Finite Group Databasebeta

← #26186379

Subgroups — #26186379

Order 12032.

598 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#20253453C211yesmaximal, characteristic
2#20252496C211yesmaximal, characteristic
3#20242257C211yesmaximal, characteristic, complemented
4#20242243C211yesmaximal, characteristic, complemented
5#20242236C211yesmaximal, characteristic, complemented
6#20242222C211yesmaximal, characteristic, complemented
7#20242205C211yesmaximal, characteristic, complemented
8#20242198C211yesmaximal, characteristic, complemented
9C8×#10625337C211yesmaximal, characteristic, complemented
10C8×#10625336C211yesmaximal, characteristic, complemented
11#20242174C211yesmaximal, characteristic, complemented
12#20241820C211yesmaximal, characteristic
13#20241791C211yesmaximal, characteristic, complemented
14#20241688C211yesmaximal, characteristic, complemented
15C47×#2985C211yesmaximal, characteristic, complemented
16C4×#10625350C2211yescharacteristic
17#15689461C411yescharacteristic
18#15689461C411yescharacteristic
19#15689405C411yescharacteristic
20#15689400C411yescharacteristic
21#15689330C411yescharacteristic
22#15689325C411yescharacteristic
23#15689283C411yescharacteristic
24#15689283C411yescharacteristic
25C8×Dic94C2211yescharacteristic
26C8×Dic94C2211yescharacteristic
27C4×#10625337C2211yescharacteristic
28C4×#10625337C2211yescharacteristic
29C4×#10625336C2211yescharacteristic
30C4×#10625336C2211yescharacteristic
31C8×D188C2211yescharacteristic
32C8×D188C2211yescharacteristic
33(C22⋊D47)⋊C8—21no—
34(C22⋊D47)⋊C8—21no—
35#15689176C2211yescharacteristic
36#15689176C2211yescharacteristic
37#15689175—21no—
38C2×C4⋊(C47⋊C8)C2211yescharacteristic
39C4⋊C8×D47—21no—
40#15688875C2211yescharacteristic
41#15688875C2211yescharacteristic
42#15688873C2211yescharacteristic
43#15688873C2211yescharacteristic
44Q8⋊(C47⋊C8)C2211yescharacteristic
45Q8⋊(C47⋊C8)C2211yescharacteristic
46Q8⋊(C47⋊C8)C2211yescharacteristic
47Q8⋊(C47⋊C8)C2211yescharacteristic
48#15688853C2211yescharacteristic
49#15688853C2211yescharacteristic
50#15688850C2211yescharacteristic
51#15688850C2211yescharacteristic
52#15688845C2211yescharacteristic
53#15688845C2211yescharacteristic
54#15688844C2211yescharacteristic
55C8×C47⋊C8C2211yescharacteristic
56C47×#533C2211yescharacteristic
57C47×#446—21no—
58C47×#446—21no—
59C47×#446C2211yescharacteristic
60C47×#446C2211yescharacteristic
61C8×Q8×C47C2211yescharacteristic
62C8×Q8×C47C2211yescharacteristic
63C47×#433—21no—
64C47×#433—21no—
65C47×#432C2211yescharacteristic
66C47×#431C2211yescharacteristic
67C2×#10625350C2×C411yescharacteristic
68C2×#10625337C2×C411yescharacteristic
69C2×#10625337C2×C411yescharacteristic
70C2×#10625336C2×C411yescharacteristic
71C2×#10625336C2×C411yescharacteristic
72C22×C47⋊C8—21no—
73C2×C8×D47—21no—
74C2×C8×D47—21no—
75C4×Dic94C2×C411yescharacteristic
76C4×Dic94C2311yescharacteristic
77C4×D188C2×C411yescharacteristic
78C4×D188C2311yescharacteristic
79C4×C22⋊D47—21no—
80C4×C22⋊D47—21no—
81#13093983C2×C411yescharacteristic
82#13093981—21no—
83C42×D47—21no—
84#13093957C2×C411yescharacteristic
85#13093957C2×C411yescharacteristic
86#13093955C2×C411yescharacteristic
87#13093955C2×C411yescharacteristic
88C4⋊(C47⋊C8)—21no—
89C4⋊(C47⋊C8)—21no—
90C4⋊(C47⋊C8)C2×C411yescharacteristic
91C4⋊(C47⋊C8)C2×C411yescharacteristic
92C4⋊(C47⋊C8)C2311yescharacteristic
93C4⋊(C47⋊C8)C2311yescharacteristic
94Dic47⋊C8—21no—
95Dic47⋊C8—21no—
96#13093948C2×C411yescharacteristic
97#13093948C2×C411yescharacteristic
98#13093948C2×C411yescharacteristic
99#13093948C2×C411yescharacteristic
100#13093947C2×C411yescharacteristic
101#13093947C2×C411yescharacteristic
102#13093944C2×C411yescharacteristic
103#13093944C2×C411yescharacteristic
104C22⋊(C47⋊C8)—21no—
105C22⋊(C47⋊C8)C2×C411yescharacteristic
106C22⋊(C47⋊C8)C2×C411yescharacteristic
107(C2×D47)⋊C8—21no—
108(C2×D47)⋊C8—21no—
109C4×C47⋊C8C2311yescharacteristic
110C4×C47⋊C8C2311yescharacteristic
111C4×C47⋊C8C2311yescharacteristic
112C4×C47⋊C8C2311yescharacteristic
113C8×Dic47—21no—
114C8×Dic47—21no—
115C4⋊Q8×C47C2×C411yescharacteristic
116C4⋊Q8×C47C2×C411yescharacteristic
117#125×C47—21no—
118C22⋊Q8×C47—21no—
119C22⋊Q8×C47C2×C411yescharacteristic
120C22⋊Q8×C47C2×C411yescharacteristic
121C4×Q8×C47—21no—
122C4×Q8×C47C2311yescharacteristic
123C4×Q8×C47C2311yescharacteristic
124#117×C47—21no—
125C2×C42×C47C2311yescharacteristic
126C4⋊C8×C47—21no—
127C4⋊C8×C47—21no—
128C4⋊C8×C47—21no—
129C4⋊C8×C47—21no—
130C4⋊C8×C47D411yescharacteristic
131C4⋊C8×C47D411yescharacteristic
132C4⋊C8×C47C2311yescharacteristic
133C4⋊C8×C47C2311yescharacteristic
134C22⋊C8×C47—21no—
135C22⋊C8×C47D411yescharacteristic
136C22⋊C8×C47D411yescharacteristic
137#97×C47—21no—
138#97×C47—21no—
139#97×C47—21no—
140C4×C8×C47—21no—
141C4×C8×C47—21no—
142C4×C8×C47C2311yescharacteristic
143C4×C8×C47C2311yescharacteristic
144C2×Q8×C47—21no—
145C2×Q8×C47C22×C411yescharacteristic
146C2×Q8×C47C22×C411yescharacteristic
147C22×C4×C47C22×C411yescharacteristic
148C22×C4×C47C2×D411yescharacteristic
149C22×C4×C47#4111yescharacteristic
150C2×C22⋊D47—21no—
151#10625350—22no—
152#10625350—22no—
153C2×D188C22×C411yescharacteristic
154C2×C4×D47—21no—
155C2×C4×D47—21no—
156C2×C4×D47—21no—
157C2×Dic94C22×C411yescharacteristic
158C2×C8×C47—21no—
159C2×C8×C47—21no—
160C2×C8×C47C2×D411yescharacteristic
161C2×C8×C47C2×D411yescharacteristic
162C2×C8×C47C2×D411yescharacteristic
163C2×C8×C47C2×D411yescharacteristic
164C4⋊C4×C47—21no—
165C4⋊C4×C47—21no—
166C4⋊C4×C47—21no—
167C4⋊C4×C47—21no—
168C4⋊C4×C47—21no—
169C4⋊C4×C47—21no—
170C4⋊C4×C47—21no—
171C4⋊C4×C47C22×C411yescharacteristic
172C4⋊C4×C47C22×C411yescharacteristic
173C22⋊C4×C47—21no—
174C22⋊C4×C47—21no—
175C42×C47—21no—
176C42×C47—21no—
177C42×C47C22×C411yescharacteristic
178C42×C47C2×D411yescharacteristic
179C42×C47#4111yescharacteristic
180C42×C47C2411yescharacteristic
181C22⋊Dic47—21no—
182#10625337—22no—
183#10625337—22no—
184#10625336—22no—
185#10625336—22no—
186(C2×D47)⋊C4—21no—
187(C2×D47)⋊C4—21no—
188C4⋊Dic47C22×C411yescharacteristic
189C4⋊Dic47C22×C411yescharacteristic
190Dic47⋊C4—21no—
191Dic47⋊C4—21no—
192C4×Dic47—21no—
193C4×Dic47—21no—
194C4×Dic47—21no—
195C2×C47⋊C8—21no—
196C2×C47⋊C8—21no—
197C2×C47⋊C8—21no—
198C2×C47⋊C8—21no—
199C2×C47⋊C8—81no—
200C2×C47⋊C8C22×C411yescharacteristic
201C2×C47⋊C8C22×C411yescharacteristic
202C2×C47⋊C8C22×C411yescharacteristic
203C2×C47⋊C8C22×C411yescharacteristic
204C8×D47—81no—
205C8×D47—81no—
206C23×C47C4×D411yescharacteristic
207C22×D47—21no—
208Q8×C47—21no—
209Q8×C47—21no—
210Q8×C47—21no—
211Q8×C47—21no—
212Q8×C47—81no—
213C2×C4×C47—21no—
214C2×C4×C47—21no—
215C2×C4×C47—21no—
216C2×C4×C47—21no—
217C2×C4×C47—21no—
218C2×C4×C47—21no—
219C2×C4×C47—21no—
220C2×C4×C47—21no—
221C2×C4×C47—41no—
222C2×C4×C47C4×D411yescharacteristic
223C2×C4×C47C4×D411yescharacteristic
224C2×C4×C47C4×D411yescharacteristic
225C2×C4×C47C23×C411yescharacteristic
226C2×C4×C47C22×D411yescharacteristic
227C2×C4×C47C2×#4111yescharacteristic
228C22⋊D47—24no—
229C2×Dic47—21no—
230C2×Dic47—21no—
231C2×Dic47—21no—
232D188—21no—
233D188#13112yes—
234C4×D47—24no—
235C4×D47—41no—
236C4×D47—41no—
237C4×D47—81no—
238Dic94—21no—
239Dic94#13112yes—
240C8×C47—21no—
241C8×C47—21no—
242C8×C47—21no—
243C8×C47—21no—
244C8×C47—41no—
245C8×C47—41no—
246C47⋊C8—21no—
247C47⋊C8—21no—
248C47⋊C8—21no—
249C47⋊C8—21no—
250C47⋊C8—41no—
251C47⋊C8—41no—
252C47⋊C8—41no—
253C47⋊C8—41no—
254#17921—471nomaximal, Sylow
255C22×C47—21no—
256C22×C47—21no—
257C22×C47—21no—
258C22×C47C2×C4×D411yescharacteristic
259C2×D47—22no—
260C2×D47—41no—
261C2×D47—41no—
262C4×C47—21no—
263C4×C47—21no—
264C4×C47—21no—
265C4×C47—21no—
266C4×C47—41no—
267C4×C47—41no—
268C4×C47—41no—
269C4×C47—41no—
270C4×C47C2×#13111yescharacteristic
271C4×C47C2×#13111yescharacteristic
272C4×C47#57711yescharacteristic
273C4×C47#57811yescharacteristic
274Dic47—22no—
275Dic47—41no—
276Dic47—41no—
277#3107—471no—
278#2988—471no—
279#2985D4711yescharacteristic, complemented
280#1587—471no—
281#1585—471no—
282#1584—471no—
283#1582—471no—
284#1579—471no—
285SD16⋊C8—471no—
286C8×Q16—471no—
287C8×SD16—471no—
288#1573—471no—
289#1488—471no—
290#1481—471no—
291#1454—471no—
292C2×C47—41no—
293C2×C47#292911yescharacteristic
294C2×C47#294711yescharacteristic
295C2×C47#295711yescharacteristic
296D47—42no—
297#533C2×D4711yescharacteristic
298C4×#41—471no—
299#485—471no—
300#485—471no—
301#475—471no—
302#474—471no—
303#462—471no—
304#461—471no—
305#452—471no—
306#452—471no—
307#446—21no—
308#446—21no—
309#446C2×D4711yescharacteristic
310#446C2×D4711yescharacteristic
311C8×Q8—471no—
312C8×Q8—471no—
313C8×Q8C2×D4711yescharacteristic
314C8×Q8C2×D4711yescharacteristic
315C4×Q16—471no—
316C4×Q16—471no—
317C4×SD16—471no—
318C4×SD16—471no—
319C8×D4—471no—
320C8×D4—471no—
321C8×D4—941no—
322C8×D4—941no—
323#433—21no—
324#433—21no—
325#432C2×D4711yescharacteristic
326#431—471no—
327#431—471no—
328#431—941no—
329#431C2×D4711yescharacteristic
330C2×C4⋊C8—471no—
331C2×C4⋊C8—941no—
332#335—471no—
333#335—471no—
334#334—471no—
335#334—471no—
336Q8⋊C8—471no—
337Q8⋊C8—471no—
338Q8⋊C8—471no—
339Q8⋊C8—471no—
340Q8⋊C8—471no—
341Q8⋊C8—471no—
342D4⋊C8—471no—
343D4⋊C8—471no—
344#322—471no—
345#322—471no—
346#322—471no—
347C82—471no—
348C47#1792111yescharacteristic, complemented, Sylow
349C2×#41—471no—
350C2×Q16—471no—
351C2×Q16—471no—
352C2×SD16—471no—
353C2×SD16—471no—
354C22×C8—941no—
355C22×C8—941no—
356C22×C8—941no—
357C4⋊Q8C4×D4711yescharacteristic
358C4⋊Q8C4×D4711yescharacteristic
359#125—21no—
360C22⋊Q8—21no—
361C22⋊Q8C4×D4711yescharacteristic
362C22⋊Q8C4×D4711yescharacteristic
363C4×Q8—21no—
364C4×Q8—471no—
365C4×Q8—471no—
366C4×Q8C22×D4711yescharacteristic
367C4×Q8C22×D4711yescharacteristic
368C4×D4—471no—
369C4×D4—471no—
370C4×D4—941no—
371C4×D4—941no—
372#117—21no—
373#117—471no—
374#117—941no—
375C2×C42—941no—
376C2×C42C22×D4711yescharacteristic
377#107—471no—
378#107—471no—
379#106—471no—
380#106—471no—
381C4⋊C8—21no—
382C4⋊C8—21no—
383C4⋊C8—21no—
384C4⋊C8—21no—
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⋊C8C22⋊D4711yescharacteristic
394C4⋊C8C22⋊D4711yescharacteristic
395C4⋊C8C22×D4711yescharacteristic
396C4⋊C8C22×D4711yescharacteristic
397#103—471no—
398#103—471no—
399#103—471no—
400#103—471no—
401#103—471no—
402#103—471no—
403#102—471no—
404#102—471no—
405C22⋊C8—21no—
406C22⋊C8—471no—
407C22⋊C8—471no—
408C22⋊C8—941no—
409C22⋊C8—941no—
410C22⋊C8—941no—
411C22⋊C8C22⋊D4711yescharacteristic
412C22⋊C8C22⋊D4711yescharacteristic
413#97—21no—
414#97—21no—
415#97—21no—
416C4×C8—21no—
417C4×C8—21no—
418C4×C8—471no—
419C4×C8—471no—
420C4×C8—471no—
421C4×C8—471no—
422C4×C8—941no—
423C4×C8—941no—
424C4×C8C22×D4711yescharacteristic
425C4×C8C22×D4711yescharacteristic
426#41—942no—
427#41—942no—
428C2×Q8—21no—
429C2×Q8—471no—
430C2×Q8C2×C4×D4711yescharacteristic
431C2×Q8C2×C4×D4711yescharacteristic
432C2×D4—471no—
433C2×D4—941no—
434C22×C4—941no—
435C22×C4—941no—
436C22×C4—941no—
437C22×C4C2×C4×D4711yescharacteristic
438C22×C4#1062535011yescharacteristic
439C22×C4C2×C22⋊D4711yescharacteristic
440Q16—942no—
441Q16—942no—
442SD16—942no—
443SD16—942no—
444C2×C8—21no—
445C2×C8—21no—
446C2×C8—471no—
447C2×C8—471no—
448C2×C8—471no—
449C2×C8—471no—
450C2×C8—941no—
451C2×C8—941no—
452C2×C8—941no—
453C2×C8—941no—
454C2×C8—3761no—
455C2×C8—3761no—
456C2×C8—3761no—
457C2×C8C2×C22⋊D4711yescharacteristic
458C2×C8C2×C22⋊D4711yescharacteristic
459C2×C8C2×C22⋊D4711yescharacteristic
460C2×C8C2×C22⋊D4711yescharacteristic
461C4⋊C4—21no—
462C4⋊C4—21no—
463C4⋊C4—21no—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—21no—
468C4⋊C4—471no—
469C4⋊C4—471no—
470C4⋊C4—941no—
471C4⋊C4—941no—
472C4⋊C4C2×C4×D4711yescharacteristic
473C4⋊C4C2×C4×D4711yescharacteristic
474C22⋊C4—21no—
475C22⋊C4—21no—
476C22⋊C4—941no—
477C22⋊C4—941no—
478C22⋊C4—941no—
479C42—21no—
480C42—21no—
481C42—941no—
482C42—941no—
483C42—941no—
484C42C2×C4×D4711yescharacteristic
485C42#1062535011yescharacteristic
486C42C2×C22⋊D4711yescharacteristic
487C42C23×D4711yescharacteristic
488C23—941no—
489C23C4×C22⋊D4711yescharacteristic
490Q8—21no—
491Q8—21no—
492Q8—21no—
493Q8—21no—
494Q8—81no—
495Q8—472no—
496Q8—941no—
497D4—472no—
498D4—944no—
499D4—941no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—41no—
509C2×C4—944no—
510C2×C4—941no—
511C2×C4—941no—
512C2×C4—941no—
513C2×C4—3761no—
514C2×C4—1881no—
515C2×C4—1881no—
516C2×C4C4×C22⋊D4711yescharacteristic
517C2×C4C4×C22⋊D4711yescharacteristic
518C2×C4C4×C22⋊D4711yescharacteristic
519C2×C4C22×C4×D4711yescharacteristic
520C2×C4C22×C22⋊D4711yescharacteristic
521C2×C4C2×#1062535011yescharacteristic
522C8—21no—
523C8—21no—
524C8—21no—
525C8—21no—
526C8—41no—
527C8—41no—
528C8—941no—
529C8—941no—
530C8—941no—
531C8—941no—
532C8—1881no—
533C8—1881no—
534C8—1881no—
535C8—1881no—
536C22—21no—
537C22—21no—
538C22—21no—
539C22—942no—
540C22—1881no—
541C22—1881no—
542C22C2×C4×C22⋊D4711yescharacteristic
543C4—21no—
544C4—21no—
545C4—21no—
546C4—21no—
547C4—41no—
548C4—41no—
549C4—41no—
550C4—41no—
551C4—942no—
552C4—1881no—
553C4—1881no—
554C4#131×D4711yescharacteristic
555C4#131×D4711yescharacteristic
556C4#1568993111yescharacteristic
557C4#1568993811yescharacteristic
558C2—41no—
559C2—1882no—
560C2#2025208011yescharacteristic
561C2#2025223111yescharacteristic
562C2#2025230311yescharacteristic
subgroupG/H
centerC2×C4C22×C22⋊D47
commutatorC2×C4×C47C23×C4
FrattiniC42C23×D47
FittingC47×#2985C2
socleC22×C47C2×C4×D4
prime radicalC2×#10625350C2×C4
cosocleC42×C47C24
hypercenter#2985D47
hypocenterC47#17921
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C2×C4 ◁ #431 ◁ #2985

Complements

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

NH ≅ G/Ntype
3. #20242257C2semidirect
4. #20242243C2semidirect
5. #20242236C2semidirect
6. #20242222C2semidirect
7. #20242205C2semidirect
8. #20242198C2semidirect
9. C8×#10625337C2semidirect
10. C8×#10625336C2semidirect
11. #20242174C2semidirect
13. #20241791C2semidirect
14. #20241688C2semidirect
15. C47×#2985C2semidirect
279. #2985D47semidirect
348. C47#17921semidirect

Maximal subgroups

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

subgroupindex
#202534532
#202524962
#202422572
#202422432
#202422362
#202422222
#202422052
#202421982
C8×#106253372
C8×#106253362
#202421742
#202418202
#202417912
#202416882
C47×#29852
#1792147