Finite Group Databasebeta

← #458⋊H(3,𝔽3) #13141261

Subgroups — #458⋊H(3,𝔽3) #13141261

Order 1728.

259 conjugacy classes of subgroups; only the 26 normal subgroups are shown.

normal subgroups only

This group has 259 conjugacy classes of subgroups — too many to draw. Only its 26 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
1C32⋊#997C211yesmaximal, characteristic
2C32×#458C311yesmaximal, characteristic, complemented
3C3×#458⋊C3C313yesmaximal, complemented
4C4×C22⋊H(3,𝔽3)—41nomaximal
5C32×H(5,𝔽2)C2×C311yescharacteristic
6C3×#997C2×C313yes—
7C32×#123—31no—
8C32×#99—31no—
9C2×C22⋊H(3,𝔽3)—41no—
10C2×C22⋊H(3,𝔽3)—81no—
11Q8⋊H(3,𝔽3)—81no—
12C3×#458—31no—
13C3×#458C3211yescharacteristic
14#458⋊C3—33no—
15C32×#41—61no—
16C2×C32×D4—31no—
17C2×C32×D4—61no—
18C22×C32×C4A411yescharacteristic, complemented
19C3×C4×A4—43no—
20C32×C4⋊C4—61no—
21C32×C22⋊C4—31no—
22C32×C22⋊C4—61no—
23C22⋊H(3,𝔽3)—41no—
24C22⋊H(3,𝔽3)—81no—
25C4×H(3,𝔽3)—161no—
26C3×H(5,𝔽2)—31no—
27C3×H(5,𝔽2)C2×C3211yescharacteristic
28#997—33no—
29C3×#123—33no—
30C3×#123—31no—
31C3×#99—33no—
32C3×#99—31no—
33C23×C32—21no—
34C23×C32—31no—
35C23×C32C2×A411yescharacteristic
36C2×C3×A4—43no—
37C2×C3×A4—83no—
38C32×Q8—21no—
39C32×D4—62no—
40C32×D4—61no—
41C2×C32×C4—31no—
42C2×C32×C4—61no—
43C2×C32×C4—61no—
44C2×C32×C4—61no—
45C3×SL(2,𝔽3)—83no—
46#458H(3,𝔽3)11yescharacteristic, complemented, Sylow
47C2×H(3,𝔽3)—161no—
48C3×#41—63no—
49C3×#41—61no—
50C2×C3×D4—33no—
51C2×C3×D4—31no—
52C2×C3×D4—63no—
53C2×C3×D4—61no—
54C22×C3×C4—31no—
55C22×C3×C4C3×A411yescharacteristic
56C4×A4—123no—
57C3×C4⋊C4—61no—
58C3×C4⋊C4—63no—
59C3×C22⋊C4—33no—
60C3×C22⋊C4—31no—
61C3×C22⋊C4—61no—
62C3×C22⋊C4—63no—
63C22×C32—31no—
64C22×C32—41no—
65C22×C32—61no—
66C22×C32—81no—
67C22×C32—121no—
68C3×A4—43no—
69C3×A4—83no—
70C32×C4—41no—
71C32×C4—61no—
72C32×C4—121no—
73C32×C4—163no—
74H(5,𝔽2)C2×H(3,𝔽3)11yescharacteristic
75#123—31no—
76#99—31no—
77H(3,𝔽3)—161noSylow
78C23×C3—21no—
79C23×C3—33no—
80C23×C3—31no—
81C23×C3—31no—
82C23×C3—61no—
83C23×C3C2×C3×A411yescharacteristic
84C2×A4—123no—
85C2×A4—243no—
86C3×Q8—21no—
87C3×Q8—61no—
88C3×D4—66no—
89C3×D4—62no—
90C3×D4—61no—
91C3×D4—63no—
92C2×C3×C4—33no—
93C2×C3×C4—31no—
94C2×C3×C4—61no—
95C2×C3×C4—63no—
96C2×C3×C4—61no—
97C2×C3×C4—63no—
98C2×C3×C4—61no—
99C2×C3×C4—63no—
100SL(2,𝔽3)—243no—
101C2×C32—61no—
102C2×C32—121no—
103C2×C32—163no—
104C2×C32#120⋊C311yescharacteristic
105#41—61no—
106C2×D4—31no—
107C2×D4—61no—
108C22×C4C22⋊H(3,𝔽3)11yescharacteristic, complemented
109C4⋊C4—61no—
110C22⋊C4—31no—
111C22⋊C4—61no—
112C22×C3—33no—
113C22×C3—31no—
114C22×C3—41no—
115C22×C3—63no—
116C22×C3—61no—
117C22×C3—81no—
118C22×C3—121no—
119C22×C3—123no—
120C22×C3—121no—
121C22×C3—241no—
122A4—123no—
123A4—243no—
124C3×C4—41no—
125C3×C4—63no—
126C3×C4—61no—
127C3×C4—121no—
128C3×C4—123no—
129C3×C4—121no—
130C3×C4—483no—
131C32—163no—
132C32#458⋊C311yescharacteristic, complemented
133C23—21no—
134C23—31no—
135C23C2×C22⋊H(3,𝔽3)11yescharacteristic
136Q8—21no—
137D4—62no—
138D4—61no—
139C2×C4—31no—
140C2×C4—61no—
141C2×C4—61no—
142C2×C4—61no—
143C2×C3—31no—
144C2×C3—63no—
145C2×C3—61no—
146C2×C3—123no—
147C2×C3—121no—
148C2×C3—483no—
149C2×C3C3×#120⋊C311yescharacteristic
150C22—31no—
151C22—41no—
152C22—61no—
153C22—81no—
154C22—121no—
155C4—41no—
156C4—61no—
157C4—121no—
158C3—31no—
159C3—483no—
160C3C3×#458⋊C311yescharacteristic
161C2—61no—
162C2—121no—
163C2#120⋊H(3,𝔽3)11yescharacteristic
subgroupG/H
center, socleC2×C3C3×#120⋊C3
commutator, cosocleC3×H(5,𝔽2)C2×C32
FrattiniC23×C3C2×C3×A4
FittingC32×#458C3
prime radicalC32⋊#997C2
hypercenterC2×C32#120⋊C3
hypocenterH(5,𝔽2)C2×H(3,𝔽3)
LCSG ▷ C3×H(5,𝔽2) ▷ H(5,𝔽2)
UCSC1 ◁ C2×C3 ◁ C2×C32

Complements

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

NH ≅ G/Ntype
2. C32×#458C3semidirect
3. C3×#458⋊C3C3semidirect
18. C22×C32×C4A4semidirect
46. #458H(3,𝔽3)semidirect
108. C22×C4C22⋊H(3,𝔽3)semidirect
132. C32#458⋊C3semidirect

Maximal subgroups

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

subgroupindex
C32⋊#9972
C32×#4583
C3×#458⋊C33
C4×C22⋊H(3,𝔽3)4