Finite Group Databasebeta

← #13165965

Subgroups — #13165965

Order 1728.

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

normal subgroups only

This group has 714 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⋊(C3⋊GL(2,𝔽3))C211yesmaximal, characteristic, complemented
2C2×C32×GL(2,𝔽3)C211yesmaximal, characteristic, complemented
3SL(2,𝔽3)×C3⋊Dic3C211yesmaximal, characteristic, complemented
4#10593215—34nomaximal
5#10592862—31nomaximal
6C2×C32×SL(2,𝔽3)C2211yescharacteristic
7C2×#6291—41nomaximal
8C3⋊(C3⋊GL(2,𝔽3))—21no—
9C32×GL(2,𝔽3)—21no—
10C2×C3⋊GL(2,𝔽3)—34no—
11C2×C3⋊GL(2,𝔽3)—64no—
12C2×C3×GL(2,𝔽3)D314yescomplemented
13C2×C32×SD16—31no—
14Q8×C3⋊Dic3D311yescharacteristic, complemented
15C2×#3755—31no—
16#64149—31no—
17Dic3×SL(2,𝔽3)—34no—
18#63659—31no—
19#63648—31no—
20#63641—31no—
21C22×C3⋊(C3⋊D3)—41no—
22C22×C32×D3—41no—
23C2×C3×C3⋊Dic3—41no—
24C32×SL(2,𝔽3)D411yescharacteristic, complemented
25#6291—44no—
26C4⋊GL(2,𝔽3)—91no—
27#5163—94no—
28C2×C32×Q8C2×D311yescharacteristic
29C2×C32×D4—31no—
30C2×#638—61no—
31C2×#638—61no—
32C2×C4⋊(C3⋊D3)—31no—
33C2×C3×SL(2,𝔽3)—24no—
34C2×C3×SL(2,𝔽3)C2×D314yes—
35C2×#626—124no—
36C3⋊GL(2,𝔽3)—68no—
37C3⋊GL(2,𝔽3)—64no—
38C3×GL(2,𝔽3)—24no—
39C32×SD16—61no—
40C32×SD16—61no—
41C2×C32×C8—31no—
42#3755—61no—
43#3755—61no—
44C4⋊(C3⋊Dic3)—31no—
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×GL(2,𝔽3)—91no—
56C2×GL(2,𝔽3)—184no—
57C2×GL(2,𝔽3)C3⋊D311yescharacteristic, complemented
58C2×C3×SD16—34no—
59Q8×Dic3—34no—
60C2×#215—94no—
61#940—94no—
62C4×SL(2,𝔽3)—91no—
63#832—94no—
64#821—94no—
65Dic3⋊C8—94no—
66C23×C32—61no—
67C22×C3⋊D3—61no—
68C22×C3⋊D3—124no—
69C22×C3⋊D3—244no—
70C22×C3×D3—44no—
71C32×Q8—31no—
72C32×Q8D1211yescharacteristic, complemented
73C32×D4—61no—
74C32×D4—61no—
75C2×C32×C4—31no—
76#638—122no—
77#638—122no—
78C2×C3⋊Dic3—31no—
79C2×C3⋊Dic3S411yescharacteristic
80C4⋊(C3⋊D3)—61no—
81C4⋊(C3⋊D3)—61no—
82C2×C3×Dic3—124no—
83C3×SL(2,𝔽3)—24no—
84C3×SL(2,𝔽3)#6714yescomplemented
85#626—1216no—
86C32×C8—61no—
87C3⋊(C3⋊C8)—61no—
88#460—271noSylow
89C2×C33—41no—
90C2×C33—41no—
91C2×C33—41no—
92C3⋊(C3⋊D3)—82no—
93C32×D3—82no—
94C2×C3×Q8D3214yes—
95C2×C3×D4—34no—
96C2×#67—184no—
97C2×#67—184no—
98C2×D12—94no—
99C2×D12—361no—
100C2×SL(2,𝔽3)—24no—
101C2×SL(2,𝔽3)C2×C3⋊D311yescharacteristic
102GL(2,𝔽3)—21no—
103GL(2,𝔽3)—188no—
104GL(2,𝔽3)—181no—
105C3×SD16—64no—
106C3×SD16—64no—
107C2×C3×C8—34no—
108#215—184no—
109#215—184no—
110C4⋊Dic3—94no—
111C4×Dic3—94no—
112C2×C3⋊C8—94no—
113C22×C32—44no—
114C22×C32—84no—
115C22×C32—121no—
116C22×C32—121no—
117C22×C32—121no—
118C22×C32C2×S411yescharacteristic
119C2×C3⋊D3—121no—
120C2×C3⋊D3—121no—
121C2×C3⋊D3—121no—
122C2×C3⋊D3—128no—
123C2×C3⋊D3—244no—
124C2×C3⋊D3—244no—
125C2×C3⋊D3—248no—
126C2×C3⋊D3—248no—
127C2×C3⋊D3—248no—
128C2×C3×D3—48no—
129C2×C3×D3—84no—
130C2×C3×D3—84no—
131C32×C4—31no—
132C32×C4—31no—
133C3⋊Dic3—61no—
134C3⋊Dic3GL(2,𝔽3)12yescomplemented
135C3×Dic3—128no—
136C2×SD16—31no—
137C2×SD16—271no—
138#127—271no—
139C4×Q8—91no—
140C4⋊C8—271no—
141#102—271no—
142#102—271no—
143C33—41noSylow
144C23×C3—64no—
145C22×D3—41no—
146C22×D3—184no—
147C22×D3—361no—
148C22×D3—724no—
149C3×Q8—34no—
150C3×Q8#62614yescomplemented
151C3×D4—64no—
152C3×D4—64no—
153C2×C3×C4—34no—
154C2×C3×C4—361no—
155#67—368no—
156#67—368no—
157C2×Dic3—34no—
158C2×Dic3—94no—
159D12—184no—
160D12—184no—
161D12—364no—
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×C32C4⋊S411yescharacteristic
174C2×C32C2×GL(2,𝔽3)11yescharacteristic
175C2×C32#98611yescharacteristic
176C3⋊D3—248no—
177C3⋊D3—2416no—
178C3⋊D3—241no—
179C3×D3—88no—
180C2×Q8D3×C3⋊D311yescharacteristic
181C2×D4—31no—
182C2×D4—271no—
183C2×D4—541no—
184C2×D4—541no—
185SD16—61no—
186SD16—61no—
187SD16—541no—
188SD16—541no—
189C2×C8—31no—
190C2×C8—271no—
191C4⋊C4—271no—
192C42—271no—
193C22×C3—41no—
194C22×C3—84no—
195C22×C3—124no—
196C22×C3—124no—
197C22×C3—124no—
198C22×C3D3×S414yes—
199C2×D3—42no—
200C2×D3—81no—
201C2×D3—81no—
202C2×D3—362no—
203C2×D3—364no—
204C2×D3—364no—
205C2×D3—364no—
206C2×D3—721no—
207C2×D3—721no—
208C2×D3—728no—
209C2×D3—728no—
210C2×D3—728no—
211C3×C4—34no—
212C3×C4—34no—
213C3×C4—362no—
214Dic3—38no—
215Dic3—184no—
216C32—44no—
217C32—84no—
218C32C4⋊GL(2,𝔽3)11yescharacteristic, complemented
219C23—61no—
220C23—541no—
221Q8—31no—
222Q8#629111yescharacteristic, complemented
223D4—61no—
224D4—61no—
225D4—541no—
226D4—541no—
227D4—1082no—
228D4—1082no—
229C2×C4—31no—
230C2×C4—91no—
231C2×C4—271no—
232C8—61no—
233C8—541no—
234C2×C3—41no—
235C2×C3—41no—
236C2×C3—41no—
237C2×C3—84no—
238C2×C3—84no—
239C2×C3—84no—
240C2×C3—244no—
241C2×C3#6419914yes—
242C2×C3D3×GL(2,𝔽3)14yes—
243C2×C3#6420714yes—
244D3—82no—
245D3—722no—
246D3—7216no—
247D3—724no—
248C22—121no—
249C22—121no—
250C22—121no—
251C22—1081no—
252C22—1081no—
253C22—1081no—
254C22C3⋊D3×S411yescharacteristic
255C4—31no—
256C4—31no—
257C4—92no—
258C4—541no—
259C3—41no—
260C3—84no—
261C3#1059321514yescomplemented
262C2—241no—
263C2—2161no—
264C2#1172417911yescharacteristic
265C2C3⋊D3×GL(2,𝔽3)11yescharacteristic
266C2#1172418711yescharacteristic
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⋊(C3⋊GL(2,𝔽3))C2semidirect
2. C2×C32×GL(2,𝔽3)C2semidirect
3. SL(2,𝔽3)×C3⋊Dic3C2semidirect
12. C2×C3×GL(2,𝔽3)D3semidirect
14. Q8×C3⋊Dic3D3semidirect
24. C32×SL(2,𝔽3)D4semidirect
57. C2×GL(2,𝔽3)C3⋊D3semidirect
72. C32×Q8D12semidirect
84. C3×SL(2,𝔽3)#67semidirect
134. C3⋊Dic3GL(2,𝔽3)semidirect
150. C3×Q8#626semidirect
163. SL(2,𝔽3)#638semidirect
218. C32C4⋊GL(2,𝔽3)semidirect
222. Q8#6291semidirect
261. C3#10593215semidirect

Maximal subgroups

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

subgroupindex
C2×C3⋊(C3⋊GL(2,𝔽3))2
C2×C32×GL(2,𝔽3)2
SL(2,𝔽3)×C3⋊Dic32
#105932153
#105928623
C2×#62914