Finite Group Databasebeta

← C2×#20256060 #27095875

Subgroups — C2×#20256060 #27095875

Order 12032.

2174 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×Q8×Dic94C211yesmaximal, characteristic, complemented
2C2×#15689795C211yesmaximal, characteristic
3#20256060C2116yesmaximal, direct product factor
4C2×#15689470C213yesmaximal, complemented
5C2×#15689414C213yesmaximal, complemented
6C2×#15689324C213yesmaximal, complemented
7C2×C4×#10625325C213yesmaximal, complemented
8C2×C8×Q8×C47C211yesmaximal, characteristic, complemented
9C22×#10625354—21no—
10C22×#10625325C2213yes—
11Q8×Dic94—24no—
12Q8×Dic94C2218yescomplemented
13#15689795—24no—
14#15689795C2218yes—
15C2×#13094024C2213yes—
16C2×#13094023—23no—
17C2×#13094021C2213yes—
18C2×#13093999—23no—
19C2×Q8×Dic47—21no—
20C2×C4×Dic94C2213yes—
21C2×C4×D188C2213yes—
22#15689470C22124yescomplemented
23#15689414C22124yescomplemented
24#15689324C22124yescomplemented
25C4×#10625325C22124yescomplemented
26C2×#13093956C2213yes—
27C2×#13093949C2211yescharacteristic
28C2×#13093946C2216yes—
29C2×#13093946C2213yes—
30C2×C4×Q8×C47C2211yescharacteristic
31C8×Q8×C47C2218yescomplemented
32C2×C4⋊C8×C47C2213yes—
33C2×C4×C8×C47C2213yes—
34C2×#10625354—28no—
35C2×#10625354—48no—
36C22×Dic94—23no—
37C22×Dic94C2311yescharacteristic
38C22×D188—23no—
39C22×D188—23no—
40C22×D188C2313yes—
41C22×C4×D47—23no—
42C2×#10625325—212no—
43C2×#10625325C23112yes—
44#13094024—26no—
45#13094024C23112yes—
46#13094023—224no—
47#13094021—26no—
48#13094021C23112yes—
49#13093999—224no—
50Q8×Dic47—28no—
51C4×Dic94—26no—
52C4×Dic94C23112yes—
53C4×D188—26no—
54C4×D188C23112yes—
55C2×C4⋊Dic47—23no—
56C2×C4⋊Dic47C2311yescharacteristic
57C2×C4⋊Dic47C2313yes—
58C2×Dic47⋊C4—23no—
59C2×(C2×D47)⋊C4—23no—
60C2×C4×Dic47—23no—
61#13093956C23112yes—
62#13093949C2314yes—
63#13093946C23124yes—
64#13093946C23112yes—
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—23no—
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—21no—
83#10625354—432no—
84C2×D188—26no—
85C2×D188—26no—
86C2×D188—23no—
87C2×D188—212no—
88C2×D188—212no—
89C2×D188—412no—
90C2×D188—86no—
91C2×D188C2416yes—
92C2×C4×D47—212no—
93C2×C4×D47—412no—
94C2×Dic94—21no—
95C2×Dic94—22no—
96C2×Dic94—212no—
97C2×Dic94—86no—
98C2×Dic94#4114yescomplemented
99C2×Dic94C2412yes—
100C2×C8×C47—26no—
101C2×C8×C47—22no—
102C2×C8×C47C2416yes—
103C2×C8×C47C2412yes—
104C4⋊C4×C47C2×D4112yes—
105C42×C47C2×D416yes—
106C42×C47C2416yes—
107(C2×D47)⋊C4—212no—
108C4⋊Dic47—212no—
109C4⋊Dic47—21no—
110C4⋊Dic47—23no—
111C4⋊Dic47C2416yes—
112C4⋊Dic47C2412yes—
113Dic47⋊C4—212no—
114C4×Dic47—212no—
115#10625325—224no—
116C23×C47C22×D411yescharacteristic
117C22×D47—26no—
118C22×D47—46no—
119C22×D47—46no—
120C22×D47—46no—
121Q8×C47C2×SD16116yescomplemented
122C2×C4×C47—26no—
123C2×C4×C47C2×SD16112yes—
124C2×C4×C47C22×D416yes—
125C2×C4×C47C22×D411yescharacteristic
126C2×C4×C47C22×D416yes—
127C2×C4×C47C2×#4112yes—
128C2×C4×C472−1+412yes—
129C2×C4×C47C2511yescharacteristic
130C2×Dic47—24no—
131C2×Dic47—22no—
132C2×Dic47—26no—
133C2×Dic47—22no—
134C2×Dic47—42no—
135C2×Dic47—46no—
136D188—26no—
137D188—26no—
138D188—46no—
139D188—424no—
140D188—812no—
141C4×D47—424no—
142Dic94—24no—
143Dic94—22no—
144Dic94—812no—
145Dic94C2×#4114yescomplemented
146C8×C47—26no—
147C8×C47—22no—
148C2×#3362—471nomaximal, Sylow
149C22×C47#54812yes—
150C22×C47C22×SD1612yes—
151C22×C47#57612yes—
152C22×C47C23×D411yescharacteristic
153C2×D47—46no—
154C2×D47—46no—
155C2×D47—46no—
156C2×D47—812no—
157C4×C47—26no—
158C4×C47#54812yes—
159C4×C47C22×SD16112yes—
160C4×C47C22×#4111yescharacteristic
161C4×C47C2×2−1+411yescharacteristic
162Dic47—24no—
163Dic47—42no—
164Dic47—46no—
165C2×Q82—471no—
166C2×#550—471no—
167#3362—4716no—
168C2×#486—473no—
169C2×#476—473no—
170C2×#460—473no—
171C2×C8×Q8D4711yescharacteristic, complemented
172C2×C4×SD16—473no—
173C2×C47#336214yescomplemented
174C2×C47C2×#54811yescharacteristic
175C2×C47C23×SD1611yescharacteristic
176C2×C47C2×#57611yescharacteristic
177D47—86no—
178C22×#41—941no—
179C22×SD16—473no—
180Q82—478no—
181Q82—944no—
182#550—478no—
183#550—944no—
184C2×C4⋊Q8—473no—
185C2×C4⋊Q8—943no—
186C2×#127—473no—
187C2×#121—943no—
188C2×C4×Q8—473no—
189C2×C4×Q8—941no—
190C2×C4×Q8C2×D4711yescharacteristic
191C2×C4×D4—473no—
192#486—4724no—
193#476—4724no—
194#460—4724no—
195C8×Q8C2×D4718yescomplemented
196C4×SD16—4724no—
197C2×#106—473no—
198C2×C4⋊C8C2×D4713yes—
199C2×#103—471no—
200C2×#102—473no—
201C2×#102—476no—
202C2×C4×C8C2×D4713yes—
203C47C2×#336211yescharacteristic, complemented, Sylow
204C2×#41—948no—
205C2×#41—1888no—
206C22×Q8—471no—
207C22×Q8—943no—
208C22×Q8D18811yescharacteristic
209C22×D4—473no—
210C22×D4—943no—
211C22×D4—943no—
212C23×C4—943no—
213C2×SD16—4712no—
214C2×SD16—9412no—
215C22×C8C22×D4713yes—
216C22×C8C22×D4711yescharacteristic
217C4⋊Q8—4712no—
218C4⋊Q8—9424no—
219C4⋊Q8—946no—
220#127—4712no—
221#127—946no—
222#121—9424no—
223C4×Q8—4712no—
224C4×Q8—948no—
225C4×Q8—946no—
226C4×Q8D18814yes—
227C4×Q8C22×D4714yes—
228C4×D4—4712no—
229C4×D4—946no—
230C2×C4⋊C4—471no—
231C2×C4⋊C4—473no—
232C2×C4⋊C4—943no—
233C2×C4⋊C4—943no—
234C2×C4⋊C4D18813yes—
235C2×C22⋊C4—943no—
236C2×C42—943no—
237C2×C42C22×D4713yes—
238#106—4712no—
239C4⋊C8C22×D47112yes—
240#103—474no—
241#102—4712no—
242#102—4724no—
243C4×C8C22×D47112yes—
244C24—943no—
245#41—18832no—
246C2×Q8—474no—
247C2×Q8—472no—
248C2×Q8—941no—
249C2×Q8—9412no—
250C2×Q8—942no—
251C2×Q8—3766no—
252C2×Q8#1062532518yescomplemented
253C2×Q8C2×D18814yes—
254C2×D4—476no—
255C2×D4—9412no—
256C2×D4—943no—
257C2×D4—9412no—
258C2×D4—946no—
259C2×D4—946no—
260C2×D4—3766no—
261C2×D4—18812no—
262C22×C4—941no—
263C22×C4—943no—
264C22×C4—941no—
265C22×C4—9412no—
266C22×C4—18812no—
267C22×C4C2×D18813yes—
268C22×C4C2×D18813yes—
269C22×C4C23×D4711yescharacteristic
270SD16—9424no—
271C2×C8—26no—
272C2×C8—22no—
273C2×C8C23×D4712yes—
274C2×C8C23×D4716yes—
275C4⋊C4—476no—
276C4⋊C4—472no—
277C4⋊C4—9412no—
278C4⋊C4—941no—
279C4⋊C4—943no—
280C4⋊C4—9412no—
281C4⋊C4C2×D188112yes—
282C22⋊C4—9412no—
283C42—9412no—
284C42C2×D18816yes—
285C42C23×D4716yes—
286C23—946no—
287C23—1886no—
288C23—1886no—
289C23—1886no—
290C23C22×D18811yescharacteristic
291Q8—474no—
292Q8—944no—
293Q8—942no—
294Q8—37612no—
295Q8C2×#10625325116yescomplemented
296D4—946no—
297D4—946no—
298D4—37612no—
299D4—1886no—
300D4—18824no—
301C2×C4—26no—
302C2×C4—944no—
303C2×C4—942no—
304C2×C4—946no—
305C2×C4—942no—
306C2×C4—1882no—
307C2×C4—1886no—
308C2×C4—18824no—
309C2×C4C2×#10625325112yes—
310C2×C4C22×D18816yes—
311C2×C4C22×D18811yescharacteristic
312C2×C4C22×D18816yes—
313C2×C4#41×D4712yes—
314C2×C4#1309407212yes—
315C2×C4C24×D4711yescharacteristic
316C8—26no—
317C8—22no—
318C22—37612no—
319C22—1886no—
320C22—1886no—
321C22—1886no—
322C22#1568978512yes—
323C22C22×#1062532512yes—
324C22#1568992812yes—
325C22C23×D18811yescharacteristic
326C4—26no—
327C4—944no—
328C4—1882no—
329C4—1886no—
330C4#1568978512yes—
331C4C22×#10625325112yes—
332C4C2×#41×D4711yescharacteristic
333C4C2×#1309407211yescharacteristic
334C2—3766no—
335C2#2025606014yesdirect product factor
336C2C2×#1568978511yescharacteristic
337C2C23×#1062532511yescharacteristic
338C2C2×#1568992811yescharacteristic
subgroupG/H
centerC23C22×D188
commutator, cosocleC2×C4×C47C25
FrattiniC2×C4C24×D47
FittingC2×C8×Q8×C47C2
socleC23×C47C22×D4
prime radicalC2×#15689795C2
hypercenterC2×C8×Q8D47
hypocenterC47C2×#3362
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×Q8×Dic94C2semidirect
3. #20256060C2direct (N × H)
4. C2×#15689470C2semidirect
5. C2×#15689414C2semidirect
6. C2×#15689324C2semidirect
7. C2×C4×#10625325C2semidirect
8. C2×C8×Q8×C47C2semidirect
12. Q8×Dic94C22semidirect
22. #15689470C22semidirect
23. #15689414C22semidirect
24. #15689324C22semidirect
25. C4×#10625325C22semidirect
31. C8×Q8×C47C22semidirect
75. C2×Q8×C47SD16semidirect
98. C2×Dic94#41semidirect
121. Q8×C47C2×SD16semidirect
145. Dic94C2×#41semidirect
171. C2×C8×Q8D47semidirect
173. C2×C47#3362semidirect
195. C8×Q8C2×D47semidirect
203. C47C2×#3362semidirect
252. C2×Q8#10625325semidirect
295. Q8C2×#10625325semidirect
335. C2#20256060direct (N × H)

Maximal subgroups

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

subgroupindex
C2×Q8×Dic942
C2×#156897952
#202560602
C2×#156894702
C2×#156894142
C2×#156893242
C2×C4×#106253252
C2×C8×Q8×C472
C2×#336247