Finite Group Databasebeta

← #27157715

Subgroups — #27157715

Order 12416.

408 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 408 conjugacy classes of subgroups — too many to draw. Only its 90 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×#13095520C211yesmaximal, characteristic, complemented
2C2×#15690867C212yesmaximal, complemented
3C2×C22⋊#11715897C211yesmaximal, characteristic, complemented
4#20261601C212yesmaximal, complemented
5C2×#15690770C211yesmaximal, characteristic
6C2×#13095520—24no—
7C2×#13095520—22no—
8C22×#11715897C2212yes—
9C22×#11715897C2212yes—
10#15690867—24no—
11C22⋊#11715897—22no—
12C22×C22⋊D97C411yescharacteristic
13C23×Dic97C2211yescharacteristic
14C22×C4×D97C2211yescharacteristic
15#15690770—22no—
16#15690770—22no—
17C2×#13095490C411yescharacteristic
18C2×#13095488C411yescharacteristic
19C2×C4×Dic97C2211yescharacteristic
20C2×C22⋊C4×C97C411yescharacteristic
21#13095520—28no—
22C2×#11715897—22no—
23C2×#11715897—22no—
24C2×#11715897—22no—
25C2×#11715897—22no—
26C2×#11715897—22no—
27C2×#11715897D414yescomplemented
28C23×D97C2×C411yescharacteristic
29C2×C22⋊D97—22no—
30C2×C22⋊D97—22no—
31C2×C22⋊D97—22no—
32C22×Dic97—22no—
33C22×Dic97—22no—
34C22×Dic97C2×C411yescharacteristic
35C22×Dic97C2×C411yescharacteristic
36C22×Dic97D412yes—
37C22×Dic97D412yes—
38C22×Dic97C2311yescharacteristic
39C2×C4×D97—21no—
40C2×C4×D97—21no—
41C2×C4×D97—22no—
42C2×C4×D97—22no—
43#13095490—22no—
44#13095488—22no—
45C4×Dic97—21no—
46C4×Dic97—21no—
47C24×C97C2×C411yescharacteristic
48C22×C4×C97C2×C411yescharacteristic
49C22×C4×C97C2×C411yescharacteristic
50C22⋊C4×C97—22no—
51C23×C97—22no—
52C23×C97—21no—
53C23×C97—21no—
54C23×C97—22no—
55C23×C97C22⋊C412yes—
56C23×C97C22×C411yescharacteristic
57C22×D97—21no—
58C22×D97—22no—
59C22×D97—21no—
60C22×D97—22no—
61C22×D97M1612yescomplemented
62C2×C4×C97—21no—
63C2×C4×C97—21no—
64C2×C4×C97—22no—
65C2×C4×C97—22no—
66C22⋊D97—28no—
67C2×Dic97—21no—
68C2×Dic97—22no—
69C2×Dic97—21no—
70C2×Dic97—22no—
71C2×Dic97—24no—
72C2×Dic97—22no—
73C2×Dic97C22⋊C412yes—
74C2×Dic97M1612yescomplemented
75C2×Dic97C2×D412yes—
76C2×Dic97C2×D411yescharacteristic
77C2×Dic97C2×D411yescharacteristic
78C2×Dic97#4112yes—
79C4×D97—41no—
80C4×D97—42no—
81C4×D97—41no—
82#11715897—24no—
83#11715897—42no—
84C22×C97—22no—
85C22×C97—22no—
86C22×C97—22no—
87C22×C97—42no—
88C22×C97—42no—
89C22×C97C2×C22⋊C411yescharacteristic
90C22×C97C4×D412yes—
91C22×C97C2×M1611yescharacteristic
92C22×C97C2×M1611yescharacteristic
93C22×C97#13112yes—
94C2×D97—22no—
95C2×D97—24no—
96C2×D97—41no—
97C2×D97—41no—
98C2×D97—42no—
99C4×C97—41no—
100C4×C97—41no—
101Dic97—22no—
102Dic97—22no—
103Dic97#12011yescharacteristic
104Dic97#12112yes—
105Dic97#12311yescharacteristic
106C2×C97—42no—
107C2×C97#38611yescharacteristic
108C2×C97#40711yescharacteristic
109C2×C97#40811yescharacteristic
110C2×C97#43512yes—
111C2×C97#43612yes—
112D97—42no—
113#1870—971nomaximal, Sylow
114C97#187011yescharacteristic, complemented, Sylow
115C22×M16—971no—
116C2×C4×D4—971no—
117C2×C22⋊C8—971no—
118C2×C22⋊C8—972no—
119#336—972no—
120C22×D4—971no—
121C23×C4—971no—
122C23×C4—971no—
123C2×M16—1944no—
124C2×M16—1942no—
125C22×C8—972no—
126C22×C8—972no—
127C4×D4—1942no—
128C4×D4—1942no—
129C2×C4⋊C4—971no—
130C2×C22⋊C4—971no—
131C2×C22⋊C4#8760411yescharacteristic
132C2×C42—971no—
133C22⋊C8—1944no—
134C22⋊C8—1942no—
135C24—971no—
136C24C2×#8760411yescharacteristic
137C2×D4—1942no—
138C2×D4—1942no—
139C2×D4—1942no—
140C22×C4—971no—
141C22×C4—972no—
142C22×C4—972no—
143C22×C4—971no—
144C22×C4—971no—
145C22×C4—1942no—
146C22×C4—1941no—
147C22×C4—1941no—
148C22×C4—1942no—
149C22×C4—1942no—
150C22×C4—1942no—
151C22×C4C2×#8760411yescharacteristic
152C22×C4C2×#8760411yescharacteristic
153M16—1948no—
154C2×C8—974no—
155C2×C8—1942no—
156C2×C8—1942no—
157C2×C8—1942no—
158C2×C8—1942no—
159C2×C8—1942no—
160C4⋊C4—1942no—
161C22⋊C4—22no—
162C22⋊C4—1942no—
163C42—1941no—
164C42—1941no—
165C23—22no—
166C23—21no—
167C23—21no—
168C23—22no—
169C23—972no—
170C23—1941no—
171C23—1942no—
172C23—1941no—
173C23—1942no—
174C23C22⋊#8760412yes—
175C23C22×#8760411yescharacteristic
176D4—1948no—
177C2×C4—21no—
178C2×C4—21no—
179C2×C4—22no—
180C2×C4—22no—
181C2×C4—971no—
182C2×C4—971no—
183C2×C4—972no—
184C2×C4—972no—
185C2×C4—972no—
186C2×C4—972no—
187C2×C4—3882no—
188C2×C4—3881no—
189C2×C4—3881no—
190C2×C4—1942no—
191C2×C4—1944no—
192C2×C4—1942no—
193C2×C4—1941no—
194C2×C4—1942no—
195C2×C4—1941no—
196C8—3882no—
197C8—1944no—
198C22—22no—
199C22—22no—
200C22—22no—
201C22—42no—
202C22—42no—
203C22—3881no—
204C22—3881no—
205C22—3882no—
206C22—1942no—
207C22—1944no—
208C22C2×C22⋊#8760411yescharacteristic
209C22D4×#8760412yes—
210C22D97⋊M1611yescharacteristic
211C22D97⋊M1611yescharacteristic
212C22#1569089712yes—
213C4—41no—
214C4—41no—
215C4—972no—
216C4—971no—
217C4—971no—
218C4—1942no—
219C4—1942no—
220C2—42no—
221C2—3882no—
222C2#2026168311yescharacteristic
223C2#2026170711yescharacteristic
224C2#2026170811yescharacteristic
225C2#2026176712yes—
226C2#2026176812yes—
subgroupG/H
center, FrattiniC23C22×#87604
commutator, socleC23×C97C22×C4
FittingC2×C22⋊C4×C97C4
prime radicalC22×C22⋊D97C4
cosocleC22×Dic97C23
hypercenterC2×C22⋊C4#87604
hypocenterC97#1870
LCSG ▷ C23×C97 ▷ C97
UCSC1 ◁ C23 ◁ C2×C22⋊C4

Complements

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

NH ≅ G/Ntype
1. C22×#13095520C2semidirect
2. C2×#15690867C2semidirect
3. C2×C22⋊#11715897C2semidirect
4. #20261601C2semidirect
27. C2×#11715897D4semidirect
61. C22×D97M16semidirect
74. C2×Dic97M16semidirect
114. C97#1870semidirect

Maximal subgroups

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

subgroupindex
C22×#130955202
C2×#156908672
C2×C22⋊#117158972
#202616012
C2×#156907702
#187097