Finite Group Databasebeta

← #26221944

Subgroups — #26221944

Order 12032.

1520 conjugacy classes of subgroups.

The subgroup lattice has 596 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×Q8×Dic47C211yesmaximal, characteristic, complemented
2C2×#15689224C212yesmaximal, complemented
3#121×Dic47C211yesmaximal, characteristic
4#20243745C212yesmaximal, complemented
5#20243706C211yesmaximal, characteristic
6#20243644C212yesmaximal, complemented
7#20243578C211yesmaximal, characteristic, complemented
8#20243323C211yesmaximal, characteristic, complemented
9#20243220C212yesmaximal, complemented
10#20243152C211yesmaximal, characteristic, complemented
11C2×C47×#448C211yesmaximal, characteristic
12C2×Q8×Dic47—28no—
13C2×Q8×Dic47—21no—
14C2×Q8×Dic47—21no—
15C2×Q8×Dic47C2212yes—
16C2×Q8×Dic47C2212yes—
17C2×D4×Dic47—21no—
18C2×D4×Dic47C2211yescharacteristic
19C22×C4⋊Dic47—21no—
20C22×C4⋊Dic47C2211yescharacteristic
21C22×C4×Dic47—21no—
22C22×C4×Dic47C2211yescharacteristic
23#15689224—28no—
24C2×#13093958C2212yes—
25C2×#13093948C2212yes—
26C2×#13093948C2212yes—
27C2×#13093945C2212yes—
28C2×C22⋊(C47⋊C8)C2211yescharacteristic
29C2×#13093934C2212yes—
30#15689026C2211yescharacteristic
31#15689022C2211yescharacteristic
32#15689016—21no—
33#15689003—21no—
34#15688983C2211yescharacteristic
35C4⋊C4×Dic47C2211yescharacteristic
36C22⋊C4×Dic47—21no—
37#15688886C2212yes—
38#15688885C2212yes—
39#15688877C2212yes—
40C23×Q8×C47C2211yescharacteristic
41C22×SD16×C47C2212yes—
42C2×#121×C47C2211yescharacteristic
43C47×#448C418yescomplemented
44C2×#103×C47C2212yes—
45C2×C22⋊C8×C47C2211yescharacteristic
46C23×Dic47—21no—
47C23×Dic47—21no—
48C23×Dic47D411yescharacteristic
49C23×Dic47D411yescharacteristic
50C22×C47⋊C8C2312yes—
51Q8×Dic47—28no—
52Q8×Dic47—28no—
53Q8×Dic47—24no—
54Q8×Dic47—24no—
55Q8×Dic47—24no—
56Q8×Dic47D418yescomplemented
57D4×Dic47—24no—
58D4×Dic47—44no—
59C2×C4⋊Dic47—28no—
60C2×C4⋊Dic47—21no—
61C2×C4⋊Dic47—22no—
62C2×C4⋊Dic47—22no—
63C2×C4⋊Dic47—24no—
64C2×C4⋊Dic47—21no—
65C2×C4⋊Dic47C2311yescharacteristic
66C2×C4⋊Dic47C2311yescharacteristic
67C2×C4⋊Dic47C2312yes—
68C2×C22⋊Dic47—21no—
69C2×C22⋊Dic47—21no—
70C2×C22⋊Dic47—21no—
71C2×C22⋊Dic47—21no—
72C2×C4×Dic47—28no—
73C2×C4×Dic47—22no—
74C2×C4×Dic47—22no—
75C2×C4×Dic47—24no—
76C2×C4×Dic47—21no—
77C2×C4×Dic47D411yescharacteristic
78C2×C4×Dic47D411yescharacteristic
79C2×C4×Dic47C2311yescharacteristic
80C2×C4×Dic47C2311yescharacteristic
81#13093958—24no—
82#13093948—24no—
83#13093948—24no—
84#13093945—24no—
85C22⋊(C47⋊C8)—22no—
86#13093934—24no—
87C47⋊#95—21no—
88C47⋊#95—21no—
89C47⋊#95—21no—
90C22×Q8×C47—24no—
91C22×Q8×C47—24no—
92C22×Q8×C47—21no—
93C22×Q8×C47—22no—
94C22×Q8×C47C2×C414yescomplemented
95C22×Q8×C47C2312yes—
96C22×D4×C47—21no—
97C22×D4×C47C2311yescharacteristic
98C23×C4×C47—21no—
99C23×C4×C47C2311yescharacteristic
100C2×SD16×C47—24no—
101C2×SD16×C47—24no—
102C2×SD16×C47C2×C418yescomplemented
103C22×C8×C47C2312yes—
104#121×C47—22no—
105#121×C47C2×C414yes—
106C2×C4⋊C4×C47C2311yescharacteristic
107C2×C22⋊C4×C47—21no—
108#103×C47C2×C418yescomplemented
109C22⋊C8×C47C2×C414yescomplemented
110C24×C47—21no—
111C24×C47C2×D411yescharacteristic
112C2×Q8×C47—216no—
113C2×Q8×C47—24no—
114C2×Q8×C47—24no—
115C2×Q8×C47—216no—
116C2×Q8×C47—24no—
117C2×Q8×C47—22no—
118C2×Q8×C47—22no—
119C2×Q8×C47—22no—
120C2×Q8×C47C22×C414yes—
121C2×Q8×C47C2×D414yes—
122C2×Q8×C47#4114yes—
123C2×D4×C47—24no—
124C2×D4×C47—22no—
125C2×D4×C47—22no—
126C2×D4×C47—21no—
127C2×D4×C47—42no—
128C2×D4×C47—42no—
129C2×D4×C47C22×C412yes—
130C22×C4×C47—24no—
131C22×C4×C47—24no—
132C22×C4×C47—24no—
133C22×C4×C47—22no—
134C22×C4×C47—22no—
135C22×C4×C47—22no—
136C22×C4×C47—21no—
137C22×C4×C47C22×C412yes—
138C22×C4×C47C2×D411yescharacteristic
139C22×C4×C47#4112yes—
140C22×C4×C47C2411yescharacteristic
141C22×Dic47—22no—
142C22×Dic47—22no—
143C22×Dic47—22no—
144C22×Dic47—24no—
145C22×Dic47—22no—
146C22×Dic47—22no—
147C22×Dic47—22no—
148C22×Dic47—21no—
149C22×Dic47—21no—
150C22×Dic47—41no—
151C22×Dic47—42no—
152C22×Dic47—42no—
153C22×Dic47—42no—
154C22×Dic47—42no—
155C22×Dic47—42no—
156C22×Dic47C2×D411yescharacteristic
157C22×Dic47C2×D411yescharacteristic
158C22×Dic47C2×D411yescharacteristic
159C22×Dic47C2×D411yescharacteristic
160SD16×C47—216no—
161C2×C8×C47—22no—
162C2×C8×C47—22no—
163C2×C8×C47C22×C414yes—
164C4⋊C4×C47—21no—
165C4⋊C4×C47C22×C412yes—
166C22⋊C4×C47—24no—
167C22⋊Dic47—42no—
168C22⋊Dic47—42no—
169C22⋊Dic47—42no—
170C22⋊Dic47—42no—
171C4⋊Dic47—28no—
172C4⋊Dic47—24no—
173C4⋊Dic47—22no—
174C4⋊Dic47—28no—
175C4⋊Dic47—22no—
176C4⋊Dic47—22no—
177C4⋊Dic47—42no—
178C4⋊Dic47C2×D414yes—
179C4×Dic47—28no—
180C4×Dic47—22no—
181C4×Dic47—28no—
182C4×Dic47—22no—
183C4×Dic47—22no—
184C4×Dic47—42no—
185C4×Dic47C2×D414yes—
186C2×C47⋊C8—22no—
187C2×C47⋊C8—22no—
188C2×C47⋊C8—22no—
189C23×C47—21no—
190C23×C47—21no—
191C23×C47—21no—
192C23×C47—21no—
193C23×C47—22no—
194C23×C47—41no—
195C23×C47—41no—
196C23×C47—41no—
197C23×C47—41no—
198C23×C47—41no—
199C23×C47—41no—
200C23×C47C4×D412yes—
201C23×C47AGL(1,ℤ/8)12yes—
202C23×C47#13712yes—
203C23×C47C22×D411yescharacteristic
204Q8×C47—216no—
205Q8×C47—24no—
206Q8×C47—24no—
207Q8×C47—24no—
208Q8×C47C4×D418yescomplemented
209D4×C47—24no—
210D4×C47—48no—
211D4×C47—42no—
212C2×C4×C47—216no—
213C2×C4×C47—24no—
214C2×C4×C47—24no—
215C2×C4×C47—21no—
216C2×C4×C47—24no—
217C2×C4×C47—21no—
218C2×C4×C47—24no—
219C2×C4×C47—24no—
220C2×C4×C47—22no—
221C2×C4×C47—41no—
222C2×C4×C47—41no—
223C2×C4×C47C4×D412yes—
224C2×C4×C47C23×C411yescharacteristic
225C2×C4×C47C22×D412yes—
226C2×C4×C47C2×#4112yes—
227C2×C4×C47C2×#4111yescharacteristic
228C2×Dic47—28no—
229C2×Dic47—21no—
230C2×Dic47—21no—
231C2×Dic47—21no—
232C2×Dic47—21no—
233C2×Dic47—21no—
234C2×Dic47—21no—
235C2×Dic47—21no—
236C2×Dic47—24no—
237C2×Dic47—21no—
238C2×Dic47—21no—
239C2×Dic47—44no—
240C2×Dic47—48no—
241C2×Dic47—42no—
242C2×Dic47—42no—
243C2×Dic47—41no—
244C2×Dic47—41no—
245C2×Dic47—41no—
246C2×Dic47—41no—
247C2×Dic47—41no—
248C2×Dic47—41no—
249C2×Dic47—84no—
250C2×Dic47#12012yes—
251C2×Dic47#12012yes—
252C2×Dic47#13712yes—
253C8×C47—24no—
254C47⋊C8—42no—
255#21454—471nomaximal, Sylow
256C22×C47—22no—
257C22×C47—22no—
258C22×C47—22no—
259C22×C47—22no—
260C22×C47—22no—
261C22×C47—22no—
262C22×C47—42no—
263C22×C47—42no—
264C22×C47—42no—
265C22×C47—81no—
266C22×C47—81no—
267C22×C47—81no—
268C22×C47—81no—
269C22×C47#44014yescomplemented
270C22×C47C2×C4×D411yescharacteristic
271C22×C47C2×#12011yescharacteristic
272C22×C47#52511yescharacteristic
273C22×C47C2×AGL(1,ℤ/8)11yescharacteristic
274C22×C47C2×#13711yescharacteristic
275C22×C47C2×#13711yescharacteristic
276C22×C47#57511yescharacteristic
277C4×C47—28no—
278C4×C47—22no—
279C4×C47—42no—
280C4×C47C2×C4×D412yes—
281C4×C47#52512yes—
282Dic47—22no—
283Dic47—22no—
284Dic47—22no—
285Dic47—41no—
286Dic47—44no—
287Dic47—81no—
288Dic47—81no—
289C22×C4×Q8—471no—
290C2×#448D4711yescharacteristic
291C2×#440—472no—
292C4×#121—471no—
293#1880—472no—
294#1874—471no—
295#1863—472no—
296#1852—471no—
297#1787—471no—
298#1763—472no—
299#1749—471no—
300C2×C47—24no—
301C2×C47—82no—
302C2×C47C4×#12011yescharacteristic
303C2×C47C2×#44011yescharacteristic
304C2×C47C4×#13711yescharacteristic
305C2×C47#300411yescharacteristic
306C2×C47#300611yescharacteristic
307C2×C47#311211yescharacteristic
308C2×C47#311711yescharacteristic
309C23×Q8C2×D4711yescharacteristic
310C22×SD16C2×D4712yes—
311C2×#121C2×D4711yescharacteristic
312C2×C4×Q8—472no—
313C2×C4×Q8—472no—
314C2×C4×Q8—941no—
315C2×C4×Q8—941no—
316C2×C4×Q8—948no—
317C2×C4×D4—471no—
318C2×C4×D4—941no—
319C22×C4⋊C4—471no—
320C22×C4⋊C4—941no—
321C22×C42—471no—
322C22×C42—941no—
323#448Dic4718yescomplemented
324#440—948no—
325C2×#107—472no—
326C2×#103—472no—
327C2×#103—472no—
328C2×#103C2×D4712yes—
329C2×#102—472no—
330C2×C22⋊C8—471no—
331C2×C22⋊C8C2×D4711yescharacteristic
332C2×#97—472no—
333#390—471no—
334#389—471no—
335#388—941no—
336#386—941no—
337#380—471no—
338C4×C4⋊C4—471no—
339C4×C22⋊C4—941no—
340#340—472no—
341#340—472no—
342#336—472no—
343C47#2145411yescharacteristic, complemented, Sylow
344C22×Q8—24no—
345C22×Q8—24no—
346C22×Q8—21no—
347C22×Q8—22no—
348C22×Q8C2×Dic4714yescomplemented
349C22×Q8C22×D4712yes—
350C22×D4—21no—
351C22×D4C22×D4711yescharacteristic
352C23×C4—21no—
353C23×C4—471no—
354C23×C4—471no—
355C23×C4—941no—
356C23×C4—941no—
357C23×C4C22×D4711yescharacteristic
358C2×SD16—24no—
359C2×SD16—24no—
360C2×SD16C2×Dic4718yescomplemented
361C22×C8—472no—
362C22×C8C22×D4712yes—
363#121—22no—
364#121C2×Dic4714yes—
365C4×Q8—478no—
366C4×Q8—944no—
367C4×Q8—944no—
368C4×Q8—948no—
369C4×Q8—948no—
370C4×Q8—944no—
371C4×D4—944no—
372C4×D4—1884no—
373C2×C4⋊C4—472no—
374C2×C4⋊C4—471no—
375C2×C4⋊C4—471no—
376C2×C4⋊C4—944no—
377C2×C4⋊C4—942no—
378C2×C4⋊C4—942no—
379C2×C4⋊C4—948no—
380C2×C4⋊C4—941no—
381C2×C4⋊C4—941no—
382C2×C4⋊C4C22×D4711yescharacteristic
383C2×C22⋊C4—21no—
384C2×C22⋊C4—941no—
385C2×C22⋊C4—941no—
386C2×C22⋊C4—941no—
387C2×C22⋊C4—941no—
388C2×C42—471no—
389C2×C42—471no—
390C2×C42—471no—
391C2×C42—471no—
392C2×C42—944no—
393C2×C42—942no—
394C2×C42—942no—
395C2×C42—948no—
396C2×C42—941no—
397#107—944no—
398#103—944no—
399#103—944no—
400#103C2×Dic4718yescomplemented
401#102—944no—
402C22⋊C8—942no—
403C22⋊C8C2×Dic4714yescomplemented
404#97—944no—
405#95—941no—
406#95—941no—
407#95—941no—
408C24—21no—
409C24D4×D4711yescharacteristic
410C2×Q8—24no—
411C2×Q8—216no—
412C2×Q8—24no—
413C2×Q8—216no—
414C2×Q8—24no—
415C2×Q8—22no—
416C2×Q8—22no—
417C2×Q8—22no—
418C2×Q8D4×D4714yes—
419C2×Q8#1062535214yes—
420C2×Q8C22×Dic4714yes—
421C2×D4—24no—
422C2×D4—22no—
423C2×D4—22no—
424C2×D4—21no—
425C2×D4—42no—
426C2×D4—42no—
427C2×D4C22×Dic4712yes—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—24no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—21no—
435C22×C4—471no—
436C22×C4—471no—
437C22×C4—471no—
438C22×C4—471no—
439C22×C4—942no—
440C22×C4—942no—
441C22×C4—942no—
442C22×C4—942no—
443C22×C4—944no—
444C22×C4—942no—
445C22×C4—942no—
446C22×C4—941no—
447C22×C4—941no—
448C22×C4—1882no—
449C22×C4—1881no—
450C22×C4—1882no—
451C22×C4—1882no—
452C22×C4—1882no—
453C22×C4—1882no—
454C22×C4D4×D4711yescharacteristic
455C22×C4#1062535212yes—
456C22×C4C22×Dic4712yes—
457C22×C4C23×D4711yescharacteristic
458SD16—216no—
459C2×C8—22no—
460C2×C8—22no—
461C2×C8—942no—
462C2×C8—942no—
463C2×C8—942no—
464C2×C8C22×Dic4714yes—
465C4⋊C4—21no—
466C4⋊C4—474no—
467C4⋊C4—948no—
468C4⋊C4—948no—
469C4⋊C4—944no—
470C4⋊C4—942no—
471C4⋊C4—942no—
472C4⋊C4—942no—
473C4⋊C4—1882no—
474C4⋊C4C22×Dic4712yes—
475C22⋊C4—24no—
476C22⋊C4—1882no—
477C22⋊C4—1882no—
478C22⋊C4—1882no—
479C22⋊C4—1882no—
480C42—474no—
481C42—942no—
482C42—942no—
483C42—948no—
484C42—948no—
485C42—942no—
486C42—1882no—
487C23—21no—
488C23—21no—
489C23—21no—
490C23—21no—
491C23—22no—
492C23—41no—
493C23—41no—
494C23—41no—
495C23—41no—
496C23—41no—
497C23—41no—
498C23D4×Dic4712yes—
499C23#1309405112yes—
500C23#1309405312yes—
501C23C2×D4×D4711yescharacteristic
502Q8—216no—
503Q8—24no—
504Q8—24no—
505Q8—24no—
506Q8D4×Dic4718yescomplemented
507D4—24no—
508D4—48no—
509D4—42no—
510C2×C4—216no—
511C2×C4—24no—
512C2×C4—24no—
513C2×C4—21no—
514C2×C4—24no—
515C2×C4—21no—
516C2×C4—24no—
517C2×C4—24no—
518C2×C4—22no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—472no—
522C2×C4—472no—
523C2×C4—472no—
524C2×C4—948no—
525C2×C4—941no—
526C2×C4—941no—
527C2×C4—941no—
528C2×C4—941no—
529C2×C4—941no—
530C2×C4—941no—
531C2×C4—944no—
532C2×C4—941no—
533C2×C4—941no—
534C2×C4—941no—
535C2×C4—3764no—
536C2×C4—1884no—
537C2×C4—1881no—
538C2×C4—1881no—
539C2×C4—1881no—
540C2×C4—1881no—
541C2×C4—1881no—
542C2×C4—1881no—
543C2×C4—1888no—
544C2×C4—1882no—
545C2×C4—1882no—
546C2×C4D4×Dic4712yes—
547C2×C4C23×Dic4711yescharacteristic
548C2×C4C2×D4×D4712yes—
549C2×C4C2×#1062535212yes—
550C2×C4C2×#1062535211yescharacteristic
551C8—24no—
552C8—1882no—
553C22—22no—
554C22—22no—
555C22—22no—
556C22—22no—
557C22—22no—
558C22—22no—
559C22—42no—
560C22—42no—
561C22—42no—
562C22—81no—
563C22—81no—
564C22—81no—
565C22—81no—
566C22#1568922414yescomplemented
567C22C2×D4×Dic4711yescharacteristic
568C22#120×D4711yescharacteristic
569C22#1568962911yescharacteristic
570C22C2×#1309405111yescharacteristic
571C22#137×D4711yescharacteristic
572C22C2×#1309405311yescharacteristic
573C22#1568992211yescharacteristic
574C4—28no—
575C4—22no—
576C4—42no—
577C4—942no—
578C4—942no—
579C4—942no—
580C4—3761no—
581C4—3761no—
582C4—1881no—
583C4—1884no—
584C4C2×D4×Dic4712yes—
585C4#1568962912yes—
586C2—24no—
587C2—82no—
588C2#120×Dic4711yescharacteristic
589C2C2×#1568922411yescharacteristic
590C2#137×Dic4711yescharacteristic
591C2#2025261911yescharacteristic
592C2#2025263911yescharacteristic
593C2#2025350511yescharacteristic
594C2#2025355411yescharacteristic
subgroupG/H
centerC23C2×D4×D47
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC2×C47×#448C2
socleC23×C47C22×D4
prime radicalC2×#121×C47C22
cosocleC22×C4×C47C24
hypercenterC2×#448D47
hypocenterC47#21454
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#448

Complements

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

NH ≅ G/Ntype
1. C22×Q8×Dic47C2semidirect
2. C2×#15689224C2semidirect
4. #20243745C2semidirect
6. #20243644C2semidirect
7. #20243578C2semidirect
8. #20243323C2semidirect
9. #20243220C2semidirect
10. #20243152C2semidirect
43. C47×#448C4semidirect
56. Q8×Dic47D4semidirect
94. C22×Q8×C47C2×C4semidirect
102. C2×SD16×C47C2×C4semidirect
108. #103×C47C2×C4semidirect
109. C22⋊C8×C47C2×C4semidirect
208. Q8×C47C4×D4semidirect
269. C22×C47#440semidirect
323. #448Dic47semidirect
343. C47#21454semidirect
348. C22×Q8C2×Dic47semidirect
360. C2×SD16C2×Dic47semidirect
400. #103C2×Dic47semidirect
403. C22⋊C8C2×Dic47semidirect
506. Q8D4×Dic47semidirect
566. C22#15689224semidirect

Maximal subgroups

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

subgroupindex
C22×Q8×Dic472
C2×#156892242
#121×Dic472
#202437452
#202437062
#202436442
#202435782
#202433232
#202432202
#202431522
C2×C47×#4482
#2145447