Finite Group Databasebeta

← C3×#10593511 #13166874

Subgroups — C3×#10593511 #13166874

Order 1728.

448 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 448 conjugacy classes of subgroups — too many to draw. Only its 98 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
1C22×C3×#622C211yesmaximal, characteristic
2C3×#63957C212yesmaximal
3C3×#63743C212yesmaximal, complemented
4C3×#63742C212yesmaximal, complemented
5#10593511C311yesmaximal, characteristic, direct product factor
6C22×C3×C3⋊Dic3C2211yescharacteristic
7C2×C3×#622C414yescomplemented
8C2×C3×#622C2211yescharacteristic
9C2×C3×#622C2211yescharacteristic
10C32×#217—22no—
11C3×#3724C2214yes—
12C3×Dic32—22no—
13C22×#622C2×C311yescharacteristic
14C3×#877—62no—
15#63957C2×C312yes—
16#63743C2×C312yescomplemented
17#63742C2×C312yescomplemented
18C23×C33D411yescharacteristic
19C2×C3×C3⋊Dic3C2×C412yes—
20C2×C3×C3⋊Dic3D411yescharacteristic
21C2×C3×C3⋊Dic3Q812yes—
22C2×C3×C3⋊Dic3C2311yescharacteristic
23C2×C32×Dic3—24no—
24C3×#622C2×C412yes—
25C3×#622C2×C412yes—
26C3×#427—91nomaximal
27C22×C3⋊Dic3C22×C311yescharacteristic
28C22×C3×Dic3—62no—
29C2×#622C3×C414yescomplemented
30C2×#622C22×C311yescharacteristic
31C2×#622C22×C311yescharacteristic
32C32×C22⋊C4—62no—
33C3×#217—22no—
34C3×#217—22no—
35C3×#217—42no—
36C3×Dic3⋊C4—64no—
37C3×C4×Dic3—64no—
38#3724C22×C314yes—
39Dic32—22no—
40C22×C33—22no—
41C22×C33C4⋊C412yes—
42C22×C33C2×D411yescharacteristic
43C3×C3⋊Dic3C4⋊C412yes—
44C3×C3⋊Dic3C22×C411yescharacteristic
45C3×C3⋊Dic3C2×Q811yescharacteristic
46C32×Dic3—44no—
47C22×C3×C8—91no—
48C3×#117—92no—
49#877—62no—
50C3×#107—92no—
51C3×#106—92no—
52C23×C32—22no—
53C23×C32—42no—
54C23×C32C3×D411yescharacteristic
55C2×C32×C4—64no—
56C2×C3⋊Dic3C2×C3×C412yes—
57C2×C3⋊Dic3C3×D411yescharacteristic
58C2×C3⋊Dic3C3×Q812yes—
59C2×C3⋊Dic3C23×C311yescharacteristic
60C2×C3×Dic3—24no—
61C2×C3×Dic3—24no—
62C2×C3×Dic3—44no—
63C2×C3×Dic3—62no—
64C2×C3×Dic3—62no—
65C2×C3×Dic3—64no—
66C2×C3×Dic3—64no—
67#622C2×C3×C412yes—
68#622C2×C3×C412yes—
69#427—91noSylow
70C2×C33—22no—
71C2×C33C2×C4⋊C411yescharacteristic
72C2×C33#13512yes—
73C22×C3×C4—91no—
74C22×Dic3—62no—
75C2×C3×C8—91no—
76C2×C3×C8—91no—
77C2×C3×C8—94no—
78C3×C4⋊C4—94no—
79C3×C22⋊C4—62no—
80C3×C22⋊C4—62no—
81C3×C22⋊C4—122no—
82C3×C42—182no—
83#217—22no—
84Dic3⋊C4—64no—
85C4×Dic3—64no—
86C22×C32—22no—
87C22×C32—24no—
88C22×C32—22no—
89C22×C32—42no—
90C22×C32—44no—
91C22×C32—44no—
92C22×C32—84no—
93C22×C32C3×C4⋊C412yes—
94C22×C32C2×C3×D411yescharacteristic
95C32×C4—124no—
96C3⋊Dic3C3×C4⋊C412yes—
97C3⋊Dic3C22×C3×C411yescharacteristic
98C3⋊Dic3C2×C3×Q811yescharacteristic
99C3×Dic3—44no—
100C3×Dic3—44no—
101C3×Dic3—64no—
102C3×Dic3—62no—
103C3×Dic3—62no—
104C3×Dic3—84no—
105C22×C8—91no—
106#117—92no—
107#107—92no—
108#106—92no—
109C33#42711yescharacteristic, complemented, Sylow
110C23×C3—22no—
111C23×C3—42no—
112C23×C3GO+(4,𝔽2)11yescharacteristic
113C2×C3×C4—64no—
114C2×C3×C4—64no—
115C2×C3×C4—91no—
116C2×C3×C4—91no—
117C2×C3×C4—92no—
118C2×C3×C4—92no—
119C2×C3×C4—124no—
120C2×Dic3—24no—
121C2×Dic3—64no—
122C2×Dic3—62no—
123C2×Dic3—64no—
124C2×Dic3—62no—
125C3×C8—92no—
126C3×C8—92no—
127C2×C32—22no—
128C2×C32—24no—
129C2×C32—22no—
130C2×C32—42no—
131C2×C32—44no—
132C2×C32—44no—
133C2×C32—84no—
134C2×C32C2×C3×C4⋊C411yescharacteristic
135C2×C32C3×#13512yes—
136C22×C4—91no—
137C2×C8—91no—
138C2×C8—91no—
139C2×C8—94no—
140C4⋊C4—94no—
141C22⋊C4—62no—
142C42—182no—
143C22×C3—22no—
144C22×C3—22no—
145C22×C3—24no—
146C22×C3—44no—
147C22×C3—42no—
148C22×C3—44no—
149C22×C3—84no—
150C22×C3#377312yes—
151C22×C3C2×GO+(4,𝔽2)11yescharacteristic
152C3×C4—91no—
153C3×C4—92no—
154C3×C4—91no—
155C3×C4—124no—
156C3×C4—124no—
157C3×C4—244no—
158Dic3—44no—
159Dic3—64no—
160Dic3—62no—
161Dic3—62no—
162C32—22no—
163C32—42no—
164C32C3×#42711yescharacteristic, complemented
165C23C3×GO+(4,𝔽2)11yescharacteristic
166C2×C4—64no—
167C2×C4—91no—
168C2×C4—91no—
169C2×C4—92no—
170C2×C4—92no—
171C8—92no—
172C8—92no—
173C2×C3—22no—
174C2×C3—22no—
175C2×C3—24no—
176C2×C3—44no—
177C2×C3—42no—
178C2×C3—44no—
179C2×C3—84no—
180C2×C3C2×#377311yescharacteristic
181C2×C3#6423712yes—
182C22—22no—
183C22C3×#377312yes—
184C22C2×C3×GO+(4,𝔽2)11yescharacteristic
185C4—91no—
186C4—92no—
187C4—91no—
188C4—124no—
189C3—22no—
190C3—42no—
191C3#1059351111yescharacteristic, direct product factor
192C2—22no—
193C2C2×C3×#377311yescharacteristic
194C2C3×#6423712yes—
subgroupG/H
centerC22×C3C2×GO+(4,𝔽2)
commutatorC3⋊Dic3C22×C3×C4
FrattiniC22C2×C3×GO+(4,𝔽2)
Fitting, prime radicalC23×C33D4
socleC22×C33C2×D4
cosocleC2×C3⋊Dic3C23×C3
hypercenterC23×C3GO+(4,𝔽2)
hypocenterC32C3×#427
LCSG ▷ C3⋊Dic3 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C23×C3

Complements

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

NH ≅ G/Ntype
3. C3×#63743C2semidirect
4. C3×#63742C2semidirect
5. #10593511C3direct (N × H)
7. C2×C3×#622C4semidirect
16. #63743C2×C3semidirect
17. #63742C2×C3semidirect
29. C2×#622C3×C4semidirect
109. C33#427semidirect
164. C32C3×#427semidirect
191. C3#10593511direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×#6222
C3×#639572
C3×#637432
C3×#637422
#105935113
C3×#4279