Finite Group Databasebeta

← C2×#20254701 #27089057

Subgroups — C2×#20254701 #27089057

Order 12032.

2178 conjugacy classes of subgroups.

The subgroup lattice has 628 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689678C211yesmaximal, characteristic, complemented
2C2×#15689670C211yesmaximal, characteristic
3#20254701C2116yesmaximal, direct product factor
4C2×#15689406C212yesmaximal, complemented
5C2×#15689414C211yesmaximal, characteristic, complemented
6C2×#15689413C211yesmaximal, characteristic, complemented
7C2×#15689355C212yesmaximal, complemented
8C2×#15689349C212yesmaximal, complemented
9C2×#15689321C212yesmaximal, complemented
10C2×#15689325C211yesmaximal, characteristic, complemented
11C2×C4⋊C8×D47C211yesmaximal, characteristic, complemented
12C2×C47×#461C211yesmaximal, characteristic, complemented
13C22×#10625350—21no—
14C22×#10625336C2212yes—
15C22×C8×D47C2212yes—
16#15689678—24no—
17#15689678C2218yescomplemented
18#15689670—24no—
19#15689670C2218yes—
20C2×#13094024C2211yescharacteristic
21C2×#13094023C2211yescharacteristic
22C2×#13094020C2211yescharacteristic
23C2×#13094015—21no—
24C2×#13094010—21no—
25C2×#13094004—21no—
26C2×#13094000C2212yes—
27C2×#13093999C2211yescharacteristic
28C2×#13093997—21no—
29C2×#13093995C2211yescharacteristic
30C2×Q8×Dic47C2212yes—
31C2×#13093989—21no—
32C2×C4×D188C2211yescharacteristic
33C2×C4×C22⋊D47—21no—
34C2×#13093982—21no—
35C2×C42×D47C2211yescharacteristic
36#15689414C2218yescomplemented
37#15689413C2218yescomplemented
38#15689406C22116yescomplemented
39#15689355C22116yescomplemented
40#15689349C22116yescomplemented
41#15689325C2218yescomplemented
42#15689321C22116yescomplemented
43C2×#13093956C2212yes—
44C2×#13093955C2212yes—
45C2×C4⋊(C47⋊C8)C2211yescharacteristic
46C2×Dic47⋊C8C2212yes—
47C4⋊C8×D47C2218yescomplemented
48C2×#13093948C2212yes—
49C2×#13093946C2212yes—
50C2×#13093945C2212yes—
51C22×SD16×C47C2212yes—
52C2×C4⋊Q8×C47C2211yescharacteristic
53C2×C4×D4×C47C2211yescharacteristic
54C47×#461C2218yescomplemented
55C2×C4⋊C8×C47C2211yescharacteristic
56C2×#103×C47C2212yes—
57C2×#10625350—28no—
58C2×#10625350—48no—
59C22×Dic94—21no—
60C22×Dic94—21no—
61C22×D188—21no—
62C22×D188C2311yescharacteristic
63C22×C22⋊D47—21no—
64C22×C22⋊D47—21no—
65C22×C22⋊D47—21no—
66C22×C4×D47—21no—
67C22×C4×D47D411yescharacteristic
68C22×C4×D47D411yescharacteristic
69C22×C4×D47C2311yescharacteristic
70C2×#10625336—44no—
71C2×#10625336C2318yes—
72C22×C47⋊C8C2312yes—
73C2×C8×D47—44no—
74C2×C8×D47C2318yes—
75#13094024—22no—
76#13094024C2314yes—
77#13094023—22no—
78#13094023C2314yes—
79#13094020—22no—
80#13094020C2314yes—
81#13094015—28no—
82#13094010—28no—
83#13094004—28no—
84#13094000—24no—
85#13094000C2318yes—
86#13093999—22no—
87#13093999C2314yes—
88#13093997—28no—
89#13093995—22no—
90#13093995C2314yes—
91Q8×Dic47—24no—
92Q8×Dic47C2318yes—
93#13093989—28no—
94C4×D188—22no—
95C4×D188C2314yes—
96C4×C22⋊D47—28no—
97#13093982—28no—
98C2×C4⋊Dic47—21no—
99C2×C4⋊Dic47—22no—
100C2×C4⋊Dic47C2311yescharacteristic
101C2×C4⋊Dic47C2312yes—
102C2×Dic47⋊C4—21no—
103C2×Dic47⋊C4—22no—
104C2×C22⋊Dic47—21no—
105C2×(C2×D47)⋊C4—22no—
106C2×(C2×D47)⋊C4—21no—
107C2×(C2×D47)⋊C4—21no—
108C2×C4×Dic47—22no—
109C2×C4×Dic47D411yescharacteristic
110C2×C4×Dic47D411yescharacteristic
111C2×C4×Dic47C2311yescharacteristic
112C42×D47D414yes—
113C42×D47C2314yes—
114#13093956C2318yes—
115#13093955C2318yes—
116C4⋊(C47⋊C8)C2314yes—
117Dic47⋊C8C2318yes—
118#13093948C2318yes—
119#13093946C2318yes—
120#13093945C2318yes—
121C22×Q8×C47C2312yes—
122C22×D4×C47C2311yescharacteristic
123C23×C4×C47—21no—
124C2×SD16×C47—44no—
125C2×SD16×C47C2318yes—
126C22×C8×C47C2312yes—
127C4⋊Q8×C47—22no—
128C4⋊Q8×C47C2314yes—
129C4×D4×C47—22no—
130C4×D4×C47C2314yes—
131C2×C4⋊C4×C47—21no—
132C2×C4⋊C4×C47—21no—
133C2×C4⋊C4×C47C2311yescharacteristic
134C2×C22⋊C4×C47—21no—
135C2×C42×C47C2311yescharacteristic
136C4⋊C8×C47C2314yes—
137#103×C47C2318yes—
138C24×C47—21no—
139C23×D47—21no—
140C23×D47C2×D411yescharacteristic
141C2×Q8×C47—22no—
142C2×Q8×C47—24no—
143C2×Q8×C47—24no—
144C2×Q8×C47C2414yes—
145C2×D4×C47—21no—
146C2×D4×C47—22no—
147C2×D4×C47—22no—
148C2×D4×C47C2412yes—
149C22×C4×C47—21no—
150C22×C4×C47—22no—
151C22×C4×C47—24no—
152C22×C4×C47—44no—
153C22×C4×C47C2×D411yescharacteristic
154C22×C4×C47C2×D411yescharacteristic
155C22×C4×C47C2411yescharacteristic
156C2×C22⋊D47—24no—
157C2×C22⋊D47—24no—
158C2×C22⋊D47—24no—
159C2×C22⋊D47—44no—
160C2×C22⋊D47—44no—
161C2×C22⋊D47—44no—
162C22×Dic47—22no—
163C22×Dic47—21no—
164C22×Dic47C2×D411yescharacteristic
165C22×Dic47C2×D411yescharacteristic
166C22×Dic47C2×D411yescharacteristic
167#10625350—432no—
168C2×D188—22no—
169C2×D188—22no—
170C2×D188—24no—
171C2×D188—21no—
172C2×D188—44no—
173C2×D188C2412yes—
174C2×C4×D47—22no—
175C2×C4×D47—22no—
176C2×C4×D47—24no—
177C2×C4×D47—42no—
178C2×C4×D47—44no—
179C2×C4×D47SD1618yescomplemented
180C2×C4×D47C2×D412yes—
181C2×C4×D47C2×D414yes—
182C2×C4×D47C2×D414yes—
183C2×C4×D47C2412yes—
184C2×Dic94—24no—
185C2×Dic94—24no—
186C2×Dic94—44no—
187C2×Dic94—44no—
188SD16×C47—48no—
189C2×C8×C47—24no—
190C2×C8×C47C2414yes—
191C4⋊C4×C47—24no—
192C4⋊C4×C47—24no—
193C4⋊C4×C47—21no—
194C4⋊C4×C47C2412yes—
195C22⋊C4×C47—24no—
196C42×C47C2×D412yes—
197C42×C47C2412yes—
198C22⋊Dic47—24no—
199#10625336—48no—
200(C2×D47)⋊C4—28no—
201(C2×D47)⋊C4—24no—
202(C2×D47)⋊C4—24no—
203C4⋊Dic47—24no—
204C4⋊Dic47—21no—
205C4⋊Dic47—22no—
206C4⋊Dic47—28no—
207C4⋊Dic47C2414yes—
208C4⋊Dic47C2412yes—
209Dic47⋊C4—24no—
210Dic47⋊C4—28no—
211C4×Dic47—28no—
212C4×Dic47C2×D412yes—
213C4×Dic47C2×D414yes—
214C4×Dic47C2×D414yes—
215C4×Dic47C2412yes—
216C2×C47⋊C8—24no—
217C2×C47⋊C8C2414yes—
218C8×D47—48no—
219C23×C47—22no—
220C23×C47—42no—
221C23×C47—42no—
222C23×C47—42no—
223C23×C47C22×D411yescharacteristic
224C22×D47—22no—
225C22×D47—22no—
226C22×D47—22no—
227C22×D47—42no—
228C22×D47—42no—
229C22×D47—42no—
230C22×D47C2×SD1614yes—
231C22×D47C22×D412yes—
232Q8×C47—24no—
233Q8×C47—24no—
234Q8×C47—44no—
235D4×C47—22no—
236D4×C47—22no—
237D4×C47—42no—
238C2×C4×C47—22no—
239C2×C4×C47—24no—
240C2×C4×C47—22no—
241C2×C4×C47—44no—
242C2×C4×C47—42no—
243C2×C4×C47—48no—
244C2×C4×C47C2×SD1614yes—
245C2×C4×C47C22×D412yes—
246C2×C4×C47C22×D411yescharacteristic
247C2×C4×C47C22×D412yes—
248C2×C4×C47H(5,𝔽2)12yes—
249C2×C4×C472−1+412yes—
250C2×C4×C47C2511yescharacteristic
251C22⋊D47—48no—
252C22⋊D47—48no—
253C22⋊D47—48no—
254C2×Dic47—22no—
255C2×Dic47—22no—
256C2×Dic47—24no—
257C2×Dic47—22no—
258C2×Dic47—42no—
259C2×Dic47—44no—
260C2×Dic47C2×SD1614yes—
261C2×Dic47C22×D412yes—
262C2×Dic47C22×D412yes—
263C2×Dic47C22×D412yes—
264D188—22no—
265D188—22no—
266D188—48no—
267D188—42no—
268C4×D47—24no—
269C4×D47—22no—
270C4×D47—22no—
271C4×D47—44no—
272C4×D47—48no—
273C4×D47C2×SD16116yescomplemented
274Dic94—48no—
275Dic94—48no—
276C8×C47—24no—
277C47⋊C8—24no—
278C2×#3246—471nomaximal, Sylow
279C22×C47—42no—
280C22×C47—42no—
281C22×C47—42no—
282C22×C47—84no—
283C22×C47#53712yes—
284C22×C47C22×SD1612yes—
285C22×C47#57512yes—
286C22×C47C23×D411yescharacteristic
287C2×D47—22no—
288C2×D47—28no—
289C2×D47—22no—
290C2×D47—42no—
291C2×D47—42no—
292C2×D47—42no—
293C2×D47—84no—
294C2×D47C22×SD1614yes—
295C4×C47—22no—
296C4×C47—44no—
297C4×C47—42no—
298C4×C47#53712yes—
299C4×C47C22×SD1614yes—
300C4×C47C2×H(5,𝔽2)11yescharacteristic
301C4×C47C2×2−1+411yescharacteristic
302Dic47—22no—
303Dic47—22no—
304Dic47—42no—
305Dic47—44no—
306Dic47C22×SD1614yes—
307C2×#533—471no—
308C2×#532—471no—
309#3246—4716no—
310C2×#475—472no—
311C2×#476—471no—
312C2×#476—471no—
313C2×#465—472no—
314C2×#465—472no—
315C2×#460—472no—
316C2×#461—471no—
317C2×#461D4711yescharacteristic, complemented
318C22×C4⋊C8—471no—
319C2×C47—82no—
320C2×C47#324614yescomplemented
321C2×C47C2×#53711yescharacteristic
322C2×C47C23×SD1611yescharacteristic
323C2×C47C2×#57511yescharacteristic
324D47—24no—
325D47—82no—
326C22×#41—941no—
327C22×SD16—472no—
328C22×SD16C2×D4712yes—
329C23×C8—472no—
330#533—478no—
331#533—944no—
332#532—478no—
333#532—944no—
334C2×C4⋊Q8—471no—
335C2×C4⋊Q8—471no—
336C2×C4⋊Q8C2×D4711yescharacteristic
337C2×#127—471no—
338C2×#125—941no—
339C2×#124—941no—
340C2×C22⋊Q8—472no—
341C2×C22⋊Q8—941no—
342C2×#121—471no—
343C2×#121—471no—
344C2×#121—941no—
345C2×C4×Q8—472no—
346C2×C4×Q8—941no—
347C2×C4×D4—471no—
348C2×C4×D4—941no—
349C2×C4×D4C2×D4711yescharacteristic
350C2×#117—941no—
351C22×C42—471no—
352#476—478no—
353#476—478no—
354#475—4716no—
355#465—4716no—
356#465—4716no—
357#461—478no—
358#461C2×D4718yescomplemented
359#460—4716no—
360C2×#106—472no—
361C2×#106—472no—
362C2×C4⋊C8—472no—
363C2×C4⋊C8—471no—
364C2×C4⋊C8—478no—
365C2×C4⋊C8C2×D4711yescharacteristic
366C2×#103—472no—
367C2×#103C2×D4712yes—
368C2×#102—472no—
369C2×#102—472no—
370C47C2×#324611yescharacteristic, complemented, Sylow
371C2×#41—948no—
372C2×#41—1888no—
373C22×Q8—941no—
374C22×Q8—941no—
375C22×Q8C22×D4712yes—
376C22×D4—471no—
377C22×D4—941no—
378C22×D4—941no—
379C22×D4—941no—
380C22×D4—941no—
381C22×D4C22×D4711yescharacteristic
382C23×C4—21no—
383C23×C4—471no—
384C23×C4—471no—
385C23×C4—471no—
386C23×C4—941no—
387C2×SD16—44no—
388C2×SD16—478no—
389C2×SD16—1884no—
390C2×SD16C22×D4718yes—
391C22×C8—472no—
392C22×C8—478no—
393C22×C8—1884no—
394C22×C8C22×D4712yes—
395C4⋊Q8—22no—
396C4⋊Q8—474no—
397C4⋊Q8—474no—
398C4⋊Q8—942no—
399C4⋊Q8—942no—
400C4⋊Q8C22×D4714yes—
401#127—474no—
402#127—942no—
403#125—948no—
404#124—948no—
405C22⋊Q8—478no—
406C22⋊Q8—948no—
407C22⋊Q8—944no—
408#121—474no—
409#121—474no—
410#121—948no—
411#121—942no—
412#121—942no—
413C4×Q8—478no—
414C4×Q8—948no—
415C4×Q8—944no—
416C4×D4—22no—
417C4×D4—474no—
418C4×D4—948no—
419C4×D4—942no—
420C4×D4C22×D4714yes—
421#117—948no—
422C2×C4⋊C4—21no—
423C2×C4⋊C4—21no—
424C2×C4⋊C4—471no—
425C2×C4⋊C4—472no—
426C2×C4⋊C4—941no—
427C2×C4⋊C4—942no—
428C2×C4⋊C4—942no—
429C2×C4⋊C4—941no—
430C2×C4⋊C4C22×D4711yescharacteristic
431C2×C22⋊C4—21no—
432C2×C22⋊C4—941no—
433C2×C22⋊C4—942no—
434C2×C22⋊C4—941no—
435C2×C22⋊C4—941no—
436C2×C42—471no—
437C2×C42—471no—
438C2×C42—471no—
439C2×C42—474no—
440C2×C42—474no—
441C2×C42—942no—
442C2×C42C22×D4711yescharacteristic
443#106—478no—
444#106—478no—
445C4⋊C8—478no—
446C4⋊C8—474no—
447C4⋊C8C22×D4714yes—
448#103—478no—
449#103C22×D4718yes—
450#102—478no—
451#102—478no—
452C24—21no—
453C24—471no—
454C24—941no—
455#41—18832no—
456C2×Q8—22no—
457C2×Q8—24no—
458C2×Q8—24no—
459C2×Q8—944no—
460C2×Q8—944no—
461C2×Q8—1884no—
462C2×Q8—1884no—
463C2×Q8C23×D4714yes—
464C2×D4—21no—
465C2×D4—22no—
466C2×D4—22no—
467C2×D4—472no—
468C2×D4—944no—
469C2×D4—944no—
470C2×D4—944no—
471C2×D4—944no—
472C2×D4—941no—
473C2×D4—942no—
474C2×D4—942no—
475C2×D4—1884no—
476C2×D4—1884no—
477C2×D4—1884no—
478C2×D4—1884no—
479C2×D4C23×D4712yes—
480C22×C4—22no—
481C22×C4—21no—
482C22×C4—24no—
483C22×C4—44no—
484C22×C4—471no—
485C22×C4—478no—
486C22×C4—474no—
487C22×C4—471no—
488C22×C4—471no—
489C22×C4—472no—
490C22×C4—472no—
491C22×C4—474no—
492C22×C4—942no—
493C22×C4—942no—
494C22×C4—942no—
495C22×C4—941no—
496C22×C4—944no—
497C22×C4—1882no—
498C22×C4—1884no—
499C22×C4D4×D4711yescharacteristic
500C22×C4D4×D4711yescharacteristic
501C22×C4C23×D4711yescharacteristic
502SD16—48no—
503SD16—1888no—
504C2×C8—24no—
505C2×C8—474no—
506C2×C8—944no—
507C2×C8—1888no—
508C2×C8C23×D4714yes—
509C4⋊C4—24no—
510C4⋊C4—24no—
511C4⋊C4—21no—
512C4⋊C4—474no—
513C4⋊C4—472no—
514C4⋊C4—944no—
515C4⋊C4—948no—
516C4⋊C4—948no—
517C4⋊C4—941no—
518C4⋊C4—942no—
519C4⋊C4—944no—
520C4⋊C4C23×D4712yes—
521C22⋊C4—24no—
522C22⋊C4—944no—
523C22⋊C4—948no—
524C22⋊C4—944no—
525C22⋊C4—944no—
526C42—474no—
527C42—474no—
528C42—472no—
529C42—472no—
530C42—948no—
531C42D4×D4712yes—
532C42C23×D4712yes—
533C23—22no—
534C23—42no—
535C23—42no—
536C23—42no—
537C23—474no—
538C23—472no—
539C23—942no—
540C23—942no—
541C23—942no—
542C23—1882no—
543C23—1882no—
544C23—1882no—
545C23C2×D4×D4711yescharacteristic
546Q8—24no—
547Q8—24no—
548Q8—44no—
549Q8—1888no—
550Q8—1888no—
551D4—22no—
552D4—22no—
553D4—42no—
554D4—942no—
555D4—942no—
556D4—1888no—
557D4—1888no—
558D4—1888no—
559D4—1888no—
560D4—1882no—
561C2×C4—22no—
562C2×C4—24no—
563C2×C4—22no—
564C2×C4—42no—
565C2×C4—44no—
566C2×C4—48no—
567C2×C4—474no—
568C2×C4—472no—
569C2×C4—4716no—
570C2×C4—472no—
571C2×C4—472no—
572C2×C4—942no—
573C2×C4—942no—
574C2×C4—942no—
575C2×C4—942no—
576C2×C4—944no—
577C2×C4—944no—
578C2×C4—942no—
579C2×C4—1884no—
580C2×C4—1882no—
581C2×C4—1884no—
582C2×C4—1888no—
583C2×C4SD16×D4714yes—
584C2×C4C2×D4×D4712yes—
585C2×C4C2×D4×D4712yes—
586C2×C4C2×D4×D4711yescharacteristic
587C2×C4#1309406912yes—
588C2×C4#1309407112yes—
589C2×C4C24×D4711yescharacteristic
590C8—24no—
591C8—944no—
592C22—42no—
593C22—42no—
594C22—42no—
595C22—84no—
596C22—474no—
597C22—942no—
598C22—942no—
599C22—948no—
600C22—3764no—
601C22—1882no—
602C22—1882no—
603C22—1882no—
604C22#1568970212yes—
605C22C2×SD16×D4712yes—
606C22#1568992212yes—
607C22C22×D4×D4711yescharacteristic
608C4—22no—
609C4—42no—
610C4—44no—
611C4—474no—
612C4—942no—
613C4—942no—
614C4—1882no—
615C4—1884no—
616C4#1568970212yes—
617C4C2×SD16×D4714yes—
618C4C2×#1309406911yescharacteristic
619C4C2×#1309407111yescharacteristic
620C2—82no—
621C2—944no—
622C2—3762no—
623C2#2025470114yesdirect product factor
624C2C2×#1568970211yescharacteristic
625C2C22×SD16×D4711yescharacteristic
626C2C2×#1568992211yescharacteristic
subgroupG/H
centerC23C2×D4×D47
commutator, cosocleC2×C4×C47C25
FrattiniC2×C4C24×D47
FittingC2×C47×#461C2
socleC23×C47C22×D4
prime radicalC2×#15689670C2
hypercenterC2×#461D47
hypocenterC47C2×#3246
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#461

Complements

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

NH ≅ G/Ntype
1. C2×#15689678C2semidirect
3. #20254701C2direct (N × H)
4. C2×#15689406C2semidirect
5. C2×#15689414C2semidirect
6. C2×#15689413C2semidirect
7. C2×#15689355C2semidirect
8. C2×#15689349C2semidirect
9. C2×#15689321C2semidirect
10. C2×#15689325C2semidirect
11. C2×C4⋊C8×D47C2semidirect
12. C2×C47×#461C2semidirect
17. #15689678C22semidirect
36. #15689414C22semidirect
37. #15689413C22semidirect
38. #15689406C22semidirect
39. #15689355C22semidirect
40. #15689349C22semidirect
41. #15689325C22semidirect
42. #15689321C22semidirect
47. C4⋊C8×D47C22semidirect
54. C47×#461C22semidirect
179. C2×C4×D47SD16semidirect
273. C4×D47C2×SD16semidirect
317. C2×#461D47semidirect
320. C2×C47#3246semidirect
358. #461C2×D47semidirect
370. C47C2×#3246semidirect
623. C2#20254701direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#156896782
C2×#156896702
#202547012
C2×#156894062
C2×#156894142
C2×#156894132
C2×#156893552
C2×#156893492
C2×#156893212
C2×#156893252
C2×C4⋊C8×D472
C2×C47×#4612
C2×#324647