Finite Group Databasebeta

← C2×#20254589 #27088387

Subgroups — C2×#20254589 #27088387

Order 12032.

1770 conjugacy classes of subgroups.

The subgroup lattice has 536 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689678C211yesmaximal, characteristic
2C2×#125×D47C211yesmaximal, characteristic
3#20254589C2116yesmaximal, direct product factor
4C2×#15689533C211yesmaximal, characteristic, complemented
5C2×#15689532C211yesmaximal, characteristic, complemented
6C2×#15689491C211yesmaximal, characteristic, complemented
7C2×#15689457C212yesmaximal
8C2×#15689452C212yesmaximal
9C2×#15689432C212yesmaximal, complemented
10C2×#15689428C212yesmaximal, complemented
11C2×#15689178C211yesmaximal, characteristic
12C2×C47×#491C211yesmaximal, characteristic, complemented
13#15689678—24no—
14#15689678C2218yes—
15C2×#13094024C2211yescharacteristic
16C2×#13094016C2211yescharacteristic
17C2×#13094015—21no—
18C2×#13094015C2211yescharacteristic
19C2×#13094015C2211yescharacteristic
20C2×#13094014—21no—
21C2×#13094014—21no—
22#125×D47—24no—
23#125×D47C2218yes—
24C2×#13094004—21no—
25C2×#13094000C2212yes—
26C2×Q8×Dic47C2212yes—
27C2×#13093989—21no—
28C2×#13093982—21no—
29C2×C4⋊C4×D47—21no—
30C2×C4⋊C4×D47—21no—
31C2×C4⋊C4×D47C2211yescharacteristic
32C2×C4⋊C4×D47C2211yescharacteristic
33C2×C42×D47C2211yescharacteristic
34#15689533C2218yescomplemented
35#15689532C2218yescomplemented
36#15689491C2218yescomplemented
37#15689457C22116yes—
38#15689452C22116yes—
39#15689432C22116yescomplemented
40#15689428C22116yescomplemented
41#15689178C2218yes—
42C2×#13093958C2212yes—
43C2×#13093957C2212yes—
44C2×#13093956C2212yes—
45C2×#13093955C2212yes—
46C2×Dic47⋊C8C2212yes—
47C2×#13093948C2212yes—
48C2×#13093948C2212yes—
49C2×(C2×D47)⋊C8C2212yes—
50C2×#13093935C2211yescharacteristic
51C2×C4⋊Q8×C47C2211yescharacteristic
52C2×#125×C47C2211yescharacteristic
53C47×#491C2218yescomplemented
54C2×#103×C47C2212yes—
55C2×#103×C47C2212yes—
56C2×#97×C47C2211yescharacteristic
57C22×Dic94—21no—
58C22×C4×D47—21no—
59C22×C4×D47—21no—
60C22×C4×D47D411yescharacteristic
61C22×C4×D47D411yescharacteristic
62C22×C4×D47C2311yescharacteristic
63C22×C47⋊C8C2312yes—
64#13094024—22no—
65#13094024C2314yes—
66#13094016—22no—
67#13094016C2314yes—
68#13094015—22no—
69#13094015—22no—
70#13094015—28no—
71#13094015C2314yes—
72#13094015C2314yes—
73#13094014—28no—
74#13094014—28no—
75#13094004—28no—
76#13094000—24no—
77#13094000C2318yes—
78Q8×Dic47—24no—
79Q8×Dic47C2318yes—
80#13093989—28no—
81#13093982—28no—
82C2×C4⋊Dic47—21no—
83C2×C4⋊Dic47—21no—
84C2×C4⋊Dic47—21no—
85C2×C4⋊Dic47—22no—
86C2×C4⋊Dic47C2311yescharacteristic
87C2×C4⋊Dic47C2311yescharacteristic
88C2×C4⋊Dic47C2312yes—
89C2×Dic47⋊C4—21no—
90C2×Dic47⋊C4—21no—
91C2×Dic47⋊C4—21no—
92C2×Dic47⋊C4—21no—
93C2×Dic47⋊C4—21no—
94C2×Dic47⋊C4—21no—
95C2×Dic47⋊C4—22no—
96C4⋊C4×D47—28no—
97C4⋊C4×D47—28no—
98C4⋊C4×D47—22no—
99C4⋊C4×D47—22no—
100C4⋊C4×D47C2314yes—
101C4⋊C4×D47C2314yes—
102C2×(C2×D47)⋊C4—22no—
103C2×C4×Dic47—22no—
104C2×C4×Dic47D411yescharacteristic
105C2×C4×Dic47D411yescharacteristic
106C2×C4×Dic47C2311yescharacteristic
107C42×D47D414yes—
108C42×D47C2314yes—
109#13093958C2318yes—
110#13093957C2318yes—
111#13093956C2318yes—
112#13093955C2318yes—
113Dic47⋊C8C2318yes—
114#13093948C2318yes—
115#13093948C2318yes—
116(C2×D47)⋊C8C2318yes—
117#13093935C2314yes—
118C22×Q8×C47C2312yes—
119C22×C8×C47C2312yes—
120C4⋊Q8×C47—22no—
121C4⋊Q8×C47C2314yes—
122#125×C47—22no—
123#125×C47C2314yes—
124C2×C4⋊C4×C47—21no—
125C2×C4⋊C4×C47—21no—
126C2×C4⋊C4×C47—21no—
127C2×C4⋊C4×C47—21no—
128C2×C4⋊C4×C47C2311yescharacteristic
129C2×C4⋊C4×C47C2311yescharacteristic
130C2×C42×C47C2311yescharacteristic
131#103×C47C2318yes—
132#103×C47C2318yes—
133#97×C47C2314yes—
134C23×D47C2×D411yescharacteristic
135C2×Q8×C47—22no—
136C2×Q8×C47—28no—
137C2×Q8×C47C2414yes—
138C22×C4×C47—21no—
139C22×C4×C47—21no—
140C22×C4×C47—22no—
141C22×C4×C47C2×D411yescharacteristic
142C22×C4×C47C2×D411yescharacteristic
143C22×C4×C47C2411yescharacteristic
144C22×Dic47—22no—
145C22×Dic47—21no—
146C22×Dic47—21no—
147C22×Dic47C2×D411yescharacteristic
148C22×Dic47C2×D411yescharacteristic
149C22×Dic47C2×D411yescharacteristic
150C2×C4×D47—24no—
151C2×C4×D47—24no—
152C2×C4×D47—24no—
153C2×C4×D47—24no—
154C2×C4×D47—42no—
155C2×C4×D47—44no—
156C2×C4×D47—44no—
157C2×C4×D47C2×D412yes—
158C2×C4×D47C2×D414yes—
159C2×C4×D47C2×D414yes—
160C2×C4×D47C2412yes—
161C2×Dic94—24no—
162C2×Dic94—44no—
163C2×C8×C47—24no—
164C2×C8×C47C2414yes—
165C4⋊C4×C47—24no—
166C4⋊C4×C47—24no—
167C4⋊C4×C47—24no—
168C4⋊C4×C47—24no—
169C4⋊C4×C47—21no—
170C4⋊C4×C47—21no—
171C4⋊C4×C47C2412yes—
172C4⋊C4×C47C2412yes—
173C42×C47C2×D412yes—
174C42×C47C2412yes—
175(C2×D47)⋊C4—28no—
176C4⋊Dic47—21no—
177C4⋊Dic47—21no—
178C4⋊Dic47—24no—
179C4⋊Dic47—22no—
180C4⋊Dic47—24no—
181C4⋊Dic47—24no—
182C4⋊Dic47—28no—
183C4⋊Dic47C2412yes—
184C4⋊Dic47C2412yes—
185C4⋊Dic47C2414yes—
186Dic47⋊C4—24no—
187Dic47⋊C4—24no—
188Dic47⋊C4—24no—
189Dic47⋊C4—24no—
190Dic47⋊C4—24no—
191Dic47⋊C4—24no—
192Dic47⋊C4—28no—
193C4×Dic47—28no—
194C4×Dic47C2×D412yes—
195C4×Dic47C2×D414yes—
196C4×Dic47C2×D414yes—
197C4×Dic47C2412yes—
198C2×C47⋊C8—24no—
199C2×C47⋊C8C2414yes—
200C23×C47C22×D411yescharacteristic
201C22×D47—22no—
202C22×D47—22no—
203C22×D47#13714yes—
204C22×D47C22×D412yes—
205Q8×C47—28no—
206Q8×C47—44no—
207C2×C4×C47—22no—
208C2×C4×C47—22no—
209C2×C4×C47—24no—
210C2×C4×C47—22no—
211C2×C4×C47—22no—
212C2×C4×C47—44no—
213C2×C4×C47—42no—
214C2×C4×C47—42no—
215C2×C4×C47C22×D412yes—
216C2×C4×C47C22×D412yes—
217C2×C4×C47C22×D411yescharacteristic
218C2×C4×C472−1+414yes—
219C2×C4×C47C2511yescharacteristic
220C2×Dic47—22no—
221C2×Dic47—22no—
222C2×Dic47—22no—
223C2×Dic47—24no—
224C2×Dic47—22no—
225C2×Dic47—22no—
226C2×Dic47—42no—
227C2×Dic47—44no—
228C2×Dic47—42no—
229C2×Dic47C22×D412yes—
230C2×Dic47C22×D412yes—
231C2×Dic47C22×D412yes—
232C4×D47—28no—
233C4×D47—24no—
234C4×D47—24no—
235C4×D47—44no—
236C4×D47—48no—
237C4×D47—48no—
238Dic94—48no—
239C8×C47—24no—
240C47⋊C8—24no—
241C2×#3236—471nomaximal, Sylow
242C22×C47#53612yes—
243C22×C47C2×#13712yes—
244C22×C47#57512yes—
245C22×C47C23×D411yescharacteristic
246C2×D47—22no—
247C2×D47—28no—
248C2×D47—22no—
249C2×D47C2×#13714yes—
250C4×C47—22no—
251C4×C47—22no—
252C4×C47—44no—
253C4×C47—42no—
254C4×C47—42no—
255C4×C47#53612yes—
256C4×C47C2×2−1+412yes—
257Dic47—22no—
258Dic47—22no—
259Dic47—22no—
260Dic47—42no—
261Dic47—44no—
262Dic47—42no—
263C2×#533—471no—
264C22×#125—471no—
265#3236—4716no—
266C2×#501—471no—
267C2×#501—471no—
268C2×#491—471no—
269C2×#491D4711yescharacteristic, complemented
270C2×#484—472no—
271C2×#483—472no—
272C2×#479—472no—
273C2×#479—472no—
274C2×#432—471no—
275C2×C47#323614yescomplemented
276C2×C47C2×#53611yescharacteristic
277C2×C47C22×#13711yescharacteristic
278C2×C47C2×#57511yescharacteristic
279D47—24no—
280#533—478no—
281#533—944no—
282C2×C4⋊Q8—471no—
283C2×C4⋊Q8C2×D4711yescharacteristic
284C2×#125—478no—
285C2×#125—471no—
286C2×#125—471no—
287C2×#125—471no—
288C2×#125—941no—
289C2×#125—941no—
290C2×#125—941no—
291C2×#125—944no—
292C2×#125C2×D4711yescharacteristic
293C2×C22⋊Q8—472no—
294C2×C22⋊Q8—941no—
295C2×C4×Q8—472no—
296C2×C4×Q8—941no—
297C2×#117—941no—
298C22×C4⋊C4—471no—
299C22×C4⋊C4—471no—
300C22×C4⋊C4—941no—
301C22×C4⋊C4—941no—
302C22×C42—471no—
303#501—478no—
304#501—478no—
305#491—478no—
306#491C2×D4718yescomplemented
307#484—4716no—
308#483—4716no—
309#479—4716no—
310#479—4716no—
311#432—478no—
312C2×#107—472no—
313C2×#107—472no—
314C2×#106—472no—
315C2×#106—472no—
316C2×C4⋊C8—472no—
317C2×#103—472no—
318C2×#103—472no—
319C2×#103C2×D4712yes—
320C2×#103C2×D4712yes—
321C2×C22⋊C8—472no—
322C2×#97—471no—
323C2×#97C2×D4711yescharacteristic
324C47C2×#323611yescharacteristic, complemented, Sylow
325C22×Q8—941no—
326C22×Q8C22×D4712yes—
327C23×C4—471no—
328C23×C4—471no—
329C23×C4—471no—
330C23×C4—941no—
331C23×C4—941no—
332C22×C8—472no—
333C22×C8C22×D4712yes—
334C4⋊Q8—22no—
335C4⋊Q8—474no—
336C4⋊Q8—942no—
337C4⋊Q8C22×D4714yes—
338#125—22no—
339#125—474no—
340#125—474no—
341#125—474no—
342#125—948no—
343#125—948no—
344#125—942no—
345#125—942no—
346#125—948no—
347#125—942no—
348#125C22×D4714yes—
349C22⋊Q8—478no—
350C22⋊Q8—948no—
351C22⋊Q8—944no—
352C4×Q8—478no—
353C4×Q8—948no—
354C4×Q8—944no—
355#117—948no—
356C2×C4⋊C4—21no—
357C2×C4⋊C4—21no—
358C2×C4⋊C4—21no—
359C2×C4⋊C4—21no—
360C2×C4⋊C4—471no—
361C2×C4⋊C4—471no—
362C2×C4⋊C4—472no—
363C2×C4⋊C4—474no—
364C2×C4⋊C4—474no—
365C2×C4⋊C4—942no—
366C2×C4⋊C4—942no—
367C2×C4⋊C4—948no—
368C2×C4⋊C4—948no—
369C2×C4⋊C4—941no—
370C2×C4⋊C4—941no—
371C2×C4⋊C4—941no—
372C2×C4⋊C4—941no—
373C2×C4⋊C4—941no—
374C2×C4⋊C4—941no—
375C2×C4⋊C4—942no—
376C2×C4⋊C4—942no—
377C2×C4⋊C4—941no—
378C2×C4⋊C4—941no—
379C2×C4⋊C4—941no—
380C2×C4⋊C4C22×D4711yescharacteristic
381C2×C4⋊C4C22×D4711yescharacteristic
382C2×C22⋊C4—942no—
383C2×C42—471no—
384C2×C42—471no—
385C2×C42—471no—
386C2×C42—474no—
387C2×C42—474no—
388C2×C42—942no—
389C2×C42C22×D4711yescharacteristic
390#107—478no—
391#107—478no—
392#106—478no—
393#106—478no—
394C4⋊C8—478no—
395#103—478no—
396#103—478no—
397#103C22×D4718yes—
398#103C22×D4718yes—
399C22⋊C8—478no—
400#97—474no—
401#97C22×D4714yes—
402C24—471no—
403C2×Q8—22no—
404C2×Q8—28no—
405C2×Q8—944no—
406C2×Q8—1884no—
407C2×Q8C23×D4714yes—
408C22×C4—22no—
409C22×C4—21no—
410C22×C4—21no—
411C22×C4—471no—
412C22×C4—471no—
413C22×C4—471no—
414C22×C4—474no—
415C22×C4—472no—
416C22×C4—472no—
417C22×C4—474no—
418C22×C4—944no—
419C22×C4—944no—
420C22×C4—944no—
421C22×C4—944no—
422C22×C4—942no—
423C22×C4—941no—
424C22×C4—941no—
425C22×C4—1882no—
426C22×C4—1884no—
427C22×C4—1884no—
428C22×C4D4×D4711yescharacteristic
429C22×C4D4×D4711yescharacteristic
430C22×C4C23×D4711yescharacteristic
431C2×C8—24no—
432C2×C8—474no—
433C2×C8—944no—
434C2×C8C23×D4714yes—
435C4⋊C4—24no—
436C4⋊C4—24no—
437C4⋊C4—24no—
438C4⋊C4—24no—
439C4⋊C4—21no—
440C4⋊C4—21no—
441C4⋊C4—472no—
442C4⋊C4—472no—
443C4⋊C4—474no—
444C4⋊C4—944no—
445C4⋊C4—944no—
446C4⋊C4—944no—
447C4⋊C4—944no—
448C4⋊C4—948no—
449C4⋊C4—944no—
450C4⋊C4—944no—
451C4⋊C4—948no—
452C4⋊C4—944no—
453C4⋊C4—944no—
454C4⋊C4—941no—
455C4⋊C4—941no—
456C4⋊C4—942no—
457C4⋊C4—944no—
458C4⋊C4C23×D4712yes—
459C4⋊C4C23×D4712yes—
460C22⋊C4—948no—
461C42—474no—
462C42—472no—
463C42—474no—
464C42—472no—
465C42—948no—
466C42D4×D4712yes—
467C42C23×D4712yes—
468C23—474no—
469C23—472no—
470C23—942no—
471C23—942no—
472C23C2×D4×D4711yescharacteristic
473Q8—28no—
474Q8—44no—
475Q8—1888no—
476C2×C4—22no—
477C2×C4—22no—
478C2×C4—24no—
479C2×C4—22no—
480C2×C4—22no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—44no—
484C2×C4—472no—
485C2×C4—472no—
486C2×C4—472no—
487C2×C4—948no—
488C2×C4—942no—
489C2×C4—942no—
490C2×C4—942no—
491C2×C4—944no—
492C2×C4—944no—
493C2×C4—944no—
494C2×C4—942no—
495C2×C4—942no—
496C2×C4—1884no—
497C2×C4—1888no—
498C2×C4—1888no—
499C2×C4—1882no—
500C2×C4—1882no—
501C2×C4—1884no—
502C2×C4C2×D4×D4712yes—
503C2×C4C2×D4×D4712yes—
504C2×C4C2×D4×D4711yescharacteristic
505C2×C4#1309407114yes—
506C2×C4C24×D4711yescharacteristic
507C8—24no—
508C8—944no—
509C22—474no—
510C22—942no—
511C22—948no—
512C22—942no—
513C22#1568969312yes—
514C22#137×D4712yes—
515C22#1568992212yes—
516C22C22×D4×D4711yescharacteristic
517C4—22no—
518C4—22no—
519C4—42no—
520C4—42no—
521C4—44no—
522C4—942no—
523C4—942no—
524C4—942no—
525C4—1882no—
526C4—1882no—
527C4—1884no—
528C4#1568969312yes—
529C4C2×#1309407112yes—
530C2—944no—
531C2#2025458914yesdirect product factor
532C2C2×#1568969311yescharacteristic
533C2C2×#137×D4711yescharacteristic
534C2C2×#1568992211yescharacteristic
subgroupG/H
centerC23C2×D4×D47
commutator, cosocleC2×C4×C47C25
FrattiniC2×C4C24×D47
FittingC2×C47×#491C2
socleC23×C47C22×D4
prime radicalC23×D47C2×D4
hypercenterC2×#491D47
hypocenterC47C2×#3236
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#491

Complements

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

NH ≅ G/Ntype
3. #20254589C2direct (N × H)
4. C2×#15689533C2semidirect
5. C2×#15689532C2semidirect
6. C2×#15689491C2semidirect
9. C2×#15689432C2semidirect
10. C2×#15689428C2semidirect
12. C2×C47×#491C2semidirect
34. #15689533C22semidirect
35. #15689532C22semidirect
36. #15689491C22semidirect
39. #15689432C22semidirect
40. #15689428C22semidirect
53. C47×#491C22semidirect
269. C2×#491D47semidirect
275. C2×C47#3236semidirect
306. #491C2×D47semidirect
324. C47C2×#3236semidirect
531. C2#20254589direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#156896782
C2×#125×D472
#202545892
C2×#156895332
C2×#156895322
C2×#156894912
C2×#156894572
C2×#156894522
C2×#156894322
C2×#156894282
C2×#156891782
C2×C47×#4912
C2×#323647