Finite Group Databasebeta

← #27150961

Subgroups — #27150961

Order 12416.

994 conjugacy classes of subgroups; only the 153 normal subgroups are shown.

normal subgroups only

This group has 994 conjugacy classes of subgroups — too many to draw. Only its 153 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
1C22×D4×D97C211yesmaximal, characteristic, complemented
2C2×#15690835C211yesmaximal, characteristic, complemented
3#20261242C211yesmaximal, characteristic, complemented
4#20260884C214yesmaximal, complemented
5#20260867C214yesmaximal, complemented
6#20260741C211yesmaximal, characteristic, complemented
7#20260740C211yesmaximal, characteristic, complemented
8#20260698C211yesmaximal, characteristic, complemented
9C2×AGL(1,ℤ/8)×C97C211yesmaximal, characteristic, complemented
10C24×D97—21no—
11C22×D388C2211yescharacteristic
12C22×C22⋊D97—21no—
13C2×D4×D97—24no—
14C2×D4×D97—22no—
15C2×D4×D97—24no—
16C2×D4×D97C2214yes—
17C22×C4×D97C2211yescharacteristic
18#15690835—24no—
19C2×#13095498C2212yescomplemented
20C2×#13095496C2212yescomplemented
21C2×#13095495C2211yescharacteristic, complemented
22#15690791—21no—
23#15690786C2212yescomplemented
24#15690781C2212yescomplemented
25#15690778—21no—
26C4×C22⋊D97—22no—
27#15690769C2211yescharacteristic, complemented
28#15690732C2214yescomplemented
29#15690731C2214yescomplemented
30#15690722C2214yescomplemented
31C2×#41×C97C2211yescharacteristic, complemented
32C22×D4×C97C2211yescharacteristic
33AGL(1,ℤ/8)×C97—24no—
34C2×SD16×C97C2212yescomplemented
35C2×D8×C97C2212yescomplemented
36C2×M16×C97C2211yescharacteristic, complemented
37C23×D97—28no—
38C23×D97—21no—
39C23×D97—24no—
40C23×D97—48no—
41C23×D97D411yescharacteristic
42C2×D388—21no—
43C2×D388—22no—
44C2×D388D414yescomplemented
45C2×D388C2312yes—
46C2×C22⋊D97—28no—
47C2×C22⋊D97—22no—
48C2×C22⋊D97—24no—
49D4×D97—216no—
50D4×D97—28no—
51D4×D97—28no—
52C22×Dic97D411yescharacteristic
53C2×C4×D97—22no—
54C2×C4×D97—22no—
55C2×C4×D97D412yes—
56C2×C4×D97C2312yes—
57#13095498—24no—
58#13095496—24no—
59#13095495—22no—
60C2×#11715895C2312yescomplemented
61C4⋊Dic97C2312yescomplemented
62#13095490—22no—
63C22⋊Dic97—21no—
64C22⋊Dic97—22no—
65#13095488—22no—
66C4×Dic97—21no—
67C24×C97—21no—
68#41×C97—22no—
69#41×C97—22no—
70C2×Q8×C97C2311yescharacteristic, complemented
71C2×D4×C97—21no—
72C2×D4×C97—21no—
73C2×D4×C97—22no—
74C2×D4×C97D414yescomplemented
75C2×D4×C97C2312yes—
76C2×D4×C97C2311yescharacteristic, complemented
77C22×C4×C97—21no—
78C22×C4×C97C2311yescharacteristic
79SD16×C97—24no—
80D8×C97—24no—
81M16×C97—22no—
82C2×C8×C97C2312yescomplemented
83C23×C97—24no—
84C23×C97—22no—
85C23×C97—21no—
86C23×C97—44no—
87C23×C97C2×D411yescharacteristic
88C22×D97—22no—
89C22×D97—24no—
90C22×D97—216no—
91C22×D97—24no—
92C22×D97—22no—
93C22×D97—432no—
94C22×D97—48no—
95C22×D97—44no—
96C22×D97—48no—
97C22×D97C2×D412yes—
98Q8×C97—21no—
99Q8×C97—21no—
100D4×C97—24no—
101D4×C97—22no—
102D4×C97—21no—
103D4×C97—21no—
104D4×C97—42no—
105D4×C97C2×D414yes—
106C2×C4×C97—21no—
107C2×C4×C97—42no—
108C2×C4×C97C2×D411yescharacteristic
109C2×C4×C97C2×D412yes—
110C2×C4×C97C2×D412yes—
111C2×C4×C97C2411yescharacteristic
112C22⋊D97—216no—
113C22⋊D97—44no—
114C2×Dic97—22no—
115C2×Dic97—22no—
116C2×Dic97C2×D412yes—
117D388—24no—
118D388—22no—
119D388C2×D414yes—
120C4×D97—24no—
121C4×D97—22no—
122C4×D97—22no—
123C8×C97—22no—
124#11715895—22no—
125C22×C97—22no—
126C22×C97—24no—
127C22×C97—48no—
128C22×C97—44no—
129C22×C97—41no—
130C22×C97—42no—
131C22×C97#12012yes—
132C22×C97C22×D411yescharacteristic
133C2×D97—28no—
134C2×D97—22no—
135C2×D97—22no—
136C2×D97—24no—
137C2×D97—432no—
138C2×D97—48no—
139C2×D97—44no—
140C2×D97AGL(1,ℤ/8)14yescomplemented
141C4×C97—41no—
142C4×C97#12012yes—
143C4×C97C22×D411yescharacteristic
144C4×C97C22×D411yescharacteristic
145Dic97—22no—
146Dic97—42no—
147C2×C97—22no—
148C2×C97—44no—
149C2×C97—81no—
150C2×C97C2×#12011yescharacteristic
151C2×C97C2×AGL(1,ℤ/8)12yes—
152D97—24no—
153D97—44no—
154#3003—971nomaximal, Sylow
155C97#300311yescharacteristic, complemented, Sylow
156C23×D4—971no—
157C2×AGL(1,ℤ/8)—971no—
158C2×AGL(1,ℤ/8)D9711yescharacteristic, complemented
159#525—971no—
160#450—974no—
161#447—974no—
162#418—971no—
163#418—971no—
164#407—971no—
165C25—1941no—
166C2×#41C2×D9711yescharacteristic, complemented
167C22×D4—974no—
168C22×D4—971no—
169C22×D4—1942no—
170C22×D4—1941no—
171C22×D4—1944no—
172C22×D4—1944no—
173C22×D4C2×D9711yescharacteristic
174C23×C4—971no—
175AGL(1,ℤ/8)—24no—
176AGL(1,ℤ/8)—1944no—
177C2×SD16—972no—
178C2×SD16C2×D9712yescomplemented
179C2×D8—972no—
180C2×D8C2×D9712yescomplemented
181C2×M16—971no—
182C2×M16C2×D9711yescharacteristic, complemented
183#123—1941no—
184C22⋊Q8—972no—
185#121—972no—
186#120—1941no—
187C4×D4—1942no—
188#117—971no—
189#102—974no—
190#102—974no—
191C22⋊C8—974no—
192C24—21no—
193C24—971no—
194C24—3888no—
195C24—1944no—
196C24—1948no—
197C24—1941no—
198#41—22no—
199#41—22no—
200C2×Q8C22×D9711yescharacteristic, complemented
201C2×D4—21no—
202C2×D4—21no—
203C2×D4—22no—
204C2×D4—974no—
205C2×D4—972no—
206C2×D4—1944no—
207C2×D4—1948no—
208C2×D4—19416no—
209C2×D4—1942no—
210C2×D4—1948no—
211C2×D4—1948no—
212C2×D4—1941no—
213C2×D4—1942no—
214C2×D4C22⋊D9714yescomplemented
215C2×D4C22×D9712yes—
216C2×D4C22×D9711yescharacteristic, complemented
217C22×C4—21no—
218C22×C4—971no—
219C22×C4—972no—
220C22×C4—972no—
221C22×C4—1942no—
222C22×C4—1942no—
223C22×C4C22×D9711yescharacteristic
224SD16—24no—
225SD16—1944no—
226D8—24no—
227D8—1944no—
228M16—22no—
229M16—1942no—
230C2×C8—972no—
231C2×C8C22×D9712yescomplemented
232C4⋊C4—972no—
233C4⋊C4—1942no—
234C22⋊C4—1942no—
235C22⋊C4—1942no—
236C22⋊C4—1941no—
237C42—1941no—
238C23—24no—
239C23—22no—
240C23—21no—
241C23—44no—
242C23—972no—
243C23—3884no—
244C23—3888no—
245C23—38832no—
246C23—3888no—
247C23—1944no—
248C23—19416no—
249C23—1944no—
250C23—1942no—
251C23—1942no—
252C23D4×D9711yescharacteristic
253Q8—21no—
254Q8—21no—
255D4—24no—
256D4—22no—
257D4—21no—
258D4—21no—
259D4—42no—
260D4—974no—
261D4—3884no—
262D4—19416no—
263D4—1944no—
264D4—1942no—
265D4C2×C22⋊D9714yes—
266C2×C4—21no—
267C2×C4—42no—
268C2×C4—972no—
269C2×C4—1942no—
270C2×C4—1942no—
271C2×C4—1944no—
272C2×C4—1942no—
273C2×C4—1942no—
274C2×C4D4×D9711yescharacteristic
275C2×C4D4×D9712yes—
276C2×C4C2×C22⋊D9712yes—
277C2×C4C23×D9711yescharacteristic
278C8—22no—
279C8—1942no—
280C22—22no—
281C22—24no—
282C22—48no—
283C22—44no—
284C22—41no—
285C22—42no—
286C22—974no—
287C22—3888no—
288C22—38832no—
289C22—3884no—
290C22—1948no—
291C22—1944no—
292C22—1942no—
293C22—1942no—
294C22#1569077812yes—
295C22C2×D4×D9711yescharacteristic
296C4—41no—
297C4—3882no—
298C4—1942no—
299C4#1569077812yes—
300C4C2×D4×D9711yescharacteristic
301C4C22×C22⋊D9711yescharacteristic
302C2—22no—
303C2—44no—
304C2—81no—
305C2—3884no—
306C2—1944no—
307C2C2×#1569077811yescharacteristic
308C2AGL(1,ℤ/8)×D9712yes—
subgroupG/H
centerC22C2×D4×D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
FittingC2×AGL(1,ℤ/8)×C97C2
socleC22×C97C22×D4
hypercenterC2×AGL(1,ℤ/8)D97
hypocenterC97#3003
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. C22×D4×D97C2semidirect
2. C2×#15690835C2semidirect
3. #20261242C2semidirect
4. #20260884C2semidirect
5. #20260867C2semidirect
6. #20260741C2semidirect
7. #20260740C2semidirect
8. #20260698C2semidirect
9. C2×AGL(1,ℤ/8)×C97C2semidirect
19. C2×#13095498C22semidirect
20. C2×#13095496C22semidirect
21. C2×#13095495C22semidirect
23. #15690786C22semidirect
24. #15690781C22semidirect
27. #15690769C22semidirect
28. #15690732C22semidirect
29. #15690731C22semidirect
30. #15690722C22semidirect
31. C2×#41×C97C22semidirect
34. C2×SD16×C97C22semidirect
35. C2×D8×C97C22semidirect
36. C2×M16×C97C22semidirect
44. C2×D388D4semidirect
60. C2×#11715895C23semidirect
61. C4⋊Dic97C23semidirect
70. C2×Q8×C97C23semidirect
74. C2×D4×C97D4semidirect
76. C2×D4×C97C23semidirect
82. C2×C8×C97C23semidirect
140. C2×D97AGL(1,ℤ/8)semidirect
155. C97#3003semidirect
158. C2×AGL(1,ℤ/8)D97semidirect
166. C2×#41C2×D97semidirect
178. C2×SD16C2×D97semidirect
180. C2×D8C2×D97semidirect
182. C2×M16C2×D97semidirect
200. C2×Q8C22×D97semidirect
214. C2×D4C22⋊D97semidirect
216. C2×D4C22×D97semidirect
231. C2×C8C22×D97semidirect

Maximal subgroups

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

subgroupindex
C22×D4×D972
C2×#156908352
#202612422
#202608842
#202608672
#202607412
#202607402
#202606982
C2×AGL(1,ℤ/8)×C972
#300397