Finite Group Databasebeta

← #13136949

Subgroups — #13136949

Order 1728.

268 conjugacy classes of subgroups; only the 31 normal subgroups are shown.

normal subgroups only

This group has 268 conjugacy classes of subgroups — too many to draw. Only its 31 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#11721863C211yesmaximal, characteristic
2#11720787C211yesmaximal, characteristic
3#11720784C211yesmaximal, characteristic
4C22×#1088—21no—
5C2×Q8×H(3,𝔽3)C411yescharacteristic, complemented
6C2×C4×#272C2211yescharacteristic
7C2×#6188—21no—
8#88542—21no—
9C4⋊#1084C411yescharacteristic, complemented
10#88540—21no—
11C3×#946—62no—
12C22×#272D411yescharacteristic
13C2×#1088—22no—
14C2×#1088—41no—
15C2×#1088—41no—
16Q8×H(3,𝔽3)—41no—
17C2×C4×H(3,𝔽3)—21no—
18C2×C4×H(3,𝔽3)C2×C411yescharacteristic
19C2×#1084—21no—
20C2×#1084D411yescharacteristic
21C4×#272—41no—
22#6188—41no—
23C3×#328—91nomaximal
24C2×C32×Q8—22no—
25C2×C3×C4×D3—62no—
26C3×(C2×D3)⋊C4—64no—
27C3×C4⋊Dic3—62no—
28C3×Dic3⋊C4—64no—
29C22×H(3,𝔽3)C22⋊C411yescharacteristic
30C2×#272—21no—
31C2×#272—21no—
32C2×#272SD1611yescharacteristic
33C2×#272Q1611yescharacteristic
34#1088—42no—
35C4×H(3,𝔽3)—21no—
36C4×H(3,𝔽3)—41no—
37#1084—21no—
38#1084—41no—
39C3×C22⋊Q8—91no—
40#946—62no—
41C3×C22⋊C8—91no—
42C3×#95—91no—
43C22×C3×D3—62no—
44C32×Q8—44no—
45C2×C32×C4—24no—
46C2×C32×C4—22no—
47C2×C3×Dic3—62no—
48C2×C3×Dic3—64no—
49C3×C4×D3—124no—
50#328—91noSylow
51C2×H(3,𝔽3)#9911yescharacteristic
52C2×H(3,𝔽3)#10311yescharacteristic
53C2×H(3,𝔽3)#10411yescharacteristic
54#272—21no—
55#272—21no—
56C2×C3×Q8—62no—
57C2×C3×Q8#15711yescharacteristic
58C22×C3×C4—91no—
59C22×C3×C4—181no—
60C2×C4×D3—62no—
61C2×C3×C8—181no—
62C3×C4⋊C4—91no—
63C3×C4⋊C4—181no—
64C3×C22⋊C4—181no—
65(C2×D3)⋊C4—64no—
66C4⋊Dic3—62no—
67Dic3⋊C4—64no—
68C22×C32—22no—
69C2×C3×D3—62no—
70C2×C3×D3—62no—
71C2×C3×D3—122no—
72C2×C3×D3—122no—
73C32×C4—24no—
74C32×C4—44no—
75C3×Dic3—122no—
76C3×Dic3—124no—
77C22⋊Q8—91no—
78C22⋊C8—91no—
79#95—91no—
80H(3,𝔽3)#32811yescharacteristic, complemented, Sylow
81C23×C3—91no—
82C22×D3—62no—
83C3×Q8—41no—
84C3×Q8—124no—
85C2×C3×C4—21no—
86C2×C3×C4—64no—
87C2×C3×C4—62no—
88C2×C3×C4—91no—
89C2×C3×C4—182no—
90C2×C3×C4—181no—
91C2×C3×C4—361no—
92C2×C3×C4—361no—
93C2×C3×C4—361no—
94C2×C3×C4C2×#15711yescharacteristic
95C2×Dic3—62no—
96C2×Dic3—64no—
97C4×D3—124no—
98C3×C8—361no—
99C2×C32—22no—
100C2×C32—22no—
101C2×C32—22no—
102C3×D3—122no—
103C3×D3—122no—
104C2×Q8#108811yescharacteristic, complemented
105C22×C4—91no—
106C22×C4—181no—
107C2×C8—181no—
108C4⋊C4—91no—
109C4⋊C4—181no—
110C22⋊C4—181no—
111C22×C3—62no—
112C22×C3—91no—
113C22×C3—91no—
114C22×C3—181no—
115C22×C3—181no—
116C22×C3C22⋊#15711yescharacteristic
117C2×D3—62no—
118C2×D3—62no—
119C2×D3—122no—
120C2×D3—122no—
121C3×C4—21no—
122C3×C4—41no—
123C3×C4—64no—
124C3×C4—124no—
125C3×C4—181no—
126C3×C4—362no—
127C3×C4—361no—
128Dic3—122no—
129Dic3—124no—
130C32—22no—
131C23—91no—
132Q8—41no—
133C2×C4—21no—
134C2×C4—91no—
135C2×C4—182no—
136C2×C4—181no—
137C2×C4—361no—
138C2×C4—361no—
139C2×C4—361no—
140C2×C4C2×#108811yescharacteristic
141C8—361no—
142C2×C3—62no—
143C2×C3—62no—
144C2×C3—62no—
145C2×C3—181no—
146C2×C3—181no—
147C2×C3#6378411yescharacteristic
148C2×C3#6378511yescharacteristic
149C2×C3#6378611yescharacteristic
150D3—122no—
151D3—122no—
152C22—91no—
153C22—91no—
154C22—181no—
155C22—181no—
156C22C22⋊#108811yescharacteristic
157C4—21no—
158C4—41no—
159C4—181no—
160C4—362no—
161C4—361no—
162C3—62no—
163C3#1059021511yescharacteristic
164C2—181no—
165C2—181no—
166C2#1172079011yescharacteristic
167C2#1172079111yescharacteristic
168C2#1172079211yescharacteristic
subgroupG/H
center, socleC22×C3C22⋊#157
commutatorC2×C4×H(3,𝔽3)C2×C4
FrattiniC2×C3×C4C2×#157
FittingC2×Q8×H(3,𝔽3)C4
prime radicalC22×#272D4
cosocleC2×C4×#272C22
hypercenterC2×C3×Q8#157
hypocenterH(3,𝔽3)#328
LCSG ▷ C2×C4×H(3,𝔽3) ▷ C22×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22×C3 ◁ C2×C3×C4 ◁ C2×C3×Q8

Complements

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

NH ≅ G/Ntype
5. C2×Q8×H(3,𝔽3)C4semidirect
9. C4⋊#1084C4semidirect
80. H(3,𝔽3)#328semidirect
104. C2×Q8#1088semidirect

Maximal subgroups

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

subgroupindex
#117218632
#117207872
#117207842
C3×#3289