Finite Group Databasebeta

← #10593637⋊C3 #13146860

Subgroups — #10593637⋊C3 #13146860

Order 1728.

241 conjugacy classes of subgroups; only the 33 normal subgroups are shown.

normal subgroups only

This group has 241 conjugacy classes of subgroups — too many to draw. Only its 33 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#64265⋊C3C212yesmaximal
2#131⋊H(3,𝔽3)C211yesmaximal, characteristic
3#10593637C311yesmaximal, characteristic, complemented
4C3×#5143—31nomaximal
5#41⋊H(3,𝔽3)C2211yescharacteristic
6(C3⋊#226)⋊C3—22no—
7#3746⋊C3—41nomaximal
8#64265C2×C312yes—
9C32×#131D311yescharacteristic
10C3×#951—32no—
11C2×C3×Dic12—31no—
12C3×#131⋊C3—21no—
13C3×#131⋊C3C2×C311yescharacteristic
14C3×#824—32no—
15C3×#819—31no—
16#634⋊C3—42no—
17Q8⋊H(3,𝔽3)D411yescharacteristic
18C8×H(3,𝔽3)—41no—
19#5403—91no—
20C3×#473—91no—
21#5143—31no—
22C32×#41C2×D311yescharacteristic
23C2×C3×Dic6—32no—
24C3×#41⋊C3—21no—
25C3×#41⋊C3C22×C311yescharacteristic
26C3⋊#226—22no—
27C32×M16—31no—
28C2×C32×C8—31no—
29#3746—41no—
30C3×#216—62no—
31C3×#214—62no—
32C3×#208—32no—
33C3×Dic12—61no—
34C3×Dic12—61no—
35C4×H(3,𝔽3)—41no—
36(C3⋊Dic3)⋊C3—82no—
37#984—92no—
38C3×#137—92no—
39C2×C3×Q16—91no—
40C3×#131—31no—
41C3×#131C3×D311yescharacteristic
42#951—32no—
43C2×Dic12—31no—
44#131⋊C3—31no—
45#131⋊C3—61no—
46C3×#108—91no—
47C3×#101—92no—
48#824—32no—
49#819—31no—
50C32×Q8D1211yescharacteristic
51C32×D4—31no—
52C2×C32×C4—31no—
53#634—42no—
54C2×C3×Dic3—62no—
55C3×Dic6—62no—
56C3×Dic6—62no—
57C3×SL(2,𝔽3)—21no—
58C3×SL(2,𝔽3)C3×D411yescharacteristic
59C32×C8—31no—
60C32×C8—41no—
61C32×C8—81no—
62C32×C8S411yescharacteristic
63C3×C3⋊C8—62no—
64#473—91noSylow
65C2×H(3,𝔽3)—41no—
66C3×#41—31no—
67C3×#41C2×C3×D311yescharacteristic
68C2×C3×Q8—92no—
69C2×Dic6—32no—
70#41⋊C3—31no—
71#41⋊C3—61no—
72#226—182no—
73C3×Q16—181no—
74C3×Q16—181no—
75C3×Q16—182no—
76C3×SD16—182no—
77C3×M16—31no—
78C3×M16—31no—
79C3×M16—61no—
80C3×M16—92no—
81C2×C3×C8—31no—
82C2×C3×C8—31no—
83C2×C3×C8—61no—
84#216—62no—
85#214—62no—
86#208—32no—
87Dic12—61no—
88Dic12—61no—
89Dic12—361no—
90C22×C32—31no—
91C32×C4—31no—
92C32×C4—41no—
93C32×C4—81no—
94C32×C4C2×S411yescharacteristic
95C3⋊Dic3—82no—
96C3×Dic3—122no—
97#137—92no—
98C2×Q16—91no—
99#131(C3⋊D3)⋊C311yescharacteristic
100#108—91no—
101#101—92no—
102H(3,𝔽3)—41noSylow
103C3×Q8—31no—
104C3×Q8—182no—
105C3×Q8—182no—
106C3×Q8C3×D1211yescharacteristic
107C3×D4—31no—
108C3×D4—31no—
109C3×D4—61no—
110C2×C3×C4—31no—
111C2×C3×C4—31no—
112C2×C3×C4—61no—
113C2×C3×C4—182no—
114C2×Dic3—62no—
115Dic6—62no—
116Dic6—62no—
117Dic6—362no—
118SL(2,𝔽3)—31no—
119SL(2,𝔽3)—61no—
120C3×C8—31no—
121C3×C8—31no—
122C3×C8—31no—
123C3×C8—61no—
124C3×C8—121no—
125C3×C8—182no—
126C3×C8—241no—
127C3×C8C3×S411yescharacteristic
128C3⋊C8—62no—
129C2×C32—41no—
130C2×C32—61no—
131C2×C32—81no—
132C2×C32C4⋊S411yescharacteristic
133#41C2×(C3⋊D3)⋊C311yescharacteristic
134C2×Q8—92no—
135Q16—181no—
136Q16—182no—
137Q16—181no—
138SD16—182no—
139M16—31no—
140M16—92no—
141C2×C8—31no—
142C22×C3—31no—
143C22×C3—31no—
144C22×C3—61no—
145C3×C4—31no—
146C3×C4—31no—
147C3×C4—31no—
148C3×C4—61no—
149C3×C4—121no—
150C3×C4—241no—
151C3×C4—362no—
152C3×C4C2×C3×S411yescharacteristic
153Dic3—122no—
154Dic3—722no—
155C32—41no—
156C32—81no—
157C32#540311yescharacteristic, complemented
158Q8—182no—
159Q8—182no—
160Q8C4⋊((C3⋊D3)⋊C3)11yescharacteristic
161D4—31no—
162C2×C4—31no—
163C2×C4—182no—
164C8—31no—
165C8—182no—
166C8(C3⋊S4)⋊C311yescharacteristic
167C2×C3—31no—
168C2×C3—61no—
169C2×C3—61no—
170C2×C3—121no—
171C2×C3—121no—
172C2×C3—241no—
173C2×C3C3×C4⋊S411yescharacteristic
174C22—31no—
175C4—31no—
176C4—362no—
177C4C2×(C3⋊S4)⋊C311yescharacteristic
178C3—31no—
179C3—121no—
180C3—241no—
181C3C3×#540311yescharacteristic
182C2—61no—
183C2C4⋊((C3⋊S4)⋊C3)11yescharacteristic
subgroupG/H
centerC2C4⋊((C3⋊S4)⋊C3)
commutator, cosocleC3×#41⋊C3C22×C3
FrattiniC3×C4C2×C3×S4
FittingC32×#131D3
socleC2×C3C3×C4⋊S4
prime radical#41⋊H(3,𝔽3)C22
hypercenterC8(C3⋊S4)⋊C3
hypocenterC3×SL(2,𝔽3)C3×D4
LCSG ▷ C3×#41⋊C3 ▷ C3×SL(2,𝔽3)
UCSC1 ◁ C2 ◁ C4 ◁ C8

Complements

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

NH ≅ G/Ntype
3. #10593637C3semidirect
157. C32#5403semidirect

Maximal subgroups

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

subgroupindex
#64265⋊C32
#131⋊H(3,𝔽3)2
#105936373
C3×#51433
#3746⋊C34