Finite Group Databasebeta

← C2×#20246955 #26384534

Subgroups — C2×#20246955 #26384534

Order 12032.

3678 conjugacy classes of subgroups.

The subgroup lattice has 656 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#13094011C211yesmaximal, characteristic, complemented
2C22×#13094004C211yesmaximal, characteristic, complemented
3C22×#13093995C211yesmaximal, characteristic
4C22×C4×Dic94C211yesmaximal, characteristic, complemented
5#20246955C2116yesmaximal, direct product factor
6C2×#15689046C212yesmaximal, complemented
7C2×#15689038C212yesmaximal, complemented
8C2×#15689023C212yesmaximal, complemented
9C2×#15689019C211yesmaximal, characteristic, complemented
10C2×#15689016C212yesmaximal, complemented
11C2×C4×(C2×D47)⋊C4C211yesmaximal, characteristic, complemented
12C2×C47×#390C211yesmaximal, characteristic, complemented
13C23×Dic94C2211yescharacteristic
14C23×C22⋊D47C2212yes—
15C23×C4×D47C2211yescharacteristic
16C2×#13094011—28no—
17C2×#13094011C2218yescomplemented
18C2×#13094004—28no—
19C2×#13094004C2218yescomplemented
20C2×#13093995—28no—
21C2×#13093995C2218yes—
22C2×C4×Dic94—28no—
23C2×C4×Dic94C2218yescomplemented
24C22×C4⋊Dic47C2212yes—
25C22×C4⋊Dic47C2211yescharacteristic
26C22×Dic47⋊C4C2212yes—
27C22×Dic47⋊C4C2212yes—
28C22×C22⋊Dic47C2212yes—
29C22×C22⋊Dic47C2212yes—
30C22×C22⋊Dic47C2212yes—
31C22×(C2×D47)⋊C4C2211yescharacteristic
32C22×(C2×D47)⋊C4C2212yes—
33C22×(C2×D47)⋊C4C2211yescharacteristic
34C22×C4×Dic47C2212yes—
35C22×C4×Dic47C2211yescharacteristic
36#15689046C22116yescomplemented
37#15689038C22116yescomplemented
38#15689023C22116yescomplemented
39#15689019C2218yescomplemented
40#15689016C22116yescomplemented
41C4×(C2×D47)⋊C4C2218yescomplemented
42C2×C47⋊#95C2211yescharacteristic
43C2×C47⋊#95C2212yes—
44C2×C47⋊#95C2211yescharacteristic
45C23×D4×C47C2211yescharacteristic
46C22×C4⋊C4×C47C2211yescharacteristic
47C22×C22⋊C4×C47C2212yes—
48C22×C22⋊C4×C47C2212yes—
49C22×C42×C47C2211yescharacteristic
50C47×#390C2218yescomplemented
51C24×D47C2311yescharacteristic
52C22×Dic94—24no—
53C22×Dic94—24no—
54C22×Dic94D418yescomplemented
55C22×Dic94C2314yes—
56C22×C22⋊D47—28no—
57C22×C22⋊D47—28no—
58C22×C22⋊D47—44no—
59C22×C22⋊D47C2318yes—
60C23×Dic47C2311yescharacteristic
61C23×Dic47C2311yescharacteristic
62C23×Dic47C2311yescharacteristic
63C23×Dic47C2312yes—
64C23×Dic47C2312yes—
65C22×C4×D47—24no—
66C22×C4×D47—24no—
67C22×C4×D47—24no—
68C22×C4×D47C2314yes—
69#13094011—232no—
70#13094004—232no—
71#13093995—232no—
72C4×Dic94—232no—
73C2×C4⋊Dic47—28no—
74C2×C4⋊Dic47D418yes—
75C2×C4⋊Dic47C2318yes—
76C2×C4⋊Dic47C2314yes—
77C2×Dic47⋊C4—28no—
78C2×Dic47⋊C4—44no—
79C2×Dic47⋊C4C2318yes—
80C2×Dic47⋊C4C2318yes—
81C2×C22⋊Dic47—28no—
82C2×C22⋊Dic47—28no—
83C2×C22⋊Dic47—28no—
84C2×C22⋊Dic47C2318yes—
85C2×C22⋊Dic47C2318yes—
86C2×C22⋊Dic47C2318yes—
87C2×(C2×D47)⋊C4—24no—
88C2×(C2×D47)⋊C4—24no—
89C2×(C2×D47)⋊C4—28no—
90C2×(C2×D47)⋊C4C2314yes—
91C2×(C2×D47)⋊C4C2314yes—
92C2×(C2×D47)⋊C4C2318yes—
93C2×C4×Dic47—24no—
94C2×C4×Dic47—28no—
95C2×C4×Dic47C2314yes—
96C2×C4×Dic47C2318yes—
97C47⋊#95C2314yes—
98C47⋊#95C2314yes—
99C47⋊#95C2318yes—
100C25×C47C2312yes—
101C22×D4×C47—24no—
102C22×D4×C47—24no—
103C22×D4×C47—24no—
104C22×D4×C47C2314yes—
105C23×C4×C47C2311yescharacteristic
106C23×C4×C47C2312yes—
107C23×C4×C47C2311yescharacteristic
108C23×C4×C47C2311yescharacteristic
109C2×C4⋊C4×C47—24no—
110C2×C4⋊C4×C47C2314yes—
111C2×C22⋊C4×C47—28no—
112C2×C22⋊C4×C47—44no—
113C2×C22⋊C4×C47C2318yes—
114C2×C22⋊C4×C47C2318yes—
115C2×C42×C47—24no—
116C2×C42×C47C2314yes—
117C24×C47—24no—
118C24×C47—24no—
119C24×C47—24no—
120C24×C47—24no—
121C24×C47—24no—
122C24×C47—24no—
123C24×C47—24no—
124C24×C47C2414yes—
125C24×C47C2411yescharacteristic
126C23×D47—22no—
127C23×D47—22no—
128C23×D47—22no—
129C23×D47—22no—
130C23×D47—22no—
131C23×D47—22no—
132C23×D47#4114yes—
133C23×D47C2412yes—
134C2×D4×C47—28no—
135C2×D4×C47—28no—
136C2×D4×C47—28no—
137C2×D4×C47—416no—
138C22×C4×C47—22no—
139C22×C4×C47—22no—
140C22×C4×C47—22no—
141C22×C4×C47—24no—
142C22×C4×C47—24no—
143C22×C4×C47—22no—
144C22×C4×C47—22no—
145C22×C4×C47—24no—
146C22×C4×C47C2×D414yes—
147C22×C4×C47C2×D414yes—
148C22×C4×C47#4114yes—
149C22×C4×C47#4114yes—
150C22×C4×C47C2412yes—
151C22×C4×C47C2414yes—
152C22×C4×C47C2412yes—
153C22×C4×C47C2412yes—
154C2×C22⋊D47—216no—
155C2×C22⋊D47—216no—
156C2×C22⋊D47—48no—
157C2×C22⋊D47—432no—
158C22×Dic47—24no—
159C22×Dic47—24no—
160C22×Dic47—24no—
161C22×Dic47—24no—
162C22×Dic47—22no—
163C22×Dic47—22no—
164C22×Dic47—22no—
165C22×Dic47—22no—
166C22×Dic47—22no—
167C22×Dic47—22no—
168C22×Dic47—22no—
169C22×Dic47—22no—
170C22×Dic47—22no—
171C22×Dic47C2×D418yes—
172C22×Dic47C2×D418yes—
173C22×Dic47C2412yes—
174C22×Dic47C2412yes—
175C22×Dic47C2412yes—
176C22×Dic47C2414yes—
177C22×Dic47C2414yes—
178C2×C4×D47—28no—
179C2×C4×D47—28no—
180C2×C4×D47—28no—
181C2×C4×D47—416no—
182C2×Dic94—232no—
183C2×Dic94—28no—
184C2×Dic94—28no—
185C2×Dic94C2×D4116yescomplemented
186C4⋊C4×C47—28no—
187C22⋊C4×C47—216no—
188C22⋊C4×C47—48no—
189C42×C47—28no—
190C22⋊Dic47—216no—
191C22⋊Dic47—216no—
192C22⋊Dic47—216no—
193(C2×D47)⋊C4—28no—
194(C2×D47)⋊C4—28no—
195(C2×D47)⋊C4—216no—
196C4⋊Dic47—216no—
197C4⋊Dic47C2×D4116yes—
198Dic47⋊C4—216no—
199Dic47⋊C4—48no—
200C4×Dic47—28no—
201C4×Dic47—216no—
202C23×C47—24no—
203C23×C47—24no—
204C23×C47—24no—
205C23×C47—24no—
206C23×C47—24no—
207C23×C47—24no—
208C23×C47—24no—
209C23×C47—48no—
210C23×C47—48no—
211C23×C47—48no—
212C23×C47—48no—
213C23×C47—48no—
214C23×C47—48no—
215C23×C47—48no—
216C23×C47C22×D412yes—
217C23×C47C22×D412yes—
218C23×C47C2×#4112yes—
219C23×C47C2×#4112yes—
220C23×C47C2×#4112yes—
221C23×C47H(5,𝔽2)12yes—
222C23×C472−1+412yes—
223C23×C47C2511yescharacteristic
224C22×D47—28no—
225C22×D47—28no—
226C22×D47—28no—
227C22×D47—22no—
228C22×D47—22no—
229C22×D47—22no—
230C22×D47—22no—
231C22×D47—22no—
232C22×D47—22no—
233C22×D47—44no—
234C22×D47—44no—
235C22×D47—44no—
236C22×D47—44no—
237C22×D47C2×#4114yes—
238D4×C47—416no—
239C2×C4×C47—28no—
240C2×C4×C47—22no—
241C2×C4×C47—22no—
242C2×C4×C47—22no—
243C2×C4×C47—24no—
244C2×C4×C47—22no—
245C2×C4×C47—24no—
246C2×C4×C47—24no—
247C2×C4×C47—22no—
248C2×C4×C47—44no—
249C2×C4×C47—48no—
250C2×C4×C47#121116yes—
251C2×C4×C47C22×D414yes—
252C2×C4×C47C22×D414yes—
253C2×C4×C47C2×#4114yes—
254C2×C4×C47C2×#4114yes—
255C22⋊D47—432no—
256C2×Dic47—216no—
257C2×Dic47—216no—
258C2×Dic47—24no—
259C2×Dic47—24no—
260C2×Dic47—24no—
261C2×Dic47—24no—
262C2×Dic47—22no—
263C2×Dic47—22no—
264C2×Dic47—22no—
265C2×Dic47—22no—
266C2×Dic47—22no—
267C2×Dic47—22no—
268C2×Dic47—22no—
269C2×Dic47—22no—
270C2×Dic47—22no—
271C2×Dic47—44no—
272C2×Dic47—44no—
273C2×Dic47—44no—
274C2×Dic47C22×D418yes—
275C2×Dic47C22×D418yes—
276C4×D47—416no—
277Dic94—232no—
278C2×#2426—471nomaximal, Sylow
279C22×C47—44no—
280C22×C47—44no—
281C22×C47—44no—
282C22×C47—44no—
283C22×C47—44no—
284C22×C47—44no—
285C22×C47—44no—
286C22×C47—816no—
287C22×C47C2×#12114yes—
288C22×C47#52914yes—
289C22×C47#54214yes—
290C22×C47#54614yes—
291C22×C47#54814yes—
292C22×C47#54814yes—
293C22×C47#55014yes—
294C22×C47C23×D411yescharacteristic
295C22×C47C23×D411yescharacteristic
296C22×C47C22×#4111yescharacteristic
297C22×C47C22×#4111yescharacteristic
298C22×C47C22×#4111yescharacteristic
299C22×C47C2×H(5,𝔽2)11yescharacteristic
300C22×C47C2×2−1+411yescharacteristic
301C2×D47—24no—
302C2×D47—24no—
303C2×D47—24no—
304C2×D47—42no—
305C2×D47—416no—
306C2×D47—42no—
307C2×D47—42no—
308C2×D47—42no—
309C4×C47—24no—
310C4×C47—42no—
311C4×C47—44no—
312C4×C47C2×#12118yes—
313Dic47—28no—
314Dic47—28no—
315Dic47—42no—
316Dic47—42no—
317Dic47—42no—
318C22×#124—471no—
319C22×C22⋊Q8—471no—
320C22×#121—471no—
321C22×C4×Q8—471no—
322#2426—4716no—
323C2×#396—472no—
324C2×#394—472no—
325C2×#390—472no—
326C2×#390D4711yescharacteristic, complemented
327C2×#389—471no—
328C2×#388—472no—
329C2×C4×C22⋊C4—471no—
330C2×C47—84no—
331C2×C47#242618yescomplemented
332C2×C47C22×#12111yescharacteristic
333C2×C47C2×#52911yescharacteristic
334C2×C47C2×#54211yescharacteristic
335C2×C47C2×#54611yescharacteristic
336C2×C47C2×#54811yescharacteristic
337C2×C47C2×#54811yescharacteristic
338C2×C47C2×#55011yescharacteristic
339D47—44no—
340C23×Q8—471no—
341C23×D4—472no—
342C23×D4C2×D4711yescharacteristic
343C24×C4—471no—
344C2×#124—478no—
345C2×#124—948no—
346C2×C22⋊Q8—478no—
347C2×C22⋊Q8—948no—
348C2×#121—478no—
349C2×#121—948no—
350C2×C4×Q8—478no—
351C2×C4×Q8—948no—
352C22×C4⋊C4—471no—
353C22×C4⋊C4—472no—
354C22×C4⋊C4—472no—
355C22×C4⋊C4—472no—
356C22×C4⋊C4C2×D4711yescharacteristic
357C22×C22⋊C4—472no—
358C22×C22⋊C4—472no—
359C22×C22⋊C4—472no—
360C22×C22⋊C4—471no—
361C22×C22⋊C4—472no—
362C22×C22⋊C4—471no—
363C22×C22⋊C4C2×D4712yes—
364C22×C22⋊C4C2×D4712yes—
365C22×C42—472no—
366C22×C42—471no—
367C22×C42C2×D4711yescharacteristic
368#396—4716no—
369#394—4716no—
370#390—4716no—
371#390C2×D4718yescomplemented
372#389—478no—
373#388—4716no—
374C4×C22⋊C4—478no—
375C2×#95—472no—
376C2×#95—471no—
377C2×#95—471no—
378C47C2×#242611yescharacteristic, complemented, Sylow
379C25—471no—
380C25C22×D4712yes—
381C22×Q8—478no—
382C22×Q8—474no—
383C22×Q8—944no—
384C22×Q8—944no—
385C22×D4—24no—
386C22×D4—24no—
387C22×D4—24no—
388C22×D4—478no—
389C22×D4—948no—
390C22×D4—948no—
391C22×D4—1884no—
392C22×D4C22×D4714yes—
393C23×C4—472no—
394C23×C4—472no—
395C23×C4—471no—
396C23×C4—471no—
397C23×C4—471no—
398C23×C4—474no—
399C23×C4—944no—
400C23×C4—944no—
401C23×C4—944no—
402C23×C4C22×D4712yes—
403C23×C4C22×D4711yescharacteristic
404C23×C4C22×D4711yescharacteristic
405C23×C4C22×D4711yescharacteristic
406#124—9432no—
407C22⋊Q8—9432no—
408#121—9432no—
409C4×Q8—9432no—
410C2×C4⋊C4—24no—
411C2×C4⋊C4—478no—
412C2×C4⋊C4—474no—
413C2×C4⋊C4—478no—
414C2×C4⋊C4—478no—
415C2×C4⋊C4—478no—
416C2×C4⋊C4—948no—
417C2×C4⋊C4—948no—
418C2×C4⋊C4—1884no—
419C2×C4⋊C4C22×D4714yes—
420C2×C22⋊C4—28no—
421C2×C22⋊C4—44no—
422C2×C22⋊C4—478no—
423C2×C22⋊C4—478no—
424C2×C22⋊C4—478no—
425C2×C22⋊C4—478no—
426C2×C22⋊C4—474no—
427C2×C22⋊C4—474no—
428C2×C22⋊C4—948no—
429C2×C22⋊C4—944no—
430C2×C22⋊C4—948no—
431C2×C22⋊C4—948no—
432C2×C22⋊C4—948no—
433C2×C22⋊C4—944no—
434C2×C22⋊C4C22×D4718yes—
435C2×C22⋊C4C22×D4718yes—
436C2×C42—24no—
437C2×C42—478no—
438C2×C42—474no—
439C2×C42—948no—
440C2×C42—944no—
441C2×C42C22×D4714yes—
442#95—474no—
443#95—474no—
444#95—478no—
445C24—24no—
446C24—24no—
447C24—24no—
448C24—24no—
449C24—24no—
450C24—24no—
451C24—24no—
452C24—474no—
453C24—472no—
454C24—942no—
455C24—942no—
456C24—942no—
457C24—942no—
458C24—942no—
459C24—942no—
460C24C23×D4714yes—
461C24C23×D4711yescharacteristic
462C2×Q8—4716no—
463C2×Q8—9432no—
464C2×Q8—948no—
465C2×Q8—948no—
466C2×D4—28no—
467C2×D4—28no—
468C2×D4—28no—
469C2×D4—416no—
470C2×D4—9416no—
471C2×D4—9416no—
472C2×D4—1888no—
473C2×D4—18832no—
474C22×C4—22no—
475C22×C4—22no—
476C22×C4—22no—
477C22×C4—24no—
478C22×C4—22no—
479C22×C4—24no—
480C22×C4—22no—
481C22×C4—24no—
482C22×C4—478no—
483C22×C4—478no—
484C22×C4—474no—
485C22×C4—474no—
486C22×C4—472no—
487C22×C4—472no—
488C22×C4—472no—
489C22×C4—944no—
490C22×C4—944no—
491C22×C4—944no—
492C22×C4—944no—
493C22×C4—942no—
494C22×C4—942no—
495C22×C4—942no—
496C22×C4—942no—
497C22×C4—942no—
498C22×C4—942no—
499C22×C4—942no—
500C22×C4—942no—
501C22×C4—942no—
502C22×C4—948no—
503C22×C4—948no—
504C22×C4—948no—
505C22×C4—18816no—
506C22×C4#1062535014yes—
507C22×C4D4×D4714yes—
508C22×C4D4×D4714yes—
509C22×C4#1062535214yes—
510C22×C4C23×D4712yes—
511C22×C4C23×D4714yes—
512C22×C4C23×D4712yes—
513C22×C4C23×D4712yes—
514C4⋊C4—28no—
515C4⋊C4—4716no—
516C4⋊C4—9416no—
517C4⋊C4—9416no—
518C4⋊C4—1888no—
519C22⋊C4—216no—
520C22⋊C4—48no—
521C22⋊C4—9416no—
522C22⋊C4—948no—
523C22⋊C4—9416no—
524C22⋊C4—9416no—
525C22⋊C4—9416no—
526C22⋊C4—948no—
527C42—28no—
528C42—9416no—
529C42—948no—
530C23—24no—
531C23—24no—
532C23—24no—
533C23—24no—
534C23—24no—
535C23—24no—
536C23—24no—
537C23—48no—
538C23—48no—
539C23—48no—
540C23—48no—
541C23—48no—
542C23—48no—
543C23—48no—
544C23—474no—
545C23—948no—
546C23—948no—
547C23—948no—
548C23—942no—
549C23—942no—
550C23—942no—
551C23—942no—
552C23—942no—
553C23—942no—
554C23—1884no—
555C23—1884no—
556C23—1884no—
557C23—1884no—
558C23C2×D4×D4712yes—
559C23C2×D4×D4712yes—
560C23#41×D4712yes—
561C23C2×#1062535212yes—
562C23C2×#1062535012yes—
563C23#1309406912yes—
564C23#1309407112yes—
565C23C24×D4711yescharacteristic
566Q8—9432no—
567D4—416no—
568D4—18832no—
569C2×C4—28no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—24no—
574C2×C4—22no—
575C2×C4—24no—
576C2×C4—22no—
577C2×C4—24no—
578C2×C4—44no—
579C2×C4—48no—
580C2×C4—478no—
581C2×C4—478no—
582C2×C4—9416no—
583C2×C4—9416no—
584C2×C4—944no—
585C2×C4—944no—
586C2×C4—944no—
587C2×C4—944no—
588C2×C4—942no—
589C2×C4—942no—
590C2×C4—942no—
591C2×C4—942no—
592C2×C4—942no—
593C2×C4—942no—
594C2×C4—942no—
595C2×C4—942no—
596C2×C4—942no—
597C2×C4—1884no—
598C2×C4—1884no—
599C2×C4—1884no—
600C2×C4—18816no—
601C2×C4#13093997116yes—
602C2×C4C2×D4×D4714yes—
603C2×C4C2×D4×D4714yes—
604C2×C4C2×#1062535214yes—
605C2×C4C2×#1062535014yes—
606C22—44no—
607C22—44no—
608C22—44no—
609C22—44no—
610C22—44no—
611C22—44no—
612C22—44no—
613C22—816no—
614C22—944no—
615C22—944no—
616C22—944no—
617C22—1882no—
618C22—1882no—
619C22—1882no—
620C22—18816no—
621C22—1882no—
622C22C2×#1309399714yes—
623C22#1568965514yes—
624C22#1568973814yes—
625C22#1568977014yes—
626C22#1568977914yes—
627C22#1568978114yes—
628C22#1568979314yes—
629C22C22×D4×D4711yescharacteristic
630C22C22×D4×D4711yescharacteristic
631C22C2×#41×D4711yescharacteristic
632C22C22×#1062535211yescharacteristic
633C22C22×#1062535011yescharacteristic
634C22C2×#1309406911yescharacteristic
635C22C2×#1309407111yescharacteristic
636C4—24no—
637C4—42no—
638C4—44no—
639C4—948no—
640C4—948no—
641C4—1882no—
642C4—1882no—
643C4—1882no—
644C4C2×#1309399718yes—
645C2—84no—
646C2—1884no—
647C2#2024695518yesdirect product factor
648C2C22×#1309399711yescharacteristic
649C2C2×#1568965511yescharacteristic
650C2C2×#1568973811yescharacteristic
651C2C2×#1568977011yescharacteristic
652C2C2×#1568977911yescharacteristic
653C2C2×#1568978111yescharacteristic
654C2C2×#1568979311yescharacteristic
subgroupG/H
centerC24C23×D47
commutator, cosocleC23×C47C25
FrattiniC23C24×D47
FittingC2×C47×#390C2
socleC24×C47C24
prime radicalC22×#13093995C2
hypercenterC2×#390D47
hypocenterC47C2×#2426
LCSG ▷ C23×C47 ▷ C47
UCSC1 ◁ C24 ◁ C2×#390

Complements

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

NH ≅ G/Ntype
1. C22×#13094011C2semidirect
2. C22×#13094004C2semidirect
4. C22×C4×Dic94C2semidirect
5. #20246955C2direct (N × H)
6. C2×#15689046C2semidirect
7. C2×#15689038C2semidirect
8. C2×#15689023C2semidirect
9. C2×#15689019C2semidirect
10. C2×#15689016C2semidirect
11. C2×C4×(C2×D47)⋊C4C2semidirect
12. C2×C47×#390C2semidirect
17. C2×#13094011C22semidirect
19. C2×#13094004C22semidirect
23. C2×C4×Dic94C22semidirect
36. #15689046C22semidirect
37. #15689038C22semidirect
38. #15689023C22semidirect
39. #15689019C22semidirect
40. #15689016C22semidirect
41. C4×(C2×D47)⋊C4C22semidirect
50. C47×#390C22semidirect
54. C22×Dic94D4semidirect
185. C2×Dic94C2×D4semidirect
326. C2×#390D47semidirect
331. C2×C47#2426semidirect
371. #390C2×D47semidirect
378. C47C2×#2426semidirect
647. C2#20246955direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#130940112
C22×#130940042
C22×#130939952
C22×C4×Dic942
#202469552
C2×#156890462
C2×#156890382
C2×#156890232
C2×#156890192
C2×#156890162
C2×C4×(C2×D47)⋊C42
C2×C47×#3902
C2×#242647