Finite Group Databasebeta

← #26113140

Subgroups — #26113140

Order 12032.

580 conjugacy classes of subgroups.

The subgroup lattice has 442 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20252499C211yesmaximal, characteristic, complemented
2#20244316C211yesmaximal, characteristic, complemented
3#20243408C211yesmaximal, characteristic, complemented
4#20242056C211yesmaximal, characteristic, complemented
5#20241289C211yesmaximal, characteristic, complemented
6#20241262C211yesmaximal, characteristic, complemented
7C47×#3038C211yesmaximal, characteristic
8Q8×C47⋊C8—21no—
9D4×C47⋊C8—22no—
10D4×C47⋊C8—21no—
11#15689176—21no—
12#15689176C2211yescharacteristic
13#15689147C2211yescharacteristic
14C2×C4⋊(C47⋊C8)—21no—
15C2×C4⋊(C47⋊C8)C2211yescharacteristic
16C2×#13093951—21no—
17C2×#13093945—22no—
18C4×C4⋊Dic47C2211yescharacteristic
19#15688882—21no—
20#15688867—21no—
21#15688867—21no—
22C2×AGL(1,ℤ/8)×C47—21no—
23C2×#127×C47C2211yescharacteristic
24C4×#41×C47C2211yescharacteristic
25C47×#460—21no—
26C47×#460C412yescomplemented
27C47×#459—21no—
28C47×#459C412yescomplemented
29C47×#423C2211yescharacteristic
30C47×#418—21no—
31C2×#10625329—21no—
32C22×C47⋊C8—22no—
33C22×C47⋊C8D412yes—
34C2×C4⋊Dic47—21no—
35C2×C4×Dic47—21no—
36C4⋊(C47⋊C8)—22no—
37C4⋊(C47⋊C8)—21no—
38C4⋊(C47⋊C8)—21no—
39C4⋊(C47⋊C8)—21no—
40C4⋊(C47⋊C8)—21no—
41C4⋊(C47⋊C8)D411yescharacteristic
42C4⋊(C47⋊C8)D411yescharacteristic
43#13093951—42no—
44#13093945—44no—
45C22⋊(C47⋊C8)—22no—
46C22⋊(C47⋊C8)—21no—
47C4×C47⋊C8—22no—
48C4×C47⋊C8—21no—
49C47⋊#95—21no—
50C2×#41×C47D411yescharacteristic
51C22×D4×C47—21no—
52C22×D4×C47—21no—
53C22×D4×C47—21no—
54AGL(1,ℤ/8)×C47—42no—
55AGL(1,ℤ/8)×C47—42no—
56C2×SD16×C47—22no—
57C2×D8×C47—22no—
58C2×M16×C47—21no—
59#127×C47—21no—
60#127×C47—21no—
61#127×C47—21no—
62#127×C47C2×C412yes—
63C4×Q8×C47C2×C411yescharacteristic
64C4×Q8×C47D411yescharacteristic
65C4×D4×C47—21no—
66C4×D4×C47—21no—
67C4×D4×C47C2×C411yescharacteristic
68C4×D4×C47D411yescharacteristic
69#117×C47—21no—
70#117×C47D411yescharacteristic
71C2×C42×C47—21no—
72C2×C42×C47C2311yescharacteristic
73C4⋊C8×C47—21no—
74C4⋊C8×C47C2×C412yes—
75#102×C47—22no—
76#102×C47—22no—
77C24×C47—41no—
78#41×C47—21no—
79#41×C47—21no—
80#41×C47D812yes—
81#41×C47SD1612yes—
82C2×Q8×C47C22⋊C411yescharacteristic
83C2×D4×C47—21no—
84C2×D4×C47—22no—
85C2×D4×C47—22no—
86C2×D4×C47—22no—
87C2×D4×C47—42no—
88C2×D4×C47—42no—
89C2×D4×C47—41no—
90C2×D4×C47—41no—
91C2×D4×C47—41no—
92C2×D4×C47—41no—
93C2×D4×C47—41no—
94C2×D4×C47—41no—
95C2×D4×C47—41no—
96C2×D4×C47—82no—
97C2×D4×C47C22⋊C411yescharacteristic
98C22×C4×C47—21no—
99C22×C4×C47—21no—
100C22×C4×C47—21no—
101C22×C4×C47C2×D411yescharacteristic
102C22×C4×C47C2×D411yescharacteristic
103C22×C4×C47#4111yescharacteristic
104C22×Dic47—41no—
105SD16×C47—44no—
106D8×C47—44no—
107M16×C47—41no—
108M16×C47—41no—
109C2×C8×C47—22no—
110C4⋊C4×C47—21no—
111C4⋊C4×C47—21no—
112C4⋊C4×C47C22⋊C411yescharacteristic
113C4⋊C4×C47C22⋊C411yescharacteristic
114C22⋊C4×C47—21no—
115C22⋊C4×C47—21no—
116C22⋊C4×C47—21no—
117C42×C47—21no—
118C42×C47—21no—
119C42×C47—21no—
120C42×C47C22×C411yescharacteristic
121C42×C47C2×D411yescharacteristic
122C42×C47C2×D411yescharacteristic
123C42×C47#4111yescharacteristic
124C4⋊Dic47—22no—
125C4⋊Dic47—41no—
126C4×Dic47—41no—
127C4×Dic47—41no—
128#10625329—41no—
129#10625329—41no—
130C2×C47⋊C8—21no—
131C2×C47⋊C8—22no—
132C2×C47⋊C8—21no—
133C2×C47⋊C8—42no—
134C2×C47⋊C8—41no—
135C2×C47⋊C8—82no—
136C23×C47—21no—
137C23×C47—42no—
138C23×C47—82no—
139C23×C47—82no—
140C23×C47—81no—
141C23×C47—81no—
142C23×C47#12111yescharacteristic
143Q8×C47—21no—
144Q8×C47#10212yescomplemented
145D4×C47—22no—
146D4×C47—22no—
147D4×C47—21no—
148D4×C47—44no—
149D4×C47—42no—
150D4×C47—42no—
151D4×C47—82no—
152D4×C47—82no—
153D4×C47—81no—
154D4×C47—81no—
155D4×C47—81no—
156D4×C47—81no—
157D4×C47#10212yescomplemented
158C2×C4×C47—21no—
159C2×C4×C47—21no—
160C2×C4×C47—21no—
161C2×C4×C47—22no—
162C2×C4×C47—22no—
163C2×C4×C47—21no—
164C2×C4×C47—21no—
165C2×C4×C47—41no—
166C2×C4×C47—41no—
167C2×C4×C47—41no—
168C2×C4×C47—81no—
169C2×C4×C47C2×C22⋊C411yescharacteristic
170C2×C4×C47C4×D411yescharacteristic
171C2×C4×C47C4×D411yescharacteristic
172C2×C4×C47#12011yescharacteristic
173C2×C4×C47#12111yescharacteristic
174C2×C4×C47#12311yescharacteristic
175C2×C4×C47C2×D811yescharacteristic
176C2×C4×C47C2×SD1611yescharacteristic
177C2×C4×C47#13511yescharacteristic
178C2×C4×C47#13511yescharacteristic
179C2×Dic47—42no—
180C2×Dic47—81no—
181C2×Dic47—81no—
182C8×C47—42no—
183C47⋊C8—42no—
184C47⋊C8—41no—
185C47⋊C8—41no—
186C47⋊C8—81no—
187#9318—471nomaximal, Sylow
188C22×C47—21no—
189C22×C47—21no—
190C22×C47—22no—
191C22×C47—41no—
192C22×C47—41no—
193C22×C47—82no—
194C22×C47—82no—
195C22×C47—82no—
196C22×C47—162no—
197C22×C47—162no—
198C22×C47#38611yescharacteristic
199C22×C47#46511yescharacteristic
200C22×C47#46611yescharacteristic
201C4×C47—21no—
202C4×C47—22no—
203C4×C47—21no—
204C4×C47—21no—
205C4×C47—41no—
206C4×C47—41no—
207C4×C47C2×#10211yescharacteristic
208C4×C47#41611yescharacteristic
209C4×C47#45311yescharacteristic
210C4×C47#45411yescharacteristic
211C4×C47#46511yescharacteristic
212C4×C47#46611yescharacteristic
213Dic47—82no—
214#3038D4711yescharacteristic
215#2988—471no—
216#1982—471no—
217#1806—471no—
218#1545—471no—
219#1303—471no—
220#1292—471no—
221C2×C47—21no—
222C2×C47—21no—
223C2×C47—42no—
224C2×C47—162no—
225C2×C47#188311yescharacteristic
226C2×C47#189311yescharacteristic
227C2×C47#190311yescharacteristic
228C2×AGL(1,ℤ/8)—21no—
229C2×#127C2×D4711yescharacteristic
230C4×#41C2×D4711yescharacteristic
231#460—21no—
232#460Dic4712yescomplemented
233#459—21no—
234#459Dic4712yescomplemented
235C8×Q8—941no—
236C8×D4—942no—
237C8×D4—941no—
238#431—471no—
239#431—941no—
240#423—471no—
241#423C2×D4711yescharacteristic
242C2×C4⋊C8—471no—
243C2×C4⋊C8—941no—
244C2×#104—941no—
245#418—21no—
246C2×#102—942no—
247C4×C4⋊C4—471no—
248#339—941no—
249#331—941no—
250#331—941no—
251C47#931811yescharacteristic, complemented, Sylow
252C2×#41C22⋊D4711yescharacteristic
253C22×D4—21no—
254C22×D4—21no—
255C22×D4—21no—
256AGL(1,ℤ/8)—42no—
257AGL(1,ℤ/8)—42no—
258C2×SD16—22no—
259C2×D8—22no—
260C2×M16—21no—
261C2×M16—941no—
262C22×C8—472no—
263C22×C8—942no—
264#127—21no—
265#127—21no—
266#127—21no—
267#127C2×Dic4712yes—
268C4×Q8C2×Dic4711yescharacteristic
269C4×Q8C22⋊D4711yescharacteristic
270C4×D4—21no—
271C4×D4—21no—
272C4×D4C2×Dic4711yescharacteristic
273C4×D4C22⋊D4711yescharacteristic
274#117—21no—
275#117C22⋊D4711yescharacteristic
276C2×C4⋊C4—941no—
277C2×C42—21no—
278C2×C42—941no—
279C2×C42C22×D4711yescharacteristic
280C4⋊C8—21no—
281C4⋊C8—471no—
282C4⋊C8—471no—
283C4⋊C8—942no—
284C4⋊C8—941no—
285C4⋊C8—941no—
286C4⋊C8—941no—
287C4⋊C8—941no—
288C4⋊C8C2×Dic4712yes—
289#104—1882no—
290#102—22no—
291#102—22no—
292#102—1884no—
293C22⋊C8—942no—
294C22⋊C8—941no—
295C4×C8—942no—
296C4×C8—941no—
297#95—941no—
298C24—41no—
299#41—21no—
300#41—21no—
301#41#1062533412yes—
302#41#1062533512yes—
303C2×Q8C22⋊Dic4711yescharacteristic
304C2×D4—21no—
305C2×D4—22no—
306C2×D4—22no—
307C2×D4—22no—
308C2×D4—42no—
309C2×D4—42no—
310C2×D4—41no—
311C2×D4—41no—
312C2×D4—41no—
313C2×D4—41no—
314C2×D4—41no—
315C2×D4—41no—
316C2×D4—41no—
317C2×D4—82no—
318C2×D4C22⋊Dic4711yescharacteristic
319C22×C4—21no—
320C22×C4—21no—
321C22×C4—21no—
322C22×C4—1881no—
323C22×C4D4×D4711yescharacteristic
324C22×C4#1062535211yescharacteristic
325C22×C4C2×C22⋊D4711yescharacteristic
326SD16—44no—
327D8—44no—
328M16—41no—
329M16—41no—
330M16—1881no—
331M16—1881no—
332C2×C8—22no—
333C2×C8—941no—
334C2×C8—941no—
335C2×C8—942no—
336C2×C8—3762no—
337C2×C8—1881no—
338C2×C8—1882no—
339C4⋊C4—21no—
340C4⋊C4—21no—
341C4⋊C4—942no—
342C4⋊C4—1881no—
343C4⋊C4C22⋊Dic4711yescharacteristic
344C4⋊C4C22⋊Dic4711yescharacteristic
345C22⋊C4—21no—
346C22⋊C4—21no—
347C22⋊C4—21no—
348C42—21no—
349C42—21no—
350C42—21no—
351C42—1881no—
352C42—1881no—
353C42D4×D4711yescharacteristic
354C42#1062535211yescharacteristic
355C42C22×Dic4711yescharacteristic
356C42C2×C22⋊D4711yescharacteristic
357C23—21no—
358C23—42no—
359C23—82no—
360C23—82no—
361C23—81no—
362C23—81no—
363C23#1309399611yescharacteristic
364Q8—21no—
365Q8#1309394512yescomplemented
366D4—22no—
367D4—22no—
368D4—21no—
369D4—44no—
370D4—42no—
371D4—42no—
372D4—82no—
373D4—82no—
374D4—81no—
375D4—81no—
376D4—81no—
377D4—81no—
378D4#1309394512yescomplemented
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—22no—
383C2×C4—22no—
384C2×C4—21no—
385C2×C4—21no—
386C2×C4—41no—
387C2×C4—41no—
388C2×C4—41no—
389C2×C4—81no—
390C2×C4—3761no—
391C2×C4—3761no—
392C2×C4—1882no—
393C2×C4C2×C22⋊Dic4711yescharacteristic
394C2×C4D4×Dic4711yescharacteristic
395C2×C4D4×Dic4711yescharacteristic
396C2×C4#1309399211yescharacteristic
397C2×C4#1309399611yescharacteristic
398C2×C4#1309400511yescharacteristic
399C2×C4C2×#1062533411yescharacteristic
400C2×C4C2×#1062533511yescharacteristic
401C2×C4#1309404411yescharacteristic
402C2×C4#1309404411yescharacteristic
403C8—42no—
404C8—3761no—
405C8—1881no—
406C8—1882no—
407C8—1881no—
408C22—21no—
409C22—21no—
410C22—22no—
411C22—41no—
412C22—41no—
413C22—82no—
414C22—82no—
415C22—82no—
416C22—162no—
417C22—162no—
418C22#1568900311yescharacteristic
419C22#1568935011yescharacteristic
420C22#1568935711yescharacteristic
421C4—21no—
422C4—22no—
423C4—21no—
424C4—21no—
425C4—41no—
426C4—41no—
427C4—3762no—
428C4C2×#1309394511yescharacteristic
429C4#1568911911yescharacteristic
430C4#1568928911yescharacteristic
431C4#1568929411yescharacteristic
432C4#1568935011yescharacteristic
433C4#1568935711yescharacteristic
434C2—21no—
435C2—21no—
436C2—42no—
437C2—162no—
438C2#2024376611yescharacteristic
439C2#2024381711yescharacteristic
440C2#2024387011yescharacteristic
subgroupG/H
centerC22#15689003
commutatorC42×C47C22×C4
FrattiniC2×C42C22×D47
Fitting, prime radicalC47×#3038C2
socleC22×C47#386
cosocleC2×C42×C47C23
hypercenter#3038D47
hypocenterC47#9318
LCSG ▷ C42×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3038

Complements

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

NH ≅ G/Ntype
1. #20252499C2semidirect
2. #20244316C2semidirect
3. #20243408C2semidirect
4. #20242056C2semidirect
5. #20241289C2semidirect
6. #20241262C2semidirect
26. C47×#460C4semidirect
28. C47×#459C4semidirect
144. Q8×C47#102semidirect
157. D4×C47#102semidirect
232. #460Dic47semidirect
234. #459Dic47semidirect
251. C47#9318semidirect
365. Q8#13093945semidirect
378. D4#13093945semidirect

Maximal subgroups

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

subgroupindex
#202524992
#202443162
#202434082
#202420562
#202412892
#202412622
C47×#30382
#931847