Finite Group Databasebeta

← #13167460

Subgroups — #13167460

Order 1728.

372 conjugacy classes of subgroups; only the 47 normal subgroups are shown.

normal subgroups only

This group has 372 conjugacy classes of subgroups — too many to draw. Only its 47 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#11723660C212yesmaximal, complemented
2C2×C3×#3708C211yesmaximal, characteristic, complemented
3#11722790C211yesmaximal, characteristic, complemented
4#11722789C212yesmaximal, complemented
5#11722788C211yesmaximal, characteristic, complemented
6#10593333—31nomaximal
7C3×#3828—22no—
8C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
9#88812C2212yes—
10#88811—22no—
11#88810C2212yes—
12C3×#3708C2211yescharacteristic
13C3×#3708C2211yescharacteristic
14#63829—62no—
15#63822—32no—
16C2×#3708D311yescharacteristic, complemented
17#63736—31no—
18#63735—32no—
19#63734—31no—
20C2×C33×C4D411yescharacteristic
21C3×#638—42no—
22C3×C4⋊(C3⋊D3)—22no—
23C3×C4×C3⋊D3—42no—
24C3×#634—22no—
25C3×C3⋊(C3⋊C8)D411yescharacteristic
26C3×C3⋊(C3⋊C8)C2311yescharacteristic
27#4904—91nomaximal
28#3828—22no—
29C3×#235—62no—
30C3×#235—62no—
31C3×#235—122no—
32C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
33C2×C3×C3⋊C8—62no—
34#3719—62no—
35#3718—32no—
36#3717—62no—
37#3716—62no—
38#3715—62no—
39#3714—62no—
40#3713—32no—
41#3708C2×D311yescharacteristic
42#3708C2×D311yescharacteristic
43C22×C33D811yescharacteristic
44C2×C3×C3⋊D3—42no—
45C33×C4D811yescharacteristic
46C33×C4C2×D411yescharacteristic
47C3×C3⋊Dic3—42no—
48#950—182no—
49#904—92no—
50C2×C3×C16—91no—
51Dic24—91no—
52#800—92no—
53D48—91no—
54C2×C32×C4—22no—
55C2×C32×C4—42no—
56C2×C32×C4D1211yescharacteristic
57#638—42no—
58C4⋊(C3⋊D3)—22no—
59C4×C3⋊D3—42no—
60#634—22no—
61C3×#67—122no—
62C3×#67—124no—
63C3×#67—122no—
64C3×D12—62no—
65C3×D12—62no—
66C3×D12—122no—
67C3×C4×D3—122no—
68C3×C4×D3—124no—
69C3×C4×D3—122no—
70C3×Dic6—62no—
71C3×Dic6—62no—
72C3×Dic6—122no—
73C3⋊(C3⋊C8)D1211yescharacteristic
74C3⋊(C3⋊C8)C22×D311yescharacteristic
75C3×C3⋊C8—62no—
76C3×C3⋊C8—62no—
77#508—271noSylow
78C2×C33—21no—
79C2×C33C2×D811yescharacteristic
80C3×C3⋊D3—82no—
81C3×#41—182no—
82#235—62no—
83#235—62no—
84#235—122no—
85C2×C3×C8—91no—
86#216—182no—
87#215—182no—
88#214—182no—
89#213—182no—
90C2×C3⋊C8—62no—
91Dic12—92no—
92D24—92no—
93#204—182no—
94C3×C16—91no—
95C3×C16—91no—
96C22×C32—22no—
97C22×C32—42no—
98C22×C32D2411yescharacteristic
99C2×C3⋊D3—42no—
100C2×C3×D3—122no—
101C2×C3×D3—124no—
102C2×C3×D3—122no—
103C32×C4—22no—
104C32×C4—22no—
105C32×C4—42no—
106C32×C4—42no—
107C32×C4D2411yescharacteristic
108C32×C4C2×D1211yescharacteristic
109C3⋊Dic3—42no—
110C3×Dic3—122no—
111C3×Dic3—124no—
112C3×Dic3—122no—
113#135—272no—
114Q32—271no—
115SD32—272no—
116D16—271no—
117C2×C16—91no—
118C33#50811yescharacteristic, complemented, Sylow
119C3×Q8—182no—
120C3×D4—182no—
121C3×D4—362no—
122C2×C3×C4—22no—
123C2×C3×C4—42no—
124C2×C3×C4—362no—
125C2×C3×C4GO+(4,𝔽2)11yescharacteristic
126#67—122no—
127#67—124no—
128#67—122no—
129D12—62no—
130D12—62no—
131D12—122no—
132C4×D3—122no—
133C4×D3—124no—
134C4×D3—122no—
135Dic6—62no—
136Dic6—62no—
137Dic6—122no—
138C3×C8—91no—
139C3×C8—91no—
140C3⋊C8—62no—
141C3⋊C8—62no—
142C2×C32—22no—
143C2×C32—21no—
144C2×C32—42no—
145C2×C32—42no—
146C2×C32—82no—
147C2×C32C2×D2411yescharacteristic
148C3⋊D3—82no—
149C3×D3—242no—
150C3×D3—244no—
151C3×D3—242no—
152#41—182no—
153Q16—272no—
154SD16—542no—
155D8—272no—
156C2×C8—91no—
157C16—91no—
158C16—91no—
159C22×C3—22no—
160C22×C3—42no—
161C22×C3—362no—
162C22×C3#377411yescharacteristic
163C2×D3—122no—
164C2×D3—124no—
165C2×D3—122no—
166C3×C4—22no—
167C3×C4—22no—
168C3×C4—42no—
169C3×C4—42no—
170C3×C4—362no—
171C3×C4#377411yescharacteristic
172C3×C4C2×GO+(4,𝔽2)11yescharacteristic
173Dic3—122no—
174Dic3—124no—
175Dic3—122no—
176C32—22no—
177C32—42no—
178C32#490411yescharacteristic, complemented
179Q8—182no—
180D4—182no—
181D4—362no—
182C2×C4—362no—
183C2×C4#157⋊D311yescharacteristic
184C8—91no—
185C8—91no—
186C2×C3—21no—
187C2×C3—22no—
188C2×C3—42no—
189C2×C3—42no—
190C2×C3—82no—
191C2×C3—722no—
192C2×C3C2×#377411yescharacteristic
193D3—242no—
194D3—244no—
195D3—242no—
196C22—362no—
197C22#8894111yescharacteristic
198C4—362no—
199C4#8894111yescharacteristic
200C4C2×#157⋊D311yescharacteristic
201C3—22no—
202C3—42no—
203C3#1059333311yescharacteristic, complemented
204C2—21no—
205C2—722no—
206C2C2×#8894111yescharacteristic
subgroupG/H
centerC4#88941
commutator, cosocleC3×C3⋊(C3⋊C8)C23
FrattiniC4C2×#157⋊D3
FittingC2×C33×C4D4
socleC2×C33C2×D8
hypercenterC2×C4#157⋊D3
hypocenterC33#508
LCSG ▷ C3×C3⋊(C3⋊C8) ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C4 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
1. #11723660C2semidirect
2. C2×C3×#3708C2semidirect
3. #11722790C2semidirect
4. #11722789C2semidirect
5. #11722788C2semidirect
16. C2×#3708D3semidirect
118. C33#508semidirect
178. C32#4904semidirect
203. C3#10593333semidirect

Maximal subgroups

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

subgroupindex
#117236602
C2×C3×#37082
#117227902
#117227892
#117227882
#105933333
#49049