Finite Group Databasebeta

← C2×#20254514 #27088317

Subgroups — C2×#20254514 #27088317

Order 12032.

2286 conjugacy classes of subgroups.

The subgroup lattice has 606 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689668C211yesmaximal, characteristic
2C22×#13094023C211yesmaximal, characteristic, complemented
3#20254514C2116yesmaximal, direct product factor
4C2×#15689520C212yesmaximal, complemented
5C2×#15689468C212yesmaximal, complemented
6C2×#15689442C211yesmaximal, characteristic, complemented
7C2×#15689441C211yesmaximal, characteristic, complemented
8C2×#15689439C211yesmaximal, characteristic, complemented
9C2×#15689412C212yesmaximal, complemented
10C2×#15689411C212yesmaximal, complemented
11C2×#15689173C211yesmaximal, characteristic, complemented
12C2×C47×#481C211yesmaximal, characteristic, complemented
13C22×#10625352—21no—
14C23×Dic94—21no—
15#15689668—24no—
16#15689668C2218yes—
17C2×#13094023—21no—
18C2×#13094023—21no—
19C2×#13094023—24no—
20C2×#13094023C2212yes—
21C2×#13094023C2212yes—
22C2×#13094023C2211yescharacteristic
23C2×#13094023C2218yescomplemented
24C2×#13094020C2211yescharacteristic
25C2×#13094010—21no—
26C2×#13093998C2211yescharacteristic
27C2×#13093996—21no—
28C2×D4×Dic47C2211yescharacteristic
29C2×#13093986C2211yescharacteristic
30C2×(C22⋊D47)⋊C4—21no—
31C22×C4⋊Dic47C2211yescharacteristic
32C22×Dic47⋊C4—21no—
33C22×C4×Dic47C2211yescharacteristic
34#15689520C22116yescomplemented
35#15689468C22116yescomplemented
36#15689442C2218yescomplemented
37#15689441C2218yescomplemented
38#15689439C2218yescomplemented
39#15689412C22116yescomplemented
40#15689411C22116yescomplemented
41#15689173C2218yescomplemented
42C2×#13093956C2212yes—
43C2×#13093955C2212yes—
44C2×#13093955C2212yes—
45C2×Dic47⋊C8C2212yes—
46C2×#13093946C2212yes—
47C2×#13093945C2212yes—
48C2×#13093944C2212yes—
49C2×C22⋊(C47⋊C8)C2211yescharacteristic
50C2×C8×Dic47C2212yes—
51C2×#121×C47C2211yescharacteristic
52C22×C4⋊C4×C47C2211yescharacteristic
53C47×#481C2218yescomplemented
54C2×#106×C47C2212yes—
55C2×#102×C47C2212yes—
56C2×C22⋊C8×C47C2211yescharacteristic
57C2×#10625352—28no—
58C2×#10625352—48no—
59C22×Dic94—21no—
60C22×Dic94—22no—
61C22×Dic94—22no—
62C22×Dic94—28no—
63C22×Dic94—48no—
64C22×D188C2311yescharacteristic
65C22×C22⋊D47—21no—
66C22×C22⋊D47—21no—
67C22×C22⋊D47—21no—
68C23×Dic47—21no—
69C23×Dic47—21no—
70C23×Dic47D411yescharacteristic
71C23×Dic47D411yescharacteristic
72C22×C4×D47—21no—
73C22×C47⋊C8C2312yes—
74#13094023—24no—
75#13094023—24no—
76#13094023—28no—
77#13094023—28no—
78#13094023—22no—
79#13094023C2314yes—
80#13094023C2318yes—
81#13094023C2318yes—
82#13094020—22no—
83#13094020C2314yes—
84#13094010—28no—
85#13093998—22no—
86#13093998C2314yes—
87#13093996—28no—
88D4×Dic47—22no—
89D4×Dic47C2314yes—
90#13093986—22no—
91#13093986C2314yes—
92(C22⋊D47)⋊C4—28no—
93C2×C4⋊Dic47—21no—
94C2×C4⋊Dic47—22no—
95C2×C4⋊Dic47C2312yes—
96C2×C4⋊Dic47C2314yes—
97C2×C4⋊Dic47C2311yescharacteristic
98C2×Dic47⋊C4—22no—
99C2×Dic47⋊C4—22no—
100C2×Dic47⋊C4—28no—
101C2×Dic47⋊C4—21no—
102C2×C22⋊Dic47—21no—
103C2×(C2×D47)⋊C4—21no—
104C2×(C2×D47)⋊C4—21no—
105C2×C4×Dic47D411yescharacteristic
106C2×C4×Dic47D411yescharacteristic
107C2×C4×Dic47D414yes—
108C2×C4×Dic47C2311yescharacteristic
109C2×C4×Dic47C2311yescharacteristic
110C2×C4×Dic47C2314yes—
111#13093956C2318yes—
112#13093955C2318yes—
113#13093955C2318yes—
114Dic47⋊C8C2318yes—
115#13093946C2318yes—
116#13093945C2318yes—
117#13093944C2318yes—
118C22⋊(C47⋊C8)C2314yes—
119C8×Dic47C2318yes—
120C22×D4×C47—21no—
121C22×D4×C47C2311yescharacteristic
122C23×C4×C47—21no—
123C23×C4×C47C2311yescharacteristic
124C22×C8×C47C2312yes—
125#121×C47—22no—
126#121×C47C2314yes—
127C2×C4⋊C4×C47—21no—
128C2×C4⋊C4×C47—22no—
129C2×C4⋊C4×C47C2314yes—
130C2×C4⋊C4×C47C2312yes—
131C2×C4⋊C4×C47C2311yescharacteristic
132C2×C22⋊C4×C47—21no—
133#106×C47C2318yes—
134#102×C47C2318yes—
135C22⋊C8×C47C2314yes—
136C24×C47—21no—
137C24×C47C2×D411yescharacteristic
138C23×D47—21no—
139C2×D4×C47—24no—
140C2×D4×C47—21no—
141C2×D4×C47—24no—
142C2×D4×C47—44no—
143C2×D4×C47C2412yes—
144C22×C4×C47—24no—
145C22×C4×C47—21no—
146C22×C4×C47—24no—
147C22×C4×C47—22no—
148C22×C4×C47—44no—
149C22×C4×C47C2×D411yescharacteristic
150C22×C4×C47C2×D412yes—
151C22×C4×C47C2412yes—
152C22×C4×C47C2411yescharacteristic
153C2×C22⋊D47—24no—
154C2×C22⋊D47—24no—
155C2×C22⋊D47—24no—
156C2×C22⋊D47—44no—
157C2×C22⋊D47—44no—
158C2×C22⋊D47—44no—
159C22×Dic47—24no—
160C22×Dic47—21no—
161C22×Dic47—24no—
162C22×Dic47—22no—
163C22×Dic47—24no—
164C22×Dic47—44no—
165C22×Dic47—44no—
166C22×Dic47SD1618yes—
167C22×Dic47C2×D411yescharacteristic
168C22×Dic47C2×D414yes—
169C22×Dic47C2×D411yescharacteristic
170C22×Dic47C2×D411yescharacteristic
171C22×Dic47C2×D414yes—
172C22×Dic47C2×D411yescharacteristic
173#10625352—432no—
174C2×D188—24no—
175C2×D188—21no—
176C2×D188C2412yes—
177C2×C4×D47—24no—
178C2×C4×D47—44no—
179C2×Dic94—24no—
180C2×Dic94—28no—
181C2×Dic94—28no—
182C2×Dic94—48no—
183C2×Dic94—432no—
184C2×Dic94—44no—
185C2×Dic94—48no—
186C2×C8×C47—24no—
187C2×C8×C47C2414yes—
188C4⋊C4×C47—24no—
189C4⋊C4×C47—22no—
190C4⋊C4×C47—21no—
191C4⋊C4×C47C2412yes—
192C4⋊C4×C47C2414yes—
193C22⋊C4×C47—24no—
194C22⋊Dic47—24no—
195(C2×D47)⋊C4—24no—
196(C2×D47)⋊C4—24no—
197C4⋊Dic47—24no—
198C4⋊Dic47—22no—
199C4⋊Dic47—21no—
200C4⋊Dic47C2414yes—
201C4⋊Dic47C2412yes—
202Dic47⋊C4—28no—
203Dic47⋊C4—28no—
204Dic47⋊C4—24no—
205C4×Dic47C2×D414yes—
206C4×Dic47C2×D412yes—
207C4×Dic47C2×D414yes—
208C4×Dic47C2×D412yes—
209C4×Dic47C2412yes—
210C4×Dic47C2412yes—
211C2×C47⋊C8—24no—
212C2×C47⋊C8C2414yes—
213C23×C47—22no—
214C23×C47—22no—
215C23×C47—22no—
216C23×C47—42no—
217C23×C47—42no—
218C23×C47—42no—
219C23×C47C2×SD1614yes—
220C23×C47C22×D412yes—
221C23×C47C22×D411yescharacteristic
222C22×D47—22no—
223C22×D47—42no—
224C22×D47—42no—
225C22×D47—42no—
226D4×C47—24no—
227D4×C47—48no—
228D4×C47—42no—
229C2×C4×C47—22no—
230C2×C4×C47—24no—
231C2×C4×C47—24no—
232C2×C4×C47—24no—
233C2×C4×C47—22no—
234C2×C4×C47—48no—
235C2×C4×C47—42no—
236C2×C4×C47—44no—
237C2×C4×C47C22×D412yes—
238C2×C4×C47C22×D411yescharacteristic
239C2×C4×C472−1+414yes—
240C2×C4×C47C2511yescharacteristic
241C22⋊D47—48no—
242C22⋊D47—48no—
243C22⋊D47—48no—
244C2×Dic47—22no—
245C2×Dic47—28no—
246C2×Dic47—24no—
247C2×Dic47—22no—
248C2×Dic47—48no—
249C2×Dic47—42no—
250C2×Dic47—44no—
251C2×Dic47—48no—
252C2×Dic47C2×SD1614yes—
253C2×Dic47C2×SD1614yes—
254C2×Dic47C2×SD16116yes—
255C2×Dic47AGL(1,ℤ/8)14yes—
256C2×Dic47C22×D412yes—
257C2×Dic47C22×D412yes—
258C2×Dic47C22×D412yes—
259C2×Dic47C22×D412yes—
260D188—24no—
261D188—42no—
262C4×D47—48no—
263Dic94—416no—
264Dic94—48no—
265Dic94—416no—
266C8×C47—24no—
267C47⋊C8—24no—
268C2×#3227—471nomaximal, Sylow
269C22×C47—22no—
270C22×C47—28no—
271C22×C47—22no—
272C22×C47—42no—
273C22×C47—42no—
274C22×C47—42no—
275C22×C47—84no—
276C22×C47#53612yes—
277C22×C47C22×SD1614yes—
278C22×C47C22×SD1612yes—
279C22×C47C2×AGL(1,ℤ/8)12yes—
280C22×C47C23×D411yescharacteristic
281C2×D47—42no—
282C2×D47—42no—
283C2×D47—42no—
284C2×D47—84no—
285C4×C47—22no—
286C4×C47—42no—
287C4×C47—44no—
288C4×C47#53612yes—
289C4×C47C2×2−1+412yes—
290Dic47—22no—
291Dic47—42no—
292Dic47—44no—
293Dic47C22×SD1614yes—
294Dic47C22×SD1614yes—
295Dic47C2×AGL(1,ℤ/8)14yes—
296C2×#532—471no—
297C22×C4⋊Q8—471no—
298#3227—4716no—
299C2×#498—472no—
300C2×#486—472no—
301C2×#481—471no—
302C2×#481—471no—
303C2×#481—471no—
304C2×#481D4711yescharacteristic, complemented
305C2×#476—472no—
306C2×#476—472no—
307C2×#431—471no—
308C2×C47—24no—
309C2×C47—82no—
310C2×C47#322714yescomplemented
311C2×C47C2×#53611yescharacteristic
312C2×C47C23×SD1611yescharacteristic
313C2×C47C22×AGL(1,ℤ/8)11yescharacteristic
314D47—82no—
315C22×#41—941no—
316C23×Q8—941no—
317#532—478no—
318#532—944no—
319C2×C4⋊Q8—472no—
320C2×C4⋊Q8—471no—
321C2×C4⋊Q8—478no—
322C2×C4⋊Q8—472no—
323C2×C4⋊Q8—941no—
324C2×C4⋊Q8—941no—
325C2×C4⋊Q8—944no—
326C2×#127—471no—
327C2×#124—941no—
328C2×#121—471no—
329C2×#121—941no—
330C2×#121C2×D4711yescharacteristic
331C2×C4×D4—471no—
332C2×C4×D4—471no—
333C2×C4×D4—941no—
334C22×C4⋊C4—471no—
335C22×C4⋊C4—941no—
336C22×C4⋊C4C2×D4711yescharacteristic
337C22×C42—471no—
338#498—4716no—
339#486—4716no—
340#481—478no—
341#481—478no—
342#481—478no—
343#481C2×D4718yescomplemented
344#476—4716no—
345#476—4716no—
346#431—478no—
347C2×#106—472no—
348C2×#106—472no—
349C2×#106—472no—
350C2×#106C2×D4712yes—
351C2×C4⋊C8—472no—
352C2×#102—472no—
353C2×#102—472no—
354C2×#102—472no—
355C2×#102C2×D4712yes—
356C2×C22⋊C8—471no—
357C2×C22⋊C8C2×D4711yescharacteristic
358C2×C4×C8—472no—
359C47C2×#322711yescharacteristic, complemented, Sylow
360C2×#41—948no—
361C2×#41—1888no—
362C22×Q8—941no—
363C22×Q8—948no—
364C22×Q8—942no—
365C22×Q8—942no—
366C22×Q8—1888no—
367C22×D4—21no—
368C22×D4—471no—
369C22×D4—941no—
370C22×D4—941no—
371C22×D4—941no—
372C22×D4C22×D4711yescharacteristic
373C23×C4—21no—
374C23×C4—471no—
375C23×C4—471no—
376C23×C4—941no—
377C23×C4—941no—
378C23×C4—941no—
379C23×C4C22×D4711yescharacteristic
380C22×C8—472no—
381C22×C8C22×D4712yes—
382C4⋊Q8—478no—
383C4⋊Q8—478no—
384C4⋊Q8—474no—
385C4⋊Q8—948no—
386C4⋊Q8—944no—
387C4⋊Q8—948no—
388C4⋊Q8—944no—
389C4⋊Q8—942no—
390#127—474no—
391#127—942no—
392#124—948no—
393#121—22no—
394#121—474no—
395#121—948no—
396#121—942no—
397#121C22×D4714yes—
398C4×D4—474no—
399C4×D4—474no—
400C4×D4—948no—
401C4×D4—942no—
402C4×D4—942no—
403C2×C4⋊C4—22no—
404C2×C4⋊C4—21no—
405C2×C4⋊C4—472no—
406C2×C4⋊C4—474no—
407C2×C4⋊C4—471no—
408C2×C4⋊C4—948no—
409C2×C4⋊C4—942no—
410C2×C4⋊C4—942no—
411C2×C4⋊C4—941no—
412C2×C4⋊C4—942no—
413C2×C4⋊C4—941no—
414C2×C4⋊C4C22×D4712yes—
415C2×C4⋊C4C22×D4714yes—
416C2×C4⋊C4C22×D4711yescharacteristic
417C2×C22⋊C4—21no—
418C2×C22⋊C4—941no—
419C2×C22⋊C4—941no—
420C2×C22⋊C4—941no—
421C2×C42—471no—
422C2×C42—471no—
423C2×C42—474no—
424C2×C42—471no—
425C2×C42—471no—
426C2×C42—474no—
427#106—478no—
428#106—478no—
429#106—478no—
430#106C22×D4718yes—
431C4⋊C8—478no—
432#102—478no—
433#102—478no—
434#102—478no—
435#102C22×D4718yes—
436C22⋊C8—474no—
437C22⋊C8C22×D4714yes—
438C4×C8—478no—
439C24—21no—
440C24—941no—
441C24D4×D4711yescharacteristic
442#41—18832no—
443C2×Q8—944no—
444C2×Q8—948no—
445C2×Q8—948no—
446C2×Q8—1888no—
447C2×Q8—1884no—
448C2×Q8—18832no—
449C2×Q8—1888no—
450C2×D4—24no—
451C2×D4—21no—
452C2×D4—24no—
453C2×D4—44no—
454C2×D4—472no—
455C2×D4—944no—
456C2×D4—944no—
457C2×D4—944no—
458C2×D4—941no—
459C2×D4—944no—
460C2×D4—1884no—
461C2×D4—1884no—
462C2×D4—1884no—
463C2×D4C23×D4712yes—
464C22×C4—24no—
465C22×C4—24no—
466C22×C4—21no—
467C22×C4—22no—
468C22×C4—44no—
469C22×C4—478no—
470C22×C4—474no—
471C22×C4—471no—
472C22×C4—471no—
473C22×C4—471no—
474C22×C4—474no—
475C22×C4—471no—
476C22×C4—944no—
477C22×C4—944no—
478C22×C4—944no—
479C22×C4—944no—
480C22×C4—941no—
481C22×C4—942no—
482C22×C4—1884no—
483C22×C4—1884no—
484C22×C4—1884no—
485C22×C4D4×D4712yes—
486C22×C4D4×D4711yescharacteristic
487C22×C4C23×D4712yes—
488C22×C4C23×D4711yescharacteristic
489C2×C8—24no—
490C2×C8—474no—
491C2×C8—944no—
492C2×C8C23×D4714yes—
493C4⋊C4—24no—
494C4⋊C4—22no—
495C4⋊C4—21no—
496C4⋊C4—474no—
497C4⋊C4—472no—
498C4⋊C4—948no—
499C4⋊C4—948no—
500C4⋊C4—944no—
501C4⋊C4—944no—
502C4⋊C4—942no—
503C4⋊C4—941no—
504C4⋊C4C23×D4714yes—
505C4⋊C4C23×D4712yes—
506C22⋊C4—24no—
507C22⋊C4—944no—
508C22⋊C4—944no—
509C22⋊C4—944no—
510C42—474no—
511C42—472no—
512C42—474no—
513C42—472no—
514C42—472no—
515C42—472no—
516C23—22no—
517C23—22no—
518C23—22no—
519C23—42no—
520C23—42no—
521C23—42no—
522C23—942no—
523C23—1882no—
524C23—1882no—
525C23—1882no—
526C23SD16×D4714yes—
527C23C2×D4×D4712yes—
528C23C2×D4×D4711yescharacteristic
529Q8—18816no—
530Q8—1888no—
531Q8—18816no—
532D4—24no—
533D4—48no—
534D4—42no—
535D4—944no—
536D4—1888no—
537D4—1888no—
538D4—1888no—
539D4—1882no—
540C2×C4—22no—
541C2×C4—24no—
542C2×C4—24no—
543C2×C4—22no—
544C2×C4—24no—
545C2×C4—48no—
546C2×C4—42no—
547C2×C4—44no—
548C2×C4—474no—
549C2×C4—474no—
550C2×C4—4716no—
551C2×C4—474no—
552C2×C4—472no—
553C2×C4—472no—
554C2×C4—472no—
555C2×C4—472no—
556C2×C4—948no—
557C2×C4—942no—
558C2×C4—942no—
559C2×C4—944no—
560C2×C4—1888no—
561C2×C4—1888no—
562C2×C4—1888no—
563C2×C4—1882no—
564C2×C4—1884no—
565C2×C4C2×D4×D4712yes—
566C2×C4C2×D4×D4711yescharacteristic
567C2×C4#1309407114yes—
568C2×C4C24×D4711yescharacteristic
569C8—24no—
570C8—944no—
571C22—28no—
572C22—22no—
573C22—22no—
574C22—42no—
575C22—42no—
576C22—42no—
577C22—84no—
578C22—3764no—
579C22—1882no—
580C22—1882no—
581C22—1882no—
582C22#1568969512yes—
583C22C2×SD16×D4714yes—
584C22C2×SD16×D4712yes—
585C22AGL(1,ℤ/8)×D4712yes—
586C22C22×D4×D4711yescharacteristic
587C4—22no—
588C4—42no—
589C4—44no—
590C4—474no—
591C4—474no—
592C4—474no—
593C4—942no—
594C4—1882no—
595C4—1884no—
596C4#1568969512yes—
597C4C2×#1309407112yes—
598C2—24no—
599C2—82no—
600C2—3762no—
601C2#2025451414yesdirect product factor
602C2C2×#1568969511yescharacteristic
603C2C22×SD16×D4711yescharacteristic
604C2C2×AGL(1,ℤ/8)×D4711yescharacteristic
subgroupG/H
centerC23C2×D4×D47
commutator, cosocleC2×C4×C47C25
FrattiniC2×C4C24×D47
FittingC2×C47×#481C2
socleC23×C47C22×D4
prime radicalC2×#15689668C2
hypercenterC2×#481D47
hypocenterC47C2×#3227
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#481

Complements

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

NH ≅ G/Ntype
2. C22×#13094023C2semidirect
3. #20254514C2direct (N × H)
4. C2×#15689520C2semidirect
5. C2×#15689468C2semidirect
6. C2×#15689442C2semidirect
7. C2×#15689441C2semidirect
8. C2×#15689439C2semidirect
9. C2×#15689412C2semidirect
10. C2×#15689411C2semidirect
11. C2×#15689173C2semidirect
12. C2×C47×#481C2semidirect
23. C2×#13094023C22semidirect
34. #15689520C22semidirect
35. #15689468C22semidirect
36. #15689442C22semidirect
37. #15689441C22semidirect
38. #15689439C22semidirect
39. #15689412C22semidirect
40. #15689411C22semidirect
41. #15689173C22semidirect
53. C47×#481C22semidirect
304. C2×#481D47semidirect
310. C2×C47#3227semidirect
343. #481C2×D47semidirect
359. C47C2×#3227semidirect
601. C2#20254514direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#156896682
C22×#130940232
#202545142
C2×#156895202
C2×#156894682
C2×#156894422
C2×#156894412
C2×#156894392
C2×#156894122
C2×#156894112
C2×#156891732
C2×C47×#4812
C2×#322747