Finite Group Databasebeta

← #27157267

Subgroups — #27157267

Order 12416.

246 conjugacy classes of subgroups; only the 58 normal subgroups are shown.

normal subgroups only

This group has 246 conjugacy classes of subgroups — too many to draw. Only its 58 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
1#20261758C211yesmaximal, characteristic, complemented
2#20261740C211yesmaximal, characteristic, complemented
3#20261594C212yesmaximal, complemented
4#20261593C212yesmaximal, complemented
5#20261282C211yesmaximal, characteristic
6C2×#13095521—21no—
7#15690877—21no—
8#15690877C2212yes—
9#15690877C2212yes—
10#15690867—21no—
11#15690865—21no—
12C2×#13095511—21no—
13#15690809C2211yescharacteristic
14#15690806C2211yescharacteristic
15#15690796—21no—
16#15690785C411yescharacteristic
17#15690783—21no—
18#15690781C411yescharacteristic
19C4×D388C411yescharacteristic
20C4×C22⋊D97—21no—
21C42×D97C2211yescharacteristic
22C4×D4×C97C411yescharacteristic
23#13095521—42no—
24C2×#11715897—22no—
25C2×#11715897—22no—
26#13095511—44no—
27C2×Dic194—21no—
28C2×D388—21no—
29C2×D388C2×C411yescharacteristic
30C2×C22⋊D97—21no—
31C2×C22⋊D97—21no—
32C2×C22⋊D97—21no—
33C2×C4×D97—21no—
34C2×C4×D97C2×C411yescharacteristic
35C2×C4×D97D411yescharacteristic
36C2×C4×D97D411yescharacteristic
37C4⋊Dic97C2×C411yescharacteristic
38#13095490—21no—
39C22⋊Dic97—21no—
40#13095488—21no—
41#13095488—21no—
42C4×Dic97D411yescharacteristic
43C4×Dic97D411yescharacteristic
44C4×Dic97C2311yescharacteristic
45C2×D4×C97C2×C411yescharacteristic
46C22×C4×C97—21no—
47C4⋊C4×C97C2×C411yescharacteristic
48C22⋊C4×C97—21no—
49C42×C97C2×C411yescharacteristic
50C23×C97—21no—
51C22×D97—21no—
52C22×D97C22⋊C411yescharacteristic
53D4×C97—41no—
54C2×C4×C97—21no—
55C2×C4×C97—42no—
56C2×C4×C97C22⋊C411yescharacteristic
57C2×C4×C97C22⋊C411yescharacteristic
58C2×C4×C97C22×C411yescharacteristic
59C22⋊D97—42no—
60C22⋊D97—41no—
61C22⋊D97—42no—
62C22⋊D97—41no—
63C2×Dic97—21no—
64C2×Dic97C22⋊C411yescharacteristic
65C2×Dic97C2×D411yescharacteristic
66C2×Dic97C2×D411yescharacteristic
67D388—41no—
68D388—42no—
69C4×D97—22no—
70C4×D97—21no—
71C4×D97—21no—
72C4×D97—41no—
73C4×D97—42no—
74Dic194—42no—
75#11715897—42no—
76#11715897—42no—
77C22×C97—41no—
78C22×C97—41no—
79C22×C97—41no—
80C22×C97C2×C22⋊C411yescharacteristic
81C2×D97—21no—
82C2×D97—21no—
83C2×D97—21no—
84C2×D97—41no—
85C2×D97—41no—
86C2×D97—41no—
87C4×C97—21no—
88C4×C97—21no—
89C4×C97—21no—
90C4×C97—41no—
91Dic97—21no—
92Dic97—41no—
93Dic97AGL(1,ℤ/8)11yescharacteristic
94Dic97AGL(1,ℤ/8)12yes—
95Dic97#13711yescharacteristic
96C2×C97—81no—
97C2×C97#41311yescharacteristic
98C2×C97#41711yescharacteristic
99C2×C97#41811yescharacteristic
100D97—41no—
101D97—81no—
102#1557—971nomaximal, Sylow
103C97#155711yescharacteristic, complemented, Sylow
104#532—971no—
105#432—971no—
106#423—971no—
107#332—972no—
108#331—972no—
109C2×#41—1941no—
110C2×M16—1941no—
111C4⋊Q8—971no—
112#127—971no—
113#124—1941no—
114#121—971no—
115#121—971no—
116#121—1941no—
117C4×D4—971no—
118C4×D4—1941no—
119C4×D4#8760411yescharacteristic
120C2×C42—971no—
121C4⋊C8—972no—
122C4⋊C8—972no—
123C4⋊C8—1941no—
124C22⋊C8—1941no—
125#97—1941no—
126#41—3884no—
127C2×Q8—1941no—
128C2×D4—971no—
129C2×D4—1941no—
130C2×D4—1941no—
131C2×D4—1941no—
132C2×D4—1941no—
133C2×D4C2×#8760411yescharacteristic
134C22×C4—21no—
135C22×C4—971no—
136C22×C4—971no—
137C22×C4—971no—
138C22×C4—1941no—
139M16—3882no—
140C2×C8—1942no—
141C2×C8—1942no—
142C4⋊C4—971no—
143C4⋊C4—1941no—
144C4⋊C4C2×#8760411yescharacteristic
145C22⋊C4—21no—
146C22⋊C4—1941no—
147C22⋊C4—1941no—
148C22⋊C4—1941no—
149C42—971no—
150C42—971no—
151C42—971no—
152C42C2×#8760411yescharacteristic
153C23—21no—
154C23—971no—
155C23—1941no—
156Q8—3882no—
157D4—41no—
158D4—3881no—
159D4—3882no—
160D4—3882no—
161D4—3881no—
162D4—3881no—
163D4—3882no—
164C2×C4—21no—
165C2×C4—42no—
166C2×C4—971no—
167C2×C4—971no—
168C2×C4—971no—
169C2×C4—3882no—
170C2×C4—3881no—
171C2×C4—1941no—
172C2×C4—1942no—
173C2×C4—1941no—
174C2×C4—1941no—
175C2×C4C22⋊#8760411yescharacteristic
176C2×C4C22⋊#8760411yescharacteristic
177C2×C4C22×#8760411yescharacteristic
178C8—3882no—
179C8—3882no—
180C22—41no—
181C22—41no—
182C22—41no—
183C22—3881no—
184C22—3881no—
185C22—3881no—
186C22—1941no—
187C22—1941no—
188C22—1941no—
189C22C2×C22⋊#8760411yescharacteristic
190C4—21no—
191C4—21no—
192C4—21no—
193C4—41no—
194C4—971no—
195C4—971no—
196C4—972no—
197C4—3881no—
198C4—1941no—
199C2—81no—
200C2—7761no—
201C2—3881no—
202C2#2026172111yescharacteristic
203C2#2026172611yescharacteristic
204C2#2026172711yescharacteristic
subgroupG/H
centerC22C2×C22⋊#87604
commutatorC2×C4×C97C22×C4
FrattiniC2×C4C22×#87604
FittingC4×D4×C97C4
socleC22×C97C2×C22⋊C4
prime radical#20261282C2
cosocleC4×Dic97C23
hypercenterC4×D4#87604
hypocenterC97#1557
LCSG ▷ C2×C4×C97 ▷ C22×C97 ▷ C97
UCSC1 ◁ C22 ◁ C42 ◁ C4×D4

Complements

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

NH ≅ G/Ntype
1. #20261758C2semidirect
2. #20261740C2semidirect
3. #20261594C2semidirect
4. #20261593C2semidirect
103. C97#1557semidirect

Maximal subgroups

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

subgroupindex
#202617582
#202617402
#202615942
#202615932
#202612822
#155797