Finite Group Databasebeta

← #10592768⋊C3 #13142953

Subgroups — #10592768⋊C3 #13142953

Order 1728.

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

normal subgroups only

This group has 371 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)×#125C211yesmaximal, characteristic
2#64078⋊C3C211yesmaximal, characteristic
3#63637⋊C3C214yesmaximal
4#63630⋊C3C211yesmaximal, characteristic
5#10592768C311yesmaximal, characteristic, complemented
6C3×#5069—31nomaximal
7C2×#634⋊C3C2212yes—
8C4⋊C4×H(3,𝔽3)—22no—
9C4⋊C4×H(3,𝔽3)C2212yes—
10C42×H(3,𝔽3)C2211yescharacteristic
11C4⋊((C3⋊Dic3)⋊C3)—22no—
12C2×(C3⋊(C3⋊C8))⋊C3C2212yes—
13C32×#125—21no—
14C32×#125D311yescharacteristic
15C32×#125C2×C311yescharacteristic
16#64078C2×C311yescharacteristic
17C3×#869—31no—
18#63637C2×C314yes—
19#63630C2×C311yescharacteristic
20C3×#810—34no—
21C3×#803—31no—
22C2×C4×H(3,𝔽3)—22no—
23C2×C4×H(3,𝔽3)D412yes—
24C2×C4×H(3,𝔽3)C2311yescharacteristic
25C2×(C3⋊Dic3)⋊C3—22no—
26#634⋊C3—24no—
27(C3⋊(C3⋊C8))⋊C3—22no—
28C3×#491—91no—
29#5069—31no—
30#5069—91no—
31C2×#634C22×C312yes—
32C2×C3×Dic6—32no—
33C32×C4⋊C4—24no—
34C32×C4⋊C4—22no—
35C32×C4⋊C4—22no—
36C32×C4⋊C4—22no—
37C32×C4⋊C4C2×D312yes—
38C32×C4⋊C4C22×C312yes—
39C32×C42—21no—
40C32×C42C2×D311yescharacteristic
41C32×C42C22×C311yescharacteristic
42C4⋊(C3⋊Dic3)—22no—
43C2×C3⋊(C3⋊C8)C22×C312yes—
44C3×C4⋊Dic3—62no—
45C2×C3×C3⋊C8—32no—
46C22×H(3,𝔽3)C2×D411yescharacteristic
47C4×H(3,𝔽3)—22no—
48C4×H(3,𝔽3)—42no—
49C4×H(3,𝔽3)#4112yes—
50(C3⋊Dic3)⋊C3—42no—
51C3×C4⋊Q8—91no—
52C3×#125—31no—
53C3×#125—31no—
54C3×#125—61no—
55C3×#125C3×D311yescharacteristic
56#869—31no—
57#869—91no—
58C3×#103—94no—
59C3×#97—91no—
60#810—34no—
61#810—94no—
62#803—31no—
63#803—91no—
64C2×C32×C4—24no—
65C2×C32×C4—22no—
66C2×C32×C4—22no—
67C2×C32×C4—22no—
68C2×C32×C4—21no—
69C2×C32×C4#6712yes—
70C2×C32×C4C3×D412yes—
71C2×C32×C4C22×D311yescharacteristic
72C2×C32×C4C23×C311yescharacteristic
73C2×C3⋊Dic3—22no—
74#634—24no—
75C2×C3×Dic3—62no—
76C3×Dic6—64no—
77C3⋊(C3⋊C8)—22no—
78C3×C3⋊C8—62no—
79#491—91noSylow
80C2×H(3,𝔽3)#12411yescharacteristic
81C2×H(3,𝔽3)#13712yes—
82C2×C3×Q8—92no—
83C2×Dic6—32no—
84C2×Dic6—92no—
85C2×C3×C8—92no—
86C3×C4⋊C4—22no—
87C3×C4⋊C4—32no—
88C3×C4⋊C4—32no—
89C3×C4⋊C4—62no—
90C3×C4⋊C4—62no—
91C3×C4⋊C4—64no—
92C3×C4⋊C4—62no—
93C3×C4⋊C4—182no—
94C3×C4⋊C4C2×C3×D312yes—
95C3×C42—31no—
96C3×C42—31no—
97C3×C42—61no—
98C3×C42C2×C3×D311yescharacteristic
99C4⋊Dic3—62no—
100C4⋊Dic3—182no—
101C2×C3⋊C8—32no—
102C2×C3⋊C8—92no—
103C22×C32—21no—
104C22×C32C2×#6711yescharacteristic
105C22×C32C2×C3×D411yescharacteristic
106C32×C4—22no—
107C32×C4—22no—
108C32×C4—22no—
109C32×C4—42no—
110C32×C4—44no—
111C32×C4—42no—
112C32×C4—42no—
113C32×C4#23912yes—
114C32×C4C3×#4112yes—
115C3⋊Dic3—42no—
116C3×Dic3—122no—
117C4⋊Q8—91no—
118#125(C3⋊D3)⋊C311yescharacteristic
119#103—94no—
120#97—91no—
121H(3,𝔽3)#49111yescharacteristic, complemented, Sylow
122C3×Q8—184no—
123C2×C3×C4—22no—
124C2×C3×C4—32no—
125C2×C3×C4—32no—
126C2×C3×C4—31no—
127C2×C3×C4—31no—
128C2×C3×C4—62no—
129C2×C3×C4—62no—
130C2×C3×C4—64no—
131C2×C3×C4—62no—
132C2×C3×C4—61no—
133C2×C3×C4—182no—
134C2×C3×C4C3×#6712yes—
135C2×C3×C4C22×C3×D311yescharacteristic
136C2×Dic3—62no—
137C2×Dic3—182no—
138Dic6—64no—
139Dic6—184no—
140C3×C8—182no—
141C3⋊C8—62no—
142C3⋊C8—182no—
143C2×C32—21no—
144C2×C32—22no—
145C2×C32#94711yescharacteristic
146C2×C32#95112yes—
147C2×C32C3×#12411yescharacteristic
148C2×C32C3×#13712yes—
149C2×Q8—92no—
150C2×C8—92no—
151C4⋊C4—22no—
152C4⋊C4—182no—
153C4⋊C4C2×(C3⋊D3)⋊C312yes—
154C42C2×(C3⋊D3)⋊C311yescharacteristic
155C22×C3—31no—
156C22×C3—31no—
157C22×C3—61no—
158C22×C3C2×C3×#6711yescharacteristic
159C3×C4—22no—
160C3×C4—32no—
161C3×C4—32no—
162C3×C4—42no—
163C3×C4—62no—
164C3×C4—62no—
165C3×C4—62no—
166C3×C4—122no—
167C3×C4—122no—
168C3×C4—122no—
169C3×C4—124no—
170C3×C4—362no—
171C3×C4C3×#23912yes—
172Dic3—122no—
173Dic3—362no—
174C32—21no—
175C32#506911yescharacteristic, complemented
176C32C3×#49111yescharacteristic, complemented
177Q8—184no—
178C2×C4—22no—
179C2×C4—182no—
180C2×C4#638⋊C312yes—
181C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
182C8—182no—
183C2×C3—31no—
184C2×C3—31no—
185C2×C3—32no—
186C2×C3—32no—
187C2×C3—61no—
188C2×C3—62no—
189C2×C3C3×#94711yescharacteristic
190C2×C3C3×#95112yes—
191C22C2×#638⋊C311yescharacteristic
192C4—22no—
193C4—42no—
194C4—362no—
195C4#8872212yes—
196C3—31no—
197C3—31no—
198C3—61no—
199C3C3×#506911yescharacteristic
200C2#64156⋊C311yescharacteristic
201C2#64160⋊C312yes—
subgroupG/H
centerC22C2×#638⋊C3
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C3×C4C22×C3×D3
FittingH(3,𝔽3)×#125C2
socleC22×C3C2×C3×#67
prime radicalC22×H(3,𝔽3)C2×D4
hypercenter#125(C3⋊D3)⋊C3
hypocenterC32C3×#491
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C42 ◁ #125

Complements

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

NH ≅ G/Ntype
5. #10592768C3semidirect
121. H(3,𝔽3)#491semidirect
175. C32#5069semidirect
176. C32C3×#491semidirect

Maximal subgroups

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

subgroupindex
H(3,𝔽3)×#1252
#64078⋊C32
#63637⋊C32
#63630⋊C32
#105927683
C3×#50693