Finite Group Databasebeta

← #27157640

Subgroups — #27157640

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—44no—
32C2×C4×D97C2×C411yescharacteristic
33C2×C4×D97D411yescharacteristic
34C2×C4×D97D411yescharacteristic
35C2×C4×D97C2311yescharacteristic
36C4⋊Dic97—21no—
37C4⋊Dic97C2×C412yes—
38#13095490—22no—
39#13095490—22no—
40C22×C4×C97—21no—
41C22×C4×C97C2×C411yescharacteristic
42C4⋊C4×C97—21no—
43C4⋊C4×C97C2×C412yes—
44C23×C97C22⋊C411yescharacteristic
45C22×D97—22no—
46C22×D97—22no—
47C22×D97C22⋊C411yescharacteristic
48C22×D97SD1612yes—
49C22×D97Q1612yes—
50C22×D97C2×D411yescharacteristic
51C2×#87604—42no—
52C2×#87604—42no—
53C2×#87604—42no—
54C2×C4×C97—22no—
55C2×C4×C97—22no—
56C2×C4×C97—42no—
57C2×C4×C97C22⋊C411yescharacteristic
58C2×C4×C97C22×C411yescharacteristic
59C2×Dic97—22no—
60C2×Dic97—22no—
61C2×Dic97—42no—
62C2×Dic97C22⋊C411yescharacteristic
63C2×Dic97C2×D411yescharacteristic
64C4×D97—22no—
65C4×D97—22no—
66C4×D97—22no—
67C4×D97—44no—
68C4×D97—44no—
69C4×D97#4114yes—
70#11715897—42no—
71C22×C97—21no—
72C22×C97—21no—
73C22×C97#10312yes—
74C22×C97C2×C22⋊C411yescharacteristic
75C2×D97—21no—
76C2×D97—21no—
77C2×D97—24no—
78C2×D97—24no—
79C2×D97#10312yes—
80C2×D97#12312yes—
81C2×D97C2×SD1611yescharacteristic
82C2×D97C2×Q1611yescharacteristic
83C2×D97AGL(1,ℤ/8)12yes—
84#87604—82no—
85C4×C97—21no—
86C4×C97—42no—
87C4×C97#11712yes—
88Dic97—21no—
89Dic97—42no—
90Dic97#12312yes—
91C2×C97—22no—
92C2×C97#38111yescharacteristic
93C2×C97C2×#10311yescharacteristic
94C2×C97#41811yescharacteristic
95D97—22no—
96D97#48112yes—
97D97#48412yes—
98#1824—971nomaximal, Sylow
99C97#182411yescharacteristic, complemented, Sylow
100C22×C4⋊C4—971no—
101C2×C22⋊C8—971no—
102#380—971no—
103#340—974no—
104C23×C4—971no—
105C23×C4—1941no—
106C22×C8—972no—
107C2×C4⋊C4—971no—
108C2×C4⋊C4—972no—
109C2×C4⋊C4—972no—
110C2×C4⋊C4—972no—
111C2×C4⋊C4—1941no—
112C2×C4⋊C4—1941no—
113C2×C4⋊C4—1941no—
114C2×C4⋊C4#8760411yescharacteristic, complemented
115C2×C22⋊C4—1941no—
116C22⋊C8—1941no—
117C22⋊C8—1941no—
118#95—1941no—
119C24—971no—
120C22×C4—21no—
121C22×C4—971no—
122C22×C4—971no—
123C22×C4—971no—
124C22×C4—971no—
125C22×C4—971no—
126C22×C4—3884no—
127C22×C4—1944no—
128C22×C4—1942no—
129C22×C4—1942no—
130C22×C4—1942no—
131C22×C4—1941no—
132C22×C4C2×#8760411yescharacteristic
133C2×C8—1942no—
134C2×C8—1944no—
135C4⋊C4—21no—
136C4⋊C4—972no—
137C4⋊C4—1942no—
138C4⋊C4—1942no—
139C4⋊C4—1944no—
140C4⋊C4—1941no—
141C4⋊C4C2×#8760412yes—
142C22⋊C4—3882no—
143C23—971no—
144C23—972no—
145C23—972no—
146C23—971no—
147C23—1942no—
148C23—1942no—
149C23C22⋊#8760411yescharacteristic
150C2×C4—22no—
151C2×C4—22no—
152C2×C4—42no—
153C2×C4—971no—
154C2×C4—974no—
155C2×C4—971no—
156C2×C4—3884no—
157C2×C4—3884no—
158C2×C4—3882no—
159C2×C4—3882no—
160C2×C4—3882no—
161C2×C4—3882no—
162C2×C4—1942no—
163C2×C4—1942no—
164C2×C4—1942no—
165C2×C4—1942no—
166C2×C4—1942no—
167C2×C4C22⋊#8760411yescharacteristic
168C2×C4C22×#8760411yescharacteristic
169C8—3882no—
170C22—21no—
171C22—21no—
172C22—972no—
173C22—971no—
174C22—971no—
175C22—972no—
176C22—972no—
177C22—1944no—
178C22—1944no—
179C22—1941no—
180C22—1941no—
181C22#1569087312yes—
182C22C2×C22⋊#8760411yescharacteristic
183C4—21no—
184C4—42no—
185C4—7762no—
186C4—972no—
187C4—3882no—
188C4—1941no—
189C4#1569088912yes—
190C2—22no—
191C2—972no—
192C2—972no—
193C2—1942no—
194C2#2026167811yescharacteristic
195C2C2×#1569087311yescharacteristic
196C2#2026172711yescharacteristic
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#1824
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
99. C97#1824semidirect
114. 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
#182497