Finite Group Databasebeta

← #13165968

Subgroups — #13165968

Order 1728.

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

normal subgroups only

This group has 738 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×C3⋊(C3⋊GL(2,𝔽3))C211yesmaximal, characteristic, complemented
3C32×#859C211yesmaximal, characteristic, complemented
4#10593218—34nomaximal
5#10592505—31nomaximal
6C2×C32×SL(2,𝔽3)C2211yescharacteristic
7C2×#6292—41nomaximal
8C3⋊D3×SL(2,𝔽3)—21no—
9C3⋊(C3⋊GL(2,𝔽3))—21no—
10C2×Q8×C3⋊D3D311yescharacteristic, complemented
11C2×D3×SL(2,𝔽3)—34no—
12C2×C3⋊GL(2,𝔽3)—34no—
13C2×C3⋊GL(2,𝔽3)—64no—
14C2×#3755—31no—
15C2×#3744—31no—
16#64104—31no—
17C3×#859D314yescomplemented
18C32×#103—31no—
19#63647—31no—
20#63637—31no—
21C22×C3⋊(C3⋊D3)—41no—
22C22×C3×C3⋊D3—41no—
23C2×C32×Dic3—41no—
24C32×SL(2,𝔽3)D411yescharacteristic, complemented
25#6292—44no—
26#5421—91no—
27#4806—94no—
28C2×C32×Q8C2×D311yescharacteristic
29Q8×C3⋊D3—21no—
30Q8×C3⋊D3—61no—
31C2×C4⋊(C3⋊D3)—31no—
32C2×C4⋊(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×#626—124no—
38D3×SL(2,𝔽3)—64no—
39C3⋊GL(2,𝔽3)—68no—
40C3⋊GL(2,𝔽3)—64no—
41C2×C32×C8—31no—
42C32×C4⋊C4—31no—
43#3755—61no—
44#3755—61no—
45#3752—61no—
46C2×C3⋊(C3⋊C8)—31no—
47#3744—62no—
48C22×C33—41no—
49C2×C3⋊(C3⋊D3)—42no—
50C2×C3⋊(C3⋊D3)—81no—
51C2×C3⋊(C3⋊D3)—81no—
52C2×C3×C3⋊D3—42no—
53C2×C3×C3⋊D3—81no—
54C2×C3×C3⋊D3—81no—
55C32×Dic3—42no—
56C2×D3×Q8—34no—
57C22×SL(2,𝔽3)—91no—
58C2×GL(2,𝔽3)—91no—
59C2×GL(2,𝔽3)—184no—
60C2×#215—94no—
61C2×#204—94no—
62#895—94no—
63#859C3⋊D311yescharacteristic, complemented
64C3×#103—34no—
65(C2×D3)⋊C8—94no—
66#810—94no—
67C22×C3⋊D3—61no—
68C22×C3⋊D3—124no—
69C22×C3⋊D3—244no—
70C22×C3⋊D3S411yescharacteristic
71C22×C3×D3—124no—
72C32×Q8—31no—
73C32×Q8#6711yescharacteristic, complemented
74C2×C32×C4—31no—
75C2×C32×C4—61no—
76C2×C3⋊Dic3—31no—
77C4⋊(C3⋊D3)—61no—
78C4⋊(C3⋊D3)—61no—
79C4⋊(C3⋊D3)—122no—
80C4×C3⋊D3—61no—
81C4×C3⋊D3—61no—
82#634—32no—
83#634—61no—
84C2×C3×Dic3—44no—
85C3×SL(2,𝔽3)—24no—
86C3×SL(2,𝔽3)D1214yescomplemented
87#626—1216no—
88C32×C8—61no—
89C3⋊(C3⋊C8)—61no—
90#448—271noSylow
91C2×C33—41no—
92C2×C33—41no—
93C2×C33—41no—
94C3⋊(C3⋊D3)—82no—
95C3×C3⋊D3—82no—
96C2×C3×Q8D3214yes—
97C2×#67—361no—
98D3×Q8—64no—
99D3×Q8—184no—
100C2×D12—94no—
101C2×D12—184no—
102C2×C4×D3—94no—
103C2×Dic6—94no—
104C2×SL(2,𝔽3)—24no—
105C2×SL(2,𝔽3)—181no—
106C2×SL(2,𝔽3)C2×C3⋊D311yescharacteristic
107GL(2,𝔽3)—188no—
108GL(2,𝔽3)—181no—
109C2×C3×C8—34no—
110C3×C4⋊C4—34no—
111#215—184no—
112#215—184no—
113(C2×D3)⋊C4—184no—
114C2×C3⋊C8—94no—
115#204—188no—
116C22×C32—44no—
117C22×C32—84no—
118C22×C32C2×S411yescharacteristic
119C2×C3⋊D3—21no—
120C2×C3⋊D3—21no—
121C2×C3⋊D3—121no—
122C2×C3⋊D3—121no—
123C2×C3⋊D3—121no—
124C2×C3⋊D3—128no—
125C2×C3⋊D3—244no—
126C2×C3⋊D3—244no—
127C2×C3⋊D3—248no—
128C2×C3⋊D3—248no—
129C2×C3⋊D3—248no—
130C2×C3⋊D3GL(2,𝔽3)12yescomplemented
131C2×C3×D3—128no—
132C2×C3×D3—244no—
133C2×C3×D3—244no—
134C32×C4—31no—
135C32×C4—31no—
136C32×C4—121no—
137C3⋊Dic3—61no—
138C3×Dic3—48no—
139C22×Q8—91no—
140C2×SD16—271no—
141C2×SD16—271no—
142#121—271no—
143#103—31no—
144#103—271no—
145C22⋊C8—271no—
146C33—41noSylow
147C23×C3—361no—
148C22×D3—34no—
149C22×D3—184no—
150C22×D3—361no—
151C22×D3—724no—
152C3×Q8—34no—
153C3×Q8#62614yescomplemented
154C2×C3×C4—34no—
155C2×C3×C4—64no—
156#67—364no—
157C2×Dic3—41no—
158C2×Dic3—94no—
159D12—184no—
160D12—184no—
161D12—368no—
162C4×D3—184no—
163C4×D3—184no—
164Dic6—98no—
165Dic6—184no—
166SL(2,𝔽3)—24no—
167SL(2,𝔽3)C4⋊(C3⋊D3)11yescharacteristic, complemented
168C3×C8—64no—
169C3⋊C8—184no—
170C2×C32—44no—
171C2×C32—44no—
172C2×C32—44no—
173C2×C32—84no—
174C2×C32—84no—
175C2×C32—84no—
176C2×C32C2×GL(2,𝔽3)11yescharacteristic
177C2×C32#98311yescharacteristic
178C2×C32GL(2,ℤ/4)11yescharacteristic
179C3⋊D3—22no—
180C3⋊D3—248no—
181C3⋊D3—2416no—
182C3⋊D3—241no—
183C3×D3—248no—
184C2×Q8—181no—
185C2×Q8—271no—
186C2×Q8—541no—
187C2×Q8D3×C3⋊D311yescharacteristic
188C2×D4—271no—
189C2×D4—541no—
190C22×C4—271no—
191SD16—542no—
192SD16—541no—
193SD16—541no—
194C2×C8—31no—
195C2×C8—271no—
196C4⋊C4—31no—
197C22⋊C4—541no—
198C22×C3—41no—
199C22×C3—84no—
200C22×C3—362no—
201C22×C3—721no—
202C22×C3—721no—
203C22×C3D3×S414yes—
204C2×D3—38no—
205C2×D3—64no—
206C2×D3—64no—
207C2×D3—362no—
208C2×D3—364no—
209C2×D3—364no—
210C2×D3—364no—
211C2×D3—721no—
212C2×D3—721no—
213C2×D3—728no—
214C2×D3—728no—
215C2×D3—728no—
216C3×C4—34no—
217C3×C4—34no—
218C3×C4—124no—
219Dic3—42no—
220Dic3—184no—
221C32—44no—
222C32—84no—
223C32#542111yescharacteristic, complemented
224C23—91no—
225C23—541no—
226Q8—31no—
227Q8—272no—
228Q8—541no—
229Q8#629211yescharacteristic, complemented
230D4—541no—
231D4—541no—
232D4—1082no—
233C2×C4—31no—
234C2×C4—61no—
235C2×C4—271no—
236C2×C4—541no—
237C2×C4—541no—
238C8—61no—
239C8—541no—
240C2×C3—41no—
241C2×C3—41no—
242C2×C3—41no—
243C2×C3—84no—
244C2×C3—84no—
245C2×C3—84no—
246C2×C3—722no—
247C2×C3#6420214yes—
248C2×C3D3×GL(2,𝔽3)14yes—
249C2×C3#6421014yes—
250D3—68no—
251D3—722no—
252D3—7216no—
253D3—724no—
254C22—92no—
255C22—181no—
256C22—181no—
257C22—1081no—
258C22—1081no—
259C22—1081no—
260C22C3⋊D3×S411yescharacteristic
261C4—31no—
262C4—31no—
263C4—121no—
264C4—541no—
265C3—41no—
266C3—84no—
267C3#1059321814yescomplemented
268C2—182no—
269C2—2161no—
270C2#1172418211yescharacteristic
271C2C3⋊D3×GL(2,𝔽3)11yescharacteristic
272C2#1172419011yescharacteristic
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×C3⋊(C3⋊GL(2,𝔽3))C2semidirect
3. C32×#859C2semidirect
10. C2×Q8×C3⋊D3D3semidirect
17. C3×#859D3semidirect
24. C32×SL(2,𝔽3)D4semidirect
63. #859C3⋊D3semidirect
73. C32×Q8#67semidirect
86. C3×SL(2,𝔽3)D12semidirect
130. C2×C3⋊D3GL(2,𝔽3)semidirect
153. C3×Q8#626semidirect
167. SL(2,𝔽3)C4⋊(C3⋊D3)semidirect
223. C32#5421semidirect
229. Q8#6292semidirect
267. C3#10593218semidirect

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×C3⋊(C3⋊GL(2,𝔽3))2
C32×#8592
#105932183
#105925053
C2×#62924