Finite Group Databasebeta

← #13146607

Subgroups — #13146607

Order 1728.

313 conjugacy classes of subgroups; only the 39 normal subgroups are shown.

normal subgroups only

This group has 313 conjugacy classes of subgroups — too many to draw. Only its 39 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×#88627C211yesmaximal, characteristic
2C2×#88589C211yesmaximal, characteristic, complemented
3#11721096C212yesmaximal
4#11720703C211yesmaximal, characteristic, complemented
5#11720702C212yesmaximal, complemented
6C2×C4×#272C2211yescharacteristic
7C2×#6196C2212yes—
8C2×#6188C2211yescharacteristic
9C2×#6188C2211yescharacteristic
10#88627—22no—
11#88589—22no—
12#3750⋊C3—22no—
13#88451—22no—
14#88449C2212yes—
15#63851—62no—
16C22×#272D411yescharacteristic
17C2×C4×H(3,𝔽3)D411yescharacteristic
18C2×#1084C2311yescharacteristic
19C4×#272—21no—
20C4×#272—21no—
21C2×(C3⋊Dic3)⋊C3—22no—
22C2×(C3⋊Dic3)⋊C3—22no—
23#6196—22no—
24#6196—22no—
25#6188—21no—
26#6188D412yes—
27#5137—91nomaximal
28C2×C3×C4×D3—62no—
29C2×#627—62no—
30#3750—22no—
31C3×Dic3⋊C4—62no—
32Dic3⋊Dic3—62no—
33(C2×D3)⋊Dic3—62no—
34(C2×D3)⋊Dic3—62no—
35C22×H(3,𝔽3)C2×D411yescharacteristic
36C2×#272—21no—
37C2×#272—22no—
38C4×H(3,𝔽3)—21no—
39#1084C2×D411yescharacteristic
40#1084#4111yescharacteristic
41(C3⋊Dic3)⋊C3—42no—
42(C3⋊Dic3)⋊C3—42no—
43C2×C3×M16—91no—
44#924—92no—
45C2×Dic12—91no—
46#896—182no—
47#817—91no—
48#816—92no—
49C22×C3×D3—62no—
50C2×C32×C4—22no—
51C2×C3⋊Dic3—22no—
52C2×C3⋊Dic3—22no—
53C2×C3×Dic3—62no—
54C2×C3×Dic3—62no—
55C2×C3×Dic3—62no—
56C3×C4×D3—122no—
57C3×C4×D3—122no—
58#627—122no—
59#627—122no—
60#470—271noSylow
61C2×H(3,𝔽3)#12111yescharacteristic
62C2×H(3,𝔽3)#13712yes—
63#272—41no—
64C22×C3×C4—91no—
65C2×C4×D3—62no—
66C2×Dic6—92no—
67C2×Dic6—182no—
68C3×M16—182no—
69C2×C3×C8—91no—
70C2×C3×C8—91no—
71C3×C4⋊C4—182no—
72#217—182no—
73(C2×D3)⋊C4—182no—
74(C2×D3)⋊C4—182no—
75C4⋊Dic3—92no—
76Dic3⋊C4—62no—
77Dic3⋊C4—182no—
78Dic3⋊C4—182no—
79Dic12—182no—
80C22×C32—22no—
81C2×C3×D3—64no—
82C2×C3×D3—122no—
83C2×C3×D3—122no—
84C32×C4—42no—
85C3⋊Dic3—42no—
86C3⋊Dic3—42no—
87C3×Dic3—62no—
88C3×Dic3—62no—
89C3×Dic3—122no—
90C3×Dic3—122no—
91C2×Q16—271no—
92C2×M16—91no—
93C22⋊Q8—272no—
94#106—271no—
95#103—272no—
96H(3,𝔽3)#47011yescharacteristic, complemented, Sylow
97C23×C3—91no—
98C22×D3—62no—
99C2×C3×C4—62no—
100C2×C3×C4—91no—
101C2×C3×C4—181no—
102C2×C3×C4—181no—
103C2×C3×C4—182no—
104C2×C3×C4—182no—
105C2×C3×C4GO+(4,𝔽2)11yescharacteristic
106C2×Dic3—62no—
107C2×Dic3—62no—
108C2×Dic3—62no—
109C2×Dic3—182no—
110C2×Dic3—182no—
111C4×D3—122no—
112C4×D3—122no—
113Dic6—182no—
114Dic6—182no—
115Dic6—362no—
116Dic6—362no—
117C3×C8—92no—
118C3×C8—181no—
119C2×C32—22no—
120C2×C32—22no—
121C2×C32—22no—
122C3×D3—124no—
123C2×Q8—272no—
124C22×C4—91no—
125Q16—542no—
126M16—182no—
127C2×C8—91no—
128C2×C8—91no—
129C4⋊C4—182no—
130C4⋊C4—272no—
131C22⋊C4—542no—
132C22×C3—62no—
133C22×C3—182no—
134C22×C3—181no—
135C22×C3C2×GO+(4,𝔽2)11yescharacteristic
136C2×D3—64no—
137C2×D3—122no—
138C2×D3—122no—
139C3×C4—21no—
140C3×C4—91no—
141C3×C4—91no—
142C3×C4—122no—
143C3×C4—362no—
144C3×C4—362no—
145Dic3—62no—
146Dic3—62no—
147Dic3—122no—
148Dic3—122no—
149Dic3—362no—
150Dic3—362no—
151C32—22no—
152C23—91no—
153Q8—542no—
154Q8—542no—
155C2×C4—91no—
156C2×C4—181no—
157C2×C4—181no—
158C2×C4—182no—
159C2×C4—182no—
160C2×C4#1088⋊C211yescharacteristic
161C8—92no—
162C8—181no—
163C2×C3—62no—
164C2×C3—62no—
165C2×C3—62no—
166C2×C3—361no—
167C2×C3#6422612yes—
168C2×C3#6424211yescharacteristic
169D3—124no—
170C22—182no—
171C22—181no—
172C22C2×#1088⋊C211yescharacteristic
173C4—21no—
174C4—91no—
175C4—91no—
176C4—362no—
177C4—362no—
178C3—62no—
179C3#1059344911yescharacteristic
180C2—361no—
181C2#1172229412yes—
182C2#1172231611yescharacteristic
subgroupG/H
centerC22C2×#1088⋊C2
commutator, cosocleC2×#1084C23
Frattini, socleC22×C3C2×GO+(4,𝔽2)
FittingC2×C4×H(3,𝔽3)D4
prime radicalC22×#272D4
hypercenterC2×C4#1088⋊C2
hypocenterH(3,𝔽3)#470
LCSG ▷ C2×#1084 ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
2. C2×#88589C2semidirect
4. #11720703C2semidirect
5. #11720702C2semidirect
96. H(3,𝔽3)#470semidirect

Maximal subgroups

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

subgroupindex
C2×#886272
C2×#885892
#117210962
#117207032
#117207022
#51379