Finite Group Databasebeta

← #27156598

Subgroups — #27156598

Order 12416.

228 conjugacy classes of subgroups; only the 54 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 54 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#20261703C211yesmaximal, characteristic
2#20261677C211yesmaximal, characteristic
3#20260594C211yesmaximal, characteristic
4C2×#13095519C412yescomplemented
5C2×C4⋊#87604—21no—
6C2×C22⋊#87604—21no—
7#15690867—21no—
8C22⋊#11715897—21no—
9(C2×#87604)⋊C4C412yescomplemented
10C22×C4×D97C2211yescharacteristic
11C2×#13095490—21no—
12C2×#13095488—21no—
13#15690717C411yescharacteristic, complemented
14#95×C97C411yescharacteristic, complemented
15C22×#87604—22no—
16#13095519—42no—
17C2×#11715897—22no—
18C4⋊#87604—42no—
19C22⋊#87604—41no—
20C22⋊#87604—41no—
21C23×D97D411yescharacteristic
22C22×Dic97—21no—
23C22×Dic97D411yescharacteristic
24C2×C4×D97—42no—
25C2×C4×D97C2×C411yescharacteristic
26C2×C4×D97C2×C411yescharacteristic
27C2×C4×D97D411yescharacteristic
28C2×C4×D97Q811yescharacteristic
29#13095490—42no—
30#13095488—41no—
31#13095488—41no—
32C22×C4×C97—21no—
33C22×C4×C97C2×C411yescharacteristic
34C23×C97C22⋊C411yescharacteristic
35C22×D97—41no—
36C22×D97—41no—
37C22×D97C22⋊C411yescharacteristic
38C22×D97C4⋊C411yescharacteristic
39C22×D97D811yescharacteristic
40C22×D97SD1611yescharacteristic
41C22×D97SD1611yescharacteristic
42C22×D97Q1611yescharacteristic
43C2×#87604—42no—
44C2×#87604—42no—
45C2×#87604—42no—
46C2×C4×C97—41no—
47C2×C4×C97—41no—
48C2×C4×C97—41no—
49C2×C4×C97—41no—
50C2×C4×C97C4211yescharacteristic
51C2×C4×C97C4⋊C411yescharacteristic
52C2×Dic97—22no—
53C2×Dic97—41no—
54C2×Dic97—41no—
55C2×Dic97—41no—
56C2×Dic97C22⋊C411yescharacteristic
57C2×Dic97C4⋊C411yescharacteristic
58C4×D97—22no—
59C4×D97—22no—
60C4×D97—41no—
61C4×D97—41no—
62#11715897—42no—
63C22×C97—21no—
64C22×C97—21no—
65C22×C97#9511yescharacteristic
66C22×C97#10211yescharacteristic
67C22×C97#10311yescharacteristic
68C2×D97—21no—
69C2×D97—21no—
70C2×D97—21no—
71C2×D97—21no—
72C2×D97—41no—
73C2×D97—41no—
74C2×D97—41no—
75C2×D97—41no—
76C2×D97#10211yescharacteristic
77C2×D97#10311yescharacteristic
78C2×D97#10611yescharacteristic
79C2×D97#10711yescharacteristic
80#87604—82no—
81C4×C97—21no—
82C4×C97—21no—
83C4×C97—81no—
84Dic97—21no—
85Dic97—81no—
86Dic97#9912yes—
87C2×C97—21no—
88C2×C97—21no—
89C2×C97H(3,ℤ/4)11yescharacteristic
90C2×C97#34011yescharacteristic
91C2×C97#34211yescharacteristic
92D97—21no—
93D97—21no—
94D97—21no—
95D97—21no—
96#1289—971nomaximal, Sylow
97C97#128911yescharacteristic, complemented, Sylow
98C2×C22⋊C8—971no—
99#380—971no—
100#380—971no—
101C23×C4—971no—
102C22×C8—972no—
103C2×C4⋊C4—1941no—
104C2×C4⋊C4—1941no—
105C2×C22⋊C4—1941no—
106C2×C22⋊C4—1941no—
107C22⋊C8—1941no—
108C22⋊C8—1941no—
109#95—972no—
110#95—971no—
111#95#8760411yescharacteristic, complemented
112C24—971no—
113C22×C4—21no—
114C22×C4—971no—
115C22×C4—971no—
116C22×C4—971no—
117C22×C4—971no—
118C22×C4—971no—
119C22×C4—3882no—
120C22×C4—1942no—
121C22×C4—1941no—
122C22×C4C2×#8760411yescharacteristic
123C2×C8—3882no—
124C2×C8—1942no—
125C4⋊C4—3882no—
126C4⋊C4—3882no—
127C22⋊C4—3881no—
128C22⋊C4—3881no—
129C22⋊C4—3881no—
130C22⋊C4—3881no—
131C23—971no—
132C23—971no—
133C23—971no—
134C23—971no—
135C23—971no—
136C23—971no—
137C23—3881no—
138C23—3881no—
139C23C22⋊#8760411yescharacteristic
140C2×C4—41no—
141C2×C4—41no—
142C2×C4—41no—
143C2×C4—41no—
144C2×C4—971no—
145C2×C4—971no—
146C2×C4—3881no—
147C2×C4—3881no—
148C2×C4—3881no—
149C2×C4—3881no—
150C2×C4—3882no—
151C2×C4—3882no—
152C2×C4—3882no—
153C2×C4—3881no—
154C2×C4—1942no—
155C2×C4—1942no—
156C2×C4—1942no—
157C2×C4C4×#8760411yescharacteristic
158C2×C4C4⋊#8760411yescharacteristic
159C8—3882no—
160C22—21no—
161C22—21no—
162C22—971no—
163C22—971no—
164C22—971no—
165C22—971no—
166C22—3881no—
167C22—3881no—
168C22—3881no—
169C22—3881no—
170C22—1941no—
171C22—1941no—
172C22—1941no—
173C22—1941no—
174C22(C2×#87604)⋊C411yescharacteristic
175C22#1569087211yescharacteristic
176C22#1569087311yescharacteristic
177C4—21no—
178C4—21no—
179C4—81no—
180C4—7761no—
181C4—7762no—
182C4—972no—
183C4—1941no—
184C2—21no—
185C2—21no—
186C2—1941no—
187C2—1941no—
188C2—1941no—
189C2—1941no—
190C2#2026160211yescharacteristic
191C2#2026160811yescharacteristic
192C2#2026161211yescharacteristic
subgroupG/H
centerC22(C2×#87604)⋊C4
commutatorC2×C4×C97C42
FrattiniC22×C4C2×#87604
Fitting#95×C97C4
socleC22×C97#95
prime radicalC23×D97D4
cosocleC22×C4×D97C22
hypercenter#95#87604
hypocenterC97#1289
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C22×C4 ◁ #95

Complements

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

NH ≅ G/Ntype
4. C2×#13095519C4semidirect
9. (C2×#87604)⋊C4C4semidirect
13. #15690717C4semidirect
14. #95×C97C4semidirect
97. C97#1289semidirect
111. #95#87604semidirect

Maximal subgroups

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

subgroupindex
#202617032
#202616772
#202605942
#128997