Finite Group Databasebeta

← C2×#20247700 #26385449

Subgroups — C2×#20247700 #26385449

Order 12032.

3374 conjugacy classes of subgroups.

The subgroup lattice has 564 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#13093986C211yesmaximal, characteristic
2C22×C4⋊C4×D47C211yesmaximal, characteristic, complemented
3#20247700C2116yesmaximal, direct product factor
4C2×#15689062C211yesmaximal, characteristic, complemented
5C2×#15689060C211yesmaximal, characteristic, complemented
6C2×#15689058C211yesmaximal, characteristic, complemented
7C2×#15689020C212yesmaximal, complemented
8C2×#15689001C212yesmaximal
9C2×#15688997C212yesmaximal, complemented
10C2×#15688982C212yesmaximal, complemented
11C2×#15688982C211yesmaximal, characteristic, complemented
12C2×C47×#400C211yesmaximal, characteristic, complemented
13C23×D188C2211yescharacteristic
14C23×C4×D47C2211yescharacteristic
15C23×C4×D47C2211yescharacteristic
16C23×C4×D47C2212yes—
17C2×#13093986—28no—
18C2×#13093986C2218yes—
19C22×C4⋊Dic47C2211yescharacteristic
20C22×C4⋊Dic47C2212yes—
21C22×Dic47⋊C4C2212yes—
22C22×Dic47⋊C4C2212yes—
23C22×Dic47⋊C4C2212yes—
24C2×C4⋊C4×D47—28no—
25C2×C4⋊C4×D47C2218yescomplemented
26C22×(C2×D47)⋊C4C2212yes—
27C22×(C2×D47)⋊C4C2211yescharacteristic
28C22×(C2×D47)⋊C4C2211yescharacteristic
29C22×(C2×D47)⋊C4C2212yes—
30C22×(C2×D47)⋊C4C2212yes—
31C22×C4×Dic47C2211yescharacteristic
32#15689062C2218yescomplemented
33#15689060C2218yescomplemented
34#15689058C2218yescomplemented
35#15689020C22116yescomplemented
36#15689001C22116yes—
37#15688997C22116yescomplemented
38#15688982C22116yescomplemented
39#15688982C2218yescomplemented
40C2×C47⋊#95C2212yes—
41C2×C47⋊#95C2211yescharacteristic
42C2×C47⋊#95C2212yes—
43C22×C4⋊C4×C47C2211yescharacteristic
44C22×C4⋊C4×C47C2211yescharacteristic
45C47×#400C2218yescomplemented
46C2×#95×C47C2212yes—
47C2×#95×C47C2211yescharacteristic
48C2×#95×C47C2212yes—
49C24×D47C2311yescharacteristic
50C24×D47C2311yescharacteristic
51C22×D188—24no—
52C22×D188—24no—
53C22×D188Q818yes—
54C22×D188C2314yes—
55C23×Dic47C2312yes—
56C23×Dic47C2312yes—
57C23×Dic47C2311yescharacteristic
58C23×Dic47C2311yescharacteristic
59C22×C4×D47—28no—
60C22×C4×D47—28no—
61C22×C4×D47—24no—
62C22×C4×D47—28no—
63C22×C4×D47—44no—
64C22×C4×D47—44no—
65C22×C4×D47D418yes—
66C22×C4×D47C2314yes—
67C22×C4×D47C2314yes—
68C22×C4×D47C2318yes—
69#13093986—232no—
70C2×C4⋊Dic47—24no—
71C2×C4⋊Dic47—44no—
72C2×C4⋊Dic47C2314yes—
73C2×C4⋊Dic47C2318yes—
74C2×Dic47⋊C4—28no—
75C2×Dic47⋊C4—28no—
76C2×Dic47⋊C4—44no—
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—24no—
84C2×(C2×D47)⋊C4—44no—
85C2×(C2×D47)⋊C4—44no—
86C2×(C2×D47)⋊C4C2318yes—
87C2×(C2×D47)⋊C4C2314yes—
88C2×(C2×D47)⋊C4C2314yes—
89C2×(C2×D47)⋊C4C2318yes—
90C2×(C2×D47)⋊C4C2318yes—
91C2×C4×Dic47—24no—
92C2×C4×Dic47C2314yes—
93C47⋊#95C2318yes—
94C47⋊#95C2314yes—
95C47⋊#95C2318yes—
96C23×C4×C47C2311yescharacteristic
97C23×C4×C47C2312yes—
98C23×C4×C47C2312yes—
99C23×C4×C47C2311yescharacteristic
100C23×C4×C47C2311yescharacteristic
101C2×C4⋊C4×C47—24no—
102C2×C4⋊C4×C47—24no—
103C2×C4⋊C4×C47C2314yes—
104C2×C4⋊C4×C47C2314yes—
105#95×C47C2318yes—
106#95×C47C2314yes—
107#95×C47C2318yes—
108C24×C47C2411yescharacteristic
109C23×D47—24no—
110C23×D47—24no—
111C23×D47—22no—
112C23×D47—24no—
113C23×D47—22no—
114C23×D47—24no—
115C23×D47C2×D414yes—
116C23×D47C2×Q814yes—
117C23×D47C2×Q814yes—
118C23×D47#4114yes—
119C23×D47C2412yes—
120C23×D47C2412yes—
121C22×C4×C47—22no—
122C22×C4×C47—24no—
123C22×C4×C47—22no—
124C22×C4×C47—22no—
125C22×C4×C47—24no—
126C22×C4×C47—24no—
127C22×C4×C47—24no—
128C22×C4×C47—24no—
129C22×C4×C47—24no—
130C22×C4×C47—24no—
131C22×C4×C47—24no—
132C22×C4×C47C2×D414yes—
133C22×C4×C47C2×Q814yes—
134C22×C4×C47C2412yes—
135C22×C4×C47C2414yes—
136C22×C4×C47C2414yes—
137C22×C4×C47C2412yes—
138C22×C4×C47C2412yes—
139C22×Dic47—22no—
140C22×Dic47—24no—
141C22×Dic47—24no—
142C22×Dic47—24no—
143C22×Dic47—24no—
144C22×Dic47—24no—
145C22×Dic47—24no—
146C22×Dic47—24no—
147C22×Dic47C2×D414yes—
148C22×Dic47#4118yes—
149C22×Dic47C2414yes—
150C22×Dic47C2414yes—
151C22×Dic47C2412yes—
152C22×Dic47C2412yes—
153C2×D188—232no—
154C2×D188—28no—
155C2×D188—28no—
156C2×D188C2×Q8116yes—
157C2×C4×D47—232no—
158C2×C4×D47—216no—
159C2×C4×D47—216no—
160C2×C4×D47—28no—
161C2×C4×D47—216no—
162C2×C4×D47—48no—
163C2×C4×D47—48no—
164C2×C4×D47—432no—
165C2×C4×D47—416no—
166C2×C4×D47C2×D4116yes—
167C4⋊C4×C47—28no—
168C4⋊C4×C47—28no—
169(C2×D47)⋊C4—28no—
170(C2×D47)⋊C4—216no—
171(C2×D47)⋊C4—28no—
172(C2×D47)⋊C4—48no—
173(C2×D47)⋊C4—48no—
174C4⋊Dic47—28no—
175C4⋊Dic47—48no—
176Dic47⋊C4—216no—
177Dic47⋊C4—216no—
178Dic47⋊C4—48no—
179C4×Dic47—28no—
180C23×C47C22×D412yes—
181C23×C47C22×Q812yes—
182C23×C47C2×#4112yes—
183C23×C47C2×#4114yes—
184C23×C47H(5,𝔽2)14yes—
185C23×C47C2511yescharacteristic
186C22×D47—28no—
187C22×D47—28no—
188C22×D47—216no—
189C22×D47—216no—
190C22×D47—24no—
191C22×D47—24no—
192C22×D47—28no—
193C22×D47—216no—
194C22×D47—22no—
195C22×D47—24no—
196C22×D47—22no—
197C22×D47—24no—
198C22×D47—44no—
199C22×D47—44no—
200C22×D47—48no—
201C22×D47C22⋊Q8116yes—
202C22×D47C22×D414yes—
203C22×D47C22×Q814yes—
204C22×D47C22×Q814yes—
205C22×D47C2×#4114yes—
206C2×C4×C47—28no—
207C2×C4×C47—28no—
208C2×C4×C47—22no—
209C2×C4×C47—24no—
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—24no—
217C2×C4×C47—24no—
218C2×C4×C47—24no—
219C2×C4×C47—44no—
220C2×C4×C47—48no—
221C2×C4×C47—48no—
222C2×C4×C47C22×D414yes—
223C2×C4×C47C22×Q814yes—
224C2×Dic47—22no—
225C2×Dic47—24no—
226C2×Dic47—24no—
227C2×Dic47—24no—
228C2×Dic47—24no—
229C2×Dic47—24no—
230C2×Dic47—24no—
231C2×Dic47—24no—
232C2×Dic47—48no—
233C2×Dic47—44no—
234C2×Dic47—48no—
235C2×Dic47#123116yes—
236C2×Dic47C22×D414yes—
237C2×Dic47C2×#4118yes—
238D188—232no—
239C4×D47—232no—
240C4×D47—432no—
241C4×D47—416no—
242C2×#2487—471nomaximal, Sylow
243C22×C47C2×C22⋊Q814yes—
244C22×C47C2×#12314yes—
245C22×C47#53514yes—
246C22×C47#55318yes—
247C22×C47#55418yes—
248C22×C47C23×D411yescharacteristic
249C22×C47C23×Q811yescharacteristic
250C22×C47C22×#4111yescharacteristic
251C22×C47C22×#4112yes—
252C22×C47C2×H(5,𝔽2)12yes—
253C2×D47—24no—
254C2×D47—24no—
255C2×D47—28no—
256C2×D47—28no—
257C2×D47—232no—
258C2×D47—24no—
259C2×D47—28no—
260C2×D47—42no—
261C2×D47—42no—
262C2×D47—44no—
263C2×D47—416no—
264C2×D47C2×C22⋊Q818yes—
265C4×C47—24no—
266C4×C47—24no—
267C4×C47—42no—
268C4×C47—44no—
269C4×C47—44no—
270Dic47—44no—
271Dic47—42no—
272Dic47—44no—
273Dic47C2×#12318yes—
274C22×C4×D4—471no—
275C23×C4⋊C4—471no—
276#2487—4716no—
277C2×#400—471no—
278C2×#400D4711yescharacteristic, complemented
279C2×#399—471no—
280C2×#399—471no—
281C2×#389—472no—
282C2×#386—472no—
283C2×#385—472no—
284C2×#380—471no—
285C2×#380—472no—
286C2×C47#248718yescomplemented
287C2×C47C22×C22⋊Q811yescharacteristic
288C2×C47C22×#12311yescharacteristic
289C2×C47C2×#53511yescharacteristic
290C2×C47C2×#55312yes—
291C2×C47C2×#55412yes—
292D47—28no—
293D47—44no—
294C23×D4—471no—
295C24×C4—471no—
296C24×C4—471no—
297C24×C4—472no—
298C2×C4×D4—478no—
299C2×C4×D4—948no—
300C22×C4⋊C4—471no—
301C22×C4⋊C4—472no—
302C22×C4⋊C4—472no—
303C22×C4⋊C4—472no—
304C22×C4⋊C4—472no—
305C22×C4⋊C4—478no—
306C22×C4⋊C4—948no—
307C22×C4⋊C4C2×D4711yescharacteristic
308C22×C4⋊C4C2×D4711yescharacteristic
309C22×C22⋊C4—471no—
310C22×C22⋊C4—471no—
311C22×C22⋊C4—472no—
312C22×C22⋊C4—472no—
313C22×C22⋊C4—472no—
314C22×C42—471no—
315#400—478no—
316#400C2×D4718yescomplemented
317#399—478no—
318#399—478no—
319#389—4716no—
320#386—4716no—
321#385—4716no—
322#380—4716no—
323#380—478no—
324C2×#95—471no—
325C2×#95—472no—
326C2×#95—472no—
327C2×#95C2×D4711yescharacteristic
328C2×#95C2×D4712yes—
329C2×#95C2×D4712yes—
330C47C2×#248711yescharacteristic, complemented, Sylow
331C25—471no—
332C25—471no—
333C22×D4—478no—
334C22×D4—474no—
335C22×D4—944no—
336C22×D4—944no—
337C23×C4—471no—
338C23×C4—478no—
339C23×C4—474no—
340C23×C4—471no—
341C23×C4—472no—
342C23×C4—472no—
343C23×C4—474no—
344C23×C4—478no—
345C23×C4—948no—
346C23×C4—944no—
347C23×C4—948no—
348C23×C4—948no—
349C23×C4—1884no—
350C23×C4—1884no—
351C23×C4C22×D4712yes—
352C23×C4C22×D4712yes—
353C23×C4C22×D4711yescharacteristic
354C23×C4C22×D4711yescharacteristic
355C23×C4C22×D4711yescharacteristic
356C4×D4—9432no—
357C2×C4⋊C4—24no—
358C2×C4⋊C4—24no—
359C2×C4⋊C4—478no—
360C2×C4⋊C4—474no—
361C2×C4⋊C4—478no—
362C2×C4⋊C4—478no—
363C2×C4⋊C4—478no—
364C2×C4⋊C4—9432no—
365C2×C4⋊C4—948no—
366C2×C4⋊C4—948no—
367C2×C4⋊C4—944no—
368C2×C4⋊C4—1884no—
369C2×C4⋊C4—1884no—
370C2×C4⋊C4C22×D4714yes—
371C2×C4⋊C4C22×D4714yes—
372C2×C22⋊C4—474no—
373C2×C22⋊C4—474no—
374C2×C22⋊C4—478no—
375C2×C22⋊C4—478no—
376C2×C22⋊C4—478no—
377C2×C22⋊C4—944no—
378C2×C22⋊C4—948no—
379C2×C22⋊C4—944no—
380C2×C22⋊C4—1884no—
381C2×C22⋊C4—1884no—
382C2×C42—474no—
383C2×C42—944no—
384#95—478no—
385#95—474no—
386#95—478no—
387#95C22×D4718yes—
388#95C22×D4714yes—
389#95C22×D4718yes—
390C24—474no—
391C24—474no—
392C24—474no—
393C24—472no—
394C24—474no—
395C24—472no—
396C24—944no—
397C24—944no—
398C24—942no—
399C24—944no—
400C24—942no—
401C24—944no—
402C24C23×D4711yescharacteristic
403C2×D4—4716no—
404C2×D4—948no—
405C2×D4—9432no—
406C2×D4—948no—
407C22×C4—22no—
408C22×C4—24no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—474no—
419C22×C4—478no—
420C22×C4—472no—
421C22×C4—4716no—
422C22×C4—474no—
423C22×C4—474no—
424C22×C4—472no—
425C22×C4—9432no—
426C22×C4—9416no—
427C22×C4—948no—
428C22×C4—9416no—
429C22×C4—9416no—
430C22×C4—942no—
431C22×C4—944no—
432C22×C4—944no—
433C22×C4—944no—
434C22×C4—944no—
435C22×C4—944no—
436C22×C4—944no—
437C22×C4—944no—
438C22×C4—1888no—
439C22×C4—1888no—
440C22×C4—18816no—
441C22×C4—18832no—
442C22×C4D4×D4714yes—
443C22×C4Q8×D4714yes—
444C22×C4C23×D4712yes—
445C22×C4C23×D4714yes—
446C22×C4C23×D4714yes—
447C22×C4C23×D4712yes—
448C22×C4C23×D4712yes—
449C4⋊C4—28no—
450C4⋊C4—28no—
451C4⋊C4—9416no—
452C4⋊C4—9416no—
453C4⋊C4—948no—
454C4⋊C4—1888no—
455C4⋊C4—1888no—
456C22⋊C4—9416no—
457C22⋊C4—948no—
458C22⋊C4—948no—
459C22⋊C4—1888no—
460C22⋊C4—1888no—
461C42—948no—
462C23—4716no—
463C23—474no—
464C23—474no—
465C23—474no—
466C23—474no—
467C23—9416no—
468C23—9416no—
469C23—948no—
470C23—948no—
471C23—944no—
472C23—944no—
473C23—9416no—
474C23—948no—
475C23—942no—
476C23—944no—
477C23—942no—
478C23—944no—
479C23—1888no—
480C23—1884no—
481C23—1884no—
482C23C2×D4×D4712yes—
483C23C2×Q8×D4712yes—
484C23#41×D4712yes—
485C23#41×D4714yes—
486C23#1309407014yes—
487C23C24×D4711yescharacteristic
488D4—9432no—
489C2×C4—28no—
490C2×C4—28no—
491C2×C4—22no—
492C2×C4—24no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—24no—
496C2×C4—24no—
497C2×C4—24no—
498C2×C4—24no—
499C2×C4—24no—
500C2×C4—24no—
501C2×C4—24no—
502C2×C4—48no—
503C2×C4—44no—
504C2×C4—48no—
505C2×C4—4716no—
506C2×C4—474no—
507C2×C4—478no—
508C2×C4—9432no—
509C2×C4—942no—
510C2×C4—944no—
511C2×C4—944no—
512C2×C4—944no—
513C2×C4—944no—
514C2×C4—944no—
515C2×C4—944no—
516C2×C4—944no—
517C2×C4—18816no—
518C2×C4—18832no—
519C2×C4—1888no—
520C2×C4—1884no—
521C2×C4—1888no—
522C2×C4C2×D4×D4714yes—
523C2×C4C2×Q8×D4714yes—
524C22—478no—
525C22—944no—
526C22—948no—
527C22—9432no—
528C22—948no—
529C22—944no—
530C22—944no—
531C22—948no—
532C22—1882no—
533C22—1884no—
534C22—18816no—
535C22—1882no—
536C22C22⋊Q8×D4714yes—
537C22#123×D4714yes—
538C22#1568969114yes—
539C22#1568982418yes—
540C22#1568982918yes—
541C22C22×D4×D4711yescharacteristic
542C22C22×Q8×D4711yescharacteristic
543C22C2×#41×D4711yescharacteristic
544C22C2×#41×D4712yes—
545C22C2×#1309407012yes—
546C4—24no—
547C4—24no—
548C4—42no—
549C4—44no—
550C4—44no—
551C4—478no—
552C4—1884no—
553C4—1884no—
554C4—1882no—
555C2—948no—
556C2—1884no—
557C2#2024770018yesdirect product factor
558C2C2×C22⋊Q8×D4711yescharacteristic
559C2C2×#123×D4711yescharacteristic
560C2C2×#1568969111yescharacteristic
561C2C2×#1568982412yes—
562C2C2×#1568982912yes—
subgroupG/H
centerC24C23×D47
commutator, cosocleC23×C47C25
FrattiniC23C24×D47
FittingC2×C47×#400C2
socleC24×C47C24
prime radicalC23×D188C22
hypercenterC2×#400D47
hypocenterC47C2×#2487
LCSG ▷ C23×C47 ▷ C47
UCSC1 ◁ C24 ◁ C2×#400

Complements

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

NH ≅ G/Ntype
2. C22×C4⋊C4×D47C2semidirect
3. #20247700C2direct (N × H)
4. C2×#15689062C2semidirect
5. C2×#15689060C2semidirect
6. C2×#15689058C2semidirect
7. C2×#15689020C2semidirect
9. C2×#15688997C2semidirect
10. C2×#15688982C2semidirect
11. C2×#15688982C2semidirect
12. C2×C47×#400C2semidirect
25. C2×C4⋊C4×D47C22semidirect
32. #15689062C22semidirect
33. #15689060C22semidirect
34. #15689058C22semidirect
35. #15689020C22semidirect
37. #15688997C22semidirect
38. #15688982C22semidirect
39. #15688982C22semidirect
45. C47×#400C22semidirect
278. C2×#400D47semidirect
286. C2×C47#2487semidirect
316. #400C2×D47semidirect
330. C47C2×#2487semidirect
557. C2#20247700direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#130939862
C22×C4⋊C4×D472
#202477002
C2×#156890622
C2×#156890602
C2×#156890582
C2×#156890202
C2×#156890012
C2×#156889972
C2×#156889822
C2×C47×#4002
C2×#248747