Finite Group Databasebeta

← C3×#10590017 #13154867

Subgroups — C3×#10590017 #13154867

Order 1728.

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

normal subgroups only

This group has 232 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×#63802C211yesmaximal, characteristic
2C3×#63769C211yesmaximal, characteristic
3C3×#63542C211yesmaximal, characteristic
4#10590017C311yesmaximal, characteristic, direct product factor
5C3×#3797—21no—
6C3×C4⋊#157—21no—
7C3×C8×C3⋊D3C2211yescharacteristic
8C32×Dic12—21no—
9C3×#3718C412yescomplemented
10C32×C3⋊C16—21no—
11C3×D3×Q16—61no—
12#63802C2×C311yescharacteristic
13#63769C2×C311yescharacteristic
14C3×D3×C16—61no—
15#63542C2×C311yescharacteristic
16C2×C3×#157—41no—
17C3×C4×C3⋊D3D411yescharacteristic
18C32×Dic6—22no—
19C3×#627—41no—
20C3×#624—41no—
21C33×C8D411yescharacteristic
22C3×C3⋊(C3⋊C8)C2×C411yescharacteristic
23C3×#358—91nomaximal
24C3×D3×Q8—62no—
25#3797—21no—
26C4⋊#157—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×C412yescomplemented
36C32×C16—61no—
37C3×C3⋊C16—21no—
38C3×C3⋊C16—21no—
39C3×C3⋊C16—41no—
40C2×C3×C3⋊D3D811yescharacteristic
41C3×#157—81no—
42C33×C4C22⋊C411yescharacteristic
43C3×C3⋊Dic3SD1611yescharacteristic
44C32×Dic3—42no—
45C2×C3×Q16—91no—
46D3×Q16—61no—
47C2×C3×C16—91no—
48C3×#107—91no—
49D3×C16—61no—
50C2×#157—41no—
51C32×Q8—62no—
52C4×C3⋊D3C3×D411yescharacteristic
53C3×C4×D3—61no—
54C3×C4×D3—61no—
55C3×C4×D3—122no—
56C3×Dic6—22no—
57C3×Dic6—22no—
58C3×Dic6—42no—
59C3×Dic6—122no—
60#627—41no—
61#624—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#358—91noSylow
71C2×C33#10211yescharacteristic
72C3×C3⋊D3SD3211yescharacteristic
73C3×C3⋊D3Q3211yescharacteristic
74C2×C3×Q8—181no—
75D3×Q8—62no—
76C3×Q16—61no—
77C3×Q16—61no—
78C3×Q16—92no—
79C3×Q16—121no—
80C2×C3×C8—91no—
81C3×C4⋊C4—181no—
82#216—62no—
83Dic12—21no—
84D3×C8—61no—
85D3×C8—61no—
86C3×C16—61no—
87C3×C16—61no—
88C3×C16—121no—
89C3⋊C16—21no—
90C2×C3⋊D3C3×D811yescharacteristic
91C2×C3×D3—61no—
92C2×C3×D3—61no—
93#157—81no—
94C32×C4—21no—
95C32×C4—21no—
96C32×C4—41no—
97C32×C4—41no—
98C32×C4—122no—
99C32×C4C3×C22⋊C411yescharacteristic
100C3⋊Dic3C3×SD1611yescharacteristic
101C3×Dic3—42no—
102C3×Dic3—42no—
103C3×Dic3—61no—
104C3×Dic3—61no—
105C3×Dic3—82no—
106C2×Q16—91no—
107C2×C16—91no—
108#107—91no—
109C33#35811yescharacteristic, complemented, Sylow
110C3×Q8—62no—
111C3×Q8—62no—
112C3×Q8—122no—
113C3×Q8—361no—
114C2×C3×C4—91no—
115C2×C3×C4—361no—
116C2×C3×C4—361no—
117C4×D3—61no—
118C4×D3—61no—
119C4×D3—122no—
120Dic6—22no—
121Dic6—122no—
122C3×C8—21no—
123C3×C8—21no—
124C3×C8—41no—
125C3×C8—41no—
126C3×C8—91no—
127C3×C8GO+(4,𝔽2)11yescharacteristic
128C3⋊C8—61no—
129C3⋊C8—61no—
130C2×C32—21no—
131C2×C32—21no—
132C2×C32—41no—
133C2×C32—41no—
134C2×C32C3×#10211yescharacteristic
135C3⋊D3C3×SD3211yescharacteristic
136C3⋊D3C3×Q3211yescharacteristic
137C3×D3—61no—
138C3×D3—61no—
139C3×D3—61no—
140C3×D3—61no—
141C2×Q8—181no—
142Q16—61no—
143Q16—92no—
144C2×C8—91no—
145C4⋊C4—181no—
146C16—61no—
147C22×C3—91no—
148C2×D3—61no—
149C2×D3—61no—
150C3×C4—21no—
151C3×C4—21no—
152C3×C4—41no—
153C3×C4—41no—
154C3×C4—91no—
155C3×C4—122no—
156C3×C4—122no—
157C3×C4—242no—
158C3×C4—721no—
159C3×C4D32⋊C411yescharacteristic
160Dic3—42no—
161Dic3—61no—
162Dic3—61no—
163C32—21no—
164C32—21no—
165C32—41no—
166C32—41no—
167C32C3×#35811yescharacteristic, complemented
168Q8—62no—
169Q8—361no—
170C2×C4—91no—
171C2×C4—361no—
172C2×C4—361no—
173C8—91no—
174C8C3×GO+(4,𝔽2)11yescharacteristic
175C2×C3—21no—
176C2×C3—21no—
177C2×C3—41no—
178C2×C3—41no—
179C2×C3—91no—
180C2×C3—91no—
181C2×C3#6372911yescharacteristic
182D3—61no—
183D3—61no—
184D3—61no—
185D3—61no—
186C22—91no—
187C4—91no—
188C4—122no—
189C4—721no—
190C4C3×D32⋊C411yescharacteristic
191C3—21no—
192C3—21no—
193C3—41no—
194C3—41no—
195C3#1059001711yescharacteristic, direct product factor
196C2—91no—
197C2—91no—
198C2C3×#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⋊D3D8
cosocleC8×C3⋊D3C22×C3
hypercenterC3×C8GO+(4,𝔽2)
hypocenterC32C3×#358
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. #10590017C3direct (N × H)
9. C3×#3718C4semidirect
35. #3718C3×C4semidirect
109. C33#358semidirect
167. C32C3×#358semidirect
195. C3#10590017direct (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×#637692
C3×#635422
#105900173
C3×#3589