Finite Group Databasebeta

← C2×#20243473 #26216052

Subgroups — C2×#20243473 #26216052

Order 12032.

1878 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×C4⋊C4×D47C211yesmaximal, characteristic
2C22×(C2×D47)⋊C8C211yesmaximal, characteristic
3C2×#15688980C211yesmaximal, characteristic
4#20243473C218yesmaximal, direct product factor
5C2×#15688886C211yesmaximal, characteristic, complemented
6C2×#15688885C211yesmaximal, characteristic, complemented
7C2×#15688885C211yesmaximal, characteristic, complemented
8C2×C47×#340C211yesmaximal, 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×D47C418yescomplemented
18C2×C4⋊C4×D47C2214yes—
19C22×(C2×D47)⋊C4—21no—
20C2×(C2×D47)⋊C8—24no—
21C2×(C2×D47)⋊C8C2214yes—
22#15688980—22no—
23#15688980C2214yes—
24C2×C47⋊#95—21no—
25#15688886C2214yescomplemented
26#15688885C2214yescomplemented
27#15688885C2214yescomplemented
28C23×C8×C47C2211yescharacteristic
29C22×C4⋊C4×C47C2211yescharacteristic
30C22×C4⋊C4×C47C2211yescharacteristic
31C47×#340C2214yescomplemented
32C24×D47D411yescharacteristic
33C23×Dic47—21no—
34C23×Dic47—21no—
35C23×Dic47D411yescharacteristic
36C22×C4×D47—22no—
37C22×C4×D47—22no—
38C22×C4×D47—22no—
39C22×C4×D47—22no—
40C22×C4×D47—28no—
41C22×C4×D47—24no—
42C22×C4×D47—44no—
43C22×C4×D47—44no—
44C22×C4×D47C2×C414yes—
45C22×C4×D47D412yes—
46C22×C4×D47D414yes—
47C22×C4×D47C2312yes—
48C22×C47⋊C8—22no—
49C22×C47⋊C8—22no—
50C22×C47⋊C8—22no—
51C22×C47⋊C8C2312yes—
52C2×C4⋊Dic47—21no—
53C2×C4⋊Dic47—22no—
54C2×C4⋊Dic47—22no—
55C2×C4⋊Dic47—22no—
56C2×C4⋊Dic47—21no—
57C2×C4⋊Dic47C2×C414yes—
58C2×C4⋊Dic47C2312yes—
59C2×C4⋊Dic47C2312yes—
60C2×Dic47⋊C4—28no—
61C2×Dic47⋊C4—24no—
62C4⋊C4×D47—216no—
63C4⋊C4×D47—28no—
64C4⋊C4×D47C2×C4116yescomplemented
65C2×(C2×D47)⋊C4—24no—
66C2×(C2×D47)⋊C4—44no—
67(C2×D47)⋊C8—28no—
68C47⋊#95—24no—
69C23×C4×C47—21no—
70C23×C4×C47—21no—
71C23×C4×C47C2311yescharacteristic
72C22×C8×C47—22no—
73C22×C8×C47—22no—
74C22×C8×C47—22no—
75C22×C8×C47C2312yes—
76C2×C4⋊C4×C47—21no—
77C2×C4⋊C4×C47—22no—
78C2×C4⋊C4×C47—22no—
79C2×C4⋊C4×C47—22no—
80C2×C4⋊C4×C47—21no—
81C2×C4⋊C4×C47C2×C414yes—
82C2×C4⋊C4×C47C2312yes—
83C2×C4⋊C4×C47C2312yes—
84C24×C47C2×D411yescharacteristic
85C23×D47—22no—
86C23×D47—22no—
87C23×D47—22no—
88C23×D47—22no—
89C23×D47C22⋊C414yes—
90C23×D47D814yes—
91C23×D47SD1614yes—
92C23×D47C2×D412yes—
93C22×C4×C47—24no—
94C22×C4×C47—22no—
95C22×C4×C47—22no—
96C22×C4×C47—42no—
97C22×C4×C47—42no—
98C22×C4×C47—42no—
99C22×C4×C47—42no—
100C22×C4×C47—42no—
101C22×C4×C47C22×C412yes—
102C22×C4×C47C2×D411yescharacteristic
103C22×C4×C47C2×D412yes—
104C22×C4×C47#4112yes—
105C22×C4×C47#4112yes—
106C22×C4×C47#4112yes—
107C22×C4×C47#4112yes—
108C22×C4×C47C2411yescharacteristic
109C22×Dic47—22no—
110C22×Dic47—22no—
111C22×Dic47—24no—
112C22×Dic47—22no—
113C22×Dic47—22no—
114C22×Dic47—42no—
115C22×Dic47—42no—
116C22×Dic47—42no—
117C22×Dic47—42no—
118C22×Dic47—42no—
119C22×Dic47C22⋊C414yes—
120C22×Dic47C2×D412yes—
121C2×C4×D47—28no—
122C2×C4×D47—28no—
123C2×C4×D47—28no—
124C2×C4×D47—28no—
125C2×C4×D47—24no—
126C2×C4×D47—24no—
127C2×C4×D47—22no—
128C2×C4×D47—22no—
129C2×C4×D47—22no—
130C2×C4×D47—22no—
131C2×C4×D47—216no—
132C2×C4×D47—432no—
133C2×C4×D47—48no—
134C2×C4×D47—48no—
135C2×C4×D47C22⋊C418yes—
136C2×C4×D47C22×C414yes—
137C2×C4×D47C2×D414yes—
138C2×C8×C47—22no—
139C2×C8×C47—22no—
140C2×C8×C47—22no—
141C2×C8×C47—44no—
142C4⋊C4×C47—24no—
143C4⋊C4×C47—22no—
144C4⋊C4×C47—22no—
145C4⋊C4×C47—22no—
146C4⋊C4×C47—42no—
147C4⋊C4×C47C22×C414yes—
148(C2×D47)⋊C4—48no—
149C4⋊Dic47—24no—
150C4⋊Dic47—22no—
151C4⋊Dic47—22no—
152C4⋊Dic47—22no—
153C4⋊Dic47—42no—
154C4⋊Dic47C22×C414yes—
155Dic47⋊C4—216no—
156C2×C47⋊C8—22no—
157C2×C47⋊C8—22no—
158C2×C47⋊C8—22no—
159C2×C47⋊C8—44no—
160C23×C47C2×C22⋊C412yes—
161C23×C47#12312yes—
162C23×C47#12312yes—
163C23×C47C2×D812yes—
164C23×C47C2×SD1612yes—
165C23×C47#13712yes—
166C23×C47#13712yes—
167C23×C47C22×D411yescharacteristic
168C22×D47—22no—
169C22×D47—22no—
170C22×D47—22no—
171C22×D47—22no—
172C22×D47—28no—
173C22×D47—28no—
174C22×D47—28no—
175C22×D47—28no—
176C22×D47—28no—
177C22×D47—28no—
178C22×D47#102116yes—
179C22×D47C2×C22⋊C414yes—
180C22×D47C2×D814yes—
181C22×D47C2×SD1614yes—
182C2×C4×C47—24no—
183C2×C4×C47—42no—
184C2×C4×C47—42no—
185C2×C4×C47—48no—
186C2×C4×C47—42no—
187C2×C4×C47—42no—
188C2×C4×C47—42no—
189C2×C4×C47—84no—
190C2×C4×C47C2×C22⋊C412yes—
191C2×C4×C47#11714yes—
192C2×C4×C47#11714yes—
193C2×C4×C47#12314yes—
194C2×C4×C47#12312yes—
195C2×C4×C47#12314yes—
196C2×C4×C47#12312yes—
197C2×C4×C47C23×C411yescharacteristic
198C2×C4×C47C22×D411yescharacteristic
199C2×C4×C47C2×#4111yescharacteristic
200C2×C4×C47C2×#4111yescharacteristic
201C2×C4×C47C2×#4111yescharacteristic
202C2×C4×C47C2×#4111yescharacteristic
203C2×Dic47—22no—
204C2×Dic47—22no—
205C2×Dic47—28no—
206C2×Dic47—24no—
207C2×Dic47—42no—
208C2×Dic47—42no—
209C2×Dic47—48no—
210C2×Dic47—42no—
211C2×Dic47—42no—
212C2×Dic47—42no—
213C2×Dic47—84no—
214C2×Dic47C2×C22⋊C414yes—
215C4×D47—24no—
216C4×D47—24no—
217C4×D47—24no—
218C4×D47—24no—
219C4×D47—216no—
220C4×D47—432no—
221C8×C47—42no—
222C47⋊C8—42no—
223C2×#1823—471nomaximal, Sylow
224C22×C47#38114yes—
225C22×C47C2×#10214yes—
226C22×C47#41914yes—
227C22×C47#48014yes—
228C22×C47#48014yes—
229C22×C47#48314yes—
230C22×C47#48314yes—
231C22×C47C22×C22⋊C411yescharacteristic
232C22×C47C2×#12311yescharacteristic
233C22×C47C2×#12311yescharacteristic
234C22×C47C22×D811yescharacteristic
235C22×C47C22×SD1611yescharacteristic
236C22×C47C2×#13711yescharacteristic
237C22×C47C2×#13711yescharacteristic
238C2×D47—24no—
239C2×D47—24no—
240C2×D47—24no—
241C2×D47—24no—
242C2×D47—232no—
243C2×D47—24no—
244C2×D47—24no—
245C2×D47C2×#10218yes—
246C4×C47—44no—
247C4×C47—82no—
248C4×C47#38114yes—
249C4×C47C2×#11711yescharacteristic
250C4×C47C2×#11711yescharacteristic
251C4×C47C2×#12311yescharacteristic
252C4×C47C2×#12311yescharacteristic
253Dic47—24no—
254Dic47—44no—
255Dic47—82no—
256C23×C4⋊C4—471no—
257C22×C22⋊C8—471no—
258C2×#380—471no—
259#1823—478no—
260C2×#340—471no—
261C2×#340—471no—
262C2×#340—471no—
263C2×#340D4711yescharacteristic, complemented
264C2×C47#182318yescomplemented
265C2×C47C2×#38111yescharacteristic
266C2×C47C22×#10211yescharacteristic
267C2×C47C2×#41911yescharacteristic
268C2×C47C2×#48011yescharacteristic
269C2×C47C2×#48011yescharacteristic
270C2×C47C2×#48311yescharacteristic
271C2×C47C2×#48311yescharacteristic
272D47—28no—
273C24×C4—471no—
274C24×C4—941no—
275C23×C8—471no—
276C23×C8C2×D4711yescharacteristic
277C22×C4⋊C4—471no—
278C22×C4⋊C4—478no—
279C22×C4⋊C4—474no—
280C22×C4⋊C4—471no—
281C22×C4⋊C4—942no—
282C22×C4⋊C4—944no—
283C22×C4⋊C4—941no—
284C22×C4⋊C4C2×D4711yescharacteristic
285C22×C4⋊C4C2×D4711yescharacteristic
286C22×C22⋊C4—941no—
287C2×C22⋊C8—474no—
288C2×C22⋊C8—944no—
289#380—474no—
290#380—942no—
291C2×#95—941no—
292#340—474no—
293#340—474no—
294#340—474no—
295#340C2×D4714yescomplemented
296C47C2×#182311yescharacteristic, complemented, Sylow
297C25—471no—
298C23×C4—21no—
299C23×C4—21no—
300C23×C4—471no—
301C23×C4—474no—
302C23×C4—472no—
303C23×C4—474no—
304C23×C4—472no—
305C23×C4—948no—
306C23×C4—944no—
307C23×C4—942no—
308C23×C4—942no—
309C23×C4—942no—
310C23×C4—942no—
311C23×C4—941no—
312C23×C4—941no—
313C23×C4—1884no—
314C23×C4—1884no—
315C23×C4C22×D4711yescharacteristic
316C22×C8—22no—
317C22×C8—22no—
318C22×C8—22no—
319C22×C8—472no—
320C22×C8—942no—
321C22×C8—942no—
322C22×C8—942no—
323C22×C8C22×D4712yes—
324C2×C4⋊C4—21no—
325C2×C4⋊C4—22no—
326C2×C4⋊C4—22no—
327C2×C4⋊C4—22no—
328C2×C4⋊C4—21no—
329C2×C4⋊C4—474no—
330C2×C4⋊C4—472no—
331C2×C4⋊C4—4716no—
332C2×C4⋊C4—472no—
333C2×C4⋊C4—9416no—
334C2×C4⋊C4—948no—
335C2×C4⋊C4—948no—
336C2×C4⋊C4—944no—
337C2×C4⋊C4—941no—
338C2×C4⋊C4—942no—
339C2×C4⋊C4—942no—
340C2×C4⋊C4—942no—
341C2×C4⋊C4—941no—
342C2×C4⋊C4C4×D4714yes—
343C2×C4⋊C4C22×D4712yes—
344C2×C4⋊C4C22×D4712yes—
345C2×C22⋊C4—944no—
346C2×C22⋊C4—1884no—
347C22⋊C8—948no—
348#95—944no—
349C24—474no—
350C24—474no—
351C24—474no—
352C24—472no—
353C24—942no—
354C24—942no—
355C24—942no—
356C24—942no—
357C24D4×D4711yescharacteristic
358C22×C4—24no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—42no—
362C22×C4—42no—
363C22×C4—42no—
364C22×C4—42no—
365C22×C4—42no—
366C22×C4—474no—
367C22×C4—472no—
368C22×C4—478no—
369C22×C4—474no—
370C22×C4—474no—
371C22×C4—9416no—
372C22×C4—942no—
373C22×C4—942no—
374C22×C4—948no—
375C22×C4—948no—
376C22×C4—948no—
377C22×C4—944no—
378C22×C4—948no—
379C22×C4—944no—
380C22×C4—942no—
381C22×C4—942no—
382C22×C4—942no—
383C22×C4—942no—
384C22×C4—944no—
385C22×C4—942no—
386C22×C4—942no—
387C22×C4—18832no—
388C22×C4—1888no—
389C22×C4—1888no—
390C22×C4—1882no—
391C22×C4—1882no—
392C22×C4—1882no—
393C22×C4—1882no—
394C22×C4—1882no—
395C22×C4C2×C4×D4712yes—
396C22×C4#1062535012yes—
397C22×C4#1062535012yes—
398C22×C4D4×D4711yescharacteristic
399C22×C4D4×D4712yes—
400C22×C4#1062535212yes—
401C22×C4#1062535412yes—
402C22×C4C23×D4711yescharacteristic
403C2×C8—22no—
404C2×C8—22no—
405C2×C8—22no—
406C2×C8—44no—
407C2×C8—942no—
408C2×C8—942no—
409C2×C8—942no—
410C2×C8—1884no—
411C4⋊C4—24no—
412C4⋊C4—22no—
413C4⋊C4—22no—
414C4⋊C4—22no—
415C4⋊C4—42no—
416C4⋊C4—474no—
417C4⋊C4—9416no—
418C4⋊C4—944no—
419C4⋊C4—942no—
420C4⋊C4—942no—
421C4⋊C4—942no—
422C4⋊C4—1882no—
423C4⋊C4C2×C4×D4714yes—
424C22⋊C4—1888no—
425C23—474no—
426C23—4716no—
427C23—474no—
428C23—474no—
429C23—948no—
430C23—942no—
431C23—942no—
432C23—948no—
433C23—942no—
434C23—942no—
435C23—948no—
436C23—948no—
437C23—948no—
438C23—948no—
439C23C22⋊C4×D4712yes—
440C23#1309400612yes—
441C23#1309400912yes—
442C23D8×D4712yes—
443C23SD16×D4712yes—
444C23#1309405312yes—
445C23#1309405612yes—
446C23C2×D4×D4711yescharacteristic
447C2×C4—24no—
448C2×C4—48no—
449C2×C4—42no—
450C2×C4—42no—
451C2×C4—42no—
452C2×C4—42no—
453C2×C4—42no—
454C2×C4—84no—
455C2×C4—474no—
456C2×C4—948no—
457C2×C4—942no—
458C2×C4—942no—
459C2×C4—9416no—
460C2×C4—944no—
461C2×C4—944no—
462C2×C4—944no—
463C2×C4—944no—
464C2×C4—944no—
465C2×C4—3764no—
466C2×C4—18832no—
467C2×C4—1882no—
468C2×C4—1882no—
469C2×C4—1882no—
470C2×C4—1888no—
471C2×C4—1882no—
472C2×C4—1882no—
473C2×C4C22⋊C4×D4712yes—
474C2×C4#1309398114yes—
475C2×C4#1309398214yes—
476C2×C4#1309400612yes—
477C2×C4#1309400614yes—
478C2×C4#1309400912yes—
479C2×C4#1309400914yes—
480C2×C4C22×C4×D4711yescharacteristic
481C2×C4C2×D4×D4711yescharacteristic
482C2×C4C2×#1062535211yescharacteristic
483C2×C4C2×#1062535011yescharacteristic
484C2×C4C2×#1062535011yescharacteristic
485C2×C4C2×#1062535411yescharacteristic
486C8—42no—
487C8—1882no—
488C22—478no—
489C22—944no—
490C22—9432no—
491C22—944no—
492C22—944no—
493C22—944no—
494C22—944no—
495C22—944no—
496C22#1568898514yes—
497C22#102×D4714yes—
498C22#1568913014yes—
499C22#1568943514yes—
500C22#1568943814yes—
501C22#1568945214yes—
502C22#1568945514yes—
503C22C2×C22⋊C4×D4711yescharacteristic
504C22C2×#1309400611yescharacteristic
505C22C2×#1309400911yescharacteristic
506C22C2×D8×D4711yescharacteristic
507C22C2×SD16×D4711yescharacteristic
508C22C2×#1309405311yescharacteristic
509C22C2×#1309405611yescharacteristic
510C4—44no—
511C4—82no—
512C4—944no—
513C4—3762no—
514C4—1884no—
515C4#1568898514yes—
516C4C2×#1309398111yescharacteristic
517C4C2×#1309398211yescharacteristic
518C4C2×#1309400611yescharacteristic
519C4C2×#1309400911yescharacteristic
520C2—948no—
521C2#2024347318yesdirect product factor
522C2C2×#1568898511yescharacteristic
523C2C2×#102×D4711yescharacteristic
524C2C2×#1568913011yescharacteristic
525C2C2×#1568943511yescharacteristic
526C2C2×#1568943811yescharacteristic
527C2C2×#1568945211yescharacteristic
528C2C2×#1568945511yescharacteristic
subgroupG/H
centerC24D4×D47
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC2×C47×#340C2
socleC24×C47C2×D4
prime radicalC24×D47D4
cosocleC22×C4×C47C24
hypercenterC2×#340D47
hypocenterC47C2×#1823
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C24 ◁ C23×C4 ◁ C2×#340

Complements

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

NH ≅ G/Ntype
4. #20243473C2direct (N × H)
5. C2×#15688886C2semidirect
6. C2×#15688885C2semidirect
7. C2×#15688885C2semidirect
8. C2×C47×#340C2semidirect
17. C2×C4⋊C4×D47C4semidirect
25. #15688886C22semidirect
26. #15688885C22semidirect
27. #15688885C22semidirect
31. C47×#340C22semidirect
64. C4⋊C4×D47C2×C4semidirect
263. C2×#340D47semidirect
264. C2×C47#1823semidirect
295. #340C2×D47semidirect
296. C47C2×#1823semidirect
521. C2#20243473direct (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×(C2×D47)⋊C82
C2×#156889802
#202434732
C2×#156888862
C2×#156888852
C2×C47×#3402
C2×#182347