Finite Group Databasebeta

← #13155171

Subgroups — #13155171

Order 1728.

250 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 250 conjugacy classes of subgroups — too many to draw. Only its 59 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#11724514C211yesmaximal, characteristic
2C2×#88989C211yesmaximal, characteristic
3C3×#64283C211yesmaximal, characteristic
4#10590095—31nomaximal
5C2×C3×C4×C3⋊D3C2211yescharacteristic
6#88994C412yes—
7C2×C3⋊#622—21no—
8C4⋊(C3⋊#157)—21no—
9C4×C3⋊#157—21no—
10#88989—21no—
11#88989C412yescomplemented
12C3×#3792—21no—
13C3×#3788—21no—
14C3×#3787C412yescomplemented
15#64286—31no—
16#64283D311yescharacteristic
17C2×#3788—31no—
18C22×C3×C3⋊D3C2×C411yescharacteristic
19C2×C3⋊#157—22no—
20C2×C33×C4Q811yescharacteristic
21C2×C3×C3⋊Dic3D411yescharacteristic
22C3×C4×C3⋊D3C2×C411yescharacteristic
23C3×C4×C3⋊D3C2×C411yescharacteristic
24C3×C4×C3⋊D3D411yescharacteristic
25C3×C4×C3⋊D3D411yescharacteristic
26C3⋊#622—22no—
27C3×#622—22no—
28#4556—91nomaximal
29C2×C4×C3⋊D3C2×D311yescharacteristic
30C2×C3×C4×D3—121no—
31C22⋊#157—32no—
32#3792—21no—
33C2×#622—61no—
34C4⋊#157—61no—
35C4×#157—61no—
36#3788—21no—
37#3788—32no—
38#3788—61no—
39#3787Dic312yescomplemented
40C22×C33C4⋊C411yescharacteristic
41C2×C3×C3⋊D3—41no—
42C2×C3×C3⋊D3C4211yescharacteristic
43C2×C3×C3⋊D3C22⋊C411yescharacteristic
44C3⋊#157—42no—
45C33×C4C4⋊C411yescharacteristic
46C33×C4C4⋊C411yescharacteristic
47C3×C3⋊Dic3C22⋊C411yescharacteristic
48C3×C3⋊Dic3C22⋊C411yescharacteristic
49C2×C3×M16—91no—
50#926—91no—
51C2×#208—91no—
52C22×C3⋊D3C4×D311yescharacteristic
53C22×C3×D3—121no—
54C2×#157—62no—
55C2×C32×C4—41no—
56C2×C32×C4—81no—
57C2×C32×C4Dic611yescharacteristic
58C2×C3⋊Dic3#6711yescharacteristic
59C4×C3⋊D3C4×D311yescharacteristic
60C4×C3⋊D3D1211yescharacteristic
61C4×C3⋊D3C2×Dic311yescharacteristic
62C4×C3⋊D3#6711yescharacteristic
63C2×C3×Dic3—121no—
64C3×C4×D3—121no—
65C3×C4×D3—121no—
66C3×C4×D3—121no—
67C3×C4×D3—121no—
68#622—22no—
69#622—62no—
70#344—271noSylow
71C2×C33—21no—
72C2×C33#9511yescharacteristic
73C3×C3⋊D3—21no—
74C3×C3⋊D3—21no—
75C22×C3×C4—91no—
76C2×C4×D3—121no—
77C3×M16—181no—
78C3×M16—181no—
79C2×C3×C8—92no—
80#217—92no—
81C4⋊Dic3—181no—
82C4×Dic3—181no—
83#208—92no—
84#208—181no—
85C2×C3⋊C8—181no—
86C22×C32—41no—
87C22×C32—81no—
88C22×C32Dic3⋊C411yescharacteristic
89C2×C3⋊D3—41no—
90C2×C3⋊D3C4×Dic311yescharacteristic
91C2×C3⋊D3(C2×D3)⋊C411yescharacteristic
92C2×C3×D3—121no—
93C2×C3×D3—121no—
94C2×C3×D3—124no—
95#157—122no—
96C32×C4—41no—
97C32×C4—41no—
98C32×C4—81no—
99C32×C4—81no—
100C32×C4Dic3⋊C411yescharacteristic
101C32×C4C4⋊Dic311yescharacteristic
102C3⋊Dic3(C2×D3)⋊C411yescharacteristic
103C3⋊Dic3#21711yescharacteristic
104C3×Dic3—121no—
105C3×Dic3—121no—
106C2×M16—91no—
107C2×M16—271no—
108#117—271no—
109C33#34411yescharacteristic, complemented, Sylow
110C23×C3—91no—
111C22×D3—121no—
112C2×C3×C4—41no—
113C2×C3×C4—81no—
114C2×C3×C4—91no—
115C2×C3×C4—91no—
116C2×C3×C4—91no—
117C2×C3×C4—91no—
118C2×C3×C4—91no—
119C2×C3×C4PSU(3,𝔽2)11yescharacteristic
120C2×Dic3—121no—
121C2×Dic3—182no—
122C4×D3—121no—
123C4×D3—121no—
124C4×D3—121no—
125C4×D3—121no—
126C3×C8—182no—
127C3⋊C8—182no—
128C2×C32—21no—
129C2×C32—42no—
130C2×C32—41no—
131C2×C32—82no—
132C2×C32—81no—
133C2×C32C3⋊#9511yescharacteristic
134C3⋊D3—21no—
135C3⋊D3—21no—
136C3×D3—122no—
137C3×D3—122no—
138C22×C4—91no—
139M16—181no—
140M16—181no—
141M16—272no—
142M16—541no—
143C2×C8—92no—
144C2×C8—541no—
145C4⋊C4—541no—
146C22⋊C4—272no—
147C42—541no—
148C22×C3—41no—
149C22×C3—81no—
150C22×C3—91no—
151C22×C3—91no—
152C22×C3—361no—
153C22×C3#377711yescharacteristic
154C2×D3—121no—
155C2×D3—121no—
156C2×D3—124no—
157C3×C4—41no—
158C3×C4—41no—
159C3×C4—81no—
160C3×C4—81no—
161C3×C4—91no—
162C3×C4—91no—
163C3×C4#377711yescharacteristic
164C3×C4#377711yescharacteristic
165Dic3—121no—
166Dic3—121no—
167Dic3—362no—
168C32—41no—
169C32—81no—
170C32#455611yescharacteristic, complemented
171C23—91no—
172C2×C4—91no—
173C2×C4—91no—
174C2×C4—91no—
175C2×C4—91no—
176C2×C4—91no—
177C2×C4—542no—
178C2×C4C3⋊PSU(3,𝔽2)11yescharacteristic
179C8—182no—
180C8—542no—
181C2×C3—21no—
182C2×C3—42no—
183C2×C3—41no—
184C2×C3—82no—
185C2×C3—81no—
186C2×C3—181no—
187C2×C3—181no—
188C2×C3#6374711yescharacteristic
189D3—122no—
190D3—122no—
191C22—91no—
192C22—91no—
193C22—361no—
194C22#8894511yescharacteristic
195C4—91no—
196C4—91no—
197C4—1082no—
198C4#8894511yescharacteristic
199C4#8894611yescharacteristic
200C3—41no—
201C3—81no—
202C3#1059009511yescharacteristic, complemented
203C2—21no—
204C2—181no—
205C2—181no—
206C2C3⋊#6374711yescharacteristic
subgroupG/H
centerC2C3⋊#63747
commutatorC2×C3×C3⋊D3C42
Frattini, hypercenterC2×C4C3⋊PSU(3,𝔽2)
FittingC2×C33×C4Q8
socleC2×C33#95
prime radicalC22×C3×C3⋊D3C2×C4
cosocleC2×C3×C4×C3⋊D3C22
hypocenterC33#344
LCSG ▷ C2×C3×C3⋊D3 ▷ C2×C33 ▷ C33
UCSC1 ◁ C2 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
11. #88989C4semidirect
14. C3×#3787C4semidirect
39. #3787Dic3semidirect
109. C33#344semidirect
170. C32#4556semidirect
202. C3#10590095semidirect

Maximal subgroups

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

subgroupindex
#117245142
C2×#889892
C3×#642832
#105900953
#45569