Finite Group Databasebeta

← #27157639

Subgroups — #27157639

Order 12416.

344 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 344 conjugacy classes of subgroups — too many to draw. Only its 84 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#20261608C214yesmaximal, complemented
4C2×C4⋊C4×D97C211yesmaximal, characteristic
5C2×#13095519C2212yes—
6C2×C4⋊#87604C2212yes—
7C2×C22⋊#87604—21no—
8#15690867—21no—
9C22⋊#11715897—21no—
10(C2×#87604)⋊C4—21no—
11C22×C4×D97—21no—
12C22×C4×D97C2211yescharacteristic
13C2×C4⋊Dic97C411yescharacteristic, complemented
14C2×#13095490—21no—
15C4⋊C4×D97—21no—
16C4⋊C4×D97—21no—
17C4⋊C4×D97C412yescomplemented
18C4⋊C4×D97C2212yes—
19C2×C4⋊C4×C97C411yescharacteristic, complemented
20C22×#87604—22no—
21#13095519—24no—
22C2×#11715897—22no—
23C4⋊#87604—24no—
24C22⋊#87604—42no—
25C23×D97D411yescharacteristic
26C22×Dic97—21no—
27C22×Dic97D411yescharacteristic
28C2×C4×D97—24no—
29C2×C4×D97—22no—
30C2×C4×D97—22no—
31C2×C4×D97—42no—
32C2×C4×D97—42no—
33C2×C4×D97C2×C411yescharacteristic
34C2×C4×D97D411yescharacteristic
35C2×C4×D97D411yescharacteristic
36C2×C4×D97C2311yescharacteristic
37C4⋊Dic97—21no—
38C4⋊Dic97C2×C412yes—
39#13095490—22no—
40#13095490—22no—
41C22×C4×C97—21no—
42C22×C4×C97C2×C411yescharacteristic
43C4⋊C4×C97—21no—
44C4⋊C4×C97C2×C412yes—
45C23×C97C22⋊C411yescharacteristic
46C22×D97—22no—
47C22×D97—22no—
48C22×D97C22⋊C411yescharacteristic
49C22×D97D812yes—
50C22×D97SD1612yes—
51C22×D97C2×D411yescharacteristic
52C2×#87604—42no—
53C2×#87604—42no—
54C2×#87604—42no—
55C2×C4×C97—22no—
56C2×C4×C97—22no—
57C2×C4×C97—42no—
58C2×C4×C97C22⋊C411yescharacteristic
59C2×C4×C97C22×C411yescharacteristic
60C2×Dic97—22no—
61C2×Dic97—22no—
62C2×Dic97—42no—
63C2×Dic97C22⋊C411yescharacteristic
64C2×Dic97C2×D411yescharacteristic
65C4×D97—22no—
66C4×D97—22no—
67C4×D97—22no—
68C4×D97—44no—
69C4×D97—42no—
70C4×D97—42no—
71C4×D97#4114yes—
72#11715897—42no—
73C22×C97—21no—
74C22×C97—21no—
75C22×C97#10212yes—
76C22×C97C2×C22⋊C411yescharacteristic
77C2×D97—21no—
78C2×D97—21no—
79C2×D97—24no—
80C2×D97—24no—
81C2×D97#10212yes—
82C2×D97#12312yes—
83C2×D97C2×D811yescharacteristic
84C2×D97C2×SD1611yescharacteristic
85C2×D97#13712yes—
86#87604—82no—
87C4×C97—21no—
88C4×C97—42no—
89C4×C97#11712yes—
90Dic97—21no—
91Dic97—42no—
92Dic97#12312yes—
93C2×C97—22no—
94C2×C97#38111yescharacteristic
95C2×C97C2×#10211yescharacteristic
96C2×C97#41911yescharacteristic
97D97—22no—
98D97#48012yes—
99D97#48312yes—
100#1823—971nomaximal, Sylow
101C97#182311yescharacteristic, complemented, Sylow
102C22×C4⋊C4—971no—
103C2×C22⋊C8—971no—
104#380—971no—
105#340—974no—
106C23×C4—971no—
107C23×C4—1941no—
108C22×C8—972no—
109C2×C4⋊C4—971no—
110C2×C4⋊C4—972no—
111C2×C4⋊C4—972no—
112C2×C4⋊C4—972no—
113C2×C4⋊C4—1941no—
114C2×C4⋊C4—1941no—
115C2×C4⋊C4—1941no—
116C2×C4⋊C4#8760411yescharacteristic, complemented
117C2×C22⋊C4—1941no—
118C22⋊C8—1941no—
119C22⋊C8—1941no—
120#95—1941no—
121C24—971no—
122C22×C4—21no—
123C22×C4—971no—
124C22×C4—971no—
125C22×C4—971no—
126C22×C4—971no—
127C22×C4—971no—
128C22×C4—3882no—
129C22×C4—3882no—
130C22×C4—1944no—
131C22×C4—1942no—
132C22×C4—1942no—
133C22×C4—1942no—
134C22×C4—1941no—
135C22×C4C2×#8760411yescharacteristic
136C2×C8—1942no—
137C2×C8—1944no—
138C4⋊C4—21no—
139C4⋊C4—972no—
140C4⋊C4—1942no—
141C4⋊C4—1942no—
142C4⋊C4—1944no—
143C4⋊C4—1941no—
144C4⋊C4C2×#8760412yes—
145C22⋊C4—3882no—
146C23—971no—
147C23—972no—
148C23—972no—
149C23—971no—
150C23—1942no—
151C23—1942no—
152C23C22⋊#8760411yescharacteristic
153C2×C4—22no—
154C2×C4—22no—
155C2×C4—42no—
156C2×C4—971no—
157C2×C4—974no—
158C2×C4—971no—
159C2×C4—3882no—
160C2×C4—3882no—
161C2×C4—3884no—
162C2×C4—3882no—
163C2×C4—3882no—
164C2×C4—3882no—
165C2×C4—3882no—
166C2×C4—1942no—
167C2×C4—1942no—
168C2×C4—1942no—
169C2×C4—1942no—
170C2×C4—1942no—
171C2×C4C22⋊#8760411yescharacteristic
172C2×C4C22×#8760411yescharacteristic
173C8—3882no—
174C22—21no—
175C22—21no—
176C22—972no—
177C22—971no—
178C22—971no—
179C22—972no—
180C22—972no—
181C22—1944no—
182C22—1944no—
183C22—1941no—
184C22—1941no—
185C22#1569087212yes—
186C22C2×C22⋊#8760411yescharacteristic
187C4—21no—
188C4—42no—
189C4—7762no—
190C4—972no—
191C4—3882no—
192C4—1941no—
193C4#1569088912yes—
194C2—22no—
195C2—972no—
196C2—972no—
197C2—1942no—
198C2#2026167811yescharacteristic
199C2C2×#1569087211yescharacteristic
200C2#2026172811yescharacteristic
subgroupG/H
centerC22C2×C22⋊#87604
commutatorC2×C4×C97C22×C4
FrattiniC2×C4C22×#87604
FittingC2×C4⋊C4×C97C4
socleC22×C97C2×C22⋊C4
prime radicalC23×D97D4
cosocleC2×C4×D97C23
hypercenterC2×C4⋊C4#87604
hypocenterC97#1823
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4⋊C4

Complements

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

NH ≅ G/Ntype
3. #20261608C2semidirect
13. C2×C4⋊Dic97C4semidirect
17. C4⋊C4×D97C4semidirect
19. C2×C4⋊C4×C97C4semidirect
101. C97#1823semidirect
116. C2×C4⋊C4#87604semidirect

Maximal subgroups

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

subgroupindex
#202617032
#202616772
#202616082
C2×C4⋊C4×D972
#182397