2×#99—474no—
289C2×#99—944no—
290#381—478no—
291#381—944no—
292C2×#95—944no—
293#342—478no—
294C47C2×#189011yescharacteristic, complemented, Sylow
295C25C22⋊D4711yescharacteristic
296C25C22×D4711yescharacteristic
297C22×D4—24no—
298C22×D4—22no—
299C22×D4—22no—
300C22×D4C4×D4714yes—
301C22×D4D18814yes—
302C22×D4C22⋊D4712yes—
303C22×D4C22×D4712yes—
304C23×C4—22no—
305C23×C4—474no—
306C23×C4—472no—
307C23×C4—942no—
308C23×C4—944no—
309C23×C4—941no—
310C23×C4—941no—
311C23×C4—941no—
312C23×C4—942no—
313C23×C4—942no—
314C23×C4—942no—
315C23×C4C22⋊D4711yescharacteristic
316C23×C4C22⋊D4714yes—
317#123—4716no—
318#123—9416no—
319#123—948no—
320C4×D4—216no—
321C4×D4—216no—
322#117—24no—
323#117—24no—
324C2×C4⋊C4—22no—
325C2×C4⋊C4—948no—
326C2×C4⋊C4—944no—
327C2×C22⋊C4—22no—
328C2×C22⋊C4—24no—
329C2×C22⋊C4—22no—
330C2×C22⋊C4—24no—
331C2×C22⋊C4—22no—
332C2×C22⋊C4—22no—
333C2×C22⋊C4—22no—
334C2×C22⋊C4—474no—
335C2×C22⋊C4—472no—
336C2×C22⋊C4—474no—
337C2×C22⋊C4—948no—
338C2×C22⋊C4—942no—
339C2×C22⋊C4—941no—
340C2×C22⋊C4—944no—
341C2×C22⋊C4—944no—
342C2×C22⋊C4—944no—
343C2×C22⋊C4—942no—
344C2×C22⋊C4C22⋊D4714yes—
345C2×C22⋊C4C22×D4714yes—
346C2×C42—22no—
347#99—948no—
348#95—9416no—
349C24—21no—
350C24—21no—
351C24—22no—
352C24—44no—
353C24—44no—
354C24(C2×D47)⋊C414yes—
355C24C2×C4×D4712yes—
356C24C2×D18812yes—
357C24#1062535014yes—
358C24C2×C22⋊D4711yescharacteristic
359C24C2×C22⋊D4712yes—
360C24C2×C22⋊D4712yes—
361C24C2×C22⋊D4711yescharacteristic
362C24C2×C22⋊D4712yes—
363C24C23×D4711yescharacteristic
364C2×D4—28no—
365C2×D4—28no—
366C2×D4—24no—
367C2×D4—28no—
368C2×D4—28no—
369C2×D4—24no—
370C2×D4—24no—
371C2×D4—24no—
372C2×D4(C2×D47)⋊C418yes—
373C2×D4C2×C4×D4714yes—
374C2×D4C2×D18814yes—
375C22×C4—21no—
376C22×C4—21no—
377C22×C4—22no—
378C22×C4—24no—
379C22×C4—28no—
380C22×C4—24no—
381C22×C4—48no—
382C22×C4—84no—
383C22×C4—474no—
384C22×C4—944no—
385C22×C4—944no—
386C22×C4—944no—
387C22×C4—948no—
388C22×C4—948no—
389C22×C4—944no—
390C22×C4—942no—
391C22×C4—942no—
392C22×C4—942no—
393C22×C4—944no—
394C22×C4—944no—
395C22×C4—1888no—
396C22×C4—1884no—
397C22×C4—1884no—
398C22×C4—1881no—
399C22×C4—1882no—
400C22×C4—1881no—
401C22×C4—1884no—
402C22×C4—1884no—
403C22×C4—1882no—
404C22×C4—1882no—
405C22×C4(C2×D47)⋊C414yes—
406C22×C4C2×C22⋊D4712yes—
407C22×C4C2×C22⋊D4718yes—
408C4⋊C4—28no—
409C4⋊C4—9416no—
410C22⋊C4—28no—
411C22⋊C4—24no—
412C22⋊C4—24no—
413C22⋊C4—24no—
414C22⋊C4—24no—
415C22⋊C4—474no—
416C22⋊C4—944no—
417C22⋊C4—9416no—
418C22⋊C4—942no—
419C22⋊C4—944no—
420C22⋊C4—944no—
421C22⋊C4—1884no—
422C42—28no—
423C23—28no—
424C23—24no—
425C23—24no—
426C23—22no—
427C23—24no—
428C23—22no—
429C23—44no—
430C23—44no—
431C23—48no—
432C23—44no—
433C23—42no—
434C23—42no—
435C23—48no—
436C23—44no—
437C23—44no—
438C23—84no—
439C23—82no—
440C23C2×(C2×D47)⋊C412yes—
441C23C2×(C2×D47)⋊C414yes—
442C23C2×(C2×D47)⋊C412yes—
443C23C4×C22⋊D4714yes—
444C23C4×C22⋊D4714yes—
445C23#1309399311yescharacteristic
446C23#1309399312yes—
447C23#1309399311yescharacteristic
448C23#1309399814yes—
449C23#1309399814yes—
450C23#1309400712yes—
451C23#1309400712yes—
452C23C22×C4×D4711yescharacteristic
453C23C22×C22⋊D4711yescharacteristic
454C23C22×C22⋊D4712yes—
455C23C22×D18811yescharacteristic
456C23C2×#1062535012yes—
457D4—216no—
458D4—28no—
459D4—28no—
460C2×C4—24no—
461C2×C4—24no—
462C2×C4—22no—
463C2×C4—22no—
464C2×C4—28no—
465C2×C4—48no—
466C2×C4—48no—
467C2×C4—84no—
468C2×C4—88no—
469C2×C4—944no—
470C2×C4—944no—
471C2×C4—948no—
472C2×C4—3768no—
473C2×C4—1888no—
474C2×C4—1884no—
475C2×C4—1888no—
476C2×C4—1884no—
477C2×C4—1884no—
478C2×C4—1888no—
479C2×C4—1884no—
480C2×C4—1882no—
481C2×C4—1884no—
482C2×C4—1882no—
483C2×C4C2×(C2×D47)⋊C414yes—
484C22—28no—
485C22—24no—
486C22—24no—
487C22—28no—
488C22—24no—
489C22—24no—
490C22—22no—
491C22—22no—
492C22—24no—
493C22—42no—
494C22—44no—
495C22—44no—
496C22—42no—
497C22—41no—
498C22—41no—
499C22—42no—
500C22—44no—
501C22—48no—
502C22—44no—
503C22—84no—
504C22—82no—
505C22#1568900212yes—
506C22#1568900214yes—
507C22#1568900212yes—
508C22#1568931314yes—
509C22C22×(C2×D47)⋊C411yescharacteristic
510C22C2×C4×C22⋊D4712yes—
511C22C2×#1309399311yescharacteristic
512C22C2×#1309399812yes—
513C22C2×#1309400711yescharacteristic
514C4—24no—
515C4—48no—
516C4—3764no—
517C4—1884no—
518C4—1884no—
519C2—22no—
520C2—21no—
521C2—22no—
522C2—21no—
523C2—24no—
524C2—22no—
525C2—44no—
526C2#2024380114yesdirect product factor
527C2C2×#1568900211yescharacteristic
528C2C2×#1568931312yes—
subgroupG/H
centerC23#13093993
commutatorC23×C47C23×C4
FrattiniC24C23×D47
FittingC2×C47×#518C2
socleC23×C47#120
prime radicalC23×D4×C47C22
cosocleC24×C47C24
hypercenterC2×#518D47
hypocenterC47C2×#1890
LCSG ▷ C23×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C25 ◁ C2×#518

Complements

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

NH ≅ G/Ntype
3. C2×#15688984C2semidirect
4. #20243801C2direct (N × H)
5. C2×#15688890C2semidirect
10. C2×#13094005C4semidirect
19. #15688984C22semidirect
21. #15688890C22semidirect
38. C23×Dic47C2×C4semidirect
42. #13094005C2×C4semidirect
52. C2×C22⋊Dic47C2×C4semidirect
142. C22×Dic47C22×C4semidirect
156. C22⋊Dic47C22×C4semidirect
268. C2×C47#1890semidirect
294. C47C2×#1890semidirect
526. C2#20243801direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#130940052
C22×#130939392
C2×#156889842
#202438012
C2×#156888902
C2×C47×#5182
C2×#189047