Finite Group Databasebeta

← #13166088

Subgroups — #13166088

Order 1728.

423 conjugacy classes of subgroups; only the 50 normal subgroups are shown.

normal subgroups only

This group has 423 conjugacy classes of subgroups — too many to draw. Only its 50 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×C3⋊(C3⋊(#41⋊C3))C211yesmaximal, characteristic
2C3×C3⋊#859C212yesmaximal, complemented
3#10593200—32nomaximal
4#10591554—31nomaximal
5C2×C32×SL(2,𝔽3)C2211yescharacteristic
6C3⋊(C3⋊(#41⋊C3))C412yescomplemented
7#88815—41nomaximal
8C2×#3832D311yescharacteristic
9C2×C3⋊(#41⋊C3)—32no—
10C2×C3⋊(#41⋊C3)—61no—
11#63900—31no—
12C2×#3712—31no—
13C3⋊#859D312yescomplemented
14C3×#859—32no—
15C3×#859—62no—
16C3×#835—32no—
17#63564—32no—
18C2×C3×C3⋊Dic3—41no—
19C2×C3×C3⋊Dic3—42no—
20C32×SL(2,𝔽3)C2×C411yescharacteristic, complemented
21#5426—91no—
22#4803—92no—
23C2×C32×Q8C2×D311yescharacteristic
24#3832—32no—
25#3832Dic312yescomplemented
26C2×C4⋊(C3⋊D3)—31no—
27C2×C4×C3⋊D3—31no—
28C2×C3×SL(2,𝔽3)—22no—
29C2×C3×SL(2,𝔽3)—21no—
30C2×C3×SL(2,𝔽3)—41no—
31C2×C3×SL(2,𝔽3)C2×D312yes—
32C2×#624—61no—
33C3⋊(#41⋊C3)—34no—
34C3⋊(#41⋊C3)—62no—
35C3×C4⋊Dic3—32no—
36C2×C3×C3⋊C8—32no—
37#3724—61no—
38Dic3⋊Dic3—122no—
39#3712—62no—
40C22×C33—41no—
41C3×C3⋊Dic3—42no—
42C3×C3⋊Dic3—82no—
43C2×#239—32no—
44C2×#239—61no—
45C2×#41⋊C3—91no—
46C2×ΓL(1,𝔽25)—92no—
47D3×C4⋊C4—92no—
48#859—32no—
49#859—62no—
50C3×#103—92no—
51#835—32no—
52#821—92no—
53#809—92no—
54C22×C3⋊D3—31no—
55C32×Q8—31no—
56C32×Q8C2×Dic311yescharacteristic
57C2×C32×C4—31no—
58C2×C3⋊Dic3—42no—
59C2×C3⋊Dic3S411yescharacteristic
60C4⋊(C3⋊D3)—32no—
61C4⋊(C3⋊D3)—61no—
62C4×C3⋊D3—32no—
63C4×C3⋊D3—32no—
64C2×C3×Dic3—62no—
65C2×C3×Dic3—122no—
66C2×C3×Dic3—122no—
67C2×C3×Dic3—241no—
68C2×C3×Dic3—242no—
69C3×SL(2,𝔽3)—22no—
70C3×SL(2,𝔽3)—21no—
71C3×SL(2,𝔽3)—41no—
72C3×SL(2,𝔽3)C4×D312yescomplemented
73#624—122no—
74C3×C3⋊C8—62no—
75#417—271noSylow
76C2×C33—41no—
77C2×C33—41no—
78C2×C33—41no—
79C2×C3×Q8—21no—
80C2×C3×Q8D3212yes—
81#239—34no—
82#239—62no—
83#239—94no—
84#239—182no—
85C2×D12—92no—
86C2×D12—181no—
87C2×C4×D3—92no—
88C2×C4×D3—181no—
89C2×C4×D3—182no—
90#41⋊C3—92no—
91C2×SL(2,𝔽3)—22no—
92C2×SL(2,𝔽3)—41no—
93C2×SL(2,𝔽3)D3211yescharacteristic
94C2×C3×C8—92no—
95C3×C4⋊C4—92no—
96C4⋊Dic3—32no—
97C4⋊Dic3—361no—
98Dic3⋊C4—182no—
99C2×C3⋊C8—32no—
100ΓL(1,𝔽25)—184no—
101C22×C32—42no—
102C22×C32—81no—
103C22×C32—82no—
104C22×C32—161no—
105C22×C32C2×S411yescharacteristic
106C2×C3⋊D3—32no—
107C2×C3⋊D3—61no—
108C2×C3⋊D3—61no—
109C32×C4—31no—
110C32×C4—31no—
111C3⋊Dic3—82no—
112C3⋊Dic3SL(2,ℤ/4)12yes—
113C3×Dic3—124no—
114C3×Dic3—122no—
115C3×Dic3—242no—
116C3×Dic3—242no—
117C3×Dic3—244no—
118C2×#41—91no—
119C2×M16—271no—
120C2×C4⋊C4—271no—
121#103—92no—
122#102—272no—
123C33—41noSylow
124C22×D3—92no—
125C22×D3—181no—
126C3×Q8—21no—
127C3×Q8—32no—
128C3×Q8—61no—
129C3×Q8D3×Dic312yes—
130C2×C3×C4—32no—
131C2×C3×C4—61no—
132C2×C3×C4—182no—
133C2×C3×C4—361no—
134C2×Dic3—32no—
135C2×Dic3—61no—
136C2×Dic3—62no—
137C2×Dic3—122no—
138C2×Dic3—242no—
139D12—94no—
140D12—182no—
141D12—182no—
142D12—182no—
143C4×D3—94no—
144C4×D3—94no—
145C4×D3—182no—
146C4×D3—182no—
147C4×D3—364no—
148SL(2,𝔽3)—22no—
149SL(2,𝔽3)—41no—
150SL(2,𝔽3)#62411yescharacteristic, complemented
151C3×C8—182no—
152C3⋊C8—62no—
153C2×C32—42no—
154C2×C32—42no—
155C2×C32—42no—
156C2×C32—81no—
157C2×C32—81no—
158C2×C32—81no—
159C2×C32—82no—
160C2×C32—82no—
161C2×C32—82no—
162C2×C32—161no—
163C2×C32—161no—
164C2×C32—161no—
165C2×C32#98411yescharacteristic
166C2×C32#98611yescharacteristic
167C2×C32C2×SL(2,ℤ/4)11yescharacteristic
168C3⋊D3—62no—
169#41—92no—
170#41—272no—
171C2×Q8#111311yescharacteristic
172C2×D4—271no—
173C22×C4—271no—
174C22×C4—541no—
175M16—542no—
176C2×C8—92no—
177C4⋊C4—92no—
178C4⋊C4—541no—
179C22×C3—21no—
180C22×C3—41no—
181C22×C3—82no—
182C22×C3—161no—
183C22×C3D3×S412yes—
184C2×D3—94no—
185C2×D3—182no—
186C2×D3—182no—
187C2×D3—182no—
188C2×D3—361no—
189C2×D3—361no—
190C3×C4—32no—
191C3×C4—32no—
192C3×C4—61no—
193C3×C4—61no—
194C3×C4—362no—
195C3×C4—362no—
196Dic3—34no—
197Dic3—62no—
198Dic3—122no—
199Dic3—242no—
200Dic3—244no—
201C32—42no—
202C32—81no—
203C32—82no—
204C32—161no—
205C32#542611yescharacteristic, complemented
206C23—271no—
207Q8—31no—
208Q8#629411yescharacteristic
209D4—272no—
210D4—541no—
211C2×C4—31no—
212C2×C4—91no—
213C2×C4—182no—
214C2×C4—272no—
215C2×C4—272no—
216C2×C4—1082no—
217C8—182no—
218C2×C3—21no—
219C2×C3—21no—
220C2×C3—21no—
221C2×C3—41no—
222C2×C3—41no—
223C2×C3—41no—
224C2×C3—82no—
225C2×C3—82no—
226C2×C3—82no—
227C2×C3—161no—
228C2×C3—161no—
229C2×C3—161no—
230C2×C3#6419612yes—
231C2×C3#6419912yes—
232C2×C3D3×SL(2,ℤ/4)12yes—
233D3—184no—
234D3—362no—
235C22—272no—
236C22—541no—
237C22—541no—
238C22#8910111yescharacteristic
239C4—31no—
240C4—31no—
241C4—92no—
242C4—362no—
243C3—21no—
244C3—41no—
245C3—82no—
246C3—161no—
247C3#1059320012yescomplemented
248C2—542no—
249C2#1172420611yescharacteristic
250C2#1172420811yescharacteristic
251C2#1172421511yescharacteristic
subgroupG/H
center, Frattini, hypercenterC22#89101
commutator, hypocenterC32×SL(2,𝔽3)C2×C4
FittingC2×C32×Q8C2×D3
socleC22×C32C2×S4
prime radicalC2×C3⋊(C3⋊(#41⋊C3))C2
cosocleC2×C32×SL(2,𝔽3)C22
LCSG ▷ C32×SL(2,𝔽3)
UCSC1 ◁ C22

Complements

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

NH ≅ G/Ntype
2. C3×C3⋊#859C2semidirect
6. C3⋊(C3⋊(#41⋊C3))C4semidirect
13. C3⋊#859D3semidirect
20. C32×SL(2,𝔽3)C2×C4semidirect
25. #3832Dic3semidirect
72. C3×SL(2,𝔽3)C4×D3semidirect
150. SL(2,𝔽3)#624semidirect
205. C32#5426semidirect
247. C3#10593200semidirect

Maximal subgroups

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

subgroupindex
C2×C3⋊(C3⋊(#41⋊C3))2
C3×C3⋊#8592
#105932003
#105915543
#888154