Finite Group Databasebeta

← #26182723

Subgroups — #26182723

Order 12032.

1214 conjugacy classes of subgroups.

The subgroup lattice has 654 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×Q8×D47C211yesmaximal, characteristic
2#20252019C211yesmaximal, characteristic
3#20252018C211yesmaximal, characteristic
4#20241836C211yesmaximal, characteristic, complemented
5#20241835C211yesmaximal, characteristic, complemented
6#20241834C211yesmaximal, characteristic, complemented
7#20241815C214yesmaximal, complemented
8#20241781C214yesmaximal, complemented
9C47×#1491C211yesmaximal, characteristic, complemented
10C22×Q8×D47C411yescharacteristic
11C2×Q8×Dic47C411yescharacteristic
12C2×C4×Dic94—21no—
13C2×C4×Dic94C2211yescharacteristic
14C2×#13093989—21no—
15C2×#13093989C411yescharacteristic
16C4×Q8×D47—21no—
17C4×Q8×D47—21no—
18C4×Q8×D47—21no—
19C4×Q8×D47—21no—
20C4×Q8×D47C412yes—
21C4×Q8×D47C412yes—
22C4×Q8×D47C2212yes—
23C4×Q8×D47C2212yes—
24C2×C4⋊C4×D47—21no—
25C2×C4⋊C4×D47C411yescharacteristic
26C2×C42×D47—21no—
27C2×C42×D47C2211yescharacteristic
28#15689180C2211yescharacteristic
29#15689178—21no—
30#15689178—21no—
31#15689178—21no—
32#15689175C2212yes—
33#15689174C2212yes—
34#15689174C2212yes—
35#15689173—21no—
36#15689146—21no—
37#15689145C2212yes—
38#15689086—21no—
39#15689085—21no—
40C4×#10625329C2211yescharacteristic
41#15689075—21no—
42#15688855C2214yescomplemented
43Q8⋊(C47⋊C8)C2214yescomplemented
44#15688853C2214yescomplemented
45C2×C4×Q8×C47C2211yescharacteristic
46C47×#432C2211yescharacteristic
47C4×M16×C47C2211yescharacteristic
48C47×Q8⋊C8C2214yescomplemented
49C22×Dic94—21no—
50C22×Dic94C2×C411yescharacteristic
51C2×Q8×D47—28no—
52C2×Q8×D47—21no—
53C2×Q8×D47—21no—
54C2×Q8×D47C2×C412yes—
55C2×Q8×D47C2×C412yes—
56C22×C4×D47—21no—
57C22×C4×D47—21no—
58C22×C4×D47C2×C411yescharacteristic
59C22×C4×D47D411yescharacteristic
60C22×C4×D47D411yescharacteristic
61C2×#10625329—21no—
62C2×#10625324—22no—
63Q8×Dic47—21no—
64Q8×Dic47—21no—
65Q8×Dic47C2×C412yes—
66Q8×Dic47C2×C412yes—
67C4×Dic94—21no—
68C4×Dic94—21no—
69C4×Dic94—22no—
70C4×Dic94—22no—
71C4×Dic94—22no—
72C4×Dic94—22no—
73C4×Dic94C2×C412yes—
74C4×Dic94C2312yes—
75#13093989—21no—
76#13093989—21no—
77#13093989—22no—
78#13093989—22no—
79#13093989—22no—
80#13093989—22no—
81#13093989C2×C412yes—
82#13093989C2×C412yes—
83C2×C4⋊Dic47—21no—
84C2×C4⋊Dic47C2×C411yescharacteristic
85C2×Dic47⋊C4—21no—
86C2×Dic47⋊C4—21no—
87C2×Dic47⋊C4—21no—
88C4⋊C4×D47—22no—
89C4⋊C4×D47—22no—
90C4⋊C4×D47—21no—
91C4⋊C4×D47—21no—
92C4⋊C4×D47—22no—
93C4⋊C4×D47—22no—
94C4⋊C4×D47C2×C412yes—
95C4⋊C4×D47C2×C412yes—
96C2×C4×Dic47—21no—
97C2×C4×Dic47—21no—
98C2×C4×Dic47—21no—
99C2×C4×Dic47C2×C411yescharacteristic
100C2×C4×Dic47D411yescharacteristic
101C2×C4×Dic47D411yescharacteristic
102C42×D47—22no—
103C42×D47—22no—
104C42×D47—22no—
105C42×D47—22no—
106C42×D47C2×C411yescharacteristic
107C42×D47C2×C411yescharacteristic
108C42×D47D411yescharacteristic
109C42×D47D411yescharacteristic
110C42×D47D411yescharacteristic
111C42×D47D411yescharacteristic
112C42×D47C2311yescharacteristic
113C42×D47C2311yescharacteristic
114C4⋊(C47⋊C8)C2312yes—
115Dic47⋊C8—22no—
116Dic47⋊C8—22no—
117Dic47⋊C8—22no—
118C22⋊(C47⋊C8)—21no—
119(C2×D47)⋊C8—22no—
120(C2×D47)⋊C8—22no—
121(C2×D47)⋊C8—22no—
122#13093935—21no—
123#13093935—21no—
124C4×C47⋊C8C2312yes—
125C8×Dic47—22no—
126C22×Q8×C47C2×C411yescharacteristic
127C2×M16×C47—21no—
128C4×Q8×C47—21no—
129C4×Q8×C47—21no—
130C4×Q8×C47C2×C412yes—
131C4×Q8×C47C2312yes—
132C2×C4⋊C4×C47—21no—
133C2×C4⋊C4×C47C2×C411yescharacteristic
134C2×C42×C47—21no—
135C2×C42×C47C2311yescharacteristic
136C4⋊C8×C47C2312yes—
137C22⋊C8×C47—21no—
138#97×C47—21no—
139#97×C47—21no—
140C4×C8×C47C2312yes—
141C23×D47C22⋊C411yescharacteristic
142C2×Q8×C47—24no—
143C2×Q8×C47—21no—
144C2×Q8×C47C22×C412yes—
145C22×C4×C47—21no—
146C22×C4×C47—21no—
147C22×C4×C47C22×C411yescharacteristic
148C22×C4×C47C2×D411yescharacteristic
149C22×C4×C47C2×D411yescharacteristic
150C22×Dic47—21no—
151C22×Dic47—21no—
152C22×Dic47C22⋊C411yescharacteristic
153C22×Dic47C22⋊C411yescharacteristic
154C22×Dic47C22⋊C411yescharacteristic
155Q8×D47—28no—
156Q8×D47—28no—
157Q8×D47—24no—
158Q8×D47—24no—
159Q8×D47—24no—
160Q8×D47M1618yescomplemented
161C2×C4×D47—28no—
162C2×C4×D47—22no—
163C2×C4×D47—22no—
164C2×C4×D47—24no—
165C2×C4×D47—24no—
166C2×C4×D47—22no—
167C2×C4×D47—22no—
168C2×C4×D47—42no—
169C2×C4×D47—44no—
170C2×C4×D47C22⋊C411yescharacteristic
171C2×C4×D47C22⋊C411yescharacteristic
172C2×C4×D47C22⋊C411yescharacteristic
173C2×C4×D47C22⋊C411yescharacteristic
174C2×C4×D47C22⋊C411yescharacteristic
175C2×C4×D47C22⋊C411yescharacteristic
176C2×C4×D47C22×C411yescharacteristic
177C2×C4×D47C22×C411yescharacteristic
178C2×C4×D47C2×D411yescharacteristic
179C2×C4×D47C2×D411yescharacteristic
180C2×C4×D47C2×D411yescharacteristic
181C2×C4×D47C2×D411yescharacteristic
182C2×Dic94—28no—
183C2×Dic94—21no—
184C2×Dic94—22no—
185C2×Dic94—22no—
186C2×Dic94—24no—
187C2×Dic94C22×C412yes—
188M16×C47—42no—
189C2×C8×C47—22no—
190C2×C8×C47—22no—
191C4⋊C4×C47—22no—
192C4⋊C4×C47—21no—
193C4⋊C4×C47—22no—
194C4⋊C4×C47C22×C412yes—
195C42×C47—22no—
196C42×C47—22no—
197C42×C47C22×C411yescharacteristic
198C42×C47C2×D411yescharacteristic
199C42×C47C2×D411yescharacteristic
200C42×C47C2411yescharacteristic
201C4⋊Dic47—21no—
202C4⋊Dic47—22no—
203C4⋊Dic47—22no—
204C4⋊Dic47C22×C412yes—
205Dic47⋊C4—22no—
206Dic47⋊C4—22no—
207Dic47⋊C4—22no—
208Dic47⋊C4—22no—
209Dic47⋊C4—22no—
210Dic47⋊C4—22no—
211C4×Dic47—22no—
212C4×Dic47—22no—
213C4×Dic47—22no—
214C4×Dic47—22no—
215C4×Dic47—22no—
216C4×Dic47—22no—
217C4×Dic47C22⋊C411yescharacteristic
218C4×Dic47C22⋊C411yescharacteristic
219C4×Dic47C22⋊C411yescharacteristic
220C4×Dic47C22⋊C411yescharacteristic
221C4×Dic47C22⋊C411yescharacteristic
222C4×Dic47C22⋊C411yescharacteristic
223C4×Dic47C22×C411yescharacteristic
224C4×Dic47C22×C411yescharacteristic
225C4×Dic47C2×D411yescharacteristic
226C4×Dic47C2×D411yescharacteristic
227C4×Dic47C2×D411yescharacteristic
228C4×Dic47C2×D411yescharacteristic
229#10625329—42no—
230C2×C47⋊C8—22no—
231C2×C47⋊C8—22no—
232#10625324—44no—
233C23×C47C2×C22⋊C411yescharacteristic
234C22×D47—22no—
235C22×D47—22no—
236C22×D47—22no—
237C22×D47C2×C22⋊C411yescharacteristic
238C22×D47C2×C22⋊C411yescharacteristic
239Q8×C47—24no—
240Q8×C47—22no—
241Q8×C47C2×M1614yes—
242C2×C4×C47—24no—
243C2×C4×C47—22no—
244C2×C4×C47—22no—
245C2×C4×C47—22no—
246C2×C4×C47—22no—
247C2×C4×C47—22no—
248C2×C4×C47—42no—
249C2×C4×C47C2×C22⋊C411yescharacteristic
250C2×C4×C47C2×C22⋊C411yescharacteristic
251C2×C4×C47C2×C22⋊C411yescharacteristic
252C2×C4×C47C23×C411yescharacteristic
253C2×C4×C47C22×D411yescharacteristic
254C2×C4×C47C22×D411yescharacteristic
255C2×Dic47—24no—
256C2×Dic47—22no—
257C2×Dic47—22no—
258C2×Dic47—22no—
259C2×Dic47—22no—
260C2×Dic47—22no—
261C2×Dic47—42no—
262C2×Dic47C2×C22⋊C411yescharacteristic
263C2×Dic47C2×C22⋊C411yescharacteristic
264C2×Dic47C2×C22⋊C411yescharacteristic
265C2×Dic47C2×C22⋊C411yescharacteristic
266C2×Dic47C2×C22⋊C411yescharacteristic
267C2×Dic47C2×C22⋊C411yescharacteristic
268C4×D47—28no—
269C4×D47—28no—
270C4×D47—22no—
271C4×D47—22no—
272C4×D47—22no—
273C4×D47—22no—
274C4×D47—22no—
275C4×D47—22no—
276C4×D47—41no—
277C4×D47—41no—
278C4×D47—41no—
279C4×D47—41no—
280C4×D47—44no—
281C4×D47—44no—
282C4×D47C2×M1614yes—
283C4×D47#13714yes—
284Dic94—28no—
285Dic94—24no—
286Dic94—22no—
287Dic94—28no—
288Dic94C2×M1614yes—
289C8×C47—42no—
290C8×C47—42no—
291C47⋊C8—42no—
292C47⋊C8—42no—
293#17449—471nomaximal, Sylow
294C22×C47—21no—
295C22×C47—21no—
296C22×C47#42112yes—
297C22×C47C22×C22⋊C411yescharacteristic
298C2×D47—28no—
299C2×D47—21no—
300C2×D47—21no—
301C2×D47—21no—
302C2×D47—21no—
303C2×D47#40712yes—
304C2×D47#41912yes—
305C4×C47—24no—
306C4×C47—21no—
307C4×C47—21no—
308C4×C47—21no—
309C4×C47—21no—
310C4×C47—42no—
311C4×C47C22×M1612yes—
312C4×C47C2×#13712yes—
313Dic47—21no—
314Dic47—24no—
315Dic47—21no—
316Dic47—21no—
317Dic47—21no—
318Dic47—42no—
319Dic47#40712yes—
320Dic47#41912yes—
321C22×C4×Q8—471no—
322#2920—471no—
323#2920—471no—
324#1491—471no—
325#1491—471no—
326#1491—471no—
327#1491D4711yescharacteristic, complemented
328#1487—474no—
329#1479—474no—
330C2×C47—22no—
331C2×C47C2×#40711yescharacteristic
332C2×C47C2×#41911yescharacteristic
333C2×C47C2×#42111yescharacteristic
334D47—22no—
335D47—22no—
336C23×Q8—471no—
337C2×C4×Q8—471no—
338C2×C4×Q8—472no—
339C2×C4×Q8—472no—
340C2×C4×Q8—472no—
341C2×C4×Q8—472no—
342C2×C4×Q8—471no—
343C2×C4×Q8—471no—
344C2×C4×Q8—941no—
345C2×C4×Q8—941no—
346C2×C4×Q8—941no—
347C2×C4×Q8—941no—
348C2×C4×Q8—941no—
349C2×C4×Q8—941no—
350C2×C4×Q8C2×D4711yescharacteristic
351C22×C4⋊C4—471no—
352C22×C4⋊C4—941no—
353C22×C42—471no—
354C22×C42—941no—
355#432—471no—
356#432—941no—
357#432—941no—
358#432—941no—
359#432C2×D4711yescharacteristic
360#431—472no—
361#431—472no—
362#431—472no—
363#431—941no—
364#423—472no—
365#423—941no—
366#407—941no—
367#407—941no—
368C4×M16—471no—
369C4×M16—941no—
370C4×M16C2×D4711yescharacteristic
371Q8⋊C8—474no—
372Q8⋊C8—474no—
373Q8⋊C8—474no—
374Q8⋊C8C2×D4714yescomplemented
375C47#1744911yescharacteristic, complemented, Sylow
376C22×Q8—472no—
377C22×Q8—472no—
378C22×Q8—471no—
379C22×Q8—941no—
380C22×Q8—941no—
381C22×Q8—941no—
382C22×Q8—948no—
383C22×Q8C4×D4711yescharacteristic
384C23×C4—471no—
385C23×C4—471no—
386C23×C4—471no—
387C23×C4—941no—
388C23×C4—941no—
389C2×M16—21no—
390C2×M16—942no—
391C2×M16—941no—
392C4×Q8—21no—
393C4×Q8—21no—
394C4×Q8—472no—
395C4×Q8—472no—
396C4×Q8—472no—
397C4×Q8—472no—
398C4×Q8—472no—
399C4×Q8—472no—
400C4×Q8—942no—
401C4×Q8—942no—
402C4×Q8—942no—
403C4×Q8—942no—
404C4×Q8—942no—
405C4×Q8—942no—
406C4×Q8—941no—
407C4×Q8—941no—
408C4×Q8—942no—
409C4×Q8—942no—
410C4×Q8—941no—
411C4×Q8—941no—
412C4×Q8—941no—
413C4×Q8—941no—
414C4×Q8C4×D4712yes—
415C4×Q8C22×D4712yes—
416C2×C4⋊C4—21no—
417C2×C4⋊C4—471no—
418C2×C4⋊C4—472no—
419C2×C4⋊C4—472no—
420C2×C4⋊C4—941no—
421C2×C4⋊C4—941no—
422C2×C4⋊C4—942no—
423C2×C4⋊C4—942no—
424C2×C4⋊C4—942no—
425C2×C4⋊C4—942no—
426C2×C4⋊C4—941no—
427C2×C4⋊C4—941no—
428C2×C4⋊C4—941no—
429C2×C4⋊C4—941no—
430C2×C4⋊C4C4×D4711yescharacteristic
431C2×C42—21no—
432C2×C42—471no—
433C2×C42—471no—
434C2×C42—471no—
435C2×C42—471no—
436C2×C42—471no—
437C2×C42—471no—
438C2×C42—471no—
439C2×C42—471no—
440C2×C42—471no—
441C2×C42—471no—
442C2×C42—471no—
443C2×C42—942no—
444C2×C42—942no—
445C2×C42—942no—
446C2×C42—942no—
447C2×C42—941no—
448C2×C42—941no—
449C2×C42—941no—
450C2×C42C22×D4711yescharacteristic
451C4⋊C8—472no—
452C4⋊C8—942no—
453C4⋊C8—942no—
454C4⋊C8—942no—
455C4⋊C8C22×D4712yes—
456C22⋊C8—21no—
457C22⋊C8—942no—
458C22⋊C8—942no—
459C22⋊C8—942no—
460C22⋊C8—941no—
461#97—21no—
462#97—21no—
463#97—941no—
464#97—941no—
465C4×C8—472no—
466C4×C8—942no—
467C4×C8C22×D4712yes—
468C24—471no—
469C2×Q8—24no—
470C2×Q8—21no—
471C2×Q8—478no—
472C2×Q8—472no—
473C2×Q8—942no—
474C2×Q8—942no—
475C2×Q8—944no—
476C2×Q8—948no—
477C2×Q8—944no—
478C2×Q8—944no—
479C2×Q8—948no—
480C2×Q8—948no—
481C2×Q8—944no—
482C2×Q8—941no—
483C2×Q8C2×C4×D4712yes—
484C22×C4—21no—
485C22×C4—21no—
486C22×C4—471no—
487C22×C4—471no—
488C22×C4—471no—
489C22×C4—471no—
490C22×C4—471no—
491C22×C4—471no—
492C22×C4—471no—
493C22×C4—471no—
494C22×C4—471no—
495C22×C4—471no—
496C22×C4—471no—
497C22×C4—471no—
498C22×C4—471no—
499C22×C4—471no—
500C22×C4—471no—
501C22×C4—944no—
502C22×C4—944no—
503C22×C4—942no—
504C22×C4—942no—
505C22×C4—948no—
506C22×C4—942no—
507C22×C4—942no—
508C22×C4—941no—
509C22×C4—941no—
510C22×C4—1882no—
511C22×C4—1884no—
512C22×C4C2×C4×D4711yescharacteristic
513C22×C4D4×D4711yescharacteristic
514C22×C4D4×D4711yescharacteristic
515M16—42no—
516M16—1884no—
517M16—1882no—
518C2×C8—22no—
519C2×C8—22no—
520C2×C8—942no—
521C2×C8—942no—
522C4⋊C4—22no—
523C4⋊C4—21no—
524C4⋊C4—22no—
525C4⋊C4—472no—
526C4⋊C4—942no—
527C4⋊C4—942no—
528C4⋊C4—942no—
529C4⋊C4—942no—
530C4⋊C4—942no—
531C4⋊C4—942no—
532C4⋊C4—942no—
533C4⋊C4—941no—
534C4⋊C4—942no—
535C4⋊C4C2×C4×D4712yes—
536C42—22no—
537C42—22no—
538C42—471no—
539C42—471no—
540C42—471no—
541C42—471no—
542C42—471no—
543C42—471no—
544C42—471no—
545C42—471no—
546C42—471no—
547C42—471no—
548C42—471no—
549C42—471no—
550C42—942no—
551C42—942no—
552C42—942no—
553C42—942no—
554C42—942no—
555C42—942no—
556C42C2×C4×D4711yescharacteristic
557C42D4×D4711yescharacteristic
558C42D4×D4711yescharacteristic
559C42C23×D4711yescharacteristic
560C23—471no—
561C23—471no—
562C23—942no—
563C23—942no—
564C23—942no—
565C23C22⋊C4×D4711yescharacteristic
566Q8—24no—
567Q8—22no—
568Q8—474no—
569Q8—948no—
570Q8—948no—
571Q8—944no—
572Q8—942no—
573Q8M16×D4714yes—
574C2×C4—24no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—42no—
581C2×C4—471no—
582C2×C4—471no—
583C2×C4—474no—
584C2×C4—474no—
585C2×C4—471no—
586C2×C4—471no—
587C2×C4—471no—
588C2×C4—471no—
589C2×C4—942no—
590C2×C4—942no—
591C2×C4—942no—
592C2×C4—942no—
593C2×C4—948no—
594C2×C4—948no—
595C2×C4—942no—
596C2×C4—942no—
597C2×C4—944no—
598C2×C4—942no—
599C2×C4—942no—
600C2×C4—942no—
601C2×C4—942no—
602C2×C4—942no—
603C2×C4—1881no—
604C2×C4—1881no—
605C2×C4—1881no—
606C2×C4—1881no—
607C2×C4—1884no—
608C2×C4—1884no—
609C2×C4—1882no—
610C2×C4C22⋊C4×D4711yescharacteristic
611C2×C4C22⋊C4×D4711yescharacteristic
612C2×C4C22⋊C4×D4711yescharacteristic
613C2×C4C22×C4×D4711yescharacteristic
614C2×C4C2×D4×D4711yescharacteristic
615C2×C4C2×D4×D4711yescharacteristic
616C8—42no—
617C8—42no—
618C8—1882no—
619C8—1882no—
620C22—21no—
621C22—21no—
622C22—472no—
623C22—472no—
624C22—948no—
625C22—941no—
626C22—941no—
627C22—941no—
628C22—941no—
629C22#1568913812yes—
630C22C2×C22⋊C4×D4711yescharacteristic
631C4—21no—
632C4—21no—
633C4—24no—
634C4—21no—
635C4—21no—
636C4—42no—
637C4—472no—
638C4—472no—
639C4—941no—
640C4—941no—
641C4—941no—
642C4—944no—
643C4—941no—
644C4—1882no—
645C4C2×M16×D4712yes—
646C4#137×D4712yes—
647C2—22no—
648C2—942no—
649C2—942no—
650C2#407×D4711yescharacteristic
651C2#419×D4711yescharacteristic
652C2C2×#1568913811yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D47
commutatorC2×C4×C47C23×C4
FrattiniC42C23×D47
FittingC47×#1491C2
socleC22×C47C22×C22⋊C4
prime radicalC23×D47C22⋊C4
cosocleC42×C47C24
hypercenter#1491D47
hypocenterC47#17449
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1491

Complements

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

NH ≅ G/Ntype
4. #20241836C2semidirect
5. #20241835C2semidirect
6. #20241834C2semidirect
7. #20241815C2semidirect
8. #20241781C2semidirect
9. C47×#1491C2semidirect
42. #15688855C22semidirect
43. Q8⋊(C47⋊C8)C22semidirect
44. #15688853C22semidirect
48. C47×Q8⋊C8C22semidirect
160. Q8×D47M16semidirect
327. #1491D47semidirect
374. Q8⋊C8C2×D47semidirect
375. C47#17449semidirect

Maximal subgroups

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

subgroupindex
C2×C4×Q8×D472
#202520192
#202520182
#202418362
#202418352
#202418342
#202418152
#202417812
C47×#14912
#1744947