Finite Group Databasebeta

← #26109717

Subgroups — #26109717

Order 12032.

514 conjugacy classes of subgroups.

The subgroup lattice has 422 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20245793C211yesmaximal, characteristic
2#20243365C211yesmaximal, characteristic
3#20243365C211yesmaximal, characteristic
4#20241272C211yesmaximal, characteristic, complemented
5#20241272C211yesmaximal, characteristic, complemented
6#20241271C211yesmaximal, characteristic, complemented
7C47×#1296C211yesmaximal, characteristic, complemented
8C2×#13093982C412yes—
9C2×C42×D47C2211yescharacteristic
10C2×C4⋊(C47⋊C8)C2211yescharacteristic
11C2×C4⋊(C47⋊C8)C2211yescharacteristic
12C2×(C2×D47)⋊C8—21no—
13C2×(C2×D47)⋊C8—21no—
14#15688993C2211yescharacteristic
15#15688988C412yes—
16#15688980—21no—
17#15688980—21no—
18C4⋊C4×Dic47—21no—
19C4⋊C4×Dic47—21no—
20#15688877—21no—
21#15688877—21no—
22C2×C4⋊C8×C47C2211yescharacteristic
23C2×C4⋊C8×C47C2211yescharacteristic
24C47×#384C2211yescharacteristic
25C22×C4×D47C2×C411yescharacteristic
26C22×C4×D47D411yescharacteristic
27C22×C4×D47D411yescharacteristic
28C22×C47⋊C8—21no—
29C22×C47⋊C8—21no—
30#13093982—44no—
31C2×C4⋊Dic47—21no—
32C2×C4⋊Dic47—21no—
33C2×C4⋊Dic47C2×C412yes—
34C2×(C2×D47)⋊C4—22no—
35C2×C4×Dic47—22no—
36C2×C4×Dic47C2×C411yescharacteristic
37C2×C4×Dic47D411yescharacteristic
38C2×C4×Dic47D411yescharacteristic
39C42×D47—21no—
40C42×D47—21no—
41C42×D47—21no—
42C42×D47—21no—
43C4⋊(C47⋊C8)—21no—
44C4⋊(C47⋊C8)—21no—
45C4⋊(C47⋊C8)—21no—
46C4⋊(C47⋊C8)—21no—
47(C2×D47)⋊C8—42no—
48(C2×D47)⋊C8—42no—
49C47⋊#95—22no—
50C47⋊#95—22no—
51C22×C8×C47—21no—
52C22×C8×C47—21no—
53C2×C4⋊C4×C47—21no—
54C2×C4⋊C4×C47—21no—
55C2×C4⋊C4×C47C2×C412yes—
56C2×C42×C47C2311yescharacteristic
57C4⋊C8×C47—21no—
58C4⋊C8×C47—21no—
59C4⋊C8×C47—21no—
60C4⋊C8×C47—21no—
61C23×D47C22⋊C411yescharacteristic
62C22×C4×C47—22no—
63C22×C4×C47C22×C411yescharacteristic
64C22×C4×C47C2×D411yescharacteristic
65C22×C4×C47C2×D411yescharacteristic
66C22×Dic47—22no—
67C22×Dic47C22⋊C411yescharacteristic
68C22×Dic47C22⋊C411yescharacteristic
69C22×Dic47C22⋊C411yescharacteristic
70C2×C4×D47—22no—
71C2×C4×D47—21no—
72C2×C4×D47—21no—
73C2×C4×D47—22no—
74C2×C4×D47—21no—
75C2×C4×D47—21no—
76C2×C4×D47—21no—
77C2×C4×D47—21no—
78C2×C4×D47—21no—
79C2×C4×D47—21no—
80C2×C4×D47—21no—
81C2×C4×D47—21no—
82C2×C4×D47—21no—
83C2×C4×D47—21no—
84C2×C4×D47—21no—
85C2×C4×D47—21no—
86C2×C8×C47—41no—
87C2×C8×C47—41no—
88C2×C8×C47—41no—
89C2×C8×C47—41no—
90C2×C8×C47—41no—
91C2×C8×C47—41no—
92C4⋊C4×C47—42no—
93C4⋊C4×C47—41no—
94C4⋊C4×C47—41no—
95C4⋊C4×C47—41no—
96C4⋊C4×C47—41no—
97C42×C47#4111yescharacteristic
98C42×C47#4111yescharacteristic
99C42×C47#4111yescharacteristic
100C42×C47#4111yescharacteristic
101(C2×D47)⋊C4—44no—
102C4⋊Dic47—41no—
103C4⋊Dic47—41no—
104C4⋊Dic47—41no—
105C4⋊Dic47—41no—
106C4⋊Dic47—42no—
107C4×Dic47—21no—
108C4×Dic47—21no—
109C4×Dic47—21no—
110C4×Dic47—21no—
111C4×Dic47—21no—
112C4×Dic47—21no—
113C4×Dic47—44no—
114C2×C47⋊C8—41no—
115C2×C47⋊C8—41no—
116C2×C47⋊C8—41no—
117C2×C47⋊C8—41no—
118C2×C47⋊C8—41no—
119C2×C47⋊C8—41no—
120C23×C47C2×C22⋊C411yescharacteristic
121C22×D47—21no—
122C22×D47—21no—
123C22×D47—21no—
124C22×D47—21no—
125C22×D47—21no—
126C22×D47—21no—
127C22×D47#10012yes—
128C2×C4×C47—21no—
129C2×C4×C47—21no—
130C2×C4×C47—42no—
131C2×C4×C47—42no—
132C2×C4×C47—42no—
133C2×C4×C47#11711yescharacteristic
134C2×C4×C47#11711yescharacteristic
135C2×C4×C47#12311yescharacteristic
136C2×C4×C47#12311yescharacteristic
137C2×C4×C47#12311yescharacteristic
138C2×C4×C47#12311yescharacteristic
139C2×C4×C47#13511yescharacteristic
140C2×C4×C47#13511yescharacteristic
141C2×C4×C47#13511yescharacteristic
142C2×C4×C47#13511yescharacteristic
143C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
144C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
145C2×C4×C47#13711yescharacteristic
146C2×C4×C47#13711yescharacteristic
147C2×Dic47—21no—
148C2×Dic47—21no—
149C2×Dic47—21no—
150C2×Dic47—21no—
151C2×Dic47—21no—
152C2×Dic47—21no—
153C2×Dic47—21no—
154C2×Dic47—21no—
155C2×Dic47—42no—
156C2×Dic47—42no—
157C2×Dic47—42no—
158C2×Dic47#10112yes—
159C4×D47—24no—
160C4×D47—24no—
161C4×D47—41no—
162C4×D47—41no—
163C4×D47—41no—
164C4×D47—41no—
165C4×D47—41no—
166C4×D47—41no—
167C4×D47—41no—
168C4×D47—41no—
169C8×C47—81no—
170C8×C47—81no—
171C47⋊C8—81no—
172C47⋊C8—81no—
173#8729—471nomaximal, Sylow
174C22×C47#38111yescharacteristic
175C22×C47C2×#10011yescharacteristic
176C22×C47C2×#10111yescharacteristic
177C22×C47#41611yescharacteristic
178C22×C47#41611yescharacteristic
179C22×C47#41711yescharacteristic
180C22×C47#41711yescharacteristic
181C2×D47—22no—
182C2×D47—22no—
183C2×D47—22no—
184C2×D47—41no—
185C2×D47—41no—
186C2×D47—41no—
187C2×D47—41no—
188C4×C47—21no—
189C4×C47—21no—
190C4×C47—82no—
191C4×C47#48211yescharacteristic
192C4×C47#48211yescharacteristic
193C4×C47#48211yescharacteristic
194C4×C47#48211yescharacteristic
195C4×C47#48511yescharacteristic
196C4×C47#48511yescharacteristic
197C4×C47#48511yescharacteristic
198C4×C47#48511yescharacteristic
199Dic47—22no—
200Dic47—41no—
201Dic47—41no—
202Dic47—82no—
203#2285—471no—
204#1797—471no—
205#1797—471no—
206#1296—471no—
207#1296—471no—
208#1296—471no—
209#1296D4711yescharacteristic, complemented
210C2×C47#154411yescharacteristic
211C2×C47#155111yescharacteristic
212C2×C47#155111yescharacteristic
213C2×C47#156111yescharacteristic
214C2×C47#182111yescharacteristic
215C2×C47#182511yescharacteristic
216C2×C47#182511yescharacteristic
217D47—42no—
218C2×#117—472no—
219C22×C42—471no—
220C2×C4⋊C8—471no—
221C2×C4⋊C8—471no—
222C2×C4⋊C8C2×D4711yescharacteristic
223C2×C4⋊C8C2×D4711yescharacteristic
224C2×C22⋊C8—941no—
225C2×C22⋊C8—941no—
226#384—471no—
227#384C2×D4711yescharacteristic
228#382—472no—
229#380—941no—
230#380—941no—
231C4×C4⋊C4—941no—
232C4×C4⋊C4—941no—
233#336—941no—
234#336—941no—
235C47#872911yescharacteristic, complemented, Sylow
236C23×C4—471no—
237C23×C4—471no—
238C23×C4—471no—
239C22×C8—21no—
240C22×C8—21no—
241C22×C8—941no—
242C22×C8—941no—
243#117—1884no—
244C2×C4⋊C4—21no—
245C2×C4⋊C4—21no—
246C2×C4⋊C4—472no—
247C2×C4⋊C4—941no—
248C2×C4⋊C4—941no—
249C2×C4⋊C4C4×D4712yes—
250C2×C22⋊C4—942no—
251C2×C42—471no—
252C2×C42—471no—
253C2×C42—471no—
254C2×C42—941no—
255C2×C42—941no—
256C2×C42—941no—
257C2×C42—941no—
258C2×C42—942no—
259C2×C42C22×D4711yescharacteristic
260C4⋊C8—21no—
261C4⋊C8—21no—
262C4⋊C8—21no—
263C4⋊C8—21no—
264C4⋊C8—941no—
265C4⋊C8—941no—
266C4⋊C8—941no—
267C4⋊C8—941no—
268C22⋊C8—1882no—
269C22⋊C8—1882no—
270#95—942no—
271#95—942no—
272C24—471no—
273C22×C4—22no—
274C22×C4—471no—
275C22×C4—471no—
276C22×C4—471no—
277C22×C4—942no—
278C22×C4—942no—
279C22×C4—941no—
280C22×C4—941no—
281C22×C4—941no—
282C22×C4—941no—
283C22×C4—941no—
284C22×C4—941no—
285C22×C4—941no—
286C22×C4—941no—
287C22×C4—941no—
288C22×C4—941no—
289C22×C4—941no—
290C22×C4—941no—
291C22×C4—941no—
292C22×C4—941no—
293C22×C4—942no—
294C22×C4C2×C4×D4711yescharacteristic
295C22×C4D4×D4711yescharacteristic
296C22×C4D4×D4711yescharacteristic
297C2×C8—41no—
298C2×C8—41no—
299C2×C8—41no—
300C2×C8—41no—
301C2×C8—41no—
302C2×C8—41no—
303C2×C8—1881no—
304C2×C8—1881no—
305C2×C8—1881no—
306C2×C8—1881no—
307C2×C8—1881no—
308C2×C8—1881no—
309C4⋊C4—42no—
310C4⋊C4—41no—
311C4⋊C4—41no—
312C4⋊C4—41no—
313C4⋊C4—41no—
314C4⋊C4—1882no—
315C4⋊C4—1881no—
316C4⋊C4—1881no—
317C4⋊C4—1881no—
318C4⋊C4—1881no—
319C22⋊C4—1884no—
320C42—941no—
321C42—941no—
322C42—941no—
323C42—941no—
324C42—941no—
325C42—941no—
326C42—1884no—
327C42#1062535011yescharacteristic
328C42#1062535011yescharacteristic
329C42#1062535211yescharacteristic
330C42#1062535411yescharacteristic
331C23—472no—
332C23—941no—
333C23—941no—
334C23—941no—
335C23—941no—
336C23—941no—
337C23—941no—
338C23C22⋊C4×D4711yescharacteristic
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—42no—
342C2×C4—42no—
343C2×C4—42no—
344C2×C4—472no—
345C2×C4—944no—
346C2×C4—944no—
347C2×C4—941no—
348C2×C4—941no—
349C2×C4—941no—
350C2×C4—941no—
351C2×C4—941no—
352C2×C4—941no—
353C2×C4—941no—
354C2×C4—941no—
355C2×C4—1882no—
356C2×C4—1882no—
357C2×C4—1882no—
358C2×C4—1881no—
359C2×C4—1881no—
360C2×C4—1881no—
361C2×C4—1881no—
362C2×C4—1881no—
363C2×C4—1881no—
364C2×C4—1881no—
365C2×C4—1881no—
366C2×C4#1309398111yescharacteristic
367C2×C4#1309398211yescharacteristic
368C2×C4#1309400611yescharacteristic
369C2×C4#1309400611yescharacteristic
370C2×C4#1309400911yescharacteristic
371C2×C4#1309400911yescharacteristic
372C2×C4#1309404311yescharacteristic
373C2×C4#1309404511yescharacteristic
374C2×C4#1309404511yescharacteristic
375C2×C4#1309404711yescharacteristic
376C2×C4#1309404811yescharacteristic
377C2×C4#1309405111yescharacteristic
378C2×C4#1309405311yescharacteristic
379C2×C4#1309405611yescharacteristic
380C8—81no—
381C8—81no—
382C8—3761no—
383C8—3761no—
384C22—942no—
385C22—942no—
386C22—942no—
387C22—1881no—
388C22—1881no—
389C22—1881no—
390C22—1881no—
391C22#1568898511yescharacteristic
392C22#100×D4711yescharacteristic
393C22#101×D4711yescharacteristic
394C22#1568911611yescharacteristic
395C22#1568911811yescharacteristic
396C22#1568912111yescharacteristic
397C22#1568912311yescharacteristic
398C4—21no—
399C4—21no—
400C4—82no—
401C4—942no—
402C4—3762no—
403C4—1881no—
404C4—1881no—
405C4#1568944511yescharacteristic
406C4#1568944611yescharacteristic
407C4#1568944911yescharacteristic
408C4#1568945011yescharacteristic
409C4#1568946211yescharacteristic
410C4#1568946311yescharacteristic
411C4#1568946611yescharacteristic
412C4#1568946711yescharacteristic
413C2—1882no—
414C2#2024205111yescharacteristic
415C2#2024208011yescharacteristic
416C2#2024208211yescharacteristic
417C2#2024212411yescharacteristic
418C2#2024346711yescharacteristic
419C2#2024348111yescharacteristic
420C2#2024348211yescharacteristic
subgroupG/H
centerC23C22⋊C4×D47
commutatorC22×C4×C47C22×C4
FrattiniC2×C42C22×D47
FittingC47×#1296C2
socleC23×C47C2×C22⋊C4
prime radicalC23×D47C22⋊C4
cosocleC2×C42×C47C23
hypercenter#1296D47
hypocenterC47#8729
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1296

Complements

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

NH ≅ G/Ntype
4. #20241272C2semidirect
5. #20241272C2semidirect
6. #20241271C2semidirect
7. C47×#1296C2semidirect
209. #1296D47semidirect
235. C47#8729semidirect

Maximal subgroups

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

subgroupindex
#202457932
#202433652
#202412722
#202412712
C47×#12962
#872947