Finite Group Databasebeta

← #13161470

Subgroups — #13161470

Order 1728.

414 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 414 conjugacy classes of subgroups — too many to draw. Only its 75 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
1#11723649C212yesmaximal
2C3×#63628C212yesmaximal, complemented
3C3×C16×C3⋊D3C211yesmaximal, characteristic
4#11722069C212yesmaximal, complemented
5#10590240—31nomaximal
6D3×Dic24—32nomaximal
7#89040—22no—
8C3×#3746C2214yes—
9C3×C8×C3⋊D3C2211yescharacteristic
10#88812—22no—
11C33×C16C2211yescharacteristic
12C3×C3⋊(C3⋊C16)C2211yescharacteristic
13#63802—32no—
14D3×Dic12—34no—
15#63628D312yescomplemented
16C16×C3⋊D3D311yescharacteristic
17C3×Dic24—32no—
18C3×Dic24—32no—
19C3×Dic24—62no—
20C3×D3×C16—32no—
21C3×D3×C16—61no—
22#63551—34no—
23#63550—32no—
24C3×C4×C3⋊D3D411yescharacteristic
25C3×#634—24no—
26#6296—42no—
27#6294—42no—
28C33×C8C2311yescharacteristic
29C3×C3⋊(C3⋊C8)D411yescharacteristic
30D3×Q32—92no—
31C2×Dic24—91no—
32#3797—62no—
33D3×Dic6—64no—
34#3746C2×D314yes—
35C8×C3⋊D3C2×D311yescharacteristic
36C3×Dic12—34no—
37C3×Dic12—34no—
38C3×Dic12—64no—
39C3×D3×C8—32no—
40C3×D3×C8—61no—
41#3719—64no—
42#3718—62no—
43C32×C16—21no—
44C32×C16—22no—
45C32×C16—41no—
46C32×C16C2×D312yes—
47C32×C16C2×D311yescharacteristic
48C3⋊(C3⋊C16)C2×D311yescharacteristic
49C3×C3⋊C16—32no—
50C3×C3⋊C16—61no—
51C2×C3×C3⋊D3D811yescharacteristic
52C33×C4C2×D411yescharacteristic
53C3×C3⋊Dic3—44no—
54C3×C3⋊Dic3D811yescharacteristic
55D3×Q16—94no—
56C2×Dic12—92no—
57C3×Q32—92no—
58C2×C3×C16—91no—
59#829—94no—
60Dic24—32no—
61Dic24—32no—
62Dic24—62no—
63Dic24—92no—
64D3×C16—32no—
65D3×C16—61no—
66C4×C3⋊D3D1211yescharacteristic
67#634—24no—
68C3×C4×D3—32no—
69C3×C4×D3—61no—
70C3×Dic6—64no—
71C3×Dic6—64no—
72C3×Dic6—124no—
73#627—124no—
74#627—122no—
75#624—122no—
76D3×Dic3—124no—
77C32×C8—21no—
78C32×C8—22no—
79C32×C8—41no—
80C32×C8C22×D311yescharacteristic
81C32×C8C22×D312yes—
82C3⋊(C3⋊C8)D1211yescharacteristic
83C3×C3⋊C8—32no—
84C3×C3⋊C8—61no—
85C2×Q32—271noSylow
86C2×C33C2×D811yescharacteristic
87C3×C3⋊D3Q3212yescomplemented
88D3×Q8—184no—
89C2×Dic6—182no—
90C3×Q16—94no—
91C2×C3×C8—91no—
92#216—184no—
93Dic12—34no—
94Dic12—34no—
95Dic12—64no—
96Dic12—182no—
97D3×C8—32no—
98D3×C8—61no—
99C3×C16—21no—
100C3×C16—22no—
101C3×C16—41no—
102C3×C16—91no—
103C3×C16D3212yes—
104C3×C16D3211yescharacteristic
105C3⋊C16—32no—
106C3⋊C16—61no—
107C2×C3⋊D3D2411yescharacteristic
108C2×C3×D3—32no—
109C2×C3×D3—61no—
110C32×C4—22no—
111C32×C4—21no—
112C32×C4—41no—
113C32×C4C2×D1211yescharacteristic
114C32×C4D3×D412yes—
115C3⋊Dic3—44no—
116C3⋊Dic3D2411yescharacteristic
117C3×Dic3—32no—
118C3×Dic3—61no—
119C3×Dic3—124no—
120C3×Dic3—124no—
121C3×Dic3—244no—
122C2×Q16—272no—
123Q32—92no—
124Q32—272no—
125C2×C16—91no—
126C33C2×Q3211yescharacteristic, complemented, Sylow
127C3×Q8—184no—
128C2×C3×C4—91no—
129C2×Dic3—362no—
130C4×D3—32no—
131C4×D3—61no—
132C4×D3—364no—
133Dic6—64no—
134Dic6—64no—
135Dic6—124no—
136Dic6—364no—
137Dic6—362no—
138C3×C8—21no—
139C3×C8—22no—
140C3×C8—41no—
141C3×C8—91no—
142C3×C8C2×D3212yes—
143C3×C8C2×D3211yescharacteristic
144C3⋊C8—32no—
145C3⋊C8—61no—
146C2×C32—22no—
147C2×C32—21no—
148C2×C32—41no—
149C2×C32C2×D2411yescharacteristic
150C2×C32D3×D812yes—
151C3⋊D3Dic2412yescomplemented
152C3×D3—34no—
153C3×D3—62no—
154C2×Q8—542no—
155Q16—94no—
156Q16—542no—
157C2×C8—91no—
158C16—91no—
159C16#111311yescharacteristic
160C22×C3—91no—
161C2×D3—32no—
162C2×D3—61no—
163C3×C4—21no—
164C3×C4—22no—
165C3×C4—41no—
166C3×C4—91no—
167C3×C4—364no—
168C3×C4D3×D1212yes—
169C3×C4#380211yescharacteristic
170Dic3—32no—
171Dic3—61no—
172Dic3—124no—
173Dic3—124no—
174Dic3—244no—
175C32—22no—
176C32—21no—
177C32—41no—
178C32C2×Dic2411yescharacteristic, complemented
179C32D3×Q3212yescomplemented
180Q8—184no—
181Q8—1082no—
182C2×C4—91no—
183C2×C4—1082no—
184C8—91no—
185C8C2×#111311yescharacteristic
186C2×C3—21no—
187C2×C3—22no—
188C2×C3—41no—
189C2×C3—92no—
190C2×C3D3×D2412yes—
191C2×C3#6379711yescharacteristic
192D3—34no—
193D3—62no—
194C22—91no—
195C4—91no—
196C4—364no—
197C4C4⋊#111311yescharacteristic
198C3—21no—
199C3—22no—
200C3—41no—
201C3D3×Dic2412yescomplemented
202C3#1059024011yescharacteristic, complemented
203C2—92no—
204C2#1172364711yescharacteristic
subgroupG/H
centerC2#11723647
commutator, cosocleC33×C8C23
FrattiniC8C2×#1113
FittingC33×C16C22
socleC2×C33C2×D8
prime radicalC2×C3×C3⋊D3D8
hypercenterC16#1113
hypocenterC33C2×Q32
LCSG ▷ C33×C8 ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C2 ◁ C4 ◁ C8 ◁ C16

Complements

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

NH ≅ G/Ntype
2. C3×#63628C2semidirect
4. #11722069C2semidirect
15. #63628D3semidirect
87. C3×C3⋊D3Q32semidirect
126. C33C2×Q32semidirect
151. C3⋊D3Dic24semidirect
178. C32C2×Dic24semidirect
179. C32D3×Q32semidirect
201. C3D3×Dic24semidirect
202. C3#10590240semidirect

Maximal subgroups

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

subgroupindex
#117236492
C3×#636282
C3×C16×C3⋊D32
#117220692
#105902403
D3×Dic243