Finite Group Databasebeta

← #13155170

Subgroups — #13155170

Order 1728.

266 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 266 conjugacy classes of subgroups — too many to draw. Only its 63 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×C4⋊(C3⋊#157)C211yesmaximal, characteristic
2#11724511C211yesmaximal, characteristic
3C3×#64283C211yesmaximal, characteristic
4#10590094—31nomaximal
5C22×C3⋊#157C412yes—
6C2×C3×C4×C3⋊D3C2211yescharacteristic
7#88993—21no—
8C4⋊(C3⋊#157)—21no—
9C4⋊(C3⋊#157)—21no—
10#88989C412yescomplemented
11#88988—21no—
12C3×#3792—21no—
13C3×#3788C412yescomplemented
14C3×#3787—21no—
15C2×C4⋊#157—31no—
16#64283—31no—
17#64283D311yescharacteristic
18C22×C3×C3⋊D3C2×C411yescharacteristic
19C2×C3⋊#157—22no—
20C2×C3⋊#157—42no—
21C2×C33×C4Q811yescharacteristic
22C2×C3×C3⋊Dic3D411yescharacteristic
23C3×C4×C3⋊D3C2×C411yescharacteristic
24C3×C4×C3⋊D3C2×C411yescharacteristic
25C3×C4×C3⋊D3D411yescharacteristic
26C3×C4×C3⋊D3D411yescharacteristic
27C3⋊#622—22no—
28C3×#622—22no—
29#4554—91nomaximal
30C22×#157—32no—
31C2×C4×C3⋊D3C2×D311yescharacteristic
32C2×C3×C4×D3—121no—
33#3792—21no—
34#3792—61no—
35C4⋊#157—61no—
36C4⋊#157—61no—
37#3788—32no—
38#3788Dic312yescomplemented
39#3787—21no—
40#3787—61no—
41C22×C33C4⋊C411yescharacteristic
42C2×C3×C3⋊D3—21no—
43C2×C3×C3⋊D3—21no—
44C2×C3×C3⋊D3C4211yescharacteristic
45C2×C3×C3⋊D3C22⋊C411yescharacteristic
46C3⋊#157—42no—
47C33×C4C4⋊C411yescharacteristic
48C33×C4C4⋊C411yescharacteristic
49C3×C3⋊Dic3C22⋊C411yescharacteristic
50C3×C3⋊Dic3C22⋊C411yescharacteristic
51C2×C3×M16—91no—
52C2×C4⋊Dic3—91no—
53C2×#208—91no—
54C22×C3⋊D3C4×D311yescharacteristic
55C22×C3×D3—121no—
56C2×#157—62no—
57C2×#157—122no—
58C2×C32×C4—41no—
59C2×C32×C4—81no—
60C2×C32×C4Dic611yescharacteristic
61C2×C3⋊Dic3#6711yescharacteristic
62C4×C3⋊D3C4×D311yescharacteristic
63C4×C3⋊D3D1211yescharacteristic
64C4×C3⋊D3C2×Dic311yescharacteristic
65C4×C3⋊D3#6711yescharacteristic
66C2×C3×Dic3—121no—
67C3×C4×D3—121no—
68C3×C4×D3—121no—
69C3×C4×D3—121no—
70C3×C4×D3—121no—
71#622—22no—
72#622—62no—
73#343—271noSylow
74C2×C33—21no—
75C2×C33#9511yescharacteristic
76C3×C3⋊D3—21no—
77C3×C3⋊D3#10011yescharacteristic
78C3×C3⋊D3#10111yescharacteristic
79C22×C3×C4—91no—
80C22×Dic3—92no—
81C2×C4×D3—121no—
82C3×M16—92no—
83C3×M16—181no—
84C2×C3×C8—181no—
85C4⋊Dic3—181no—
86C4⋊Dic3—181no—
87#208—92no—
88#208—181no—
89C2×C3⋊C8—181no—
90C22×C32—41no—
91C22×C32—81no—
92C22×C32Dic3⋊C411yescharacteristic
93C2×C3⋊D3—21no—
94C2×C3⋊D3—21no—
95C2×C3⋊D3C4×Dic311yescharacteristic
96C2×C3⋊D3(C2×D3)⋊C411yescharacteristic
97C2×C3×D3—121no—
98C2×C3×D3—121no—
99C2×C3×D3—122no—
100C2×C3×D3—122no—
101#157—122no—
102C32×C4—41no—
103C32×C4—41no—
104C32×C4—81no—
105C32×C4—81no—
106C32×C4Dic3⋊C411yescharacteristic
107C32×C4C4⋊Dic311yescharacteristic
108C3⋊Dic3(C2×D3)⋊C411yescharacteristic
109C3⋊Dic3#21711yescharacteristic
110C3×Dic3—121no—
111C3×Dic3—121no—
112C2×M16—91no—
113C2×M16—271no—
114C2×C4⋊C4—271no—
115C33#34311yescharacteristic, complemented, Sylow
116C23×C3—91no—
117C22×D3—121no—
118C2×C3×C4—41no—
119C2×C3×C4—81no—
120C2×C3×C4—91no—
121C2×C3×C4—91no—
122C2×C3×C4—91no—
123C2×C3×C4—91no—
124C2×C3×C4—91no—
125C2×C3×C4PSU(3,𝔽2)11yescharacteristic
126C2×Dic3—121no—
127C2×Dic3—182no—
128C2×Dic3—362no—
129C4×D3—121no—
130C4×D3—121no—
131C4×D3—121no—
132C4×D3—121no—
133C3×C8—182no—
134C3⋊C8—182no—
135C2×C32—21no—
136C2×C32—42no—
137C2×C32—41no—
138C2×C32—82no—
139C2×C32—81no—
140C2×C32C3⋊#9511yescharacteristic
141C3⋊D3—21no—
142C3⋊D3#82311yescharacteristic
143C3⋊D3#82411yescharacteristic
144C3×D3—121no—
145C3×D3—121no—
146C3×D3—122no—
147C22×C4—91no—
148C22×C4—272no—
149M16—92no—
150M16—181no—
151M16—272no—
152M16—541no—
153C2×C8—181no—
154C2×C8—541no—
155C4⋊C4—541no—
156C4⋊C4—541no—
157C22×C3—41no—
158C22×C3—81no—
159C22×C3—91no—
160C22×C3—91no—
161C22×C3—181no—
162C22×C3—181no—
163C22×C3#377711yescharacteristic
164C2×D3—121no—
165C2×D3—121no—
166C2×D3—122no—
167C2×D3—122no—
168C3×C4—41no—
169C3×C4—41no—
170C3×C4—81no—
171C3×C4—81no—
172C3×C4—91no—
173C3×C4—91no—
174C3×C4#377711yescharacteristic
175C3×C4#377711yescharacteristic
176Dic3—121no—
177Dic3—121no—
178Dic3—362no—
179C32—41no—
180C32—81no—
181C32#455411yescharacteristic, complemented
182C23—91no—
183C2×C4—91no—
184C2×C4—91no—
185C2×C4—91no—
186C2×C4—91no—
187C2×C4—91no—
188C2×C4—542no—
189C2×C4—1082no—
190C2×C4C3⋊PSU(3,𝔽2)11yescharacteristic
191C8—182no—
192C8—542no—
193C2×C3—21no—
194C2×C3—42no—
195C2×C3—41no—
196C2×C3—82no—
197C2×C3—81no—
198C2×C3—91no—
199C2×C3—91no—
200C2×C3—181no—
201C2×C3#6374711yescharacteristic
202D3—121no—
203D3—121no—
204D3—122no—
205C22—91no—
206C22—91no—
207C22—181no—
208C22—181no—
209C22#8894511yescharacteristic
210C4—91no—
211C4—91no—
212C4—1082no—
213C4#8894511yescharacteristic
214C4#8894611yescharacteristic
215C3—41no—
216C3—81no—
217C3#1059009411yescharacteristic, complemented
218C2—21no—
219C2—91no—
220C2—91no—
221C2—181no—
222C2C3⋊#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#343
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
10. #88989C4semidirect
13. C3×#3788C4semidirect
38. #3788Dic3semidirect
115. C33#343semidirect
181. C32#4554semidirect
217. C3#10590094semidirect

Maximal subgroups

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

subgroupindex
C2×C4⋊(C3⋊#157)2
#117245112
C3×#642832
#105900943
#45549