Finite Group Databasebeta

← #13181519

Subgroups — #13181519

Order 1728.

920 conjugacy classes of subgroups; only the 221 normal subgroups are shown.

normal subgroups only

This group has 920 conjugacy classes of subgroups — too many to draw. Only its 221 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×#3832C211yesmaximal, characteristic
2#11724517C218yesmaximal, complemented
3#11724511C216yesmaximal, complemented
4#10596644—31nomaximal
5C2×C33×Q8C411yescharacteristic
6C3×#3832C414yes—
7C3×#3832C2214yes—
8C2×C3×C4⋊(C3⋊D3)C413yes—
9C2×C3×C4×C3⋊D3C2213yes—
10#88993C2214yes—
11#88989C22112yescomplemented
12#88988C22112yescomplemented
13C2×#3832D311yescharacteristic
14C2×C3×#239—62no—
15#64289—38no—
16#64283—36no—
17C22×C3×C3⋊D3C2×C413yes—
18C33×Q8C2×C414yes—
19C2×C33×C4C2×C413yes—
20C2×C3×C3⋊Dic3C2311yescharacteristic
21C3×C4⋊(C3⋊D3)C2×C4112yes—
22C3×C4×C3⋊D3C2×C416yes—
23C3×C4×C3⋊D3C2316yes—
24C3⋊#622C2318yes—
25#5819—91nomaximal
26C2×C32×Q8—22no—
27C2×C32×Q8—42no—
28C2×C32×Q8Dic311yescharacteristic
29#3832Dic314yes—
30#3832C2×D314yes—
31C2×C4⋊(C3⋊D3)Dic313yes—
32C2×C4×C3⋊D3C2×D313yes—
33C3×#239—68no—
34C3×#239—68no—
35C2×C3×D12—66no—
36C2×C3×C4×D3—66no—
37#3792—34no—
38#3788—312no—
39#3787—312no—
40C22×C33C22×C411yescharacteristic
41C2×C3×C3⋊D3—43no—
42C2×C3×C3⋊D3C22×C416yes—
43C33×C4C22×C416yes—
44C3×C3⋊Dic3C22×C411yescharacteristic
45C3×C3⋊Dic3C2411yescharacteristic
46C2×C3×#41—91no—
47C2×#239—62no—
48#948—98no—
49C2×#208—96no—
50C22×C3⋊D3C2×Dic313yes—
51C22×C3×D3—66no—
52C32×Q8—28no—
53C32×Q8—48no—
54C32×Q8C2×Dic314yes—
55C2×C32×C4—26no—
56C2×C32×C4—46no—
57C2×C32×C4C2×Dic313yes—
58C2×C3⋊Dic3C22×D311yescharacteristic
59C4⋊(C3⋊D3)C2×Dic3112yes—
60C4×C3⋊D3C2×Dic316yes—
61C4×C3⋊D3C22×D316yes—
62C2×C3×Dic3—62no—
63C3×D12—624no—
64C3×C4×D3—612no—
65C3×C4×D3—612no—
66#622—38no—
67#568—271noSylow
68C2×C33—21no—
69C2×C33C23×C411yescharacteristic
70C3×C3⋊D3—26no—
71C3×#41—94no—
72C3×#41—94no—
73C2×C3×Q8—22no—
74C2×C3×Q8—42no—
75C2×C3×Q8#15711yescharacteristic
76C2×C3×D4—93no—
77C22×C3×C4—93no—
78#239—68no—
79#239—68no—
80C2×D12—66no—
81C2×C4×D3—66no—
82#208—94no—
83#208—912no—
84C2×C3⋊C8—912no—
85C22×C32—22no—
86C22×C32—42no—
87C22×C32C22×Dic311yescharacteristic
88C2×C3⋊D3—43no—
89C2×C3⋊D3C22×Dic316yes—
90C2×C3×D3—612no—
91C2×C3×D3—1212no—
92C32×C4—212no—
93C32×C4—412no—
94C32×C4C22×Dic316yes—
95C3⋊Dic3C22×Dic311yescharacteristic
96C3⋊Dic3C23×D311yescharacteristic
97C3×Dic3—62no—
98C3×Dic3—62no—
99C2×#41—91no—
100#131—278no—
101C2×M16—276no—
102C33#56811yescharacteristic, complemented, Sylow
103C23×C3—93no—
104C22×D3—66no—
105C3×Q8—28no—
106C3×Q8—48no—
107C3×Q8C2×#15714yes—
108C3×D4—912no—
109C2×C3×C4—26no—
110C2×C3×C4—46no—
111C2×C3×C4—91no—
112C2×C3×C4—96no—
113C2×C3×C4—96no—
114C2×C3×C4C2×#15713yes—
115C2×Dic3—62no—
116D12—624no—
117C4×D3—612no—
118C4×D3—612no—
119C3⋊C8—98no—
120C2×C32—21no—
121C2×C32—24no—
122C2×C32—22no—
123C2×C32—44no—
124C2×C32—42no—
125C2×C32C23×Dic311yescharacteristic
126C3⋊D3—26no—
127C3×D3—1212no—
128#41—94no—
129#41—94no—
130C2×Q8C3⋊#15711yescharacteristic
131C2×D4—93no—
132C22×C4—93no—
133M16—274no—
134M16—2712no—
135C2×C8—2712no—
136C22×C3—22no—
137C22×C3—42no—
138C22×C3—96no—
139C22×C3—363no—
140C22×C3C22×#15711yescharacteristic
141C2×D3—612no—
142C2×D3—1212no—
143C3×C4—212no—
144C3×C4—412no—
145C3×C4—91no—
146C3×C4—91no—
147C3×C4C22×#15716yes—
148Dic3—62no—
149Dic3—62no—
150C32—22no—
151C32—42no—
152C32#581911yescharacteristic, complemented
153C23—93no—
154Q8C2×C3⋊#15714yes—
155D4—912no—
156C2×C4—91no—
157C2×C4—96no—
158C2×C4—96no—
159C2×C4C2×C3⋊#15713yes—
160C8—278no—
161C2×C3—24no—
162C2×C3—21no—
163C2×C3—22no—
164C2×C3—44no—
165C2×C3—42no—
166C2×C3—186no—
167C2×C3C23×#15711yescharacteristic
168D3—1212no—
169C22—96no—
170C22—363no—
171C22C22×C3⋊#15711yescharacteristic
172C4—91no—
173C4—91no—
174C4C22×C3⋊#15716yes—
175C3—22no—
176C3—42no—
177C3#1059664411yescharacteristic, complemented
178C2—21no—
179C2—186no—
180C2C23×C3⋊#15711yescharacteristic
subgroupG/H
center, FrattiniC2C23×C3⋊#157
commutator, socleC2×C33C23×C4
FittingC2×C33×Q8C4
prime radicalC2×C3×#3832C2
cosocleC3×C3⋊Dic3C24
hypercenterC2×Q8C3⋊#157
hypocenterC33#568
LCSG ▷ C2×C33 ▷ C33
UCSC1 ◁ C2 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
2. #11724517C2semidirect
3. #11724511C2semidirect
11. #88989C22semidirect
12. #88988C22semidirect
102. C33#568semidirect
152. C32#5819semidirect
177. C3#10596644semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#38322
#117245172
#117245112
#105966443
#58199