Finite Group Databasebeta

← C2×#20241881 #26183375

Subgroups — C2×#20241881 #26183375

Order 12032.

1474 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
1C2×#15689668C211yesmaximal, characteristic
2C2×#15689173C211yesmaximal, characteristic, complemented
3C2×#15688974C211yesmaximal, characteristic, complemented
4#20241881C218yesmaximal, direct product factor
5C2×#15688858C211yesmaximal, characteristic, complemented
6C2×#15688857C211yesmaximal, characteristic, complemented
7C2×#15688856C211yesmaximal, characteristic, complemented
8C2×C47×#327C211yesmaximal, characteristic, complemented
9C22×#10625352—21no—
10#15689668—22no—
11#15689668—22no—
12#15689668C414yescomplemented
13#15689668C2214yes—
14C2×#13094023C411yescharacteristic, complemented
15C2×#13094020C411yescharacteristic, complemented
16C2×#13094010—21no—
17C2×#13093998C2211yescharacteristic
18C2×#13093996—21no—
19C2×D4×Dic47C411yescharacteristic, complemented
20C2×#13093986C411yescharacteristic, complemented
21C2×(C22⋊D47)⋊C4—21no—
22C22×Dic47⋊C4—21no—
23C22×C4×Dic47—21no—
24C22×C4×Dic47C2211yescharacteristic
25#15689173—22no—
26#15689173C2214yescomplemented
27C2×Dic47⋊C8—21no—
28C2×C22⋊(C47⋊C8)C2211yescharacteristic
29C2×C8×Dic47—21no—
30#15688974—22no—
31#15688974C2214yescomplemented
32C2×C47⋊#95C2211yescharacteristic
33#15688858C2214yescomplemented
34#15688857C2214yescomplemented
35#15688856C2214yescomplemented
36C2×#121×C47C2211yescharacteristic
37C2×C22⋊C8×C47C2211yescharacteristic
38C2×#95×C47C2211yescharacteristic
39C47×#327C2214yescomplemented
40C2×#10625352—24no—
41C2×#10625352—24no—
42C2×#10625352—44no—
43C2×#10625352—44no—
44C22×Dic94—21no—
45C22×D188C2×C411yescharacteristic
46C22×C22⋊D47—21no—
47C22×C22⋊D47—21no—
48C22×C22⋊D47—21no—
49C23×Dic47—21no—
50C23×Dic47—21no—
51C23×Dic47D411yescharacteristic
52C23×Dic47D411yescharacteristic
53C22×C4×D47—21no—
54C22×C47⋊C8—21no—
55#13094023—22no—
56#13094023C2×C414yescomplemented
57#13094020—22no—
58#13094020C2×C414yescomplemented
59#13094010—24no—
60#13094010—24no—
61#13093998—21no—
62#13093998—21no—
63#13093998C2×C412yes—
64#13093998C2312yes—
65#13093996—24no—
66#13093996—24no—
67D4×Dic47—22no—
68D4×Dic47C2×C414yescomplemented
69#13093986—22no—
70#13093986C2×C414yescomplemented
71(C22⋊D47)⋊C4—24no—
72(C22⋊D47)⋊C4—24no—
73C2×C4⋊Dic47C2×C411yescharacteristic
74C2×Dic47⋊C4—28no—
75C2×Dic47⋊C4—21no—
76C2×Dic47⋊C4—24no—
77C2×C22⋊Dic47—21no—
78C2×(C2×D47)⋊C4—21no—
79C2×(C2×D47)⋊C4—21no—
80C2×C4×Dic47—24no—
81C2×C4×Dic47—44no—
82C2×C4×Dic47C2×C411yescharacteristic
83C2×C4×Dic47C2×C411yescharacteristic
84C2×C4×Dic47C2×C412yes—
85C2×C4×Dic47D412yes—
86C2×C4×Dic47D411yescharacteristic
87C2×C4×Dic47D411yescharacteristic
88C2×C4×Dic47D412yes—
89C2×C4×Dic47C2312yes—
90Dic47⋊C8—24no—
91C22⋊(C47⋊C8)—21no—
92C22⋊(C47⋊C8)C2312yes—
93C8×Dic47—24no—
94C47⋊#95—21no—
95C47⋊#95C2312yes—
96C22×D4×C47—21no—
97C22×D4×C47C2×C411yescharacteristic
98C23×C4×C47—21no—
99C23×C4×C47C2311yescharacteristic
100C22×C8×C47—21no—
101#121×C47—21no—
102#121×C47—21no—
103#121×C47C2×C412yes—
104#121×C47C2312yes—
105C2×C4⋊C4×C47C2×C411yescharacteristic
106C2×C22⋊C4×C47—21no—
107C22⋊C8×C47—21no—
108C22⋊C8×C47C2312yes—
109#95×C47—21no—
110#95×C47C2312yes—
111C24×C47—21no—
112C24×C47C2×D411yescharacteristic
113C23×D47—21no—
114C2×D4×C47—22no—
115C2×D4×C47—22no—
116C2×D4×C47—21no—
117C2×D4×C47—42no—
118C2×D4×C47—42no—
119C2×D4×C47—42no—
120C2×D4×C47C22×C412yes—
121C22×C4×C47—24no—
122C22×C4×C47—21no—
123C22×C4×C47—22no—
124C22×C4×C47—42no—
125C22×C4×C47—42no—
126C22×C4×C47—42no—
127C22×C4×C47C22×C411yescharacteristic
128C22×C4×C47C22×C411yescharacteristic
129C22×C4×C47C2×D411yescharacteristic
130C22×C4×C47C2×D411yescharacteristic
131C22×C4×C47C2×D411yescharacteristic
132C22×C4×C47C2411yescharacteristic
133C2×C22⋊D47—22no—
134C2×C22⋊D47—22no—
135C2×C22⋊D47—24no—
136C2×C22⋊D47—24no—
137C2×C22⋊D47—42no—
138C2×C22⋊D47—42no—
139C2×C22⋊D47—44no—
140C2×C22⋊D47—44no—
141C22×Dic47—24no—
142C22×Dic47—24no—
143C22×Dic47—21no—
144C22×Dic47—22no—
145C22×Dic47—24no—
146C22×Dic47—42no—
147C22×Dic47—42no—
148C22×Dic47—44no—
149C22×Dic47C22⋊C411yescharacteristic
150C22×Dic47C22⋊C412yes—
151C22×Dic47C22⋊C411yescharacteristic
152C22×Dic47C22⋊C411yescharacteristic
153C22×Dic47C22⋊C412yes—
154C22×Dic47C22⋊C411yescharacteristic
155C22×Dic47D814yes—
156C22×Dic47SD1614yes—
157C22×Dic47C2×D412yes—
158C22×Dic47C2×D412yes—
159#10625352—416no—
160#10625352—48no—
161#10625352—48no—
162C2×D188—21no—
163C2×D188—42no—
164C2×D188C22×C412yes—
165C2×C4×D47—24no—
166C2×C4×D47—44no—
167C2×Dic94—24no—
168C2×Dic94—44no—
169C2×C8×C47—22no—
170C2×C8×C47—42no—
171C4⋊C4×C47—21no—
172C4⋊C4×C47C22×C412yes—
173C22⋊C4×C47—22no—
174C22⋊C4×C47—22no—
175C22⋊Dic47—24no—
176(C2×D47)⋊C4—22no—
177(C2×D47)⋊C4—22no—
178(C2×D47)⋊C4—24no—
179C4⋊Dic47—21no—
180C4⋊Dic47C22×C412yes—
181Dic47⋊C4—216no—
182Dic47⋊C4—24no—
183C4×Dic47—48no—
184C4×Dic47C22⋊C412yes—
185C4×Dic47C22⋊C412yes—
186C4×Dic47C22⋊C412yes—
187C4×Dic47C22⋊C412yes—
188C4×Dic47C22×C412yes—
189C4×Dic47C22×C412yes—
190C4×Dic47C2×D412yes—
191C4×Dic47C2×D412yes—
192C2×C47⋊C8—22no—
193C2×C47⋊C8—42no—
194C23×C47—22no—
195C23×C47—22no—
196C23×C47—22no—
197C23×C47—42no—
198C23×C47—42no—
199C23×C47—42no—
200C23×C47C2×C22⋊C411yescharacteristic
201C23×C47C2×C22⋊C411yescharacteristic
202C23×C47C2×D812yes—
203C23×C47C2×SD1612yes—
204C23×C47C22×D411yescharacteristic
205C22×D47—22no—
206C22×D47—42no—
207C22×D47—42no—
208C22×D47—42no—
209D4×C47—42no—
210D4×C47—42no—
211D4×C47—42no—
212D4×C47—42no—
213D4×C47—44no—
214C2×C4×C47—22no—
215C2×C4×C47—22no—
216C2×C4×C47—24no—
217C2×C4×C47—22no—
218C2×C4×C47—41no—
219C2×C4×C47—41no—
220C2×C4×C47—41no—
221C2×C4×C47—41no—
222C2×C4×C47—42no—
223C2×C4×C47—48no—
224C2×C4×C47—42no—
225C2×C4×C47—42no—
226C2×C4×C47C2×C22⋊C411yescharacteristic
227C2×C4×C47C2×C22⋊C411yescharacteristic
228C2×C4×C47C23×C411yescharacteristic
229C2×C4×C47C22×D411yescharacteristic
230C22⋊D47—42no—
231C22⋊D47—42no—
232C22⋊D47—44no—
233C22⋊D47—48no—
234C22⋊D47—48no—
235C2×Dic47—22no—
236C2×Dic47—24no—
237C2×Dic47—24no—
238C2×Dic47—24no—
239C2×Dic47—22no—
240C2×Dic47—42no—
241C2×Dic47—48no—
242C2×Dic47—42no—
243C2×Dic47—42no—
244C2×Dic47—48no—
245C2×Dic47#10214yes—
246C2×Dic47#10214yes—
247C2×Dic47#10218yes—
248C2×Dic47#10414yes—
249C2×Dic47C2×C22⋊C412yes—
250C2×Dic47C2×C22⋊C412yes—
251C2×Dic47C2×C22⋊C412yes—
252C2×Dic47C2×C22⋊C412yes—
253C2×Dic47C2×D814yes—
254C2×Dic47C2×SD1614yes—
255D188—42no—
256D188—42no—
257C4×D47—48no—
258Dic94—48no—
259C8×C47—42no—
260C47⋊C8—42no—
261C2×#1502—471nomaximal, Sylow
262C22×C47—21no—
263C22×C47—21no—
264C22×C47—21no—
265C22×C47—21no—
266C22×C47—24no—
267C22×C47—24no—
268C22×C47—42no—
269C22×C47—42no—
270C22×C47—42no—
271C22×C47—84no—
272C22×C47#41012yes—
273C22×C47C2×#10212yes—
274C22×C47C2×#10212yes—
275C22×C47C2×#10412yes—
276C22×C47C22×C22⋊C411yescharacteristic
277C22×C47C22×D811yescharacteristic
278C22×C47C22×SD1611yescharacteristic
279C2×D47—42no—
280C2×D47—42no—
281C2×D47—42no—
282C2×D47—84no—
283C4×C47—21no—
284C4×C47—21no—
285C4×C47—21no—
286C4×C47—21no—
287C4×C47—44no—
288C4×C47—42no—
289Dic47—22no—
290Dic47—44no—
291Dic47—42no—
292Dic47C2×#10214yes—
293Dic47C2×#10214yes—
294Dic47C2×#10414yes—
295C2×#532—471no—
296C2×#431—471no—
297C2×C4×C4⋊C4—471no—
298#1502—478no—
299C2×#327—471no—
300C2×#327—471no—
301C2×#327—471no—
302C2×#327D4711yescharacteristic, complemented
303C2×C47—21no—
304C2×C47—21no—
305C2×C47—22no—
306C2×C47—82no—
307C2×C47#150214yescomplemented
308C2×C47C2×#41011yescharacteristic
309C2×C47C22×#10211yescharacteristic
310C2×C47C22×#10411yescharacteristic
311D47—82no—
312C22×#41—941no—
313#532—474no—
314#532—474no—
315#532—942no—
316#532—942no—
317C2×C4⋊Q8—471no—
318C2×#127—471no—
319C2×#124—941no—
320C2×#121—471no—
321C2×#121—941no—
322C2×#121C2×D4711yescharacteristic
323C2×C4×D4—471no—
324C2×C4×D4—471no—
325C2×C4×D4—941no—
326C22×C4⋊C4—941no—
327C22×C42—471no—
328C22×C42—941no—
329#431—474no—
330#431—942no—
331C2×C4⋊C8—941no—
332C2×C22⋊C8—471no—
333C2×C22⋊C8C2×D4711yescharacteristic
334C2×C4×C8—941no—
335C4×C4⋊C4—474no—
336C4×C4⋊C4—942no—
337C2×#95—471no—
338C2×#95C2×D4711yescharacteristic
339#327—474no—
340#327—474no—
341#327—474no—
342#327C2×D4714yescomplemented
343C47C2×#150211yescharacteristic, complemented, Sylow
344C2×#41—944no—
345C2×#41—944no—
346C2×#41—1884no—
347C2×#41—1884no—
348C22×Q8—941no—
349C22×D4—21no—
350C22×D4—471no—
351C22×D4—941no—
352C22×D4—941no—
353C22×D4—941no—
354C22×D4C4×D4711yescharacteristic
355C23×C4—21no—
356C23×C4—471no—
357C23×C4—471no—
358C23×C4—941no—
359C23×C4—941no—
360C23×C4—941no—
361C23×C4C22×D4711yescharacteristic
362C22×C8—21no—
363C22×C8—941no—
364C4⋊Q8—474no—
365C4⋊Q8—942no—
366#127—474no—
367#127—942no—
368#124—944no—
369#124—944no—
370#121—21no—
371#121—21no—
372#121—472no—
373#121—472no—
374#121—944no—
375#121—944no—
376#121—941no—
377#121—941no—
378#121C4×D4712yes—
379#121C22×D4712yes—
380C4×D4—474no—
381C4×D4—474no—
382C4×D4—944no—
383C4×D4—944no—
384C4×D4—942no—
385C4×D4—942no—
386C2×C4⋊C4—471no—
387C2×C4⋊C4—944no—
388C2×C4⋊C4—941no—
389C2×C4⋊C4—948no—
390C2×C4⋊C4C4×D4711yescharacteristic
391C2×C22⋊C4—21no—
392C2×C22⋊C4—941no—
393C2×C22⋊C4—941no—
394C2×C22⋊C4—941no—
395C2×C42—471no—
396C2×C42—471no—
397C2×C42—472no—
398C2×C42—471no—
399C2×C42—471no—
400C2×C42—472no—
401C2×C42—472no—
402C2×C42—472no—
403C2×C42—944no—
404C2×C42—1884no—
405C4⋊C8—944no—
406C22⋊C8—21no—
407C22⋊C8—472no—
408C22⋊C8—941no—
409C22⋊C8C22×D4712yes—
410C4×C8—944no—
411#95—21no—
412#95—472no—
413#95—941no—
414#95C22×D4712yes—
415C24—21no—
416C24—941no—
417C24D4×D4711yescharacteristic
418#41—18816no—
419#41—1888no—
420#41—1888no—
421C2×Q8—944no—
422C2×Q8—1884no—
423C2×D4—22no—
424C2×D4—22no—
425C2×D4—21no—
426C2×D4—42no—
427C2×D4—42no—
428C2×D4—42no—
429C2×D4—472no—
430C2×D4—944no—
431C2×D4—944no—
432C2×D4—942no—
433C2×D4—942no—
434C2×D4—941no—
435C2×D4—1882no—
436C2×D4—1884no—
437C2×D4—1884no—
438C2×D4—1882no—
439C2×D4—1882no—
440C2×D4C2×C4×D4712yes—
441C22×C4—24no—
442C22×C4—21no—
443C22×C4—22no—
444C22×C4—42no—
445C22×C4—42no—
446C22×C4—42no—
447C22×C4—472no—
448C22×C4—471no—
449C22×C4—471no—
450C22×C4—471no—
451C22×C4—474no—
452C22×C4—474no—
453C22×C4—472no—
454C22×C4—472no—
455C22×C4—471no—
456C22×C4—472no—
457C22×C4—944no—
458C22×C4—944no—
459C22×C4—944no—
460C22×C4—944no—
461C22×C4—941no—
462C22×C4—942no—
463C22×C4—1884no—
464C22×C4—1884no—
465C22×C4—1882no—
466C22×C4—1882no—
467C22×C4C2×C4×D4711yescharacteristic
468C22×C4C2×C4×D4711yescharacteristic
469C22×C4D4×D4711yescharacteristic
470C22×C4D4×D4711yescharacteristic
471C22×C4D4×D4711yescharacteristic
472C22×C4C23×D4711yescharacteristic
473C2×C8—22no—
474C2×C8—42no—
475C2×C8—942no—
476C2×C8—1882no—
477C4⋊C4—21no—
478C4⋊C4—472no—
479C4⋊C4—944no—
480C4⋊C4—9416no—
481C4⋊C4—941no—
482C4⋊C4C2×C4×D4712yes—
483C22⋊C4—22no—
484C22⋊C4—22no—
485C22⋊C4—944no—
486C22⋊C4—944no—
487C22⋊C4—942no—
488C22⋊C4—942no—
489C42—472no—
490C42—472no—
491C42—472no—
492C42—472no—
493C42—472no—
494C42—472no—
495C42—472no—
496C42—472no—
497C42—1888no—
498C23—22no—
499C23—22no—
500C23—22no—
501C23—42no—
502C23—42no—
503C23—42no—
504C23—942no—
505C23—1882no—
506C23—1882no—
507C23—1882no—
508C23C22⋊C4×D4711yescharacteristic
509C23C22⋊C4×D4711yescharacteristic
510C23D8×D4712yes—
511C23SD16×D4712yes—
512C23C2×D4×D4711yescharacteristic
513Q8—1888no—
514D4—42no—
515D4—42no—
516D4—44no—
517D4—42no—
518D4—42no—
519D4—1882no—
520D4—1882no—
521D4—1888no—
522D4—1888no—
523D4—1884no—
524D4—1882no—
525D4—1882no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—24no—
529C2×C4—22no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—42no—
535C2×C4—48no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—472no—
539C2×C4—472no—
540C2×C4—472no—
541C2×C4—474no—
542C2×C4—474no—
543C2×C4—478no—
544C2×C4—474no—
545C2×C4—474no—
546C2×C4—474no—
547C2×C4—472no—
548C2×C4—944no—
549C2×C4—944no—
550C2×C4—942no—
551C2×C4—944no—
552C2×C4—942no—
553C2×C4—1888no—
554C2×C4—1888no—
555C2×C4—1882no—
556C2×C4—1888no—
557C2×C4—1882no—
558C2×C4—1882no—
559C2×C4C22⋊C4×D4711yescharacteristic
560C2×C4C22⋊C4×D4711yescharacteristic
561C2×C4C22×C4×D4711yescharacteristic
562C2×C4C2×D4×D4711yescharacteristic
563C8—42no—
564C8—1882no—
565C22—21no—
566C22—21no—
567C22—21no—
568C22—21no—
569C22—24no—
570C22—24no—
571C22—42no—
572C22—42no—
573C22—42no—
574C22—84no—
575C22—3764no—
576C22—1882no—
577C22—1882no—
578C22—1882no—
579C22#1568909512yes—
580C22#102×D4712yes—
581C22#102×D4712yes—
582C22#104×D4712yes—
583C22C2×C22⋊C4×D4711yescharacteristic
584C22C2×D8×D4711yescharacteristic
585C22C2×SD16×D4711yescharacteristic
586C4—21no—
587C4—21no—
588C4—21no—
589C4—21no—
590C4—44no—
591C4—42no—
592C4—474no—
593C4—474no—
594C4—474no—
595C4—942no—
596C4—1884no—
597C4—1882no—
598C2—21no—
599C2—22no—
600C2—21no—
601C2—82no—
602C2—3762no—
603C2#2024188114yesdirect product factor
604C2C2×#1568909511yescharacteristic
605C2C2×#102×D4711yescharacteristic
606C2C2×#104×D4711yescharacteristic
subgroupG/H
centerC23C22⋊C4×D47
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC2×C47×#327C2
socleC23×C47C2×C22⋊C4
prime radicalC2×#15689668C2
cosocleC22×C4×C47C24
hypercenterC2×#327D47
hypocenterC47C2×#1502
LCSG ▷ C2×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#327

Complements

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

NH ≅ G/Ntype
2. C2×#15689173C2semidirect
3. C2×#15688974C2semidirect
4. #20241881C2direct (N × H)
5. C2×#15688858C2semidirect
6. C2×#15688857C2semidirect
7. C2×#15688856C2semidirect
8. C2×C47×#327C2semidirect
12. #15689668C4semidirect
14. C2×#13094023C4semidirect
15. C2×#13094020C4semidirect
19. C2×D4×Dic47C4semidirect
20. C2×#13093986C4semidirect
26. #15689173C22semidirect
31. #15688974C22semidirect
33. #15688858C22semidirect
34. #15688857C22semidirect
35. #15688856C22semidirect
39. C47×#327C22semidirect
56. #13094023C2×C4semidirect
58. #13094020C2×C4semidirect
68. D4×Dic47C2×C4semidirect
70. #13093986C2×C4semidirect
302. C2×#327D47semidirect
307. C2×C47#1502semidirect
342. #327C2×D47semidirect
343. C47C2×#1502semidirect
603. C2#20241881direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#156896682
C2×#156891732
C2×#156889742
#202418812
C2×#156888582
C2×#156888572
C2×#156888562
C2×C47×#3272
C2×#150247