Finite Group Databasebeta

← #64266⋊C3 #11722358

Subgroups — #64266⋊C3 #11722358

Order 864.

158 conjugacy classes of subgroups; only the 25 normal subgroups are shown.

normal subgroups only

This group has 158 conjugacy classes of subgroups — too many to draw. Only its 25 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
1#41⋊H(3,𝔽3)C211yesmaximal, characteristic, complemented
2(C3⋊GL(2,𝔽3))⋊C3C211yesmaximal, characteristic, complemented
3(C3⋊#226)⋊C3C211yesmaximal, characteristic, complemented
4#64266C311yesmaximal, characteristic, complemented
5C3×#950—31nomaximal
6C4×(C3⋊D3)⋊C3—41nomaximal
7Q8⋊H(3,𝔽3)C2211yescharacteristic
8C32×#41D311yescharacteristic, complemented
9C3×#235—31no—
10C3×#41⋊C3—21no—
11C3×#41⋊C3C2×C311yescharacteristic, complemented
12C3⋊GL(2,𝔽3)C2×C311yescharacteristic, complemented
13C3⋊#226C2×C311yescharacteristic, complemented
14C3×#216—31no—
15C3×#215—31no—
16C3×#214—31no—
17C3×#213—31no—
18C2×C3×C3⋊C8—31no—
19C2×(C3⋊D3)⋊C3—41no—
20C4×H(3,𝔽3)—41no—
21(C3⋊Dic3)⋊C3—41no—
22#985—91no—
23C3×#135—91no—
24#950—31no—
25C32×Q8C2×D311yescharacteristic
26C32×D4—31no—
27C2×C32×C4—31no—
28C4×C3⋊D3—41no—
29C3×#67—61no—
30C3×D12—31no—
31C3×C4×D3—61no—
32C3×Dic6—31no—
33C3×SL(2,𝔽3)—21no—
34C3×SL(2,𝔽3)C22×C311yescharacteristic
35C3×C3⋊C8—31no—
36C3×C3⋊C8—31no—
37C2×H(3,𝔽3)—41no—
38(C3⋊D3)⋊C3—42no—
39C3×#41—31no—
40C3×#41—91no—
41C3×#41C3×D311yescharacteristic
42#235—31no—
43#41⋊C3—31no—
44#41⋊C3—61no—
45GL(2,𝔽3)—91no—
46#226—91no—
47C3×Q16—91no—
48C3×SD16—91no—
49C3×SD16—91no—
50C3×D8—91no—
51C2×C3×C8—91no—
52#216—31no—
53#215—31no—
54#214—31no—
55#213—31no—
56C2×C3⋊C8—31no—
57C22×C32—31no—
58C2×C3⋊D3—41no—
59C2×C3×D3—61no—
60C32×C4—31no—
61C32×C4—41no—
62C32×C4—81no—
63C32×C4S411yescharacteristic
64C3⋊Dic3—41no—
65C3×Dic3—61no—
66#135—91noSylow
67H(3,𝔽3)—41noSylow
68C3×Q8—31no—
69C3×Q8—91no—
70C3×Q8C2×C3×D311yescharacteristic
71C3×D4—31no—
72C3×D4—31no—
73C3×D4—61no—
74C3×D4—91no—
75C3×D4—181no—
76C2×C3×C4—31no—
77C2×C3×C4—31no—
78C2×C3×C4—61no—
79C2×C3×C4—181no—
80#67—61no—
81D12—31no—
82C4×D3—61no—
83C4×D3—361no—
84Dic6—31no—
85SL(2,𝔽3)—31no—
86SL(2,𝔽3)—61no—
87C3×C8—91no—
88C3×C8—91no—
89C3⋊C8—31no—
90C3⋊C8—31no—
91C2×C32—41no—
92C2×C32—61no—
93C2×C32—81no—
94C2×C32C2×S411yescharacteristic
95C3⋊D3—42no—
96C3×D3—121no—
97#41—91no—
98#41(C3⋊D3)⋊C311yescharacteristic, complemented
99Q16—91no—
100SD16—91no—
101SD16—91no—
102D8—91no—
103C2×C8—91no—
104C22×C3—31no—
105C22×C3—31no—
106C22×C3—61no—
107C22×C3—181no—
108C2×D3—61no—
109C2×D3—361no—
110C3×C4—31no—
111C3×C4—31no—
112C3×C4—31no—
113C3×C4—61no—
114C3×C4—121no—
115C3×C4—181no—
116C3×C4—241no—
117C3×C4C3×S411yescharacteristic
118Dic3—61no—
119Dic3—361no—
120C32—41no—
121C32—81no—
122C32#98511yescharacteristic, complemented
123Q8—91no—
124Q8C2×(C3⋊D3)⋊C311yescharacteristic
125D4—31no—
126D4—91no—
127D4—181no—
128C2×C4—31no—
129C2×C4—181no—
130C8—91no—
131C8—91no—
132C2×C3—31no—
133C2×C3—61no—
134C2×C3—61no—
135C2×C3—121no—
136C2×C3—121no—
137C2×C3—241no—
138C2×C3—361no—
139C2×C3C2×C3×S411yescharacteristic
140D3—121no—
141D3—362no—
142C22—31no—
143C22—181no—
144C4—31no—
145C4—181no—
146C4(C3⋊S4)⋊C311yescharacteristic
147C3—31no—
148C3—121no—
149C3—241no—
150C3C3×#98511yescharacteristic
151C2—61no—
152C2—361no—
153C2C2×(C3⋊S4)⋊C311yescharacteristic
subgroupG/H
center, hypercenterC4(C3⋊S4)⋊C3
commutator, cosocle, hypocenterC3×SL(2,𝔽3)C22×C3
Frattini, socleC2×C3C2×C3×S4
FittingC32×#41D3
LCSG ▷ C3×SL(2,𝔽3)
UCSC1 ◁ C4

Complements

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

NH ≅ G/Ntype
1. #41⋊H(3,𝔽3)C2semidirect
2. (C3⋊GL(2,𝔽3))⋊C3C2semidirect
3. (C3⋊#226)⋊C3C2semidirect
4. #64266C3semidirect
8. C32×#41D3semidirect
11. C3×#41⋊C3C2×C3semidirect
12. C3⋊GL(2,𝔽3)C2×C3semidirect
13. C3⋊#226C2×C3semidirect
98. #41(C3⋊D3)⋊C3semidirect
122. C32#985semidirect

Maximal subgroups

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

subgroupindex
#41⋊H(3,𝔽3)2
(C3⋊GL(2,𝔽3))⋊C32
(C3⋊#226)⋊C32
#642663
C3×#9503
C4×(C3⋊D3)⋊C34