Finite Group Databasebeta

← #13173069

Subgroups — #13173069

Order 1728.

880 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 880 conjugacy classes of subgroups — too many to draw. Only its 135 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
1C22×C32×SL(2,𝔽3)C211yesmaximal, characteristic
2C2×C3⋊(C3⋊#226)C211yesmaximal, characteristic, complemented
3C3⋊(C3⋊#859)C211yesmaximal, characteristic, complemented
4#10593653—312nomaximal
5#10592927—31nomaximal
6C2×C32×SL(2,𝔽3)—21no—
7C2×C32×SL(2,𝔽3)C2211yescharacteristic
8C3⋊(C3⋊#226)—21no—
9#88864—41nomaximal
10C22×C32×Q8D311yescharacteristic
11C22×C3×SL(2,𝔽3)D3112yes—
12C2×C3⋊#226—312no—
13C2×#3756—31no—
14C2×#3756—31no—
15#64133—31no—
16C3⋊#859—312no—
17#63662—31no—
18#63662—31no—
19#63657—31no—
20C23×C33—41no—
21C2×C3⋊(C3⋊Dic3)—41no—
22C2×C3⋊(C3⋊Dic3)—41no—
23C32×SL(2,𝔽3)D411yescharacteristic
24#5419—99no—
25#5228—94no—
26C2×C32×Q8—21no—
27C2×C32×Q8—31no—
28C2×C32×Q8—61no—
29C2×C32×Q8C2×D311yescharacteristic
30C22×C32×C4—31no—
31C2×#634—31no—
32C2×C3×SL(2,𝔽3)—212no—
33C2×C3×SL(2,𝔽3)C2×D3112yes—
34C3⋊#226—612no—
35#3757—61no—
36#3757—1212no—
37#3756—61no—
38#3756—62no—
39#3756—61no—
40C4⋊(C3⋊Dic3)—31no—
41#3750—61no—
42C2×C3⋊(C3⋊C8)—31no—
43C2×C3⋊(C3⋊C8)—31no—
44C22×C33—41no—
45C22×C33—42no—
46C22×C33—81no—
47C22×C33—81no—
48C3⋊(C3⋊Dic3)—81no—
49C3⋊(C3⋊Dic3)—81no—
50C22×C3×Q8C3⋊D314yes—
51C22×SL(2,𝔽3)C3⋊D319yes—
52C2×#226—99no—
53C2×#216—94no—
54C2×#216—94no—
55#924—94no—
56#859—99no—
57#835—94no—
58#835—94no—
59#830—94no—
60C23×C32—412no—
61C23×C32S411yescharacteristic
62C32×Q8—31no—
63C32×Q8—32no—
64C32×Q8—61no—
65C32×Q8#6711yescharacteristic
66C2×C32×C4—31no—
67C2×C32×C4—31no—
68C2×C32×C4—61no—
69C2×C32×C4—61no—
70C2×C3⋊Dic3—61no—
71C2×C3⋊Dic3—61no—
72C2×C3⋊Dic3—1212no—
73C2×C3⋊Dic3—1212no—
74#634—61no—
75#634—61no—
76C3×SL(2,𝔽3)#67112yes—
77C3⋊(C3⋊C8)—61no—
78C3⋊(C3⋊C8)—61no—
79#451—271noSylow
80C2×C33—41no—
81C2×C33—41no—
82C2×C33—41no—
83C2×C33—82no—
84C2×C3×Q8—24no—
85C2×C3×Q8—34no—
86C2×C3×Q8—64no—
87C2×C3×Q8C2×C3⋊D314yes—
88C22×C3×C4—34no—
89C2×Dic6—94no—
90C2×SL(2,𝔽3)—29no—
91C2×SL(2,𝔽3)C2×C3⋊D319yes—
92#226—189no—
93#217—184no—
94#217—369no—
95#216—184no—
96#216—188no—
97#216—184no—
98C4⋊Dic3—94no—
99Dic3⋊C4—184no—
100C2×C3⋊C8—94no—
101C2×C3⋊C8—94no—
102C22×C32—21no—
103C22×C32—21no—
104C22×C32—412no—
105C22×C32—424no—
106C22×C32—812no—
107C22×C32—812no—
108C22×C32#22612yescomplemented
109C22×C32C2×S411yescharacteristic
110C32×C4—31no—
111C32×C4—31no—
112C32×C4—61no—
113C3⋊Dic3—121no—
114C3⋊Dic3—121no—
115C3⋊Dic3—2412no—
116C3⋊Dic3—2412no—
117C22×Q8C3⋊(C3⋊D3)11yescharacteristic
118C2×Q16—271no—
119C2×Q16—271no—
120C22⋊Q8—271no—
121#103—271no—
122#103—271no—
123C22⋊C8—271no—
124C33—41noSylow
125C23×C3—49no—
126C23×C3C3⋊S414yes—
127C3×Q8—34no—
128C3×Q8—38no—
129C3×Q8—64no—
130C3×Q8#63814yes—
131C2×C3×C4—34no—
132C2×C3×C4—34no—
133C2×C3×C4—64no—
134C2×C3×C4—64no—
135C2×Dic3—184no—
136C2×Dic3—184no—
137C2×Dic3—369no—
138C2×Dic3—369no—
139Dic6—184no—
140Dic6—184no—
141SL(2,𝔽3)#63819yes—
142C3⋊C8—184no—
143C3⋊C8—184no—
144C2×C32—22no—
145C2×C32—412no—
146C2×C32—412no—
147C2×C32—412no—
148C2×C32—824no—
149C2×C32C2×#22611yescharacteristic
150C2×C32#98311yescharacteristic
151C2×C32GL(2,ℤ/4)11yescharacteristic
152C2×Q8—21no—
153C2×Q8—31no—
154C2×Q8—61no—
155C2×Q8—271no—
156C2×Q8C2×C3⋊(C3⋊D3)11yescharacteristic
157C22×C4—31no—
158Q16—541no—
159Q16—542no—
160Q16—541no—
161C2×C8—271no—
162C2×C8—271no—
163C4⋊C4—271no—
164C4⋊C4—541no—
165C22⋊C4—541no—
166C22×C3—24no—
167C22×C3—24no—
168C22×C3—49no—
169C22×C3—418no—
170C22×C3—89no—
171C22×C3—89no—
172C22×C3C3⋊#22618yescomplemented
173C22×C3C2×C3⋊S414yes—
174C3×C4—34no—
175C3×C4—34no—
176C3×C4—64no—
177Dic3—364no—
178Dic3—364no—
179Dic3—729no—
180Dic3—729no—
181C32—412no—
182C32#541911yescharacteristic, complemented
183C23C3⋊(C3⋊S4)11yescharacteristic
184Q8—31no—
185Q8—32no—
186Q8—61no—
187Q8—541no—
188Q8—541no—
189Q8#631211yescharacteristic
190C2×C4—31no—
191C2×C4—31no—
192C2×C4—61no—
193C2×C4—61no—
194C2×C4—541no—
195C2×C4—541no—
196C8—541no—
197C8—541no—
198C2×C3—28no—
199C2×C3—49no—
200C2×C3—49no—
201C2×C3—49no—
202C2×C3—818no—
203C2×C3C2×C3⋊#22614yes—
204C2×C3#6426414yes—
205C2×C3#6426914yes—
206C22—21no—
207C22—21no—
208C22C3⋊(C3⋊#226)12yescomplemented
209C22C2×C3⋊(C3⋊S4)11yescharacteristic
210C4—31no—
211C4—31no—
212C4—61no—
213C4—1081no—
214C4—1081no—
215C3—49no—
216C3#1059365314yescomplemented
217C2—22no—
218C2C2×C3⋊(C3⋊#226)11yescharacteristic
219C2#1172446811yescharacteristic
220C2#1172447311yescharacteristic
subgroupG/H
center, FrattiniC22C2×C3⋊(C3⋊S4)
commutator, cosocleC2×C32×SL(2,𝔽3)C22
FittingC22×C32×Q8D3
socleC22×C32C2×S4
prime radicalC22×C32×SL(2,𝔽3)C2
hypercenterC23C3⋊(C3⋊S4)
hypocenterC32×SL(2,𝔽3)D4
LCSG ▷ C2×C32×SL(2,𝔽3) ▷ C32×SL(2,𝔽3)
UCSC1 ◁ C22 ◁ C23

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
108. C22×C32#226semidirect
172. C22×C3C3⋊#226semidirect
182. C32#5419semidirect
208. C22C3⋊(C3⋊#226)semidirect
216. C3#10593653semidirect

Maximal subgroups

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

subgroupindex
C22×C32×SL(2,𝔽3)2
C2×C3⋊(C3⋊#226)2
C3⋊(C3⋊#859)2
#105936533
#105929273
#888644