Finite Group Databasebeta

← #489×D97 #27152062

Subgroups — #489×D97 #27152062

Order 12416.

436 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 436 conjugacy classes of subgroups — too many to draw. Only its 125 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4⋊Q8×D97C211yesmaximal, characteristic, complemented
2#124×D97C211yesmaximal, characteristic
3#20261097C211yesmaximal, characteristic, complemented
4#20261096C212yesmaximal, complemented
5#20261095C211yesmaximal, characteristic, complemented
6#20261094C211yesmaximal, characteristic, complemented
7#20261093C212yesmaximal, complemented
8#103×D97C212yesmaximal, complemented
9#102×D97C212yesmaximal
10#97×D97C211yesmaximal, characteristic, complemented
11#489×C97C211yesmaximal, characteristic, complemented
12C2×Q8×D97—21no—
13C2×Q8×D97C2211yescharacteristic
14C2×D4×D97C2211yescharacteristic
15C2×C8×D97C2212yes—
16#15690810C2211yescharacteristic, complemented
17#15690809—21no—
18#15690809C2212yescomplemented
19#15690808—21no—
20#15690799C2211yescharacteristic
21#15690798C2211yescharacteristic
22#15690797C2211yescharacteristic
23#15690796—21no—
24#15690796—21no—
25C4⋊C4×D97—21no—
26C4⋊C4×D97C2212yes—
27C22⋊C4×D97—21no—
28C22⋊C4×D97—21no—
29C42×D97C2211yescharacteristic
30#15690735C2212yescomplemented
31#15690734C2212yescomplemented
32#15690733C2212yescomplemented
33#15690732C2212yes—
34#15690731C2212yes—
35#15690730C2212yes—
36#15690721C2211yescharacteristic, complemented
37#15690720C2212yescomplemented
38C4⋊Q8×C97C2211yescharacteristic, complemented
39#124×C97C2211yescharacteristic
40#103×C97C2212yescomplemented
41#102×C97C2212yes—
42#97×C97C2211yescharacteristic, complemented
43C23×D97—21no—
44C2×Dic194—21no—
45C2×Dic194—21no—
46C2×Dic194—22no—
47C2×Dic194C2311yescharacteristic
48Q8×D97—24no—
49Q8×D97—44no—
50C2×D388C2311yescharacteristic
51C2×C22⋊D97—21no—
52D4×D97—24no—
53C2×C4×D97—21no—
54C2×C4×D97—22no—
55C2×C4×D97D411yescharacteristic
56C2×C4×D97D411yescharacteristic
57C2×C4×D97C2311yescharacteristic
58C2×#11715895C2312yes—
59C8×D97—24no—
60C4⋊Dic97—21no—
61C4⋊Dic97C2312yes—
62#13095490—22no—
63#13095490—22no—
64C22⋊Dic97—21no—
65C22⋊Dic97—21no—
66#13095488—21no—
67#13095488—21no—
68#13095488—21no—
69#13095488—21no—
70C4×Dic97D411yescharacteristic
71C4×Dic97D411yescharacteristic
72C4×Dic97C2311yescharacteristic
73C2×Q8×C97—21no—
74C2×Q8×C97C2311yescharacteristic
75C2×D4×C97C2311yescharacteristic
76C2×C8×C97C2312yes—
77C4⋊C4×C97—21no—
78C4⋊C4×C97C2312yes—
79C22⋊C4×C97—21no—
80C22⋊C4×C97—21no—
81C42×C97C2311yescharacteristic
82C23×C97—21no—
83C22×D97—21no—
84C22×D97—42no—
85C22×D97—42no—
86C22×D97—42no—
87C22×D97C2×D411yescharacteristic
88Q8×C97—22no—
89Q8×C97—42no—
90D4×C97—22no—
91C2×C4×C97—21no—
92C2×C4×C97—22no—
93C2×C4×C97C2×D411yescharacteristic
94C2×C4×C97C2×D411yescharacteristic
95C2×C4×C97C2411yescharacteristic
96C22⋊D97—42no—
97C2×Dic97—21no—
98C2×Dic97—22no—
99C2×Dic97C2×D411yescharacteristic
100C2×Dic97C2×D411yescharacteristic
101C2×Dic97C2×D411yescharacteristic
102D388—22no—
103C4×D97—22no—
104C4×D97—22no—
105C4×D97—42no—
106C4×D97—44no—
107C4×D97#4114yes—
108Dic194—22no—
109Dic194—44no—
110Dic194—42no—
111Dic194—42no—
112C8×C97—22no—
113#11715895—22no—
114C22×C97—41no—
115C22×C97—41no—
116C22×C97—41no—
117C22×C97C22×D411yescharacteristic
118C2×D97—41no—
119C2×D97—41no—
120C2×D97—41no—
121C2×D97—84no—
122C2×D97#12412yes—
123C2×D97AGL(1,ℤ/8)12yes—
124C2×D97#13712yes—
125C4×C97—21no—
126C4×C97—21no—
127C4×C97—41no—
128C4×C97—42no—
129C4×C97C2×#4112yes—
130Dic97—21no—
131Dic97—21no—
132Dic97—41no—
133Dic97—42no—
134Dic97#12412yes—
135C2×C97—81no—
136C2×C97C2×#12411yescharacteristic
137C2×C97C2×AGL(1,ℤ/8)11yescharacteristic
138C2×C97C2×#13711yescharacteristic
139D97—81no—
140D97#48914yesdirect product factor
141C2×#489—971nomaximal, Sylow
142C97C2×#48911yescharacteristic, complemented, Sylow
143C2×C4⋊Q8—971no—
144C2×#124—971no—
145#489—971no—
146#489—971no—
147#489—972no—
148#489—972no—
149#489—971no—
150#489D9711yescharacteristic, direct product factor
151C2×#103—972no—
152C2×#102—972no—
153C2×#97—971no—
154C22×Q8—971no—
155C22×Q8—1941no—
156C22×D4—971no—
157C22×C8—972no—
158C4⋊Q8—972no—
159C4⋊Q8—971no—
160C4⋊Q8—1941no—
161C4⋊Q8—1941no—
162C4⋊Q8C2×D9711yescharacteristic, complemented
163#124—971no—
164#124—971no—
165#124—971no—
166#124—1941no—
167#124—1941no—
168#124C2×D9711yescharacteristic
169C2×C4⋊C4—972no—
170C2×C4⋊C4—1941no—
171C2×C22⋊C4—1941no—
172C2×C22⋊C4—1941no—
173C2×C42—971no—
174#103—972no—
175#103—972no—
176#103—972no—
177#103C2×D9712yescomplemented
178#102—972no—
179#102—972no—
180#102—972no—
181#102C2×D9712yes—
182#97—972no—
183#97—971no—
184#97C2×D9711yescharacteristic, complemented
185C24—1941no—
186C2×Q8—21no—
187C2×Q8—971no—
188C2×Q8—3884no—
189C2×Q8—1941no—
190C2×Q8—1942no—
191C2×Q8—1944no—
192C2×Q8—1941no—
193C2×Q8C22×D9711yescharacteristic
194C2×D4—971no—
195C2×D4—1944no—
196C2×D4—1941no—
197C2×D4C22×D9711yescharacteristic
198C22×C4—971no—
199C22×C4—971no—
200C22×C4—971no—
201C22×C4—1941no—
202C22×C4—1942no—
203C2×C8—972no—
204C2×C8—1944no—
205C2×C8C22×D9712yes—
206C4⋊C4—21no—
207C4⋊C4—972no—
208C4⋊C4—1942no—
209C4⋊C4—1942no—
210C4⋊C4—1941no—
211C4⋊C4C22×D9712yes—
212C22⋊C4—21no—
213C22⋊C4—21no—
214C22⋊C4—1941no—
215C22⋊C4—1941no—
216C22⋊C4—1941no—
217C22⋊C4—1941no—
218C22⋊C4—1941no—
219C22⋊C4—1941no—
220C42—971no—
221C42—971no—
222C42—971no—
223C42C22×D9711yescharacteristic
224C23—21no—
225C23—971no—
226C23—3882no—
227C23—3882no—
228C23—3882no—
229C23—1941no—
230Q8—22no—
231Q8—42no—
232Q8—3882no—
233Q8—3882no—
234Q8—3884no—
235Q8—1942no—
236D4—22no—
237D4—3882no—
238D4—1942no—
239C2×C4—21no—
240C2×C4—22no—
241C2×C4—971no—
242C2×C4—974no—
243C2×C4—971no—
244C2×C4—971no—
245C2×C4—3884no—
246C2×C4—3882no—
247C2×C4—1942no—
248C2×C4—1942no—
249C2×C4—1941no—
250C2×C4—1942no—
251C2×C4D4×D9711yescharacteristic
252C2×C4D4×D9711yescharacteristic
253C2×C4C23×D9711yescharacteristic
254C8—22no—
255C8—1942no—
256C22—41no—
257C22—41no—
258C22—41no—
259C22—7764no—
260C22—972no—
261C22—972no—
262C22—972no—
263C22—3881no—
264C22—3881no—
265C22—3881no—
266C22C2×D4×D9711yescharacteristic
267C4—21no—
268C4—21no—
269C4—41no—
270C4—42no—
271C4—972no—
272C4—3881no—
273C4—3882no—
274C4—1941no—
275C4—1941no—
276C4#41×D9712yes—
277C2—81no—
278C2—7761no—
279C2—974no—
280C2#124×D9711yescharacteristic
281C2AGL(1,ℤ/8)×D9711yescharacteristic
282C2#137×D9711yescharacteristic
subgroupG/H
centerC22C2×D4×D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
Fitting#489×C97C2
socleC22×C97C22×D4
prime radicalC2×D4×D97C22
hypercenter#489D97
hypocenterC97C2×#489
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C42 ◁ #489

Complements

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

NH ≅ G/Ntype
1. C4⋊Q8×D97C2semidirect
3. #20261097C2semidirect
4. #20261096C2semidirect
5. #20261095C2semidirect
6. #20261094C2semidirect
7. #20261093C2semidirect
8. #103×D97C2semidirect
10. #97×D97C2semidirect
11. #489×C97C2semidirect
16. #15690810C22semidirect
18. #15690809C22semidirect
30. #15690735C22semidirect
31. #15690734C22semidirect
32. #15690733C22semidirect
36. #15690721C22semidirect
37. #15690720C22semidirect
38. C4⋊Q8×C97C22semidirect
40. #103×C97C22semidirect
42. #97×C97C22semidirect
140. D97#489direct (N × H)
142. C97C2×#489semidirect
150. #489D97direct (N × H)
162. C4⋊Q8C2×D97semidirect
177. #103C2×D97semidirect
184. #97C2×D97semidirect

Maximal subgroups

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

subgroupindex
C4⋊Q8×D972
#124×D972
#202610972
#202610962
#202610952
#202610942
#202610932
#103×D972
#102×D972
#97×D972
#489×C972
C2×#48997