Finite Group Databasebeta

← #428×D97 #27150317

Subgroups — #428×D97 #27150317

Order 12416.

536 conjugacy classes of subgroups.

The subgroup lattice has 296 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×M16×D97C211yesmaximal, characteristic
2#117×D97C211yesmaximal, characteristic
3C2×C4⋊C4×D97C211yesmaximal, characteristic
4#20260778C211yesmaximal, characteristic, complemented
5#20260777C211yesmaximal, characteristic, complemented
6#20260776C212yesmaximal, complemented
7#20260775C211yesmaximal, characteristic, complemented
8#20260774C212yesmaximal, complemented
9#107×D97C212yesmaximal, complemented
10#106×D97C212yesmaximal, complemented
11#428×C97C211yesmaximal, characteristic, complemented
12C22×C4×D97—21no—
13C22×C4×D97C2211yescharacteristic
14C2×#13095495C2211yescharacteristic
15C2×#13095494C2212yes—
16M16×D97C418yescomplemented
17C2×C8×D97C2212yes—
18#15690769C2211yescharacteristic
19#15690768C2211yescharacteristic
20#15690768C2211yescharacteristic
21#15690767—21no—
22#15690766—21no—
23C2×C4⋊Dic97C2211yescharacteristic
24C2×#13095490—21no—
25C4⋊C4×D97—21no—
26C4⋊C4×D97—21no—
27C4⋊C4×D97C2212yes—
28C4⋊C4×D97C2212yes—
29C4⋊C4×D97C2211yescharacteristic
30C4⋊C4×D97C2211yescharacteristic
31C22⋊C4×D97—21no—
32C42×D97—21no—
33#15690744C2212yescomplemented
34#15690743C2212yescomplemented
35#15690743C2212yescomplemented
36#15690742C2212yescomplemented
37#15690741C2212yescomplemented
38#15690741C2212yescomplemented
39C2×M16×C97C2211yescharacteristic
40#117×C97C2211yescharacteristic
41C2×C4⋊C4×C97C2211yescharacteristic
42#107×C97C2212yescomplemented
43#106×C97C2212yescomplemented
44C23×D97D411yescharacteristic
45C22×Dic97—21no—
46C22×Dic97D411yescharacteristic
47C2×C4×D97—21no—
48C2×C4×D97—21no—
49C2×C4×D97—22no—
50C2×C4×D97—22no—
51C2×C4×D97—44no—
52C2×C4×D97C2×C414yes—
53C2×C4×D97D411yescharacteristic
54C2×C4×D97D411yescharacteristic
55C2×C4×D97Q814yes—
56C2×C4×D97C2311yescharacteristic
57C2×C4×D97C2311yescharacteristic
58#13095495C2×C414yescomplemented
59#13095494C2×C418yescomplemented
60C2×#11715895C2312yes—
61C8×D97C2×C418yescomplemented
62C4⋊Dic97—21no—
63C4⋊Dic97C2312yes—
64C4⋊Dic97C2311yescharacteristic
65C4⋊Dic97C2311yescharacteristic
66#13095490—22no—
67#13095490—22no—
68#13095490—21no—
69#13095490—21no—
70C22⋊Dic97—21no—
71#13095488—21no—
72#13095488—21no—
73C4×Dic97—21no—
74C4×Dic97—21no—
75C4×Dic97—21no—
76C22×C4×C97—21no—
77C22×C4×C97C2311yescharacteristic
78M16×C97C2×C414yescomplemented
79C2×C8×C97C2312yes—
80C4⋊C4×C97—21no—
81C4⋊C4×C97C2312yes—
82C4⋊C4×C97C2311yescharacteristic
83C4⋊C4×C97C2311yescharacteristic
84C22⋊C4×C97—21no—
85C42×C97—21no—
86C23×C97C2×D411yescharacteristic
87C22×D97—22no—
88C22×D97—22no—
89C22×D97C4⋊C414yes—
90C22×D97C2×D411yescharacteristic
91C22×D97C2×D411yescharacteristic
92C2×C4×C97—21no—
93C2×C4×C97—21no—
94C2×C4×C97—22no—
95C2×C4×C97—42no—
96C2×C4×C97C22×C412yes—
97C2×C4×C97C2×D411yescharacteristic
98C2×C4×C97C2×Q812yes—
99C2×C4×C97C2411yescharacteristic
100C2×Dic97—21no—
101C2×Dic97—21no—
102C2×Dic97—22no—
103C2×Dic97—42no—
104C2×Dic97C4⋊C414yes—
105C2×Dic97C2×D411yescharacteristic
106C2×Dic97C2×D411yescharacteristic
107C4×D97—44no—
108C4×D97—42no—
109C4×D97—44no—
110C4×D97—42no—
111C4×D97C4⋊C414yes—
112C4×D97C4⋊C414yes—
113C4×D97C22×C412yes—
114C4×D97C22×C412yes—
115C4×D97C2×Q812yes—
116C4×D97C2×Q812yes—
117C8×C97C22×C414yescomplemented
118#11715895C22×C414yescomplemented
119C22×C97—21no—
120C22×C97—21no—
121C22×C97C2×C4⋊C412yes—
122C22×C97C22×D411yescharacteristic
123C2×D97—21no—
124C2×D97—21no—
125C2×D97—28no—
126C2×D97C2×C4⋊C412yes—
127C2×D97C2×C4⋊C412yes—
128C2×D97AGL(1,ℤ/8)12yes—
129C2×D97#13712yes—
130C4×C97—42no—
131C4×C97—41no—
132C4×C97—41no—
133C4×C97C2×C4⋊C412yes—
134C4×C97C23×C411yescharacteristic
135C4×C97C22×Q811yescharacteristic
136Dic97—42no—
137Dic97—41no—
138Dic97—41no—
139Dic97C2×C4⋊C412yes—
140Dic97C2×C4⋊C412yes—
141C2×C97—22no—
142C2×C97C22×C4⋊C411yescharacteristic
143C2×C97C2×AGL(1,ℤ/8)11yescharacteristic
144C2×C97C2×#13711yescharacteristic
145D97—22no—
146D97#42814yesdirect product factor
147C2×#428—971nomaximal, Sylow
148C97C2×#42811yescharacteristic, complemented, Sylow
149C22×M16—971no—
150C2×#117—971no—
151C22×C4⋊C4—971no—
152#428—971no—
153#428—971no—
154#428—971no—
155#428—972no—
156#428—972no—
157#428D9711yescharacteristic, direct product factor
158C2×#107—972no—
159C2×#106—972no—
160C23×C4—971no—
161C23×C4—1941no—
162C2×M16—971no—
163C2×M16—978no—
164C2×M16—972no—
165C2×M16C2×D9711yescharacteristic
166C22×C8—972no—
167#117—971no—
168#117—971no—
169#117—971no—
170#117—1941no—
171#117—1941no—
172#117C2×D9711yescharacteristic
173C2×C4⋊C4—971no—
174C2×C4⋊C4—972no—
175C2×C4⋊C4—972no—
176C2×C4⋊C4—971no—
177C2×C4⋊C4—971no—
178C2×C4⋊C4—1941no—
179C2×C4⋊C4—1941no—
180C2×C4⋊C4—1941no—
181C2×C4⋊C4C2×D9711yescharacteristic
182C2×C22⋊C4—1941no—
183C2×C42—1941no—
184#107—972no—
185#107—972no—
186#107—972no—
187#107C2×D9712yescomplemented
188#106—972no—
189#106—972no—
190#106—972no—
191#106C2×D9712yescomplemented
192C24—971no—
193C22×C4—21no—
194C22×C4—971no—
195C22×C4—974no—
196C22×C4—971no—
197C22×C4—971no—
198C22×C4—974no—
199C22×C4—971no—
200C22×C4—971no—
201C22×C4—3884no—
202C22×C4—1941no—
203C22×C4—1941no—
204C22×C4—1942no—
205C22×C4—1942no—
206C22×C4—1941no—
207C22×C4C22×D9711yescharacteristic
208M16—974no—
209M16—978no—
210M16C4×D9714yescomplemented
211C2×C8—972no—
212C2×C8—978no—
213C2×C8C22×D9712yes—
214C4⋊C4—21no—
215C4⋊C4—971no—
216C4⋊C4—972no—
217C4⋊C4—971no—
218C4⋊C4—1942no—
219C4⋊C4—1942no—
220C4⋊C4—1941no—
221C4⋊C4—1941no—
222C4⋊C4—1941no—
223C4⋊C4C22×D9711yescharacteristic
224C4⋊C4C22×D9712yes—
225C4⋊C4C22×D9711yescharacteristic
226C22⋊C4—21no—
227C22⋊C4—1941no—
228C22⋊C4—1941no—
229C22⋊C4—1941no—
230C42—21no—
231C42—1941no—
232C42—1941no—
233C42—1941no—
234C23—971no—
235C23—974no—
236C23—971no—
237C23—1942no—
238C23—1942no—
239C23D4×D9711yescharacteristic
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—22no—
243C2×C4—42no—
244C2×C4—974no—
245C2×C4—971no—
246C2×C4—971no—
247C2×C4—974no—
248C2×C4—974no—
249C2×C4—972no—
250C2×C4—972no—
251C2×C4—972no—
252C2×C4—972no—
253C2×C4—3884no—
254C2×C4—3882no—
255C2×C4—3882no—
256C2×C4—3884no—
257C2×C4—3882no—
258C2×C4—1941no—
259C2×C4—1941no—
260C2×C4—1942no—
261C2×C4C2×C4×D9712yes—
262C2×C4D4×D9711yescharacteristic
263C2×C4Q8×D9712yes—
264C2×C4C23×D9711yescharacteristic
265C8—974no—
266C8C2×C4×D9714yescomplemented
267C22—21no—
268C22—21no—
269C22—972no—
270C22—972no—
271C22—972no—
272C22—972no—
273C22—1948no—
274C22—1941no—
275C22—1941no—
276C22C4⋊C4×D9712yes—
277C22C2×D4×D9711yescharacteristic
278C4—42no—
279C4—41no—
280C4—41no—
281C4—972no—
282C4—972no—
283C4—3882no—
284C4—3881no—
285C4—3881no—
286C4C4⋊C4×D9712yes—
287C4C22×C4×D9711yescharacteristic
288C4C2×Q8×D9711yescharacteristic
289C2—22no—
290C2—974no—
291C2—1942no—
292C2C2×C4⋊C4×D9711yescharacteristic
293C2AGL(1,ℤ/8)×D9711yescharacteristic
294C2#137×D9711yescharacteristic
subgroupG/H
centerC22C2×D4×D97
commutatorC4×C97C23×C4
FrattiniC2×C4C23×D97
Fitting#428×C97C2
socleC22×C97C22×D4
prime radicalC23×D97D4
cosocleC2×C4×C97C24
hypercenter#428D97
hypocenterC97C2×#428
LCSG ▷ C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C22×C4 ◁ #428

Complements

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

NH ≅ G/Ntype
4. #20260778C2semidirect
5. #20260777C2semidirect
6. #20260776C2semidirect
7. #20260775C2semidirect
8. #20260774C2semidirect
9. #107×D97C2semidirect
10. #106×D97C2semidirect
11. #428×C97C2semidirect
16. M16×D97C4semidirect
33. #15690744C22semidirect
34. #15690743C22semidirect
35. #15690743C22semidirect
36. #15690742C22semidirect
37. #15690741C22semidirect
38. #15690741C22semidirect
42. #107×C97C22semidirect
43. #106×C97C22semidirect
58. #13095495C2×C4semidirect
59. #13095494C2×C4semidirect
61. C8×D97C2×C4semidirect
78. M16×C97C2×C4semidirect
117. C8×C97C22×C4semidirect
118. #11715895C22×C4semidirect
146. D97#428direct (N × H)
148. C97C2×#428semidirect
157. #428D97direct (N × H)
187. #107C2×D97semidirect
191. #106C2×D97semidirect
210. M16C4×D97semidirect
266. C8C2×C4×D97semidirect

Maximal subgroups

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

subgroupindex
C2×M16×D972
#117×D972
C2×C4⋊C4×D972
#202607782
#202607772
#202607762
#202607752
#202607742
#107×D972
#106×D972
#428×C972
C2×#42897