Finite Group Databasebeta

← C2×#20251987 #27079950

Subgroups — C2×#20251987 #27079950

Order 12032.

2766 conjugacy classes of subgroups.

The subgroup lattice has 552 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#13094032C211yesmaximal, characteristic
2C2×#15689604C211yesmaximal, characteristic
3C2×#15689603C211yesmaximal, characteristic
4#20251987C2116yesmaximal, direct product factor
5C2×#15689165C211yesmaximal, characteristic, complemented
6C2×#15689164C212yesmaximal, complemented
7C2×#15689162C212yesmaximal, complemented
8C2×#15689159C211yesmaximal, characteristic, complemented
9C2×#15689158C212yesmaximal, complemented
10C22×#13093957C211yesmaximal, characteristic, complemented
11C2×#107×D47C212yesmaximal, complemented
12C2×C47×#428C211yesmaximal, characteristic, complemented
13C22×#10625350C2211yescharacteristic
14C2×#13094032C4116yescomplemented
15C2×#13094032C2218yes—
16C22×#10625324C2212yes—
17C23×C47⋊C8C2211yescharacteristic
18C22×C8×D47C2212yes—
19#15689604—24no—
20#15689604C2218yes—
21#15689603—24no—
22#15689603C2218yes—
23C2×C4×Dic94—21no—
24C2×#13093989—21no—
25C2×C4×D188—21no—
26C2×#13093986—21no—
27C2×C4×C22⋊D47—22no—
28C2×(C22⋊D47)⋊C4—22no—
29C2×#13093983C2211yescharacteristic
30C2×#13093982C2212yes—
31C2×#13093982C2212yes—
32C22×C4⋊Dic47C2211yescharacteristic
33C2×C4⋊C4×D47C2212yes—
34C2×C4⋊C4×D47C2212yes—
35#15689165C2218yescomplemented
36#15689164C22116yescomplemented
37#15689162C22116yescomplemented
38#15689159C2218yescomplemented
39#15689158C22116yescomplemented
40C2×#13093958C2212yes—
41C2×#13093957C2211yescharacteristic
42C2×#13093957C2212yes—
43C2×#13093957C2212yes—
44C2×#13093957C2211yescharacteristic
45C2×#13093957C2218yescomplemented
46#107×D47C22116yescomplemented
47C2×#13093956C2212yes—
48C2×#13093955C2211yescharacteristic
49C2×#13093955C2211yescharacteristic
50C22×M16×C47C2211yescharacteristic
51C2×#117×C47C2211yescharacteristic
52C22×C4⋊C4×C47C2211yescharacteristic
53C47×#428C2218yescomplemented
54C2×#107×C47C2212yes—
55C2×#106×C47C2212yes—
56C2×#10625350C2×C418yes—
57C2×#10625350D414yes—
58C2×#10625350Q818yes—
59C2×#10625350C2314yes—
60C22×Dic94D411yescharacteristic
61C22×D188D411yescharacteristic
62C22×C22⋊D47D412yes—
63C23×Dic47—21no—
64C22×C4×D47—22no—
65C22×C4×D47—22no—
66C22×C4×D47C2312yes—
67#13094032C2×C4164yescomplemented
68C2×#10625324C2×C4116yescomplemented
69C2×#10625324C2318yes—
70C22×C47⋊C8C2×C418yescomplemented
71C22×C47⋊C8C2311yescharacteristic
72C22×C47⋊C8C2311yescharacteristic
73C22×C47⋊C8C2314yes—
74C2×C8×D47C2×C4116yescomplemented
75C2×C8×D47C2318yes—
76C4×Dic94—28no—
77#13093989—28no—
78C4×D188—28no—
79#13093986—28no—
80C4×C22⋊D47—216no—
81(C22⋊D47)⋊C4—216no—
82#13093983—22no—
83#13093983C2314yes—
84#13093982—24no—
85#13093982—24no—
86#13093982C2318yes—
87#13093982C2318yes—
88C2×C4⋊Dic47—21no—
89C2×C4⋊Dic47—22no—
90C2×C4⋊Dic47C2311yescharacteristic
91C2×C4⋊Dic47C2312yes—
92C2×C4⋊Dic47C2314yes—
93C2×C4⋊Dic47C2311yescharacteristic
94C2×Dic47⋊C4—22no—
95C2×Dic47⋊C4—22no—
96C4⋊C4×D47—24no—
97C4⋊C4×D47—24no—
98C4⋊C4×D47C2318yes—
99C4⋊C4×D47C2318yes—
100C2×C22⋊Dic47—21no—
101C2×(C2×D47)⋊C4—22no—
102C2×(C2×D47)⋊C4—22no—
103C2×C4×Dic47—21no—
104C2×C4×Dic47—22no—
105C2×C4×Dic47—22no—
106#13093958C2318yes—
107#13093957C2314yes—
108#13093957C2318yes—
109#13093957C2318yes—
110#13093957C2314yes—
111#13093956C2318yes—
112#13093955C2314yes—
113#13093955C2314yes—
114C23×C4×C47—21no—
115C23×C4×C47C2311yescharacteristic
116C2×M16×C47C2×C418yescomplemented
117C2×M16×C47C2314yes—
118C22×C8×C47C2312yes—
119#117×C47—22no—
120#117×C47C2314yes—
121C2×C4⋊C4×C47—21no—
122C2×C4⋊C4×C47—22no—
123C2×C4⋊C4×C47C2312yes—
124C2×C4⋊C4×C47C2314yes—
125C2×C4⋊C4×C47C2311yescharacteristic
126C2×C4⋊C4×C47C2311yescharacteristic
127C2×C22⋊C4×C47—21no—
128C2×C42×C47—21no—
129#107×C47C2318yes—
130#106×C47C2318yes—
131C24×C47C2×D411yescharacteristic
132C23×D47C2×D412yes—
133C22×C4×C47—21no—
134C22×C4×C47—21no—
135C22×C4×C47—22no—
136C22×C4×C47—24no—
137C22×C4×C47—82no—
138C22×C4×C47C22×C414yes—
139C22×C4×C47C2×D411yescharacteristic
140C22×C4×C47C2×D412yes—
141C22×C4×C47C2×Q814yes—
142C22×C4×C47C2411yescharacteristic
143C22×C4×C47C2412yes—
144C2×C22⋊D47C4⋊C4116yes—
145C2×C22⋊D47C2×D418yes—
146C22×Dic47—21no—
147C22×Dic47—22no—
148C22×Dic47—21no—
149C22×Dic47—24no—
150C22×Dic47—82no—
151C22×Dic47C2×D412yes—
152#10625350C4⋊C4132yes—
153#10625350C22×C4116yes—
154#10625350C2×Q8116yes—
155C2×D188C4⋊C418yes—
156C2×D188C2×D414yes—
157C2×C4×D47—28no—
158C2×C4×D47—28no—
159C2×C4×D47—84no—
160C2×C4×D47—84no—
161C2×C4×D47C22×C418yes—
162C2×C4×D47C2×D414yes—
163C2×C4×D47C2×Q818yes—
164C2×C4×D47C2414yes—
165C2×Dic94C4⋊C418yes—
166C2×Dic94C2×D414yes—
167M16×C47C22×C4116yescomplemented
168C2×C8×C47C22×C418yes—
169C2×C8×C47C2414yes—
170C4⋊C4×C47—22no—
171C4⋊C4×C47—24no—
172C4⋊C4×C47—21no—
173C4⋊C4×C47—21no—
174C4⋊C4×C47C2412yes—
175C4⋊C4×C47C2414yes—
176C4⋊C4×C47C2412yes—
177C22⋊C4×C47—24no—
178C42×C47—24no—
179C22⋊Dic47—24no—
180(C2×D47)⋊C4—28no—
181(C2×D47)⋊C4—28no—
182C4⋊Dic47—21no—
183C4⋊Dic47—22no—
184C4⋊Dic47—24no—
185C4⋊Dic47—21no—
186C4⋊Dic47C2412yes—
187C4⋊Dic47C2412yes—
188C4⋊Dic47C2414yes—
189Dic47⋊C4—28no—
190Dic47⋊C4—28no—
191C4×Dic47—24no—
192C4×Dic47—28no—
193C4×Dic47—28no—
194C2×C47⋊C8C22×C414yes—
195C2×C47⋊C8C22×C414yes—
196C2×C47⋊C8C22×C4116yescomplemented
197C2×C47⋊C8C2412yes—
198C2×C47⋊C8C2412yes—
199#10625324C22×C4132yescomplemented
200C8×D47C22×C4132yescomplemented
201C23×C47—22no—
202C23×C47—22no—
203C23×C47C2×C4⋊C414yes—
204C23×C47C22×D412yes—
205C23×C47C22×D411yescharacteristic
206C22×D47—24no—
207C22×D47—24no—
208C22×D47C2×C4⋊C418yes—
209C22×D47C22×D414yes—
210C2×C4×C47—22no—
211C2×C4×C47—24no—
212C2×C4×C47—22no—
213C2×C4×C47—42no—
214C2×C4×C47—44no—
215C2×C4×C47—42no—
216C2×C4×C47—84no—
217C2×C4×C47C2×C4⋊C414yes—
218C2×C4×C47C2×C4⋊C418yes—
219C2×C4×C47C23×C412yes—
220C2×C4×C47C23×C414yes—
221C2×C4×C47C22×D412yes—
222C2×C4×C47C22×D411yescharacteristic
223C2×C4×C47C22×Q812yes—
224C2×C4×C47C22×Q814yes—
225C2×C4×C47C2511yescharacteristic
226C22⋊D47C2×C4⋊C4132yes—
227C2×Dic47—22no—
228C2×Dic47—22no—
229C2×Dic47—24no—
230C2×Dic47—42no—
231C2×Dic47—44no—
232C2×Dic47—42no—
233C2×Dic47—84no—
234C2×Dic47C2×C4⋊C418yes—
235C2×Dic47C22×D414yes—
236D188C2×C4⋊C4116yes—
237C4×D47—88no—
238C4×D47—88no—
239C4×D47C2×C4⋊C4116yes—
240C4×D47C23×C418yes—
241C4×D47C22×Q818yes—
242Dic94C2×C4⋊C4116yes—
243C8×C47C23×C418yes—
244C47⋊C8C23×C414yes—
245C47⋊C8C23×C414yes—
246C2×#2913—471nomaximal, Sylow
247C22×C47—28no—
248C22×C47—22no—
249C22×C47—22no—
250C22×C47C22×C4⋊C414yes—
251C22×C47C22×C4⋊C412yes—
252C22×C47#57612yes—
253C22×C47#57812yes—
254C22×C47C23×D411yescharacteristic
255C2×D47—216no—
256C2×D47—24no—
257C2×D47—24no—
258C2×D47C22×C4⋊C418yes—
259C4×C47—42no—
260C4×C47—44no—
261C4×C47—42no—
262C4×C47C22×C4⋊C414yes—
263C4×C47C22×C4⋊C412yes—
264C4×C47C24×C411yescharacteristic
265C4×C47C23×Q811yescharacteristic
266Dic47—42no—
267Dic47—44no—
268Dic47—42no—
269Dic47C22×C4⋊C418yes—
270C22×#131—471no—
271C2×#520—471no—
272C2×#520—471no—
273#2913—4716no—
274C2×#428—472no—
275C2×#428—472no—
276C2×#428—471no—
277C2×#428D4711yescharacteristic, complemented
278C2×#427—472no—
279C2×#427—471no—
280C22×#107—472no—
281C22×#107—471no—
282C2×C47—24no—
283C2×C47#291314yescomplemented
284C2×C47C23×C4⋊C411yescharacteristic
285C2×C47C2×#57611yescharacteristic
286C2×C47C2×#57811yescharacteristic
287D47—28no—
288C22×#41—471no—
289C2×#131—4716no—
290C2×#131—478no—
291C22×M16—472no—
292C22×M16C2×D4711yescharacteristic
293C23×C8—471no—
294C23×C8—472no—
295#520—478no—
296#520—478no—
297#520—944no—
298#520—944no—
299C2×C4×Q8—941no—
300C2×C4×Q8—941no—
301C2×C4×D4—941no—
302C2×C4×D4—942no—
303C2×C4×D4—941no—
304C2×C4×D4—942no—
305C2×#117—471no—
306C2×#117—472no—
307C2×#117—472no—
308C2×#117C2×D4711yescharacteristic
309C22×C4⋊C4—471no—
310C22×C4⋊C4—472no—
311C22×C4⋊C4—472no—
312C22×C4⋊C4C2×D4711yescharacteristic
313#428—478no—
314#428—4716no—
315#428—4716no—
316#428C2×D4718yescomplemented
317#427—478no—
318#427—4716no—
319C2×#107—472no—
320C2×#107—471no—
321C2×#107—472no—
322C2×#107—472no—
323C2×#107—478no—
324C2×#107—471no—
325C2×#107—4716no—
326C2×#107C2×D4712yes—
327C2×#106—472no—
328C2×#106—471no—
329C2×#106—471no—
330C2×#106C2×D4712yes—
331C47C2×#291311yescharacteristic, complemented, Sylow
332C2×#41—478no—
333C2×#41—474no—
334C2×#41—478no—
335C2×#41—474no—
336C22×Q8—471no—
337C22×D4—472no—
338C22×D4—471no—
339C23×C4—21no—
340C23×C4—472no—
341C23×C4—942no—
342C23×C4—942no—
343C23×C4—941no—
344C23×C4C22×D4711yescharacteristic
345#131—4764no—
346C2×M16—4716no—
347C2×M16—478no—
348C2×M16C4×D4718yescomplemented
349C2×M16C22×D4714yes—
350C22×C8—478no—
351C22×C8—471no—
352C22×C8—471no—
353C22×C8—474no—
354C22×C8—4716no—
355C22×C8—478no—
356C22×C8C22×D4712yes—
357C4×Q8—948no—
358C4×Q8—948no—
359C4×D4—948no—
360C4×D4—9416no—
361C4×D4—948no—
362C4×D4—9416no—
363#117—22no—
364#117—474no—
365#117—478no—
366#117—478no—
367#117—944no—
368#117—944no—
369#117—942no—
370#117C22×D4714yes—
371C2×C4⋊C4—21no—
372C2×C4⋊C4—22no—
373C2×C4⋊C4—472no—
374C2×C4⋊C4—471no—
375C2×C4⋊C4—474no—
376C2×C4⋊C4—471no—
377C2×C4⋊C4—478no—
378C2×C4⋊C4—478no—
379C2×C4⋊C4—944no—
380C2×C4⋊C4—944no—
381C2×C4⋊C4—942no—
382C2×C4⋊C4—942no—
383C2×C4⋊C4—941no—
384C2×C4⋊C4—942no—
385C2×C4⋊C4C22×D4711yescharacteristic
386C2×C4⋊C4C22×D4712yes—
387C2×C4⋊C4C22×D4714yes—
388C2×C4⋊C4C22×D4711yescharacteristic
389C2×C22⋊C4—21no—
390C2×C22⋊C4—942no—
391C2×C22⋊C4—942no—
392C2×C22⋊C4—941no—
393C2×C42—21no—
394C2×C42—942no—
395C2×C42—942no—
396C2×C42—941no—
397#107—478no—
398#107—474no—
399#107—478no—
400#107—478no—
401#107—474no—
402#107C22×D4718yes—
403#106—478no—
404#106—474no—
405#106—474no—
406#106C22×D4718yes—
407C24—472no—
408C24D4×D4711yescharacteristic
409#41—4732no—
410#41—4716no—
411#41—4716no—
412C2×Q8—478no—
413C2×Q8—474no—
414C2×D4—4716no—
415C2×D4—478no—
416C2×D4—478no—
417C2×D4—474no—
418C22×C4—21no—
419C22×C4—21no—
420C22×C4—22no—
421C22×C4—24no—
422C22×C4—82no—
423C22×C4—472no—
424C22×C4—478no—
425C22×C4—474no—
426C22×C4—478no—
427C22×C4—474no—
428C22×C4—948no—
429C22×C4—948no—
430C22×C4—944no—
431C22×C4—941no—
432C22×C4—942no—
433C22×C4—941no—
434C22×C4—3764no—
435C22×C4—3764no—
436C22×C4—3762no—
437C22×C4C2×C4×D4714yes—
438C22×C4D4×D4712yes—
439C22×C4D4×D4711yescharacteristic
440C22×C4Q8×D4714yes—
441C22×C4C23×D4712yes—
442C22×C4C23×D4711yescharacteristic
443M16—4732no—
444M16C2×C4×D47116yescomplemented
445C2×C8—474no—
446C2×C8—474no—
447C2×C8—4716no—
448C2×C8—472no—
449C2×C8—472no—
450C2×C8—4732no—
451C2×C8C2×C4×D4718yes—
452C2×C8C23×D4714yes—
453C4⋊C4—22no—
454C4⋊C4—24no—
455C4⋊C4—21no—
456C4⋊C4—21no—
457C4⋊C4—472no—
458C4⋊C4—472no—
459C4⋊C4—474no—
460C4⋊C4—948no—
461C4⋊C4—948no—
462C4⋊C4—941no—
463C4⋊C4—942no—
464C4⋊C4—944no—
465C4⋊C4—941no—
466C4⋊C4C23×D4712yes—
467C4⋊C4C23×D4712yes—
468C4⋊C4C23×D4714yes—
469C22⋊C4—24no—
470C22⋊C4—948no—
471C22⋊C4—948no—
472C22⋊C4—944no—
473C42—24no—
474C42—948no—
475C42—948no—
476C42—944no—
477C23—22no—
478C23—22no—
479C23—478no—
480C23—474no—
481C23—944no—
482C23—944no—
483C23C4⋊C4×D4714yes—
484C23C2×D4×D4712yes—
485C23C2×D4×D4711yescharacteristic
486Q8—4716no—
487D4—4732no—
488D4—4716no—
489C2×C4—22no—
490C2×C4—22no—
491C2×C4—24no—
492C2×C4—44no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—84no—
496C2×C4—478no—
497C2×C4—474no—
498C2×C4—4716no—
499C2×C4—478no—
500C2×C4—478no—
501C2×C4—942no—
502C2×C4—942no—
503C2×C4—944no—
504C2×C4—3768no—
505C2×C4—3768no—
506C2×C4—3764no—
507C2×C4—1882no—
508C2×C4—1884no—
509C2×C4—1882no—
510C2×C4C4⋊C4×D4714yes—
511C2×C4C4⋊C4×D4718yes—
512C2×C4C22×C4×D4712yes—
513C2×C4C22×C4×D4714yes—
514C2×C4C2×D4×D4712yes—
515C2×C4C2×D4×D4711yescharacteristic
516C2×C4C2×Q8×D4712yes—
517C2×C4C2×Q8×D4714yes—
518C2×C4C24×D4711yescharacteristic
519C8—474no—
520C8—474no—
521C8C22×C4×D4718yes—
522C22—28no—
523C22—22no—
524C22—22no—
525C22—478no—
526C22—9416no—
527C22—944no—
528C22—944no—
529C22C2×C4⋊C4×D4714yes—
530C22C2×C4⋊C4×D4712yes—
531C22#1568992712yes—
532C22#1568993912yes—
533C22C22×D4×D4711yescharacteristic
534C4—44no—
535C4—42no—
536C4—42no—
537C4—478no—
538C4—1882no—
539C4—1884no—
540C4—1882no—
541C4C2×C4⋊C4×D4712yes—
542C4C2×C4⋊C4×D4714yes—
543C4C23×C4×D4711yescharacteristic
544C4C22×Q8×D4711yescharacteristic
545C2—24no—
546C2—948no—
547C2#2025198714yesdirect product factor
548C2C22×C4⋊C4×D4711yescharacteristic
549C2C2×#1568992711yescharacteristic
550C2C2×#1568993911yescharacteristic
subgroupG/H
centerC23C2×D4×D47
commutatorC4×C47C24×C4
FrattiniC2×C4C24×D47
FittingC2×C47×#428C2
socleC23×C47C22×D4
prime radicalC22×#10625350C22
cosocleC2×C4×C47C25
hypercenterC2×#428D47
hypocenterC47C2×#2913
LCSG ▷ C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#428

Complements

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

NH ≅ G/Ntype
4. #20251987C2direct (N × H)
5. C2×#15689165C2semidirect
6. C2×#15689164C2semidirect
7. C2×#15689162C2semidirect
8. C2×#15689159C2semidirect
9. C2×#15689158C2semidirect
10. C22×#13093957C2semidirect
11. C2×#107×D47C2semidirect
12. C2×C47×#428C2semidirect
14. C2×#13094032C4semidirect
35. #15689165C22semidirect
36. #15689164C22semidirect
37. #15689162C22semidirect
38. #15689159C22semidirect
39. #15689158C22semidirect
45. C2×#13093957C22semidirect
46. #107×D47C22semidirect
53. C47×#428C22semidirect
67. #13094032C2×C4semidirect
68. C2×#10625324C2×C4semidirect
70. C22×C47⋊C8C2×C4semidirect
74. C2×C8×D47C2×C4semidirect
116. C2×M16×C47C2×C4semidirect
167. M16×C47C22×C4semidirect
196. C2×C47⋊C8C22×C4semidirect
199. #10625324C22×C4semidirect
200. C8×D47C22×C4semidirect
277. C2×#428D47semidirect
283. C2×C47#2913semidirect
316. #428C2×D47semidirect
331. C47C2×#2913semidirect
348. C2×M16C4×D47semidirect
444. M16C2×C4×D47semidirect
547. C2#20251987direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#130940322
C2×#156896042
C2×#156896032
#202519872
C2×#156891652
C2×#156891642
C2×#156891622
C2×#156891592
C2×#156891582
C22×#130939572
C2×#107×D472
C2×C47×#4282
C2×#291347