Finite Group Databasebeta

← C2×#20243682 #26217399

Subgroups — C2×#20243682 #26217399

Order 12032.

2286 conjugacy classes of subgroups.

The subgroup lattice has 646 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#10625324C211yesmaximal, characteristic, complemented
2C22×C4×C22⋊D47C211yesmaximal, characteristic
3C22×C22⋊(C47⋊C8)C211yesmaximal, characteristic, complemented
4C22×(C2×D47)⋊C8C211yesmaximal, characteristic, complemented
5#20243682C218yesmaximal, direct product factor
6C2×#15688877C211yesmaximal, characteristic, complemented
7C2×#15688877C211yesmaximal, characteristic, complemented
8C22×C22⋊C8×C47C211yesmaximal, characteristic, complemented
9C23×C22⋊D47C411yescharacteristic
10C23×C4×D47C2211yescharacteristic
11C22×#10625324—24no—
12C22×#10625324—24no—
13C22×#10625324—24no—
14C22×#10625324C2214yescomplemented
15C23×C47⋊C8C2211yescharacteristic
16C23×C47⋊C8C2211yescharacteristic
17C2×C4×C22⋊D47—24no—
18C2×C4×C22⋊D47—24no—
19C2×C4×C22⋊D47C414yes—
20C2×C4×C22⋊D47C2214yes—
21C22×Dic47⋊C4C411yescharacteristic
22C22×C22⋊Dic47C411yescharacteristic
23C22×(C2×D47)⋊C4C411yescharacteristic
24C22×C4×Dic47C2211yescharacteristic
25C2×C22⋊(C47⋊C8)—24no—
26C2×C22⋊(C47⋊C8)C2214yescomplemented
27C2×(C2×D47)⋊C8—24no—
28C2×(C2×D47)⋊C8C2214yescomplemented
29#15688877C2214yescomplemented
30#15688877C2214yescomplemented
31C24×C4×C47C2211yescharacteristic
32C23×C8×C47C2211yescharacteristic
33C23×C8×C47C2211yescharacteristic
34C2×C22⋊C8×C47—24no—
35C2×C22⋊C8×C47C2214yescomplemented
36C24×D47C2×C411yescharacteristic
37C22×C22⋊D47—24no—
38C22×C22⋊D47—24no—
39C22×C22⋊D47—24no—
40C22×C22⋊D47C2×C414yes—
41C23×Dic47C2×C411yescharacteristic
42C23×Dic47C2×C411yescharacteristic
43C23×Dic47C2×C411yescharacteristic
44C22×C4×D47—22no—
45C22×C4×D47—22no—
46C22×C4×D47—22no—
47C22×C4×D47—22no—
48C22×C4×D47C2×C412yes—
49C22×C4×D47D414yes—
50C22×C4×D47D414yes—
51C22×C4×D47C2312yes—
52C2×#10625324—232no—
53C2×#10625324—28no—
54C2×#10625324—28no—
55C2×#10625324—28no—
56C22×C47⋊C8—22no—
57C22×C47⋊C8—22no—
58C22×C47⋊C8—22no—
59C22×C47⋊C8—22no—
60C22×C47⋊C8—22no—
61C22×C47⋊C8D414yescomplemented
62C22×C47⋊C8C2312yes—
63C22×C47⋊C8C2312yes—
64C4×C22⋊D47—216no—
65C4×C22⋊D47—28no—
66C4×C22⋊D47—28no—
67C2×Dic47⋊C4—24no—
68C2×Dic47⋊C4C2×C414yes—
69C2×C22⋊Dic47—24no—
70C2×C22⋊Dic47C2×C414yes—
71C2×(C2×D47)⋊C4—24no—
72C2×(C2×D47)⋊C4C2×C414yes—
73C2×C4×Dic47—22no—
74C2×C4×Dic47—22no—
75C2×C4×Dic47C2×C412yes—
76C2×C4×Dic47C2312yes—
77C22⋊(C47⋊C8)—28no—
78(C2×D47)⋊C8—28no—
79C25×C47C2×C411yescharacteristic
80C23×C4×C47—22no—
81C23×C4×C47—22no—
82C23×C4×C47—22no—
83C23×C4×C47—22no—
84C23×C4×C47—22no—
85C23×C4×C47—22no—
86C23×C4×C47C2×C411yescharacteristic
87C23×C4×C47C2×C412yes—
88C23×C4×C47C2312yes—
89C23×C4×C47C2311yescharacteristic
90C22×C8×C47—22no—
91C22×C8×C47—22no—
92C22×C8×C47—22no—
93C22×C8×C47—22no—
94C22×C8×C47—22no—
95C22×C8×C47D414yescomplemented
96C22×C8×C47C2312yes—
97C22×C8×C47C2312yes—
98C22⋊C8×C47—28no—
99C24×C47—22no—
100C24×C47—22no—
101C24×C47—22no—
102C24×C47—22no—
103C24×C47—22no—
104C24×C47—22no—
105C24×C47M1614yescomplemented
106C24×C47C22×C412yes—
107C24×C47C22×C411yescharacteristic
108C23×D47—22no—
109C23×D47—22no—
110C23×D47—22no—
111C23×D47—22no—
112C23×D47—22no—
113C23×D47—22no—
114C23×D47C22⋊C414yes—
115C23×D47C22×C412yes—
116C22×C4×C47—22no—
117C22×C4×C47—22no—
118C22×C4×C47—22no—
119C22×C4×C47—24no—
120C22×C4×C47—24no—
121C22×C4×C47—24no—
122C22×C4×C47—22no—
123C22×C4×C47—22no—
124C22×C4×C47—22no—
125C22×C4×C47—22no—
126C22×C4×C47—22no—
127C22×C4×C47—22no—
128C22×C4×C47—44no—
129C22×C4×C47—44no—
130C22×C4×C47—44no—
131C22×C4×C47—44no—
132C22×C4×C47C22×C412yes—
133C22×C4×C47C22×C411yescharacteristic
134C22×C4×C47C2×D412yes—
135C22×C4×C47C2×D412yes—
136C22×C4×C47C2×D412yes—
137C22×C4×C47C2×D412yes—
138C22×C4×C47#4112yes—
139C22×C4×C47#4112yes—
140C22×C4×C47C2411yescharacteristic
141C2×C22⋊D47—232no—
142C2×C22⋊D47—28no—
143C2×C22⋊D47—28no—
144C2×C22⋊D47—28no—
145C22×Dic47—22no—
146C22×Dic47—22no—
147C22×Dic47—22no—
148C22×Dic47—22no—
149C22×Dic47—22no—
150C22×Dic47—22no—
151C22×Dic47—22no—
152C22×Dic47C22⋊C414yes—
153C22×Dic47M1614yescomplemented
154C22×Dic47C22×C412yes—
155C22×Dic47C22×C412yes—
156C22×Dic47C22×C412yes—
157C2×C4×D47—24no—
158C2×C4×D47—24no—
159C2×C4×D47—28no—
160C2×C4×D47—28no—
161C2×C4×D47—28no—
162C2×C4×D47—28no—
163C2×C4×D47—22no—
164C2×C4×D47—22no—
165C2×C4×D47—22no—
166C2×C4×D47—22no—
167C2×C4×D47C22⋊C418yes—
168C2×C4×D47C2×D414yes—
169C2×C4×D47C2×D414yes—
170C2×C8×C47—28no—
171C2×C8×C47—22no—
172C2×C8×C47—22no—
173C2×C8×C47—22no—
174C2×C8×C47—22no—
175C2×C8×C47—22no—
176C2×C8×C47—44no—
177C2×C8×C47C2×D414yescomplemented
178C22⋊Dic47—28no—
179(C2×D47)⋊C4—28no—
180Dic47⋊C4—28no—
181C4×Dic47—24no—
182C4×Dic47—22no—
183C4×Dic47—22no—
184C2×C47⋊C8—28no—
185C2×C47⋊C8—22no—
186C2×C47⋊C8—22no—
187C2×C47⋊C8—22no—
188C2×C47⋊C8—22no—
189C2×C47⋊C8—22no—
190C2×C47⋊C8—44no—
191C2×C47⋊C8C2×D414yescomplemented
192#10625324—232no—
193C23×C47—22no—
194C23×C47—22no—
195C23×C47—22no—
196C23×C47—22no—
197C23×C47—22no—
198C23×C47—22no—
199C23×C47—28no—
200C23×C47—28no—
201C23×C47—28no—
202C23×C47—44no—
203C23×C47—44no—
204C23×C47—44no—
205C23×C47—44no—
206C23×C47C2×C22⋊C412yes—
207C23×C47C4×D412yes—
208C23×C47C4×D412yes—
209C23×C47C2×M1612yes—
210C23×C47C2×M1612yes—
211C23×C47C2×M1614yescomplemented
212C23×C47#13112yes—
213C23×C47#13112yes—
214C23×C47C23×C411yescharacteristic
215C22×D47—22no—
216C22×D47—22no—
217C22×D47—22no—
218C22×D47—22no—
219C22×D47—22no—
220C22×D47—22no—
221C22×D47—28no—
222C22×D47—28no—
223C22×D47—28no—
224C22×D47—44no—
225C22×D47—44no—
226C22×D47—44no—
227C22×D47—44no—
228C22×D47C2×C22⋊C414yes—
229C2×C4×C47—22no—
230C2×C4×C47—22no—
231C2×C4×C47—22no—
232C2×C4×C47—44no—
233C2×C4×C47—42no—
234C2×C4×C47—42no—
235C2×C4×C47—42no—
236C2×C4×C47—42no—
237C2×C4×C47—48no—
238C2×C4×C47C2×C22⋊C412yes—
239C2×C4×C47C4×D412yes—
240C2×C4×C47C4×D412yes—
241C2×C4×C47#12014yes—
242C2×C4×C47#12114yes—
243C2×C4×C47#12114yes—
244C2×C4×C47#12314yes—
245C2×C4×C47C22×D411yescharacteristic
246C2×C4×C47C22×D411yescharacteristic
247C2×C4×C47C22×D411yescharacteristic
248C2×C4×C47C22×D411yescharacteristic
249C2×C4×C47C2×#4111yescharacteristic
250C2×C4×C47C2×#4111yescharacteristic
251C22⋊D47—232no—
252C2×Dic47—22no—
253C2×Dic47—22no—
254C2×Dic47—22no—
255C2×Dic47—22no—
256C2×Dic47—28no—
257C2×Dic47—28no—
258C2×Dic47—22no—
259C2×Dic47—22no—
260C2×Dic47—22no—
261C2×Dic47—44no—
262C2×Dic47C2×C22⋊C414yes—
263C2×Dic47C2×M1614yescomplemented
264C4×D47—24no—
265C4×D47—24no—
266C4×D47—24no—
267C4×D47—24no—
268C4×D47—216no—
269C8×C47—24no—
270C8×C47—42no—
271C47⋊C8—24no—
272C47⋊C8—42no—
273C2×#1870—471nomaximal, Sylow
274C22×C47—24no—
275C22×C47—24no—
276C22×C47—24no—
277C22×C47—42no—
278C22×C47—416no—
279C22×C47—42no—
280C22×C47—42no—
281C22×C47—42no—
282C22×C47#38614yes—
283C22×C47#40714yes—
284C22×C47#40814yes—
285C22×C47#43514yes—
286C22×C47#43514yes—
287C22×C47#43614yes—
288C22×C47#43614yes—
289C22×C47C22×C22⋊C411yescharacteristic
290C22×C47C2×C4×D411yescharacteristic
291C22×C47C2×C4×D411yescharacteristic
292C22×C47C22×M1611yescharacteristic
293C22×C47C22×M1611yescharacteristic
294C22×C47C2×#13111yescharacteristic
295C22×C47C2×#13111yescharacteristic
296C2×D47—24no—
297C2×D47—24no—
298C2×D47—24no—
299C2×D47—416no—
300C2×D47—42no—
301C2×D47—42no—
302C2×D47—42no—
303C2×D47—42no—
304C4×C47—42no—
305C4×C47#38614yes—
306C4×C47C2×#12011yescharacteristic
307C4×C47C2×#12111yescharacteristic
308C4×C47C2×#12111yescharacteristic
309C4×C47C2×#12311yescharacteristic
310Dic47—24no—
311Dic47—24no—
312Dic47—42no—
313C23×M16—471no—
314C22×C4×D4—471no—
315C22×C22⋊C8—471no—
316C22×C22⋊C8—471no—
317C22×C22⋊C8D4711yescharacteristic, complemented
318#1870—478no—
319C2×#336—471no—
320C2×#336—471no—
321C2×C47—44no—
322C2×C47#187018yescomplemented
323C2×C47C2×#38611yescharacteristic
324C2×C47C2×#40711yescharacteristic
325C2×C47C2×#40811yescharacteristic
326C2×C47C2×#43511yescharacteristic
327C2×C47C2×#43511yescharacteristic
328C2×C47C2×#43611yescharacteristic
329C2×C47C2×#43611yescharacteristic
330D47—44no—
331C23×D4—471no—
332C24×C4—471no—
333C24×C4C2×D4711yescharacteristic
334C22×M16—474no—
335C22×M16—944no—
336C22×M16—944no—
337C22×M16—944no—
338C23×C8—471no—
339C23×C8—471no—
340C23×C8C2×D4711yescharacteristic
341C23×C8C2×D4711yescharacteristic
342C2×C4×D4—474no—
343C2×C4×D4—474no—
344C2×C4×D4—944no—
345C2×C4×D4—944no—
346C22×C4⋊C4—471no—
347C22×C22⋊C4—471no—
348C22×C22⋊C4—471no—
349C22×C42—471no—
350C2×C22⋊C8—24no—
351C2×C22⋊C8—474no—
352C2×C22⋊C8—474no—
353C2×C22⋊C8—944no—
354C2×C22⋊C8—944no—
355C2×C22⋊C8C2×D4714yescomplemented
356#336—474no—
357#336—474no—
358C47C2×#187011yescharacteristic, complemented, Sylow
359C25—471no—
360C25C4×D4711yescharacteristic
361C22×D4—474no—
362C22×D4—944no—
363C22×D4—944no—
364C22×D4—944no—
365C23×C4—22no—
366C23×C4—22no—
367C23×C4—22no—
368C23×C4—22no—
369C23×C4—22no—
370C23×C4—22no—
371C23×C4—471no—
372C23×C4—471no—
373C23×C4—472no—
374C23×C4—474no—
375C23×C4—474no—
376C23×C4—472no—
377C23×C4—471no—
378C23×C4—942no—
379C23×C4—942no—
380C23×C4—942no—
381C23×C4—942no—
382C23×C4C4×D4711yescharacteristic
383C23×C4C4×D4712yes—
384C23×C4C22×D4712yes—
385C23×C4C22×D4711yescharacteristic
386C2×M16—948no—
387C2×M16—948no—
388C2×M16—9432no—
389C2×M16—948no—
390C22×C8—22no—
391C22×C8—22no—
392C22×C8—22no—
393C22×C8—22no—
394C22×C8—22no—
395C22×C8—474no—
396C22×C8—472no—
397C22×C8—472no—
398C22×C8—942no—
399C22×C8—942no—
400C22×C8—942no—
401C22×C8—942no—
402C22×C8—942no—
403C22×C8C22⋊D4714yescomplemented
404C22×C8C22×D4712yes—
405C22×C8C22×D4712yes—
406C4×D4—9416no—
407C4×D4—948no—
408C4×D4—948no—
409C2×C4⋊C4—474no—
410C2×C4⋊C4—944no—
411C2×C22⋊C4—474no—
412C2×C22⋊C4—474no—
413C2×C22⋊C4—944no—
414C2×C22⋊C4—944no—
415C2×C42—472no—
416C2×C42—472no—
417C2×C42—942no—
418C2×C42—942no—
419C22⋊C8—28no—
420C22⋊C8—948no—
421C22⋊C8—948no—
422C24—22no—
423C24—22no—
424C24—22no—
425C24—22no—
426C24—22no—
427C24—22no—
428C24—474no—
429C24—472no—
430C24—942no—
431C24—942no—
432C24—942no—
433C24—942no—
434C24—942no—
435C24—942no—
436C24#1062532414yescomplemented
437C24C2×C4×D4712yes—
438C24C2×C4×D4711yescharacteristic
439C2×D4—948no—
440C2×D4—948no—
441C2×D4—9432no—
442C2×D4—948no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—24no—
447C22×C4—24no—
448C22×C4—24no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—44no—
456C22×C4—44no—
457C22×C4—44no—
458C22×C4—44no—
459C22×C4—474no—
460C22×C4—474no—
461C22×C4—472no—
462C22×C4—472no—
463C22×C4—478no—
464C22×C4—474no—
465C22×C4—474no—
466C22×C4—472no—
467C22×C4—942no—
468C22×C4—942no—
469C22×C4—942no—
470C22×C4—942no—
471C22×C4—944no—
472C22×C4—944no—
473C22×C4—948no—
474C22×C4—948no—
475C22×C4—948no—
476C22×C4—948no—
477C22×C4—942no—
478C22×C4—942no—
479C22×C4—942no—
480C22×C4—942no—
481C22×C4—942no—
482C22×C4—942no—
483C22×C4—942no—
484C22×C4C2×C4×D4712yes—
485C22×C4C2×C4×D4711yescharacteristic
486C22×C4#1062535012yes—
487C22×C4D4×D4712yes—
488C22×C4D4×D4712yes—
489C22×C4D4×D4712yes—
490C22×C4#1062535212yes—
491C22×C4C2×C22⋊D4712yes—
492C22×C4C23×D4711yescharacteristic
493M16—9432no—
494C2×C8—28no—
495C2×C8—22no—
496C2×C8—22no—
497C2×C8—22no—
498C2×C8—22no—
499C2×C8—22no—
500C2×C8—44no—
501C2×C8—474no—
502C2×C8—948no—
503C2×C8—942no—
504C2×C8—942no—
505C2×C8—942no—
506C2×C8—942no—
507C2×C8—942no—
508C2×C8—1884no—
509C2×C8C2×C22⋊D4714yescomplemented
510C4⋊C4—948no—
511C22⋊C4—948no—
512C22⋊C4—948no—
513C42—944no—
514C42—942no—
515C42—942no—
516C23—22no—
517C23—22no—
518C23—22no—
519C23—28no—
520C23—22no—
521C23—22no—
522C23—22no—
523C23—28no—
524C23—28no—
525C23—44no—
526C23—44no—
527C23—44no—
528C23—44no—
529C23—474no—
530C23—948no—
531C23—942no—
532C23—942no—
533C23—948no—
534C23—948no—
535C23—942no—
536C23—942no—
537C23—942no—
538C23—942no—
539C23—1884no—
540C23—1884no—
541C23—1884no—
542C23—1884no—
543C23C22⋊C4×D4712yes—
544C23(C22⋊D47)⋊C412yes—
545C23C4×C22⋊D4712yes—
546C23M16×D4712yes—
547C23C2×#1062532412yes—
548C23C2×#1062532414yescomplemented
549C23#1309403012yes—
550C23#1309403212yes—
551C23C22×C4×D4711yescharacteristic
552D4—9432no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—44no—
557C2×C4—48no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—474no—
563C2×C4—474no—
564C2×C4—948no—
565C2×C4—942no—
566C2×C4—942no—
567C2×C4—942no—
568C2×C4—948no—
569C2×C4—942no—
570C2×C4—9416no—
571C2×C4—944no—
572C2×C4—944no—
573C2×C4—944no—
574C2×C4—944no—
575C2×C4—942no—
576C2×C4—942no—
577C2×C4—942no—
578C2×C4—1884no—
579C2×C4C22⋊C4×D4712yes—
580C2×C4(C22⋊D47)⋊C412yes—
581C2×C4C4×C22⋊D4712yes—
582C2×C4#1309399214yes—
583C2×C4#1309399514yes—
584C2×C4#1309399714yes—
585C2×C4#1309400614yes—
586C2×C4C2×D4×D4711yescharacteristic
587C2×C4C2×D4×D4711yescharacteristic
588C2×C4C2×D4×D4711yescharacteristic
589C2×C4C22×C22⋊D4711yescharacteristic
590C2×C4C2×#1062535211yescharacteristic
591C2×C4C2×#1062535011yescharacteristic
592C8—24no—
593C8—42no—
594C8—944no—
595C8—1882no—
596C22—24no—
597C22—24no—
598C22—24no—
599C22—42no—
600C22—42no—
601C22—42no—
602C22—42no—
603C22—416no—
604C22—944no—
605C22—944no—
606C22—944no—
607C22—1882no—
608C22—1882no—
609C22—1882no—
610C22—1882no—
611C22—18816no—
612C22#1568899914yes—
613C22#1568908514yes—
614C22#1568908814yes—
615C22#1568919114yes—
616C22#1568919414yes—
617C22#1568919814yes—
618C22#1568919914yes—
619C22C2×C22⋊C4×D4711yescharacteristic
620C22C2×(C22⋊D47)⋊C411yescharacteristic
621C22C2×C4×C22⋊D4711yescharacteristic
622C22C2×M16×D4711yescharacteristic
623C22C22×#1062532411yescharacteristic
624C22C2×#1309403011yescharacteristic
625C22C2×#1309403211yescharacteristic
626C4—42no—
627C4—944no—
628C4—944no—
629C4—1882no—
630C4#1568899914yes—
631C4C2×#1309399211yescharacteristic
632C4C2×#1309399511yescharacteristic
633C4C2×#1309399711yescharacteristic
634C4C2×#1309400611yescharacteristic
635C2—44no—
636C2—1884no—
637C2#2024368218yesdirect product factor
638C2C2×#1568899911yescharacteristic
639C2C2×#1568908511yescharacteristic
640C2C2×#1568908811yescharacteristic
641C2C2×#1568919111yescharacteristic
642C2C2×#1568919411yescharacteristic
643C2C2×#1568919811yescharacteristic
644C2C2×#1568919911yescharacteristic
subgroupG/H
centerC23×C4C22×D47
commutatorC23×C47C23×C4
FrattiniC22×C4C23×D47
FittingC22×C22⋊C8×C47C2
socleC24×C47C22×C4
prime radicalC23×C22⋊D47C4
cosocleC22×C4×C47C24
hypercenterC22×C22⋊C8D47
hypocenterC47C2×#1870
LCSG ▷ C23×C47 ▷ C47
UCSC1 ◁ C23×C4 ◁ C22×C22⋊C8

Complements

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

NH ≅ G/Ntype
1. C23×#10625324C2semidirect
3. C22×C22⋊(C47⋊C8)C2semidirect
4. C22×(C2×D47)⋊C8C2semidirect
5. #20243682C2direct (N × H)
6. C2×#15688877C2semidirect
7. C2×#15688877C2semidirect
8. C22×C22⋊C8×C47C2semidirect
14. C22×#10625324C22semidirect
26. C2×C22⋊(C47⋊C8)C22semidirect
28. C2×(C2×D47)⋊C8C22semidirect
29. #15688877C22semidirect
30. #15688877C22semidirect
35. C2×C22⋊C8×C47C22semidirect
61. C22×C47⋊C8D4semidirect
95. C22×C8×C47D4semidirect
105. C24×C47M16semidirect
153. C22×Dic47M16semidirect
177. C2×C8×C47C2×D4semidirect
191. C2×C47⋊C8C2×D4semidirect
211. C23×C47C2×M16semidirect
263. C2×Dic47C2×M16semidirect
317. C22×C22⋊C8D47semidirect
322. C2×C47#1870semidirect
355. C2×C22⋊C8C2×D47semidirect
358. C47C2×#1870semidirect
403. C22×C8C22⋊D47semidirect
436. C24#10625324semidirect
509. C2×C8C2×C22⋊D47semidirect
548. C23C2×#10625324semidirect
637. C2#20243682direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#106253242
C22×C4×C22⋊D472
C22×C22⋊(C47⋊C8)2
C22×(C2×D47)⋊C82
#202436822
C2×#156888772
C22×C22⋊C8×C472
C2×#187047