Finite Group Databasebeta

← C3×C3⋊C8×SL(2,𝔽3) #13157032

Subgroups — C3×C3⋊C8×SL(2,𝔽3) #13157032

Order 1728.

324 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 324 conjugacy classes of subgroups — too many to draw. Only its 80 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×C4×SL(2,𝔽3)C211yesmaximal, characteristic
2C3×Q8×C3⋊C8C311yesmaximal, characteristic, complemented
3C3×C8×SL(2,𝔽3)—31nomaximal
4C3⋊C8×SL(2,𝔽3)C313yesmaximal, direct product factor
5C2×C32×SL(2,𝔽3)C411yescharacteristic
6C2×C32×C3⋊C8—41nomaximal
7C32×C4×Q8C2×C311yescharacteristic
8C3×C4×SL(2,𝔽3)—21no—
9C3×C4×SL(2,𝔽3)—23no—
10C3×C4×SL(2,𝔽3)D311yescharacteristic
11C3×C4×SL(2,𝔽3)C2×C313yes—
12C3×C4⋊(C3⋊C8)—31no—
13C3×C4×C3⋊C8—31no—
14C2×C33×C4—41no—
15C32×SL(2,𝔽3)C811yescharacteristic, complemented
16C32×C3⋊C8—42no—
17C3×C8×Q8—31no—
18Q8×C3⋊C8C3211yescharacteristic, complemented
19C8×SL(2,𝔽3)—33no—
20C2×C32×Q8C3×C411yescharacteristic
21C2×C3×SL(2,𝔽3)—21no—
22C2×C3×SL(2,𝔽3)—23no—
23C2×C3×SL(2,𝔽3)Dic311yescharacteristic
24C2×C3×SL(2,𝔽3)C3×C413yes—
25C2×C32×C8—121no—
26C32×C4⋊C4—31no—
27C32×C42—31no—
28C2×C3×C3⋊C8—31no—
29C2×C3×C3⋊C8—43no—
30C2×C3×C3⋊C8A411yescharacteristic
31C22×C33—41no—
32C33×C4—41no—
33C33×C4—41no—
34C3×C4×Q8—21no—
35C3×C4×Q8C3×D311yescharacteristic
36C3×C4×Q8C2×C3211yescharacteristic
37C4×SL(2,𝔽3)—23no—
38C4×SL(2,𝔽3)C3×D313yes—
39C3×C4⋊C8—91no—
40C3×C4×C8—91no—
41C4⋊(C3⋊C8)—31no—
42C4×C3⋊C8—31no—
43C32×Q8—31no—
44C32×Q8C3×C811yescharacteristic, complemented
45C2×C32×C4—31no—
46C2×C32×C4—31no—
47C2×C32×C4—41no—
48C2×C32×C4—43no—
49C2×C32×C4—83no—
50C2×C32×C4—81no—
51C2×C32×C4C2×A411yescharacteristic
52C3×SL(2,𝔽3)—21no—
53C3×SL(2,𝔽3)—23no—
54C3×SL(2,𝔽3)C3⋊C811yescharacteristic, direct product factor
55C3×SL(2,𝔽3)C3×C813yescomplemented
56C32×C8—122no—
57C3×C3⋊C8—46no—
58C3×C3⋊C8—61no—
59C3×C3⋊C8SL(2,𝔽3)12yesdirect product factor
60C8×Q8—31noSylow
61C2×C33—41no—
62C2×C33—41no—
63C2×C33—41no—
64C2×C3×Q8—21no—
65C2×C3×Q8C3×Dic311yescharacteristic
66C2×C3×Q8C32×C411yescharacteristic
67C2×SL(2,𝔽3)—23no—
68C2×SL(2,𝔽3)C3×Dic313yes—
69C2×C3×C8—31no—
70C2×C3×C8—91no—
71C2×C3×C8—123no—
72C3×C4⋊C4—31no—
73C3×C4⋊C4—31no—
74C3×C4⋊C4—61no—
75C3×C42—31no—
76C3×C42—31no—
77C3×C42—61no—
78C2×C3⋊C8—31no—
79C2×C3⋊C8C3×A411yescharacteristic
80C22×C32—41no—
81C22×C32—43no—
82C22×C32—83no—
83C22×C32—81no—
84C22×C32C4×A411yescharacteristic
85C32×C4—31no—
86C32×C4—31no—
87C32×C4—41no—
88C32×C4—41no—
89C32×C4—43no—
90C32×C4—43no—
91C32×C4—61no—
92C32×C4—83no—
93C32×C4—83no—
94C32×C4—81no—
95C32×C4—81no—
96C32×C4C2×SL(2,𝔽3)11yescharacteristic
97C32×C4#41⋊C311yescharacteristic
98C4×Q8C32×D311yescharacteristic
99C4⋊C8—91no—
100C4×C8—91no—
101C33—41noSylow
102C3×Q8—21no—
103C3×Q8—31no—
104C3×Q8—31no—
105C3×Q8—61no—
106C3×Q8C3×C3⋊C811yescharacteristic, complemented
107C3×Q8C32×C811yescharacteristic, complemented
108C2×C3×C4—21no—
109C2×C3×C4—31no—
110C2×C3×C4—31no—
111C2×C3×C4—31no—
112C2×C3×C4—31no—
113C2×C3×C4—43no—
114C2×C3×C4—61no—
115C2×C3×C4—61no—
116C2×C3×C4—83no—
117C2×C3×C4D3×A411yescharacteristic
118C2×C3×C4C2×C3×A411yescharacteristic
119SL(2,𝔽3)—23no—
120SL(2,𝔽3)C3×C3⋊C813yesdirect product factor
121C3×C8—32no—
122C3×C8—126no—
123C3×C8—181no—
124C3⋊C8—61no—
125C3⋊C8C3×SL(2,𝔽3)12yesdirect product factor
126C2×C32—41no—
127C2×C32—41no—
128C2×C32—41no—
129C2×C32—43no—
130C2×C32—43no—
131C2×C32—43no—
132C2×C32—83no—
133C2×C32—83no—
134C2×C32—83no—
135C2×C32—81no—
136C2×C32—81no—
137C2×C32—81no—
138C2×C32C4×SL(2,𝔽3)11yescharacteristic
139C2×C32C8×A411yescharacteristic
140C2×C32#131⋊C311yescharacteristic
141C2×Q8C32×Dic311yescharacteristic
142C2×C8—31no—
143C2×C8—91no—
144C4⋊C4—31no—
145C42—31no—
146C22×C3—21no—
147C22×C3—43no—
148C22×C3—83no—
149C22×C3Dic3×A411yescharacteristic
150C22×C3C3×C4×A411yescharacteristic
151C3×C4—21no—
152C3×C4—21no—
153C3×C4—31no—
154C3×C4—31no—
155C3×C4—31no—
156C3×C4—31no—
157C3×C4—43no—
158C3×C4—43no—
159C3×C4—61no—
160C3×C4—61no—
161C3×C4—61no—
162C3×C4—61no—
163C3×C4—83no—
164C3×C4—83no—
165C3×C4—121no—
166C3×C4C3⋊(#41⋊C3)11yescharacteristic
167C3×C4D3×SL(2,𝔽3)11yescharacteristic
168C3×C4C2×C3×SL(2,𝔽3)11yescharacteristic
169C3×C4C3×#41⋊C311yescharacteristic
170C32—41no—
171C32—43no—
172C32—83no—
173C32—81no—
174C32C8×SL(2,𝔽3)11yescharacteristic, complemented
175Q8—31no—
176Q8C32×C3⋊C811yescharacteristic, complemented
177C2×C4—31no—
178C2×C4—31no—
179C2×C4C3×D3×A411yescharacteristic
180C8—32no—
181C8—181no—
182C2×C3—21no—
183C2×C3—21no—
184C2×C3—21no—
185C2×C3—43no—
186C2×C3—43no—
187C2×C3—43no—
188C2×C3—83no—
189C2×C3—83no—
190C2×C3—83no—
191C2×C3A4×C3⋊C811yescharacteristic
192C2×C3Dic3×SL(2,𝔽3)11yescharacteristic
193C2×C3C3⋊(#131⋊C3)11yescharacteristic
194C2×C3C3×C4×SL(2,𝔽3)11yescharacteristic
195C2×C3C3×C8×A411yescharacteristic
196C2×C3C3×#131⋊C311yescharacteristic
197C22C3×Dic3×A411yescharacteristic
198C4—31no—
199C4—31no—
200C4—61no—
201C4C3×C3⋊(#41⋊C3)11yescharacteristic
202C4C3×D3×SL(2,𝔽3)11yescharacteristic
203C3—21no—
204C3—43no—
205C3—83no—
206C3C3⋊C8×SL(2,𝔽3)11yescharacteristic, direct product factor
207C3C3×C8×SL(2,𝔽3)11yescharacteristic, complemented
208C2C3×A4×C3⋊C811yescharacteristic
209C2C3×Dic3×SL(2,𝔽3)11yescharacteristic
210C2C3×C3⋊(#131⋊C3)11yescharacteristic
subgroupG/H
center, hypercenterC2×C3×C4D3×A4
commutator, hypocenterC3×Q8C32×C8
FrattiniC2×C4C3×D3×A4
FittingC32×C4×Q8C2×C3
socleC22×C32C4×A4
prime radicalC2×C32×SL(2,𝔽3)C4
cosocleC3×C4×Q8C2×C32
LCSG ▷ C3×Q8
UCSC1 ◁ C2×C3×C4

Complements

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

NH ≅ G/Ntype
2. C3×Q8×C3⋊C8C3semidirect
4. C3⋊C8×SL(2,𝔽3)C3direct (N × H)
15. C32×SL(2,𝔽3)C8semidirect
18. Q8×C3⋊C8C32semidirect
44. C32×Q8C3×C8semidirect
54. C3×SL(2,𝔽3)C3⋊C8direct (N × H)
55. C3×SL(2,𝔽3)C3×C8semidirect
59. C3×C3⋊C8SL(2,𝔽3)direct (N × H)
106. C3×Q8C3×C3⋊C8semidirect
107. C3×Q8C32×C8semidirect
120. SL(2,𝔽3)C3×C3⋊C8direct (N × H)
125. C3⋊C8C3×SL(2,𝔽3)direct (N × H)
174. C32C8×SL(2,𝔽3)semidirect
176. Q8C32×C3⋊C8semidirect
206. C3C3⋊C8×SL(2,𝔽3)direct (N × H)
207. C3C3×C8×SL(2,𝔽3)semidirect

Maximal subgroups

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

subgroupindex
C32×C4×SL(2,𝔽3)2
C3×Q8×C3⋊C83
C3×C8×SL(2,𝔽3)3
C3⋊C8×SL(2,𝔽3)3
C2×C32×C3⋊C84