Finite Group Databasebeta

← #26234624

Subgroups — #26234624

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×#15689223C211yesmaximal, characteristic
2C2×#15689223C211yesmaximal, characteristic
3C2×C4×#10625335C211yesmaximal, characteristic
4#20246308C211yesmaximal, characteristic, complemented
5#20244123C211yesmaximal, characteristic, complemented
6#20244118C211yesmaximal, characteristic, complemented
7#20243655C211yesmaximal, characteristic, complemented
8#20243649C211yesmaximal, characteristic
9#20243641C211yesmaximal, characteristic, complemented
10#20243635C211yesmaximal, characteristic
11#20243554C211yesmaximal, characteristic, complemented
12#20243398C211yesmaximal, characteristic
13#20243390C211yesmaximal, characteristic, complemented
14#20243263C211yesmaximal, characteristic, complemented
15D4×C4⋊C4×C47C211yesmaximal, characteristic
16C22×#10625335C2211yescharacteristic
17C2×C4×Dic94C2211yescharacteristic
18C2×C4×Dic94C2211yescharacteristic
19C2×C4×Dic94C2211yescharacteristic
20#15689223—24no—
21#15689223C418yescomplemented
22C4×#10625335—24no—
23C2×#13093957C2211yescharacteristic
24C2×#13093957C2211yescharacteristic
25C2×#13093957C2211yescharacteristic
26C2×#13093955C2211yescharacteristic
27C2×#13093955C2211yescharacteristic
28C2×C4⋊(C47⋊C8)C2211yescharacteristic
29C2×#13093947C2211yescharacteristic
30C2×#13093947C2211yescharacteristic
31C2×#13093947C2211yescharacteristic
32C2×#13093945C2211yescharacteristic
33C2×#13093945C2211yescharacteristic
34C2×#13093945C2211yescharacteristic
35C2×#13093935C2211yescharacteristic
36C2×#13093935C2211yescharacteristic
37C2×C4×C47⋊C8C2211yescharacteristic
38#15689021C2211yescharacteristic
39#15689021C2211yescharacteristic
40#15689018—21no—
41#15689018—21no—
42#15688989C2211yescharacteristic
43#15688987—21no—
44#15688974—21no—
45#15688885C2211yescharacteristic
46#15688885C2211yescharacteristic
47#15688885C2211yescharacteristic
48#15688885C2211yescharacteristic
49#15688878C2211yescharacteristic
50#15688878C2211yescharacteristic
51C2×C4×D4×C47C2211yescharacteristic
52C2×C4×D4×C47C2211yescharacteristic
53C2×C4×D4×C47C2211yescharacteristic
54C22×C4⋊C4×C47—21no—
55C47×#389C2211yescharacteristic
56C47×#389C2211yescharacteristic
57C47×#385—21no—
58C47×#385—21no—
59C47×#384C2211yescharacteristic
60C47×#380—21no—
61C4×C4⋊C4×C47C2211yescharacteristic
62C22×Dic94C2311yescharacteristic
63C2×#10625335C2×C414yes—
64C2×#10625335D414yes—
65C2×#10625335Q814yes—
66C22×C47⋊C8C2311yescharacteristic
67C22×C47⋊C8C2311yescharacteristic
68C22×C47⋊C8C2311yescharacteristic
69C22×C47⋊C8C2311yescharacteristic
70C4×Dic94—22no—
71C4×Dic94—22no—
72C4×Dic94—22no—
73C4×Dic94—41no—
74C4×Dic94—41no—
75C4×Dic94C2×C414yescomplemented
76C2×C4⋊Dic47C2311yescharacteristic
77C2×C4⋊Dic47C2311yescharacteristic
78C2×C4⋊Dic47C2311yescharacteristic
79C2×Dic47⋊C4—21no—
80C2×Dic47⋊C4—21no—
81C2×Dic47⋊C4—21no—
82C2×Dic47⋊C4—21no—
83C2×Dic47⋊C4—21no—
84C2×C4×Dic47—21no—
85C2×C4×Dic47—21no—
86C2×C4×Dic47—21no—
87#13093957—22no—
88#13093957—41no—
89#13093957C2×C414yescomplemented
90#13093955—22no—
91#13093955—41no—
92C4⋊(C47⋊C8)—22no—
93#13093947—22no—
94#13093947—22no—
95#13093947C2×C414yescomplemented
96#13093945—22no—
97#13093945—22no—
98#13093945C2×C414yes—
99#13093935—22no—
100#13093935C2×C414yescomplemented
101C4×C47⋊C8—22no—
102C47⋊#95—21no—
103C47⋊#95—21no—
104C47⋊#95—21no—
105C47⋊#95—21no—
106C22×D4×C47C2311yescharacteristic
107C23×C4×C47—21no—
108C23×C4×C47—21no—
109C23×C4×C47—21no—
110C4×D4×C47—22no—
111C4×D4×C47—22no—
112C4×D4×C47—22no—
113C4×D4×C47—41no—
114C4×D4×C47—41no—
115C4×D4×C47C2×C414yes—
116C2×C4⋊C4×C47—21no—
117C2×C4⋊C4×C47—21no—
118C2×C4⋊C4×C47—21no—
119C2×C4⋊C4×C47—21no—
120C2×C4⋊C4×C47—44no—
121C2×C4⋊C4×C47D411yescharacteristic
122C2×C4⋊C4×C47D411yescharacteristic
123C2×C4⋊C4×C47C2311yescharacteristic
124C2×C4⋊C4×C47C2311yescharacteristic
125C2×C4⋊C4×C47C2311yescharacteristic
126C2×C22⋊C4×C47—21no—
127C2×C22⋊C4×C47—21no—
128C2×C22⋊C4×C47—21no—
129C2×C42×C47C2311yescharacteristic
130C2×C42×C47C2311yescharacteristic
131C2×C42×C47C2311yescharacteristic
132#95×C47—21no—
133#95×C47—21no—
134#95×C47D411yescharacteristic
135#95×C47D411yescharacteristic
136C24×C47—21no—
137C2×D4×C47—21no—
138C2×D4×C47—21no—
139C2×D4×C47—21no—
140C2×D4×C47C22×C412yes—
141C2×D4×C47C2×D412yes—
142C2×D4×C47C2×Q812yes—
143C22×C4×C47—21no—
144C22×C4×C47—24no—
145C22×C4×C47—21no—
146C22×C4×C47—21no—
147C22×C4×C47—42no—
148C22×C4×C47—42no—
149C22×C4×C47—42no—
150C22×C4×C47—42no—
151C22×C4×C47—42no—
152C22×C4×C47—42no—
153C22×C4×C47—81no—
154C22×C4×C47—81no—
155C22×C4×C47C2×D411yescharacteristic
156C22×C4×C47C2×D411yescharacteristic
157C22×C4×C47C2×D411yescharacteristic
158C22×C4×C47C2×D411yescharacteristic
159C22×C4×C47C2×D411yescharacteristic
160C22×C4×C47C2×D411yescharacteristic
161C22×C4×C47C2411yescharacteristic
162C22×Dic47—21no—
163C22×Dic47—21no—
164C22×Dic47—21no—
165C22×Dic47—21no—
166C2×Dic94—21no—
167C2×Dic94—21no—
168C2×Dic94—21no—
169C2×Dic94C22×C412yes—
170C2×Dic94C2×D412yes—
171C2×Dic94C2×Q812yes—
172C4⋊C4×C47—22no—
173C4⋊C4×C47—21no—
174C4⋊C4×C47—21no—
175C4⋊C4×C47—21no—
176C4⋊C4×C47—21no—
177C4⋊C4×C47—21no—
178C4⋊C4×C47—41no—
179C4⋊C4×C47—42no—
180C4⋊C4×C47—42no—
181C4⋊C4×C47—42no—
182C4⋊C4×C47—81no—
183C4⋊C4×C47C22×C412yes—
184C22⋊C4×C47—24no—
185C22⋊C4×C47—42no—
186C22⋊C4×C47—42no—
187C42×C47—21no—
188C42×C47—21no—
189C42×C47—21no—
190C42×C47—21no—
191C42×C47C22×C412yes—
192C42×C47#4112yes—
193#10625335C4⋊C4116yescomplemented
194C4⋊Dic47—21no—
195C4⋊Dic47—21no—
196C4⋊Dic47—21no—
197C4⋊Dic47—21no—
198C4⋊Dic47—21no—
199C4⋊Dic47C22×C412yes—
200Dic47⋊C4—24no—
201Dic47⋊C4—42no—
202Dic47⋊C4—42no—
203Dic47⋊C4—42no—
204Dic47⋊C4—42no—
205C4×Dic47—24no—
206C4×Dic47—42no—
207C4×Dic47—42no—
208C2×C47⋊C8—21no—
209C2×C47⋊C8—21no—
210C2×C47⋊C8—21no—
211C2×C47⋊C8—21no—
212C2×C47⋊C8—21no—
213C2×C47⋊C8—21no—
214C2×C47⋊C8—21no—
215C2×C47⋊C8—21no—
216C2×C47⋊C8C22×C412yes—
217C2×C47⋊C8C22×C412yes—
218C2×C47⋊C8C2×D412yes—
219C2×C47⋊C8C2×Q812yes—
220C23×C47—22no—
221C23×C47—22no—
222C23×C47—22no—
223C23×C47—41no—
224C23×C47—41no—
225C23×C47—41no—
226C23×C47—41no—
227C23×C47C22×D411yescharacteristic
228D4×C47—22no—
229D4×C47—22no—
230D4×C47—22no—
231D4×C47C2×C4⋊C414yes—
232C2×C4×C47—21no—
233C2×C4×C47—21no—
234C2×C4×C47—21no—
235C2×C4×C47—21no—
236C2×C4×C47—21no—
237C2×C4×C47—21no—
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—21no—
246C2×C4×C47—21no—
247C2×C4×C47—21no—
248C2×C4×C47—48no—
249C2×C4×C47—41no—
250C2×C4×C47—41no—
251C2×C4×C47—41no—
252C2×C4×C47—41no—
253C2×C4×C47—41no—
254C2×C4×C47—41no—
255C2×C4×C47—41no—
256C2×C4×C47—41no—
257C2×C4×C47—84no—
258C2×C4×C47—84no—
259C2×C4×C47C4×D412yes—
260C2×C4×C47C4×D412yes—
261C2×C4×C47AGL(1,ℤ/8)12yes—
262C2×C4×C47C23×C411yescharacteristic
263C2×C4×C47C22×D411yescharacteristic
264C2×C4×C47C22×Q811yescharacteristic
265C2×C4×C47C2×#4111yescharacteristic
266C2×C4×C47H(5,𝔽2)11yescharacteristic
267C2×C4×C472−1+411yescharacteristic
268C2×Dic47—22no—
269C2×Dic47—22no—
270C2×Dic47—22no—
271C2×Dic47—22no—
272C2×Dic47—41no—
273C2×Dic47—41no—
274C2×Dic47—41no—
275C2×Dic47—41no—
276C2×Dic47—41no—
277C2×Dic47—41no—
278C2×Dic47—41no—
279C2×Dic47—41no—
280Dic94—22no—
281Dic94—22no—
282Dic94—22no—
283Dic94C2×C4⋊C414yescomplemented
284C47⋊C8—22no—
285C47⋊C8—41no—
286C47⋊C8—41no—
287C47⋊C8C2×C4⋊C414yescomplemented
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—22no—
304C4×C47—22no—
305C4×C47—22no—
306C4×C47—42no—
307C4×C47—41no—
308C4×C47—41no—
309C4×C47—41no—
310C4×C47—81no—
311C4×C47—81no—
312C4×C47C22×C4⋊C411yescharacteristic
313C4×C47#52011yescharacteristic
314C4×C47#54711yescharacteristic
315C4×C47#55411yescharacteristic
316Dic47—24no—
317Dic47—42no—
318Dic47—81no—
319Dic47—81no—
320C2×#440—471no—
321C2×#440—471no—
322C2×C4×SD16—471no—
323#2351—471no—
324D4×C4⋊C4D4711yescharacteristic
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×SD16—471no—
344C2×C4×Q8—471no—
345C2×C4×Q8—471no—
346C2×C4×Q8—471no—
347C2×C4×D4C2×D4711yescharacteristic
348C2×C4×D4C2×D4711yescharacteristic
349C2×C4×D4C2×D4711yescharacteristic
350C22×C4⋊C4—21no—
351#440—478no—
352#440—944no—
353C4×SD16—944no—
354C2×#107—471no—
355C2×#107—471no—
356C2×#107—471no—
357C2×#106—471no—
358C2×#106—471no—
359C2×C4⋊C8—471no—
360C2×#103—471no—
361C2×#103—471no—
362C2×#103—471no—
363C2×#102—471no—
364C2×#102—471no—
365C2×#102—471no—
366C2×#97—471no—
367C2×#97—471no—
368C2×C4×C8—471no—
369#389—471no—
370#389—471no—
371#389—941no—
372#389—941no—
373#389C2×D4711yescharacteristic
374#389C2×D4711yescharacteristic
375#385—21no—
376#385—21no—
377#384C2×D4711yescharacteristic
378#382—471no—
379#382—941no—
380#380—21no—
381C4×C4⋊C4—941no—
382C4×C4⋊C4C2×D4711yescharacteristic
383#340—471no—
384#340—471no—
385#340—471no—
386#340—471no—
387#336—471no—
388#336—471no—
389C47#2267711yescharacteristic, complemented, Sylow
390C22×Q8—471no—
391C22×D4C22×D4711yescharacteristic
392C23×C4—21no—
393C23×C4—21no—
394C23×C4—21no—
395C2×SD16—474no—
396C2×SD16—474no—
397C2×SD16—474no—
398C22×C8—471no—
399C22×C8—471no—
400C22×C8—471no—
401C22×C8—471no—
402C4×Q8—474no—
403C4×Q8—942no—
404C4×Q8—942no—
405C4×Q8—942no—
406C4×Q8—1881no—
407C4×Q8—1881no—
408C4×D4—22no—
409C4×D4—22no—
410C4×D4—22no—
411C4×D4—41no—
412C4×D4—41no—
413C4×D4C4×D4714yes—
414C2×C4⋊C4—21no—
415C2×C4⋊C4—21no—
416C2×C4⋊C4—21no—
417C2×C4⋊C4—21no—
418C2×C4⋊C4—44no—
419C2×C4⋊C4—471no—
420C2×C4⋊C4—471no—
421C2×C4⋊C4—471no—
422C2×C4⋊C4—941no—
423C2×C4⋊C4—941no—
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×C4⋊C4C22×D4711yescharacteristic
431C2×C4⋊C4C22×D4711yescharacteristic
432C2×C22⋊C4—21no—
433C2×C22⋊C4—21no—
434C2×C22⋊C4—21no—
435C2×C42—941no—
436C2×C42—941no—
437C2×C42—941no—
438C2×C42C22×D4711yescharacteristic
439C2×C42C22×D4711yescharacteristic
440C2×C42C22×D4711yescharacteristic
441#107—474no—
442#107—942no—
443#107—1881no—
444#106—942no—
445#106—1881no—
446C4⋊C8—942no—
447#103—474no—
448#103—942no—
449#103—942no—
450#102—474no—
451#102—942no—
452#102—942no—
453#97—474no—
454#97—942no—
455C4×C8—942no—
456#95—21no—
457#95—21no—
458#95—941no—
459#95—941no—
460#95—941no—
461#95—941no—
462#95C22⋊D4711yescharacteristic
463#95C22⋊D4711yescharacteristic
464C24—21no—
465C2×Q8—472no—
466C2×Q8—472no—
467C2×Q8—472no—
468C2×Q8—941no—
469C2×Q8—941no—
470C2×Q8—941no—
471C2×D4—21no—
472C2×D4—21no—
473C2×D4—21no—
474C2×D4C2×C4×D4712yes—
475C2×D4D4×D4712yes—
476C2×D4Q8×D4712yes—
477C22×C4—21no—
478C22×C4—24no—
479C22×C4—21no—
480C22×C4—21no—
481C22×C4—42no—
482C22×C4—42no—
483C22×C4—42no—
484C22×C4—42no—
485C22×C4—42no—
486C22×C4—42no—
487C22×C4—81no—
488C22×C4—81no—
489C22×C4—941no—
490C22×C4—941no—
491C22×C4—941no—
492C22×C4—941no—
493C22×C4C2×C22⋊D4711yescharacteristic
494C22×C4C2×C22⋊D4711yescharacteristic
495C22×C4C2×C22⋊D4711yescharacteristic
496C22×C4C2×C22⋊D4711yescharacteristic
497C22×C4C2×C22⋊D4711yescharacteristic
498C22×C4C2×C22⋊D4711yescharacteristic
499C22×C4C23×D4711yescharacteristic
500SD16—4716no—
501C2×C8—472no—
502C2×C8—472no—
503C2×C8—472no—
504C2×C8—472no—
505C2×C8—941no—
506C2×C8—941no—
507C2×C8—941no—
508C2×C8—941no—
509C2×C8—941no—
510C2×C8—941no—
511C2×C8—941no—
512C2×C8—941no—
513C4⋊C4—22no—
514C4⋊C4—21no—
515C4⋊C4—21no—
516C4⋊C4—21no—
517C4⋊C4—21no—
518C4⋊C4—21no—
519C4⋊C4—41no—
520C4⋊C4—42no—
521C4⋊C4—42no—
522C4⋊C4—42no—
523C4⋊C4—81no—
524C4⋊C4—472no—
525C4⋊C4—944no—
526C4⋊C4—941no—
527C4⋊C4—941no—
528C4⋊C4—941no—
529C4⋊C4—941no—
530C4⋊C4—941no—
531C4⋊C4—1882no—
532C4⋊C4—1882no—
533C4⋊C4—1882no—
534C4⋊C4—1882no—
535C4⋊C4C2×C4×D4712yes—
536C22⋊C4—24no—
537C22⋊C4—42no—
538C22⋊C4—42no—
539C42—21no—
540C42—21no—
541C42—21no—
542C42—21no—
543C42—944no—
544C42—1882no—
545C42—1882no—
546C42C2×C4×D4712yes—
547C42#1062535012yes—
548C23—22no—
549C23—22no—
550C23—22no—
551C23—41no—
552C23—41no—
553C23—41no—
554C23—41no—
555C23C22×C22⋊D4711yescharacteristic
556Q8—474no—
557Q8—942no—
558Q8—942no—
559Q8—942no—
560D4—22no—
561D4—22no—
562D4—22no—
563D4C4⋊C4×D4714yes—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—22no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—48no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—84no—
590C2×C4—84no—
591C2×C4—942no—
592C2×C4—942no—
593C2×C4—942no—
594C2×C4—942no—
595C2×C4—1881no—
596C2×C4—1881no—
597C2×C4—1881no—
598C2×C4—1881no—
599C2×C4—1881no—
600C2×C4—1881no—
601C2×C4—1881no—
602C2×C4—1881no—
603C2×C4C4×C22⋊D4712yes—
604C2×C4C4×C22⋊D4712yes—
605C2×C4#1309405012yes—
606C2×C4C22×C4×D4711yescharacteristic
607C2×C4C2×D4×D4711yescharacteristic
608C2×C4C2×Q8×D4711yescharacteristic
609C2×C4C2×#1062535011yescharacteristic
610C2×C4#1309406911yescharacteristic
611C2×C4#1309407211yescharacteristic
612C8—474no—
613C8—942no—
614C8—1881no—
615C8—1881no—
616C22—24no—
617C22—42no—
618C22—42no—
619C22—42no—
620C22—42no—
621C22—42no—
622C22—42no—
623C22C2×C4×C22⋊D4711yescharacteristic
624C22D4×C22⋊D4711yescharacteristic
625C22Q8×C22⋊D4711yescharacteristic
626C22C2×#1309405011yescharacteristic
627C22#1568992111yescharacteristic
628C22#1568993411yescharacteristic
629C22#1568993811yescharacteristic
630C4—22no—
631C4—22no—
632C4—22no—
633C4—42no—
634C4—41no—
635C4—41no—
636C4—41no—
637C4—81no—
638C4—81no—
639C4—944no—
640C4—3761no—
641C4—3761no—
642C4—1882no—
643C4C2×C4⋊C4×D4711yescharacteristic
644C4#1568960411yescharacteristic
645C4#1568977411yescharacteristic
646C4#1568982811yescharacteristic
647C2—44no—
648C2C4⋊C4×C22⋊D4711yescharacteristic
649C2C4×#1309405011yescharacteristic
650C2#2025222711yescharacteristic
651C2#2025487911yescharacteristic
652C2#2025493911yescharacteristic
653C2#2025630911yescharacteristic
654C2#2025633311yescharacteristic
subgroupG/H
centerC23C22×C22⋊D47
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingD4×C4⋊C4×C47C2
socleC23×C47C22×D4
prime radicalC22×D4×C47C23
cosocleC22×C4×C47C24
hypercenterD4×C4⋊C4D47
hypocenterC47#22677
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ D4×C4⋊C4

Complements

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

NH ≅ G/Ntype
4. #20246308C2semidirect
5. #20244123C2semidirect
6. #20244118C2semidirect
7. #20243655C2semidirect
9. #20243641C2semidirect
11. #20243554C2semidirect
13. #20243390C2semidirect
14. #20243263C2semidirect
21. #15689223C4semidirect
75. C4×Dic94C2×C4semidirect
89. #13093957C2×C4semidirect
95. #13093947C2×C4semidirect
100. #13093935C2×C4semidirect
193. #10625335C4⋊C4semidirect
283. Dic94C2×C4⋊C4semidirect
287. C47⋊C8C2×C4⋊C4semidirect
389. C47#22677semidirect

Maximal subgroups

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

subgroupindex
C2×#156892232
C2×C4×#106253352
#202463082
#202441232
#202441182
#202436552
#202436492
#202436412
#202436352
#202435542
#202433982
#202433902
#202432632
D4×C4⋊C4×C472
#2267747