Finite Group Databasebeta

← C2×#20244615 #26243917

Subgroups — C2×#20244615 #26243917

Order 12032.

1498 conjugacy classes of subgroups.

The subgroup lattice has 678 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689701C211yesmaximal, characteristic
2C22×#13093951C211yesmaximal, characteristic, complemented
3C22×#13093950C211yesmaximal, characteristic, complemented
4C2×#15689122C211yesmaximal, characteristic, complemented
5C2×#15688988C211yesmaximal, characteristic, complemented
6#20244615C218yesmaximal, direct product factor
7C2×#15688891C211yesmaximal, characteristic, complemented
8C2×C47×#417C211yesmaximal, characteristic, complemented
9C22×#10625350C2211yescharacteristic
10C22×#10625329C2211yescharacteristic
11#15689701—22no—
12#15689701—22no—
13#15689701—22no—
14#15689701C2214yes—
15C2×#13094003—21no—
16C2×#13094000—21no—
17C2×#13093998—21no—
18C2×#13093996—22no—
19C2×#13093995—21no—
20C22×C4⋊Dic47C2211yescharacteristic
21C22×C4×Dic47C2211yescharacteristic
22C2×#13093951—24no—
23C2×#13093951C2214yescomplemented
24C2×#13093950—24no—
25C2×#13093950C2214yescomplemented
26#15689122—22no—
27#15689122C2214yescomplemented
28C2×#13093947—21no—
29C2×#13093944—21no—
30#15688988—22no—
31#15688988C2214yescomplemented
32C2×C47⋊#95—22no—
33#15688891C2214yescomplemented
34C22×#41×C47C2211yescharacteristic
35C22×M16×C47C2211yescharacteristic
36C22×C4⋊C4×C47C2211yescharacteristic
37C47×#417—22no—
38C47×#417C2214yescomplemented
39C2×#103×C47—21no—
40C2×#102×C47—21no—
41C2×#10625350—21no—
42C2×#10625350—21no—
43C2×#10625350—22no—
44C2×#10625350—22no—
45C2×#10625350—22no—
46C2×#10625350—22no—
47C2×#10625350D412yes—
48C2×#10625350C2312yes—
49C22×Dic94D411yescharacteristic
50C22×D188D411yescharacteristic
51C22×C22⋊D47—21no—
52C22×C22⋊D47—22no—
53C23×Dic47—21no—
54C23×Dic47D412yes—
55C22×C4×D47—21no—
56C2×#10625329—21no—
57C2×#10625329—22no—
58C2×#10625329—22no—
59C2×#10625329C2312yes—
60C22×C47⋊C8—21no—
61#13094003—24no—
62#13094003—24no—
63#13094000—24no—
64#13094000—24no—
65#13093998—24no—
66#13093998—24no—
67#13093996—28no—
68#13093996—28no—
69#13093995—24no—
70#13093995—24no—
71C2×C4⋊Dic47—21no—
72C2×C4⋊Dic47—21no—
73C2×C4⋊Dic47—21no—
74C2×C4⋊Dic47—21no—
75C2×C4⋊Dic47D412yes—
76C2×C4⋊Dic47C2312yes—
77C2×Dic47⋊C4—22no—
78C2×C22⋊Dic47—22no—
79C2×(C2×D47)⋊C4—22no—
80C2×C4×Dic47—21no—
81C2×C4×Dic47—22no—
82C2×C4×Dic47D414yescomplemented
83C2×C4×Dic47C2312yes—
84#13093951—28no—
85#13093950—28no—
86#13093947—24no—
87#13093944—24no—
88C47⋊#95—28no—
89C2×#41×C47—21no—
90C2×#41×C47—21no—
91C2×#41×C47—22no—
92C2×#41×C47—22no—
93C2×#41×C47—22no—
94C2×#41×C47—22no—
95C2×#41×C47D412yes—
96C2×#41×C47C2312yes—
97C22×Q8×C47D411yescharacteristic
98C22×D4×C47—21no—
99C22×D4×C47D411yescharacteristic
100C23×C4×C47—21no—
101C23×C4×C47—21no—
102C23×C4×C47C2311yescharacteristic
103C2×M16×C47—21no—
104C2×M16×C47—22no—
105C2×M16×C47—22no—
106C2×M16×C47C2312yes—
107C22×C8×C47—21no—
108C2×C4⋊C4×C47—21no—
109C2×C4⋊C4×C47—21no—
110C2×C4⋊C4×C47—21no—
111C2×C4⋊C4×C47—21no—
112C2×C4⋊C4×C47—21no—
113C2×C4⋊C4×C47C2312yes—
114#103×C47—24no—
115#102×C47—24no—
116C24×C47—21no—
117C24×C47C2×D411yescharacteristic
118C23×D47—21no—
119#41×C47—24no—
120#41×C47—24no—
121#41×C47—22no—
122#41×C47—22no—
123#41×C47—22no—
124#41×C47—22no—
125#41×C47—42no—
126#41×C47—42no—
127#41×C47—42no—
128#41×C47—42no—
129C2×Q8×C47—21no—
130C2×Q8×C47—22no—
131C2×Q8×C47—22no—
132C2×Q8×C47C2×D412yes—
133C2×D4×C47—22no—
134C2×D4×C47—22no—
135C2×D4×C47—21no—
136C2×D4×C47—22no—
137C2×D4×C47—22no—
138C2×D4×C47—42no—
139C2×D4×C47—42no—
140C2×D4×C47C2×D412yes—
141C22×C4×C47—21no—
142C22×C4×C47—22no—
143C22×C4×C47—22no—
144C22×C4×C47—22no—
145C22×C4×C47—42no—
146C22×C4×C47—42no—
147C22×C4×C47—44no—
148C22×C4×C47—41no—
149C22×C4×C47—41no—
150C22×C4×C47C2×D411yescharacteristic
151C22×C4×C47C2×D411yescharacteristic
152C22×C4×C47C2×D411yescharacteristic
153C22×C4×C47C2×D411yescharacteristic
154C22×C4×C47C2×D411yescharacteristic
155C22×C4×C47C2×D412yes—
156C22×C4×C47#4112yes—
157C22×C4×C47#4112yes—
158C22×C4×C47#4112yes—
159C22×C4×C47C2411yescharacteristic
160C2×C22⋊D47—22no—
161C2×C22⋊D47—22no—
162C2×C22⋊D47—28no—
163C2×C22⋊D47—42no—
164C2×C22⋊D47—42no—
165C2×C22⋊D47—84no—
166C22×Dic47—22no—
167C22×Dic47—21no—
168C22×Dic47—24no—
169C22×Dic47—22no—
170C22×Dic47—22no—
171C22×Dic47—42no—
172C22×Dic47—42no—
173C22×Dic47—44no—
174C22×Dic47C2×D414yes—
175#10625350—24no—
176#10625350—24no—
177#10625350—22no—
178#10625350—22no—
179#10625350—22no—
180#10625350—22no—
181#10625350—42no—
182#10625350—42no—
183#10625350—42no—
184#10625350—42no—
185C2×D188—22no—
186C2×D188—22no—
187C2×D188—21no—
188C2×D188C2×D412yes—
189C2×C4×D47—22no—
190C2×C4×D47—22no—
191C2×C4×D47—42no—
192C2×C4×D47—42no—
193C2×Dic94—22no—
194C2×Dic94—22no—
195C2×Dic94—21no—
196C2×Dic94C2×D412yes—
197M16×C47—22no—
198M16×C47—22no—
199M16×C47—42no—
200C2×C8×C47—22no—
201C2×C8×C47—42no—
202C4⋊C4×C47—22no—
203C4⋊C4×C47—22no—
204C4⋊C4×C47—41no—
205C4⋊C4×C47—41no—
206C22⋊Dic47—28no—
207(C2×D47)⋊C4—28no—
208C4⋊Dic47—22no—
209C4⋊Dic47—22no—
210C4⋊Dic47—22no—
211C4⋊Dic47—22no—
212Dic47⋊C4—28no—
213C4×Dic47—24no—
214C4×Dic47—22no—
215C4×Dic47C2×D414yescomplemented
216#10625329—22no—
217#10625329—22no—
218#10625329—42no—
219C2×C47⋊C8—22no—
220C2×C47⋊C8—42no—
221C23×C47—22no—
222C23×C47—22no—
223C23×C47—22no—
224C23×C47—42no—
225C23×C47—42no—
226C23×C47—42no—
227C23×C47#12011yescharacteristic
228C23×C47#12011yescharacteristic
229C23×C47#12112yes—
230C23×C47#12412yes—
231C23×C47C22×D411yescharacteristic
232C22×D47—22no—
233C22×D47—42no—
234C22×D47—42no—
235C22×D47—42no—
236Q8×C47—22no—
237Q8×C47—22no—
238Q8×C47—42no—
239D4×C47—22no—
240D4×C47—22no—
241D4×C47—42no—
242D4×C47—42no—
243D4×C47—44no—
244D4×C47—42no—
245C2×C4×C47—22no—
246C2×C4×C47—42no—
247C2×C4×C47—42no—
248C2×C4×C47—42no—
249C2×C4×C47—44no—
250C2×C4×C47—44no—
251C2×C4×C47—42no—
252C2×C4×C47—82no—
253C2×C4×C47—82no—
254C2×C4×C47#12011yescharacteristic
255C2×C4×C47#12011yescharacteristic
256C2×C4×C47#12112yes—
257C2×C4×C47#12112yes—
258C2×C4×C47#12112yes—
259C2×C4×C47#12112yes—
260C2×C4×C47#12112yes—
261C2×C4×C47#12312yes—
262C2×C4×C47#12312yes—
263C2×C4×C47#12312yes—
264C2×C4×C47#12312yes—
265C2×C4×C47#12412yes—
266C2×C4×C47C22×D411yescharacteristic
267C2×C4×C47C22×D411yescharacteristic
268C2×C4×C47C22×D411yescharacteristic
269C2×C4×C47C2×#4111yescharacteristic
270C2×C4×C47C2×#4111yescharacteristic
271C2×C4×C47C2×#4111yescharacteristic
272C22⋊D47—42no—
273C22⋊D47—42no—
274C22⋊D47—44no—
275C22⋊D47—88no—
276C2×Dic47—22no—
277C2×Dic47—24no—
278C2×Dic47—24no—
279C2×Dic47—42no—
280C2×Dic47—42no—
281C2×Dic47—48no—
282C2×Dic47—42no—
283C2×Dic47—44no—
284C2×Dic47—44no—
285C2×Dic47—82no—
286D188—22no—
287D188—22no—
288D188—42no—
289C4×D47—42no—
290C4×D47—42no—
291C4×D47—44no—
292Dic94—22no—
293Dic94—22no—
294Dic94—42no—
295C8×C47—42no—
296C47⋊C8—42no—
297C2×#2040—471nomaximal, Sylow
298C22×C47—21no—
299C22×C47—21no—
300C22×C47—21no—
301C22×C47—21no—
302C22×C47—24no—
303C22×C47—24no—
304C22×C47—42no—
305C22×C47—42no—
306C22×C47—42no—
307C22×C47—84no—
308C22×C47#39412yes—
309C22×C47#39412yes—
310C22×C47#45412yes—
311C22×C47#45512yes—
312C22×C47C2×#12011yescharacteristic
313C22×C47C2×#12111yescharacteristic
314C22×C47C2×#12411yescharacteristic
315C2×D47—42no—
316C2×D47—42no—
317C2×D47—42no—
318C2×D47—84no—
319C4×C47—42no—
320C4×C47—82no—
321C4×C47#39412yes—
322C4×C47#39412yes—
323C4×C47C2×#12111yescharacteristic
324C4×C47C2×#12111yescharacteristic
325C4×C47C2×#12311yescharacteristic
326C4×C47C2×#12311yescharacteristic
327Dic47—44no—
328Dic47—42no—
329Dic47—44no—
330C2×#537—471no—
331C22×#104—471no—
332C22×#104—471no—
333C2×#417—471no—
334C2×#417D4711yescharacteristic, complemented
335C2×#382—471no—
336#2040—478no—
337C2×#343—471no—
338C2×C47—21no—
339C2×C47—21no—
340C2×C47—22no—
341C2×C47—82no—
342C2×C47#204014yescomplemented
343C2×C47C2×#39411yescharacteristic
344C2×C47C2×#45511yescharacteristic
345C2×C47C2×#45411yescharacteristic
346D47—82no—
347C22×#41—471no—
348C22×#41C2×D4711yescharacteristic
349C22×M16—471no—
350C22×M16C2×D4711yescharacteristic
351#537—474no—
352#537—942no—
353#537—942no—
354#537—942no—
355C2×C22⋊Q8—941no—
356C2×C22⋊Q8—941no—
357C2×#121—941no—
358C2×#121—942no—
359C2×#121—941no—
360C22×C4⋊C4—471no—
361C22×C4⋊C4C2×D4711yescharacteristic
362C22×C42—471no—
363C2×#104—474no—
364C2×#104—474no—
365C2×#104—944no—
366C2×#104—944no—
367#417—22no—
368#417—474no—
369#417—942no—
370#417C2×D4714yescomplemented
371C2×#103—21no—
372C2×#103—941no—
373C2×#102—21no—
374C2×#102—941no—
375#382—474no—
376#382—942no—
377C2×#95—942no—
378#343—474no—
379C47C2×#204011yescharacteristic, complemented, Sylow
380C2×#41—21no—
381C2×#41—21no—
382C2×#41—22no—
383C2×#41—22no—
384C2×#41—22no—
385C2×#41—22no—
386C2×#41—472no—
387C2×#41—472no—
388C2×#41—941no—
389C2×#41—941no—
390C2×#41—942no—
391C2×#41—942no—
392C2×#41—942no—
393C2×#41—942no—
394C2×#41C22⋊D4712yes—
395C2×#41C22×D4712yes—
396C22×Q8—471no—
397C22×Q8C22⋊D4711yescharacteristic
398C22×D4—21no—
399C22×D4—471no—
400C22×D4—942no—
401C22×D4—941no—
402C22×D4C22⋊D4711yescharacteristic
403C23×C4—21no—
404C23×C4—21no—
405C23×C4—472no—
406C23×C4—941no—
407C23×C4—941no—
408C23×C4C22×D4711yescharacteristic
409C2×M16—21no—
410C2×M16—22no—
411C2×M16—22no—
412C2×M16—472no—
413C2×M16—941no—
414C2×M16—942no—
415C2×M16—942no—
416C2×M16C22×D4712yes—
417C22×C8—21no—
418C22×C8—941no—
419C22⋊Q8—944no—
420C22⋊Q8—944no—
421C22⋊Q8—944no—
422C22⋊Q8—944no—
423#121—944no—
424#121—944no—
425#121—948no—
426#121—944no—
427#121—948no—
428#121—944no—
429C2×C4⋊C4—21no—
430C2×C4⋊C4—21no—
431C2×C4⋊C4—21no—
432C2×C4⋊C4—21no—
433C2×C4⋊C4—21no—
434C2×C4⋊C4—472no—
435C2×C4⋊C4—472no—
436C2×C4⋊C4—942no—
437C2×C4⋊C4—941no—
438C2×C4⋊C4—941no—
439C2×C4⋊C4—941no—
440C2×C4⋊C4—941no—
441C2×C4⋊C4C22×D4712yes—
442C2×C22⋊C4—942no—
443C2×C22⋊C4—942no—
444C2×C42—474no—
445C2×C42—472no—
446C2×C42—942no—
447C2×C42—941no—
448#104—948no—
449#104—948no—
450#103—24no—
451#103—944no—
452#102—24no—
453#102—944no—
454#95—948no—
455C24—21no—
456C24—941no—
457C24D4×D4711yescharacteristic
458#41—24no—
459#41—24no—
460#41—22no—
461#41—22no—
462#41—22no—
463#41—22no—
464#41—42no—
465#41—42no—
466#41—42no—
467#41—42no—
468#41—944no—
469#41—944no—
470#41—942no—
471#41—942no—
472#41—942no—
473#41—942no—
474#41—1882no—
475#41—1882no—
476#41—1882no—
477#41—1882no—
478C2×Q8—21no—
479C2×Q8—22no—
480C2×Q8—22no—
481C2×Q8—472no—
482C2×Q8—941no—
483C2×Q8—942no—
484C2×Q8—942no—
485C2×Q8C2×C22⋊D4712yes—
486C2×D4—22no—
487C2×D4—22no—
488C2×D4—21no—
489C2×D4—22no—
490C2×D4—22no—
491C2×D4—42no—
492C2×D4—42no—
493C2×D4—472no—
494C2×D4—948no—
495C2×D4—942no—
496C2×D4—942no—
497C2×D4—941no—
498C2×D4—942no—
499C2×D4—942no—
500C2×D4—3764no—
501C2×D4—1882no—
502C2×D4—1882no—
503C2×D4C2×C22⋊D4712yes—
504C22×C4—21no—
505C22×C4—22no—
506C22×C4—22no—
507C22×C4—22no—
508C22×C4—42no—
509C22×C4—42no—
510C22×C4—44no—
511C22×C4—41no—
512C22×C4—41no—
513C22×C4—474no—
514C22×C4—942no—
515C22×C4—942no—
516C22×C4—941no—
517C22×C4—944no—
518C22×C4—942no—
519C22×C4—942no—
520C22×C4—942no—
521C22×C4—1884no—
522C22×C4—1882no—
523C22×C4—1882no—
524C22×C4—1882no—
525C22×C4—1882no—
526C22×C4#1062535012yes—
527C22×C4#1062535012yes—
528C22×C4D4×D4711yescharacteristic
529C22×C4D4×D4711yescharacteristic
530C22×C4D4×D4711yescharacteristic
531C22×C4D4×D4712yes—
532C22×C4#1062535212yes—
533C22×C4C2×C22⋊D4711yescharacteristic
534C22×C4C2×C22⋊D4711yescharacteristic
535C22×C4C23×D4711yescharacteristic
536M16—22no—
537M16—22no—
538M16—42no—
539M16—942no—
540M16—942no—
541M16—1882no—
542C2×C8—22no—
543C2×C8—42no—
544C2×C8—942no—
545C2×C8—1882no—
546C4⋊C4—22no—
547C4⋊C4—22no—
548C4⋊C4—41no—
549C4⋊C4—41no—
550C4⋊C4—948no—
551C4⋊C4—942no—
552C4⋊C4—942no—
553C4⋊C4—942no—
554C4⋊C4—942no—
555C22⋊C4—948no—
556C22⋊C4—948no—
557C42—474no—
558C42—944no—
559C42—942no—
560C23—22no—
561C23—22no—
562C23—22no—
563C23—42no—
564C23—42no—
565C23—42no—
566C23—942no—
567C23—1882no—
568C23—1882no—
569C23—1882no—
570C23#1309399211yescharacteristic
571C23#1309399211yescharacteristic
572C23#1309399712yes—
573C23#1309401012yes—
574C23C2×D4×D4711yescharacteristic
575Q8—22no—
576Q8—22no—
577Q8—42no—
578Q8—942no—
579Q8—942no—
580Q8—1882no—
581D4—22no—
582D4—22no—
583D4—44no—
584D4—42no—
585D4—42no—
586D4—42no—
587D4—942no—
588D4—942no—
589D4—3768no—
590D4—1884no—
591D4—1882no—
592D4—1882no—
593D4—1882no—
594C2×C4—22no—
595C2×C4—42no—
596C2×C4—44no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—44no—
600C2×C4—42no—
601C2×C4—82no—
602C2×C4—82no—
603C2×C4—944no—
604C2×C4—942no—
605C2×C4—944no—
606C2×C4—3762no—
607C2×C4—1882no—
608C2×C4—1882no—
609C2×C4—1888no—
610C2×C4—1882no—
611C2×C4—1884no—
612C2×C4—1884no—
613C2×C4—1882no—
614C2×C4—1882no—
615C2×C4—1884no—
616C2×C4#1309399211yescharacteristic
617C2×C4#1309399211yescharacteristic
618C2×C4#1309399612yes—
619C2×C4#1309399612yes—
620C2×C4#1309399712yes—
621C2×C4#1309399712yes—
622C2×C4#1309399712yes—
623C2×C4#1309400612yes—
624C2×C4#1309400612yes—
625C2×C4#1309400712yes—
626C2×C4#1309400712yes—
627C2×C4#1309401012yes—
628C2×C4C2×D4×D4711yescharacteristic
629C2×C4C2×D4×D4711yescharacteristic
630C2×C4C22×C22⋊D4711yescharacteristic
631C2×C4C2×#1062535211yescharacteristic
632C2×C4C2×#1062535011yescharacteristic
633C2×C4C2×#1062535011yescharacteristic
634C8—42no—
635C8—1882no—
636C22—21no—
637C22—21no—
638C22—21no—
639C22—21no—
640C22—24no—
641C22—24no—
642C22—42no—
643C22—42no—
644C22—42no—
645C22—84no—
646C22—3764no—
647C22—1882no—
648C22—1882no—
649C22—1882no—
650C22#1568903812yes—
651C22#1568903812yes—
652C22#1568929412yes—
653C22#1568929912yes—
654C22C2×#1309399211yescharacteristic
655C22C2×#1309399711yescharacteristic
656C22C2×#1309401011yescharacteristic
657C4—42no—
658C4—82no—
659C4—1884no—
660C4—1884no—
661C4—1882no—
662C4#1568903812yes—
663C4#1568903812yes—
664C4C2×#1309399611yescharacteristic
665C4C2×#1309399711yescharacteristic
666C4C2×#1309400611yescharacteristic
667C4C2×#1309400711yescharacteristic
668C2—21no—
669C2—22no—
670C2—21no—
671C2—82no—
672C2—3762no—
673C2#2024461514yesdirect product factor
674C2C2×#1568903811yescharacteristic
675C2C2×#1568929911yescharacteristic
676C2C2×#1568929411yescharacteristic
subgroupG/H
centerC23#13093992
commutator, cosocleC22×C4×C47C24
FrattiniC22×C4C23×D47
FittingC2×C47×#417C2
socleC23×C47#120
prime radicalC2×#15689701C2
hypercenterC2×#417D47
hypocenterC47C2×#2040
LCSG ▷ C22×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#417

Complements

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

NH ≅ G/Ntype
2. C22×#13093951C2semidirect
3. C22×#13093950C2semidirect
4. C2×#15689122C2semidirect
5. C2×#15688988C2semidirect
6. #20244615C2direct (N × H)
7. C2×#15688891C2semidirect
8. C2×C47×#417C2semidirect
23. C2×#13093951C22semidirect
25. C2×#13093950C22semidirect
27. #15689122C22semidirect
31. #15688988C22semidirect
33. #15688891C22semidirect
38. C47×#417C22semidirect
82. C2×C4×Dic47D4semidirect
215. C4×Dic47C2×D4semidirect
334. C2×#417D47semidirect
342. C2×C47#2040semidirect
370. #417C2×D47semidirect
379. C47C2×#2040semidirect
673. C2#20244615direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#156897012
C22×#130939512
C22×#130939502
C2×#156891222
C2×#156889882
#202446152
C2×#156888912
C2×C47×#4172
C2×#204047