Finite Group Databasebeta

← C3×#10593396 #13166747

Subgroups — C3×#10593396 #13166747

Order 1728.

496 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 496 conjugacy classes of subgroups — too many to draw. Only its 114 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
1C2×C3×#3787C211yesmaximal, characteristic
2C3×#63900C212yesmaximal
3C3×#63743C214yesmaximal, complemented
4#10593396C311yesmaximal, characteristic, direct product factor
5C2×C3×C4×C3⋊D3C2211yescharacteristic
6C2×C3×#624—22no—
7C2×C3×#622C2212yes—
8C3×#3787C414yescomplemented
9C32×C4⋊Dic3—22no—
10C3×#3724C2214yes—
11C2×#3787C2×C311yescharacteristic
12C3×D3×C4⋊C4—62no—
13#63900C2×C312yes—
14#63743C2×C314yescomplemented
15C22×C3×C3⋊D3D411yescharacteristic
16C2×C33×C4D411yescharacteristic
17C2×C3×C3⋊Dic3C2311yescharacteristic
18C3×C4×C3⋊D3C2×C412yes—
19C3×C4×C3⋊D3Q812yes—
20C2×C32×Dic3—24no—
21C3×#624—44no—
22C3×#622C2×C414yescomplemented
23C2×C3×#107—91nomaximal
24C2×C4×C3⋊D3C22×C311yescharacteristic
25C2×C3×C4×D3—62no—
26C2×C3×C4×D3—64no—
27C2×#624—22no—
28C2×#622C22×C312yes—
29#3787C3×C414yescomplemented
30C32×C4⋊C4—62no—
31C3×C4⋊Dic3—22no—
32C3×C4⋊Dic3—22no—
33C3×C4⋊Dic3—42no—
34C3×Dic3⋊C4—64no—
35#3724C22×C314yes—
36C22×C33C2×D411yescharacteristic
37C2×C3×C3⋊D3C4⋊C412yes—
38C2×C3×C3⋊D3D812yes—
39C2×C3×C3⋊D3Q1612yes—
40C33×C4C4⋊C412yes—
41C3×C3⋊Dic3C22×C411yescharacteristic
42C3×C3⋊Dic3C2×Q811yescharacteristic
43C32×Dic3—44no—
44C22×C3×C8—91no—
45C2×C3×C4⋊C4—92no—
46D3×C4⋊C4—62no—
47C3×#107—94no—
48C22×C3⋊D3C3×D411yescharacteristic
49C22×C3×D3—62no—
50C2×C32×C4—22no—
51C2×C32×C4—42no—
52C2×C32×C4—64no—
53C2×C32×C4C3×D411yescharacteristic
54C2×C3⋊Dic3C23×C311yescharacteristic
55C4×C3⋊D3C2×C3×C412yes—
56C4×C3⋊D3C3×Q812yes—
57C2×C3×Dic3—24no—
58C2×C3×Dic3—24no—
59C2×C3×Dic3—44no—
60C2×C3×Dic3—62no—
61C3×C4×D3—64no—
62C3×C4×D3—64no—
63C3×C4×D3—128no—
64#624—44no—
65#622C2×C3×C414yescomplemented
66C2×#107—91noSylow
67C2×C33C2×C4⋊C411yescharacteristic
68C2×C33C2×D811yescharacteristic
69C2×C33C2×Q1611yescharacteristic
70C3×C3⋊D3#10714yescomplemented
71C22×C3×C4—91no—
72C22×C3×C4—182no—
73C2×C4×D3—62no—
74C2×C4×D3—64no—
75C2×C3×C8—92no—
76C2×C3×C8—94no—
77C3×C4⋊C4—62no—
78C3×C4⋊C4—62no—
79C3×C4⋊C4—94no—
80C3×C4⋊C4—122no—
81C4⋊Dic3—22no—
82Dic3⋊C4—64no—
83C22×C32—22no—
84C22×C32—42no—
85C22×C32C2×C3×D411yescharacteristic
86C2×C3⋊D3C3×C4⋊C412yes—
87C2×C3⋊D3C3×D812yes—
88C2×C3⋊D3C3×Q1612yes—
89C2×C3×D3—64no—
90C2×C3×D3—64no—
91C2×C3×D3—64no—
92C32×C4—24no—
93C32×C4—44no—
94C32×C4—124no—
95C32×C4C3×C4⋊C412yes—
96C3⋊Dic3C22×C3×C411yescharacteristic
97C3⋊Dic3C2×C3×Q811yescharacteristic
98C3×Dic3—44no—
99C3×Dic3—44no—
100C3×Dic3—62no—
101C3×Dic3—62no—
102C3×Dic3—84no—
103C22×C8—91no—
104C2×C4⋊C4—92no—
105#107—94no—
106C33C2×#10711yescharacteristic, complemented, Sylow
107C23×C3—91no—
108C22×D3—62no—
109C2×C3×C4—22no—
110C2×C3×C4—42no—
111C2×C3×C4—64no—
112C2×C3×C4—64no—
113C2×C3×C4—91no—
114C2×C3×C4—92no—
115C2×C3×C4—92no—
116C2×C3×C4—124no—
117C2×C3×C4—364no—
118C2×C3×C4GO+(4,𝔽2)11yescharacteristic
119C2×Dic3—24no—
120C2×Dic3—62no—
121C4×D3—64no—
122C4×D3—64no—
123C4×D3—128no—
124C3×C8—94no—
125C2×C32—22no—
126C2×C32—22no—
127C2×C32—22no—
128C2×C32—42no—
129C2×C32—42no—
130C2×C32—42no—
131C2×C32C2×C3×C4⋊C411yescharacteristic
132C2×C32C2×C3×D811yescharacteristic
133C2×C32C2×C3×Q1611yescharacteristic
134C3⋊D3C3×#10714yescomplemented
135C3×D3—68no—
136C22×C4—91no—
137C22×C4—182no—
138C2×C8—92no—
139C2×C8—94no—
140C4⋊C4—62no—
141C4⋊C4—94no—
142C22×C3—22no—
143C22×C3—42no—
144C22×C3—92no—
145C22×C3—92no—
146C22×C3—92no—
147C22×C3C2×GO+(4,𝔽2)11yescharacteristic
148C2×D3—64no—
149C2×D3—64no—
150C2×D3—64no—
151C3×C4—24no—
152C3×C4—44no—
153C3×C4—91no—
154C3×C4—91no—
155C3×C4—124no—
156C3×C4—124no—
157C3×C4—244no—
158C3×C4#377312yes—
159Dic3—44no—
160Dic3—62no—
161Dic3—62no—
162C32—22no—
163C32—42no—
164C32C2×C3×#10711yescharacteristic, complemented
165C23—91no—
166C2×C4—64no—
167C2×C4—91no—
168C2×C4—92no—
169C2×C4—92no—
170C2×C4—364no—
171C2×C4C3×GO+(4,𝔽2)11yescharacteristic
172C8—94no—
173C2×C3—22no—
174C2×C3—22no—
175C2×C3—22no—
176C2×C3—42no—
177C2×C3—42no—
178C2×C3—42no—
179C2×C3—94no—
180C2×C3#6422511yescharacteristic
181C2×C3#6422811yescharacteristic
182C2×C3C2×#377311yescharacteristic
183D3—68no—
184C22—92no—
185C22—92no—
186C22—92no—
187C22C2×C3×GO+(4,𝔽2)11yescharacteristic
188C4—91no—
189C4—91no—
190C4—124no—
191C4C3×#377312yes—
192C3—22no—
193C3—42no—
194C3#1059339611yescharacteristic, direct product factor
195C2—94no—
196C2C3×#6422511yescharacteristic
197C2C3×#6422811yescharacteristic
198C2C2×C3×#377311yescharacteristic
subgroupG/H
centerC22×C3C2×GO+(4,𝔽2)
commutatorC3⋊Dic3C22×C3×C4
FrattiniC22C2×C3×GO+(4,𝔽2)
FittingC2×C33×C4D4
socleC22×C33C2×D4
prime radicalC22×C3×C3⋊D3D4
cosocleC2×C3⋊Dic3C23×C3
hypercenterC2×C3×C4GO+(4,𝔽2)
hypocenterC32C2×C3×#107
LCSG ▷ C3⋊Dic3 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C2×C3×C4

Complements

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

NH ≅ G/Ntype
3. C3×#63743C2semidirect
4. #10593396C3direct (N × H)
8. C3×#3787C4semidirect
14. #63743C2×C3semidirect
22. C3×#622C2×C4semidirect
29. #3787C3×C4semidirect
65. #622C2×C3×C4semidirect
70. C3×C3⋊D3#107semidirect
106. C33C2×#107semidirect
134. C3⋊D3C3×#107semidirect
164. C32C2×C3×#107semidirect
194. C3#10593396direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#37872
C3×#639002
C3×#637432
#105933963
C2×C3×#1079