Finite Group Databasebeta

← #27140065

Subgroups — #27140065

Order 12416.

312 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 312 conjugacy classes of subgroups — too many to draw. Only its 81 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×#13095490C211yesmaximal, characteristic
2#20260699C211yesmaximal, characteristic
3#20260460C214yesmaximal, complemented
4#407×C97C211yesmaximal, characteristic
5C23×Dic97C411yescharacteristic
6C23×Dic97C411yescharacteristic
7C2×#13095495—21no—
8C2×#13095490—21no—
9C2×#13095490—21no—
10C2×#13095490—21no—
11C2×#13095490—21no—
12C2×#13095490C412yes—
13C2×#13095490C2212yes—
14C22⋊#11715895—21no—
15C22⋊#11715895C2212yes—
16C23×C4×C97C2211yescharacteristic
17C2×M16×C97—21no—
18C22⋊C8×C97—21no—
19C22⋊C8×C97C2212yes—
20C22×Dic97—21no—
21C22×Dic97—21no—
22C22×Dic97—24no—
23C22×Dic97—21no—
24C22×Dic97—21no—
25C22×Dic97—42no—
26C22×Dic97C2×C412yes—
27C22×Dic97C2×C412yes—
28#13095495—42no—
29C2×#11715895—22no—
30#13095490—22no—
31#13095490—22no—
32#13095490—22no—
33#13095490—22no—
34C24×C97C2×C411yescharacteristic
35C22×C4×C97—42no—
36C22×C4×C97C2×C411yescharacteristic
37C22×C4×C97D411yescharacteristic
38C22×C4×C97D411yescharacteristic
39C22×C4×C97D411yescharacteristic
40C22×C4×C97D411yescharacteristic
41C22×C4×C97C2311yescharacteristic
42M16×C97—42no—
43C2×C8×C97—22no—
44C23×C97—22no—
45C23×C97—22no—
46C23×C97—22no—
47C23×C97C22⋊C411yescharacteristic
48C23×C97C22⋊C411yescharacteristic
49C23×C97C22×C411yescharacteristic
50C2×C4×C97—22no—
51C2×C4×C97—22no—
52C2×C4×C97—41no—
53C2×C4×C97—41no—
54C2×C4×C97—41no—
55C2×C4×C97—41no—
56C2×C4×C97C22⋊C411yescharacteristic
57C2×C4×C97C22⋊C411yescharacteristic
58C2×C4×C97C2×D411yescharacteristic
59C2×C4×C97C2×D411yescharacteristic
60C2×Dic97—24no—
61C2×Dic97—24no—
62C2×Dic97—22no—
63C2×Dic97—22no—
64C2×Dic97—22no—
65C2×Dic97—42no—
66C2×Dic97—42no—
67C2×Dic97—42no—
68C2×Dic97M1614yescomplemented
69C8×C97—42no—
70#11715895—42no—
71C22×C97—28no—
72C22×C97—21no—
73C22×C97—21no—
74C22×C97—21no—
75C22×C97—21no—
76C22×C97—21no—
77C22×C97—21no—
78C22×C97#9912yes—
79C22×C97#10112yes—
80C22×C97C2×C22⋊C411yescharacteristic
81C22×C97C2×M1612yes—
82C4×C97—21no—
83C4×C97—21no—
84C4×C97—21no—
85C4×C97—21no—
86Dic97—24no—
87Dic97—42no—
88C2×C97—22no—
89C2×C97—22no—
90C2×C97—22no—
91C2×C97#40711yescharacteristic
92C2×C97C2×#9911yescharacteristic
93C2×C97C2×#10111yescharacteristic
94#1463—971nomaximal, Sylow
95C97#146311yescharacteristic, complemented, Sylow
96C22×C4⋊C4—971no—
97#407—971no—
98#407D9711yescharacteristic
99#324—974no—
100C23×C4—971no—
101C23×C4—971no—
102C23×C4C2×D9711yescharacteristic
103C2×M16—21no—
104C2×M16—1941no—
105C2×C4⋊C4—972no—
106C2×C4⋊C4—972no—
107C2×C4⋊C4—1941no—
108C2×C4⋊C4—1941no—
109C2×C4⋊C4—1941no—
110C2×C4⋊C4—1941no—
111C22⋊C8—21no—
112C22⋊C8—972no—
113C22⋊C8—1941no—
114C22⋊C8C2×D9712yes—
115C24C4×D9711yescharacteristic
116C22×C4—42no—
117C22×C4—972no—
118C22×C4—972no—
119C22×C4—3882no—
120C22×C4—1944no—
121C22×C4—1941no—
122C22×C4—1941no—
123C22×C4—1941no—
124C22×C4—1941no—
125C22×C4C4×D9711yescharacteristic
126C22×C4D38811yescharacteristic
127C22×C4D38811yescharacteristic
128C22×C4C22⋊D9711yescharacteristic
129C22×C4C22⋊D9711yescharacteristic
130C22×C4C22×D9711yescharacteristic
131M16—42no—
132M16—3882no—
133C2×C8—22no—
134C2×C8—1942no—
135C4⋊C4—1942no—
136C4⋊C4—1942no—
137C4⋊C4—1942no—
138C4⋊C4—1942no—
139C23—22no—
140C23—22no—
141C23—22no—
142C23#1309548811yescharacteristic
143C23#1309548811yescharacteristic
144C23C2×C4×D9711yescharacteristic
145C2×C4—22no—
146C2×C4—22no—
147C2×C4—41no—
148C2×C4—41no—
149C2×C4—41no—
150C2×C4—41no—
151C2×C4—974no—
152C2×C4—3882no—
153C2×C4—3882no—
154C2×C4—3882no—
155C2×C4—1942no—
156C2×C4—1944no—
157C2×C4—1944no—
158C2×C4—1942no—
159C2×C4—1942no—
160C2×C4#1309548811yescharacteristic
161C2×C4#1309548811yescharacteristic
162C2×C4C2×C22⋊D9711yescharacteristic
163C2×C4C2×D38811yescharacteristic
164C8—42no—
165C8—3882no—
166C22—28no—
167C22—21no—
168C22—21no—
169C22—21no—
170C22—21no—
171C22—21no—
172C22—21no—
173C22#1569072412yes—
174C22#1569072812yes—
175C22C2×#1309548811yescharacteristic
176C22M16×D9712yes—
177C4—21no—
178C4—21no—
179C4—21no—
180C4—21no—
181C4—3882no—
182C4—1944no—
183C2—22no—
184C2—22no—
185C2—22no—
186C2#2026069811yescharacteristic
187C2C2×#1569072411yescharacteristic
188C2C2×#1569072811yescharacteristic
subgroupG/H
centerC22C2×#13095488
commutatorC23×C97C22×C4
FrattiniC22×C4C22×D97
Fitting#407×C97C2
socleC22×C97C2×C22⋊C4
prime radicalC24×C97C2×C4
cosocleC22×C4×C97C23
hypercenter#407D97
hypocenterC97#1463
LCSG ▷ C23×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C23×C4 ◁ #407

Complements

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

NH ≅ G/Ntype
3. #20260460C2semidirect
68. C2×Dic97M16semidirect
95. C97#1463semidirect

Maximal subgroups

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

subgroupindex
C22×#130954902
#202606992
#202604602
#407×C972
#146397