Finite Group Databasebeta

← #13048674

Subgroups — #13048674

Order 1296.

576 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 576 conjugacy classes of subgroups — too many to draw. Only its 77 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×C3⋊#272C211yesmaximal, characteristic, complemented
2C3×C4×#272C211yesmaximal, characteristic, complemented
3C4×D3×H(3,𝔽3)C211yesmaximal, characteristic, complemented
4#10619894C211yesmaximal, characteristic, complemented
5#10619893C211yesmaximal, characteristic, complemented
6#10619892C211yesmaximal, characteristic, complemented
7#10619891C211yesmaximal, characteristic, complemented
8C3×#3798—34nomaximal
9#88740—31nomaximal
10C2×C3⋊#272C2211yescharacteristic
11C2×C3×#272C2211yescharacteristic
12C2×D3×H(3,𝔽3)C2211yescharacteristic
13C3×C4×H(3,𝔽3)C2211yescharacteristic
14C3⋊#1084C2211yescharacteristic
15C3×#1084C2211yescharacteristic
16Dic3×H(3,𝔽3)C2211yescharacteristic
17C3×C4×C3⋊D3—34no—
18C32×C4×D3—34no—
19C32×C4×D3D314yescomplemented
20C3×#627—34no—
21C3×#626—34no—
22C3×#626—34no—
23C3×#625—34no—
24C2×C4×H(3,𝔽3)—31no—
25#6235—31no—
26#6235—31no—
27C4⋊#272—31no—
28C4×#272—31no—
29C4×#272—64no—
30C4×#272D311yescharacteristic, complemented
31#6229—31no—
32C2×C3×H(3,𝔽3)C2311yescharacteristic
33C3⋊#272—21no—
34C3×#272—21no—
35D3×H(3,𝔽3)—21no—
36C3×#235—94no—
37C3×#235—91no—
38#3798—34no—
39C2×C3×C3⋊D3—34no—
40C2×C32×D3—34no—
41C2×C32×D3C2×D314yes—
42C33×C4C2×D314yes—
43C3×C3⋊Dic3—34no—
44C32×Dic3—34no—
45C32×Dic3C2×D314yes—
46C22×H(3,𝔽3)—31no—
47C2×#272—31no—
48C2×#272—64no—
49C2×#272C2×D311yescharacteristic
50C4×H(3,𝔽3)—24no—
51C4×H(3,𝔽3)—31no—
52C4×H(3,𝔽3)C2×D311yescharacteristic
53#1084—31no—
54#1084—64no—
55#1084C2×D311yescharacteristic
56C3×H(3,𝔽3)#4111yescharacteristic, complemented, Sylow
57C2×C32×C4—34no—
58C2×C32×C4—91no—
59C4×C3⋊D3—34no—
60C3×#67—94no—
61C3×#67—94no—
62C3×#67—91no—
63C3×#67—91no—
64C3×D12—94no—
65C3×D12—91no—
66C3×C4×D3—34no—
67C3×C4×D3—34no—
68C3×C4×D3—34no—
69C3×C4×D3—64no—
70C3×C4×D3—94no—
71C3×C4×D3—91no—
72C3×C4×D3—184no—
73C3×C4×D3C3⋊D311yescharacteristic
74C3×Dic6—94no—
75C3×Dic6—91no—
76#627—34no—
77#626—34no—
78#626—34no—
79#625—34no—
80C2×C33C22×D314yes—
81C3×C3⋊D3—64no—
82C32×D3—24no—
83C32×D3—64no—
84C2×H(3,𝔽3)—24no—
85C2×H(3,𝔽3)—61no—
86C2×H(3,𝔽3)C22×D311yescharacteristic
87#272—21no—
88#272—68no—
89#272—61no—
90C3×#41—271no—
91#235—94no—
92#235—91no—
93C22×C32—34no—
94C22×C32—91no—
95C2×C3⋊D3—34no—
96C2×C3×D3—34no—
97C2×C3×D3—34no—
98C2×C3×D3—34no—
99C2×C3×D3—64no—
100C2×C3×D3—94no—
101C2×C3×D3—91no—
102C2×C3×D3—184no—
103C2×C3×D3C2×C3⋊D311yescharacteristic
104C32×C4—24no—
105C32×C4—34no—
106C32×C4—34no—
107C32×C4—64no—
108C32×C4—91no—
109C32×C4D3214yes—
110C32×C4C2×C3⋊D311yescharacteristic
111C3⋊Dic3—34no—
112C3×Dic3—34no—
113C3×Dic3—34no—
114C3×Dic3—34no—
115C3×Dic3—64no—
116C3×Dic3—94no—
117C3×Dic3—91no—
118C3×Dic3—184no—
119C3×Dic3C2×C3⋊D311yescharacteristic
120C33#23514yescomplemented
121H(3,𝔽3)—24no—
122H(3,𝔽3)#23511yescharacteristic, complemented
123C3×Q8—271no—
124C3×D4—271no—
125C3×D4—271no—
126C3×D4—271no—
127C2×C3×C4—31no—
128C2×C3×C4—94no—
129C2×C3×C4—91no—
130C2×C3×C4—91no—
131C2×C3×C4—181no—
132C2×C3×C4—271no—
133#67—94no—
134#67—94no—
135#67—91no—
136#67—91no—
137D12—94no—
138D12—91no—
139C4×D3—34no—
140C4×D3—94no—
141C4×D3—91no—
142C4×D3—184no—
143C4×D3#27211yescharacteristic, complemented
144Dic6—94no—
145Dic6—91no—
146C2×C32—24no—
147C2×C32—34no—
148C2×C32—64no—
149C2×C32—64no—
150C2×C32—181no—
151C2×C32C2×D3214yes—
152C2×C32C22×C3⋊D311yescharacteristic
153C3⋊D3—64no—
154C3×D3—21no—
155C3×D3—64no—
156C3×D3—64no—
157C3×D3—68no—
158C3×D3—64no—
159C3×D3—188no—
160C3×D3—184no—
161C3×D3—181no—
162#41—271noSylow
163C22×C3—31no—
164C22×C3—94no—
165C22×C3—91no—
166C22×C3—91no—
167C22×C3—181no—
168C22×C3—271no—
169C2×D3—34no—
170C2×D3—94no—
171C2×D3—91no—
172C2×D3—184no—
173C2×D3C2×#27211yescharacteristic
174C3×C4—21no—
175C3×C4—31no—
176C3×C4—34no—
177C3×C4—64no—
178C3×C4—94no—
179C3×C4—91no—
180C3×C4—91no—
181C3×C4—181no—
182C3×C4—271no—
183C3×C4C2×#27211yescharacteristic
184C3×C4D3×C3⋊D311yescharacteristic
185Dic3—34no—
186Dic3—94no—
187Dic3—91no—
188Dic3—184no—
189Dic3C2×#27211yescharacteristic
190C32—24no—
191C32—34no—
192C32—64no—
193C32#379814yescomplemented
194C32#382811yescharacteristic
195Q8—271no—
196D4—271no—
197D4—271no—
198D4—271no—
199C2×C4—31no—
200C2×C4—91no—
201C2×C4—271no—
202C2×C3—21no—
203C2×C3—34no—
204C2×C3—61no—
205C2×C3—64no—
206C2×C3—184no—
207C2×C3—182no—
208C2×C3—181no—
209C2×C3—181no—
210C2×C3—541no—
211C2×C3C22×#27211yescharacteristic
212C2×C3C2×D3×C3⋊D311yescharacteristic
213D3—21no—
214D3—64no—
215D3—184no—
216D3—188no—
217D3—181no—
218C22—31no—
219C22—91no—
220C22—271no—
221C4—31no—
222C4—91no—
223C4—271no—
224C4D3×#27211yescharacteristic
225C3—21no—
226C3—34no—
227C3—64no—
228C3#8874011yescharacteristic, complemented
229C3#8902011yescharacteristic
230C2—61no—
231C2—181no—
232C2—541no—
233C2C2×D3×#27211yescharacteristic
subgroupG/H
center, hypercenterC3×C4D3×C3⋊D3
commutator, cosocleC2×C3×H(3,𝔽3)C23
FrattiniC2×C3C2×D3×C3⋊D3
FittingC3×C4×H(3,𝔽3)C22
socleC2×C32C22×C3⋊D3
hypocenterC3×H(3,𝔽3)#41
LCSG ▷ C2×C3×H(3,𝔽3) ▷ C3×H(3,𝔽3)
UCSC1 ◁ C3×C4

Complements

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

NH ≅ G/Ntype
1. C4×C3⋊#272C2semidirect
2. C3×C4×#272C2semidirect
3. C4×D3×H(3,𝔽3)C2semidirect
4. #10619894C2semidirect
5. #10619893C2semidirect
6. #10619892C2semidirect
7. #10619891C2semidirect
19. C32×C4×D3D3semidirect
30. C4×#272D3semidirect
56. C3×H(3,𝔽3)#41semidirect
120. C33#235semidirect
122. H(3,𝔽3)#235semidirect
143. C4×D3#272semidirect
193. C32#3798semidirect
228. C3#88740semidirect

Maximal subgroups

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

subgroupindex
C4×C3⋊#2722
C3×C4×#2722
C4×D3×H(3,𝔽3)2
#106198942
#106198932
#106198922
#106198912
C3×#37983
#887403