Finite Group Databasebeta

← C3×#10590016 #13154866

Subgroups — C3×#10590016 #13154866

Order 1728.

268 conjugacy classes of subgroups; only the 42 normal subgroups are shown.

normal subgroups only

This group has 268 conjugacy classes of subgroups — too many to draw. Only its 42 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
2C3×#63769C211yesmaximal, characteristic, complemented
3C3×#63542C211yesmaximal, characteristic, complemented
4#10590016C311yesmaximal, characteristic, direct product factor
5C3×#3802—21no—
6C3×C4⋊#157—21no—
7C3×C8×C3⋊D3C2211yescharacteristic
8C32×D24—21no—
9C3×#3713C412yescomplemented
10C32×C3⋊C16—21no—
11C3×D3×D8—61no—
12#63797C2×C311yescharacteristic
13#63769C2×C311yescharacteristic, complemented
14C3×D3×C16—61no—
15#63542C2×C311yescharacteristic, complemented
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
23C3×#357—91nomaximal
24C3×D3×D4—62no—
25#3802—21no—
26C4⋊#157—21no—
27C32×D8—61no—
28C8×C3⋊D3C22×C311yescharacteristic
29C3×#213—62no—
30C3×D24—21no—
31C3×D24—21no—
32C3×D24—41no—
33C3×D3×C8—61no—
34C3×D3×C8—61no—
35#3713C3×C412yescomplemented
36C32×C16—61no—
37C3×C3⋊C16—21no—
38C3×C3⋊C16—21no—
39C3×C3⋊C16—41no—
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—
49C2×C3×C16—91no—
50C3×#107—91no—
51D3×C16—61no—
52C22×C3×D3—122no—
53C2×D32—41no—
54C2×#157—41no—
55C32×D4—62no—
56C4×C3⋊D3C3×D411yescharacteristic
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×C8C3×D411yescharacteristic
69C3⋊(C3⋊C8)C2×C3×C411yescharacteristic
70C3×C3⋊C8—61no—
71C3×C3⋊C8—61no—
72#357—91noSylow
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—
84C3×C4⋊C4—181no—
85#213—62no—
86D24—21no—
87D3×C8—61no—
88D3×C8—61no—
89C3×C16—61no—
90C3×C16—61no—
91C3×C16—121no—
92C3⋊C16—21no—
93C22×C32—122no—
94C2×C3⋊D3C3×D811yescharacteristic
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—81no—
105C32×C4—21no—
106C32×C4—21no—
107C32×C4—41no—
108C32×C4—41no—
109C32×C4C3×C22⋊C411yescharacteristic
110C3⋊Dic3C3×SD1611yescharacteristic
111C3×Dic3—61no—
112C3×Dic3—61no—
113C2×D8—91no—
114C2×C16—91no—
115#107—91no—
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—
124C2×C3×C4—361no—
125#67—122no—
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×C32C3×#10211yescharacteristic
143C3⋊D3C3×D1611yescharacteristic
144C3⋊D3C3×SD3211yescharacteristic
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—181no—
157C16—61no—
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×C4—721no—
175C3×C4D32⋊C411yescharacteristic
176Dic3—61no—
177Dic3—61no—
178C32—21no—
179C32—21no—
180C32—41no—
181C32—41no—
182C32C3×#35711yescharacteristic, complemented
183C23—361no—
184D4—62no—
185D4—361no—
186C2×C4—91no—
187C2×C4—361no—
188C8—91no—
189C8C3×GO+(4,𝔽2)11yescharacteristic
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—721no—
211C4C3×D32⋊C411yescharacteristic
212C3—21no—
213C3—21no—
214C3—41no—
215C3—41no—
216C3#1059001611yescharacteristic, direct product factor
217C2—91no—
218C2—91no—
219C2—242no—
220C2C3×#6372911yescharacteristic
subgroupG/H
centerC2×C3#63729
commutatorC3⋊(C3⋊C8)C2×C3×C4
FrattiniC8C3×GO+(4,𝔽2)
FittingC33×C8D4
socleC2×C33#102
prime radicalC3×#63797C2
cosocleC8×C3⋊D3C22×C3
hypercenterC3×C8GO+(4,𝔽2)
hypocenterC32C3×#357
LCSG ▷ C3⋊(C3⋊C8) ▷ C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2×C3 ◁ C3×C4 ◁ C3×C8

Complements

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

NH ≅ G/Ntype
2. C3×#63769C2semidirect
3. C3×#63542C2semidirect
4. #10590016C3direct (N × H)
9. C3×#3713C4semidirect
13. #63769C2×C3semidirect
15. #63542C2×C3semidirect
35. #3713C3×C4semidirect
116. C33#357semidirect
182. C32C3×#357semidirect
216. C3#10590016direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#637972
C3×#637692
C3×#635422
#105900163
C3×#3579