Finite Group Databasebeta

← C2×#20254622 #27088414

Subgroups — C2×#20254622 #27088414

Order 12032.

2018 conjugacy classes of subgroups.

The subgroup lattice has 586 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4⋊Q8×D47C211yesmaximal, characteristic, complemented
2C2×#15689660C211yesmaximal, characteristic
3#20254622C2116yesmaximal, direct product factor
4C2×#15689530C212yesmaximal, complemented
5C2×#15689484C212yesmaximal, complemented
6C2×#15689457C212yesmaximal, complemented
7C2×#15689443C212yesmaximal
8C2×#15689408C211yesmaximal, characteristic, complemented
9C2×#15689404C211yesmaximal, characteristic, complemented
10C2×#15689400C211yesmaximal, characteristic, complemented
11C2×#15689179C211yesmaximal, characteristic, complemented
12C2×C47×#475C211yesmaximal, characteristic, complemented
13C22×Q8×D47—21no—
14C2×#13094024C2211yescharacteristic
15C2×#13094023—21no—
16C2×#13094023C2212yes—
17C2×#13094022—21no—
18C4⋊Q8×D47—24no—
19C4⋊Q8×D47C2218yescomplemented
20#15689660—24no—
21#15689660C2218yes—
22C2×#13094019—21no—
23C2×#13094015C2211yescharacteristic
24C2×#13094013C2211yescharacteristic
25C2×#13094009—21no—
26C2×#13094000C2211yescharacteristic
27C2×#13093999C2211yescharacteristic
28C2×C4×D188—21no—
29C2×C4×D188C2211yescharacteristic
30C2×#13093981—21no—
31C2×C4⋊C4×D47—21no—
32C2×C4⋊C4×D47C2212yes—
33C2×C42×D47C2211yescharacteristic
34#15689530C22116yescomplemented
35#15689484C22116yescomplemented
36#15689457C22116yescomplemented
37#15689443C22116yes—
38#15689408C2218yescomplemented
39#15689404C2218yescomplemented
40#15689400C2218yescomplemented
41#15689179C2218yescomplemented
42C2×#13093958C2212yes—
43C2×#13093957C2212yes—
44C2×#13093955C2212yes—
45C2×C4⋊(C47⋊C8)C2211yescharacteristic
46C2×#13093948C2212yes—
47C2×#13093946C2212yes—
48C2×#13093944C2212yes—
49C2×(C2×D47)⋊C8C2212yes—
50C2×#13093934C2212yes—
51C2×C4⋊Q8×C47C2211yescharacteristic
52C2×C4×Q8×C47C2211yescharacteristic
53C47×#475C2218yescomplemented
54C2×#106×C47C2212yes—
55C2×C4⋊C8×C47C2211yescharacteristic
56C2×#103×C47C2212yes—
57C22×Dic94—21no—
58C22×Dic94—22no—
59C2×Q8×D47—28no—
60C2×Q8×D47—48no—
61C22×D188—21no—
62C22×D188C2311yescharacteristic
63C22×C4×D47—22no—
64C22×C4×D47—21no—
65C22×C4×D47D411yescharacteristic
66C22×C4×D47D411yescharacteristic
67C22×C4×D47C2311yescharacteristic
68C22×C47⋊C8C2312yes—
69#13094024—22no—
70#13094024C2314yes—
71#13094023—24no—
72#13094023—28no—
73#13094023C2318yes—
74#13094022—28no—
75#13094019—28no—
76#13094015—22no—
77#13094015C2314yes—
78#13094013—22no—
79#13094013C2314yes—
80#13094009—28no—
81#13094000—22no—
82#13094000C2314yes—
83#13093999—22no—
84#13093999C2314yes—
85C4×D188—28no—
86C4×D188—22no—
87C4×D188C2314yes—
88#13093981—28no—
89C2×C4⋊Dic47—21no—
90C2×C4⋊Dic47—21no—
91C2×C4⋊Dic47C2312yes—
92C2×C4⋊Dic47C2311yescharacteristic
93C2×Dic47⋊C4—22no—
94C2×Dic47⋊C4—22no—
95C2×Dic47⋊C4—21no—
96C2×Dic47⋊C4—21no—
97C4⋊C4×D47—28no—
98C4⋊C4×D47—24no—
99C4⋊C4×D47C2318yes—
100C2×(C2×D47)⋊C4—21no—
101C2×(C2×D47)⋊C4—21no—
102C2×(C2×D47)⋊C4—21no—
103C2×(C2×D47)⋊C4—21no—
104C2×(C2×D47)⋊C4—21no—
105C2×C4×Dic47D411yescharacteristic
106C2×C4×Dic47D411yescharacteristic
107C2×C4×Dic47C2311yescharacteristic
108C42×D47D414yes—
109C42×D47C2314yes—
110#13093958C2318yes—
111#13093957C2318yes—
112#13093955C2318yes—
113C4⋊(C47⋊C8)C2314yes—
114#13093948C2318yes—
115#13093946C2318yes—
116#13093944C2318yes—
117(C2×D47)⋊C8C2318yes—
118#13093934C2318yes—
119C22×Q8×C47—21no—
120C22×Q8×C47C2311yescharacteristic
121C22×C8×C47C2312yes—
122C4⋊Q8×C47—22no—
123C4⋊Q8×C47C2314yes—
124C4×Q8×C47—22no—
125C4×Q8×C47C2314yes—
126C2×C4⋊C4×C47—21no—
127C2×C4⋊C4×C47—21no—
128C2×C4⋊C4×C47C2311yescharacteristic
129C2×C4⋊C4×C47C2312yes—
130C2×C42×C47—21no—
131C2×C42×C47C2311yescharacteristic
132#106×C47C2318yes—
133C4⋊C8×C47C2314yes—
134#103×C47C2318yes—
135C23×D47—21no—
136C23×D47C2×D411yescharacteristic
137C2×Q8×C47—21no—
138C2×Q8×C47—24no—
139C2×Q8×C47—22no—
140C2×Q8×C47—22no—
141C2×Q8×C47—44no—
142C2×Q8×C47C2412yes—
143C22×C4×C47—21no—
144C22×C4×C47—21no—
145C22×C4×C47—22no—
146C22×C4×C47C2×D411yescharacteristic
147C22×C4×C47C2×D411yescharacteristic
148C22×C4×C47C2411yescharacteristic
149C22×Dic47—21no—
150C22×Dic47—22no—
151C22×Dic47C2×D411yescharacteristic
152C22×Dic47C2×D411yescharacteristic
153C22×Dic47C2×D411yescharacteristic
154Q8×D47—432no—
155C2×D188—22no—
156C2×D188—22no—
157C2×D188—24no—
158C2×D188—21no—
159C2×D188—44no—
160C2×D188C2412yes—
161C2×C4×D47—24no—
162C2×C4×D47—24no—
163C2×C4×D47—22no—
164C2×C4×D47—22no—
165C2×C4×D47—28no—
166C2×C4×D47—24no—
167C2×C4×D47—48no—
168C2×C4×D47—44no—
169C2×C4×D47C2×D412yes—
170C2×C4×D47C2×D414yes—
171C2×C4×D47C2×D414yes—
172C2×C4×D47C2412yes—
173C2×Dic94—24no—
174C2×Dic94—28no—
175C2×Dic94—48no—
176C2×Dic94—44no—
177C2×C8×C47—24no—
178C2×C8×C47C2414yes—
179C4⋊C4×C47—24no—
180C4⋊C4×C47—24no—
181C4⋊C4×C47—21no—
182C4⋊C4×C47—22no—
183C4⋊C4×C47C2414yes—
184C4⋊C4×C47C2412yes—
185C42×C47—24no—
186C42×C47C2×D412yes—
187C42×C47C2412yes—
188(C2×D47)⋊C4—24no—
189(C2×D47)⋊C4—24no—
190(C2×D47)⋊C4—24no—
191(C2×D47)⋊C4—24no—
192(C2×D47)⋊C4—24no—
193C4⋊Dic47—24no—
194C4⋊Dic47—22no—
195C4⋊Dic47—21no—
196C4⋊Dic47—24no—
197C4⋊Dic47C2414yes—
198C4⋊Dic47C2412yes—
199Dic47⋊C4—28no—
200Dic47⋊C4—28no—
201Dic47⋊C4—24no—
202Dic47⋊C4—24no—
203C4×Dic47C2×D412yes—
204C4×Dic47C2×D414yes—
205C4×Dic47C2×D414yes—
206C4×Dic47C2412yes—
207C2×C47⋊C8—24no—
208C2×C47⋊C8C2414yes—
209C23×C47C22×D411yescharacteristic
210C22×D47—22no—
211C22×D47—22no—
212C22×D47—22no—
213C22×D47—42no—
214C22×D47—42no—
215C22×D47—42no—
216C22×D47#13714yes—
217C22×D47C22×D412yes—
218Q8×C47—22no—
219Q8×C47—22no—
220Q8×C47—42no—
221Q8×C47—48no—
222C2×C4×C47—22no—
223C2×C4×C47—24no—
224C2×C4×C47—22no—
225C2×C4×C47—22no—
226C2×C4×C47—42no—
227C2×C4×C47—44no—
228C2×C4×C47—42no—
229C2×C4×C47AGL(1,ℤ/8)14yes—
230C2×C4×C47C22×D412yes—
231C2×C4×C47C22×D411yescharacteristic
232C2×C4×C47C22×D412yes—
233C2×C4×C472−1+412yes—
234C2×C4×C472−1+412yes—
235C2×C4×C47C2511yescharacteristic
236C2×Dic47—22no—
237C2×Dic47—22no—
238C2×Dic47—22no—
239C2×Dic47—22no—
240C2×Dic47—24no—
241C2×Dic47—44no—
242C2×Dic47—42no—
243C2×Dic47C22×D412yes—
244C2×Dic47C22×D412yes—
245C2×Dic47C22×D412yes—
246D188—22no—
247D188—22no—
248D188—48no—
249D188—42no—
250C4×D47—28no—
251C4×D47—24no—
252C4×D47—28no—
253C4×D47—22no—
254C4×D47—22no—
255C4×D47—416no—
256C4×D47—48no—
257Dic94—416no—
258Dic94—48no—
259C8×C47—24no—
260C47⋊C8—24no—
261C2×#3239—471nomaximal, Sylow
262C22×C47#53612yes—
263C22×C47C2×AGL(1,ℤ/8)12yes—
264C22×C47C2×#13712yes—
265C22×C47C23×D411yescharacteristic
266C2×D47—22no—
267C2×D47—28no—
268C2×D47—22no—
269C2×D47—42no—
270C2×D47—42no—
271C2×D47—42no—
272C2×D47—84no—
273C2×D47C2×#13714yes—
274C4×C47—22no—
275C4×C47—42no—
276C4×C47—44no—
277C4×C47—42no—
278C4×C47#53612yes—
279C4×C47C2×AGL(1,ℤ/8)14yes—
280C4×C47C2×2−1+411yescharacteristic
281C4×C47C2×2−1+411yescharacteristic
282Dic47—22no—
283Dic47—22no—
284Dic47—22no—
285Dic47—44no—
286Dic47—42no—
287C22×C4⋊Q8—471no—
288C2×#529—471no—
289#3239—4716no—
290C2×#501—472no—
291C2×#489—472no—
292C2×#484—472no—
293C2×#481—472no—
294C2×#475—471no—
295C2×#475D4711yescharacteristic, complemented
296C2×#474—471no—
297C2×#474—471no—
298C2×#432—471no—
299C2×C47#323914yescomplemented
300C2×C47C2×#53611yescharacteristic
301C2×C47C22×AGL(1,ℤ/8)11yescharacteristic
302C2×C47C22×#13711yescharacteristic
303D47—24no—
304D47—82no—
305C23×Q8—941no—
306C2×C4⋊Q8—478no—
307C2×C4⋊Q8—472no—
308C2×C4⋊Q8—471no—
309C2×C4⋊Q8—941no—
310C2×C4⋊Q8—944no—
311C2×C4⋊Q8—941no—
312C2×C4⋊Q8C2×D4711yescharacteristic
313#529—478no—
314#529—944no—
315C2×#126—941no—
316C2×#125—471no—
317C2×#124—471no—
318C2×#123—941no—
319C2×C22⋊Q8—471no—
320C2×#121—471no—
321C2×C4×Q8C2×D4711yescharacteristic
322C2×C4×D4—471no—
323C2×C4×D4—941no—
324C2×#117—941no—
325C22×C4⋊C4—472no—
326C22×C4⋊C4—941no—
327C22×C42—471no—
328#501—4716no—
329#489—4716no—
330#484—4716no—
331#481—4716no—
332#475—478no—
333#475C2×D4718yescomplemented
334#474—478no—
335#474—478no—
336#432—478no—
337C2×#107—472no—
338C2×#107—472no—
339C2×#106—472no—
340C2×#106C2×D4712yes—
341C2×C4⋊C8—471no—
342C2×C4⋊C8C2×D4711yescharacteristic
343C2×#103—472no—
344C2×#103C2×D4712yes—
345C2×#102—472no—
346C2×#102—472no—
347C2×C22⋊C8—472no—
348C2×#97—472no—
349C47C2×#323911yescharacteristic, complemented, Sylow
350C22×Q8—21no—
351C22×Q8—942no—
352C22×Q8—948no—
353C22×Q8—941no—
354C22×Q8—1888no—
355C22×Q8C22×D4711yescharacteristic
356C22×D4—471no—
357C22×D4—941no—
358C23×C4—471no—
359C23×C4—471no—
360C23×C4—471no—
361C23×C4—942no—
362C23×C4—941no—
363C22×C8—472no—
364C22×C8C22×D4712yes—
365C4⋊Q8—22no—
366C4⋊Q8—478no—
367C4⋊Q8—474no—
368C4⋊Q8—944no—
369C4⋊Q8—948no—
370C4⋊Q8—948no—
371C4⋊Q8—942no—
372C4⋊Q8C22×D4714yes—
373#126—948no—
374#125—474no—
375#125—942no—
376#124—474no—
377#124—942no—
378#123—948no—
379C22⋊Q8—474no—
380C22⋊Q8—942no—
381#121—474no—
382#121—942no—
383C4×Q8—22no—
384C4×Q8C22×D4714yes—
385C4×D4—474no—
386C4×D4—948no—
387C4×D4—942no—
388#117—948no—
389C2×C4⋊C4—21no—
390C2×C4⋊C4—21no—
391C2×C4⋊C4—472no—
392C2×C4⋊C4—471no—
393C2×C4⋊C4—478no—
394C2×C4⋊C4—944no—
395C2×C4⋊C4—948no—
396C2×C4⋊C4—942no—
397C2×C4⋊C4—942no—
398C2×C4⋊C4—941no—
399C2×C4⋊C4—941no—
400C2×C4⋊C4—941no—
401C2×C4⋊C4—941no—
402C2×C4⋊C4C22×D4712yes—
403C2×C4⋊C4C22×D4711yescharacteristic
404C2×C22⋊C4—941no—
405C2×C22⋊C4—941no—
406C2×C22⋊C4—941no—
407C2×C22⋊C4—941no—
408C2×C22⋊C4—941no—
409C2×C42—21no—
410C2×C42—471no—
411C2×C42—471no—
412C2×C42—471no—
413C2×C42—474no—
414C2×C42—474no—
415C2×C42C22×D4711yescharacteristic
416#107—478no—
417#107—478no—
418#106—478no—
419#106C22×D4718yes—
420C4⋊C8—474no—
421C4⋊C8C22×D4714yes—
422#103—478no—
423#103C22×D4718yes—
424#102—478no—
425#102—478no—
426C22⋊C8—478no—
427#97—478no—
428C24—471no—
429C24—941no—
430C2×Q8—21no—
431C2×Q8—24no—
432C2×Q8—22no—
433C2×Q8—22no—
434C2×Q8—44no—
435C2×Q8—948no—
436C2×Q8—944no—
437C2×Q8—1888no—
438C2×Q8—18832no—
439C2×Q8—1884no—
440C2×Q8C23×D4712yes—
441C2×D4—472no—
442C2×D4—944no—
443C2×D4—941no—
444C2×D4—942no—
445C2×D4—942no—
446C2×D4—1884no—
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—22no—
450C22×C4—471no—
451C22×C4—471no—
452C22×C4—471no—
453C22×C4—472no—
454C22×C4—474no—
455C22×C4—474no—
456C22×C4—472no—
457C22×C4—944no—
458C22×C4—948no—
459C22×C4—944no—
460C22×C4—942no—
461C22×C4—942no—
462C22×C4—941no—
463C22×C4—942no—
464C22×C4—944no—
465C22×C4—1888no—
466C22×C4—1884no—
467C22×C4D4×D4711yescharacteristic
468C22×C4D4×D4711yescharacteristic
469C22×C4C23×D4711yescharacteristic
470C2×C8—24no—
471C2×C8—474no—
472C2×C8—944no—
473C2×C8C23×D4714yes—
474C4⋊C4—24no—
475C4⋊C4—24no—
476C4⋊C4—22no—
477C4⋊C4—21no—
478C4⋊C4—474no—
479C4⋊C4—472no—
480C4⋊C4—948no—
481C4⋊C4—948no—
482C4⋊C4—944no—
483C4⋊C4—944no—
484C4⋊C4—944no—
485C4⋊C4—942no—
486C4⋊C4—941no—
487C4⋊C4—944no—
488C4⋊C4C23×D4714yes—
489C4⋊C4C23×D4712yes—
490C22⋊C4—944no—
491C22⋊C4—944no—
492C22⋊C4—944no—
493C22⋊C4—944no—
494C22⋊C4—944no—
495C42—24no—
496C42—474no—
497C42—472no—
498C42—474no—
499C42—472no—
500C42D4×D4712yes—
501C42C23×D4712yes—
502C23—474no—
503C23—472no—
504C23—942no—
505C23—942no—
506C23—942no—
507C23—1882no—
508C23—1882no—
509C23—1882no—
510C23C2×D4×D4711yescharacteristic
511Q8—22no—
512Q8—22no—
513Q8—48no—
514Q8—42no—
515Q8—18816no—
516Q8—1888no—
517D4—942no—
518D4—942no—
519D4—1888no—
520D4—1882no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—24no—
525C2×C4—44no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—472no—
529C2×C4—472no—
530C2×C4—472no—
531C2×C4—948no—
532C2×C4—942no—
533C2×C4—942no—
534C2×C4—942no—
535C2×C4—948no—
536C2×C4—944no—
537C2×C4—942no—
538C2×C4—942no—
539C2×C4—942no—
540C2×C4—944no—
541C2×C4—18816no—
542C2×C4—1882no—
543C2×C4—1884no—
544C2×C4—1888no—
545C2×C4#1309405114yes—
546C2×C4C2×D4×D4712yes—
547C2×C4C2×D4×D4712yes—
548C2×C4C2×D4×D4711yescharacteristic
549C2×C4#1309407112yes—
550C2×C4#1309407212yes—
551C2×C4C24×D4711yescharacteristic
552C8—24no—
553C8—944no—
554C22—474no—
555C22—942no—
556C22—948no—
557C22—942no—
558C22—3764no—
559C22—1882no—
560C22—1882no—
561C22—1882no—
562C22#1568969612yes—
563C22C2×#1309405112yes—
564C22#137×D4712yes—
565C22C22×D4×D4711yescharacteristic
566C4—22no—
567C4—44no—
568C4—42no—
569C4—42no—
570C4—942no—
571C4—942no—
572C4—942no—
573C4—1884no—
574C4—1882no—
575C4#1568969612yes—
576C4C2×#1309405114yes—
577C4C2×#1309407111yescharacteristic
578C4C2×#1309407211yescharacteristic
579C2—944no—
580C2—3762no—
581C2#2025462214yesdirect product factor
582C2C2×#1568969611yescharacteristic
583C2C22×#1309405111yescharacteristic
584C2C2×#137×D4711yescharacteristic
subgroupG/H
centerC23C2×D4×D47
commutator, cosocleC2×C4×C47C25
FrattiniC2×C4C24×D47
FittingC2×C47×#475C2
socleC23×C47C22×D4
prime radicalC2×#13093999C22
hypercenterC2×#475D47
hypocenterC47C2×#3239
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#475

Complements

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

NH ≅ G/Ntype
1. C2×C4⋊Q8×D47C2semidirect
3. #20254622C2direct (N × H)
4. C2×#15689530C2semidirect
5. C2×#15689484C2semidirect
6. C2×#15689457C2semidirect
8. C2×#15689408C2semidirect
9. C2×#15689404C2semidirect
10. C2×#15689400C2semidirect
11. C2×#15689179C2semidirect
12. C2×C47×#475C2semidirect
19. C4⋊Q8×D47C22semidirect
34. #15689530C22semidirect
35. #15689484C22semidirect
36. #15689457C22semidirect
38. #15689408C22semidirect
39. #15689404C22semidirect
40. #15689400C22semidirect
41. #15689179C22semidirect
53. C47×#475C22semidirect
295. C2×#475D47semidirect
299. C2×C47#3239semidirect
333. #475C2×D47semidirect
349. C47C2×#3239semidirect
581. C2#20254622direct (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×#156896602
#202546222
C2×#156895302
C2×#156894842
C2×#156894572
C2×#156894432
C2×#156894082
C2×#156894042
C2×#156894002
C2×#156891792
C2×C47×#4752
C2×#323947