Finite Group Databasebeta

← #26216193

Subgroups — #26216193

Order 12032.

1342 conjugacy classes of subgroups.

The subgroup lattice has 672 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×#10625350C211yesmaximal, characteristic
2C2×#15689181C212yesmaximal, complemented
3#20243456C211yesmaximal, characteristic, complemented
4#20243371C212yesmaximal, complemented
5#20243367C211yesmaximal, characteristic, complemented
6#20243367C211yesmaximal, characteristic, complemented
7#20243299C212yesmaximal
8#20243153C212yesmaximal, complemented
9#20243149C211yesmaximal, characteristic, complemented
10#20243145C211yesmaximal, characteristic, complemented
11C47×#1817C211yesmaximal, characteristic, complemented
12C22×#10625350C2211yescharacteristic
13C4×#10625350—28no—
14C2×C4×Dic94C411yescharacteristic
15C2×C4×Dic94C411yescharacteristic
16C2×C4×D188C411yescharacteristic
17C2×C4×D188C411yescharacteristic
18C2×C4×C22⋊D47C411yescharacteristic
19C2×C4×C22⋊D47C411yescharacteristic
20C2×C4×C22⋊D47C411yescharacteristic
21C2×C4×C22⋊D47C411yescharacteristic
22C2×#13093983C2211yescharacteristic
23C2×#13093981C2211yescharacteristic
24C2×#13093981C2211yescharacteristic
25C2×C42×D47C2211yescharacteristic
26C2×C42×D47C2211yescharacteristic
27#15689181—28no—
28C2×Dic47⋊C8C2212yes—
29C2×Dic47⋊C8C2212yes—
30C2×C22⋊(C47⋊C8)C2212yes—
31C2×(C2×D47)⋊C8C2212yes—
32C2×(C2×D47)⋊C8C2212yes—
33C2×(C2×D47)⋊C8C2212yes—
34C2×(C2×D47)⋊C8C2212yes—
35C2×#13093934C2212yes—
36C2×C8×Dic47C2212yes—
37#15688878C2211yescharacteristic
38#15688878C2211yescharacteristic
39#15688878C2212yes—
40C22×C42×C47C2211yescharacteristic
41C2×C22⋊C8×C47C2212yes—
42C47×#336C2211yescharacteristic
43C47×#336C2211yescharacteristic
44C47×#336C2212yes—
45C2×#10625350—24no—
46C2×#10625350—24no—
47C2×#10625350—22no—
48C2×#10625350—22no—
49C22×Dic94C2×C411yescharacteristic
50C22×D188C2×C411yescharacteristic
51C22×C22⋊D47C2×C411yescharacteristic
52C22×C22⋊D47C2×C411yescharacteristic
53C22×C4×D47C2×C411yescharacteristic
54C22×C4×D47C2×C411yescharacteristic
55C22×C4×D47C2×C411yescharacteristic
56C22×C4×D47C2×C411yescharacteristic
57C22×C4×D47C2311yescharacteristic
58C22×C4×D47C2311yescharacteristic
59C22×C47⋊C8C2312yes—
60C22×C47⋊C8C2312yes—
61C4×Dic94—24no—
62C4×Dic94—24no—
63C4×D188—24no—
64C4×D188—24no—
65C4×C22⋊D47—24no—
66C4×C22⋊D47—24no—
67C4×C22⋊D47—24no—
68C4×C22⋊D47—24no—
69#13093983D418yes—
70#13093981—24no—
71#13093981—24no—
72C2×C4⋊Dic47C2×C411yescharacteristic
73C2×C4⋊Dic47C2×C411yescharacteristic
74C2×Dic47⋊C4C2×C411yescharacteristic
75C2×Dic47⋊C4C2×C411yescharacteristic
76C2×Dic47⋊C4C2×C411yescharacteristic
77C2×Dic47⋊C4C2×C411yescharacteristic
78C2×C22⋊Dic47C2×C411yescharacteristic
79C2×C22⋊Dic47C2×C411yescharacteristic
80C2×(C2×D47)⋊C4C2×C411yescharacteristic
81C2×(C2×D47)⋊C4C2×C411yescharacteristic
82C2×(C2×D47)⋊C4C2×C411yescharacteristic
83C2×(C2×D47)⋊C4C2×C411yescharacteristic
84C2×C4×Dic47C2×C411yescharacteristic
85C2×C4×Dic47C2×C411yescharacteristic
86C2×C4×Dic47C2×C411yescharacteristic
87C2×C4×Dic47C2×C411yescharacteristic
88C2×C4×Dic47C2311yescharacteristic
89C2×C4×Dic47C2311yescharacteristic
90C42×D47—24no—
91C42×D47—24no—
92Dic47⋊C8—24no—
93Dic47⋊C8—24no—
94C22⋊(C47⋊C8)—24no—
95(C2×D47)⋊C8—24no—
96(C2×D47)⋊C8—24no—
97(C2×D47)⋊C8—24no—
98(C2×D47)⋊C8—24no—
99#13093934—24no—
100C8×Dic47—24no—
101C23×C4×C47C2×C411yescharacteristic
102C23×C4×C47C2×C411yescharacteristic
103C23×C4×C47C2311yescharacteristic
104C22×C8×C47C2312yes—
105C22×C8×C47C2312yes—
106C2×C42×C47—24no—
107C2×C42×C47C2×C411yescharacteristic
108C2×C42×C47C2×C411yescharacteristic
109C2×C42×C47C2311yescharacteristic
110C2×C42×C47C2311yescharacteristic
111C22⋊C8×C47—24no—
112C24×C47C22×C411yescharacteristic
113C23×D47C22×C411yescharacteristic
114C23×D47C22×C411yescharacteristic
115C22×C4×C47—22no—
116C22×C4×C47—21no—
117C22×C4×C47—21no—
118C22×C4×C47—22no—
119C22×C4×C47—22no—
120C22×C4×C47—22no—
121C22×C4×C47—22no—
122C22×C4×C47—41no—
123C22×C4×C47—41no—
124C22×C4×C47C22×C411yescharacteristic
125C22×C4×C47C22×C411yescharacteristic
126C22×C4×C47C22×C411yescharacteristic
127C22×C4×C47C22×C411yescharacteristic
128C22×C4×C47C22×C411yescharacteristic
129C22×C4×C47C2×D414yes—
130C22×C4×C47C2411yescharacteristic
131C2×C22⋊D47—22no—
132C2×C22⋊D47—22no—
133C2×C22⋊D47—22no—
134C2×C22⋊D47—22no—
135C2×C22⋊D47—22no—
136C2×C22⋊D47—22no—
137C22×Dic47C22×C411yescharacteristic
138C22×Dic47C22×C411yescharacteristic
139C22×Dic47C22×C411yescharacteristic
140C22×Dic47C22×C411yescharacteristic
141C22×Dic47C22×C411yescharacteristic
142C22×Dic47C22×C411yescharacteristic
143#10625350—216no—
144#10625350—216no—
145C2×D188—22no—
146C2×D188—22no—
147C2×D188—22no—
148C2×C4×D47—22no—
149C2×C4×D47—22no—
150C2×C4×D47—22no—
151C2×C4×D47—21no—
152C2×C4×D47—21no—
153C2×C4×D47—22no—
154C2×C4×D47—22no—
155C2×C4×D47—22no—
156C2×C4×D47—22no—
157C2×C4×D47—22no—
158C2×C4×D47—22no—
159C2×C4×D47—22no—
160C2×C4×D47—22no—
161C2×C4×D47—41no—
162C2×C4×D47—41no—
163C2×C4×D47—41no—
164C2×C4×D47—41no—
165C2×C4×D47#4114yes—
166C2×C4×D47#4112yes—
167C2×C4×D47#4112yes—
168C2×Dic94—22no—
169C2×Dic94—22no—
170C2×Dic94—22no—
171C2×C8×C47—22no—
172C2×C8×C47—22no—
173C2×C8×C47—22no—
174C2×C8×C47—22no—
175C2×C8×C47—22no—
176C2×C8×C47—22no—
177C42×C47—22no—
178C42×C47—22no—
179C42×C47—22no—
180C42×C47—22no—
181C22⋊Dic47C22⋊C414yes—
182C22⋊Dic47C22⋊C414yes—
183(C2×D47)⋊C4—22no—
184(C2×D47)⋊C4—22no—
185(C2×D47)⋊C4—22no—
186(C2×D47)⋊C4—22no—
187C4⋊Dic47C22⋊C414yes—
188C4⋊Dic47C22⋊C414yes—
189Dic47⋊C4—22no—
190Dic47⋊C4—22no—
191Dic47⋊C4—22no—
192Dic47⋊C4—22no—
193C4×Dic47—22no—
194C4×Dic47—22no—
195C4×Dic47—22no—
196C4×Dic47—22no—
197C4×Dic47C2×D414yes—
198C4×Dic47C2×D414yes—
199C2×C47⋊C8—22no—
200C2×C47⋊C8—22no—
201C2×C47⋊C8—22no—
202C2×C47⋊C8—22no—
203C2×C47⋊C8—22no—
204C2×C47⋊C8—22no—
205C23×C47—21no—
206C23×C47—21no—
207C23×C47—21no—
208C23×C47—21no—
209C23×C47—21no—
210C23×C47—21no—
211C23×C47C2×C22⋊C412yes—
212C23×C47C23×C411yescharacteristic
213C22×D47—21no—
214C22×D47—21no—
215C22×D47—21no—
216C22×D47—21no—
217C22×D47—21no—
218C22×D47—21no—
219C22×D47—21no—
220C22×D47—21no—
221C22×D47—21no—
222C22×D47—21no—
223C22×D47—21no—
224C22×D47—21no—
225C22×D47#11712yes—
226C22×D47#11712yes—
227C2×C4×C47—21no—
228C2×C4×C47—21no—
229C2×C4×C47—21no—
230C2×C4×C47—21no—
231C2×C4×C47—21no—
232C2×C4×C47—21no—
233C2×C4×C47—24no—
234C2×C4×C47—24no—
235C2×C4×C47—21no—
236C2×C4×C47—21no—
237C2×C4×C47—21no—
238C2×C4×C47—21no—
239C2×C4×C47—21no—
240C2×C4×C47—21no—
241C2×C4×C47—44no—
242C2×C4×C47—44no—
243C2×C4×C47C2×C22⋊C412yes—
244C2×C4×C47#13112yes—
245C2×C4×C47#13112yes—
246C2×C4×C47C22×D412yes—
247C2×C4×C47C2×#4112yes—
248C2×C4×C47C2×#4111yescharacteristic
249C2×C4×C47C2×#4111yescharacteristic
250C22⋊D47—28no—
251C22⋊D47—28no—
252C2×Dic47—21no—
253C2×Dic47—21no—
254C2×Dic47—21no—
255C2×Dic47—21no—
256C2×Dic47—21no—
257C2×Dic47—21no—
258C2×Dic47—21no—
259C2×Dic47—21no—
260C2×Dic47—21no—
261C2×Dic47—21no—
262C2×Dic47—21no—
263C2×Dic47—21no—
264C2×Dic47C2×C22⋊C412yes—
265C2×Dic47C2×C22⋊C412yes—
266C2×Dic47C2×C22⋊C412yes—
267C2×Dic47C2×C22⋊C412yes—
268C2×Dic47#11712yes—
269C2×Dic47#11712yes—
270D188—28no—
271C4×D47—24no—
272C4×D47—24no—
273C4×D47—24no—
274C4×D47—24no—
275C4×D47—44no—
276C4×D47—44no—
277C4×D47—44no—
278C4×D47—44no—
279Dic94—28no—
280C8×C47—42no—
281C8×C47—42no—
282C47⋊C8—42no—
283C47⋊C8—42no—
284#20720—471nomaximal, Sylow
285C22×C47—22no—
286C22×C47—22no—
287C22×C47—22no—
288C22×C47—41no—
289C22×C47—41no—
290C22×C47—41no—
291C22×C47—41no—
292C22×C47C22×C22⋊C411yescharacteristic
293C22×C47C2×#11711yescharacteristic
294C22×C47C2×#11711yescharacteristic
295C22×C47C2×#13111yescharacteristic
296C22×C47C2×#13111yescharacteristic
297C22×C47#56811yescharacteristic
298C22×C47#56811yescharacteristic
299C2×D47—22no—
300C2×D47—22no—
301C2×D47—22no—
302C2×D47—22no—
303C2×D47—22no—
304C2×D47—22no—
305C2×D47—41no—
306C2×D47—41no—
307C2×D47—41no—
308C2×D47—41no—
309C2×D47—41no—
310C2×D47—41no—
311C2×D47—41no—
312C2×D47—41no—
313C4×C47—22no—
314C4×C47—22no—
315C4×C47—22no—
316C4×C47—41no—
317C4×C47—41no—
318C4×C47#52512yes—
319C4×C47#52512yes—
320Dic47—22no—
321Dic47—22no—
322Dic47—22no—
323Dic47—22no—
324Dic47—22no—
325Dic47—22no—
326C2×C4×#41—471no—
327C2×#433—472no—
328#1817—471no—
329#1817D4711yescharacteristic, complemented
330#1798—472no—
331#1797—471no—
332#1797—471no—
333#1780—472no—
334#1750—472no—
335#1749—471no—
336C4×C22⋊C8—471no—
337C2×C47—42no—
338C2×C47#228011yescharacteristic
339C2×C47#288011yescharacteristic
340C2×C47#288011yescharacteristic
341C2×C47#292211yescharacteristic
342C2×C47#292311yescharacteristic
343C2×C47#292511yescharacteristic
344C2×C47#292511yescharacteristic
345D47—42no—
346D47—42no—
347C22×#41—471no—
348C4×#41—948no—
349C2×C4×Q8—471no—
350C2×C4×Q8—471no—
351C2×C4×D4—471no—
352C2×C4×D4—471no—
353C2×C4×D4—471no—
354C2×C4×D4—471no—
355C2×C4×D4—471no—
356C2×C4×D4—471no—
357C2×#117—471no—
358C2×#117—471no—
359C2×#117—471no—
360C22×C42—471no—
361C22×C42—471no—
362C22×C42C2×D4711yescharacteristic
363#433—948no—
364C2×C4⋊C8—472no—
365C2×C4⋊C8—472no—
366C2×C22⋊C8—472no—
367C2×C22⋊C8—472no—
368C2×C22⋊C8—472no—
369C2×C22⋊C8—472no—
370C2×C22⋊C8—472no—
371C2×C22⋊C8C2×D4712yes—
372C2×#97—472no—
373C2×C4×C8—472no—
374#336—471no—
375#336—471no—
376#336—472no—
377#336C2×D4711yescharacteristic
378#336C2×D4711yescharacteristic
379#336C2×D4712yes—
380C47#2072011yescharacteristic, complemented, Sylow
381C2×#41—944no—
382C2×#41—942no—
383C2×#41—942no—
384C2×#41—944no—
385C22×Q8—471no—
386C22×D4—471no—
387C22×D4—471no—
388C22×D4—471no—
389C23×C4—471no—
390C23×C4—471no—
391C23×C4—471no—
392C23×C4—471no—
393C23×C4—471no—
394C23×C4—471no—
395C23×C4C4×D4711yescharacteristic
396C23×C4C4×D4711yescharacteristic
397C23×C4C22×D4711yescharacteristic
398C22×C8—472no—
399C22×C8—472no—
400C22×C8C22×D4712yes—
401C22×C8C22×D4712yes—
402C4×Q8—944no—
403C4×Q8—944no—
404C4×D4—944no—
405C4×D4—944no—
406C4×D4—944no—
407C4×D4—944no—
408C4×D4—944no—
409C4×D4—944no—
410#117—478no—
411#117—944no—
412#117—944no—
413C2×C4⋊C4—471no—
414C2×C4⋊C4—471no—
415C2×C4⋊C4—471no—
416C2×C4⋊C4—471no—
417C2×C4⋊C4—471no—
418C2×C4⋊C4—471no—
419C2×C22⋊C4—471no—
420C2×C22⋊C4—471no—
421C2×C22⋊C4—471no—
422C2×C22⋊C4—471no—
423C2×C22⋊C4—471no—
424C2×C22⋊C4—471no—
425C2×C42—24no—
426C2×C42—471no—
427C2×C42—471no—
428C2×C42—471no—
429C2×C42—471no—
430C2×C42—471no—
431C2×C42—471no—
432C2×C42—944no—
433C2×C42—944no—
434C2×C42C4×D4711yescharacteristic
435C2×C42C4×D4711yescharacteristic
436C2×C42C22×D4711yescharacteristic
437C2×C42C22×D4711yescharacteristic
438C4⋊C8—944no—
439C4⋊C8—944no—
440C22⋊C8—24no—
441C22⋊C8—944no—
442C22⋊C8—944no—
443C22⋊C8—944no—
444C22⋊C8—944no—
445C22⋊C8—944no—
446#97—944no—
447C4×C8—944no—
448C24—471no—
449C24—471no—
450C24C2×C4×D4711yescharacteristic
451#41—9416no—
452#41—9416no—
453C2×Q8—942no—
454C2×Q8—942no—
455C2×Q8—942no—
456C2×D4—942no—
457C2×D4—942no—
458C2×D4—942no—
459C2×D4—942no—
460C2×D4—942no—
461C2×D4—942no—
462C2×D4—942no—
463C2×D4—942no—
464C2×D4—942no—
465C22×C4—22no—
466C22×C4—21no—
467C22×C4—21no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—41no—
473C22×C4—41no—
474C22×C4—471no—
475C22×C4—471no—
476C22×C4—471no—
477C22×C4—471no—
478C22×C4—471no—
479C22×C4—471no—
480C22×C4—474no—
481C22×C4—472no—
482C22×C4—472no—
483C22×C4—942no—
484C22×C4—942no—
485C22×C4—942no—
486C22×C4—942no—
487C22×C4—942no—
488C22×C4—942no—
489C22×C4—942no—
490C22×C4—941no—
491C22×C4—941no—
492C22×C4—942no—
493C22×C4—942no—
494C22×C4—942no—
495C22×C4—942no—
496C22×C4—1881no—
497C22×C4—1881no—
498C22×C4—1881no—
499C22×C4—1881no—
500C22×C4C2×C4×D4711yescharacteristic
501C22×C4C2×C4×D4711yescharacteristic
502C22×C4C2×C4×D4711yescharacteristic
503C22×C4C2×C4×D4711yescharacteristic
504C22×C4C2×C4×D4711yescharacteristic
505C22×C4D4×D4714yes—
506C22×C4C23×D4711yescharacteristic
507C2×C8—22no—
508C2×C8—22no—
509C2×C8—22no—
510C2×C8—22no—
511C2×C8—22no—
512C2×C8—22no—
513C2×C8—942no—
514C2×C8—942no—
515C2×C8—942no—
516C2×C8—942no—
517C2×C8—942no—
518C2×C8—942no—
519C4⋊C4—474no—
520C4⋊C4—474no—
521C4⋊C4—942no—
522C4⋊C4—942no—
523C4⋊C4—942no—
524C4⋊C4—942no—
525C22⋊C4—474no—
526C22⋊C4—474no—
527C22⋊C4—942no—
528C22⋊C4—942no—
529C22⋊C4—942no—
530C22⋊C4—942no—
531C42—22no—
532C42—22no—
533C42—22no—
534C42—22no—
535C42—474no—
536C42—474no—
537C42—942no—
538C42—942no—
539C42—942no—
540C42—942no—
541C23—21no—
542C23—21no—
543C23—21no—
544C23—21no—
545C23—21no—
546C23—21no—
547C23—472no—
548C23—472no—
549C23—941no—
550C23—941no—
551C23—941no—
552C23—941no—
553C23—941no—
554C23—941no—
555C23—941no—
556C23—941no—
557C23—941no—
558C23—941no—
559C23—941no—
560C23—941no—
561C23C22⋊C4×D4712yes—
562C23C22×C4×D4711yescharacteristic
563Q8—948no—
564D4—948no—
565D4—948no—
566D4—948no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—24no—
574C2×C4—24no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—44no—
582C2×C4—44no—
583C2×C4—472no—
584C2×C4—472no—
585C2×C4—472no—
586C2×C4—472no—
587C2×C4—472no—
588C2×C4—472no—
589C2×C4—941no—
590C2×C4—941no—
591C2×C4—941no—
592C2×C4—941no—
593C2×C4—941no—
594C2×C4—941no—
595C2×C4—941no—
596C2×C4—941no—
597C2×C4—941no—
598C2×C4—941no—
599C2×C4—941no—
600C2×C4—941no—
601C2×C4—944no—
602C2×C4—944no—
603C2×C4—944no—
604C2×C4—944no—
605C2×C4—1884no—
606C2×C4—1884no—
607C2×C4—1884no—
608C2×C4—1884no—
609C2×C4C22⋊C4×D4712yes—
610C2×C4#1309403012yes—
611C2×C4#1309403212yes—
612C2×C4C2×D4×D4712yes—
613C2×C4#41×D4712yes—
614C2×C4#41×D4711yescharacteristic
615C2×C4#41×D4711yescharacteristic
616C8—42no—
617C8—42no—
618C8—1882no—
619C8—1882no—
620C22—22no—
621C22—22no—
622C22—22no—
623C22—41no—
624C22—41no—
625C22—41no—
626C22—41no—
627C22—942no—
628C22—942no—
629C22—942no—
630C22—942no—
631C22—942no—
632C22—942no—
633C22—1881no—
634C22—1881no—
635C22—1881no—
636C22—1881no—
637C22—1881no—
638C22—1881no—
639C22—1881no—
640C22—1881no—
641C22C2×C22⋊C4×D4711yescharacteristic
642C22#117×D4711yescharacteristic
643C22#117×D4711yescharacteristic
644C22C2×#1309403011yescharacteristic
645C22C2×#1309403211yescharacteristic
646C22#1568988911yescharacteristic
647C22#1568988911yescharacteristic
648C4—22no—
649C4—22no—
650C4—22no—
651C4—41no—
652C4—41no—
653C4—942no—
654C4—942no—
655C4—942no—
656C4—942no—
657C4—942no—
658C4—942no—
659C4#1568963112yes—
660C4#1568963112yes—
661C2—42no—
662C2—1882no—
663C2—1882no—
664C2#2024576811yescharacteristic
665C2#2025183411yescharacteristic
666C2#2025183811yescharacteristic
667C2#2025202711yescharacteristic
668C2#2025203411yescharacteristic
669C2#2025204911yescharacteristic
670C2#2025205511yescharacteristic
subgroupG/H
center, FrattiniC22×C4C23×D47
commutator, socleC23×C47C23×C4
FittingC47×#1817C2
prime radicalC22×#10625350C22
cosocleC22×C4×C47C24
hypercenter#1817D47
hypocenterC47#20720
LCSG ▷ C23×C47 ▷ C47
UCSC1 ◁ C22×C4 ◁ #1817

Complements

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

NH ≅ G/Ntype
2. C2×#15689181C2semidirect
3. #20243456C2semidirect
4. #20243371C2semidirect
5. #20243367C2semidirect
6. #20243367C2semidirect
8. #20243153C2semidirect
9. #20243149C2semidirect
10. #20243145C2semidirect
11. C47×#1817C2semidirect
329. #1817D47semidirect
380. C47#20720semidirect

Maximal subgroups

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

subgroupindex
C2×C4×#106253502
C2×#156891812
#202434562
#202433712
#202433672
#202432992
#202431532
#202431492
#202431452
C47×#18172
#2072047