Finite Group Databasebeta

← C2×#20246811 #26384688

Subgroups — C2×#20246811 #26384688

Order 12032.

3526 conjugacy classes of subgroups.

The subgroup lattice has 652 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#13094004C211yesmaximal, characteristic, complemented
2C22×#13094003C211yesmaximal, characteristic, complemented
3C22×C4×C22⋊D47C211yesmaximal, characteristic
4#20246811C2116yesmaximal, direct product factor
5C2×#15689048C212yesmaximal, complemented
6C2×#15689039C212yesmaximal
7C2×#15689028C211yesmaximal, characteristic, complemented
8C2×#15689022C211yesmaximal, characteristic, complemented
9C2×#15689021C212yesmaximal, complemented
10C2×#15689014C212yesmaximal, complemented
11C2×#15688980C211yesmaximal, characteristic, complemented
12C22×#117×C47C211yesmaximal, characteristic, complemented
13C23×Dic94C2211yescharacteristic
14C23×C22⋊D47C2211yescharacteristic
15C23×C4×D47C2211yescharacteristic
16C2×#13094004—28no—
17C2×#13094004C2218yescomplemented
18C2×#13094003—28no—
19C2×#13094003C2218yescomplemented
20C2×C4×C22⋊D47—28no—
21C2×C4×C22⋊D47C2218yes—
22C22×C4⋊Dic47C2212yes—
23C22×C4⋊Dic47C2212yes—
24C22×C4⋊Dic47C2211yescharacteristic
25C22×Dic47⋊C4C2211yescharacteristic
26C22×Dic47⋊C4C2212yes—
27C22×C22⋊Dic47C2211yescharacteristic
28C22×C22⋊Dic47C2212yes—
29C22×(C2×D47)⋊C4C2211yescharacteristic
30C22×(C2×D47)⋊C4C2212yes—
31C22×(C2×D47)⋊C4C2212yes—
32C22×C4×Dic47C2211yescharacteristic
33#15689048C22116yescomplemented
34#15689039C22116yes—
35#15689028C2218yescomplemented
36#15689022C2218yescomplemented
37#15689021C22116yescomplemented
38#15689014C22116yescomplemented
39#15688980C2218yescomplemented
40C2×C47⋊#95C2212yes—
41C2×C47⋊#95C2212yes—
42C2×C47⋊#95C2212yes—
43C2×C47⋊#95C2212yes—
44C24×C4×C47C2211yescharacteristic
45C2×#117×C47—28no—
46C2×#117×C47C2218yescomplemented
47C22×C4⋊C4×C47C2211yescharacteristic
48C22×C4⋊C4×C47C2211yescharacteristic
49C22×C22⋊C4×C47C2212yes—
50C22×C42×C47C2212yes—
51C24×D47C2311yescharacteristic
52C22×Dic94—24no—
53C22×Dic94—24no—
54C22×Dic94—24no—
55C22×Dic94C2314yes—
56C22×C22⋊D47—24no—
57C22×C22⋊D47—24no—
58C22×C22⋊D47—24no—
59C22×C22⋊D47C2314yes—
60C23×Dic47C2311yescharacteristic
61C23×Dic47C2312yes—
62C23×Dic47C2312yes—
63C23×Dic47C2311yescharacteristic
64C23×Dic47C2311yescharacteristic
65C22×C4×D47—24no—
66C22×C4×D47—24no—
67C22×C4×D47—24no—
68C22×C4×D47C2314yes—
69#13094004—232no—
70#13094003—232no—
71C4×C22⋊D47—232no—
72C2×C4⋊Dic47—24no—
73C2×C4⋊Dic47—28no—
74C2×C4⋊Dic47—28no—
75C2×C4⋊Dic47C2314yes—
76C2×C4⋊Dic47C2318yes—
77C2×C4⋊Dic47C2318yes—
78C2×Dic47⋊C4—24no—
79C2×Dic47⋊C4—28no—
80C2×Dic47⋊C4C2318yes—
81C2×Dic47⋊C4C2314yes—
82C2×C22⋊Dic47—24no—
83C2×C22⋊Dic47—28no—
84C2×C22⋊Dic47C2318yes—
85C2×C22⋊Dic47C2314yes—
86C2×(C2×D47)⋊C4—28no—
87C2×(C2×D47)⋊C4—24no—
88C2×(C2×D47)⋊C4—28no—
89C2×(C2×D47)⋊C4C2318yes—
90C2×(C2×D47)⋊C4C2314yes—
91C2×(C2×D47)⋊C4C2318yes—
92C2×C4×Dic47—24no—
93C2×C4×Dic47C2314yes—
94C47⋊#95C2318yes—
95C47⋊#95C2318yes—
96C47⋊#95C2318yes—
97C47⋊#95C2318yes—
98C25×C47C2311yescharacteristic
99C23×C4×C47—24no—
100C23×C4×C47—24no—
101C23×C4×C47D418yes—
102C23×C4×C47C2312yes—
103C23×C4×C47C2312yes—
104C23×C4×C47C2311yescharacteristic
105C23×C4×C47C2314yes—
106C23×C4×C47C2311yescharacteristic
107#117×C47—232no—
108C2×C4⋊C4×C47—24no—
109C2×C4⋊C4×C47D418yescomplemented
110C2×C4⋊C4×C47C2314yes—
111C2×C4⋊C4×C47C2314yes—
112C2×C22⋊C4×C47—28no—
113C2×C22⋊C4×C47C2318yes—
114C2×C42×C47—28no—
115C2×C42×C47C2318yes—
116C24×C47—22no—
117C24×C47—22no—
118C24×C47—22no—
119C24×C47—22no—
120C24×C47—22no—
121C24×C47—22no—
122C24×C47C2×D414yes—
123C24×C47C2412yes—
124C24×C47C2411yescharacteristic
125C23×D47—22no—
126C23×D47—22no—
127C23×D47—22no—
128C23×D47—22no—
129C23×D47—22no—
130C23×D47—22no—
131C23×D47#4114yes—
132C23×D47C2412yes—
133C22×C4×C47—22no—
134C22×C4×C47—22no—
135C22×C4×C47—24no—
136C22×C4×C47—24no—
137C22×C4×C47—232no—
138C22×C4×C47—28no—
139C22×C4×C47—28no—
140C22×C4×C47—24no—
141C22×C4×C47—24no—
142C22×C4×C47—24no—
143C22×C4×C47C2×D414yes—
144C22×C4×C47C2×D4116yes—
145C22×C4×C47C2×D418yes—
146C22×C4×C47C2×D414yes—
147C22×C4×C47C2×D414yes—
148C22×C4×C47#4114yes—
149C22×C4×C47C2414yes—
150C22×C4×C47C2412yes—
151C22×C4×C47C2414yes—
152C22×C4×C47C2412yes—
153C2×C22⋊D47—232no—
154C2×C22⋊D47—28no—
155C2×C22⋊D47—28no—
156C2×C22⋊D47—28no—
157C22×Dic47—22no—
158C22×Dic47—22no—
159C22×Dic47—22no—
160C22×Dic47—24no—
161C22×Dic47—24no—
162C22×Dic47—22no—
163C22×Dic47—24no—
164C22×Dic47—22no—
165C22×Dic47—24no—
166C22×Dic47—24no—
167C22×Dic47—22no—
168C22×Dic47—24no—
169C22×Dic47—22no—
170C22×Dic47—22no—
171C22×Dic47#4114yes—
172C22×Dic47C2412yes—
173C22×Dic47C2414yes—
174C22×Dic47C2414yes—
175C22×Dic47C2412yes—
176C22×Dic47C2412yes—
177C2×C4×D47—28no—
178C2×C4×D47—28no—
179C2×C4×D47—28no—
180C2×C4×D47—416no—
181C2×Dic94—28no—
182C2×Dic94—28no—
183C2×Dic94—28no—
184C2×Dic94—416no—
185C4⋊C4×C47—28no—
186C4⋊C4×C47C2×D4116yescomplemented
187C22⋊C4×C47—216no—
188C42×C47—216no—
189C22⋊Dic47—28no—
190C22⋊Dic47—216no—
191(C2×D47)⋊C4—216no—
192(C2×D47)⋊C4—28no—
193(C2×D47)⋊C4—216no—
194C4⋊Dic47—28no—
195C4⋊Dic47—216no—
196C4⋊Dic47—216no—
197Dic47⋊C4—28no—
198Dic47⋊C4—216no—
199C4×Dic47—28no—
200C23×C47—28no—
201C23×C47—28no—
202C23×C47—28no—
203C23×C47—22no—
204C23×C47—22no—
205C23×C47—22no—
206C23×C47—22no—
207C23×C47—22no—
208C23×C47—22no—
209C23×C47—44no—
210C23×C47—44no—
211C23×C47—44no—
212C23×C47—44no—
213C23×C47C22×D412yes—
214C23×C47C22×D414yes—
215C23×C47C22×D412yes—
216C23×C47C2×#4112yes—
217C23×C47C2×#4112yes—
218C23×C47C2×#4112yes—
219C23×C472−1+412yes—
220C23×C472−1+412yes—
221C23×C47C2511yescharacteristic
222C22×D47—28no—
223C22×D47—28no—
224C22×D47—28no—
225C22×D47—22no—
226C22×D47—22no—
227C22×D47—22no—
228C22×D47—22no—
229C22×D47—22no—
230C22×D47—22no—
231C22×D47—44no—
232C22×D47—44no—
233C22×D47—44no—
234C22×D47—44no—
235C22×D47C2×#4114yes—
236C2×C4×C47—28no—
237C2×C4×C47—216no—
238C2×C4×C47—22no—
239C2×C4×C47—24no—
240C2×C4×C47—22no—
241C2×C4×C47—24no—
242C2×C4×C47—232no—
243C2×C4×C47—24no—
244C2×C4×C47—24no—
245C2×C4×C47—24no—
246C2×C4×C47—48no—
247C2×C4×C47#121116yes—
248C2×C4×C47C22×D418yes—
249C2×C4×C47C22×D414yes—
250C2×C4×C47C22×D414yes—
251C2×C4×C47C22×D414yes—
252C2×C4×C47C2×#4114yes—
253C22⋊D47—232no—
254C2×Dic47—28no—
255C2×Dic47—22no—
256C2×Dic47—22no—
257C2×Dic47—22no—
258C2×Dic47—24no—
259C2×Dic47—24no—
260C2×Dic47—22no—
261C2×Dic47—22no—
262C2×Dic47—24no—
263C2×Dic47—24no—
264C2×Dic47—24no—
265C2×Dic47—24no—
266C2×Dic47—22no—
267C2×Dic47—22no—
268C2×Dic47—22no—
269C2×Dic47—48no—
270C2×Dic47—44no—
271C2×Dic47—48no—
272C2×Dic47—44no—
273C2×Dic47C2×#4114yes—
274C4×D47—416no—
275Dic94—416no—
276C2×#2415—471nomaximal, Sylow
277C22×C47—24no—
278C22×C47—24no—
279C22×C47—24no—
280C22×C47—416no—
281C22×C47—42no—
282C22×C47—42no—
283C22×C47—42no—
284C22×C47—42no—
285C22×C47C2×#12114yes—
286C22×C47#52514yes—
287C22×C47#53614yes—
288C22×C47#54714yes—
289C22×C47#54814yes—
290C22×C47#55214yes—
291C22×C47#55614yes—
292C22×C47C23×D411yescharacteristic
293C22×C47C23×D411yescharacteristic
294C22×C47C22×#4111yescharacteristic
295C22×C47C22×#4111yescharacteristic
296C22×C47C22×#4111yescharacteristic
297C22×C47C2×2−1+411yescharacteristic
298C22×C47C2×2−1+411yescharacteristic
299C2×D47—24no—
300C2×D47—24no—
301C2×D47—24no—
302C2×D47—42no—
303C2×D47—416no—
304C2×D47—42no—
305C2×D47—42no—
306C2×D47—42no—
307C4×C47—24no—
308C4×C47—28no—
309C4×C47—44no—
310C4×C47C2×#12118yes—
311Dic47—24no—
312Dic47—44no—
313Dic47—42no—
314Dic47—44no—
315Dic47—42no—
316C22×C22⋊Q8—471no—
317C22×C22⋊Q8—471no—
318C22×C4×D4—471no—
319C22×#117D4711yescharacteristic, complemented
320#2415—4716no—
321C2×#397—472no—
322C2×#394—472no—
323C2×#391—471no—
324C2×#389—472no—
325C2×#389—471no—
326C2×#388—472no—
327C2×#380—471no—
328C2×C47—44no—
329C2×C47#241518yescomplemented
330C2×C47C22×#12111yescharacteristic
331C2×C47C2×#52511yescharacteristic
332C2×C47C2×#53611yescharacteristic
333C2×C47C2×#54711yescharacteristic
334C2×C47C2×#54811yescharacteristic
335C2×C47C2×#55211yescharacteristic
336C2×C47C2×#55611yescharacteristic
337D47—44no—
338C23×Q8—471no—
339C23×D4—471no—
340C24×C4—471no—
341C24×C4C2×D4711yescharacteristic
342C2×C22⋊Q8—478no—
343C2×C22⋊Q8—478no—
344C2×C22⋊Q8—948no—
345C2×C22⋊Q8—948no—
346C2×C4×D4—478no—
347C2×C4×D4—948no—
348C2×#117—28no—
349C2×#117C2×D4718yescomplemented
350C22×C4⋊C4—472no—
351C22×C4⋊C4—472no—
352C22×C4⋊C4—471no—
353C22×C4⋊C4—472no—
354C22×C4⋊C4—471no—
355C22×C4⋊C4C2×D4711yescharacteristic
356C22×C4⋊C4C2×D4711yescharacteristic
357C22×C22⋊C4—472no—
358C22×C22⋊C4—471no—
359C22×C22⋊C4—472no—
360C22×C22⋊C4—472no—
361C22×C22⋊C4—471no—
362C22×C22⋊C4C2×D4712yes—
363C22×C42—471no—
364C22×C42C2×D4712yes—
365#397—4716no—
366#394—4716no—
367#391—478no—
368#389—478no—
369#389—4716no—
370#388—4716no—
371#380—478no—
372C2×#95—472no—
373C2×#95—472no—
374C2×#95—472no—
375C2×#95—472no—
376C47C2×#241511yescharacteristic, complemented, Sylow
377C25—471no—
378C25C22×D4711yescharacteristic
379C22×Q8—474no—
380C22×Q8—944no—
381C22×Q8—944no—
382C22×Q8—944no—
383C22×D4—474no—
384C22×D4—944no—
385C22×D4—944no—
386C22×D4—944no—
387C23×C4—24no—
388C23×C4—24no—
389C23×C4—471no—
390C23×C4—472no—
391C23×C4—472no—
392C23×C4—471no—
393C23×C4—471no—
394C23×C4—474no—
395C23×C4—944no—
396C23×C4—944no—
397C23×C4—944no—
398C23×C4D18818yes—
399C23×C4C22×D4712yes—
400C23×C4C22×D4714yes—
401C23×C4C22×D4711yescharacteristic
402C23×C4C22×D4712yes—
403C23×C4C22×D4711yescharacteristic
404C22⋊Q8—9432no—
405C22⋊Q8—9432no—
406C4×D4—9432no—
407#117—232no—
408C2×C4⋊C4—24no—
409C2×C4⋊C4—474no—
410C2×C4⋊C4—478no—
411C2×C4⋊C4—474no—
412C2×C4⋊C4—478no—
413C2×C4⋊C4—478no—
414C2×C4⋊C4—948no—
415C2×C4⋊C4—944no—
416C2×C4⋊C4—948no—
417C2×C4⋊C4—944no—
418C2×C4⋊C4—948no—
419C2×C4⋊C4C22⋊D4718yescomplemented
420C2×C4⋊C4C22×D4714yes—
421C2×C4⋊C4C22×D4714yes—
422C2×C22⋊C4—28no—
423C2×C22⋊C4—474no—
424C2×C22⋊C4—478no—
425C2×C22⋊C4—474no—
426C2×C22⋊C4—478no—
427C2×C22⋊C4—478no—
428C2×C22⋊C4—948no—
429C2×C22⋊C4—944no—
430C2×C22⋊C4—948no—
431C2×C22⋊C4—944no—
432C2×C22⋊C4—948no—
433C2×C22⋊C4C22×D4718yes—
434C2×C42—28no—
435C2×C42—474no—
436C2×C42—944no—
437C2×C42C22×D4718yes—
438#95—478no—
439#95—478no—
440#95—478no—
441#95—478no—
442C24—22no—
443C24—22no—
444C24—22no—
445C24—22no—
446C24—22no—
447C24—22no—
448C24—474no—
449C24—472no—
450C24—942no—
451C24—942no—
452C24—942no—
453C24—942no—
454C24—942no—
455C24—942no—
456C24C2×D18814yes—
457C24C23×D4712yes—
458C24C23×D4711yescharacteristic
459C2×Q8—948no—
460C2×Q8—948no—
461C2×Q8—948no—
462C2×Q8—18816no—
463C2×D4—948no—
464C2×D4—948no—
465C2×D4—9432no—
466C2×D4—948no—
467C22×C4—22no—
468C22×C4—24no—
469C22×C4—22no—
470C22×C4—24no—
471C22×C4—232no—
472C22×C4—28no—
473C22×C4—28no—
474C22×C4—24no—
475C22×C4—24no—
476C22×C4—24no—
477C22×C4—474no—
478C22×C4—472no—
479C22×C4—472no—
480C22×C4—474no—
481C22×C4—474no—
482C22×C4—472no—
483C22×C4—942no—
484C22×C4—942no—
485C22×C4—944no—
486C22×C4—942no—
487C22×C4—942no—
488C22×C4—944no—
489C22×C4—942no—
490C22×C4—944no—
491C22×C4—944no—
492C22×C4—944no—
493C22×C4—942no—
494C22×C4—944no—
495C22×C4—942no—
496C22×C4—942no—
497C22×C4—948no—
498C22×C4—948no—
499C22×C4—948no—
500C22×C4—18816no—
501C22×C4C2×D18814yes—
502C22×C4C2×D188116yes—
503C22×C4C2×D18814yes—
504C22×C4#1062535014yes—
505C22×C4C2×C22⋊D4718yes—
506C22×C4C2×C22⋊D4714yes—
507C22×C4C23×D4714yes—
508C22×C4C23×D4714yes—
509C22×C4C23×D4712yes—
510C22×C4C23×D4712yes—
511C4⋊C4—28no—
512C4⋊C4—9416no—
513C4⋊C4—9416no—
514C4⋊C4—948no—
515C4⋊C4—9416no—
516C4⋊C4—948no—
517C4⋊C4C2×C22⋊D47116yescomplemented
518C22⋊C4—216no—
519C22⋊C4—9416no—
520C22⋊C4—948no—
521C22⋊C4—9416no—
522C22⋊C4—9416no—
523C22⋊C4—948no—
524C42—216no—
525C42—948no—
526C23—28no—
527C23—28no—
528C23—22no—
529C23—22no—
530C23—22no—
531C23—22no—
532C23—28no—
533C23—22no—
534C23—22no—
535C23—44no—
536C23—44no—
537C23—44no—
538C23—44no—
539C23—474no—
540C23—948no—
541C23—948no—
542C23—948no—
543C23—942no—
544C23—942no—
545C23—942no—
546C23—942no—
547C23—942no—
548C23—942no—
549C23—1884no—
550C23—1884no—
551C23—1884no—
552C23—1884no—
553C23C22×C22⋊D4712yes—
554C23C22×D18814yes—
555C23C22×D18812yes—
556C23#41×D4712yes—
557C23#41×D4712yes—
558C23C2×#1062535012yes—
559C23#1309407112yes—
560C23#1309407112yes—
561C23C24×D4711yescharacteristic
562Q8—18816no—
563D4—9432no—
564C2×C4—216no—
565C2×C4—28no—
566C2×C4—22no—
567C2×C4—24no—
568C2×C4—22no—
569C2×C4—24no—
570C2×C4—232no—
571C2×C4—24no—
572C2×C4—24no—
573C2×C4—24no—
574C2×C4—48no—
575C2×C4—474no—
576C2×C4—948no—
577C2×C4—942no—
578C2×C4—942no—
579C2×C4—944no—
580C2×C4—942no—
581C2×C4—942no—
582C2×C4—944no—
583C2×C4—942no—
584C2×C4—944no—
585C2×C4—944no—
586C2×C4—944no—
587C2×C4—942no—
588C2×C4—944no—
589C2×C4—942no—
590C2×C4—942no—
591C2×C4—1884no—
592C2×C4—1888no—
593C2×C4—1888no—
594C2×C4—1884no—
595C2×C4—18816no—
596C2×C4#13093998116yes—
597C2×C4C22×C22⋊D4718yes—
598C2×C4C22×C22⋊D4714yes—
599C2×C4C22×D18814yes—
600C2×C4C22×D18814yes—
601C2×C4C2×#1062535014yes—
602C22—24no—
603C22—24no—
604C22—24no—
605C22—416no—
606C22—42no—
607C22—42no—
608C22—42no—
609C22—42no—
610C22—944no—
611C22—944no—
612C22—944no—
613C22—18816no—
614C22—1882no—
615C22—1882no—
616C22—1882no—
617C22—1882no—
618C22C2×#1309399814yes—
619C22#1568963514yes—
620C22#1568969914yes—
621C22#1568977214yes—
622C22#1568978014yes—
623C22#1568981214yes—
624C22#1568984114yes—
625C22C23×C22⋊D4711yescharacteristic
626C22C23×D18811yescharacteristic
627C22C2×#41×D4711yescharacteristic
628C22C2×#41×D4711yescharacteristic
629C22C22×#1062535011yescharacteristic
630C22C2×#1309407111yescharacteristic
631C22C2×#1309407111yescharacteristic
632C4—28no—
633C4—24no—
634C4—44no—
635C4—944no—
636C4—1882no—
637C4—1884no—
638C4—1884no—
639C4—1882no—
640C4C2×#1309399818yes—
641C2—44no—
642C2—1884no—
643C2#2024681118yesdirect product factor
644C2C22×#1309399811yescharacteristic
645C2C2×#1568963511yescharacteristic
646C2C2×#1568969911yescharacteristic
647C2C2×#1568977211yescharacteristic
648C2C2×#1568978011yescharacteristic
649C2C2×#1568981211yescharacteristic
650C2C2×#1568984111yescharacteristic
subgroupG/H
centerC24C23×D47
commutator, cosocleC23×C47C25
FrattiniC23C24×D47
FittingC22×#117×C47C2
socleC24×C47C24
prime radicalC23×C22⋊D47C22
hypercenterC22×#117D47
hypocenterC47C2×#2415
LCSG ▷ C23×C47 ▷ C47
UCSC1 ◁ C24 ◁ C22×#117

Complements

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

NH ≅ G/Ntype
1. C22×#13094004C2semidirect
2. C22×#13094003C2semidirect
4. #20246811C2direct (N × H)
5. C2×#15689048C2semidirect
7. C2×#15689028C2semidirect
8. C2×#15689022C2semidirect
9. C2×#15689021C2semidirect
10. C2×#15689014C2semidirect
11. C2×#15688980C2semidirect
12. C22×#117×C47C2semidirect
17. C2×#13094004C22semidirect
19. C2×#13094003C22semidirect
33. #15689048C22semidirect
35. #15689028C22semidirect
36. #15689022C22semidirect
37. #15689021C22semidirect
38. #15689014C22semidirect
39. #15688980C22semidirect
46. C2×#117×C47C22semidirect
109. C2×C4⋊C4×C47D4semidirect
186. C4⋊C4×C47C2×D4semidirect
319. C22×#117D47semidirect
329. C2×C47#2415semidirect
349. C2×#117C2×D47semidirect
376. C47C2×#2415semidirect
419. C2×C4⋊C4C22⋊D47semidirect
517. C4⋊C4C2×C22⋊D47semidirect
643. C2#20246811direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#130940042
C22×#130940032
C22×C4×C22⋊D472
#202468112
C2×#156890482
C2×#156890392
C2×#156890282
C2×#156890222
C2×#156890212
C2×#156890142
C2×#156889802
C22×#117×C472
C2×#241547