Finite Group Databasebeta

← #13048751

Subgroups — #13048751

Order 1296.

505 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 505 conjugacy classes of subgroups — too many to draw. Only its 59 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
1C3×C4×#272C211yesmaximal, characteristic, complemented
2C4×C3⋊((C3⋊D3)⋊C3)C212yesmaximal, complemented
3#10619904C212yesmaximal, complemented
4#10619902C211yesmaximal, characteristic, complemented
5#10619900C211yesmaximal, characteristic, complemented
6#89020—32nomaximal
7#88650—31nomaximal
8C2×C3×#272C2211yescharacteristic
9C2×C3⋊((C3⋊D3)⋊C3)C2212yes—
10C3×C4×H(3,𝔽3)C2211yescharacteristic
11C3×#1084C2211yescharacteristic
12C3⋊((C3⋊Dic3)⋊C3)C2212yes—
13C4×C3⋊(C3⋊D3)D312yescomplemented
14C3×C4×C3⋊D3—32no—
15C32×C4×D3—32no—
16C32×C4×D3—61no—
17#6293—32no—
18#6292—32no—
19#6291—32no—
20#6290—32no—
21C4×#272D311yescharacteristic, complemented
22C4×(C3⋊D3)⋊C3—32no—
23C4×(C3⋊D3)⋊C3—62no—
24#6200—32no—
25#6198—31no—
26#6196—31no—
27C2×C3×H(3,𝔽3)C2311yescharacteristic
28C3×#272—21no—
29C3⋊((C3⋊D3)⋊C3)—22no—
30#3828—91no—
31#3798—92no—
32#3798—96no—
33C2×C3⋊(C3⋊D3)C2×D312yes—
34C2×C3×C3⋊D3—32no—
35C2×C32×D3—32no—
36C2×C32×D3—61no—
37C33×C4—21no—
38C33×C4C2×D312yes—
39C3⋊(C3⋊Dic3)C2×D312yes—
40C3×C3⋊Dic3—32no—
41C32×Dic3—32no—
42C32×Dic3—61no—
43C2×#272C2×D311yescharacteristic
44C2×(C3⋊D3)⋊C3—32no—
45C2×(C3⋊D3)⋊C3—62no—
46C4×H(3,𝔽3)—22no—
47C4×H(3,𝔽3)—41no—
48C4×H(3,𝔽3)C2×D311yescharacteristic
49#1084C2×D311yescharacteristic
50(C3⋊Dic3)⋊C3—32no—
51(C3⋊Dic3)⋊C3—62no—
52C3×H(3,𝔽3)#4111yescharacteristic, complemented, Sylow
53C2×C32×C4—91no—
54#638—92no—
55C4⋊(C3⋊D3)—91no—
56C4×C3⋊D3—32no—
57C4×C3⋊D3—32no—
58C4×C3⋊D3—62no—
59C4×C3⋊D3—96no—
60#634—91no—
61C3×C4×D3—36no—
62C3×C4×D3—32no—
63C3×C4×D3—61no—
64C3×C4×D3—63no—
65C3×C4×D3—96no—
66C3×C4×D3—92no—
67#627—92no—
68#627—96no—
69#626—92no—
70#626—96no—
71#626—92no—
72#626—96no—
73#625—92no—
74#625—96no—
75C2×C33—21no—
76C2×C33C22×D312yes—
77C3⋊(C3⋊D3)—22no—
78C3×C3⋊D3—62no—
79C32×D3—62no—
80C32×D3—62no—
81C2×H(3,𝔽3)—22no—
82C2×H(3,𝔽3)—41no—
83C2×H(3,𝔽3)C22×D311yescharacteristic
84#272—21no—
85(C3⋊D3)⋊C3—64no—
86(C3⋊D3)⋊C3—62no—
87#235—272no—
88#235—273no—
89#235—271no—
90C22×C32—91no—
91C2×C3⋊D3—32no—
92C2×C3⋊D3—32no—
93C2×C3⋊D3—62no—
94C2×C3⋊D3—96no—
95C2×C3×D3—36no—
96C2×C3×D3—32no—
97C2×C3×D3—61no—
98C2×C3×D3—63no—
99C2×C3×D3—96no—
100C2×C3×D3—92no—
101C32×C4—22no—
102C32×C4—21no—
103C32×C4—36no—
104C32×C4—41no—
105C32×C4—63no—
106C32×C4—91no—
107C32×C4D3212yes—
108C32×C4D3211yescharacteristic
109C3⋊Dic3—32no—
110C3⋊Dic3—32no—
111C3⋊Dic3—62no—
112C3⋊Dic3—96no—
113C3×Dic3—36no—
114C3×Dic3—32no—
115C3×Dic3—61no—
116C3×Dic3—63no—
117C3×Dic3—96no—
118C3×Dic3—92no—
119C33—21no—
120C33#23512yescomplemented
121H(3,𝔽3)—22no—
122H(3,𝔽3)—41no—
123H(3,𝔽3)#23511yescharacteristic, complemented
124C2×C3×C4—93no—
125C2×C3×C4—91no—
126C2×C3×C4—272no—
127#67—272no—
128#67—276no—
129#67—272no—
130#67—272no—
131D12—273no—
132D12—271no—
133D12—272no—
134C4×D3—32no—
135C4×D3—61no—
136C4×D3—96no—
137C4×D3—92no—
138C4×D3—272no—
139C4×D3—542no—
140Dic6—272no—
141Dic6—273no—
142Dic6—271no—
143C2×C32—22no—
144C2×C32—21no—
145C2×C32—36no—
146C2×C32—41no—
147C2×C32—63no—
148C2×C32—181no—
149C2×C32C2×D3212yes—
150C2×C32C2×D3211yescharacteristic
151C3⋊D3—62no—
152C3⋊D3—64no—
153C3⋊D3—62no—
154C3⋊D3—186no—
155C3×D3—62no—
156C3×D3—66no—
157C3×D3—66no—
158C3×D3—62no—
159C3×D3—186no—
160C3×D3—182no—
161#41—811noSylow
162C22×C3—93no—
163C22×C3—91no—
164C22×C3—272no—
165C2×D3—32no—
166C2×D3—61no—
167C2×D3—96no—
168C2×D3—92no—
169C2×D3—272no—
170C2×D3—542no—
171C3×C4—32no—
172C3×C4—61no—
173C3×C4—62no—
174C3×C4—93no—
175C3×C4—91no—
176C3×C4—121no—
177C3×C4—272no—
178C3×C4((C3⋊D3)⋊C3)⋊C213yes—
179C3×C4#111311yescharacteristic
180Dic3—32no—
181Dic3—61no—
182Dic3—96no—
183Dic3—92no—
184Dic3—272no—
185Dic3—542no—
186C32—21no—
187C32—22no—
188C32—36no—
189C32—41no—
190C32—63no—
191C32#379811yescharacteristic
192C32#379812yescomplemented
193Q8—811no—
194D4—811no—
195D4—812no—
196C2×C4—91no—
197C2×C4—272no—
198C2×C3—32no—
199C2×C3—61no—
200C2×C3—62no—
201C2×C3—121no—
202C2×C3—183no—
203C2×C3—181no—
204C2×C3—542no—
205C2×C3C2×((C3⋊D3)⋊C3)⋊C213yes—
206C2×C3C2×#111311yescharacteristic
207D3—62no—
208D3—62no—
209D3—186no—
210D3—182no—
211D3—542no—
212D3—544no—
213C22—91no—
214C22—272no—
215C4—91no—
216C4—272no—
217C4#6643211yescharacteristic
218C3—32no—
219C3—61no—
220C3—62no—
221C3—121no—
222C3#8865013yescomplemented
223C3#8904211yescharacteristic
224C2—181no—
225C2—542no—
226C2C2×#6643211yescharacteristic
subgroupG/H
center, hypercenterC4#66432
commutator, cosocleC2×C3×H(3,𝔽3)C23
FrattiniC2×C3C2×#1113
FittingC3×C4×H(3,𝔽3)C22
socleC2×C32C2×D32
hypocenterC3×H(3,𝔽3)#41
LCSG ▷ C2×C3×H(3,𝔽3) ▷ C3×H(3,𝔽3)
UCSC1 ◁ C4

Complements

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

NH ≅ G/Ntype
1. C3×C4×#272C2semidirect
2. C4×C3⋊((C3⋊D3)⋊C3)C2semidirect
3. #10619904C2semidirect
4. #10619902C2semidirect
5. #10619900C2semidirect
13. C4×C3⋊(C3⋊D3)D3semidirect
21. C4×#272D3semidirect
52. C3×H(3,𝔽3)#41semidirect
120. C33#235semidirect
123. H(3,𝔽3)#235semidirect
192. C32#3798semidirect
222. C3#88650semidirect

Maximal subgroups

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

subgroupindex
C3×C4×#2722
C4×C3⋊((C3⋊D3)⋊C3)2
#106199042
#106199022
#106199002
#890203
#886503