Finite Group Databasebeta

← #13165953

Subgroups — #13165953

Order 1728.

586 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 586 conjugacy classes of subgroups — too many to draw. Only its 65 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, complemented
2C2×C32×GL(2,𝔽3)C211yesmaximal, characteristic, complemented
3C3⋊(C3⋊#859)C211yesmaximal, characteristic, complemented
4#10593203—34nomaximal
5#10592871—31nomaximal
6C2×C32×SL(2,𝔽3)C2211yescharacteristic
7C3⋊(C3⋊(#41⋊C3))—21no—
8C32×GL(2,𝔽3)—21no—
9(C2×D3)⋊(C3⋊Dic3)—41nomaximal
10C2×#3832D311yescharacteristic, complemented
11C2×C3⋊(#41⋊C3)—34no—
12C2×C3×GL(2,𝔽3)D314yescomplemented
13C2×C32×SD16—31no—
14#64147—31no—
15C2×#3753—31no—
16C3⋊#859—34no—
17C3⋊#859—64no—
18#63662—31no—
19#63648—31no—
20#63647—31no—
21C22×C32×D3—41no—
22C2×C3⋊(C3⋊Dic3)—41no—
23C2×C3×C3⋊Dic3—41no—
24C32×SL(2,𝔽3)D411yescharacteristic, complemented
25#5427—91no—
26#5172—94no—
27C2×C32×Q8C2×D311yescharacteristic
28C2×C32×D4—31no—
29C2×#638—61no—
30#3832—21no—
31#3832—61no—
32C2×C4⋊(C3⋊D3)—31no—
33C2×C4×C3⋊D3—31no—
34C2×C3×SL(2,𝔽3)—24no—
35C2×C3×SL(2,𝔽3)C2×D314yes—
36C3⋊(#41⋊C3)—64no—
37C3×GL(2,𝔽3)—24no—
38C32×SD16—61no—
39C32×SD16—61no—
40C2×C32×C8—31no—
41#3757—61no—
42#3753—62no—
43C4⋊(C3⋊Dic3)—31no—
44C2×C3⋊(C3⋊C8)—31no—
45(C2×D3)⋊Dic3—124no—
46C22×C33—41no—
47C2×C32×D3—42no—
48C2×C32×D3—81no—
49C2×C32×D3—81no—
50C3⋊(C3⋊Dic3)—81no—
51C3×C3⋊Dic3—81no—
52C2×#239—34no—
53C2×#41⋊C3—91no—
54C2×GL(2,𝔽3)C3⋊D311yescharacteristic, complemented
55C2×C3×SD16—34no—
56#938—94no—
57C2×#213—94no—
58#859—91no—
59#859—184no—
60#835—94no—
61#821—94no—
62(C2×D3)⋊C8—94no—
63C23×C32—61no—
64C22×C3⋊D3—31no—
65C22×C3×D3—44no—
66C32×Q8—31no—
67C32×Q8#6711yescharacteristic
68C32×D4—61no—
69C32×D4—61no—
70C2×C32×C4—31no—
71#638—122no—
72C2×C3⋊Dic3—61no—
73C2×C3⋊Dic3—124no—
74C2×C3⋊Dic3—244no—
75C2×C3⋊Dic3S411yescharacteristic
76C4⋊(C3⋊D3)—32no—
77C4⋊(C3⋊D3)—61no—
78C4×C3⋊D3—61no—
79C4×C3⋊D3—61no—
80C2×C3×Dic3—124no—
81C3×SL(2,𝔽3)—24no—
82C3×SL(2,𝔽3)#6714yescomplemented
83C32×C8—61no—
84C3⋊(C3⋊C8)—61no—
85#449—271noSylow
86C2×C33—41no—
87C2×C33—41no—
88C2×C33—41no—
89C32×D3—82no—
90C2×C3×Q8D3214yes—
91C2×C3×D4—34no—
92C2×#67—184no—
93#239—64no—
94#239—184no—
95C2×D12—94no—
96C2×C4×D3—94no—
97#41⋊C3—181no—
98C2×SL(2,𝔽3)—24no—
99C2×SL(2,𝔽3)C2×C3⋊D311yescharacteristic
100GL(2,𝔽3)—21no—
101C3×SD16—64no—
102C3×SD16—64no—
103C2×C3×C8—34no—
104#217—184no—
105#213—188no—
106(C2×D3)⋊C4—361no—
107C4⋊Dic3—94no—
108C2×C3⋊C8—94no—
109C22×C32—44no—
110C22×C32—84no—
111C22×C32—121no—
112C22×C32—121no—
113C22×C32—121no—
114C22×C32C2×S411yescharacteristic
115C2×C3⋊D3—61no—
116C2×C3⋊D3—61no—
117C2×C3⋊D3—61no—
118C2×C3×D3—48no—
119C2×C3×D3—84no—
120C2×C3×D3—84no—
121C32×C4—31no—
122C32×C4—31no—
123C3⋊Dic3—21no—
124C3⋊Dic3—121no—
125C3⋊Dic3—244no—
126C3⋊Dic3—248no—
127C3×Dic3—244no—
128C2×#41—91no—
129C2×SD16—31no—
130C2×D8—271no—
131#121—271no—
132#103—271no—
133#102—271no—
134C22⋊C8—271no—
135C33—41noSylow
136C23×C3—64no—
137C22×D3—41no—
138C22×D3—94no—
139C3×Q8—34no—
140C3×Q8#62514yes—
141C3×D4—64no—
142C3×D4—64no—
143C2×C3×C4—34no—
144C2×C3×C4—361no—
145#67—368no—
146C2×Dic3—34no—
147C2×Dic3—184no—
148C2×Dic3—361no—
149C2×Dic3—724no—
150D12—98no—
151D12—184no—
152C4×D3—184no—
153C4×D3—184no—
154SL(2,𝔽3)—24no—
155SL(2,𝔽3)#63811yescharacteristic, complemented
156C3×C8—64no—
157C3⋊C8—184no—
158C2×C32—44no—
159C2×C32—44no—
160C2×C32—44no—
161C2×C32—84no—
162C2×C32—84no—
163C2×C32—84no—
164C2×C32—241no—
165C2×C32#98511yescharacteristic
166C2×C32#98611yescharacteristic
167C2×C32GL(2,ℤ/4)11yescharacteristic
168C3⋊D3—121no—
169C3×D3—88no—
170#41—181no—
171#41—541no—
172C2×Q8D3×C3⋊D311yescharacteristic
173C2×D4—31no—
174C2×D4—271no—
175C2×D4—541no—
176C22×C4—271no—
177SD16—61no—
178SD16—61no—
179D8—542no—
180C2×C8—31no—
181C2×C8—271no—
182C4⋊C4—271no—
183C22⋊C4—541no—
184C22×C3—41no—
185C22×C3—84no—
186C22×C3—124no—
187C22×C3—124no—
188C22×C3—124no—
189C22×C3D3×S414yes—
190C2×D3—42no—
191C2×D3—81no—
192C2×D3—81no—
193C2×D3—184no—
194C2×D3—184no—
195C2×D3—184no—
196C3×C4—34no—
197C3×C4—34no—
198C3×C4—721no—
199Dic3—64no—
200Dic3—364no—
201Dic3—721no—
202Dic3—728no—
203C32—44no—
204C32—84no—
205C32#542711yescharacteristic, complemented
206C23—61no—
207C23—271no—
208Q8—31no—
209Q8#629011yescharacteristic
210D4—61no—
211D4—61no—
212D4—272no—
213D4—541no—
214D4—1082no—
215C2×C4—31no—
216C2×C4—91no—
217C2×C4—541no—
218C2×C4—541no—
219C2×C4—541no—
220C8—61no—
221C8—541no—
222C2×C3—41no—
223C2×C3—41no—
224C2×C3—41no—
225C2×C3—84no—
226C2×C3—84no—
227C2×C3—84no—
228C2×C3—244no—
229C2×C3#6419714yes—
230C2×C3#6419914yes—
231C2×C3#6420914yes—
232D3—82no—
233D3—364no—
234C22—121no—
235C22—121no—
236C22—121no—
237C22—541no—
238C22—541no—
239C22—541no—
240C22C3⋊D3×S411yescharacteristic
241C4—31no—
242C4—31no—
243C4—181no—
244C4—1081no—
245C3—41no—
246C3—84no—
247C3#1059320314yescomplemented
248C2—241no—
249C2—1081no—
250C2#1172417711yescharacteristic
251C2#1172417911yescharacteristic
252C2#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⋊(C3⋊(#41⋊C3))C2semidirect
2. C2×C32×GL(2,𝔽3)C2semidirect
3. C3⋊(C3⋊#859)C2semidirect
10. C2×#3832D3semidirect
12. C2×C3×GL(2,𝔽3)D3semidirect
24. C32×SL(2,𝔽3)D4semidirect
54. C2×GL(2,𝔽3)C3⋊D3semidirect
82. C3×SL(2,𝔽3)#67semidirect
155. SL(2,𝔽3)#638semidirect
205. C32#5427semidirect
247. C3#10593203semidirect

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
C2×C32×GL(2,𝔽3)2
C3⋊(C3⋊#859)2
#105932033
#105928713
(C2×D3)⋊(C3⋊Dic3)4