Finite Group Databasebeta

← #13154765

Subgroups — #13154765

Order 1728.

404 conjugacy classes of subgroups; only the 54 normal subgroups are shown.

normal subgroups only

This group has 404 conjugacy classes of subgroups — too many to draw. Only its 54 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
1D12×C3⋊Dic3C211yesmaximal, characteristic
2C4×C3⋊#622C211yesmaximal, characteristic, complemented
3C3×C4⋊#622C211yesmaximal, characteristic, complemented
4D4⋊#622—31nomaximal
5C2×C32×D12C411yescharacteristic
6C2×D3×C3⋊Dic3—21no—
7C2×C3⋊#622—21no—
8C2×C3×#622—21no—
9C4⋊(C3⋊(C3⋊Dic3))C411yescharacteristic
10C3×C4×C3⋊Dic3C2211yescharacteristic
11(C2×D3)⋊(C3⋊Dic3)—21no—
12D4×C3⋊Dic3—31no—
13Dic3×D12—62no—
14C4⋊#622D311yescharacteristic, complemented
15C4×#622—31no—
16C22×C32×D3—21no—
17C2×C33×C4C2×C411yescharacteristic
18C2×C3⋊(C3⋊Dic3)—21no—
19C2×C3×C3⋊Dic3D411yescharacteristic
20C2×C3×C3⋊Dic3D411yescharacteristic
21C32×D12—21no—
22C32×D12C812yescomplemented
23D3×C3⋊Dic3—41no—
24D3×C3⋊Dic3—41no—
25C3⋊#622—22no—
26C3×#622—41no—
27#4483—91nomaximal
28C2×C32×D4—31no—
29C22×C3⋊Dic3—61no—
30C2×C3×D12—22no—
31C2×D3×Dic3—64no—
32C2×#622—21no—
33C2×#622—61no—
34#3757—61no—
35C4⋊(C3⋊Dic3)—31no—
36C4⋊(C3⋊Dic3)—62no—
37C4⋊(C3⋊Dic3)—122no—
38C4×C3⋊Dic3C2×D311yescharacteristic
39C3×C4×Dic3—62no—
40(C2×D3)⋊Dic3—64no—
41C22×C33C22⋊C411yescharacteristic
42C2×C32×D3—22no—
43C2×C32×D3—41no—
44C2×C32×D3—41no—
45C33×C4C2×C811yescharacteristic
46C33×C4M1611yescharacteristic
47C3⋊(C3⋊Dic3)—41no—
48C3×C3⋊Dic3—21no—
49C3×C3⋊Dic3D811yescharacteristic
50C3×C3⋊Dic3SD1611yescharacteristic
51D4×Dic3—182no—
52C4×D12—91no—
53C3×C4⋊C8—91no—
54C4×C3⋊C8—91no—
55C23×C32—61no—
56C22×C3×D3—24no—
57C32×D4—32no—
58C32×D4—61no—
59C2×C32×C4—22no—
60C2×C32×C4—42no—
61C2×C32×C4C4×D311yescharacteristic
62C2×C3⋊Dic3—61no—
63C2×C3⋊Dic3—64no—
64C2×C3⋊Dic3—124no—
65C2×C3⋊Dic3—121no—
66C2×C3⋊Dic3—121no—
67C2×C3⋊Dic3D1211yescharacteristic
68C2×C3⋊Dic3#6711yescharacteristic
69C2×C3×Dic3—62no—
70C2×C3×Dic3—62no—
71C3×D12—24no—
72C3×D12—24no—
73D3×Dic3—124no—
74D3×Dic3—124no—
75#622—41no—
76#622—62no—
77D4⋊C8—271noSylow
78C2×C33C22⋊C811yescharacteristic
79C2×C33#10211yescharacteristic
80C2×C33#10411yescharacteristic
81C32×D3—42no—
82C2×C3×D4—62no—
83C22×Dic3—184no—
84C2×D12#15711yescharacteristic
85C2×C4×D3—181no—
86C2×C3×C8—181no—
87C3×C42—91no—
88#217—184no—
89(C2×D3)⋊C4—181no—
90C4⋊Dic3—91no—
91C4⋊Dic3—182no—
92C4⋊Dic3—362no—
93C4×Dic3—62no—
94C2×C3⋊C8—181no—
95C22×C32—22no—
96C22×C32—42no—
97C22×C32—62no—
98C22×C32—121no—
99C22×C32—121no—
100C22×C32(C2×D3)⋊C411yescharacteristic
101C2×C3×D3—28no—
102C2×C3×D3—44no—
103C2×C3×D3—44no—
104C32×C4—22no—
105C32×C4—22no—
106C32×C4—42no—
107C32×C4—42no—
108C32×C4D3×C811yescharacteristic
109C32×C4ΓL(1,𝔽25)11yescharacteristic
110C3⋊Dic3—21no—
111C3⋊Dic3—121no—
112C3⋊Dic3—124no—
113C3⋊Dic3—244no—
114C3⋊Dic3#21311yescharacteristic
115C3⋊Dic3#21511yescharacteristic
116C3×Dic3—62no—
117C3×Dic3—62no—
118C3×Dic3—122no—
119C4×D4—271no—
120C4⋊C8—91no—
121C4×C8—271no—
122C33D4⋊C811yescharacteristic, complemented, Sylow
123C23×C3—64no—
124C22×D3—21no—
125C3×D4—64no—
126C3×D4—64no—
127C2×C3×C4—22no—
128C2×C3×C4—42no—
129C2×C3×C4—91no—
130C2×C3×C4—91no—
131C2×C3×C4C2×#15711yescharacteristic
132C2×Dic3—62no—
133C2×Dic3—62no—
134C2×Dic3—184no—
135C2×Dic3—181no—
136C2×Dic3—364no—
137C2×Dic3—364no—
138C2×Dic3—364no—
139D12—21no—
140D12#62212yescomplemented
141C4×D3—361no—
142C4×D3—361no—
143C3×C8—361no—
144C3⋊C8—182no—
145C2×C32—22no—
146C2×C32—22no—
147C2×C32—22no—
148C2×C32—42no—
149C2×C32—42no—
150C2×C32—42no—
151C2×C32—122no—
152C2×C32#80911yescharacteristic
153C2×C32(C2×D3)⋊C811yescharacteristic
154C2×C32#82511yescharacteristic
155C3×D3—48no—
156C2×D4—31no—
157C22×C4—541no—
158C2×C8—181no—
159C2×C8—541no—
160C4⋊C4—271no—
161C22⋊C4—541no—
162C42—91no—
163C22×C3—22no—
164C22×C3—42no—
165C22×C3—68no—
166C22×C3—124no—
167C22×C3—124no—
168C22×C3C22⋊#15711yescharacteristic
169C2×D3—22no—
170C2×D3—41no—
171C2×D3—41no—
172C3×C4—22no—
173C3×C4—22no—
174C3×C4—42no—
175C3×C4—42no—
176C3×C4—91no—
177C3×C4—91no—
178C3×C4—181no—
179C3×C4C2×#62211yescharacteristic
180C3×C4#379211yescharacteristic
181Dic3—62no—
182Dic3—62no—
183Dic3—122no—
184Dic3—364no—
185Dic3—361no—
186Dic3—724no—
187C32—22no—
188C32—42no—
189C32#448311yescharacteristic, complemented
190C23—61no—
191D4—32no—
192D4—61no—
193C2×C4—91no—
194C2×C4—91no—
195C2×C4—541no—
196C2×C4—1081no—
197C2×C4—1081no—
198C2×C4D3×#15711yescharacteristic
199C8—361no—
200C8—542no—
201C2×C3—22no—
202C2×C3—22no—
203C2×C3—22no—
204C2×C3—42no—
205C2×C3—42no—
206C2×C3—42no—
207C2×C3—128no—
208C2×C3#6378211yescharacteristic
209C2×C3#6378311yescharacteristic
210C2×C3C22⋊#62211yescharacteristic
211D3—42no—
212C22—62no—
213C22—121no—
214C22—121no—
215C22(C2×D3)⋊#15711yescharacteristic
216C4—91no—
217C4—91no—
218C4—181no—
219C4—1081no—
220C4D3×#62211yescharacteristic
221C4#8892511yescharacteristic
222C3—22no—
223C3—42no—
224C3D4⋊#62211yescharacteristic, complemented
225C2—122no—
226C2#1172270611yescharacteristic
227C2#1172271011yescharacteristic
228C2(C2×D3)⋊#62211yescharacteristic
subgroupG/H
centerC22(C2×D3)⋊#157
commutatorC33×C4C2×C8
Frattini, hypercenterC2×C4D3×#157
FittingC2×C33×C4C2×C4
socleC22×C33C22⋊C4
prime radicalC2×C32×D12C4
cosocleC3×C4×C3⋊Dic3C22
hypocenterC33D4⋊C8
LCSG ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
2. C4×C3⋊#622C2semidirect
3. C3×C4⋊#622C2semidirect
14. C4⋊#622D3semidirect
22. C32×D12C8semidirect
122. C33D4⋊C8semidirect
140. D12#622semidirect
189. C32#4483semidirect
224. C3D4⋊#622semidirect

Maximal subgroups

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

subgroupindex
D12×C3⋊Dic32
C4×C3⋊#6222
C3×C4⋊#6222
D4⋊#6223
#44839