Finite Group Databasebeta

← #11921024

Subgroups — #11921024

Order 1152.

320 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 320 conjugacy classes of subgroups — too many to draw. Only its 81 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
1C3×C4×#41⋊C3C211yesmaximal, characteristic
2C3⋊(SL(2,𝔽3)⋊C8)C212yesmaximal, complemented
3#73068—33nomaximal
4#71638—31nomaximal
5C2×C3×#41⋊C3C411yescharacteristic
6C3×C4×SL(2,𝔽3)C411yescharacteristic
7C3×C4×SL(2,𝔽3)C2211yescharacteristic
8C4×C3⋊(C3⋊C8)—41nomaximal
9C3×C4×#41D311yescharacteristic
10C4×#41⋊C3D313yes—
11#4925—31no—
12C2×C4×C3⋊C8—31no—
13SL(2,𝔽3)⋊C8—36no—
14#4535—32no—
15D4⋊(C3⋊C8)—32no—
16C3×#41⋊C3C812yescomplemented
17C2×C3×SL(2,𝔽3)C2×C411yescharacteristic
18C32×C42—41no—
19C2×C3⋊(C3⋊C8)—42no—
20#1476—91noSylow
21C2×C3×#41Dic311yescharacteristic
22C2×#41⋊C3Dic313yes—
23C3×C4×Q8Dic311yescharacteristic
24C3×C4×Q8C2×D311yescharacteristic
25C3×C4×D4—31no—
26C3×C4×D4—31no—
27C3×#117—31no—
28C2×C3×C42—31no—
29C22×C3⋊C8—61no—
30C4×SL(2,𝔽3)Dic313yes—
31C4×SL(2,𝔽3)C2×D313yes—
32#830—61no—
33C4⋊(C3⋊C8)—32no—
34C4×C3⋊C8—32no—
35C4×C3⋊C8—61no—
36C4×C3⋊C8—61no—
37C4×C3⋊C8—123no—
38C2×C32×C4—41no—
39C2×C32×C4—41no—
40C2×C32×C4—41no—
41C3×SL(2,𝔽3)C2×C811yescharacteristic, complemented
42C3⋊(C3⋊C8)—44no—
43C4×#41C3⋊D311yescharacteristic
44#431—91no—
45C2×C4×C8—91no—
46Q8⋊C8—92no—
47D4⋊C8—92no—
48C3×#41—32no—
49C3×#41C3⋊C812yescomplemented
50C2×C3×Q8C2×Dic311yescharacteristic
51C2×C3×D4—31no—
52C22×C3×C4—31no—
53C22×C3×C4—31no—
54C22×C3×C4—31no—
55#41⋊C3C3⋊C816yescomplemented
56C2×SL(2,𝔽3)C2×Dic313yes—
57C3×C4⋊C4—31no—
58C3×C4⋊C4—31no—
59C3×C22⋊C4—61no—
60C3×C42—31no—
61C3×C42—31no—
62C3×C42—31no—
63C3×C42—43no—
64C3×C42S411yescharacteristic
65C2×C3⋊C8—62no—
66C2×C3⋊C8—64no—
67C2×C3⋊C8—62no—
68C2×C3⋊C8—126no—
69C22×C32—41no—
70C32×C4—42no—
71C32×C4—42no—
72C32×C4—42no—
73C2×#41C3⋊Dic311yescharacteristic
74C22×C8—181no—
75C4×Q8C3⋊Dic311yescharacteristic
76C4×Q8C2×C3⋊D311yescharacteristic
77C4×D4—31no—
78C4×D4—31no—
79#117—31no—
80C2×C42—31no—
81C4⋊C8—92no—
82C22⋊C8—181no—
83C4×C8—92no—
84C4×C8—181no—
85C4×C8—181no—
86C23×C3—31no—
87C3×Q8—31no—
88C3×Q8C2×C3⋊C811yescharacteristic
89C3×D4—32no—
90C3×D4—61no—
91C2×C3×C4—31no—
92C2×C3×C4—32no—
93C2×C3×C4—32no—
94C2×C3×C4—31no—
95C2×C3×C4—31no—
96C2×C3×C4—43no—
97C2×C3×C4—43no—
98C2×C3×C4—43no—
99C2×C3×C4—62no—
100C2×C3×C4—62no—
101C2×C3×C4SL(2,ℤ/4)11yescharacteristic
102C2×C3×C4SL(2,ℤ/4)11yescharacteristic
103C2×C3×C4C2×S411yescharacteristic
104SL(2,𝔽3)C2×C3⋊C813yescomplemented
105C3⋊C8—64no—
106C3⋊C8—1212no—
107C3⋊C8—122no—
108C2×C32—41no—
109C2×C32—41no—
110C2×C32—41no—
111#41—32no—
112#41C3⋊(C3⋊C8)12yescomplemented
113C2×Q8C2×C3⋊Dic311yescharacteristic
114C2×D4—31no—
115C22×C4—31no—
116C22×C4—31no—
117C22×C4—31no—
118C2×C8—182no—
119C2×C8—184no—
120C2×C8—182no—
121C4⋊C4—31no—
122C4⋊C4—31no—
123C22⋊C4—61no—
124C42—31no—
125C42—31no—
126C42—31no—
127C42C3⋊S411yescharacteristic
128C22×C3—32no—
129C22×C3—43no—
130C22×C3—61no—
131C22×C3—61no—
132C22×C3C2×SL(2,ℤ/4)11yescharacteristic
133C3×C4—31no—
134C3×C4—31no—
135C3×C4—46no—
136C3×C4—46no—
137C3×C4—46no—
138C3×C4—61no—
139C3×C4—61no—
140C3×C4A4⋊C812yes—
141C3×C4GU(2,𝔽3)12yes—
142C3×C4#98512yes—
143C32—41noSylow
144C23—31no—
145Q8—31no—
146Q8C2×C3⋊(C3⋊C8)11yescharacteristic
147D4—32no—
148D4—61no—
149C2×C4—31no—
150C2×C4—32no—
151C2×C4—32no—
152C2×C4—31no—
153C2×C4—31no—
154C2×C4—62no—
155C2×C4—62no—
156C2×C4C3⋊SL(2,ℤ/4)11yescharacteristic
157C2×C4C3⋊SL(2,ℤ/4)11yescharacteristic
158C2×C4C2×C3⋊S411yescharacteristic
159C8—184no—
160C8—362no—
161C2×C3—43no—
162C2×C3—43no—
163C2×C3—43no—
164C2×C3—62no—
165C2×C3C2×A4⋊C811yescharacteristic
166C2×C3C2×GU(2,𝔽3)11yescharacteristic
167C2×C3#542411yescharacteristic
168C22—32no—
169C22—61no—
170C22—61no—
171C22C2×C3⋊SL(2,ℤ/4)11yescharacteristic
172C4—31no—
173C4—31no—
174C4—61no—
175C4—61no—
176C4C3⋊(A4⋊C8)12yes—
177C4C3⋊GU(2,𝔽3)12yes—
178C4#6426612yes—
179C3—43no—
180C3#7306811yescharacteristic, complemented
181C2—62no—
182C2C2×C3⋊(A4⋊C8)11yescharacteristic
183C2C2×C3⋊GU(2,𝔽3)11yescharacteristic
184C2#1059365811yescharacteristic
subgroupG/H
center, hypercenterC42C3⋊S4
commutator, hypocenterC3×SL(2,𝔽3)C2×C8
FrattiniC2×C4C2×C3⋊S4
FittingC3×C4×#41D3
socleC22×C3C2×SL(2,ℤ/4)
prime radicalC2×C3×#41⋊C3C4
cosocleC3×C4×SL(2,𝔽3)C22
LCSG ▷ C3×SL(2,𝔽3)
UCSC1 ◁ C42

Complements

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

NH ≅ G/Ntype
2. C3⋊(SL(2,𝔽3)⋊C8)C2semidirect
16. C3×#41⋊C3C8semidirect
41. C3×SL(2,𝔽3)C2×C8semidirect
49. C3×#41C3⋊C8semidirect
55. #41⋊C3C3⋊C8semidirect
104. SL(2,𝔽3)C2×C3⋊C8semidirect
112. #41C3⋊(C3⋊C8)semidirect
180. C3#73068semidirect

Maximal subgroups

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

subgroupindex
C3×C4×#41⋊C32
C3⋊(SL(2,𝔽3)⋊C8)2
#730683
#716383
C4×C3⋊(C3⋊C8)4