Finite Group Databasebeta

← C2×#20243579 #26216987

Subgroups — C2×#20243579 #26216987

Order 12032.

1966 conjugacy classes of subgroups.

The subgroup lattice has 538 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4⋊C4×D47C211yesmaximal, characteristic
2C22×#13093955C211yesmaximal, characteristic, complemented
3C22×(C2×D47)⋊C8C211yesmaximal, characteristic
4C2×#15688980C211yesmaximal, characteristic
5#20243579C218yesmaximal, direct product factor
6C2×#15688886C211yesmaximal, characteristic, complemented
7C2×#15688885C211yesmaximal, characteristic, complemented
8C22×#106×C47C211yesmaximal, characteristic, complemented
9C23×C4×D47—21no—
10C23×C4×D47C2211yescharacteristic
11C23×C47⋊C8C2211yescharacteristic
12C22×C4⋊Dic47C2211yescharacteristic
13C22×C4⋊Dic47C2211yescharacteristic
14C22×Dic47⋊C4—21no—
15C2×C4⋊C4×D47—22no—
16C2×C4⋊C4×D47—24no—
17C2×C4⋊C4×D47—24no—
18C2×C4⋊C4×D47C2214yes—
19C22×(C2×D47)⋊C4—21no—
20C2×#13093955—24no—
21C2×#13093955C2214yescomplemented
22C2×(C2×D47)⋊C8C418yescomplemented
23C2×(C2×D47)⋊C8C2214yes—
24#15688980—22no—
25#15688980C2214yes—
26C2×C47⋊#95—21no—
27#15688886C2214yescomplemented
28#15688885C2214yescomplemented
29C23×C8×C47C2211yescharacteristic
30C22×C4⋊C4×C47C2211yescharacteristic
31C22×C4⋊C4×C47C2211yescharacteristic
32C2×#106×C47—24no—
33C2×#106×C47C2214yescomplemented
34C24×D47D411yescharacteristic
35C23×Dic47—21no—
36C23×Dic47—21no—
37C23×Dic47D411yescharacteristic
38C22×C4×D47—22no—
39C22×C4×D47—22no—
40C22×C4×D47—22no—
41C22×C4×D47—22no—
42C22×C4×D47—28no—
43C22×C4×D47—24no—
44C22×C4×D47—44no—
45C22×C4×D47—44no—
46C22×C4×D47C2×C414yes—
47C22×C4×D47D412yes—
48C22×C4×D47Q814yes—
49C22×C4×D47C2312yes—
50C22×C47⋊C8—22no—
51C22×C47⋊C8—22no—
52C22×C47⋊C8C2×C414yescomplemented
53C22×C47⋊C8C2312yes—
54C2×C4⋊Dic47—21no—
55C2×C4⋊Dic47—22no—
56C2×C4⋊Dic47—22no—
57C2×C4⋊Dic47—22no—
58C2×C4⋊Dic47—21no—
59C2×C4⋊Dic47D414yes—
60C2×C4⋊Dic47C2312yes—
61C2×C4⋊Dic47C2312yes—
62C2×Dic47⋊C4—28no—
63C2×Dic47⋊C4—24no—
64C4⋊C4×D47—216no—
65C4⋊C4×D47—28no—
66C4⋊C4×D47—28no—
67C2×(C2×D47)⋊C4—24no—
68C2×(C2×D47)⋊C4—44no—
69#13093955—28no—
70(C2×D47)⋊C8C2×C4116yescomplemented
71C47⋊#95—24no—
72C23×C4×C47—21no—
73C23×C4×C47—21no—
74C23×C4×C47C2311yescharacteristic
75C22×C8×C47—22no—
76C22×C8×C47—22no—
77C22×C8×C47C2×C414yescomplemented
78C22×C8×C47C2312yes—
79C2×C4⋊C4×C47—21no—
80C2×C4⋊C4×C47—22no—
81C2×C4⋊C4×C47—22no—
82C2×C4⋊C4×C47—22no—
83C2×C4⋊C4×C47—21no—
84C2×C4⋊C4×C47D414yes—
85C2×C4⋊C4×C47C2312yes—
86C2×C4⋊C4×C47C2312yes—
87#106×C47—28no—
88C24×C47C2×D411yescharacteristic
89C23×D47—22no—
90C23×D47—22no—
91C23×D47—22no—
92C23×D47—22no—
93C23×D47C4⋊C414yes—
94C23×D47SD1614yes—
95C23×D47SD1614yes—
96C23×D47C2×D412yes—
97C22×C4×C47—24no—
98C22×C4×C47—22no—
99C22×C4×C47—22no—
100C22×C4×C47—42no—
101C22×C4×C47—42no—
102C22×C4×C47—42no—
103C22×C4×C47—42no—
104C22×C4×C47—42no—
105C22×C4×C47C22×C412yes—
106C22×C4×C47C2×D411yescharacteristic
107C22×C4×C47C2×D412yes—
108C22×C4×C47C2×D412yes—
109C22×C4×C47C2×Q812yes—
110C22×C4×C47#4112yes—
111C22×C4×C47#4112yes—
112C22×C4×C47C2411yescharacteristic
113C22×Dic47—22no—
114C22×Dic47—22no—
115C22×Dic47—24no—
116C22×Dic47—22no—
117C22×Dic47—22no—
118C22×Dic47—42no—
119C22×Dic47—42no—
120C22×Dic47—42no—
121C22×Dic47—42no—
122C22×Dic47—42no—
123C22×Dic47C4⋊C414yes—
124C22×Dic47C2×D412yes—
125C2×C4×D47—28no—
126C2×C4×D47—28no—
127C2×C4×D47—24no—
128C2×C4×D47—28no—
129C2×C4×D47—28no—
130C2×C4×D47—24no—
131C2×C4×D47—22no—
132C2×C4×D47—22no—
133C2×C4×D47—22no—
134C2×C4×D47—22no—
135C2×C4×D47—216no—
136C2×C4×D47—48no—
137C2×C4×D47—432no—
138C2×C4×D47—48no—
139C2×C4×D47C4⋊C418yes—
140C2×C4×D47C22×C414yes—
141C2×C4×D47C2×Q814yes—
142C2×C8×C47—28no—
143C2×C8×C47—22no—
144C2×C8×C47—22no—
145C2×C8×C47C22×C414yescomplemented
146C4⋊C4×C47—24no—
147C4⋊C4×C47—22no—
148C4⋊C4×C47—22no—
149C4⋊C4×C47—22no—
150C4⋊C4×C47—42no—
151C4⋊C4×C47C2×D414yes—
152(C2×D47)⋊C4—48no—
153C4⋊Dic47—24no—
154C4⋊Dic47—22no—
155C4⋊Dic47—22no—
156C4⋊Dic47—22no—
157C4⋊Dic47—42no—
158C4⋊Dic47C2×D414yes—
159Dic47⋊C4—216no—
160C2×C47⋊C8—28no—
161C2×C47⋊C8—22no—
162C2×C47⋊C8—22no—
163C2×C47⋊C8C22×C414yescomplemented
164C23×C47C2×C4⋊C412yes—
165C23×C47#12012yes—
166C23×C47#12312yes—
167C23×C47C2×SD1612yes—
168C23×C47C2×SD1612yes—
169C23×C47AGL(1,ℤ/8)12yes—
170C23×C47#13712yes—
171C23×C47C22×D411yescharacteristic
172C22×D47—22no—
173C22×D47—22no—
174C22×D47—22no—
175C22×D47—22no—
176C22×D47—28no—
177C22×D47—28no—
178C22×D47—28no—
179C22×D47—28no—
180C22×D47—28no—
181C22×D47—28no—
182C22×D47#106116yescomplemented
183C22×D47C2×C4⋊C414yes—
184C22×D47C2×SD1614yes—
185C22×D47C2×SD1614yes—
186C2×C4×C47—24no—
187C2×C4×C47—42no—
188C2×C4×C47—42no—
189C2×C4×C47—48no—
190C2×C4×C47—42no—
191C2×C4×C47—42no—
192C2×C4×C47—42no—
193C2×C4×C47—84no—
194C2×C4×C47C2×C4⋊C412yes—
195C2×C4×C47C4×D414yes—
196C2×C4×C47C4×D414yes—
197C2×C4×C47#12012yes—
198C2×C4×C47C22⋊Q814yes—
199C2×C4×C47C22⋊Q814yes—
200C2×C4×C47#12312yes—
201C2×C4×C47C23×C411yescharacteristic
202C2×C4×C47C22×D411yescharacteristic
203C2×C4×C47C22×D411yescharacteristic
204C2×C4×C47C22×Q811yescharacteristic
205C2×C4×C47C2×#4111yescharacteristic
206C2×C4×C47C2×#4111yescharacteristic
207C2×Dic47—22no—
208C2×Dic47—22no—
209C2×Dic47—28no—
210C2×Dic47—24no—
211C2×Dic47—42no—
212C2×Dic47—42no—
213C2×Dic47—48no—
214C2×Dic47—42no—
215C2×Dic47—42no—
216C2×Dic47—42no—
217C2×Dic47—84no—
218C2×Dic47C2×C4⋊C414yes—
219C4×D47—24no—
220C4×D47—24no—
221C4×D47—24no—
222C4×D47—24no—
223C4×D47—216no—
224C4×D47—432no—
225C8×C47—24no—
226C47⋊C8—24no—
227C2×#1853—471nomaximal, Sylow
228C22×C47#38514yes—
229C22×C47C2×#10614yes—
230C22×C47#42814yes—
231C22×C47#44814yes—
232C22×C47#45014yes—
233C22×C47#48114yes—
234C22×C47#48314yes—
235C22×C47C22×C4⋊C411yescharacteristic
236C22×C47C2×#12011yescharacteristic
237C22×C47C2×#12311yescharacteristic
238C22×C47C22×SD1611yescharacteristic
239C22×C47C22×SD1611yescharacteristic
240C22×C47C2×AGL(1,ℤ/8)11yescharacteristic
241C22×C47C2×#13711yescharacteristic
242C2×D47—24no—
243C2×D47—24no—
244C2×D47—232no—
245C2×D47—24no—
246C2×D47—24no—
247C2×D47—24no—
248C2×D47—24no—
249C2×D47C2×#10618yescomplemented
250C4×C47—44no—
251C4×C47—82no—
252C4×C47#38514yes—
253C4×C47C2×C4×D411yescharacteristic
254C4×C47C2×C4×D411yescharacteristic
255C4×C47C2×C22⋊Q811yescharacteristic
256C4×C47C2×C22⋊Q811yescharacteristic
257Dic47—24no—
258Dic47—44no—
259Dic47—82no—
260C23×C4⋊C4—471no—
261C22×#106—471no—
262C22×#106D4711yescharacteristic, complemented
263C22×C22⋊C8—471no—
264C2×#380—471no—
265#1853—478no—
266C2×#340—471no—
267C2×#340—471no—
268C2×C47#185318yescomplemented
269C2×C47C2×#38511yescharacteristic
270C2×C47C22×#10611yescharacteristic
271C2×C47C2×#42811yescharacteristic
272C2×C47C2×#45011yescharacteristic
273C2×C47C2×#44811yescharacteristic
274C2×C47C2×#48311yescharacteristic
275C2×C47C2×#48111yescharacteristic
276D47—28no—
277C24×C4—471no—
278C24×C4—941no—
279C23×C8—471no—
280C23×C8C2×D4711yescharacteristic
281C22×C4⋊C4—471no—
282C22×C4⋊C4—471no—
283C22×C4⋊C4—474no—
284C22×C4⋊C4—944no—
285C22×C4⋊C4—942no—
286C22×C4⋊C4—941no—
287C22×C4⋊C4—944no—
288C22×C4⋊C4C2×D4711yescharacteristic
289C22×C4⋊C4C2×D4711yescharacteristic
290C22×C22⋊C4—941no—
291C2×#106—24no—
292C2×#106—474no—
293C2×#106—944no—
294C2×#106C2×D4714yescomplemented
295C2×C22⋊C8—478no—
296C2×C22⋊C8—474no—
297#380—474no—
298#380—942no—
299C2×#95—941no—
300#340—474no—
301#340—474no—
302C47C2×#185311yescharacteristic, complemented, Sylow
303C25—471no—
304C23×C4—21no—
305C23×C4—21no—
306C23×C4—471no—
307C23×C4—474no—
308C23×C4—472no—
309C23×C4—474no—
310C23×C4—472no—
311C23×C4—944no—
312C23×C4—948no—
313C23×C4—942no—
314C23×C4—942no—
315C23×C4—942no—
316C23×C4—942no—
317C23×C4—941no—
318C23×C4—941no—
319C23×C4—1884no—
320C23×C4—1884no—
321C23×C4C22×D4711yescharacteristic
322C22×C8—22no—
323C22×C8—22no—
324C22×C8—474no—
325C22×C8—472no—
326C22×C8—942no—
327C22×C8—942no—
328C22×C8C4×D4714yescomplemented
329C22×C8C22×D4712yes—
330C2×C4⋊C4—21no—
331C2×C4⋊C4—22no—
332C2×C4⋊C4—22no—
333C2×C4⋊C4—22no—
334C2×C4⋊C4—21no—
335C2×C4⋊C4—474no—
336C2×C4⋊C4—472no—
337C2×C4⋊C4—472no—
338C2×C4⋊C4—948no—
339C2×C4⋊C4—9416no—
340C2×C4⋊C4—944no—
341C2×C4⋊C4—948no—
342C2×C4⋊C4—941no—
343C2×C4⋊C4—942no—
344C2×C4⋊C4—948no—
345C2×C4⋊C4—942no—
346C2×C4⋊C4—942no—
347C2×C4⋊C4—941no—
348C2×C4⋊C4C22⋊D4714yes—
349C2×C4⋊C4C22×D4712yes—
350C2×C4⋊C4C22×D4712yes—
351C2×C22⋊C4—944no—
352C2×C22⋊C4—1884no—
353#106—28no—
354#106—948no—
355C22⋊C8—4716no—
356#95—944no—
357C24—474no—
358C24—474no—
359C24—474no—
360C24—472no—
361C24—942no—
362C24—942no—
363C24—942no—
364C24—942no—
365C24D4×D4711yescharacteristic
366C22×C4—24no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—42no—
370C22×C4—42no—
371C22×C4—42no—
372C22×C4—42no—
373C22×C4—42no—
374C22×C4—474no—
375C22×C4—472no—
376C22×C4—478no—
377C22×C4—474no—
378C22×C4—474no—
379C22×C4—9416no—
380C22×C4—942no—
381C22×C4—942no—
382C22×C4—948no—
383C22×C4—948no—
384C22×C4—944no—
385C22×C4—948no—
386C22×C4—948no—
387C22×C4—944no—
388C22×C4—942no—
389C22×C4—942no—
390C22×C4—942no—
391C22×C4—942no—
392C22×C4—944no—
393C22×C4—942no—
394C22×C4—942no—
395C22×C4—1888no—
396C22×C4—18832no—
397C22×C4—1888no—
398C22×C4—1882no—
399C22×C4—1882no—
400C22×C4—1882no—
401C22×C4—1882no—
402C22×C4—1882no—
403C22×C4C2×C4×D4712yes—
404C22×C4#1062535012yes—
405C22×C4D4×D4711yescharacteristic
406C22×C4D4×D4712yes—
407C22×C4Q8×D4712yes—
408C22×C4#1062535412yes—
409C22×C4C2×C22⋊D4712yes—
410C22×C4C23×D4711yescharacteristic
411C2×C8—28no—
412C2×C8—22no—
413C2×C8—22no—
414C2×C8—474no—
415C2×C8—948no—
416C2×C8—942no—
417C2×C8—942no—
418C2×C8C2×C4×D4714yescomplemented
419C4⋊C4—24no—
420C4⋊C4—22no—
421C4⋊C4—22no—
422C4⋊C4—22no—
423C4⋊C4—42no—
424C4⋊C4—474no—
425C4⋊C4—9416no—
426C4⋊C4—944no—
427C4⋊C4—942no—
428C4⋊C4—942no—
429C4⋊C4—942no—
430C4⋊C4—1882no—
431C4⋊C4C2×C22⋊D4714yes—
432C22⋊C4—1888no—
433C23—474no—
434C23—474no—
435C23—474no—
436C23—4716no—
437C23—948no—
438C23—948no—
439C23—948no—
440C23—942no—
441C23—942no—
442C23—942no—
443C23—942no—
444C23—948no—
445C23—948no—
446C23—948no—
447C23C4⋊C4×D4712yes—
448C23#1309399212yes—
449C23#1309400912yes—
450C23SD16×D4712yes—
451C23SD16×D4712yes—
452C23#1309405112yes—
453C23#1309405312yes—
454C23C2×D4×D4711yescharacteristic
455C2×C4—24no—
456C2×C4—48no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—84no—
463C2×C4—474no—
464C2×C4—948no—
465C2×C4—942no—
466C2×C4—942no—
467C2×C4—9416no—
468C2×C4—944no—
469C2×C4—944no—
470C2×C4—944no—
471C2×C4—944no—
472C2×C4—944no—
473C2×C4—3764no—
474C2×C4—18832no—
475C2×C4—1882no—
476C2×C4—1882no—
477C2×C4—1882no—
478C2×C4—1888no—
479C2×C4—1882no—
480C2×C4—1882no—
481C2×C4C4⋊C4×D4712yes—
482C2×C4C4×C22⋊D4714yes—
483C2×C4#1309398614yes—
484C2×C4#1309399212yes—
485C2×C4#1309400014yes—
486C2×C4#1309400214yes—
487C2×C4#1309400912yes—
488C2×C4C22×C4×D4711yescharacteristic
489C2×C4C2×D4×D4711yescharacteristic
490C2×C4C22×C22⋊D4711yescharacteristic
491C2×C4C2×Q8×D4711yescharacteristic
492C2×C4C2×#1062535011yescharacteristic
493C2×C4C2×#1062535411yescharacteristic
494C8—24no—
495C8—944no—
496C22—478no—
497C22—944no—
498C22—944no—
499C22—944no—
500C22—944no—
501C22—9432no—
502C22—944no—
503C22—944no—
504C22#1568899414yes—
505C22#106×D4714yes—
506C22#1568916214yes—
507C22#1568926014yes—
508C22#1568927214yes—
509C22#1568944314yes—
510C22#1568945514yes—
511C22C2×C4⋊C4×D4711yescharacteristic
512C22C2×#1309399211yescharacteristic
513C22C2×#1309400911yescharacteristic
514C22C2×SD16×D4711yescharacteristic
515C22C2×SD16×D4711yescharacteristic
516C22C2×#1309405111yescharacteristic
517C22C2×#1309405311yescharacteristic
518C4—44no—
519C4—82no—
520C4—944no—
521C4—3762no—
522C4—1884no—
523C4#1568899414yes—
524C4C2×C4×C22⋊D4711yescharacteristic
525C4C2×#1309398611yescharacteristic
526C4C2×#1309400011yescharacteristic
527C4C2×#1309400211yescharacteristic
528C2—948no—
529C2#2024357918yesdirect product factor
530C2C2×#1568899411yescharacteristic
531C2C2×#106×D4711yescharacteristic
532C2C2×#1568916211yescharacteristic
533C2C2×#1568927211yescharacteristic
534C2C2×#1568926011yescharacteristic
535C2C2×#1568945511yescharacteristic
536C2C2×#1568944311yescharacteristic
subgroupG/H
centerC24D4×D47
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC22×#106×C47C2
socleC24×C47C2×D4
prime radicalC24×D47D4
cosocleC22×C4×C47C24
hypercenterC22×#106D47
hypocenterC47C2×#1853
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C24 ◁ C23×C4 ◁ C22×#106

Complements

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

NH ≅ G/Ntype
2. C22×#13093955C2semidirect
5. #20243579C2direct (N × H)
6. C2×#15688886C2semidirect
7. C2×#15688885C2semidirect
8. C22×#106×C47C2semidirect
21. C2×#13093955C22semidirect
22. C2×(C2×D47)⋊C8C4semidirect
27. #15688886C22semidirect
28. #15688885C22semidirect
33. C2×#106×C47C22semidirect
52. C22×C47⋊C8C2×C4semidirect
70. (C2×D47)⋊C8C2×C4semidirect
77. C22×C8×C47C2×C4semidirect
145. C2×C8×C47C22×C4semidirect
163. C2×C47⋊C8C22×C4semidirect
182. C22×D47#106semidirect
249. C2×D47C2×#106semidirect
262. C22×#106D47semidirect
268. C2×C47#1853semidirect
294. C2×#106C2×D47semidirect
302. C47C2×#1853semidirect
328. C22×C8C4×D47semidirect
418. C2×C8C2×C4×D47semidirect
529. C2#20243579direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C4⋊C4×D472
C22×#130939552
C22×(C2×D47)⋊C82
C2×#156889802
#202435792
C2×#156888862
C2×#156888852
C22×#106×C472
C2×#185347