Finite Group Databasebeta

← #26111999

Subgroups — #26111999

Order 12032.

738 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×C4⋊Dic47C211yesmaximal, characteristic
2#20243456C211yesmaximal, characteristic
3C2×#15688860C212yesmaximal
4#20241266C212yesmaximal
5C47×#2855C211yesmaximal, characteristic
6C22×C4⋊Dic47—21no—
7C22×C4×Dic47—21no—
8C2×C22⋊(C47⋊C8)C2212yes—
9C4×C4⋊Dic47—24no—
10C2×C47⋊#95C2212yes—
11#15688878—22no—
12#15688860—24no—
13C2×C22⋊Q8×C47C2211yescharacteristic
14C22×C42×C47C2211yescharacteristic
15C47×#391C412yescomplemented
16C47×#389C412yescomplemented
17C47×#387—22no—
18C47×#385—21no—
19C47×#381—21no—
20C47×#380C2211yescharacteristic
21C23×Dic47—22no—
22C22×C47⋊C8—22no—
23C2×C4⋊Dic47—24no—
24C2×C4⋊Dic47—28no—
25C2×C4×Dic47—24no—
26C2×C4×Dic47—42no—
27C2×C4×Dic47—42no—
28C22⋊(C47⋊C8)—22no—
29C22⋊(C47⋊C8)—22no—
30C47⋊#95—22no—
31C47⋊#95D414yes—
32C22×Q8×C47C2×C411yescharacteristic
33C23×C4×C47D411yescharacteristic
34C23×C4×C47D411yescharacteristic
35C23×C4×C47C2311yescharacteristic
36C22⋊Q8×C47—22no—
37C22⋊Q8×C47—22no—
38C2×C4⋊C4×C47—22no—
39C2×C4⋊C4×C47—21no—
40C2×C4⋊C4×C47C2×C411yescharacteristic
41C2×C4⋊C4×C47C2×C412yes—
42C2×C22⋊C4×C47—21no—
43C2×C22⋊C4×C47—21no—
44C2×C42×C47—22no—
45C2×C42×C47—22no—
46C2×C42×C47C2×C412yes—
47C2×C42×C47D412yes—
48#95×C47—22no—
49#95×C47—22no—
50#95×C47—22no—
51#95×C47—21no—
52C24×C47C2×D411yescharacteristic
53C2×Q8×C47—22no—
54C2×Q8×C47—42no—
55C22×C4×C47—21no—
56C22×C4×C47—21no—
57C22×C4×C47—21no—
58C22×C4×C47—21no—
59C22×C4×C47—21no—
60C22×C4×C47—21no—
61C22×C4×C47—24no—
62C22×C4×C47—21no—
63C22×C4×C47—22no—
64C22×C4×C47—21no—
65C22×C4×C47—42no—
66C22×C4×C47C22⋊C412yes—
67C22×C4×C47C22⋊C412yes—
68C22×C4×C47SD1612yes—
69C22×C4×C47Q1612yes—
70C22×C4×C47C22×C411yescharacteristic
71C22×C4×C47C2×D411yescharacteristic
72C22×C4×C47C2×D412yes—
73C22×C4×C47#4112yes—
74C22×Dic47—28no—
75C22×Dic47—24no—
76C22×Dic47—44no—
77C22×Dic47—42no—
78C22×Dic47—42no—
79C4⋊C4×C47—22no—
80C4⋊C4×C47—42no—
81C4⋊C4×C47—44no—
82C4⋊C4×C47—42no—
83C22⋊C4×C47—41no—
84C22⋊C4×C47—41no—
85C22⋊C4×C47—41no—
86C22⋊C4×C47—41no—
87C42×C47—24no—
88C42×C47—24no—
89C4⋊Dic47—216no—
90C4×Dic47—48no—
91C2×C47⋊C8—44no—
92C2×C47⋊C8—42no—
93C23×C47—22no—
94C23×C47—22no—
95C23×C47C2×C22⋊C411yescharacteristic
96C23×C47#12011yescharacteristic
97C23×C47#12311yescharacteristic
98C23×C47C2×SD1611yescharacteristic
99C23×C47C2×Q1611yescharacteristic
100C23×C47AGL(1,ℤ/8)11yescharacteristic
101C23×C47#13711yescharacteristic
102Q8×C47—82no—
103C2×C4×C47—24no—
104C2×C4×C47—24no—
105C2×C4×C47—22no—
106C2×C4×C47—22no—
107C2×C4×C47—22no—
108C2×C4×C47—22no—
109C2×C4×C47—22no—
110C2×C4×C47—22no—
111C2×C4×C47—22no—
112C2×C4×C47—22no—
113C2×C4×C47—22no—
114C2×C4×C47—22no—
115C2×C4×C47—42no—
116C2×C4×C47—42no—
117C2×C4×C47—41no—
118C2×C4×C47—41no—
119C2×C4×C47—42no—
120C2×C4×C47—42no—
121C2×C4×C47—42no—
122C2×C4×C47—42no—
123C2×C4×C47—42no—
124C2×C4×C47#10314yes—
125C2×C4×C47#10414yes—
126C2×C4×C47C4×D412yes—
127C2×C4×C47#12112yes—
128C2×Dic47—28no—
129C2×Dic47—44no—
130C2×Dic47—48no—
131C2×Dic47—48no—
132C2×Dic47—44no—
133C47⋊C8—82no—
134#9144—471nomaximal, Sylow
135C22×C47—21no—
136C22×C47—21no—
137C22×C47—21no—
138C22×C47—21no—
139C22×C47—22no—
140C22×C47—22no—
141C22×C47—22no—
142C22×C47—22no—
143C22×C47#38611yescharacteristic
144C22×C47#41012yes—
145C22×C47C2×#10311yescharacteristic
146C22×C47#41711yescharacteristic
147C22×C47C2×#10412yes—
148C22×C47#45011yescharacteristic
149C22×C47#45111yescharacteristic
150C22×C47#48311yescharacteristic
151C22×C47#48411yescharacteristic
152C4×C47—24no—
153C4×C47—24no—
154C4×C47—22no—
155C4×C47—22no—
156C4×C47—81no—
157C4×C47—81no—
158C4×C47—82no—
159Dic47—48no—
160#2855D4711yescharacteristic
161C2×C4×C4⋊C4—471no—
162#1817—471no—
163C2×#328—472no—
164#1294—472no—
165C2×C47—21no—
166C2×C47—21no—
167C2×C47—21no—
168C2×C47—21no—
169C2×C47#150312yes—
170C2×C47#150912yes—
171C2×C47#187211yescharacteristic
172C2×C47#189611yescharacteristic
173C2×C47#189911yescharacteristic
174C2×C22⋊Q8C2×D4711yescharacteristic
175C22×C4⋊C4—941no—
176C22×C42—941no—
177C22×C42C2×D4711yescharacteristic
178C2×C22⋊C8—472no—
179#391Dic4712yescomplemented
180#389Dic4712yescomplemented
181#387—22no—
182#385—21no—
183#381—21no—
184#380C2×D4711yescharacteristic
185C4×C4⋊C4—944no—
186C2×#95—472no—
187#336—942no—
188#328—944no—
189C47#914411yescharacteristic, complemented, Sylow
190C22×Q8C2×Dic4711yescharacteristic
191C23×C4—942no—
192C23×C4C22⋊D4711yescharacteristic
193C23×C4C22⋊D4711yescharacteristic
194C23×C4C22×D4711yescharacteristic
195C22×C8—942no—
196C22⋊Q8—22no—
197C22⋊Q8—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—948no—
201C2×C4⋊C4—944no—
202C2×C4⋊C4C2×Dic4711yescharacteristic
203C2×C4⋊C4C2×Dic4712yes—
204C2×C22⋊C4—21no—
205C2×C22⋊C4—21no—
206C2×C42—22no—
207C2×C42—22no—
208C2×C42—944no—
209C2×C42—1882no—
210C2×C42—1882no—
211C2×C42C2×Dic4712yes—
212C2×C42C22⋊D4712yes—
213C22⋊C8—942no—
214C22⋊C8—942no—
215#95—22no—
216#95—22no—
217#95—22no—
218#95—21no—
219#95—474no—
220#95—942no—
221C24C2×C22⋊D4711yescharacteristic
222C2×Q8—22no—
223C2×Q8—42no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—21no—
228C22×C4—21no—
229C22×C4—21no—
230C22×C4—24no—
231C22×C4—22no—
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—42no—
235C22×C4—948no—
236C22×C4—944no—
237C22×C4—1884no—
238C22×C4—1882no—
239C22×C4—1882no—
240C22×C4#1062533612yes—
241C22×C4#1062533712yes—
242C22×C4C22⋊Dic4712yes—
243C22×C4C22⋊Dic4712yes—
244C22×C4D4×D4712yes—
245C22×C4#1062535212yes—
246C22×C4C22×Dic4711yescharacteristic
247C22×C4C2×C22⋊D4711yescharacteristic
248C2×C8—1884no—
249C2×C8—1882no—
250C4⋊C4—22no—
251C4⋊C4—42no—
252C4⋊C4—44no—
253C4⋊C4—42no—
254C4⋊C4—9416no—
255C22⋊C4—41no—
256C22⋊C4—41no—
257C22⋊C4—41no—
258C22⋊C4—41no—
259C42—24no—
260C42—24no—
261C42—1888no—
262C23—22no—
263C23—22no—
264C23C2×C22⋊Dic4711yescharacteristic
265C23#1309399211yescharacteristic
266C23#1309400511yescharacteristic
267C23C2×#1062533611yescharacteristic
268C23C2×#1062533711yescharacteristic
269C23#1309404911yescharacteristic
270C23#1309405411yescharacteristic
271Q8—82no—
272C2×C4—24no—
273C2×C4—24no—
274C2×C4—22no—
275C2×C4—22no—
276C2×C4—22no—
277C2×C4—22no—
278C2×C4—22no—
279C2×C4—22no—
280C2×C4—22no—
281C2×C4—22no—
282C2×C4—22no—
283C2×C4—22no—
284C2×C4—42no—
285C2×C4—42no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—42no—
289C2×C4—42no—
290C2×C4—42no—
291C2×C4—42no—
292C2×C4—42no—
293C2×C4—948no—
294C2×C4—1884no—
295C2×C4—1888no—
296C2×C4—1888no—
297C2×C4—1884no—
298C2×C4#1309394814yes—
299C2×C4#1309395114yes—
300C2×C4D4×Dic4712yes—
301C2×C4#1309399612yes—
302C8—3762no—
303C22—21no—
304C22—21no—
305C22—21no—
306C22—21no—
307C22—22no—
308C22—22no—
309C22—22no—
310C22—22no—
311C22#1568900311yescharacteristic
312C22#1568909712yes—
313C22C2×#1309394811yescharacteristic
314C22#1568912411yescharacteristic
315C22C2×#1309395112yes—
316C22#1568927311yescharacteristic
317C22#1568927811yescharacteristic
318C22#1568945111yescharacteristic
319C22#1568945611yescharacteristic
320C4—24no—
321C4—22no—
322C4—22no—
323C4—24no—
324C4—81no—
325C4—81no—
326C4—82no—
327C4—1888no—
328C2—21no—
329C2—21no—
330C2—21no—
331C2—21no—
332C2#2024188712yes—
333C2#2024191312yes—
334C2#2024369611yescharacteristic
335C2#2024383211yescharacteristic
336C2#2024384711yescharacteristic
subgroupG/H
centerC23C2×C22⋊Dic47
commutatorC22×C4×C47C22×C4
FrattiniC23×C4C22×D47
FittingC47×#2855C2
socleC23×C47C2×C22⋊C4
prime radicalC24×C47C2×D4
cosocleC23×C4×C47C23
hypercenter#2855D47
hypocenterC47#9144
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C23 ◁ C22×C42 ◁ #2855

Complements

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

NH ≅ G/Ntype
15. C47×#391C4semidirect
16. C47×#389C4semidirect
179. #391Dic47semidirect
180. #389Dic47semidirect
189. C47#9144semidirect

Maximal subgroups

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

subgroupindex
C2×C4×C4⋊Dic472
#202434562
C2×#156888602
#202412662
C47×#28552
#914447