Finite Group Databasebeta

← #26217658

Subgroups — #26217658

Order 12032.

1830 conjugacy classes of subgroups.

The subgroup lattice has 578 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#41×Dic47C211yesmaximal, characteristic
2C2×#15689124C211yesmaximal, characteristic
3C2×#15689132C211yesmaximal, characteristic, complemented
4C2×#15689030C211yesmaximal, characteristic, complemented
5#20243731C214yesmaximal, complemented
6#20243724C214yesmaximal, complemented
7#20243108C211yesmaximal, characteristic, complemented
8#20243081C211yesmaximal, characteristic, complemented
9C22×#137×C47C211yesmaximal, characteristic
10C22×#10625329C2211yescharacteristic
11#41×Dic47—24no—
12#41×Dic47—24no—
13C2×Q8×Dic47C2212yes—
14C2×D4×Dic47—22no—
15C2×D4×Dic47C2212yes—
16C2×#13093983—21no—
17C2×#13093983C2211yescharacteristic
18C22×C4⋊Dic47C2211yescharacteristic
19C22×C4×Dic47—21no—
20C22×C4×Dic47C2211yescharacteristic
21#15689132—24no—
22#15689124—24no—
23C2×#13093948C2214yes—
24C2×#13093948C2212yes—
25C2×#13093945C2212yes—
26#15689030—21no—
27#15689030—21no—
28#15689030C2214yes—
29C2×C47⋊#95—22no—
30#15688886C2214yes—
31#15688877C2214yes—
32C22×#41×C47C2211yescharacteristic
33C23×Q8×C47C2211yescharacteristic
34C2×#137×C47—24no—
35C2×#137×C47—24no—
36C2×#137×C47C418yescomplemented
37C22×Q16×C47C2212yes—
38C22×SD16×C47C2212yes—
39C22×M16×C47C2211yescharacteristic
40C23×Dic47—22no—
41C23×Dic47—21no—
42C23×Dic47D412yes—
43C2×#10625329—22no—
44C2×#10625329—22no—
45C2×#10625329—22no—
46C22×C47⋊C8C2312yes—
47Q8×Dic47—24no—
48Q8×Dic47—24no—
49D4×Dic47—216no—
50D4×Dic47—24no—
51D4×Dic47—24no—
52#13093983—28no—
53#13093983—22no—
54#13093983D414yes—
55C2×C4⋊Dic47—22no—
56C2×C4⋊Dic47—22no—
57C2×C4⋊Dic47—22no—
58C2×C4⋊Dic47—21no—
59C2×C4⋊Dic47C2312yes—
60C2×C4⋊Dic47C2312yes—
61C2×C22⋊Dic47—21no—
62C2×C22⋊Dic47—22no—
63C2×C4×Dic47—28no—
64C2×C4×Dic47—21no—
65C2×C4×Dic47—22no—
66C2×C4×Dic47—22no—
67C2×C4×Dic47—22no—
68C2×C4×Dic47D411yescharacteristic
69C2×C4×Dic47D411yescharacteristic
70C2×C4×Dic47C2312yes—
71#13093948—24no—
72#13093948—28no—
73#13093945—24no—
74C47⋊#95—24no—
75C47⋊#95—24no—
76C2×#41×C47—22no—
77C2×#41×C47—22no—
78C2×#41×C47—22no—
79C2×#41×C47C2×C414yes—
80C2×#41×C47D414yes—
81C2×#41×C47D414yes—
82C22×Q8×C47—21no—
83C22×Q8×C47—22no—
84C22×Q8×C47—22no—
85C22×Q8×C47—22no—
86C22×Q8×C47—22no—
87C22×Q8×C47—22no—
88C22×Q8×C47C2×C414yescomplemented
89C22×Q8×C47C2312yes—
90C22×Q8×C47C2311yescharacteristic
91C22×D4×C47—21no—
92C22×D4×C47C2311yescharacteristic
93C23×C4×C47—21no—
94C23×C4×C47—21no—
95C23×C4×C47C2311yescharacteristic
96#137×C47—232no—
97C2×Q16×C47—24no—
98C2×Q16×C47—24no—
99C2×Q16×C47C2×C418yescomplemented
100C2×SD16×C47—24no—
101C2×SD16×C47—24no—
102C2×SD16×C47C2×C418yescomplemented
103C2×M16×C47—22no—
104C2×M16×C47—22no—
105C2×M16×C47C2×C414yescomplemented
106C22×C8×C47C2312yes—
107C24×C47—21no—
108C24×C47C2×D411yescharacteristic
109#41×C47—28no—
110#41×C47—28no—
111#41×C47—28no—
112#41×C47C22⋊C4116yes—
113C2×Q8×C47—24no—
114C2×Q8×C47—21no—
115C2×Q8×C47—21no—
116C2×Q8×C47—21no—
117C2×Q8×C47—22no—
118C2×Q8×C47—28no—
119C2×Q8×C47—28no—
120C2×Q8×C47—22no—
121C2×Q8×C47—22no—
122C2×Q8×C47—22no—
123C2×Q8×C47—22no—
124C2×Q8×C47—42no—
125C2×Q8×C47—42no—
126C2×Q8×C47—44no—
127C2×Q8×C47—44no—
128C2×Q8×C47C22×C414yes—
129C2×Q8×C47C22×C412yes—
130C2×Q8×C47C2×D412yes—
131C2×Q8×C47C2×D412yes—
132C2×D4×C47—24no—
133C2×D4×C47—24no—
134C2×D4×C47—24no—
135C2×D4×C47—21no—
136C2×D4×C47—21no—
137C2×D4×C47—21no—
138C2×D4×C47C22×C412yes—
139C2×D4×C47C2×D412yes—
140C2×D4×C47C2×D412yes—
141C22×C4×C47—21no—
142C22×C4×C47—24no—
143C22×C4×C47—24no—
144C22×C4×C47—24no—
145C22×C4×C47—24no—
146C22×C4×C47—22no—
147C22×C4×C47—42no—
148C22×C4×C47—42no—
149C22×C4×C47C22×C412yes—
150C22×C4×C47C2×D412yes—
151C22×C4×C47C2×D412yes—
152C22×C4×C47C2×D411yescharacteristic
153C22×C4×C47C2×D412yes—
154C22×C4×C47#4112yes—
155C22×C4×C47#4112yes—
156C22×C4×C47C2411yescharacteristic
157C22×Dic47—28no—
158C22×Dic47—24no—
159C22×Dic47—22no—
160C22×Dic47—24no—
161C22×Dic47—22no—
162C22×Dic47—42no—
163C22×Dic47—44no—
164C22×Dic47—44no—
165C22×Dic47—44no—
166C22×Dic47—42no—
167C22×Dic47C2×D412yes—
168C22×Dic47C2×D412yes—
169Q16×C47—216no—
170SD16×C47—216no—
171M16×C47—28no—
172C2×C8×C47—22no—
173C2×C8×C47—22no—
174C2×C8×C47C22×C414yes—
175C22⋊Dic47—28no—
176C22⋊Dic47—24no—
177C4⋊Dic47—28no—
178C4⋊Dic47—22no—
179C4⋊Dic47—22no—
180C4⋊Dic47—22no—
181C4⋊Dic47—42no—
182C4⋊Dic47C2×D414yes—
183C4×Dic47—28no—
184C4×Dic47—22no—
185C4×Dic47—24no—
186C4×Dic47—22no—
187C4×Dic47—22no—
188C4×Dic47—22no—
189#10625329—44no—
190C2×C47⋊C8—22no—
191C2×C47⋊C8—22no—
192C2×C47⋊C8—22no—
193C23×C47—22no—
194C23×C47—22no—
195C23×C47—22no—
196C23×C47—22no—
197C23×C47—22no—
198C23×C47—42no—
199C23×C47—42no—
200C23×C47C4×D412yes—
201C23×C47#12012yes—
202C23×C47#12112yes—
203C23×C47C22×D411yescharacteristic
204Q8×C47—22no—
205Q8×C47—22no—
206Q8×C47—22no—
207Q8×C47—24no—
208Q8×C47—24no—
209Q8×C47—42no—
210Q8×C47—42no—
211Q8×C47—48no—
212Q8×C47C2×C22⋊C414yes—
213D4×C47—216no—
214D4×C47—22no—
215D4×C47—22no—
216D4×C47—22no—
217D4×C47C2×C22⋊C414yes—
218C2×C4×C47—22no—
219C2×C4×C47—22no—
220C2×C4×C47—22no—
221C2×C4×C47—216no—
222C2×C4×C47—24no—
223C2×C4×C47—42no—
224C2×C4×C47—42no—
225C2×C4×C47—48no—
226C2×C4×C47C2×C22⋊C414yes—
227C2×C4×C47C4×D414yes—
228C2×C4×C47C4×D412yes—
229C2×C4×C47#12012yes—
230C2×C4×C47#12114yes—
231C2×C4×C47#12112yes—
232C2×C4×C47#12314yes—
233C2×C4×C47C23×C411yescharacteristic
234C2×C4×C47C22×D411yescharacteristic
235C2×C4×C47C22×D411yescharacteristic
236C2×C4×C47C22×D411yescharacteristic
237C2×C4×C47C2×#4111yescharacteristic
238C2×C4×C47C2×#4111yescharacteristic
239C2×Dic47—22no—
240C2×Dic47—22no—
241C2×Dic47—22no—
242C2×Dic47—24no—
243C2×Dic47—22no—
244C2×Dic47—22no—
245C2×Dic47—48no—
246C2×Dic47—416no—
247C2×Dic47—42no—
248C2×Dic47—42no—
249C2×Dic47—42no—
250C2×Dic47—44no—
251C2×Dic47—84no—
252C2×Dic47#13714yescomplemented
253C8×C47—24no—
254C47⋊C8—42no—
255#20927—471nomaximal, Sylow
256C22×C47—24no—
257C22×C47—24no—
258C22×C47—22no—
259C22×C47—22no—
260C22×C47—24no—
261C22×C47—44no—
262C22×C47—44no—
263C22×C47—42no—
264C22×C47—42no—
265C22×C47#38614yes—
266C22×C47C2×C4×D411yescharacteristic
267C22×C47C2×#12011yescharacteristic
268C22×C47C2×#12111yescharacteristic
269C22×C47C2×#13712yes—
270C22×C47#57512yes—
271C4×C47—24no—
272C4×C47—44no—
273C4×C47#38614yes—
274C4×C47C22×C22⋊C411yescharacteristic
275C4×C47C2×C4×D411yescharacteristic
276C4×C47C2×#12111yescharacteristic
277C4×C47C2×#12311yescharacteristic
278Dic47—24no—
279Dic47—42no—
280Dic47—44no—
281Dic47—82no—
282C22×#137D4711yescharacteristic
283C2×C4×#41—471no—
284C2×#417—471no—
285C2×#419—471no—
286C2×#391—471no—
287#1878—474no—
288#1877—474no—
289#1736—471no—
290#1728—471no—
291C2×C47—24no—
292C2×C47—44no—
293C2×C47C2×#38611yescharacteristic
294C2×C47C4×#13712yes—
295C2×C47#300712yes—
296C2×C47#303911yescharacteristic
297C2×C47#304011yescharacteristic
298C22×#41C2×D4711yescharacteristic
299C23×Q8C2×D4711yescharacteristic
300C2×#137—24no—
301C2×#137—24no—
302C2×#137Dic4718yescomplemented
303C22×Q16C2×D4712yes—
304C22×SD16C2×D4712yes—
305C22×M16—471no—
306C22×M16C2×D4711yescharacteristic
307C4×#41—944no—
308C4×#41—944no—
309C2×C4×Q8—472no—
310C2×C4×D4—472no—
311C2×C4×D4—942no—
312C2×#117—471no—
313C2×#117—941no—
314C22×C4⋊C4—471no—
315C22×C42—471no—
316C22×C42—941no—
317#419—944no—
318#417—944no—
319C2×#103—474no—
320C2×#103—472no—
321C2×#102—472no—
322#391—474no—
323#391—941no—
324#391—941no—
325C2×#95—942no—
326#340—474no—
327#336—474no—
328C47#2092711yescharacteristic, complemented, Sylow
329C2×#41—22no—
330C2×#41—22no—
331C2×#41—22no—
332C2×#41C2×Dic4714yes—
333C2×#41C22⋊D4714yes—
334C2×#41C22⋊D4714yes—
335C22×Q8—21no—
336C22×Q8—22no—
337C22×Q8—22no—
338C22×Q8—22no—
339C22×Q8—22no—
340C22×Q8—22no—
341C22×Q8C2×Dic4714yescomplemented
342C22×Q8C22×D4712yes—
343C22×Q8C22×D4711yescharacteristic
344C22×D4—21no—
345C22×D4C22×D4711yescharacteristic
346C23×C4—21no—
347C23×C4—21no—
348C23×C4—472no—
349C23×C4—942no—
350C23×C4—941no—
351C23×C4C22×D4711yescharacteristic
352#137—232no—
353C2×Q16—24no—
354C2×Q16—24no—
355C2×Q16C2×Dic4718yescomplemented
356C2×SD16—24no—
357C2×SD16—24no—
358C2×SD16C2×Dic4718yescomplemented
359C2×M16—22no—
360C2×M16—22no—
361C2×M16—942no—
362C2×M16—942no—
363C2×M16—942no—
364C2×M16C2×Dic4714yescomplemented
365C22×C8—472no—
366C22×C8C22×D4712yes—
367C4×Q8—944no—
368C4×Q8—944no—
369C4×D4—944no—
370C4×D4—9416no—
371C4×D4—944no—
372#117—474no—
373#117—948no—
374#117—942no—
375C2×C4⋊C4—472no—
376C2×C4⋊C4—472no—
377C2×C4⋊C4—942no—
378C2×C4⋊C4—942no—
379C2×C4⋊C4—942no—
380C2×C4⋊C4—941no—
381C2×C22⋊C4—942no—
382C2×C22⋊C4—941no—
383C2×C42—471no—
384C2×C42—471no—
385C2×C42—472no—
386C2×C42—942no—
387C2×C42—948no—
388C2×C42—941no—
389C2×C42—942no—
390C2×C42—942no—
391#103—948no—
392#103—944no—
393#102—944no—
394#95—944no—
395#95—944no—
396C24—21no—
397C24D4×D4711yescharacteristic
398#41—28no—
399#41—28no—
400#41—28no—
401#41C22⋊Dic47116yes—
402C2×Q8—24no—
403C2×Q8—21no—
404C2×Q8—21no—
405C2×Q8—21no—
406C2×Q8—22no—
407C2×Q8—28no—
408C2×Q8—28no—
409C2×Q8—22no—
410C2×Q8—22no—
411C2×Q8—22no—
412C2×Q8—22no—
413C2×Q8—42no—
414C2×Q8—44no—
415C2×Q8—42no—
416C2×Q8—44no—
417C2×Q8C22×Dic4714yes—
418C2×Q8C22×Dic4712yes—
419C2×Q8C2×C22⋊D4712yes—
420C2×Q8C2×C22⋊D4712yes—
421C2×D4—24no—
422C2×D4—24no—
423C2×D4—24no—
424C2×D4—21no—
425C2×D4—21no—
426C2×D4—21no—
427C2×D4C22×Dic4712yes—
428C2×D4C2×C22⋊D4712yes—
429C2×D4C2×C22⋊D4712yes—
430C22×C4—21no—
431C22×C4—24no—
432C22×C4—24no—
433C22×C4—24no—
434C22×C4—24no—
435C22×C4—22no—
436C22×C4—42no—
437C22×C4—42no—
438C22×C4—472no—
439C22×C4—472no—
440C22×C4—944no—
441C22×C4—948no—
442C22×C4—944no—
443C22×C4—942no—
444C22×C4—942no—
445C22×C4—1884no—
446C22×C4—1882no—
447C22×C4—1884no—
448C22×C4—1884no—
449C22×C4—1882no—
450C22×C4D4×D4711yescharacteristic
451C22×C4D4×D4712yes—
452C22×C4#1062535212yes—
453C22×C4#1062535212yes—
454C22×C4C22×Dic4712yes—
455C22×C4C2×C22⋊D4712yes—
456C22×C4C2×C22⋊D4712yes—
457C22×C4C23×D4711yescharacteristic
458Q16—216no—
459SD16—216no—
460M16—28no—
461M16—1884no—
462C2×C8—22no—
463C2×C8—22no—
464C2×C8—942no—
465C2×C8—942no—
466C2×C8—942no—
467C2×C8C22×Dic4714yes—
468C4⋊C4—474no—
469C4⋊C4—948no—
470C4⋊C4—942no—
471C4⋊C4—942no—
472C4⋊C4—942no—
473C4⋊C4—1882no—
474C22⋊C4—948no—
475C22⋊C4—944no—
476C42—948no—
477C42—942no—
478C42—942no—
479C42—944no—
480C42—942no—
481C42—942no—
482C23—22no—
483C23—22no—
484C23—22no—
485C23—22no—
486C23—22no—
487C23—42no—
488C23—42no—
489C23D4×Dic4712yes—
490C23#1309399212yes—
491C23#1309399612yes—
492C23C2×D4×D4711yescharacteristic
493Q8—22no—
494Q8—22no—
495Q8—22no—
496Q8—24no—
497Q8—24no—
498Q8—48no—
499Q8—42no—
500Q8—42no—
501Q8C2×C22⋊Dic4714yes—
502D4—216no—
503D4—22no—
504D4—22no—
505D4—22no—
506D4C2×C22⋊Dic4714yes—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—216no—
511C2×C4—24no—
512C2×C4—48no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—474no—
516C2×C4—942no—
517C2×C4—942no—
518C2×C4—944no—
519C2×C4—942no—
520C2×C4—942no—
521C2×C4—942no—
522C2×C4—3764no—
523C2×C4—1888no—
524C2×C4—18816no—
525C2×C4—1882no—
526C2×C4—1884no—
527C2×C4—1882no—
528C2×C4—1882no—
529C2×C4C2×C22⋊Dic4714yes—
530C2×C4D4×Dic4712yes—
531C2×C4D4×Dic4714yes—
532C2×C4#1309399212yes—
533C2×C4#1309399612yes—
534C2×C4#1309399614yes—
535C2×C4#1309400514yes—
536C2×C4C23×Dic4711yescharacteristic
537C2×C4C2×D4×D4711yescharacteristic
538C2×C4C22×C22⋊D4711yescharacteristic
539C2×C4C22×C22⋊D4711yescharacteristic
540C2×C4C2×#1062535211yescharacteristic
541C2×C4C2×#1062535211yescharacteristic
542C8—24no—
543C8—1882no—
544C22—24no—
545C22—22no—
546C22—22no—
547C22—24no—
548C22—24no—
549C22—44no—
550C22—42no—
551C22—42no—
552C22—44no—
553C22#1568900314yes—
554C22C2×D4×Dic4711yescharacteristic
555C22C2×#1309399211yescharacteristic
556C22C2×#1309399611yescharacteristic
557C22#137×D4712yes—
558C22#1568992212yes—
559C4—24no—
560C4—44no—
561C4—944no—
562C4—3762no—
563C4—1882no—
564C4—1884no—
565C4#1568900314yes—
566C4C22×C22⋊Dic4711yescharacteristic
567C4C2×D4×Dic4711yescharacteristic
568C4C2×#1309399611yescharacteristic
569C4C2×#1309400511yescharacteristic
570C2—24no—
571C2—44no—
572C2C2×#1568900311yescharacteristic
573C2#137×Dic4712yes—
574C2#2025264412yes—
575C2#2025287311yescharacteristic
576C2#2025288411yescharacteristic
subgroupG/H
centerC23C2×D4×D47
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC22×#137×C47C2
socleC23×C47C22×D4
prime radicalC22×#41×C47C22
cosocleC22×C4×C47C24
hypercenterC22×#137D47
hypocenterC47#20927
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×#137

Complements

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

NH ≅ G/Ntype
3. C2×#15689132C2semidirect
4. C2×#15689030C2semidirect
5. #20243731C2semidirect
6. #20243724C2semidirect
7. #20243108C2semidirect
8. #20243081C2semidirect
36. C2×#137×C47C4semidirect
88. C22×Q8×C47C2×C4semidirect
99. C2×Q16×C47C2×C4semidirect
102. C2×SD16×C47C2×C4semidirect
105. C2×M16×C47C2×C4semidirect
252. C2×Dic47#137semidirect
302. C2×#137Dic47semidirect
328. C47#20927semidirect
341. C22×Q8C2×Dic47semidirect
355. C2×Q16C2×Dic47semidirect
358. C2×SD16C2×Dic47semidirect
364. C2×M16C2×Dic47semidirect

Maximal subgroups

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

subgroupindex
C2×#41×Dic472
C2×#156891242
C2×#156891322
C2×#156890302
#202437312
#202437242
#202431082
#202430812
C22×#137×C472
#2092747