Finite Group Databasebeta

← C2×#20253184 #27083788

Subgroups — C2×#20253184 #27083788

Order 12032.

2530 conjugacy classes of subgroups.

The subgroup lattice has 610 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689676C211yesmaximal, characteristic, complemented
2C22×#13093986C211yesmaximal, characteristic
3#20253184C2116yesmaximal, direct product factor
4C2×#15689426C212yesmaximal, complemented
5C2×#15689422C212yesmaximal, complemented
6C2×#15689412C212yesmaximal, complemented
7C2×#15689402C212yesmaximal, complemented
8C2×#15689163C211yesmaximal, characteristic, complemented
9C2×#15689146C211yesmaximal, characteristic, complemented
10C2×#15689129C211yesmaximal, characteristic
11C2×#15689127C211yesmaximal, characteristic
12C2×C47×#428C211yesmaximal, characteristic, complemented
13C23×D188C2211yescharacteristic
14C23×C4×D47—21no—
15C22×#10625329C2211yescharacteristic
16#15689676—24no—
17#15689676C2218yescomplemented
18C2×#13094023C2211yescharacteristic, complemented
19C2×#13094023C2211yescharacteristic, complemented
20C2×#13094015C2211yescharacteristic, complemented
21C2×#13094015C2211yescharacteristic, complemented
22C2×#13094001—21no—
23C2×#13094001—21no—
24C2×C4×Dic94—21no—
25C2×C4×Dic94—21no—
26C2×#13093986—21no—
27C2×#13093986—21no—
28C2×#13093986—24no—
29C2×#13093986C2218yes—
30C2×#13093986C2212yes—
31C2×#13093986C2212yes—
32C2×#13093983C2211yescharacteristic
33C22×(C2×D47)⋊C4—21no—
34C22×C4×Dic47C2211yescharacteristic
35#15689426C22116yescomplemented
36#15689422C22116yescomplemented
37#15689412C22116yescomplemented
38#15689402C22116yescomplemented
39#15689163C2218yescomplemented
40C2×#13093957C2212yescomplemented
41C2×#13093955C2212yescomplemented
42#15689146C2218yescomplemented
43C2×Dic47⋊C8C2212yescomplemented
44C2×Dic47⋊C8C2212yescomplemented
45#15689129C2218yes—
46#15689127C2218yes—
47C2×#13093946C2212yes—
48C2×#13093946C2212yes—
49C2×#13093944C2212yes—
50C2×#13093944C2212yes—
51C22×M16×C47C2211yescharacteristic
52C2×#117×C47C2211yescharacteristic
53C22×C4⋊C4×C47C2211yescharacteristic
54C47×#428C2218yescomplemented
55C2×#107×C47C2212yescomplemented
56C2×#106×C47C2212yescomplemented
57C24×D47—21no—
58C22×Dic94—21no—
59C22×Dic94—21no—
60C22×D188—21no—
61C22×D188—22no—
62C22×D188—24no—
63C22×D188Q818yescomplemented
64C22×D188C2312yes—
65C22×D188C2314yes—
66C23×Dic47D411yescharacteristic
67C23×Dic47D411yescharacteristic
68C22×C4×D47—22no—
69C22×C4×D47—28no—
70C22×C4×D47—22no—
71C22×C4×D47—48no—
72C2×#10625329—24no—
73C2×#10625329C2314yes—
74C22×C47⋊C8C2312yes—
75#13094023—22no—
76#13094023—22no—
77#13094023C2314yescomplemented
78#13094023C2314yescomplemented
79#13094015—22no—
80#13094015—22no—
81#13094015C2314yescomplemented
82#13094015C2314yescomplemented
83#13094001—28no—
84#13094001—28no—
85C4×Dic94—28no—
86C4×Dic94—28no—
87#13093986—28no—
88#13093986—28no—
89#13093986—24no—
90#13093986—24no—
91#13093986C2318yes—
92#13093986C2318yes—
93#13093983—22no—
94#13093983C2314yes—
95C2×C4⋊Dic47—21no—
96C2×C4⋊Dic47—21no—
97C2×C4⋊Dic47C2311yescharacteristic
98C2×C4⋊Dic47C2311yescharacteristic
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×Dic47D411yescharacteristic
109C2×C4×Dic47D411yescharacteristic
110C2×C4×Dic47D414yes—
111C2×C4×Dic47C2311yescharacteristic
112C2×C4×Dic47C2311yescharacteristic
113C2×C4×Dic47C2314yes—
114#13093957C2318yescomplemented
115#13093955C2318yescomplemented
116Dic47⋊C8C2318yescomplemented
117Dic47⋊C8C2318yescomplemented
118#13093946C2318yes—
119#13093946C2318yes—
120#13093944C2318yes—
121#13093944C2318yes—
122C23×C4×C47—21no—
123C23×C4×C47C2311yescharacteristic
124C2×M16×C47—24no—
125C2×M16×C47C2314yes—
126C22×C8×C47C2312yes—
127#117×C47—22no—
128#117×C47C2314yes—
129C2×C4⋊C4×C47—21no—
130C2×C4⋊C4×C47—22no—
131C2×C4⋊C4×C47C2312yes—
132C2×C4⋊C4×C47C2314yes—
133C2×C4⋊C4×C47C2311yescharacteristic
134C2×C4⋊C4×C47C2311yescharacteristic
135C2×C22⋊C4×C47—21no—
136C2×C42×C47—21no—
137#107×C47C2318yescomplemented
138#106×C47C2318yescomplemented
139C24×C47C2×D411yescharacteristic
140C23×D47—28no—
141C23×D47—22no—
142C23×D47—24no—
143C23×D47—44no—
144C23×D47—44no—
145C22×C4×C47—21no—
146C22×C4×C47—21no—
147C22×C4×C47—24no—
148C22×C4×C47—22no—
149C22×C4×C47—44no—
150C22×C4×C47C2×D411yescharacteristic
151C22×C4×C47C2×D412yes—
152C22×C4×C47C2×Q814yes—
153C22×C4×C47#4114yes—
154C22×C4×C47C2411yescharacteristic
155C22×C4×C47C2412yes—
156C22×Dic47—24no—
157C22×Dic47—21no—
158C22×Dic47—21no—
159C22×Dic47—42no—
160C22×Dic47C2×D411yescharacteristic
161C22×Dic47C2×D411yescharacteristic
162C22×Dic47C2×D411yescharacteristic
163C22×Dic47C2×D414yes—
164C22×Dic47C2×D414yes—
165C22×Dic47C2×D411yescharacteristic
166C2×D188—216no—
167C2×D188—28no—
168C2×D188—22no—
169C2×D188—24no—
170C2×D188—24no—
171C2×D188—28no—
172C2×D188C2×Q818yes—
173C2×D188C2×Q8116yescomplemented
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—24no—
182C2×Dic94—44no—
183C2×Dic94—44no—
184M16×C47—28no—
185C2×C8×C47—24no—
186C2×C8×C47C2414yes—
187C4⋊C4×C47—24no—
188C4⋊C4×C47—22no—
189C4⋊C4×C47—21no—
190C4⋊C4×C47—21no—
191C4⋊C4×C47C2412yes—
192C4⋊C4×C47C2414yes—
193C4⋊C4×C47C2412yes—
194C22⋊C4×C47—24no—
195C42×C47—24no—
196C22⋊Dic47—24no—
197(C2×D47)⋊C4—28no—
198(C2×D47)⋊C4—28no—
199C4⋊Dic47—21no—
200C4⋊Dic47—21no—
201C4⋊Dic47—24no—
202C4⋊Dic47—24no—
203C4⋊Dic47C2412yes—
204C4⋊Dic47C2412yes—
205Dic47⋊C4—24no—
206Dic47⋊C4—24no—
207Dic47⋊C4—24no—
208Dic47⋊C4—24no—
209C4×Dic47—24no—
210C4×Dic47C2×D414yes—
211C4×Dic47C2×D414yes—
212C4×Dic47C2×D412yes—
213C4×Dic47C2×D412yes—
214C4×Dic47C2412yes—
215C4×Dic47C2412yes—
216#10625329—28no—
217C2×C47⋊C8—24no—
218C2×C47⋊C8C2414yes—
219C23×C47—22no—
220C23×C47—22no—
221C23×C47C22⋊Q814yes—
222C23×C47C22×D412yes—
223C23×C47C22×D411yescharacteristic
224C22×D47—28no—
225C22×D47—216no—
226C22×D47—24no—
227C22×D47—48no—
228C22×D47—432no—
229C22×D47—48no—
230C22×D47—44no—
231C22×D47—44no—
232C2×C4×C47—22no—
233C2×C4×C47—22no—
234C2×C4×C47—24no—
235C2×C4×C47—48no—
236C2×C4×C47—42no—
237C2×C4×C47—44no—
238C2×C4×C47—42no—
239C2×C4×C47C22⋊Q818yes—
240C2×C4×C47C22⋊Q814yes—
241C2×C4×C47C22×D412yes—
242C2×C4×C47C22×D411yescharacteristic
243C2×C4×C47C22×Q812yes—
244C2×C4×C47C22×Q814yes—
245C2×C4×C47C2×#4112yes—
246C2×C4×C47C2×#4114yes—
247C2×C4×C47C2511yescharacteristic
248C2×Dic47—22no—
249C2×Dic47—22no—
250C2×Dic47—22no—
251C2×Dic47—28no—
252C2×Dic47—22no—
253C2×Dic47—22no—
254C2×Dic47—44no—
255C2×Dic47—42no—
256C2×Dic47—42no—
257C2×Dic47AGL(1,ℤ/8)14yes—
258C2×Dic47C22×D412yes—
259C2×Dic47C22×D412yes—
260C2×Dic47C22×D412yes—
261C2×Dic47C22×D412yes—
262D188—28no—
263D188—216no—
264D188—24no—
265D188C22×Q818yes—
266C4×D47—416no—
267C4×D47—416no—
268Dic94—48no—
269Dic94—48no—
270C8×C47—24no—
271C47⋊C8—24no—
272C2×#3077—471nomaximal, Sylow
273C22×C47—28no—
274C22×C47—22no—
275C22×C47—22no—
276C22×C47C2×C22⋊Q814yes—
277C22×C47C2×C22⋊Q812yes—
278C22×C47C2×AGL(1,ℤ/8)12yes—
279C22×C47#57512yes—
280C22×C47C23×D411yescharacteristic
281C2×D47—28no—
282C2×D47—432no—
283C2×D47—416no—
284C2×D47—44no—
285C2×D47—44no—
286C4×C47—42no—
287C4×C47—44no—
288C4×C47—42no—
289C4×C47C2×C22⋊Q812yes—
290C4×C47C2×C22⋊Q814yes—
291C4×C47C23×Q811yescharacteristic
292C4×C47C22×#4111yescharacteristic
293Dic47—22no—
294Dic47—22no—
295Dic47—22no—
296Dic47—42no—
297Dic47—42no—
298Dic47C2×AGL(1,ℤ/8)14yes—
299C2×#533—471no—
300C22×C4×D4—471no—
301#3077—4716no—
302C2×#478—472no—
303C2×#478—472no—
304C2×#476—472no—
305C2×#474—472no—
306C2×#428—471no—
307C2×#428D4711yescharacteristic, complemented
308C2×#423—471no—
309C2×#418—471no—
310C2×#418—471no—
311C2×C47—24no—
312C2×C47#307714yescomplemented
313C2×C47C22×C22⋊Q811yescharacteristic
314C2×C47C22×AGL(1,ℤ/8)11yescharacteristic
315C2×C47C2×#57511yescharacteristic
316D47—48no—
317C23×D4—471no—
318C24×C4—941no—
319C22×M16—471no—
320C22×M16C2×D4711yescharacteristic
321#533—478no—
322#533—944no—
323C2×C4⋊Q8—471no—
324C2×C4⋊Q8—471no—
325C2×#125—471no—
326C2×#125—471no—
327C2×C22⋊Q8—941no—
328C2×C22⋊Q8—941no—
329C2×C4×Q8—941no—
330C2×C4×Q8—941no—
331C2×C4×D4—472no—
332C2×C4×D4—478no—
333C2×C4×D4—472no—
334C2×C4×D4—941no—
335C2×C4×D4—941no—
336C2×C4×D4—944no—
337C2×#117—471no—
338C2×#117C2×D4711yescharacteristic
339C22×C4⋊C4C2×D4711yescharacteristic
340C22×C22⋊C4—941no—
341C22×C42—471no—
342#478—4716no—
343#478—4716no—
344#476—4716no—
345#474—4716no—
346#428—478no—
347#428C2×D4718yescomplemented
348C2×#107—472no—
349C2×#107C2×D4712yescomplemented
350C2×#106—472no—
351C2×#106C2×D4712yescomplemented
352#423—478no—
353C2×C4⋊C8—472no—
354C2×C4⋊C8—472no—
355#418—478no—
356#418—478no—
357C2×#102—472no—
358C2×#102—472no—
359C2×#102—472no—
360C2×#102—472no—
361C47C2×#307711yescharacteristic, complemented, Sylow
362C25—941no—
363C22×Q8—941no—
364C22×Q8—941no—
365C22×D4—478no—
366C22×D4—472no—
367C22×D4—474no—
368C22×D4—944no—
369C22×D4—942no—
370C22×D4—941no—
371C23×C4—21no—
372C23×C4—471no—
373C23×C4—471no—
374C23×C4—948no—
375C23×C4—942no—
376C23×C4—942no—
377C23×C4—1888no—
378C23×C4C22×D4711yescharacteristic
379C2×M16—24no—
380C2×M16—474no—
381C2×M16—944no—
382C2×M16C22×D4714yes—
383C22×C8—472no—
384C22×C8C22×D4712yes—
385C4⋊Q8—474no—
386C4⋊Q8—474no—
387C4⋊Q8—942no—
388C4⋊Q8—942no—
389#125—474no—
390#125—474no—
391#125—942no—
392#125—942no—
393C22⋊Q8—948no—
394C22⋊Q8—948no—
395C4×Q8—948no—
396C4×Q8—948no—
397C4×D4—478no—
398C4×D4—478no—
399C4×D4—948no—
400C4×D4—948no—
401C4×D4—944no—
402C4×D4—944no—
403#117—22no—
404#117—474no—
405#117—942no—
406#117C22×D4714yes—
407C2×C4⋊C4—22no—
408C2×C4⋊C4—21no—
409C2×C4⋊C4—471no—
410C2×C4⋊C4—471no—
411C2×C4⋊C4—941no—
412C2×C4⋊C4—941no—
413C2×C4⋊C4—941no—
414C2×C4⋊C4—941no—
415C2×C4⋊C4—941no—
416C2×C4⋊C4—941no—
417C2×C4⋊C4C22×D4712yes—
418C2×C4⋊C4C22×D4714yes—
419C2×C4⋊C4C22×D4711yescharacteristic
420C2×C4⋊C4C22×D4711yescharacteristic
421C2×C22⋊C4—21no—
422C2×C22⋊C4—948no—
423C2×C22⋊C4—942no—
424C2×C22⋊C4—942no—
425C2×C22⋊C4—941no—
426C2×C42—21no—
427C2×C42—471no—
428C2×C42—471no—
429C2×C42—474no—
430C2×C42—471no—
431C2×C42—471no—
432C2×C42—474no—
433C2×C42—941no—
434#107—478no—
435#107C22×D4718yescomplemented
436#106—478no—
437#106C22×D4718yescomplemented
438C4⋊C8—478no—
439C4⋊C8—478no—
440#102—478no—
441#102—478no—
442#102—478no—
443#102—478no—
444C24—948no—
445C24—944no—
446C24—942no—
447C24—1884no—
448C24—1884no—
449C24D4×D4711yescharacteristic
450C2×Q8—944no—
451C2×Q8—944no—
452C2×Q8—1884no—
453C2×Q8—1884no—
454C2×D4—478no—
455C2×D4—4716no—
456C2×D4—474no—
457C2×D4—9416no—
458C2×D4—948no—
459C2×D4—944no—
460C2×D4—942no—
461C2×D4—944no—
462C2×D4—948no—
463C22×C4—24no—
464C22×C4—21no—
465C22×C4—21no—
466C22×C4—22no—
467C22×C4—44no—
468C22×C4—471no—
469C22×C4—474no—
470C22×C4—471no—
471C22×C4—471no—
472C22×C4—471no—
473C22×C4—474no—
474C22×C4—948no—
475C22×C4—948no—
476C22×C4—944no—
477C22×C4—941no—
478C22×C4—941no—
479C22×C4—1888no—
480C22×C4—18832no—
481C22×C4—1888no—
482C22×C4—1882no—
483C22×C4#1062535014yes—
484C22×C4D4×D4712yes—
485C22×C4D4×D4711yescharacteristic
486C22×C4Q8×D4714yes—
487C22×C4C23×D4712yes—
488C22×C4C23×D4711yescharacteristic
489M16—28no—
490M16—948no—
491C2×C8—24no—
492C2×C8—474no—
493C2×C8—944no—
494C2×C8C23×D4714yes—
495C4⋊C4—24no—
496C4⋊C4—22no—
497C4⋊C4—21no—
498C4⋊C4—21no—
499C4⋊C4—472no—
500C4⋊C4—472no—
501C4⋊C4—944no—
502C4⋊C4—944no—
503C4⋊C4—944no—
504C4⋊C4—944no—
505C4⋊C4—944no—
506C4⋊C4—944no—
507C4⋊C4—941no—
508C4⋊C4—941no—
509C4⋊C4C23×D4712yes—
510C4⋊C4C23×D4712yes—
511C4⋊C4C23×D4714yes—
512C22⋊C4—24no—
513C22⋊C4—948no—
514C22⋊C4—948no—
515C22⋊C4—944no—
516C42—24no—
517C42—474no—
518C42—474no—
519C42—472no—
520C42—472no—
521C42—472no—
522C42—472no—
523C42—944no—
524C23—22no—
525C23—22no—
526C23—9416no—
527C23—948no—
528C23—944no—
529C23—1888no—
530C23—18832no—
531C23—1888no—
532C23—1884no—
533C23—1884no—
534C23#1309400214yes—
535C23C2×D4×D4712yes—
536C23C2×D4×D4711yescharacteristic
537Q8—1888no—
538Q8—1888no—
539D4—478no—
540D4—948no—
541D4—9416no—
542D4—944no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—24no—
546C2×C4—48no—
547C2×C4—44no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—474no—
551C2×C4—472no—
552C2×C4—472no—
553C2×C4—472no—
554C2×C4—472no—
555C2×C4—948no—
556C2×C4—942no—
557C2×C4—942no—
558C2×C4—942no—
559C2×C4—942no—
560C2×C4—942no—
561C2×C4—18816no—
562C2×C4—18816no—
563C2×C4—1884no—
564C2×C4—1882no—
565C2×C4—1882no—
566C2×C4#1309400218yes—
567C2×C4#1309400214yes—
568C2×C4C2×D4×D4712yes—
569C2×C4C2×D4×D4711yescharacteristic
570C2×C4C2×Q8×D4714yes—
571C2×C4C2×Q8×D4712yes—
572C2×C4C2×#1062535012yes—
573C2×C4C2×#1062535014yes—
574C2×C4C24×D4711yescharacteristic
575C8—24no—
576C8—944no—
577C22—28no—
578C22—22no—
579C22—22no—
580C22—948no—
581C22—18832no—
582C22—18816no—
583C22—1884no—
584C22—1884no—
585C22C2×#1309400214yes—
586C22C2×#1309400212yes—
587C22AGL(1,ℤ/8)×D4712yes—
588C22#1568992212yes—
589C22C22×D4×D4711yescharacteristic
590C4—44no—
591C4—42no—
592C4—42no—
593C4—474no—
594C4—942no—
595C4—942no—
596C4—942no—
597C4—1882no—
598C4—1882no—
599C4C2×#1309400214yes—
600C4C2×#1309400212yes—
601C4C22×Q8×D4711yescharacteristic
602C4C22×#1062535011yescharacteristic
603C2—24no—
604C2—1888no—
605C2#2025318414yesdirect product factor
606C2C22×#1309400211yescharacteristic
607C2C2×AGL(1,ℤ/8)×D4711yescharacteristic
608C2C2×#1568992211yescharacteristic
subgroupG/H
centerC23C2×D4×D47
commutator, cosocleC2×C4×C47C25
FrattiniC2×C4C24×D47
FittingC2×C47×#428C2
socleC23×C47C22×D4
prime radicalC23×D188C22
hypercenterC2×#428D47
hypocenterC47C2×#3077
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#428

Complements

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

NH ≅ G/Ntype
1. C2×#15689676C2semidirect
3. #20253184C2direct (N × H)
4. C2×#15689426C2semidirect
5. C2×#15689422C2semidirect
6. C2×#15689412C2semidirect
7. C2×#15689402C2semidirect
8. C2×#15689163C2semidirect
9. C2×#15689146C2semidirect
12. C2×C47×#428C2semidirect
17. #15689676C22semidirect
18. C2×#13094023C22semidirect
19. C2×#13094023C22semidirect
20. C2×#13094015C22semidirect
21. C2×#13094015C22semidirect
35. #15689426C22semidirect
36. #15689422C22semidirect
37. #15689412C22semidirect
38. #15689402C22semidirect
39. #15689163C22semidirect
40. C2×#13093957C22semidirect
41. C2×#13093955C22semidirect
42. #15689146C22semidirect
43. C2×Dic47⋊C8C22semidirect
44. C2×Dic47⋊C8C22semidirect
54. C47×#428C22semidirect
55. C2×#107×C47C22semidirect
56. C2×#106×C47C22semidirect
63. C22×D188Q8semidirect
77. #13094023C23semidirect
78. #13094023C23semidirect
81. #13094015C23semidirect
82. #13094015C23semidirect
114. #13093957C23semidirect
115. #13093955C23semidirect
116. Dic47⋊C8C23semidirect
117. Dic47⋊C8C23semidirect
137. #107×C47C23semidirect
138. #106×C47C23semidirect
173. C2×D188C2×Q8semidirect
307. C2×#428D47semidirect
312. C2×C47#3077semidirect
347. #428C2×D47semidirect
349. C2×#107C2×D47semidirect
351. C2×#106C2×D47semidirect
361. C47C2×#3077semidirect
435. #107C22×D47semidirect
437. #106C22×D47semidirect
605. C2#20253184direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#156896762
C22×#130939862
#202531842
C2×#156894262
C2×#156894222
C2×#156894122
C2×#156894022
C2×#156891632
C2×#156891462
C2×#156891292
C2×#156891272
C2×C47×#4282
C2×#307747