Finite Group Databasebeta

← #13157177

Subgroups — #13157177

Order 1728.

404 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 404 conjugacy classes of subgroups — too many to draw. Only its 57 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×#64102C211yesmaximal, characteristic
2C4⋊(C3⋊#622)C211yesmaximal, characteristic, complemented
3C4×C3⋊#622C211yesmaximal, characteristic, complemented
4#10590160—31nomaximal
5C2×C3×C4⋊(C3⋊D3)C411yescharacteristic
6C2×C3×C4×C3⋊D3—21no—
7C2×C3⋊#622—21no—
8C2×C3⋊#622—21no—
9C33×C4⋊C4C411yescharacteristic
10C3×#3752—21no—
11C3×C4×C3⋊Dic3C2211yescharacteristic
12#64102D311yescharacteristic
13C3×#893—62no—
14C4⋊#622—31no—
15C4×#622—31no—
16C22×C3×C3⋊D3—21no—
17C2×C33×C4—21no—
18C2×C33×C4C2×C411yescharacteristic
19C2×C3×C3⋊Dic3D411yescharacteristic
20C2×C3×C3⋊Dic3D411yescharacteristic
21C3×C4⋊(C3⋊D3)—21no—
22C3×C4⋊(C3⋊D3)C812yescomplemented
23C3×C4×C3⋊D3—41no—
24C3×C4×C3⋊D3—41no—
25C3⋊#622—22no—
26C3⋊#622—41no—
27D4⋊(C3⋊C8)—91nomaximal
28C2×C4⋊(C3⋊D3)Dic311yescharacteristic
29C2×C4×C3⋊D3—21no—
30C2×C3×D12—62no—
31C2×C3×C4×D3—64no—
32C2×#622—61no—
33C2×#622—61no—
34C32×C4⋊C4—22no—
35C32×C4⋊C4—42no—
36C32×C4⋊C4Dic311yescharacteristic
37#3752—21no—
38C4×C3⋊Dic3C2×D311yescharacteristic
39C3×(C2×D3)⋊C4—64no—
40C3×C4×Dic3—62no—
41C22×C33C22⋊C411yescharacteristic
42C2×C3×C3⋊D3—22no—
43C2×C3×C3⋊D3—41no—
44C2×C3×C3⋊D3—41no—
45C33×C4—41no—
46C33×C4C2×C811yescharacteristic
47C33×C4M1611yescharacteristic
48C3×C3⋊Dic3—21no—
49C3×C3⋊Dic3D811yescharacteristic
50C3×C3⋊Dic3SD1611yescharacteristic
51C3×C4×D4—91no—
52#893—62no—
53C4⋊(C3⋊C8)—91no—
54C4×C3⋊C8—91no—
55C22×C3⋊D3—21no—
56C22×C3×D3—64no—
57C2×C32×C4—21no—
58C2×C32×C4—24no—
59C2×C32×C4—22no—
60C2×C32×C4—44no—
61C2×C32×C4—42no—
62C2×C32×C4C2×Dic311yescharacteristic
63C2×C3⋊Dic3#6711yescharacteristic
64C2×C3⋊Dic3#6711yescharacteristic
65C4⋊(C3⋊D3)—21no—
66C4⋊(C3⋊D3)C3⋊C812yescomplemented
67C4×C3⋊D3—41no—
68C4×C3⋊D3—41no—
69C2×C3×Dic3—62no—
70C2×C3×Dic3—62no—
71C3×D12—64no—
72C3×D12—64no—
73C3×C4×D3—124no—
74C3×C4×D3—124no—
75#622—62no—
76#622—121no—
77D4⋊C8—271noSylow
78C2×C33C22⋊C811yescharacteristic
79C2×C33#10211yescharacteristic
80C2×C33#10411yescharacteristic
81C3×C3⋊D3—42no—
82C2×C3×D4—91no—
83C22×C3×C4—181no—
84C2×D12—62no—
85C2×C4×D3—64no—
86C3×C4⋊C4—22no—
87C3×C4⋊C4—42no—
88C3×C4⋊C4#15711yescharacteristic
89C3×C22⋊C4—181no—
90C3×C42—91no—
91(C2×D3)⋊C4—64no—
92C4×Dic3—62no—
93C2×C3⋊C8—181no—
94C2×C3⋊C8—181no—
95C22×C32—22no—
96C22×C32—42no—
97C22×C32#21711yescharacteristic
98C2×C3⋊D3—22no—
99C2×C3⋊D3—41no—
100C2×C3⋊D3—41no—
101C2×C3×D3—68no—
102C2×C3×D3—124no—
103C2×C3×D3—124no—
104C32×C4—22no—
105C32×C4—22no—
106C32×C4—42no—
107C32×C4—42no—
108C32×C4—41no—
109C32×C4—44no—
110C32×C4—84no—
111C32×C4C2×C3⋊C811yescharacteristic
112C32×C4#20811yescharacteristic
113C3⋊Dic3—21no—
114C3⋊Dic3#21311yescharacteristic
115C3⋊Dic3#21411yescharacteristic
116C3×Dic3—62no—
117C3×Dic3—62no—
118C3×Dic3—122no—
119C4×D4—91no—
120C4⋊C8—271no—
121C4×C8—271no—
122C33D4⋊C811yescharacteristic, complemented, Sylow
123C23×C3—181no—
124C22×D3—64no—
125C3×D4—92no—
126C3×D4—181no—
127C2×C3×C4—24no—
128C2×C3×C4—21no—
129C2×C3×C4—22no—
130C2×C3×C4—44no—
131C2×C3×C4—42no—
132C2×C3×C4—91no—
133C2×C3×C4—91no—
134C2×C3×C4—361no—
135C2×C3×C4—361no—
136C2×C3×C4C2×#15711yescharacteristic
137C2×Dic3—62no—
138C2×Dic3—62no—
139D12—64no—
140D12—64no—
141C4×D3—124no—
142C4×D3—124no—
143C3⋊C8—182no—
144C3⋊C8—361no—
145C2×C32—22no—
146C2×C32—22no—
147C2×C32—22no—
148C2×C32—42no—
149C2×C32—42no—
150C2×C32—42no—
151C2×C32#83011yescharacteristic
152C2×C32#83211yescharacteristic
153C2×C32#83711yescharacteristic
154C3⋊D3—42no—
155C3×D3—128no—
156C2×D4—91no—
157C22×C4—181no—
158C2×C8—541no—
159C2×C8—541no—
160C4⋊C4C3⋊#15711yescharacteristic
161C22⋊C4—181no—
162C42—91no—
163C22×C3—22no—
164C22×C3—42no—
165C22×C3—182no—
166C22×C3—361no—
167C22×C3—361no—
168C22×C3C22⋊#15711yescharacteristic
169C2×D3—68no—
170C2×D3—124no—
171C2×D3—124no—
172C3×C4—22no—
173C3×C4—22no—
174C3×C4—44no—
175C3×C4—41no—
176C3×C4—42no—
177C3×C4—42no—
178C3×C4—84no—
179C3×C4—91no—
180C3×C4—91no—
181C3×C4—181no—
182C3×C4#378711yescharacteristic
183C3×C4#378811yescharacteristic
184Dic3—62no—
185Dic3—62no—
186Dic3—122no—
187C32—22no—
188C32—42no—
189C32D4⋊(C3⋊C8)11yescharacteristic, complemented
190C23—181no—
191D4—92no—
192D4—181no—
193C2×C4—21no—
194C2×C4—91no—
195C2×C4—91no—
196C2×C4—361no—
197C2×C4—361no—
198C2×C4C2×C3⋊#15711yescharacteristic
199C8—542no—
200C8—1081no—
201C2×C3—22no—
202C2×C3—22no—
203C2×C3—22no—
204C2×C3—42no—
205C2×C3—42no—
206C2×C3—42no—
207C2×C3—362no—
208C2×C3#6377811yescharacteristic
209C2×C3#6378211yescharacteristic
210C2×C3#6378511yescharacteristic
211D3—128no—
212C22—182no—
213C22—361no—
214C22—361no—
215C22#8899411yescharacteristic
216C4—41no—
217C4—91no—
218C4—91no—
219C4—181no—
220C4#8898811yescharacteristic
221C4#8898911yescharacteristic
222C3—22no—
223C3—42no—
224C3#1059016011yescharacteristic, complemented
225C2—362no—
226C2#1172310311yescharacteristic
227C2#1172310711yescharacteristic
228C2#1172311011yescharacteristic
subgroupG/H
centerC22#88994
commutatorC33×C4C2×C8
FrattiniC2×C4C2×C3⋊#157
FittingC33×C4⋊C4C4
socleC22×C33C22⋊C4
prime radicalC2×C3×C4⋊(C3⋊D3)C4
cosocleC3×C4×C3⋊Dic3C22
hypercenterC4⋊C4C3⋊#157
hypocenterC33D4⋊C8
LCSG ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
2. C4⋊(C3⋊#622)C2semidirect
3. C4×C3⋊#622C2semidirect
22. C3×C4⋊(C3⋊D3)C8semidirect
66. C4⋊(C3⋊D3)C3⋊C8semidirect
122. C33D4⋊C8semidirect
189. C32D4⋊(C3⋊C8)semidirect
224. C3#10590160semidirect

Maximal subgroups

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

subgroupindex
C3×#641022
C4⋊(C3⋊#622)2
C4×C3⋊#6222
#105901603
D4⋊(C3⋊C8)9