Finite Group Databasebeta

← C2×#20246902 #26384153

Subgroups — C2×#20246902 #26384153

Order 12032.

3830 conjugacy classes of subgroups.

The subgroup lattice has 648 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#13094006C212yesmaximal, complemented
2C22×#13093998C211yesmaximal, characteristic
3C22×C4×D188C211yesmaximal, characteristic, complemented
4#20246902C2116yesmaximal, direct product factor
5C2×#15689050C212yesmaximal, complemented
6C2×#15689039C212yesmaximal, complemented
7C2×#15689025C211yesmaximal, characteristic, complemented
8C2×#15689021C211yesmaximal, characteristic, complemented
9C2×#15688997C212yesmaximal, complemented
10C2×#15688993C211yesmaximal, characteristic, complemented
11C2×#15688983C211yesmaximal, characteristic, complemented
12C2×C4×C22⋊C4×C47C211yesmaximal, characteristic, complemented
13C23×D188C2211yescharacteristic
14C23×C22⋊D47C2212yes—
15C23×C4×D47C2212yes—
16C2×#13094006—216no—
17C2×#13094006C22116yescomplemented
18C2×#13093998—28no—
19C2×#13093998C2218yes—
20C2×C4×D188—28no—
21C2×C4×D188C2218yescomplemented
22C22×C4⋊Dic47C2211yescharacteristic
23C22×C4⋊Dic47C2211yescharacteristic
24C22×C4⋊Dic47C2211yescharacteristic
25C22×C4⋊Dic47C2212yes—
26C22×C4⋊Dic47C2212yes—
27C22×Dic47⋊C4C2212yes—
28C22×C22⋊Dic47C2212yes—
29C22×(C2×D47)⋊C4C2212yes—
30C22×(C2×D47)⋊C4C2212yes—
31C22×(C2×D47)⋊C4C2212yes—
32C22×(C2×D47)⋊C4C2212yes—
33#15689050C22116yescomplemented
34#15689039C22116yescomplemented
35#15689025C2218yescomplemented
36#15689021C2218yescomplemented
37#15688997C22116yescomplemented
38#15688993C2218yescomplemented
39#15688983C2218yescomplemented
40C2×C47⋊#95C2212yes—
41C2×C47⋊#95C2212yes—
42C24×C4×C47C2211yescharacteristic
43C22×C22⋊C4×C47C2211yescharacteristic
44C22×C22⋊C4×C47C2211yescharacteristic
45C22×C42×C47C2211yescharacteristic
46C22×C42×C47C2211yescharacteristic
47C4×C22⋊C4×C47C2218yescomplemented
48C2×#95×C47C2211yescharacteristic
49C2×#95×C47C2211yescharacteristic
50C24×D47C2312yes—
51C22×D188—24no—
52C22×D188—24no—
53C22×D188D418yes—
54C22×D188C2314yes—
55C22×C22⋊D47—28no—
56C22×C22⋊D47—28no—
57C22×C22⋊D47—44no—
58C22×C22⋊D47C2318yes—
59C23×Dic47C2312yes—
60C23×Dic47C2312yes—
61C23×Dic47C2312yes—
62C22×C4×D47—28no—
63C22×C4×D47—28no—
64C22×C4×D47—44no—
65C22×C4×D47C2318yes—
66#13094006—264no—
67#13093998—232no—
68C4×D188—232no—
69C2×C4⋊Dic47—24no—
70C2×C4⋊Dic47—24no—
71C2×C4⋊Dic47—28no—
72C2×C4⋊Dic47—24no—
73C2×C4⋊Dic47—44no—
74C2×C4⋊Dic47C2314yes—
75C2×C4⋊Dic47C2314yes—
76C2×C4⋊Dic47C2314yes—
77C2×C4⋊Dic47C2318yes—
78C2×C4⋊Dic47C2318yes—
79C2×Dic47⋊C4—28no—
80C2×Dic47⋊C4C2318yes—
81C2×C22⋊Dic47—28no—
82C2×C22⋊Dic47C2318yes—
83C2×(C2×D47)⋊C4—28no—
84C2×(C2×D47)⋊C4—28no—
85C2×(C2×D47)⋊C4—28no—
86C2×(C2×D47)⋊C4—28no—
87C2×(C2×D47)⋊C4C2318yes—
88C2×(C2×D47)⋊C4C2318yes—
89C2×(C2×D47)⋊C4C2318yes—
90C2×(C2×D47)⋊C4C2318yes—
91C47⋊#95C2318yes—
92C47⋊#95C2318yes—
93C25×C47C2311yescharacteristic
94C23×C4×C47—24no—
95C23×C4×C47—24no—
96C23×C4×C47—24no—
97C23×C4×C47C2311yescharacteristic
98C23×C4×C47C2311yescharacteristic
99C23×C4×C47C2314yes—
100C23×C4×C47C2311yescharacteristic
101C23×C4×C47C2311yescharacteristic
102C23×C4×C47C2311yescharacteristic
103C23×C4×C47C2311yescharacteristic
104C2×C22⋊C4×C47—24no—
105C2×C22⋊C4×C47D418yescomplemented
106C2×C22⋊C4×C47C2314yes—
107C2×C22⋊C4×C47C2314yes—
108C2×C42×C47—24no—
109C2×C42×C47—24no—
110C2×C42×C47C2314yes—
111C2×C42×C47C2314yes—
112#95×C47C2314yes—
113#95×C47C2314yes—
114C24×C47—22no—
115C24×C47—22no—
116C24×C47—22no—
117C24×C47—22no—
118C24×C47—22no—
119C24×C47—22no—
120C24×C47C2×D414yes—
121C24×C47C2412yes—
122C24×C47C2411yescharacteristic
123C23×D47—24no—
124C23×D47—24no—
125C23×D47—24no—
126C23×D47—24no—
127C23×D47—24no—
128C23×D47—24no—
129C23×D47C2×D418yes—
130C23×D47C2414yes—
131C22×C4×C47—22no—
132C22×C4×C47—22no—
133C22×C4×C47—22no—
134C22×C4×C47—22no—
135C22×C4×C47—22no—
136C22×C4×C47—22no—
137C22×C4×C47—22no—
138C22×C4×C47—22no—
139C22×C4×C47—22no—
140C22×C4×C47—22no—
141C22×C4×C47—22no—
142C22×C4×C47—22no—
143C22×C4×C47—28no—
144C22×C4×C47—28no—
145C22×C4×C47—28no—
146C22×C4×C47—416no—
147C22×C4×C47C2×D414yes—
148C22×C4×C47C2×D414yes—
149C22×C4×C47C2×D414yes—
150C22×C4×C47#4114yes—
151C22×C4×C47#4114yes—
152C22×C4×C47#4114yes—
153C22×C4×C47C2412yes—
154C22×C4×C47C2412yes—
155C22×C4×C47C2412yes—
156C22×C4×C47C2412yes—
157C22×C4×C47C2412yes—
158C22×C4×C47C2412yes—
159C2×C22⋊D47—216no—
160C2×C22⋊D47—216no—
161C2×C22⋊D47—48no—
162C2×C22⋊D47—432no—
163C22×Dic47—24no—
164C22×Dic47—24no—
165C22×Dic47—24no—
166C22×Dic47—24no—
167C22×Dic47—24no—
168C22×Dic47—24no—
169C22×Dic47—24no—
170C22×Dic47—24no—
171C22×Dic47—24no—
172C22×Dic47C2414yes—
173C22×Dic47C2414yes—
174C22×Dic47C2414yes—
175C2×D188—232no—
176C2×D188—28no—
177C2×D188—28no—
178C2×D188C2×D4116yes—
179C2×C4×D47—216no—
180C2×C4×D47—216no—
181C2×C4×D47—48no—
182C2×C4×D47—432no—
183C22⋊C4×C47—28no—
184C22⋊C4×C47C2×D4116yescomplemented
185C42×C47—28no—
186C42×C47—28no—
187C22⋊Dic47—216no—
188(C2×D47)⋊C4—216no—
189(C2×D47)⋊C4—216no—
190(C2×D47)⋊C4—216no—
191(C2×D47)⋊C4—216no—
192C4⋊Dic47—28no—
193C4⋊Dic47—28no—
194C4⋊Dic47—216no—
195C4⋊Dic47—28no—
196C4⋊Dic47—48no—
197Dic47⋊C4—216no—
198C23×C47—28no—
199C23×C47—28no—
200C23×C47—28no—
201C23×C47—22no—
202C23×C47—22no—
203C23×C47—22no—
204C23×C47—22no—
205C23×C47—22no—
206C23×C47—22no—
207C23×C47—44no—
208C23×C47—44no—
209C23×C47—44no—
210C23×C47—44no—
211C23×C47C22×D412yes—
212C23×C47C22×D414yes—
213C23×C47C22×D412yes—
214C23×C47C2×#4112yes—
215C23×C47C2×#4112yes—
216C23×C47C2×#4112yes—
217C23×C47H(5,𝔽2)12yes—
218C23×C472−1+412yes—
219C23×C47C2511yescharacteristic
220C22×D47—216no—
221C22×D47—216no—
222C22×D47—216no—
223C22×D47—24no—
224C22×D47—24no—
225C22×D47—24no—
226C22×D47—24no—
227C22×D47—24no—
228C22×D47—24no—
229C22×D47—48no—
230C22×D47—48no—
231C22×D47—48no—
232C22×D47—48no—
233C22×D47C22×D418yes—
234C2×C4×C47—28no—
235C2×C4×C47—28no—
236C2×C4×C47—28no—
237C2×C4×C47—22no—
238C2×C4×C47—22no—
239C2×C4×C47—22no—
240C2×C4×C47—22no—
241C2×C4×C47—22no—
242C2×C4×C47—22no—
243C2×C4×C47—22no—
244C2×C4×C47—22no—
245C2×C4×C47—22no—
246C2×C4×C47—22no—
247C2×C4×C47—22no—
248C2×C4×C47—22no—
249C2×C4×C47—44no—
250C2×C4×C47—44no—
251C2×C4×C47—416no—
252C2×C4×C47#123116yescomplemented
253C2×C4×C47C22×D414yes—
254C2×C4×C47C22×D414yes—
255C2×C4×C47C22×D414yes—
256C2×C4×C47C2×#4114yes—
257C2×C4×C47C2×#4114yes—
258C2×C4×C47C2×#4114yes—
259C22⋊D47—432no—
260C2×Dic47—24no—
261C2×Dic47—24no—
262C2×Dic47—24no—
263C2×Dic47—24no—
264C2×Dic47—24no—
265C2×Dic47—24no—
266C2×Dic47—24no—
267C2×Dic47—24no—
268C2×Dic47—24no—
269C2×Dic47—48no—
270C2×Dic47—48no—
271C2×Dic47—48no—
272D188—232no—
273C4×D47—432no—
274C2×#2422—471nomaximal, Sylow
275C22×C47—24no—
276C22×C47—24no—
277C22×C47—24no—
278C22×C47—42no—
279C22×C47—416no—
280C22×C47—42no—
281C22×C47—42no—
282C22×C47—42no—
283C22×C47C2×#12314yes—
284C22×C47#53214yes—
285C22×C47#54214yes—
286C22×C47D4214yes—
287C22×C47#54714yes—
288C22×C47#54714yes—
289C22×C47#55314yes—
290C22×C47C23×D411yescharacteristic
291C22×C47C23×D411yescharacteristic
292C22×C47C22×#4111yescharacteristic
293C22×C47C22×#4111yescharacteristic
294C22×C47C22×#4111yescharacteristic
295C22×C47C2×H(5,𝔽2)11yescharacteristic
296C22×C47C2×2−1+411yescharacteristic
297C2×D47—28no—
298C2×D47—28no—
299C2×D47—28no—
300C2×D47—44no—
301C2×D47—432no—
302C2×D47—44no—
303C2×D47—44no—
304C2×D47—44no—
305C4×C47—24no—
306C4×C47—24no—
307C4×C47—24no—
308C4×C47—42no—
309C4×C47—42no—
310C4×C47C2×#12318yescomplemented
311Dic47—44no—
312Dic47—44no—
313Dic47—44no—
314C22×#123—472no—
315C22×#121—471no—
316C22×C4×D4—471no—
317#2422—4716no—
318C2×#397—472no—
319C2×#394—472no—
320C2×#390—471no—
321C2×#389—471no—
322C2×#385—472no—
323C2×#384—471no—
324C2×#380—471no—
325C2×C4×C22⋊C4D4711yescharacteristic, complemented
326C2×C47—44no—
327C2×C47#242218yescomplemented
328C2×C47C22×#12311yescharacteristic
329C2×C47C2×#53211yescharacteristic
330C2×C47C2×#54211yescharacteristic
331C2×C47C2×D4211yescharacteristic
332C2×C47C2×#54711yescharacteristic
333C2×C47C2×#54711yescharacteristic
334C2×C47C2×#55311yescharacteristic
335D47—48no—
336C23×D4—472no—
337C23×D4—471no—
338C24×C4—472no—
339C24×C4C2×D4711yescharacteristic
340C2×#123—4716no—
341C2×#123—9416no—
342C2×#121—478no—
343C2×#121—948no—
344C2×C4×D4—478no—
345C2×C4×D4—948no—
346C22×C4⋊C4—472no—
347C22×C4⋊C4—471no—
348C22×C4⋊C4—472no—
349C22×C4⋊C4—471no—
350C22×C4⋊C4—471no—
351C22×C4⋊C4—472no—
352C22×C22⋊C4—472no—
353C22×C22⋊C4—472no—
354C22×C22⋊C4—472no—
355C22×C22⋊C4—472no—
356C22×C22⋊C4—472no—
357C22×C22⋊C4C2×D4711yescharacteristic
358C22×C22⋊C4C2×D4711yescharacteristic
359C22×C42C2×D4711yescharacteristic
360C22×C42C2×D4711yescharacteristic
361#397—4716no—
362#394—4716no—
363#390—478no—
364#389—478no—
365#385—4716no—
366#384—478no—
367#380—478no—
368C4×C22⋊C4C2×D4718yescomplemented
369C2×#95—472no—
370C2×#95—472no—
371C2×#95C2×D4711yescharacteristic
372C2×#95C2×D4711yescharacteristic
373C47C2×#242211yescharacteristic, complemented, Sylow
374C25—472no—
375C25C22×D4711yescharacteristic
376C22×D4—478no—
377C22×D4—478no—
378C22×D4—474no—
379C22×D4—948no—
380C22×D4—948no—
381C22×D4—944no—
382C22×D4—944no—
383C22×D4—1884no—
384C23×C4—24no—
385C23×C4—24no—
386C23×C4—24no—
387C23×C4—472no—
388C23×C4—472no—
389C23×C4—472no—
390C23×C4—478no—
391C23×C4—948no—
392C23×C4—948no—
393C23×C4—1884no—
394C23×C4C22×D4714yes—
395C23×C4C22×D4711yescharacteristic
396C23×C4C22×D4711yescharacteristic
397C23×C4C22×D4711yescharacteristic
398C23×C4C22×D4711yescharacteristic
399C23×C4C22×D4711yescharacteristic
400C23×C4C22×D4711yescharacteristic
401#123—9464no—
402#121—9432no—
403C4×D4—9432no—
404C2×C4⋊C4—474no—
405C2×C4⋊C4—474no—
406C2×C4⋊C4—478no—
407C2×C4⋊C4—478no—
408C2×C4⋊C4—474no—
409C2×C4⋊C4—478no—
410C2×C4⋊C4—948no—
411C2×C4⋊C4—948no—
412C2×C4⋊C4—944no—
413C2×C4⋊C4—944no—
414C2×C4⋊C4—944no—
415C2×C4⋊C4—1884no—
416C2×C22⋊C4—24no—
417C2×C22⋊C4—478no—
418C2×C22⋊C4—478no—
419C2×C22⋊C4—478no—
420C2×C22⋊C4—478no—
421C2×C22⋊C4—478no—
422C2×C22⋊C4—948no—
423C2×C22⋊C4—948no—
424C2×C22⋊C4—948no—
425C2×C22⋊C4—948no—
426C2×C22⋊C4—948no—
427C2×C22⋊C4D18818yescomplemented
428C2×C22⋊C4C22×D4714yes—
429C2×C22⋊C4C22×D4714yes—
430C2×C42—24no—
431C2×C42—24no—
432C2×C42C22×D4714yes—
433C2×C42C22×D4714yes—
434#95—478no—
435#95—478no—
436#95C22×D4714yes—
437#95C22×D4714yes—
438C24—22no—
439C24—22no—
440C24—22no—
441C24—22no—
442C24—22no—
443C24—22no—
444C24—478no—
445C24—474no—
446C24—944no—
447C24—944no—
448C24—944no—
449C24—944no—
450C24—944no—
451C24—944no—
452C24C2×D18814yes—
453C24C23×D4712yes—
454C24C23×D4711yescharacteristic
455C2×D4—4716no—
456C2×D4—9416no—
457C2×D4—9416no—
458C2×D4—9432no—
459C2×D4—948no—
460C2×D4—948no—
461C2×D4—1888no—
462C2×D4—18832no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—22no—
474C22×C4—22no—
475C22×C4—28no—
476C22×C4—28no—
477C22×C4—28no—
478C22×C4—416no—
479C22×C4—474no—
480C22×C4—474no—
481C22×C4—474no—
482C22×C4—9416no—
483C22×C4—944no—
484C22×C4—944no—
485C22×C4—944no—
486C22×C4—944no—
487C22×C4—944no—
488C22×C4—944no—
489C22×C4—944no—
490C22×C4—944no—
491C22×C4—944no—
492C22×C4—9416no—
493C22×C4—1888no—
494C22×C4—18832no—
495C22×C4C2×D18814yes—
496C22×C4C2×D18814yes—
497C22×C4#1062535014yes—
498C22×C4#1062535014yes—
499C22×C4D4×D4714yes—
500C22×C4#1062535214yes—
501C22×C4C23×D4712yes—
502C22×C4C23×D4712yes—
503C22×C4C23×D4712yes—
504C22×C4C23×D4712yes—
505C22×C4C23×D4712yes—
506C22×C4C23×D4712yes—
507C4⋊C4—948no—
508C4⋊C4—948no—
509C4⋊C4—9416no—
510C4⋊C4—9416no—
511C4⋊C4—948no—
512C4⋊C4—1888no—
513C22⋊C4—28no—
514C22⋊C4—9416no—
515C22⋊C4—9416no—
516C22⋊C4—9416no—
517C22⋊C4—9416no—
518C22⋊C4—9416no—
519C22⋊C4C2×D188116yescomplemented
520C42—28no—
521C42—28no—
522C23—28no—
523C23—28no—
524C23—28no—
525C23—22no—
526C23—22no—
527C23—22no—
528C23—22no—
529C23—22no—
530C23—22no—
531C23—44no—
532C23—44no—
533C23—44no—
534C23—44no—
535C23—478no—
536C23—9416no—
537C23—9416no—
538C23—9416no—
539C23—944no—
540C23—944no—
541C23—944no—
542C23—944no—
543C23—944no—
544C23—944no—
545C23—1888no—
546C23—1888no—
547C23—1888no—
548C23—1888no—
549C23C2×D4×D4712yes—
550C23C22×D18814yes—
551C23C22×D18812yes—
552C23C2×#1062535212yes—
553C23C2×#1062535012yes—
554C23C2×#1062535012yes—
555C23#1309407012yes—
556C23#1309407112yes—
557C23C24×D4711yescharacteristic
558D4—9432no—
559D4—18832no—
560C2×C4—28no—
561C2×C4—28no—
562C2×C4—28no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—44no—
576C2×C4—44no—
577C2×C4—416no—
578C2×C4—944no—
579C2×C4—944no—
580C2×C4—944no—
581C2×C4—944no—
582C2×C4—944no—
583C2×C4—944no—
584C2×C4—944no—
585C2×C4—944no—
586C2×C4—944no—
587C2×C4—18832no—
588C2×C4—1888no—
589C2×C4—1888no—
590C2×C4—1888no—
591C2×C4#13094006116yescomplemented
592C2×C4C2×D4×D4714yes—
593C2×C4C22×D18814yes—
594C2×C4C22×D18814yes—
595C2×C4C2×#1062535214yes—
596C2×C4C2×#1062535014yes—
597C2×C4C2×#1062535014yes—
598C22—24no—
599C22—24no—
600C22—24no—
601C22—42no—
602C22—42no—
603C22—416no—
604C22—42no—
605C22—42no—
606C22—948no—
607C22—948no—
608C22—948no—
609C22—1884no—
610C22—18832no—
611C22—1884no—
612C22—1884no—
613C22—1884no—
614C22C2×#1309400614yes—
615C22#1568967314yes—
616C22#1568974414yes—
617C22D4×D18814yes—
618C22#1568977714yes—
619C22#1568977814yes—
620C22#1568981914yes—
621C22C22×D4×D4711yescharacteristic
622C22C23×D18811yescharacteristic
623C22C22×#1062535211yescharacteristic
624C22C22×#1062535011yescharacteristic
625C22C22×#1062535011yescharacteristic
626C22C2×#1309407011yescharacteristic
627C22C2×#1309407111yescharacteristic
628C4—24no—
629C4—24no—
630C4—24no—
631C4—42no—
632C4—42no—
633C4—1884no—
634C4—1884no—
635C4—1884no—
636C4C2×#1309400618yescomplemented
637C2—44no—
638C2—1888no—
639C2#2024690218yesdirect product factor
640C2C22×#1309400611yescharacteristic
641C2C2×#1568967311yescharacteristic
642C2C2×#1568974411yescharacteristic
643C2C2×D4×D18811yescharacteristic
644C2C2×#1568977711yescharacteristic
645C2C2×#1568977811yescharacteristic
646C2C2×#1568981911yescharacteristic
subgroupG/H
centerC24C23×D47
commutator, cosocleC23×C47C25
FrattiniC23C24×D47
FittingC2×C4×C22⋊C4×C47C2
socleC24×C47C24
prime radicalC22×#13093998C2
hypercenterC2×C4×C22⋊C4D47
hypocenterC47C2×#2422
LCSG ▷ C23×C47 ▷ C47
UCSC1 ◁ C24 ◁ C2×C4×C22⋊C4

Complements

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

NH ≅ G/Ntype
1. C22×#13094006C2semidirect
3. C22×C4×D188C2semidirect
4. #20246902C2direct (N × H)
5. C2×#15689050C2semidirect
6. C2×#15689039C2semidirect
7. C2×#15689025C2semidirect
8. C2×#15689021C2semidirect
9. C2×#15688997C2semidirect
10. C2×#15688993C2semidirect
11. C2×#15688983C2semidirect
12. C2×C4×C22⋊C4×C47C2semidirect
17. C2×#13094006C22semidirect
21. C2×C4×D188C22semidirect
33. #15689050C22semidirect
34. #15689039C22semidirect
35. #15689025C22semidirect
36. #15689021C22semidirect
37. #15688997C22semidirect
38. #15688993C22semidirect
39. #15688983C22semidirect
47. C4×C22⋊C4×C47C22semidirect
105. C2×C22⋊C4×C47D4semidirect
184. C22⋊C4×C47C2×D4semidirect
252. C2×C4×C47#123semidirect
310. C4×C47C2×#123semidirect
325. C2×C4×C22⋊C4D47semidirect
327. C2×C47#2422semidirect
368. C4×C22⋊C4C2×D47semidirect
373. C47C2×#2422semidirect
427. C2×C22⋊C4D188semidirect
519. C22⋊C4C2×D188semidirect
591. C2×C4#13094006semidirect
636. C4C2×#13094006semidirect
639. C2#20246902direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#130940062
C22×#130939982
C22×C4×D1882
#202469022
C2×#156890502
C2×#156890392
C2×#156890252
C2×#156890212
C2×#156889972
C2×#156889932
C2×#156889832
C2×C4×C22⋊C4×C472
C2×#242247