Finite Group Databasebeta

← C2×#20249199 #26382623

Subgroups — C2×#20249199 #26382623

Order 12032.

4142 conjugacy classes of subgroups.

The subgroup lattice has 608 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#13094001C211yesmaximal, characteristic, complemented
2C23×(C2×D47)⋊C4C211yesmaximal, characteristic
3#20249199C2116yesmaximal, direct product factor
4C2×#15689058C212yesmaximal, complemented
5C2×#15689047C212yesmaximal, complemented
6C2×#15689045C212yesmaximal, complemented
7C2×#15689036C212yesmaximal, complemented
8C2×#15688996C211yesmaximal, characteristic, complemented
9C2×#15688981C211yesmaximal, characteristic, complemented
10C2×#15688978C211yesmaximal, characteristic
11C2×#15688978C211yesmaximal, characteristic
12C2×C47×#385C211yesmaximal, characteristic, complemented
13C25×D47C2211yescharacteristic
14C23×Dic94C2211yescharacteristic, complemented
15C24×Dic47C2211yescharacteristic
16C2×#13094001—28no—
17C2×#13094001C2218yescomplemented
18C22×C4⋊Dic47C2211yescharacteristic, complemented
19C22×Dic47⋊C4C2211yescharacteristic, complemented
20C22×Dic47⋊C4C2211yescharacteristic, complemented
21C22×Dic47⋊C4C2212yescomplemented
22C22×Dic47⋊C4C2212yescomplemented
23C22×C22⋊Dic47C2211yescharacteristic
24C22×C22⋊Dic47C2211yescharacteristic
25C22×(C2×D47)⋊C4—28no—
26C22×(C2×D47)⋊C4C2212yes—
27C22×(C2×D47)⋊C4C2212yes—
28C22×(C2×D47)⋊C4C2218yes—
29C22×(C2×D47)⋊C4C2212yes—
30C22×(C2×D47)⋊C4C2212yes—
31C22×(C2×D47)⋊C4C2212yes—
32C22×(C2×D47)⋊C4C2212yes—
33#15689058C22116yescomplemented
34#15689047C22116yescomplemented
35#15689045C22116yescomplemented
36#15689036C22116yescomplemented
37#15688996C2218yescomplemented
38#15688981C2218yescomplemented
39#15688978C2218yes—
40#15688978C2218yes—
41C2×C47⋊#95C2212yescomplemented
42C2×C47⋊#95C2212yescomplemented
43C24×C4×C47C2211yescharacteristic
44C22×C4⋊C4×C47C2212yescomplemented
45C22×C22⋊C4×C47C2211yescharacteristic
46C22×C22⋊C4×C47C2211yescharacteristic
47C47×#385C2218yescomplemented
48C2×#95×C47C2212yescomplemented
49C24×D47—24no—
50C24×D47—24no—
51C24×D47—44no—
52C24×D47—44no—
53C24×D47Q818yescomplemented
54C24×D47C2312yes—
55C24×D47C2314yes—
56C22×Dic94—24no—
57C22×Dic94—44no—
58C22×Dic94C2314yescomplemented
59C23×Dic47—44no—
60C23×Dic47D418yes—
61C23×Dic47C2312yes—
62C23×Dic47C2311yescharacteristic
63C23×Dic47C2311yescharacteristic
64C23×Dic47C2311yescharacteristic
65C23×Dic47C2311yescharacteristic
66C23×Dic47C2314yes—
67#13094001—232no—
68C2×C4⋊Dic47—24no—
69C2×C4⋊Dic47C2314yescomplemented
70C2×Dic47⋊C4—24no—
71C2×Dic47⋊C4—24no—
72C2×Dic47⋊C4—44no—
73C2×Dic47⋊C4—44no—
74C2×Dic47⋊C4C2314yescomplemented
75C2×Dic47⋊C4C2314yescomplemented
76C2×Dic47⋊C4C2318yescomplemented
77C2×Dic47⋊C4C2318yescomplemented
78C2×C22⋊Dic47—24no—
79C2×C22⋊Dic47—24no—
80C2×C22⋊Dic47C2314yes—
81C2×C22⋊Dic47C2314yes—
82C2×(C2×D47)⋊C4—28no—
83C2×(C2×D47)⋊C4—28no—
84C2×(C2×D47)⋊C4—232no—
85C2×(C2×D47)⋊C4—44no—
86C2×(C2×D47)⋊C4—44no—
87C2×(C2×D47)⋊C4—44no—
88C2×(C2×D47)⋊C4—44no—
89C2×(C2×D47)⋊C4C2318yes—
90C2×(C2×D47)⋊C4C2318yes—
91C2×(C2×D47)⋊C4C2318yes—
92C2×(C2×D47)⋊C4C2318yes—
93C2×(C2×D47)⋊C4C2318yes—
94C2×(C2×D47)⋊C4C2318yes—
95C47⋊#95C2318yescomplemented
96C47⋊#95C2318yescomplemented
97C25×C47C2311yescharacteristic
98C23×C4×C47—24no—
99C23×C4×C47—44no—
100C23×C4×C47C2312yes—
101C23×C4×C47C2312yes—
102C23×C4×C47C2311yescharacteristic
103C23×C4×C47C2311yescharacteristic
104C23×C4×C47C2314yes—
105C2×C4⋊C4×C47—44no—
106C2×C4⋊C4×C47C2318yescomplemented
107C2×C22⋊C4×C47—24no—
108C2×C22⋊C4×C47—24no—
109C2×C22⋊C4×C47C2314yes—
110C2×C22⋊C4×C47C2314yes—
111#95×C47C2318yescomplemented
112C24×C47—24no—
113C24×C47—24no—
114C24×C47C2×D414yes—
115C24×C47C2×Q814yes—
116C24×C47#4114yes—
117C24×C47C2412yes—
118C24×C47C2411yescharacteristic
119C23×D47—232no—
120C23×D47—24no—
121C23×D47—28no—
122C23×D47—24no—
123C23×D47—28no—
124C23×D47—24no—
125C23×D47—24no—
126C23×D47—24no—
127C23×D47—24no—
128C23×D47—432no—
129C23×D47—48no—
130C23×D47—48no—
131C23×D47—416no—
132C23×D47—416no—
133C23×D47—432no—
134C23×D47C2×Q818yes—
135C23×D47C2×Q8116yescomplemented
136C23×D47C2414yes—
137C22×C4×C47—22no—
138C22×C4×C47—22no—
139C22×C4×C47—28no—
140C22×C4×C47—24no—
141C22×C4×C47—24no—
142C22×C4×C47—24no—
143C22×C4×C47—24no—
144C22×C4×C47—24no—
145C22×C4×C47—24no—
146C22×C4×C47—24no—
147C22×C4×C47—24no—
148C22×C4×C47—48no—
149C22×C4×C47—416no—
150C22×C4×C47C2412yes—
151C22×C4×C47C2414yes—
152C22×C4×C47C2412yes—
153C22×C4×C47C2414yes—
154C22×Dic47—24no—
155C22×Dic47—24no—
156C22×Dic47—22no—
157C22×Dic47—22no—
158C22×Dic47—24no—
159C22×Dic47—24no—
160C22×Dic47—24no—
161C22×Dic47—24no—
162C22×Dic47—24no—
163C22×Dic47—416no—
164C22×Dic47—48no—
165C22×Dic47C2×D414yes—
166C22×Dic47C2×D414yes—
167C22×Dic47C2×D4116yes—
168C22×Dic47C2414yes—
169C22×Dic47C2412yes—
170C22×Dic47C2412yes—
171C22×Dic47C2412yes—
172C22×Dic47C2412yes—
173C2×Dic94—28no—
174C2×Dic94—48no—
175C2×Dic94—416no—
176C4⋊C4×C47—48no—
177C22⋊C4×C47—28no—
178C22⋊C4×C47—28no—
179C22⋊Dic47—28no—
180C22⋊Dic47—28no—
181(C2×D47)⋊C4—216no—
182(C2×D47)⋊C4—216no—
183(C2×D47)⋊C4—48no—
184(C2×D47)⋊C4—48no—
185(C2×D47)⋊C4—48no—
186(C2×D47)⋊C4—48no—
187C4⋊Dic47—28no—
188Dic47⋊C4—28no—
189Dic47⋊C4—28no—
190Dic47⋊C4—48no—
191Dic47⋊C4—48no—
192C23×C47—28no—
193C23×C47—28no—
194C23×C47—216no—
195C23×C47—216no—
196C23×C47—24no—
197C23×C47—24no—
198C23×C47C22⋊Q8116yes—
199C23×C47C22×D414yes—
200C23×C47C22×D412yes—
201C23×C47C22×Q814yes—
202C23×C47C22×Q812yes—
203C23×C47C2×#4114yes—
204C23×C47C2×#4112yes—
205C23×C47H(5,𝔽2)14yes—
206C23×C47H(5,𝔽2)14yes—
207C23×C47C2511yescharacteristic
208C22×D47—216no—
209C22×D47—232no—
210C22×D47—216no—
211C22×D47—216no—
212C22×D47—24no—
213C22×D47—24no—
214C22×D47—24no—
215C22×D47—24no—
216C22×D47—24no—
217C22×D47—24no—
218C22×D47—432no—
219C22×D47—4128no—
220C22×D47—416no—
221C22×D47—416no—
222C22×D47—432no—
223C22×D47—48no—
224C22×D47—48no—
225C22×D47—48no—
226C22×D47—48no—
227C22×D47C22×Q818yes—
228C2×C4×C47—22no—
229C2×C4×C47—22no—
230C2×C4×C47—24no—
231C2×C4×C47—24no—
232C2×C4×C47—24no—
233C2×C4×C47—24no—
234C2×C4×C47—24no—
235C2×C4×C47—24no—
236C2×C4×C47—24no—
237C2×C4×C47—24no—
238C2×C4×C47—416no—
239C2×C4×C47—44no—
240C2×C4×C47—44no—
241C2×C4×C47—48no—
242C2×C4×C47—48no—
243C2×Dic47—28no—
244C2×Dic47—28no—
245C2×Dic47—24no—
246C2×Dic47—24no—
247C2×Dic47—22no—
248C2×Dic47—22no—
249C2×Dic47—24no—
250C2×Dic47—24no—
251C2×Dic47—24no—
252C2×Dic47—24no—
253C2×Dic47—24no—
254C2×Dic47—416no—
255C2×Dic47—44no—
256C2×Dic47—44no—
257C2×Dic47—48no—
258C2×Dic47C22×D414yes—
259C2×Dic47C22×D414yes—
260Dic94—416no—
261C2×#2626—471nomaximal, Sylow
262C22×C47—232no—
263C22×C47—24no—
264C22×C47—24no—
265C22×C47—28no—
266C22×C47—28no—
267C22×C47C2×C22⋊Q818yes—
268C22×C47C2×C22⋊Q814yes—
269C22×C47#53414yes—
270C22×C47#53514yes—
271C22×C47#53818yes—
272C22×C47#54318yes—
273C22×C47C23×D411yescharacteristic
274C22×C47C23×Q811yescharacteristic
275C22×C47C22×#4111yescharacteristic
276C22×C47C2×H(5,𝔽2)12yes—
277C22×C47C2×H(5,𝔽2)12yes—
278C2×D47—28no—
279C2×D47—28no—
280C2×D47—28no—
281C2×D47—464no—
282C2×D47—432no—
283C2×D47—44no—
284C2×D47—44no—
285C2×D47—44no—
286C2×D47—44no—
287C4×C47—42no—
288C4×C47—42no—
289C4×C47—44no—
290C4×C47—44no—
291Dic47—24no—
292Dic47—24no—
293Dic47—42no—
294Dic47—42no—
295Dic47—44no—
296C22×C22⋊Q8—471no—
297C23×C22⋊C4—471no—
298#2626—4716no—
299C2×#399—472no—
300C2×#396—472no—
301C2×#396—472no—
302C2×#393—472no—
303C2×#385—471no—
304C2×#385D4711yescharacteristic, complemented
305C2×#380—471no—
306C2×#379—471no—
307C2×#379—471no—
308C2×C47—28no—
309C2×C47#262618yescomplemented
310C2×C47C22×C22⋊Q811yescharacteristic
311C2×C47C2×#53411yescharacteristic
312C2×C47C2×#53511yescharacteristic
313C2×C47C2×#53812yes—
314C2×C47C2×#54312yes—
315D47—48no—
316C26—471no—
317C23×Q8—471no—
318C24×C4—471no—
319C24×C4C2×D4711yescharacteristic
320C2×C22⋊Q8—478no—
321C2×C22⋊Q8—948no—
322C22×C4⋊C4—471no—
323C22×C4⋊C4—471no—
324C22×C4⋊C4—471no—
325C22×C4⋊C4—472no—
326C22×C4⋊C4—472no—
327C22×C4⋊C4C2×D4712yescomplemented
328C22×C22⋊C4—471no—
329C22×C22⋊C4—471no—
330C22×C22⋊C4—478no—
331C22×C22⋊C4—472no—
332C22×C22⋊C4—472no—
333C22×C22⋊C4—472no—
334C22×C22⋊C4—472no—
335C22×C22⋊C4—472no—
336C22×C22⋊C4—472no—
337C22×C22⋊C4—948no—
338C22×C22⋊C4C2×D4711yescharacteristic
339C22×C22⋊C4C2×D4711yescharacteristic
340#399—4716no—
341#396—4716no—
342#396—4716no—
343#393—4716no—
344#385—478no—
345#385C2×D4718yescomplemented
346#380—478no—
347#379—478no—
348#379—478no—
349C2×#95—472no—
350C2×#95—472no—
351C2×#95C2×D4712yescomplemented
352C47C2×#262611yescharacteristic, complemented, Sylow
353C25—472no—
354C25—478no—
355C25—474no—
356C25—944no—
357C25—944no—
358C25—1884no—
359C25—1884no—
360C25C22×D4711yescharacteristic
361C22×Q8—474no—
362C22×Q8—944no—
363C22×Q8—1884no—
364C23×C4—24no—
365C23×C4—44no—
366C23×C4—471no—
367C23×C4—471no—
368C23×C4—478no—
369C23×C4—474no—
370C23×C4—471no—
371C23×C4—472no—
372C23×C4—471no—
373C23×C4—1884no—
374C23×C4C22×D4711yescharacteristic
375C23×C4C22×D4712yes—
376C23×C4C22×D4711yescharacteristic
377C23×C4C22×D4712yes—
378C23×C4C22×D4714yes—
379C22⋊Q8—9432no—
380C2×C4⋊C4—44no—
381C2×C4⋊C4—474no—
382C2×C4⋊C4—474no—
383C2×C4⋊C4—474no—
384C2×C4⋊C4—478no—
385C2×C4⋊C4—478no—
386C2×C4⋊C4—944no—
387C2×C4⋊C4—944no—
388C2×C4⋊C4—944no—
389C2×C4⋊C4—1884no—
390C2×C4⋊C4—1884no—
391C2×C4⋊C4C22×D4718yescomplemented
392C2×C22⋊C4—24no—
393C2×C22⋊C4—24no—
394C2×C22⋊C4—474no—
395C2×C22⋊C4—474no—
396C2×C22⋊C4—478no—
397C2×C22⋊C4—478no—
398C2×C22⋊C4—478no—
399C2×C22⋊C4—478no—
400C2×C22⋊C4—478no—
401C2×C22⋊C4—478no—
402C2×C22⋊C4—948no—
403C2×C22⋊C4—948no—
404C2×C22⋊C4—9432no—
405C2×C22⋊C4—944no—
406C2×C22⋊C4—944no—
407C2×C22⋊C4—1884no—
408C2×C22⋊C4—1884no—
409C2×C22⋊C4—1884no—
410C2×C22⋊C4—1884no—
411C2×C22⋊C4C22×D4714yes—
412C2×C22⋊C4C22×D4714yes—
413#95—478no—
414#95—478no—
415#95C22×D4718yescomplemented
416C24—24no—
417C24—24no—
418C24—478no—
419C24—474no—
420C24—4716no—
421C24—9432no—
422C24—948no—
423C24—944no—
424C24—948no—
425C24—944no—
426C24—944no—
427C24—944no—
428C24—944no—
429C24—944no—
430C24—18832no—
431C24—1888no—
432C24—1888no—
433C24—18816no—
434C24—18816no—
435C24—18832no—
436C24#1062535014yes—
437C24D4×D4714yes—
438C24Q8×D4714yes—
439C24C23×D4712yes—
440C24C23×D4711yescharacteristic
441C2×Q8—948no—
442C2×Q8—1888no—
443C2×Q8—18816no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—28no—
447C22×C4—24no—
448C22×C4—24no—
449C22×C4—24no—
450C22×C4—24no—
451C22×C4—24no—
452C22×C4—24no—
453C22×C4—24no—
454C22×C4—24no—
455C22×C4—48no—
456C22×C4—416no—
457C22×C4—474no—
458C22×C4—474no—
459C22×C4—472no—
460C22×C4—472no—
461C22×C4—4716no—
462C22×C4—474no—
463C22×C4—472no—
464C22×C4—472no—
465C22×C4—944no—
466C22×C4—944no—
467C22×C4—942no—
468C22×C4—942no—
469C22×C4—944no—
470C22×C4—944no—
471C22×C4—944no—
472C22×C4—944no—
473C22×C4—944no—
474C22×C4—18816no—
475C22×C4—1888no—
476C22×C4C23×D4714yes—
477C22×C4C23×D4714yes—
478C22×C4C23×D4712yes—
479C22×C4C23×D4712yes—
480C4⋊C4—48no—
481C4⋊C4—948no—
482C4⋊C4—948no—
483C4⋊C4—948no—
484C4⋊C4—1888no—
485C4⋊C4—1888no—
486C22⋊C4—28no—
487C22⋊C4—28no—
488C22⋊C4—9416no—
489C22⋊C4—9416no—
490C22⋊C4—948no—
491C22⋊C4—948no—
492C22⋊C4—1888no—
493C22⋊C4—1888no—
494C22⋊C4—1888no—
495C22⋊C4—1888no—
496C23—216no—
497C23—28no—
498C23—28no—
499C23—24no—
500C23—24no—
501C23—216no—
502C23—478no—
503C23—9416no—
504C23—9416no—
505C23—9432no—
506C23—9416no—
507C23—944no—
508C23—944no—
509C23—944no—
510C23—944no—
511C23—944no—
512C23—944no—
513C23—18832no—
514C23—188128no—
515C23—18816no—
516C23—18816no—
517C23—18832no—
518C23—1888no—
519C23—1888no—
520C23—1888no—
521C23—1888no—
522C23#13094002116yes—
523C23C2×D4×D4714yes—
524C23C2×D4×D4712yes—
525C23C2×Q8×D4714yes—
526C23C2×Q8×D4712yes—
527C23C2×#1062535014yes—
528C23C2×#1062535012yes—
529C23#1309406914yes—
530C23#1309407014yes—
531C23C24×D4711yescharacteristic
532Q8—18816no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—24no—
536C2×C4—24no—
537C2×C4—24no—
538C2×C4—24no—
539C2×C4—24no—
540C2×C4—24no—
541C2×C4—24no—
542C2×C4—24no—
543C2×C4—416no—
544C2×C4—44no—
545C2×C4—44no—
546C2×C4—48no—
547C2×C4—48no—
548C2×C4—474no—
549C2×C4—474no—
550C2×C4—948no—
551C2×C4—948no—
552C2×C4—944no—
553C2×C4—944no—
554C2×C4—942no—
555C2×C4—942no—
556C2×C4—944no—
557C2×C4—944no—
558C2×C4—944no—
559C2×C4—944no—
560C2×C4—944no—
561C2×C4—18816no—
562C2×C4—1884no—
563C2×C4—1884no—
564C2×C4—1888no—
565C22—28no—
566C22—232no—
567C22—24no—
568C22—24no—
569C22—28no—
570C22—948no—
571C22—948no—
572C22—948no—
573C22—18864no—
574C22—18832no—
575C22—1884no—
576C22—1884no—
577C22—1884no—
578C22—1884no—
579C22C2×#1309400218yes—
580C22C2×#1309400214yes—
581C22#1568968414yes—
582C22#1568968814yes—
583C22#1568970618yes—
584C22#1568974818yes—
585C22C22×D4×D4711yescharacteristic
586C22C22×Q8×D4711yescharacteristic
587C22C22×#1062535011yescharacteristic
588C22C2×#1309406912yes—
589C22C2×#1309407012yes—
590C4—42no—
591C4—44no—
592C4—42no—
593C4—44no—
594C4—944no—
595C4—944no—
596C4—1882no—
597C4—1882no—
598C4—1884no—
599C2—28no—
600C2—1888no—
601C2#2024919918yesdirect product factor
602C2C22×#1309400211yescharacteristic
603C2C2×#1568968411yescharacteristic
604C2C2×#1568968811yescharacteristic
605C2C2×#1568970612yes—
606C2C2×#1568974812yes—
subgroupG/H
centerC24C23×D47
commutator, cosocleC23×C47C25
FrattiniC23C24×D47
FittingC2×C47×#385C2
socleC24×C47C24
prime radicalC25×D47C22
hypercenterC2×#385D47
hypocenterC47C2×#2626
LCSG ▷ C23×C47 ▷ C47
UCSC1 ◁ C24 ◁ C2×#385

Complements

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

NH ≅ G/Ntype
1. C22×#13094001C2semidirect
3. #20249199C2direct (N × H)
4. C2×#15689058C2semidirect
5. C2×#15689047C2semidirect
6. C2×#15689045C2semidirect
7. C2×#15689036C2semidirect
8. C2×#15688996C2semidirect
9. C2×#15688981C2semidirect
12. C2×C47×#385C2semidirect
14. C23×Dic94C22semidirect
17. C2×#13094001C22semidirect
18. C22×C4⋊Dic47C22semidirect
19. C22×Dic47⋊C4C22semidirect
20. C22×Dic47⋊C4C22semidirect
21. C22×Dic47⋊C4C22semidirect
22. C22×Dic47⋊C4C22semidirect
33. #15689058C22semidirect
34. #15689047C22semidirect
35. #15689045C22semidirect
36. #15689036C22semidirect
37. #15688996C22semidirect
38. #15688981C22semidirect
41. C2×C47⋊#95C22semidirect
42. C2×C47⋊#95C22semidirect
44. C22×C4⋊C4×C47C22semidirect
47. C47×#385C22semidirect
48. C2×#95×C47C22semidirect
53. C24×D47Q8semidirect
58. C22×Dic94C23semidirect
69. C2×C4⋊Dic47C23semidirect
74. C2×Dic47⋊C4C23semidirect
75. C2×Dic47⋊C4C23semidirect
76. C2×Dic47⋊C4C23semidirect
77. C2×Dic47⋊C4C23semidirect
95. C47⋊#95C23semidirect
96. C47⋊#95C23semidirect
106. C2×C4⋊C4×C47C23semidirect
111. #95×C47C23semidirect
135. C23×D47C2×Q8semidirect
304. C2×#385D47semidirect
309. C2×C47#2626semidirect
327. C22×C4⋊C4C2×D47semidirect
345. #385C2×D47semidirect
351. C2×#95C2×D47semidirect
352. C47C2×#2626semidirect
391. C2×C4⋊C4C22×D47semidirect
415. #95C22×D47semidirect
601. C2#20249199direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#130940012
C23×(C2×D47)⋊C42
#202491992
C2×#156890582
C2×#156890472
C2×#156890452
C2×#156890362
C2×#156889962
C2×#156889812
C2×#156889782
C2×C47×#3852
C2×#262647