Finite Group Databasebeta

← #13166576

Subgroups — #13166576

Order 1728.

480 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 480 conjugacy classes of subgroups — too many to draw. Only its 106 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
1C22×C3⋊#622C211yesmaximal, characteristic
2C2×C3×#3792C211yesmaximal, characteristic
3#11723595C211yesmaximal, characteristic
4#11722724C212yesmaximal, complemented
5Dic3×#622C212yesmaximal, complemented
6#10593738—31nomaximal
7C22×C3×C3⋊Dic3C2211yescharacteristic
8C2×C3⋊#622C414yescomplemented
9C2×C3⋊#622C2211yescharacteristic
10C2×C3⋊#622C2211yescharacteristic
11C3×#3792C414yescomplemented
12C2×C3×#622C2212yes—
13#88864C411yescharacteristic
14C32×#217C411yescharacteristic
15#88807C412yes—
16Dic3×C3⋊Dic3C2212yes—
17C2×#3792D311yescharacteristic
18C22×#622—31no—
19#64086—31no—
20#63955—62no—
21#63777—32no—
22C4×#622—32no—
23C23×C33C2×C411yescharacteristic
24C2×C3⋊(C3⋊Dic3)C2×C412yes—
25C2×C3×C3⋊Dic3C2×C411yescharacteristic
26C2×C3×C3⋊Dic3C2×C412yes—
27C2×C3×C3⋊Dic3C2×C412yes—
28C2×C3×C3⋊Dic3C2311yescharacteristic
29C2×C32×Dic3C2×C412yes—
30C3⋊#622C2×C412yes—
31C3⋊#622C2×C412yes—
32C3×#622C2×C414yes—
33#5122—91nomaximal
34C22×C3⋊Dic3C2×D311yescharacteristic
35C22×C3×Dic3—62no—
36#3792Dic314yescomplemented
37C2×#622—34no—
38C2×#622—31no—
39C2×#622—31no—
40C2×#622C2×D312yes—
41C32×C22⋊C4—31no—
42#3757—31no—
43#3757—62no—
44#3757—122no—
45#3750—32no—
46C4×C3⋊Dic3—32no—
47C3×#217—22no—
48Dic3⋊Dic3—64no—
49Dic32—64no—
50C22×C33—22no—
51C22×C33C4212yes—
52C22×C33C22×C411yescharacteristic
53C3⋊(C3⋊Dic3)—22no—
54C3×C3⋊Dic3C4212yes—
55C3×C3⋊Dic3C22×C411yescharacteristic
56C3×C3⋊Dic3C22×C411yescharacteristic
57C32×Dic3—22no—
58C2×C3×M16—91no—
59#926—91no—
60C22×C3⋊C8—91no—
61#877—182no—
62#815—92no—
63C8×Dic3—92no—
64C23×C32—22no—
65C23×C32—42no—
66C23×C32C4×D311yescharacteristic
67C2×C32×C4—32no—
68C2×C3⋊Dic3—32no—
69C2×C3⋊Dic3—64no—
70C2×C3⋊Dic3—124no—
71C2×C3⋊Dic3C4×D311yescharacteristic
72C2×C3⋊Dic3C4×D312yes—
73C2×C3⋊Dic3C2×Dic312yes—
74C2×C3⋊Dic3C22×D311yescharacteristic
75C2×C3×Dic3—24no—
76C2×C3×Dic3—62no—
77C2×C3×Dic3—62no—
78C2×C3×Dic3—64no—
79C2×C3×Dic3—64no—
80#622—32no—
81#622—32no—
82#622C2×Dic314yes—
83#405—271noSylow
84C2×C33—22no—
85C2×C33C2×C4211yescharacteristic
86C2×C33#13112yes—
87C22×C3×C4—91no—
88C22×Dic3—62no—
89C3×M16—94no—
90C2×C3×C8—92no—
91C3×C22⋊C4—62no—
92#217—91no—
93#217—182no—
94#217—362no—
95#217#15711yescharacteristic
96C4⋊Dic3—92no—
97Dic3⋊C4—184no—
98C4×Dic3—92no—
99C4×Dic3—184no—
100C2×C3⋊C8—91no—
101C2×C3⋊C8—91no—
102C2×C3⋊C8—94no—
103C22×C32—22no—
104C22×C32—24no—
105C22×C32—22no—
106C22×C32—42no—
107C22×C32—44no—
108C22×C32—44no—
109C22×C32—84no—
110C22×C32C4×Dic312yes—
111C22×C32C2×C4×D311yescharacteristic
112C32×C4—62no—
113C3⋊Dic3—62no—
114C3⋊Dic3—124no—
115C3⋊Dic3—244no—
116C3⋊Dic3C4×Dic312yes—
117C3⋊Dic3C2×C4×D311yescharacteristic
118C3⋊Dic3C22×Dic311yescharacteristic
119C3×Dic3—44no—
120C3×Dic3—64no—
121C3×Dic3—62no—
122C3×Dic3—62no—
123C2×M16—91no—
124C22×C8—271no—
125#117—271no—
126#97—272no—
127C4×C8—272no—
128C33#40511yescharacteristic, complemented, Sylow
129C23×C3—22no—
130C23×C3—42no—
131C23×C3C2×#15711yescharacteristic
132C2×C3×C4—64no—
133C2×C3×C4—91no—
134C2×C3×C4—91no—
135C2×C3×C4—92no—
136C2×C3×C4—92no—
137C2×Dic3—62no—
138C2×Dic3—64no—
139C2×Dic3—64no—
140C2×Dic3—62no—
141C2×Dic3—92no—
142C2×Dic3—184no—
143C2×Dic3—364no—
144C2×Dic3C2×#15712yes—
145C3×C8—94no—
146C3⋊C8—92no—
147C3⋊C8—92no—
148C2×C32—22no—
149C2×C32—24no—
150C2×C32—22no—
151C2×C32—42no—
152C2×C32—44no—
153C2×C32—44no—
154C2×C32—84no—
155C2×C32#90712yes—
156C2×C32C2×C4×Dic311yescharacteristic
157C22×C4—91no—
158M16—94no—
159C2×C8—92no—
160C2×C8—271no—
161C2×C8—271no—
162C2×C8—274no—
163C4⋊C4—272no—
164C22⋊C4—31no—
165C22⋊C4—271no—
166C42—272no—
167C22×C3—22no—
168C22×C3—22no—
169C22×C3—24no—
170C22×C3—44no—
171C22×C3—42no—
172C22×C3—44no—
173C22×C3—84no—
174C22×C3C4×#15712yes—
175C22×C3C22×#15711yescharacteristic
176C3×C4—91no—
177C3×C4—91no—
178C3×C4—92no—
179C3×C4—124no—
180Dic3—22no—
181Dic3—64no—
182Dic3—62no—
183Dic3—62no—
184Dic3—182no—
185Dic3—364no—
186Dic3—724no—
187C32—22no—
188C32—42no—
189C32#512211yescharacteristic, complemented
190C23D3×#15711yescharacteristic
191C2×C4—32no—
192C2×C4—91no—
193C2×C4—91no—
194C2×C4—92no—
195C2×C4—92no—
196C2×C4—272no—
197C8—94no—
198C8—272no—
199C8—272no—
200C2×C3—22no—
201C2×C3—22no—
202C2×C3—24no—
203C2×C3—44no—
204C2×C3—42no—
205C2×C3—44no—
206C2×C3—84no—
207C2×C3C2×C4×#15711yescharacteristic
208C2×C3#6428712yes—
209C22—22no—
210C22Dic3×#15712yes—
211C22C2×D3×#15711yescharacteristic
212C4—62no—
213C4—91no—
214C4—91no—
215C4—92no—
216C4—542no—
217C3—22no—
218C3—42no—
219C3#1059373811yescharacteristic, complemented
220C2—22no—
221C2C2×Dic3×#15711yescharacteristic
222C2#1172426712yes—
subgroupG/H
center, FrattiniC22C2×D3×#157
commutatorC2×C33C2×C42
Fitting, prime radicalC23×C33C2×C4
socleC22×C33C22×C4
cosocleC2×C3×C3⋊Dic3C23
hypercenterC23D3×#157
hypocenterC33#405
LCSG ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C23

Complements

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

NH ≅ G/Ntype
4. #11722724C2semidirect
5. Dic3×#622C2semidirect
8. C2×C3⋊#622C4semidirect
11. C3×#3792C4semidirect
36. #3792Dic3semidirect
128. C33#405semidirect
189. C32#5122semidirect
219. C3#10593738semidirect

Maximal subgroups

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

subgroupindex
C22×C3⋊#6222
C2×C3×#37922
#117235952
#117227242
Dic3×#6222
#105937383
#51229