Finite Group Databasebeta

← #13157202

Subgroups — #13157202

Order 1728.

376 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 376 conjugacy classes of subgroups — too many to draw. Only its 69 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
1C2×#88989C212yesmaximal
2C2×C3×C8×C3⋊D3C211yesmaximal, characteristic
3#10590185—31nomaximal
4C2×C3×C4×C3⋊D3C2211yescharacteristic
5C2×C3⋊#622—22no—
6#88989—22no—
7#88989C414yes—
8C2×C33×C8C411yescharacteristic
9C2×C3×C3⋊(C3⋊C8)C411yescharacteristic
10C3×C8×C3⋊D3—22no—
11C2×#3788—32no—
12C2×C8×C3⋊D3D311yescharacteristic
13C2×C3×D3×C8—62no—
14C22×C3×C3⋊D3Q811yescharacteristic
15C2×C33×C4C2×C411yescharacteristic
16C2×C3×C3⋊Dic3D411yescharacteristic
17C3×C4×C3⋊D3C2×C412yes—
18C3×C4×C3⋊D3D412yes—
19C3⋊#622—24no—
20C33×C8—21no—
21C3×C3⋊(C3⋊C8)—21no—
22#4552—91nomaximal
23C2×C4×C3⋊D3C2×D311yescharacteristic
24C2×C3×C4×D3—62no—
25C2×#622—62no—
26#3788—34no—
27#3788—62no—
28C2×C32×C8—22no—
29C2×C32×C8—42no—
30C2×C32×C8Dic311yescharacteristic
31C2×C3⋊(C3⋊C8)Dic311yescharacteristic
32C8×C3⋊D3—22no—
33C2×C3×C3⋊C8—62no—
34C3×D3×C8—68no—
35C22×C33C4⋊C411yescharacteristic
36C2×C3×C3⋊D3—21no—
37C2×C3×C3⋊D3—21no—
38C2×C3×C3⋊D3C4⋊C412yes—
39C33×C4C4211yescharacteristic
40C33×C4C22⋊C411yescharacteristic
41C3×C3⋊Dic3C22⋊C412yes—
42C22×C3×C8—91no—
43C2×#208—92no—
44C2×D3×C8—62no—
45C22×C3⋊D3Dic611yescharacteristic
46C22×C3×D3—62no—
47C2×C32×C4—22no—
48C2×C32×C4—42no—
49C2×C32×C4C2×Dic311yescharacteristic
50C2×C3⋊Dic3D1211yescharacteristic
51C4×C3⋊D3C4×D312yes—
52C4×C3⋊D3#6712yes—
53C2×C3×Dic3—62no—
54C3×C4×D3—64no—
55C3×C4×D3—64no—
56#622—64no—
57C32×C8—21no—
58C32×C8—24no—
59C32×C8—44no—
60C3⋊(C3⋊C8)—21no—
61C3×C3⋊C8—64no—
62#341—271noSylow
63C2×C33#9511yescharacteristic
64C2×C33#10811yescharacteristic
65C2×C33#10811yescharacteristic
66C3×C3⋊D3—22no—
67C22×C3×C4—91no—
68C2×C4×D3—62no—
69C2×C3×C8—22no—
70C2×C3×C8—42no—
71C2×C3×C8—91no—
72C2×C3×C8—182no—
73C2×C3×C8#15711yescharacteristic
74#208—94no—
75#208—182no—
76C2×C3⋊C8—62no—
77C2×C3⋊C8—182no—
78D3×C8—68no—
79C22×C32—22no—
80C22×C32—42no—
81C22×C32C4⋊Dic311yescharacteristic
82C2×C3⋊D3—21no—
83C2×C3⋊D3—21no—
84C2×C3⋊D3Dic3⋊C412yes—
85C2×C3×D3—64no—
86C2×C3×D3—64no—
87C2×C3×D3—64no—
88C32×C4—22no—
89C32×C4—22no—
90C32×C4—42no—
91C32×C4—42no—
92C32×C4C4×Dic311yescharacteristic
93C32×C4#21711yescharacteristic
94C3⋊Dic3(C2×D3)⋊C412yes—
95C3×Dic3—64no—
96C2×M16—272no—
97C22×C8—91no—
98C33#34111yescharacteristic, complemented, Sylow
99C23×C3—91no—
100C22×D3—62no—
101C2×C3×C4—22no—
102C2×C3×C4—42no—
103C2×C3×C4—91no—
104C2×C3×C4—92no—
105C2×C3×C4—92no—
106C2×C3×C4C2×#15711yescharacteristic
107C2×Dic3—62no—
108C4×D3—64no—
109C4×D3—64no—
110C3×C8—24no—
111C3×C8—21no—
112C3×C8—44no—
113C3×C8—181no—
114C3⋊C8—64no—
115C3⋊C8—184no—
116C2×C32—22no—
117C2×C32—22no—
118C2×C32—22no—
119C2×C32—42no—
120C2×C32—42no—
121C2×C32—42no—
122C2×C32#81911yescharacteristic
123C2×C32#81911yescharacteristic
124C2×C32C3⋊#9511yescharacteristic
125C3⋊D3—22no—
126C3×D3—68no—
127C22×C4—91no—
128M16—274no—
129M16—542no—
130C2×C8—91no—
131C2×C8—182no—
132C2×C8—542no—
133C2×C8C3⋊#15711yescharacteristic
134C22×C3—22no—
135C22×C3—42no—
136C22×C3—92no—
137C22×C3—181no—
138C22×C3—181no—
139C22×C3C4⋊#15711yescharacteristic
140C2×D3—64no—
141C2×D3—64no—
142C2×D3—64no—
143C3×C4—22no—
144C3×C4—22no—
145C3×C4—42no—
146C3×C4—42no—
147C3×C4—92no—
148C3×C4C4×#15711yescharacteristic
149C3×C4C22⋊#15711yescharacteristic
150Dic3—64no—
151C32—22no—
152C32—42no—
153C32#455211yescharacteristic, complemented
154C23—91no—
155C2×C4—91no—
156C2×C4—92no—
157C2×C4—92no—
158C2×C4C2×C3⋊#15711yescharacteristic
159C8—21no—
160C8—181no—
161C8—544no—
162C2×C3—22no—
163C2×C3—22no—
164C2×C3—22no—
165C2×C3—42no—
166C2×C3—42no—
167C2×C3—42no—
168C2×C3—182no—
169C2×C3#6377011yescharacteristic
170C2×C3#6377111yescharacteristic
171C2×C3#6378111yescharacteristic
172D3—68no—
173C22—92no—
174C22—181no—
175C22—181no—
176C22C4⋊(C3⋊#157)11yescharacteristic
177C4—92no—
178C4C4×C3⋊#15711yescharacteristic
179C4#8899411yescharacteristic
180C3—22no—
181C3—42no—
182C3#1059018511yescharacteristic, complemented
183C2—182no—
184C2#1172309511yescharacteristic
185C2#1172309611yescharacteristic
186C2#1172310611yescharacteristic
subgroupG/H
centerC22C4⋊(C3⋊#157)
commutatorC33×C4C42
FrattiniC2×C4C2×C3⋊#157
FittingC2×C33×C8C4
socleC22×C33C4⋊C4
prime radicalC22×C3×C3⋊D3Q8
cosocleC2×C3×C4×C3⋊D3C22
hypercenterC2×C8C3⋊#157
hypocenterC33#341
LCSG ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×C8

Complements

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

NH ≅ G/Ntype
98. C33#341semidirect
153. C32#4552semidirect
182. C3#10590185semidirect

Maximal subgroups

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

subgroupindex
C2×#889892
C2×C3×C8×C3⋊D32
#105901853
#45529