Finite Group Databasebeta

← #13155059

Subgroups — #13155059

Order 1728.

232 conjugacy classes of subgroups; only the 35 normal subgroups are shown.

normal subgroups only

This group has 232 conjugacy classes of subgroups — too many to draw. Only its 35 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#11723651C211yesmaximal, characteristic
2C3×#63764C211yesmaximal, characteristic, complemented
3#11722066C211yesmaximal, characteristic, complemented
4#10590015—31nomaximal
5#89042C411yescharacteristic
6C3×C8×C3⋊D3—21no—
7C3×C8×C3⋊D3C2211yescharacteristic
8#88809—21no—
9#88809C411yescharacteristic
10C3×#3708—21no—
11C3×C3⋊(C3⋊C16)—21no—
12#63803—31no—
13#63803—61no—
14#63764D311yescharacteristic, complemented
15#63544—31no—
16#63543—61no—
17C3×C4×C3⋊D3—21no—
18C3×C4×C3⋊D3C2×C411yescharacteristic
19#6296C811yescharacteristic
20#6295—21no—
21#6295C811yescharacteristic
22C33×C8D411yescharacteristic
23C3×C3⋊(C3⋊C8)—21no—
24C3×C3⋊(C3⋊C8)D411yescharacteristic
25#4508—91nomaximal
26#3798—31no—
27#3798—61no—
28C8×C3⋊D3—21no—
29C8×C3⋊D3C2×D311yescharacteristic
30C3×D3×C8—61no—
31C3×D3×C8—61no—
32C3×D3×C8—61no—
33C3×D3×C8—61no—
34C3×D3×C8—121no—
35#3712—31no—
36#3712—61no—
37#3711—61no—
38#3711—61no—
39#3711—61no—
40#3708—21no—
41C3⋊(C3⋊C16)—21no—
42C3×C3⋊C16—61no—
43C3×C3⋊C16—61no—
44C3×C3⋊C16—121no—
45C2×C3×C3⋊D3—21no—
46C2×C3×C3⋊D3M1611yescharacteristic
47C33×C4C22⋊C411yescharacteristic
48C3×C3⋊Dic3—21no—
49C3×C3⋊Dic3C2×C811yescharacteristic
50#901—91no—
51#901—181no—
52C2×C3×C16—91no—
53#812—91no—
54#798—181no—
55C4×C3⋊D3—21no—
56C4×C3⋊D3C4×D311yescharacteristic
57C3×C4×D3—61no—
58C3×C4×D3—61no—
59C3×C4×D3—61no—
60C3×C4×D3—61no—
61C3×C4×D3—121no—
62#627—31no—
63#627—61no—
64#626—61no—
65#626—61no—
66#626—61no—
67#625—31no—
68#625—61no—
69C32×C8—21no—
70C32×C8—21no—
71C32×C8—41no—
72C32×C8—41no—
73C32×C8D1211yescharacteristic
74C3⋊(C3⋊C8)—21no—
75C3⋊(C3⋊C8)#6711yescharacteristic
76C3×C3⋊C8—61no—
77C3×C3⋊C8—61no—
78C3×C3⋊C8—61no—
79C3×C3⋊C8—61no—
80C3×C3⋊C8—121no—
81#350—271noSylow
82C2×C33C22⋊C811yescharacteristic
83C3×C3⋊D3—21no—
84C3×C3⋊D3—41no—
85#235—91no—
86#235—181no—
87C2×C3×C8—91no—
88C2×C3×C8—181no—
89#208—91no—
90#208—181no—
91ΓL(1,𝔽25)—181no—
92ΓL(1,𝔽25)—181no—
93ΓL(1,𝔽25)—181no—
94D3×C8—61no—
95D3×C8—61no—
96D3×C8—61no—
97D3×C8—61no—
98D3×C8—121no—
99C3×C16—181no—
100C3×C16—181no—
101C3⋊C16—61no—
102C3⋊C16—61no—
103C3⋊C16—121no—
104C2×C3⋊D3—21no—
105C2×C3⋊D3ΓL(1,𝔽25)11yescharacteristic
106C2×C3×D3—61no—
107C2×C3×D3—61no—
108C2×C3×D3—61no—
109C2×C3×D3—61no—
110C2×C3×D3—121no—
111C32×C4—21no—
112C32×C4—21no—
113C32×C4—41no—
114C32×C4—41no—
115C32×C4(C2×D3)⋊C411yescharacteristic
116C3⋊Dic3—21no—
117C3⋊Dic3D3×C811yescharacteristic
118C3×Dic3—61no—
119C3×Dic3—61no—
120C3×Dic3—61no—
121C3×Dic3—61no—
122C3×Dic3—121no—
123#131—271no—
124M32—271no—
125C2×C16—91no—
126C33#35011yescharacteristic, complemented, Sylow
127C2×C3×C4—91no—
128C2×C3×C4—181no—
129#67—181no—
130#67—181no—
131#67—181no—
132D12—91no—
133D12—181no—
134C4×D3—61no—
135C4×D3—61no—
136C4×D3—61no—
137C4×D3—61no—
138C4×D3—121no—
139Dic6—91no—
140Dic6—181no—
141C3×C8—21no—
142C3×C8—21no—
143C3×C8—41no—
144C3×C8—41no—
145C3×C8—91no—
146C3×C8—181no—
147C3×C8GO+(4,𝔽2)11yescharacteristic
148C3⋊C8—61no—
149C3⋊C8—61no—
150C3⋊C8—61no—
151C3⋊C8—61no—
152C3⋊C8—121no—
153C2×C32—21no—
154C2×C32—21no—
155C2×C32—41no—
156C2×C32—41no—
157C2×C32(C2×D3)⋊C811yescharacteristic
158C3⋊D3—21no—
159C3⋊D3—41no—
160C3×D3—121no—
161C3×D3—121no—
162C3×D3—121no—
163C3×D3—122no—
164C3×D3—121no—
165#41—271no—
166M16—271no—
167M16—541no—
168C2×C8—91no—
169C2×C8—181no—
170C16—181no—
171C16—181no—
172C22×C3—91no—
173C22×C3—181no—
174C2×D3—61no—
175C2×D3—61no—
176C2×D3—61no—
177C2×D3—61no—
178C2×D3—121no—
179C3×C4—21no—
180C3×C4—21no—
181C3×C4—41no—
182C3×C4—41no—
183C3×C4—91no—
184C3×C4—181no—
185C3×C4D32⋊C411yescharacteristic
186Dic3—61no—
187Dic3—61no—
188Dic3—61no—
189Dic3—61no—
190Dic3—121no—
191C32—21no—
192C32—21no—
193C32—41no—
194C32—41no—
195C32#450811yescharacteristic, complemented
196Q8—271no—
197D4—271no—
198D4—541no—
199C2×C4—91no—
200C2×C4—181no—
201C8—91no—
202C8—181no—
203C8#157⋊D311yescharacteristic
204C2×C3—21no—
205C2×C3—21no—
206C2×C3—41no—
207C2×C3—41no—
208C2×C3—181no—
209C2×C3—361no—
210C2×C3D32⋊C811yescharacteristic
211D3—121no—
212D3—121no—
213D3—121no—
214D3—122no—
215D3—121no—
216C22—91no—
217C22—181no—
218C4—91no—
219C4—181no—
220C4#8893911yescharacteristic
221C3—21no—
222C3—21no—
223C3—41no—
224C3—41no—
225C3#1059001511yescharacteristic, complemented
226C2—181no—
227C2—361no—
228C2#1172278011yescharacteristic
subgroupG/H
center, Frattini, hypercenterC8#157⋊D3
commutatorC3×C3⋊Dic3C2×C8
FittingC33×C8D4
socleC2×C33C22⋊C8
prime radical#89042C4
cosocleC3×C8×C3⋊D3C22
hypocenterC33#350
LCSG ▷ C3×C3⋊Dic3 ▷ C2×C33 ▷ C33
UCSC1 ◁ C8

Complements

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

NH ≅ G/Ntype
2. C3×#63764C2semidirect
3. #11722066C2semidirect
14. #63764D3semidirect
126. C33#350semidirect
195. C32#4508semidirect
225. C3#10590015semidirect

Maximal subgroups

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

subgroupindex
#117236512
C3×#637642
#117220662
#105900153
#45089