Finite Group Databasebeta

← #13048327

Subgroups — #13048327

Order 1296.

520 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 520 conjugacy classes of subgroups — too many to draw. Only its 67 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×#66432C211yesmaximal, characteristic
2Dic3×#272C211yesmaximal, characteristic, complemented
3#10619883C212yesmaximal, complemented
4C3×#6199C211yesmaximal, characteristic, complemented
5#6289⋊C3C212yesmaximal, complemented
6D3×#624—32nomaximal
7C4×((C3⋊D3)⋊C3)⋊C2—31nomaximal
8C2×C3×#272C2211yescharacteristic
9C2×C3⋊((C3⋊D3)⋊C3)C2212yes—
10#66432C414yescomplemented
11C3⋊#1084C2211yescharacteristic, complemented
12Dic3×H(3,𝔽3)C2211yescharacteristic, complemented
13C3×(C3⋊Dic3)⋊C3C2212yescomplemented
14C2×D3×C3⋊D3—32no—
15#6294—32no—
16#6289—32no—
17#6289D312yescomplemented
18C3×#624—32no—
19C3×D3×Dic3—32no—
20C3×D3×Dic3—32no—
21C3×D3×Dic3—61no—
22C2×((C3⋊D3)⋊C3)⋊C2—31no—
23C4×#272—31no—
24C4×(C3⋊D3)⋊C3—32no—
25#6199D311yescharacteristic, complemented
26((C3⋊D3)⋊C3)⋊C4—32no—
27((C3⋊D3)⋊C3)⋊C4—62no—
28C2×C3×H(3,𝔽3)C2311yescharacteristic
29C3×#272C2×C412yescomplemented
30C3⋊((C3⋊D3)⋊C3)C2×C414yescomplemented
31C2×#624—91no—
32C4×D32—92no—
33C4×D32—92no—
34C2×C3⋊(C3⋊D3)C2×D312yes—
35C2×C3×C3⋊D3—32no—
36C2×C32×D3—32no—
37C2×C32×D3—61no—
38D3×C3⋊D3—38no—
39C3×C3⋊Dic3—32no—
40C3×C3⋊Dic3—61no—
41C3×C3⋊Dic3C2×D312yescomplemented
42C32×Dic3—21no—
43C32×Dic3—32no—
44C32×Dic3C2×D312yescomplemented
45C2×#272C2×D311yescharacteristic
46C2×(C3⋊D3)⋊C3—32no—
47C2×(C3⋊D3)⋊C3—62no—
48((C3⋊D3)⋊C3)⋊C2—34no—
49C4×H(3,𝔽3)—31no—
50#1084—31no—
51#1084—62no—
52#1084—121no—
53(C3⋊Dic3)⋊C3—22no—
54(C3⋊Dic3)⋊C3C2×D312yescomplemented
55C3×H(3,𝔽3)C22×C411yescharacteristic, complemented, Sylow
56C22×C3⋊D3—91no—
57C2×D32—92no—
58C2×D32—92no—
59C2×D32—182no—
60C4×C3⋊D3—32no—
61C4×C3⋊D3—92no—
62C2×C3×Dic3—91no—
63C2×C3×Dic3—91no—
64C3×C4×D3—92no—
65C3×C4×D3—92no—
66C3×C4×D3—92no—
67C3×C4×D3—92no—
68C3×C4×D3—181no—
69#624—32no—
70#624—92no—
71#624—92no—
72#624—92no—
73#624—182no—
74D3×Dic3—32no—
75D3×Dic3—32no—
76D3×Dic3—61no—
77D3×Dic3—92no—
78D3×Dic3—92no—
79C2×C33—21no—
80C2×C33C22×D312yes—
81C3⋊(C3⋊D3)C4×D314yescomplemented
82C3×C3⋊D3—34no—
83C32×D3—34no—
84C32×D3—62no—
85C2×H(3,𝔽3)—22no—
86C2×H(3,𝔽3)—41no—
87C2×H(3,𝔽3)C22×D311yescharacteristic
88#272C4×D312yescomplemented
89(C3⋊D3)⋊C3—34no—
90(C3⋊D3)⋊C3—64no—
91C2×C4×D3—271no—
92C2×C4×D3—271no—
93C2×C4×D3—272no—
94C22×C32—91no—
95C2×C3⋊D3—32no—
96C2×C3⋊D3—32no—
97C2×C3⋊D3—62no—
98C2×C3⋊D3—92no—
99C2×C3⋊D3—94no—
100C2×C3⋊D3—182no—
101C2×C3×D3—32no—
102C2×C3×D3—32no—
103C2×C3×D3—62no—
104C2×C3×D3—61no—
105C2×C3×D3—61no—
106C2×C3×D3—92no—
107C2×C3×D3—92no—
108C2×C3×D3—121no—
109C2×C3×D3—182no—
110D32—98no—
111D32—98no—
112D32—188no—
113C32×C4—32no—
114C32×C4—61no—
115C32×C4—92no—
116C3⋊Dic3—32no—
117C3⋊Dic3—61no—
118C3⋊Dic3D3212yescomplemented
119C3×Dic3—32no—
120C3×Dic3—32no—
121C3×Dic3—32no—
122C3×Dic3—62no—
123C3×Dic3—61no—
124C3×Dic3—92no—
125C3×Dic3—92no—
126C3×Dic3—91no—
127C3×Dic3—181no—
128C3×Dic3—182no—
129C3×Dic3—361no—
130C3×Dic3D3211yescharacteristic
131C33—21no—
132C33C2×C4×D312yescomplemented
133H(3,𝔽3)—22no—
134H(3,𝔽3)—41no—
135H(3,𝔽3)C2×C4×D311yescharacteristic, complemented
136C22×D3—272no—
137C22×D3—271no—
138C22×D3—271no—
139C22×D3—541no—
140C2×C3×C4—271no—
141C2×C3×C4—271no—
142C2×C3×C4—272no—
143C2×Dic3—91no—
144C2×Dic3—91no—
145C2×Dic3—272no—
146C4×D3—92no—
147C4×D3—92no—
148C4×D3—92no—
149C4×D3—92no—
150C4×D3—181no—
151C4×D3—272no—
152C4×D3—272no—
153C4×D3—272no—
154C4×D3—272no—
155C4×D3—542no—
156C2×C32—22no—
157C2×C32—21no—
158C2×C32—32no—
159C2×C32—41no—
160C2×C32—62no—
161C2×C32—61no—
162C2×C32—92no—
163C2×C32—121no—
164C2×C32C2×D3211yescharacteristic
165C2×C32C2×D3212yes—
166C3⋊D3—34no—
167C3⋊D3—34no—
168C3⋊D3—64no—
169C3⋊D3—94no—
170C3⋊D3—184no—
171C3×D3—34no—
172C3×D3—34no—
173C3×D3—64no—
174C3×D3—62no—
175C3×D3—62no—
176C3×D3—94no—
177C3×D3—94no—
178C3×D3—122no—
179C3×D3—184no—
180C22×C4—811noSylow
181C22×C3—91no—
182C22×C3—91no—
183C22×C3—181no—
184C22×C3—272no—
185C2×D3—32no—
186C2×D3—61no—
187C2×D3—92no—
188C2×D3—92no—
189C2×D3—182no—
190C2×D3—272no—
191C2×D3—278no—
192C2×D3—274no—
193C2×D3—274no—
194C2×D3—542no—
195C2×D3—544no—
196C3×C4—31no—
197C3×C4—92no—
198C3×C4—92no—
199C3×C4—92no—
200C3×C4—181no—
201C3×C4—182no—
202C3×C4—271no—
203Dic3—32no—
204Dic3—92no—
205Dic3—92no—
206Dic3—91no—
207Dic3—181no—
208Dic3—182no—
209Dic3—361no—
210Dic3((C3⋊D3)⋊C3)⋊C211yescharacteristic, complemented
211C32—21no—
212C32—22no—
213C32—32no—
214C32—41no—
215C32—62no—
216C32—61no—
217C32—121no—
218C32C4×D3211yescharacteristic
219C32C4×D3212yescomplemented
220C23—811no—
221C2×C4—272no—
222C2×C4—271no—
223C2×C4—271no—
224C2×C4—812no—
225C2×C3—21no—
226C2×C3—32no—
227C2×C3—61no—
228C2×C3—62no—
229C2×C3—92no—
230C2×C3—92no—
231C2×C3—121no—
232C2×C3—182no—
233C2×C3—274no—
234C2×C3C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
235C2×C3D3311yescharacteristic
236D3—34no—
237D3—62no—
238D3—94no—
239D3—94no—
240D3—184no—
241D3—274no—
242D3—544no—
243C22—91no—
244C22—272no—
245C22—814no—
246C4—31no—
247C4—92no—
248C4—271no—
249C3—21no—
250C3—32no—
251C3—61no—
252C3—62no—
253C3—121no—
254C3C4×((C3⋊D3)⋊C3)⋊C211yescharacteristic, complemented
255C3#8894911yescharacteristic
256C2—92no—
257C2—274no—
258C2D3×((C3⋊D3)⋊C3)⋊C211yescharacteristic
subgroupG/H
center, hypercenterC2D3×((C3⋊D3)⋊C3)⋊C2
commutator, hypocenterC3×H(3,𝔽3)C22×C4
FrattiniC2×C3D33
Fitting, cosocleC2×C3×H(3,𝔽3)C23
socleC2×C32C2×D32
prime radicalC2×#66432C2
LCSG ▷ C3×H(3,𝔽3)
UCSC1 ◁ C2

Complements

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

NH ≅ G/Ntype
2. Dic3×#272C2semidirect
3. #10619883C2semidirect
4. C3×#6199C2semidirect
5. #6289⋊C3C2semidirect
10. #66432C4semidirect
11. C3⋊#1084C22semidirect
12. Dic3×H(3,𝔽3)C22semidirect
13. C3×(C3⋊Dic3)⋊C3C22semidirect
17. #6289D3semidirect
25. #6199D3semidirect
29. C3×#272C2×C4semidirect
30. C3⋊((C3⋊D3)⋊C3)C2×C4semidirect
41. C3×C3⋊Dic3C2×D3semidirect
44. C32×Dic3C2×D3semidirect
54. (C3⋊Dic3)⋊C3C2×D3semidirect
55. C3×H(3,𝔽3)C22×C4semidirect
81. C3⋊(C3⋊D3)C4×D3semidirect
88. #272C4×D3semidirect
118. C3⋊Dic3D32semidirect
132. C33C2×C4×D3semidirect
135. H(3,𝔽3)C2×C4×D3semidirect
210. Dic3((C3⋊D3)⋊C3)⋊C2semidirect
219. C32C4×D32semidirect
254. C3C4×((C3⋊D3)⋊C3)⋊C2semidirect

Maximal subgroups

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

subgroupindex
C2×#664322
Dic3×#2722
#106198832
C3×#61992
#6289⋊C32
D3×#6243
C4×((C3⋊D3)⋊C3)⋊C23