Finite Group Databasebeta

← #10592770⋊C3 #13142984

Subgroups — #10592770⋊C3 #13142984

Order 1728.

493 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 493 conjugacy classes of subgroups — too many to draw. Only its 92 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
1H(3,𝔽3)×#127C211yesmaximal, characteristic
2#64079⋊C3C211yesmaximal, characteristic, complemented
3#63659⋊C3C214yesmaximal, complemented
4#63630⋊C3C211yesmaximal, characteristic, complemented
5#10592770C311yesmaximal, characteristic, complemented
6C3×#5071—31nomaximal
7C2×D4×H(3,𝔽3)—22no—
8C2×D4×H(3,𝔽3)C2212yes—
9C42×H(3,𝔽3)C2211yescharacteristic
10C4⋊((C3⋊Dic3)⋊C3)C2212yes—
11#3750⋊C3—22no—
12C2×(C3⋊(C3⋊C8))⋊C3C2212yes—
13C32×#127—21no—
14C32×#127D311yescharacteristic
15C32×#127C2×C311yescharacteristic
16#64079C2×C311yescharacteristic, complemented
17C3×#870—31no—
18#63659C2×C314yescomplemented
19#63630C2×C311yescharacteristic, complemented
20C3×#832—34no—
21C3×#803—31no—
22C23×H(3,𝔽3)—22no—
23D4×H(3,𝔽3)—24no—
24D4×H(3,𝔽3)—44no—
25C2×C4×H(3,𝔽3)D412yes—
26C2×C4×H(3,𝔽3)C2311yescharacteristic
27C2×(C3⋊Dic3)⋊C3—22no—
28(C3⋊(C3⋊C8))⋊C3—22no—
29C3×#490—91no—
30#5071—31no—
31#5071—91no—
32C2×C32×D4—24no—
33C2×C32×D4—22no—
34C2×C32×D4—22no—
35C2×C32×D4—22no—
36C2×C32×D4C2×D312yes—
37C2×C32×D4C22×C312yes—
38C32×C42—21no—
39C32×C42C2×D311yescharacteristic
40C32×C42C22×C311yescharacteristic
41C4⋊(C3⋊Dic3)C22×C312yes—
42#3750—22no—
43C2×C3⋊(C3⋊C8)C22×C312yes—
44C3×C4⋊Dic3—32no—
45C3×Dic3⋊C4—62no—
46C2×C3×C3⋊C8—32no—
47C22×H(3,𝔽3)—42no—
48C22×H(3,𝔽3)—44no—
49C22×H(3,𝔽3)C2×D411yescharacteristic
50C4×H(3,𝔽3)—22no—
51C4×H(3,𝔽3)#4112yes—
52(C3⋊Dic3)⋊C3—42no—
53C3×#127—31no—
54C3×#127—31no—
55C3×#127—61no—
56C3×#127C3×D311yescharacteristic
57C3×#125—91no—
58#870—31no—
59#870—91no—
60C3×#102—94no—
61C3×#97—91no—
62#832—34no—
63#832—94no—
64#803—31no—
65#803—91no—
66C23×C32—24no—
67C23×C32—22no—
68C23×C32—22no—
69C32×D4—24no—
70C32×D4—24no—
71C32×D4—44no—
72C32×D4—48no—
73C32×D4—44no—
74C32×D4—44no—
75C2×C32×C4—22no—
76C2×C32×C4—21no—
77C2×C32×C4#6712yes—
78C2×C32×C4C3×D412yes—
79C2×C32×C4C22×D311yescharacteristic
80C2×C32×C4C23×C311yescharacteristic
81C2×C3⋊Dic3—22no—
82C2×C3×Dic3—62no—
83C3⋊(C3⋊C8)—22no—
84C3×C3⋊C8—62no—
85#490—91noSylow
86C2×H(3,𝔽3)—82no—
87C2×H(3,𝔽3)#12411yescharacteristic
88C2×H(3,𝔽3)AGL(1,ℤ/8)12yes—
89C2×C3×D4—22no—
90C2×C3×D4—32no—
91C2×C3×D4—32no—
92C2×C3×D4—62no—
93C2×C3×D4—62no—
94C2×C3×D4—64no—
95C2×C3×D4—62no—
96C2×C3×D4C2×C3×D312yes—
97C2×C3×C8—92no—
98C3×C4⋊C4—92no—
99C3×C4⋊C4—182no—
100C3×C42—31no—
101C3×C42—31no—
102C3×C42—61no—
103C3×C42C2×C3×D311yescharacteristic
104C4⋊Dic3—32no—
105C4⋊Dic3—92no—
106Dic3⋊C4—62no—
107Dic3⋊C4—182no—
108C2×C3⋊C8—32no—
109C2×C3⋊C8—92no—
110C22×C32—21no—
111C22×C32—48no—
112C22×C32—44no—
113C22×C32—44no—
114C22×C32—42no—
115C22×C32—42no—
116C22×C32—44no—
117C22×C32C2×#6711yescharacteristic
118C22×C32C2×C3×D411yescharacteristic
119C32×C4—24no—
120C32×C4—22no—
121C32×C4—22no—
122C32×C4—22no—
123C32×C4#23712yes—
124C32×C4C3×#4112yes—
125C3⋊Dic3—42no—
126C3×Dic3—122no—
127#127(C3⋊D3)⋊C311yescharacteristic
128#125—91no—
129#102—94no—
130#97—91no—
131H(3,𝔽3)#49011yescharacteristic, complemented, Sylow
132C23×C3—22no—
133C23×C3—62no—
134C23×C3—62no—
135C23×C3—64no—
136C3×D4—24no—
137C3×D4—44no—
138C3×D4—64no—
139C3×D4—64no—
140C3×D4—124no—
141C3×D4—124no—
142C3×D4—124no—
143C3×D4—128no—
144C2×C3×C4—32no—
145C2×C3×C4—32no—
146C2×C3×C4—31no—
147C2×C3×C4—31no—
148C2×C3×C4—62no—
149C2×C3×C4—61no—
150C2×C3×C4—182no—
151C2×C3×C4C3×#6712yes—
152C2×C3×C4C22×C3×D311yescharacteristic
153C2×Dic3—62no—
154C2×Dic3—182no—
155C3×C8—182no—
156C3⋊C8—62no—
157C3⋊C8—182no—
158C2×C32—22no—
159C2×C32—21no—
160C2×C32—84no—
161C2×C32—82no—
162C2×C32—82no—
163C2×C32#93212yes—
164C2×C32#93611yescharacteristic
165C2×C32C3×#12411yescharacteristic
166C2×C32C3×AGL(1,ℤ/8)12yes—
167C2×D4—22no—
168C2×D4C2×(C3⋊D3)⋊C312yes—
169C2×C8—92no—
170C4⋊C4—92no—
171C4⋊C4—182no—
172C42C2×(C3⋊D3)⋊C311yescharacteristic
173C22×C3—31no—
174C22×C3—31no—
175C22×C3—44no—
176C22×C3—42no—
177C22×C3—61no—
178C22×C3—122no—
179C22×C3—122no—
180C22×C3—124no—
181C22×C3—124no—
182C22×C3—128no—
183C22×C3—124no—
184C22×C3C2×C3×#6711yescharacteristic
185C3×C4—22no—
186C3×C4—32no—
187C3×C4—32no—
188C3×C4—62no—
189C3×C4—62no—
190C3×C4—64no—
191C3×C4—62no—
192C3×C4—362no—
193C3×C4C3×#23712yes—
194Dic3—122no—
195Dic3—362no—
196C32—21no—
197C32#507111yescharacteristic, complemented
198C32C3×#49011yescharacteristic, complemented
199C23—22no—
200D4—24no—
201D4—44no—
202C2×C4—182no—
203C2×C4#638⋊C312yes—
204C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
205C8—182no—
206C2×C3—32no—
207C2×C3—32no—
208C2×C3—31no—
209C2×C3—31no—
210C2×C3—62no—
211C2×C3—61no—
212C2×C3—82no—
213C2×C3—242no—
214C2×C3—242no—
215C2×C3—244no—
216C2×C3C3×#93212yes—
217C2×C3C3×#93611yescharacteristic
218C22—44no—
219C22—42no—
220C22C2×#638⋊C311yescharacteristic
221C4—22no—
222C4—362no—
223C4#3830⋊C312yes—
224C3—31no—
225C3—31no—
226C3—61no—
227C3C3×#507111yescharacteristic
228C2—82no—
229C2#64141⋊C312yes—
230C2#64145⋊C311yescharacteristic
subgroupG/H
centerC22C2×#638⋊C3
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C3×C4C22×C3×D3
Fitting, prime radicalH(3,𝔽3)×#127C2
socleC22×C3C2×C3×#67
hypercenter#127(C3⋊D3)⋊C3
hypocenterC32C3×#490
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C42 ◁ #127

Complements

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

NH ≅ G/Ntype
2. #64079⋊C3C2semidirect
3. #63659⋊C3C2semidirect
4. #63630⋊C3C2semidirect
5. #10592770C3semidirect
16. #64079C2×C3semidirect
18. #63659C2×C3semidirect
19. #63630C2×C3semidirect
131. H(3,𝔽3)#490semidirect
197. C32#5071semidirect
198. C32C3×#490semidirect

Maximal subgroups

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

subgroupindex
H(3,𝔽3)×#1272
#64079⋊C32
#63659⋊C32
#63630⋊C32
#105927703
C3×#50713