Finite Group Databasebeta

← #27151226

Subgroups — #27151226

Order 12416.

500 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 500 conjugacy classes of subgroups — too many to draw. Only its 141 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
1C2×#15690841C211yesmaximal, characteristic, complemented
2C2×#15690836C211yesmaximal, characteristic
3C4×#13095511C211yesmaximal, characteristic, complemented
4#20260954C214yesmaximal, complemented
5#20260947C214yesmaximal, complemented
6#20260759C211yesmaximal, characteristic, complemented
7#20260744C211yesmaximal, characteristic, complemented
8#20260740C211yesmaximal, characteristic, complemented
9C2×#124×C97C211yesmaximal, characteristic, complemented
10C2×#13095511C2211yescharacteristic
11#15690841—24no—
12#15690836—24no—
13C2×#13095501C2212yescomplemented
14C2×#13095499C2212yes—
15C2×#13095498C2212yescomplemented
16C2×#13095496C2212yes—
17C2×#13095495C2211yescharacteristic
18C2×#13095495C2211yescharacteristic
19C4×Dic194C2212yescomplemented
20C4×D388C2212yescomplemented
21C4×C22⋊D97—22no—
22#15690769C2211yescharacteristic
23#15690767—21no—
24C42×D97—21no—
25C4⋊#11715895C2214yescomplemented
26#15690734C2214yescomplemented
27#15690731C2214yes—
28C22×Q8×C97C2211yescharacteristic
29C22×D4×C97C2211yescharacteristic
30#124×C97—21no—
31#124×C97—21no—
32#124×C97C2214yes—
33C2×C22⋊C4×C97—22no—
34C2×C42×C97C2211yescharacteristic
35#13095511—22no—
36#13095511D414yes—
37C2×Dic194C2311yescharacteristic
38C2×D388C2311yescharacteristic
39C2×C22⋊D97—21no—
40C2×C4×D97—21no—
41C2×C4×D97—22no—
42#13095501—24no—
43#13095499—24no—
44#13095498—24no—
45#13095496—24no—
46#13095495—22no—
47#13095495—22no—
48C2×#11715895C2312yes—
49C2×#11715895C2312yes—
50C4⋊Dic97C2312yes—
51#13095490—22no—
52C22⋊Dic97—21no—
53#13095488—22no—
54C4×Dic97—21no—
55C4×Dic97—22no—
56C24×C97—21no—
57C2×Q8×C97—21no—
58C2×Q8×C97—22no—
59C2×Q8×C97—22no—
60C2×Q8×C97C2312yes—
61C2×D4×C97—22no—
62C2×D4×C97—21no—
63C2×D4×C97—22no—
64C2×D4×C97C2312yes—
65C22×C4×C97—21no—
66C22×C4×C97D412yes—
67C22×C4×C97C2311yescharacteristic
68C22⋊C4×C97—24no—
69C22⋊C4×C97—24no—
70C42×C97D411yescharacteristic
71C42×C97D411yescharacteristic
72C42×C97C2312yes—
73C23×C97—22no—
74C23×C97—44no—
75C23×C97—42no—
76C23×C97C2×D411yescharacteristic
77C22×D97—21no—
78Q8×C97—22no—
79Q8×C97—22no—
80Q8×C97—42no—
81D4×C97—22no—
82D4×C97—22no—
83D4×C97—42no—
84C2×C4×C97—22no—
85C2×C4×C97—42no—
86C2×C4×C97—42no—
87C2×C4×C97C2×D412yes—
88C2×C4×C97C2×D412yes—
89C2×C4×C97C2×D412yes—
90C2×C4×C97C2×D411yescharacteristic
91C2×C4×C97#4112yes—
92C2×C4×C97C2411yescharacteristic
93C22⋊D97—24no—
94C2×Dic97—21no—
95C2×Dic97—22no—
96D388—21no—
97D388C2×D412yes—
98C4×D97—24no—
99C4×D97—44no—
100Dic194—21no—
101Dic194C2×D412yes—
102#11715895—22no—
103#11715895—22no—
104C22×C97—22no—
105C22×C97—42no—
106C22×C97—44no—
107C22×C97—84no—
108C22×C97#12112yes—
109C22×C97C22×D411yescharacteristic
110C2×D97—22no—
111C2×D97—42no—
112C4×C97—22no—
113C4×C97—22no—
114C4×C97—42no—
115C4×C97#12112yes—
116C4×C97C22×D411yescharacteristic
117C4×C97C2×#4111yescharacteristic
118Dic97—22no—
119Dic97—42no—
120C2×C97—22no—
121C2×C97—82no—
122C2×C97C2×#12111yescharacteristic
123C2×C97#57512yes—
124D97—42no—
125#3040—971nomaximal, Sylow
126C97#304011yescharacteristic, complemented, Sylow
127C2×#137—971no—
128C2×AGL(1,ℤ/8)—971no—
129C2×#124D9711yescharacteristic, complemented
130C4×#41—971no—
131#464—974no—
132#463—974no—
133#423—971no—
134#419—971no—
135#418—971no—
136C2×#41—971no—
137C22×Q8C2×D9711yescharacteristic
138C22×D4C2×D9711yescharacteristic
139#137—1944no—
140AGL(1,ℤ/8)—1944no—
141C2×Q16—972no—
142C2×SD16—972no—
143C2×SD16—972no—
144C2×D8—972no—
145C2×M16—971no—
146C2×M16—971no—
147#124—21no—
148#124—21no—
149#124C2×D9714yes—
150C4×Q8—972no—
151C4×D4—972no—
152C4×D4—1942no—
153#117—971no—
154#117—1941no—
155C2×C22⋊C4—22no—
156C2×C42—1941no—
157C2×C42C2×D9711yescharacteristic
158C4⋊C8—974no—
159#103—974no—
160#102—974no—
161C24—21no—
162#41—974no—
163#41—1942no—
164C2×Q8—21no—
165C2×Q8—22no—
166C2×Q8—22no—
167C2×Q8—971no—
168C2×Q8C22×D9712yes—
169C2×D4—22no—
170C2×D4—21no—
171C2×D4—22no—
172C2×D4—971no—
173C2×D4—1941no—
174C2×D4C22×D9712yes—
175C22×C4—21no—
176C22×C4—1942no—
177C22×C4—1941no—
178C22×C4C22⋊D9712yes—
179C22×C4C22×D9711yescharacteristic
180Q16—1944no—
181SD16—1944no—
182SD16—1944no—
183D8—1944no—
184M16—1942no—
185M16—1942no—
186C2×C8—972no—
187C2×C8—972no—
188C4⋊C4—972no—
189C4⋊C4—1942no—
190C22⋊C4—24no—
191C22⋊C4—24no—
192C22⋊C4—1941no—
193C22⋊C4—1942no—
194C42—1941no—
195C42—1942no—
196C42C22⋊D9711yescharacteristic
197C42C22⋊D9711yescharacteristic
198C42C22×D9712yes—
199C23—22no—
200C23—44no—
201C23—42no—
202C23—1941no—
203C23C2×C22⋊D9711yescharacteristic
204Q8—22no—
205Q8—22no—
206Q8—42no—
207Q8—972no—
208Q8—1941no—
209D4—22no—
210D4—22no—
211D4—42no—
212D4—972no—
213D4—1944no—
214D4—1941no—
215C2×C4—22no—
216C2×C4—42no—
217C2×C4—42no—
218C2×C4—3884no—
219C2×C4—1942no—
220C2×C4—1941no—
221C2×C4—1944no—
222C2×C4D4×D9712yes—
223C2×C4C2×C22⋊D9712yes—
224C2×C4C2×C22⋊D9712yes—
225C2×C4C2×C22⋊D9711yescharacteristic
226C2×C4#1309551012yes—
227C2×C4C23×D9711yescharacteristic
228C8—1942no—
229C8—1942no—
230C22—22no—
231C22—42no—
232C22—44no—
233C22—84no—
234C22—3882no—
235C22—1942no—
236C22#1569078112yes—
237C22C22×C22⋊D9711yescharacteristic
238C4—22no—
239C4—22no—
240C4—42no—
241C4—3882no—
242C4—1942no—
243C4#1569078112yes—
244C4C2×D4×D9711yescharacteristic
245C4C2×#1309551011yescharacteristic
246C2—22no—
247C2—82no—
248C2—3882no—
249C2C2×#1569078111yescharacteristic
250C2#2026153312yes—
subgroupG/H
centerC22C22×C22⋊D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
FittingC2×#124×C97C2
socleC22×C97C22×D4
prime radicalC2×#15690836C2
hypercenterC2×#124D97
hypocenterC97#3040
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#124

Complements

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

NH ≅ G/Ntype
1. C2×#15690841C2semidirect
3. C4×#13095511C2semidirect
4. #20260954C2semidirect
5. #20260947C2semidirect
6. #20260759C2semidirect
7. #20260744C2semidirect
8. #20260740C2semidirect
9. C2×#124×C97C2semidirect
13. C2×#13095501C22semidirect
15. C2×#13095498C22semidirect
19. C4×Dic194C22semidirect
20. C4×D388C22semidirect
25. C4⋊#11715895C22semidirect
26. #15690734C22semidirect
126. C97#3040semidirect
129. C2×#124D97semidirect

Maximal subgroups

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

subgroupindex
C2×#156908412
C2×#156908362
C4×#130955112
#202609542
#202609472
#202607592
#202607442
#202607402
C2×#124×C972
#304097