Finite Group Databasebeta

← #13136943

Subgroups — #13136943

Order 1728.

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

normal subgroups only

This group has 323 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#11721848C211yesmaximal, characteristic
2#11720787C211yesmaximal, characteristic, complemented
3#11720784C211yesmaximal, characteristic, complemented
4C22×#1088—21no—
5C2×D4×H(3,𝔽3)C411yescharacteristic, complemented
6C2×#6235—21no—
7C2×C4×#272C2211yescharacteristic
8C2×#6188—21no—
9#88552—21no—
10C4⋊#1084C411yescharacteristic, complemented
11C3×#938—62no—
12C23×H(3,𝔽3)—21no—
13C22×#272D411yescharacteristic
14C2×#1088—22no—
15C2×#1088—41no—
16C2×#1088—41no—
17D4×H(3,𝔽3)—41no—
18C2×C4×H(3,𝔽3)C2×C411yescharacteristic
19#6235—41no—
20#6235—41no—
21C2×#1084—21no—
22C2×#1084D411yescharacteristic
23C4×#272—41no—
24#6188—41no—
25C3×#327—91nomaximal
26C2×C32×D4—22no—
27C2×C3×#67—64no—
28C2×C3×C4×D3—62no—
29C3×#217—64no—
30C3×C4⋊Dic3—62no—
31C22×H(3,𝔽3)—41no—
32C22×H(3,𝔽3)—41no—
33C22×H(3,𝔽3)—41no—
34C22×H(3,𝔽3)C22⋊C411yescharacteristic
35C2×#272—21no—
36C2×#272—21no—
37C2×#272D811yescharacteristic
38C2×#272SD1611yescharacteristic
39#1088—42no—
40C4×H(3,𝔽3)—21no—
41#1084—21no—
42#1084—41no—
43C3×#121—91no—
44#938—62no—
45C3×C22⋊C8—91no—
46C3×#95—91no—
47C23×C32—24no—
48C22×C3×D3—62no—
49C32×D4—44no—
50C2×C32×C4—22no—
51C3×#67—124no—
52C3×#67—124no—
53C2×C3×Dic3—62no—
54C2×C3×Dic3—64no—
55C3×C4×D3—124no—
56#327—91noSylow
57C2×H(3,𝔽3)—81no—
58C2×H(3,𝔽3)#9911yescharacteristic
59C2×H(3,𝔽3)#10211yescharacteristic
60C2×H(3,𝔽3)#10411yescharacteristic
61#272—21no—
62#272—21no—
63C2×C3×D4—62no—
64C2×C3×D4—181no—
65C2×C3×D4#15711yescharacteristic
66C22×C3×C4—91no—
67C22×C3×C4—181no—
68C2×#67—64no—
69C2×C4×D3—62no—
70C2×C3×C8—181no—
71C3×C4⋊C4—91no—
72C3×C22⋊C4—181no—
73#217—64no—
74C4⋊Dic3—62no—
75C22×C32—22no—
76C22×C32—44no—
77C22×C32—44no—
78C22×C32—44no—
79C2×C3×D3—62no—
80C2×C3×D3—62no—
81C2×C3×D3—122no—
82C2×C3×D3—122no—
83C32×C4—24no—
84C3×Dic3—122no—
85C3×Dic3—124no—
86#121—91no—
87C22⋊C8—91no—
88#95—91no—
89H(3,𝔽3)#32711yescharacteristic, complemented, Sylow
90C23×C3—21no—
91C23×C3—64no—
92C23×C3—91no—
93C22×D3—62no—
94C3×D4—41no—
95C3×D4—124no—
96C3×D4—361no—
97C3×D4—361no—
98C2×C3×C4—62no—
99C2×C3×C4—91no—
100C2×C3×C4—182no—
101C2×C3×C4—181no—
102C2×C3×C4—361no—
103C2×C3×C4—361no—
104C2×C3×C4—361no—
105C2×C3×C4C2×#15711yescharacteristic
106#67—124no—
107#67—124no—
108C2×Dic3—62no—
109C2×Dic3—64no—
110C4×D3—124no—
111C3×C8—361no—
112C2×C32—22no—
113C2×C32—22no—
114C2×C32—22no—
115C2×C32—84no—
116C3×D3—122no—
117C3×D3—122no—
118C2×D4—181no—
119C2×D4#108811yescharacteristic, complemented
120C22×C4—91no—
121C22×C4—181no—
122C2×C8—181no—
123C4⋊C4—91no—
124C22⋊C4—181no—
125C22×C3—41no—
126C22×C3—41no—
127C22×C3—41no—
128C22×C3—62no—
129C22×C3—91no—
130C22×C3—91no—
131C22×C3—124no—
132C22×C3—124no—
133C22×C3—124no—
134C22×C3—181no—
135C22×C3—181no—
136C22×C3C22⋊#15711yescharacteristic
137C2×D3—62no—
138C2×D3—62no—
139C2×D3—122no—
140C2×D3—122no—
141C3×C4—21no—
142C3×C4—64no—
143C3×C4—181no—
144C3×C4—362no—
145C3×C4—361no—
146Dic3—122no—
147Dic3—124no—
148C32—22no—
149C23—21no—
150C23—91no—
151D4—41no—
152D4—361no—
153D4—361no—
154C2×C4—91no—
155C2×C4—182no—
156C2×C4—181no—
157C2×C4—361no—
158C2×C4—361no—
159C2×C4—361no—
160C2×C4C2×#108811yescharacteristic
161C8—361no—
162C2×C3—62no—
163C2×C3—62no—
164C2×C3—62no—
165C2×C3—81no—
166C2×C3—181no—
167C2×C3—181no—
168C2×C3—244no—
169C2×C3#6378211yescharacteristic
170C2×C3#6378311yescharacteristic
171C2×C3#6378611yescharacteristic
172D3—122no—
173D3—122no—
174C22—41no—
175C22—41no—
176C22—41no—
177C22—91no—
178C22—91no—
179C22—181no—
180C22—181no—
181C22C22⋊#108811yescharacteristic
182C4—21no—
183C4—181no—
184C4—362no—
185C4—361no—
186C3—62no—
187C3#1059020911yescharacteristic
188C2—81no—
189C2—181no—
190C2—181no—
191C2#1172078811yescharacteristic
192C2#1172078911yescharacteristic
193C2#1172079211yescharacteristic
subgroupG/H
center, socleC22×C3C22⋊#157
commutatorC2×C4×H(3,𝔽3)C2×C4
FrattiniC2×C3×C4C2×#157
FittingC2×D4×H(3,𝔽3)C4
prime radical#11721848C2
cosocleC2×C4×#272C22
hypercenterC2×C3×D4#157
hypocenterH(3,𝔽3)#327
LCSG ▷ C2×C4×H(3,𝔽3) ▷ C22×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22×C3 ◁ C2×C3×C4 ◁ C2×C3×D4

Complements

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

NH ≅ G/Ntype
2. #11720787C2semidirect
3. #11720784C2semidirect
5. C2×D4×H(3,𝔽3)C4semidirect
10. C4⋊#1084C4semidirect
89. H(3,𝔽3)#327semidirect
119. C2×D4#1088semidirect

Maximal subgroups

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

subgroupindex
#117218482
#117207872
#117207842
C3×#3279