Finite Group Databasebeta

← #13166104

Subgroups — #13166104

Order 1728.

312 conjugacy classes of subgroups; only the 36 normal subgroups are shown.

normal subgroups only

This group has 312 conjugacy classes of subgroups — too many to draw. Only its 36 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
1C3×#64265C212yesmaximal
2C3⋊(C3⋊(#131⋊C3))C211yesmaximal, characteristic
3#10593226—32nomaximal
4#10591683—31nomaximal
5C32×#41⋊C3C2211yescharacteristic
6C3×C3⋊#226—22no—
7#88812—41nomaximal
8#64265D312yes—
9C3×#984—32no—
10C3×#984—62no—
11#64157D311yescharacteristic
12C3×#951—32no—
13C2×#3718—31no—
14C3⋊(#131⋊C3)—32no—
15C3⋊(#131⋊C3)—61no—
16#63578—31no—
17#63560—32no—
18C3×#634—42no—
19C32×SL(2,𝔽3)D411yescharacteristic
20C3×C3⋊(C3⋊C8)—41no—
21#5403—91no—
22#5204—92no—
23C32×#41C2×D311yescharacteristic
24C2×C3×Dic6—32no—
25C3×#41⋊C3—22no—
26C3×#41⋊C3—21no—
27C3×#41⋊C3—41no—
28C3×#41⋊C3C2×D312yes—
29C3⋊#226—22no—
30C3×#226—64no—
31C3×#226—62no—
32#3748—31no—
33C2×C3⋊(C3⋊C8)—31no—
34C3×#216—62no—
35C3×#214—62no—
36C3×#208—32no—
37#3719—122no—
38#3718—61no—
39#3718—61no—
40C33×C4—41no—
41C3×C3⋊Dic3—82no—
42#984—32no—
43#984—62no—
44C3×#137—92no—
45#951—32no—
46#948—32no—
47#948—61no—
48C2×#216—92no—
49#131⋊C3—91no—
50#836—92no—
51#824—92no—
52#822—92no—
53C32×Q8D1211yescharacteristic
54C32×D4—31no—
55C2×C32×C4—31no—
56#634—42no—
57C2×C3×Dic3—62no—
58C3×Dic6—62no—
59C3×Dic6—62no—
60C3×Dic6—122no—
61C3×Dic6—242no—
62C3×SL(2,𝔽3)—22no—
63C3×SL(2,𝔽3)—21no—
64C3×SL(2,𝔽3)—41no—
65C3×SL(2,𝔽3)#6712yes—
66C3⋊(C3⋊C8)—31no—
67C3⋊(C3⋊C8)S411yescharacteristic
68C3×C3⋊C8—62no—
69C3×C3⋊C8—122no—
70C3×C3⋊C8—241no—
71#473—271noSylow
72C2×C33—41no—
73C3×#41—21no—
74C3×#41D3212yes—
75C2×C3×Q8—92no—
76C2×Dic6—32no—
77#41⋊C3—22no—
78#41⋊C3—41no—
79#41⋊C3D3211yescharacteristic
80#226—64no—
81#226—62no—
82C3×Q16—182no—
83C3×SD16—182no—
84C3×M16—92no—
85#216—62no—
86#216—182no—
87#216—182no—
88#214—62no—
89#208—32no—
90#208—92no—
91#208—181no—
92C2×C3⋊C8—92no—
93C2×C3⋊C8—181no—
94Dic12—361no—
95C22×C32—31no—
96C32×C4—31no—
97C32×C4—42no—
98C32×C4—81no—
99C32×C4—82no—
100C32×C4—161no—
101C32×C4C2×S411yescharacteristic
102C3⋊Dic3—82no—
103C3×Dic3—122no—
104C3×Dic3—242no—
105C3×Dic3—244no—
106#137—92no—
107C2×Q16—271no—
108#131—91no—
109#108—271no—
110#101—272no—
111C33—41noSylow
112C3×Q8—21no—
113C3×Q8—182no—
114C3×Q8—182no—
115C3×Q8#62612yes—
116C3×D4—32no—
117C3×D4—61no—
118C2×C3×C4—32no—
119C2×C3×C4—61no—
120C2×C3×C4—182no—
121C2×Dic3—62no—
122Dic6—62no—
123Dic6—62no—
124Dic6—122no—
125Dic6—242no—
126SL(2,𝔽3)—22no—
127SL(2,𝔽3)—41no—
128SL(2,𝔽3)#62511yescharacteristic
129C3×C8—182no—
130C3×C8—361no—
131C3⋊C8—32no—
132C3⋊C8—61no—
133C3⋊C8—62no—
134C3⋊C8—92no—
135C3⋊C8—181no—
136C2×C32—42no—
137C2×C32—61no—
138C2×C32—81no—
139C2×C32—82no—
140C2×C32—161no—
141C2×C32C4⋊S411yescharacteristic
142#41#111311yescharacteristic
143C2×Q8—92no—
144Q16—182no—
145Q16—541no—
146Q16—541no—
147SD16—182no—
148M16—92no—
149M16—271no—
150C2×C8—271no—
151C22×C3—32no—
152C22×C3—61no—
153C3×C4—21no—
154C3×C4—32no—
155C3×C4—41no—
156C3×C4—61no—
157C3×C4—82no—
158C3×C4—161no—
159C3×C4—362no—
160C3×C4D3×S412yes—
161Dic3—122no—
162Dic3—242no—
163Dic3—244no—
164C32—42no—
165C32—81no—
166C32—82no—
167C32—161no—
168C32#540311yescharacteristic, complemented
169Q8—182no—
170Q8—182no—
171Q8#629511yescharacteristic
172D4—31no—
173C2×C4—31no—
174C2×C4—182no—
175C8—91no—
176C8—182no—
177C8—271no—
178C2×C3—21no—
179C2×C3—41no—
180C2×C3—62no—
181C2×C3—82no—
182C2×C3—121no—
183C2×C3—161no—
184C2×C3#6420712yes—
185C22—31no—
186C4—31no—
187C4—362no—
188C4#8910111yescharacteristic
189C3—21no—
190C3—41no—
191C3—82no—
192C3—161no—
193C3#1059322612yescomplemented
194C2—61no—
195C2#1172421411yescharacteristic
subgroupG/H
centerC2#11724214
commutator, prime radical, cosocleC32×#41⋊C3C22
Frattini, hypercenterC4#89101
FittingC32×#41C2×D3
socleC2×C32C4⋊S4
hypocenterC32×SL(2,𝔽3)D4
LCSG ▷ C32×#41⋊C3 ▷ C32×SL(2,𝔽3)
UCSC1 ◁ C2 ◁ C4

Complements

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

NH ≅ G/Ntype
168. C32#5403semidirect
193. C3#10593226semidirect

Maximal subgroups

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

subgroupindex
C3×#642652
C3⋊(C3⋊(#131⋊C3))2
#105932263
#105916833
#888124