Finite Group Databasebeta

← #27151233

Subgroups — #27151233

Order 12416.

500 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 500 conjugacy classes of subgroups — too many to draw. Only its 133 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×#15690838C211yesmaximal, characteristic
3C2×#15690799C211yesmaximal, characteristic, complemented
4#20261203C211yesmaximal, characteristic, complemented
5#20260960C214yesmaximal, complemented
6#20260953C214yesmaximal, complemented
7#20260758C211yesmaximal, characteristic, complemented
8#20260739C211yesmaximal, characteristic, complemented
9#419×C97C211yesmaximal, characteristic, complemented
10C2×#13095511C2211yescharacteristic
11C22×D388C2211yescharacteristic
12#15690841—24no—
13#15690838—24no—
14C2×#13095501C2212yescomplemented
15C2×#13095500C2212yes—
16C2×#13095498C2212yescomplemented
17C2×D776C2212yes—
18C2×#13095495C2211yescharacteristic
19#15690799—21no—
20#15690799—21no—
21#15690799C2214yes—
22#15690775C2212yescomplemented
23#15690772C2212yescomplemented
24#15690770—22no—
25#15690767—21no—
26C2×#13095488—22no—
27C2×C4×Dic97C2211yescharacteristic
28C42×D97—21no—
29#15690739C2214yescomplemented
30#15690730C2214yes—
31C22×Q8×C97C2211yescharacteristic
32C2×M16×C97C2211yescharacteristic
33#117×C97C2211yescharacteristic
34#103×C97C2214yescomplemented
35C23×D97—21no—
36#13095511—22no—
37#13095511D414yes—
38C2×Dic194C2311yescharacteristic
39C2×D388—22no—
40C2×D388—22no—
41C2×D388—21no—
42C2×D388C2312yes—
43C2×D388C2311yescharacteristic
44C2×C22⋊D97—21no—
45C22×Dic97D412yes—
46C2×C4×D97—21no—
47C2×C4×D97—22no—
48#13095501—24no—
49#13095500—24no—
50#13095498—24no—
51D776—24no—
52#13095495—22no—
53C2×#11715895C2312yes—
54#13095490—22no—
55#13095488—24no—
56#13095488—24no—
57#13095488—22no—
58C4×Dic97—22no—
59C4×Dic97D411yescharacteristic
60C4×Dic97D411yescharacteristic
61C4×Dic97C2312yes—
62C2×Q8×C97—21no—
63C2×Q8×C97—22no—
64C2×Q8×C97—22no—
65C2×Q8×C97C2312yes—
66C22×C4×C97—21no—
67C22×C4×C97C2311yescharacteristic
68M16×C97—22no—
69C2×C8×C97C2312yes—
70C4⋊C4×C97C2312yes—
71C22⋊C4×C97—21no—
72C42×C97—21no—
73C23×C97C2×D411yescharacteristic
74C22×D97—21no—
75C22×D97—22no—
76C22×D97—44no—
77C22×D97—42no—
78Q8×C97—22no—
79Q8×C97—22no—
80Q8×C97—42no—
81C2×C4×C97—22no—
82C2×C4×C97—22no—
83C2×C4×C97—42no—
84C2×C4×C97C2×D411yescharacteristic
85C2×C4×C97C2×D412yes—
86C2×C4×C97#4112yes—
87C2×C4×C97C2411yescharacteristic
88C22⋊D97—24no—
89C2×Dic97—21no—
90C2×Dic97—42no—
91C2×Dic97C2×D412yes—
92C2×Dic97C2×D412yes—
93D388—22no—
94D388—21no—
95D388—22no—
96D388—42no—
97D388C2×D412yes—
98C4×D97—24no—
99C4×D97—44no—
100Dic194—21no—
101Dic194C2×D412yes—
102C8×C97—22no—
103#11715895—22no—
104C22×C97—22no—
105C22×C97#12112yes—
106C22×C97C22×D411yescharacteristic
107C2×D97—22no—
108C2×D97—42no—
109C2×D97—42no—
110C2×D97—44no—
111C2×D97—84no—
112C4×C97—42no—
113C4×C97—42no—
114C4×C97#12112yes—
115C4×C97C22×D411yescharacteristic
116C4×C97C2×#4111yescharacteristic
117Dic97—22no—
118Dic97—22no—
119Dic97—22no—
120C2×C97—22no—
121C2×C97C2×#12111yescharacteristic
122C2×C97#57512yes—
123D97—42no—
124D97—82no—
125#3040—971nomaximal, Sylow
126C97#304011yescharacteristic, complemented, Sylow
127C2×#137—971no—
128C2×AGL(1,ℤ/8)—971no—
129C2×#124—971no—
130C4×#41—971no—
131#464—974no—
132#463—974no—
133#423—971no—
134#419D9711yescharacteristic, complemented
135#418—971no—
136C2×#41—971no—
137C22×Q8C2×D9711yescharacteristic
138C22×D4—971no—
139#137—1944no—
140AGL(1,ℤ/8)—1944no—
141C2×Q16—972no—
142C2×SD16—972no—
143C2×SD16—972no—
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—1941no—
154#117C2×D9711yescharacteristic
155C2×C22⋊C4—1942no—
156C2×C42—971no—
157C2×C42—1941no—
158C4⋊C8—974no—
159#103C2×D9714yescomplemented
160#102—974no—
161C24—1941no—
162#41—974no—
163#41—1942no—
164C2×Q8—21no—
165C2×Q8—22no—
166C2×Q8—22no—
167C2×Q8—971no—
168C2×Q8C22×D9712yes—
169C2×D4—971no—
170C2×D4—972no—
171C2×D4—1941no—
172C2×D4—1941no—
173C2×D4—1942no—
174C2×D4—1942no—
175C22×C4—21no—
176C22×C4—972no—
177C22×C4—1942no—
178C22×C4—1941no—
179C22×C4C22×D9711yescharacteristic
180Q16—1944no—
181SD16—1944no—
182SD16—1944no—
183D8—1944no—
184M16—22no—
185M16—1942no—
186C2×C8—972no—
187C2×C8C22×D9712yes—
188C4⋊C4—1942no—
189C4⋊C4C22×D9712yes—
190C22⋊C4—21no—
191C22⋊C4—1944no—
192C22⋊C4—1944no—
193C22⋊C4—1942no—
194C42—21no—
195C42—971no—
196C42—971no—
197C42—972no—
198C42—1942no—
199C23—3884no—
200C23—3882no—
201C23—1942no—
202C23—1941no—
203C23D4×D9711yescharacteristic
204Q8—22no—
205Q8—22no—
206Q8—42no—
207Q8—972no—
208Q8—1941no—
209D4—972no—
210D4—3882no—
211D4—1944no—
212D4—1942no—
213D4—1942no—
214D4—1941no—
215C2×C4—22no—
216C2×C4—22no—
217C2×C4—42no—
218C2×C4—972no—
219C2×C4—972no—
220C2×C4—3884no—
221C2×C4—3882no—
222C2×C4—1941no—
223C2×C4—1944no—
224C2×C4D4×D9711yescharacteristic
225C2×C4D4×D9712yes—
226C2×C4#1309551112yes—
227C2×C4C23×D9711yescharacteristic
228C8—22no—
229C8—1942no—
230C22—22no—
231C22—7764no—
232C22—3882no—
233C22—3884no—
234C22—3882no—
235C22—1942no—
236C22#1569078312yes—
237C22C2×D4×D9711yescharacteristic
238C4—42no—
239C4—42no—
240C4—1942no—
241C4—1942no—
242C4—1942no—
243C4#1569078312yes—
244C4C2×D4×D9711yescharacteristic
245C4C2×#1309551111yescharacteristic
246C2—22no—
247C2—7762no—
248C2—3882no—
249C2C2×#1569078311yescharacteristic
250C2#2026153412yes—
subgroupG/H
centerC22C2×D4×D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
Fitting#419×C97C2
socleC22×C97C22×D4
prime radicalC2×#15690838C2
hypercenter#419D97
hypocenterC97#3040
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C22×C4 ◁ #419

Complements

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

NH ≅ G/Ntype
1. C2×#15690841C2semidirect
3. C2×#15690799C2semidirect
4. #20261203C2semidirect
5. #20260960C2semidirect
6. #20260953C2semidirect
7. #20260758C2semidirect
8. #20260739C2semidirect
9. #419×C97C2semidirect
14. C2×#13095501C22semidirect
16. C2×#13095498C22semidirect
22. #15690775C22semidirect
23. #15690772C22semidirect
29. #15690739C22semidirect
34. #103×C97C22semidirect
126. C97#3040semidirect
134. #419D97semidirect
159. #103C2×D97semidirect

Maximal subgroups

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

subgroupindex
C2×#156908412
C2×#156908382
C2×#156907992
#202612032
#202609602
#202609532
#202607582
#202607392
#419×C972
#304097