Finite Group Databasebeta

← C3×#64264 #11724457

Subgroups — C3×#64264 #11724457

Order 864.

268 conjugacy classes of subgroups; only the 42 normal subgroups are shown.

normal subgroups only

This group has 268 conjugacy classes of subgroups — too many to draw. Only its 42 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
1C2×C32×SL(2,𝔽3)C211yesmaximal, characteristic, complemented
2C3×C3⋊GL(2,𝔽3)C211yesmaximal, characteristic, complemented
3C3×C3⋊#226C211yesmaximal, characteristic, complemented
4#64264C311yesmaximal, characteristic, direct product factor
5C3×#983—33nomaximal
6C3×#942—31nomaximal
7C3×#638—41nomaximal
8C32×SL(2,𝔽3)C2211yescharacteristic
9C2×C32×Q8D311yescharacteristic, complemented
10C3×#235—31no—
11C2×C3×SL(2,𝔽3)—21no—
12C2×C3×SL(2,𝔽3)—23no—
13C2×C3×SL(2,𝔽3)D313yescomplemented
14C2×C3×SL(2,𝔽3)C2×C311yescharacteristic, complemented
15C3⋊GL(2,𝔽3)C2×C311yescharacteristic, complemented
16C3⋊#226C2×C311yescharacteristic, complemented
17C3×GL(2,𝔽3)—33no—
18C3×#226—33no—
19C3×#216—31no—
20C3×#216—31no—
21C3×#215—31no—
22C3×#215—31no—
23C3×#208—31no—
24C22×C33—41no—
25C2×C3×C3⋊D3—41no—
26C3×C3⋊Dic3—41no—
27#983—33no—
28C3×#137—91no—
29#942—31no—
30C32×Q8—31no—
31C32×Q8C2×D311yescharacteristic
32C2×C32×C4—31no—
33#638—41no—
34C3×#67—61no—
35C3×#67—123no—
36C3×D12—31no—
37C3×C4×D3—61no—
38C3×Dic6—31no—
39C3×SL(2,𝔽3)—21no—
40C3×SL(2,𝔽3)—23no—
41C3×SL(2,𝔽3)C2×D313yes—
42C3×SL(2,𝔽3)C22×C311yescharacteristic
43C3×C3⋊C8—31no—
44C3×C3⋊C8—31no—
45C2×C33—41no—
46C2×C33—81no—
47C3×C3⋊D3—81no—
48C3×#41—91no—
49C2×C3×Q8—21no—
50C2×C3×Q8C3×D311yescharacteristic, complemented
51C2×C3×Q8C3⋊D311yescharacteristic, complemented
52#235—31no—
53C2×SL(2,𝔽3)—23no—
54C2×SL(2,𝔽3)C3×D313yescomplemented
55GL(2,𝔽3)—33no—
56#226—33no—
57C3×Q16—91no—
58C3×Q16—91no—
59C3×SD16—91no—
60C3×SD16—91no—
61C3×M16—91no—
62#216—31no—
63#216—31no—
64#215—31no—
65#215—31no—
66#208—31no—
67C22×C32—41no—
68C22×C32—43no—
69C22×C32—83no—
70C22×C32—81no—
71C22×C32S411yescharacteristic
72C2×C3⋊D3—41no—
73C2×C3×D3—61no—
74C2×C3×D3—123no—
75C32×C4—31no—
76C32×C4—31no—
77C3⋊Dic3—41no—
78C3×Dic3—61no—
79C3×Dic3—123no—
80#137—91noSylow
81C33—41noSylow
82C3×Q8—21no—
83C3×Q8—31no—
84C3×Q8—31no—
85C3×Q8—61no—
86C3×Q8—91no—
87C3×Q8C2×C3×D311yescharacteristic
88C3×Q8C2×C3⋊D311yescharacteristic
89C3×D4—91no—
90C3×D4—181no—
91C2×C3×C4—31no—
92C2×C3×C4—31no—
93C2×C3×C4—61no—
94C2×C3×C4—181no—
95#67—61no—
96#67—123no—
97D12—31no—
98C4×D3—61no—
99Dic6—31no—
100SL(2,𝔽3)—23no—
101SL(2,𝔽3)C2×C3×D313yes—
102C3×C8—91no—
103C3×C8—91no—
104C3⋊C8—31no—
105C3⋊C8—31no—
106C2×C32—21no—
107C2×C32—41no—
108C2×C32—43no—
109C2×C32—83no—
110C2×C32—83no—
111C2×C32—81no—
112C2×C32—86no—
113C2×C32—82no—
114C2×C32—81no—
115C2×C32C2×S411yescharacteristic
116C3⋊D3—81no—
117C3×D3—121no—
118C3×D3—243no—
119#41—91no—
120C2×Q8C3×C3⋊D311yescharacteristic, complemented
121Q16—91no—
122Q16—91no—
123SD16—91no—
124SD16—91no—
125M16—91no—
126C22×C3—21no—
127C22×C3—43no—
128C22×C3—83no—
129C22×C3—181no—
130C22×C3C3×S411yescharacteristic
131C22×C3C3⋊S411yescharacteristic
132C2×D3—61no—
133C2×D3—123no—
134C3×C4—31no—
135C3×C4—31no—
136C3×C4—31no—
137C3×C4—31no—
138C3×C4—61no—
139C3×C4—61no—
140C3×C4—181no—
141Dic3—61no—
142Dic3—123no—
143C32—41no—
144C32—43no—
145C32—83no—
146C32—81no—
147C32#98311yescharacteristic, complemented
148Q8—31no—
149Q8—91no—
150Q8C2×C3×C3⋊D311yescharacteristic
151D4—91no—
152D4—181no—
153C2×C4—31no—
154C2×C4—181no—
155C8—91no—
156C8—91no—
157C2×C3—22no—
158C2×C3—21no—
159C2×C3—21no—
160C2×C3—21no—
161C2×C3—43no—
162C2×C3—83no—
163C2×C3—86no—
164C2×C3—83no—
165C2×C3—361no—
166C2×C3C2×C3×S411yescharacteristic
167C2×C3C2×C3⋊S411yescharacteristic
168D3—121no—
169D3—243no—
170C22—181no—
171C22C3×C3⋊S411yescharacteristic
172C4—31no—
173C4—31no—
174C4—181no—
175C3—21no—
176C3—43no—
177C3—83no—
178C3C3×#98311yescharacteristic, complemented
179C3#6426411yescharacteristic, direct product factor
180C2—21no—
181C2—361no—
182C2C2×C3×C3⋊S411yescharacteristic
subgroupG/H
centerC2×C3C2×C3⋊S4
commutator, cosocle, hypocenterC3×SL(2,𝔽3)C22×C3
FrattiniC2C2×C3×C3⋊S4
FittingC2×C32×Q8D3
socleC2×C32C2×S4
hypercenterC22×C3C3⋊S4
LCSG ▷ C3×SL(2,𝔽3)
UCSC1 ◁ C2×C3 ◁ C22×C3

Complements

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

NH ≅ G/Ntype
1. C2×C32×SL(2,𝔽3)C2semidirect
2. C3×C3⋊GL(2,𝔽3)C2semidirect
3. C3×C3⋊#226C2semidirect
4. #64264C3direct (N × H)
9. C2×C32×Q8D3semidirect
13. C2×C3×SL(2,𝔽3)D3semidirect
14. C2×C3×SL(2,𝔽3)C2×C3semidirect
15. C3⋊GL(2,𝔽3)C2×C3semidirect
16. C3⋊#226C2×C3semidirect
50. C2×C3×Q8C3×D3semidirect
51. C2×C3×Q8C3⋊D3semidirect
54. C2×SL(2,𝔽3)C3×D3semidirect
120. C2×Q8C3×C3⋊D3semidirect
147. C32#983semidirect
178. C3C3×#983semidirect
179. C3#64264direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C32×SL(2,𝔽3)2
C3×C3⋊GL(2,𝔽3)2
C3×C3⋊#2262
#642643
C3×#9833
C3×#9423
C3×#6384