Finite Group Databasebeta

← #26106907

Subgroups — #26106907

Order 12032.

578 conjugacy classes of subgroups.

The subgroup lattice has 330 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×Q8×Dic47C211yesmaximal, characteristic
2#20243243C211yesmaximal, characteristic
3#20243133C211yesmaximal, characteristic
4C2×Q8⋊(C47⋊C8)C211yesmaximal, characteristic
5#20241228C211yesmaximal, characteristic
6#20241227C211yesmaximal, characteristic
7C2×C47×Q8⋊C8C211yesmaximal, characteristic
8C2×Q8×Dic47C412yes—
9C2×C4⋊(C47⋊C8)C2211yescharacteristic
10C2×#13093934—21no—
11C2×C4×C47⋊C8C2211yescharacteristic
12C4⋊C4×Dic47—22no—
13C4⋊C4×Dic47C412yes—
14C4×C4⋊Dic47—21no—
15C4×C4⋊Dic47C2211yescharacteristic
16C42×Dic47—21no—
17C42×Dic47C2211yescharacteristic
18#15688877—21no—
19#15688877—22no—
20Q8⋊(C47⋊C8)C414yescomplemented
21C2×C4×Q8×C47C2211yescharacteristic
22C2×C4⋊C8×C47C2211yescharacteristic
23C2×C4×C8×C47C2211yescharacteristic
24C47×Q8⋊C8C414yescomplemented
25C22×C47⋊C8—21no—
26C22×C47⋊C8—21no—
27Q8×Dic47—24no—
28Q8×Dic47—24no—
29C2×C4⋊Dic47—22no—
30C2×C4⋊Dic47C2×C412yes—
31C2×C4×Dic47—22no—
32C2×C4×Dic47—21no—
33C2×C4×Dic47—22no—
34C2×C4×Dic47C2×C412yes—
35C2×C4×Dic47D411yescharacteristic
36C2×C4×Dic47D412yes—
37C2×C4×Dic47D411yescharacteristic
38C4⋊(C47⋊C8)—21no—
39C4⋊(C47⋊C8)C2×C412yes—
40#13093934—42no—
41C4×C47⋊C8—21no—
42C4×C47⋊C8C2×C412yes—
43C47⋊#95—21no—
44C47⋊#95—22no—
45C22×Q8×C47C2×C411yescharacteristic
46C22×C8×C47—21no—
47C22×C8×C47—21no—
48C4×Q8×C47—21no—
49C4×Q8×C47—21no—
50C4×Q8×C47C2×C412yes—
51C4×Q8×C47C2×C412yes—
52C2×C4⋊C4×C47—21no—
53C2×C4⋊C4×C47C2×C411yescharacteristic
54C2×C42×C47—21no—
55C2×C42×C47C2311yescharacteristic
56C4⋊C8×C47—21no—
57C4⋊C8×C47C2×C412yes—
58C4×C8×C47—21no—
59C4×C8×C47C2×C412yes—
60C2×Q8×C47—21no—
61C2×Q8×C47—22no—
62C2×Q8×C47C4212yes—
63C2×Q8×C47M1614yes—
64C22×C4×C47—21no—
65C22×C4×C47—21no—
66C22×C4×C47C22×C411yescharacteristic
67C22×C4×C47C2×D411yescharacteristic
68C22×C4×C47C2×D411yescharacteristic
69C22×Dic47—22no—
70C22×Dic47C22⋊C412yes—
71C22×Dic47C22⋊C412yes—
72C2×C8×C47—22no—
73C2×C8×C47—22no—
74C2×C8×C47—42no—
75C2×C8×C47—41no—
76C2×C8×C47—41no—
77C4⋊C4×C47—21no—
78C4⋊C4×C47—22no—
79C4⋊C4×C47—22no—
80C4⋊C4×C47C4212yes—
81C42×C47—22no—
82C42×C47—22no—
83C42×C47C22×C411yescharacteristic
84C42×C47C22×C411yescharacteristic
85C42×C47#4111yescharacteristic
86C42×C47#4111yescharacteristic
87C4⋊Dic47—28no—
88C4⋊Dic47—22no—
89C4⋊Dic47M1614yes—
90C4×Dic47—28no—
91C4×Dic47—22no—
92C4×Dic47—22no—
93C4×Dic47—21no—
94C4×Dic47—21no—
95C4×Dic47—42no—
96C4×Dic47—44no—
97C4×Dic47—41no—
98C4×Dic47—41no—
99C4×Dic47SD1612yes—
100C4×Dic47Q1612yes—
101C2×C47⋊C8—22no—
102C2×C47⋊C8—22no—
103C2×C47⋊C8—42no—
104C2×C47⋊C8—41no—
105C2×C47⋊C8—41no—
106C23×C47C2×C22⋊C411yescharacteristic
107Q8×C47—22no—
108Q8×C47—24no—
109Q8×C47#9714yescomplemented
110C2×C4×C47—22no—
111C2×C4×C47—24no—
112C2×C4×C47—22no—
113C2×C4×C47—22no—
114C2×C4×C47—42no—
115C2×C4×C47C2×C4211yescharacteristic
116C2×C4×C47#11711yescharacteristic
117C2×C4×C47C4×D411yescharacteristic
118C2×C4×C47C4×D411yescharacteristic
119C2×C4×C47C4×D411yescharacteristic
120C2×C4×C47C4×D411yescharacteristic
121C2×C4×C47C2×M1612yes—
122C2×C4×C47C2×M1612yes—
123C2×C4×C47C2×SD1611yescharacteristic
124C2×C4×C47C2×Q1611yescharacteristic
125C2×C4×C47#13511yescharacteristic
126C2×C4×C47#13511yescharacteristic
127C2×Dic47—22no—
128C2×Dic47—22no—
129C2×Dic47—22no—
130C2×Dic47—24no—
131C2×Dic47—22no—
132C2×Dic47—22no—
133C2×Dic47—42no—
134C2×Dic47—42no—
135C2×Dic47#10314yes—
136C8×C47—42no—
137C8×C47—42no—
138C47⋊C8—42no—
139C47⋊C8—42no—
140#8175—471nomaximal, Sylow
141C22×C47C4×C22⋊C411yescharacteristic
142C22×C47#40712yes—
143C22×C47C2×#10311yescharacteristic
144C22×C47#41611yescharacteristic
145C22×C47#42112yes—
146C4×C47—24no—
147C4×C47—21no—
148C4×C47—21no—
149C4×C47—42no—
150C4×C47C2×#9711yescharacteristic
151C4×C47C4×M1611yescharacteristic
152C4×C47#43212yes—
153C4×C47C4×SD1611yescharacteristic
154C4×C47C4×SD1611yescharacteristic
155C4×C47C4×Q1611yescharacteristic
156C4×C47C4×Q1611yescharacteristic
157Dic47—24no—
158Dic47—42no—
159Dic47—44no—
160C42×Q8—471no—
161#1767—471no—
162#1744—471no—
163C2×Q8⋊C8—471no—
164C2×Q8⋊C8D4711yescharacteristic
165#1279—471no—
166#1279—471no—
167C2×C47#148712yes—
168C2×C47#148812yes—
169C2×C47#174911yescharacteristic
170C2×C47#175911yescharacteristic
171C2×C47C4×#10311yescharacteristic
172C2×C4×Q8—472no—
173C2×C4×Q8C2×D4711yescharacteristic
174C2×C4⋊C8—471no—
175C2×C4⋊C8C2×D4711yescharacteristic
176C2×#97—941no—
177C2×C4×C8—471no—
178C2×C4×C8C2×D4711yescharacteristic
179C4×C4⋊C4—472no—
180C4×C4⋊C4—471no—
181C4×C4⋊C4—941no—
182C4×C4⋊C4—942no—
183C43—471no—
184C43—941no—
185#336—942no—
186#336—941no—
187Q8⋊C8—474no—
188Q8⋊C8Dic4714yescomplemented
189C47#817511yescharacteristic, complemented, Sylow
190C22×Q8C4×D4711yescharacteristic
191C22×C8—21no—
192C22×C8—21no—
193C22×C8—941no—
194C22×C8—941no—
195C4×Q8—21no—
196C4×Q8—21no—
197C4×Q8—944no—
198C4×Q8—944no—
199C4×Q8C4×D4712yes—
200C4×Q8C2×Dic4712yes—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—472no—
203C2×C4⋊C4—942no—
204C2×C4⋊C4C4×D4711yescharacteristic
205C2×C42—21no—
206C2×C42—471no—
207C2×C42—472no—
208C2×C42—471no—
209C2×C42—472no—
210C2×C42—942no—
211C2×C42—942no—
212C2×C42—941no—
213C2×C42C22×D4711yescharacteristic
214C4⋊C8—21no—
215C4⋊C8—472no—
216C4⋊C8—941no—
217C4⋊C8C2×Dic4712yes—
218#97—1882no—
219C4×C8—21no—
220C4×C8—472no—
221C4×C8—941no—
222C4×C8C2×Dic4712yes—
223#95—942no—
224#95—941no—
225C2×Q8—21no—
226C2×Q8—22no—
227C2×Q8#1062532414yes—
228C2×Q8C4×Dic4712yes—
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—472no—
232C22×C4—472no—
233C22×C4—942no—
234C22×C4C2×C4×D4711yescharacteristic
235C22×C4D4×D4711yescharacteristic
236C22×C4D4×D4711yescharacteristic
237C2×C8—22no—
238C2×C8—22no—
239C2×C8—42no—
240C2×C8—41no—
241C2×C8—41no—
242C2×C8—942no—
243C2×C8—942no—
244C2×C8—1882no—
245C2×C8—1881no—
246C2×C8—1881no—
247C4⋊C4—21no—
248C4⋊C4—22no—
249C4⋊C4—22no—
250C4⋊C4—474no—
251C4⋊C4—948no—
252C4⋊C4—942no—
253C4⋊C4C4×Dic4712yes—
254C42—22no—
255C42—22no—
256C42—472no—
257C42—472no—
258C42—948no—
259C42—942no—
260C42—942no—
261C42—941no—
262C42—941no—
263C42—1882no—
264C42—1884no—
265C42—1881no—
266C42—1881no—
267C42C2×C4×D4711yescharacteristic
268C42#1062535211yescharacteristic
269C42#1062535211yescharacteristic
270C42C22×Dic4711yescharacteristic
271C23C22⋊C4×D4711yescharacteristic
272Q8—22no—
273Q8—24no—
274Q8#1309393414yescomplemented
275C2×C4—22no—
276C2×C4—24no—
277C2×C4—22no—
278C2×C4—22no—
279C2×C4—42no—
280C2×C4—474no—
281C2×C4—942no—
282C2×C4—942no—
283C2×C4—944no—
284C2×C4—942no—
285C2×C4—942no—
286C2×C4—942no—
287C2×C4—1882no—
288C2×C4—1882no—
289C2×C4C2×C4×Dic4711yescharacteristic
290C2×C4#1309398011yescharacteristic
291C2×C4(C22⋊D47)⋊C411yescharacteristic
292C2×C4(C22⋊D47)⋊C411yescharacteristic
293C2×C4D4×Dic4711yescharacteristic
294C2×C4D4×Dic4711yescharacteristic
295C2×C4M16×D4712yes—
296C2×C4C2×#1062532412yes—
297C2×C4SD16×D4711yescharacteristic
298C2×C4Q16×D4711yescharacteristic
299C2×C4#1309404511yescharacteristic
300C2×C4#1309404711yescharacteristic
301C8—42no—
302C8—42no—
303C8—1882no—
304C8—1882no—
305C22C22⋊C4×Dic4711yescharacteristic
306C22#1568908512yes—
307C22#103×D4711yescharacteristic
308C22#1568911811yescharacteristic
309C22#1568913812yes—
310C4—21no—
311C4—21no—
312C4—24no—
313C4—42no—
314C4—944no—
315C4—1882no—
316C4—1884no—
317C4C2×#1309393411yescharacteristic
318C4M16×Dic4711yescharacteristic
319C4#1568917712yes—
320C4#1568920811yescharacteristic
321C4SD16×Dic4711yescharacteristic
322C4#1568921511yescharacteristic
323C4Q16×Dic4711yescharacteristic
324C2#2024181512yes—
325C2#2024182112yes—
326C2#2024315211yescharacteristic
327C2#2024320011yescharacteristic
328C2#103×Dic4711yescharacteristic
subgroupG/H
centerC22×C4D4×D47
commutatorC2×C4×C47C2×C42
FrattiniC2×C42C22×D47
FittingC2×C47×Q8⋊C8C2
socle, prime radicalC23×C47C2×C22⋊C4
cosocleC2×C42×C47C23
hypercenterC2×Q8⋊C8D47
hypocenterC47#8175
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22×C4 ◁ C2×C42 ◁ C2×Q8⋊C8

Complements

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

NH ≅ G/Ntype
20. Q8⋊(C47⋊C8)C4semidirect
24. C47×Q8⋊C8C4semidirect
109. Q8×C47#97semidirect
188. Q8⋊C8Dic47semidirect
189. C47#8175semidirect
274. Q8#13093934semidirect

Maximal subgroups

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

subgroupindex
C4×Q8×Dic472
#202432432
#202431332
C2×Q8⋊(C47⋊C8)2
#202412282
#202412272
C2×C47×Q8⋊C82
#817547