Finite Group Databasebeta

← #26138686

Subgroups — #26138686

Order 12032.

1710 conjugacy classes of subgroups.

The subgroup lattice has 366 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15688989C211yesmaximal, characteristic
2C2×C4×C4⋊Dic47C211yesmaximal, characteristic
3#20241659C212yesmaximal
4#20241656C212yesmaximal
5#20241620C214yesmaximal
6#20241618C214yesmaximal
7C22×C4×Q8×C47C211yesmaximal, characteristic
8C22×C4⋊Dic47C2212yes—
9C22×C4⋊Dic47C2212yes—
10C22×C4×Dic47C2212yes—
11#15688989—24no—
12C4×C4⋊Dic47—24no—
13C2×C47⋊#95C2212yes—
14C2×C47⋊#95C2212yes—
15C2×C47⋊#95C2218yes—
16C2×C47⋊#95C2212yes—
17C2×C47⋊#95C2218yes—
18C23×Q8×C47C2211yescharacteristic
19C2×C4×Q8×C47—24no—
20C2×C4×Q8×C47—24no—
21C2×C4×Q8×C47C418yescomplemented
22C22×C4⋊C4×C47C2212yes—
23C22×C4⋊C4×C47C2211yescharacteristic
24C22×C42×C47C2212yes—
25C22×C42×C47C2211yescharacteristic
26C23×Dic47C2314yes—
27C23×Dic47C2314yes—
28C2×C4⋊Dic47—24no—
29C2×C4⋊Dic47—24no—
30C2×C4⋊Dic47—24no—
31C2×C4⋊Dic47—28no—
32C2×C4⋊Dic47Q818yescomplemented
33C2×C4×Dic47—24no—
34C2×C4×Dic47—44no—
35C47⋊#95—24no—
36C47⋊#95—24no—
37C47⋊#95—24no—
38C47⋊#95—216no—
39C47⋊#95—48no—
40C22×Q8×C47—22no—
41C22×Q8×C47—22no—
42C22×Q8×C47—28no—
43C22×Q8×C47C2×C414yes—
44C23×C4×C47C2312yes—
45C23×C4×C47C2311yescharacteristic
46C23×C4×C47C2311yescharacteristic
47C23×C4×C47C2312yes—
48C23×C4×C47C2311yescharacteristic
49C4×Q8×C47—232no—
50C2×C4⋊C4×C47—24no—
51C2×C4⋊C4×C47—24no—
52C2×C4⋊C4×C47—22no—
53C2×C4⋊C4×C47—22no—
54C2×C4⋊C4×C47C2×C414yes—
55C2×C4⋊C4×C47C2×C418yes—
56C2×C42×C47—24no—
57C2×C42×C47—24no—
58C2×C42×C47C2×C418yes—
59C2×C42×C47C2×C414yes—
60C2×C42×C47D414yes—
61C2×C42×C47D414yes—
62C24×C47C2411yescharacteristic
63C2×Q8×C47—216no—
64C2×Q8×C47—216no—
65C2×Q8×C47—28no—
66C2×Q8×C47—216no—
67C22×C4×C47—22no—
68C22×C4×C47—22no—
69C22×C4×C47—24no—
70C22×C4×C47—24no—
71C22×C4×C47—22no—
72C22×C4×C47—22no—
73C22×C4×C47—22no—
74C22×C4×C47—22no—
75C22×C4×C47—24no—
76C22×C4×C47—24no—
77C22×C4×C47C22×C414yes—
78C22×C4×C47C22×C412yes—
79C22×C4×C47C22×C412yes—
80C22×C4×C47C22×C414yes—
81C22×C4×C47C22×C412yes—
82C22×C4×C47C2×D412yes—
83C22×C4×C47C2×D412yes—
84C22×C4×C47C2×D412yes—
85C22×C4×C47C2×D412yes—
86C22×C4×C47C2×D412yes—
87C22×C4×C47C2×D412yes—
88C22×C4×C47C2×Q814yes—
89C22×C4×C47#4118yes—
90C22×C4×C47#4114yes—
91C22×Dic47—24no—
92C22×Dic47—28no—
93C22×Dic47—24no—
94C22×Dic47—28no—
95C22×Dic47—24no—
96C22×Dic47—28no—
97C22×Dic47—28no—
98C22×Dic47—24no—
99C22×Dic47—24no—
100C22×Dic47C2×Q818yes—
101C4⋊C4×C47—216no—
102C4⋊C4×C47—28no—
103C42×C47—216no—
104C42×C47C22⋊C4116yes—
105C4⋊Dic47—216no—
106C4⋊Dic47—48no—
107C4×Dic47—48no—
108C23×C47C23×C411yescharacteristic
109C23×C47C22×D411yescharacteristic
110C23×C47C22×D411yescharacteristic
111C23×C47C22×Q812yes—
112C23×C47C2×#4114yes—
113C23×C47C2×#4112yes—
114C23×C47H(5,𝔽2)12yes—
115C23×C472−1+412yes—
116Q8×C47—232no—
117C2×C4×C47—24no—
118C2×C4×C47—24no—
119C2×C4×C47—24no—
120C2×C4×C47—24no—
121C2×C4×C47—24no—
122C2×C4×C47—24no—
123C2×C4×C47—28no—
124C2×C4×C47—28no—
125C2×C4×C47—22no—
126C2×C4×C47—22no—
127C2×C4×C47—24no—
128C2×C4×C47—24no—
129C2×C4×C47—28no—
130C2×C4×C47—42no—
131C2×C4×C47—42no—
132C2×C4×C47—44no—
133C2×C4×C47C2×C22⋊C414yes—
134C2×C4×C47C2×C22⋊C414yes—
135C2×C4×C47C2×C22⋊C414yes—
136C2×C4×C47#11718yes—
137C2×C4×C47C4×Q818yes—
138C2×C4×C47C22⋊Q818yes—
139C2×C4×C47#12414yes—
140C2×C4×C47C4⋊Q814yes—
141C2×Dic47—28no—
142C2×Dic47—216no—
143C2×Dic47—44no—
144C2×Dic47—44no—
145C2×Dic47—48no—
146C2×Dic47—44no—
147C2×Dic47—44no—
148C2×Dic47—44no—
149C2×Dic47—48no—
150C2×Dic47—48no—
151#13925—471nomaximal, Sylow
152C22×C47C22×C22⋊C411yescharacteristic
153C22×C47C2×#11712yes—
154C22×C47C2×C4×Q812yes—
155C22×C47#52012yes—
156C22×C47C2×C22⋊Q812yes—
157C22×C47#52512yes—
158C22×C47#52512yes—
159C22×C47C2×#12411yescharacteristic
160C22×C47C2×C4⋊Q811yescharacteristic
161C22×C47#53511yescharacteristic
162C22×C47#53611yescharacteristic
163C22×C47#53712yes—
164C22×C47#55414yes—
165C22×C47#55614yes—
166C22×C47#55714yes—
167C22×C47#55914yes—
168C4×C47—24no—
169C4×C47—24no—
170C4×C47—28no—
171C4×C47—44no—
172C4×C47#39118yes—
173Dic47—48no—
174Dic47—84no—
175C22×C4×Q8D4711yescharacteristic
176C2×#382—471no—
177C2×C4×C4⋊C4—471no—
178#1446—472no—
179#1445—472no—
180#1436—474no—
181C4⋊#95—474no—
182C2×C47#228011yescharacteristic
183C2×C47C2×#39111yescharacteristic
184C2×C47#231111yescharacteristic
185C2×C47#235514yes—
186C2×C47#258012yes—
187C2×C47#258112yes—
188C2×C47#258412yes—
189C2×C47#259212yes—
190C23×Q8C2×D4711yescharacteristic
191C2×C4×Q8—24no—
192C2×C4×Q8—24no—
193C2×C4×Q8Dic4718yescomplemented
194C22×C4⋊C4—472no—
195C22×C4⋊C4—472no—
196C22×C4⋊C4C2×D4712yes—
197C22×C4⋊C4C2×D4711yescharacteristic
198C22×C42—472no—
199C22×C42C2×D4712yes—
200C22×C42C2×D4711yescharacteristic
201#382—944no—
202C4×C4⋊C4—944no—
203C2×#95—478no—
204C2×#95—472no—
205C2×#95—472no—
206C2×#95—472no—
207C2×#95—478no—
208C47#1392511yescharacteristic, complemented, Sylow
209C22×Q8—28no—
210C22×Q8—22no—
211C22×Q8—22no—
212C22×Q8C2×Dic4714yes—
213C23×C4—474no—
214C23×C4—474no—
215C23×C4C22×D4712yes—
216C23×C4C22×D4712yes—
217C23×C4C22×D4711yescharacteristic
218C23×C4C22×D4711yescharacteristic
219C23×C4C22×D4711yescharacteristic
220C4×Q8—232no—
221C2×C4⋊C4—24no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—478no—
226C2×C4⋊C4—944no—
227C2×C4⋊C4—944no—
228C2×C4⋊C4—948no—
229C2×C4⋊C4—944no—
230C2×C4⋊C4C2×Dic4714yes—
231C2×C4⋊C4C2×Dic4718yes—
232C2×C42—24no—
233C2×C42—24no—
234C2×C42—944no—
235C2×C42—1884no—
236C2×C42C2×Dic4718yes—
237C2×C42C2×Dic4714yes—
238C2×C42C22⋊D4714yes—
239C2×C42C22⋊D4714yes—
240#95—9416no—
241#95—944no—
242#95—944no—
243#95—944no—
244#95—1888no—
245C24C23×D4711yescharacteristic
246C2×Q8—216no—
247C2×Q8—216no—
248C2×Q8—28no—
249C2×Q8—216no—
250C22×C4—22no—
251C22×C4—22no—
252C22×C4—24no—
253C22×C4—24no—
254C22×C4—22no—
255C22×C4—22no—
256C22×C4—22no—
257C22×C4—22no—
258C22×C4—24no—
259C22×C4—24no—
260C22×C4—478no—
261C22×C4—944no—
262C22×C4—948no—
263C22×C4—944no—
264C22×C4—948no—
265C22×C4—944no—
266C22×C4—948no—
267C22×C4—948no—
268C22×C4—944no—
269C22×C4—944no—
270C22×C4#1062535018yes—
271C22×C4Q8×D4714yes—
272C22×C4#1062535414yes—
273C22×C4C22×Dic4714yes—
274C22×C4C22×Dic4712yes—
275C22×C4C22×Dic4712yes—
276C22×C4C22×Dic4714yes—
277C22×C4C22×Dic4712yes—
278C22×C4C2×C22⋊D4712yes—
279C22×C4C2×C22⋊D4712yes—
280C22×C4C2×C22⋊D4712yes—
281C22×C4C2×C22⋊D4712yes—
282C22×C4C2×C22⋊D4712yes—
283C22×C4C2×C22⋊D4712yes—
284C4⋊C4—216no—
285C4⋊C4—28no—
286C4⋊C4—9416no—
287C4⋊C4—1888no—
288C42—216no—
289C42—1888no—
290C42C22⋊Dic47116yes—
291C23C23×Dic4711yescharacteristic
292C23C22×C22⋊D4711yescharacteristic
293C23C22×C22⋊D4711yescharacteristic
294C23C2×Q8×D4712yes—
295C23C2×#1062535014yes—
296C23C2×#1062535412yes—
297C23#1309407012yes—
298C23#1309407112yes—
299Q8—232no—
300C2×C4—24no—
301C2×C4—24no—
302C2×C4—24no—
303C2×C4—24no—
304C2×C4—28no—
305C2×C4—28no—
306C2×C4—24no—
307C2×C4—24no—
308C2×C4—22no—
309C2×C4—22no—
310C2×C4—24no—
311C2×C4—24no—
312C2×C4—28no—
313C2×C4—42no—
314C2×C4—42no—
315C2×C4—44no—
316C2×C4—9416no—
317C2×C4—948no—
318C2×C4—1888no—
319C2×C4—1888no—
320C2×C4—1884no—
321C2×C4—1884no—
322C2×C4—1884no—
323C2×C4—1888no—
324C2×C4—1884no—
325C2×C4—1884no—
326C2×C4C2×C22⋊Dic4714yes—
327C2×C4C2×C22⋊Dic4714yes—
328C2×C4C2×C22⋊Dic4714yes—
329C2×C4#1309398318yes—
330C2×C4Q8×Dic4718yes—
331C2×C4#1309400018yes—
332C2×C4#1309401314yes—
333C2×C4#1309402214yes—
334C22C22×C22⋊Dic4711yescharacteristic
335C22C2×#1309398312yes—
336C22C2×Q8×Dic4712yes—
337C22#1568960612yes—
338C22C2×#1309400012yes—
339C22#1568963412yes—
340C22#1568963412yes—
341C22C2×#1309401311yescharacteristic
342C22C2×#1309402211yescharacteristic
343C22#1568968711yescharacteristic
344C22#1568969411yescharacteristic
345C22#1568970112yes—
346C22#1568983314yes—
347C22#1568984614yes—
348C22#1568985114yes—
349C22#1568985814yes—
350C4—24no—
351C4—28no—
352C4—24no—
353C4—44no—
354C4—3764no—
355C4—1888no—
356C4#1568903018yes—
357C2#2024577111yescharacteristic
358C2C2×#1568903011yescharacteristic
359C2#2024598011yescharacteristic
360C2#2024634514yes—
361C2#2024874612yes—
362C2#2024875412yes—
363C2#2024877612yes—
364C2#2024887612yes—
subgroupG/H
center, FrattiniC24C23×D47
commutatorC23×C47C23×C4
FittingC22×C4×Q8×C47C2
socle, prime radical, cosocleC24×C47C24
hypercenterC22×C4×Q8D47
hypocenterC47#13925
LCSG ▷ C23×C47 ▷ C47
UCSC1 ◁ C24 ◁ C22×C4×Q8

Complements

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

NH ≅ G/Ntype
21. C2×C4×Q8×C47C4semidirect
32. C2×C4⋊Dic47Q8semidirect
193. C2×C4×Q8Dic47semidirect
208. C47#13925semidirect

Maximal subgroups

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

subgroupindex
C2×#156889892
C2×C4×C4⋊Dic472
#202416592
#202416562
#202416202
#202416182
C22×C4×Q8×C472
#1392547