Finite Group Databasebeta

← #27140756

Subgroups — #27140756

Order 12416.

288 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 288 conjugacy classes of subgroups — too many to draw. Only its 57 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#20261051C212yesmaximal, complemented
2#20260671C211yesmaximal, characteristic, complemented
3#20260469C212yesmaximal, complemented
4#20260459C211yesmaximal, characteristic, complemented
5#534×C97C211yesmaximal, characteristic
6C2×C4⋊Dic97C2211yescharacteristic, complemented
7C2×#13095490—21no—
8C2×C22⋊Dic97—21no—
9#15690741—22no—
10#15690731—22no—
11C22⋊#11715895C2212yescomplemented
12#15690717C2211yescharacteristic, complemented
13C22×D4×C97—21no—
14#123×C97—22no—
15#121×C97—21no—
16#121×C97C2212yes—
17#120×C97—22no—
18C2×C22⋊C4×C97C2211yescharacteristic
19C22×Dic97—21no—
20C22×Dic97—21no—
21C2×#11715895—22no—
22C4⋊Dic97—22no—
23#13095490—42no—
24C22⋊Dic97—42no—
25C24×C97—21no—
26C24×C97D411yescharacteristic
27C2×D4×C97—22no—
28C2×D4×C97—22no—
29C2×D4×C97—21no—
30C2×D4×C97—42no—
31C2×D4×C97—42no—
32C2×D4×C97D412yes—
33C22×C4×C97—21no—
34C22×C4×C97D411yescharacteristic
35C22×C4×C97C2311yescharacteristic
36C4⋊C4×C97—21no—
37C4⋊C4×C97D412yes—
38C22⋊C4×C97—22no—
39C22⋊C4×C97—22no—
40C22⋊C4×C97—22no—
41C23×C97—21no—
42C23×C97—21no—
43C23×C97—22no—
44C23×C97—41no—
45C23×C97—41no—
46C23×C97—42no—
47C23×C97—81no—
48C23×C97—81no—
49C23×C97SD1612yes—
50C23×C97C2×D411yescharacteristic
51D4×C97—42no—
52D4×C97—44no—
53D4×C97—42no—
54D4×C97—42no—
55D4×C97—81no—
56C2×C4×C97—21no—
57C2×C4×C97—22no—
58C2×C4×C97—41no—
59C2×C4×C97—42no—
60C2×C4×C97—41no—
61C2×C4×C97C2×D412yes—
62C2×Dic97—22no—
63C2×Dic97—41no—
64C2×Dic97—41no—
65C2×Dic97—41no—
66C2×Dic97—41no—
67C2×Dic97—41no—
68#11715895—42no—
69C22×C97—21no—
70C22×C97—21no—
71C22×C97—22no—
72C22×C97—42no—
73C22×C97—41no—
74C22×C97—41no—
75C22×C97—42no—
76C22×C97—42no—
77C22×C97—42no—
78C22×C97—81no—
79C22×C97—84no—
80C22×C97#12011yescharacteristic
81C22×C97C2×SD1611yescharacteristic
82C22×C97AGL(1,ℤ/8)11yescharacteristic
83C4×C97—22no—
84C4×C97—42no—
85C4×C97—41no—
86Dic97—42no—
87Dic97—81no—
88C2×C97—21no—
89C2×C97—21no—
90C2×C97—42no—
91C2×C97—82no—
92C2×C97#45011yescharacteristic
93C2×C97#45311yescharacteristic
94C2×C97#45811yescharacteristic
95#1613—971nomaximal, Sylow
96C97#161311yescharacteristic, complemented, Sylow
97#534D9711yescharacteristic
98#481—972no—
99#399—971no—
100#327—972no—
101#323—971no—
102C22×D4—21no—
103#123—22no—
104#121—21no—
105#121C2×D9712yes—
106#120—22no—
107C2×C4⋊C4—971no—
108C2×C4⋊C4—1941no—
109C2×C22⋊C4—1941no—
110C2×C22⋊C4C2×D9711yescharacteristic
111#106—1942no—
112#102—1942no—
113C22⋊C8—972no—
114#95—971no—
115C24—21no—
116C24C22⋊D9711yescharacteristic
117C2×D4—22no—
118C2×D4—22no—
119C2×D4—21no—
120C2×D4—42no—
121C2×D4—42no—
122C2×D4C22⋊D9712yes—
123C22×C4—21no—
124C22×C4—1941no—
125C22×C4—1941no—
126C22×C4C22⋊D9711yescharacteristic
127C22×C4C22×D9711yescharacteristic
128C2×C8—1942no—
129C4⋊C4—21no—
130C4⋊C4—3882no—
131C4⋊C4—1942no—
132C4⋊C4C22⋊D9712yes—
133C22⋊C4—22no—
134C22⋊C4—22no—
135C22⋊C4—22no—
136C22⋊C4—3882no—
137C23—21no—
138C23—21no—
139C23—22no—
140C23—41no—
141C23—41no—
142C23—42no—
143C23—81no—
144C23—81no—
145C23#1309549912yes—
146C23C2×C22⋊D9711yescharacteristic
147D4—42no—
148D4—44no—
149D4—42no—
150D4—42no—
151D4—81no—
152C2×C4—21no—
153C2×C4—22no—
154C2×C4—41no—
155C2×C4—42no—
156C2×C4—41no—
157C2×C4—3881no—
158C2×C4—3881no—
159C2×C4—3881no—
160C2×C4—3881no—
161C2×C4—3881no—
162C2×C4—1942no—
163C2×C4C2×C22⋊D9712yes—
164C8—3882no—
165C22—21no—
166C22—21no—
167C22—22no—
168C22—42no—
169C22—41no—
170C22—41no—
171C22—42no—
172C22—42no—
173C22—42no—
174C22—81no—
175C22—84no—
176C22#1569077911yescharacteristic
177C22C2×#1309549911yescharacteristic
178C22#1569083611yescharacteristic
179C4—22no—
180C4—42no—
181C4—41no—
182C4—7761no—
183C4—3882no—
184C2—21no—
185C2—21no—
186C2—42no—
187C2—82no—
188C2#2026088611yescharacteristic
189C2#2026090311yescharacteristic
190C2#2026092511yescharacteristic
subgroupG/H
centerC22#15690779
commutator, cosocleC22×C4×C97C23
FrattiniC22×C4C22×D97
Fitting, prime radical#534×C97C2
socleC22×C97#120
hypercenter#534D97
hypocenterC97#1613
LCSG ▷ C22×C4×C97 ▷ C22×C97 ▷ C97
UCSC1 ◁ C22 ◁ C22×C4 ◁ #534

Complements

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

NH ≅ G/Ntype
1. #20261051C2semidirect
2. #20260671C2semidirect
3. #20260469C2semidirect
4. #20260459C2semidirect
6. C2×C4⋊Dic97C22semidirect
11. C22⋊#11715895C22semidirect
12. #15690717C22semidirect
96. C97#1613semidirect

Maximal subgroups

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

subgroupindex
#202610512
#202606712
#202604692
#202604592
#534×C972
#161397