Finite Group Databasebeta

← #27151100

Subgroups — #27151100

Order 12416.

698 conjugacy classes of subgroups; only the 161 normal subgroups are shown.

normal subgroups only

This group has 698 conjugacy classes of subgroups — too many to draw. Only its 161 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
1#20261533C212yesmaximal, complemented
2#20261309C211yesmaximal, characteristic, complemented
3#20260918C214yesmaximal, complemented
4#20260908C214yesmaximal, complemented
5#20260752C212yesmaximal, complemented
6C2×#15690729C211yesmaximal, characteristic, complemented
7C2×2−1+4×C97C211yesmaximal, characteristic, complemented
8C2×#13095511C2211yescharacteristic
9#15690841—22no—
10#15690840C2214yescomplemented
11#15690836—22no—
12#15690835C2214yescomplemented
13#15690830—24no—
14C2×#13095495C2212yes—
15#15690808C2212yescomplemented
16#15690799C2212yescomplemented
17#15690793—22no—
18#15690786—22no—
19#15690769C2211yescharacteristic
20#15690737C2218yescomplemented
21#15690729C2214yescomplemented
222−1+4×C97—24no—
232−1+4×C97—22no—
242−1+4×C97C2214yes—
25C2×#41×C97—22no—
26C2×#41×C97—22no—
27C2×#41×C97C2212yes—
28C22×Q8×C97—22no—
29C22×Q8×C97C2211yescharacteristic
30#13095511—22no—
31#13095511—22no—
32C2×Dic194C2311yescharacteristic
33C2×D388C2311yescharacteristic
34C2×C22⋊D97—21no—
35C2×C4×D97—21no—
36#13095501—24no—
37#13095499—24no—
38#13095498—24no—
39#13095496—24no—
40#13095495—22no—
41#13095495C2314yes—
42C2×#11715895—22no—
43C4⋊Dic97—21no—
44#13095490—24no—
45C22⋊Dic97—21no—
46#13095488—24no—
47C4×Dic97C2312yes—
48#41×C97—28no—
49#41×C97—28no—
50#41×C97—28no—
51#41×C97—28no—
52#41×C97—22no—
53#41×C97—22no—
54#41×C97D414yes—
55#41×C97C2314yes—
56C2×Q8×C97—22no—
57C2×Q8×C97—28no—
58C2×Q8×C97—28no—
59C2×Q8×C97—22no—
60C2×Q8×C97—22no—
61C2×Q8×C97D412yes—
62C2×Q8×C97D412yes—
63C2×Q8×C97C2312yes—
64C2×D4×C97—22no—
65C2×D4×C97—22no—
66C2×D4×C97D412yes—
67C22×C4×C97—24no—
68C22×C4×C97—22no—
69C22×C4×C97D412yes—
70C22×C4×C97C2311yescharacteristic
71C23×C97—22no—
72C23×C97C2×D411yescharacteristic
73C22×D97—21no—
74Q8×C97—28no—
75Q8×C97—22no—
76Q8×C97—24no—
77Q8×C97—24no—
78Q8×C97C2×D414yes—
79D4×C97—28no—
80D4×C97—24no—
81D4×C97—24no—
82D4×C97—22no—
83D4×C97C2×D414yes—
84C2×C4×C97—216no—
85C2×C4×C97—22no—
86C2×C4×C97—22no—
87C2×C4×C97—22no—
88C2×C4×C97—24no—
89C2×C4×C97—24no—
90C2×C4×C97C2×D414yes—
91C2×C4×C97C2×D411yescharacteristic
92C2×C4×C97C2×D412yes—
93C2×C4×C97C2×D412yes—
94C2×C4×C97C2411yescharacteristic
95C22⋊D97—41no—
96C22⋊D97—41no—
97C2×Dic97—22no—
98C2×Dic97—21no—
99D388—22no—
100C4×D97—42no—
101Dic194—22no—
102#11715895—24no—
103C22×C97—24no—
104C22×C97—41no—
105C22×C97—44no—
106C22×C97#12011yescharacteristic
107C22×C97#12011yescharacteristic
108C22×C97C22×D411yescharacteristic
109C2×D97—42no—
110C2×D97—41no—
111C4×C97—24no—
112C4×C97—24no—
113C4×C97#12012yes—
114C4×C97C22×D412yes—
115Dic97—42no—
116Dic97—41no—
117C2×C97—21no—
118C2×C97—21no—
119C2×C97—21no—
120C2×C97—44no—
121C2×C97C2×#12011yescharacteristic
122D97—81no—
123#3020—971nomaximal, Sylow
124C97#302011yescharacteristic, complemented, Sylow
125C2×2−1+4D9711yescharacteristic, complemented
126#575—972no—
127#536—971no—
128#456—974no—
129#454—974no—
130#421—972no—
131C2×#101—971no—
1322−1+4—24no—
1332−1+4—22no—
1342−1+4C2×D9714yes—
135C2×#41—22no—
136C2×#41—22no—
137C2×#41—971no—
138C2×#41C2×D9712yes—
139C22×Q8—22no—
140C22×Q8C2×D9711yescharacteristic
141#137—974no—
142#137—1942no—
143AGL(1,ℤ/8)—974no—
144AGL(1,ℤ/8)—1942no—
145#135—1944no—
146C2×M16—972no—
147C4⋊Q8—972no—
148#124—972no—
149#123—1942no—
150C22⋊Q8—1942no—
151#117—971no—
152#104—978no—
153#101—974no—
154#41—28no—
155#41—28no—
156#41—28no—
157#41—28no—
158#41—22no—
159#41—22no—
160#41—1942no—
161#41—1942no—
162#41C22⋊D9714yes—
163#41C22×D9714yes—
164C2×Q8—22no—
165C2×Q8—28no—
166C2×Q8—28no—
167C2×Q8—22no—
168C2×Q8—22no—
169C2×Q8—971no—
170C2×Q8C22⋊D9712yes—
171C2×Q8C22⋊D9712yes—
172C2×Q8C22×D9712yes—
173C2×D4—22no—
174C2×D4—22no—
175C2×D4—971no—
176C2×D4—1941no—
177C2×D4C22⋊D9712yes—
178C22×C4—24no—
179C22×C4—22no—
180C22×C4—1941no—
181C22×C4C22⋊D9712yes—
182C22×C4C22×D9711yescharacteristic
183Q16—1944no—
184SD16—1944no—
185SD16—1944no—
186D8—1944no—
187M16—974no—
188M16—1942no—
189C2×C8—1942no—
190C4⋊C4—1944no—
191C4⋊C4—1941no—
192C22⋊C4—1941no—
193C22⋊C4—1944no—
194C42—972no—
195C23—22no—
196C23—1941no—
197C23C2×C22⋊D9711yescharacteristic
198Q8—28no—
199Q8—22no—
200Q8—24no—
201Q8—24no—
202Q8—1942no—
203Q8C2×C22⋊D9714yes—
204D4—28no—
205D4—24no—
206D4—24no—
207D4—22no—
208D4—3881no—
209D4—3881no—
210D4—1942no—
211D4C2×C22⋊D9714yes—
212C2×C4—216no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—22no—
216C2×C4—24no—
217C2×C4—24no—
218C2×C4—3882no—
219C2×C4—1942no—
220C2×C4—1941no—
221C2×C4C2×C22⋊D9714yes—
222C2×C4C2×C22⋊D9711yescharacteristic
223C2×C4C2×C22⋊D9712yes—
224C2×C4C2×C22⋊D9712yes—
225C2×C4C23×D9711yescharacteristic
226C8—1944no—
227C22—24no—
228C22—41no—
229C22—44no—
230C22—3882no—
231C22—3881no—
232C22#1569077911yescharacteristic
233C22#1569077911yescharacteristic
234C22C22×C22⋊D9711yescharacteristic
235C4—24no—
236C4—24no—
237C4—3882no—
238C4—3881no—
239C4#1569077912yes—
240C4C22×C22⋊D9712yes—
241C2—21no—
242C2—21no—
243C2—21no—
244C2—44no—
245C2—7761no—
246C2C2×#1569077911yescharacteristic
subgroupG/H
centerC2C2×#15690779
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
FittingC2×2−1+4×C97C2
socleC2×C97C2×#120
hypercenterC2×2−1+4D97
hypocenterC97#3020
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C2 ◁ C22×C4 ◁ C2×2−1+4

Complements

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

NH ≅ G/Ntype
1. #20261533C2semidirect
2. #20261309C2semidirect
3. #20260918C2semidirect
4. #20260908C2semidirect
5. #20260752C2semidirect
6. C2×#15690729C2semidirect
7. C2×2−1+4×C97C2semidirect
10. #15690840C22semidirect
12. #15690835C22semidirect
15. #15690808C22semidirect
16. #15690799C22semidirect
20. #15690737C22semidirect
21. #15690729C22semidirect
124. C97#3020semidirect
125. C2×2−1+4D97semidirect

Maximal subgroups

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

subgroupindex
#202615332
#202613092
#202609182
#202609082
#202607522
C2×#156907292
C2×2−1+4×C972
#302097