Finite Group Databasebeta

← #10593631⋊C3 #13146834

Subgroups — #10593631⋊C3 #13146834

Order 1728.

345 conjugacy classes of subgroups; only the 25 normal subgroups are shown.

normal subgroups only

This group has 345 conjugacy classes of subgroups — too many to draw. Only its 25 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#127⋊H(3,𝔽3)C211yesmaximal, characteristic, complemented
2(C3⋊#857)⋊C3C212yesmaximal, complemented
3#10593631C311yesmaximal, characteristic, complemented
4C3×#5077—31nomaximal
5C2×(C3⋊S4)⋊C3—41nomaximal
6C32⋊(C42⋊C3)C2211yescharacteristic, complemented
7C3×#1004—31no—
8C32×#127D311yescharacteristic, complemented
9C3×#932—32no—
10C3×#127⋊C3—21no—
11C3×#127⋊C3C2×C311yescharacteristic, complemented
12C3⋊#857C2×C312yescomplemented
13C3×#833—31no—
14C3×#805—32no—
15C2×C22⋊H(3,𝔽3)—41no—
16(C3⋊S4)⋊C3—42no—
17#857⋊C2—91no—
18C3×#453—91no—
19#5077—31no—
20C2×C3⋊S4—41no—
21C2×C32×D4—31no—
22C2×C32×D4—31no—
23C2×C3×#67—61no—
24C2×C3×#67—61no—
25C3×#237—62no—
26C3×D3×D4—62no—
27C3×#235—32no—
28C32×C42C2×D311yescharacteristic, complemented
29C3×#214—62no—
30C3×#213—62no—
31C3×#208—32no—
32C3×C42⋊C3—21no—
33C3×C42⋊C3C22×C311yescharacteristic, complemented
34C2×(C3⋊D3)⋊C3—161no—
35C22⋊H(3,𝔽3)—41no—
36C3×H(5,𝔽2)—91no—
37#1004—31no—
38C3×AGL(1,ℤ/8)—92no—
39C3×#127—31no—
40C3×#127C3×D311yescharacteristic
41#932—32no—
42#127⋊C3—31no—
43#127⋊C3—61no—
44#857—92no—
45C3×#104—92no—
46C3×#100—91no—
47#833—31no—
48#805—32no—
49C23×C32—31no—
50C23×C32S411yescharacteristic
51C22×C3×D3—61no—
52C22×C3×D3—61no—
53C2×C3×A4—41no—
54C2×C3×A4—81no—
55C3⋊S4—42no—
56C32×D4—62no—
57C32×D4—62no—
58C2×C32×C4—31no—
59C3×#67—62no—
60C3×#67—62no—
61C3×#67—62no—
62C2×C3×Dic3—61no—
63C2×C3×Dic3—61no—
64C3×D12—32no—
65C3×C4×D3—62no—
66C3×Dic6—32no—
67C3×C3⋊C8—62no—
68#453—91noSylow
69C2×H(3,𝔽3)—161no—
70(C3⋊D3)⋊C3—162no—
71C2×S4—361no—
72C3×#41—92no—
73C3×#41—182no—
74C2×C3×D4—31no—
75C2×C3×D4—31no—
76C2×C3×D4—31no—
77C2×C3×D4—31no—
78C2×C3×D4—61no—
79C2×C3×D4—61no—
80C2×C3×D4—182no—
81C2×C3×D4—181no—
82C2×C3×D4—181no—
83C2×#67—61no—
84C2×#67—61no—
85#237—62no—
86D3×D4—62no—
87#235—32no—
88C3×SD16—182no—
89C3×D8—182no—
90C3×M16—92no—
91C3×C42—31no—
92C3×C42C2×C3×D311yescharacteristic
93#214—62no—
94#213—62no—
95#208—32no—
96C42⋊C3—31no—
97C42⋊C3—61no—
98C22×C32—61no—
99C22×C32—61no—
100C22×C32—121no—
101C22×C32C2×S411yescharacteristic
102C2×C3⋊D3—161no—
103C2×C3×D3—62no—
104C2×C3×D3—241no—
105C2×C3×D3—241no—
106C3×A4—41no—
107C3×A4—81no—
108C32×C4—32no—
109C3×Dic3—62no—
110H(5,𝔽2)—91no—
111AGL(1,ℤ/8)—92no—
112#127(C3⋊D3)⋊C311yescharacteristic, complemented
113#104—92no—
114#100—91no—
115H(3,𝔽3)—161noSylow
116C23×C3—31no—
117C23×C3—31no—
118C23×C3—31no—
119C23×C3—61no—
120C23×C3—181no—
121C23×C3—181no—
122C23×C3C3×S411yescharacteristic
123C22×D3—61no—
124C22×D3—61no—
125C2×A4—121no—
126C2×A4—241no—
127S4—362no—
128C3×Q8—92no—
129C3×D4—62no—
130C3×D4—62no—
131C3×D4—62no—
132C3×D4—62no—
133C3×D4—92no—
134C3×D4—122no—
135C3×D4—122no—
136C3×D4—182no—
137C3×D4—182no—
138C3×D4—182no—
139C2×C3×C4—31no—
140C2×C3×C4—31no—
141C2×C3×C4—61no—
142C2×C3×C4—182no—
143C2×C3×C4—181no—
144C2×C3×C4—181no—
145#67—62no—
146#67—62no—
147#67—62no—
148C2×Dic3—61no—
149C2×Dic3—61no—
150D12—32no—
151C4×D3—62no—
152Dic6—32no—
153C3×C8—182no—
154C3⋊C8—62no—
155C2×C32—31no—
156C2×C32—41no—
157C2×C32—121no—
158C2×C32—161no—
159C2×C32—321no—
160C3⋊D3—162no—
161C3×D3—122no—
162#41—92no—
163#41—182no—
164C2×D4—31no—
165C2×D4—31no—
166C2×D4—182no—
167C2×D4—181no—
168C2×D4—181no—
169SD16—182no—
170D8—182no—
171M16—92no—
172C42C2×(C3⋊D3)⋊C311yescharacteristic, complemented
173C22×C3—31no—
174C22×C3—61no—
175C22×C3—61no—
176C22×C3—61no—
177C22×C3—61no—
178C22×C3—121no—
179C22×C3—121no—
180C22×C3—121no—
181C22×C3—121no—
182C22×C3—121no—
183C22×C3—121no—
184C22×C3—182no—
185C22×C3—721no—
186C22×C3—721no—
187C22×C3C2×C3×S411yescharacteristic
188C2×D3—62no—
189C2×D3—241no—
190C2×D3—241no—
191C2×D3—1441no—
192A4—121no—
193A4—241no—
194C3×C4—32no—
195C3×C4—32no—
196C3×C4—62no—
197C3×C4—182no—
198Dic3—62no—
199C32—161no—
200C32—321no—
201C32#857⋊C211yescharacteristic, complemented
202C23—31no—
203C23—181no—
204C23—181no—
205C23(C3⋊S4)⋊C311yescharacteristic
206Q8—92no—
207D4—62no—
208D4—62no—
209D4—92no—
210D4—182no—
211D4—182no—
212D4—182no—
213C2×C4—31no—
214C2×C4—182no—
215C2×C4—181no—
216C2×C4—181no—
217C8—182no—
218C2×C3—31no—
219C2×C3—31no—
220C2×C3—41no—
221C2×C3—61no—
222C2×C3—121no—
223C2×C3—121no—
224C2×C3—121no—
225C2×C3—241no—
226C2×C3—362no—
227C2×C3—481no—
228C2×C3—961no—
229D3—122no—
230D3—1442no—
231C22—61no—
232C22—61no—
233C22—121no—
234C22—182no—
235C22—721no—
236C22—721no—
237C22C2×(C3⋊S4)⋊C311yescharacteristic
238C4—32no—
239C4—182no—
240C3—31no—
241C3—481no—
242C3—961no—
243C3C3×#857⋊C211yescharacteristic
244C2—31no—
245C2—41no—
246C2—121no—
247C2—362no—
subgroupG/H
center, hypercenterC1G
commutator, cosocle, hypocenterC3×C42⋊C3C22×C3
Frattini, socleC22×C3C2×C3×S4
FittingC32×#127D3
LCSG ▷ C3×C42⋊C3
UCSC1

Complements

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

NH ≅ G/Ntype
1. #127⋊H(3,𝔽3)C2semidirect
2. (C3⋊#857)⋊C3C2semidirect
3. #10593631C3semidirect
6. C32⋊(C42⋊C3)C22semidirect
8. C32×#127D3semidirect
11. C3×#127⋊C3C2×C3semidirect
12. C3⋊#857C2×C3semidirect
28. C32×C42C2×D3semidirect
33. C3×C42⋊C3C22×C3semidirect
112. #127(C3⋊D3)⋊C3semidirect
172. C42C2×(C3⋊D3)⋊C3semidirect
201. C32#857⋊C2semidirect

Maximal subgroups

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

subgroupindex
#127⋊H(3,𝔽3)2
(C3⋊#857)⋊C32
#105936313
C3×#50773
C2×(C3⋊S4)⋊C34