Finite Group Databasebeta

← #26111569

Subgroups — #26111569

Order 12032.

738 conjugacy classes of subgroups.

The subgroup lattice has 458 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689497C211yesmaximal, characteristic
2#20246452C211yesmaximal, characteristic
3#20244337C211yesmaximal, characteristic, complemented
4C2×#15688866C211yesmaximal, characteristic
5#20241229C211yesmaximal, characteristic, complemented
6#20241227C211yesmaximal, characteristic, complemented
7C47×#1987C211yesmaximal, characteristic, complemented
8C22×#10625336—22no—
9C2×#13094021C2211yescharacteristic
10C2×C4×D188—21no—
11C2×#13093986—21no—
12#15689497—22no—
13#15689497C414yescomplemented
14C2×C4⋊(C47⋊C8)C2211yescharacteristic
15C2×C4×C47⋊C8C2211yescharacteristic
16#15689025—21no—
17#15689024—21no—
18#15689001—22no—
19C4×C4⋊Dic47C2211yescharacteristic
20#15688885—21no—
21#15688885—21no—
22#15688866—22no—
23C2×C4⋊Q8×C47C2211yescharacteristic
24C2×C4⋊C8×C47C2211yescharacteristic
25C4×C4⋊C4×C47C2211yescharacteristic
26C47×#340—21no—
27C47×#340—21no—
28C22×D188—21no—
29C22×D188—21no—
30C22×D188D412yes—
31C22×C4×D47—22no—
32C2×#10625336—28no—
33C2×#10625336—44no—
34C2×#10625336—44no—
35C22×C47⋊C8—21no—
36C22×C47⋊C8—21no—
37#13094021—21no—
38#13094021—21no—
39#13094021C2×C412yes—
40#13094021D412yes—
41C4×D188—42no—
42C4×D188—42no—
43#13093986—42no—
44#13093986—42no—
45C2×C4⋊Dic47D411yescharacteristic
46C2×C4⋊Dic47D411yescharacteristic
47C2×(C2×D47)⋊C4—22no—
48C2×(C2×D47)⋊C4—22no—
49C2×(C2×D47)⋊C4—22no—
50C2×C4×Dic47—21no—
51C4⋊(C47⋊C8)—21no—
52C4⋊(C47⋊C8)—21no—
53C4×C47⋊C8—21no—
54C4×C47⋊C8C2×C412yescomplemented
55C47⋊#95—21no—
56C22×Q8×C47—21no—
57C22×C8×C47—21no—
58C4⋊Q8×C47—21no—
59C4⋊Q8×C47—21no—
60C4⋊Q8×C47—21no—
61C4⋊Q8×C47C2×C412yescomplemented
62C2×C4⋊C4×C47—21no—
63C2×C4⋊C4×C47—21no—
64C2×C4⋊C4×C47D411yescharacteristic
65C2×C4⋊C4×C47D411yescharacteristic
66C2×C42×C47—21no—
67C2×C42×C47C2311yescharacteristic
68C4⋊C8×C47—21no—
69C4⋊C8×C47—21no—
70#95×C47—21no—
71C23×D47—22no—
72C2×Q8×C47—22no—
73C2×Q8×C47—42no—
74C2×Q8×C47—42no—
75C2×Q8×C47—41no—
76C22×C4×C47—21no—
77C22×C4×C47—21no—
78C22×C4×C47—41no—
79C22×C4×C47C2×D411yescharacteristic
80C22×C4×C47C2×D411yescharacteristic
81C22×C4×C47C2×D411yescharacteristic
82C22×Dic47—21no—
83C22×Dic47—21no—
84C2×D188—22no—
85C2×D188—22no—
86C2×D188—22no—
87C2×D188—22no—
88C2×D188—22no—
89C2×D188—41no—
90C2×D188—41no—
91C2×D188—41no—
92C2×D188—41no—
93C2×D188—41no—
94C2×D188—41no—
95C2×D188—41no—
96C2×D188—41no—
97C2×D188—42no—
98C2×D188—42no—
99C2×D188C22⋊C414yes—
100C2×C4×D47—44no—
101C2×C4×D47—44no—
102C2×C4×D47—82no—
103C2×C8×C47—41no—
104C2×C8×C47—41no—
105C2×C8×C47—41no—
106C4⋊C4×C47—21no—
107C4⋊C4×C47—22no—
108C4⋊C4×C47—22no—
109C4⋊C4×C47—21no—
110C4⋊C4×C47—21no—
111C4⋊C4×C47—41no—
112C4⋊C4×C47—41no—
113C4⋊C4×C47SD1612yescomplemented
114C42×C47—41no—
115C42×C47—41no—
116C42×C47C22×C411yescharacteristic
117C42×C47C2×D411yescharacteristic
118C42×C47#4111yescharacteristic
119C42×C47#4111yescharacteristic
120#10625336—416no—
121(C2×D47)⋊C4—44no—
122(C2×D47)⋊C4—44no—
123(C2×D47)⋊C4—82no—
124C4⋊Dic47—21no—
125C4⋊Dic47—21no—
126C4⋊Dic47—21no—
127C4⋊Dic47SD1612yescomplemented
128C4×Dic47—41no—
129C4×Dic47—41no—
130C2×C47⋊C8—22no—
131C2×C47⋊C8—22no—
132C2×C47⋊C8—22no—
133C2×C47⋊C8—41no—
134C2×C47⋊C8—41no—
135C2×C47⋊C8—41no—
136C23×C47#12011yescharacteristic
137C22×D47—24no—
138C22×D47—42no—
139C22×D47—42no—
140C22×D47—42no—
141C22×D47—42no—
142C22×D47—42no—
143C22×D47—42no—
144Q8×C47—44no—
145Q8×C47—82no—
146C2×C4×C47—21no—
147C2×C4×C47—21no—
148C2×C4×C47—22no—
149C2×C4×C47—41no—
150C2×C4×C47—41no—
151C2×C4×C47—42no—
152C2×C4×C47—41no—
153C2×C4×C47—41no—
154C2×C4×C47—41no—
155C2×C4×C47—81no—
156C2×C4×C47—81no—
157C2×C4×C47C2×C22⋊C411yescharacteristic
158C2×C4×C47C4×D411yescharacteristic
159C2×C4×C47C4×D411yescharacteristic
160C2×C4×C47#12111yescharacteristic
161C2×C4×C47#12111yescharacteristic
162C2×C4×C47#12311yescharacteristic
163C2×C4×C47C2×SD1611yescharacteristic
164C2×C4×C47C2×SD1611yescharacteristic
165C2×C4×C47#13511yescharacteristic
166C2×C4×C47#13511yescharacteristic
167C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
168C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
169C2×C4×C47#13711yescharacteristic
170C2×C4×C47#13711yescharacteristic
171C2×Dic47—22no—
172C2×Dic47—41no—
173C2×Dic47—41no—
174C2×Dic47—41no—
175C2×Dic47—41no—
176C2×Dic47—41no—
177D188—44no—
178D188—42no—
179D188—42no—
180D188—42no—
181D188—42no—
182D188—42no—
183D188—42no—
184D188—42no—
185D188—42no—
186D188—82no—
187C4×D47—88no—
188C8×C47—81no—
189C47⋊C8—24no—
190C47⋊C8—81no—
191#9064—471nomaximal, Sylow
192C22×C47#38611yescharacteristic
193C22×C47#45011yescharacteristic
194C22×C47#45011yescharacteristic
195C22×C47#45211yescharacteristic
196C22×C47#45211yescharacteristic
197C22×C47#45311yescharacteristic
198C22×C47#45311yescharacteristic
199C2×D47—44no—
200C2×D47—44no—
201C2×D47—44no—
202C2×D47—82no—
203C2×D47—82no—
204C2×D47—82no—
205C2×D47—82no—
206C4×C47—21no—
207C4×C47—21no—
208C4×C47—42no—
209C4×C47—81no—
210C4×C47—82no—
211C4×C47C4×SD1611yescharacteristic
212C4×C47C4×SD1611yescharacteristic
213C4×C47#44011yescharacteristic
214C4×C47#44011yescharacteristic
215C4×C47#46111yescharacteristic
216C4×C47#46111yescharacteristic
217C4×C47#46311yescharacteristic
218C4×C47#46311yescharacteristic
219Dic47—42no—
220Dic47—81no—
221C2×#492—471no—
222#2366—471no—
223#1987—471no—
224#1987D4711yescharacteristic, complemented
225C2×#331—471no—
226#1279—471no—
227#1279—471no—
228C2×C47#162911yescharacteristic
229C2×C47#163011yescharacteristic
230C2×C47#163111yescharacteristic
231C2×C47#163111yescharacteristic
232C2×C47#187711yescharacteristic
233C2×C47#187711yescharacteristic
234C2×C47#188711yescharacteristic
235D47—84no—
236C22×SD16—942no—
237C2×C4⋊Q8C2×D4711yescharacteristic
238C2×#127—471no—
239C2×C4×D4—941no—
240C2×C4×D4—941no—
241#492—474no—
242#492—942no—
243C2×C4⋊C8—471no—
244C2×C4⋊C8C2×D4711yescharacteristic
245C2×C4×C8—471no—
246#390—941no—
247#390—941no—
248#386—942no—
249C4×C4⋊C4—471no—
250C4×C4⋊C4C2×D4711yescharacteristic
251#340—21no—
252#340—21no—
253#340—941no—
254#340—941no—
255#331—942no—
256C47#906411yescharacteristic, complemented, Sylow
257C22×Q8—21no—
258C22×D4—472no—
259C22×D4—941no—
260C22×D4—941no—
261C23×C4—942no—
262C2×SD16—948no—
263C2×SD16—1884no—
264C2×SD16—1884no—
265C22×C8—21no—
266C22×C8—941no—
267C22×C8—941no—
268C4⋊Q8—21no—
269C4⋊Q8—21no—
270C4⋊Q8—21no—
271C4⋊Q8C4×D4712yescomplemented
272#127—472no—
273#127—472no—
274#127—941no—
275#127—941no—
276C4×D4—1882no—
277C4×D4—1882no—
278C4×D4—1882no—
279C4×D4—1882no—
280C2×C4⋊C4—21no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—471no—
283C2×C4⋊C4—471no—
284C2×C4⋊C4C22⋊D4711yescharacteristic
285C2×C4⋊C4C22⋊D4711yescharacteristic
286C2×C22⋊C4—942no—
287C2×C22⋊C4—942no—
288C2×C22⋊C4—942no—
289C2×C42—21no—
290C2×C42—941no—
291C2×C42C22×D4711yescharacteristic
292C4⋊C8—21no—
293C4⋊C8—21no—
294C4⋊C8—941no—
295C4⋊C8—941no—
296C4×C8—472no—
297C4×C8—941no—
298#95—21no—
299#95—941no—
300C24—942no—
301C2×Q8—22no—
302C2×Q8—42no—
303C2×Q8—42no—
304C2×Q8—41no—
305C2×D4—474no—
306C2×D4—942no—
307C2×D4—942no—
308C2×D4—942no—
309C2×D4—942no—
310C2×D4—942no—
311C2×D4—1882no—
312C2×D4—1882no—
313C2×D4—1881no—
314C2×D4—1881no—
315C2×D4—1881no—
316C2×D4—1881no—
317C2×D4—1881no—
318C2×D4—1881no—
319C2×D4—1881no—
320C2×D4—1881no—
321C22×C4—21no—
322C22×C4—21no—
323C22×C4—41no—
324C22×C4—941no—
325C22×C4—941no—
326C22×C4—3762no—
327C22×C4—1884no—
328C22×C4—1884no—
329C22×C4D4×D4711yescharacteristic
330C22×C4D4×D4711yescharacteristic
331C22×C4C2×C22⋊D4711yescharacteristic
332SD16—18816no—
333C2×C8—41no—
334C2×C8—41no—
335C2×C8—41no—
336C2×C8—942no—
337C2×C8—942no—
338C2×C8—942no—
339C2×C8—1881no—
340C2×C8—1881no—
341C2×C8—1881no—
342C4⋊C4—21no—
343C4⋊C4—22no—
344C4⋊C4—22no—
345C4⋊C4—21no—
346C4⋊C4—21no—
347C4⋊C4—41no—
348C4⋊C4—41no—
349C4⋊C4—472no—
350C4⋊C4—941no—
351C4⋊C4—941no—
352C4⋊C4—941no—
353C4⋊C4#1062533612yescomplemented
354C22⋊C4—3762no—
355C22⋊C4—1884no—
356C22⋊C4—1884no—
357C42—41no—
358C42—41no—
359C42—1881no—
360C42—1881no—
361C42C2×C4×D4711yescharacteristic
362C42#1062535011yescharacteristic
363C42D4×D4711yescharacteristic
364C42#1062535211yescharacteristic
365C23—944no—
366C23—1882no—
367C23—1882no—
368C23—1882no—
369C23—1882no—
370C23—1882no—
371C23—1882no—
372C23#1309399211yescharacteristic
373Q8—44no—
374Q8—82no—
375D4—3762no—
376D4—1884no—
377D4—1882no—
378D4—1882no—
379D4—1882no—
380D4—1882no—
381D4—1882no—
382D4—1882no—
383D4—1882no—
384D4—1882no—
385C2×C4—21no—
386C2×C4—21no—
387C2×C4—22no—
388C2×C4—41no—
389C2×C4—41no—
390C2×C4—42no—
391C2×C4—41no—
392C2×C4—41no—
393C2×C4—41no—
394C2×C4—81no—
395C2×C4—81no—
396C2×C4—942no—
397C2×C4—3768no—
398C2×C4—1881no—
399C2×C4—1881no—
400C2×C4—1881no—
401C2×C4—1881no—
402C2×C4—1881no—
403C2×C4C22⋊C4×D4711yescharacteristic
404C2×C4(C22⋊D47)⋊C411yescharacteristic
405C2×C4C4×C22⋊D4711yescharacteristic
406C2×C4#1309399511yescharacteristic
407C2×C4#1309399711yescharacteristic
408C2×C4#1309400611yescharacteristic
409C2×C4SD16×D4711yescharacteristic
410C2×C4C2×#1062533611yescharacteristic
411C2×C4#1309404411yescharacteristic
412C2×C4#1309404711yescharacteristic
413C2×C4#1309404911yescharacteristic
414C2×C4#1309405111yescharacteristic
415C2×C4#1309405511yescharacteristic
416C2×C4#1309405611yescharacteristic
417C8—81no—
418C8—944no—
419C8—3761no—
420C22—3762no—
421C22—3762no—
422C22—3762no—
423C22—3762no—
424C22—1884no—
425C22—1884no—
426C22—1884no—
427C22#1568899911yescharacteristic
428C22#1568927211yescharacteristic
429C22#1568927311yescharacteristic
430C22#1568928311yescharacteristic
431C22#1568928411yescharacteristic
432C22#1568928911yescharacteristic
433C22#1568928911yescharacteristic
434C4—21no—
435C4—21no—
436C4—42no—
437C4—82no—
438C4—81no—
439C4—3761no—
440C4—1882no—
441C4#1568920811yescharacteristic
442C4C4×#1062533611yescharacteristic
443C4#1568922111yescharacteristic
444C4#1568922611yescharacteristic
445C4#1568932511yescharacteristic
446C4#1568932711yescharacteristic
447C4#1568933511yescharacteristic
448C4#1568934111yescharacteristic
449C2—3764no—
450C2#2024250311yescharacteristic
451C2#2024250811yescharacteristic
452C2#2024251311yescharacteristic
453C2#2024251411yescharacteristic
454C2#2024371911yescharacteristic
455C2#2024372511yescharacteristic
456C2#2024378311yescharacteristic
subgroupG/H
centerC23#13093992
commutatorC42×C47C22×C4
FrattiniC2×C42C22×D47
FittingC47×#1987C2
socleC23×C47#120
prime radicalC2×#13094021C22
cosocleC2×C42×C47C23
hypercenter#1987D47
hypocenterC47#9064
LCSG ▷ C42×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1987

Complements

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

NH ≅ G/Ntype
3. #20244337C2semidirect
5. #20241229C2semidirect
6. #20241227C2semidirect
7. C47×#1987C2semidirect
13. #15689497C4semidirect
54. C4×C47⋊C8C2×C4semidirect
61. C4⋊Q8×C47C2×C4semidirect
113. C4⋊C4×C47SD16semidirect
127. C4⋊Dic47SD16semidirect
224. #1987D47semidirect
256. C47#9064semidirect
271. C4⋊Q8C4×D47semidirect
353. C4⋊C4#10625336semidirect

Maximal subgroups

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

subgroupindex
C2×#156894972
#202464522
#202443372
C2×#156888662
#202412292
#202412272
C47×#19872
#906447