Finite Group Databasebeta

← #26222506

Subgroups — #26222506

Order 12032.

1016 conjugacy classes of subgroups.

The subgroup lattice has 666 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689345C211yesmaximal, characteristic
2C2×#15689215C211yesmaximal, characteristic, complemented
3C2×#15689208C211yesmaximal, characteristic
4#20245927C211yesmaximal, characteristic
5C4×Q8×Dic47C211yesmaximal, characteristic, complemented
6#20244318C211yesmaximal, characteristic
7#20244154C211yesmaximal, characteristic, complemented
8#20244020C211yesmaximal, characteristic, complemented
9#20244006C211yesmaximal, characteristic
10#20243741C211yesmaximal, characteristic
11#20243727C211yesmaximal, characteristic, complemented
12C4⋊C8×Dic47C211yesmaximal, characteristic, complemented
13#20243207C211yesmaximal, characteristic, complemented
14#20243202C211yesmaximal, characteristic
15C47×#1805C211yesmaximal, characteristic, complemented
16C22×#10625337C2211yescharacteristic
17C22×#10625336C2211yescharacteristic
18C2×#13094012C2211yescharacteristic
19C2×Q8×Dic47C2211yescharacteristic
20C2×Q8×Dic47C2211yescharacteristic
21C2×#13093989C2211yescharacteristic
22C2×#13093986C2211yescharacteristic
23#15689345C418yescomplemented
24#15689215—24no—
25#15689208—24no—
26C2×#13093957C2211yescharacteristic
27C2×#13093955C2211yescharacteristic
28C2×C4⋊(C47⋊C8)C2211yescharacteristic
29C2×#13093949C2211yescharacteristic
30C2×#13093949C2211yescharacteristic
31C2×#13093947C2211yescharacteristic
32C2×#13093946C2211yescharacteristic
33C2×#13093946C2211yescharacteristic
34C2×#13093944C2211yescharacteristic
35C2×C8×Dic47C2211yescharacteristic
36C2×C8×Dic47C2211yescharacteristic
37#15689028C2211yescharacteristic
38#15689024C2211yescharacteristic
39#15689012—21no—
40#15689012—21no—
41C4⋊C4×Dic47—21no—
42C4⋊C4×Dic47—21no—
43C4⋊C4×Dic47C2211yescharacteristic
44C4⋊C4×Dic47C2211yescharacteristic
45C4×C4⋊Dic47—21no—
46C4×C4⋊Dic47C2211yescharacteristic
47#15688970—21no—
48#15688970—21no—
49C42×Dic47—21no—
50C42×Dic47C2211yescharacteristic
51#15688885C2211yescharacteristic
52#15688885C2211yescharacteristic
53#15688877C2211yescharacteristic
54#15688877C2211yescharacteristic
55C2×C4×Q8×C47C2211yescharacteristic
56C2×C4⋊C8×C47C2211yescharacteristic
57C2×#103×C47C2211yescharacteristic
58C2×#103×C47C2211yescharacteristic
59C47×#382C2211yescharacteristic
60C47×#340C2211yescharacteristic
61C47×#340C2211yescharacteristic
62C22×Dic94C2311yescharacteristic
63C22×D188C2311yescharacteristic
64C22×C4×D47—21no—
65C2×#10625337—22no—
66C2×#10625337—22no—
67C2×#10625337C2×C414yescomplemented
68C2×#10625336—22no—
69C2×#10625336—22no—
70C2×#10625336C2×C414yes—
71C22×C47⋊C8C2311yescharacteristic
72C22×C47⋊C8C2311yescharacteristic
73#13094012—22no—
74#13094012C2×C414yes—
75Q8×Dic47—22no—
76Q8×Dic47—22no—
77Q8×Dic47—22no—
78Q8×Dic47D414yes—
79#13093989—22no—
80#13093989—22no—
81#13093986—22no—
82#13093986—22no—
83C2×C4⋊Dic47—21no—
84C2×C4⋊Dic47—21no—
85C2×C4⋊Dic47C2311yescharacteristic
86C2×C4⋊Dic47C2311yescharacteristic
87C2×Dic47⋊C4—21no—
88C2×(C2×D47)⋊C4—21no—
89C2×(C2×D47)⋊C4—21no—
90C2×(C2×D47)⋊C4—21no—
91C2×(C2×D47)⋊C4—21no—
92C2×(C2×D47)⋊C4—21no—
93C2×C4×Dic47—21no—
94C2×C4×Dic47—21no—
95C2×C4×Dic47—21no—
96C2×C4×Dic47—21no—
97C2×C4×Dic47—21no—
98C2×C4×Dic47—21no—
99C2×C4×Dic47D411yescharacteristic
100C2×C4×Dic47D411yescharacteristic
101C2×C4×Dic47D411yescharacteristic
102C2×C4×Dic47D411yescharacteristic
103C2×C4×Dic47C2311yescharacteristic
104C2×C4×Dic47C2311yescharacteristic
105#13093957—22no—
106#13093955—22no—
107C4⋊(C47⋊C8)C2×C414yescomplemented
108#13093949—22no—
109#13093949—22no—
110#13093947C2×C414yescomplemented
111#13093946—22no—
112#13093946—22no—
113#13093944C2×C414yes—
114C8×Dic47—22no—
115C8×Dic47—22no—
116C47⋊#95—21no—
117C47⋊#95—21no—
118C47⋊#95—21no—
119C47⋊#95—21no—
120C47⋊#95—21no—
121C22×Q8×C47C2311yescharacteristic
122C22×C8×C47C2311yescharacteristic
123C22×C8×C47C2311yescharacteristic
124C4×Q8×C47—22no—
125C4×Q8×C47C2×C414yescomplemented
126C2×C4⋊C4×C47—21no—
127C2×C4⋊C4×C47C2311yescharacteristic
128C2×C4⋊C4×C47C2311yescharacteristic
129C2×C4⋊C4×C47C2311yescharacteristic
130C2×C42×C47—21no—
131C2×C42×C47C2311yescharacteristic
132C4⋊C8×C47—22no—
133#103×C47—22no—
134#103×C47—22no—
135#95×C47—21no—
136#95×C47—21no—
137C23×D47—21no—
138C2×Q8×C47—21no—
139C2×Q8×C47—21no—
140C2×Q8×C47—21no—
141C2×Q8×C47C22×C412yes—
142C2×Q8×C47C2×D412yes—
143C2×Q8×C47#4112yes—
144C22×C4×C47—21no—
145C22×C4×C47—21no—
146C22×C4×C47—21no—
147C22×C4×C47—21no—
148C22×C4×C47C2×D411yescharacteristic
149C22×C4×C47C2×D411yescharacteristic
150C22×C4×C47C2411yescharacteristic
151C22×Dic47—21no—
152C22×Dic47—21no—
153C22×Dic47—21no—
154C22×Dic47C2×D411yescharacteristic
155C22×Dic47C2×D411yescharacteristic
156C22×Dic47C2×D411yescharacteristic
157C22×Dic47C2×D411yescharacteristic
158C2×D188—21no—
159C2×D188—21no—
160C2×D188—21no—
161C2×D188—21no—
162C2×D188—21no—
163C2×D188C22×C412yes—
164C2×C4×D47—42no—
165C2×C4×D47—42no—
166C2×C4×D47—42no—
167C2×Dic94—21no—
168C2×Dic94—21no—
169C2×Dic94—21no—
170C2×Dic94—21no—
171C2×Dic94—21no—
172C2×Dic94C22×C412yes—
173C2×C8×C47—21no—
174C2×C8×C47—21no—
175C2×C8×C47—21no—
176C2×C8×C47—21no—
177C2×C8×C47—21no—
178C2×C8×C47—21no—
179C4⋊C4×C47—24no—
180C4⋊C4×C47—21no—
181C4⋊C4×C47—21no—
182C4⋊C4×C47—21no—
183C4⋊C4×C47—21no—
184C4⋊C4×C47—21no—
185C4⋊C4×C47C22×C412yes—
186C42×C47—24no—
187C42×C47C22×C412yes—
188C42×C47#4112yes—
189#10625337—28no—
190#10625336—28no—
191(C2×D47)⋊C4—24no—
192(C2×D47)⋊C4—24no—
193(C2×D47)⋊C4—42no—
194(C2×D47)⋊C4—42no—
195(C2×D47)⋊C4—42no—
196C4⋊Dic47—24no—
197C4⋊Dic47—21no—
198C4⋊Dic47—21no—
199C4⋊Dic47—24no—
200C4⋊Dic47C2×D412yes—
201C4⋊Dic47#4112yes—
202Dic47⋊C4—42no—
203C4×Dic47—24no—
204C4×Dic47—21no—
205C4×Dic47—21no—
206C4×Dic47—24no—
207C4×Dic47—22no—
208C4×Dic47—22no—
209C4×Dic47—22no—
210C4×Dic47—41no—
211C4×Dic47—42no—
212C4×Dic47—42no—
213C4×Dic47—42no—
214C4×Dic47—42no—
215C4×Dic47C2×D412yes—
216C4×Dic47#4112yes—
217C2×C47⋊C8—21no—
218C2×C47⋊C8—21no—
219C2×C47⋊C8—21no—
220C2×C47⋊C8—21no—
221C2×C47⋊C8C22×C412yes—
222C2×C47⋊C8C22×C412yes—
223C23×C47C22×D411yescharacteristic
224C22×D47—22no—
225C22×D47—41no—
226C22×D47—41no—
227C22×D47—41no—
228C22×D47—41no—
229C22×D47—41no—
230C22×D47—41no—
231Q8×C47—22no—
232Q8×C47—22no—
233Q8×C47—22no—
234Q8×C47C4×D414yes—
235C2×C4×C47—22no—
236C2×C4×C47—22no—
237C2×C4×C47—22no—
238C2×C4×C47—21no—
239C2×C4×C47—21no—
240C2×C4×C47—21no—
241C2×C4×C47—21no—
242C2×C4×C47—22no—
243C2×C4×C47—41no—
244C2×C4×C47—41no—
245C2×C4×C47—41no—
246C2×C4×C47—41no—
247C2×C4×C47—41no—
248C2×C4×C47—41no—
249C2×C4×C47—41no—
250C2×C4×C47—41no—
251C2×C4×C47C4×D412yes—
252C2×C4×C47C4×D412yes—
253C2×C4×C47C23×C411yescharacteristic
254C2×C4×C47C22×D411yescharacteristic
255C2×C4×C47C2×#4111yescharacteristic
256C2×C4×C47C2×#4111yescharacteristic
257C2×C4×C47C2×#4111yescharacteristic
258C2×C4×C47C2×#4111yescharacteristic
259C2×Dic47—21no—
260C2×Dic47—21no—
261C2×Dic47—21no—
262C2×Dic47—21no—
263C2×Dic47—22no—
264C2×Dic47—22no—
265C2×Dic47—21no—
266C2×Dic47—22no—
267C2×Dic47—21no—
268C2×Dic47—21no—
269C2×Dic47—41no—
270C2×Dic47—41no—
271C2×Dic47—41no—
272C2×Dic47—41no—
273C2×Dic47—41no—
274C2×Dic47—41no—
275C2×Dic47#12112yes—
276C2×Dic47#12112yes—
277C2×Dic47#13512yes—
278C2×Dic47#13512yes—
279C2×Dic47#13712yes—
280D188—22no—
281D188—22no—
282D188—41no—
283D188—41no—
284C4×D47—84no—
285Dic94—22no—
286Dic94—22no—
287Dic94—41no—
288Dic94—41no—
289C8×C47—41no—
290C8×C47—41no—
291C47⋊C8—22no—
292C47⋊C8—22no—
293C4×#464—471nomaximal, Sylow
294C22×C47C2×C4×D411yescharacteristic
295C22×C47C2×#12111yescharacteristic
296C22×C47#52511yescharacteristic
297C22×C47C2×#13511yescharacteristic
298C22×C47C2×#13511yescharacteristic
299C22×C47C2×#13711yescharacteristic
300C22×C47#57511yescharacteristic
301C2×D47—42no—
302C2×D47—42no—
303C2×D47—42no—
304C2×D47—81no—
305C2×D47—81no—
306C2×D47—81no—
307C2×D47—81no—
308C4×C47—22no—
309C4×C47—22no—
310C4×C47—24no—
311C4×C47—42no—
312C4×C47—81no—
313C4×C47—81no—
314C4×C47C2×C4×D411yescharacteristic
315C4×C47C4×#4111yescharacteristic
316C4×C47#52911yescharacteristic
317C4×C47#53211yescharacteristic
318Dic47—22no—
319Dic47—22no—
320Dic47—41no—
321Dic47—42no—
322Dic47—42no—
323Dic47—42no—
324Dic47#46414yescomplemented
325C2×#464—471no—
326C2×C4×Q16—471no—
327C2×C4×SD16—471no—
328C4×#124—471no—
329C42×Q8—471no—
330#1983—471no—
331#1949—471no—
332#1927—471no—
333#1925—471no—
334#1880—471no—
335#1878—471no—
336#1805D4711yescharacteristic, complemented
337C4×C4⋊C8—471no—
338C4×#103—471no—
339C4×#102—471no—
340C2×C47C4×#12111yescharacteristic
341C2×C47C4×#13511yescharacteristic
342C2×C47C4×#13711yescharacteristic
343C2×C47C2×#46411yescharacteristic
344C2×C47#303511yescharacteristic
345C2×C47#310211yescharacteristic
346C2×C47#310811yescharacteristic
347D47—82no—
348C22×Q16—471no—
349C22×SD16—471no—
350C2×#124—471no—
351C2×C4×Q8—471no—
352C2×C4×Q8—471no—
353C2×C4×Q8—471no—
354C2×C4×Q8C2×D4711yescharacteristic
355C2×C4×D4—471no—
356#464—478no—
357C4×Q16—944no—
358C4×SD16—944no—
359C2×#107—471no—
360C2×#106—471no—
361C2×C4⋊C8—471no—
362C2×C4⋊C8C2×D4711yescharacteristic
363C2×#103—471no—
364C2×#103—471no—
365C2×#103—471no—
366C2×#103C2×D4711yescharacteristic
367C2×#103C2×D4711yescharacteristic
368C2×#102—471no—
369C2×#102—471no—
370C2×#102—471no—
371C2×C4×C8—471no—
372C2×C4×C8—471no—
373#391—471no—
374#390—471no—
375#388—941no—
376#388—941no—
377#382C2×D4711yescharacteristic
378C4×C4⋊C4—471no—
379C4×C4⋊C4—471no—
380C4×C4⋊C4—471no—
381C4×C4⋊C4—941no—
382C4×C4⋊C4—941no—
383C4×C4⋊C4—941no—
384C4×C22⋊C4—941no—
385C4×C22⋊C4—941no—
386C43—471no—
387C43—941no—
388#340—471no—
389#340—471no—
390#340C2×D4711yescharacteristic
391#340C2×D4711yescharacteristic
392#336—471no—
393#336—471no—
394C47C4×#46411yescharacteristic, complemented, Sylow
395C22×Q8—471no—
396C22×Q8C22×D4711yescharacteristic
397C22×D4—471no—
398C23×C4—941no—
399C2×Q16—474no—
400C2×Q16—942no—
401C2×Q16—942no—
402C2×SD16—474no—
403C2×SD16—942no—
404C2×SD16—942no—
405C22×C8—471no—
406C22×C8—471no—
407C22×C8C22×D4711yescharacteristic
408C22×C8C22×D4711yescharacteristic
409#124—474no—
410#124—942no—
411C4×Q8—22no—
412C4×Q8—474no—
413C4×Q8—942no—
414C4×Q8—942no—
415C4×Q8—942no—
416C4×Q8—942no—
417C4×Q8—942no—
418C4×Q8C4×D4714yescomplemented
419C4×D4—942no—
420C4×D4—942no—
421C2×C4⋊C4—21no—
422C2×C4⋊C4—471no—
423C2×C4⋊C4—471no—
424C2×C4⋊C4—941no—
425C2×C4⋊C4—941no—
426C2×C4⋊C4—941no—
427C2×C4⋊C4C22×D4711yescharacteristic
428C2×C4⋊C4C22×D4711yescharacteristic
429C2×C4⋊C4C22×D4711yescharacteristic
430C2×C22⋊C4—941no—
431C2×C22⋊C4—941no—
432C2×C22⋊C4—941no—
433C2×C22⋊C4—941no—
434C2×C22⋊C4—941no—
435C2×C42—21no—
436C2×C42—471no—
437C2×C42—471no—
438C2×C42—471no—
439C2×C42—471no—
440C2×C42—471no—
441C2×C42—471no—
442C2×C42—941no—
443C2×C42—941no—
444C2×C42—941no—
445C2×C42—941no—
446C2×C42—941no—
447C2×C42—941no—
448C2×C42C22×D4711yescharacteristic
449#107—942no—
450#106—942no—
451C4⋊C8—22no—
452C4⋊C8—474no—
453#103—22no—
454#103—22no—
455#103—474no—
456#103—942no—
457#103—942no—
458#102—474no—
459#102—942no—
460#102—942no—
461C4×C8—942no—
462C4×C8—942no—
463#95—21no—
464#95—21no—
465#95—941no—
466#95—941no—
467#95—941no—
468#95—941no—
469#95—941no—
470C24—941no—
471C2×Q8—21no—
472C2×Q8—21no—
473C2×Q8—21no—
474C2×Q8—472no—
475C2×Q8—941no—
476C2×Q8—941no—
477C2×Q8—941no—
478C2×Q8—941no—
479C2×Q8—941no—
480C2×Q8C2×C4×D4712yes—
481C2×Q8D4×D4712yes—
482C2×Q8#1062535412yes—
483C2×D4—472no—
484C2×D4—941no—
485C2×D4—941no—
486C2×D4—941no—
487C2×D4—941no—
488C2×D4—941no—
489C22×C4—21no—
490C22×C4—21no—
491C22×C4—21no—
492C22×C4—21no—
493C22×C4—471no—
494C22×C4—471no—
495C22×C4—471no—
496C22×C4—471no—
497C22×C4—941no—
498C22×C4—941no—
499C22×C4—941no—
500C22×C4—1882no—
501C22×C4—1882no—
502C22×C4—1882no—
503C22×C4D4×D4711yescharacteristic
504C22×C4D4×D4711yescharacteristic
505C22×C4C23×D4711yescharacteristic
506Q16—948no—
507SD16—948no—
508C2×C8—21no—
509C2×C8—21no—
510C2×C8—21no—
511C2×C8—21no—
512C2×C8—21no—
513C2×C8—21no—
514C2×C8—472no—
515C2×C8—472no—
516C2×C8—941no—
517C2×C8—941no—
518C2×C8—941no—
519C2×C8—941no—
520C4⋊C4—24no—
521C4⋊C4—21no—
522C4⋊C4—21no—
523C4⋊C4—21no—
524C4⋊C4—21no—
525C4⋊C4—21no—
526C4⋊C4—472no—
527C4⋊C4—472no—
528C4⋊C4—944no—
529C4⋊C4—944no—
530C4⋊C4—941no—
531C4⋊C4—941no—
532C4⋊C4—1882no—
533C4⋊C4C2×C4×D4712yes—
534C22⋊C4—944no—
535C22⋊C4—944no—
536C22⋊C4—1882no—
537C22⋊C4—1882no—
538C22⋊C4—1882no—
539C42—24no—
540C42—472no—
541C42—472no—
542C42—942no—
543C42—942no—
544C42—942no—
545C42—944no—
546C42—944no—
547C42—941no—
548C42—941no—
549C42—1881no—
550C42—1882no—
551C42—1882no—
552C42—1882no—
553C42—1882no—
554C42C2×C4×D4712yes—
555C42#1062535212yes—
556C23—942no—
557C23—1881no—
558C23—1881no—
559C23—1881no—
560C23—1881no—
561C23—1881no—
562C23—1881no—
563C23C2×D4×D4711yescharacteristic
564Q8—22no—
565Q8—22no—
566Q8—22no—
567Q8—942no—
568Q8—942no—
569Q8—1881no—
570Q8—1881no—
571Q8#1309398614yes—
572D4—942no—
573D4—942no—
574D4—1881no—
575D4—1881no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—22no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—472no—
593C2×C4—472no—
594C2×C4—472no—
595C2×C4—472no—
596C2×C4—472no—
597C2×C4—941no—
598C2×C4—941no—
599C2×C4—941no—
600C2×C4—941no—
601C2×C4—942no—
602C2×C4—942no—
603C2×C4—942no—
604C2×C4—941no—
605C2×C4—941no—
606C2×C4—941no—
607C2×C4—3764no—
608C2×C4—1881no—
609C2×C4—1881no—
610C2×C4—1881no—
611C2×C4—1881no—
612C2×C4—1881no—
613C2×C4—1881no—
614C2×C4(C22⋊D47)⋊C412yes—
615C2×C4(C22⋊D47)⋊C412yes—
616C2×C4C22×C4×D4711yescharacteristic
617C2×C4C2×D4×D4711yescharacteristic
618C2×C4#41×D4711yescharacteristic
619C2×C4#41×D4711yescharacteristic
620C2×C4C2×#1062535211yescharacteristic
621C2×C4C2×#1062535411yescharacteristic
622C8—41no—
623C8—41no—
624C8—942no—
625C8—942no—
626C22—3761no—
627C22—3761no—
628C22—3761no—
629C22—3761no—
630C22—1882no—
631C22—1882no—
632C22—1882no—
633C22C2×(C22⋊D47)⋊C411yescharacteristic
634C22#121×D4711yescharacteristic
635C22#1568963611yescharacteristic
636C22#135×D4711yescharacteristic
637C22#135×D4711yescharacteristic
638C22#137×D4711yescharacteristic
639C22#1568992211yescharacteristic
640C4—22no—
641C4—24no—
642C4—22no—
643C4—42no—
644C4—81no—
645C4—81no—
646C4—474no—
647C4—942no—
648C4—942no—
649C4—1882no—
650C4—1882no—
651C4—1881no—
652C4—1882no—
653C4C2×#1309398611yescharacteristic
654C4#1568959111yescharacteristic
655C4#1568965911yescharacteristic
656C4#1568966811yescharacteristic
657C2—3762no—
658C2#2024589411yescharacteristic
659C2#2025215611yescharacteristic
660C2#2025220311yescharacteristic
661C2#464×D4711yescharacteristic
662C2#2025284211yescharacteristic
663C2#2025341311yescharacteristic
664C2#2025347311yescharacteristic
subgroupG/H
centerC23C2×D4×D47
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC47×#1805C2
socleC23×C47C22×D4
prime radicalC22×D188C23
cosocleC22×C4×C47C24
hypercenter#1805D47
hypocenterC47C4×#464
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1805

Complements

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

NH ≅ G/Ntype
2. C2×#15689215C2semidirect
5. C4×Q8×Dic47C2semidirect
7. #20244154C2semidirect
8. #20244020C2semidirect
11. #20243727C2semidirect
12. C4⋊C8×Dic47C2semidirect
13. #20243207C2semidirect
15. C47×#1805C2semidirect
23. #15689345C4semidirect
67. C2×#10625337C2×C4semidirect
107. C4⋊(C47⋊C8)C2×C4semidirect
110. #13093947C2×C4semidirect
125. C4×Q8×C47C2×C4semidirect
324. Dic47#464semidirect
336. #1805D47semidirect
394. C47C4×#464semidirect
418. C4×Q8C4×D47semidirect

Maximal subgroups

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

subgroupindex
C2×#156893452
C2×#156892152
C2×#156892082
#202459272
C4×Q8×Dic472
#202443182
#202441542
#202440202
#202440062
#202437412
#202437272
C4⋊C8×Dic472
#202432072
#202432022
C47×#18052
C4×#46447