Finite Group Databasebeta

← #11921028

Subgroups — #11921028

Order 1152.

294 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 294 conjugacy classes of subgroups — too many to draw. Only its 65 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#73072—33nomaximal
4#71641—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#4924—31no—
12C4×#208—31no—
13SL(2,𝔽3)⋊C8—36no—
14#4535—32no—
15D4⋊(C3⋊C8)—32no—
16C3×#41⋊C3—21no—
17C2×C3×SL(2,𝔽3)C2×C411yescharacteristic
18C32×C42—41no—
19C2×C3⋊(C3⋊C8)—42no—
20#1489—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—
29C2×#208—61no—
30C2×#208—61no—
31C4×SL(2,𝔽3)Dic313yes—
32C4×SL(2,𝔽3)C2×D313yes—
33C4⋊(C3⋊C8)—32no—
34C4⋊(C3⋊C8)—61no—
35C4⋊(C3⋊C8)—123no—
36#803—61no—
37C4×C3⋊C8—32no—
38C2×C32×C4—41no—
39C2×C32×C4—41no—
40C2×C32×C4—41no—
41C3×SL(2,𝔽3)M1611yescharacteristic, complemented
42C3⋊(C3⋊C8)—82no—
43C4×#41C3⋊D311yescharacteristic
44#423—91no—
45C4×M16—91no—
46Q8⋊C8—92no—
47D4⋊C8—92no—
48C3×#41—21no—
49C3×#41—61no—
50C2×C3×Q8C2×Dic311yescharacteristic
51C2×C3×D4—31no—
52C22×C3×C4—31no—
53C22×C3×C4—31no—
54C22×C3×C4—31no—
55#41⋊C3—23no—
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
65#208—64no—
66#208—122no—
67C2×C3⋊C8—62no—
68C2×C3⋊C8—62no—
69C2×C3⋊C8—126no—
70C22×C32—41no—
71C32×C4—42no—
72C32×C4—41no—
73C32×C4—41no—
74C32×C4—81no—
75C2×#41C3⋊Dic311yescharacteristic
76C2×M16—181no—
77C2×M16—181no—
78C4×Q8C3⋊Dic311yescharacteristic
79C4×Q8C2×C3⋊D311yescharacteristic
80C4×D4—31no—
81C4×D4—31no—
82#117—31no—
83C2×C42—31no—
84C4⋊C8—92no—
85C4⋊C8—181no—
86#97—181no—
87C4×C8—92no—
88C23×C3—31no—
89C3×Q8—31no—
90C3×Q8#20811yescharacteristic
91C3×D4—32no—
92C3×D4—61no—
93C2×C3×C4—31no—
94C2×C3×C4—31no—
95C2×C3×C4—31no—
96C2×C3×C4—43no—
97C2×C3×C4—43no—
98C2×C3×C4—43no—
99C2×C3×C4—61no—
100C2×C3×C4—61no—
101C2×C3×C4—62no—
102C2×C3×C4—61no—
103C2×C3×C4—61no—
104C2×C3×C4SL(2,ℤ/4)11yescharacteristic
105C2×C3×C4SL(2,ℤ/4)11yescharacteristic
106C2×C3×C4C2×S411yescharacteristic
107SL(2,𝔽3)#20813yescomplemented
108C3⋊C8—64no—
109C3⋊C8—122no—
110C3⋊C8—246no—
111C2×C32—41no—
112C2×C32—41no—
113C2×C32—41no—
114#41—21no—
115#41—61no—
116C2×Q8C2×C3⋊Dic311yescharacteristic
117C2×D4—31no—
118C22×C4—31no—
119C22×C4—31no—
120C22×C4—31no—
121M16—184no—
122M16—362no—
123C2×C8—182no—
124C2×C8—182no—
125C4⋊C4—31no—
126C4⋊C4—31no—
127C22⋊C4—61no—
128C42—31no—
129C42—31no—
130C42—31no—
131C42C3⋊S411yescharacteristic
132C22×C3—43no—
133C22×C3—61no—
134C22×C3—61no—
135C22×C3—61no—
136C22×C3C2×SL(2,ℤ/4)11yescharacteristic
137C3×C4—21no—
138C3×C4—31no—
139C3×C4—31no—
140C3×C4—43no—
141C3×C4—43no—
142C3×C4—46no—
143C3×C4—61no—
144C3×C4—61no—
145C3×C4—83no—
146C3×C4GU(2,𝔽3)12yes—
147C3×C4#98411yescharacteristic
148C3×C4#98611yescharacteristic
149C32—41noSylow
150C23—31no—
151Q8—31no—
152Q8#374811yescharacteristic
153D4—32no—
154D4—61no—
155C2×C4—31no—
156C2×C4—31no—
157C2×C4—31no—
158C2×C4—61no—
159C2×C4—61no—
160C2×C4—61no—
161C2×C4—61no—
162C2×C4—62no—
163C2×C4C3⋊SL(2,ℤ/4)11yescharacteristic
164C2×C4C3⋊SL(2,ℤ/4)11yescharacteristic
165C2×C4C2×C3⋊S411yescharacteristic
166C8—184no—
167C8—362no—
168C2×C3—43no—
169C2×C3—43no—
170C2×C3—43no—
171C2×C3—121no—
172C2×C3#540911yescharacteristic
173C2×C3C2×GU(2,𝔽3)11yescharacteristic
174C2×C3#542611yescharacteristic
175C22—61no—
176C22—61no—
177C22—61no—
178C22C2×C3⋊SL(2,ℤ/4)11yescharacteristic
179C4—21no—
180C4—31no—
181C4—31no—
182C4—61no—
183C4—61no—
184C4C3⋊GU(2,𝔽3)12yes—
185C4#6426511yescharacteristic
186C4#6426711yescharacteristic
187C3—43no—
188C3#7307211yescharacteristic, complemented
189C2—121no—
190C2#1059364311yescharacteristic
191C2C2×C3⋊GU(2,𝔽3)11yescharacteristic
192C2#1059366011yescharacteristic
subgroupG/H
center, FrattiniC2×C4C2×C3⋊S4
commutatorC2×C3×SL(2,𝔽3)C2×C4
FittingC3×C4×#41D3
socleC22×C3C2×SL(2,ℤ/4)
prime radicalC2×C3×#41⋊C3C4
cosocleC3×C4×SL(2,𝔽3)C22
hypercenterC42C3⋊S4
hypocenterC3×SL(2,𝔽3)M16
LCSG ▷ C2×C3×SL(2,𝔽3) ▷ C3×SL(2,𝔽3)
UCSC1 ◁ C2×C4 ◁ C42

Complements

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

NH ≅ G/Ntype
2. C3⋊(SL(2,𝔽3)⋊C8)C2semidirect
41. C3×SL(2,𝔽3)M16semidirect
107. SL(2,𝔽3)#208semidirect
188. C3#73072semidirect

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
#730723
#716413
C4⋊(C3⋊(C3⋊C8))4