Finite Group Databasebeta

← C2×#20247691 #26385328

Subgroups — C2×#20247691 #26385328

Order 12032.

3678 conjugacy classes of subgroups.

The subgroup lattice has 568 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#13094009C212yesmaximal
2C22×#13093986C211yesmaximal, characteristic
3C22×C4⋊C4×D47C211yesmaximal, characteristic, complemented
4#20247691C2116yesmaximal, direct product factor
5C2×#15689053C211yesmaximal, characteristic, complemented
6C2×#15689050C211yesmaximal, characteristic, complemented
7C2×#15689049C211yesmaximal, characteristic, complemented
8C2×#15689018C212yesmaximal, complemented
9C2×#15688985C212yesmaximal, complemented
10C2×#15688982C212yesmaximal, complemented
11C2×#15688982C211yesmaximal, characteristic, complemented
12C2×C47×#398C211yesmaximal, characteristic, complemented
13C23×D188C2211yescharacteristic
14C23×C4×D47C2211yescharacteristic
15C23×C4×D47C2212yes—
16C23×C4×D47C2211yescharacteristic
17C2×#13094009—216no—
18C2×#13094009C22116yes—
19C2×#13093986—28no—
20C2×#13093986C2218yes—
21C22×C4⋊Dic47C2211yescharacteristic
22C22×Dic47⋊C4C2212yes—
23C22×Dic47⋊C4C2212yes—
24C22×Dic47⋊C4C2212yes—
25C2×C4⋊C4×D47—28no—
26C2×C4⋊C4×D47C2218yescomplemented
27C22×(C2×D47)⋊C4C2212yes—
28C22×(C2×D47)⋊C4C2212yes—
29C22×(C2×D47)⋊C4C2212yes—
30C22×(C2×D47)⋊C4C2211yescharacteristic
31C22×(C2×D47)⋊C4C2211yescharacteristic
32C22×C4×Dic47C2211yescharacteristic
33#15689053C2218yescomplemented
34#15689050C2218yescomplemented
35#15689049C2218yescomplemented
36#15689018C22116yescomplemented
37#15688985C22116yescomplemented
38#15688982C22116yescomplemented
39#15688982C2218yescomplemented
40C2×C47⋊#95C2212yes—
41C2×C47⋊#95C2211yescharacteristic
42C2×C47⋊#95C2212yes—
43C2×C47⋊#95C2212yes—
44C22×C4⋊C4×C47C2211yescharacteristic
45C22×C4⋊C4×C47C2211yescharacteristic
46C22×C4⋊C4×C47C2212yes—
47C47×#398C2218yescomplemented
48C2×#95×C47C2212yes—
49C2×#95×C47C2211yescharacteristic
50C24×D47C2311yescharacteristic
51C24×D47C2311yescharacteristic
52C22×D188—24no—
53C22×D188—24no—
54C22×D188D418yes—
55C22×D188C2314yes—
56C23×Dic47C2311yescharacteristic
57C23×Dic47C2312yes—
58C23×Dic47C2312yes—
59C23×Dic47C2311yescharacteristic
60C22×C4×D47—28no—
61C22×C4×D47—28no—
62C22×C4×D47—28no—
63C22×C4×D47—28no—
64C22×C4×D47—24no—
65C22×C4×D47—28no—
66C22×C4×D47D418yes—
67C22×C4×D47C2314yes—
68C22×C4×D47C2318yes—
69C22×C4×D47C2314yes—
70#13094009—264no—
71#13093986—232no—
72C2×C4⋊Dic47—24no—
73C2×C4⋊Dic47C2314yes—
74C2×Dic47⋊C4—28no—
75C2×Dic47⋊C4—28no—
76C2×Dic47⋊C4—28no—
77C2×Dic47⋊C4C2318yes—
78C2×Dic47⋊C4C2318yes—
79C2×Dic47⋊C4C2318yes—
80C4⋊C4×D47—232no—
81C2×(C2×D47)⋊C4—24no—
82C2×(C2×D47)⋊C4—28no—
83C2×(C2×D47)⋊C4—28no—
84C2×(C2×D47)⋊C4—28no—
85C2×(C2×D47)⋊C4—24no—
86C2×(C2×D47)⋊C4C2318yes—
87C2×(C2×D47)⋊C4C2318yes—
88C2×(C2×D47)⋊C4C2318yes—
89C2×(C2×D47)⋊C4C2314yes—
90C2×(C2×D47)⋊C4C2314yes—
91C2×C4×Dic47—24no—
92C2×C4×Dic47C2314yes—
93C47⋊#95C2318yes—
94C47⋊#95C2314yes—
95C47⋊#95C2318yes—
96C47⋊#95C2318yes—
97C23×C4×C47C2311yescharacteristic
98C23×C4×C47C2312yes—
99C23×C4×C47C2312yes—
100C23×C4×C47C2311yescharacteristic
101C23×C4×C47C2311yescharacteristic
102C2×C4⋊C4×C47—24no—
103C2×C4⋊C4×C47—24no—
104C2×C4⋊C4×C47—28no—
105C2×C4⋊C4×C47C2318yes—
106C2×C4⋊C4×C47C2314yes—
107C2×C4⋊C4×C47C2314yes—
108#95×C47C2318yes—
109#95×C47C2314yes—
110C24×C47C2411yescharacteristic
111C23×D47—24no—
112C23×D47—24no—
113C23×D47—22no—
114C23×D47—24no—
115C23×D47—22no—
116C23×D47—24no—
117C23×D47C2×D414yes—
118C23×D47C2×D414yes—
119C23×D47C2×D414yes—
120C23×D47#4114yes—
121C23×D47C2412yes—
122C23×D47C2412yes—
123C22×C4×C47—22no—
124C22×C4×C47—24no—
125C22×C4×C47—22no—
126C22×C4×C47—24no—
127C22×C4×C47—24no—
128C22×C4×C47—22no—
129C22×C4×C47—24no—
130C22×C4×C47—24no—
131C22×C4×C47—24no—
132C22×C4×C47—24no—
133C22×C4×C47—24no—
134C22×C4×C47C2×D414yes—
135C22×C4×C47C2×D414yes—
136C22×C4×C47C2414yes—
137C22×C4×C47C2412yes—
138C22×C4×C47C2414yes—
139C22×C4×C47C2412yes—
140C22×C4×C47C2412yes—
141C22×Dic47—22no—
142C22×Dic47—24no—
143C22×Dic47—24no—
144C22×Dic47—24no—
145C22×Dic47—24no—
146C22×Dic47—24no—
147C22×Dic47—24no—
148C22×Dic47—24no—
149C22×Dic47C2×D414yes—
150C22×Dic47#4118yes—
151C22×Dic47C2412yes—
152C22×Dic47C2414yes—
153C22×Dic47C2414yes—
154C22×Dic47C2412yes—
155C2×D188—232no—
156C2×D188—28no—
157C2×D188—28no—
158C2×D188C2×D4116yes—
159C2×C4×D47—232no—
160C2×C4×D47—216no—
161C2×C4×D47—216no—
162C2×C4×D47—216no—
163C2×C4×D47—216no—
164C2×C4×D47—28no—
165C2×C4×D47—216no—
166C2×C4×D47—432no—
167C2×C4×D47—416no—
168C2×C4×D47C2×D4116yes—
169C4⋊C4×C47—28no—
170C4⋊C4×C47—28no—
171C4⋊C4×C47—216no—
172(C2×D47)⋊C4—28no—
173(C2×D47)⋊C4—216no—
174(C2×D47)⋊C4—216no—
175(C2×D47)⋊C4—216no—
176(C2×D47)⋊C4—28no—
177C4⋊Dic47—28no—
178Dic47⋊C4—216no—
179Dic47⋊C4—216no—
180Dic47⋊C4—216no—
181C4×Dic47—28no—
182C23×C47C22×D412yes—
183C23×C47C22×D412yes—
184C23×C47C2×#4112yes—
185C23×C47C2×#4114yes—
186C23×C472−1+414yes—
187C23×C47C2511yescharacteristic
188C22×D47—28no—
189C22×D47—28no—
190C22×D47—216no—
191C22×D47—216no—
192C22×D47—24no—
193C22×D47—24no—
194C22×D47—28no—
195C22×D47—216no—
196C22×D47—22no—
197C22×D47—24no—
198C22×D47—22no—
199C22×D47—24no—
200C22×D47—44no—
201C22×D47—44no—
202C22×D47—48no—
203C22×D47#121116yes—
204C22×D47C22×D414yes—
205C22×D47C22×D414yes—
206C22×D47C22×D414yes—
207C22×D47C2×#4114yes—
208C2×C4×C47—28no—
209C2×C4×C47—28no—
210C2×C4×C47—22no—
211C2×C4×C47—24no—
212C2×C4×C47—22no—
213C2×C4×C47—24no—
214C2×C4×C47—24no—
215C2×C4×C47—24no—
216C2×C4×C47—22no—
217C2×C4×C47—24no—
218C2×C4×C47—24no—
219C2×C4×C47—24no—
220C2×C4×C47—24no—
221C2×C4×C47—48no—
222C2×C4×C47—44no—
223C2×C4×C47—48no—
224C2×C4×C47C22×D414yes—
225C2×C4×C47C22×D414yes—
226C2×Dic47—22no—
227C2×Dic47—24no—
228C2×Dic47—24no—
229C2×Dic47—24no—
230C2×Dic47—24no—
231C2×Dic47—24no—
232C2×Dic47—24no—
233C2×Dic47—24no—
234C2×Dic47—48no—
235C2×Dic47—44no—
236C2×Dic47—48no—
237C2×Dic47#123116yescomplemented
238C2×Dic47C22×D414yes—
239C2×Dic47C2×#4118yes—
240D188—232no—
241C4×D47—232no—
242C4×D47—432no—
243C4×D47—416no—
244C2×#2486—471nomaximal, Sylow
245C22×C47C2×#12114yes—
246C22×C47C2×#12314yes—
247C22×C47#53614yes—
248C22×C47#54718yes—
249C22×C47#55218yes—
250C22×C47C23×D411yescharacteristic
251C22×C47C23×D411yescharacteristic
252C22×C47C22×#4111yescharacteristic
253C22×C47C22×#4112yes—
254C22×C47C2×2−1+412yes—
255C2×D47—24no—
256C2×D47—24no—
257C2×D47—28no—
258C2×D47—28no—
259C2×D47—232no—
260C2×D47—24no—
261C2×D47—28no—
262C2×D47—42no—
263C2×D47—42no—
264C2×D47—44no—
265C2×D47—416no—
266C2×D47C2×#12118yes—
267C4×C47—24no—
268C4×C47—24no—
269C4×C47—44no—
270C4×C47—42no—
271C4×C47—44no—
272Dic47—44no—
273Dic47—42no—
274Dic47—44no—
275Dic47C2×#12318yescomplemented
276C22×#123—472no—
277C22×C4×D4—471no—
278C23×C4⋊C4—471no—
279#2486—4716no—
280C2×#398—471no—
281C2×#398D4711yescharacteristic, complemented
282C2×#397—471no—
283C2×#397—471no—
284C2×#389—472no—
285C2×#381—472no—
286C2×#380—472no—
287C2×#380—471no—
288C2×C47#248618yescomplemented
289C2×C47C22×#12111yescharacteristic
290C2×C47C22×#12311yescharacteristic
291C2×C47C2×#53611yescharacteristic
292C2×C47C2×#54712yes—
293C2×C47C2×#55212yes—
294D47—28no—
295D47—44no—
296C23×D4—471no—
297C24×C4—471no—
298C24×C4—472no—
299C24×C4—471no—
300C2×#123—4716no—
301C2×#123—9416no—
302C2×C4×D4—478no—
303C2×C4×D4—948no—
304C22×C4⋊C4—471no—
305C22×C4⋊C4—472no—
306C22×C4⋊C4—472no—
307C22×C4⋊C4—472no—
308C22×C4⋊C4—478no—
309C22×C4⋊C4—948no—
310C22×C4⋊C4C2×D4711yescharacteristic
311C22×C4⋊C4C2×D4711yescharacteristic
312C22×C4⋊C4C2×D4712yes—
313C22×C22⋊C4—471no—
314C22×C22⋊C4—472no—
315C22×C22⋊C4—472no—
316C22×C22⋊C4—471no—
317C22×C22⋊C4—472no—
318C22×C42—471no—
319#398—478no—
320#398C2×D4718yescomplemented
321#397—478no—
322#397—478no—
323#389—4716no—
324#381—4716no—
325#380—4716no—
326#380—478no—
327C2×#95—471no—
328C2×#95—472no—
329C2×#95—472no—
330C2×#95—472no—
331C2×#95C2×D4711yescharacteristic
332C2×#95C2×D4712yes—
333C47C2×#248611yescharacteristic, complemented, Sylow
334C25—471no—
335C25—471no—
336C22×D4—478no—
337C22×D4—474no—
338C22×D4—944no—
339C22×D4—944no—
340C23×C4—471no—
341C23×C4—478no—
342C23×C4—474no—
343C23×C4—472no—
344C23×C4—471no—
345C23×C4—472no—
346C23×C4—478no—
347C23×C4—474no—
348C23×C4—948no—
349C23×C4—944no—
350C23×C4—948no—
351C23×C4—948no—
352C23×C4—948no—
353C23×C4—948no—
354C23×C4C22×D4712yes—
355C23×C4C22×D4712yes—
356C23×C4C22×D4711yescharacteristic
357C23×C4C22×D4711yescharacteristic
358C23×C4C22×D4711yescharacteristic
359#123—9464no—
360C4×D4—9432no—
361C2×C4⋊C4—24no—
362C2×C4⋊C4—24no—
363C2×C4⋊C4—28no—
364C2×C4⋊C4—478no—
365C2×C4⋊C4—474no—
366C2×C4⋊C4—478no—
367C2×C4⋊C4—478no—
368C2×C4⋊C4—9432no—
369C2×C4⋊C4—948no—
370C2×C4⋊C4—948no—
371C2×C4⋊C4—944no—
372C2×C4⋊C4—948no—
373C2×C4⋊C4C22×D4714yes—
374C2×C4⋊C4C22×D4714yes—
375C2×C4⋊C4C22×D4718yes—
376C2×C22⋊C4—474no—
377C2×C22⋊C4—474no—
378C2×C22⋊C4—478no—
379C2×C22⋊C4—478no—
380C2×C22⋊C4—478no—
381C2×C22⋊C4—944no—
382C2×C22⋊C4—948no—
383C2×C22⋊C4—948no—
384C2×C22⋊C4—948no—
385C2×C22⋊C4—944no—
386C2×C42—474no—
387C2×C42—944no—
388#95—478no—
389#95—474no—
390#95—478no—
391#95—478no—
392#95C22×D4718yes—
393#95C22×D4714yes—
394C24—474no—
395C24—474no—
396C24—474no—
397C24—472no—
398C24—474no—
399C24—472no—
400C24—944no—
401C24—944no—
402C24—942no—
403C24—944no—
404C24—942no—
405C24—944no—
406C24C23×D4711yescharacteristic
407C2×D4—4716no—
408C2×D4—948no—
409C2×D4—9432no—
410C2×D4—948no—
411C22×C4—22no—
412C22×C4—24no—
413C22×C4—22no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—22no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—474no—
423C22×C4—478no—
424C22×C4—472no—
425C22×C4—4716no—
426C22×C4—472no—
427C22×C4—474no—
428C22×C4—474no—
429C22×C4—9432no—
430C22×C4—9416no—
431C22×C4—948no—
432C22×C4—9416no—
433C22×C4—9416no—
434C22×C4—9416no—
435C22×C4—9416no—
436C22×C4—942no—
437C22×C4—944no—
438C22×C4—944no—
439C22×C4—944no—
440C22×C4—944no—
441C22×C4—944no—
442C22×C4—944no—
443C22×C4—944no—
444C22×C4—18816no—
445C22×C4—18832no—
446C22×C4D4×D4714yes—
447C22×C4D4×D4714yes—
448C22×C4C23×D4712yes—
449C22×C4C23×D4714yes—
450C22×C4C23×D4714yes—
451C22×C4C23×D4712yes—
452C22×C4C23×D4712yes—
453C4⋊C4—28no—
454C4⋊C4—28no—
455C4⋊C4—216no—
456C4⋊C4—9416no—
457C4⋊C4—9416no—
458C4⋊C4—9416no—
459C4⋊C4—948no—
460C22⋊C4—948no—
461C22⋊C4—9416no—
462C22⋊C4—9416no—
463C22⋊C4—9416no—
464C22⋊C4—948no—
465C42—948no—
466C23—4716no—
467C23—474no—
468C23—474no—
469C23—474no—
470C23—474no—
471C23—9416no—
472C23—9416no—
473C23—948no—
474C23—944no—
475C23—948no—
476C23—944no—
477C23—9416no—
478C23—948no—
479C23—942no—
480C23—944no—
481C23—942no—
482C23—944no—
483C23—1888no—
484C23—1884no—
485C23—1884no—
486C23C2×D4×D4712yes—
487C23C2×D4×D4712yes—
488C23#41×D4714yes—
489C23#41×D4712yes—
490C23#1309407214yes—
491C23C24×D4711yescharacteristic
492D4—9432no—
493C2×C4—28no—
494C2×C4—28no—
495C2×C4—22no—
496C2×C4—24no—
497C2×C4—22no—
498C2×C4—24no—
499C2×C4—24no—
500C2×C4—22no—
501C2×C4—24no—
502C2×C4—24no—
503C2×C4—24no—
504C2×C4—24no—
505C2×C4—24no—
506C2×C4—48no—
507C2×C4—48no—
508C2×C4—44no—
509C2×C4—4716no—
510C2×C4—474no—
511C2×C4—478no—
512C2×C4—9432no—
513C2×C4—942no—
514C2×C4—944no—
515C2×C4—944no—
516C2×C4—944no—
517C2×C4—944no—
518C2×C4—944no—
519C2×C4—944no—
520C2×C4—944no—
521C2×C4—18816no—
522C2×C4—18832no—
523C2×C4—1888no—
524C2×C4—1888no—
525C2×C4—1884no—
526C2×C4C2×D4×D4714yes—
527C2×C4C2×D4×D4714yes—
528C22—478no—
529C22—948no—
530C22—944no—
531C22—948no—
532C22—9432no—
533C22—944no—
534C22—948no—
535C22—944no—
536C22—1882no—
537C22—1884no—
538C22—18816no—
539C22—1882no—
540C22#121×D4714yes—
541C22#123×D4714yes—
542C22#1568969814yes—
543C22#1568977518yes—
544C22#1568980518yes—
545C22C22×D4×D4711yescharacteristic
546C22C22×D4×D4711yescharacteristic
547C22C2×#41×D4712yes—
548C22C2×#41×D4711yescharacteristic
549C22C2×#1309407212yes—
550C4—24no—
551C4—24no—
552C4—44no—
553C4—44no—
554C4—42no—
555C4—478no—
556C4—1884no—
557C4—1884no—
558C4—1882no—
559C2—948no—
560C2—1884no—
561C2#2024769118yesdirect product factor
562C2C2×#121×D4711yescharacteristic
563C2C2×#123×D4711yescharacteristic
564C2C2×#1568969811yescharacteristic
565C2C2×#1568977512yes—
566C2C2×#1568980512yes—
subgroupG/H
centerC24C23×D47
commutator, cosocleC23×C47C25
FrattiniC23C24×D47
FittingC2×C47×#398C2
socleC24×C47C24
prime radicalC23×D188C22
hypercenterC2×#398D47
hypocenterC47C2×#2486
LCSG ▷ C23×C47 ▷ C47
UCSC1 ◁ C24 ◁ C2×#398

Complements

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

NH ≅ G/Ntype
3. C22×C4⋊C4×D47C2semidirect
4. #20247691C2direct (N × H)
5. C2×#15689053C2semidirect
6. C2×#15689050C2semidirect
7. C2×#15689049C2semidirect
8. C2×#15689018C2semidirect
9. C2×#15688985C2semidirect
10. C2×#15688982C2semidirect
11. C2×#15688982C2semidirect
12. C2×C47×#398C2semidirect
26. C2×C4⋊C4×D47C22semidirect
33. #15689053C22semidirect
34. #15689050C22semidirect
35. #15689049C22semidirect
36. #15689018C22semidirect
37. #15688985C22semidirect
38. #15688982C22semidirect
39. #15688982C22semidirect
47. C47×#398C22semidirect
237. C2×Dic47#123semidirect
275. Dic47C2×#123semidirect
281. C2×#398D47semidirect
288. C2×C47#2486semidirect
320. #398C2×D47semidirect
333. C47C2×#2486semidirect
561. C2#20247691direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#130940092
C22×#130939862
C22×C4⋊C4×D472
#202476912
C2×#156890532
C2×#156890502
C2×#156890492
C2×#156890182
C2×#156889852
C2×#156889822
C2×C47×#3982
C2×#248647