Finite Group Databasebeta

← #13147023

Subgroups — #13147023

Order 1728.

392 conjugacy classes of subgroups; only the 43 normal subgroups are shown.

normal subgroups only

This group has 392 conjugacy classes of subgroups — too many to draw. Only its 43 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
1C4⋊#6258C212yesmaximal, complemented
2C8×C22⋊H(3,𝔽3)C211yesmaximal, characteristic, complemented
3C3×#5402—33nomaximal
4C3×#5115—31nomaximal
5C2×#6258—22no—
6C4×C22⋊H(3,𝔽3)C2211yescharacteristic
7#88529—41nomaximal
8C3×C4⋊S4—36no—
9C22×C32×C8D311yescharacteristic, complemented
10C3×#930—32no—
11C2×C3×D24—31no—
12C3×C8×A4D313yescomplemented
13C3×#821—32no—
14C3×#818—31no—
15C2×C22⋊H(3,𝔽3)D411yescharacteristic
16#6258—42no—
17C4⋊#272—42no—
18C8×H(3,𝔽3)—41no—
19#5402—33no—
20C3×#466—91no—
21#5115—31no—
22C2×C3×S4—66no—
23C22×C32×C4C2×D311yescharacteristic
24C2×C3×#67—62no—
25C2×C3×D12—32no—
26C3×C4×A4C2×D313yes—
27C2×C32×C8—31no—
28C2×C32×C8—31no—
29C3×(C2×D3)⋊C4—62no—
30C3×C4⋊Dic3—32no—
31C3×D24—61no—
32C3×D24—61no—
33C3×D24—123no—
34C2×#272—82no—
35C22⋊H(3,𝔽3)D811yescharacteristic, complemented
36C4×H(3,𝔽3)—41no—
37C4⋊S4—36no—
38C2×C3×D8—91no—
39C22×C3×C8—31no—
40C22×C3×C8C3⋊D311yescharacteristic
41C3×#121—92no—
42#930—32no—
43C2×D24—31no—
44C8×A4—33no—
45C3×#107—91no—
46C3×#102—92no—
47#821—32no—
48#818—31no—
49C23×C32D1211yescharacteristic
50C22×C3×D3—62no—
51C2×C3×A4D1213yes—
52C3×S4—126no—
53C2×C32×C4—31no—
54C2×C32×C4—31no—
55C3×#67—122no—
56C3×#67—122no—
57C2×C3×Dic3—62no—
58C3×D12—62no—
59C3×D12—62no—
60C3×D12—126no—
61C32×C8—31no—
62C32×C8—43no—
63C32×C8S411yescharacteristic, complemented
64#466—91noSylow
65C2×H(3,𝔽3)—41no—
66#272—162no—
67C2×S4—66no—
68C2×C3×D4—92no—
69C2×C3×D4—182no—
70C22×C3×C4—31no—
71C22×C3×C4C2×C3⋊D311yescharacteristic
72C2×#67—62no—
73C2×D12—32no—
74C4×A4—33no—
75C3×D8—181no—
76C3×D8—181no—
77C2×C3×C8—31no—
78C2×C3×C8—31no—
79C2×C3×C8—31no—
80C2×C3×C8—31no—
81C2×C3×C8—61no—
82C2×C3×C8—61no—
83C3×C4⋊C4—92no—
84C3×C22⋊C4—182no—
85(C2×D3)⋊C4—62no—
86C4⋊Dic3—32no—
87D24—61no—
88D24—61no—
89D24—123no—
90C22×C32—31no—
91C22×C32—31no—
92C22×C32D2411yescharacteristic, complemented
93C2×C3×D3—122no—
94C2×C3×D3—122no—
95C2×C3×D3—122no—
96C2×C3×D3—246no—
97C3×A4D2413yescomplemented
98C32×C4—31no—
99C32×C4—43no—
100C32×C4C2×S411yescharacteristic
101C3×Dic3—122no—
102C2×D8—91no—
103C22×C8#27211yescharacteristic, complemented
104#121—92no—
105#107—91no—
106#102—92no—
107H(3,𝔽3)—41noSylow
108C23×C3—31no—
109C23×C3—182no—
110C23×C3C4⋊(C3⋊D3)11yescharacteristic
111C22×D3—62no—
112C2×A4—33no—
113S4—126no—
114C3×D4—182no—
115C3×D4—182no—
116C3×D4—362no—
117C3×D4—362no—
118C2×C3×C4—31no—
119C2×C3×C4—31no—
120C2×C3×C4—31no—
121C2×C3×C4—31no—
122C2×C3×C4—61no—
123C2×C3×C4—61no—
124C2×C3×C4—182no—
125#67—122no—
126#67—122no—
127C2×Dic3—62no—
128D12—62no—
129D12—62no—
130D12—126no—
131C3×C8—31no—
132C3×C8—31no—
133C3×C8—31no—
134C3×C8—61no—
135C3×C8—123no—
136C3×C8C3⋊S411yescharacteristic
137C2×C32—31no—
138C2×C32—31no—
139C2×C32—43no—
140C2×C32C4⋊S411yescharacteristic
141C3×D3—242no—
142C3×D3—486no—
143C2×D4—92no—
144C2×D4—182no—
145C22×C4C2×#27211yescharacteristic
146D8—181no—
147D8—181no—
148C2×C8—31no—
149C2×C8—31no—
150C4⋊C4—92no—
151C22⋊C4—182no—
152C22×C3—31no—
153C22×C3—31no—
154C22×C3—31no—
155C22×C3—31no—
156C22×C3—31no—
157C22×C3—61no—
158C22×C3—61no—
159C22×C3—362no—
160C22×C3—362no—
161C22×C3—362no—
162C22×C3#374511yescharacteristic
163C2×D3—122no—
164C2×D3—122no—
165C2×D3—122no—
166C2×D3—246no—
167A4—33no—
168C3×C4—31no—
169C3×C4—31no—
170C3×C4—31no—
171C3×C4—61no—
172C3×C4—123no—
173C3×C4—362no—
174C3×C4C2×C3⋊S411yescharacteristic
175Dic3—122no—
176C32—43no—
177C32#540211yescharacteristic, complemented
178C23—182no—
179C23C4⋊#27211yescharacteristic
180D4—182no—
181D4—182no—
182D4—362no—
183D4—362no—
184C2×C4—31no—
185C2×C4—31no—
186C2×C4—182no—
187C8—31no—
188C8#625811yescharacteristic, complemented
189C2×C3—31no—
190C2×C3—31no—
191C2×C3—31no—
192C2×C3—31no—
193C2×C3—31no—
194C2×C3—61no—
195C2×C3—61no—
196C2×C3—123no—
197C2×C3—722no—
198C2×C3C4⋊(C3⋊S4)11yescharacteristic
199D3—242no—
200D3—486no—
201C22—31no—
202C22—31no—
203C22—362no—
204C22—362no—
205C22—362no—
206C22#8852911yescharacteristic, complemented
207C4—31no—
208C4—362no—
209C4C2×#625811yescharacteristic
210C3—31no—
211C3—123no—
212C3#1059363611yescharacteristic
213C2—31no—
214C2—31no—
215C2—722no—
216C2C4⋊#625811yescharacteristic
subgroupG/H
centerC2×C3C4⋊(C3⋊S4)
commutator, cosocleC4×C22⋊H(3,𝔽3)C22
FrattiniC3×C4C2×C3⋊S4
FittingC22×C32×C8D3
socleC23×C3C4⋊(C3⋊D3)
hypercenterC3×C8C3⋊S4
hypocenterC22⋊H(3,𝔽3)D8
LCSG ▷ C4×C22⋊H(3,𝔽3) ▷ C2×C22⋊H(3,𝔽3) ▷ C22⋊H(3,𝔽3)
UCSC1 ◁ C2×C3 ◁ C3×C4 ◁ C3×C8

Complements

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

NH ≅ G/Ntype
1. C4⋊#6258C2semidirect
2. C8×C22⋊H(3,𝔽3)C2semidirect
9. C22×C32×C8D3semidirect
12. C3×C8×A4D3semidirect
35. C22⋊H(3,𝔽3)D8semidirect
63. C32×C8S4semidirect
92. C22×C32D24semidirect
97. C3×A4D24semidirect
103. C22×C8#272semidirect
177. C32#5402semidirect
188. C8#6258semidirect
206. C22#88529semidirect

Maximal subgroups

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

subgroupindex
C4⋊#62582
C8×C22⋊H(3,𝔽3)2
C3×#54023
C3×#51153
#885294