Finite Group Databasebeta

← C2×#20254419 #27087556

Subgroups — C2×#20254419 #27087556

Order 12032.

2146 conjugacy classes of subgroups.

The subgroup lattice has 620 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4⋊Q8×D47C211yesmaximal, characteristic, complemented
2C2×#15689660C211yesmaximal, characteristic
3#20254419C2116yesmaximal, direct product factor
4C2×#15689516C212yesmaximal, complemented
5C2×#15689490C212yesmaximal, complemented
6C2×#15689333C211yesmaximal, characteristic, complemented
7C2×#15689329C211yesmaximal, characteristic, complemented
8C2×#15689325C211yesmaximal, characteristic, complemented
9C2×#15689277C212yesmaximal, complemented
10C2×#15689263C212yesmaximal
11C2×#15689179C211yesmaximal, characteristic, complemented
12C2×C47×#462C211yesmaximal, characteristic, complemented
13C22×Q8×D47C2212yes—
14C22×#10625337C2212yes—
15C22×#10625325C2212yes—
16C22×#10625336C2212yes—
17C2×#13094024C2211yescharacteristic
18C2×#13094023—21no—
19C2×#13094023—21no—
20C2×#13094022C2212yes—
21C4⋊Q8×D47—24no—
22C4⋊Q8×D47C2218yescomplemented
23#15689660—24no—
24#15689660C2218yes—
25C2×#13094019—21no—
26C2×#13094015C2211yescharacteristic
27C2×#13094013C2211yescharacteristic
28C2×#13094009—21no—
29C2×#13094000C2211yescharacteristic
30C2×#13093999C2211yescharacteristic
31C2×C4×D188—21no—
32C2×C4×D188C2211yescharacteristic
33C2×#13093981—21no—
34C2×C4⋊C4×D47—21no—
35C2×C4⋊C4×D47—21no—
36C2×C42×D47C2211yescharacteristic
37#15689516C22116yescomplemented
38#15689490C22116yescomplemented
39#15689333C2218yescomplemented
40#15689329C2218yescomplemented
41#15689325C2218yescomplemented
42#15689277C22116yescomplemented
43#15689263C22116yes—
44#15689179C2218yescomplemented
45C2×C4⋊(C47⋊C8)C2211yescharacteristic
46C2×#13093949C2212yes—
47C2×#13093948C2212yes—
48C2×#13093947C2212yes—
49C2×(C2×D47)⋊C8C2212yes—
50C2×#13093934C2212yes—
51C22×Q16×C47C2212yes—
52C2×C4⋊Q8×C47C2211yescharacteristic
53C2×C4×Q8×C47C2211yescharacteristic
54C47×#462C2218yescomplemented
55C2×C4⋊C8×C47C2211yescharacteristic
56C2×#103×C47C2212yes—
57C22×Dic94—22no—
58C22×Dic94C2312yes—
59C2×Q8×D47—24no—
60C2×Q8×D47—44no—
61C2×Q8×D47—44no—
62C2×Q8×D47C2318yes—
63C22×D188—21no—
64C22×D188C2311yescharacteristic
65C22×C4×D47—22no—
66C22×C4×D47—21no—
67C22×C4×D47D411yescharacteristic
68C22×C4×D47D411yescharacteristic
69C22×C4×D47C2311yescharacteristic
70C2×#10625337—44no—
71C2×#10625337C2318yes—
72C2×#10625325—44no—
73C2×#10625325C2318yes—
74C2×#10625336—44no—
75C2×#10625336C2318yes—
76C22×C47⋊C8C2312yes—
77#13094024—22no—
78#13094024C2314yes—
79#13094023—28no—
80#13094023—28no—
81#13094022—24no—
82#13094022C2318yes—
83#13094019—28no—
84#13094015—22no—
85#13094015C2314yes—
86#13094013—22no—
87#13094013C2314yes—
88#13094009—28no—
89#13094000—22no—
90#13094000C2314yes—
91#13093999—22no—
92#13093999C2314yes—
93C4×D188—28no—
94C4×D188—22no—
95C4×D188C2314yes—
96#13093981—28no—
97C2×C4⋊Dic47—21no—
98C2×C4⋊Dic47—21no—
99C2×C4⋊Dic47—21no—
100C2×C4⋊Dic47C2311yescharacteristic
101C2×Dic47⋊C4—22no—
102C2×Dic47⋊C4—22no—
103C2×Dic47⋊C4—21no—
104C2×Dic47⋊C4—21no—
105C4⋊C4×D47—28no—
106C4⋊C4×D47—28no—
107C2×(C2×D47)⋊C4—21no—
108C2×(C2×D47)⋊C4—21no—
109C2×(C2×D47)⋊C4—21no—
110C2×(C2×D47)⋊C4—21no—
111C2×(C2×D47)⋊C4—21no—
112C2×C4×Dic47D411yescharacteristic
113C2×C4×Dic47D411yescharacteristic
114C2×C4×Dic47C2311yescharacteristic
115C42×D47D414yes—
116C42×D47C2314yes—
117C4⋊(C47⋊C8)C2314yes—
118#13093949C2318yes—
119#13093948C2318yes—
120#13093947C2318yes—
121(C2×D47)⋊C8C2318yes—
122#13093934C2318yes—
123C22×Q8×C47C2311yescharacteristic
124C22×Q8×C47C2312yes—
125C2×Q16×C47—44no—
126C2×Q16×C47C2318yes—
127C22×C8×C47C2312yes—
128C4⋊Q8×C47—22no—
129C4⋊Q8×C47C2314yes—
130C4×Q8×C47—22no—
131C4×Q8×C47C2314yes—
132C2×C4⋊C4×C47—21no—
133C2×C4⋊C4×C47—21no—
134C2×C4⋊C4×C47—21no—
135C2×C4⋊C4×C47C2311yescharacteristic
136C2×C42×C47—21no—
137C2×C42×C47C2311yescharacteristic
138C4⋊C8×C47C2314yes—
139#103×C47C2318yes—
140C23×D47—21no—
141C23×D47C2×D411yescharacteristic
142C2×Q8×C47—21no—
143C2×Q8×C47—22no—
144C2×Q8×C47—22no—
145C2×Q8×C47—24no—
146C2×Q8×C47—24no—
147C2×Q8×C47—22no—
148C2×Q8×C47C2414yes—
149C2×Q8×C47C2412yes—
150C22×C4×C47—21no—
151C22×C4×C47—21no—
152C22×C4×C47—22no—
153C22×C4×C47C2×D411yescharacteristic
154C22×C4×C47C2×D411yescharacteristic
155C22×C4×C47C2411yescharacteristic
156C22×Dic47—21no—
157C22×Dic47—22no—
158C22×Dic47C2×D411yescharacteristic
159C22×Dic47C2×D411yescharacteristic
160C22×Dic47C2×D411yescharacteristic
161Q8×D47—48no—
162Q8×D47—48no—
163Q8×D47—416no—
164C2×D188—22no—
165C2×D188—22no—
166C2×D188—24no—
167C2×D188—21no—
168C2×D188—44no—
169C2×D188C2412yes—
170C2×C4×D47—24no—
171C2×C4×D47—24no—
172C2×C4×D47—22no—
173C2×C4×D47—22no—
174C2×C4×D47—28no—
175C2×C4×D47—24no—
176C2×C4×D47—48no—
177C2×C4×D47—44no—
178C2×C4×D47C2×D412yes—
179C2×C4×D47C2×D414yes—
180C2×C4×D47C2×D414yes—
181C2×C4×D47C2412yes—
182C2×Dic94—24no—
183C2×Dic94—24no—
184C2×Dic94—22no—
185C2×Dic94—28no—
186C2×Dic94—48no—
187C2×Dic94C2414yes—
188Q16×C47—48no—
189C2×C8×C47—24no—
190C2×C8×C47C2414yes—
191C4⋊C4×C47—24no—
192C4⋊C4×C47—24no—
193C4⋊C4×C47—24no—
194C4⋊C4×C47—21no—
195C4⋊C4×C47C2412yes—
196C42×C47—24no—
197C42×C47C2×D412yes—
198C42×C47C2412yes—
199#10625337—48no—
200#10625336—48no—
201(C2×D47)⋊C4—24no—
202(C2×D47)⋊C4—24no—
203(C2×D47)⋊C4—24no—
204(C2×D47)⋊C4—24no—
205(C2×D47)⋊C4—24no—
206C4⋊Dic47—24no—
207C4⋊Dic47—24no—
208C4⋊Dic47—21no—
209C4⋊Dic47—24no—
210C4⋊Dic47C2412yes—
211Dic47⋊C4—28no—
212Dic47⋊C4—28no—
213Dic47⋊C4—24no—
214Dic47⋊C4—24no—
215C4×Dic47C2×D412yes—
216C4×Dic47C2×D414yes—
217C4×Dic47C2×D414yes—
218C4×Dic47C2412yes—
219C2×C47⋊C8—24no—
220C2×C47⋊C8C2414yes—
221#10625325—48no—
222C23×C47C22×D411yescharacteristic
223C22×D47—22no—
224C22×D47—22no—
225C22×D47—22no—
226C22×D47—42no—
227C22×D47—42no—
228C22×D47—42no—
229C22×D47#13714yes—
230C22×D47C22×D412yes—
231Q8×C47—24no—
232Q8×C47—22no—
233Q8×C47—22no—
234Q8×C47—24no—
235Q8×C47—44no—
236Q8×C47—42no—
237C2×C4×C47—22no—
238C2×C4×C47—24no—
239C2×C4×C47—22no—
240C2×C4×C47—22no—
241C2×C4×C47—44no—
242C2×C4×C47—42no—
243C2×C4×C47—42no—
244C2×C4×C47#13714yes—
245C2×C4×C47C22×D412yes—
246C2×C4×C47C22×D411yescharacteristic
247C2×C4×C47C22×D412yes—
248C2×C4×C47H(5,𝔽2)12yes—
249C2×C4×C47H(5,𝔽2)12yes—
250C2×C4×C47C2511yescharacteristic
251C2×Dic47—22no—
252C2×Dic47—22no—
253C2×Dic47—22no—
254C2×Dic47—24no—
255C2×Dic47—22no—
256C2×Dic47—44no—
257C2×Dic47—42no—
258C2×Dic47C22×D412yes—
259C2×Dic47C22×D412yes—
260C2×Dic47C22×D412yes—
261D188—22no—
262D188—22no—
263D188—48no—
264D188—42no—
265C4×D47—28no—
266C4×D47—24no—
267C4×D47—28no—
268C4×D47—22no—
269C4×D47—22no—
270C4×D47—416no—
271C4×D47—48no—
272Dic94—24no—
273Dic94—24no—
274Dic94—416no—
275Dic94—44no—
276C8×C47—24no—
277C47⋊C8—24no—
278C2×#3218—471nomaximal, Sylow
279C22×C47#53512yes—
280C22×C47C2×#13712yes—
281C22×C47C2×#13712yes—
282C22×C47C23×D411yescharacteristic
283C2×D47—22no—
284C2×D47—28no—
285C2×D47—22no—
286C2×D47—42no—
287C2×D47—42no—
288C2×D47—42no—
289C2×D47—84no—
290C2×D47C2×#13714yes—
291C4×C47—22no—
292C4×C47—44no—
293C4×C47—42no—
294C4×C47—42no—
295C4×C47#53512yes—
296C4×C47C2×#13714yes—
297C4×C47C2×H(5,𝔽2)11yescharacteristic
298C4×C47C2×H(5,𝔽2)11yescharacteristic
299Dic47—22no—
300Dic47—22no—
301Dic47—22no—
302Dic47—44no—
303Dic47—42no—
304C22×C4⋊Q8—471no—
305C2×#529—471no—
306#3218—4716no—
307C2×#497—472no—
308C2×#491—472no—
309C2×#462—471no—
310C2×#462D4711yescharacteristic, complemented
311C2×#461—471no—
312C2×#461—471no—
313C2×#451—472no—
314C2×#448—472no—
315C2×#432—471no—
316C2×C47#321814yescomplemented
317C2×C47C2×#53511yescharacteristic
318C2×C47C22×#13711yescharacteristic
319C2×C47C22×#13711yescharacteristic
320D47—24no—
321D47—82no—
322C23×Q8—472no—
323C22×Q16—472no—
324C22×Q16C2×D4712yes—
325C22×SD16—472no—
326C22×SD16—472no—
327C2×C4⋊Q8—472no—
328C2×C4⋊Q8—478no—
329C2×C4⋊Q8—471no—
330C2×C4⋊Q8—941no—
331C2×C4⋊Q8—941no—
332C2×C4⋊Q8—944no—
333C2×C4⋊Q8C2×D4711yescharacteristic
334#529—478no—
335#529—944no—
336C2×#126—941no—
337C2×#125—471no—
338C2×#124—471no—
339C2×#123—941no—
340C2×C22⋊Q8—471no—
341C2×#121—471no—
342C2×C4×Q8C2×D4711yescharacteristic
343C2×C4×D4—471no—
344C2×C4×D4—941no—
345C2×#117—941no—
346C22×C4⋊C4—941no—
347C22×C4⋊C4—941no—
348C22×C42—471no—
349#497—4716no—
350#491—4716no—
351#462—478no—
352#462C2×D4718yescomplemented
353#461—478no—
354#461—478no—
355#451—4716no—
356#448—4716no—
357#432—478no—
358C2×C4⋊C8—471no—
359C2×C4⋊C8C2×D4711yescharacteristic
360C2×#103—472no—
361C2×#103—472no—
362C2×#103—472no—
363C2×#103C2×D4712yes—
364C2×C22⋊C8—472no—
365C2×#97—472no—
366C47C2×#321811yescharacteristic, complemented, Sylow
367C22×Q8—478no—
368C22×Q8—472no—
369C22×Q8—942no—
370C22×Q8—944no—
371C22×Q8—1884no—
372C22×Q8—1884no—
373C22×Q8C22×D4711yescharacteristic
374C22×Q8C22×D4712yes—
375C22×D4—471no—
376C22×D4—941no—
377C23×C4—471no—
378C23×C4—471no—
379C23×C4—471no—
380C23×C4—942no—
381C23×C4—941no—
382C2×Q16—44no—
383C2×Q16—478no—
384C2×Q16—1884no—
385C2×Q16C22×D4718yes—
386C2×SD16—478no—
387C2×SD16—478no—
388C2×SD16—1884no—
389C2×SD16—1884no—
390C22×C8—472no—
391C22×C8C22×D4712yes—
392C4⋊Q8—22no—
393C4⋊Q8—478no—
394C4⋊Q8—474no—
395C4⋊Q8—948no—
396C4⋊Q8—948no—
397C4⋊Q8—944no—
398C4⋊Q8—942no—
399C4⋊Q8C22×D4714yes—
400#126—948no—
401#125—474no—
402#125—942no—
403#124—474no—
404#124—942no—
405#123—948no—
406C22⋊Q8—474no—
407C22⋊Q8—942no—
408#121—474no—
409#121—942no—
410C4×Q8—22no—
411C4×Q8C22×D4714yes—
412C4×D4—474no—
413C4×D4—948no—
414C4×D4—942no—
415#117—948no—
416C2×C4⋊C4—21no—
417C2×C4⋊C4—21no—
418C2×C4⋊C4—21no—
419C2×C4⋊C4—471no—
420C2×C4⋊C4—948no—
421C2×C4⋊C4—948no—
422C2×C4⋊C4—942no—
423C2×C4⋊C4—942no—
424C2×C4⋊C4—941no—
425C2×C4⋊C4—941no—
426C2×C4⋊C4—941no—
427C2×C4⋊C4—941no—
428C2×C4⋊C4—941no—
429C2×C4⋊C4C22×D4711yescharacteristic
430C2×C22⋊C4—941no—
431C2×C22⋊C4—941no—
432C2×C22⋊C4—941no—
433C2×C22⋊C4—941no—
434C2×C22⋊C4—941no—
435C2×C42—21no—
436C2×C42—471no—
437C2×C42—471no—
438C2×C42—471no—
439C2×C42—474no—
440C2×C42—474no—
441C2×C42C22×D4711yescharacteristic
442C4⋊C8—474no—
443C4⋊C8C22×D4714yes—
444#103—478no—
445#103—478no—
446#103—478no—
447#103C22×D4718yes—
448C22⋊C8—478no—
449#97—478no—
450C24—471no—
451C24—941no—
452C2×Q8—21no—
453C2×Q8—22no—
454C2×Q8—22no—
455C2×Q8—24no—
456C2×Q8—24no—
457C2×Q8—22no—
458C2×Q8—474no—
459C2×Q8—948no—
460C2×Q8—942no—
461C2×Q8—944no—
462C2×Q8—944no—
463C2×Q8—1888no—
464C2×Q8—1888no—
465C2×Q8—1888no—
466C2×Q8—18816no—
467C2×Q8C23×D4714yes—
468C2×Q8C23×D4712yes—
469C2×D4—472no—
470C2×D4—944no—
471C2×D4—941no—
472C2×D4—942no—
473C2×D4—942no—
474C2×D4—1884no—
475C22×C4—21no—
476C22×C4—21no—
477C22×C4—22no—
478C22×C4—471no—
479C22×C4—471no—
480C22×C4—471no—
481C22×C4—474no—
482C22×C4—474no—
483C22×C4—472no—
484C22×C4—472no—
485C22×C4—944no—
486C22×C4—948no—
487C22×C4—944no—
488C22×C4—942no—
489C22×C4—942no—
490C22×C4—941no—
491C22×C4—942no—
492C22×C4—944no—
493C22×C4—1888no—
494C22×C4—1884no—
495C22×C4D4×D4711yescharacteristic
496C22×C4D4×D4711yescharacteristic
497C22×C4C23×D4711yescharacteristic
498Q16—48no—
499Q16—1888no—
500SD16—1888no—
501SD16—1888no—
502C2×C8—24no—
503C2×C8—474no—
504C2×C8—944no—
505C2×C8C23×D4714yes—
506C4⋊C4—24no—
507C4⋊C4—24no—
508C4⋊C4—24no—
509C4⋊C4—21no—
510C4⋊C4—472no—
511C4⋊C4—948no—
512C4⋊C4—948no—
513C4⋊C4—944no—
514C4⋊C4—944no—
515C4⋊C4—944no—
516C4⋊C4—941no—
517C4⋊C4—944no—
518C4⋊C4—944no—
519C4⋊C4C23×D4712yes—
520C22⋊C4—944no—
521C22⋊C4—944no—
522C22⋊C4—944no—
523C22⋊C4—944no—
524C22⋊C4—944no—
525C42—24no—
526C42—474no—
527C42—472no—
528C42—474no—
529C42—472no—
530C42D4×D4712yes—
531C42C23×D4712yes—
532C23—474no—
533C23—472no—
534C23—942no—
535C23—942no—
536C23—942no—
537C23—1882no—
538C23—1882no—
539C23—1882no—
540C23C2×D4×D4711yescharacteristic
541Q8—22no—
542Q8—24no—
543Q8—24no—
544Q8—22no—
545Q8—42no—
546Q8—44no—
547Q8—944no—
548Q8—944no—
549Q8—18816no—
550Q8—1884no—
551D4—942no—
552D4—942no—
553D4—1888no—
554D4—1882no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—24no—
559C2×C4—42no—
560C2×C4—44no—
561C2×C4—42no—
562C2×C4—472no—
563C2×C4—472no—
564C2×C4—472no—
565C2×C4—948no—
566C2×C4—942no—
567C2×C4—942no—
568C2×C4—942no—
569C2×C4—948no—
570C2×C4—944no—
571C2×C4—942no—
572C2×C4—942no—
573C2×C4—942no—
574C2×C4—944no—
575C2×C4—18816no—
576C2×C4—1882no—
577C2×C4—1884no—
578C2×C4—1888no—
579C2×C4#1309405614yes—
580C2×C4C2×D4×D4712yes—
581C2×C4C2×D4×D4712yes—
582C2×C4C2×D4×D4711yescharacteristic
583C2×C4#1309406912yes—
584C2×C4#1309407012yes—
585C2×C4C24×D4711yescharacteristic
586C8—24no—
587C8—944no—
588C22—474no—
589C22—942no—
590C22—942no—
591C22—948no—
592C22—3764no—
593C22—1882no—
594C22—1882no—
595C22—1882no—
596C22#1568968912yes—
597C22#137×D4712yes—
598C22C2×#1309405612yes—
599C22C22×D4×D4711yescharacteristic
600C4—22no—
601C4—42no—
602C4—44no—
603C4—42no—
604C4—942no—
605C4—942no—
606C4—942no—
607C4—1882no—
608C4—1884no—
609C4#1568968912yes—
610C4C2×#1309405614yes—
611C4C2×#1309406911yescharacteristic
612C4C2×#1309407011yescharacteristic
613C2—944no—
614C2—3762no—
615C2#2025441914yesdirect product factor
616C2C2×#1568968911yescharacteristic
617C2C2×#137×D4711yescharacteristic
618C2C22×#1309405611yescharacteristic
subgroupG/H
centerC23C2×D4×D47
commutator, cosocleC2×C4×C47C25
FrattiniC2×C4C24×D47
FittingC2×C47×#462C2
socleC23×C47C22×D4
prime radicalC2×#13093999C22
hypercenterC2×#462D47
hypocenterC47C2×#3218
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#462

Complements

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

NH ≅ G/Ntype
1. C2×C4⋊Q8×D47C2semidirect
3. #20254419C2direct (N × H)
4. C2×#15689516C2semidirect
5. C2×#15689490C2semidirect
6. C2×#15689333C2semidirect
7. C2×#15689329C2semidirect
8. C2×#15689325C2semidirect
9. C2×#15689277C2semidirect
11. C2×#15689179C2semidirect
12. C2×C47×#462C2semidirect
22. C4⋊Q8×D47C22semidirect
37. #15689516C22semidirect
38. #15689490C22semidirect
39. #15689333C22semidirect
40. #15689329C22semidirect
41. #15689325C22semidirect
42. #15689277C22semidirect
44. #15689179C22semidirect
54. C47×#462C22semidirect
310. C2×#462D47semidirect
316. C2×C47#3218semidirect
352. #462C2×D47semidirect
366. C47C2×#3218semidirect
615. C2#20254419direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C4⋊Q8×D472
C2×#156896602
#202544192
C2×#156895162
C2×#156894902
C2×#156893332
C2×#156893292
C2×#156893252
C2×#156892772
C2×#156892632
C2×#156891792
C2×C47×#4622
C2×#321847