Finite Group Databasebeta

← C2×#20254567 #27088356

Subgroups — C2×#20254567 #27088356

Order 12032.

1810 conjugacy classes of subgroups.

The subgroup lattice has 574 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689657C211yesmaximal, characteristic
2#20254567C2116yesmaximal, direct product factor
3C2×#15689522C212yesmaximal, complemented
4C2×#15689480C212yesmaximal, complemented
5C2×#15689464C212yesmaximal, complemented
6C2×#15689447C212yesmaximal
7C2×#15689415C211yesmaximal, characteristic, complemented
8C2×#15689405C211yesmaximal, characteristic, complemented
9C2×#15689410C211yesmaximal, characteristic, complemented
10C2×#15689400C211yesmaximal, characteristic, complemented
11C2×#15689176C211yesmaximal, characteristic, complemented
12C2×C47×#533C211yesmaximal, characteristic, complemented
13#15689657—24no—
14#15689657C2218yes—
15C2×#13094018—21no—
16C2×#13094016C2211yescharacteristic
17C2×#13094012C2211yescharacteristic
18C2×#13094007—21no—
19C2×#13094003C2211yescharacteristic
20C2×#13093998C2211yescharacteristic
21C2×C4×Dic94C2211yescharacteristic
22C2×C4×D188C2211yescharacteristic
23C2×C4×C22⋊D47—21no—
24C2×#13093981—21no—
25#15689522C22116yescomplemented
26#15689480C22116yescomplemented
27#15689464C22116yescomplemented
28#15689447C22116yes—
29#15689415C2218yescomplemented
30#15689410C2218yescomplemented
31#15689405C2218yescomplemented
32#15689400C2218yescomplemented
33#15689176C2218yescomplemented
34C2×#13093957C2212yes—
35C2×#13093957C2212yes—
36C2×#13093955C2212yes—
37C2×#13093955C2212yes—
38C2×C4⋊(C47⋊C8)C2211yescharacteristic
39C2×C4⋊(C47⋊C8)C2211yescharacteristic
40C2×#13093947C2212yes—
41C2×#13093947C2212yes—
42C2×#13093944C2212yes—
43C2×#13093944C2212yes—
44C2×C22⋊(C47⋊C8)C2212yes—
45C2×C4×C47⋊C8C2212yes—
46C47×#533—24no—
47C47×#533C2218yescomplemented
48C2×C4⋊Q8×C47C2211yescharacteristic
49C2×C4⋊Q8×C47C2211yescharacteristic
50C2×#125×C47C2212yes—
51C2×C22⋊Q8×C47—21no—
52C2×C22⋊Q8×C47—21no—
53C2×C4×Q8×C47—21no—
54C2×C4×Q8×C47—21no—
55C2×#117×C47C2212yes—
56C22×C42×C47C2211yescharacteristic
57C22×Dic94C2311yescharacteristic
58C22×D188C2311yescharacteristic
59C22×C22⋊D47—21no—
60C22×C4×D47—21no—
61C22×C47⋊C8C2312yes—
62C22×C47⋊C8C2312yes—
63#13094018—28no—
64#13094016—22no—
65#13094016C2314yes—
66#13094012—22no—
67#13094012C2314yes—
68#13094007—28no—
69#13094003—22no—
70#13094003C2314yes—
71#13093998—22no—
72#13093998C2314yes—
73C4×Dic94—22no—
74C4×Dic94C2314yes—
75C4×D188—22no—
76C4×D188C2314yes—
77C4×C22⋊D47—28no—
78#13093981—28no—
79C2×C4⋊Dic47C2311yescharacteristic
80C2×C4⋊Dic47C2311yescharacteristic
81C2×Dic47⋊C4—21no—
82C2×Dic47⋊C4—21no—
83C2×Dic47⋊C4—21no—
84C2×Dic47⋊C4—21no—
85C2×C22⋊Dic47—21no—
86C2×(C2×D47)⋊C4—21no—
87C2×(C2×D47)⋊C4—21no—
88C2×(C2×D47)⋊C4—21no—
89C2×(C2×D47)⋊C4—21no—
90C2×C4×Dic47—21no—
91#13093957C2318yes—
92#13093957C2318yes—
93#13093955C2318yes—
94#13093955C2318yes—
95C4⋊(C47⋊C8)C2314yes—
96C4⋊(C47⋊C8)C2314yes—
97#13093947C2318yes—
98#13093947C2318yes—
99#13093944C2318yes—
100#13093944C2318yes—
101C22⋊(C47⋊C8)C2318yes—
102C4×C47⋊C8C2318yes—
103C22×Q8×C47—21no—
104C22×Q8×C47—21no—
105C23×C4×C47D411yescharacteristic
106C23×C4×C47D411yescharacteristic
107C23×C4×C47C2311yescharacteristic
108C4⋊Q8×C47—22no—
109C4⋊Q8×C47—22no—
110C4⋊Q8×C47C2314yes—
111C4⋊Q8×C47C2314yes—
112#125×C47—24no—
113#125×C47C2318yes—
114C22⋊Q8×C47—28no—
115C22⋊Q8×C47—28no—
116C4×Q8×C47—28no—
117C4×Q8×C47—28no—
118#117×C47—24no—
119#117×C47C2318yes—
120C2×C4⋊C4×C47—22no—
121C2×C4⋊C4×C47—22no—
122C2×C4⋊C4×C47—21no—
123C2×C4⋊C4×C47—21no—
124C2×C4⋊C4×C47C2312yes—
125C2×C4⋊C4×C47C2312yes—
126C2×C22⋊C4×C47—22no—
127C2×C42×C47—22no—
128C2×C42×C47D411yescharacteristic
129C2×C42×C47D411yescharacteristic
130C2×C42×C47D414yes—
131C2×C42×C47C2311yescharacteristic
132C2×C42×C47C2311yescharacteristic
133C2×C42×C47C2314yes—
134C24×C47C2×D411yescharacteristic
135C23×D47—21no—
136C2×Q8×C47—24no—
137C2×Q8×C47—24no—
138C2×Q8×C47—44no—
139C2×Q8×C47—44no—
140C22×C4×C47—24no—
141C22×C4×C47—22no—
142C22×C4×C47—22no—
143C22×C4×C47—22no—
144C22×C4×C47—22no—
145C22×C4×C47—42no—
146C22×C4×C47C2×D411yescharacteristic
147C22×C4×C47C2×D412yes—
148C22×C4×C47C2×D411yescharacteristic
149C22×C4×C47C2×D411yescharacteristic
150C22×C4×C47C2×D411yescharacteristic
151C22×C4×C47C2×D414yes—
152C22×C4×C47C2×D414yes—
153C22×C4×C47C2×D411yescharacteristic
154C22×C4×C47C2412yes—
155C22×C4×C47C2411yescharacteristic
156C2×C22⋊D47—24no—
157C2×C22⋊D47—44no—
158C22×Dic47—21no—
159C22×Dic47—21no—
160C22×Dic47—21no—
161C2×D188—22no—
162C2×D188—22no—
163C2×D188—21no—
164C2×D188C2412yes—
165C2×C4×D47—24no—
166C2×C4×D47—44no—
167C2×Dic94—22no—
168C2×Dic94—22no—
169C2×Dic94—21no—
170C2×Dic94C2412yes—
171C4⋊C4×C47—28no—
172C4⋊C4×C47—28no—
173C4⋊C4×C47—24no—
174C4⋊C4×C47—24no—
175C4⋊C4×C47—22no—
176C4⋊C4×C47—22no—
177C4⋊C4×C47C2414yes—
178C4⋊C4×C47C2414yes—
179C22⋊C4×C47—28no—
180C42×C47—28no—
181C42×C47C2×D414yes—
182C42×C47C2×D414yes—
183C42×C47C2×D412yes—
184C42×C47C2×D412yes—
185C42×C47C2412yes—
186C42×C47C2412yes—
187C22⋊Dic47—24no—
188(C2×D47)⋊C4—24no—
189(C2×D47)⋊C4—24no—
190(C2×D47)⋊C4—24no—
191(C2×D47)⋊C4—24no—
192C4⋊Dic47—21no—
193C4⋊Dic47—21no—
194C4⋊Dic47C2412yes—
195C4⋊Dic47C2412yes—
196Dic47⋊C4—24no—
197Dic47⋊C4—24no—
198Dic47⋊C4—24no—
199Dic47⋊C4—24no—
200C4×Dic47—24no—
201C2×C47⋊C8—24no—
202C2×C47⋊C8—24no—
203C2×C47⋊C8C2414yes—
204C2×C47⋊C8C2414yes—
205C23×C47—22no—
206C23×C47—22no—
207C23×C47—22no—
208C23×C47C22×D412yes—
209C23×C47C22×D411yescharacteristic
210C22×D47—22no—
211C22×D47—42no—
212C22×D47—42no—
213C22×D47—42no—
214Q8×C47—48no—
215Q8×C47—48no—
216C2×C4×C47—22no—
217C2×C4×C47—22no—
218C2×C4×C47—22no—
219C2×C4×C47—22no—
220C2×C4×C47—24no—
221C2×C4×C47—28no—
222C2×C4×C47—22no—
223C2×C4×C47—22no—
224C2×C4×C47—24no—
225C2×C4×C47—24no—
226C2×C4×C47—44no—
227C2×C4×C47—44no—
228C2×C4×C47—44no—
229C2×C4×C47#13514yes—
230C2×C4×C47C22×D412yes—
231C2×C4×C47C22×D412yes—
232C2×C4×C47C22×D412yes—
233C2×C4×C47C22×D412yes—
234C2×C4×C47C22×D411yescharacteristic
235C2×C4×C47C22×D412yes—
236C2×C4×C472−1+412yes—
237C2×C4×C472−1+412yes—
238C2×C4×C47C2511yescharacteristic
239C22⋊D47—48no—
240C2×Dic47—22no—
241C2×Dic47—22no—
242C2×Dic47—22no—
243C2×Dic47—42no—
244C2×Dic47—42no—
245C2×Dic47—42no—
246D188—22no—
247D188—22no—
248D188—42no—
249C4×D47—48no—
250Dic94—22no—
251Dic94—22no—
252Dic94—42no—
253C47⋊C8—24no—
254C47⋊C8—24no—
255C2×#3233—471nomaximal, Sylow
256C22×C47—22no—
257C22×C47—22no—
258C22×C47—22no—
259C22×C47—44no—
260C22×C47#53612yes—
261C22×C47C2×#13512yes—
262C22×C47#57512yes—
263C22×C47C23×D411yescharacteristic
264C2×D47—42no—
265C2×D47—42no—
266C2×D47—42no—
267C2×D47—84no—
268C4×C47—22no—
269C4×C47—22no—
270C4×C47—22no—
271C4×C47—22no—
272C4×C47—44no—
273C4×C47—44no—
274C4×C47#53612yes—
275C4×C47C2×#13514yes—
276C4×C47C2×2−1+411yescharacteristic
277C4×C47C2×2−1+411yescharacteristic
278Dic47—42no—
279Dic47—42no—
280Dic47—42no—
281C2×#533D4711yescharacteristic, complemented
282C2×#529—471no—
283#3233—4716no—
284C2×#499—472no—
285C2×#488—472no—
286C2×#485—472no—
287C2×#482—472no—
288C2×#477—471no—
289C2×#475—471no—
290C2×#476—471no—
291C2×#474—471no—
292C2×#431—471no—
293C2×C47—42no—
294C2×C47#323314yescomplemented
295C2×C47C2×#53611yescharacteristic
296C2×C47C22×#13511yescharacteristic
297C2×C47C2×#57511yescharacteristic
298D47—82no—
299#533—24no—
300#533C2×D4718yescomplemented
301C2×C4⋊Q8C2×D4711yescharacteristic
302C2×C4⋊Q8C2×D4711yescharacteristic
303#529—478no—
304#529—944no—
305C2×#126—941no—
306C2×#125—471no—
307C2×#125C2×D4712yes—
308C2×#124—471no—
309C2×#123—941no—
310C2×C22⋊Q8—21no—
311C2×C22⋊Q8—21no—
312C2×C22⋊Q8—471no—
313C2×#121—471no—
314C2×C4×Q8—21no—
315C2×C4×Q8—21no—
316C2×C4×Q8—471no—
317C2×C4×D4—471no—
318C2×C4×D4—941no—
319C2×#117—941no—
320C2×#117C2×D4712yes—
321C22×C42C2×D4711yescharacteristic
322#499—4716no—
323#488—4716no—
324#485—4716no—
325#482—4716no—
326#477—478no—
327#476—478no—
328#475—478no—
329#474—478no—
330#431—478no—
331C2×#107—472no—
332C2×#107—472no—
333C2×#106—472no—
334C2×#106—472no—
335C2×C4⋊C8—471no—
336C2×C4⋊C8—471no—
337C2×#103—472no—
338C2×#103—472no—
339C2×#102—472no—
340C2×#102—472no—
341C2×C22⋊C8—472no—
342C2×C4×C8—472no—
343C47C2×#323311yescharacteristic, complemented, Sylow
344C22×Q8—21no—
345C22×Q8—21no—
346C22×Q8—471no—
347C22×D4—471no—
348C22×D4—941no—
349C23×C4—941no—
350C23×C4C22⋊D4711yescharacteristic
351C23×C4C22⋊D4711yescharacteristic
352C23×C4C22×D4711yescharacteristic
353C22×C8—472no—
354C22×C8—472no—
355C4⋊Q8—22no—
356C4⋊Q8—22no—
357C4⋊Q8C22×D4714yes—
358C4⋊Q8C22×D4714yes—
359#126—948no—
360#125—24no—
361#125—474no—
362#125—942no—
363#125C22×D4718yes—
364#124—474no—
365#124—942no—
366#123—948no—
367C22⋊Q8—28no—
368C22⋊Q8—28no—
369C22⋊Q8—474no—
370C22⋊Q8—942no—
371#121—474no—
372#121—942no—
373C4×Q8—28no—
374C4×Q8—28no—
375C4×Q8—474no—
376C4×Q8—942no—
377C4×D4—474no—
378C4×D4—948no—
379C4×D4—942no—
380#117—24no—
381#117—948no—
382#117C22×D4718yes—
383C2×C4⋊C4—22no—
384C2×C4⋊C4—22no—
385C2×C4⋊C4—21no—
386C2×C4⋊C4—21no—
387C2×C4⋊C4—471no—
388C2×C4⋊C4—471no—
389C2×C4⋊C4—941no—
390C2×C4⋊C4—941no—
391C2×C4⋊C4—941no—
392C2×C4⋊C4—941no—
393C2×C4⋊C4C22×D4712yes—
394C2×C4⋊C4C22×D4712yes—
395C2×C22⋊C4—22no—
396C2×C22⋊C4—941no—
397C2×C22⋊C4—941no—
398C2×C22⋊C4—941no—
399C2×C22⋊C4—941no—
400C2×C22⋊C4—941no—
401C2×C42—22no—
402C2×C42—941no—
403C2×C42C22⋊D4711yescharacteristic
404C2×C42C22⋊D4711yescharacteristic
405C2×C42C22⋊D4714yes—
406C2×C42C22×D4711yescharacteristic
407C2×C42C22×D4711yescharacteristic
408C2×C42C22×D4714yes—
409#107—478no—
410#107—478no—
411#106—478no—
412#106—478no—
413C4⋊C8—474no—
414C4⋊C8—474no—
415#103—478no—
416#103—478no—
417#102—478no—
418#102—478no—
419C22⋊C8—478no—
420C4×C8—478no—
421C24—941no—
422C24C2×C22⋊D4711yescharacteristic
423C2×Q8—24no—
424C2×Q8—24no—
425C2×Q8—44no—
426C2×Q8—44no—
427C2×Q8—472no—
428C2×Q8—941no—
429C2×Q8—942no—
430C2×Q8—942no—
431C2×D4—472no—
432C2×D4—944no—
433C2×D4—941no—
434C2×D4—942no—
435C2×D4—942no—
436C2×D4—1884no—
437C22×C4—24no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—42no—
443C22×C4—941no—
444C22×C4—941no—
445C22×C4—941no—
446C22×C4—944no—
447C22×C4—1884no—
448C22×C4C2×C22⋊D4712yes—
449C22×C4C2×C22⋊D4714yes—
450C22×C4C2×C22⋊D4711yescharacteristic
451C22×C4C2×C22⋊D4711yescharacteristic
452C22×C4C2×C22⋊D4711yescharacteristic
453C22×C4C2×C22⋊D4711yescharacteristic
454C22×C4C2×C22⋊D4714yes—
455C22×C4C2×C22⋊D4711yescharacteristic
456C22×C4C23×D4712yes—
457C22×C4C23×D4711yescharacteristic
458C2×C8—474no—
459C2×C8—474no—
460C2×C8—944no—
461C2×C8—944no—
462C4⋊C4—28no—
463C4⋊C4—28no—
464C4⋊C4—24no—
465C4⋊C4—24no—
466C4⋊C4—22no—
467C4⋊C4—22no—
468C4⋊C4—472no—
469C4⋊C4—472no—
470C4⋊C4—944no—
471C4⋊C4—944no—
472C4⋊C4—944no—
473C4⋊C4—944no—
474C4⋊C4—941no—
475C4⋊C4—941no—
476C4⋊C4C23×D4714yes—
477C4⋊C4C23×D4714yes—
478C22⋊C4—28no—
479C22⋊C4—944no—
480C22⋊C4—944no—
481C22⋊C4—944no—
482C22⋊C4—944no—
483C22⋊C4—944no—
484C42—28no—
485C42—944no—
486C42C2×C22⋊D4714yes—
487C42C2×C22⋊D4712yes—
488C42C2×C22⋊D4714yes—
489C42C2×C22⋊D4712yes—
490C42C23×D4712yes—
491C42C23×D4712yes—
492C23—22no—
493C23—22no—
494C23—22no—
495C23—942no—
496C23—1882no—
497C23—1882no—
498C23—1882no—
499C23C22×C22⋊D4712yes—
500C23C22×C22⋊D4711yescharacteristic
501Q8—48no—
502Q8—48no—
503Q8—942no—
504Q8—942no—
505Q8—1882no—
506D4—942no—
507D4—942no—
508D4—1888no—
509D4—1882no—
510C2×C4—22no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—28no—
515C2×C4—24no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—24no—
519C2×C4—24no—
520C2×C4—44no—
521C2×C4—44no—
522C2×C4—44no—
523C2×C4—942no—
524C2×C4—942no—
525C2×C4—942no—
526C2×C4—1882no—
527C2×C4—1882no—
528C2×C4—1882no—
529C2×C4—1888no—
530C2×C4#1309404414yes—
531C2×C4C22×C22⋊D4712yes—
532C2×C4C22×C22⋊D4712yes—
533C2×C4C22×C22⋊D4712yes—
534C2×C4C22×C22⋊D4712yes—
535C2×C4C22×C22⋊D4712yes—
536C2×C4C22×C22⋊D4711yescharacteristic
537C2×C4#1309407212yes—
538C2×C4#1309407212yes—
539C2×C4C24×D4711yescharacteristic
540C8—944no—
541C8—944no—
542C22—22no—
543C22—22no—
544C22—22no—
545C22—44no—
546C22—3764no—
547C22—1882no—
548C22—1882no—
549C22—1882no—
550C22#1568969712yes—
551C22C2×#1309404412yes—
552C22#1568992112yes—
553C22C23×C22⋊D4711yescharacteristic
554C4—22no—
555C4—22no—
556C4—22no—
557C4—22no—
558C4—44no—
559C4—44no—
560C4—1882no—
561C4—1882no—
562C4—1882no—
563C4#1568969712yes—
564C4C2×#1309404414yes—
565C4C2×#1309407211yescharacteristic
566C4C2×#1309407211yescharacteristic
567C2—42no—
568C2—3762no—
569C2#2025456714yesdirect product factor
570C2C2×#1568969711yescharacteristic
571C2C22×#1309404411yescharacteristic
572C2C2×#1568992111yescharacteristic
subgroupG/H
centerC23C22×C22⋊D47
commutator, cosocleC2×C4×C47C25
FrattiniC2×C4C24×D47
FittingC2×C47×#533C2
socleC23×C47C22×D4
prime radicalC2×#13093998C22
hypercenterC2×#533D47
hypocenterC47C2×#3233
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C22×C42 ◁ C2×#533

Complements

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

NH ≅ G/Ntype
2. #20254567C2direct (N × H)
3. C2×#15689522C2semidirect
4. C2×#15689480C2semidirect
5. C2×#15689464C2semidirect
7. C2×#15689415C2semidirect
8. C2×#15689405C2semidirect
9. C2×#15689410C2semidirect
10. C2×#15689400C2semidirect
11. C2×#15689176C2semidirect
12. C2×C47×#533C2semidirect
25. #15689522C22semidirect
26. #15689480C22semidirect
27. #15689464C22semidirect
29. #15689415C22semidirect
30. #15689410C22semidirect
31. #15689405C22semidirect
32. #15689400C22semidirect
33. #15689176C22semidirect
47. C47×#533C22semidirect
281. C2×#533D47semidirect
294. C2×C47#3233semidirect
300. #533C2×D47semidirect
343. C47C2×#3233semidirect
569. C2#20254567direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#156896572
#202545672
C2×#156895222
C2×#156894802
C2×#156894642
C2×#156894472
C2×#156894152
C2×#156894052
C2×#156894102
C2×#156894002
C2×#156891762
C2×C47×#5332
C2×#323347