Finite Group Databasebeta

← C2×#20254220 #27086898

Subgroups — C2×#20254220 #27086898

Order 12032.

2342 conjugacy classes of subgroups.

The subgroup lattice has 666 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689634C211yesmaximal, characteristic
2C22×#13094003C211yesmaximal, characteristic, complemented
3#20254220C2116yesmaximal, direct product factor
4C2×#15689459C211yesmaximal, characteristic, complemented
5C2×#15689442C211yesmaximal, characteristic, complemented
6C2×#15689465C212yesmaximal, complemented
7C2×#15689381C212yesmaximal, complemented
8C2×#15689369C212yesmaximal, complemented
9C2×#15689287C212yesmaximal, complemented
10C2×#15689281C211yesmaximal, characteristic, complemented
11C2×#15689264C211yesmaximal, characteristic, complemented
12C2×C47×#407C211yesmaximal, characteristic, complemented
13C22×#10625350C2211yescharacteristic
14C23×Dic94C2211yescharacteristic
15C22×Dic188C2212yescomplemented
16C22×#10625325C2212yescomplemented
17#15689634—24no—
18#15689634C2218yes—
19C2×#13094007—21no—
20C2×#13094003—21no—
21C2×#13094003—21no—
22C2×#13094003—24no—
23C2×#13094003C2212yes—
24C2×#13094003C2211yescharacteristic
25C2×#13094003C2212yes—
26C2×#13094003C2211yescharacteristic
27C2×#13094003C2218yescomplemented
28C2×#13093998C2211yescharacteristic
29C2×#13093998C2211yescharacteristic
30C2×#13093993—21no—
31C2×C4×C22⋊D47—21no—
32C2×C4×C22⋊D47—21no—
33C2×#13093983C2211yescharacteristic
34C22×C4⋊Dic47C2211yescharacteristic
35C22×Dic47⋊C4—21no—
36C22×C22⋊Dic47—21no—
37#15689465C22116yescomplemented
38#15689459C2218yescomplemented
39#15689442C2218yescomplemented
40#15689381C22116yescomplemented
41#15689369C22116yescomplemented
42#15689287C22116yescomplemented
43#15689281C2218yescomplemented
44#15689264C2218yescomplemented
45C2×#13093958C2212yescomplemented
46C2×#13093956C2212yescomplemented
47C2×#13093949C2212yescomplemented
48C2×#13093949C2212yescomplemented
49C2×#13093949C2212yescomplemented
50C2×#13093946C2212yescomplemented
51C24×C4×C47C2211yescharacteristic
52C22×M16×C47C2212yes—
53C47×#407C2218yescomplemented
54C2×C22⋊C8×C47C2212yes—
55C2×C22⋊C8×C47C2211yescharacteristic
56C2×C22⋊C8×C47C2211yescharacteristic
57C2×#10625350—22no—
58C2×#10625350—44no—
59C2×#10625350C2314yes—
60C22×Dic94—21no—
61C22×Dic94—22no—
62C22×Dic94—44no—
63C22×Dic94C2312yes—
64C22×Dic94C2311yescharacteristic
65C22×Dic94C2314yes—
66C22×D188C2311yescharacteristic
67C22×C22⋊D47—21no—
68C22×C22⋊D47—21no—
69C22×C22⋊D47—21no—
70C23×Dic47—21no—
71C23×Dic47—21no—
72C22×C4×D47—21no—
73C2×Dic188—44no—
74C2×Dic188C2318yescomplemented
75C2×#10625325—44no—
76C2×#10625325C2318yescomplemented
77#13094007—28no—
78#13094003—24no—
79#13094003—24no—
80#13094003—28no—
81#13094003—28no—
82#13094003—22no—
83#13094003—22no—
84#13094003C2314yes—
85#13094003C2318yes—
86#13094003C2318yes—
87#13094003C2314yes—
88#13093998—22no—
89#13093998—22no—
90#13093998C2314yes—
91#13093998C2314yes—
92#13093993—28no—
93C4×C22⋊D47—28no—
94C4×C22⋊D47—28no—
95#13093983—22no—
96#13093983C2314yes—
97C2×C4⋊Dic47—21no—
98C2×C4⋊Dic47—22no—
99C2×C4⋊Dic47C2312yes—
100C2×C4⋊Dic47C2314yes—
101C2×C4⋊Dic47C2311yescharacteristic
102C2×C4⋊Dic47C2311yescharacteristic
103C2×Dic47⋊C4—22no—
104C2×Dic47⋊C4—22no—
105C2×Dic47⋊C4—28no—
106C2×Dic47⋊C4—21no—
107C2×Dic47⋊C4—21no—
108C2×C22⋊Dic47—22no—
109C2×C22⋊Dic47—22no—
110C2×C22⋊Dic47—28no—
111C2×C22⋊Dic47—21no—
112C2×C22⋊Dic47—21no—
113C2×C22⋊Dic47—21no—
114C2×(C2×D47)⋊C4—21no—
115C2×(C2×D47)⋊C4—21no—
116C2×C4×Dic47—21no—
117#13093958C2318yescomplemented
118#13093956C2318yescomplemented
119#13093949C2318yescomplemented
120#13093949C2318yescomplemented
121#13093949C2318yescomplemented
122#13093946C2318yescomplemented
123C25×C47D411yescharacteristic
124C23×C4×C47—44no—
125C23×C4×C47D411yescharacteristic
126C23×C4×C47D411yescharacteristic
127C23×C4×C47D411yescharacteristic
128C23×C4×C47D414yes—
129C23×C4×C47C2311yescharacteristic
130C23×C4×C47C2311yescharacteristic
131C23×C4×C47C2311yescharacteristic
132C23×C4×C47C2314yes—
133C2×M16×C47—44no—
134C2×M16×C47C2318yes—
135C22×C8×C47C2312yes—
136C22×C8×C47C2312yes—
137C22⋊C8×C47C2314yes—
138C22⋊C8×C47C2314yes—
139C22⋊C8×C47C2318yes—
140C24×C47—24no—
141C24×C47—42no—
142C24×C47—42no—
143C24×C47C2×D411yescharacteristic
144C24×C47C2×D411yescharacteristic
145C24×C47C2×D411yescharacteristic
146C24×C47C2×D414yes—
147C23×D47—21no—
148C22×C4×C47—24no—
149C22×C4×C47—24no—
150C22×C4×C47—24no—
151C22×C4×C47—48no—
152C22×C4×C47—48no—
153C22×C4×C47—42no—
154C22×C4×C47—42no—
155C22×C4×C47—42no—
156C22×C4×C47C2×D411yescharacteristic
157C22×C4×C47C2×D411yescharacteristic
158C22×C4×C47C2×D411yescharacteristic
159C22×C4×C47C2×D412yes—
160C22×C4×C47C2×D412yes—
161C22×C4×C47C2×D412yes—
162C22×C4×C47C2×D414yes—
163C22×C4×C47C2×D414yes—
164C22×C4×C47C2×D414yes—
165C22×C4×C47C2412yes—
166C22×C4×C47C2412yes—
167C22×C4×C47C2412yes—
168C22×C4×C47C2411yescharacteristic
169C2×C22⋊D47—24no—
170C2×C22⋊D47—24no—
171C2×C22⋊D47—24no—
172C2×C22⋊D47—44no—
173C2×C22⋊D47—44no—
174C2×C22⋊D47—44no—
175C22×Dic47—21no—
176C22×Dic47—21no—
177C22×Dic47—24no—
178C22×Dic47—22no—
179C22×Dic47—21no—
180C22×Dic47—24no—
181C22×Dic47—22no—
182C22×Dic47—44no—
183C22×Dic47—44no—
184#10625350—48no—
185#10625350—48no—
186C2×D188—24no—
187C2×D188—21no—
188C2×D188C2412yes—
189C2×C4×D47—24no—
190C2×C4×D47—44no—
191C2×Dic94—24no—
192C2×Dic94—24no—
193C2×Dic94—24no—
194C2×Dic94—24no—
195C2×Dic94—22no—
196C2×Dic94—21no—
197C2×Dic94—48no—
198C2×Dic94—44no—
199C2×Dic94—48no—
200C2×Dic94C2414yes—
201C2×Dic94C2412yes—
202M16×C47—48no—
203C2×C8×C47—24no—
204C2×C8×C47—24no—
205C2×C8×C47C2414yes—
206C2×C8×C47C2414yes—
207C22⋊Dic47—28no—
208C22⋊Dic47—28no—
209C22⋊Dic47—24no—
210C22⋊Dic47—24no—
211C22⋊Dic47—24no—
212(C2×D47)⋊C4—24no—
213(C2×D47)⋊C4—24no—
214C4⋊Dic47—24no—
215C4⋊Dic47—22no—
216C4⋊Dic47—21no—
217C4⋊Dic47—21no—
218C4⋊Dic47C2414yes—
219C4⋊Dic47C2412yes—
220C4⋊Dic47C2412yes—
221Dic47⋊C4—28no—
222Dic47⋊C4—28no—
223Dic47⋊C4—24no—
224Dic47⋊C4—24no—
225C4×Dic47—24no—
226Dic188—48no—
227#10625325—48no—
228C23×C47—22no—
229C23×C47—22no—
230C23×C47—22no—
231C23×C47—22no—
232C23×C47—28no—
233C23×C47—22no—
234C23×C47—22no—
235C23×C47—22no—
236C23×C47—22no—
237C23×C47—44no—
238C23×C47—44no—
239C23×C47—416no—
240C23×C47#13714yes—
241C23×C47C22×D412yes—
242C23×C47C22×D412yes—
243C23×C47C22×D412yes—
244C23×C47C22×D411yescharacteristic
245C22×D47—22no—
246C22×D47—42no—
247C22×D47—42no—
248C22×D47—42no—
249C2×C4×C47—22no—
250C2×C4×C47—22no—
251C2×C4×C47—22no—
252C2×C4×C47—24no—
253C2×C4×C47—24no—
254C2×C4×C47—24no—
255C2×C4×C47—24no—
256C2×C4×C47—24no—
257C2×C4×C47—24no—
258C2×C4×C47—44no—
259C2×C4×C47—44no—
260C2×C4×C47—44no—
261C2×C4×C47C22×D412yes—
262C2×C4×C47C22×D412yes—
263C2×C4×C47C22×D412yes—
264C2×C4×C47C22×D411yescharacteristic
265C2×C4×C47H(5,𝔽2)14yes—
266C2×C4×C47C2511yescharacteristic
267C22⋊D47—48no—
268C22⋊D47—48no—
269C22⋊D47—48no—
270C2×Dic47—22no—
271C2×Dic47—22no—
272C2×Dic47—24no—
273C2×Dic47—24no—
274C2×Dic47—22no—
275C2×Dic47—48no—
276C2×Dic47—48no—
277C2×Dic47—44no—
278C2×Dic47—42no—
279C2×Dic47—42no—
280C2×Dic47—44no—
281C2×Dic47—42no—
282D188—24no—
283D188—42no—
284C4×D47—48no—
285Dic94—24no—
286Dic94—24no—
287Dic94—24no—
288Dic94—48no—
289Dic94—44no—
290Dic94—42no—
291C8×C47—24no—
292C8×C47—24no—
293C2×#3196—471nomaximal, Sylow
294C22×C47—22no—
295C22×C47—28no—
296C22×C47—22no—
297C22×C47—22no—
298C22×C47—22no—
299C22×C47—22no—
300C22×C47—22no—
301C22×C47—22no—
302C22×C47—22no—
303C22×C47—416no—
304C22×C47—44no—
305C22×C47—44no—
306C22×C47#53412yes—
307C22×C47C2×#13714yes—
308C22×C47C2×#13712yes—
309C22×C47#57512yes—
310C22×C47C23×D411yescharacteristic
311C2×D47—42no—
312C2×D47—42no—
313C2×D47—42no—
314C2×D47—84no—
315C4×C47—22no—
316C4×C47—22no—
317C4×C47—22no—
318C4×C47#53412yes—
319C4×C47C2×H(5,𝔽2)12yes—
320Dic47—44no—
321Dic47—42no—
322Dic47—42no—
323Dic47—44no—
324Dic47—42no—
325C2×#525—471no—
326C22×C22⋊Q8—471no—
327#3196—4716no—
328C2×#484—471no—
329C2×#481—471no—
330C2×#485—472no—
331C2×#470—472no—
332C2×#468—472no—
333C2×#452—472no—
334C2×#451—471no—
335C2×#448—471no—
336C2×#407D4711yescharacteristic, complemented
337C2×C47—24no—
338C2×C47—42no—
339C2×C47—42no—
340C2×C47#319614yescomplemented
341C2×C47C2×#53411yescharacteristic
342C2×C47C22×#13711yescharacteristic
343C2×C47C2×#57511yescharacteristic
344D47—82no—
345C22×#41—471no—
346C23×Q8—471no—
347C24×C4C2×D4711yescharacteristic
348C22×Q16—472no—
349C22×SD16—472no—
350C22×M16C2×D4712yes—
351#525—478no—
352#525—944no—
353C2×#123—941no—
354C2×C22⋊Q8—472no—
355C2×C22⋊Q8—471no—
356C2×C22⋊Q8—478no—
357C2×C22⋊Q8—472no—
358C2×C22⋊Q8—471no—
359C2×C22⋊Q8—941no—
360C2×C22⋊Q8—941no—
361C2×C22⋊Q8—944no—
362C2×#121—471no—
363C2×#121—471no—
364C2×#120—941no—
365C2×C4×D4—941no—
366C2×C4×D4—941no—
367C2×#117—471no—
368C22×C4⋊C4—471no—
369C22×C4⋊C4—941no—
370C22×C22⋊C4—941no—
371#485—4716no—
372#484—478no—
373#481—478no—
374#470—4716no—
375#468—4716no—
376#452—4716no—
377#451—478no—
378#448—478no—
379C2×#107—472no—
380C2×#106—472no—
381C2×#103—472no—
382C2×#103—472no—
383C2×#103—472no—
384C2×#102—472no—
385#407C2×D4718yescomplemented
386C2×C22⋊C8C2×D4712yes—
387C2×C22⋊C8C2×D4711yescharacteristic
388C2×C22⋊C8C2×D4711yescharacteristic
389C47C2×#319611yescharacteristic, complemented, Sylow
390C25D18811yescharacteristic
391C2×#41—474no—
392C2×#41—942no—
393C2×#41—1884no—
394C22×Q8—472no—
395C22×Q8—471no—
396C22×Q8—474no—
397C22×Q8—942no—
398C22×Q8—941no—
399C22×Q8—1884no—
400C22×D4—471no—
401C22×D4—941no—
402C22×D4—941no—
403C22×D4—941no—
404C23×C4—44no—
405C23×C4—941no—
406C23×C4—941no—
407C23×C4—941no—
408C23×C4D18811yescharacteristic
409C23×C4D18811yescharacteristic
410C23×C4D18811yescharacteristic
411C23×C4D18814yes—
412C23×C4C22×D4711yescharacteristic
413C23×C4C22×D4711yescharacteristic
414C23×C4C22×D4711yescharacteristic
415C23×C4C22×D4714yes—
416C2×Q16—478no—
417C2×Q16—1884no—
418C2×SD16—478no—
419C2×SD16—1884no—
420C2×M16—44no—
421C2×M16C22×D4718yes—
422C22×C8C22×D4712yes—
423C22×C8C22×D4712yes—
424#123—948no—
425C22⋊Q8—478no—
426C22⋊Q8—478no—
427C22⋊Q8—474no—
428C22⋊Q8—474no—
429C22⋊Q8—944no—
430C22⋊Q8—944no—
431C22⋊Q8—948no—
432C22⋊Q8—948no—
433C22⋊Q8—942no—
434C22⋊Q8—942no—
435#121—474no—
436#121—474no—
437#121—942no—
438#121—942no—
439#120—948no—
440C4×D4—948no—
441C4×D4—948no—
442#117—474no—
443#117—942no—
444C2×C4⋊C4—472no—
445C2×C4⋊C4—474no—
446C2×C4⋊C4—471no—
447C2×C4⋊C4—471no—
448C2×C4⋊C4—942no—
449C2×C4⋊C4—941no—
450C2×C4⋊C4—948no—
451C2×C4⋊C4—942no—
452C2×C4⋊C4—942no—
453C2×C4⋊C4—941no—
454C2×C4⋊C4—941no—
455C2×C22⋊C4—948no—
456C2×C22⋊C4—942no—
457C2×C22⋊C4—942no—
458C2×C22⋊C4—941no—
459C2×C22⋊C4—941no—
460C2×C22⋊C4—941no—
461C2×C22⋊C4—941no—
462C2×C22⋊C4—941no—
463C2×C42—941no—
464#107—478no—
465#106—478no—
466#103—478no—
467#103—478no—
468#103—478no—
469#102—478no—
470C22⋊C8C22×D4714yes—
471C22⋊C8C22×D4714yes—
472C22⋊C8C22×D4718yes—
473C24—24no—
474C24—42no—
475C24—42no—
476C24—941no—
477C24C2×D18811yescharacteristic
478C24C2×D18811yescharacteristic
479C24C2×D18811yescharacteristic
480C24C2×D18814yes—
481#41—1888no—
482#41—1888no—
483C2×Q8—474no—
484C2×Q8—472no—
485C2×Q8—944no—
486C2×Q8—942no—
487C2×Q8—941no—
488C2×Q8—944no—
489C2×Q8—944no—
490C2×Q8—944no—
491C2×Q8—1888no—
492C2×Q8—1884no—
493C2×Q8—1888no—
494C2×D4—472no—
495C2×D4—944no—
496C2×D4—944no—
497C2×D4—944no—
498C2×D4—941no—
499C2×D4—944no—
500C2×D4—1884no—
501C2×D4—1884no—
502C2×D4—1884no—
503C22×C4—24no—
504C22×C4—24no—
505C22×C4—24no—
506C22×C4—48no—
507C22×C4—48no—
508C22×C4—42no—
509C22×C4—42no—
510C22×C4—42no—
511C22×C4—944no—
512C22×C4—944no—
513C22×C4—941no—
514C22×C4—941no—
515C22×C4—942no—
516C22×C4—941no—
517C22×C4—942no—
518C22×C4—944no—
519C22×C4—1884no—
520C22×C4—1884no—
521C22×C4—1884no—
522C22×C4C2×D18812yes—
523C22×C4C2×D18812yes—
524C22×C4C2×D18812yes—
525C22×C4C2×D18811yescharacteristic
526C22×C4C2×D18811yescharacteristic
527C22×C4C2×D18811yescharacteristic
528C22×C4C2×D18814yes—
529C22×C4C2×D18814yes—
530C22×C4C2×D18814yes—
531C22×C4C23×D4712yes—
532C22×C4C23×D4712yes—
533C22×C4C23×D4712yes—
534C22×C4C23×D4711yescharacteristic
535Q16—1888no—
536SD16—1888no—
537M16—48no—
538C2×C8—24no—
539C2×C8—24no—
540C2×C8C23×D4714yes—
541C2×C8C23×D4714yes—
542C4⋊C4—474no—
543C4⋊C4—472no—
544C4⋊C4—472no—
545C4⋊C4—948no—
546C4⋊C4—948no—
547C4⋊C4—944no—
548C4⋊C4—944no—
549C4⋊C4—944no—
550C4⋊C4—942no—
551C4⋊C4—941no—
552C4⋊C4—941no—
553C22⋊C4—948no—
554C22⋊C4—948no—
555C22⋊C4—944no—
556C22⋊C4—944no—
557C22⋊C4—944no—
558C22⋊C4—944no—
559C22⋊C4—944no—
560C42—944no—
561C23—22no—
562C23—22no—
563C23—28no—
564C23—22no—
565C23—22no—
566C23—22no—
567C23—22no—
568C23—22no—
569C23—22no—
570C23—44no—
571C23—44no—
572C23—416no—
573C23—942no—
574C23—1882no—
575C23—1882no—
576C23—1882no—
577C23#1309405714yes—
578C23C22×D18812yes—
579C23C22×D18812yes—
580C23C22×D18812yes—
581C23C22×D18811yescharacteristic
582Q8—944no—
583Q8—944no—
584Q8—944no—
585Q8—1888no—
586Q8—1884no—
587Q8—1882no—
588D4—944no—
589D4—1888no—
590D4—1888no—
591D4—1888no—
592D4—1882no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—24no—
597C2×C4—24no—
598C2×C4—24no—
599C2×C4—24no—
600C2×C4—24no—
601C2×C4—24no—
602C2×C4—44no—
603C2×C4—44no—
604C2×C4—44no—
605C2×C4—942no—
606C2×C4—942no—
607C2×C4—944no—
608C2×C4—942no—
609C2×C4—944no—
610C2×C4—1888no—
611C2×C4—1888no—
612C2×C4—1888no—
613C2×C4—1882no—
614C2×C4—1884no—
615C2×C4—1882no—
616C2×C4—1882no—
617C2×C4—1884no—
618C2×C4C22×D18812yes—
619C2×C4C22×D18812yes—
620C2×C4C22×D18812yes—
621C2×C4C22×D18811yescharacteristic
622C2×C4#1309406914yes—
623C2×C4C24×D4711yescharacteristic
624C8—24no—
625C8—24no—
626C22—28no—
627C22—22no—
628C22—22no—
629C22—22no—
630C22—22no—
631C22—22no—
632C22—22no—
633C22—22no—
634C22—22no—
635C22—416no—
636C22—44no—
637C22—44no—
638C22—3764no—
639C22—1882no—
640C22—1882no—
641C22—1882no—
642C22#1568968512yes—
643C22C2×#1309405714yes—
644C22C2×#1309405712yes—
645C22#1568992312yes—
646C22C23×D18811yescharacteristic
647C4—22no—
648C4—22no—
649C4—22no—
650C4—1882no—
651C4—1884no—
652C4—1882no—
653C4—1882no—
654C4—1884no—
655C4#1568968512yes—
656C4C2×#1309406912yes—
657C2—24no—
658C2—42no—
659C2—42no—
660C2—3762no—
661C2#2025422014yesdirect product factor
662C2C2×#1568968511yescharacteristic
663C2C22×#1309405711yescharacteristic
664C2C2×#1568992311yescharacteristic
subgroupG/H
centerC23C22×D188
commutator, cosocleC2×C4×C47C25
FrattiniC2×C4C24×D47
FittingC2×C47×#407C2
socleC23×C47C22×D4
prime radicalC2×#15689634C2
hypercenterC2×#407D47
hypocenterC47C2×#3196
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C24×C4 ◁ C2×#407

Complements

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

NH ≅ G/Ntype
2. C22×#13094003C2semidirect
3. #20254220C2direct (N × H)
4. C2×#15689459C2semidirect
5. C2×#15689442C2semidirect
6. C2×#15689465C2semidirect
7. C2×#15689381C2semidirect
8. C2×#15689369C2semidirect
9. C2×#15689287C2semidirect
10. C2×#15689281C2semidirect
11. C2×#15689264C2semidirect
12. C2×C47×#407C2semidirect
15. C22×Dic188C22semidirect
16. C22×#10625325C22semidirect
27. C2×#13094003C22semidirect
37. #15689465C22semidirect
38. #15689459C22semidirect
39. #15689442C22semidirect
40. #15689381C22semidirect
41. #15689369C22semidirect
42. #15689287C22semidirect
43. #15689281C22semidirect
44. #15689264C22semidirect
45. C2×#13093958C22semidirect
46. C2×#13093956C22semidirect
47. C2×#13093949C22semidirect
48. C2×#13093949C22semidirect
49. C2×#13093949C22semidirect
50. C2×#13093946C22semidirect
53. C47×#407C22semidirect
74. C2×Dic188C23semidirect
76. C2×#10625325C23semidirect
117. #13093958C23semidirect
118. #13093956C23semidirect
119. #13093949C23semidirect
120. #13093949C23semidirect
121. #13093949C23semidirect
122. #13093946C23semidirect
336. C2×#407D47semidirect
340. C2×C47#3196semidirect
385. #407C2×D47semidirect
389. C47C2×#3196semidirect
661. C2#20254220direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#156896342
C22×#130940032
#202542202
C2×#156894592
C2×#156894422
C2×#156894652
C2×#156893812
C2×#156893692
C2×#156892872
C2×#156892812
C2×#156892642
C2×C47×#4072
C2×#319647