Finite Group Databasebeta

← #26186352

Subgroups — #26186352

Order 12032.

596 conjugacy classes of subgroups.

The subgroup lattice has 564 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20253472C211yesmaximal, characteristic
2#20252540C211yesmaximal, characteristic
3#20252467C211yesmaximal, characteristic, complemented
4#20242268C211yesmaximal, characteristic, complemented
5#20242236C211yesmaximal, characteristic, complemented
6#20242231C211yesmaximal, characteristic, complemented
7#20242204C211yesmaximal, characteristic, complemented
8#20242203C211yesmaximal, characteristic, complemented
9Q16×C47⋊C8C211yesmaximal, characteristic, complemented
10#20242187C211yesmaximal, characteristic, complemented
11#20242175C211yesmaximal, characteristic, complemented
12#20241815C211yesmaximal, characteristic, complemented
13#20241795C211yesmaximal, characteristic
14#20241689C211yesmaximal, characteristic, complemented
15C47×#1587C211yesmaximal, characteristic, complemented
16#15689660C2211yescharacteristic
17C4×Q8×D47C2211yescharacteristic
18#15689452C411yescharacteristic
19#15689438C411yescharacteristic
20#15689429C411yescharacteristic
21#15689428C411yescharacteristic
22#15689348C411yescharacteristic
23#15689347C411yescharacteristic
24#15689277C411yescharacteristic
25#15689263C411yescharacteristic
26Q8×C47⋊C8C2211yescharacteristic
27Q8×C47⋊C8C2211yescharacteristic
28C8×Dic94—21no—
29C8×Dic94C2211yescharacteristic
30#15689247—21no—
31#15689247C2211yescharacteristic
32C4×#10625337C2211yescharacteristic
33C4×#10625325C2211yescharacteristic
34C4×#10625336C2211yescharacteristic
35#15689202C2211yescharacteristic
36#15689200—21no—
37#15689196—21no—
38#15689188C2211yescharacteristic
39#15689185—21no—
40#15689175—21no—
41#15689175C2211yescharacteristic
42#15689174—21no—
43#15689174C2211yescharacteristic
44#15689174C2211yescharacteristic
45#15689174C2211yescharacteristic
46#15689148—21no—
47#15689145C2211yescharacteristic
48#15688876C2211yescharacteristic
49#15688875C2211yescharacteristic
50#15688873C2211yescharacteristic
51#15688855C2211yescharacteristic
52Q8⋊(C47⋊C8)C2211yescharacteristic
53Q8⋊(C47⋊C8)C2211yescharacteristic
54#15688853C2211yescharacteristic
55#15688852C2211yescharacteristic
56#15688850C2211yescharacteristic
57#15688845C2211yescharacteristic
58#15688844C2211yescharacteristic
59C8×C47⋊C8C2211yescharacteristic
60C47×#446C2211yescharacteristic
61C8×Q8×C47C2211yescharacteristic
62C4×Q16×C47C2211yescharacteristic
63C47×#334C2211yescharacteristic
64C47×Q8⋊C8C2211yescharacteristic
65C47×Q8⋊C8C2211yescharacteristic
66C47×#322C2211yescharacteristic
67C2×Q8×D47C2×C411yescharacteristic
68C2×#10625337C2×C411yescharacteristic
69C2×#10625325C2×C411yescharacteristic
70C2×#10625336C2×C411yescharacteristic
71C2×#10625324—21no—
72C2×#10625324—21no—
73C2×C8×D47—21no—
74#13094019—21no—
75#13094015C2×C411yescharacteristic
76#13094013C2×C411yescharacteristic
77#13094009—21no—
78#13094000C2×C411yescharacteristic
79#13093999C2×C411yescharacteristic
80Q8×Dic47C2×C411yescharacteristic
81C4×Dic94—21no—
82C4×Dic94C2311yescharacteristic
83#13093989—21no—
84#13093989C2×C411yescharacteristic
85C4×D188—21no—
86C4×D188C2311yescharacteristic
87#13093981—21no—
88C4⋊C4×D47—21no—
89C4⋊C4×D47C2×C411yescharacteristic
90C42×D47—21no—
91C42×D47C2311yescharacteristic
92#13093957C2×C411yescharacteristic
93#13093956C2×C411yescharacteristic
94#13093955C2×C411yescharacteristic
95C4⋊(C47⋊C8)—21no—
96C4⋊(C47⋊C8)—21no—
97C4⋊(C47⋊C8)C2311yescharacteristic
98C4⋊(C47⋊C8)C2311yescharacteristic
99Dic47⋊C8—21no—
100Dic47⋊C8—21no—
101Dic47⋊C8—21no—
102Dic47⋊C8—21no—
103Dic47⋊C8C2×C411yescharacteristic
104Dic47⋊C8C2×C411yescharacteristic
105Dic47⋊C8D411yescharacteristic
106Dic47⋊C8D411yescharacteristic
107#13093949C2×C411yescharacteristic
108#13093948C2×C411yescharacteristic
109#13093948C2×C411yescharacteristic
110#13093947C2×C411yescharacteristic
111#13093946C2×C411yescharacteristic
112#13093944C2×C411yescharacteristic
113(C2×D47)⋊C8—21no—
114(C2×D47)⋊C8—21no—
115(C2×D47)⋊C8—21no—
116(C2×D47)⋊C8—21no—
117(C2×D47)⋊C8C2×C411yescharacteristic
118(C2×D47)⋊C8C2×C411yescharacteristic
119(C2×D47)⋊C8D411yescharacteristic
120(C2×D47)⋊C8D411yescharacteristic
121C4×C47⋊C8—21no—
122C4×C47⋊C8—21no—
123C4×C47⋊C8C2311yescharacteristic
124C4×C47⋊C8C2311yescharacteristic
125C4×C47⋊C8C2311yescharacteristic
126C8×Dic47—21no—
127C8×Dic47—21no—
128C8×Dic47—21no—
129C2×Q16×C47C2×C411yescharacteristic
130C4×Q8×C47C2311yescharacteristic
131C4×Q8×C47C2311yescharacteristic
132#107×C47C2×C411yescharacteristic
133C4⋊C8×C47—21no—
134C4⋊C8×C47—21no—
135C4⋊C8×C47C2311yescharacteristic
136C4⋊C8×C47C2311yescharacteristic
137#103×C47C2×C411yescharacteristic
138#103×C47C2×C411yescharacteristic
139#97×C47—21no—
140C4×C8×C47—21no—
141C4×C8×C47C2311yescharacteristic
142C4×C8×C47C2311yescharacteristic
143C4×C8×C47C2311yescharacteristic
144C2×Q8×C47C22×C411yescharacteristic
145C2×Q8×C47C22×C411yescharacteristic
146Q8×D47—22no—
147Q8×D47—22no—
148C2×D188—21no—
149C2×D188C22×C411yescharacteristic
150C2×C4×D47—21no—
151C2×C4×D47—21no—
152C2×C4×D47—21no—
153C2×C4×D47C22×C411yescharacteristic
154C2×C4×D47C2×D411yescharacteristic
155C2×C4×D47#4111yescharacteristic
156C2×Dic94—21no—
157C2×Dic94C22×C411yescharacteristic
158Q16×C47—22no—
159C2×C8×C47—21no—
160C2×C8×C47—21no—
161C2×C8×C47—21no—
162C2×C8×C47C22×C411yescharacteristic
163C2×C8×C47C22×C411yescharacteristic
164C2×C8×C47C2×D411yescharacteristic
165C2×C8×C47C2×D411yescharacteristic
166C4⋊C4×C47—21no—
167C4⋊C4×C47—21no—
168C4⋊C4×C47C22×C411yescharacteristic
169C4⋊C4×C47C22×C411yescharacteristic
170C42×C47—21no—
171C42×C47—21no—
172C42×C47C2411yescharacteristic
173#10625337—22no—
174#10625336—22no—
175(C2×D47)⋊C4—21no—
176(C2×D47)⋊C4—21no—
177(C2×D47)⋊C4—21no—
178(C2×D47)⋊C4—21no—
179(C2×D47)⋊C4—21no—
180C4⋊Dic47—21no—
181C4⋊Dic47—21no—
182C4⋊Dic47C22×C411yescharacteristic
183C4⋊Dic47C22×C411yescharacteristic
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×Dic47C22×C411yescharacteristic
193C4×Dic47C2×D411yescharacteristic
194C4×Dic47#4111yescharacteristic
195C2×C47⋊C8—21no—
196C2×C47⋊C8—21no—
197C2×C47⋊C8—21no—
198C2×C47⋊C8C22×C411yescharacteristic
199C2×C47⋊C8C22×C411yescharacteristic
200C2×C47⋊C8C2×D411yescharacteristic
201C2×C47⋊C8C2×D411yescharacteristic
202#10625325—22no—
203#10625324—81no—
204#10625324—81no—
205C8×D47—81no—
206C22×D47—21no—
207C22×D47C4×D411yescharacteristic
208Q8×C47—21no—
209Q8×C47—21no—
210Q8×C47—21no—
211Q8×C47#13112yes—
212C2×C4×C47—21no—
213C2×C4×C47—21no—
214C2×C4×C47—21no—
215C2×C4×C47—21no—
216C2×C4×C47C23×C411yescharacteristic
217C2×C4×C47C22×D411yescharacteristic
218C2×C4×C47C2×#4111yescharacteristic
219C2×Dic47—21no—
220C2×Dic47—21no—
221C2×Dic47—21no—
222C2×Dic47C4×D411yescharacteristic
223C2×Dic47C4×D411yescharacteristic
224C2×Dic47C4×D411yescharacteristic
225D188—21no—
226D188—21no—
227D188—81no—
228C4×D47—24no—
229C4×D47—21no—
230C4×D47—21no—
231C4×D47—21no—
232C4×D47—21no—
233C4×D47—41no—
234C4×D47—41no—
235C4×D47—41no—
236C4×D47—81no—
237Dic94—24no—
238Dic94—21no—
239Dic94#13112yes—
240C8×C47—21no—
241C8×C47—21no—
242C8×C47—21no—
243C8×C47—21no—
244C8×C47—41no—
245C8×C47—41no—
246C8×C47—41no—
247C47⋊C8—21no—
248C47⋊C8—21no—
249C47⋊C8—21no—
250C47⋊C8—21no—
251C47⋊C8—41no—
252C47⋊C8—41no—
253C47⋊C8—41no—
254#17919—471nomaximal, Sylow
255C22×C47C2×C4×D411yescharacteristic
256C2×D47—21no—
257C2×D47—21no—
258C2×D47—21no—
259C2×D47—41no—
260C2×D47—41no—
261C2×D47—41no—
262C4×C47—22no—
263C4×C47—21no—
264C4×C47—41no—
265C4×C47—41no—
266C4×C47—41no—
267C4×C47C2×#13111yescharacteristic
268C4×C47C2×#13111yescharacteristic
269C4×C47#57711yescharacteristic
270C4×C47#57811yescharacteristic
271Dic47—21no—
272Dic47—22no—
273Dic47—21no—
274Dic47—21no—
275Dic47—41no—
276Dic47—41no—
277#3108—471no—
278#2993—471no—
279#2982—471no—
280#1588—471no—
281#1587D4711yescharacteristic, complemented
282#1584—471no—
283#1583—471no—
284#1579—471no—
285SD16⋊C8—471no—
286C8×Q16—471no—
287C8×SD16—471no—
288#1574—471no—
289#1487—471no—
290#1482—471no—
291#1454—471no—
292C2×C47#292911yescharacteristic
293C2×C47#294711yescharacteristic
294C2×C47#295711yescharacteristic
295D47—41no—
296D47—81no—
297#529—471no—
298C2×C4×Q8—471no—
299#483—471no—
300#480—471no—
301#479—471no—
302#479—471no—
303#464—471no—
304#464—471no—
305#451—471no—
306#448—471no—
307#446C2×D4711yescharacteristic
308C8×Q8—471no—
309C8×Q8—471no—
310C8×Q8—471no—
311C8×Q8—471no—
312C8×Q8—941no—
313C8×Q8—941no—
314C8×Q8C2×D4711yescharacteristic
315C4×Q16—471no—
316C4×Q16C2×D4711yescharacteristic
317C4×SD16—471no—
318C4×SD16—471no—
319#436—471no—
320#436—941no—
321#435—941no—
322C8×D4—471no—
323#433—941no—
324#431—471no—
325#431—471no—
326#431—471no—
327#431—471no—
328#431—941no—
329#431—941no—
330#424—941no—
331#423—471no—
332#335—471no—
333#335—471no—
334#334—471no—
335#334C2×D4711yescharacteristic
336Q8⋊C8—471no—
337Q8⋊C8—471no—
338Q8⋊C8—471no—
339Q8⋊C8—471no—
340Q8⋊C8C2×D4711yescharacteristic
341Q8⋊C8C2×D4711yescharacteristic
342D4⋊C8—471no—
343D4⋊C8—471no—
344#322—471no—
345#322—471no—
346#322C2×D4711yescharacteristic
347C82—471no—
348C47#1791911yescharacteristic, complemented, Sylow
349C22×Q8—471no—
350C2×Q16—471no—
351C2×Q16C4×D4711yescharacteristic
352C2×SD16—471no—
353C2×SD16—471no—
354C2×M16—941no—
355C2×M16—941no—
356C22×C8—941no—
357#126—941no—
358#125—471no—
359#124—471no—
360#123—941no—
361C22⋊Q8—471no—
362#121—471no—
363C4×Q8—471no—
364C4×Q8—471no—
365C4×Q8—471no—
366C4×Q8—941no—
367C4×Q8—941no—
368C4×Q8C22×D4711yescharacteristic
369C4×Q8C22×D4711yescharacteristic
370C4×D4—471no—
371C4×D4—941no—
372#117—941no—
373C2×C4⋊C4—471no—
374C2×C4⋊C4—941no—
375C2×C42—471no—
376C2×C42—941no—
377#107—471no—
378#107C4×D4711yescharacteristic
379#106—471no—
380#106—471no—
381C4⋊C8—21no—
382C4⋊C8—21no—
383C4⋊C8—471no—
384C4⋊C8—471no—
385C4⋊C8—471no—
386C4⋊C8—471no—
387C4⋊C8—471no—
388C4⋊C8—471no—
389C4⋊C8—941no—
390C4⋊C8—941no—
391C4⋊C8—941no—
392C4⋊C8—941no—
393C4⋊C8—941no—
394C4⋊C8—941no—
395C4⋊C8C22×D4711yescharacteristic
396C4⋊C8C22×D4711yescharacteristic
397#103—471no—
398#103—471no—
399#103—471no—
400#103—471no—
401#103C4×D4711yescharacteristic
402#103C4×D4711yescharacteristic
403#102—471no—
404#102—471no—
405C22⋊C8—471no—
406C22⋊C8—471no—
407C22⋊C8—471no—
408C22⋊C8—471no—
409C22⋊C8—941no—
410C22⋊C8—941no—
411C22⋊C8—941no—
412C22⋊C8—941no—
413#97—21no—
414C4×C8—21no—
415C4×C8—471no—
416C4×C8—471no—
417C4×C8—471no—
418C4×C8—941no—
419C4×C8—941no—
420C4×C8—941no—
421C4×C8—941no—
422C4×C8—941no—
423C4×C8C22×D4711yescharacteristic
424C4×C8C22×D4711yescharacteristic
425C4×C8C22×D4711yescharacteristic
426C2×Q8—471no—
427C2×Q8—941no—
428C2×Q8—942no—
429C2×Q8—942no—
430C2×Q8C2×C4×D4711yescharacteristic
431C2×Q8C2×C4×D4711yescharacteristic
432C2×D4—471no—
433C2×D4—941no—
434C22×C4—471no—
435C22×C4—471no—
436C22×C4—471no—
437C22×C4—941no—
438C22×C4—941no—
439C22×C4—941no—
440Q16—22no—
441Q16—942no—
442SD16—942no—
443SD16—942no—
444M16—3761no—
445M16—3761no—
446C2×C8—21no—
447C2×C8—21no—
448C2×C8—21no—
449C2×C8—471no—
450C2×C8—471no—
451C2×C8—471no—
452C2×C8—471no—
453C2×C8—941no—
454C2×C8—941no—
455C2×C8—941no—
456C2×C8—3761no—
457C2×C8C2×C4×D4711yescharacteristic
458C2×C8C2×C4×D4711yescharacteristic
459C2×C8D4×D4711yescharacteristic
460C2×C8D4×D4711yescharacteristic
461C4⋊C4—21no—
462C4⋊C4—21no—
463C4⋊C4—471no—
464C4⋊C4—471no—
465C4⋊C4—941no—
466C4⋊C4—941no—
467C4⋊C4—941no—
468C4⋊C4—941no—
469C4⋊C4—941no—
470C4⋊C4—941no—
471C4⋊C4—941no—
472C4⋊C4C2×C4×D4711yescharacteristic
473C4⋊C4C2×C4×D4711yescharacteristic
474C22⋊C4—941no—
475C22⋊C4—941no—
476C22⋊C4—941no—
477C22⋊C4—941no—
478C22⋊C4—941no—
479C42—21no—
480C42—21no—
481C42—471no—
482C42—471no—
483C42—471no—
484C42—941no—
485C42—941no—
486C42—941no—
487C42C23×D4711yescharacteristic
488C23—471no—
489C23—941no—
490Q8—21no—
491Q8—21no—
492Q8—21no—
493Q8—472no—
494Q8—944no—
495Q8—941no—
496Q8#1309403212yes—
497D4—941no—
498D4—941no—
499D4—3761no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—471no—
505C2×C4—471no—
506C2×C4—471no—
507C2×C4—944no—
508C2×C4—941no—
509C2×C4—941no—
510C2×C4—941no—
511C2×C4—941no—
512C2×C4—941no—
513C2×C4—941no—
514C2×C4—941no—
515C2×C4—3761no—
516C2×C4—1881no—
517C2×C4—1881no—
518C2×C4—1881no—
519C2×C4C22×C4×D4711yescharacteristic
520C2×C4C2×D4×D4711yescharacteristic
521C2×C4#41×D4711yescharacteristic
522C8—21no—
523C8—21no—
524C8—21no—
525C8—21no—
526C8—41no—
527C8—41no—
528C8—41no—
529C8—941no—
530C8—941no—
531C8—941no—
532C8—941no—
533C8—1881no—
534C8—1881no—
535C8—1881no—
536C22—941no—
537C22—941no—
538C22—941no—
539C22—1881no—
540C22—1881no—
541C22—1881no—
542C22C4×D4×D4711yescharacteristic
543C4—21no—
544C4—22no—
545C4—41no—
546C4—41no—
547C4—41no—
548C4—941no—
549C4—941no—
550C4—941no—
551C4—942no—
552C4—1881no—
553C4—1881no—
554C4#131×D4711yescharacteristic
555C4C2×#1309403211yescharacteristic
556C4#1568993211yescharacteristic
557C4#1568993911yescharacteristic
558C2—3761no—
559C2—1881no—
560C2#2025207911yescharacteristic
561C2#2025223011yescharacteristic
562C2#2025230011yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D47
commutatorC2×C4×C47C23×C4
FrattiniC42C23×D47
FittingC47×#1587C2
socleC22×C47C2×C4×D4
prime radical#13093999C2×C4
cosocleC42×C47C24
hypercenter#1587D47
hypocenterC47#17919
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C2×C4 ◁ C4×C8 ◁ #1587

Complements

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

NH ≅ G/Ntype
3. #20252467C2semidirect
4. #20242268C2semidirect
5. #20242236C2semidirect
6. #20242231C2semidirect
7. #20242204C2semidirect
8. #20242203C2semidirect
9. Q16×C47⋊C8C2semidirect
10. #20242187C2semidirect
11. #20242175C2semidirect
12. #20241815C2semidirect
14. #20241689C2semidirect
15. C47×#1587C2semidirect
281. #1587D47semidirect
348. C47#17919semidirect

Maximal subgroups

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

subgroupindex
#202534722
#202525402
#202524672
#202422682
#202422362
#202422312
#202422042
#202422032
Q16×C47⋊C82
#202421872
#202421752
#202418152
#202417952
#202416892
C47×#15872
#1791947