Finite Group Databasebeta

← C4×#15689494 #26224695

Subgroups — C4×#15689494 #26224695

Order 12032.

1254 conjugacy classes of subgroups.

The subgroup lattice has 648 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689494C211yesmaximal, characteristic
2C2×C4×#10625325C211yesmaximal, characteristic, complemented
3C2×C4×#10625336C211yesmaximal, characteristic, complemented
4C2×C4×#10625335C211yesmaximal, characteristic, complemented
5C4×#13094022C211yesmaximal, characteristic, complemented
6C4×#13094020C211yesmaximal, characteristic
7#20244244C211yesmaximal, characteristic, complemented
8#20244243C211yesmaximal, characteristic, complemented
9#20244240C211yesmaximal, characteristic, complemented
10#20244239C211yesmaximal, characteristic, complemented
11#20244192C211yesmaximal, characteristic
12#20244188C211yesmaximal, characteristic, complemented
13#20243543C211yesmaximal, characteristic, complemented
14C4×C8×Dic47C211yesmaximal, characteristic, complemented
15C2×C4×SD16×C47C211yesmaximal, characteristic, complemented
16C22×#10625325C2211yescharacteristic
17C22×#10625335C2211yescharacteristic
18C22×#10625336C2211yescharacteristic
19C2×#13094022C2211yescharacteristic
20C2×#13094020C2211yescharacteristic
21C2×C4×Dic94C2211yescharacteristic
22C2×C4×D188C2211yescharacteristic
23C2×C4×C22⋊D47—22no—
24#15689494C418yesdirect product factor
25C4×#10625325—24no—
26C4×#10625336—24no—
27C4×#10625335—24no—
28C2×#13093956C2211yescharacteristic
29C2×#13093955C2211yescharacteristic
30C2×#13093955C2211yescharacteristic
31C2×#13093949C2211yescharacteristic
32C2×#13093948C2211yescharacteristic
33C2×#13093947C2211yescharacteristic
34C2×#13093946C2211yescharacteristic
35C2×#13093945C2211yescharacteristic
36C2×#13093944C2211yescharacteristic
37C2×C4×C47⋊C8C2211yescharacteristic
38C2×C8×Dic47C2211yescharacteristic
39C2×C8×Dic47C2211yescharacteristic
40C2×C8×Dic47C2211yescharacteristic
41C2×C8×Dic47C2211yescharacteristic
42#15689030C2211yescharacteristic
43#15689028C2211yescharacteristic
44#15689026C2211yescharacteristic
45#15689024C2211yescharacteristic
46#15689023—22no—
47#15689018—22no—
48#15688992C2211yescharacteristic
49#15688992C2211yescharacteristic
50C4×Dic47⋊C4—22no—
51C42×Dic47C2211yescharacteristic
52C22×SD16×C47C2211yescharacteristic
53C2×C4×Q8×C47C2211yescharacteristic
54C2×C4×D4×C47C2211yescharacteristic
55C4×SD16×C47—24no—
56C2×#106×C47C2211yescharacteristic
57C2×#103×C47C2211yescharacteristic
58C2×#102×C47C2211yescharacteristic
59C2×C4×C8×C47C2211yescharacteristic
60C22×Dic94C2311yescharacteristic
61C22×D188C2311yescharacteristic
62C22×C22⋊D47—22no—
63C22×C4×D47—21no—
64C2×#10625325—22no—
65C2×#10625325—22no—
66C2×#10625325C2×C414yescomplemented
67C2×#10625335—22no—
68C2×#10625335—22no—
69C2×#10625335C2×C414yescomplemented
70C2×#10625336—22no—
71C2×#10625336—22no—
72C2×#10625336C2×C414yescomplemented
73C22×C47⋊C8C2311yescharacteristic
74C22×C47⋊C8C2311yescharacteristic
75#13094022—22no—
76#13094022C2×C414yescomplemented
77#13094020—22no—
78#13094020C2×C414yes—
79C4×Dic94—22no—
80C4×Dic94—22no—
81C4×D188—22no—
82C4×D188—22no—
83C4×C22⋊D47—48no—
84C2×C4⋊Dic47C2311yescharacteristic
85C2×C4⋊Dic47C2311yescharacteristic
86C2×Dic47⋊C4—22no—
87C2×Dic47⋊C4—21no—
88C2×Dic47⋊C4—22no—
89C2×Dic47⋊C4—22no—
90C2×C22⋊Dic47—22no—
91C2×(C2×D47)⋊C4—21no—
92C2×(C2×D47)⋊C4—22no—
93C2×C4×Dic47—21no—
94C2×C4×Dic47D412yes—
95C2×C4×Dic47D412yes—
96C2×C4×Dic47C2311yescharacteristic
97C2×C4×Dic47C2311yescharacteristic
98#13093956—22no—
99#13093955—22no—
100#13093955—22no—
101#13093949—22no—
102#13093948—22no—
103#13093947—22no—
104#13093946—22no—
105#13093945—22no—
106#13093944—22no—
107C4×C47⋊C8D414yescomplemented
108C8×Dic47—22no—
109C8×Dic47—22no—
110C8×Dic47—22no—
111C8×Dic47C2×C414yescomplemented
112C47⋊#95—22no—
113C47⋊#95—22no—
114C22×Q8×C47C2311yescharacteristic
115C22×D4×C47C2311yescharacteristic
116C23×C4×C47—21no—
117C2×SD16×C47—22no—
118C2×SD16×C47—22no—
119C2×SD16×C47C2×C414yescomplemented
120C22×C8×C47C2311yescharacteristic
121C22×C8×C47C2311yescharacteristic
122C4×Q8×C47—22no—
123C4×Q8×C47—22no—
124C4×D4×C47—22no—
125C4×D4×C47—22no—
126C2×C4⋊C4×C47—21no—
127C2×C4⋊C4×C47C2311yescharacteristic
128C2×C4⋊C4×C47C2311yescharacteristic
129C2×C22⋊C4×C47—21no—
130C2×C42×C47—21no—
131C2×C42×C47C2311yescharacteristic
132#106×C47—22no—
133#103×C47—22no—
134#102×C47—22no—
135C4×C8×C47D414yescomplemented
136C24×C47—21no—
137C23×D47—21no—
138C2×Q8×C47—21no—
139C2×Q8×C47—21no—
140C2×Q8×C47—21no—
141C2×Q8×C47—21no—
142C2×Q8×C47—21no—
143C2×Q8×C47C22×C412yes—
144C2×D4×C47—21no—
145C2×D4×C47—21no—
146C2×D4×C47—21no—
147C2×D4×C47—21no—
148C2×D4×C47—21no—
149C2×D4×C47C22×C412yes—
150C22×C4×C47—21no—
151C22×C4×C47—21no—
152C22×C4×C47—21no—
153C22×C4×C47—42no—
154C22×C4×C47—44no—
155C22×C4×C47C2×D411yescharacteristic
156C22×C4×C47C2×D411yescharacteristic
157C22×C4×C47C2411yescharacteristic
158C2×C22⋊D47—28no—
159C2×C22⋊D47—44no—
160C2×C22⋊D47—44no—
161C22×Dic47—21no—
162C22×Dic47—21no—
163C22×Dic47—21no—
164C22×Dic47C2×D412yes—
165C22×Dic47C2×D412yes—
166C2×D188—21no—
167C2×D188—21no—
168C2×D188—21no—
169C2×D188—21no—
170C2×D188—21no—
171C2×D188C22×C412yes—
172C2×C4×D47—42no—
173C2×C4×D47—44no—
174C2×Dic94—21no—
175C2×Dic94—21no—
176C2×Dic94—21no—
177C2×Dic94—21no—
178C2×Dic94—21no—
179C2×Dic94C22×C412yes—
180SD16×C47—28no—
181C2×C8×C47—21no—
182C2×C8×C47—21no—
183C2×C8×C47C22×C412yes—
184C2×C8×C47C2×D412yes—
185C2×C8×C47C2×D412yes—
186C2×C8×C47#4112yes—
187C4⋊C4×C47—21no—
188C4⋊C4×C47—21no—
189C4⋊C4×C47—21no—
190C4⋊C4×C47—21no—
191C4⋊C4×C47—42no—
192C22⋊C4×C47—42no—
193C42×C47—42no—
194C42×C47C2×D412yes—
195C42×C47C2×D412yes—
196C22⋊Dic47—44no—
197#10625336—28no—
198#10625335—28no—
199(C2×D47)⋊C4—42no—
200(C2×D47)⋊C4—44no—
201C4⋊Dic47—21no—
202C4⋊Dic47—21no—
203C4⋊Dic47—21no—
204C4⋊Dic47—21no—
205Dic47⋊C4—28no—
206Dic47⋊C4—42no—
207Dic47⋊C4—44no—
208Dic47⋊C4—44no—
209C4×Dic47—21no—
210C4×Dic47—21no—
211C4×Dic47—24no—
212C4×Dic47—42no—
213C4×Dic47SD1618yescomplemented
214C4×Dic47C22×C412yes—
215C4×Dic47#4112yes—
216C2×C47⋊C8—21no—
217C2×C47⋊C8—21no—
218C2×C47⋊C8C22×C412yes—
219C2×C47⋊C8C2×D412yes—
220C2×C47⋊C8C2×D412yes—
221C2×C47⋊C8#4112yes—
222#10625325—28no—
223C23×C47—22no—
224C23×C47—41no—
225C23×C47—41no—
226C23×C47—42no—
227C23×C47—42no—
228C23×C47C22×D411yescharacteristic
229C22×D47—22no—
230C22×D47—41no—
231C22×D47—41no—
232C22×D47—42no—
233C22×D47—42no—
234Q8×C47—22no—
235Q8×C47—22no—
236Q8×C47—41no—
237Q8×C47—41no—
238D4×C47—22no—
239D4×C47—22no—
240D4×C47—41no—
241D4×C47—41no—
242C2×C4×C47—22no—
243C2×C4×C47—22no—
244C2×C4×C47—41no—
245C2×C4×C47—41no—
246C2×C4×C47—41no—
247C2×C4×C47—41no—
248C2×C4×C47—42no—
249C2×C4×C47—42no—
250C2×C4×C47—84no—
251C2×C4×C47#12712yes—
252C2×C4×C47#12712yes—
253C2×C4×C47C2×SD1614yes—
254C2×C4×C47C23×C411yescharacteristic
255C2×C4×C47C22×D411yescharacteristic
256C2×C4×C47C22×D411yescharacteristic
257C2×C4×C47C2×#4111yescharacteristic
258C2×C4×C47C2×#4111yescharacteristic
259C2×C4×C47C2×#4111yescharacteristic
260C22⋊D47—416no—
261C2×Dic47—22no—
262C2×Dic47—22no—
263C2×Dic47—22no—
264C2×Dic47—41no—
265C2×Dic47—41no—
266C2×Dic47—41no—
267C2×Dic47—41no—
268C2×Dic47—42no—
269C2×Dic47—42no—
270C2×Dic47C4×D414yes—
271C2×Dic47C2×SD1614yes—
272C2×Dic47C2×SD1614yes—
273C2×Dic47#13514yes—
274D188—22no—
275D188—22no—
276D188—41no—
277D188—41no—
278C4×D47—84no—
279Dic94—22no—
280Dic94—22no—
281Dic94—41no—
282Dic94—41no—
283C8×C47—22no—
284C8×C47C4×D414yescomplemented
285C47⋊C8—22no—
286C47⋊C8C4×D414yescomplemented
287C4×#492—471nomaximal, Sylow
288C22×C47—42no—
289C22×C47—44no—
290C22×C47—81no—
291C22×C47—81no—
292C22×C47—82no—
293C22×C47C2×C4×D411yescharacteristic
294C22×C47C2×#12711yescharacteristic
295C22×C47#53211yescharacteristic
296C22×C47C22×SD1612yes—
297C22×C47C2×#13512yes—
298C2×D47—42no—
299C2×D47—44no—
300C2×D47—81no—
301C2×D47—81no—
302C2×D47—82no—
303C4×C47—22no—
304C4×C47—42no—
305C4×C47—81no—
306C4×C47—81no—
307C4×C47#49214yescomplemented
308C4×C47C2×C4×D411yescharacteristic
309C4×C47C2×C4×D411yescharacteristic
310C4×C47#53211yescharacteristic
311C4×C47#53211yescharacteristic
312Dic47—24no—
313Dic47—42no—
314Dic47—81no—
315Dic47—81no—
316Dic47C4×SD1618yescomplemented
317C2×#492—471no—
318C2×C4×SD16—471no—
319C2×C4×SD16—471no—
320C2×C4×SD16—471no—
321C2×C4×SD16D4711yescharacteristic, complemented
322C4×C4⋊Q8—471no—
323C4×#127—471no—
324#1968—471no—
325#1968—471no—
326#1968—471no—
327#1968—471no—
328#1957—471no—
329#1956—471no—
330#1844—471no—
331C42×C8—471no—
332C2×C47—82no—
333C2×C47C4×#12711yescharacteristic
334C2×C47C2×C4×SD1612yes—
335C2×C47C2×#49211yescharacteristic
336C2×C47#314711yescharacteristic
337C2×C47#316712yes—
338D47—82no—
339C22×SD16—471no—
340C22×SD16—471no—
341C22×SD16—471no—
342C22×SD16C2×D4711yescharacteristic
343C2×C4⋊Q8—471no—
344C2×#127—471no—
345C2×C4×Q8—471no—
346C2×C4×Q8C2×D4711yescharacteristic
347C2×C4×D4—471no—
348C2×C4×D4—942no—
349C2×C4×D4C2×D4711yescharacteristic
350#492—478no—
351C4×SD16—24no—
352C4×SD16—944no—
353C4×SD16—944no—
354C4×SD16—944no—
355C2×#106—471no—
356C2×#106—471no—
357C2×#106—471no—
358C2×#106C2×D4711yescharacteristic
359C2×#103—471no—
360C2×#103—471no—
361C2×#103—471no—
362C2×#103C2×D4711yescharacteristic
363C2×#102—471no—
364C2×#102—471no—
365C2×#102—471no—
366C2×#102C2×D4711yescharacteristic
367C2×C4×C8—471no—
368C2×C4×C8—471no—
369C2×C4×C8—471no—
370C2×C4×C8—471no—
371C2×C4×C8—471no—
372C2×C4×C8C2×D4711yescharacteristic
373#391—471no—
374#391—471no—
375#390—471no—
376#390—471no—
377#390—942no—
378#389—942no—
379#384—471no—
380#384—471no—
381C4×C4⋊C4—942no—
382C43—471no—
383C47C4×#49211yescharacteristic, complemented, Sylow
384C22×Q8—471no—
385C22×Q8C22×D4711yescharacteristic
386C22×D4—471no—
387C22×D4—942no—
388C22×D4C22×D4711yescharacteristic
389C23×C4—21no—
390C23×C4—941no—
391C2×SD16—22no—
392C2×SD16—22no—
393C2×SD16—474no—
394C2×SD16—474no—
395C2×SD16—474no—
396C2×SD16—942no—
397C2×SD16—942no—
398C2×SD16—942no—
399C2×SD16—942no—
400C2×SD16—942no—
401C2×SD16—942no—
402C2×SD16C4×D4714yescomplemented
403C22×C8—471no—
404C22×C8—471no—
405C22×C8C22×D4711yescharacteristic
406C22×C8C22×D4711yescharacteristic
407C4⋊Q8—474no—
408C4⋊Q8—942no—
409#127—474no—
410#127—942no—
411C4×Q8—22no—
412C4×Q8—22no—
413C4×Q8—942no—
414C4×Q8—942no—
415C4×D4—22no—
416C4×D4—22no—
417C4×D4—942no—
418C4×D4—942no—
419C4×D4—1888no—
420C2×C4⋊C4—21no—
421C2×C4⋊C4—471no—
422C2×C4⋊C4—471no—
423C2×C4⋊C4—942no—
424C2×C4⋊C4—942no—
425C2×C4⋊C4—942no—
426C2×C4⋊C4—941no—
427C2×C4⋊C4C22×D4711yescharacteristic
428C2×C4⋊C4C22×D4711yescharacteristic
429C2×C22⋊C4—21no—
430C2×C22⋊C4—942no—
431C2×C22⋊C4—942no—
432C2×C22⋊C4—941no—
433C2×C42—21no—
434C2×C42—472no—
435C2×C42—471no—
436C2×C42—472no—
437C2×C42—471no—
438C2×C42—941no—
439C2×C42C22×D4711yescharacteristic
440#106—22no—
441#106—942no—
442#106—942no—
443#106—942no—
444#103—22no—
445#103—942no—
446#103—942no—
447#103—942no—
448#102—22no—
449#102—942no—
450#102—942no—
451#102—942no—
452C4×C8—474no—
453C4×C8—474no—
454C4×C8—942no—
455C4×C8—942no—
456C4×C8—942no—
457C4×C8C22⋊D4714yescomplemented
458#95—942no—
459#95—942no—
460C24—21no—
461C24—941no—
462C2×Q8—21no—
463C2×Q8—21no—
464C2×Q8—21no—
465C2×Q8—21no—
466C2×Q8—21no—
467C2×Q8—472no—
468C2×Q8—941no—
469C2×Q8—941no—
470C2×Q8—941no—
471C2×Q8—941no—
472C2×Q8—941no—
473C2×Q8C2×C4×D4712yes—
474C2×D4—21no—
475C2×D4—21no—
476C2×D4—21no—
477C2×D4—21no—
478C2×D4—21no—
479C2×D4—472no—
480C2×D4—948no—
481C2×D4—941no—
482C2×D4—941no—
483C2×D4—941no—
484C2×D4—941no—
485C2×D4—941no—
486C2×D4—1884no—
487C2×D4—1884no—
488C2×D4C2×C4×D4712yes—
489C22×C4—21no—
490C22×C4—21no—
491C22×C4—21no—
492C22×C4—42no—
493C22×C4—44no—
494C22×C4—472no—
495C22×C4—472no—
496C22×C4—941no—
497C22×C4—941no—
498C22×C4—941no—
499C22×C4—1882no—
500C22×C4—1884no—
501C22×C4D4×D4711yescharacteristic
502C22×C4D4×D4711yescharacteristic
503C22×C4C23×D4711yescharacteristic
504SD16—28no—
505SD16—948no—
506SD16—948no—
507SD16—948no—
508C2×C8—21no—
509C2×C8—21no—
510C2×C8—472no—
511C2×C8—472no—
512C2×C8—472no—
513C2×C8—472no—
514C2×C8—941no—
515C2×C8—941no—
516C2×C8C2×C4×D4712yes—
517C2×C8#1062535012yes—
518C2×C8C2×C22⋊D4712yes—
519C2×C8C2×C22⋊D4712yes—
520C4⋊C4—21no—
521C4⋊C4—21no—
522C4⋊C4—21no—
523C4⋊C4—21no—
524C4⋊C4—42no—
525C4⋊C4—948no—
526C4⋊C4—941no—
527C4⋊C4—941no—
528C4⋊C4—941no—
529C4⋊C4—941no—
530C4⋊C4—1884no—
531C4⋊C4—1884no—
532C4⋊C4—1882no—
533C22⋊C4—42no—
534C22⋊C4—1884no—
535C22⋊C4—1884no—
536C22⋊C4—1882no—
537C42—42no—
538C42—478no—
539C42—472no—
540C42—472no—
541C42—944no—
542C42—941no—
543C42—941no—
544C42—1882no—
545C42D4×D4712yes—
546C42C2×C22⋊D4712yes—
547C23—22no—
548C23—42no—
549C23—42no—
550C23—41no—
551C23—41no—
552C23—942no—
553C23—1882no—
554C23—1882no—
555C23—1881no—
556C23—1881no—
557C23C2×D4×D4711yescharacteristic
558Q8—22no—
559Q8—22no—
560Q8—41no—
561Q8—41no—
562Q8—942no—
563Q8—942no—
564Q8—1881no—
565Q8—1881no—
566D4—22no—
567D4—22no—
568D4—41no—
569D4—41no—
570D4—942no—
571D4—942no—
572D4—18816no—
573D4—1881no—
574D4—1881no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—84no—
584C2×C4—474no—
585C2×C4—474no—
586C2×C4—474no—
587C2×C4—474no—
588C2×C4—942no—
589C2×C4—942no—
590C2×C4—942no—
591C2×C4—3764no—
592C2×C4—1882no—
593C2×C4—1882no—
594C2×C4—1881no—
595C2×C4—1881no—
596C2×C4—1881no—
597C2×C4—1881no—
598C2×C4#1309402012yes—
599C2×C4#1309402012yes—
600C2×C4SD16×D4714yes—
601C2×C4C22×C4×D4711yescharacteristic
602C2×C4C2×D4×D4711yescharacteristic
603C2×C4C22×C22⋊D4711yescharacteristic
604C2×C4#41×D4711yescharacteristic
605C2×C4#41×D4711yescharacteristic
606C2×C4C2×#1062535011yescharacteristic
607C8—22no—
608C8—474no—
609C8—942no—
610C8C4×C22⋊D4714yescomplemented
611C22—42no—
612C22—44no—
613C22—81no—
614C22—81no—
615C22—82no—
616C22—3761no—
617C22—3761no—
618C22—3762no—
619C22—1882no—
620C22—1884no—
621C22C4×D4×D4711yescharacteristic
622C22C2×#1309402011yescharacteristic
623C22#1568967011yescharacteristic
624C22C2×SD16×D4712yes—
625C22#135×D4712yes—
626C4—22no—
627C4—42no—
628C4—81no—
629C4—81no—
630C4—478no—
631C4—944no—
632C4—3761no—
633C4—3761no—
634C4—1882no—
635C4#1568949414yesdirect product factor
636C4C4×D4×D4711yescharacteristic
637C4C2×C4×C22⋊D4711yescharacteristic
638C4#1568966911yescharacteristic
639C4#1568967011yescharacteristic
640C2—82no—
641C2—3762no—
642C2C4×#1309402011yescharacteristic
643C2C4×SD16×D4712yes—
644C2C2×#1568949411yescharacteristic
645C2#2025381211yescharacteristic
646C2#2025394412yes—
subgroupG/H
centerC22×C4D4×D47
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC2×C4×SD16×C47C2
socleC23×C47C22×D4
prime radicalC2×#13094020C22
cosocleC22×C4×C47C24
hypercenterC2×C4×SD16D47
hypocenterC47C4×#492
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22×C4 ◁ C2×C42 ◁ C2×C4×SD16

Complements

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

NH ≅ G/Ntype
2. C2×C4×#10625325C2semidirect
3. C2×C4×#10625336C2semidirect
4. C2×C4×#10625335C2semidirect
5. C4×#13094022C2semidirect
7. #20244244C2semidirect
8. #20244243C2semidirect
9. #20244240C2semidirect
10. #20244239C2semidirect
12. #20244188C2semidirect
13. #20243543C2semidirect
14. C4×C8×Dic47C2semidirect
15. C2×C4×SD16×C47C2semidirect
24. #15689494C4direct (N × H)
66. C2×#10625325C2×C4semidirect
69. C2×#10625335C2×C4semidirect
72. C2×#10625336C2×C4semidirect
76. #13094022C2×C4semidirect
107. C4×C47⋊C8D4semidirect
111. C8×Dic47C2×C4semidirect
119. C2×SD16×C47C2×C4semidirect
135. C4×C8×C47D4semidirect
213. C4×Dic47SD16semidirect
284. C8×C47C4×D4semidirect
286. C47⋊C8C4×D4semidirect
307. C4×C47#492semidirect
316. Dic47C4×SD16semidirect
321. C2×C4×SD16D47semidirect
383. C47C4×#492semidirect
402. C2×SD16C4×D47semidirect
457. C4×C8C22⋊D47semidirect
610. C8C4×C22⋊D47semidirect
635. C4#15689494direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#156894942
C2×C4×#106253252
C2×C4×#106253362
C2×C4×#106253352
C4×#130940222
C4×#130940202
#202442442
#202442432
#202442402
#202442392
#202441922
#202441882
#202435432
C4×C8×Dic472
C2×C4×SD16×C472
C4×#49247