Finite Group Databasebeta

← C2×#20255779 #27094738

Subgroups — C2×#20255779 #27094738

Order 12032.

1942 conjugacy classes of subgroups.

The subgroup lattice has 520 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689834C211yesmaximal, characteristic
2C2×#15689833C211yesmaximal, characteristic
3#20255779C2116yesmaximal, direct product factor
4C2×#15689526C211yesmaximal, characteristic, complemented
5C2×#15689460C212yesmaximal, complemented
6C2×#15689457C212yesmaximal, complemented
7C2×#15689419C211yesmaximal, characteristic, complemented
8C2×#15689363C212yesmaximal, complemented
9C2×C4×#10625337C211yesmaximal, characteristic, complemented
10C2×#15689188C211yesmaximal, characteristic
11C2×#107×D47C212yesmaximal, complemented
12C2×C47×#477C211yesmaximal, characteristic, complemented
13C22×#10625337C2211yescharacteristic
14C22×C8×D47C2212yes—
15#15689834—24no—
16#15689834C2218yes—
17#15689833—24no—
18#15689833C2218yes—
19C2×#13094024C2211yescharacteristic
20C2×#13094016—21no—
21C2×#13094015—21no—
22C2×#13094004C2212yes—
23C2×#13094002—22no—
24C2×#13094000—22no—
25C2×#13094000C2212yes—
26C2×C4×Dic94C2211yescharacteristic
27C2×C4×D188—21no—
28C2×C4×D188C2211yescharacteristic
29C2×#13093986—21no—
30C2×C4⋊C4×D47C2212yes—
31C2×C4⋊C4×D47C2212yes—
32#15689526C2218yescomplemented
33#15689460C22116yescomplemented
34#15689457C22116yescomplemented
35#15689419C2218yescomplemented
36#15689363C22116yescomplemented
37C4×#10625337C2218yescomplemented
38#15689188C2218yes—
39C2×#13093958C2212yes—
40C2×#13093957C2212yes—
41C2×#13093957C2211yescharacteristic
42C2×#13093957C2212yes—
43#107×D47C22116yescomplemented
44C2×#13093949C2212yes—
45C2×#13093948C2211yescharacteristic
46C2×#13093947C2211yescharacteristic
47C2×(C2×D47)⋊C8C2212yes—
48C2×C4×C47⋊C8C2211yescharacteristic
49C2×C4⋊Q8×C47C2211yescharacteristic
50C2×C4×Q8×C47C2211yescharacteristic
51C47×#477C2218yescomplemented
52C2×#107×C47C2212yes—
53C2×C4⋊C8×C47C2211yescharacteristic
54C2×#103×C47C2212yes—
55C22×Dic94—21no—
56C22×Dic94C2311yescharacteristic
57C22×D188D411yescharacteristic
58C22×C4×D47—22no—
59C22×C4×D47—22no—
60C22×C4×D47C2312yes—
61C2×#10625337—24no—
62C2×#10625337C2314yes—
63C22×C47⋊C8C2311yescharacteristic
64C22×C47⋊C8C2311yescharacteristic
65C2×C8×D47—28no—
66C2×C8×D47C2318yes—
67#13094024—22no—
68#13094024C2314yes—
69#13094016—28no—
70#13094015—28no—
71#13094004—24no—
72#13094004C2318yes—
73#13094002—216no—
74#13094000—24no—
75#13094000—216no—
76#13094000C2318yes—
77C4×Dic94—22no—
78C4×Dic94C2314yes—
79C4×D188—28no—
80C4×D188D414yes—
81C4×D188C2314yes—
82#13093986—28no—
83C2×C4⋊Dic47—21no—
84C2×C4⋊Dic47—22no—
85C2×C4⋊Dic47D411yescharacteristic
86C2×C4⋊Dic47C2311yescharacteristic
87C2×C4⋊Dic47C2312yes—
88C2×Dic47⋊C4—21no—
89C2×Dic47⋊C4—22no—
90C2×Dic47⋊C4—22no—
91C2×Dic47⋊C4—22no—
92C4⋊C4×D47—24no—
93C4⋊C4×D47—24no—
94C4⋊C4×D47C2318yes—
95C4⋊C4×D47C2318yes—
96C2×(C2×D47)⋊C4—22no—
97C2×(C2×D47)⋊C4—22no—
98C2×(C2×D47)⋊C4D412yes—
99C2×C4×Dic47—21no—
100#13093958C2318yes—
101#13093957C2314yes—
102#13093957C2318yes—
103#13093957C2318yes—
104#13093949C2318yes—
105#13093948C2314yes—
106#13093947C2314yes—
107(C2×D47)⋊C8C2318yes—
108C4×C47⋊C8C2314yes—
109C22×Q8×C47—21no—
110C22×Q8×C47C2311yescharacteristic
111C22×C8×C47C2312yes—
112C4⋊Q8×C47—22no—
113C4⋊Q8×C47C2314yes—
114C4×Q8×C47—22no—
115C4×Q8×C47C2314yes—
116C2×C4⋊C4×C47—21no—
117C2×C4⋊C4×C47—21no—
118C2×C4⋊C4×C47C2311yescharacteristic
119C2×C4⋊C4×C47C2312yes—
120C2×C42×C47—21no—
121C2×C42×C47C2311yescharacteristic
122#107×C47C2318yes—
123C4⋊C8×C47C2314yes—
124#103×C47C2318yes—
125C23×D47C2×D412yes—
126C2×Q8×C47—21no—
127C2×Q8×C47—24no—
128C2×Q8×C47—22no—
129C2×Q8×C47—22no—
130C2×Q8×C47—82no—
131C2×Q8×C47C2412yes—
132C22×C4×C47—21no—
133C22×C4×C47—21no—
134C22×C4×C47—22no—
135C22×C4×C47C2×D411yescharacteristic
136C22×C4×C47C2×D411yescharacteristic
137C22×C4×C47C2411yescharacteristic
138C22×Dic47—21no—
139C22×Dic47—21no—
140C22×Dic47—22no—
141C22×Dic47C2×D412yes—
142C2×D188Q1618yescomplemented
143C2×D188C2×D414yes—
144C2×C4×D47—24no—
145C2×C4×D47—24no—
146C2×C4×D47—28no—
147C2×C4×D47—28no—
148C2×C4×D47—84no—
149C2×C4×D47—84no—
150C2×C4×D47C2×D414yes—
151C2×C4×D47C2414yes—
152C2×Dic94—22no—
153C2×Dic94—22no—
154C2×Dic94—21no—
155C2×Dic94—24no—
156C2×Dic94—82no—
157C2×Dic94C2412yes—
158C2×C8×C47—24no—
159C2×C8×C47C2414yes—
160C4⋊C4×C47—24no—
161C4⋊C4×C47—24no—
162C4⋊C4×C47—21no—
163C4⋊C4×C47—22no—
164C4⋊C4×C47C2414yes—
165C4⋊C4×C47C2412yes—
166C42×C47—24no—
167C42×C47C2×D412yes—
168C42×C47C2412yes—
169#10625337—28no—
170(C2×D47)⋊C4—28no—
171(C2×D47)⋊C4—28no—
172(C2×D47)⋊C4C2×D418yes—
173C4⋊Dic47—21no—
174C4⋊Dic47—22no—
175C4⋊Dic47—24no—
176C4⋊Dic47—28no—
177C4⋊Dic47C2×D414yes—
178C4⋊Dic47C2412yes—
179C4⋊Dic47C2414yes—
180Dic47⋊C4—24no—
181Dic47⋊C4—28no—
182Dic47⋊C4—28no—
183Dic47⋊C4—28no—
184C4×Dic47—24no—
185C2×C47⋊C8—22no—
186C2×C47⋊C8#4114yes—
187C2×C47⋊C8C2412yes—
188C2×C47⋊C8C2412yes—
189C8×D47—216no—
190C23×C47C22×D411yescharacteristic
191C22×D47—24no—
192C22×D47—24no—
193C22×D47C2×Q1618yes—
194C22×D47C22×D414yes—
195Q8×C47—22no—
196Q8×C47—22no—
197Q8×C47—42no—
198Q8×C47—84no—
199C2×C4×C47—22no—
200C2×C4×C47—22no—
201C2×C4×C47—22no—
202C2×C4×C47—24no—
203C2×C4×C47—42no—
204C2×C4×C47—42no—
205C2×C4×C47—44no—
206C2×C4×C47C2×Q1614yes—
207C2×C4×C47C22×D412yes—
208C2×C4×C47C22×D411yescharacteristic
209C2×C4×C47C22×D412yes—
210C2×C4×C47C2×#4112yes—
211C2×C4×C472−1+412yes—
212C2×C4×C47C2511yescharacteristic
213C2×Dic47—24no—
214C2×Dic47—22no—
215C2×Dic47—22no—
216C2×Dic47—24no—
217C2×Dic47—42no—
218C2×Dic47—44no—
219C2×Dic47—42no—
220C2×Dic47C22×D414yes—
221D188C2×Q16116yescomplemented
222C4×D47—28no—
223C4×D47—24no—
224C4×D47—24no—
225C4×D47—88no—
226C4×D47—88no—
227Dic94—22no—
228Dic94—22no—
229Dic94—42no—
230Dic94—84no—
231C8×C47—24no—
232C47⋊C8—22no—
233C47⋊C8C2×#4114yes—
234C2×#3338—471nomaximal, Sylow
235C22×C47#54712yes—
236C22×C47C22×Q1612yes—
237C22×C47#57612yes—
238C22×C47C23×D411yescharacteristic
239C2×D47—216no—
240C2×D47—24no—
241C2×D47—24no—
242C2×D47C22×Q1618yes—
243C4×C47—22no—
244C4×C47—42no—
245C4×C47—42no—
246C4×C47—44no—
247C4×C47#54712yes—
248C4×C47C22×Q1614yes—
249C4×C47C22×#4111yescharacteristic
250C4×C47C2×2−1+411yescharacteristic
251Dic47—24no—
252Dic47—42no—
253Dic47—44no—
254Dic47—42no—
255C2×#554—471no—
256C2×#554—471no—
257#3338—4716no—
258C2×#500—471no—
259C2×#484—472no—
260C2×#484—472no—
261C2×#477—471no—
262C2×#477D4711yescharacteristic, complemented
263C2×#467—472no—
264C2×C4×Q16—471no—
265C2×C8×D4—471no—
266C22×#107—472no—
267C2×C47#333814yescomplemented
268C2×C47C2×#54711yescharacteristic
269C2×C47C23×Q1611yescharacteristic
270C2×C47C2×#57611yescharacteristic
271D47—28no—
272C22×Q16—471no—
273C23×C8—472no—
274#554—478no—
275#554—478no—
276#554—944no—
277#554—944no—
278C2×C4⋊Q8—471no—
279C2×C4⋊Q8C2×D4711yescharacteristic
280C2×#125—941no—
281C2×#125—941no—
282C2×C22⋊Q8—472no—
283C2×C22⋊Q8—472no—
284C2×C22⋊Q8—942no—
285C2×C22⋊Q8—942no—
286C2×C4×Q8—471no—
287C2×C4×Q8C2×D4711yescharacteristic
288C2×C4×D4—471no—
289C2×C4×D4—941no—
290C2×C4×D4—941no—
291C22×C4⋊C4—472no—
292C22×C4⋊C4—472no—
293#500—478no—
294#484—4716no—
295#484—4716no—
296#477—478no—
297#477C2×D4718yescomplemented
298#467—4716no—
299C4×Q16—478no—
300C8×D4—478no—
301C2×#107—472no—
302C2×#107—472no—
303C2×#107—472no—
304C2×#107—471no—
305C2×#107—4716no—
306C2×#107C2×D4712yes—
307C2×C4⋊C8C2×D4711yescharacteristic
308C2×#103—471no—
309C2×#103—472no—
310C2×#103—471no—
311C2×#103C2×D4712yes—
312C2×C22⋊C8—472no—
313C2×C4×C8—471no—
314C47C2×#333811yescharacteristic, complemented, Sylow
315C22×Q8—21no—
316C22×Q8—471no—
317C22×Q8—941no—
318C22×Q8C22×D4711yescharacteristic
319C22×D4—471no—
320C23×C4—472no—
321C23×C4—942no—
322C23×C4—942no—
323C2×Q16—474no—
324C2×Q16—944no—
325C22×C8—471no—
326C22×C8—471no—
327C22×C8—478no—
328C22×C8—948no—
329C22×C8C22×D4712yes—
330C4⋊Q8—22no—
331C4⋊Q8—474no—
332C4⋊Q8—942no—
333C4⋊Q8C22×D4714yes—
334#125—948no—
335#125—948no—
336C22⋊Q8—478no—
337C22⋊Q8—478no—
338C22⋊Q8—9416no—
339C22⋊Q8—9416no—
340C22⋊Q8—944no—
341C22⋊Q8—944no—
342C4×Q8—22no—
343C4×Q8—474no—
344C4×Q8—942no—
345C4×Q8C22×D4714yes—
346C4×D4—474no—
347C4×D4—474no—
348C4×D4—948no—
349C4×D4—948no—
350C2×C4⋊C4—21no—
351C2×C4⋊C4—21no—
352C2×C4⋊C4—471no—
353C2×C4⋊C4—471no—
354C2×C4⋊C4—472no—
355C2×C4⋊C4—478no—
356C2×C4⋊C4—478no—
357C2×C4⋊C4—944no—
358C2×C4⋊C4—944no—
359C2×C4⋊C4—942no—
360C2×C4⋊C4—942no—
361C2×C4⋊C4—942no—
362C2×C4⋊C4—942no—
363C2×C4⋊C4—941no—
364C2×C4⋊C4—941no—
365C2×C4⋊C4C22×D4711yescharacteristic
366C2×C4⋊C4C22×D4712yes—
367C2×C22⋊C4—472no—
368C2×C22⋊C4—942no—
369C2×C22⋊C4—942no—
370C2×C42—21no—
371C2×C42—941no—
372C2×C42C22×D4711yescharacteristic
373#107—478no—
374#107—474no—
375#107—478no—
376#107—478no—
377#107C22×D4718yes—
378C4⋊C8C22×D4714yes—
379#103—474no—
380#103—478no—
381#103—474no—
382#103C22×D4718yes—
383C22⋊C8—478no—
384C4×C8—474no—
385C24—472no—
386C2×Q8—21no—
387C2×Q8—24no—
388C2×Q8—22no—
389C2×Q8—22no—
390C2×Q8—82no—
391C2×Q8—472no—
392C2×Q8—944no—
393C2×Q8—941no—
394C2×Q8—942no—
395C2×Q8—942no—
396C2×Q8—3762no—
397C2×Q8C23×D4712yes—
398C2×D4—478no—
399C2×D4—474no—
400C22×C4—21no—
401C22×C4—21no—
402C22×C4—22no—
403C22×C4—472no—
404C22×C4—474no—
405C22×C4—474no—
406C22×C4—948no—
407C22×C4—948no—
408C22×C4—944no—
409C22×C4—944no—
410C22×C4—941no—
411C22×C4—941no—
412C22×C4—942no—
413C22×C4—3764no—
414C22×C4—3764no—
415C22×C4D4×D4711yescharacteristic
416C22×C4D4×D4711yescharacteristic
417C22×C4C23×D4711yescharacteristic
418Q16—948no—
419C2×C8—24no—
420C2×C8—474no—
421C2×C8—472no—
422C2×C8—472no—
423C2×C8—9416no—
424C2×C8—942no—
425C2×C8C23×D4714yes—
426C4⋊C4—24no—
427C4⋊C4—24no—
428C4⋊C4—21no—
429C4⋊C4—22no—
430C4⋊C4—474no—
431C4⋊C4—472no—
432C4⋊C4—474no—
433C4⋊C4—948no—
434C4⋊C4—948no—
435C4⋊C4—948no—
436C4⋊C4—948no—
437C4⋊C4—944no—
438C4⋊C4—941no—
439C4⋊C4—942no—
440C4⋊C4—944no—
441C4⋊C4C23×D4712yes—
442C4⋊C4C23×D4714yes—
443C22⋊C4—478no—
444C22⋊C4—948no—
445C22⋊C4—948no—
446C42—24no—
447C42—944no—
448C42D4×D4712yes—
449C42C23×D4712yes—
450C23—478no—
451C23—474no—
452C23—944no—
453C23—944no—
454C23C2×D4×D4711yescharacteristic
455Q8—22no—
456Q8—22no—
457Q8—42no—
458Q8—84no—
459Q8—942no—
460Q8—942no—
461Q8—3764no—
462Q8—1882no—
463D4—4716no—
464C2×C4—22no—
465C2×C4—22no—
466C2×C4—22no—
467C2×C4—24no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—44no—
471C2×C4—474no—
472C2×C4—944no—
473C2×C4—948no—
474C2×C4—944no—
475C2×C4—944no—
476C2×C4—942no—
477C2×C4—942no—
478C2×C4—944no—
479C2×C4—3768no—
480C2×C4—3768no—
481C2×C4—1882no—
482C2×C4—1882no—
483C2×C4—1884no—
484C2×C4Q16×D4714yes—
485C2×C4C2×D4×D4712yes—
486C2×C4C2×D4×D4711yescharacteristic
487C2×C4C2×D4×D4712yes—
488C2×C4#41×D4712yes—
489C2×C4#1309407212yes—
490C2×C4C24×D4711yescharacteristic
491C8—24no—
492C8—474no—
493C8—942no—
494C22—478no—
495C22—9416no—
496C22—944no—
497C22—944no—
498C22#1568977512yes—
499C22C2×Q16×D4712yes—
500C22#1568992712yes—
501C22C22×D4×D4711yescharacteristic
502C4—22no—
503C4—42no—
504C4—42no—
505C4—44no—
506C4—944no—
507C4—1882no—
508C4—1882no—
509C4—1884no—
510C4#1568977512yes—
511C4C2×Q16×D4714yes—
512C4C2×#41×D4711yescharacteristic
513C4C2×#1309407211yescharacteristic
514C2—948no—
515C2#2025577914yesdirect product factor
516C2C2×#1568977511yescharacteristic
517C2C22×Q16×D4711yescharacteristic
518C2C2×#1568992711yescharacteristic
subgroupG/H
centerC23C2×D4×D47
commutator, cosocleC2×C4×C47C25
FrattiniC2×C4C24×D47
FittingC2×C47×#477C2
socleC23×C47C22×D4
prime radicalC22×D188D4
hypercenterC2×#477D47
hypocenterC47C2×#3338
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. #20255779C2direct (N × H)
4. C2×#15689526C2semidirect
5. C2×#15689460C2semidirect
6. C2×#15689457C2semidirect
7. C2×#15689419C2semidirect
8. C2×#15689363C2semidirect
9. C2×C4×#10625337C2semidirect
11. C2×#107×D47C2semidirect
12. C2×C47×#477C2semidirect
32. #15689526C22semidirect
33. #15689460C22semidirect
34. #15689457C22semidirect
35. #15689419C22semidirect
36. #15689363C22semidirect
37. C4×#10625337C22semidirect
43. #107×D47C22semidirect
51. C47×#477C22semidirect
142. C2×D188Q16semidirect
221. D188C2×Q16semidirect
262. C2×#477D47semidirect
267. C2×C47#3338semidirect
297. #477C2×D47semidirect
314. C47C2×#3338semidirect
515. C2#20255779direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#156898342
C2×#156898332
#202557792
C2×#156895262
C2×#156894602
C2×#156894572
C2×#156894192
C2×#156893632
C2×C4×#106253372
C2×#156891882
C2×#107×D472
C2×C47×#4772
C2×#333847