Finite Group Databasebeta

← #27142192

Subgroups — #27142192

Order 12416.

378 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 378 conjugacy classes of subgroups — too many to draw. Only its 89 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×#15690793C211yesmaximal, characteristic
2C2×#15690769C211yesmaximal, characteristic, complemented
3C2×#15690724C212yesmaximal
4#20260501C212yesmaximal, complemented
5#381×C97C211yesmaximal, characteristic, complemented
6C22×C22⋊D97C2211yescharacteristic
7#15690793—21no—
8#15690793—21no—
9#15690793—22no—
10#15690769—24no—
11C2×C4⋊Dic97—21no—
12C2×#13095490C412yescomplemented
13C2×C22⋊Dic97C2211yescharacteristic
14C2×C22⋊Dic97C2212yes—
15C2×#13095488C412yescomplemented
16C2×C4×Dic97—21no—
17#15690724—24no—
18C23×C4×C97C2211yescharacteristic
19C2×C22⋊C4×C97C2212yes—
20#95×C97—22no—
21C23×D97C2×C411yescharacteristic
22C2×C22⋊D97—21no—
23C2×C22⋊D97—21no—
24C2×C22⋊D97—22no—
25C2×C22⋊D97—22no—
26C22×Dic97—22no—
27C22×Dic97C2×C411yescharacteristic
28C22×Dic97C2×C412yes—
29C4⋊Dic97—24no—
30#13095490—22no—
31#13095490—22no—
32C22⋊Dic97—22no—
33C22⋊Dic97—21no—
34C22⋊Dic97—21no—
35C22⋊Dic97D414yes—
36#13095488—22no—
37#13095488—22no—
38C4×Dic97—24no—
39C24×C97C2311yescharacteristic
40C22×C4×C97—22no—
41C22×C4×C97—22no—
42C22×C4×C97—22no—
43C22×C4×C97C2×C412yes—
44C22×C4×C97D411yescharacteristic
45C22×C4×C97D411yescharacteristic
46C22×C4×C97D412yes—
47C22⋊C4×C97—22no—
48C22⋊C4×C97—22no—
49C23×C97—42no—
50C23×C97C22×C411yescharacteristic
51C23×C97C2×D412yes—
52C23×C97C2×D411yescharacteristic
53C23×C97C2×D411yescharacteristic
54C23×C97#4112yes—
55C22×D97—21no—
56C22×D97—22no—
57C22×D97—42no—
58C2×C4×C97—24no—
59C2×C4×C97—21no—
60C2×C4×C97—21no—
61C2×C4×C97—24no—
62C2×C4×C97—21no—
63C2×C4×C97—21no—
64C2×C4×C97—44no—
65C2×C4×C97—42no—
66C2×C4×C97C22⋊C412yes—
67C2×C4×C97C22⋊C412yes—
68C22⋊D97—41no—
69C22⋊D97—42no—
70C22⋊D97—41no—
71C2×Dic97—22no—
72C2×Dic97—21no—
73C2×Dic97—22no—
74C2×Dic97—24no—
75C2×Dic97—44no—
76C2×Dic97—42no—
77C22×C97—22no—
78C22×C97—22no—
79C22×C97—22no—
80C22×C97—42no—
81C22×C97—42no—
82C22×C97C2×C22⋊C411yescharacteristic
83C22×C97C4×D412yes—
84C22×C97#12011yescharacteristic
85C22×C97#12112yes—
86C22×C97#12311yescharacteristic
87C2×D97—41no—
88C2×D97—42no—
89C2×D97—82no—
90C4×C97—22no—
91C4×C97—22no—
92C4×C97—82no—
93Dic97—44no—
94Dic97—41no—
95Dic97—42no—
96C2×C97—21no—
97C2×C97—22no—
98C2×C97—22no—
99C2×C97—21no—
100C2×C97#38611yescharacteristic
101C2×C97#41012yes—
102D97—81no—
103#1899—971nomaximal, Sylow
104C97#189911yescharacteristic, complemented, Sylow
105C2×#123—971no—
106C2×#117—971no—
107C2×#99—972no—
108#381D9711yescharacteristic, complemented
109#342—972no—
110C22×D4—971no—
111C23×C4C2×D9711yescharacteristic
112#123—1941no—
113#123—1941no—
114#123—1942no—
115#117—1944no—
116C2×C4⋊C4—972no—
117C2×C4⋊C4—1941no—
118C2×C22⋊C4—972no—
119C2×C22⋊C4—971no—
120C2×C22⋊C4—972no—
121C2×C22⋊C4C2×D9712yes—
122C2×C42—1941no—
123#99—1944no—
124#95—22no—
125C24—971no—
126C24C22×D9711yescharacteristic
127C2×D4—1942no—
128C2×D4—1941no—
129C2×D4—1941no—
130C2×D4—1942no—
131C22×C4—22no—
132C22×C4—22no—
133C22×C4—22no—
134C22×C4—972no—
135C22×C4—971no—
136C22×C4—1942no—
137C22×C4C4×D9712yes—
138C22×C4D38811yescharacteristic
139C22×C4D38811yescharacteristic
140C22×C4C22⋊D9712yes—
141C4⋊C4—1944no—
142C4⋊C4—1942no—
143C4⋊C4—1942no—
144C22⋊C4—22no—
145C22⋊C4—22no—
146C22⋊C4—974no—
147C22⋊C4—1942no—
148C22⋊C4—1941no—
149C22⋊C4—1941no—
150C22⋊C4—1942no—
151C22⋊C4—1942no—
152C42—1944no—
153C23—42no—
154C23—3882no—
155C23—1942no—
156C23—1941no—
157C23C2×C4×D9711yescharacteristic
158C23D4×D9712yes—
159C23C2×C22⋊D9711yescharacteristic
160C23C2×D38811yescharacteristic
161C23#1309551012yes—
162D4—3881no—
163D4—3882no—
164D4—3881no—
165C2×C4—24no—
166C2×C4—21no—
167C2×C4—21no—
168C2×C4—24no—
169C2×C4—21no—
170C2×C4—21no—
171C2×C4—44no—
172C2×C4—42no—
173C2×C4—3884no—
174C2×C4—3882no—
175C2×C4—1942no—
176C2×C4—1941no—
177C2×C4—1942no—
178C2×C4—1944no—
179C2×C4#1309548812yes—
180C2×C4#1309548812yes—
181C22—22no—
182C22—22no—
183C22—22no—
184C22—42no—
185C22—42no—
186C22—7762no—
187C22—3881no—
188C22—3882no—
189C22C2×#1309548811yescharacteristic
190C22#1569077012yes—
191C22#1569078011yescharacteristic
192C22#1569078212yes—
193C22#1569079411yescharacteristic
194C4—22no—
195C4—22no—
196C4—82no—
197C4—3884no—
198C4—3881no—
199C4—3882no—
200C2—22no—
201C2—21no—
202C2—22no—
203C2—21no—
204C2—7761no—
205C2#2026061211yescharacteristic
206C2#2026070812yes—
subgroupG/H
centerC22C2×#13095488
commutatorC23×C97C22×C4
FrattiniC24C22×D97
Fitting#381×C97C2
socleC22×C97C2×C22⋊C4
prime radicalC22×C22⋊D97C22
cosocleC24×C97C23
hypercenter#381D97
hypocenterC97#1899
LCSG ▷ C23×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C23×C4 ◁ #381

Complements

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

NH ≅ G/Ntype
2. C2×#15690769C2semidirect
4. #20260501C2semidirect
5. #381×C97C2semidirect
12. C2×#13095490C4semidirect
15. C2×#13095488C4semidirect
104. C97#1899semidirect
108. #381D97semidirect

Maximal subgroups

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

subgroupindex
C2×#156907932
C2×#156907692
C2×#156907242
#202605012
#381×C972
#189997