Finite Group Databasebeta

← #13137332

Subgroups — #13137332

Order 1728.

371 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 371 conjugacy classes of subgroups — too many to draw. Only its 75 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
1#11721716C211yesmaximal, characteristic
2#64078⋊C3C211yesmaximal, characteristic
3#63630⋊C3C211yesmaximal, characteristic
4#11720309C214yesmaximal
5#10590295—31nomaximal
6#10590291—31nomaximal
7C2×#634⋊C3C2212yes—
8C42×H(3,𝔽3)C2211yescharacteristic
9C4⋊#1084C2212yes—
10#88540—22no—
11C4⋊((C3⋊Dic3)⋊C3)—22no—
12C2×(C3⋊(C3⋊C8))⋊C3C2212yes—
13#64078D311yescharacteristic
14C3×#870—31no—
15C3×#870—31no—
16C3×#870—61no—
17C3×#869—31no—
18#63630D311yescharacteristic
19C3×#803—31no—
20#63567—34no—
21#63565—34no—
22C2×C4×H(3,𝔽3)D412yes—
23C2×C4×H(3,𝔽3)C2311yescharacteristic
24C2×#1084—22no—
25C2×(C3⋊Dic3)⋊C3—22no—
26#634⋊C3—24no—
27(C3⋊(C3⋊C8))⋊C3—22no—
28#5082—91no—
29#5069—91no—
30#4703—91no—
31C2×#634C2×D312yes—
32C2×C3×Dic6—32no—
33C32×C42—21no—
34C32×C42C2×D311yescharacteristic
35C32×C42C2×D311yescharacteristic
36C4⋊(C3⋊Dic3)—22no—
37C2×C3⋊(C3⋊C8)C2×D312yes—
38C3×C4⋊Dic3—32no—
39C3×C4⋊Dic3—32no—
40C3×C4⋊Dic3—62no—
41C3×C4⋊Dic3—62no—
42C3×Dic3⋊C4—64no—
43C3×Dic3⋊C4—62no—
44C3×Dic3⋊C4—62no—
45C2×C3×C3⋊C8—32no—
46C22×H(3,𝔽3)C2×D411yescharacteristic
47C4×H(3,𝔽3)—22no—
48C4×H(3,𝔽3)#4112yes—
49#1084—42no—
50(C3⋊Dic3)⋊C3—42no—
51C3×C4⋊Q8—91no—
52C3×#125—91no—
53#870—31no—
54#870—31no—
55#870—61no—
56#869—31no—
57#869—91no—
58C3×#97—91no—
59#835—94no—
60#816—94no—
61#810—94no—
62#803—31no—
63#803—91no—
64C2×C32×C4—22no—
65C2×C32×C4—21no—
66C2×C32×C4D1212yes—
67C2×C32×C4#6712yes—
68C2×C32×C4C22×D311yescharacteristic
69C2×C32×C4C22×D311yescharacteristic
70C2×C3⋊Dic3—22no—
71#634—24no—
72C2×C3×Dic3—64no—
73C2×C3×Dic3—62no—
74C2×C3×Dic3—62no—
75C2×C3×Dic3—62no—
76C3×Dic6—64no—
77C3⋊(C3⋊C8)—22no—
78C3×C3⋊C8—62no—
79#491—271noSylow
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—92no—
87C3×C4⋊C4—182no—
88C3×C4⋊C4—182no—
89C3×C42—31no—
90C3×C42—31no—
91C3×C42—61no—
92C3×C42D3211yescharacteristic
93C4⋊Dic3—32no—
94C4⋊Dic3—32no—
95C4⋊Dic3—62no—
96C4⋊Dic3—62no—
97C4⋊Dic3—182no—
98Dic3⋊C4—62no—
99Dic3⋊C4—62no—
100Dic3⋊C4—64no—
101C2×C3⋊C8—32no—
102C2×C3⋊C8—92no—
103C22×C32—21no—
104C22×C32C2×D1211yescharacteristic
105C22×C32C2×#6711yescharacteristic
106C32×C4—22no—
107C32×C4—22no—
108C32×C4—22no—
109C32×C4—42no—
110C32×C4#23512yes—
111C32×C4#23712yes—
112C3⋊Dic3—42no—
113C3×Dic3—124no—
114C3×Dic3—122no—
115C3×Dic3—122no—
116C3×Dic3—122no—
117C4⋊Q8—91no—
118#125—91no—
119#103—274no—
120#97—91no—
121H(3,𝔽3)#49111yescharacteristic, complemented, Sylow
122C3×Q8—184no—
123C2×C3×C4—32no—
124C2×C3×C4—32no—
125C2×C3×C4—31no—
126C2×C3×C4—31no—
127C2×C3×C4—62no—
128C2×C3×C4—61no—
129C2×C3×C4—182no—
130C2×C3×C4—182no—
131C2×C3×C4#62612yes—
132C2×C3×C4C2×D3211yescharacteristic
133C2×Dic3—62no—
134C2×Dic3—62no—
135C2×Dic3—64no—
136C2×Dic3—62no—
137C2×Dic3—182no—
138Dic6—64no—
139Dic6—184no—
140C3×C8—182no—
141C3⋊C8—62no—
142C3⋊C8—182no—
143C2×C32—22no—
144C2×C32—21no—
145C2×C32#87511yescharacteristic
146C2×C32#90912yes—
147C2×C32#93611yescharacteristic
148C2×C32#94212yes—
149C2×Q8—92no—
150C2×C8—92no—
151C4⋊C4—92no—
152C4⋊C4—182no—
153C4⋊C4—182no—
154C42((C3⋊D3)⋊C3)⋊C211yescharacteristic
155C22×C3—31no—
156C22×C3—31no—
157C22×C3—61no—
158C22×C3C2×#62611yescharacteristic
159C3×C4—22no—
160C3×C4—32no—
161C3×C4—32no—
162C3×C4—62no—
163C3×C4—62no—
164C3×C4—62no—
165C3×C4—122no—
166C3×C4—362no—
167C3×C4—362no—
168C3×C4#379512yes—
169Dic3—122no—
170Dic3—122no—
171Dic3—124no—
172Dic3—122no—
173Dic3—362no—
174C32—21no—
175C32#470311yescharacteristic, complemented
176C32#508211yescharacteristic, complemented
177Q8—184no—
178C2×C4—182no—
179C2×C4—182no—
180C2×C4#620012yes—
181C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
182C8—182no—
183C2×C3—32no—
184C2×C3—32no—
185C2×C3—31no—
186C2×C3—31no—
187C2×C3—62no—
188C2×C3—61no—
189C2×C3#6383312yes—
190C2×C3#6386011yescharacteristic
191C22C2×#620011yescharacteristic
192C4—22no—
193C4—362no—
194C4—362no—
195C4#8864812yes—
196C3—31no—
197C3—31no—
198C3—61no—
199C3#1059028911yescharacteristic
200C2#1172106912yes—
201C2#1172111011yescharacteristic
subgroupG/H
centerC22C2×#6200
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C2×D32
FittingC42×H(3,𝔽3)C22
socleC22×C3C2×#626
prime radicalC22×H(3,𝔽3)C2×D4
hypercenterC42((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#491
LCSG ▷ C2×C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C42

Complements

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

NH ≅ G/Ntype
121. H(3,𝔽3)#491semidirect
175. C32#4703semidirect
176. C32#5082semidirect

Maximal subgroups

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

subgroupindex
#117217162
#64078⋊C32
#63630⋊C32
#117203092
#105902953
#105902913