Finite Group Databasebeta

← C3×#10590020 #13154870

Subgroups — C3×#10590020 #13154870

Order 1728.

258 conjugacy classes of subgroups; only the 38 normal subgroups are shown.

normal subgroups only

This group has 258 conjugacy classes of subgroups — too many to draw. Only its 38 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×#63771C211yesmaximal, characteristic, complemented
3C3×#63544C211yesmaximal, characteristic, complemented
4#10590020C311yesmaximal, characteristic, direct product factor
5C3×#3802—21no—
6C3×#3788—21no—
7C3×C8×C3⋊D3C2211yescharacteristic
8C32×D24—21no—
9C3×#3713C412yes—
10C32×C3⋊C16—21no—
11C3×D3×D8—61no—
12#63797C2×C311yescharacteristic
13#63771C2×C311yescharacteristic, complemented
14C3×#798—61no—
15#63544C2×C311yescharacteristic, complemented
16C2×C3×D32—41no—
17C3×C4×C3⋊D3D411yescharacteristic
18C32×D12—22no—
19C3×#625—41no—
20C3×#622—41no—
21C33×C8D411yescharacteristic
22C3×C3⋊(C3⋊C8)C2×C411yescharacteristic
23C3×#361—91nomaximal
24C3×D3×D4—62no—
25#3802—21no—
26#3788—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×C412yes—
36C32×C16—61no—
37C3×C3⋊C16—21no—
38C3×C3⋊C16—21no—
39C3×C3⋊C16—41no—
40C2×C3×C3⋊D3SD1611yescharacteristic
41C2×C32×D3—42no—
42C3×D32—82no—
43C33×C4C22⋊C411yescharacteristic
44C3×C3⋊Dic3D811yescharacteristic
45C2×C3×D8—91no—
46D3×D8—61no—
47C3×M32—91no—
48C3×#108—91no—
49#798—61no—
50C22×C3×D3—122no—
51C2×D32—41no—
52C32×D4—62no—
53C4×C3⋊D3C3×D411yescharacteristic
54C3×#67—122no—
55C3×D12—22no—
56C3×D12—22no—
57C3×D12—42no—
58C3×C4×D3—61no—
59C3×C4×D3—61no—
60#625—41no—
61#622—41no—
62C32×C8—21no—
63C32×C8—21no—
64C32×C8—41no—
65C32×C8—41no—
66C32×C8C3×D411yescharacteristic
67C3⋊(C3⋊C8)C2×C3×C411yescharacteristic
68C3×C3⋊C8—61no—
69C3×C3⋊C8—61no—
70#361—91noSylow
71C2×C33#10211yescharacteristic
72C3×C3⋊D3—21no—
73C32×D3—82no—
74C2×C3×D4—181no—
75D3×D4—62no—
76C3×D8—61no—
77C3×D8—61no—
78C3×D8—92no—
79C3×D8—121no—
80C3×M16—181no—
81C2×C3×C8—91no—
82#213—62no—
83D24—21no—
84D3×C8—61no—
85D3×C8—61no—
86C3×C16—61no—
87C3×C16—61no—
88C3×C16—121no—
89C3⋊C16—21no—
90C22×C32—122no—
91C2×C3⋊D3C3×SD1611yescharacteristic
92C2×C3×D3—42no—
93C2×C3×D3—42no—
94C2×C3×D3—61no—
95C2×C3×D3—61no—
96C2×C3×D3—82no—
97C2×C3×D3—244no—
98D32—82no—
99C32×C4—21no—
100C32×C4—21no—
101C32×C4—41no—
102C32×C4—41no—
103C32×C4C3×C22⋊C411yescharacteristic
104C3⋊Dic3C3×D811yescharacteristic
105C3×Dic3—61no—
106C3×Dic3—61no—
107C2×D8—91no—
108M32—91no—
109#108—91no—
110C33#36111yescharacteristic, complemented, Sylow
111C23×C3—361no—
112C22×D3—122no—
113C3×D4—62no—
114C3×D4—62no—
115C3×D4—122no—
116C3×D4—361no—
117C2×C3×C4—91no—
118#67—122no—
119D12—22no—
120C4×D3—61no—
121C4×D3—61no—
122C3×C8—21no—
123C3×C8—21no—
124C3×C8—41no—
125C3×C8—41no—
126C3×C8—91no—
127C3×C8—361no—
128C3×C8GO+(4,𝔽2)11yescharacteristic
129C3⋊C8—61no—
130C3⋊C8—61no—
131C2×C32—21no—
132C2×C32—21no—
133C2×C32—41no—
134C2×C32—41no—
135C2×C32—242no—
136C2×C32C3×#10211yescharacteristic
137C3⋊D3—21no—
138C3×D3—62no—
139C3×D3—82no—
140C3×D3—82no—
141C3×D3—121no—
142C3×D3—162no—
143C2×D4—181no—
144D8—61no—
145D8—92no—
146M16—181no—
147C2×C8—91no—
148C16—61no—
149C22×C3—91no—
150C22×C3—122no—
151C22×C3—122no—
152C22×C3—242no—
153C22×C3—722no—
154C2×D3—42no—
155C2×D3—61no—
156C2×D3—61no—
157C2×D3—244no—
158C3×C4—21no—
159C3×C4—21no—
160C3×C4—41no—
161C3×C4—41no—
162C3×C4—91no—
163C3×C4D32⋊C411yescharacteristic
164Dic3—61no—
165Dic3—61no—
166C32—21no—
167C32—21no—
168C32—41no—
169C32—41no—
170C32C3×#36111yescharacteristic, complemented
171C23—361no—
172D4—62no—
173D4—361no—
174C2×C4—91no—
175C8—91no—
176C8—361no—
177C8C3×GO+(4,𝔽2)11yescharacteristic
178C2×C3—21no—
179C2×C3—21no—
180C2×C3—41no—
181C2×C3—41no—
182C2×C3—181no—
183C2×C3—242no—
184C2×C3—242no—
185C2×C3—482no—
186C2×C3#6372911yescharacteristic
187D3—62no—
188D3—82no—
189D3—121no—
190C22—91no—
191C22—122no—
192C22—722no—
193C4—91no—
194C4C3×D32⋊C411yescharacteristic
195C3—21no—
196C3—21no—
197C3—41no—
198C3—41no—
199C3#1059002011yescharacteristic, direct product factor
200C2—181no—
201C2—242no—
202C2C3×#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×#361
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×#63771C2semidirect
3. C3×#63544C2semidirect
4. #10590020C3direct (N × H)
13. #63771C2×C3semidirect
15. #63544C2×C3semidirect
110. C33#361semidirect
170. C32C3×#361semidirect
199. C3#10590020direct (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×#637712
C3×#635442
#105900203
C3×#3619