Finite Group Databasebeta

← C2×#20256031 #27095843

Subgroups — C2×#20256031 #27095843

Order 12032.

2058 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689852C211yesmaximal, characteristic, complemented
2C2×Q8×D188C211yesmaximal, characteristic
3#20256031C2116yesmaximal, direct product factor
4C2×#15689474C213yesmaximal, complemented
5C2×#15689409C213yesmaximal, complemented
6C2×#15689329C213yesmaximal
7C2×C4×#10625325C213yesmaximal
8C2×C8×Q8×C47C211yesmaximal, characteristic, complemented
9C22×Q8×D47—21no—
10C22×#10625325C2213yes—
11#15689852—24no—
12#15689852C2218yescomplemented
13Q8×D188—24no—
14Q8×D188C2218yes—
15C2×#13094024C2213yes—
16C2×#13094024C2213yes—
17C2×#13094015—23no—
18C2×#13094004—23no—
19C2×Q8×Dic47—21no—
20C2×C4×Dic94C2213yes—
21C2×C4×D188C2213yes—
22#15689474C22124yescomplemented
23#15689409C22124yescomplemented
24#15689329C22124yes—
25C4×#10625325C22124yes—
26C2×#13093956C2213yes—
27C2×#13093949C2213yes—
28C2×#13093949C2216yes—
29C2×#13093946C2211yescharacteristic
30C2×C4×Q8×C47C2211yescharacteristic
31C8×Q8×C47C2218yescomplemented
32C2×C4⋊C8×C47C2213yes—
33C2×C4×C8×C47C2213yes—
34C22×Dic94—23no—
35C22×Dic94C2313yes—
36C2×Q8×D47—28no—
37C2×Q8×D47—48no—
38C22×D188C2311yescharacteristic
39C22×C4×D47—23no—
40C2×#10625325—212no—
41C2×#10625325C23112yes—
42#13094024—26no—
43#13094024—26no—
44#13094024C23112yes—
45#13094024C23112yes—
46#13094015—224no—
47#13094004—224no—
48Q8×Dic47—28no—
49C4×Dic94—26no—
50C4×Dic94C23112yes—
51C4×D188—26no—
52C4×D188C23112yes—
53C2×C4⋊Dic47—23no—
54C2×C4⋊Dic47—23no—
55C2×C4⋊Dic47—23no—
56C2×C4⋊Dic47C2313yes—
57C2×C4⋊Dic47C2311yescharacteristic
58C2×Dic47⋊C4—23no—
59C2×(C2×D47)⋊C4—23no—
60C2×C4×Dic47—23no—
61#13093956C23112yes—
62#13093949C23112yes—
63#13093949C23124yes—
64#13093946C2314yes—
65C22×Q8×C47D411yescharacteristic
66C22×C8×C47C2311yescharacteristic
67C22×C8×C47C2313yes—
68C4×Q8×C47D414yes—
69C4×Q8×C47C2314yes—
70C2×C4⋊C4×C47D413yes—
71C2×C42×C47C2313yes—
72C4⋊C8×C47C23112yes—
73C4×C8×C47C23112yes—
74C23×D47—21no—
75C2×Q8×C47SD1618yescomplemented
76C2×Q8×C47C2×D414yes—
77C22×C4×C47C2×D413yes—
78C22×C4×C47C2×D413yes—
79C22×C4×C47C2411yescharacteristic
80C22×Dic47—21no—
81C22×Dic47—23no—
82C22×Dic47—23no—
83Q8×D47—432no—
84C2×D188—21no—
85C2×D188—22no—
86C2×D188#4114yes—
87C2×D188C2412yes—
88C2×C4×D47—212no—
89C2×C4×D47—412no—
90C2×Dic94—26no—
91C2×Dic94—26no—
92C2×Dic94—23no—
93C2×Dic94—212no—
94C2×Dic94—412no—
95C2×Dic94C2416yes—
96C2×C8×C47—26no—
97C2×C8×C47—22no—
98C2×C8×C47C2416yes—
99C2×C8×C47C2412yes—
100C4⋊C4×C47C2×D4112yes—
101C42×C47C2×D416yes—
102C42×C47C2416yes—
103(C2×D47)⋊C4—212no—
104C4⋊Dic47—212no—
105C4⋊Dic47—212no—
106C4⋊Dic47—212no—
107C4⋊Dic47—23no—
108C4⋊Dic47—21no—
109C4⋊Dic47C2412yes—
110C4⋊Dic47C2416yes—
111Dic47⋊C4—212no—
112C4×Dic47—212no—
113#10625325—224no—
114C23×C47C22×D411yescharacteristic
115C22×D47—24no—
116C22×D47—22no—
117C22×D47—42no—
118C22×D47—42no—
119Q8×C47C2×SD16116yescomplemented
120C2×C4×C47—26no—
121C2×C4×C47C2×SD16112yes—
122C2×C4×C47C22×D416yes—
123C2×C4×C47C22×D411yescharacteristic
124C2×C4×C47C22×D416yes—
125C2×C4×C47C2×#4112yes—
126C2×C4×C472−1+412yes—
127C2×C4×C47C2511yescharacteristic
128C2×Dic47—22no—
129C2×Dic47—26no—
130C2×Dic47—26no—
131C2×Dic47—46no—
132C2×Dic47—46no—
133C2×Dic47—42no—
134D188—24no—
135D188—22no—
136D188C2×#4114yes—
137C4×D47—424no—
138Dic94—26no—
139Dic94—26no—
140Dic94—46no—
141Dic94—424no—
142C8×C47—26no—
143C8×C47—22no—
144C2×#3359—471nomaximal, Sylow
145C22×C47#54812yes—
146C22×C47C22×SD1612yes—
147C22×C47#57812yes—
148C22×C47C23×D411yescharacteristic
149C2×D47—24no—
150C2×D47—48no—
151C2×D47—42no—
152C2×D47—42no—
153C4×C47—26no—
154C4×C47#54812yes—
155C4×C47C22×SD16112yes—
156C4×C47C22×#4111yescharacteristic
157C4×C47C2×2−1+411yescharacteristic
158Dic47—46no—
159Dic47—46no—
160Dic47—42no—
161C2×#557—471no—
162C2×D4×Q8—471no—
163#3359—4716no—
164C2×#487—473no—
165C2×#475—473no—
166C2×#461—473no—
167C2×C8×Q8D4711yescharacteristic, complemented
168C2×C4×SD16—473no—
169C2×C47#335914yescomplemented
170C2×C47C2×#54811yescharacteristic
171C2×C47C23×SD1611yescharacteristic
172C2×C47C2×#57811yescharacteristic
173D47—44no—
174C23×Q8—941no—
175C22×SD16—473no—
176#557—478no—
177#557—944no—
178D4×Q8—478no—
179D4×Q8—944no—
180C2×C4⋊Q8—473no—
181C2×C4⋊Q8—473no—
182C2×#125—943no—
183C2×C22⋊Q8—943no—
184C2×C4×Q8—473no—
185C2×C4×Q8—941no—
186C2×C4×Q8C2×D4711yescharacteristic
187C2×C4×D4—473no—
188#487—4724no—
189#475—4724no—
190#461—4724no—
191C8×Q8C2×D4718yescomplemented
192C4×SD16—4724no—
193C2×#106—473no—
194C2×C4⋊C8C2×D4713yes—
195C2×#103—476no—
196C2×#103—473no—
197C2×#102—471no—
198C2×C4×C8C2×D4713yes—
199C47C2×#335911yescharacteristic, complemented, Sylow
200C22×Q8—473no—
201C22×Q8—948no—
202C22×Q8—943no—
203C22×Q8—1888no—
204C22×Q8D18811yescharacteristic
205C22×D4—471no—
206C23×C4—943no—
207C2×SD16—4712no—
208C2×SD16—9412no—
209C22×C8C22×D4713yes—
210C22×C8C22×D4711yescharacteristic
211C4⋊Q8—4712no—
212C4⋊Q8—4712no—
213C4⋊Q8—946no—
214C4⋊Q8—946no—
215#125—9424no—
216C22⋊Q8—9424no—
217C4×Q8—4712no—
218C4×Q8—948no—
219C4×Q8—946no—
220C4×Q8D18814yes—
221C4×Q8C22×D4714yes—
222C4×D4—4712no—
223C4×D4—946no—
224C2×C4⋊C4—473no—
225C2×C4⋊C4—471no—
226C2×C4⋊C4—943no—
227C2×C4⋊C4—943no—
228C2×C4⋊C4—943no—
229C2×C4⋊C4—943no—
230C2×C4⋊C4D18813yes—
231C2×C22⋊C4—943no—
232C2×C42—943no—
233C2×C42C22×D4713yes—
234#106—4712no—
235C4⋊C8C22×D47112yes—
236#103—4712no—
237#103—4724no—
238#102—474no—
239C4×C8C22×D47112yes—
240C24—941no—
241C2×Q8—476no—
242C2×Q8—943no—
243C2×Q8—9412no—
244C2×Q8—946no—
245C2×Q8—946no—
246C2×Q8—18832no—
247C2×Q8—18812no—
248C2×Q8#1062532518yescomplemented
249C2×Q8C2×D18814yes—
250C2×D4—474no—
251C2×D4—472no—
252C2×D4—941no—
253C2×D4—942no—
254C22×C4—941no—
255C22×C4—943no—
256C22×C4—943no—
257C22×C4—9412no—
258C22×C4—18812no—
259C22×C4C2×D18813yes—
260C22×C4C2×D18813yes—
261C22×C4C23×D4711yescharacteristic
262SD16—9424no—
263C2×C8—26no—
264C2×C8—22no—
265C2×C8C23×D4712yes—
266C2×C8C23×D4716yes—
267C4⋊C4—472no—
268C4⋊C4—476no—
269C4⋊C4—9412no—
270C4⋊C4—943no—
271C4⋊C4—941no—
272C4⋊C4—9412no—
273C4⋊C4—9412no—
274C4⋊C4—9412no—
275C4⋊C4C2×D188112yes—
276C22⋊C4—9412no—
277C42—9412no—
278C42C2×D18816yes—
279C42C23×D4716yes—
280C23—944no—
281C23—942no—
282C23—1882no—
283C23—1882no—
284C23C22×D18811yescharacteristic
285Q8—946no—
286Q8—946no—
287Q8—1886no—
288Q8—18824no—
289Q8C2×#10625325116yescomplemented
290D4—474no—
291D4—944no—
292D4—942no—
293C2×C4—26no—
294C2×C4—942no—
295C2×C4—946no—
296C2×C4—946no—
297C2×C4—1886no—
298C2×C4—1886no—
299C2×C4—1882no—
300C2×C4—18824no—
301C2×C4C2×#10625325112yes—
302C2×C4C22×D18816yes—
303C2×C4C22×D18811yescharacteristic
304C2×C4C22×D18816yes—
305C2×C4#41×D4712yes—
306C2×C4#1309407212yes—
307C2×C4C24×D4711yescharacteristic
308C8—26no—
309C8—22no—
310C22—944no—
311C22—1888no—
312C22—1882no—
313C22—1882no—
314C22#1568978512yes—
315C22C22×#1062532512yes—
316C22#1568994012yes—
317C22C23×D18811yescharacteristic
318C4—26no—
319C4—1886no—
320C4—1886no—
321C4—1882no—
322C4#1568978512yes—
323C4C22×#10625325112yes—
324C4C2×#41×D4711yescharacteristic
325C4C2×#1309407211yescharacteristic
326C2—1884no—
327C2#2025603114yesdirect product factor
328C2C2×#1568978511yescharacteristic
329C2C23×#1062532511yescharacteristic
330C2C2×#1568994011yescharacteristic
subgroupG/H
centerC23C22×D188
commutator, cosocleC2×C4×C47C25
FrattiniC2×C4C24×D47
FittingC2×C8×Q8×C47C2
socleC23×C47C22×D4
prime radicalC22×D188C23
hypercenterC2×C8×Q8D47
hypocenterC47C2×#3359
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ C2×C8×Q8

Complements

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

NH ≅ G/Ntype
1. C2×#15689852C2semidirect
3. #20256031C2direct (N × H)
4. C2×#15689474C2semidirect
5. C2×#15689409C2semidirect
8. C2×C8×Q8×C47C2semidirect
12. #15689852C22semidirect
22. #15689474C22semidirect
23. #15689409C22semidirect
31. C8×Q8×C47C22semidirect
75. C2×Q8×C47SD16semidirect
119. Q8×C47C2×SD16semidirect
167. C2×C8×Q8D47semidirect
169. C2×C47#3359semidirect
191. C8×Q8C2×D47semidirect
199. C47C2×#3359semidirect
248. C2×Q8#10625325semidirect
289. Q8C2×#10625325semidirect
327. C2#20256031direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#156898522
C2×Q8×D1882
#202560312
C2×#156894742
C2×#156894092
C2×#156893292
C2×C4×#106253252
C2×C8×Q8×C472
C2×#335947