Finite Group Databasebeta

← #26225069

Subgroups — #26225069

Order 12032.

854 conjugacy classes of subgroups.

The subgroup lattice has 380 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689519C211yesmaximal, characteristic
2C4⋊Q8×Dic47C211yesmaximal, characteristic
3#20245955C211yesmaximal, characteristic
4#20244349C211yesmaximal, characteristic
5#20244130C214yesmaximal
6#20243547C211yesmaximal, characteristic
7#20243284C214yesmaximal
8C4×#13093934C211yesmaximal, characteristic
9C47×#1990C211yesmaximal, characteristic
10C2×#13094024C2211yescharacteristic
11C2×Q8×Dic47—21no—
12C2×#13093989—21no—
13#15689519C418yescomplemented
14C2×#13093957C2214yes—
15C2×#13093955C2214yes—
16C2×#13093934C2212yes—
17C2×#13093934C2212yes—
18C2×C4×C47⋊C8C2211yescharacteristic
19#15689030—21no—
20#15689028—21no—
21#15689022—21no—
22#15689022C2212yes—
23#15689019—21no—
24#15689019C2212yes—
25#15688993C2211yescharacteristic
26C4⋊C4×Dic47—21no—
27C4⋊C4×Dic47—21no—
28C4⋊C4×Dic47C2212yes—
29C4⋊C4×Dic47C2212yes—
30C42×Dic47C2211yescharacteristic
31#15688886C2214yes—
32C2×C4⋊Q8×C47C2211yescharacteristic
33C2×C4×C8×C47C2211yescharacteristic
34C47×#384C2211yescharacteristic
35C47×#340C2214yes—
36C22×Dic94—21no—
37C22×C47⋊C8C2312yes—
38#13094024—22no—
39#13094024C2×C414yes—
40Q8×Dic47—44no—
41#13093989—44no—
42C2×C4⋊Dic47—21no—
43C2×C4⋊Dic47—21no—
44C2×C4⋊Dic47—21no—
45C2×C4⋊Dic47—22no—
46C2×C4⋊Dic47C2312yes—
47C2×C4⋊Dic47C2312yes—
48C2×Dic47⋊C4—22no—
49C2×C4×Dic47—22no—
50C2×C4×Dic47—22no—
51C2×C4×Dic47D411yescharacteristic
52C2×C4×Dic47D411yescharacteristic
53C2×C4×Dic47D411yescharacteristic
54C2×C4×Dic47D411yescharacteristic
55C2×C4×Dic47C2311yescharacteristic
56C2×C4×Dic47C2311yescharacteristic
57#13093957—28no—
58#13093955C2×C4116yes—
59#13093934—24no—
60#13093934Q818yescomplemented
61C4×C47⋊C8C2×C414yes—
62C47⋊#95—22no—
63C47⋊#95—22no—
64C47⋊#95—22no—
65C47⋊#95—22no—
66C22×Q8×C47—21no—
67C22×C8×C47C2312yes—
68C4⋊Q8×C47—22no—
69C4⋊Q8×C47C2×C414yes—
70C2×C4⋊C4×C47—21no—
71C2×C4⋊C4×C47—21no—
72C2×C4⋊C4×C47—21no—
73C2×C4⋊C4×C47C2312yes—
74C2×C4⋊C4×C47C2312yes—
75C2×C42×C47C2311yescharacteristic
76C4×C8×C47—22no—
77C2×Q8×C47—24no—
78C2×Q8×C47—42no—
79C2×Q8×C47—42no—
80C22×C4×C47—22no—
81C22×C4×C47—22no—
82C22×C4×C47C2×D411yescharacteristic
83C22×C4×C47C2×D411yescharacteristic
84C22×C4×C47C2411yescharacteristic
85C22×Dic47—22no—
86C22×Dic47—22no—
87C22×Dic47C2×D411yescharacteristic
88C22×Dic47C2×D411yescharacteristic
89C22×Dic47C2×D411yescharacteristic
90C22×Dic47C2×D411yescharacteristic
91C2×Dic94—24no—
92C2×Dic94—42no—
93C2×Dic94—42no—
94C2×C8×C47—22no—
95C2×C8×C47—22no—
96C2×C8×C47C2×Q814yes—
97C4⋊C4×C47—24no—
98C4⋊C4×C47—22no—
99C4⋊C4×C47—24no—
100C4⋊C4×C47—42no—
101C4⋊C4×C47—42no—
102C4⋊C4×C47C22×C414yes—
103C42×C47C22×C412yes—
104C42×C47#4112yes—
105C4⋊Dic47—24no—
106C4⋊Dic47—22no—
107C4⋊Dic47—24no—
108C4⋊Dic47—44no—
109C4⋊Dic47—42no—
110C4⋊Dic47—42no—
111C4⋊Dic47C22×C414yes—
112Dic47⋊C4—44no—
113C4×Dic47—22no—
114C4×Dic47—22no—
115C4×Dic47—22no—
116C4×Dic47—22no—
117C4×Dic47—22no—
118C4×Dic47—44no—
119C4×Dic47—44no—
120C4×Dic47C2×Q814yes—
121C2×C47⋊C8C22×C414yes—
122C2×C47⋊C8C2×Q814yes—
123C2×C47⋊C8#4114yes—
124C23×C47C22×D411yescharacteristic
125Q8×C47—48no—
126C2×C4×C47—24no—
127C2×C4×C47—42no—
128C2×C4×C47—42no—
129C2×C4×C47—42no—
130C2×C4×C47—42no—
131C2×C4×C47—42no—
132C2×C4×C47C4×D412yes—
133C2×C4×C47C4×D412yes—
134C2×C4×C47AGL(1,ℤ/8)12yes—
135C2×C4×C47AGL(1,ℤ/8)12yes—
136C2×C4×C47#13712yes—
137C2×C4×C47#13712yes—
138C2×C4×C47C23×C411yescharacteristic
139C2×C4×C47C22×Q812yes—
140C2×C4×C47C2×#4112yes—
141C2×C4×C47C2×#4111yescharacteristic
142C2×Dic47—21no—
143C2×Dic47—21no—
144C2×Dic47—21no—
145C2×Dic47—21no—
146C2×Dic47—24no—
147C2×Dic47—21no—
148C2×Dic47—21no—
149C2×Dic47—21no—
150C2×Dic47—21no—
151C2×Dic47—21no—
152C2×Dic47—21no—
153C2×Dic47—42no—
154C2×Dic47—42no—
155C2×Dic47—42no—
156C2×Dic47—42no—
157C2×Dic47—42no—
158C2×Dic47C4⋊Q812yes—
159C2×Dic47C4⋊Q812yes—
160Dic94—48no—
161C8×C47—24no—
162C47⋊C8C4×Q818yescomplemented
163#21727—471nomaximal, Sylow
164C22×C47C2×C4×D411yescharacteristic
165C22×C47C2×C4⋊Q811yescharacteristic
166C22×C47#53211yescharacteristic
167C22×C47C2×AGL(1,ℤ/8)11yescharacteristic
168C22×C47C2×AGL(1,ℤ/8)11yescharacteristic
169C22×C47C2×#13711yescharacteristic
170C22×C47C2×#13711yescharacteristic
171C4×C47—44no—
172C4×C47—82no—
173C4×C47#44014yes—
174C4×C47#44014yes—
175C4×C47C2×C4×Q812yes—
176C4×C47#53312yes—
177Dic47—22no—
178Dic47—22no—
179Dic47—41no—
180Dic47—41no—
181Dic47—44no—
182Dic47—82no—
183C2×#498—471no—
184C4×C4⋊Q8—471no—
185C4×C4⋊Q8—471no—
186#1990—471no—
187#1990D4711yescharacteristic
188#1944—474no—
189#1845—471no—
190#1776—474no—
191C4×#97—471no—
192C2×C47C4×C4⋊Q811yescharacteristic
193C2×C47C2×#44011yescharacteristic
194C2×C47C2×#44011yescharacteristic
195C2×C47#316511yescharacteristic
196C2×C47#316511yescharacteristic
197C2×C47#318411yescharacteristic
198C2×C47#318511yescharacteristic
199C2×C4⋊Q8—471no—
200C2×C4⋊Q8C2×D4711yescharacteristic
201C2×C4×Q8—941no—
202C2×C4×Q8—941no—
203#498—478no—
204C2×#107—474no—
205C2×#106—474no—
206C2×#97—472no—
207C2×#97—472no—
208C2×C4×C8—471no—
209C2×C4×C8C2×D4711yescharacteristic
210#391—941no—
211#391—941no—
212#389—472no—
213#389—472no—
214#389—941no—
215#389—941no—
216#384—471no—
217#384C2×D4711yescharacteristic
218C4×C4⋊C4—472no—
219C4×C4⋊C4—472no—
220C4×C4⋊C4—941no—
221C4×C4⋊C4—941no—
222C43—471no—
223#340—474no—
224#340C2×D4714yes—
225C47#2172711yescharacteristic, complemented, Sylow
226C22×Q8—21no—
227C22×Q8—941no—
228C22×C8—472no—
229C22×C8C22×D4712yes—
230C4⋊Q8—22no—
231C4⋊Q8—474no—
232C4⋊Q8—942no—
233C4⋊Q8C4×D4714yes—
234C4×Q8—1884no—
235C4×Q8—1884no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4—21no—
238C2×C4⋊C4—21no—
239C2×C4⋊C4—472no—
240C2×C4⋊C4—472no—
241C2×C4⋊C4—941no—
242C2×C4⋊C4—942no—
243C2×C4⋊C4—942no—
244C2×C4⋊C4—941no—
245C2×C4⋊C4—941no—
246C2×C4⋊C4C22×D4712yes—
247C2×C4⋊C4C22×D4712yes—
248C2×C42—471no—
249C2×C42—471no—
250C2×C42—471no—
251C2×C42—471no—
252C2×C42—471no—
253C2×C42—471no—
254C2×C42—942no—
255C2×C42—942no—
256C2×C42C22×D4711yescharacteristic
257#107—948no—
258#106—4716no—
259#97—478no—
260#97—944no—
261C4×C8—22no—
262C4×C8—474no—
263#95—942no—
264#95—942no—
265#95—942no—
266#95—942no—
267C2×Q8—24no—
268C2×Q8—42no—
269C2×Q8—42no—
270C2×Q8—944no—
271C2×Q8—1882no—
272C2×Q8—1882no—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—471no—
276C22×C4—471no—
277C22×C4—471no—
278C22×C4—471no—
279C22×C4—942no—
280C22×C4—942no—
281C22×C4D4×D4711yescharacteristic
282C22×C4D4×D4711yescharacteristic
283C22×C4C23×D4711yescharacteristic
284C2×C8—22no—
285C2×C8—22no—
286C2×C8—474no—
287C2×C8—474no—
288C2×C8—474no—
289C2×C8Q8×D4714yes—
290C4⋊C4—24no—
291C4⋊C4—22no—
292C4⋊C4—24no—
293C4⋊C4—42no—
294C4⋊C4—42no—
295C4⋊C4—474no—
296C4⋊C4—944no—
297C4⋊C4—942no—
298C4⋊C4—944no—
299C4⋊C4—1884no—
300C4⋊C4—1884no—
301C4⋊C4—1882no—
302C4⋊C4—1882no—
303C4⋊C4C2×C4×D4714yes—
304C42—474no—
305C42—942no—
306C42—942no—
307C42—942no—
308C42—942no—
309C42—942no—
310C42—1884no—
311C42—1884no—
312C42C2×C4×D4712yes—
313C42#1062535212yes—
314C23C2×D4×D4711yescharacteristic
315Q8—48no—
316Q8—1888no—
317C2×C4—24no—
318C2×C4—42no—
319C2×C4—42no—
320C2×C4—42no—
321C2×C4—42no—
322C2×C4—42no—
323C2×C4—472no—
324C2×C4—472no—
325C2×C4—941no—
326C2×C4—941no—
327C2×C4—941no—
328C2×C4—941no—
329C2×C4—944no—
330C2×C4—941no—
331C2×C4—941no—
332C2×C4—941no—
333C2×C4—941no—
334C2×C4—941no—
335C2×C4—941no—
336C2×C4—1882no—
337C2×C4—1882no—
338C2×C4—1882no—
339C2×C4—1882no—
340C2×C4—1882no—
341C2×C4(C22⋊D47)⋊C412yes—
342C2×C4(C22⋊D47)⋊C412yes—
343C2×C4#1309404812yes—
344C2×C4#1309405112yes—
345C2×C4#1309405312yes—
346C2×C4#1309405612yes—
347C2×C4C22×C4×D4711yescharacteristic
348C2×C4C2×Q8×D4712yes—
349C2×C4#41×D4712yes—
350C2×C4C2×#1062535211yescharacteristic
351C8—24no—
352C8—478no—
353C22C2×(C22⋊D47)⋊C411yescharacteristic
354C22C4⋊Q8×D4711yescharacteristic
355C22#1568967211yescharacteristic
356C22C2×#1309404811yescharacteristic
357C22C2×#1309405111yescharacteristic
358C22C2×#1309405311yescharacteristic
359C22C2×#1309405611yescharacteristic
360C4—44no—
361C4—82no—
362C4—942no—
363C4—942no—
364C4—3762no—
365C4—1884no—
366C4—1881no—
367C4—1881no—
368C4#1568922014yes—
369C4#1568922614yes—
370C4C4×Q8×D4712yes—
371C4#1568967512yes—
372C2#2024595711yescharacteristic
373C2C2×#1568922011yescharacteristic
374C2C2×#1568922611yescharacteristic
375C2#2025392911yescharacteristic
376C2#2025393111yescharacteristic
377C2#2025412511yescharacteristic
378C2#2025413211yescharacteristic
subgroupG/H
centerC23C2×D4×D47
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC47×#1990C2
socle, prime radicalC23×C47C22×D4
cosocleC22×C4×C47C24
hypercenter#1990D47
hypocenterC47#21727
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1990

Complements

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

NH ≅ G/Ntype
13. #15689519C4semidirect
60. #13093934Q8semidirect
162. C47⋊C8C4×Q8semidirect
225. C47#21727semidirect

Maximal subgroups

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

subgroupindex
C2×#156895192
C4⋊Q8×Dic472
#202459552
#202443492
#202441302
#202435472
#202432842
C4×#130939342
C47×#19902
#2172747