Finite Group Databasebeta

← C4×#15689514 #26224825

Subgroups — C4×#15689514 #26224825

Order 12032.

1094 conjugacy classes of subgroups.

The subgroup lattice has 618 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689514C211yesmaximal, characteristic
2C2×C4×Dic188C211yesmaximal, characteristic, complemented
3C2×C4×#10625337C211yesmaximal, characteristic, complemented
4C2×C4×#10625335C211yesmaximal, characteristic
5C4×#13094022C211yesmaximal, characteristic, complemented
6C4×#13094011C211yesmaximal, characteristic
7#20244271C211yesmaximal, characteristic
8#20244270C211yesmaximal, characteristic
9#20244257C211yesmaximal, characteristic, complemented
10#20244256C211yesmaximal, characteristic, complemented
11#20244206C211yesmaximal, characteristic, complemented
12#20244203C211yesmaximal, characteristic
13#20243553C211yesmaximal, characteristic, complemented
14C4×#13093934C211yesmaximal, characteristic, complemented
15C2×C4×SD16×C47C211yesmaximal, characteristic
16C22×Dic188C2211yescharacteristic
17C22×#10625337C2211yescharacteristic
18C22×#10625335C2211yescharacteristic
19C2×#13094022C2211yescharacteristic
20C2×#13094011C2211yescharacteristic
21C2×C4×Dic94C2211yescharacteristic
22C2×C4×Dic94C2211yescharacteristic
23#15689514C418yesdirect product factor
24C4×Dic188—24no—
25C4×#10625337—24no—
26C4×#10625335—24no—
27C2×#13093958C2211yescharacteristic
28C2×#13093957C2211yescharacteristic
29C2×#13093955C2211yescharacteristic
30C2×#13093949C2211yescharacteristic
31C2×#13093949C2211yescharacteristic
32C2×#13093948C2211yescharacteristic
33C2×#13093947C2211yescharacteristic
34C2×#13093947C2211yescharacteristic
35C2×#13093945C2211yescharacteristic
36C2×#13093934C2211yescharacteristic
37C2×#13093934C2211yescharacteristic
38C2×#13093934C2211yescharacteristic
39C2×#13093934C2211yescharacteristic
40C2×C4×C47⋊C8C2211yescharacteristic
41#15689030C2211yescharacteristic
42#15689028C2211yescharacteristic
43#15689028C2211yescharacteristic
44#15689026C2211yescharacteristic
45#15689018—22no—
46#15689017—22no—
47#15688992C2211yescharacteristic
48#15688988C2211yescharacteristic
49C4×Dic47⋊C4—22no—
50C4×C22⋊Dic47—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×Dic94C2311yescharacteristic
62C2×Dic188—22no—
63C2×Dic188—22no—
64C2×Dic188C2×C414yescomplemented
65C2×#10625337—22no—
66C2×#10625337—22no—
67C2×#10625337C2×C414yescomplemented
68C2×#10625335—22no—
69C2×#10625335—22no—
70C2×#10625335C2×C414yes—
71C22×C47⋊C8C2311yescharacteristic
72C22×C47⋊C8C2311yescharacteristic
73#13094022—22no—
74#13094022C2×C414yescomplemented
75#13094011—22no—
76#13094011C2×C414yes—
77C4×Dic94—22no—
78C4×Dic94—22no—
79C4×Dic94—22no—
80C4×Dic94—22no—
81C2×C4⋊Dic47C2311yescharacteristic
82C2×C4⋊Dic47C2311yescharacteristic
83C2×Dic47⋊C4—22no—
84C2×Dic47⋊C4—21no—
85C2×Dic47⋊C4—21no—
86C2×Dic47⋊C4—22no—
87C2×C22⋊Dic47—22no—
88C2×C22⋊Dic47—22no—
89C2×C4×Dic47—21no—
90C2×C4×Dic47—21no—
91C2×C4×Dic47D412yes—
92C2×C4×Dic47D412yes—
93C2×C4×Dic47C2311yescharacteristic
94C2×C4×Dic47C2311yescharacteristic
95#13093958—22no—
96#13093957—22no—
97#13093955—22no—
98#13093949—22no—
99#13093949—22no—
100#13093948—22no—
101#13093947—22no—
102#13093947—22no—
103#13093945—22no—
104#13093934—22no—
105#13093934—22no—
106#13093934—22no—
107#13093934C2×C414yescomplemented
108C4×C47⋊C8D414yes—
109C47⋊#95—22no—
110C47⋊#95—22no—
111C47⋊#95—22no—
112C47⋊#95—22no—
113C22×Q8×C47C2311yescharacteristic
114C22×D4×C47C2311yescharacteristic
115C23×C4×C47—21no—
116C2×SD16×C47—22no—
117C2×SD16×C47—22no—
118C2×SD16×C47C2×C414yes—
119C22×C8×C47C2311yescharacteristic
120C22×C8×C47C2311yescharacteristic
121C4×Q8×C47—22no—
122C4×Q8×C47—22no—
123C4×D4×C47—22no—
124C4×D4×C47—22no—
125C2×C4⋊C4×C47—21no—
126C2×C4⋊C4×C47C2311yescharacteristic
127C2×C4⋊C4×C47C2311yescharacteristic
128C2×C22⋊C4×C47—21no—
129C2×C42×C47—21no—
130C2×C42×C47C2311yescharacteristic
131#106×C47—22no—
132#103×C47—22no—
133#102×C47—22no—
134C4×C8×C47D414yes—
135C24×C47—21no—
136C2×Q8×C47—21no—
137C2×Q8×C47—21no—
138C2×Q8×C47—21no—
139C2×Q8×C47—21no—
140C2×Q8×C47—21no—
141C2×Q8×C47C22×C412yes—
142C2×D4×C47—21no—
143C2×D4×C47—21no—
144C2×D4×C47—21no—
145C2×D4×C47—21no—
146C2×D4×C47—21no—
147C2×D4×C47C22×C412yes—
148C22×C4×C47—21no—
149C22×C4×C47—21no—
150C22×C4×C47—21no—
151C22×C4×C47—42no—
152C22×C4×C47—44no—
153C22×C4×C47C2×D411yescharacteristic
154C22×C4×C47C2×D411yescharacteristic
155C22×C4×C47C2411yescharacteristic
156C22×Dic47—21no—
157C22×Dic47—21no—
158C22×Dic47—21no—
159C22×Dic47—21no—
160C22×Dic47C2×D412yes—
161C22×Dic47C2×D412yes—
162C2×Dic94—21no—
163C2×Dic94—21no—
164C2×Dic94—21no—
165C2×Dic94—21no—
166C2×Dic94—21no—
167C2×Dic94—21no—
168C2×Dic94—21no—
169C2×Dic94—21no—
170C2×Dic94—21no—
171C2×Dic94—21no—
172C2×Dic94C22×C412yes—
173C2×Dic94C22×C412yes—
174SD16×C47—28no—
175C2×C8×C47—21no—
176C2×C8×C47—21no—
177C2×C8×C47C22×C412yes—
178C2×C8×C47C2×D412yes—
179C2×C8×C47C2×D412yes—
180C2×C8×C47#4112yes—
181C4⋊C4×C47—21no—
182C4⋊C4×C47—21no—
183C4⋊C4×C47—21no—
184C4⋊C4×C47—21no—
185C4⋊C4×C47—42no—
186C22⋊C4×C47—42no—
187C42×C47—42no—
188C42×C47C2×D412yes—
189C42×C47C2×D412yes—
190C22⋊Dic47—28no—
191C22⋊Dic47—44no—
192#10625337—28no—
193#10625335—28no—
194C4⋊Dic47—21no—
195C4⋊Dic47—21no—
196C4⋊Dic47—21no—
197C4⋊Dic47—21no—
198Dic47⋊C4—28no—
199Dic47⋊C4—44no—
200Dic47⋊C4—42no—
201Dic47⋊C4—42no—
202C4×Dic47—24no—
203C4×Dic47—21no—
204C4×Dic47—21no—
205C4×Dic47—42no—
206C4×Dic47—42no—
207C4×Dic47—42no—
208C4×Dic47C22×C412yes—
209C4×Dic47#4112yes—
210C2×C47⋊C8—21no—
211C2×C47⋊C8—21no—
212C2×C47⋊C8C22×C412yes—
213C2×C47⋊C8C2×D412yes—
214C2×C47⋊C8C2×D412yes—
215C2×C47⋊C8#4112yes—
216Dic188—28no—
217C23×C47—22no—
218C23×C47—41no—
219C23×C47—41no—
220C23×C47—42no—
221C23×C47—42no—
222C23×C47C22×D411yescharacteristic
223Q8×C47—22no—
224Q8×C47—22no—
225Q8×C47—41no—
226Q8×C47—41no—
227D4×C47—22no—
228D4×C47—22no—
229D4×C47—41no—
230D4×C47—41no—
231C2×C4×C47—22no—
232C2×C4×C47—22no—
233C2×C4×C47—41no—
234C2×C4×C47—41no—
235C2×C4×C47—41no—
236C2×C4×C47—41no—
237C2×C4×C47—42no—
238C2×C4×C47—42no—
239C2×C4×C47—84no—
240C2×C4×C47#12712yes—
241C2×C4×C47#12712yes—
242C2×C4×C47#13714yes—
243C2×C4×C47C23×C411yescharacteristic
244C2×C4×C47C22×D411yescharacteristic
245C2×C4×C47C22×D411yescharacteristic
246C2×C4×C47C2×#4111yescharacteristic
247C2×C4×C47C2×#4111yescharacteristic
248C2×C4×C47C2×#4111yescharacteristic
249C2×Dic47—22no—
250C2×Dic47—22no—
251C2×Dic47—22no—
252C2×Dic47—22no—
253C2×Dic47—22no—
254C2×Dic47—22no—
255C2×Dic47—22no—
256C2×Dic47—41no—
257C2×Dic47—41no—
258C2×Dic47—41no—
259C2×Dic47—41no—
260C2×Dic47—41no—
261C2×Dic47—41no—
262C2×Dic47—42no—
263C2×Dic47—42no—
264C2×Dic47C4×D414yes—
265Dic94—22no—
266Dic94—22no—
267Dic94—22no—
268Dic94—22no—
269Dic94—41no—
270Dic94—41no—
271Dic94—41no—
272Dic94—41no—
273C8×C47—22no—
274C8×C47C4×D414yes—
275C47⋊C8—22no—
276C47⋊C8C4×D414yes—
277C4×#497—471nomaximal, Sylow
278C22×C47—42no—
279C22×C47—44no—
280C22×C47—81no—
281C22×C47—81no—
282C22×C47—82no—
283C22×C47C2×C4×D411yescharacteristic
284C22×C47C2×#12711yescharacteristic
285C22×C47#53211yescharacteristic
286C22×C47C2×#13712yes—
287C22×C47#57512yes—
288C4×C47—22no—
289C4×C47—42no—
290C4×C47—81no—
291C4×C47—81no—
292C4×C47#49714yescomplemented
293C4×C47C2×C4×D411yescharacteristic
294C4×C47C2×C4×D411yescharacteristic
295C4×C47#53211yescharacteristic
296C4×C47#53211yescharacteristic
297Dic47—24no—
298Dic47—42no—
299Dic47—42no—
300Dic47—42no—
301Dic47—81no—
302Dic47—81no—
303C2×#497—471no—
304C2×C4×Q16—471no—
305C2×C4×Q16—471no—
306C2×C4×SD16—471no—
307C2×C4×SD16D4711yescharacteristic
308C4×C4⋊Q8—471no—
309C4×#124—471no—
310#1973—471no—
311#1973—471no—
312#1971—471no—
313#1971—471no—
314#1960—471no—
315#1959—471no—
316#1847—471no—
317C4×#97—471no—
318C2×C47—82no—
319C2×C47C4×#12711yescharacteristic
320C2×C47C4×#13712yes—
321C2×C47C2×#49711yescharacteristic
322C2×C47#315011yescharacteristic
323C2×C47#317212yes—
324C22×Q16—471no—
325C22×Q16—471no—
326C22×SD16—471no—
327C22×SD16C2×D4711yescharacteristic
328C2×C4⋊Q8—471no—
329C2×#124—471no—
330C2×C4×Q8—471no—
331C2×C4×Q8—471no—
332C2×C4×Q8C2×D4711yescharacteristic
333C2×C4×D4C2×D4711yescharacteristic
334#497—478no—
335C4×Q16—944no—
336C4×Q16—944no—
337C4×SD16—24no—
338C4×SD16—944no—
339C2×#107—471no—
340C2×#107—471no—
341C2×#106—471no—
342C2×#106C2×D4711yescharacteristic
343C2×#103—471no—
344C2×#103—471no—
345C2×#103—471no—
346C2×#103—471no—
347C2×#103—471no—
348C2×#103C2×D4711yescharacteristic
349C2×#102—471no—
350C2×#102C2×D4711yescharacteristic
351C2×#97—471no—
352C2×#97—471no—
353C2×#97—471no—
354C2×#97—471no—
355C2×C4×C8—471no—
356C2×C4×C8C2×D4711yescharacteristic
357#391—471no—
358#391—471no—
359#391—471no—
360#390—471no—
361#389—942no—
362#388—942no—
363#384—471no—
364#382—471no—
365C4×C4⋊C4—942no—
366C4×C22⋊C4—942no—
367C43—471no—
368C47C4×#49711yescharacteristic, complemented, Sylow
369C22×Q8—471no—
370C22×Q8—471no—
371C22×Q8C22×D4711yescharacteristic
372C22×D4C22×D4711yescharacteristic
373C23×C4—21no—
374C2×Q16—474no—
375C2×Q16—474no—
376C2×Q16—942no—
377C2×Q16—942no—
378C2×Q16—942no—
379C2×Q16—942no—
380C2×SD16—22no—
381C2×SD16—22no—
382C2×SD16—474no—
383C2×SD16—942no—
384C2×SD16—942no—
385C2×SD16C4×D4714yes—
386C22×C8—471no—
387C22×C8—471no—
388C22×C8C22×D4711yescharacteristic
389C22×C8C22×D4711yescharacteristic
390C4⋊Q8—474no—
391C4⋊Q8—942no—
392#124—474no—
393#124—942no—
394C4×Q8—22no—
395C4×Q8—22no—
396C4×Q8—942no—
397C4×Q8—942no—
398C4×Q8—942no—
399C4×Q8—942no—
400C4×D4—22no—
401C4×D4—22no—
402C2×C4⋊C4—21no—
403C2×C4⋊C4—471no—
404C2×C4⋊C4—471no—
405C2×C4⋊C4—942no—
406C2×C4⋊C4—942no—
407C2×C4⋊C4—941no—
408C2×C4⋊C4—941no—
409C2×C4⋊C4C22×D4711yescharacteristic
410C2×C4⋊C4C22×D4711yescharacteristic
411C2×C22⋊C4—21no—
412C2×C22⋊C4—942no—
413C2×C22⋊C4—942no—
414C2×C42—21no—
415C2×C42—471no—
416C2×C42—472no—
417C2×C42—471no—
418C2×C42—472no—
419C2×C42—941no—
420C2×C42—941no—
421C2×C42C22×D4711yescharacteristic
422#107—942no—
423#107—942no—
424#106—22no—
425#106—942no—
426#103—22no—
427#103—942no—
428#103—942no—
429#103—942no—
430#103—942no—
431#103—942no—
432#102—22no—
433#102—942no—
434#97—474no—
435#97—942no—
436#97—942no—
437#97—942no—
438C4×C8—474no—
439C4×C8C22⋊D4714yes—
440#95—942no—
441#95—942no—
442#95—942no—
443#95—942no—
444C24—21no—
445C2×Q8—21no—
446C2×Q8—21no—
447C2×Q8—21no—
448C2×Q8—21no—
449C2×Q8—21no—
450C2×Q8—472no—
451C2×Q8—472no—
452C2×Q8—941no—
453C2×Q8—941no—
454C2×Q8—941no—
455C2×Q8—941no—
456C2×Q8—941no—
457C2×Q8—941no—
458C2×Q8—941no—
459C2×Q8—941no—
460C2×Q8—941no—
461C2×Q8—941no—
462C2×Q8C2×C4×D4712yes—
463C2×D4—21no—
464C2×D4—21no—
465C2×D4—21no—
466C2×D4—21no—
467C2×D4—21no—
468C2×D4C2×C4×D4712yes—
469C22×C4—21no—
470C22×C4—21no—
471C22×C4—21no—
472C22×C4—42no—
473C22×C4—44no—
474C22×C4—472no—
475C22×C4—472no—
476C22×C4—941no—
477C22×C4—941no—
478C22×C4—941no—
479C22×C4—941no—
480C22×C4D4×D4711yescharacteristic
481C22×C4D4×D4711yescharacteristic
482C22×C4C23×D4711yescharacteristic
483Q16—948no—
484Q16—948no—
485SD16—28no—
486SD16—948no—
487C2×C8—21no—
488C2×C8—21no—
489C2×C8—472no—
490C2×C8—472no—
491C2×C8—472no—
492C2×C8—472no—
493C2×C8—941no—
494C2×C8—941no—
495C2×C8C2×C4×D4712yes—
496C2×C8#1062535012yes—
497C2×C8C2×C22⋊D4712yes—
498C2×C8C2×C22⋊D4712yes—
499C4⋊C4—21no—
500C4⋊C4—21no—
501C4⋊C4—21no—
502C4⋊C4—21no—
503C4⋊C4—42no—
504C4⋊C4—948no—
505C4⋊C4—941no—
506C4⋊C4—941no—
507C4⋊C4—941no—
508C4⋊C4—941no—
509C4⋊C4—1884no—
510C4⋊C4—1882no—
511C4⋊C4—1882no—
512C22⋊C4—42no—
513C22⋊C4—948no—
514C22⋊C4—1884no—
515C42—42no—
516C42—472no—
517C42—472no—
518C42—944no—
519C42—941no—
520C42—941no—
521C42—1882no—
522C42—1882no—
523C42—1882no—
524C42D4×D4712yes—
525C42C2×C22⋊D4712yes—
526C23—22no—
527C23—42no—
528C23—41no—
529C23—42no—
530C23—41no—
531C23C2×D4×D4711yescharacteristic
532Q8—22no—
533Q8—22no—
534Q8—41no—
535Q8—41no—
536Q8—942no—
537Q8—942no—
538Q8—942no—
539Q8—942no—
540Q8—1881no—
541Q8—1881no—
542Q8—1881no—
543Q8—1881no—
544D4—22no—
545D4—22no—
546D4—41no—
547D4—41no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—42no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—42no—
556C2×C4—84no—
557C2×C4—474no—
558C2×C4—942no—
559C2×C4—942no—
560C2×C4—942no—
561C2×C4—942no—
562C2×C4—942no—
563C2×C4—942no—
564C2×C4—942no—
565C2×C4—1881no—
566C2×C4—1881no—
567C2×C4—1882no—
568C2×C4—1882no—
569C2×C4—1881no—
570C2×C4—1881no—
571C2×C4—1881no—
572C2×C4—1881no—
573C2×C4#1309402012yes—
574C2×C4#1309402012yes—
575C2×C4#1309405314yes—
576C2×C4C22×C4×D4711yescharacteristic
577C2×C4C2×D4×D4711yescharacteristic
578C2×C4C22×C22⋊D4711yescharacteristic
579C2×C4#41×D4711yescharacteristic
580C2×C4#41×D4711yescharacteristic
581C2×C4C2×#1062535011yescharacteristic
582C8—22no—
583C8—474no—
584C8—942no—
585C8C4×C22⋊D4714yes—
586C22—44no—
587C22—42no—
588C22—82no—
589C22—81no—
590C22—81no—
591C22C4×D4×D4711yescharacteristic
592C22C2×#1309402011yescharacteristic
593C22#1568967011yescharacteristic
594C22C2×#1309405312yes—
595C22#1568992012yes—
596C4—22no—
597C4—42no—
598C4—81no—
599C4—81no—
600C4—944no—
601C4—3761no—
602C4—3761no—
603C4—1882no—
604C4—1882no—
605C4—1882no—
606C4#1568951414yesdirect product factor
607C4C4×D4×D4711yescharacteristic
608C4C2×C4×C22⋊D4711yescharacteristic
609C4#1568966911yescharacteristic
610C4#1568967011yescharacteristic
611C2—82no—
612C2C4×#1309402011yescharacteristic
613C2C4×#1309405312yes—
614C2C2×#1568951411yescharacteristic
615C2#2025383111yescharacteristic
616C2#2025399012yes—
subgroupG/H
centerC22×C4D4×D47
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC2×C4×SD16×C47C2
socleC23×C47C22×D4
prime radicalC22×D4×C47C23
cosocleC22×C4×C47C24
hypercenterC2×C4×SD16D47
hypocenterC47C4×#497
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×Dic188C2semidirect
3. C2×C4×#10625337C2semidirect
5. C4×#13094022C2semidirect
9. #20244257C2semidirect
10. #20244256C2semidirect
11. #20244206C2semidirect
13. #20243553C2semidirect
14. C4×#13093934C2semidirect
23. #15689514C4direct (N × H)
64. C2×Dic188C2×C4semidirect
67. C2×#10625337C2×C4semidirect
74. #13094022C2×C4semidirect
107. #13093934C2×C4semidirect
292. C4×C47#497semidirect
368. C47C4×#497semidirect
606. C4#15689514direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#156895142
C2×C4×Dic1882
C2×C4×#106253372
C2×C4×#106253352
C4×#130940222
C4×#130940112
#202442712
#202442702
#202442572
#202442562
#202442062
#202442032
#202435532
C4×#130939342
C2×C4×SD16×C472
C4×#49747