Finite Group Databasebeta

← #11921025

Subgroups — #11921025

Order 1152.

278 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 278 conjugacy classes of subgroups — too many to draw. Only its 61 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#73069—33nomaximal
4#71640—31nomaximal
5C2×C3×#41⋊C3C411yescharacteristic
6C3×C4×SL(2,𝔽3)C411yescharacteristic
7C3×C4×SL(2,𝔽3)C2211yescharacteristic
8#63630—41nomaximal
9C3×C4×#41D311yescharacteristic
10C4×#41⋊C3D313yes—
11#4926—31no—
12#4925—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#1488—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—
29C4×SL(2,𝔽3)Dic313yes—
30C4×SL(2,𝔽3)C2×D313yes—
31#830—61no—
32#830—61no—
33C4⋊(C3⋊C8)—32no—
34C4⋊(C3⋊C8)—61no—
35#803—61no—
36#803—123no—
37C4×C3⋊C8—32no—
38C2×C32×C4—41no—
39C2×C32×C4—41no—
40C2×C32×C4—41no—
41C3×SL(2,𝔽3)M1611yescharacteristic
42C3⋊(C3⋊C8)—44no—
43C4×#41C3⋊D311yescharacteristic
44#432—91no—
45#431—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
65C2×C3⋊C8—62no—
66C2×C3⋊C8—62no—
67C2×C3⋊C8—126no—
68C22×C32—41no—
69C32×C4—42no—
70C32×C4—81no—
71C32×C4—81no—
72C2×#41C3⋊Dic311yescharacteristic
73C4×Q8C3⋊Dic311yescharacteristic
74C4×Q8C2×C3⋊D311yescharacteristic
75C4×D4—31no—
76C4×D4—31no—
77#117—31no—
78C2×C42—31no—
79C4⋊C8—92no—
80C4⋊C8—181no—
81C22⋊C8—181no—
82C22⋊C8—181no—
83#97—181no—
84C4×C8—92no—
85C23×C3—31no—
86C3×Q8—31no—
87C3×Q8#20811yescharacteristic
88C3×D4—32no—
89C3×D4—61no—
90C2×C3×C4—31no—
91C2×C3×C4—31no—
92C2×C3×C4—31no—
93C2×C3×C4—43no—
94C2×C3×C4—43no—
95C2×C3×C4—43no—
96C2×C3×C4—61no—
97C2×C3×C4—61no—
98C2×C3×C4—62no—
99C2×C3×C4—121no—
100C2×C3×C4SL(2,ℤ/4)11yescharacteristic
101C2×C3×C4SL(2,ℤ/4)11yescharacteristic
102C2×C3×C4C2×S411yescharacteristic
103SL(2,𝔽3)#20813yes—
104C3⋊C8—1212no—
105C3⋊C8—122no—
106C3⋊C8—122no—
107C2×C32—41no—
108C2×C32—41no—
109C2×C32—41no—
110#41—21no—
111#41—61no—
112C2×Q8C2×C3⋊Dic311yescharacteristic
113C2×D4—31no—
114C22×C4—31no—
115C22×C4—31no—
116C22×C4—31no—
117C2×C8—182no—
118C2×C8—182no—
119C4⋊C4—31no—
120C4⋊C4—31no—
121C22⋊C4—61no—
122C42—31no—
123C42—31no—
124C42—31no—
125C42C3⋊S411yescharacteristic
126C22×C3—43no—
127C22×C3—61no—
128C22×C3—61no—
129C22×C3—61no—
130C22×C3C2×SL(2,ℤ/4)11yescharacteristic
131C3×C4—21no—
132C3×C4—21no—
133C3×C4—31no—
134C3×C4—31no—
135C3×C4—46no—
136C3×C4—61no—
137C3×C4—61no—
138C3×C4—83no—
139C3×C4—83no—
140C3×C4#98512yes—
141C32—41noSylow
142C23—31no—
143Q8—31no—
144Q8#374811yescharacteristic
145D4—32no—
146D4—61no—
147C2×C4—31no—
148C2×C4—31no—
149C2×C4—31no—
150C2×C4—61no—
151C2×C4—61no—
152C2×C4—62no—
153C2×C4—121no—
154C2×C4C3⋊SL(2,ℤ/4)11yescharacteristic
155C2×C4C3⋊SL(2,ℤ/4)11yescharacteristic
156C2×C4C2×C3⋊S411yescharacteristic
157C8—362no—
158C8—362no—
159C2×C3—43no—
160C2×C3—43no—
161C2×C3—43no—
162C2×C3—121no—
163C2×C3#540911yescharacteristic
164C2×C3#542311yescharacteristic
165C2×C3#542411yescharacteristic
166C22—61no—
167C22—61no—
168C22—61no—
169C22C2×C3⋊SL(2,ℤ/4)11yescharacteristic
170C4—21no—
171C4—21no—
172C4—31no—
173C4—31no—
174C4—61no—
175C4—61no—
176C4#6426612yes—
177C3—43no—
178C3#7306911yescharacteristic, complemented
179C2—121no—
180C2#1059364311yescharacteristic
181C2#1059365711yescharacteristic
182C2#1059365811yescharacteristic
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
178. C3#73069semidirect

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
#730693
#716403
#636304