Finite Group Databasebeta

← #13154766

Subgroups — #13154766

Order 1728.

460 conjugacy classes of subgroups; only the 52 normal subgroups are shown.

normal subgroups only

This group has 460 conjugacy classes of subgroups — too many to draw. Only its 52 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
1#11723494C211yesmaximal, characteristic
2C4×C3⋊#622C211yesmaximal, characteristic, complemented
3C3×C4⋊#622C211yesmaximal, characteristic, complemented
4#10590160—31nomaximal
5C2×C4⋊(C3⋊(C3⋊D3))C411yescharacteristic
6C2×#6289—21no—
7C2×C3⋊#622—21no—
8C2×C3×#622—21no—
9C3×C4×C3⋊Dic3C2211yescharacteristic
10C32×C4⋊Dic3C411yescharacteristic
11#88804—21no—
12#64102—31no—
13#63894—62no—
14C4⋊#622D311yescharacteristic, complemented
15C4×#622—31no—
16C22×C3⋊(C3⋊D3)—21no—
17C2×C33×C4C2×C411yescharacteristic
18C4⋊(C3⋊(C3⋊D3))—21no—
19C4⋊(C3⋊(C3⋊D3))C812yescomplemented
20C2×C3×C3⋊Dic3D411yescharacteristic
21C2×C3×C3⋊Dic3D411yescharacteristic
22C2×C32×Dic3—21no—
23#6289—41no—
24#6289—41no—
25C3⋊#622—22no—
26C3×#622—41no—
27#4483—91nomaximal
28C2×C4⋊(C3⋊D3)—31no—
29C2×C4⋊(C3⋊D3)—62no—
30C2×C4⋊(C3⋊D3)—122no—
31C2×C4×C3⋊D3—61no—
32C2×#624—64no—
33C2×#622—21no—
34C2×#622—61no—
35C32×C4⋊C4—31no—
36#3752—61no—
37C4×C3⋊Dic3C2×D311yescharacteristic
38C3×C4⋊Dic3—22no—
39C3×C4×Dic3—62no—
40#3722—64no—
41C22×C33C22⋊C411yescharacteristic
42C2×C3⋊(C3⋊D3)—22no—
43C2×C3⋊(C3⋊D3)—41no—
44C2×C3⋊(C3⋊D3)—41no—
45C33×C4C2×C811yescharacteristic
46C33×C4M1611yescharacteristic
47C3×C3⋊Dic3—21no—
48C3×C3⋊Dic3D811yescharacteristic
49C3×C3⋊Dic3SD1611yescharacteristic
50C32×Dic3—41no—
51#893—182no—
52C4×D12—91no—
53C3×C4⋊C8—91no—
54C4×C3⋊C8—91no—
55C22×C3⋊D3—61no—
56C22×C3⋊D3—64no—
57C22×C3⋊D3—124no—
58C2×C32×C4—22no—
59C2×C32×C4—42no—
60C2×C32×C4—61no—
61C2×C32×C4C4×D311yescharacteristic
62C2×C3⋊Dic3D1211yescharacteristic
63C2×C3⋊Dic3#6711yescharacteristic
64C4⋊(C3⋊D3)—32no—
65C4⋊(C3⋊D3)—64no—
66C4⋊(C3⋊D3)—61no—
67C4⋊(C3⋊D3)—64no—
68C4⋊(C3⋊D3)—124no—
69C4⋊(C3⋊D3)—124no—
70C4×C3⋊D3—121no—
71C4×C3⋊D3—121no—
72C2×C3×Dic3—24no—
73C2×C3×Dic3—62no—
74C2×C3×Dic3—62no—
75#624—124no—
76#624—124no—
77#622—41no—
78#622—62no—
79D4⋊C8—271noSylow
80C2×C33C22⋊C811yescharacteristic
81C2×C33#10211yescharacteristic
82C2×C33#10411yescharacteristic
83C3⋊(C3⋊D3)—42no—
84C2×D12—91no—
85C2×D12—182no—
86C2×D12—362no—
87C2×C4×D3—181no—
88C2×C4×D3—184no—
89C2×C3×C8—181no—
90C3×C4⋊C4—62no—
91C3×C42—91no—
92(C2×D3)⋊C4—181no—
93(C2×D3)⋊C4—184no—
94C4⋊Dic3#15711yescharacteristic
95C4×Dic3—62no—
96C2×C3⋊C8—181no—
97C22×C32—22no—
98C22×C32—42no—
99C22×C32(C2×D3)⋊C411yescharacteristic
100C2×C3⋊D3—62no—
101C2×C3⋊D3—68no—
102C2×C3⋊D3—128no—
103C2×C3⋊D3—121no—
104C2×C3⋊D3—121no—
105C2×C3⋊D3—124no—
106C2×C3⋊D3—124no—
107C2×C3⋊D3—244no—
108C2×C3⋊D3—244no—
109C32×C4—22no—
110C32×C4—22no—
111C32×C4—42no—
112C32×C4—42no—
113C32×C4—121no—
114C32×C4D3×C811yescharacteristic
115C32×C4ΓL(1,𝔽25)11yescharacteristic
116C3⋊Dic3—21no—
117C3⋊Dic3#21311yescharacteristic
118C3⋊Dic3#21511yescharacteristic
119C3×Dic3—44no—
120C3×Dic3—62no—
121C3×Dic3—62no—
122C3×Dic3—122no—
123C4×D4—271no—
124C4⋊C8—91no—
125C4×C8—271no—
126C33D4⋊C811yescharacteristic, complemented, Sylow
127C22×D3—184no—
128C22×D3—181no—
129C22×D3—364no—
130C2×C3×C4—22no—
131C2×C3×C4—42no—
132C2×C3×C4—64no—
133C2×C3×C4—91no—
134C2×C3×C4—91no—
135C2×C3×C4C2×#15711yescharacteristic
136C2×Dic3—21no—
137C2×Dic3—62no—
138C2×Dic3—62no—
139D12—92no—
140D12—184no—
141D12—184no—
142D12—181no—
143D12—364no—
144D12—364no—
145C4×D3—361no—
146C4×D3—361no—
147C4×D3—364no—
148C4×D3—364no—
149C3×C8—361no—
150C3⋊C8—182no—
151C2×C32—22no—
152C2×C32—22no—
153C2×C32—22no—
154C2×C32—42no—
155C2×C32—42no—
156C2×C32—42no—
157C2×C32#80911yescharacteristic
158C2×C32(C2×D3)⋊C811yescharacteristic
159C2×C32#82511yescharacteristic
160C3⋊D3—122no—
161C3⋊D3—128no—
162C3⋊D3—248no—
163C2×D4—271no—
164C22×C4—541no—
165C2×C8—181no—
166C2×C8—541no—
167C4⋊C4—31no—
168C22⋊C4—541no—
169C42—91no—
170C22×C3—22no—
171C22×C3—42no—
172C22×C3C22⋊#15711yescharacteristic
173C2×D3—188no—
174C2×D3—182no—
175C2×D3—368no—
176C2×D3—364no—
177C2×D3—364no—
178C2×D3—361no—
179C2×D3—361no—
180C2×D3—724no—
181C2×D3—724no—
182C3×C4—22no—
183C3×C4—22no—
184C3×C4—42no—
185C3×C4—42no—
186C3×C4—91no—
187C3×C4—91no—
188C3×C4—124no—
189C3×C4—181no—
190C3×C4#378711yescharacteristic
191C3×C4#378811yescharacteristic
192Dic3—41no—
193Dic3—62no—
194Dic3—62no—
195Dic3—122no—
196C32—22no—
197C32—42no—
198C32#448311yescharacteristic, complemented
199C23—541no—
200D4—272no—
201D4—541no—
202C2×C4—61no—
203C2×C4—91no—
204C2×C4—91no—
205C2×C4—1081no—
206C2×C4—1081no—
207C2×C4D3×#15711yescharacteristic
208C8—361no—
209C8—542no—
210C2×C3—22no—
211C2×C3—22no—
212C2×C3—22no—
213C2×C3—42no—
214C2×C3—42no—
215C2×C3—42no—
216C2×C3#6377811yescharacteristic
217C2×C3#6378211yescharacteristic
218C2×C3#6378511yescharacteristic
219D3—368no—
220D3—362no—
221D3—728no—
222C22—542no—
223C22—1081no—
224C22—1081no—
225C22(C2×D3)⋊#15711yescharacteristic
226C4—91no—
227C4—91no—
228C4—121no—
229C4—181no—
230C4#8892411yescharacteristic
231C4#8892611yescharacteristic
232C3—22no—
233C3—42no—
234C3#1059016011yescharacteristic, complemented
235C2—1082no—
236C2#1172270611yescharacteristic
237C2#1172271211yescharacteristic
238C2#1172272611yescharacteristic
subgroupG/H
centerC22(C2×D3)⋊#157
commutatorC33×C4C2×C8
Frattini, hypercenterC2×C4D3×#157
FittingC2×C33×C4C2×C4
socleC22×C33C22⋊C4
prime radicalC2×C4⋊(C3⋊(C3⋊D3))C4
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
19. C4⋊(C3⋊(C3⋊D3))C8semidirect
126. C33D4⋊C8semidirect
198. C32#4483semidirect
234. C3#10590160semidirect

Maximal subgroups

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

subgroupindex
#117234942
C4×C3⋊#6222
C3×C4⋊#6222
#105901603
#44839