Finite Group Databasebeta

← #13136678

Subgroups — #13136678

Order 1728.

226 conjugacy classes of subgroups; only the 21 normal subgroups are shown.

normal subgroups only

This group has 226 conjugacy classes of subgroups — too many to draw. Only its 21 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#11721151C211yesmaximal, characteristic
2#11720780C211yesmaximal, characteristic
3#11720325C211yesmaximal, characteristic
4C2×C4⋊#272C2211yescharacteristic
5C2×#6199C411yescharacteristic, complemented
6C22⋊#1088—21no—
7#3750⋊C3—21no—
8#3748⋊C3—21no—
9#88451—21no—
10#88451C411yescharacteristic, complemented
11#63884—61no—
12#63558—61no—
13C22×#272C2×C411yescharacteristic
14C22×#272D411yescharacteristic
15C2×#1088—41no—
16C2×C4×H(3,𝔽3)D411yescharacteristic
17C4⋊#272—41no—
18C2×(C3⋊Dic3)⋊C3—21no—
19C2×(C3⋊Dic3)⋊C3—21no—
20#6199—41no—
21(C3⋊(C3⋊C8))⋊C3—41no—
22#4472—91nomaximal
23C2×C3×D12—61no—
24C2×C3×D12—61no—
25C2×D3×Dic3—61no—
26#3750—21no—
27#3748—21no—
28C3×Dic3⋊C4—61no—
29C3×#208—61no—
30(C2×D3)⋊Dic3—61no—
31(C2×D3)⋊Dic3—61no—
32(C2×D3)⋊Dic3—61no—
33C22×H(3,𝔽3)C22⋊C411yescharacteristic
34C2×#272—21no—
35C2×#272—21no—
36C2×#272—41no—
37C2×#272—41no—
38#1088—81no—
39C4×H(3,𝔽3)—21no—
40(C3⋊Dic3)⋊C3—41no—
41(C3⋊Dic3)⋊C3—41no—
42#935—91no—
43#894—181no—
44C3×#99—91no—
45#833—91no—
46#823—181no—
47C22×C3×D3—61no—
48C22×C3×D3—61no—
49C22×C3×D3—61no—
50C22×C3×D3—61no—
51C2×C32×C4—21no—
52C2×C32×C4—21no—
53C2×C3⋊Dic3—21no—
54C2×C3⋊Dic3—21no—
55C2×C3×Dic3—61no—
56C2×C3×Dic3—61no—
57C3×D12—121no—
58C3×D12—121no—
59C3×D12—122no—
60D3×Dic3—122no—
61C3⋊(C3⋊C8)—41no—
62C3×C3⋊C8—121no—
63#352—271noSylow
64C2×H(3,𝔽3)—21no—
65C2×H(3,𝔽3)#9911yescharacteristic
66#272—41no—
67#272—41no—
68C2×C3×D4—91no—
69C22×Dic3—91no—
70C2×D12—61no—
71C2×D12—61no—
72C2×C4×D3—181no—
73C3×M16—181no—
74C3×C4⋊C4—181no—
75C3×C22⋊C4—181no—
76#217—91no—
77#217—181no—
78(C2×D3)⋊C4—181no—
79(C2×D3)⋊C4—181no—
80(C2×D3)⋊C4—181no—
81Dic3⋊C4—61no—
82Dic3⋊C4—181no—
83#208—61no—
84#208—181no—
85C22×C32—21no—
86C22×C32—21no—
87C2×C3×D3—62no—
88C2×C3×D3—121no—
89C2×C3×D3—121no—
90C2×C3×D3—121no—
91C2×C3×D3—121no—
92C2×C3×D3—121no—
93C2×C3×D3—121no—
94C2×C3×D3—121no—
95C2×C3×D3—241no—
96C2×C3×D3—241no—
97C32×C4—21no—
98C32×C4—21no—
99C32×C4—41no—
100C3⋊Dic3—41no—
101C3⋊Dic3—41no—
102C3×Dic3—121no—
103C3×Dic3—121no—
104#123—271no—
105#100—271no—
106#99—91no—
107H(3,𝔽3)#35211yescharacteristic, complemented, Sylow
108C23×C3—91no—
109C23×C3—91no—
110C22×D3—61no—
111C22×D3—61no—
112C22×D3—61no—
113C22×D3—61no—
114C3×D4—361no—
115C2×C3×C4—61no—
116C2×C3×C4—61no—
117C2×C3×C4—181no—
118C2×C3×C4—181no—
119C2×C3×C4—361no—
120C2×C3×C4GO+(4,𝔽2)11yescharacteristic
121C2×Dic3—61no—
122C2×Dic3—61no—
123C2×Dic3—181no—
124C2×Dic3—181no—
125C2×Dic3—361no—
126D12—121no—
127D12—121no—
128D12—122no—
129C4×D3—362no—
130C3×C8—361no—
131C3⋊C8—121no—
132C3⋊C8—361no—
133C2×C32—21no—
134C2×C32—21no—
135C2×C32—21no—
136C2×C32—21no—
137C2×C32—41no—
138C3×D3—122no—
139C3×D3—241no—
140C3×D3—241no—
141C3×D3—241no—
142C2×D4—91no—
143C22×C4—271no—
144M16—181no—
145C4⋊C4—181no—
146C22⋊C4—181no—
147C22⋊C4—271no—
148C22⋊C4—541no—
149C22×C3—61no—
150C22×C3—61no—
151C22×C3—181no—
152C22×C3—181no—
153C22×C3—361no—
154C22×C3—361no—
155C22×C3D32⋊C411yescharacteristic
156C2×D3—62no—
157C2×D3—121no—
158C2×D3—121no—
159C2×D3—121no—
160C2×D3—121no—
161C2×D3—121no—
162C2×D3—121no—
163C2×D3—121no—
164C2×D3—241no—
165C2×D3—241no—
166C3×C4—21no—
167C3×C4—61no—
168C3×C4—61no—
169C3×C4—121no—
170C3×C4—361no—
171C3×C4—361no—
172C3×C4—721no—
173Dic3—121no—
174Dic3—121no—
175Dic3—361no—
176Dic3—361no—
177C32—21no—
178C32—21no—
179C23—91no—
180C23—91no—
181D4—361no—
182C2×C4—181no—
183C2×C4—181no—
184C2×C4—361no—
185C2×C4—1081no—
186C2×C4#1088⋊C211yescharacteristic
187C8—361no—
188C2×C3—21no—
189C2×C3—61no—
190C2×C3—61no—
191C2×C3—61no—
192C2×C3—61no—
193C2×C3—121no—
194C2×C3—361no—
195C2×C3—361no—
196C2×C3#6373811yescharacteristic
197D3—122no—
198D3—241no—
199D3—241no—
200D3—241no—
201C22—181no—
202C22—181no—
203C22—361no—
204C22—361no—
205C22(((C3⋊D3)⋊C3)⋊C2)⋊C411yescharacteristic
206C4—21no—
207C4—361no—
208C4—361no—
209C4—721no—
210C3—61no—
211C3—61no—
212C3#1059005811yescharacteristic
213C2—21no—
214C2—361no—
215C2—361no—
216C2#1172070111yescharacteristic
subgroupG/H
centerC2#11720701
commutatorC22×#272C2×C4
FrattiniC2×C3×C4GO+(4,𝔽2)
FittingC2×C4×H(3,𝔽3)D4
socleC2×C3#63738
prime radical, cosocleC2×C4⋊#272C22
hypercenterC2×C4#1088⋊C2
hypocenterH(3,𝔽3)#352
LCSG ▷ C22×#272 ▷ C22×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C2 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
5. C2×#6199C4semidirect
10. #88451C4semidirect
107. H(3,𝔽3)#352semidirect

Maximal subgroups

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

subgroupindex
#117211512
#117207802
#117203252
#44729