Finite Group Databasebeta

← #26234622

Subgroups — #26234622

Order 12032.

1010 conjugacy classes of subgroups.

The subgroup lattice has 656 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689224C211yesmaximal, characteristic
2C2×SD16×Dic47C211yesmaximal, characteristic
3#20246306C211yesmaximal, characteristic, complemented
4D4×Dic47⋊C4C211yesmaximal, characteristic
5#20244126C211yesmaximal, characteristic, complemented
6#20244119C211yesmaximal, characteristic, complemented
7#20243654C211yesmaximal, characteristic, complemented
8#20243649C211yesmaximal, characteristic
9#20243642C211yesmaximal, characteristic, complemented
10#20243633C211yesmaximal, characteristic
11#20243555C211yesmaximal, characteristic, complemented
12#20243396C211yesmaximal, characteristic
13#20243391C211yesmaximal, characteristic, complemented
14#20243264C211yesmaximal, characteristic, complemented
15C2×C47×#440C211yesmaximal, characteristic
16C2×Q8×Dic47C2211yescharacteristic
17C2×Q8×Dic47C2211yescharacteristic
18C2×D4×Dic47C2211yescharacteristic
19C2×D4×Dic47C2211yescharacteristic
20C22×Dic47⋊C4—21no—
21#15689224C418yescomplemented
22SD16×Dic47—24no—
23C2×#13093958C2211yescharacteristic
24C2×#13093957C2211yescharacteristic
25C2×#13093956C2211yescharacteristic
26C2×#13093955C2211yescharacteristic
27C2×Dic47⋊C8C2211yescharacteristic
28C2×#13093948C2211yescharacteristic
29C2×#13093948C2211yescharacteristic
30C2×#13093945C2211yescharacteristic
31C2×#13093945C2211yescharacteristic
32C2×#13093934C2211yescharacteristic
33C2×C8×Dic47C2211yescharacteristic
34#15689021—21no—
35#15689021C2211yescharacteristic
36#15689021C2211yescharacteristic
37#15689019—21no—
38#15689019C2211yescharacteristic
39#15689019C2211yescharacteristic
40#15688996—21no—
41#15688995—21no—
42#15688992C2211yescharacteristic
43#15688988—21no—
44#15688988C2211yescharacteristic
45#15688981—21no—
46C4×Dic47⋊C4—21no—
47C4×Dic47⋊C4C2211yescharacteristic
48#15688886C2211yescharacteristic
49#15688886C2211yescharacteristic
50#15688885C2211yescharacteristic
51#15688885C2211yescharacteristic
52#15688878C2211yescharacteristic
53#15688877C2211yescharacteristic
54C22×SD16×C47C2211yescharacteristic
55C2×C4×Q8×C47C2211yescharacteristic
56C2×C4×D4×C47C2211yescharacteristic
57C47×#440—24no—
58C2×#107×C47C2211yescharacteristic
59C2×#103×C47C2211yescharacteristic
60C2×#102×C47C2211yescharacteristic
61C2×#97×C47C2211yescharacteristic
62C23×Dic47—21no—
63C23×Dic47—21no—
64C22×C47⋊C8C2311yescharacteristic
65C22×C47⋊C8C2311yescharacteristic
66Q8×Dic47—22no—
67Q8×Dic47—22no—
68Q8×Dic47—41no—
69Q8×Dic47C2×C414yescomplemented
70D4×Dic47—22no—
71D4×Dic47—22no—
72D4×Dic47—41no—
73D4×Dic47C2×C414yes—
74C2×C4⋊Dic47—21no—
75C2×C4⋊Dic47—21no—
76C2×C4⋊Dic47C2311yescharacteristic
77C2×C4⋊Dic47C2311yescharacteristic
78C2×C4⋊Dic47C2311yescharacteristic
79C2×C4⋊Dic47C2311yescharacteristic
80C2×Dic47⋊C4—21no—
81C2×Dic47⋊C4—21no—
82C2×Dic47⋊C4—21no—
83C2×Dic47⋊C4—21no—
84C2×Dic47⋊C4—21no—
85C2×Dic47⋊C4—21no—
86C2×Dic47⋊C4—44no—
87C2×Dic47⋊C4D411yescharacteristic
88C2×Dic47⋊C4D411yescharacteristic
89C2×C22⋊Dic47—21no—
90C2×C22⋊Dic47—21no—
91C2×C4×Dic47—21no—
92C2×C4×Dic47—21no—
93C2×C4×Dic47C2311yescharacteristic
94C2×C4×Dic47C2311yescharacteristic
95#13093958C2×C414yescomplemented
96#13093957—41no—
97#13093956—22no—
98#13093955—41no—
99Dic47⋊C8—22no—
100#13093948—22no—
101#13093948C2×C414yescomplemented
102#13093945—22no—
103#13093945C2×C414yes—
104#13093934C2×C414yescomplemented
105C8×Dic47—22no—
106C47⋊#95—21no—
107C47⋊#95—21no—
108C47⋊#95—21no—
109C47⋊#95—21no—
110C47⋊#95—21no—
111C47⋊#95—21no—
112C47⋊#95D411yescharacteristic
113C47⋊#95D411yescharacteristic
114C22×Q8×C47C2311yescharacteristic
115C22×D4×C47C2311yescharacteristic
116C23×C4×C47—21no—
117C2×SD16×C47C2×C414yes—
118C2×SD16×C47D414yes—
119C2×SD16×C47Q814yes—
120C22×C8×C47C2311yescharacteristic
121C22×C8×C47C2311yescharacteristic
122C4×Q8×C47—22no—
123C4×Q8×C47—41no—
124C4×D4×C47—22no—
125C4×D4×C47—41no—
126C2×C4⋊C4×C47—21no—
127C2×C4⋊C4×C47C2311yescharacteristic
128C2×C4⋊C4×C47C2311yescharacteristic
129C2×C22⋊C4×C47—21no—
130C2×C42×C47—21no—
131C2×C42×C47C2311yescharacteristic
132#107×C47—22no—
133#103×C47—22no—
134#102×C47—22no—
135#97×C47—22no—
136C24×C47—21no—
137C2×Q8×C47—21no—
138C2×Q8×C47—21no—
139C2×Q8×C47—21no—
140C2×Q8×C47C22×C412yes—
141C2×Q8×C47C2×D412yes—
142C2×Q8×C47C2×Q812yes—
143C2×D4×C47—21no—
144C2×D4×C47—21no—
145C2×D4×C47—21no—
146C2×D4×C47C22×C412yes—
147C2×D4×C47C2×D412yes—
148C2×D4×C47C2×Q812yes—
149C22×C4×C47—21no—
150C22×C4×C47—21no—
151C22×C4×C47—21no—
152C22×C4×C47—42no—
153C22×C4×C47—42no—
154C22×C4×C47—81no—
155C22×C4×C47C2×D411yescharacteristic
156C22×C4×C47C2×D411yescharacteristic
157C22×C4×C47C2411yescharacteristic
158C22×Dic47—21no—
159C22×Dic47—21no—
160C22×Dic47—24no—
161C22×Dic47—21no—
162C22×Dic47—21no—
163C22×Dic47—42no—
164C22×Dic47—42no—
165C22×Dic47—42no—
166C22×Dic47—42no—
167C22×Dic47—81no—
168C22×Dic47C2×D411yescharacteristic
169C22×Dic47C2×D411yescharacteristic
170C22×Dic47C2×D411yescharacteristic
171C22×Dic47C2×D411yescharacteristic
172SD16×C47C4⋊C4116yescomplemented
173C2×C8×C47—21no—
174C2×C8×C47—21no—
175C2×C8×C47—21no—
176C2×C8×C47C22×C412yes—
177C2×C8×C47C2×D412yes—
178C2×C8×C47C2×Q812yes—
179C4⋊C4×C47—21no—
180C4⋊C4×C47—21no—
181C4⋊C4×C47—21no—
182C4⋊C4×C47—21no—
183C4⋊C4×C47—42no—
184C22⋊C4×C47—42no—
185C42×C47—21no—
186C42×C47—21no—
187C42×C47—42no—
188C22⋊Dic47—24no—
189C22⋊Dic47—42no—
190C4⋊Dic47—21no—
191C4⋊Dic47—21no—
192C4⋊Dic47—24no—
193C4⋊Dic47—21no—
194C4⋊Dic47—21no—
195C4⋊Dic47—21no—
196C4⋊Dic47—21no—
197C4⋊Dic47—42no—
198C4⋊Dic47C22×C412yes—
199C4⋊Dic47C22×C412yes—
200Dic47⋊C4—22no—
201Dic47⋊C4—41no—
202Dic47⋊C4—42no—
203Dic47⋊C4—42no—
204Dic47⋊C4—42no—
205Dic47⋊C4—42no—
206Dic47⋊C4—42no—
207Dic47⋊C4—81no—
208C4×Dic47—21no—
209C4×Dic47—21no—
210C4×Dic47—24no—
211C4×Dic47—42no—
212C4×Dic47C22×C412yes—
213C4×Dic47#4112yes—
214C2×C47⋊C8—21no—
215C2×C47⋊C8—21no—
216C2×C47⋊C8—21no—
217C2×C47⋊C8—21no—
218C2×C47⋊C8—21no—
219C2×C47⋊C8C22×C412yes—
220C23×C47—22no—
221C23×C47—22no—
222C23×C47—22no—
223C23×C47—41no—
224C23×C47—41no—
225C23×C47—41no—
226C23×C47—41no—
227C23×C47C22×D411yescharacteristic
228Q8×C47—22no—
229Q8×C47—22no—
230Q8×C47—22no—
231Q8×C47C2×C4⋊C414yescomplemented
232D4×C47—22no—
233D4×C47—22no—
234D4×C47—22no—
235D4×C47C2×C4⋊C414yes—
236C2×C4×C47—22no—
237C2×C4×C47—22no—
238C2×C4×C47—22no—
239C2×C4×C47—21no—
240C2×C4×C47—21no—
241C2×C4×C47—21no—
242C2×C4×C47—21no—
243C2×C4×C47—21no—
244C2×C4×C47—21no—
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×C47—84no—
252C2×C4×C47C23×C411yescharacteristic
253C2×C4×C47C22×D411yescharacteristic
254C2×C4×C47C22×Q811yescharacteristic
255C2×C4×C47C2×#4111yescharacteristic
256C2×C4×C47H(5,𝔽2)11yescharacteristic
257C2×C4×C472−1+411yescharacteristic
258C2×Dic47—21no—
259C2×Dic47—21no—
260C2×Dic47—21no—
261C2×Dic47—21no—
262C2×Dic47—21no—
263C2×Dic47—21no—
264C2×Dic47—22no—
265C2×Dic47—22no—
266C2×Dic47—21no—
267C2×Dic47—21no—
268C2×Dic47—21no—
269C2×Dic47—48no—
270C2×Dic47—41no—
271C2×Dic47—41no—
272C2×Dic47—41no—
273C2×Dic47—41no—
274C2×Dic47—41no—
275C2×Dic47—41no—
276C2×Dic47—41no—
277C2×Dic47—41no—
278C2×Dic47—41no—
279C2×Dic47—41no—
280C2×Dic47—84no—
281C2×Dic47C4×D412yes—
282C2×Dic47C4×D412yes—
283C2×Dic47AGL(1,ℤ/8)12yes—
284C8×C47—41no—
285C8×C47C2×C4⋊C414yescomplemented
286C47⋊C8—22no—
287C47⋊C8—41no—
288#22677—471nomaximal, Sylow
289C22×C47—24no—
290C22×C47—42no—
291C22×C47—42no—
292C22×C47—42no—
293C22×C47—42no—
294C22×C47—42no—
295C22×C47—42no—
296C22×C47C2×C4×D411yescharacteristic
297C22×C47D4211yescharacteristic
298C22×C47D4×Q811yescharacteristic
299C22×C47C2×AGL(1,ℤ/8)11yescharacteristic
300C22×C47#57511yescharacteristic
301C22×C47#57711yescharacteristic
302C22×C47#57811yescharacteristic
303C4×C47—24no—
304C4×C47—41no—
305C4×C47—41no—
306C4×C47—81no—
307C4×C47—81no—
308C4×C47C22×C4⋊C411yescharacteristic
309C4×C47#52011yescharacteristic
310C4×C47#54711yescharacteristic
311C4×C47#55411yescharacteristic
312Dic47—22no—
313Dic47—22no—
314Dic47—22no—
315Dic47—42no—
316Dic47—42no—
317Dic47—41no—
318Dic47—81no—
319Dic47—81no—
320C2×#440—471no—
321C2×#440D4711yescharacteristic
322C2×C4×SD16—471no—
323#2351—471no—
324D4×C4⋊C4—471no—
325#1943—471no—
326#1942—471no—
327#1865—471no—
328#1864—471no—
329#1863—471no—
330#1862—471no—
331#1847—471no—
332#1804—471no—
333#1803—471no—
334#1771—471no—
335C2×C47—44no—
336C2×C47D4×C4⋊C411yescharacteristic
337C2×C47C4×AGL(1,ℤ/8)11yescharacteristic
338C2×C47#294711yescharacteristic
339C2×C47#326511yescharacteristic
340C2×C47#326911yescharacteristic
341C2×C47#338611yescharacteristic
342C2×C47#338811yescharacteristic
343C22×SD16C2×D4711yescharacteristic
344C2×C4×Q8—471no—
345C2×C4×Q8—471no—
346C2×C4×Q8C2×D4711yescharacteristic
347C2×C4×D4—471no—
348C2×C4×D4—471no—
349C2×C4×D4C2×D4711yescharacteristic
350C22×C4⋊C4—941no—
351#440—24no—
352#440—478no—
353C4×SD16—944no—
354C2×#107—471no—
355C2×#107—471no—
356C2×#107C2×D4711yescharacteristic
357C2×#106—471no—
358C2×#106—471no—
359C2×C4⋊C8—471no—
360C2×#103—471no—
361C2×#103—471no—
362C2×#103C2×D4711yescharacteristic
363C2×#102—471no—
364C2×#102—471no—
365C2×#102C2×D4711yescharacteristic
366C2×#97—471no—
367C2×#97C2×D4711yescharacteristic
368C2×C4×C8—471no—
369#389—471no—
370#389—471no—
371#389—471no—
372#389—471no—
373#389—941no—
374#389—941no—
375#385—941no—
376#385—941no—
377#384—471no—
378#382—471no—
379#382—941no—
380#380—941no—
381C4×C4⋊C4—471no—
382C4×C4⋊C4—941no—
383#340—471no—
384#340—471no—
385#340—471no—
386#340—471no—
387#336—471no—
388#336—471no—
389C47#2267711yescharacteristic, complemented, Sylow
390C22×Q8C22×D4711yescharacteristic
391C22×D4C22×D4711yescharacteristic
392C23×C4—21no—
393C23×C4—941no—
394C23×C4—941no—
395C2×SD16Dic9414yes—
396C2×SD16C4×D4714yes—
397C2×SD16C22⋊D4714yes—
398C22×C8—471no—
399C22×C8—471no—
400C22×C8C22×D4711yescharacteristic
401C22×C8C22×D4711yescharacteristic
402C4×Q8—22no—
403C4×Q8—41no—
404C4×Q8—474no—
405C4×Q8—942no—
406C4×Q8—942no—
407C4×Q8—1881no—
408C4×D4—22no—
409C4×D4—41no—
410C4×D4—474no—
411C4×D4—942no—
412C4×D4—942no—
413C4×D4—1881no—
414C2×C4⋊C4—21no—
415C2×C4⋊C4—471no—
416C2×C4⋊C4—471no—
417C2×C4⋊C4—471no—
418C2×C4⋊C4—471no—
419C2×C4⋊C4—471no—
420C2×C4⋊C4—471no—
421C2×C4⋊C4—941no—
422C2×C4⋊C4—941no—
423C2×C4⋊C4—941no—
424C2×C4⋊C4—941no—
425C2×C4⋊C4—941no—
426C2×C4⋊C4—941no—
427C2×C4⋊C4—941no—
428C2×C4⋊C4—941no—
429C2×C4⋊C4—1884no—
430C2×C4⋊C4C22×D4711yescharacteristic
431C2×C4⋊C4C22×D4711yescharacteristic
432C2×C22⋊C4—21no—
433C2×C22⋊C4—941no—
434C2×C22⋊C4—941no—
435C2×C42—21no—
436C2×C42—471no—
437C2×C42—471no—
438C2×C42—941no—
439C2×C42—941no—
440C2×C42C22×D4711yescharacteristic
441#107—22no—
442#107—474no—
443#107—1881no—
444#106—942no—
445#106—1881no—
446C4⋊C8—942no—
447#103—22no—
448#103—474no—
449#103—942no—
450#102—22no—
451#102—474no—
452#102—942no—
453#97—22no—
454#97—474no—
455C4×C8—942no—
456#95—471no—
457#95—471no—
458#95—941no—
459#95—941no—
460#95—941no—
461#95—941no—
462#95—941no—
463#95—941no—
464C24—21no—
465C2×Q8—21no—
466C2×Q8—21no—
467C2×Q8—21no—
468C2×Q8C2×Dic9412yes—
469C2×Q8C2×C4×D4712yes—
470C2×Q8C2×C22⋊D4712yes—
471C2×D4—21no—
472C2×D4—21no—
473C2×D4—21no—
474C2×D4C2×Dic9412yes—
475C2×D4C2×C4×D4712yes—
476C2×D4C2×C22⋊D4712yes—
477C22×C4—21no—
478C22×C4—21no—
479C22×C4—21no—
480C22×C4—42no—
481C22×C4—42no—
482C22×C4—81no—
483C22×C4—471no—
484C22×C4—471no—
485C22×C4—471no—
486C22×C4—471no—
487C22×C4—944no—
488C22×C4—941no—
489C22×C4—941no—
490C22×C4—941no—
491C22×C4—941no—
492C22×C4—3761no—
493C22×C4—1882no—
494C22×C4—1882no—
495C22×C4—1882no—
496C22×C4—1882no—
497C22×C4D4×D4711yescharacteristic
498C22×C4D4×D4711yescharacteristic
499C22×C4C23×D4711yescharacteristic
500SD16Dic47⋊C4116yescomplemented
501C2×C8—21no—
502C2×C8—21no—
503C2×C8—21no—
504C2×C8—472no—
505C2×C8—941no—
506C2×C8—941no—
507C2×C8—941no—
508C2×C8—941no—
509C2×C8—941no—
510C2×C8C2×Dic9412yes—
511C2×C8C2×C4×D4712yes—
512C2×C8C2×C22⋊D4712yes—
513C4⋊C4—21no—
514C4⋊C4—21no—
515C4⋊C4—21no—
516C4⋊C4—21no—
517C4⋊C4—42no—
518C4⋊C4—472no—
519C4⋊C4—472no—
520C4⋊C4—942no—
521C4⋊C4—944no—
522C4⋊C4—941no—
523C4⋊C4—941no—
524C4⋊C4—941no—
525C4⋊C4—941no—
526C4⋊C4—941no—
527C4⋊C4—941no—
528C4⋊C4—3761no—
529C4⋊C4—1881no—
530C4⋊C4—1882no—
531C4⋊C4—1882no—
532C4⋊C4—1882no—
533C4⋊C4—1882no—
534C4⋊C4—1882no—
535C4⋊C4—1882no—
536C22⋊C4—42no—
537C22⋊C4—944no—
538C22⋊C4—1882no—
539C42—21no—
540C42—21no—
541C42—42no—
542C42—472no—
543C42—472no—
544C42—944no—
545C42—941no—
546C42—941no—
547C42—1882no—
548C23—22no—
549C23—22no—
550C23—22no—
551C23—41no—
552C23—41no—
553C23—41no—
554C23—41no—
555C23C2×D4×D4711yescharacteristic
556Q8—22no—
557Q8—22no—
558Q8—22no—
559Q8C2×Dic47⋊C414yescomplemented
560D4—22no—
561D4—22no—
562D4—22no—
563D4C2×Dic47⋊C414yes—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—84no—
580C2×C4—472no—
581C2×C4—472no—
582C2×C4—472no—
583C2×C4—941no—
584C2×C4—941no—
585C2×C4—941no—
586C2×C4—941no—
587C2×C4—941no—
588C2×C4—941no—
589C2×C4—941no—
590C2×C4—942no—
591C2×C4—942no—
592C2×C4—941no—
593C2×C4—941no—
594C2×C4—3764no—
595C2×C4—1888no—
596C2×C4—1881no—
597C2×C4—1881no—
598C2×C4—1881no—
599C2×C4—1881no—
600C2×C4—1881no—
601C2×C4—1881no—
602C2×C4—1881no—
603C2×C4—1881no—
604C2×C4—1881no—
605C2×C4—1881no—
606C2×C4C22×C4×D4711yescharacteristic
607C2×C4C22×C22⋊D4711yescharacteristic
608C2×C4C22×Dic9411yescharacteristic
609C2×C4#41×D4711yescharacteristic
610C2×C4#1309406911yescharacteristic
611C2×C4#1309407111yescharacteristic
612C8—41no—
613C8—942no—
614C8—1881no—
615C8C2×Dic47⋊C414yescomplemented
616C22—24no—
617C22—42no—
618C22—42no—
619C22—42no—
620C22—42no—
621C22—42no—
622C22—42no—
623C22C4×D4×D4711yescharacteristic
624C22D4×C22⋊D4711yescharacteristic
625C22D4×Dic9411yescharacteristic
626C22AGL(1,ℤ/8)×D4711yescharacteristic
627C22#1568992211yescharacteristic
628C22#1568993311yescharacteristic
629C22#1568993911yescharacteristic
630C4—24no—
631C4—41no—
632C4—41no—
633C4—81no—
634C4—81no—
635C4—942no—
636C4—942no—
637C4—942no—
638C4—3761no—
639C4—3761no—
640C4—1882no—
641C4—1882no—
642C4—1881no—
643C4C22×Dic47⋊C411yescharacteristic
644C4#1568960211yescharacteristic
645C4#1568977211yescharacteristic
646C4#1568983211yescharacteristic
647C2—44no—
648C2D4×Dic47⋊C411yescharacteristic
649C2#2025219311yescharacteristic
650C2#2025222811yescharacteristic
651C2#2025487711yescharacteristic
652C2#2025493711yescharacteristic
653C2#2025631011yescharacteristic
654C2#2025633411yescharacteristic
subgroupG/H
centerC23C2×D4×D47
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC2×C47×#440C2
socleC23×C47C22×D4
prime radicalC22×D4×C47C23
cosocleC22×C4×C47C24
hypercenterC2×#440D47
hypocenterC47#22677
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#440

Complements

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

NH ≅ G/Ntype
3. #20246306C2semidirect
5. #20244126C2semidirect
6. #20244119C2semidirect
7. #20243654C2semidirect
9. #20243642C2semidirect
11. #20243555C2semidirect
13. #20243391C2semidirect
14. #20243264C2semidirect
21. #15689224C4semidirect
69. Q8×Dic47C2×C4semidirect
95. #13093958C2×C4semidirect
101. #13093948C2×C4semidirect
104. #13093934C2×C4semidirect
172. SD16×C47C4⋊C4semidirect
231. Q8×C47C2×C4⋊C4semidirect
285. C8×C47C2×C4⋊C4semidirect
389. C47#22677semidirect
500. SD16Dic47⋊C4semidirect
559. Q8C2×Dic47⋊C4semidirect
615. C8C2×Dic47⋊C4semidirect

Maximal subgroups

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

subgroupindex
C2×#156892242
C2×SD16×Dic472
#202463062
D4×Dic47⋊C42
#202441262
#202441192
#202436542
#202436492
#202436422
#202436332
#202435552
#202433962
#202433912
#202432642
C2×C47×#4402
#2267747