Finite Group Databasebeta

← #26112180

Subgroups — #26112180

Order 12032.

516 conjugacy classes of subgroups.

The subgroup lattice has 426 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20252054C211yesmaximal, characteristic, complemented
2#20242056C211yesmaximal, characteristic, complemented
3C2×#15688867C211yesmaximal, characteristic, complemented
4#20241775C211yesmaximal, characteristic, complemented
5#20241283C211yesmaximal, characteristic, complemented
6#20241262C211yesmaximal, characteristic, complemented
7C47×#3038C211yesmaximal, characteristic
8#15689184—22no—
9#15689180—21no—
10#15689176—21no—
11#15689176C2211yescharacteristic
12#15689147C2211yescharacteristic
13C2×C4⋊(C47⋊C8)C2211yescharacteristic
14C2×C4⋊(C47⋊C8)C2211yescharacteristic
15#15689090—21no—
16#15689079—21no—
17#15688867—21no—
18#15688867—21no—
19#15688867—22no—
20Q8⋊(C47⋊C8)—21no—
21D4⋊(C47⋊C8)—21no—
22C2×AGL(1,ℤ/8)×C47—21no—
23C2×#127×C47C2211yescharacteristic
24C4×#41×C47C2211yescharacteristic
25C47×#460—21no—
26C47×#460C412yes—
27C47×#459—21no—
28C47×#459C412yes—
29C47×#423C2211yescharacteristic
30C47×#418—21no—
31C2×#10625329—21no—
32C22×C47⋊C8—21no—
33C22×C47⋊C8—21no—
34C4⋊(C47⋊C8)—21no—
35C4⋊(C47⋊C8)—21no—
36C4⋊(C47⋊C8)—21no—
37C4⋊(C47⋊C8)—21no—
38C4⋊(C47⋊C8)—21no—
39C4⋊(C47⋊C8)D411yescharacteristic
40C4⋊(C47⋊C8)D411yescharacteristic
41C4⋊(C47⋊C8)D412yes—
42C22⋊(C47⋊C8)—22no—
43C22⋊(C47⋊C8)—22no—
44C22⋊(C47⋊C8)—21no—
45#13093935—22no—
46C4×C47⋊C8—22no—
47C4×C47⋊C8—21no—
48C2×#41×C47D411yescharacteristic
49C22×D4×C47—21no—
50C22×D4×C47—21no—
51C22×D4×C47—21no—
52AGL(1,ℤ/8)×C47—42no—
53AGL(1,ℤ/8)×C47—42no—
54C2×SD16×C47—22no—
55C2×D8×C47—22no—
56C2×M16×C47—21no—
57#127×C47—21no—
58#127×C47—21no—
59#127×C47—21no—
60#127×C47C2×C412yes—
61C4×Q8×C47C2×C411yescharacteristic
62C4×Q8×C47D411yescharacteristic
63C4×D4×C47—21no—
64C4×D4×C47—21no—
65C4×D4×C47C2×C411yescharacteristic
66C4×D4×C47D411yescharacteristic
67#117×C47—21no—
68#117×C47D411yescharacteristic
69C2×C42×C47—21no—
70C2×C42×C47C2311yescharacteristic
71C4⋊C8×C47—21no—
72C4⋊C8×C47C2×C412yes—
73#102×C47—22no—
74#102×C47—22no—
75C24×C47—41no—
76#41×C47—21no—
77#41×C47—21no—
78#41×C47—21no—
79#41×C47—21no—
80C2×Q8×C47C22⋊C411yescharacteristic
81C2×D4×C47—21no—
82C2×D4×C47—22no—
83C2×D4×C47—22no—
84C2×D4×C47—22no—
85C2×D4×C47—41no—
86C2×D4×C47—41no—
87C2×D4×C47—41no—
88C2×D4×C47—41no—
89C2×D4×C47—41no—
90C2×D4×C47—41no—
91C2×D4×C47—41no—
92C2×D4×C47—81no—
93C2×D4×C47—81no—
94C2×D4×C47—82no—
95C2×D4×C47C22⋊C411yescharacteristic
96C22×C4×C47—21no—
97C22×C4×C47—21no—
98C22×C4×C47—21no—
99C22×C4×C47C2×D411yescharacteristic
100C22×C4×C47#4111yescharacteristic
101C22×C4×C47#4111yescharacteristic
102SD16×C47—44no—
103D8×C47—44no—
104M16×C47—41no—
105M16×C47—41no—
106C2×C8×C47—22no—
107C4⋊C4×C47—21no—
108C4⋊C4×C47—21no—
109C4⋊C4×C47C22⋊C411yescharacteristic
110C4⋊C4×C47C22⋊C411yescharacteristic
111C22⋊C4×C47—21no—
112C22⋊C4×C47—21no—
113C22⋊C4×C47—21no—
114C42×C47—21no—
115C42×C47—21no—
116C42×C47—21no—
117C42×C47C22×C411yescharacteristic
118C42×C47C2×D411yescharacteristic
119C42×C47C2×D411yescharacteristic
120C42×C47C2×D411yescharacteristic
121#10625329—41no—
122#10625329—41no—
123C2×C47⋊C8—21no—
124C2×C47⋊C8—22no—
125C2×C47⋊C8—21no—
126C2×C47⋊C8—41no—
127C2×C47⋊C8—41no—
128C2×C47⋊C8—41no—
129C2×C47⋊C8—41no—
130C2×C47⋊C8—81no—
131C23×C47—21no—
132C23×C47—42no—
133C23×C47—82no—
134C23×C47—82no—
135C23×C47—81no—
136C23×C47—81no—
137C23×C47#12311yescharacteristic
138Q8×C47—21no—
139Q8×C47—21no—
140D4×C47—21no—
141D4×C47—21no—
142D4×C47—41no—
143D4×C47—41no—
144D4×C47—42no—
145D4×C47—42no—
146D4×C47—82no—
147D4×C47—82no—
148D4×C47—82no—
149D4×C47—81no—
150D4×C47—81no—
151D4×C47—81no—
152D4×C47—81no—
153C2×C4×C47—21no—
154C2×C4×C47—21no—
155C2×C4×C47—21no—
156C2×C4×C47—21no—
157C2×C4×C47—21no—
158C2×C4×C47—41no—
159C2×C4×C47—41no—
160C2×C4×C47—41no—
161C2×C4×C47—41no—
162C2×C4×C47—41no—
163C2×C4×C47—81no—
164C2×C4×C47C2×C22⋊C411yescharacteristic
165C2×C4×C47C4×D411yescharacteristic
166C2×C4×C47C4×D411yescharacteristic
167C2×C4×C47#12011yescharacteristic
168C2×C4×C47#12111yescharacteristic
169C2×C4×C47#12111yescharacteristic
170C2×C4×C47#13511yescharacteristic
171C2×C4×C47#13511yescharacteristic
172C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
173C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
174C8×C47—42no—
175C47⋊C8—41no—
176C47⋊C8—41no—
177C47⋊C8—42no—
178C47⋊C8—81no—
179C47⋊C8—81no—
180#9174—471nomaximal, Sylow
181C22×C47—21no—
182C22×C47—21no—
183C22×C47—41no—
184C22×C47—41no—
185C22×C47—41no—
186C22×C47—82no—
187C22×C47—82no—
188C22×C47—82no—
189C22×C47—162no—
190C22×C47—162no—
191C22×C47#38611yescharacteristic
192C22×C47#48211yescharacteristic
193C22×C47#48211yescharacteristic
194C4×C47—21no—
195C4×C47—21no—
196C4×C47—21no—
197C4×C47—41no—
198C4×C47—41no—
199C4×C47—41no—
200C4×C47#41611yescharacteristic
201C4×C47#41811yescharacteristic
202C4×C47#44911yescharacteristic
203C4×C47#44911yescharacteristic
204C4×C47#47111yescharacteristic
205C4×C47#47211yescharacteristic
206#3038D4711yescharacteristic
207#2925—471no—
208#1545—471no—
209C2×#331—471no—
210#1477—471no—
211#1301—471no—
212#1292—471no—
213C2×C47—21no—
214C2×C47—21no—
215C2×C47—81no—
216C2×C47—162no—
217C2×C47#187511yescharacteristic
218C2×C47#190311yescharacteristic
219C2×C47#190711yescharacteristic
220C2×AGL(1,ℤ/8)—21no—
221C2×#127C2×D4711yescharacteristic
222C4×#41C2×D4711yescharacteristic
223#460—21no—
224#460Dic4712yes—
225#459—21no—
226#459Dic4712yes—
227#433—942no—
228#432—941no—
229#431—471no—
230#431—941no—
231#423—471no—
232#423C2×D4711yescharacteristic
233C2×C4⋊C8—471no—
234C2×C4⋊C8—471no—
235#418—21no—
236#408—941no—
237#405—941no—
238#331—941no—
239#331—941no—
240#331—942no—
241Q8⋊C8—941no—
242D4⋊C8—941no—
243C47#917411yescharacteristic, complemented, Sylow
244C2×#41C22⋊D4711yescharacteristic
245C22×D4—21no—
246C22×D4—21no—
247C22×D4—21no—
248AGL(1,ℤ/8)—42no—
249AGL(1,ℤ/8)—42no—
250C2×SD16—22no—
251C2×D8—22no—
252C2×M16—21no—
253C2×M16—941no—
254C22×C8—941no—
255C22×C8—941no—
256#127—21no—
257#127—21no—
258#127—21no—
259#127C2×Dic4712yes—
260C4×Q8C2×Dic4711yescharacteristic
261C4×Q8C22⋊D4711yescharacteristic
262C4×D4—21no—
263C4×D4—21no—
264C4×D4C2×Dic4711yescharacteristic
265C4×D4C22⋊D4711yescharacteristic
266#117—21no—
267#117C22⋊D4711yescharacteristic
268C2×C42—21no—
269C2×C42C22×D4711yescharacteristic
270C4⋊C8—21no—
271C4⋊C8—471no—
272C4⋊C8—471no—
273C4⋊C8—472no—
274C4⋊C8—941no—
275C4⋊C8—941no—
276C4⋊C8—941no—
277C4⋊C8—941no—
278C4⋊C8—941no—
279C4⋊C8C2×Dic4712yes—
280#102—22no—
281#102—22no—
282C22⋊C8—942no—
283C22⋊C8—942no—
284C22⋊C8—941no—
285#97—942no—
286C4×C8—942no—
287C4×C8—941no—
288C24—41no—
289#41—21no—
290#41—21no—
291#41—21no—
292#41—21no—
293C2×Q8C22⋊Dic4711yescharacteristic
294C2×D4—21no—
295C2×D4—22no—
296C2×D4—22no—
297C2×D4—22no—
298C2×D4—41no—
299C2×D4—41no—
300C2×D4—41no—
301C2×D4—41no—
302C2×D4—41no—
303C2×D4—41no—
304C2×D4—41no—
305C2×D4—81no—
306C2×D4—81no—
307C2×D4—82no—
308C2×D4C22⋊Dic4711yescharacteristic
309C22×C4—21no—
310C22×C4—21no—
311C22×C4—21no—
312C22×C4#1062535211yescharacteristic
313C22×C4#1062535211yescharacteristic
314C22×C4C2×C22⋊D4711yescharacteristic
315SD16—44no—
316D8—44no—
317M16—41no—
318M16—41no—
319M16—1881no—
320M16—1881no—
321C2×C8—22no—
322C2×C8—941no—
323C2×C8—942no—
324C2×C8—941no—
325C2×C8—3761no—
326C2×C8—1881no—
327C2×C8—1881no—
328C2×C8—1881no—
329C2×C8—1881no—
330C4⋊C4—21no—
331C4⋊C4—21no—
332C4⋊C4C22⋊Dic4711yescharacteristic
333C4⋊C4C22⋊Dic4711yescharacteristic
334C22⋊C4—21no—
335C22⋊C4—21no—
336C22⋊C4—21no—
337C42—21no—
338C42—21no—
339C42—21no—
340C42D4×D4711yescharacteristic
341C42D4×D4711yescharacteristic
342C42C22×Dic4711yescharacteristic
343C42C2×C22⋊D4711yescharacteristic
344C23—21no—
345C23—42no—
346C23—82no—
347C23—82no—
348C23—81no—
349C23—81no—
350C23#1309400511yescharacteristic
351Q8—21no—
352Q8—21no—
353D4—21no—
354D4—21no—
355D4—41no—
356D4—41no—
357D4—42no—
358D4—42no—
359D4—82no—
360D4—82no—
361D4—82no—
362D4—81no—
363D4—81no—
364D4—81no—
365D4—81no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—41no—
374C2×C4—41no—
375C2×C4—41no—
376C2×C4—81no—
377C2×C4C2×C22⋊Dic4711yescharacteristic
378C2×C4D4×Dic4711yescharacteristic
379C2×C4D4×Dic4711yescharacteristic
380C2×C4#1309399211yescharacteristic
381C2×C4#1309399611yescharacteristic
382C2×C4#1309399611yescharacteristic
383C2×C4#1309404411yescharacteristic
384C2×C4#1309404411yescharacteristic
385C2×C4#1309405011yescharacteristic
386C2×C4#1309405011yescharacteristic
387C8—42no—
388C8—3761no—
389C8—3761no—
390C8—1881no—
391C8—1881no—
392C8—1882no—
393C22—21no—
394C22—21no—
395C22—41no—
396C22—41no—
397C22—41no—
398C22—82no—
399C22—82no—
400C22—82no—
401C22—162no—
402C22—162no—
403C22#1568900311yescharacteristic
404C22#1568944411yescharacteristic
405C22#1568944411yescharacteristic
406C4—21no—
407C4—21no—
408C4—21no—
409C4—41no—
410C4—41no—
411C4—41no—
412C4#1568911911yescharacteristic
413C4#1568912811yescharacteristic
414C4#1568926611yescharacteristic
415C4#1568926611yescharacteristic
416C4#1568938411yescharacteristic
417C4#1568939111yescharacteristic
418C2—21no—
419C2—21no—
420C2—81no—
421C2—162no—
422C2#2024371111yescharacteristic
423C2#2024387011yescharacteristic
424C2#2024389811yescharacteristic
subgroupG/H
centerC22#15689003
commutatorC42×C47C22×C4
FrattiniC2×C42C22×D47
Fitting, prime radicalC47×#3038C2
socleC22×C47#386
cosocleC2×C42×C47C23
hypercenter#3038D47
hypocenterC47#9174
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. #20252054C2semidirect
2. #20242056C2semidirect
3. C2×#15688867C2semidirect
4. #20241775C2semidirect
5. #20241283C2semidirect
6. #20241262C2semidirect
243. C47#9174semidirect

Maximal subgroups

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

subgroupindex
#202520542
#202420562
C2×#156888672
#202417752
#202412832
#202412622
C47×#30382
#917447