Finite Group Databasebeta

← #26219828

Subgroups — #26219828

Order 12032.

874 conjugacy classes of subgroups.

The subgroup lattice has 506 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#108×D47C211yesmaximal, characteristic
2#20251975C212yesmaximal
3C2×#15689160C211yesmaximal, characteristic
4#405×D47C211yesmaximal, characteristic
5#20244174C211yesmaximal, characteristic, complemented
6#20244173C211yesmaximal, characteristic, complemented
7#20244171C211yesmaximal, characteristic, complemented
8#20244167C212yesmaximal, complemented
9#20244165C212yesmaximal, complemented
10#20243125C212yesmaximal
11C47×#1952C211yesmaximal, characteristic, complemented
12C2×M16×D47C2212yes—
13C2×M16×D47C2211yescharacteristic
14C22×C8×D47C2211yescharacteristic
15C2×#13093982C2212yes—
16#117×D47C2211yescharacteristic
17C2×#13093960C2211yescharacteristic
18C2×#13093959C2212yes—
19#108×D47C418yescomplemented
20#15689166C2212yes—
21#15689165C2212yes—
22#15689164—22no—
23#15689163C2212yes—
24#15689162—22no—
25#15689160—21no—
26#15689160—21no—
27#15689160—21no—
28#15689160—21no—
29#15689160—22no—
30#15689158—22no—
31C2×#13093958C2211yescharacteristic
32C2×#13093955C2212yes—
33#15689081C2211yescharacteristic
34#15689080—21no—
35#15689079C2211yescharacteristic
36#15689078C2211yescharacteristic
37#15689077—21no—
38#97×D47—21no—
39C4×C8×D47—21no—
40#15688892C2212yes—
41#15688891C2212yes—
42#15688891C2212yes—
43C2×#108×C47C2211yescharacteristic
44C47×#428C2212yes—
45C2×#107×C47C2211yescharacteristic
46C47×#405C2211yescharacteristic
47C47×#343C2212yes—
48C22×C4×D47C2311yescharacteristic
49C2×#10625329C2312yes—
50C2×#10625329C2311yescharacteristic
51C2×#10625324—21no—
52C2×#10625324—22no—
53M16×D47—24no—
54M16×D47—22no—
55M16×D47C2×C418yes—
56M16×D47Q814yes—
57C22×C47⋊C8C2311yescharacteristic
58C2×C8×D47—21no—
59C2×C8×D47—22no—
60C2×C8×D47C2×C414yes—
61C2×C8×D47D411yescharacteristic
62C2×C8×D47D411yescharacteristic
63C2×C8×D47D411yescharacteristic
64C2×C8×D47D411yescharacteristic
65C2×C8×D47Q814yes—
66#13093983C2311yescharacteristic
67#13093982—22no—
68#13093982—22no—
69#13093982—22no—
70#13093982—22no—
71#13093982D411yescharacteristic
72#13093982D411yescharacteristic
73#13093981—21no—
74#13093980—21no—
75C2×C4⋊Dic47C2312yes—
76C4⋊C4×D47D411yescharacteristic
77C4⋊C4×D47D411yescharacteristic
78C2×(C2×D47)⋊C4—22no—
79C22⋊C4×D47—21no—
80C2×C4×Dic47—22no—
81C42×D47—21no—
82#13093960C2×C414yescomplemented
83#13093959C2×C418yescomplemented
84#13093958—21no—
85#13093958—21no—
86#13093958—22no—
87#13093957—22no—
88#13093957—22no—
89#13093956—22no—
90#13093955—22no—
91#13093955—22no—
92#13093955—22no—
93#13093955—22no—
94#13093935—21no—
95#13093934—21no—
96#13093934—21no—
97C4×C47⋊C8—21no—
98C8×Dic47—21no—
99C8×Dic47—21no—
100C2×M16×C47C2312yes—
101C2×M16×C47C2311yescharacteristic
102C22×C8×C47C2311yescharacteristic
103#117×C47C2311yescharacteristic
104C2×C4⋊C4×C47C2312yes—
105#108×C47C2×C414yescomplemented
106#107×C47—21no—
107#107×C47—21no—
108#107×C47—22no—
109#106×C47—22no—
110#97×C47—21no—
111C4×C8×C47—21no—
112C23×D47#4111yescharacteristic
113C22×C4×C47—22no—
114C22×C4×C47C2411yescharacteristic
115C22×Dic47—22no—
116C22×Dic47#4111yescharacteristic
117C2×C4×D47—21no—
118C2×C4×D47—21no—
119C2×C4×D47C22×C412yes—
120C2×C4×D47C2×D411yescharacteristic
121C2×C4×D47C2×D411yescharacteristic
122C2×C4×D47C2×D411yescharacteristic
123C2×C4×D47C2×D411yescharacteristic
124C2×C4×D47C2×Q812yes—
125C2×C4×D47C2×Q812yes—
126C2×C4×D47#4112yes—
127M16×C47—22no—
128M16×C47—21no—
129M16×C47C22×C414yes—
130M16×C47C2×Q812yes—
131C2×C8×C47—22no—
132C2×C8×C47—21no—
133C2×C8×C47C22×C412yes—
134C2×C8×C47C2×D411yescharacteristic
135C2×C8×C47C2×D411yescharacteristic
136C2×C8×C47C2×Q812yes—
137C4⋊C4×C47—22no—
138C4⋊C4×C47—22no—
139C4⋊C4×C47C2×D411yescharacteristic
140C4⋊C4×C47C2×D411yescharacteristic
141C22⋊C4×C47—21no—
142C42×C47—21no—
143C22⋊Dic47—21no—
144(C2×D47)⋊C4—21no—
145(C2×D47)⋊C4—21no—
146(C2×D47)⋊C4—42no—
147(C2×D47)⋊C4—42no—
148C4⋊Dic47—22no—
149C4⋊Dic47—22no—
150C4⋊Dic47C2×D411yescharacteristic
151C4⋊Dic47C2×D411yescharacteristic
152Dic47⋊C4—21no—
153Dic47⋊C4—21no—
154C4×Dic47—21no—
155C4×Dic47—21no—
156C4×Dic47—21no—
157C4×Dic47—42no—
158C4×Dic47—42no—
159#10625329—22no—
160#10625329—21no—
161#10625329C22×C414yes—
162#10625329C2×Q812yes—
163C2×C47⋊C8—21no—
164C2×C47⋊C8—22no—
165C2×C47⋊C8C22×C412yes—
166C2×C47⋊C8C2×D411yescharacteristic
167C2×C47⋊C8C2×D411yescharacteristic
168C2×C47⋊C8C2×Q812yes—
169#10625324—28no—
170#10625324—24no—
171C8×D47—28no—
172C8×D47—24no—
173C8×D47C4⋊C414yes—
174C8×D47C4⋊C414yes—
175C8×D47C4⋊C414yes—
176C8×D47C4⋊C414yes—
177C23×C47C2×#4111yescharacteristic
178C22×D47—41no—
179C22×D47—41no—
180C22×D47C4×Q812yes—
181C22×D47#12111yescharacteristic
182C22×D47#12111yescharacteristic
183C22×D47#12512yes—
184C2×C4×C47—21no—
185C2×C4×C47—21no—
186C2×C4×C47—42no—
187C2×C4×C47—82no—
188C2×C4×C47C23×C411yescharacteristic
189C2×C4×C47C22×D411yescharacteristic
190C2×C4×C47C22×D411yescharacteristic
191C2×C4×C47C22×Q811yescharacteristic
192C2×C4×C47C22×Q811yescharacteristic
193C2×C4×C47C2×#4111yescharacteristic
194C2×Dic47—21no—
195C2×Dic47—21no—
196C2×Dic47—42no—
197C2×Dic47—82no—
198C2×Dic47C4×Q812yes—
199C2×Dic47#12111yescharacteristic
200C2×Dic47#12111yescharacteristic
201C2×Dic47#12512yes—
202C4×D47—42no—
203C4×D47—42no—
204C4×D47C2×C4⋊C412yes—
205C4×D47C2×C4⋊C412yes—
206C4×D47C4×D412yes—
207C4×D47C4×D412yes—
208C4×D47C22⋊Q812yes—
209C4×D47C22⋊Q812yes—
210C4×D47C4⋊Q812yes—
211C4×D47C4⋊Q812yes—
212C8×C47—24no—
213C8×C47—22no—
214C8×C47C2×C4⋊C412yes—
215C8×C47C2×C4⋊C412yes—
216C47⋊C8—24no—
217C47⋊C8—22no—
218C47⋊C8C2×C4⋊C412yes—
219C47⋊C8C2×C4⋊C412yes—
220#21181—471nomaximal, Sylow
221C22×C47—21no—
222C22×C47—21no—
223C22×C47C2×C4×Q811yescharacteristic
224C22×C47C2×#12111yescharacteristic
225C22×C47C2×#12511yescharacteristic
226C2×D47—21no—
227C2×D47—21no—
228C2×D47—21no—
229C2×D47—21no—
230C2×D47—42no—
231C2×D47—42no—
232C2×D47#38912yes—
233C2×D47#38912yes—
234C4×C47—41no—
235C4×C47—41no—
236C4×C47—82no—
237C4×C47C22×C4⋊C411yescharacteristic
238C4×C47C2×C4×D411yescharacteristic
239C4×C47C2×C22⋊Q811yescharacteristic
240C4×C47C2×C4⋊Q811yescharacteristic
241Dic47—41no—
242Dic47—41no—
243Dic47—82no—
244Dic47#38912yes—
245Dic47#38912yes—
246C22×#108—471no—
247#2911—472no—
248C2×#427—471no—
249C2×#405—471no—
250#1952—471no—
251#1952—471no—
252#1952—471no—
253#1952D4711yescharacteristic, complemented
254#1951—472no—
255#1951—472no—
256#1742—472no—
257C2×C47—21no—
258C2×C47—21no—
259C2×C47C2×#38911yescharacteristic
260C2×C47#306711yescharacteristic
261C2×C47#306711yescharacteristic
262D47—22no—
263D47—22no—
264C22×M16—471no—
265C22×M16—472no—
266C23×C8—471no—
267C2×#117—471no—
268C2×#117—472no—
269C2×#108—472no—
270C2×#108—471no—
271C2×#108—478no—
272C2×#108C2×D4711yescharacteristic
273#428—472no—
274#428—472no—
275#428—472no—
276#428—942no—
277#428—942no—
278#428C2×D4712yes—
279#427—942no—
280#427—942no—
281#427—941no—
282#427—941no—
283#427—941no—
284#427—941no—
285C2×#107—471no—
286C2×#107C2×D4711yescharacteristic
287C2×#106—472no—
288#405—471no—
289#405—471no—
290#405—471no—
291#405—941no—
292#405—941no—
293#405C2×D4711yescharacteristic
294C2×#97—941no—
295C2×C4×C8—941no—
296#343—472no—
297#343—472no—
298#343—472no—
299#343C2×D4712yes—
300C47#2118111yescharacteristic, complemented, Sylow
301C23×C4—471no—
302C2×M16—471no—
303C2×M16—472no—
304C2×M16—478no—
305C2×M16—474no—
306C2×M16—941no—
307C2×M16—942no—
308C2×M16—942no—
309C2×M16—944no—
310C2×M16C22×D4712yes—
311C2×M16C22×D4711yescharacteristic
312C22×C8—471no—
313C22×C8—474no—
314C22×C8—471no—
315C22×C8—471no—
316C22×C8—471no—
317C22×C8—471no—
318C22×C8—474no—
319C22×C8—941no—
320C22×C8—942no—
321C22×C8C22×D4711yescharacteristic
322#117—471no—
323#117—471no—
324#117—471no—
325#117—941no—
326#117—941no—
327#117—942no—
328#117—942no—
329#117—942no—
330#117—942no—
331#117C22×D4711yescharacteristic
332C2×C4⋊C4—471no—
333C2×C4⋊C4—471no—
334C2×C4⋊C4—472no—
335C2×C4⋊C4C22×D4712yes—
336C2×C22⋊C4—941no—
337C2×C22⋊C4—942no—
338C2×C42—941no—
339C2×C42—942no—
340#108—478no—
341#108—474no—
342#108C4×D4714yescomplemented
343#107—21no—
344#107—21no—
345#107—22no—
346#107—942no—
347#107—942no—
348#107—941no—
349#107—941no—
350#107—942no—
351#106—22no—
352#106—942no—
353#106—942no—
354#106—942no—
355#106—942no—
356#106—942no—
357#97—21no—
358#97—941no—
359#97—941no—
360#97—941no—
361C4×C8—21no—
362C4×C8—941no—
363C4×C8—941no—
364C4×C8—941no—
365C24—471no—
366C22×C4—22no—
367C22×C4—471no—
368C22×C4—472no—
369C22×C4—471no—
370C22×C4—471no—
371C22×C4—471no—
372C22×C4—471no—
373C22×C4—472no—
374C22×C4—472no—
375C22×C4—472no—
376C22×C4—941no—
377C22×C4—941no—
378C22×C4—942no—
379C22×C4C23×D4711yescharacteristic
380M16—22no—
381M16—21no—
382M16—474no—
383M16—472no—
384M16—948no—
385M16—944no—
386M16—942no—
387M16—941no—
388M16C2×C4×D4714yes—
389M16Q8×D4712yes—
390C2×C8—21no—
391C2×C8—22no—
392C2×C8—472no—
393C2×C8—471no—
394C2×C8—471no—
395C2×C8—472no—
396C2×C8—474no—
397C2×C8—474no—
398C2×C8—474no—
399C2×C8—474no—
400C2×C8—948no—
401C2×C8—944no—
402C2×C8—941no—
403C2×C8—942no—
404C2×C8C2×C4×D4712yes—
405C2×C8D4×D4711yescharacteristic
406C2×C8D4×D4711yescharacteristic
407C2×C8Q8×D4712yes—
408C4⋊C4—22no—
409C4⋊C4—22no—
410C4⋊C4—471no—
411C4⋊C4—471no—
412C4⋊C4—941no—
413C4⋊C4—941no—
414C4⋊C4—942no—
415C4⋊C4—942no—
416C4⋊C4D4×D4711yescharacteristic
417C4⋊C4D4×D4711yescharacteristic
418C22⋊C4—21no—
419C22⋊C4—941no—
420C22⋊C4—941no—
421C22⋊C4—941no—
422C22⋊C4—1882no—
423C22⋊C4—1882no—
424C42—21no—
425C42—941no—
426C42—941no—
427C42—941no—
428C42—1882no—
429C42—1882no—
430C23—471no—
431C23—471no—
432C23—472no—
433C23—472no—
434C23—1881no—
435C23—1881no—
436C23#41×D4711yescharacteristic
437C2×C4—21no—
438C2×C4—21no—
439C2×C4—42no—
440C2×C4—82no—
441C2×C4—472no—
442C2×C4—471no—
443C2×C4—471no—
444C2×C4—472no—
445C2×C4—472no—
446C2×C4—472no—
447C2×C4—472no—
448C2×C4—472no—
449C2×C4—472no—
450C2×C4—472no—
451C2×C4—472no—
452C2×C4—472no—
453C2×C4—941no—
454C2×C4—941no—
455C2×C4—3762no—
456C2×C4—1882no—
457C2×C4—1882no—
458C2×C4—1882no—
459C2×C4C22×C4×D4711yescharacteristic
460C2×C4C2×D4×D4711yescharacteristic
461C2×C4C2×D4×D4711yescharacteristic
462C2×C4C2×Q8×D4711yescharacteristic
463C2×C4C2×Q8×D4711yescharacteristic
464C2×C4#41×D4711yescharacteristic
465C8—24no—
466C8—22no—
467C8—472no—
468C8—472no—
469C8—944no—
470C8—942no—
471C8C4⋊C4×D4712yes—
472C8C4⋊C4×D4712yes—
473C22—21no—
474C22—21no—
475C22—472no—
476C22—472no—
477C22—941no—
478C22—941no—
479C22—941no—
480C22—941no—
481C22—1882no—
482C22—1882no—
483C22C4×Q8×D4711yescharacteristic
484C22#121×D4711yescharacteristic
485C22#125×D4711yescharacteristic
486C4—41no—
487C4—41no—
488C4—82no—
489C4—472no—
490C4—472no—
491C4—3762no—
492C4—1881no—
493C4—1881no—
494C4C2×C4⋊C4×D4711yescharacteristic
495C4C4×D4×D4711yescharacteristic
496C4C22⋊Q8×D4711yescharacteristic
497C4C4⋊Q8×D4711yescharacteristic
498C2—21no—
499C2—21no—
500C2—942no—
501C2—942no—
502C2#389×D4711yescharacteristic
503C2#2025310611yescharacteristic
504C2#2025311011yescharacteristic
subgroupG/H
centerC22#121×D47
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC47×#1952C2
socleC22×C47C2×#121
prime radicalC23×D47#41
cosocleC22×C4×C47C24
hypercenter#1952D47
hypocenterC47#21181
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C22×C4 ◁ #1952

Complements

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

NH ≅ G/Ntype
5. #20244174C2semidirect
6. #20244173C2semidirect
7. #20244171C2semidirect
8. #20244167C2semidirect
9. #20244165C2semidirect
11. C47×#1952C2semidirect
19. #108×D47C4semidirect
82. #13093960C2×C4semidirect
83. #13093959C2×C4semidirect
105. #108×C47C2×C4semidirect
253. #1952D47semidirect
300. C47#21181semidirect
342. #108C4×D47semidirect

Maximal subgroups

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

subgroupindex
C2×#108×D472
#202519752
C2×#156891602
#405×D472
#202441742
#202441732
#202441712
#202441672
#202441652
#202431252
C47×#19522
#2118147