Finite Group Databasebeta

← #13154939

Subgroups — #13154939

Order 1728.

232 conjugacy classes of subgroups; only the 39 normal subgroups are shown.

normal subgroups only

This group has 232 conjugacy classes of subgroups — too many to draw. Only its 39 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×#63803C211yesmaximal, characteristic
2#11723089C211yesmaximal, characteristic, complemented
3#11722066C211yesmaximal, characteristic, complemented
4#10590015—31nomaximal
5C3×#3798C411yescharacteristic
6C3×C8×C3⋊D3C2211yescharacteristic
7C32×D3×C8—21no—
8C3×#3712C411yescharacteristic
9C3×#3711—21no—
10C3⋊#3708—21no—
11C3×C3⋊(C3⋊C16)—21no—
12C3×#901—61no—
13#63803D311yescharacteristic
14#63764—31no—
15#63544—31no—
16#63543—61no—
17C3×C4×C3⋊D3C2×C411yescharacteristic
18C32×C4×D3—21no—
19C3×#627C811yescharacteristic
20C3×#626—21no—
21C3×#625C811yescharacteristic
22C33×C8D411yescharacteristic
23C3×C3⋊(C3⋊C8)D411yescharacteristic
24C32×C3⋊C8—21no—
25#4561—91nomaximal
26C3×#235—61no—
27#3798Dic311yescharacteristic
28C2×C32×C8—61no—
29C8×C3⋊D3C2×D311yescharacteristic
30C3×#208—61no—
31C3×ΓL(1,𝔽25)—61no—
32C3×ΓL(1,𝔽25)—61no—
33C3×D3×C8—21no—
34C3×D3×C8—21no—
35C3×D3×C8—41no—
36C3×D3×C8—61no—
37C3×D3×C8—61no—
38#3712Dic311yescharacteristic
39#3711—21no—
40#3708—61no—
41C3⋊(C3⋊C16)—21no—
42C3×C3⋊C16—61no—
43C3×C3⋊C16—61no—
44C3×C3⋊C16—121no—
45C2×C3×C3⋊D3M1611yescharacteristic
46C2×C32×D3—21no—
47C33×C4C22⋊C411yescharacteristic
48C3×C3⋊Dic3C2×C811yescharacteristic
49C32×Dic3—21no—
50C3×#131—91no—
51#901—61no—
52#812—91no—
53C2×C3⋊C16—91no—
54#798—181no—
55C2×C32×C4—61no—
56C4×C3⋊D3C2×Dic311yescharacteristic
57C3×#67—61no—
58C3×#67—61no—
59C3×D12—61no—
60C3×C4×D3—21no—
61C3×C4×D3—21no—
62C3×C4×D3—41no—
63C3×C4×D3—61no—
64C3×C4×D3—61no—
65C3×Dic6—61no—
66#627C3⋊C811yescharacteristic
67#626—21no—
68#625C3⋊C811yescharacteristic
69C32×C8—21no—
70C32×C8—21no—
71C32×C8—41no—
72C32×C8—41no—
73C32×C8—61no—
74C32×C8#6711yescharacteristic
75C3⋊(C3⋊C8)#6711yescharacteristic
76C3×C3⋊C8—21no—
77C3×C3⋊C8—21no—
78C3×C3⋊C8—41no—
79C3×C3⋊C8—61no—
80C3×C3⋊C8—61no—
81#350—271noSylow
82C2×C33C22⋊C811yescharacteristic
83C3×C3⋊D3—21no—
84C32×D3—41no—
85C3×#41—91no—
86#235—61no—
87C3×M16—91no—
88C3×M16—181no—
89C2×C3×C8—61no—
90C2×C3×C8—61no—
91C2×C3×C8—91no—
92C2×C3×C8—121no—
93#208—61no—
94ΓL(1,𝔽25)—61no—
95ΓL(1,𝔽25)—61no—
96D3×C8—21no—
97D3×C8—61no—
98D3×C8—61no—
99C3×C16—181no—
100C3⋊C16—61no—
101C3⋊C16—61no—
102C3⋊C16—121no—
103C3⋊C16—181no—
104C22×C32—61no—
105C2×C3⋊D3#20811yescharacteristic
106C2×C3×D3—21no—
107C2×C3×D3—21no—
108C2×C3×D3—41no—
109C2×C3×D3—61no—
110C2×C3×D3—61no—
111C32×C4—21no—
112C32×C4—21no—
113C32×C4—41no—
114C32×C4—41no—
115C32×C4—61no—
116C32×C4#21711yescharacteristic
117C3⋊Dic3C2×C3⋊C811yescharacteristic
118C3×Dic3—21no—
119C3×Dic3—21no—
120C3×Dic3—41no—
121C3×Dic3—61no—
122C3×Dic3—61no—
123#131—91no—
124M32—271no—
125C2×C16—271no—
126C33#35011yescharacteristic, complemented, Sylow
127C3×Q8—91no—
128C3×D4—91no—
129C3×D4—181no—
130C2×C3×C4—61no—
131C2×C3×C4—61no—
132C2×C3×C4—91no—
133C2×C3×C4—121no—
134#67—61no—
135#67—61no—
136D12—61no—
137C4×D3—21no—
138C4×D3—61no—
139C4×D3—61no—
140Dic6—61no—
141C3×C8—21no—
142C3×C8—21no—
143C3×C8—41no—
144C3×C8—41no—
145C3×C8—61no—
146C3×C8—61no—
147C3×C8—91no—
148C3×C8—121no—
149C3×C8GO+(4,𝔽2)11yescharacteristic
150C3⋊C8—21no—
151C3⋊C8—61no—
152C3⋊C8—61no—
153C2×C32—21no—
154C2×C32—21no—
155C2×C32—41no—
156C2×C32—41no—
157C2×C32—121no—
158C2×C32#83011yescharacteristic
159C3⋊D3—21no—
160C3×D3—41no—
161C3×D3—42no—
162C3×D3—41no—
163C3×D3—121no—
164C3×D3—121no—
165#41—91no—
166M16—91no—
167M16—181no—
168C2×C8—61no—
169C2×C8—91no—
170C16—181no—
171C16—541no—
172C22×C3—61no—
173C22×C3—61no—
174C22×C3—91no—
175C22×C3—121no—
176C2×D3—21no—
177C2×D3—61no—
178C2×D3—61no—
179C3×C4—21no—
180C3×C4—21no—
181C3×C4—41no—
182C3×C4—41no—
183C3×C4—61no—
184C3×C4—61no—
185C3×C4—91no—
186C3×C4—121no—
187C3×C4D32⋊C411yescharacteristic
188Dic3—21no—
189Dic3—61no—
190Dic3—61no—
191C32—21no—
192C32—21no—
193C32—41no—
194C32—41no—
195C32#456111yescharacteristic, complemented
196Q8—91no—
197D4—91no—
198D4—181no—
199C2×C4—61no—
200C2×C4—91no—
201C8—61no—
202C8—91no—
203C8D32⋊D311yescharacteristic
204C2×C3—21no—
205C2×C3—21no—
206C2×C3—41no—
207C2×C3—41no—
208C2×C3—121no—
209C2×C3—122no—
210C2×C3—121no—
211C2×C3—181no—
212C2×C3D32⋊C811yescharacteristic
213D3—41no—
214D3—121no—
215D3—121no—
216C22—61no—
217C22—91no—
218C4—61no—
219C4—91no—
220C4D32⋊Dic311yescharacteristic
221C3—21no—
222C3—21no—
223C3—41no—
224C3—41no—
225C3#1059001511yescharacteristic, complemented
226C2—121no—
227C2—181no—
228C2D32⋊(C3⋊C8)11yescharacteristic
subgroupG/H
center, Frattini, hypercenterC8D32⋊D3
commutatorC3×C3⋊Dic3C2×C8
FittingC33×C8D4
socleC2×C33C22⋊C8
prime radicalC3×#3798C4
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. #11723089C2semidirect
3. #11722066C2semidirect
126. C33#350semidirect
195. C32#4561semidirect
225. C3#10590015semidirect

Maximal subgroups

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

subgroupindex
C3×#638032
#117230892
#117220662
#105900153
#45619