Finite Group Databasebeta

← #27150179

Subgroups — #27150179

Order 12416.

764 conjugacy classes of subgroups; only the 213 normal subgroups are shown.

normal subgroups only

This group has 764 conjugacy classes of subgroups — too many to draw. Only its 213 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×#13095511C211yesmaximal, characteristic
2C2×M16×D97C212yesmaximal, complemented
3#20260700C218yesmaximal, complemented
4#20260699C211yesmaximal, characteristic, complemented
5#20260698C212yesmaximal, complemented
6#407×C97C211yesmaximal, characteristic, complemented
7C2×#13095511—28no—
8C2×#13095511C414yes—
9C2×#13095511C2214yes—
10C22×Dic194C411yescharacteristic
11C22×D388C411yescharacteristic
12C22×C22⋊D97C411yescharacteristic
13C22×C22⋊D97C411yescharacteristic
14C22×C4×D97C2211yescharacteristic
15C22×C4×D97C2211yescharacteristic
16C2×#13095495C2212yes—
17C2×#13095494C2214yescomplemented
18M16×D97—28no—
19C2×C8×D97C2214yescomplemented
20C22⋊#11715895C2214yescomplemented
21#15690722C2218yescomplemented
22C23×C4×C97C2211yescharacteristic
23C2×M16×C97C2212yes—
24C22⋊C8×C97C2214yescomplemented
25C23×D97C2×C411yescharacteristic
26C23×D97C2×C411yescharacteristic
27#13095511—216no—
28#13095511—28no—
29#13095511—28no—
30C2×Dic194—24no—
31C2×Dic194C2×C414yes—
32C2×D388—24no—
33C2×D388C2×C414yes—
34C2×C22⋊D97—24no—
35C2×C22⋊D97—24no—
36C2×C22⋊D97C2×C414yes—
37C2×C22⋊D97C2×C414yes—
38C22×Dic97C2×C411yescharacteristic
39C22×Dic97C2×C411yescharacteristic
40C2×C4×D97—24no—
41C2×C4×D97C2×C412yes—
42C2×C4×D97C2×C412yes—
43C2×C4×D97D418yes—
44C2×C4×D97C2312yes—
45C2×C4×D97C2312yes—
46#13095495—24no—
47#13095494—28no—
48C2×#11715895C2314yes—
49C8×D97—28no—
50C24×C97C2×C411yescharacteristic
51C22×C4×C97—24no—
52C22×C4×C97C2×C411yescharacteristic
53C22×C4×C97C2×C412yes—
54C22×C4×C97C2312yes—
55C22×C4×C97C2311yescharacteristic
56M16×C97—24no—
57C2×C8×C97C2314yes—
58C23×C97—22no—
59C23×C97—42no—
60C23×C97C22×C412yes—
61C23×C97C22×C411yescharacteristic
62C22×D97—22no—
63C22×D97—42no—
64C22×D97—42no—
65C22×D97C22⋊C414yes—
66C22×D97C22×C412yes—
67C22×D97C22×C412yes—
68C2×C4×C97—22no—
69C2×C4×C97—24no—
70C2×C4×C97—24no—
71C2×C4×C97C22×C412yes—
72C2×C4×C97C22×C411yescharacteristic
73C2×C4×C97C2×D414yes—
74C2×C4×C97C2411yescharacteristic
75C22⋊D97—28no—
76C22⋊D97—28no—
77C2×Dic97—22no—
78C2×Dic97C22⋊C414yes—
79C2×Dic97C22×C412yes—
80C2×Dic97C22×C412yes—
81D388—28no—
82C4×D97—24no—
83C4×D97—24no—
84C4×D97C22⋊C418yes—
85C4×D97C2×D414yes—
86C4×D97C2×D414yes—
87Dic194—28no—
88C8×C97—24no—
89#11715895—24no—
90C22×C97—22no—
91C22×C97—22no—
92C22×C97—24no—
93C22×C97—44no—
94C22×C97C2×C22⋊C412yes—
95C22×C97C23×C411yescharacteristic
96C2×D97—24no—
97C2×D97—22no—
98C2×D97—24no—
99C2×D97—44no—
100C2×D97—44no—
101C2×D97C2×C22⋊C414yes—
102C4×C97—22no—
103C4×C97C2×C22⋊C412yes—
104C4×C97C22×D412yes—
105Dic97—22no—
106Dic97C2×C22⋊C414yes—
107C2×C97—22no—
108C2×C97—42no—
109C2×C97C22×C22⋊C411yescharacteristic
110C2×C97#56812yes—
111D97—24no—
112D97—42no—
113#2879—971nomaximal, Sylow
114C97#287911yescharacteristic, complemented, Sylow
115C22×#41—971no—
116C22×M16—972no—
117#408—978no—
118#407—971no—
119#407—972no—
120#407D9711yescharacteristic, complemented
121C2×#41—974no—
122C2×#41—974no—
123C2×#41—1948no—
124C22×Q8—971no—
125C22×D4—971no—
126C22×D4—971no—
127C22×D4—971no—
128C23×C4—971no—
129C23×C4—971no—
130C23×C4C2×D9711yescharacteristic
131C2×M16—972no—
132C2×M16—974no—
133C2×M16—1948no—
134C2×M16C2×D9712yes—
135C22×C8—974no—
136C22⋊C8—974no—
137C22⋊C8—978no—
138C22⋊C8C2×D9714yescomplemented
139C24—971no—
140C24—971no—
141C24C4×D9711yescharacteristic
142#41—19416no—
143#41—1948no—
144#41—1948no—
145C2×Q8—974no—
146C2×Q8—1944no—
147C2×D4—974no—
148C2×D4—974no—
149C2×D4—974no—
150C2×D4—1944no—
151C2×D4—1944no—
152C2×D4—1944no—
153C22×C4—24no—
154C22×C4—971no—
155C22×C4—972no—
156C22×C4—978no—
157C22×C4—972no—
158C22×C4—971no—
159C22×C4—972no—
160C22×C4—972no—
161C22×C4—1944no—
162C22×C4C4×D9711yescharacteristic
163C22×C4C4×D9712yes—
164C22×C4C22×D9712yes—
165C22×C4C22×D9711yescharacteristic
166M16—24no—
167M16—1948no—
168M16—1944no—
169C2×C8—974no—
170C2×C8—1948no—
171C2×C8C22×D9714yes—
172C23—22no—
173C23—42no—
174C23—974no—
175C23—972no—
176C23—972no—
177C23—3882no—
178C23—3882no—
179C23—1942no—
180C23C2×C4×D9712yes—
181C23C2×C4×D9711yescharacteristic
182Q8—1948no—
183D4—1948no—
184D4—1948no—
185D4—1948no—
186C2×C4—22no—
187C2×C4—24no—
188C2×C4—24no—
189C2×C4—974no—
190C2×C4—972no—
191C2×C4—978no—
192C2×C4—974no—
193C2×C4—974no—
194C2×C4—972no—
195C2×C4—1942no—
196C2×C4—1944no—
197C2×C4—1944no—
198C2×C4C2×C4×D9712yes—
199C2×C4C2×C4×D9711yescharacteristic
200C2×C4D4×D9714yes—
201C2×C4C23×D9711yescharacteristic
202C8—24no—
203C8—1944no—
204C22—22no—
205C22—22no—
206C22—24no—
207C22—44no—
208C22—974no—
209C22—3884no—
210C22—3884no—
211C22—1942no—
212C22—1944no—
213C22—1944no—
214C22C22⋊C4×D9712yes—
215C22C22×C4×D9711yescharacteristic
216C4—22no—
217C4—974no—
218C4—1942no—
219C4C22⋊C4×D9712yes—
220C4C2×D4×D9712yes—
221C2—22no—
222C2—42no—
223C2—3882no—
224C2—1944no—
225C2C2×C22⋊C4×D9711yescharacteristic
226C2#2026150112yes—
subgroupG/H
center, FrattiniC2×C4C23×D97
commutator, socleC22×C97C23×C4
Fitting#407×C97C2
prime radicalC22×#13095511C2
cosocleC2×C4×C97C24
hypercenter#407D97
hypocenterC97#2879
LCSG ▷ C22×C97 ▷ C97
UCSC1 ◁ C2×C4 ◁ #407

Complements

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

NH ≅ G/Ntype
2. C2×M16×D97C2semidirect
3. #20260700C2semidirect
4. #20260699C2semidirect
5. #20260698C2semidirect
6. #407×C97C2semidirect
17. C2×#13095494C22semidirect
19. C2×C8×D97C22semidirect
20. C22⋊#11715895C22semidirect
21. #15690722C22semidirect
24. C22⋊C8×C97C22semidirect
114. C97#2879semidirect
120. #407D97semidirect
138. C22⋊C8C2×D97semidirect

Maximal subgroups

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

subgroupindex
C22×#130955112
C2×M16×D972
#202607002
#202606992
#202606982
#407×C972
#287997