Finite Group Databasebeta

← #13157220

Subgroups — #13157220

Order 1728.

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

normal subgroups only

This group has 400 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
1C3×#64138C211yesmaximal, characteristic
2#11723113C211yesmaximal, characteristic
3#11723111C211yesmaximal, characteristic
4#10590205—31nomaximal
5C22×C33×C4C411yescharacteristic, complemented
6C2×C3×#638C2211yescharacteristic
7#88994—21no—
8#88993—21no—
9C3×#3757C411yescharacteristic, complemented
10C3×#3752—21no—
11C3×#3750—21no—
12#64138D311yescharacteristic
13C3×#929—62no—
14#63788—31no—
15#63786—31no—
16C23×C33C2×C411yescharacteristic
17C22×C3×C3⋊D3D411yescharacteristic
18C2×C3⋊#157—41no—
19C2×C33×C4—21no—
20C2×C33×C4—41no—
21C3×#638—41no—
22C2×C3×C3⋊Dic3—21no—
23C2×C3×C3⋊Dic3D411yescharacteristic
24C3⋊#622—41no—
25#4539—91nomaximal
26C22×C32×C4—22no—
27C22×C32×C4—42no—
28C22×C32×C4Dic311yescharacteristic, complemented
29C2×#638C2×D311yescharacteristic
30C2×C3×#67—62no—
31C22⋊#157—61no—
32#3792—61no—
33#3757Dic311yescharacteristic, complemented
34#3752—21no—
35#3750—21no—
36C3×#217—62no—
37C3×(C2×D3)⋊C4—64no—
38C3×Dic3⋊C4—64no—
39C22×C33—21no—
40C22×C33—41no—
41C22×C33C22⋊C411yescharacteristic
42C2×C3×C3⋊D3—21no—
43C2×C3×C3⋊D3—41no—
44C3⋊#157—81no—
45C33×C4—41no—
46C3×C3⋊Dic3—21no—
47C3×C3⋊Dic3—41no—
48C3×#123—91no—
49#929—62no—
50#834—91no—
51#833—91no—
52C23×C32—22no—
53C23×C32—42no—
54C23×C32C2×Dic311yescharacteristic
55C22×C3⋊D3#6711yescharacteristic
56C22×C3×D3—62no—
57C2×#157—121no—
58C2×C32×C4—21no—
59C2×C32×C4—24no—
60C2×C32×C4—44no—
61C2×C32×C4—44no—
62C2×C32×C4—48no—
63C2×C32×C4—41no—
64#638—41no—
65C2×C3⋊Dic3—21no—
66C2×C3⋊Dic3#6711yescharacteristic
67C3×#67—124no—
68C2×C3×Dic3—62no—
69C2×C3×Dic3—64no—
70#622—121no—
71#352—271noSylow
72C2×C33—21no—
73C2×C33—41no—
74C2×C33#9911yescharacteristic
75C3×C3⋊D3—41no—
76C2×C3×D4—91no—
77C22×C3×C4—22no—
78C22×C3×C4—42no—
79C22×C3×C4#15711yescharacteristic, complemented
80C2×#67—62no—
81C3×C4⋊C4—181no—
82C3×C22⋊C4—91no—
83C3×C22⋊C4—181no—
84#217—62no—
85#217—181no—
86(C2×D3)⋊C4—64no—
87Dic3⋊C4—64no—
88#208—181no—
89C22×C32—21no—
90C22×C32—24no—
91C22×C32—22no—
92C22×C32—44no—
93C22×C32—42no—
94C22×C32—41no—
95C22×C32—44no—
96C22×C32—48no—
97C22×C32#21711yescharacteristic
98C2×C3⋊D3—21no—
99C2×C3⋊D3—41no—
100C2×C3×D3—122no—
101C2×C3×D3—124no—
102#157—241no—
103C32×C4—44no—
104C32×C4—48no—
105C32×C4—41no—
106C3⋊Dic3—21no—
107C3⋊Dic3—41no—
108C3×Dic3—122no—
109C3×Dic3—124no—
110#123—91no—
111#100—271no—
112#99—271no—
113C33#35211yescharacteristic, complemented, Sylow
114C23×C3—22no—
115C23×C3—42no—
116C23×C3—91no—
117C23×C3C2×#15711yescharacteristic
118C22×D3—62no—
119C3×D4—361no—
120C2×C3×C4—24no—
121C2×C3×C4—21no—
122C2×C3×C4—44no—
123C2×C3×C4—44no—
124C2×C3×C4—48no—
125C2×C3×C4—41no—
126C2×C3×C4—91no—
127C2×C3×C4—181no—
128#67—124no—
129C2×Dic3—62no—
130C2×Dic3—64no—
131C2×Dic3—361no—
132C3⋊C8—361no—
133C2×C32—21no—
134C2×C32—24no—
135C2×C32—22no—
136C2×C32—44no—
137C2×C32—42no—
138C2×C32—44no—
139C2×C32—48no—
140C2×C32—41no—
141C2×C32#83411yescharacteristic
142C3⋊D3—41no—
143C3×D3—242no—
144C2×D4—91no—
145C22×C4C3⋊#15711yescharacteristic, complemented
146M16—541no—
147C4⋊C4—181no—
148C22⋊C4—91no—
149C22⋊C4—181no—
150C22⋊C4—541no—
151C22×C3—24no—
152C22×C3—21no—
153C22×C3—22no—
154C22×C3—44no—
155C22×C3—41no—
156C22×C3—44no—
157C22×C3—48no—
158C22×C3—42no—
159C22×C3—181no—
160C22×C3—361no—
161C22×C3C22⋊#15711yescharacteristic
162C2×D3—122no—
163C2×D3—124no—
164C3×C4—44no—
165C3×C4—48no—
166C3×C4—41no—
167C3×C4—181no—
168C3×C4—361no—
169Dic3—122no—
170Dic3—124no—
171Dic3—721no—
172C32—22no—
173C32—42no—
174C32#453911yescharacteristic, complemented
175C23—91no—
176C23C2×C3⋊#15711yescharacteristic
177D4—361no—
178C2×C4—21no—
179C2×C4—41no—
180C2×C4—91no—
181C2×C4—181no—
182C2×C4—1081no—
183C8—1081no—
184C2×C3—24no—
185C2×C3—21no—
186C2×C3—22no—
187C2×C3—44no—
188C2×C3—44no—
189C2×C3—48no—
190C2×C3—41no—
191C2×C3—42no—
192C2×C3—361no—
193C2×C3#6378611yescharacteristic
194D3—242no—
195C22—21no—
196C22—41no—
197C22—181no—
198C22—361no—
199C22#8899411yescharacteristic
200C4—41no—
201C4—181no—
202C4—361no—
203C4—2161no—
204C3—22no—
205C3—42no—
206C3#1059020511yescharacteristic, complemented
207C2—21no—
208C2—41no—
209C2—361no—
210C2#1172311111yescharacteristic
subgroupG/H
centerC2#11723111
commutatorC23×C33C2×C4
FrattiniC23C2×C3⋊#157
FittingC22×C33×C4C4
socleC2×C33#99
prime radical, cosocleC2×C3×#638C22
hypercenterC22×C4C3⋊#157
hypocenterC33#352
LCSG ▷ C23×C33 ▷ C22×C33 ▷ C2×C33 ▷ C33
UCSC1 ◁ C2 ◁ C22 ◁ C23 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
5. C22×C33×C4C4semidirect
9. C3×#3757C4semidirect
28. C22×C32×C4Dic3semidirect
33. #3757Dic3semidirect
79. C22×C3×C4#157semidirect
113. C33#352semidirect
145. C22×C4C3⋊#157semidirect
174. C32#4539semidirect
206. C3#10590205semidirect

Maximal subgroups

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

subgroupindex
C3×#641382
#117231132
#117231112
#105902053
#45399