Finite Group Databasebeta

← #13165967

Subgroups — #13165967

Order 1728.

658 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 658 conjugacy classes of subgroups — too many to draw. Only its 67 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⋊D3×SL(2,𝔽3)C211yesmaximal, characteristic, complemented
2C2×C32×GL(2,𝔽3)C211yesmaximal, characteristic, complemented
3C3⋊(C3⋊#859)C211yesmaximal, characteristic, complemented
4#10593217—34nomaximal
5#10592869—31nomaximal
6C2×C32×SL(2,𝔽3)C2211yescharacteristic
7C2×#6290—41nomaximal
8C3⋊D3×SL(2,𝔽3)—21no—
9C32×GL(2,𝔽3)—21no—
10C2×Q8×C3⋊D3D311yescharacteristic, complemented
11C2×D3×SL(2,𝔽3)—34no—
12C2×C3×GL(2,𝔽3)D314yescomplemented
13C2×C32×SD16—31no—
14#64147—31no—
15C2×#3754—31no—
16C3⋊#859—34no—
17C3⋊#859—64no—
18#63662—31no—
19#63647—31no—
20#63643—31no—
21C22×C3×C3⋊D3—41no—
22C22×C32×D3—41no—
23C2×C3⋊(C3⋊Dic3)—41no—
24C32×SL(2,𝔽3)D411yescharacteristic, complemented
25#6290—44no—
26#5421—91no—
27#5170—94no—
28C2×C32×Q8C2×D311yescharacteristic
29C2×C32×D4—31no—
30C2×#638—61no—
31Q8×C3⋊D3—21no—
32Q8×C3⋊D3—61no—
33C2×C4×C3⋊D3—31no—
34C2×#634—31no—
35C2×C3×SL(2,𝔽3)—24no—
36C2×C3×SL(2,𝔽3)C2×D314yes—
37C2×#625—124no—
38D3×SL(2,𝔽3)—64no—
39C3×GL(2,𝔽3)—24no—
40C32×SD16—61no—
41C32×SD16—61no—
42C2×C32×C8—31no—
43#3757—61no—
44#3754—62no—
45C4⋊(C3⋊Dic3)—31no—
46C2×C3⋊(C3⋊C8)—31no—
47C22×C33—41no—
48C2×C3×C3⋊D3—42no—
49C2×C3×C3⋊D3—81no—
50C2×C3×C3⋊D3—81no—
51C2×C32×D3—42no—
52C2×C32×D3—81no—
53C2×C32×D3—81no—
54C3⋊(C3⋊Dic3)—42no—
55C2×D3×Q8—34no—
56C22×SL(2,𝔽3)—91no—
57C2×GL(2,𝔽3)C3⋊D311yescharacteristic, complemented
58C2×C3×SD16—34no—
59#938—94no—
60C2×#214—94no—
61#859—91no—
62#859—184no—
63#835—94no—
64(C2×D3)⋊C8—94no—
65#816—94no—
66C23×C32—61no—
67C22×C3⋊D3S411yescharacteristic
68C22×C3×D3—44no—
69C22×C3×D3—124no—
70C32×Q8—31no—
71C32×Q8#6711yescharacteristic, complemented
72C32×D4—61no—
73C32×D4—61no—
74C2×C32×C4—31no—
75#638—122no—
76C2×C3⋊Dic3—31no—
77C2×C3⋊Dic3—61no—
78C2×C3⋊Dic3—124no—
79C2×C3⋊Dic3—244no—
80C4×C3⋊D3—61no—
81C4×C3⋊D3—61no—
82#634—32no—
83#634—61no—
84C3×SL(2,𝔽3)—24no—
85C3×SL(2,𝔽3)#6714yescomplemented
86#625—1216no—
87C32×C8—61no—
88C3⋊(C3⋊C8)—61no—
89#448—271noSylow
90C2×C33—41no—
91C2×C33—41no—
92C2×C33—41no—
93C3×C3⋊D3—82no—
94C32×D3—82no—
95C2×C3×Q8D3214yes—
96C2×C3×D4—34no—
97C2×#67—184no—
98C2×#67—361no—
99D3×Q8—64no—
100D3×Q8—184no—
101C2×C4×D3—94no—
102C2×Dic6—94no—
103C2×SL(2,𝔽3)—24no—
104C2×SL(2,𝔽3)—181no—
105C2×SL(2,𝔽3)C2×C3⋊D311yescharacteristic
106GL(2,𝔽3)—21no—
107C3×SD16—64no—
108C3×SD16—64no—
109C2×C3×C8—34no—
110#217—184no—
111#214—188no—
112C4⋊Dic3—94no—
113C2×C3⋊C8—94no—
114C22×C32—44no—
115C22×C32—84no—
116C22×C32—121no—
117C22×C32—121no—
118C22×C32—121no—
119C22×C32C2×S411yescharacteristic
120C2×C3⋊D3—21no—
121C2×C3⋊D3—21no—
122C2×C3⋊D3GL(2,𝔽3)12yescomplemented
123C2×C3×D3—48no—
124C2×C3×D3—84no—
125C2×C3×D3—84no—
126C2×C3×D3—128no—
127C2×C3×D3—244no—
128C2×C3×D3—244no—
129C32×C4—31no—
130C32×C4—31no—
131C3⋊Dic3—61no—
132C3⋊Dic3—121no—
133C3⋊Dic3—128no—
134C3⋊Dic3—248no—
135C22×Q8—91no—
136C2×SD16—31no—
137C2×SD16—271no—
138#121—271no—
139#103—271no—
140#103—271no—
141C22⋊C8—271no—
142C33—41noSylow
143C23×C3—64no—
144C23×C3—361no—
145C22×D3—34no—
146C22×D3—41no—
147C3×Q8—34no—
148C3×Q8#62514yescomplemented
149C3×D4—64no—
150C3×D4—64no—
151C2×C3×C4—34no—
152#67—368no—
153#67—364no—
154C2×Dic3—94no—
155C2×Dic3—184no—
156C2×Dic3—361no—
157C2×Dic3—724no—
158C4×D3—184no—
159C4×D3—184no—
160Dic6—98no—
161Dic6—184no—
162SL(2,𝔽3)—24no—
163SL(2,𝔽3)#63811yescharacteristic, complemented
164C3×C8—64no—
165C3⋊C8—184no—
166C2×C32—44no—
167C2×C32—44no—
168C2×C32—44no—
169C2×C32—84no—
170C2×C32—84no—
171C2×C32—84no—
172C2×C32—241no—
173C2×C32C2×GL(2,𝔽3)11yescharacteristic
174C2×C32#98311yescharacteristic
175C2×C32GL(2,ℤ/4)11yescharacteristic
176C3⋊D3—22no—
177C3×D3—88no—
178C3×D3—248no—
179C2×Q8—181no—
180C2×Q8—271no—
181C2×Q8—541no—
182C2×Q8D3×C3⋊D311yescharacteristic
183C2×D4—31no—
184C2×D4—541no—
185C22×C4—271no—
186SD16—61no—
187SD16—61no—
188SD16—542no—
189C2×C8—31no—
190C2×C8—271no—
191C4⋊C4—271no—
192C22⋊C4—541no—
193C22×C3—41no—
194C22×C3—84no—
195C22×C3—124no—
196C22×C3—124no—
197C22×C3—124no—
198C22×C3—362no—
199C22×C3—721no—
200C22×C3—721no—
201C22×C3D3×S414yes—
202C2×D3—38no—
203C2×D3—42no—
204C2×D3—64no—
205C2×D3—64no—
206C2×D3—81no—
207C2×D3—81no—
208C3×C4—34no—
209C3×C4—34no—
210Dic3—184no—
211Dic3—362no—
212Dic3—364no—
213Dic3—728no—
214C32—44no—
215C32—84no—
216C32#542111yescharacteristic, complemented
217C23—61no—
218C23—91no—
219Q8—31no—
220Q8—272no—
221Q8—541no—
222Q8#629011yescharacteristic, complemented
223D4—61no—
224D4—61no—
225D4—1082no—
226C2×C4—31no—
227C2×C4—271no—
228C2×C4—541no—
229C2×C4—541no—
230C2×C4—541no—
231C8—61no—
232C8—541no—
233C2×C3—41no—
234C2×C3—41no—
235C2×C3—41no—
236C2×C3—84no—
237C2×C3—84no—
238C2×C3—84no—
239C2×C3—244no—
240C2×C3—722no—
241C2×C3#6420114yes—
242C2×C3D3×GL(2,𝔽3)14yes—
243C2×C3#6420914yes—
244D3—68no—
245D3—82no—
246C22—92no—
247C22—121no—
248C22—121no—
249C22—121no—
250C22—181no—
251C22—181no—
252C22C3⋊D3×S411yescharacteristic
253C4—31no—
254C4—31no—
255C4—541no—
256C4—1081no—
257C3—41no—
258C3—84no—
259C3#1059321714yescomplemented
260C2—182no—
261C2—241no—
262C2#1172418111yescharacteristic
263C2C3⋊D3×GL(2,𝔽3)11yescharacteristic
264C2#1172418911yescharacteristic
subgroupG/H
center, Frattini, hypercenterC22C3⋊D3×S4
commutator, cosocleC2×C32×SL(2,𝔽3)C22
FittingC2×C32×Q8C2×D3
socleC22×C32C2×S4
hypocenterC32×SL(2,𝔽3)D4
LCSG ▷ C2×C32×SL(2,𝔽3) ▷ C32×SL(2,𝔽3)
UCSC1 ◁ C22

Complements

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

NH ≅ G/Ntype
1. C2×C3⋊D3×SL(2,𝔽3)C2semidirect
2. C2×C32×GL(2,𝔽3)C2semidirect
3. C3⋊(C3⋊#859)C2semidirect
10. C2×Q8×C3⋊D3D3semidirect
12. C2×C3×GL(2,𝔽3)D3semidirect
24. C32×SL(2,𝔽3)D4semidirect
57. C2×GL(2,𝔽3)C3⋊D3semidirect
71. C32×Q8#67semidirect
85. C3×SL(2,𝔽3)#67semidirect
122. C2×C3⋊D3GL(2,𝔽3)semidirect
148. C3×Q8#625semidirect
163. SL(2,𝔽3)#638semidirect
216. C32#5421semidirect
222. Q8#6290semidirect
259. C3#10593217semidirect

Maximal subgroups

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

subgroupindex
C2×C3⋊D3×SL(2,𝔽3)2
C2×C32×GL(2,𝔽3)2
C3⋊(C3⋊#859)2
#105932173
#105928693
C2×#62904