Finite Group Databasebeta

← C2×#20244698 #26244206

Subgroups — C2×#20244698 #26244206

Order 12032.

1918 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
1C23×#10625334C211yesmaximal, characteristic
2C22×C4⋊(C47⋊C8)C211yesmaximal, characteristic, complemented
3C22×#13093945C211yesmaximal, characteristic, complemented
4C22×#13093944C211yesmaximal, characteristic, complemented
5C2×#15689025C211yesmaximal, characteristic, complemented
6#20244698C218yesmaximal, direct product factor
7C2×#15688885C211yesmaximal, characteristic, complemented
8C2×C47×#390C211yesmaximal, characteristic, complemented
9C23×D188C2211yescharacteristic
10C22×#10625334—24no—
11C22×#10625334—24no—
12C22×#10625334—24no—
13C22×#10625334C2214yes—
14C23×C47⋊C8C2211yescharacteristic
15C23×C47⋊C8C2211yescharacteristic
16C22×C4⋊Dic47C2211yescharacteristic
17C22×(C2×D47)⋊C4—21no—
18C22×(C2×D47)⋊C4—21no—
19C2×C4⋊(C47⋊C8)—24no—
20C2×C4⋊(C47⋊C8)C2214yescomplemented
21C2×#13093945—24no—
22C2×#13093945C2214yescomplemented
23C2×#13093944—24no—
24C2×#13093944C2214yescomplemented
25#15689025—22no—
26#15689025C2214yescomplemented
27#15688885C2214yescomplemented
28C23×D4×C47C2211yescharacteristic
29C22×C4⋊C4×C47C2211yescharacteristic
30C22×C22⋊C4×C47—21no—
31C22×C22⋊C4×C47—21no—
32C22×C42×C47C2211yescharacteristic
33C47×#390—22no—
34C47×#390C2214yescomplemented
35C24×D47—21no—
36C22×D188—22no—
37C22×D188—22no—
38C22×D188—22no—
39C22×D188—22no—
40C22×D188—21no—
41C22×D188—22no—
42C22×D188D414yes—
43C22×D188C2312yes—
44C23×Dic47—21no—
45C2×#10625334—232no—
46C2×#10625334—28no—
47C2×#10625334—28no—
48C2×#10625334—28no—
49C22×C47⋊C8—22no—
50C22×C47⋊C8—22no—
51C22×C47⋊C8—22no—
52C22×C47⋊C8—22no—
53C22×C47⋊C8—22no—
54C22×C47⋊C8D414yes—
55C22×C47⋊C8C2312yes—
56C22×C47⋊C8C2312yes—
57C2×C4⋊Dic47—22no—
58C2×C4⋊Dic47—22no—
59C2×C4⋊Dic47—21no—
60C2×C4⋊Dic47C2312yes—
61C2×(C2×D47)⋊C4—24no—
62C2×(C2×D47)⋊C4—24no—
63C2×(C2×D47)⋊C4—44no—
64C2×(C2×D47)⋊C4—44no—
65C4⋊(C47⋊C8)—28no—
66#13093945—28no—
67#13093944—28no—
68C25×C47—21no—
69C22×D4×C47—22no—
70C22×D4×C47—22no—
71C22×D4×C47—21no—
72C22×D4×C47—22no—
73C22×D4×C47—22no—
74C22×D4×C47—22no—
75C22×D4×C47D414yes—
76C22×D4×C47C2312yes—
77C23×C4×C47—21no—
78C23×C4×C47D411yescharacteristic
79C23×C4×C47D411yescharacteristic
80C23×C4×C47C2311yescharacteristic
81C2×C4⋊C4×C47—22no—
82C2×C4⋊C4×C47—22no—
83C2×C4⋊C4×C47—21no—
84C2×C4⋊C4×C47C2312yes—
85C2×C22⋊C4×C47—24no—
86C2×C22⋊C4×C47—24no—
87C2×C22⋊C4×C47—44no—
88C2×C22⋊C4×C47—44no—
89C2×C42×C47—22no—
90C2×C42×C47—22no—
91C2×C42×C47D412yes—
92C2×C42×C47C2312yes—
93C24×C47—24no—
94C24×C47—22no—
95C24×C47—42no—
96C24×C47—42no—
97C24×C47—42no—
98C24×C47—42no—
99C24×C47—42no—
100C24×C47—42no—
101C24×C47C2×D411yescharacteristic
102C23×D47—24no—
103C23×D47—22no—
104C23×D47—42no—
105C23×D47—42no—
106C23×D47—42no—
107C23×D47—42no—
108C23×D47—42no—
109C23×D47—42no—
110C2×D4×C47—28no—
111C2×D4×C47—24no—
112C2×D4×C47—28no—
113C2×D4×C47—22no—
114C2×D4×C47—22no—
115C2×D4×C47—22no—
116C2×D4×C47—22no—
117C2×D4×C47—22no—
118C2×D4×C47—42no—
119C2×D4×C47—42no—
120C2×D4×C47—44no—
121C2×D4×C47—44no—
122C2×D4×C47C2×D414yes—
123C22×C4×C47—22no—
124C22×C4×C47—22no—
125C22×C4×C47—22no—
126C22×C4×C47—22no—
127C22×C4×C47—22no—
128C22×C4×C47—22no—
129C22×C4×C47—42no—
130C22×C4×C47—42no—
131C22×C4×C47—42no—
132C22×C4×C47D814yescomplemented
133C22×C4×C47C2×D412yes—
134C22×C4×C47C2×D412yes—
135C22×C4×C47C2×D412yes—
136C22×C4×C47C2×D412yes—
137C22×C4×C47C2×D412yes—
138C22×C4×C47C2×D411yescharacteristic
139C22×C4×C47#4112yes—
140C22×C4×C47#4112yes—
141C22×C4×C47#4112yes—
142C22×C4×C47C2411yescharacteristic
143C22×Dic47—22no—
144C22×Dic47—42no—
145C22×Dic47—42no—
146C22×Dic47—42no—
147C2×D188—28no—
148C2×D188—28no—
149C2×D188—22no—
150C2×D188—22no—
151C2×D188—22no—
152C2×D188—22no—
153C2×D188—24no—
154C2×D188—22no—
155C2×D188—42no—
156C2×D188—42no—
157C2×D188—44no—
158C2×D188—44no—
159C2×D188C2×D414yes—
160C4⋊C4×C47—22no—
161C4⋊C4×C47—22no—
162C4⋊C4×C47—42no—
163C22⋊C4×C47—48no—
164C22⋊C4×C47—48no—
165C42×C47—24no—
166C42×C47—22no—
167C42×C47—22no—
168#10625334—232no—
169(C2×D47)⋊C4—48no—
170(C2×D47)⋊C4—48no—
171C4⋊Dic47—22no—
172C4⋊Dic47—22no—
173C4⋊Dic47—42no—
174C2×C47⋊C8—28no—
175C2×C47⋊C8—22no—
176C2×C47⋊C8—22no—
177C2×C47⋊C8—22no—
178C2×C47⋊C8—22no—
179C2×C47⋊C8—22no—
180C2×C47⋊C8—44no—
181C2×C47⋊C8C2×D414yes—
182C23×C47—24no—
183C23×C47—42no—
184C23×C47—42no—
185C23×C47—42no—
186C23×C47—42no—
187C23×C47—42no—
188C23×C47—42no—
189C23×C47—48no—
190C23×C47—48no—
191C23×C47—48no—
192C23×C47—84no—
193C23×C47—84no—
194C23×C47—84no—
195C23×C47—84no—
196C23×C47#12112yes—
197C23×C47#12112yes—
198C23×C47#12112yes—
199C23×C47C2×D812yes—
200C23×C47#13512yes—
201C23×C47AGL(1,ℤ/8)12yes—
202C23×C47AGL(1,ℤ/8)12yes—
203C23×C47C22×D411yescharacteristic
204C22×D47—24no—
205C22×D47—42no—
206C22×D47—42no—
207C22×D47—42no—
208C22×D47—42no—
209C22×D47—42no—
210C22×D47—42no—
211C22×D47—48no—
212C22×D47—48no—
213C22×D47—48no—
214C22×D47—84no—
215C22×D47—84no—
216C22×D47—84no—
217C22×D47—84no—
218D4×C47—24no—
219D4×C47—24no—
220D4×C47—42no—
221D4×C47—42no—
222D4×C47—48no—
223C2×C4×C47—22no—
224C2×C4×C47—22no—
225C2×C4×C47—28no—
226C2×C4×C47—22no—
227C2×C4×C47—22no—
228C2×C4×C47—22no—
229C2×C4×C47—44no—
230C2×C4×C47—42no—
231C2×C4×C47—42no—
232C2×C4×C47—42no—
233C2×C4×C47—84no—
234C2×C4×C47#12014yes—
235C2×C4×C47#12112yes—
236C2×C4×C47#12112yes—
237C2×C4×C47#12112yes—
238C2×C4×C47#12314yes—
239C2×C4×C47#12314yes—
240C2×C4×C47#12414yes—
241C2×C4×C47C2×D814yescomplemented
242C2×C4×C47C22×D411yescharacteristic
243C2×C4×C47C22×D411yescharacteristic
244C2×C4×C47C22×D411yescharacteristic
245C2×C4×C47C2×#4111yescharacteristic
246C2×C4×C47C2×#4111yescharacteristic
247C2×C4×C47C2×#4111yescharacteristic
248C2×Dic47—42no—
249C2×Dic47—42no—
250C2×Dic47—42no—
251C2×Dic47—84no—
252D188—24no—
253D188—24no—
254D188—42no—
255D188—42no—
256D188—48no—
257C47⋊C8—24no—
258C47⋊C8—42no—
259C2×#2054—471nomaximal, Sylow
260C22×C47—44no—
261C22×C47—44no—
262C22×C47—44no—
263C22×C47—816no—
264C22×C47—82no—
265C22×C47—82no—
266C22×C47—82no—
267C22×C47—82no—
268C22×C47#39414yes—
269C22×C47#45914yes—
270C22×C47#45914yes—
271C22×C47#46314yes—
272C22×C47#46314yes—
273C22×C47#46614yes—
274C22×C47#46914yes—
275C22×C47C2×#12111yescharacteristic
276C22×C47C2×#12111yescharacteristic
277C22×C47C2×#12111yescharacteristic
278C22×C47C22×D811yescharacteristic
279C22×C47C2×#13511yescharacteristic
280C22×C47C2×AGL(1,ℤ/8)11yescharacteristic
281C22×C47C2×AGL(1,ℤ/8)11yescharacteristic
282C2×D47—44no—
283C2×D47—44no—
284C2×D47—44no—
285C2×D47—816no—
286C2×D47—82no—
287C2×D47—82no—
288C2×D47—82no—
289C2×D47—82no—
290C4×C47—24no—
291C4×C47—42no—
292C4×C47—82no—
293C4×C47#39414yes—
294C4×C47C2×#12011yescharacteristic
295C4×C47C2×#12311yescharacteristic
296C4×C47C2×#12311yescharacteristic
297C4×C47C2×#12411yescharacteristic
298Dic47—82no—
299C23×D8—471no—
300C22×C4⋊C8—471no—
301C22×#102—471no—
302C22×#102—471no—
303C2×#390—471no—
304C2×#390D4711yescharacteristic, complemented
305#2054—478no—
306C2×#340—471no—
307C2×C47—84no—
308C2×C47#205418yescomplemented
309C2×C47C2×#39411yescharacteristic
310C2×C47C2×#45911yescharacteristic
311C2×C47C2×#45911yescharacteristic
312C2×C47C2×#46311yescharacteristic
313C2×C47C2×#46311yescharacteristic
314C2×C47C2×#46611yescharacteristic
315C2×C47C2×#46911yescharacteristic
316D47—84no—
317C23×D4—471no—
318C23×D4C2×D4711yescharacteristic
319C22×D8—474no—
320C22×D8—944no—
321C22×D8—944no—
322C22×D8—944no—
323C23×C8—471no—
324C23×C8—471no—
325C22×C4⋊C4—471no—
326C22×C4⋊C4C2×D4711yescharacteristic
327C22×C22⋊C4—21no—
328C22×C22⋊C4—21no—
329C22×C22⋊C4—941no—
330C22×C22⋊C4—941no—
331C22×C42C2×D4711yescharacteristic
332C2×C4⋊C8—474no—
333C2×C4⋊C8—944no—
334C2×#102—474no—
335C2×#102—474no—
336C2×#102—944no—
337C2×#102—944no—
338#390—22no—
339#390—474no—
340#390—942no—
341#390C2×D4714yescomplemented
342#340—474no—
343C47C2×#205411yescharacteristic, complemented, Sylow
344C25—21no—
345C25—941no—
346C22×D4—22no—
347C22×D4—22no—
348C22×D4—22no—
349C22×D4—22no—
350C22×D4—22no—
351C22×D4—21no—
352C22×D4—474no—
353C22×D4—472no—
354C22×D4—942no—
355C22×D4—942no—
356C22×D4—942no—
357C22×D4—942no—
358C22×D4—942no—
359C22×D4—941no—
360C22×D4D18814yes—
361C22×D4C22×D4712yes—
362C23×C4—21no—
363C23×C4—941no—
364C23×C4C22⋊D4711yescharacteristic
365C23×C4C22⋊D4711yescharacteristic
366C23×C4C22×D4711yescharacteristic
367C2×D8—948no—
368C2×D8—948no—
369C2×D8—9432no—
370C2×D8—948no—
371C22×C8—474no—
372C22×C8—472no—
373C22×C8—472no—
374C22×C8—942no—
375C22×C8—942no—
376C22×C8—942no—
377C22×C8—942no—
378C22×C8—942no—
379C2×C4⋊C4—22no—
380C2×C4⋊C4—22no—
381C2×C4⋊C4—21no—
382C2×C4⋊C4—472no—
383C2×C4⋊C4—942no—
384C2×C4⋊C4—942no—
385C2×C4⋊C4—941no—
386C2×C4⋊C4C22×D4712yes—
387C2×C22⋊C4—24no—
388C2×C22⋊C4—24no—
389C2×C22⋊C4—44no—
390C2×C22⋊C4—44no—
391C2×C22⋊C4—944no—
392C2×C22⋊C4—944no—
393C2×C22⋊C4—1884no—
394C2×C22⋊C4—1884no—
395C2×C42—22no—
396C2×C42—22no—
397C2×C42C22⋊D4712yes—
398C2×C42C22×D4712yes—
399C4⋊C8—948no—
400#102—948no—
401#102—948no—
402C24—24no—
403C24—22no—
404C24—42no—
405C24—42no—
406C24—42no—
407C24—42no—
408C24—42no—
409C24—42no—
410C24—944no—
411C24—942no—
412C24—1882no—
413C24—1882no—
414C24—1882no—
415C24—1882no—
416C24—1882no—
417C24—1882no—
418C24C2×C22⋊D4711yescharacteristic
419C2×D4—28no—
420C2×D4—24no—
421C2×D4—28no—
422C2×D4—22no—
423C2×D4—22no—
424C2×D4—22no—
425C2×D4—22no—
426C2×D4—22no—
427C2×D4—42no—
428C2×D4—42no—
429C2×D4—44no—
430C2×D4—44no—
431C2×D4—474no—
432C2×D4—948no—
433C2×D4—944no—
434C2×D4—948no—
435C2×D4—942no—
436C2×D4—942no—
437C2×D4—942no—
438C2×D4—942no—
439C2×D4—942no—
440C2×D4—1882no—
441C2×D4—1882no—
442C2×D4—1884no—
443C2×D4—1884no—
444C2×D4C2×D18814yes—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—42no—
452C22×C4—42no—
453C22×C4—42no—
454C22×C4—942no—
455C22×C4—1882no—
456C22×C4—1882no—
457C22×C4—1882no—
458C22×C4#1062533414yescomplemented
459C22×C4C2×D18812yes—
460C22×C4#1062535012yes—
461C22×C4D4×D4712yes—
462C22×C4D4×D4712yes—
463C22×C4#1062535212yes—
464C22×C4#1062535212yes—
465C22×C4C2×C22⋊D4712yes—
466C22×C4C2×C22⋊D4712yes—
467C22×C4C2×C22⋊D4711yescharacteristic
468C22×C4C23×D4711yescharacteristic
469D8—9432no—
470C2×C8—474no—
471C2×C8—948no—
472C2×C8—942no—
473C2×C8—942no—
474C2×C8—942no—
475C2×C8—942no—
476C2×C8—942no—
477C2×C8—1884no—
478C4⋊C4—22no—
479C4⋊C4—22no—
480C4⋊C4—42no—
481C4⋊C4—942no—
482C4⋊C4—942no—
483C4⋊C4—1882no—
484C22⋊C4—48no—
485C22⋊C4—48no—
486C22⋊C4—1888no—
487C22⋊C4—1888no—
488C42—24no—
489C42—22no—
490C42—22no—
491C23—24no—
492C23—48no—
493C23—48no—
494C23—48no—
495C23—42no—
496C23—42no—
497C23—42no—
498C23—42no—
499C23—42no—
500C23—42no—
501C23—84no—
502C23—84no—
503C23—84no—
504C23—84no—
505C23—944no—
506C23—3764no—
507C23—3764no—
508C23—3764no—
509C23—3764no—
510C23—1888no—
511C23—1888no—
512C23—1888no—
513C23—1882no—
514C23—1882no—
515C23—1882no—
516C23—1882no—
517C23—1882no—
518C23—1882no—
519C23#1309399512yes—
520C23#1309399612yes—
521C23#1309399812yes—
522C23C2×#1062533412yes—
523C23#1309404412yes—
524C23#1309404912yes—
525C23#1309405012yes—
526C23C22×C22⋊D4711yescharacteristic
527D4—24no—
528D4—24no—
529D4—48no—
530D4—42no—
531D4—42no—
532D4—944no—
533D4—944no—
534D4—1888no—
535D4—1882no—
536D4—1882no—
537C2×C4—28no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—44no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—84no—
548C2×C4—3764no—
549C2×C4—1882no—
550C2×C4—1882no—
551C2×C4—1882no—
552C2×C4#1309399414yes—
553C2×C4#1309399512yes—
554C2×C4#1309399612yes—
555C2×C4#1309399812yes—
556C2×C4#1309400614yes—
557C2×C4#1309400814yes—
558C2×C4#1309401014yes—
559C2×C4C2×#1062533414yescomplemented
560C2×C4C2×D4×D4711yescharacteristic
561C2×C4C2×D4×D4711yescharacteristic
562C2×C4C22×D18811yescharacteristic
563C2×C4C2×#1062535211yescharacteristic
564C2×C4C2×#1062535211yescharacteristic
565C2×C4C2×#1062535011yescharacteristic
566C8—944no—
567C8—1882no—
568C22—44no—
569C22—44no—
570C22—44no—
571C22—82no—
572C22—82no—
573C22—82no—
574C22—82no—
575C22—816no—
576C22—3762no—
577C22—3762no—
578C22—3762no—
579C22—3762no—
580C22—37616no—
581C22—1884no—
582C22—1884no—
583C22—1884no—
584C22#1568903914yes—
585C22#1568931514yes—
586C22#1568931814yes—
587C22#1568933514yes—
588C22#1568933814yes—
589C22#1568935714yes—
590C22#1568937414yes—
591C22C2×#1309399511yescharacteristic
592C22C2×#1309399611yescharacteristic
593C22C2×#1309399811yescharacteristic
594C22C22×#1062533411yescharacteristic
595C22C2×#1309404411yescharacteristic
596C22C2×#1309404911yescharacteristic
597C22C2×#1309405011yescharacteristic
598C4—24no—
599C4—42no—
600C4—82no—
601C4—3762no—
602C4#1568903914yes—
603C4C2×#1309399411yescharacteristic
604C4C2×#1309400611yescharacteristic
605C4C2×#1309400811yescharacteristic
606C4C2×#1309401011yescharacteristic
607C2—84no—
608C2—3764no—
609C2#2024469818yesdirect product factor
610C2C2×#1568903911yescharacteristic
611C2C2×#1568931511yescharacteristic
612C2C2×#1568931811yescharacteristic
613C2C2×#1568933511yescharacteristic
614C2C2×#1568933811yescharacteristic
615C2C2×#1568935711yescharacteristic
616C2C2×#1568937411yescharacteristic
subgroupG/H
centerC24C2×C22⋊D47
commutator, cosocleC22×C4×C47C24
FrattiniC22×C4C23×D47
FittingC2×C47×#390C2
socleC24×C47C2×D4
prime radicalC23×#10625334C2
hypercenterC2×#390D47
hypocenterC47C2×#2054
LCSG ▷ C22×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C24 ◁ C22×C42 ◁ C2×#390

Complements

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

NH ≅ G/Ntype
2. C22×C4⋊(C47⋊C8)C2semidirect
3. C22×#13093945C2semidirect
4. C22×#13093944C2semidirect
5. C2×#15689025C2semidirect
6. #20244698C2direct (N × H)
7. C2×#15688885C2semidirect
8. C2×C47×#390C2semidirect
20. C2×C4⋊(C47⋊C8)C22semidirect
22. C2×#13093945C22semidirect
24. C2×#13093944C22semidirect
26. #15689025C22semidirect
27. #15688885C22semidirect
34. C47×#390C22semidirect
132. C22×C4×C47D8semidirect
241. C2×C4×C47C2×D8semidirect
304. C2×#390D47semidirect
308. C2×C47#2054semidirect
341. #390C2×D47semidirect
343. C47C2×#2054semidirect
458. C22×C4#10625334semidirect
559. C2×C4C2×#10625334semidirect
609. C2#20244698direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#106253342
C22×C4⋊(C47⋊C8)2
C22×#130939452
C22×#130939442
C2×#156890252
#202446982
C2×#156888852
C2×C47×#3902
C2×#205447