Finite Group Databasebeta

← #13154941

Subgroups — #13154941

Order 1728.

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

normal subgroups only

This group has 268 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×#63797C211yesmaximal, characteristic
2#11723094C211yesmaximal, characteristic, complemented
3#11722065C211yesmaximal, characteristic, complemented
4#10590016—31nomaximal
5C3×#3802—21no—
6C4⋊(C3⋊#157)—21no—
7C3×C8×C3⋊D3C2211yescharacteristic
8C32×D24—21no—
9C3×#3713C412yescomplemented
10C3×C3⋊(C3⋊C16)—21no—
11C3×D3×D8—61no—
12#63797D311yescharacteristic
13#63769—31no—
14#63542—31no—
15D3×C3⋊C16—61no—
16C2×C3×D32—41no—
17C2×C3⋊#157—41no—
18C3×C4×C3⋊D3D411yescharacteristic
19C32×D12—22no—
20C3×#625—41no—
21C33×C8D411yescharacteristic
22C3×C3⋊(C3⋊C8)C2×C411yescharacteristic
23#4562—91nomaximal
24C3×D3×D4—62no—
25#3802—21no—
26C4⋊#157—61no—
27C32×D8—61no—
28C8×C3⋊D3C2×D311yescharacteristic
29C3×#213—62no—
30C3×D24—21no—
31C3×D24—21no—
32C3×D24—41no—
33C3×D3×C8—61no—
34C3×D3×C8—61no—
35#3713Dic312yescomplemented
36C3⋊(C3⋊C16)—21no—
37C3×C3⋊C16—61no—
38C3×C3⋊C16—61no—
39C3×C3⋊C16—121no—
40C2×C3×C3⋊D3D811yescharacteristic
41C2×C32×D3—42no—
42C3×D32—81no—
43C3×D32—81no—
44C3⋊#157—81no—
45C33×C4C22⋊C411yescharacteristic
46C3×C3⋊Dic3SD1611yescharacteristic
47C2×C3×D8—91no—
48D3×D8—61no—
49#818—91no—
50C2×C3⋊C16—91no—
51D3×C16—181no—
52C22×C3×D3—122no—
53C2×D32—41no—
54C2×#157—121no—
55C32×D4—62no—
56C4×C3⋊D3#6711yescharacteristic
57C3×#67—122no—
58C3×D12—22no—
59C3×D12—22no—
60C3×D12—42no—
61C3×C4×D3—61no—
62C3×C4×D3—61no—
63#625—41no—
64C32×C8—21no—
65C32×C8—21no—
66C32×C8—41no—
67C32×C8—41no—
68C32×C8#6711yescharacteristic
69C3⋊(C3⋊C8)C2×Dic311yescharacteristic
70C3×C3⋊C8—61no—
71C3×C3⋊C8—61no—
72#357—271noSylow
73C2×C33#10211yescharacteristic
74C3×C3⋊D3D1611yescharacteristic
75C3×C3⋊D3SD3211yescharacteristic
76C32×D3—82no—
77C2×C3×D4—181no—
78D3×D4—62no—
79C3×D8—61no—
80C3×D8—61no—
81C3×D8—92no—
82C3×D8—121no—
83C2×C3×C8—91no—
84#213—62no—
85C4⋊Dic3—181no—
86D24—21no—
87D3×C8—61no—
88D3×C8—61no—
89C3×C16—181no—
90C3⋊C16—61no—
91C3⋊C16—61no—
92C3⋊C16—121no—
93C22×C32—122no—
94C2×C3⋊D3#21311yescharacteristic
95C2×C3×D3—42no—
96C2×C3×D3—42no—
97C2×C3×D3—61no—
98C2×C3×D3—61no—
99C2×C3×D3—82no—
100C2×C3×D3—242no—
101C2×C3×D3—242no—
102D32—81no—
103D32—81no—
104#157—241no—
105C32×C4—21no—
106C32×C4—21no—
107C32×C4—41no—
108C32×C4—41no—
109C32×C4#21711yescharacteristic
110C3⋊Dic3#21411yescharacteristic
111C3×Dic3—61no—
112C3×Dic3—61no—
113C2×D8—91no—
114C2×C16—271no—
115#107—271no—
116C33#35711yescharacteristic, complemented, Sylow
117C23×C3—361no—
118C22×D3—122no—
119C3×D4—62no—
120C3×D4—62no—
121C3×D4—122no—
122C3×D4—361no—
123C2×C3×C4—91no—
124#67—122no—
125C2×Dic3—361no—
126D12—22no—
127C4×D3—61no—
128C4×D3—61no—
129C3×C8—21no—
130C3×C8—21no—
131C3×C8—41no—
132C3×C8—41no—
133C3×C8—91no—
134C3×C8GO+(4,𝔽2)11yescharacteristic
135C3⋊C8—61no—
136C3⋊C8—61no—
137C2×C32—21no—
138C2×C32—21no—
139C2×C32—41no—
140C2×C32—41no—
141C2×C32—242no—
142C2×C32#83211yescharacteristic
143C3⋊D3#82611yescharacteristic
144C3⋊D3#82711yescharacteristic
145C3×D3—61no—
146C3×D3—61no—
147C3×D3—61no—
148C3×D3—61no—
149C3×D3—82no—
150C3×D3—82no—
151C3×D3—162no—
152C2×D4—181no—
153D8—61no—
154D8—92no—
155C2×C8—91no—
156C4⋊C4—541no—
157C16—181no—
158C22×C3—91no—
159C22×C3—122no—
160C22×C3—122no—
161C22×C3—242no—
162C22×C3—721no—
163C22×C3—721no—
164C2×D3—42no—
165C2×D3—61no—
166C2×D3—61no—
167C2×D3—242no—
168C2×D3—242no—
169C3×C4—21no—
170C3×C4—21no—
171C3×C4—41no—
172C3×C4—41no—
173C3×C4—91no—
174C3×C4D32⋊C411yescharacteristic
175Dic3—61no—
176Dic3—61no—
177Dic3—721no—
178C32—21no—
179C32—21no—
180C32—41no—
181C32—41no—
182C32#456211yescharacteristic, complemented
183C23—361no—
184D4—62no—
185D4—361no—
186C2×C4—91no—
187C2×C4—1081no—
188C8—91no—
189C8D32⋊D311yescharacteristic
190C2×C3—21no—
191C2×C3—21no—
192C2×C3—41no—
193C2×C3—41no—
194C2×C3—91no—
195C2×C3—91no—
196C2×C3—242no—
197C2×C3—242no—
198C2×C3—482no—
199C2×C3#6372911yescharacteristic
200D3—61no—
201D3—61no—
202D3—61no—
203D3—61no—
204D3—82no—
205C22—91no—
206C22—122no—
207C22—721no—
208C22—721no—
209C4—91no—
210C4—2161no—
211C4D32⋊Dic311yescharacteristic
212C3—21no—
213C3—21no—
214C3—41no—
215C3—41no—
216C3#1059001611yescharacteristic, complemented
217C2—91no—
218C2—91no—
219C2—242no—
220C2#1172275811yescharacteristic
subgroupG/H
centerC2#11722758
commutatorC3×C3⋊(C3⋊C8)C2×C4
Frattini, hypercenterC8D32⋊D3
FittingC33×C8D4
socleC2×C33#102
prime radicalC3×#63797C2
cosocleC3×C8×C3⋊D3C22
hypocenterC33#357
LCSG ▷ C3×C3⋊(C3⋊C8) ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C2 ◁ C4 ◁ C8

Complements

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

NH ≅ G/Ntype
2. #11723094C2semidirect
3. #11722065C2semidirect
9. C3×#3713C4semidirect
35. #3713Dic3semidirect
116. C33#357semidirect
182. C32#4562semidirect
216. C3#10590016semidirect

Maximal subgroups

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

subgroupindex
C3×#637972
#117230942
#117220652
#105900163
#45629