Finite Group Databasebeta

← C2×#20253484 #27084444

Subgroups — C2×#20253484 #27084444

Order 12032.

2550 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×#15689675C211yesmaximal, characteristic, complemented
2C22×#13093986C211yesmaximal, characteristic
3#20253484C2116yesmaximal, direct product factor
4C2×#15689458C211yesmaximal, characteristic, complemented
5C2×#15689402C212yesmaximal, complemented
6C2×#15689327C212yesmaximal, complemented
7C2×#15689276C211yesmaximal, characteristic
8C2×#15689273C211yesmaximal, characteristic
9C2×#15689226C212yesmaximal, complemented
10C2×#15689222C212yesmaximal, complemented
11C2×#15689177C211yesmaximal, characteristic, complemented
12C2×C47×#484C211yesmaximal, characteristic, complemented
13C23×D188C2211yescharacteristic
14C23×C4×D47—21no—
15C22×#10625325C2212yes—
16C22×#10625336C2212yes—
17#15689675—24no—
18#15689675C2218yescomplemented
19C2×#13094023C2211yescharacteristic, complemented
20C2×#13094022C2211yescharacteristic, complemented
21C2×#13094014—21no—
22C2×#13094003C2211yescharacteristic
23C2×#13094001—21no—
24C2×Q8×Dic47C2211yescharacteristic, complemented
25C2×#13093989—21no—
26C2×#13093989C2211yescharacteristic, complemented
27C2×#13093986—21no—
28C2×#13093986—21no—
29C2×#13093986—24no—
30C2×#13093986C2212yes—
31C2×#13093986C2218yes—
32C2×#13093986C2212yes—
33C2×#13093980—21no—
34C22×(C2×D47)⋊C4—21no—
35C22×C4×Dic47C2211yescharacteristic
36#15689458C2218yescomplemented
37#15689402C22116yescomplemented
38#15689327C22116yescomplemented
39#15689276C2218yes—
40#15689273C2218yes—
41#15689226C22116yescomplemented
42#15689222C22116yescomplemented
43#15689177C2218yescomplemented
44C2×#13093957C2212yescomplemented
45C2×Dic47⋊C8C2212yescomplemented
46C2×#13093947C2212yescomplemented
47C2×#13093946C2212yes—
48C2×#13093944C2212yes—
49C2×C22⋊(C47⋊C8)C2211yescharacteristic
50C2×#13093934C2212yescomplemented
51C2×C22⋊Q8×C47C2211yescharacteristic
52C22×C4⋊C4×C47C2211yescharacteristic
53C47×#484C2218yescomplemented
54C2×#107×C47C2212yescomplemented
55C2×#103×C47C2212yescomplemented
56C2×C22⋊C8×C47C2211yescharacteristic
57C24×D47—21no—
58C22×Dic94—21no—
59C22×Dic94C2311yescharacteristic
60C22×D188—21no—
61C22×D188—22no—
62C22×D188—28no—
63C22×D188C2314yes—
64C22×D188C2312yes—
65C23×Dic47D411yescharacteristic
66C23×Dic47D411yescharacteristic
67C22×C4×D47—22no—
68C22×C4×D47—28no—
69C22×C4×D47—22no—
70C22×C4×D47—48no—
71C2×#10625325—28no—
72C2×#10625325C2318yes—
73C2×#10625336—28no—
74C2×#10625336C2318yes—
75C22×C47⋊C8C2312yes—
76#13094023—22no—
77#13094023C2314yescomplemented
78#13094022—22no—
79#13094022C2314yescomplemented
80#13094014—28no—
81#13094003—22no—
82#13094003C2314yes—
83#13094001—28no—
84Q8×Dic47—22no—
85Q8×Dic47C2314yescomplemented
86#13093989—28no—
87#13093989—22no—
88#13093989C2314yescomplemented
89#13093986—28no—
90#13093986—24no—
91#13093986—28no—
92#13093986—24no—
93#13093986C2318yes—
94#13093986C2318yes—
95#13093980—28no—
96C2×C4⋊Dic47—21no—
97C2×C4⋊Dic47C2311yescharacteristic
98C2×Dic47⋊C4—21no—
99C2×Dic47⋊C4—21no—
100C2×Dic47⋊C4—21no—
101C2×Dic47⋊C4—21no—
102C2×Dic47⋊C4—21no—
103C2×C22⋊Dic47—21no—
104C2×(C2×D47)⋊C4—22no—
105C2×(C2×D47)⋊C4—22no—
106C2×(C2×D47)⋊C4—28no—
107C2×C4×Dic47—21no—
108C2×C4×Dic47—21no—
109C2×C4×Dic47D411yescharacteristic
110C2×C4×Dic47D411yescharacteristic
111C2×C4×Dic47D414yes—
112C2×C4×Dic47C2311yescharacteristic
113C2×C4×Dic47C2311yescharacteristic
114C2×C4×Dic47C2314yes—
115#13093957C2318yescomplemented
116Dic47⋊C8C2318yescomplemented
117#13093947C2318yescomplemented
118#13093946C2318yes—
119#13093944C2318yes—
120C22⋊(C47⋊C8)C2314yes—
121#13093934C2318yescomplemented
122C22×Q8×C47C2311yescharacteristic
123C23×C4×C47—21no—
124C23×C4×C47C2311yescharacteristic
125C22×C8×C47C2312yes—
126C22⋊Q8×C47—22no—
127C22⋊Q8×C47C2314yes—
128C2×C4⋊C4×C47—21no—
129C2×C4⋊C4×C47—22no—
130C2×C4⋊C4×C47—21no—
131C2×C4⋊C4×C47C2314yes—
132C2×C4⋊C4×C47C2312yes—
133C2×C4⋊C4×C47C2311yescharacteristic
134C2×C22⋊C4×C47—21no—
135#107×C47C2318yescomplemented
136#103×C47C2318yescomplemented
137C22⋊C8×C47C2314yes—
138C24×C47C2×D411yescharacteristic
139C23×D47—28no—
140C23×D47—24no—
141C23×D47—22no—
142C23×D47—44no—
143C23×D47—44no—
144C2×Q8×C47—21no—
145C2×Q8×C47—24no—
146C2×Q8×C47C2412yes—
147C22×C4×C47—24no—
148C22×C4×C47—21no—
149C22×C4×C47—21no—
150C22×C4×C47—24no—
151C22×C4×C47—22no—
152C22×C4×C47—44no—
153C22×C4×C47C2×D411yescharacteristic
154C22×C4×C47C2×D412yes—
155C22×C4×C47C2412yes—
156C22×C4×C47C2411yescharacteristic
157C22×Dic47—24no—
158C22×Dic47—24no—
159C22×Dic47—21no—
160C22×Dic47—21no—
161C22×Dic47C2×D411yescharacteristic
162C22×Dic47C2×D411yescharacteristic
163C22×Dic47C2×D414yes—
164C22×Dic47C2×D411yescharacteristic
165C22×Dic47C2×D414yes—
166C22×Dic47C2×D411yescharacteristic
167C2×D188—28no—
168C2×D188—216no—
169C2×D188—24no—
170C2×D188—22no—
171C2×D188—24no—
172C2×D188—216no—
173C2×D188#4118yes—
174C2×D188C2414yes—
175C2×C4×D47—28no—
176C2×C4×D47—28no—
177C2×C4×D47—48no—
178C2×C4×D47—432no—
179C2×C4×D47—48no—
180C2×Dic94—24no—
181C2×Dic94—21no—
182C2×Dic94—24no—
183C2×Dic94—44no—
184C2×Dic94C2412yes—
185C2×C8×C47—24no—
186C2×C8×C47C2414yes—
187C4⋊C4×C47—24no—
188C4⋊C4×C47—22no—
189C4⋊C4×C47—24no—
190C4⋊C4×C47—21no—
191C4⋊C4×C47C2414yes—
192C4⋊C4×C47C2412yes—
193C22⋊C4×C47—24no—
194C22⋊Dic47—24no—
195#10625336—216no—
196(C2×D47)⋊C4—28no—
197(C2×D47)⋊C4—28no—
198C4⋊Dic47—21no—
199C4⋊Dic47—24no—
200C4⋊Dic47C2412yes—
201Dic47⋊C4—24no—
202Dic47⋊C4—24no—
203Dic47⋊C4—24no—
204Dic47⋊C4—24no—
205Dic47⋊C4—24no—
206C4×Dic47—24no—
207C4×Dic47—24no—
208C4×Dic47C2×D414yes—
209C4×Dic47C2×D412yes—
210C4×Dic47C2×D414yes—
211C4×Dic47C2×D412yes—
212C4×Dic47C2412yes—
213C4×Dic47C2412yes—
214C2×C47⋊C8—24no—
215C2×C47⋊C8C2414yes—
216#10625325—216no—
217C23×C47—22no—
218C23×C47—22no—
219C23×C47#13714yes—
220C23×C47C22×D412yes—
221C23×C47C22×D411yescharacteristic
222C22×D47—216no—
223C22×D47—28no—
224C22×D47—24no—
225C22×D47—48no—
226C22×D47—432no—
227C22×D47—44no—
228C22×D47—48no—
229C22×D47—44no—
230Q8×C47—24no—
231Q8×C47—42no—
232C2×C4×C47—22no—
233C2×C4×C47—24no—
234C2×C4×C47—24no—
235C2×C4×C47—22no—
236C2×C4×C47—22no—
237C2×C4×C47—24no—
238C2×C4×C47—48no—
239C2×C4×C47—42no—
240C2×C4×C47—42no—
241C2×C4×C47—44no—
242C2×C4×C47C22×D412yes—
243C2×C4×C47C22×D411yescharacteristic
244C2×C4×C47C2×#4114yes—
245C2×C4×C47C2511yescharacteristic
246C2×Dic47—22no—
247C2×Dic47—28no—
248C2×Dic47—22no—
249C2×Dic47—22no—
250C2×Dic47—28no—
251C2×Dic47—22no—
252C2×Dic47—22no—
253C2×Dic47—42no—
254C2×Dic47—42no—
255C2×Dic47AGL(1,ℤ/8)14yes—
256C2×Dic47C22×D412yes—
257C2×Dic47C22×D412yes—
258C2×Dic47C22×D412yes—
259C2×Dic47C22×D412yes—
260D188—216no—
261D188—28no—
262D188—24no—
263D188C2×#4118yes—
264C4×D47—416no—
265C4×D47—416no—
266Dic94—24no—
267Dic94—42no—
268Dic94—48no—
269C8×C47—24no—
270C47⋊C8—24no—
271C2×#3110—471nomaximal, Sylow
272C22×C47—28no—
273C22×C47—22no—
274C22×C47—22no—
275C22×C47#52512yes—
276C22×C47C2×AGL(1,ℤ/8)12yes—
277C22×C47C2×#13714yes—
278C22×C47C2×#13712yes—
279C22×C47C23×D411yescharacteristic
280C2×D47—28no—
281C2×D47—432no—
282C2×D47—416no—
283C2×D47—44no—
284C2×D47—44no—
285C4×C47—22no—
286C4×C47—42no—
287C4×C47—42no—
288C4×C47—44no—
289C4×C47#52512yes—
290C4×C47C22×#4112yes—
291Dic47—22no—
292Dic47—22no—
293Dic47—22no—
294Dic47—42no—
295Dic47—42no—
296Dic47C2×AGL(1,ℤ/8)14yes—
297C2×#533—471no—
298C22×C4×D4—471no—
299#3110—4716no—
300C2×#484—471no—
301C2×#484D4711yescharacteristic, complemented
302C2×#474—472no—
303C2×#461—472no—
304C2×#450—471no—
305C2×#450—471no—
306C2×#440—472no—
307C2×#440—472no—
308C2×#432—471no—
309C2×C47—24no—
310C2×C47#311014yescomplemented
311C2×C47C2×#52511yescharacteristic
312C2×C47C22×AGL(1,ℤ/8)11yescharacteristic
313C2×C47C22×#13711yescharacteristic
314D47—48no—
315C23×D4—471no—
316C24×C4—941no—
317C22×SD16—472no—
318C22×SD16—472no—
319#533—478no—
320#533—944no—
321C2×C4⋊Q8—471no—
322C2×C4⋊Q8—471no—
323C2×#125—941no—
324C2×C22⋊Q8—471no—
325C2×C22⋊Q8—941no—
326C2×C22⋊Q8C2×D4711yescharacteristic
327C2×C4×Q8—471no—
328C2×C4×Q8—471no—
329C2×C4×Q8—941no—
330C2×C4×D4—478no—
331C2×C4×D4—472no—
332C2×C4×D4—472no—
333C2×C4×D4—941no—
334C2×C4×D4—941no—
335C2×C4×D4—944no—
336C2×#117—941no—
337C22×C4⋊C4C2×D4711yescharacteristic
338C22×C22⋊C4—941no—
339C22×C42—471no—
340#484—478no—
341#484C2×D4718yescomplemented
342#474—4716no—
343#461—4716no—
344#450—478no—
345#450—478no—
346#440—4716no—
347#440—4716no—
348#432—478no—
349C2×#107—472no—
350C2×#107C2×D4712yescomplemented
351C2×C4⋊C8—472no—
352C2×#103—472no—
353C2×#103C2×D4712yescomplemented
354C2×#102—472no—
355C2×#102—472no—
356C2×C22⋊C8—471no—
357C2×C22⋊C8C2×D4711yescharacteristic
358C2×#97—472no—
359C47C2×#311011yescharacteristic, complemented, Sylow
360C25—941no—
361C22×Q8—471no—
362C22×Q8—941no—
363C22×Q8C22×D4711yescharacteristic
364C22×D4—472no—
365C22×D4—474no—
366C22×D4—948no—
367C22×D4—942no—
368C22×D4—941no—
369C23×C4—21no—
370C23×C4—471no—
371C23×C4—471no—
372C23×C4—948no—
373C23×C4—942no—
374C23×C4—942no—
375C23×C4—1888no—
376C23×C4C22×D4711yescharacteristic
377C2×SD16—478no—
378C2×SD16—478no—
379C2×SD16—948no—
380C2×SD16—948no—
381C22×C8—472no—
382C22×C8C22×D4712yes—
383C4⋊Q8—474no—
384C4⋊Q8—474no—
385C4⋊Q8—942no—
386C4⋊Q8—942no—
387#125—948no—
388C22⋊Q8—22no—
389C22⋊Q8—474no—
390C22⋊Q8—948no—
391C22⋊Q8—942no—
392C22⋊Q8C22×D4714yes—
393C4×Q8—474no—
394C4×Q8—474no—
395C4×Q8—942no—
396C4×Q8—948no—
397C4×Q8—942no—
398C4×D4—478no—
399C4×D4—478no—
400C4×D4—948no—
401C4×D4—944no—
402C4×D4—948no—
403C4×D4—944no—
404#117—948no—
405C2×C4⋊C4—22no—
406C2×C4⋊C4—21no—
407C2×C4⋊C4—21no—
408C2×C4⋊C4—471no—
409C2×C4⋊C4—941no—
410C2×C4⋊C4—941no—
411C2×C4⋊C4—941no—
412C2×C4⋊C4—941no—
413C2×C4⋊C4—941no—
414C2×C4⋊C4—941no—
415C2×C4⋊C4C22×D4712yes—
416C2×C4⋊C4C22×D4714yes—
417C2×C4⋊C4C22×D4711yescharacteristic
418C2×C22⋊C4—21no—
419C2×C22⋊C4—948no—
420C2×C22⋊C4—942no—
421C2×C22⋊C4—942no—
422C2×C22⋊C4—941no—
423C2×C42—471no—
424C2×C42—471no—
425C2×C42—474no—
426C2×C42—471no—
427C2×C42—471no—
428C2×C42—474no—
429C2×C42—941no—
430C2×C42—941no—
431#107—478no—
432#107C22×D4718yescomplemented
433C4⋊C8—478no—
434#103—478no—
435#103C22×D4718yescomplemented
436#102—478no—
437#102—478no—
438C22⋊C8—474no—
439C22⋊C8C22×D4714yes—
440#97—478no—
441C24—948no—
442C24—944no—
443C24—942no—
444C24—1884no—
445C24—1884no—
446C24D4×D4711yescharacteristic
447C2×Q8—21no—
448C2×Q8—24no—
449C2×Q8—472no—
450C2×Q8—944no—
451C2×Q8—941no—
452C2×Q8—944no—
453C2×Q8—1884no—
454C2×Q8C23×D4712yes—
455C2×D4—478no—
456C2×D4—474no—
457C2×D4—9416no—
458C2×D4—948no—
459C2×D4—9416no—
460C2×D4—944no—
461C2×D4—942no—
462C2×D4—944no—
463C22×C4—24no—
464C22×C4—24no—
465C22×C4—21no—
466C22×C4—21no—
467C22×C4—22no—
468C22×C4—44no—
469C22×C4—471no—
470C22×C4—474no—
471C22×C4—474no—
472C22×C4—471no—
473C22×C4—471no—
474C22×C4—471no—
475C22×C4—948no—
476C22×C4—948no—
477C22×C4—944no—
478C22×C4—944no—
479C22×C4—941no—
480C22×C4—941no—
481C22×C4—1888no—
482C22×C4—18832no—
483C22×C4—1888no—
484C22×C4D4×D4712yes—
485C22×C4D4×D4711yescharacteristic
486C22×C4C23×D4712yes—
487C22×C4C23×D4711yescharacteristic
488SD16—9416no—
489SD16—9416no—
490C2×C8—24no—
491C2×C8—474no—
492C2×C8—944no—
493C2×C8C23×D4714yes—
494C4⋊C4—24no—
495C4⋊C4—22no—
496C4⋊C4—24no—
497C4⋊C4—21no—
498C4⋊C4—472no—
499C4⋊C4—944no—
500C4⋊C4—944no—
501C4⋊C4—944no—
502C4⋊C4—944no—
503C4⋊C4—944no—
504C4⋊C4—944no—
505C4⋊C4—941no—
506C4⋊C4C23×D4712yes—
507C4⋊C4C23×D4714yes—
508C22⋊C4—24no—
509C22⋊C4—948no—
510C22⋊C4—948no—
511C22⋊C4—944no—
512C42—474no—
513C42—472no—
514C42—474no—
515C42—472no—
516C42—472no—
517C42—472no—
518C42—944no—
519C42—944no—
520C23—22no—
521C23—22no—
522C23—9416no—
523C23—948no—
524C23—944no—
525C23—1888no—
526C23—18832no—
527C23—1884no—
528C23—1888no—
529C23—1884no—
530C23#1309405614yes—
531C23C2×D4×D4712yes—
532C23C2×D4×D4711yescharacteristic
533Q8—24no—
534Q8—42no—
535Q8—944no—
536Q8—1888no—
537Q8—1882no—
538D4—478no—
539D4—9416no—
540D4—948no—
541D4—944no—
542C2×C4—22no—
543C2×C4—24no—
544C2×C4—24no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—24no—
548C2×C4—48no—
549C2×C4—42no—
550C2×C4—44no—
551C2×C4—42no—
552C2×C4—474no—
553C2×C4—472no—
554C2×C4—472no—
555C2×C4—472no—
556C2×C4—472no—
557C2×C4—948no—
558C2×C4—942no—
559C2×C4—948no—
560C2×C4—942no—
561C2×C4—942no—
562C2×C4—942no—
563C2×C4—942no—
564C2×C4—18816no—
565C2×C4—18816no—
566C2×C4—1882no—
567C2×C4—1882no—
568C2×C4C2×D4×D4712yes—
569C2×C4C2×D4×D4711yescharacteristic
570C2×C4#41×D4714yes—
571C2×C4C24×D4711yescharacteristic
572C8—24no—
573C8—944no—
574C22—28no—
575C22—22no—
576C22—22no—
577C22—948no—
578C22—18832no—
579C22—18816no—
580C22—1884no—
581C22—1884no—
582C22#1568963112yes—
583C22AGL(1,ℤ/8)×D4712yes—
584C22C2×#1309405614yes—
585C22C2×#1309405612yes—
586C22C22×D4×D4711yescharacteristic
587C4—22no—
588C4—44no—
589C4—42no—
590C4—42no—
591C4—474no—
592C4—942no—
593C4—942no—
594C4—942no—
595C4—1882no—
596C4—1882no—
597C4#1568963112yes—
598C4C2×#41×D4712yes—
599C2—24no—
600C2—1888no—
601C2#2025348414yesdirect product factor
602C2C2×#1568963111yescharacteristic
603C2C2×AGL(1,ℤ/8)×D4711yescharacteristic
604C2C22×#1309405611yescharacteristic
subgroupG/H
centerC23C2×D4×D47
commutator, cosocleC2×C4×C47C25
FrattiniC2×C4C24×D47
FittingC2×C47×#484C2
socleC23×C47C22×D4
prime radicalC23×D188C22
hypercenterC2×#484D47
hypocenterC47C2×#3110
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#484

Complements

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

NH ≅ G/Ntype
1. C2×#15689675C2semidirect
3. #20253484C2direct (N × H)
4. C2×#15689458C2semidirect
5. C2×#15689402C2semidirect
6. C2×#15689327C2semidirect
9. C2×#15689226C2semidirect
10. C2×#15689222C2semidirect
11. C2×#15689177C2semidirect
12. C2×C47×#484C2semidirect
18. #15689675C22semidirect
19. C2×#13094023C22semidirect
20. C2×#13094022C22semidirect
24. C2×Q8×Dic47C22semidirect
26. C2×#13093989C22semidirect
36. #15689458C22semidirect
37. #15689402C22semidirect
38. #15689327C22semidirect
41. #15689226C22semidirect
42. #15689222C22semidirect
43. #15689177C22semidirect
44. C2×#13093957C22semidirect
45. C2×Dic47⋊C8C22semidirect
46. C2×#13093947C22semidirect
50. C2×#13093934C22semidirect
53. C47×#484C22semidirect
54. C2×#107×C47C22semidirect
55. C2×#103×C47C22semidirect
77. #13094023C23semidirect
79. #13094022C23semidirect
85. Q8×Dic47C23semidirect
88. #13093989C23semidirect
115. #13093957C23semidirect
116. Dic47⋊C8C23semidirect
117. #13093947C23semidirect
121. #13093934C23semidirect
135. #107×C47C23semidirect
136. #103×C47C23semidirect
301. C2×#484D47semidirect
310. C2×C47#3110semidirect
341. #484C2×D47semidirect
350. C2×#107C2×D47semidirect
353. C2×#103C2×D47semidirect
359. C47C2×#3110semidirect
432. #107C22×D47semidirect
435. #103C22×D47semidirect
601. C2#20253484direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#156896752
C22×#130939862
#202534842
C2×#156894582
C2×#156894022
C2×#156893272
C2×#156892762
C2×#156892732
C2×#156892262
C2×#156892222
C2×#156891772
C2×C47×#4842
C2×#311047