Finite Group Databasebeta

← #26389179

Subgroups — #26389179

Order 12032.

2322 conjugacy classes of subgroups.

The subgroup lattice has 650 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689718C212yesmaximal, complemented
2#20251714C212yesmaximal, complemented
3#20251687C212yesmaximal, complemented
4#20249502C211yesmaximal, characteristic, complemented
5#20249132C212yesmaximal, complemented
6#20248473C214yesmaximal, complemented
7#20248449C214yesmaximal, complemented
8#20248056C214yesmaximal, complemented
9#20247350C214yesmaximal, complemented
10#20246093C212yesmaximal, complemented
11#121×Dic47C212yesmaximal, complemented
12C2×#15688974C211yesmaximal, characteristic, complemented
13C47×#2656C211yesmaximal, characteristic
14#15689718—216no—
15C2×#13094017C2214yes—
16C2×#13094014C2214yes—
17C2×#13094011C2212yes—
18C2×#13094005C2214yes—
19C2×#13094005C2214yes—
20C2×#13094003C2212yes—
21C2×#13094001C2214yes—
22C2×D4×Dic47C2214yes—
23C2×D4×Dic47C2212yes—
24C22×Dic47⋊C4C2211yescharacteristic
25C22×Dic47⋊C4C2211yescharacteristic
26C22×C4×Dic47C2211yescharacteristic
27C22×C4×Dic47C2211yescharacteristic
28C22×C4×Dic47C2211yescharacteristic
29#15689055C2214yes—
30#15689054C2214yes—
31#15689046C2214yes—
32#15689046C2214yes—
33#15689044C2214yes—
34#15689037C2214yes—
35#15689028C2212yes—
36#15689026C2212yes—
37#15689026C2212yes—
38#15689026C2212yes—
39#15689026C2212yes—
40#15689022C2212yes—
41#15689019C2212yes—
42#15689019C2212yes—
43#15689017C2214yes—
44#15689017C2214yes—
45#15689016C2214yes—
46#15689016C2214yes—
47#15689011C2214yes—
48#15689011C2214yes—
49#15689009C2214yes—
50#15689003C2214yes—
51#15689003C2214yes—
52#15689003C2214yes—
53#15688998C2214yes—
54#15688988C2212yes—
55#15688983C2212yes—
56#15688983C2212yes—
57C4⋊C4×Dic47C2212yes—
58#15688974C2214yes—
59#15688974C2214yes—
60C22⋊C4×Dic47C2214yes—
61C2×C47⋊#95C2211yescharacteristic
62C2×#121×C47C2212yes—
63C2×#121×C47C2212yes—
64C2×#121×C47C2212yes—
65C47×#399C2214yes—
66C47×#392C2214yes—
67C2×#95×C47C2211yescharacteristic
68C22×Dic94C2312yes—
69C23×Dic47C2311yescharacteristic
70C23×Dic47C2311yescharacteristic
71C23×Dic47C2311yescharacteristic
72C23×Dic47C2314yes—
73C23×Dic47C2311yescharacteristic
74#13094017—216no—
75#13094014—216no—
76#13094011—44no—
77#13094005—216no—
78#13094005—216no—
79#13094003—28no—
80#13094001—216no—
81D4×Dic47—216no—
82D4×Dic47—44no—
83C2×C4⋊Dic47C2312yes—
84C2×C4⋊Dic47C2314yes—
85C2×C4⋊Dic47C2312yes—
86C2×C4⋊Dic47C2312yes—
87C2×C4⋊Dic47C2312yes—
88C2×C4⋊Dic47C2312yes—
89C2×Dic47⋊C4—24no—
90C2×Dic47⋊C4D418yescomplemented
91C2×Dic47⋊C4C2314yes—
92C2×Dic47⋊C4C2314yes—
93C2×Dic47⋊C4C2314yes—
94C2×Dic47⋊C4C2314yes—
95C2×Dic47⋊C4C2314yes—
96C2×C22⋊Dic47C2314yes—
97C2×C22⋊Dic47C2314yes—
98C2×C22⋊Dic47C2314yes—
99C2×C22⋊Dic47C2314yes—
100C2×C22⋊Dic47C2314yes—
101C2×C22⋊Dic47C2314yes—
102C2×C22⋊Dic47C2314yes—
103C2×C22⋊Dic47C2314yes—
104C2×C4×Dic47—24no—
105C2×C4×Dic47—42no—
106C2×C4×Dic47—42no—
107C2×C4×Dic47C2314yes—
108C2×C4×Dic47C2312yes—
109C2×C4×Dic47C2312yes—
110C2×C4×Dic47C2312yes—
111C2×C4×Dic47C2312yes—
112C2×C4×Dic47C2312yes—
113C2×C4×Dic47C2312yes—
114C47⋊#95C2314yes—
115C47⋊#95C2314yes—
116C47⋊#95C2314yes—
117C47⋊#95C2314yes—
118C47⋊#95C2314yes—
119C47⋊#95C2314yes—
120C47⋊#95C2314yes—
121C22×D4×C47C2314yes—
122C22×D4×C47C2312yes—
123C22×D4×C47C2312yes—
124C22×D4×C47C2312yes—
125C23×C4×C47C2311yescharacteristic
126C23×C4×C47C2311yescharacteristic
127C23×C4×C47C2311yescharacteristic
128#121×C47—28no—
129#121×C47—28no—
130#121×C47—28no—
131C2×C4⋊C4×C47C2312yes—
132C2×C4⋊C4×C47C2312yes—
133C2×C4⋊C4×C47C2312yes—
134C2×C22⋊C4×C47C2314yes—
135C2×C22⋊C4×C47C2314yes—
136C2×C22⋊C4×C47C2314yes—
137#95×C47C2314yes—
138C24×C47C2414yes—
139C24×C47C2411yescharacteristic
140C2×D4×C47—28no—
141C2×D4×C47—28no—
142C2×D4×C47—28no—
143C2×D4×C47—24no—
144C2×D4×C47—24no—
145C2×D4×C47—24no—
146C2×D4×C47—42no—
147C2×D4×C47—42no—
148C2×D4×C47—42no—
149C2×D4×C47—42no—
150C2×D4×C47—42no—
151C2×D4×C47—42no—
152C22×C4×C47—41no—
153C22×C4×C47—41no—
154C22×C4×C47—41no—
155C22×C4×C47—41no—
156C22×C4×C47—41no—
157C22×C4×C47—41no—
158C22×C4×C47C2×D414yes—
159C22×C4×C47#4114yes—
160C22×C4×C47#4114yes—
161C22×C4×C47C2414yes—
162C22×C4×C47C2412yes—
163C22×C4×C47C2412yes—
164C22×C4×C47C2412yes—
165C22×Dic47—22no—
166C22×Dic47—22no—
167C22×Dic47—22no—
168C22×Dic47—22no—
169C22×Dic47—22no—
170C22×Dic47—22no—
171C22×Dic47—28no—
172C22×Dic47—41no—
173C22×Dic47—41no—
174C22×Dic47—41no—
175C22×Dic47—41no—
176C22×Dic47—44no—
177C22×Dic47—44no—
178C22×Dic47C2×D414yes—
179C22×Dic47C2×D414yes—
180C22×Dic47C2412yes—
181C22×Dic47C2414yes—
182C22×Dic47C2414yes—
183C22×Dic47C2412yes—
184C22×Dic47C2412yes—
185C22×Dic47C2411yescharacteristic
186C22×Dic47C2411yescharacteristic
187C2×Dic94—24no—
188C2×Dic94—42no—
189C2×Dic94—42no—
190C4⋊C4×C47—24no—
191C4⋊C4×C47—24no—
192C4⋊C4×C47—24no—
193C22⋊C4×C47—28no—
194C22⋊C4×C47—28no—
195C22⋊C4×C47—28no—
196C22⋊Dic47—28no—
197C22⋊Dic47—28no—
198C22⋊Dic47—28no—
199C22⋊Dic47—28no—
200C22⋊Dic47—28no—
201C22⋊Dic47—28no—
202C22⋊Dic47—44no—
203C22⋊Dic47—44no—
204C4⋊Dic47—24no—
205C4⋊Dic47—28no—
206C4⋊Dic47—42no—
207C4⋊Dic47—42no—
208C4⋊Dic47—42no—
209C4⋊Dic47—42no—
210Dic47⋊C4—28no—
211Dic47⋊C4—28no—
212Dic47⋊C4—28no—
213Dic47⋊C4—28no—
214Dic47⋊C4C2×D4116yes—
215C4×Dic47—24no—
216C4×Dic47—24no—
217C4×Dic47—28no—
218C4×Dic47—24no—
219C4×Dic47—24no—
220C4×Dic47—42no—
221C4×Dic47—42no—
222C23×C47—21no—
223C23×C47—21no—
224C23×C47—21no—
225C23×C47—21no—
226C23×C47—24no—
227C23×C47—24no—
228C23×C47—24no—
229C23×C47—24no—
230C23×C47—24no—
231C23×C47—24no—
232C23×C47—24no—
233C23×C47C22×D412yes—
234C23×C47C2×#4112yes—
235C23×C47C2×#4112yes—
236C23×C47C2511yescharacteristic
237D4×C47—48no—
238D4×C47—416no—
239D4×C47—84no—
240D4×C47—84no—
241C2×C4×C47—22no—
242C2×C4×C47—22no—
243C2×C4×C47—22no—
244C2×C4×C47—22no—
245C2×C4×C47—22no—
246C2×C4×C47—22no—
247C2×C4×C47—24no—
248C2×C4×C47—24no—
249C2×C4×C47—24no—
250C2×C4×C47—44no—
251C2×C4×C47—44no—
252C2×C4×C47—44no—
253C2×C4×C47C22×D414yes—
254C2×C4×C47C2×#4114yes—
255C2×C4×C47C2×#4114yes—
256C2×Dic47—28no—
257C2×Dic47—28no—
258C2×Dic47—22no—
259C2×Dic47—21no—
260C2×Dic47—21no—
261C2×Dic47—22no—
262C2×Dic47—22no—
263C2×Dic47—21no—
264C2×Dic47—21no—
265C2×Dic47—22no—
266C2×Dic47—24no—
267C2×Dic47—24no—
268C2×Dic47—22no—
269C2×Dic47—22no—
270C2×Dic47—24no—
271C2×Dic47—24no—
272C2×Dic47—24no—
273C2×Dic47—24no—
274C2×Dic47—22no—
275C2×Dic47—44no—
276C2×Dic47—44no—
277C2×Dic47—88no—
278C2×Dic47C22×D414yes—
279C2×Dic47C22×D414yes—
280C2×Dic47H(5,𝔽2)12yes—
281C2×Dic472−1+412yes—
282Dic94—84no—
283#35004—471nomaximal, Sylow
284C22×C47—22no—
285C22×C47—22no—
286C22×C47—22no—
287C22×C47—22no—
288C22×C47—22no—
289C22×C47—22no—
290C22×C47—44no—
291C22×C47—44no—
292C22×C47—44no—
293C22×C47—44no—
294C22×C47—44no—
295C22×C47—44no—
296C22×C47—44no—
297C22×C47#53214yes—
298C22×C47C23×D411yescharacteristic
299C22×C47C22×#4111yescharacteristic
300C22×C47C22×#4111yescharacteristic
301C22×C47C2×H(5,𝔽2)11yescharacteristic
302C22×C47C2×2−1+411yescharacteristic
303C22×C47#58511yescharacteristic
304C22×C47#58511yescharacteristic
305C4×C47—24no—
306C4×C47—24no—
307C4×C47—24no—
308C4×C47—44no—
309Dic47—24no—
310Dic47—22no—
311Dic47—22no—
312Dic47—24no—
313Dic47—42no—
314Dic47—44no—
315Dic47—44no—
316C2×#539—472no—
317#2854—472no—
318#2850—472no—
319#2659—471no—
320#2656D4711yescharacteristic
321#2618—472no—
322#2556—474no—
323#2554—474no—
324#2519—474no—
325#2461—474no—
326#2325—472no—
327C4×#121—472no—
328C2×C4×C4⋊C4—471no—
329C2×C47—24no—
330C2×C47—84no—
331C2×C47C2×#53211yescharacteristic
332C2×C47#345911yescharacteristic
333C2×C47#346111yescharacteristic
334C2×C47#349111yescharacteristic
335C2×C47#349211yescharacteristic
336C2×C47#349611yescharacteristic
337C2×C47#349711yescharacteristic
338#539—9416no—
339C2×#126—474no—
340C2×#125—474no—
341C2×#124—472no—
342C2×#123—474no—
343C2×#123—474no—
344C2×C22⋊Q8—472no—
345C2×C22⋊Q8—474no—
346C2×#121C2×D4712yes—
347C2×#121C2×D4712yes—
348C2×#121C2×D4712yes—
349C2×C4×D4—474no—
350C2×C4×D4—472no—
351C22×C4⋊C4—471no—
352C22×C4⋊C4—471no—
353C22×C42—471no—
354C22×C42—471no—
355C22×C42—471no—
356#399C2×D4714yes—
357#398—474no—
358#398—474no—
359#396—474no—
360#396—474no—
361#395—474no—
362#394—474no—
363#392C2×D4714yes—
364#391—472no—
365#390—472no—
366#390—472no—
367#390—472no—
368#390—472no—
369#389—472no—
370#389—472no—
371#389—472no—
372#388—474no—
373#388—474no—
374#388—474no—
375#388—474no—
376#388—474no—
377#388—474no—
378#387—474no—
379#386—474no—
380#386—474no—
381#386—474no—
382#385—474no—
383#382—472no—
384#380—472no—
385#380—472no—
386C4×C4⋊C4—472no—
387C4×C4⋊C4—474no—
388C4×C4⋊C4—474no—
389C4×C22⋊C4—474no—
390C2×#95—471no—
391C2×#95C2×D4711yescharacteristic
392C47#3500411yescharacteristic, complemented, Sylow
393C22×Q8—472no—
394C22×D4C22×D4714yes—
395C22×D4C22×D4712yes—
396C22×D4C22×D4712yes—
397C22×D4C22×D4712yes—
398C23×C4—471no—
399C23×C4—471no—
400C23×C4—471no—
401C23×C4—471no—
402C23×C4—474no—
403C23×C4C22×D4711yescharacteristic
404C23×C4C22×D4711yescharacteristic
405C23×C4C22×D4711yescharacteristic
406#126—9416no—
407#125—9416no—
408#124—1884no—
409#123—9416no—
410#123—9416no—
411C22⋊Q8—9416no—
412C22⋊Q8—948no—
413#121—28no—
414#121—28no—
415#121—28no—
416C4×D4—9416no—
417C4×D4—1884no—
418C2×C4⋊C4—478no—
419C2×C4⋊C4—474no—
420C2×C4⋊C4—474no—
421C2×C4⋊C4—474no—
422C2×C4⋊C4—474no—
423C2×C4⋊C4—474no—
424C2×C4⋊C4—472no—
425C2×C4⋊C4—474no—
426C2×C4⋊C4—472no—
427C2×C4⋊C4—472no—
428C2×C4⋊C4—472no—
429C2×C4⋊C4—472no—
430C2×C4⋊C4—944no—
431C2×C4⋊C4C22×D4712yes—
432C2×C4⋊C4C22×D4712yes—
433C2×C4⋊C4C22×D4712yes—
434C2×C22⋊C4—474no—
435C2×C22⋊C4—474no—
436C2×C22⋊C4—474no—
437C2×C22⋊C4—474no—
438C2×C22⋊C4—474no—
439C2×C22⋊C4—474no—
440C2×C22⋊C4—474no—
441C2×C22⋊C4—474no—
442C2×C22⋊C4C22×D4714yes—
443C2×C22⋊C4C22×D4714yes—
444C2×C22⋊C4C22×D4714yes—
445C2×C42—474no—
446C2×C42—472no—
447C2×C42—472no—
448C2×C42—472no—
449C2×C42—472no—
450C2×C42—472no—
451C2×C42—472no—
452C2×C42—944no—
453C2×C42—1882no—
454C2×C42—1882no—
455#95—474no—
456#95—474no—
457#95—474no—
458#95—474no—
459#95—474no—
460#95—474no—
461#95—474no—
462#95C22×D4714yes—
463C24C23×D4714yes—
464C24C23×D4711yescharacteristic
465C2×Q8—944no—
466C2×Q8—1882no—
467C2×Q8—1882no—
468C2×D4—28no—
469C2×D4—28no—
470C2×D4—28no—
471C2×D4—24no—
472C2×D4—24no—
473C2×D4—24no—
474C2×D4—42no—
475C2×D4—42no—
476C2×D4—42no—
477C2×D4—42no—
478C2×D4—42no—
479C2×D4—42no—
480C22×C4—41no—
481C22×C4—41no—
482C22×C4—41no—
483C22×C4—41no—
484C22×C4—41no—
485C22×C4—41no—
486C22×C4—474no—
487C22×C4—471no—
488C22×C4—471no—
489C22×C4—472no—
490C22×C4—472no—
491C22×C4—474no—
492C22×C4—472no—
493C22×C4—474no—
494C22×C4—474no—
495C22×C4—942no—
496C22×C4—942no—
497C22×C4—948no—
498C22×C4—942no—
499C22×C4—942no—
500C22×C4—942no—
501C22×C4—942no—
502C22×C4—1884no—
503C22×C4—1884no—
504C22×C4—1881no—
505C22×C4—1881no—
506C22×C4—1881no—
507C22×C4—1881no—
508C22×C4D4×D4714yes—
509C22×C4#1062535214yes—
510C22×C4#1062535214yes—
511C22×C4C23×D4714yes—
512C22×C4C23×D4712yes—
513C22×C4C23×D4712yes—
514C22×C4C23×D4712yes—
515C4⋊C4—24no—
516C4⋊C4—24no—
517C4⋊C4—24no—
518C4⋊C4—4716no—
519C4⋊C4—948no—
520C4⋊C4—948no—
521C4⋊C4—948no—
522C4⋊C4—948no—
523C4⋊C4—944no—
524C4⋊C4—948no—
525C4⋊C4—1882no—
526C4⋊C4—1882no—
527C4⋊C4—1882no—
528C4⋊C4—1882no—
529C22⋊C4—28no—
530C22⋊C4—28no—
531C22⋊C4—28no—
532C22⋊C4—948no—
533C22⋊C4—948no—
534C22⋊C4—948no—
535C22⋊C4—948no—
536C22⋊C4—948no—
537C22⋊C4—948no—
538C22⋊C4—1884no—
539C22⋊C4—1884no—
540C42—944no—
541C42—944no—
542C42—948no—
543C42—944no—
544C42—944no—
545C42—1882no—
546C42—1882no—
547C23—21no—
548C23—21no—
549C23—21no—
550C23—21no—
551C23—24no—
552C23—24no—
553C23—24no—
554C23—24no—
555C23—24no—
556C23—24no—
557C23—24no—
558C23C2×D4×D4712yes—
559C23C2×#1062535212yes—
560C23C2×#1062535212yes—
561C23C24×D4711yescharacteristic
562Q8—3764no—
563D4—48no—
564D4—416no—
565D4—84no—
566D4—84no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—24no—
574C2×C4—24no—
575C2×C4—24no—
576C2×C4—44no—
577C2×C4—44no—
578C2×C4—44no—
579C2×C4—474no—
580C2×C4—474no—
581C2×C4—472no—
582C2×C4—472no—
583C2×C4—948no—
584C2×C4—948no—
585C2×C4—941no—
586C2×C4—941no—
587C2×C4—942no—
588C2×C4—942no—
589C2×C4—942no—
590C2×C4—941no—
591C2×C4—941no—
592C2×C4—942no—
593C2×C4—944no—
594C2×C4—944no—
595C2×C4—942no—
596C2×C4—942no—
597C2×C4—944no—
598C2×C4—944no—
599C2×C4—942no—
600C2×C4—944no—
601C2×C4—944no—
602C2×C4—3768no—
603C2×C4—1884no—
604C2×C4—1884no—
605C2×C4C2×D4×D4714yes—
606C2×C4C2×#1062535214yes—
607C2×C4C2×#1062535214yes—
608C22—22no—
609C22—22no—
610C22—22no—
611C22—22no—
612C22—22no—
613C22—22no—
614C22—44no—
615C22—44no—
616C22—44no—
617C22—44no—
618C22—44no—
619C22—44no—
620C22—44no—
621C22#1568967114yes—
622C22C22×D4×D4711yescharacteristic
623C22C22×#1062535211yescharacteristic
624C22C22×#1062535211yescharacteristic
625C22H(5,𝔽2)×D4711yescharacteristic
626C222−1+4×D4711yescharacteristic
627C22#1568995811yescharacteristic
628C22#1568995811yescharacteristic
629C4—24no—
630C4—24no—
631C4—24no—
632C4—44no—
633C4—942no—
634C4—942no—
635C4—944no—
636C4—944no—
637C4—1882no—
638C4—1884no—
639C4—1884no—
640C2—24no—
641C2—84no—
642C2C2×#1568967111yescharacteristic
643C2#2025679411yescharacteristic
644C2#2025681611yescharacteristic
645C2#2025708511yescharacteristic
646C2#2025709511yescharacteristic
647C2#2025714911yescharacteristic
648C2#2025715711yescharacteristic
subgroupG/H
center, FrattiniC23C24×D47
commutator, socle, cosocleC23×C47C25
Fitting, prime radicalC47×#2656C2
hypercenter#2656D47
hypocenterC47#35004
LCSG ▷ C23×C47 ▷ C47
UCSC1 ◁ C23 ◁ #2656

Complements

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

NH ≅ G/Ntype
1. C2×#15689718C2semidirect
2. #20251714C2semidirect
3. #20251687C2semidirect
4. #20249502C2semidirect
5. #20249132C2semidirect
6. #20248473C2semidirect
7. #20248449C2semidirect
8. #20248056C2semidirect
9. #20247350C2semidirect
10. #20246093C2semidirect
11. #121×Dic47C2semidirect
12. C2×#15688974C2semidirect
90. C2×Dic47⋊C4D4semidirect
392. C47#35004semidirect

Maximal subgroups

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

subgroupindex
C2×#156897182
#202517142
#202516872
#202495022
#202491322
#202484732
#202484492
#202480562
#202473502
#202460932
#121×Dic472
C2×#156889742
C47×#26562
#3500447