Finite Group Databasebeta

← #27142033

Subgroups — #27142033

Order 12416.

408 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 408 conjugacy classes of subgroups — too many to draw. Only its 113 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×C4×C22⋊D97C211yesmaximal, characteristic
2C2×C22⋊#11715895C211yesmaximal, characteristic, complemented
3C2×#15690722C212yesmaximal, complemented
4#20260489C212yesmaximal, complemented
5C22×M16×C97C211yesmaximal, characteristic, complemented
6C22×C22⋊D97C411yescharacteristic
7C22×C4×D97C2211yescharacteristic
8C22×#11715895C2212yes—
9C4×C22⋊D97—22no—
10C4×C22⋊D97—22no—
11C2×#13095490C411yescharacteristic
12C2×C22⋊Dic97C411yescharacteristic
13C2×#13095488C411yescharacteristic
14C2×C4×Dic97C2211yescharacteristic
15C22⋊#11715895—22no—
16#15690722—24no—
17C23×C4×C97C2211yescharacteristic
18C2×M16×C97—22no—
19C2×M16×C97—24no—
20C22×C8×C97C2212yes—
21C23×D97C2×C411yescharacteristic
22C2×C22⋊D97—22no—
23C2×C22⋊D97—22no—
24C2×C22⋊D97—22no—
25C22×Dic97C2×C411yescharacteristic
26C22×Dic97C2×C411yescharacteristic
27C22×Dic97C2×C411yescharacteristic
28C2×C4×D97—21no—
29C2×C4×D97—21no—
30C2×C4×D97—22no—
31C2×C4×D97—22no—
32C2×#11715895—22no—
33C2×#11715895—22no—
34C2×#11715895—22no—
35#13095490—22no—
36C22⋊Dic97—22no—
37#13095488—22no—
38C4×Dic97—21no—
39C4×Dic97—21no—
40C24×C97C2×C411yescharacteristic
41C22×C4×C97—22no—
42C22×C4×C97—22no—
43C22×C4×C97C2×C411yescharacteristic
44C22×C4×C97D412yes—
45C22×C4×C97D412yes—
46C22×C4×C97C2311yescharacteristic
47M16×C97—28no—
48C2×C8×C97—22no—
49C2×C8×C97—22no—
50C2×C8×C97D414yescomplemented
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—22no—
63C2×C4×C97—22no—
64C2×C4×C97—24no—
65C2×C4×C97—22no—
66C2×C4×C97C22⋊C412yes—
67C2×C4×C97C2×D412yes—
68C2×C4×C97C2×D411yescharacteristic
69C2×C4×C97C2×D411yescharacteristic
70C2×C4×C97#4112yes—
71C22⋊D97—28no—
72C2×Dic97—21no—
73C2×Dic97—21no—
74C2×Dic97—21no—
75C2×Dic97—21no—
76C2×Dic97—22no—
77C2×Dic97—22no—
78C2×Dic97M1612yescomplemented
79C4×D97—41no—
80C4×D97—42no—
81C4×D97—41no—
82C8×C97—24no—
83#11715895—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—22no—
100C4×C97#12011yescharacteristic
101C4×C97#12112yes—
102C4×C97#12311yescharacteristic
103Dic97—22no—
104Dic97—41no—
105Dic97—41no—
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×M16D9711yescharacteristic, complemented
116C2×C4×D4—971no—
117C2×C22⋊C8—971no—
118C2×C22⋊C8—972no—
119#336—972no—
120C22×D4—971no—
121C23×C4—971no—
122C23×C4C2×D9711yescharacteristic
123C2×M16—22no—
124C2×M16—24no—
125C22×C8—972no—
126C22×C8C2×D9712yes—
127C4×D4—1942no—
128C4×D4—1942no—
129C2×C4⋊C4—971no—
130C2×C22⋊C4—971no—
131C2×C22⋊C4—971no—
132C2×C42—971no—
133C22⋊C8—1942no—
134C22⋊C8—1944no—
135C24—971no—
136C24C4×D9711yescharacteristic
137C2×D4—1942no—
138C2×D4—1942no—
139C2×D4—1942no—
140C22×C4—22no—
141C22×C4—22no—
142C22×C4—971no—
143C22×C4—971no—
144C22×C4—971no—
145C22×C4—1942no—
146C22×C4—1941no—
147C22×C4—1941no—
148C22×C4—1942no—
149C22×C4C4×D9711yescharacteristic
150C22×C4D38812yes—
151C22×C4C22⋊D9712yes—
152C22×C4C22×D9711yescharacteristic
153M16—28no—
154C2×C8—22no—
155C2×C8—22no—
156C2×C8—1942no—
157C2×C8—1942no—
158C2×C8—1942no—
159C2×C8C22⋊D9714yescomplemented
160C4⋊C4—1942no—
161C22⋊C4—1942no—
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—
174C23#1309548812yes—
175C23C2×C4×D9711yescharacteristic
176D4—1948no—
177C2×C4—22no—
178C2×C4—22no—
179C2×C4—24no—
180C2×C4—22no—
181C2×C4—972no—
182C2×C4—3881no—
183C2×C4—3882no—
184C2×C4—3881no—
185C2×C4—1941no—
186C2×C4—1941no—
187C2×C4—1942no—
188C2×C4—1942no—
189C2×C4—1941no—
190C2×C4—1941no—
191C2×C4#1309548812yes—
192C2×C4C2×C22⋊D9712yes—
193C2×C4C2×C22⋊D9711yescharacteristic
194C2×C4C2×D38811yescharacteristic
195C2×C4#1309551112yes—
196C8—24no—
197C8—3882no—
198C22—22no—
199C22—22no—
200C22—22no—
201C22—42no—
202C22—42no—
203C22—3881no—
204C22—3881no—
205C22—3882no—
206C22—1942no—
207C22—1944no—
208C22C2×#1309548811yescharacteristic
209C22C4×C22⋊D9712yes—
210C22M16×D9711yescharacteristic
211C22M16×D9711yescharacteristic
212C22#1569081812yes—
213C4—22no—
214C4—3881no—
215C4—3881no—
216C4—1942no—
217C4#1569077911yescharacteristic
218C4#1569078412yes—
219C4#1569079311yescharacteristic
220C2—42no—
221C2—3882no—
222C2#2026061411yescharacteristic
223C2#2026069811yescharacteristic
224C2#2026070311yescharacteristic
225C2#2026080412yes—
226C2#2026081112yes—
subgroupG/H
center, FrattiniC22×C4C22×D97
commutator, socleC23×C97C22×C4
FittingC22×M16×C97C2
prime radicalC22×C22⋊D97C4
cosocleC22×C4×C97C23
hypercenterC22×M16D97
hypocenterC97#1870
LCSG ▷ C23×C97 ▷ C97
UCSC1 ◁ C22×C4 ◁ C22×M16

Complements

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

NH ≅ G/Ntype
2. C2×C22⋊#11715895C2semidirect
3. C2×#15690722C2semidirect
4. #20260489C2semidirect
5. C22×M16×C97C2semidirect
50. C2×C8×C97D4semidirect
61. C22×D97M16semidirect
78. C2×Dic97M16semidirect
114. C97#1870semidirect
115. C22×M16D97semidirect
159. C2×C8C22⋊D97semidirect

Maximal subgroups

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

subgroupindex
C2×C4×C22⋊D972
C2×C22⋊#117158952
C2×#156907222
#202604892
C22×M16×C972
#187097