Finite Group Databasebeta

← C3×#10590021 #13154871

Subgroups — C3×#10590021 #13154871

Order 1728.

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

normal subgroups only

This group has 222 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×#63802C211yesmaximal, characteristic
2C3×#63771C211yesmaximal, characteristic
3C3×#63544C211yesmaximal, characteristic
4#10590021C311yesmaximal, characteristic, direct product factor
5C3×#3797—21no—
6C3×#3788—21no—
7C3×C8×C3⋊D3C2211yescharacteristic
8C32×Dic12—21no—
9C3×#3718C412yes—
10C32×C3⋊C16—21no—
11C3×D3×Q16—61no—
12#63802C2×C311yescharacteristic
13#63771C2×C311yescharacteristic
14C3×#798—61no—
15#63544C2×C311yescharacteristic
16C3×C4×C3⋊D3D411yescharacteristic
17C32×Dic6—22no—
18C3×#627—41no—
19C3×#624—41no—
20C3×#622—41no—
21C33×C8D411yescharacteristic
22C3×C3⋊(C3⋊C8)C2×C411yescharacteristic
23C3×#362—91nomaximal
24C3×D3×Q8—62no—
25#3797—21no—
26#3788—21no—
27C32×Q16—61no—
28C8×C3⋊D3C22×C311yescharacteristic
29C3×#216—62no—
30C3×Dic12—21no—
31C3×Dic12—21no—
32C3×Dic12—41no—
33C3×D3×C8—61no—
34C3×D3×C8—61no—
35#3718C3×C412yes—
36C32×C16—61no—
37C3×C3⋊C16—21no—
38C3×C3⋊C16—21no—
39C3×C3⋊C16—41no—
40C2×C3×C3⋊D3SD1611yescharacteristic
41C33×C4C22⋊C411yescharacteristic
42C3×C3⋊Dic3D811yescharacteristic
43C32×Dic3—42no—
44C2×C3×Q16—91no—
45D3×Q16—61no—
46C3×M32—91no—
47C3×#108—91no—
48#798—61no—
49C32×Q8—62no—
50C4×C3⋊D3C3×D411yescharacteristic
51C3×C4×D3—61no—
52C3×C4×D3—61no—
53C3×C4×D3—122no—
54C3×Dic6—22no—
55C3×Dic6—22no—
56C3×Dic6—42no—
57C3×Dic6—122no—
58#627—41no—
59#624—41no—
60#622—41no—
61C32×C8—21no—
62C32×C8—21no—
63C32×C8—41no—
64C32×C8—41no—
65C32×C8C3×D411yescharacteristic
66C3⋊(C3⋊C8)C2×C3×C411yescharacteristic
67C3×C3⋊C8—61no—
68C3×C3⋊C8—61no—
69#362—91noSylow
70C2×C33#10211yescharacteristic
71C3×C3⋊D3—21no—
72C2×C3×Q8—181no—
73D3×Q8—62no—
74C3×Q16—61no—
75C3×Q16—61no—
76C3×Q16—92no—
77C3×Q16—121no—
78C3×M16—181no—
79C2×C3×C8—91no—
80#216—62no—
81Dic12—21no—
82D3×C8—61no—
83D3×C8—61no—
84C3×C16—61no—
85C3×C16—61no—
86C3×C16—121no—
87C3⋊C16—21no—
88C2×C3⋊D3C3×SD1611yescharacteristic
89C2×C3×D3—61no—
90C2×C3×D3—61no—
91C32×C4—21no—
92C32×C4—21no—
93C32×C4—41no—
94C32×C4—41no—
95C32×C4—122no—
96C32×C4C3×C22⋊C411yescharacteristic
97C3⋊Dic3C3×D811yescharacteristic
98C3×Dic3—42no—
99C3×Dic3—42no—
100C3×Dic3—61no—
101C3×Dic3—61no—
102C3×Dic3—82no—
103C2×Q16—91no—
104M32—91no—
105#108—91no—
106C33#36211yescharacteristic, complemented, Sylow
107C3×Q8—62no—
108C3×Q8—62no—
109C3×Q8—122no—
110C3×Q8—361no—
111C2×C3×C4—91no—
112C2×C3×C4—361no—
113C4×D3—61no—
114C4×D3—61no—
115C4×D3—122no—
116Dic6—22no—
117Dic6—122no—
118C3×C8—21no—
119C3×C8—21no—
120C3×C8—41no—
121C3×C8—41no—
122C3×C8—91no—
123C3×C8—361no—
124C3×C8GO+(4,𝔽2)11yescharacteristic
125C3⋊C8—61no—
126C3⋊C8—61no—
127C2×C32—21no—
128C2×C32—21no—
129C2×C32—41no—
130C2×C32—41no—
131C2×C32C3×#10211yescharacteristic
132C3⋊D3—21no—
133C3×D3—62no—
134C3×D3—121no—
135C2×Q8—181no—
136Q16—61no—
137Q16—92no—
138M16—181no—
139C2×C8—91no—
140C16—61no—
141C22×C3—91no—
142C2×D3—61no—
143C2×D3—61no—
144C3×C4—21no—
145C3×C4—21no—
146C3×C4—41no—
147C3×C4—41no—
148C3×C4—91no—
149C3×C4—122no—
150C3×C4—122no—
151C3×C4—242no—
152C3×C4D32⋊C411yescharacteristic
153Dic3—42no—
154Dic3—61no—
155Dic3—61no—
156C32—21no—
157C32—21no—
158C32—41no—
159C32—41no—
160C32C3×#36211yescharacteristic, complemented
161Q8—62no—
162Q8—361no—
163C2×C4—91no—
164C2×C4—361no—
165C8—91no—
166C8—361no—
167C8C3×GO+(4,𝔽2)11yescharacteristic
168C2×C3—21no—
169C2×C3—21no—
170C2×C3—41no—
171C2×C3—41no—
172C2×C3—181no—
173C2×C3#6372911yescharacteristic
174D3—62no—
175D3—121no—
176C22—91no—
177C4—91no—
178C4—122no—
179C4C3×D32⋊C411yescharacteristic
180C3—21no—
181C3—21no—
182C3—41no—
183C3—41no—
184C3#1059002111yescharacteristic, direct product factor
185C2—181no—
186C2C3×#6372911yescharacteristic
subgroupG/H
centerC2×C3#63729
commutatorC3⋊(C3⋊C8)C2×C3×C4
FrattiniC8C3×GO+(4,𝔽2)
FittingC33×C8D4
socleC2×C33#102
prime radicalC2×C3×C3⋊D3SD16
cosocleC8×C3⋊D3C22×C3
hypercenterC3×C8GO+(4,𝔽2)
hypocenterC32C3×#362
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
4. #10590021C3direct (N × H)
106. C33#362semidirect
160. C32C3×#362semidirect
184. C3#10590021direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#638022
C3×#637712
C3×#635442
#105900213
C3×#3629