Finite Group Databasebeta

← #26233552

Subgroups — #26233552

Order 12032.

934 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
1#20246281C211yesmaximal, characteristic
2Q8×C22⋊Dic47C211yesmaximal, characteristic
3#20244080C211yesmaximal, characteristic, complemented
4#20244066C211yesmaximal, characteristic, complemented
5#20244031C211yesmaximal, characteristic, complemented
6#20244024C211yesmaximal, characteristic, complemented
7#20243658C211yesmaximal, characteristic, complemented
8#20243644C211yesmaximal, characteristic, complemented
9#20243483C211yesmaximal, characteristic
10#20243421C211yesmaximal, characteristic
11#20243323C211yesmaximal, characteristic
12#20243220C211yesmaximal, characteristic, complemented
13#103×Dic47C211yesmaximal, characteristic, complemented
14#20243169C211yesmaximal, characteristic
15C2×C47×#419C211yesmaximal, characteristic
16C2×Q8×Dic47C2211yescharacteristic
17C2×Q8×Dic47C2211yescharacteristic
18C2×#13093958C2211yescharacteristic
19C2×#13093956C2211yescharacteristic
20C2×#13093948C2211yescharacteristic
21C2×#13093948C2211yescharacteristic
22C2×#13093948C2211yescharacteristic
23C2×#13093948C2211yescharacteristic
24C2×C22⋊(C47⋊C8)C2211yescharacteristic
25C2×#13093934C2211yescharacteristic
26C2×C8×Dic47C2211yescharacteristic
27#15689030—21no—
28#15689030—21no—
29#15689030C2211yescharacteristic
30#15689030C2211yescharacteristic
31#15689022C2211yescharacteristic
32#15689022C2211yescharacteristic
33#15689016—21no—
34#15689016—21no—
35#15688991—21no—
36#15688983—21no—
37#15688983C2211yescharacteristic
38#15688983C2211yescharacteristic
39C4⋊C4×Dic47C2211yescharacteristic
40C4⋊C4×Dic47C2211yescharacteristic
41C4×C4⋊Dic47—21no—
42C4×C22⋊Dic47—21no—
43C4×C22⋊Dic47C2211yescharacteristic
44#15688886C2211yescharacteristic
45#15688886C2211yescharacteristic
46#15688885C2211yescharacteristic
47#15688885C2211yescharacteristic
48#15688885C2211yescharacteristic
49#15688885C2211yescharacteristic
50#15688877C2211yescharacteristic
51#15688877C2211yescharacteristic
52C23×Q8×C47C2211yescharacteristic
53C22×M16×C47C2211yescharacteristic
54C2×#117×C47C2211yescharacteristic
55C47×#419C418yescomplemented
56C2×#103×C47C2211yescharacteristic
57C2×#103×C47C2211yescharacteristic
58C2×#103×C47C2211yescharacteristic
59C2×#103×C47C2211yescharacteristic
60C22×C47⋊C8C2311yescharacteristic
61C22×C47⋊C8C2311yescharacteristic
62Q8×Dic47—22no—
63Q8×Dic47—22no—
64Q8×Dic47—41no—
65Q8×Dic47—41no—
66C2×C4⋊Dic47—21no—
67C2×C4⋊Dic47—21no—
68C2×C4⋊Dic47—21no—
69C2×C4⋊Dic47—21no—
70C2×C4⋊Dic47C2311yescharacteristic
71C2×C4⋊Dic47C2311yescharacteristic
72C2×C4⋊Dic47C2311yescharacteristic
73C2×C4⋊Dic47C2311yescharacteristic
74C2×C22⋊Dic47—21no—
75C2×C22⋊Dic47—21no—
76C2×C22⋊Dic47D411yescharacteristic
77C2×C22⋊Dic47D411yescharacteristic
78C2×C4×Dic47—21no—
79C2×C4×Dic47—21no—
80C2×C4×Dic47—21no—
81C2×C4×Dic47—21no—
82C2×C4×Dic47C2311yescharacteristic
83C2×C4×Dic47C2311yescharacteristic
84#13093958—41no—
85#13093956—41no—
86#13093948—22no—
87#13093948—22no—
88#13093948—41no—
89#13093948—41no—
90C22⋊(C47⋊C8)—22no—
91#13093934—41no—
92C8×Dic47—41no—
93C47⋊#95—21no—
94C47⋊#95—21no—
95C47⋊#95—21no—
96C47⋊#95—21no—
97C47⋊#95—21no—
98C47⋊#95—21no—
99C47⋊#95—21no—
100C47⋊#95—21no—
101C47⋊#95—21no—
102C47⋊#95—21no—
103C47⋊#95D411yescharacteristic
104C47⋊#95D411yescharacteristic
105C22×Q8×C47—21no—
106C22×Q8×C47—22no—
107C22×Q8×C47—22no—
108C22×Q8×C47—22no—
109C22×Q8×C47—41no—
110C22×Q8×C47—41no—
111C22×Q8×C47C2×C414yes—
112C22×Q8×C47C2311yescharacteristic
113C22×Q8×C47C2311yescharacteristic
114C23×C4×C47—21no—
115C23×C4×C47C2311yescharacteristic
116C2×M16×C47—41no—
117C2×M16×C47—41no—
118C2×M16×C47C2×C414yes—
119C22×C8×C47C2311yescharacteristic
120C22×C8×C47C2311yescharacteristic
121#117×C47—22no—
122#117×C47C2×C414yes—
123C2×C4⋊C4×C47C2311yescharacteristic
124C2×C4⋊C4×C47C2311yescharacteristic
125C2×C22⋊C4×C47—21no—
126C2×C42×C47—21no—
127#103×C47C2×C414yes—
128#103×C47C2×C414yes—
129#103×C47C2×C414yes—
130#103×C47C2×C414yes—
131C24×C47C2×D411yescharacteristic
132C2×Q8×C47—24no—
133C2×Q8×C47—21no—
134C2×Q8×C47—21no—
135C2×Q8×C47—21no—
136C2×Q8×C47—21no—
137C2×Q8×C47—28no—
138C2×Q8×C47—21no—
139C2×Q8×C47—21no—
140C2×Q8×C47—21no—
141C2×Q8×C47—21no—
142C2×Q8×C47—42no—
143C2×Q8×C47—42no—
144C2×Q8×C47—44no—
145C2×Q8×C47—44no—
146C2×Q8×C47—44no—
147C2×Q8×C47C22×C412yes—
148C2×Q8×C47C22×C412yes—
149C2×Q8×C47#4112yes—
150C2×Q8×C47#4112yes—
151C22×C4×C47—21no—
152C22×C4×C47—21no—
153C22×C4×C47—21no—
154C22×C4×C47—21no—
155C22×C4×C47—21no—
156C22×C4×C47—24no—
157C22×C4×C47—21no—
158C22×C4×C47—21no—
159C22×C4×C47—21no—
160C22×C4×C47—42no—
161C22×C4×C47—42no—
162C22×C4×C47C22×C412yes—
163C22×C4×C47C2×D411yescharacteristic
164C22×C4×C47#4112yes—
165C22×C4×C47C2411yescharacteristic
166C22×Dic47—21no—
167C22×Dic47—21no—
168C22×Dic47—21no—
169C22×Dic47—21no—
170C22×Dic47C2×D411yescharacteristic
171C22×Dic47C2×D411yescharacteristic
172C22×Dic47C2×D411yescharacteristic
173C22×Dic47C2×D411yescharacteristic
174M16×C47—44no—
175C2×C8×C47—21no—
176C2×C8×C47—21no—
177C2×C8×C47—21no—
178C2×C8×C47—21no—
179C2×C8×C47C22×C412yes—
180C2×C8×C47C22×C412yes—
181C4⋊C4×C47—21no—
182C4⋊C4×C47—21no—
183C4⋊C4×C47C22×C412yes—
184C4⋊C4×C47C22×C412yes—
185C22⋊C4×C47—24no—
186C42×C47—24no—
187C22⋊Dic47—22no—
188C22⋊Dic47—41no—
189C22⋊Dic47—42no—
190C22⋊Dic47—81no—
191C4⋊Dic47—21no—
192C4⋊Dic47—21no—
193C4⋊Dic47—21no—
194C4⋊Dic47—21no—
195C4⋊Dic47—21no—
196C4⋊Dic47—21no—
197C4⋊Dic47—21no—
198C4⋊Dic47—21no—
199C4⋊Dic47—42no—
200C4⋊Dic47—42no—
201C4⋊Dic47—42no—
202C4⋊Dic47—42no—
203C4×Dic47—21no—
204C4×Dic47—21no—
205C4×Dic47—21no—
206C4×Dic47—21no—
207C4×Dic47—42no—
208C4×Dic47—42no—
209C4×Dic47—81no—
210C4×Dic47—81no—
211C2×C47⋊C8—21no—
212C2×C47⋊C8—21no—
213C2×C47⋊C8—21no—
214C2×C47⋊C8—21no—
215C2×C47⋊C8—21no—
216C2×C47⋊C8—21no—
217C23×C47—21no—
218C23×C47—21no—
219C23×C47—21no—
220C23×C47—21no—
221C23×C47—21no—
222C23×C47—21no—
223C23×C47C4×D412yes—
224C23×C47C22×D411yescharacteristic
225Q8×C47—22no—
226Q8×C47—22no—
227Q8×C47—22no—
228Q8×C47—22no—
229Q8×C47—22no—
230Q8×C47—48no—
231Q8×C47—41no—
232Q8×C47—41no—
233Q8×C47#11714yes—
234C2×C4×C47—22no—
235C2×C4×C47—22no—
236C2×C4×C47—22no—
237C2×C4×C47—21no—
238C2×C4×C47—21no—
239C2×C4×C47—22no—
240C2×C4×C47—22no—
241C2×C4×C47—22no—
242C2×C4×C47—22no—
243C2×C4×C47—22no—
244C2×C4×C47—22no—
245C2×C4×C47—22no—
246C2×C4×C47—48no—
247C2×C4×C47—41no—
248C2×C4×C47—41no—
249C2×C4×C47—41no—
250C2×C4×C47—41no—
251C2×C4×C47—41no—
252C2×C4×C47—41no—
253C2×C4×C47C4×D412yes—
254C2×C4×C47C23×C411yescharacteristic
255C2×C4×C47C2×#4111yescharacteristic
256C2×C4×C47C2×#4111yescharacteristic
257C2×C4×C47C2×#4111yescharacteristic
258C2×C4×C47H(5,𝔽2)11yescharacteristic
259C2×C4×C47H(5,𝔽2)11yescharacteristic
260C2×Dic47—21no—
261C2×Dic47—21no—
262C2×Dic47—21no—
263C2×Dic47—21no—
264C2×Dic47—21no—
265C2×Dic47—21no—
266C2×Dic47—21no—
267C2×Dic47—21no—
268C2×Dic47—21no—
269C2×Dic47—21no—
270C2×Dic47—21no—
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—41no—
281C2×Dic47—41no—
282C2×Dic47—41no—
283C2×Dic47#13712yes—
284C8×C47—22no—
285C8×C47—22no—
286C47⋊C8—41no—
287C47⋊C8—41no—
288#22581—471nomaximal, Sylow
289C22×C47—22no—
290C22×C47—22no—
291C22×C47—22no—
292C22×C47—41no—
293C22×C47—41no—
294C22×C47—41no—
295C22×C47—41no—
296C22×C47C2×C4×D411yescharacteristic
297C22×C47#54611yescharacteristic
298C22×C47#54611yescharacteristic
299C22×C47C2×#13711yescharacteristic
300C22×C47#57511yescharacteristic
301C22×C47#57711yescharacteristic
302C22×C47#57811yescharacteristic
303C4×C47—22no—
304C4×C47—24no—
305C4×C47—42no—
306C4×C47—42no—
307C4×C47—42no—
308C4×C47C2×#11711yescharacteristic
309C4×C47#51811yescharacteristic
310C4×C47#55111yescharacteristic
311C4×C47#55111yescharacteristic
312Dic47—22no—
313Dic47—41no—
314Dic47—41no—
315Dic47—41no—
316Dic47—81no—
317Dic47—81no—
318Dic47—81no—
319Dic47—81no—
320C2×#419D4711yescharacteristic
321#2347—471no—
322Q8×C22⋊C4—471no—
323#1935—471no—
324#1933—471no—
325#1928—471no—
326#1927—471no—
327#1865—471no—
328#1863—471no—
329#1825—471no—
330#1809—471no—
331#1787—471no—
332#1763—471no—
333C4×#103—471no—
334#1753—471no—
335C2×C47—42no—
336C2×C47#234211yescharacteristic
337C2×C47C4×#13711yescharacteristic
338C2×C47#294711yescharacteristic
339C2×C47#331311yescharacteristic
340C2×C47#332011yescharacteristic
341C2×C47#332611yescharacteristic
342C2×C47#332811yescharacteristic
343C23×Q8C2×D4711yescharacteristic
344C22×M16C2×D4711yescharacteristic
345C2×C4×Q8—471no—
346C2×C4×Q8—471no—
347C2×#117C2×D4711yescharacteristic
348C2×#107—471no—
349C2×#106—471no—
350#419Dic4718yescomplemented
351C2×#103—471no—
352C2×#103—471no—
353C2×#103—471no—
354C2×#103—471no—
355C2×#103C2×D4711yescharacteristic
356C2×#103C2×D4711yescharacteristic
357C2×#103C2×D4711yescharacteristic
358C2×#103C2×D4711yescharacteristic
359C2×C22⋊C8—471no—
360C2×#97—471no—
361C2×C4×C8—471no—
362#391—471no—
363#391—471no—
364#391—941no—
365#391—941no—
366#389—471no—
367#389—471no—
368#388—941no—
369#388—941no—
370#383—941no—
371#380—471no—
372#380—471no—
373#380—941no—
374C4×C4⋊C4—471no—
375C4×C4⋊C4—471no—
376C4×C4⋊C4—941no—
377C4×C22⋊C4—471no—
378C4×C22⋊C4—941no—
379#340—471no—
380#340—471no—
381#340—471no—
382#340—471no—
383#340—471no—
384#340—471no—
385#336—471no—
386#336—471no—
387C47#2258111yescharacteristic, complemented, Sylow
388C22×Q8—21no—
389C22×Q8—22no—
390C22×Q8—22no—
391C22×Q8—22no—
392C22×Q8—41no—
393C22×Q8—41no—
394C22×Q8C2×Dic4714yes—
395C22×Q8C22×D4711yescharacteristic
396C22×Q8C22×D4711yescharacteristic
397C23×C4—21no—
398C23×C4C22×D4711yescharacteristic
399C2×M16—41no—
400C2×M16—41no—
401C2×M16C2×Dic4714yes—
402C22×C8—471no—
403C22×C8—471no—
404C22×C8C22×D4711yescharacteristic
405C22×C8C22×D4711yescharacteristic
406C4×Q8—942no—
407C4×Q8—942no—
408C4×Q8—1881no—
409C4×Q8—1881no—
410#117—22no—
411#117C2×Dic4714yes—
412C2×C4⋊C4—471no—
413C2×C4⋊C4—471no—
414C2×C4⋊C4—471no—
415C2×C4⋊C4—471no—
416C2×C4⋊C4—941no—
417C2×C4⋊C4—941no—
418C2×C4⋊C4—941no—
419C2×C4⋊C4—941no—
420C2×C4⋊C4C22×D4711yescharacteristic
421C2×C4⋊C4C22×D4711yescharacteristic
422C2×C22⋊C4—21no—
423C2×C22⋊C4—471no—
424C2×C22⋊C4—471no—
425C2×C22⋊C4—941no—
426C2×C22⋊C4—941no—
427C2×C42—21no—
428C2×C42—471no—
429C2×C42—471no—
430C2×C42—941no—
431C2×C42—941no—
432C2×C42—941no—
433C2×C42—941no—
434#107—1881no—
435#106—1881no—
436#103—942no—
437#103—942no—
438#103—1881no—
439#103—1881no—
440#103C2×Dic4714yes—
441#103C2×Dic4714yes—
442#103C2×Dic4714yes—
443#103C2×Dic4714yes—
444C22⋊C8—942no—
445#97—1881no—
446C4×C8—1881no—
447#95—471no—
448#95—471no—
449#95—941no—
450#95—941no—
451#95—941no—
452#95—941no—
453#95—941no—
454#95—941no—
455#95—941no—
456#95—941no—
457#95—941no—
458#95—941no—
459C24D4×D4711yescharacteristic
460C2×Q8—24no—
461C2×Q8—21no—
462C2×Q8—21no—
463C2×Q8—21no—
464C2×Q8—21no—
465C2×Q8—28no—
466C2×Q8—21no—
467C2×Q8—21no—
468C2×Q8—21no—
469C2×Q8—21no—
470C2×Q8—42no—
471C2×Q8—42no—
472C2×Q8—44no—
473C2×Q8—44no—
474C2×Q8—44no—
475C2×Q8#1062535012yes—
476C2×Q8#1062535012yes—
477C2×Q8C22×Dic4712yes—
478C2×Q8C22×Dic4712yes—
479C22×C4—21no—
480C22×C4—21no—
481C22×C4—21no—
482C22×C4—21no—
483C22×C4—21no—
484C22×C4—24no—
485C22×C4—21no—
486C22×C4—21no—
487C22×C4—21no—
488C22×C4—42no—
489C22×C4—42no—
490C22×C4—471no—
491C22×C4—471no—
492C22×C4—471no—
493C22×C4—471no—
494C22×C4—941no—
495C22×C4—941no—
496C22×C4—941no—
497C22×C4—941no—
498C22×C4D4×D4711yescharacteristic
499C22×C4#1062535212yes—
500C22×C4C22×Dic4712yes—
501C22×C4C23×D4711yescharacteristic
502M16—44no—
503C2×C8—21no—
504C2×C8—21no—
505C2×C8—21no—
506C2×C8—21no—
507C2×C8—941no—
508C2×C8—941no—
509C2×C8—941no—
510C2×C8—941no—
511C2×C8—941no—
512C2×C8—941no—
513C2×C8C22×Dic4712yes—
514C2×C8C22×Dic4712yes—
515C4⋊C4—21no—
516C4⋊C4—21no—
517C4⋊C4—941no—
518C4⋊C4—941no—
519C4⋊C4—941no—
520C4⋊C4—941no—
521C4⋊C4—941no—
522C4⋊C4—941no—
523C4⋊C4—941no—
524C4⋊C4—941no—
525C4⋊C4—1882no—
526C4⋊C4—1882no—
527C4⋊C4—1882no—
528C4⋊C4—1882no—
529C4⋊C4C22×Dic4712yes—
530C4⋊C4C22×Dic4712yes—
531C22⋊C4—24no—
532C22⋊C4—942no—
533C22⋊C4—3761no—
534C22⋊C4—1881no—
535C22⋊C4—1882no—
536C42—24no—
537C42—941no—
538C42—941no—
539C42—941no—
540C42—941no—
541C42—3761no—
542C42—3761no—
543C42—1882no—
544C42—1882no—
545C23—21no—
546C23—21no—
547C23—21no—
548C23—21no—
549C23—21no—
550C23—21no—
551C23D4×Dic4712yes—
552C23C2×D4×D4711yescharacteristic
553Q8—22no—
554Q8—22no—
555Q8—22no—
556Q8—22no—
557Q8—22no—
558Q8—48no—
559Q8—41no—
560Q8—41no—
561Q8#1309398314yes—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—48no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—472no—
582C2×C4—941no—
583C2×C4—941no—
584C2×C4—941no—
585C2×C4—941no—
586C2×C4—941no—
587C2×C4—941no—
588C2×C4—941no—
589C2×C4—941no—
590C2×C4—941no—
591C2×C4—941no—
592C2×C4—941no—
593C2×C4—1881no—
594C2×C4—1881no—
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×C4—1881no—
604C2×C4—1881no—
605C2×C4D4×Dic4712yes—
606C2×C4C23×Dic4711yescharacteristic
607C2×C4C2×#1062535211yescharacteristic
608C2×C4C2×#1062535011yescharacteristic
609C2×C4C2×#1062535011yescharacteristic
610C2×C4#1309406911yescharacteristic
611C2×C4#1309406911yescharacteristic
612C8—22no—
613C8—22no—
614C8—1881no—
615C8—1881no—
616C22—22no—
617C22—22no—
618C22—22no—
619C22—41no—
620C22—41no—
621C22—41no—
622C22—41no—
623C22C2×D4×Dic4711yescharacteristic
624C22#1568976311yescharacteristic
625C22#1568976311yescharacteristic
626C22#137×D4711yescharacteristic
627C22#1568992211yescharacteristic
628C22#1568993211yescharacteristic
629C22#1568993911yescharacteristic
630C4—22no—
631C4—24no—
632C4—42no—
633C4—42no—
634C4—42no—
635C4—942no—
636C4—3761no—
637C4—3761no—
638C4—3761no—
639C4—3761no—
640C4—1881no—
641C4—1881no—
642C4—1881no—
643C4C2×#1309398311yescharacteristic
644C4#1568959811yescharacteristic
645C4#1568979911yescharacteristic
646C4#1568979911yescharacteristic
647C2—42no—
648C2#2024623711yescharacteristic
649C2#137×Dic4711yescharacteristic
650C2#2025223211yescharacteristic
651C2#2025545511yescharacteristic
652C2#2025556011yescharacteristic
653C2#2025563711yescharacteristic
654C2#2025566711yescharacteristic
subgroupG/H
centerC23C2×D4×D47
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC2×C47×#419C2
socleC23×C47C22×D4
prime radicalC24×C47C2×D4
cosocleC22×C4×C47C24
hypercenterC2×#419D47
hypocenterC47#22581
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#419

Complements

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

NH ≅ G/Ntype
3. #20244080C2semidirect
4. #20244066C2semidirect
5. #20244031C2semidirect
6. #20244024C2semidirect
7. #20243658C2semidirect
8. #20243644C2semidirect
12. #20243220C2semidirect
13. #103×Dic47C2semidirect
55. C47×#419C4semidirect
350. #419Dic47semidirect
387. C47#22581semidirect

Maximal subgroups

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

subgroupindex
#202462812
Q8×C22⋊Dic472
#202440802
#202440662
#202440312
#202440242
#202436582
#202436442
#202434832
#202434212
#202433232
#202432202
#103×Dic472
#202431692
C2×C47×#4192
#2258147