Finite Group Databasebeta

← #13173075

Subgroups — #13173075

Order 1728.

780 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 780 conjugacy classes of subgroups — too many to draw. Only its 123 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
1C2×C32×#41⋊C3C211yesmaximal, characteristic
2C2×C3⋊(C3⋊#226)C211yesmaximal, characteristic, complemented
3C3⋊(C3⋊#859)C211yesmaximal, characteristic, complemented
4#10593659—312nomaximal
5#10592938—31nomaximal
6C32×#41⋊C3—21no—
7C2×C32×SL(2,𝔽3)C2211yescharacteristic
8C3⋊(C3⋊#226)—21no—
9#88857—41nomaximal
10C2×C32×#41D311yescharacteristic
11C2×C3×#41⋊C3D3112yes—
12C2×C3⋊#226—312no—
13C2×#3756—31no—
14C2×#3754—31no—
15#64133—31no—
16C3⋊#859—312no—
17#63662—31no—
18#63659—31no—
19#63657—31no—
20C2×C33×C4—41no—
21C2×C3⋊(C3⋊Dic3)—41no—
22C2×C3⋊(C3⋊Dic3)—41no—
23C32×SL(2,𝔽3)D411yescharacteristic
24#5425—99no—
25#5239—94no—
26C32×#41—21no—
27C32×#41—61no—
28C2×C32×Q8C2×D311yescharacteristic
29C2×C32×D4—31no—
30C22×C32×C4—31no—
31C2×#634—31no—
32C3×#41⋊C3—212no—
33C2×C3×SL(2,𝔽3)C2×D3112yes—
34C3⋊#226—612no—
35#3757—61no—
36#3756—61no—
37#3756—61no—
38#3754—62no—
39C4⋊(C3⋊Dic3)—31no—
40#3750—61no—
41#3750—1212no—
42C2×C3⋊(C3⋊C8)—31no—
43C2×C3⋊(C3⋊C8)—31no—
44C22×C33—41no—
45C33×C4—81no—
46C3⋊(C3⋊Dic3)—42no—
47C3⋊(C3⋊Dic3)—81no—
48C2×C3×#41C3⋊D314yes—
49C2×#41⋊C3C3⋊D319yes—
50C2×#226—99no—
51C2×#216—94no—
52C2×#214—94no—
53#924—94no—
54#859—99no—
55#835—94no—
56#832—94no—
57#830—94no—
58C23×C32—31no—
59C32×Q8—31no—
60C32×Q8#6711yescharacteristic
61C32×D4—32no—
62C32×D4—61no—
63C2×C32×C4—31no—
64C2×C32×C4—412no—
65C2×C32×C4—61no—
66C2×C32×C4—61no—
67C2×C32×C4S411yescharacteristic
68C2×C3⋊Dic3—61no—
69C2×C3⋊Dic3—61no—
70C2×C3⋊Dic3—1212no—
71C2×C3⋊Dic3—1212no—
72#634—61no—
73#634—61no—
74C3×SL(2,𝔽3)#67112yes—
75C3⋊(C3⋊C8)—61no—
76C3⋊(C3⋊C8)—61no—
77#452—271noSylow
78C2×C33—41no—
79C2×C33—41no—
80C2×C33—41no—
81C3×#41—24no—
82C3×#41—64no—
83C2×C3×Q8C2×C3⋊D314yes—
84C2×C3×D4—34no—
85C22×C3×C4—34no—
86C2×Dic6—94no—
87#41⋊C3—29no—
88C2×SL(2,𝔽3)C2×C3⋊D319yes—
89#226—189no—
90#217—184no—
91#216—184no—
92#216—184no—
93#214—188no—
94C4⋊Dic3—94no—
95Dic3⋊C4—184no—
96Dic3⋊C4—369no—
97C2×C3⋊C8—94no—
98C2×C3⋊C8—94no—
99C22×C32—412no—
100C22×C32—61no—
101C22×C32—61no—
102C22×C32—61no—
103C22×C32C2×S411yescharacteristic
104C32×C4—21no—
105C32×C4—31no—
106C32×C4—31no—
107C32×C4—812no—
108C3⋊Dic3—1224no—
109C3⋊Dic3—121no—
110C3⋊Dic3—121no—
111C3⋊Dic3—2412no—
112C2×#41C3⋊(C3⋊D3)11yescharacteristic
113C2×Q16—271no—
114C2×SD16—271no—
115C22⋊Q8—271no—
116#103—271no—
117#102—271no—
118C22⋊C8—271no—
119C33—41noSylow
120C23×C3—34no—
121C3×Q8—34no—
122C3×Q8#63814yes—
123C3×D4—38no—
124C3×D4—64no—
125C2×C3×C4—34no—
126C2×C3×C4—49no—
127C2×C3×C4—64no—
128C2×C3×C4—64no—
129C2×C3×C4C3⋊S414yes—
130C2×Dic3—184no—
131C2×Dic3—184no—
132C2×Dic3—369no—
133C2×Dic3—369no—
134Dic6—184no—
135Dic6—184no—
136SL(2,𝔽3)#63819yes—
137C3⋊C8—184no—
138C3⋊C8—184no—
139C2×C32—412no—
140C2×C32—412no—
141C2×C32—412no—
142C2×C32—121no—
143C2×C32#98411yescharacteristic
144C2×C32#98511yescharacteristic
145C2×C32GL(2,ℤ/4)11yescharacteristic
146#41—21no—
147#41—61no—
148C2×Q8—271no—
149C2×Q8C2×C3⋊(C3⋊D3)11yescharacteristic
150C2×D4—31no—
151C22×C4—31no—
152Q16—541no—
153Q16—541no—
154SD16—542no—
155C2×C8—271no—
156C2×C8—271no—
157C4⋊C4—271no—
158C4⋊C4—541no—
159C22⋊C4—541no—
160C22×C3—49no—
161C22×C3—64no—
162C22×C3—64no—
163C22×C3—64no—
164C22×C3C2×C3⋊S414yes—
165C3×C4—24no—
166C3×C4—34no—
167C3×C4—34no—
168C3×C4—89no—
169Dic3—3618no—
170Dic3—364no—
171Dic3—364no—
172Dic3—729no—
173C32—412no—
174C32#542511yescharacteristic, complemented
175C23—31no—
176Q8—31no—
177Q8—541no—
178Q8—541no—
179Q8#631211yescharacteristic
180D4—32no—
181D4—61no—
182C2×C4—31no—
183C2×C4—61no—
184C2×C4—61no—
185C2×C4—541no—
186C2×C4—541no—
187C2×C4C3⋊(C3⋊S4)11yescharacteristic
188C8—541no—
189C8—541no—
190C2×C3—49no—
191C2×C3—49no—
192C2×C3—49no—
193C2×C3—124no—
194C2×C3#6426514yes—
195C2×C3#6426614yes—
196C2×C3#6426914yes—
197C22—61no—
198C22—61no—
199C22—61no—
200C22C2×C3⋊(C3⋊S4)11yescharacteristic
201C4—21no—
202C4—31no—
203C4—31no—
204C4—1081no—
205C4—1081no—
206C3—49no—
207C3#1059365914yescomplemented
208C2—121no—
209C2#1172446911yescharacteristic
210C2#1172447011yescharacteristic
211C2#1172447311yescharacteristic
subgroupG/H
center, FrattiniC22C2×C3⋊(C3⋊S4)
commutator, cosocleC2×C32×SL(2,𝔽3)C22
FittingC2×C32×#41D3
socleC22×C32C2×S4
prime radicalC2×C32×#41⋊C3C2
hypercenterC2×C4C3⋊(C3⋊S4)
hypocenterC32×SL(2,𝔽3)D4
LCSG ▷ C2×C32×SL(2,𝔽3) ▷ C32×SL(2,𝔽3)
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
2. C2×C3⋊(C3⋊#226)C2semidirect
3. C3⋊(C3⋊#859)C2semidirect
174. C32#5425semidirect
207. C3#10593659semidirect

Maximal subgroups

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

subgroupindex
C2×C32×#41⋊C32
C2×C3⋊(C3⋊#226)2
C3⋊(C3⋊#859)2
#105936593
#105929383
#888574