Finite Group Databasebeta

← C2×#20254619 #27088411

Subgroups — C2×#20254619 #27088411

Order 12032.

2018 conjugacy classes of subgroups.

The subgroup lattice has 570 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4⋊Q8×D47C211yesmaximal, characteristic, complemented
2C2×#15689658C211yesmaximal, characteristic
3#20254619C2116yesmaximal, direct product factor
4C2×#15689533C211yesmaximal, characteristic, complemented
5C2×#15689485C211yesmaximal, characteristic, complemented
6C2×#15689483C211yesmaximal, characteristic, complemented
7C2×#15689460C212yesmaximal, complemented
8C2×#15689443C212yesmaximal
9C2×#15689407C212yesmaximal, complemented
10C2×#15689402C212yesmaximal, complemented
11C2×#15689178C211yesmaximal, characteristic, complemented
12C2×C47×#501C211yesmaximal, characteristic, complemented
13C22×Q8×D47—21no—
14C2×#13094024C2211yescharacteristic
15C2×#13094023—21no—
16C2×#13094023C2212yes—
17C2×#13094022—21no—
18C4⋊Q8×D47—24no—
19C4⋊Q8×D47C2218yescomplemented
20#15689658—24no—
21#15689658C2218yes—
22C2×#13094019—21no—
23C2×#13094012C2211yescharacteristic
24C2×#13094009—21no—
25C2×#13094004C2211yescharacteristic
26C2×#13093999C2211yescharacteristic
27C2×#13093989C2211yescharacteristic
28C2×#13093986—21no—
29C2×#13093986C2211yescharacteristic
30C2×#13093982—21no—
31C2×C4⋊C4×D47—21no—
32C2×C4⋊C4×D47C2212yes—
33C2×C42×D47C2211yescharacteristic
34#15689533C2218yescomplemented
35#15689485C2218yescomplemented
36#15689483C2218yescomplemented
37#15689460C22116yescomplemented
38#15689443C22116yes—
39#15689407C22116yescomplemented
40#15689402C22116yescomplemented
41#15689178C2218yescomplemented
42C2×#13093957C2212yes—
43C2×#13093955C2212yes—
44C2×Dic47⋊C8C2212yes—
45C2×#13093949C2212yes—
46C2×#13093947C2212yes—
47C2×#13093946C2212yes—
48C2×#13093944C2212yes—
49C2×(C2×D47)⋊C8C2212yes—
50C2×#13093935C2211yescharacteristic
51C2×C4⋊Q8×C47C2211yescharacteristic
52C2×#125×C47C2211yescharacteristic
53C47×#501C2218yescomplemented
54C2×#107×C47C2212yes—
55C2×#106×C47C2212yes—
56C2×#97×C47C2211yescharacteristic
57C22×Dic94—21no—
58C22×Dic94—22no—
59C22×Dic94C2311yescharacteristic
60C2×Q8×D47—28no—
61C2×Q8×D47—48no—
62C22×D188—21no—
63C22×D188C2311yescharacteristic
64C22×C4×D47—22no—
65C22×C4×D47—21no—
66C22×C4×D47D411yescharacteristic
67C22×C4×D47D411yescharacteristic
68C22×C4×D47C2311yescharacteristic
69C22×C47⋊C8C2312yes—
70#13094024—22no—
71#13094024C2314yes—
72#13094023—24no—
73#13094023—28no—
74#13094023C2318yes—
75#13094022—28no—
76#13094019—28no—
77#13094012—22no—
78#13094012C2314yes—
79#13094009—28no—
80#13094004—22no—
81#13094004C2314yes—
82#13093999—22no—
83#13093999C2314yes—
84#13093989—22no—
85#13093989C2314yes—
86#13093986—28no—
87#13093986—22no—
88#13093986C2314yes—
89#13093982—28no—
90C2×C4⋊Dic47—21no—
91C2×C4⋊Dic47—21no—
92C2×C4⋊Dic47C2312yes—
93C2×Dic47⋊C4—22no—
94C2×Dic47⋊C4—22no—
95C2×Dic47⋊C4—21no—
96C4⋊C4×D47—28no—
97C4⋊C4×D47—24no—
98C4⋊C4×D47C2318yes—
99C2×(C2×D47)⋊C4—21no—
100C2×(C2×D47)⋊C4—21no—
101C2×(C2×D47)⋊C4—21no—
102C2×(C2×D47)⋊C4—21no—
103C2×(C2×D47)⋊C4—21no—
104C2×C4×Dic47—21no—
105C2×C4×Dic47D411yescharacteristic
106C2×C4×Dic47D411yescharacteristic
107C2×C4×Dic47C2311yescharacteristic
108C42×D47D414yes—
109C42×D47C2314yes—
110#13093957C2318yes—
111#13093955C2318yes—
112Dic47⋊C8C2318yes—
113#13093949C2318yes—
114#13093947C2318yes—
115#13093946C2318yes—
116#13093944C2318yes—
117(C2×D47)⋊C8C2318yes—
118#13093935C2314yes—
119C22×Q8×C47—21no—
120C22×C8×C47C2312yes—
121C4⋊Q8×C47—22no—
122C4⋊Q8×C47C2314yes—
123#125×C47—22no—
124#125×C47C2314yes—
125C2×C4⋊C4×C47—21no—
126C2×C4⋊C4×C47—21no—
127C2×C4⋊C4×C47—21no—
128C2×C4⋊C4×C47C2311yescharacteristic
129C2×C4⋊C4×C47C2311yescharacteristic
130C2×C4⋊C4×C47C2312yes—
131C2×C42×C47C2311yescharacteristic
132#107×C47C2318yes—
133#106×C47C2318yes—
134#97×C47C2314yes—
135C23×D47—21no—
136C23×D47C2×D411yescharacteristic
137C2×Q8×C47—24no—
138C2×Q8×C47—44no—
139C22×C4×C47—21no—
140C22×C4×C47—21no—
141C22×C4×C47—22no—
142C22×C4×C47C2×D411yescharacteristic
143C22×C4×C47C2×D411yescharacteristic
144C22×C4×C47C2411yescharacteristic
145C22×Dic47—21no—
146C22×Dic47—22no—
147C22×Dic47C2×D411yescharacteristic
148C22×Dic47C2×D411yescharacteristic
149C22×Dic47C2×D411yescharacteristic
150Q8×D47—432no—
151C2×D188—24no—
152C2×D188—24no—
153C2×D188—21no—
154C2×D188—44no—
155C2×D188C2412yes—
156C2×C4×D47—24no—
157C2×C4×D47—24no—
158C2×C4×D47—24no—
159C2×C4×D47—28no—
160C2×C4×D47—24no—
161C2×C4×D47—48no—
162C2×C4×D47—44no—
163C2×C4×D47C2×D412yes—
164C2×C4×D47C2×D414yes—
165C2×C4×D47C2×D414yes—
166C2×C4×D47C2412yes—
167C2×Dic94—24no—
168C2×Dic94—21no—
169C2×Dic94—24no—
170C2×Dic94—28no—
171C2×Dic94—44no—
172C2×Dic94—48no—
173C2×Dic94C2412yes—
174C2×C8×C47—24no—
175C2×C8×C47C2414yes—
176C4⋊C4×C47—24no—
177C4⋊C4×C47—24no—
178C4⋊C4×C47—24no—
179C4⋊C4×C47—21no—
180C4⋊C4×C47—21no—
181C4⋊C4×C47—22no—
182C4⋊C4×C47C2412yes—
183C4⋊C4×C47C2414yes—
184C4⋊C4×C47C2412yes—
185C42×C47C2×D412yes—
186C42×C47C2412yes—
187(C2×D47)⋊C4—24no—
188(C2×D47)⋊C4—24no—
189(C2×D47)⋊C4—24no—
190(C2×D47)⋊C4—24no—
191(C2×D47)⋊C4—24no—
192C4⋊Dic47—24no—
193C4⋊Dic47—22no—
194C4⋊Dic47—24no—
195C4⋊Dic47C2414yes—
196Dic47⋊C4—28no—
197Dic47⋊C4—28no—
198Dic47⋊C4—24no—
199C4×Dic47—24no—
200C4×Dic47C2×D412yes—
201C4×Dic47C2×D414yes—
202C4×Dic47C2×D414yes—
203C4×Dic47C2412yes—
204C2×C47⋊C8—24no—
205C2×C47⋊C8C2414yes—
206C23×C47C22×D411yescharacteristic
207C22×D47—22no—
208C22×D47—22no—
209C22×D47—22no—
210C22×D47—42no—
211C22×D47—42no—
212C22×D47—42no—
213C22×D47#13714yes—
214C22×D47C22×D412yes—
215Q8×C47—48no—
216C2×C4×C47—22no—
217C2×C4×C47—22no—
218C2×C4×C47—22no—
219C2×C4×C47—24no—
220C2×C4×C47—22no—
221C2×C4×C47—44no—
222C2×C4×C47—42no—
223C2×C4×C47—42no—
224C2×C4×C47C22×D412yes—
225C2×C4×C47C22×D412yes—
226C2×C4×C47C22×D411yescharacteristic
227C2×C4×C472−1+414yes—
228C2×C4×C47C2511yescharacteristic
229C2×Dic47—22no—
230C2×Dic47—22no—
231C2×Dic47—22no—
232C2×Dic47—24no—
233C2×Dic47—42no—
234C2×Dic47—44no—
235C2×Dic47AGL(1,ℤ/8)14yes—
236C2×Dic47C22×D412yes—
237C2×Dic47C22×D412yes—
238C2×Dic47C22×D412yes—
239D188—24no—
240D188—48no—
241D188—42no—
242C4×D47—28no—
243C4×D47—24no—
244C4×D47—24no—
245C4×D47—28no—
246C4×D47—416no—
247C4×D47—48no—
248Dic94—24no—
249Dic94—48no—
250Dic94—42no—
251Dic94—416no—
252C8×C47—24no—
253C47⋊C8—24no—
254C2×#3239—471nomaximal, Sylow
255C22×C47#53612yes—
256C22×C47C2×AGL(1,ℤ/8)12yes—
257C22×C47C2×#13712yes—
258C22×C47C23×D411yescharacteristic
259C2×D47—22no—
260C2×D47—28no—
261C2×D47—22no—
262C2×D47—42no—
263C2×D47—42no—
264C2×D47—42no—
265C2×D47—84no—
266C2×D47C2×#13714yes—
267C4×C47—22no—
268C4×C47—22no—
269C4×C47—44no—
270C4×C47—42no—
271C4×C47—42no—
272C4×C47#53612yes—
273C4×C47C2×2−1+412yes—
274Dic47—22no—
275Dic47—22no—
276Dic47—42no—
277Dic47—44no—
278Dic47C2×AGL(1,ℤ/8)14yes—
279C22×C4⋊Q8—471no—
280C2×#529—471no—
281#3239—4716no—
282C2×#501—471no—
283C2×#501D4711yescharacteristic, complemented
284C2×#489—471no—
285C2×#489—471no—
286C2×#484—472no—
287C2×#481—472no—
288C2×#475—472no—
289C2×#474—472no—
290C2×#432—471no—
291C2×C47#323914yescomplemented
292C2×C47C2×#53611yescharacteristic
293C2×C47C22×AGL(1,ℤ/8)11yescharacteristic
294C2×C47C22×#13711yescharacteristic
295D47—24no—
296D47—82no—
297C23×Q8—941no—
298C2×C4⋊Q8—478no—
299C2×C4⋊Q8—472no—
300C2×C4⋊Q8—471no—
301C2×C4⋊Q8—941no—
302C2×C4⋊Q8—944no—
303C2×C4⋊Q8—941no—
304C2×C4⋊Q8C2×D4711yescharacteristic
305#529—478no—
306#529—944no—
307C2×#126—941no—
308C2×#125C2×D4711yescharacteristic
309C2×#124—471no—
310C2×#123—941no—
311C2×C22⋊Q8—471no—
312C2×#121—471no—
313C2×C4×Q8—471no—
314C2×C4×D4—471no—
315C2×C4×D4—941no—
316C2×#117—941no—
317C22×C4⋊C4—472no—
318C22×C4⋊C4—941no—
319C22×C42—471no—
320#501—478no—
321#501C2×D4718yescomplemented
322#489—478no—
323#489—478no—
324#484—4716no—
325#481—4716no—
326#475—4716no—
327#474—4716no—
328#432—478no—
329C2×#107—472no—
330C2×#107C2×D4712yes—
331C2×#106—472no—
332C2×#106C2×D4712yes—
333C2×C4⋊C8—472no—
334C2×#103—472no—
335C2×#103—472no—
336C2×#102—472no—
337C2×#102—472no—
338C2×C22⋊C8—472no—
339C2×#97—471no—
340C2×#97C2×D4711yescharacteristic
341C47C2×#323911yescharacteristic, complemented, Sylow
342C22×Q8—21no—
343C22×Q8—471no—
344C22×Q8—942no—
345C22×Q8—948no—
346C22×Q8—941no—
347C22×Q8—1888no—
348C22×D4—471no—
349C22×D4—941no—
350C23×C4—471no—
351C23×C4—471no—
352C23×C4—471no—
353C23×C4—941no—
354C23×C4—942no—
355C22×C8—472no—
356C22×C8C22×D4712yes—
357C4⋊Q8—22no—
358C4⋊Q8—478no—
359C4⋊Q8—474no—
360C4⋊Q8—948no—
361C4⋊Q8—944no—
362C4⋊Q8—948no—
363C4⋊Q8—942no—
364C4⋊Q8C22×D4714yes—
365#126—948no—
366#125—22no—
367#125C22×D4714yes—
368#124—474no—
369#124—942no—
370#123—948no—
371C22⋊Q8—474no—
372C22⋊Q8—942no—
373#121—474no—
374#121—942no—
375C4×Q8—474no—
376C4×Q8—942no—
377C4×D4—474no—
378C4×D4—942no—
379C4×D4—948no—
380#117—948no—
381C2×C4⋊C4—21no—
382C2×C4⋊C4—21no—
383C2×C4⋊C4—21no—
384C2×C4⋊C4—472no—
385C2×C4⋊C4—478no—
386C2×C4⋊C4—944no—
387C2×C4⋊C4—948no—
388C2×C4⋊C4—942no—
389C2×C4⋊C4—942no—
390C2×C4⋊C4—941no—
391C2×C4⋊C4—941no—
392C2×C4⋊C4—941no—
393C2×C4⋊C4C22×D4712yes—
394C2×C4⋊C4C22×D4711yescharacteristic
395C2×C4⋊C4C22×D4711yescharacteristic
396C2×C22⋊C4—941no—
397C2×C22⋊C4—941no—
398C2×C22⋊C4—941no—
399C2×C22⋊C4—941no—
400C2×C22⋊C4—941no—
401C2×C42—471no—
402C2×C42—471no—
403C2×C42—471no—
404C2×C42—474no—
405C2×C42—474no—
406C2×C42—941no—
407C2×C42C22×D4711yescharacteristic
408#107—478no—
409#107C22×D4718yes—
410#106—478no—
411#106C22×D4718yes—
412C4⋊C8—478no—
413#103—478no—
414#103—478no—
415#102—478no—
416#102—478no—
417C22⋊C8—478no—
418#97—474no—
419#97C22×D4714yes—
420C24—471no—
421C24—941no—
422C2×Q8—24no—
423C2×Q8—44no—
424C2×Q8—472no—
425C2×Q8—948no—
426C2×Q8—941no—
427C2×Q8—944no—
428C2×Q8—944no—
429C2×Q8—1884no—
430C2×Q8—1888no—
431C2×Q8—18832no—
432C2×D4—472no—
433C2×D4—944no—
434C2×D4—941no—
435C2×D4—944no—
436C2×D4—1884no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—22no—
440C22×C4—471no—
441C22×C4—471no—
442C22×C4—471no—
443C22×C4—472no—
444C22×C4—474no—
445C22×C4—474no—
446C22×C4—472no—
447C22×C4—944no—
448C22×C4—944no—
449C22×C4—944no—
450C22×C4—948no—
451C22×C4—944no—
452C22×C4—941no—
453C22×C4—942no—
454C22×C4—1884no—
455C22×C4—1888no—
456C22×C4D4×D4711yescharacteristic
457C22×C4D4×D4711yescharacteristic
458C22×C4C23×D4711yescharacteristic
459C2×C8—24no—
460C2×C8—474no—
461C2×C8—944no—
462C2×C8C23×D4714yes—
463C4⋊C4—24no—
464C4⋊C4—24no—
465C4⋊C4—24no—
466C4⋊C4—22no—
467C4⋊C4—21no—
468C4⋊C4—21no—
469C4⋊C4—474no—
470C4⋊C4—948no—
471C4⋊C4—948no—
472C4⋊C4—944no—
473C4⋊C4—944no—
474C4⋊C4—942no—
475C4⋊C4—944no—
476C4⋊C4C23×D4714yes—
477C4⋊C4C23×D4712yes—
478C4⋊C4C23×D4712yes—
479C22⋊C4—944no—
480C22⋊C4—944no—
481C22⋊C4—944no—
482C22⋊C4—944no—
483C22⋊C4—944no—
484C42—474no—
485C42—472no—
486C42—474no—
487C42—472no—
488C42—944no—
489C42D4×D4712yes—
490C42C23×D4712yes—
491C23—474no—
492C23—472no—
493C23—942no—
494C23—942no—
495C23—942no—
496C23—1882no—
497C23—1882no—
498C23—1882no—
499C23C2×D4×D4711yescharacteristic
500Q8—48no—
501Q8—944no—
502Q8—1888no—
503Q8—18816no—
504Q8—1882no—
505D4—944no—
506D4—1888no—
507D4—1882no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—22no—
512C2×C4—24no—
513C2×C4—42no—
514C2×C4—44no—
515C2×C4—42no—
516C2×C4—474no—
517C2×C4—472no—
518C2×C4—472no—
519C2×C4—472no—
520C2×C4—948no—
521C2×C4—948no—
522C2×C4—942no—
523C2×C4—942no—
524C2×C4—944no—
525C2×C4—944no—
526C2×C4—942no—
527C2×C4—944no—
528C2×C4—1888no—
529C2×C4—18816no—
530C2×C4—1884no—
531C2×C4—1882no—
532C2×C4C2×D4×D4712yes—
533C2×C4C2×D4×D4712yes—
534C2×C4C2×D4×D4711yescharacteristic
535C2×C4#1309407214yes—
536C2×C4C24×D4711yescharacteristic
537C8—24no—
538C8—944no—
539C22—474no—
540C22—942no—
541C22—948no—
542C22—942no—
543C22—3764no—
544C22—1882no—
545C22—1882no—
546C22—1882no—
547C22#1568969812yes—
548C22AGL(1,ℤ/8)×D4712yes—
549C22#137×D4712yes—
550C22C22×D4×D4711yescharacteristic
551C4—22no—
552C4—22no—
553C4—44no—
554C4—42no—
555C4—42no—
556C4—474no—
557C4—942no—
558C4—942no—
559C4—1884no—
560C4—1882no—
561C4#1568969812yes—
562C4C2×#1309407212yes—
563C2—944no—
564C2—3762no—
565C2#2025461914yesdirect product factor
566C2C2×#1568969811yescharacteristic
567C2C2×AGL(1,ℤ/8)×D4711yescharacteristic
568C2C2×#137×D4711yescharacteristic
subgroupG/H
centerC23C2×D4×D47
commutator, cosocleC2×C4×C47C25
FrattiniC2×C4C24×D47
FittingC2×C47×#501C2
socleC23×C47C22×D4
prime radicalC2×#13093999C22
hypercenterC2×#501D47
hypocenterC47C2×#3239
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#501

Complements

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

NH ≅ G/Ntype
1. C2×C4⋊Q8×D47C2semidirect
3. #20254619C2direct (N × H)
4. C2×#15689533C2semidirect
5. C2×#15689485C2semidirect
6. C2×#15689483C2semidirect
7. C2×#15689460C2semidirect
9. C2×#15689407C2semidirect
10. C2×#15689402C2semidirect
11. C2×#15689178C2semidirect
12. C2×C47×#501C2semidirect
19. C4⋊Q8×D47C22semidirect
34. #15689533C22semidirect
35. #15689485C22semidirect
36. #15689483C22semidirect
37. #15689460C22semidirect
39. #15689407C22semidirect
40. #15689402C22semidirect
41. #15689178C22semidirect
53. C47×#501C22semidirect
283. C2×#501D47semidirect
291. C2×C47#3239semidirect
321. #501C2×D47semidirect
341. C47C2×#3239semidirect
565. C2#20254619direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C4⋊Q8×D472
C2×#156896582
#202546192
C2×#156895332
C2×#156894852
C2×#156894832
C2×#156894602
C2×#156894432
C2×#156894072
C2×#156894022
C2×#156891782
C2×C47×#5012
C2×#323947