Finite Group Databasebeta

← #13147071

Subgroups — #13147071

Order 1728.

408 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 408 conjugacy classes of subgroups — too many to draw. Only its 62 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×#41⋊H(3,𝔽3)C211yesmaximal, characteristic
2#11720738C212yesmaximal, complemented
3C3×#5424—33nomaximal
4C3×#5233—31nomaximal
5#41⋊H(3,𝔽3)C412yescomplemented
6C2×Q8⋊H(3,𝔽3)C2211yescharacteristic
7C4×#1084—41nomaximal
8C2×C32×#41D311yescharacteristic
9C2×C3×#41⋊C3D313yes—
10C3×#926—31no—
11C22×C3×C3⋊C8—31no—
12C3×#859—36no—
13C3×#835—32no—
14C3×#832—32no—
15C2×C4×H(3,𝔽3)—41no—
16C2×#1084—42no—
17Q8⋊H(3,𝔽3)C2×C411yescharacteristic
18#5424—33no—
19C3×#416—91no—
20#5233—31no—
21C32×#41—32no—
22C32×#41Dic312yescomplemented
23C2×C32×Q8C2×D311yescharacteristic
24C2×C32×D4—31no—
25C22×C32×C4—31no—
26C3×#41⋊C3Dic316yescomplemented
27C2×C3×SL(2,𝔽3)C2×D313yes—
28C3×#217—61no—
29C3×C4⋊Dic3—32no—
30C3×C4×Dic3—61no—
31C3×C4×Dic3—123no—
32C2×C3×C3⋊C8—32no—
33C2×C3×C3⋊C8—62no—
34C22×H(3,𝔽3)—41no—
35C4×H(3,𝔽3)—42no—
36#1084—44no—
37C2×C3×#41—31no—
38C2×C3×#41C3⋊D311yescharacteristic
39C2×#41⋊C3—33no—
40C22×C3×C8—91no—
41C3×#117—91no—
42#926—31no—
43C22×C3⋊C8—31no—
44#859—36no—
45C3×#103—92no—
46C3×#102—92no—
47#835—32no—
48#832—32no—
49C23×C32—31no—
50C32×Q8—31no—
51C32×Q8C2×Dic311yescharacteristic
52C32×D4—32no—
53C32×D4—61no—
54C2×C32×C4—31no—
55C2×C32×C4—32no—
56C2×C32×C4—32no—
57C2×C32×C4—43no—
58C2×C32×C4S411yescharacteristic
59C2×C3×Dic3—62no—
60C2×C3×Dic3—126no—
61C3×SL(2,𝔽3)C2×Dic313yes—
62C3×C3⋊C8—62no—
63#416—91noSylow
64C2×H(3,𝔽3)—42no—
65C2×H(3,𝔽3)—41no—
66C3×#41—32no—
67C3×#41—32no—
68C3×#41—32no—
69C3×#41—62no—
70C3×#41C3⋊Dic312yes—
71C2×C3×Q8—31no—
72C2×C3×Q8C2×C3⋊D311yescharacteristic
73C2×C3×D4—31no—
74C2×C3×D4—31no—
75C2×C3×D4—61no—
76C22×C3×C4—31no—
77C22×C3×C4—31no—
78C22×C3×C4—61no—
79#41⋊C3—36no—
80C2×SL(2,𝔽3)—33no—
81C2×C3×C8—92no—
82C2×C3×C8—182no—
83C3×C4⋊C4—92no—
84C3×C22⋊C4—181no—
85C3×C42—181no—
86#217—61no—
87C4⋊Dic3—32no—
88C4×Dic3—61no—
89C4×Dic3—123no—
90C2×C3⋊C8—32no—
91C2×C3⋊C8—62no—
92C22×C32—32no—
93C22×C32—43no—
94C22×C32—62no—
95C22×C32C2×S411yescharacteristic
96C32×C4—31no—
97C32×C4—31no—
98C32×C4—46no—
99C32×C4SL(2,ℤ/4)12yes—
100C3×Dic3—1212no—
101C3×Dic3—122no—
102C2×#41#27211yescharacteristic
103C22×C8—91no—
104#117—91no—
105#103—92no—
106#102—92no—
107H(3,𝔽3)—41noSylow
108C23×C3—31no—
109C23×C3—31no—
110C23×C3—61no—
111C3×Q8—31no—
112C3×Q8—31no—
113C3×Q8—31no—
114C3×Q8—61no—
115C3×Q8C2×C3⋊Dic311yescharacteristic
116C3×D4—32no—
117C3×D4—32no—
118C3×D4—62no—
119C3×D4—61no—
120C3×D4—61no—
121C3×D4—62no—
122C2×C3×C4—31no—
123C2×C3×C4—32no—
124C2×C3×C4—32no—
125C2×C3×C4—31no—
126C2×C3×C4—32no—
127C2×C3×C4—32no—
128C2×C3×C4—31no—
129C2×C3×C4—62no—
130C2×C3×C4—62no—
131C2×C3×C4—61no—
132C2×C3×C4—123no—
133C2×C3×C4—182no—
134C2×C3×C4C3⋊S411yescharacteristic
135C2×Dic3—62no—
136C2×Dic3—126no—
137SL(2,𝔽3)—33no—
138C3×C8—182no—
139C3⋊C8—62no—
140C2×C32—46no—
141C2×C32—43no—
142C2×C32—62no—
143C2×C32#98512yes—
144C2×C32C2×SL(2,ℤ/4)11yescharacteristic
145#41—32no—
146#41#108412yescomplemented
147C2×Q8C2×#27211yescharacteristic
148C2×D4—31no—
149C22×C4—31no—
150C2×C8—92no—
151C2×C8—182no—
152C4⋊C4—92no—
153C22⋊C4—181no—
154C42—181no—
155C22×C3—32no—
156C22×C3—32no—
157C22×C3—31no—
158C22×C3—62no—
159C22×C3—62no—
160C22×C3—62no—
161C22×C3—122no—
162C22×C3—123no—
163C22×C3C2×C3⋊S411yescharacteristic
164C3×C4—31no—
165C3×C4—31no—
166C3×C4—31no—
167C3×C4—31no—
168C3×C4—32no—
169C3×C4—61no—
170C3×C4—61no—
171C3×C4—126no—
172C3×C4—362no—
173C3×C4C3⋊SL(2,ℤ/4)12yes—
174Dic3—122no—
175Dic3—1212no—
176C32—43no—
177C32#542411yescharacteristic, complemented
178C23—31no—
179Q8—31no—
180Q8C2×#108411yescharacteristic
181D4—32no—
182D4—61no—
183C2×C4—31no—
184C2×C4—32no—
185C2×C4—32no—
186C2×C4—182no—
187C2×C4#625811yescharacteristic
188C8—182no—
189C2×C3—32no—
190C2×C3—31no—
191C2×C3—62no—
192C2×C3—62no—
193C2×C3—122no—
194C2×C3—126no—
195C2×C3—123no—
196C2×C3#6426612yes—
197C2×C3C2×C3⋊SL(2,ℤ/4)11yescharacteristic
198C22—32no—
199C22—62no—
200C22C2×#625811yescharacteristic
201C4—31no—
202C4—31no—
203C4—362no—
204C4#8861312yes—
205C3—31no—
206C3—123no—
207C3#1059365811yescharacteristic
208C2—62no—
209C2#1172239312yes—
210C2C2×#8861311yescharacteristic
subgroupG/H
center, hypercenterC2×C3×C4C3⋊S4
commutator, hypocenterQ8⋊H(3,𝔽3)C2×C4
Frattini, socleC22×C3C2×C3⋊S4
FittingC2×C32×#41D3
prime radicalC2×#41⋊H(3,𝔽3)C2
cosocleC2×Q8⋊H(3,𝔽3)C22
LCSG ▷ Q8⋊H(3,𝔽3)
UCSC1 ◁ C2×C3×C4

Complements

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

NH ≅ G/Ntype
2. #11720738C2semidirect
5. #41⋊H(3,𝔽3)C4semidirect
22. C32×#41Dic3semidirect
26. C3×#41⋊C3Dic3semidirect
146. #41#1084semidirect
177. C32#5424semidirect

Maximal subgroups

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

subgroupindex
C2×#41⋊H(3,𝔽3)2
#117207382
C3×#54243
C3×#52333
C4×#10844