Finite Group Databasebeta

← #13167581

Subgroups — #13167581

Order 1728.

448 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 448 conjugacy classes of subgroups — too many to draw. Only its 87 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
2#11723715C212yesmaximal
3#11722797C212yesmaximal, complemented
4#11722796C212yesmaximal, complemented
5#10593511—31nomaximal
6C22×C3×C3⋊Dic3C2211yescharacteristic
7C2×C3×#622C414yescomplemented
8C2×C3×#622C2211yescharacteristic
9C2×C3×#622C2211yescharacteristic
10C3×#3757—22no—
11#88815C2214yes—
12#88813—22no—
13C22×#622D311yescharacteristic
14#63957—32no—
15#63955—62no—
16#63743—32no—
17#63742—32no—
18C23×C33D411yescharacteristic
19C2×C3×C3⋊Dic3—24no—
20C2×C3×C3⋊Dic3C2×C412yes—
21C2×C3×C3⋊Dic3D411yescharacteristic
22C2×C3×C3⋊Dic3Q812yes—
23C2×C3×C3⋊Dic3C2311yescharacteristic
24C3×#622C2×C412yes—
25C3×#622C2×C412yes—
26#5106—91nomaximal
27C22×C3⋊Dic3C2×D311yescharacteristic
28C22×C3×Dic3—62no—
29C2×#622Dic314yescomplemented
30C2×#622C2×D311yescharacteristic
31C2×#622C2×D311yescharacteristic
32#3757—22no—
33C3×#217—62no—
34C3×#217—62no—
35C3×#217—122no—
36#3724—34no—
37Dic3⋊Dic3—64no—
38Dic32—64no—
39Dic32—62no—
40C22×C33—22no—
41C22×C33C4⋊C412yes—
42C22×C33C2×D411yescharacteristic
43C3×C3⋊Dic3—44no—
44C3×C3⋊Dic3C4⋊C412yes—
45C3×C3⋊Dic3C22×C411yescharacteristic
46C3×C3⋊Dic3C2×Q811yescharacteristic
47C22×C3×C8—91no—
48#926—92no—
49#877—182no—
50#818—92no—
51#817—92no—
52C23×C32—22no—
53C23×C32—42no—
54C23×C32D1211yescharacteristic
55C2×C3⋊Dic3—24no—
56C2×C3⋊Dic3Dic612yes—
57C2×C3⋊Dic3D1211yescharacteristic
58C2×C3⋊Dic3C2×Dic312yes—
59C2×C3⋊Dic3C22×D311yescharacteristic
60C2×C3×Dic3—62no—
61C2×C3×Dic3—62no—
62C2×C3×Dic3—64no—
63C2×C3×Dic3—64no—
64C2×C3×Dic3—64no—
65C2×C3×Dic3—64no—
66C2×C3×Dic3—124no—
67#622C2×Dic312yes—
68#622C2×Dic312yes—
69#427—271noSylow
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×C22⋊C4—182no—
79#217—62no—
80#217—62no—
81#217—122no—
82C4⋊Dic3—94no—
83Dic3⋊C4—184no—
84C4×Dic3—182no—
85C4×Dic3—184no—
86C22×C32—22no—
87C22×C32—24no—
88C22×C32—22no—
89C22×C32—42no—
90C22×C32—44no—
91C22×C32—44no—
92C22×C32—84no—
93C22×C32C4⋊Dic312yes—
94C22×C32C2×D1211yescharacteristic
95C3⋊Dic3—44no—
96C3⋊Dic3C4⋊Dic312yes—
97C3⋊Dic3C2×Dic611yescharacteristic
98C3⋊Dic3C22×Dic311yescharacteristic
99C3×Dic3—64no—
100C3×Dic3—62no—
101C3×Dic3—62no—
102C3×Dic3—124no—
103C3×Dic3—124no—
104C3×Dic3—244no—
105C22×C8—91no—
106#117—272no—
107#107—272no—
108#106—272no—
109C33#42711yescharacteristic, complemented, Sylow
110C23×C3—22no—
111C23×C3—42no—
112C23×C3GO+(4,𝔽2)11yescharacteristic
113C2×C3×C4—91no—
114C2×C3×C4—91no—
115C2×C3×C4—92no—
116C2×C3×C4—92no—
117C2×C3×C4—184no—
118C2×Dic3—64no—
119C2×Dic3—62no—
120C2×Dic3—64no—
121C2×Dic3—62no—
122C2×Dic3—64no—
123C2×Dic3—64no—
124C2×Dic3—124no—
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×C32#90412yes—
135C2×C32C2×C4⋊Dic311yescharacteristic
136C22×C4—91no—
137C2×C8—91no—
138C2×C8—91no—
139C2×C8—94no—
140C4⋊C4—274no—
141C22⋊C4—182no—
142C42—542no—
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—364no—
156Dic3—64no—
157Dic3—62no—
158Dic3—62no—
159Dic3—124no—
160Dic3—124no—
161Dic3—244no—
162C32—22no—
163C32—42no—
164C32#510611yescharacteristic, complemented
165C23#157⋊D311yescharacteristic
166C2×C4—91no—
167C2×C4—91no—
168C2×C4—92no—
169C2×C4—92no—
170C2×C4—184no—
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—
183C22#8894012yes—
184C22C2×#157⋊D311yescharacteristic
185C4—91no—
186C4—92no—
187C4—91no—
188C4—364no—
189C3—22no—
190C3—42no—
191C3#1059351111yescharacteristic, complemented
192C2—22no—
193C2C2×#8894011yescharacteristic
194C2#1172435112yes—
subgroupG/H
center, FrattiniC22C2×#157⋊D3
commutatorC3×C3⋊Dic3C22×C4
Fitting, prime radicalC23×C33D4
socleC22×C33C2×D4
cosocleC2×C3×C3⋊Dic3C23
hypercenterC23#157⋊D3
hypocenterC33#427
LCSG ▷ C3×C3⋊Dic3 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C23

Complements

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

NH ≅ G/Ntype
3. #11722797C2semidirect
4. #11722796C2semidirect
7. C2×C3×#622C4semidirect
29. C2×#622Dic3semidirect
109. C33#427semidirect
164. C32#5106semidirect
191. C3#10593511semidirect

Maximal subgroups

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

subgroupindex
C22×C3×#6222
#117237152
#117227972
#117227962
#105935113
#51069