Finite Group Databasebeta

← C2×#20254538 #27088326

Subgroups — C2×#20254538 #27088326

Order 12032.

2006 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×#15689677C211yesmaximal, characteristic
2C2×C4⋊Q8×D47C211yesmaximal, characteristic
3#20254538C2116yesmaximal, direct product factor
4C2×#15689527C212yesmaximal, complemented
5C2×#15689472C212yesmaximal, complemented
6C2×#15689460C212yesmaximal
7C2×#15689457C212yesmaximal
8C2×#15689419C211yesmaximal, characteristic, complemented
9C2×#15689418C211yesmaximal, characteristic, complemented
10C2×#15689415C211yesmaximal, characteristic, complemented
11C2×#15689175C211yesmaximal, characteristic
12C2×C47×#477C211yesmaximal, characteristic, complemented
13C22×Q8×D47—21no—
14#15689677—24no—
15#15689677C2218yes—
16C2×#13094024C2211yescharacteristic
17C2×#13094023—21no—
18C2×#13094023C2212yes—
19C2×#13094023C2211yescharacteristic
20C2×#13094022—21no—
21C2×#13094022C2211yescharacteristic
22C4⋊Q8×D47—24no—
23C4⋊Q8×D47C2218yes—
24C2×#13094014—21no—
25C2×#13094004C2211yescharacteristic
26C2×#13094002—21no—
27C2×#13094000C2211yescharacteristic
28C2×C4×Dic94—21no—
29C2×C4×Dic94C2211yescharacteristic
30C2×#13093981—21no—
31C2×C4⋊C4×D47—21no—
32C2×C4⋊C4×D47C2212yes—
33C2×C42×D47C2211yescharacteristic
34#15689527C22116yescomplemented
35#15689472C22116yescomplemented
36#15689460C22116yes—
37#15689457C22116yes—
38#15689419C2218yescomplemented
39#15689418C2218yescomplemented
40#15689415C2218yescomplemented
41#15689175C2218yes—
42C2×#13093958C2212yes—
43C2×#13093957C2212yes—
44C2×#13093957C2212yes—
45C2×C4⋊(C47⋊C8)C2211yescharacteristic
46C2×#13093949C2212yes—
47C2×#13093948C2212yes—
48C2×#13093947C2212yes—
49C2×(C2×D47)⋊C8C2212yes—
50C2×C8×Dic47C2212yes—
51C2×C4⋊Q8×C47C2211yescharacteristic
52C2×C4×Q8×C47C2211yescharacteristic
53C47×#477C2218yescomplemented
54C2×#107×C47C2212yes—
55C2×C4⋊C8×C47C2211yescharacteristic
56C2×#103×C47C2212yes—
57C22×Dic94—21no—
58C22×Dic94—21no—
59C22×Dic94—22no—
60C22×Dic94C2311yescharacteristic
61C2×Q8×D47—28no—
62C2×Q8×D47—48no—
63C22×C4×D47—22no—
64C22×C4×D47D411yescharacteristic
65C22×C4×D47D411yescharacteristic
66C22×C4×D47C2311yescharacteristic
67C22×C47⋊C8C2312yes—
68#13094024—22no—
69#13094024C2314yes—
70#13094023—24no—
71#13094023—28no—
72#13094023—22no—
73#13094023C2318yes—
74#13094023C2314yes—
75#13094022—28no—
76#13094022—22no—
77#13094022C2314yes—
78#13094014—28no—
79#13094004—22no—
80#13094004C2314yes—
81#13094002—28no—
82#13094000—22no—
83#13094000C2314yes—
84C4×Dic94—22no—
85C4×Dic94—28no—
86C4×Dic94C2314yes—
87#13093981—28no—
88C2×C4⋊Dic47—21no—
89C2×C4⋊Dic47—21no—
90C2×C4⋊Dic47C2312yes—
91C2×C4⋊Dic47C2311yescharacteristic
92C2×Dic47⋊C4—21no—
93C2×Dic47⋊C4—22no—
94C2×Dic47⋊C4—22no—
95C2×Dic47⋊C4—21no—
96C2×Dic47⋊C4—21no—
97C2×Dic47⋊C4—21no—
98C2×Dic47⋊C4—21no—
99C4⋊C4×D47—28no—
100C4⋊C4×D47—24no—
101C4⋊C4×D47C2318yes—
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#13093958C2318yes—
111#13093957C2318yes—
112#13093957C2318yes—
113C4⋊(C47⋊C8)C2314yes—
114#13093949C2318yes—
115#13093948C2318yes—
116#13093947C2318yes—
117(C2×D47)⋊C8C2318yes—
118C8×Dic47C2318yes—
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#107×C47C2318yes—
133C4⋊C8×C47C2314yes—
134#103×C47C2318yes—
135C23×D47C2×D411yescharacteristic
136C2×Q8×C47—21no—
137C2×Q8×C47—24no—
138C2×Q8×C47—22no—
139C2×Q8×C47—22no—
140C2×Q8×C47—44no—
141C2×Q8×C47C2412yes—
142C22×C4×C47—21no—
143C22×C4×C47—21no—
144C22×C4×C47—22no—
145C22×C4×C47C2×D411yescharacteristic
146C22×C4×C47C2×D411yescharacteristic
147C22×C4×C47C2411yescharacteristic
148C22×Dic47—21no—
149C22×Dic47—21no—
150C22×Dic47—22no—
151C22×Dic47C2×D411yescharacteristic
152C22×Dic47C2×D411yescharacteristic
153C22×Dic47C2×D411yescharacteristic
154Q8×D47—432no—
155C2×C4×D47—22no—
156C2×C4×D47—22no—
157C2×C4×D47—24no—
158C2×C4×D47—28no—
159C2×C4×D47—48no—
160C2×C4×D47Q1618yes—
161C2×C4×D47C2×D412yes—
162C2×C4×D47C2×D414yes—
163C2×C4×D47C2×D414yes—
164C2×C4×D47C2412yes—
165C2×Dic94—22no—
166C2×Dic94—22no—
167C2×Dic94—21no—
168C2×Dic94—24no—
169C2×Dic94—24no—
170C2×Dic94—28no—
171C2×Dic94—44no—
172C2×Dic94—48no—
173C2×Dic94—44no—
174C2×Dic94C2412yes—
175C2×C8×C47—24no—
176C2×C8×C47C2414yes—
177C4⋊C4×C47—24no—
178C4⋊C4×C47—24no—
179C4⋊C4×C47—21no—
180C4⋊C4×C47—22no—
181C4⋊C4×C47C2414yes—
182C4⋊C4×C47C2412yes—
183C42×C47—24no—
184C42×C47C2×D412yes—
185C42×C47C2412yes—
186(C2×D47)⋊C4—24no—
187(C2×D47)⋊C4—24no—
188C4⋊Dic47—24no—
189C4⋊Dic47—22no—
190C4⋊Dic47—21no—
191C4⋊Dic47—24no—
192C4⋊Dic47C2414yes—
193C4⋊Dic47C2412yes—
194Dic47⋊C4—24no—
195Dic47⋊C4—28no—
196Dic47⋊C4—28no—
197Dic47⋊C4—24no—
198Dic47⋊C4—24no—
199Dic47⋊C4—24no—
200Dic47⋊C4—24no—
201C4×Dic47—24no—
202C4×Dic47C2×D412yes—
203C4×Dic47C2×D414yes—
204C4×Dic47C2×D414yes—
205C4×Dic47C2412yes—
206C2×C47⋊C8—24no—
207C2×C47⋊C8C2414yes—
208C23×C47C22×D411yescharacteristic
209C22×D47—22no—
210C22×D47—22no—
211C22×D47C2×Q1614yes—
212C22×D47C22×D412yes—
213Q8×C47—22no—
214Q8×C47—22no—
215Q8×C47—48no—
216Q8×C47—42no—
217C2×C4×C47—22no—
218C2×C4×C47—22no—
219C2×C4×C47—24no—
220C2×C4×C47—22no—
221C2×C4×C47—42no—
222C2×C4×C47—44no—
223C2×C4×C47—42no—
224C2×C4×C47C2×Q1614yes—
225C2×C4×C47C22×D412yes—
226C2×C4×C47C22×D411yescharacteristic
227C2×C4×C47C22×D412yes—
228C2×C4×C472−1+412yes—
229C2×C4×C472−1+412yes—
230C2×C4×C47C2511yescharacteristic
231C2×Dic47—22no—
232C2×Dic47—22no—
233C2×Dic47—22no—
234C2×Dic47—24no—
235C2×Dic47—42no—
236C2×Dic47—44no—
237C2×Dic47—42no—
238C2×Dic47C2×Q1614yes—
239C2×Dic47AGL(1,ℤ/8)14yes—
240C2×Dic47C22×D412yes—
241C2×Dic47C22×D412yes—
242C2×Dic47C22×D412yes—
243C4×D47—24no—
244C4×D47—22no—
245C4×D47—22no—
246C4×D47—28no—
247C4×D47—416no—
248C4×D47C2×Q16116yes—
249Dic94—22no—
250Dic94—22no—
251Dic94—48no—
252Dic94—42no—
253Dic94—48no—
254Dic94—416no—
255C8×C47—24no—
256C47⋊C8—24no—
257C2×#3230—471nomaximal, Sylow
258C22×C47#53612yes—
259C22×C47C22×Q1612yes—
260C22×C47C2×AGL(1,ℤ/8)12yes—
261C22×C47C23×D411yescharacteristic
262C2×D47—22no—
263C2×D47—28no—
264C2×D47—22no—
265C2×D47C22×Q1614yes—
266C4×C47—22no—
267C4×C47—42no—
268C4×C47—44no—
269C4×C47—42no—
270C4×C47#53612yes—
271C4×C47C22×Q1614yes—
272C4×C47C2×2−1+411yescharacteristic
273C4×C47C2×2−1+411yescharacteristic
274Dic47—22no—
275Dic47—42no—
276Dic47—44no—
277Dic47—42no—
278Dic47C22×Q1614yes—
279Dic47C2×AGL(1,ℤ/8)14yes—
280C2×#533—471no—
281C22×C4⋊Q8—471no—
282#3230—4716no—
283C2×#500—472no—
284C2×#487—472no—
285C2×#484—472no—
286C2×#484—472no—
287C2×#477—471no—
288C2×#477—471no—
289C2×#477—471no—
290C2×#477D4711yescharacteristic, complemented
291C2×#431—471no—
292C2×C47#323014yescomplemented
293C2×C47C2×#53611yescharacteristic
294C2×C47C23×Q1611yescharacteristic
295C2×C47C22×AGL(1,ℤ/8)11yescharacteristic
296D47—24no—
297C23×Q8—941no—
298#533—478no—
299#533—944no—
300C2×C4⋊Q8—478no—
301C2×C4⋊Q8—472no—
302C2×C4⋊Q8—471no—
303C2×C4⋊Q8—471no—
304C2×C4⋊Q8—471no—
305C2×C4⋊Q8—941no—
306C2×C4⋊Q8—944no—
307C2×C4⋊Q8—941no—
308C2×C4⋊Q8C2×D4711yescharacteristic
309C2×#125—941no—
310C2×C22⋊Q8—471no—
311C2×C22⋊Q8—471no—
312C2×C22⋊Q8—941no—
313C2×C4×Q8—471no—
314C2×C4×Q8—941no—
315C2×C4×Q8C2×D4711yescharacteristic
316C2×#117—941no—
317C22×C4⋊C4—472no—
318C22×C4⋊C4—941no—
319C22×C42—471no—
320#500—4716no—
321#487—4716no—
322#484—4716no—
323#484—4716no—
324#477—478no—
325#477—478no—
326#477—478no—
327#477C2×D4718yescomplemented
328#431—478no—
329C2×#107—472no—
330C2×#107—472no—
331C2×#107—472no—
332C2×#107C2×D4712yes—
333C2×C4⋊C8—471no—
334C2×C4⋊C8C2×D4711yescharacteristic
335C2×#103—472no—
336C2×#103—472no—
337C2×#103—472no—
338C2×#103C2×D4712yes—
339C2×C22⋊C8—472no—
340C2×C4×C8—472no—
341C47C2×#323011yescharacteristic, complemented, Sylow
342C22×Q8—21no—
343C22×Q8—471no—
344C22×Q8—941no—
345C22×Q8—942no—
346C22×Q8—948no—
347C22×Q8—941no—
348C22×Q8—1888no—
349C22×Q8C22×D4711yescharacteristic
350C23×C4—471no—
351C23×C4—471no—
352C23×C4—471no—
353C23×C4—942no—
354C22×C8—472no—
355C22×C8C22×D4712yes—
356C4⋊Q8—22no—
357C4⋊Q8—478no—
358C4⋊Q8—474no—
359C4⋊Q8—474no—
360C4⋊Q8—474no—
361C4⋊Q8—948no—
362C4⋊Q8—944no—
363C4⋊Q8—942no—
364C4⋊Q8—948no—
365C4⋊Q8—942no—
366C4⋊Q8—942no—
367C4⋊Q8C22×D4714yes—
368#125—948no—
369C22⋊Q8—474no—
370C22⋊Q8—474no—
371C22⋊Q8—942no—
372C22⋊Q8—948no—
373C22⋊Q8—942no—
374C4×Q8—22no—
375C4×Q8—474no—
376C4×Q8—948no—
377C4×Q8—942no—
378C4×Q8C22×D4714yes—
379#117—948no—
380C2×C4⋊C4—21no—
381C2×C4⋊C4—21no—
382C2×C4⋊C4—472no—
383C2×C4⋊C4—471no—
384C2×C4⋊C4—478no—
385C2×C4⋊C4—944no—
386C2×C4⋊C4—948no—
387C2×C4⋊C4—942no—
388C2×C4⋊C4—942no—
389C2×C4⋊C4—941no—
390C2×C4⋊C4—941no—
391C2×C4⋊C4—941no—
392C2×C4⋊C4—941no—
393C2×C4⋊C4—941no—
394C2×C4⋊C4—941no—
395C2×C4⋊C4—941no—
396C2×C4⋊C4C22×D4712yes—
397C2×C4⋊C4C22×D4711yescharacteristic
398C2×C22⋊C4—941no—
399C2×C22⋊C4—941no—
400C2×C42—21no—
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#107—478no—
410#107—478no—
411#107C22×D4718yes—
412C4⋊C8—474no—
413C4⋊C8C22×D4714yes—
414#103—478no—
415#103—478no—
416#103—478no—
417#103C22×D4718yes—
418C22⋊C8—478no—
419C4×C8—478no—
420C24—471no—
421C2×Q8—21no—
422C2×Q8—24no—
423C2×Q8—22no—
424C2×Q8—22no—
425C2×Q8—44no—
426C2×Q8—472no—
427C2×Q8—944no—
428C2×Q8—948no—
429C2×Q8—941no—
430C2×Q8—944no—
431C2×Q8—942no—
432C2×Q8—942no—
433C2×Q8—1884no—
434C2×Q8—1884no—
435C2×Q8—1888no—
436C2×Q8—18832no—
437C2×Q8C23×D4712yes—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—22no—
441C22×C4—471no—
442C22×C4—471no—
443C22×C4—478no—
444C22×C4—474no—
445C22×C4—471no—
446C22×C4—472no—
447C22×C4—474no—
448C22×C4—472no—
449C22×C4—944no—
450C22×C4—948no—
451C22×C4—942no—
452C22×C4—942no—
453C22×C4—941no—
454C22×C4—941no—
455C22×C4—942no—
456C22×C4—1888no—
457C22×C4D4×D4711yescharacteristic
458C22×C4D4×D4711yescharacteristic
459C22×C4C23×D4711yescharacteristic
460C2×C8—24no—
461C2×C8—474no—
462C2×C8—944no—
463C2×C8C23×D4714yes—
464C4⋊C4—24no—
465C4⋊C4—24no—
466C4⋊C4—22no—
467C4⋊C4—21no—
468C4⋊C4—474no—
469C4⋊C4—472no—
470C4⋊C4—948no—
471C4⋊C4—948no—
472C4⋊C4—944no—
473C4⋊C4—944no—
474C4⋊C4—944no—
475C4⋊C4—944no—
476C4⋊C4—944no—
477C4⋊C4—944no—
478C4⋊C4—942no—
479C4⋊C4—941no—
480C4⋊C4—944no—
481C4⋊C4C23×D4714yes—
482C4⋊C4C23×D4712yes—
483C22⋊C4—944no—
484C22⋊C4—944no—
485C42—24no—
486C42—474no—
487C42—474no—
488C42—472no—
489C42—472no—
490C42—944no—
491C42D4×D4712yes—
492C42C23×D4712yes—
493C23—474no—
494C23—472no—
495C23—942no—
496C23—942no—
497C23C2×D4×D4711yescharacteristic
498Q8—22no—
499Q8—22no—
500Q8—48no—
501Q8—42no—
502Q8—942no—
503Q8—942no—
504Q8—1888no—
505Q8—1888no—
506Q8—18816no—
507Q8—1882no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—24no—
512C2×C4—44no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—474no—
516C2×C4—474no—
517C2×C4—472no—
518C2×C4—472no—
519C2×C4—4716no—
520C2×C4—472no—
521C2×C4—948no—
522C2×C4—942no—
523C2×C4—944no—
524C2×C4—942no—
525C2×C4—942no—
526C2×C4—942no—
527C2×C4—942no—
528C2×C4—944no—
529C2×C4—18816no—
530C2×C4—1884no—
531C2×C4—1882no—
532C2×C4—1882no—
533C2×C4Q16×D4714yes—
534C2×C4C2×D4×D4712yes—
535C2×C4C2×D4×D4712yes—
536C2×C4C2×D4×D4711yescharacteristic
537C2×C4#1309407112yes—
538C2×C4#1309407212yes—
539C2×C4C24×D4711yescharacteristic
540C8—24no—
541C8—944no—
542C22—474no—
543C22—942no—
544C22—948no—
545C22—942no—
546C22#1568969612yes—
547C22C2×Q16×D4712yes—
548C22AGL(1,ℤ/8)×D4712yes—
549C22C22×D4×D4711yescharacteristic
550C4—22no—
551C4—44no—
552C4—42no—
553C4—42no—
554C4—474no—
555C4—474no—
556C4—942no—
557C4—1884no—
558C4—1882no—
559C4—1882no—
560C4#1568969612yes—
561C4C2×Q16×D4714yes—
562C4C2×#1309407111yescharacteristic
563C4C2×#1309407211yescharacteristic
564C2—944no—
565C2#2025453814yesdirect product factor
566C2C2×#1568969611yescharacteristic
567C2C22×Q16×D4711yescharacteristic
568C2C2×AGL(1,ℤ/8)×D4711yescharacteristic
subgroupG/H
centerC23C2×D4×D47
commutator, cosocleC2×C4×C47C25
FrattiniC2×C4C24×D47
FittingC2×C47×#477C2
socleC23×C47C22×D4
prime radicalC23×D47C2×D4
hypercenterC2×#477D47
hypocenterC47C2×#3230
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#477

Complements

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

NH ≅ G/Ntype
3. #20254538C2direct (N × H)
4. C2×#15689527C2semidirect
5. C2×#15689472C2semidirect
8. C2×#15689419C2semidirect
9. C2×#15689418C2semidirect
10. C2×#15689415C2semidirect
12. C2×C47×#477C2semidirect
34. #15689527C22semidirect
35. #15689472C22semidirect
38. #15689419C22semidirect
39. #15689418C22semidirect
40. #15689415C22semidirect
53. C47×#477C22semidirect
290. C2×#477D47semidirect
292. C2×C47#3230semidirect
327. #477C2×D47semidirect
341. C47C2×#3230semidirect
565. C2#20254538direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#156896772
C2×C4⋊Q8×D472
#202545382
C2×#156895272
C2×#156894722
C2×#156894602
C2×#156894572
C2×#156894192
C2×#156894182
C2×#156894152
C2×#156891752
C2×C47×#4772
C2×#323047