Finite Group Databasebeta

← #26187735

Subgroups — #26187735

Order 12032.

638 conjugacy classes of subgroups.

The subgroup lattice has 570 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20252340C211yesmaximal, characteristic
2Q8×#10625324C211yesmaximal, characteristic
3C4×Q16×D47C211yesmaximal, characteristic
4#20242283C211yesmaximal, characteristic, complemented
5#20242282C211yesmaximal, characteristic, complemented
6#20242240C211yesmaximal, characteristic, complemented
7#20242239C211yesmaximal, characteristic, complemented
8#20242237C211yesmaximal, characteristic, complemented
9C8×#10625337C211yesmaximal, characteristic, complemented
10#20242192C211yesmaximal, characteristic, complemented
11#20242145C211yesmaximal, characteristic
12#20241815C211yesmaximal, characteristic
13#20241815C211yesmaximal, characteristic
14#20241667C211yesmaximal, characteristic
15C47×#1584C211yesmaximal, characteristic, complemented
16C2×Q16×D47C411yescharacteristic
17C4×Q8×D47C2211yescharacteristic
18C4×Q8×D47C2211yescharacteristic
19#15689254C2211yescharacteristic
20#15689253—21no—
21#15689253C2211yescharacteristic
22#15689252—21no—
23#15689251—21no—
24#15689250—21no—
25#15689250C2211yescharacteristic
26Q8×C47⋊C8C2211yescharacteristic
27C8×Dic94C2211yescharacteristic
28#15689247C2211yescharacteristic
29C4×Dic188C2211yescharacteristic
30Q16×Dic47C411yescharacteristic
31#15689217C411yescharacteristic
32C4×#10625337C2211yescharacteristic
33C4×#10625337C2211yescharacteristic
34#15689215C411yescharacteristic
35#15689215C411yescharacteristic
36#15689174C2211yescharacteristic
37#107×D47C411yescharacteristic
38#15689145—21no—
39#15689145—21no—
40#15689145C2211yescharacteristic
41#15689145C2211yescharacteristic
42#103×D47C411yescharacteristic
43#103×D47C411yescharacteristic
44#15689075—21no—
45C4×#10625324—21no—
46C4×#10625324C2211yescharacteristic
47C4×C8×D47C2211yescharacteristic
48#15688876C2211yescharacteristic
49#15688875C2211yescharacteristic
50#15688875C2211yescharacteristic
51#15688855C2211yescharacteristic
52#15688855C2211yescharacteristic
53Q8⋊(C47⋊C8)C2211yescharacteristic
54Q8⋊(C47⋊C8)C2211yescharacteristic
55#15688853C2211yescharacteristic
56#15688853C2211yescharacteristic
57#15688845C2211yescharacteristic
58C8×C47⋊C8C2211yescharacteristic
59C8×C47⋊C8C2211yescharacteristic
60C47×#446C2211yescharacteristic
61C8×Q8×C47C2211yescharacteristic
62C4×Q16×C47C2211yescharacteristic
63C47×#335C2211yescharacteristic
64C47×Q8⋊C8C2211yescharacteristic
65C47×Q8⋊C8C2211yescharacteristic
66C47×#322C2211yescharacteristic
67C2×Q8×D47C2×C411yescharacteristic
68C2×Q8×D47C2×C411yescharacteristic
69C2×Dic188C2×C411yescharacteristic
70C2×#10625337C2×C411yescharacteristic
71C2×#10625337C2×C411yescharacteristic
72Q16×D47—24no—
73C2×#10625324—21no—
74C2×#10625324—21no—
75C2×#10625324D411yescharacteristic
76C2×#10625324D411yescharacteristic
77C2×C8×D47C2×C411yescharacteristic
78C2×C8×D47C2×C411yescharacteristic
79Q8×Dic47C2×C411yescharacteristic
80Q8×Dic47C2×C411yescharacteristic
81C4×Dic94—21no—
82C4×Dic94—21no—
83C4×Dic94C2311yescharacteristic
84C4×Dic94C2311yescharacteristic
85#13093989—21no—
86#13093989—21no—
87#13093989C2×C411yescharacteristic
88#13093989C2×C411yescharacteristic
89C4⋊C4×D47—21no—
90C4⋊C4×D47—21no—
91C4⋊C4×D47C2×C411yescharacteristic
92C4⋊C4×D47C2×C411yescharacteristic
93C42×D47—21no—
94C42×D47—21no—
95C42×D47C2311yescharacteristic
96#13093958C2×C411yescharacteristic
97#13093957C2×C411yescharacteristic
98#13093957C2×C411yescharacteristic
99C4⋊(C47⋊C8)—21no—
100C4⋊(C47⋊C8)—21no—
101C4⋊(C47⋊C8)C2311yescharacteristic
102C4⋊(C47⋊C8)C2311yescharacteristic
103Dic47⋊C8—21no—
104Dic47⋊C8—21no—
105Dic47⋊C8—21no—
106Dic47⋊C8—21no—
107Dic47⋊C8—21no—
108Dic47⋊C8—21no—
109Dic47⋊C8—21no—
110#13093949C2×C411yescharacteristic
111#13093949C2×C411yescharacteristic
112#13093948C2×C411yescharacteristic
113#13093948C2×C411yescharacteristic
114#13093947C2×C411yescharacteristic
115#13093947C2×C411yescharacteristic
116(C2×D47)⋊C8—21no—
117#13093935—21no—
118#13093934—21no—
119#13093934—21no—
120#13093934D411yescharacteristic
121#13093934D411yescharacteristic
122C4×C47⋊C8—21no—
123C4×C47⋊C8C2311yescharacteristic
124C4×C47⋊C8C2311yescharacteristic
125C4×C47⋊C8C2311yescharacteristic
126C8×Dic47—21no—
127C8×Dic47—21no—
128C8×Dic47C2×C411yescharacteristic
129C8×Dic47C2×C411yescharacteristic
130C2×Q16×C47C2×C411yescharacteristic
131C4×Q8×C47C2311yescharacteristic
132C4×Q8×C47C2311yescharacteristic
133#107×C47C2×C411yescharacteristic
134C4⋊C8×C47—21no—
135C4⋊C8×C47—21no—
136C4⋊C8×C47C2311yescharacteristic
137C4⋊C8×C47C2311yescharacteristic
138#103×C47C2×C411yescharacteristic
139#103×C47C2×C411yescharacteristic
140#97×C47—21no—
141C4×C8×C47—21no—
142C4×C8×C47C2311yescharacteristic
143C4×C8×C47C2311yescharacteristic
144C4×C8×C47C2311yescharacteristic
145C2×Q8×C47C22×C411yescharacteristic
146C2×Q8×C47C22×C411yescharacteristic
147Q8×D47—22no—
148Q8×D47—22no—
149Q8×D47—41no—
150Q8×D47—41no—
151C2×C4×D47—21no—
152C2×C4×D47—21no—
153C2×C4×D47—21no—
154C2×C4×D47—21no—
155C2×C4×D47C22×C411yescharacteristic
156C2×C4×D47C2×D411yescharacteristic
157C2×C4×D47#4111yescharacteristic
158C2×Dic94—21no—
159C2×Dic94—21no—
160C2×Dic94C22×C411yescharacteristic
161C2×Dic94C22×C411yescharacteristic
162Q16×C47—22no—
163C2×C8×C47—21no—
164C2×C8×C47—21no—
165C2×C8×C47—21no—
166C2×C8×C47C22×C411yescharacteristic
167C2×C8×C47C22×C411yescharacteristic
168C2×C8×C47C2×D411yescharacteristic
169C2×C8×C47C2×D411yescharacteristic
170C4⋊C4×C47—21no—
171C4⋊C4×C47—21no—
172C4⋊C4×C47C22×C411yescharacteristic
173C4⋊C4×C47C22×C411yescharacteristic
174C42×C47—21no—
175C42×C47—21no—
176C42×C47C2411yescharacteristic
177#10625337—22no—
178#10625337M1614yescomplemented
179C4⋊Dic47—21no—
180C4⋊Dic47—21no—
181C4⋊Dic47C22×C411yescharacteristic
182C4⋊Dic47C22×C411yescharacteristic
183Dic47⋊C4—21no—
184Dic47⋊C4—21no—
185Dic47⋊C4—21no—
186Dic47⋊C4—21no—
187Dic47⋊C4—21no—
188Dic47⋊C4—21no—
189C4×Dic47—21no—
190C4×Dic47—21no—
191C4×Dic47—21no—
192C4×Dic47—21no—
193C4×Dic47—21no—
194C4×Dic47—21no—
195C4×Dic47C22×C411yescharacteristic
196C4×Dic47C2×D411yescharacteristic
197C4×Dic47#4111yescharacteristic
198C2×C47⋊C8—21no—
199C2×C47⋊C8—21no—
200C2×C47⋊C8—21no—
201C2×C47⋊C8C22×C411yescharacteristic
202C2×C47⋊C8C22×C411yescharacteristic
203C2×C47⋊C8C2×D411yescharacteristic
204C2×C47⋊C8C2×D411yescharacteristic
205Dic188—22no—
206#10625324—22no—
207#10625324—42no—
208#10625324—42no—
209#10625324Q1614yescomplemented
210C8×D47—22no—
211C8×D47—41no—
212C22×D47C4×D411yescharacteristic
213Q8×C47—21no—
214Q8×C47—21no—
215Q8×C47—21no—
216Q8×C47C2×M1612yes—
217C2×C4×C47—21no—
218C2×C4×C47—21no—
219C2×C4×C47—21no—
220C2×C4×C47—21no—
221C2×C4×C47C23×C411yescharacteristic
222C2×C4×C47C22×D411yescharacteristic
223C2×C4×C47C2×#4111yescharacteristic
224C2×Dic47—21no—
225C2×Dic47—21no—
226C2×Dic47—21no—
227C2×Dic47—21no—
228C2×Dic47C4×D411yescharacteristic
229C2×Dic47C4×D411yescharacteristic
230C2×Dic47C4×D411yescharacteristic
231C4×D47—21no—
232C4×D47—21no—
233C4×D47—41no—
234C4×D47—42no—
235C4×D47—42no—
236C4×D47—81no—
237C4×D47—81no—
238C4×D47C2×Q1612yes—
239C4×D47#13512yes—
240Dic94—21no—
241Dic94—21no—
242Dic94—21no—
243Dic94—42no—
244Dic94—42no—
245Dic94C2×M1612yes—
246C8×C47—21no—
247C8×C47—21no—
248C8×C47—21no—
249C8×C47—41no—
250C8×C47—41no—
251C8×C47—41no—
252C8×C47C2×Q1612yescomplemented
253C47⋊C8—21no—
254C47⋊C8—21no—
255C47⋊C8—41no—
256C47⋊C8—41no—
257C47⋊C8—41no—
258C47⋊C8C2×M1612yes—
259C47⋊C8C2×Q1612yescomplemented
260M16×Q16—471nomaximal, Sylow
261C22×C47C2×C4×D411yescharacteristic
262C2×D47—21no—
263C2×D47—21no—
264C2×D47C4×Q1612yes—
265C4×C47—22no—
266C4×C47—21no—
267C4×C47—41no—
268C4×C47—41no—
269C4×C47—41no—
270C4×C47C22×M1611yescharacteristic
271C4×C47#56811yescharacteristic
272C4×C47C22×Q1611yescharacteristic
273C4×C47C2×#13511yescharacteristic
274Dic47—22no—
275Dic47—21no—
276Dic47—21no—
277Dic47—41no—
278Dic47—41no—
279Dic47—41no—
280Dic47C4×Q1612yes—
281Q8×M16—471no—
282Q8×M16—471no—
283C2×C4×Q16—471no—
284#1591—471no—
285#1591—471no—
286#1584—471no—
287#1584—471no—
288#1584—471no—
289#1584D4711yescharacteristic, complemented
290C8×Q16—471no—
291C8×Q16—471no—
292#1567—471no—
293#1487—471no—
294#1487—471no—
295C8×M16—471no—
296C2×C47D4×M1611yescharacteristic
297C2×C47C2×C4×Q1611yescharacteristic
298C2×C47#295711yescharacteristic
299D47—22no—
300C22×Q16—471no—
301C2×C4×Q8—471no—
302C2×C4×Q8—471no—
303#446—471no—
304#446—471no—
305#446—471no—
306#446—941no—
307#446—941no—
308#446—941no—
309#446—941no—
310#446C2×D4711yescharacteristic
311C8×Q8—471no—
312C8×Q8—471no—
313C8×Q8—471no—
314C8×Q8C2×D4711yescharacteristic
315C4×Q16—471no—
316C4×Q16—471no—
317C4×Q16—471no—
318C4×Q16—471no—
319C4×Q16—471no—
320C4×Q16—471no—
321C4×Q16—471no—
322C4×Q16C2×D4711yescharacteristic
323#431—471no—
324C2×#107—471no—
325#423—471no—
326#423—471no—
327#423—941no—
328#423—941no—
329C2×#103—471no—
330C2×#103—471no—
331C4×M16—471no—
332C4×M16—941no—
333C4×M16—941no—
334C2×C4×C8—471no—
335#335—471no—
336#335—471no—
337#335—471no—
338#335C2×D4711yescharacteristic
339Q8⋊C8—471no—
340Q8⋊C8—471no—
341Q8⋊C8—471no—
342Q8⋊C8—471no—
343Q8⋊C8—471no—
344Q8⋊C8—471no—
345Q8⋊C8C2×D4711yescharacteristic
346Q8⋊C8C2×D4711yescharacteristic
347#322—471no—
348#322C2×D4711yescharacteristic
349C82—471no—
350C82—471no—
351C47M16×Q1611yescharacteristic, complemented, Sylow
352C22×Q8—471no—
353C22×Q8—471no—
354C2×Q16—471no—
355C2×Q16—471no—
356C2×Q16—471no—
357C2×Q16—944no—
358C2×Q16C4×D4711yescharacteristic
359C2×M16—471no—
360C2×M16—471no—
361C2×M16—941no—
362C2×M16—941no—
363C22×C8—471no—
364C22×C8—471no—
365C4×Q8—471no—
366C4×Q8—471no—
367C4×Q8—471no—
368C4×Q8—471no—
369C4×Q8—471no—
370C4×Q8—471no—
371C4×Q8—941no—
372C4×Q8—941no—
373C4×Q8—941no—
374C4×Q8—941no—
375C4×Q8C22×D4711yescharacteristic
376C4×Q8C22×D4711yescharacteristic
377C2×C4⋊C4—471no—
378C2×C4⋊C4—471no—
379C2×C4⋊C4—941no—
380C2×C4⋊C4—941no—
381C2×C42—471no—
382C2×C42—941no—
383C2×C42—941no—
384#107—471no—
385#107—471no—
386#107—471no—
387#107C4×D4711yescharacteristic
388C4⋊C8—21no—
389C4⋊C8—21no—
390C4⋊C8—471no—
391C4⋊C8—471no—
392C4⋊C8—941no—
393C4⋊C8—941no—
394C4⋊C8—941no—
395C4⋊C8—941no—
396C4⋊C8—941no—
397C4⋊C8—941no—
398C4⋊C8—941no—
399C4⋊C8—941no—
400C4⋊C8—941no—
401C4⋊C8C22×D4711yescharacteristic
402C4⋊C8C22×D4711yescharacteristic
403#103—471no—
404#103—471no—
405#103—471no—
406#103—471no—
407#103—471no—
408#103—471no—
409#103C4×D4711yescharacteristic
410#103C4×D4711yescharacteristic
411C22⋊C8—941no—
412#97—21no—
413#97—471no—
414#97—471no—
415#97—941no—
416#97—941no—
417#97—941no—
418C4×C8—21no—
419C4×C8—471no—
420C4×C8—471no—
421C4×C8—471no—
422C4×C8—471no—
423C4×C8—471no—
424C4×C8—941no—
425C4×C8—941no—
426C4×C8—941no—
427C4×C8C22×D4711yescharacteristic
428C4×C8C22×D4711yescharacteristic
429C4×C8C22×D4711yescharacteristic
430C2×Q8—471no—
431C2×Q8—471no—
432C2×Q8—941no—
433C2×Q8—941no—
434C2×Q8—942no—
435C2×Q8—942no—
436C2×Q8—1881no—
437C2×Q8—1881no—
438C2×Q8C2×C4×D4711yescharacteristic
439C2×Q8C2×C4×D4711yescharacteristic
440C22×C4—471no—
441C22×C4—471no—
442C22×C4—471no—
443C22×C4—941no—
444C22×C4—941no—
445C22×C4—941no—
446C22×C4—941no—
447Q16—22no—
448Q16—474no—
449Q16—942no—
450Q16—942no—
451M16—474no—
452M16—942no—
453M16—1882no—
454M16—1882no—
455C2×C8—21no—
456C2×C8—21no—
457C2×C8—21no—
458C2×C8—471no—
459C2×C8—471no—
460C2×C8—471no—
461C2×C8—471no—
462C2×C8—942no—
463C2×C8—941no—
464C2×C8—941no—
465C2×C8—941no—
466C2×C8—1881no—
467C2×C8C2×C4×D4711yescharacteristic
468C2×C8C2×C4×D4711yescharacteristic
469C2×C8D4×D4711yescharacteristic
470C2×C8D4×D4711yescharacteristic
471C4⋊C4—21no—
472C4⋊C4—21no—
473C4⋊C4—471no—
474C4⋊C4—471no—
475C4⋊C4—941no—
476C4⋊C4—941no—
477C4⋊C4—941no—
478C4⋊C4—941no—
479C4⋊C4—941no—
480C4⋊C4—941no—
481C4⋊C4—941no—
482C4⋊C4—941no—
483C4⋊C4C2×C4×D4711yescharacteristic
484C4⋊C4C2×C4×D4711yescharacteristic
485C42—21no—
486C42—21no—
487C42—471no—
488C42—471no—
489C42—471no—
490C42—941no—
491C42—941no—
492C42—941no—
493C42—941no—
494C42—941no—
495C42—941no—
496C42C23×D4711yescharacteristic
497C23—471no—
498Q8—21no—
499Q8—21no—
500Q8—21no—
501Q8—472no—
502Q8—941no—
503Q8—941no—
504Q8—941no—
505Q8—1882no—
506Q8—1882no—
507Q8M16×D4712yes—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—471no—
513C2×C4—472no—
514C2×C4—472no—
515C2×C4—471no—
516C2×C4—471no—
517C2×C4—941no—
518C2×C4—941no—
519C2×C4—941no—
520C2×C4—941no—
521C2×C4—941no—
522C2×C4—941no—
523C2×C4—3761no—
524C2×C4—3761no—
525C2×C4—1881no—
526C2×C4—1882no—
527C2×C4—1882no—
528C2×C4C22×C4×D4711yescharacteristic
529C2×C4C2×D4×D4711yescharacteristic
530C2×C4#41×D4711yescharacteristic
531C8—21no—
532C8—21no—
533C8—21no—
534C8—41no—
535C8—41no—
536C8—41no—
537C8—472no—
538C8—472no—
539C8—941no—
540C8—941no—
541C8—1881no—
542C8—1881no—
543C8—1881no—
544C8Q16×D4712yescomplemented
545C22—472no—
546C22—941no—
547C22—941no—
548C22C4×D4×D4711yescharacteristic
549C4—21no—
550C4—22no—
551C4—41no—
552C4—41no—
553C4—41no—
554C4—472no—
555C4—941no—
556C4—941no—
557C4—942no—
558C4—1881no—
559C4—1881no—
560C4—1881no—
561C4C2×M16×D4711yescharacteristic
562C4#1568988811yescharacteristic
563C4C2×Q16×D4711yescharacteristic
564C4#135×D4711yescharacteristic
565C2—942no—
566C2#2025212111yescharacteristic
567C2C4×Q16×D4711yescharacteristic
568C2#2025230211yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D47
commutatorC2×C4×C47C23×C4
FrattiniC42C23×D47
FittingC47×#1584C2
socleC22×C47C2×C4×D4
prime radicalC22×D47C4×D4
cosocleC42×C47C24
hypercenter#1584D47
hypocenterC47M16×Q16
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C2×C4 ◁ C4×C8 ◁ #1584

Complements

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

NH ≅ G/Ntype
4. #20242283C2semidirect
5. #20242282C2semidirect
6. #20242240C2semidirect
7. #20242239C2semidirect
8. #20242237C2semidirect
9. C8×#10625337C2semidirect
10. #20242192C2semidirect
15. C47×#1584C2semidirect
178. #10625337M16semidirect
209. #10625324Q16semidirect
252. C8×C47C2×Q16semidirect
259. C47⋊C8C2×Q16semidirect
289. #1584D47semidirect
351. C47M16×Q16semidirect
544. C8Q16×D47semidirect

Maximal subgroups

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

subgroupindex
#202523402
Q8×#106253242
C4×Q16×D472
#202422832
#202422822
#202422402
#202422392
#202422372
C8×#106253372
#202421922
#202421452
#202418152
#202416672
C47×#15842
M16×Q1647