Finite Group Databasebeta

← #27151232

Subgroups — #27151232

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×#15690835C211yesmaximal, characteristic
2C2×#15690799C211yesmaximal, characteristic, complemented
3#41×Dic97C211yesmaximal, characteristic, complemented
4#20260959C214yesmaximal, complemented
5#20260952C214yesmaximal, complemented
6#20260758C211yesmaximal, characteristic, complemented
7#20260744C211yesmaximal, characteristic, complemented
8#20260741C211yesmaximal, characteristic, complemented
9C2×#137×C97C211yesmaximal, characteristic, complemented
10C22×D388C2211yescharacteristic
11#15690835—24no—
12C2×#13095498C2212yes—
13C2×#13095496C2212yes—
14C2×#13095495C2211yescharacteristic
15#15690799—21no—
16#15690799—21no—
17#15690799C2214yes—
18Q8×Dic97C2212yescomplemented
19D4×Dic97—22no—
20D4×Dic97C2212yescomplemented
21#15690769—21no—
22#15690769C2211yescharacteristic
23C2×#13095488—22no—
24C2×C4×Dic97—21no—
25C2×C4×Dic97C2211yescharacteristic
26#15690739C2214yescomplemented
27#15690734C2214yescomplemented
28#15690732C2214yes—
29C2×#41×C97C2211yescharacteristic
30C22×Q8×C97C2211yescharacteristic
31#137×C97—24no—
32C2×Q16×C97C2212yescomplemented
33C2×SD16×C97C2212yescomplemented
34C2×M16×C97C2211yescharacteristic
35C23×D97—21no—
36C2×D388—22no—
37C2×D388—22no—
38C2×D388—21no—
39C2×D388C2312yes—
40C22×Dic97—22no—
41C22×Dic97D412yes—
42#13095498—24no—
43#13095496—24no—
44#13095495—22no—
45C2×#11715895C2312yes—
46C4⋊Dic97—22no—
47C4⋊Dic97C2312yes—
48C22⋊Dic97—21no—
49C22⋊Dic97—22no—
50#13095488—24no—
51#13095488—24no—
52C4×Dic97—21no—
53C4×Dic97—22no—
54C4×Dic97D411yescharacteristic
55C4×Dic97D411yescharacteristic
56C4×Dic97C2312yes—
57#41×C97—22no—
58#41×C97D414yes—
59C2×Q8×C97—21no—
60C2×Q8×C97—22no—
61C2×Q8×C97—22no—
62C2×Q8×C97C2312yes—
63C2×Q8×C97C2311yescharacteristic
64C2×D4×C97—21no—
65C2×D4×C97C2311yescharacteristic
66C22×C4×C97—21no—
67C22×C4×C97—21no—
68C22×C4×C97C2311yescharacteristic
69Q16×C97—24no—
70SD16×C97—24no—
71M16×C97—22no—
72C2×C8×C97C2312yes—
73C23×C97—21no—
74C23×C97C2×D411yescharacteristic
75C22×D97—22no—
76C22×D97—44no—
77C22×D97—42no—
78Q8×C97—21no—
79Q8×C97—22no—
80Q8×C97—22no—
81Q8×C97—42no—
82Q8×C97C2×D412yes—
83D4×C97—24no—
84D4×C97—21no—
85D4×C97C2×D412yes—
86C2×C4×C97—21no—
87C2×C4×C97—24no—
88C2×C4×C97—22no—
89C2×C4×C97—42no—
90C2×C4×C97C2×D411yescharacteristic
91C2×C4×C97C2×D412yes—
92C2×C4×C97#4112yes—
93C2×C4×C97C2411yescharacteristic
94C2×Dic97—22no—
95C2×Dic97—42no—
96C2×Dic97—44no—
97C2×Dic97C2×D412yes—
98C2×Dic97C2×D412yes—
99D388—22no—
100D388—22no—
101D388—42no—
102C8×C97—22no—
103#11715895—22no—
104C22×C97—22no—
105C22×C97—22no—
106C22×C97—42no—
107C22×C97#12112yes—
108C22×C97C22×D411yescharacteristic
109C2×D97—42no—
110C2×D97—44no—
111C2×D97—84no—
112C4×C97—22no—
113C4×C97—42no—
114C4×C97#12112yes—
115C4×C97C22×D411yescharacteristic
116C4×C97C2×#4111yescharacteristic
117Dic97—22no—
118Dic97—22no—
119Dic97—42no—
120C2×C97—22no—
121C2×C97—42no—
122C2×C97C2×#12111yescharacteristic
123C2×C97#57512yes—
124D97—82no—
125#3040—971nomaximal, Sylow
126C97#304011yescharacteristic, complemented, Sylow
127C2×#137D9711yescharacteristic, complemented
128C2×AGL(1,ℤ/8)—971no—
129C2×#124—971no—
130C4×#41—971no—
131#464—974no—
132#463—974no—
133#423—971no—
134#419—971no—
135#418—971no—
136C2×#41C2×D9711yescharacteristic
137C22×Q8C2×D9711yescharacteristic
138C22×D4—971no—
139#137—24no—
140AGL(1,ℤ/8)—1944no—
141C2×Q16C2×D9712yescomplemented
142C2×SD16—972no—
143C2×SD16C2×D9712yescomplemented
144C2×D8—972no—
145C2×M16—971no—
146C2×M16C2×D9711yescharacteristic
147#124—974no—
148#124—1941no—
149#124—1941no—
150C4×Q8—972no—
151C4×D4—972no—
152C4×D4—1942no—
153#117—971no—
154#117—1941no—
155C2×C22⋊C4—1942no—
156C2×C42—971no—
157C2×C42—1941no—
158C4⋊C8—974no—
159#103—974no—
160#102—974no—
161C24—1941no—
162#41—22no—
163#41C22⋊D9714yes—
164C2×Q8—21no—
165C2×Q8—22no—
166C2×Q8—22no—
167C2×Q8C22×D9712yes—
168C2×Q8C22×D9711yescharacteristic
169C2×D4—21no—
170C2×D4—972no—
171C2×D4—1941no—
172C2×D4—1942no—
173C2×D4—1942no—
174C2×D4C22×D9711yescharacteristic
175C22×C4—21no—
176C22×C4—21no—
177C22×C4—972no—
178C22×C4—1942no—
179C22×C4C22×D9711yescharacteristic
180Q16—24no—
181SD16—24no—
182SD16—1944no—
183D8—1944no—
184M16—22no—
185M16—1942no—
186C2×C8—972no—
187C2×C8C22×D9712yes—
188C4⋊C4—972no—
189C4⋊C4—1942no—
190C22⋊C4—1944no—
191C22⋊C4—1944no—
192C22⋊C4—1942no—
193C22⋊C4—1941no—
194C42—971no—
195C42—971no—
196C42—972no—
197C42—1942no—
198C42—1941no—
199C23—21no—
200C23—3884no—
201C23—3882no—
202C23—1942no—
203C23D4×D9711yescharacteristic
204Q8—21no—
205Q8—22no—
206Q8—22no—
207Q8—42no—
208Q8C2×C22⋊D9712yes—
209D4—24no—
210D4—21no—
211D4—3882no—
212D4—1942no—
213D4—1942no—
214D4C2×C22⋊D9712yes—
215C2×C4—21no—
216C2×C4—24no—
217C2×C4—22no—
218C2×C4—42no—
219C2×C4—972no—
220C2×C4—972no—
221C2×C4—3884no—
222C2×C4—3882no—
223C2×C4—1942no—
224C2×C4D4×D9711yescharacteristic
225C2×C4C2×C22⋊D9712yes—
226C2×C4#1309551012yes—
227C2×C4C23×D9711yescharacteristic
228C8—22no—
229C8—1942no—
230C22—22no—
231C22—22no—
232C22—42no—
233C22—7764no—
234C22—3882no—
235C22—3884no—
236C22#1569078212yes—
237C22C2×D4×D9711yescharacteristic
238C4—22no—
239C4—42no—
240C4—3882no—
241C4—1942no—
242C4—1942no—
243C4#1569078212yes—
244C4C22×C22⋊D9711yescharacteristic
245C4C2×#1309551011yescharacteristic
246C2—22no—
247C2—42no—
248C2—7762no—
249C2C2×#1569078211yescharacteristic
250C2#2026153412yes—
subgroupG/H
centerC22C2×D4×D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
FittingC2×#137×C97C2
socleC22×C97C22×D4
prime radicalC2×#15690835C2
hypercenterC2×#137D97
hypocenterC97#3040
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#137

Complements

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

NH ≅ G/Ntype
2. C2×#15690799C2semidirect
3. #41×Dic97C2semidirect
4. #20260959C2semidirect
5. #20260952C2semidirect
6. #20260758C2semidirect
7. #20260744C2semidirect
8. #20260741C2semidirect
9. C2×#137×C97C2semidirect
18. Q8×Dic97C22semidirect
20. D4×Dic97C22semidirect
26. #15690739C22semidirect
27. #15690734C22semidirect
32. C2×Q16×C97C22semidirect
33. C2×SD16×C97C22semidirect
126. C97#3040semidirect
127. C2×#137D97semidirect
141. C2×Q16C2×D97semidirect
143. C2×SD16C2×D97semidirect

Maximal subgroups

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

subgroupindex
C2×#156908352
C2×#156907992
#41×Dic972
#202609592
#202609522
#202607582
#202607442
#202607412
C2×#137×C972
#304097