Finite Group Databasebeta

← #13165959

Subgroups — #13165959

Order 1728.

574 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 574 conjugacy classes of subgroups — too many to draw. Only its 71 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
2C3⋊(C3⋊#859)C211yesmaximal, characteristic, complemented
3C32×#859C211yesmaximal, characteristic, complemented
4#10593209—34nomaximal
5#10592501—31nomaximal
6C2×C32×SL(2,𝔽3)C2211yescharacteristic
7C3⋊D3×SL(2,𝔽3)C412yescomplemented
8#88803—41nomaximal
9C2×Q8×C3⋊D3D311yescharacteristic
10C2×D3×SL(2,𝔽3)—34no—
11C2×#3743—31no—
12#64101—31no—
13C3⋊#859—34no—
14C3⋊#859—64no—
15C3×#859D314yescomplemented
16C32×#103—31no—
17#63662—31no—
18#63643—31no—
19#63637—31no—
20C22×C3×C3⋊D3—41no—
21C2×C3⋊(C3⋊Dic3)—41no—
22C2×C32×Dic3—41no—
23C32×SL(2,𝔽3)C2×C411yescharacteristic
24#5420—91no—
25#4802—94no—
26C2×C32×Q8C2×D311yescharacteristic
27Q8×C3⋊D3—32no—
28Q8×C3⋊D3Dic312yescomplemented
29C2×C4×C3⋊D3—31no—
30C2×#634—31no—
31C2×C3×SL(2,𝔽3)—24no—
32C2×C3×SL(2,𝔽3)C2×D314yes—
33D3×SL(2,𝔽3)—38no—
34C2×C32×C8—31no—
35C32×C4⋊C4—31no—
36#3752—61no—
37C4⋊(C3⋊Dic3)—31no—
38C4×C3⋊Dic3—61no—
39C2×C3⋊(C3⋊C8)—31no—
40#3743—62no—
41(C2×D3)⋊Dic3—124no—
42C22×C33—41no—
43C2×C3×C3⋊D3—42no—
44C2×C3×C3⋊D3—81no—
45C2×C3×C3⋊D3—81no—
46C3⋊(C3⋊Dic3)—81no—
47C32×Dic3—81no—
48C2×D3×Q8—34no—
49C22×SL(2,𝔽3)—91no—
50C2×ΓL(1,𝔽25)—94no—
51#892—94no—
52#859—91no—
53#859—184no—
54#859C3⋊D311yescharacteristic, complemented
55C3×#103—34no—
56#835—94no—
57#816—94no—
58#810—94no—
59C22×C3⋊D3S411yescharacteristic
60C22×C3×D3—124no—
61C32×Q8—31no—
62C32×Q8C2×Dic311yescharacteristic
63C2×C32×C4—31no—
64C2×C32×C4—61no—
65C2×C3⋊Dic3—31no—
66C2×C3⋊Dic3—61no—
67C2×C3⋊Dic3—124no—
68C2×C3⋊Dic3—244no—
69C4×C3⋊D3—32no—
70C4×C3⋊D3—32no—
71#634—32no—
72#634—61no—
73C2×C3×Dic3—44no—
74C3×SL(2,𝔽3)—24no—
75C3×SL(2,𝔽3)C4×D314yes—
76C32×C8—61no—
77C3⋊(C3⋊C8)—61no—
78#419—271noSylow
79C2×C33—41no—
80C2×C33—41no—
81C2×C33—41no—
82C3×C3⋊D3—82no—
83C2×C3×Q8D3214yes—
84D3×Q8—38no—
85D3×Q8—98no—
86C2×C4×D3—94no—
87C2×Dic6—94no—
88C2×SL(2,𝔽3)—24no—
89C2×SL(2,𝔽3)—92no—
90C2×SL(2,𝔽3)C2×C3⋊D311yescharacteristic
91C2×C3×C8—34no—
92C3×C4⋊C4—34no—
93#217—361no—
94(C2×D3)⋊C4—184no—
95C4⋊Dic3—94no—
96C4×Dic3—184no—
97C2×C3⋊C8—94no—
98ΓL(1,𝔽25)—188no—
99C22×C32—44no—
100C22×C32—84no—
101C22×C32C2×S411yescharacteristic
102C2×C3⋊D3—21no—
103C2×C3⋊D3—21no—
104C2×C3⋊D3SL(2,ℤ/4)12yes—
105C2×C3×D3—128no—
106C2×C3×D3—244no—
107C2×C3×D3—244no—
108C32×C4—31no—
109C32×C4—31no—
110C32×C4—121no—
111C3⋊Dic3—32no—
112C3⋊Dic3—121no—
113C3⋊Dic3—244no—
114C3⋊Dic3—248no—
115C3×Dic3—84no—
116C22×Q8—91no—
117C2×M16—271no—
118#117—271no—
119#103—31no—
120#103—271no—
121#103—271no—
122#103—271no—
123C33—41noSylow
124C23×C3—361no—
125C22×D3—34no—
126C3×Q8—34no—
127C3×Q8D3×Dic314yes—
128C2×C3×C4—34no—
129C2×C3×C4—64no—
130C2×Dic3—41no—
131C2×Dic3—94no—
132C2×Dic3—184no—
133C2×Dic3—361no—
134C2×Dic3—724no—
135C4×D3—98no—
136C4×D3—98no—
137Dic6—98no—
138Dic6—184no—
139SL(2,𝔽3)—24no—
140SL(2,𝔽3)C4×C3⋊D311yescharacteristic
141C3×C8—64no—
142C3⋊C8—184no—
143C2×C32—44no—
144C2×C32—44no—
145C2×C32—44no—
146C2×C32—84no—
147C2×C32—84no—
148C2×C32—84no—
149C2×C32#98311yescharacteristic
150C2×C32#98311yescharacteristic
151C2×C32C2×SL(2,ℤ/4)11yescharacteristic
152C3⋊D3—22no—
153C3×D3—248no—
154C2×Q8—92no—
155C2×Q8—271no—
156C2×Q8—272no—
157C2×Q8D3×C3⋊D311yescharacteristic
158C22×C4—271no—
159M16—542no—
160C2×C8—31no—
161C2×C8—271no—
162C4⋊C4—31no—
163C4⋊C4—271no—
164C22⋊C4—541no—
165C42—541no—
166C22×C3—41no—
167C22×C3—84no—
168C22×C3—362no—
169C22×C3—721no—
170C22×C3—721no—
171C22×C3D3×S414yes—
172C2×D3—38no—
173C2×D3—64no—
174C2×D3—64no—
175C3×C4—34no—
176C3×C4—34no—
177C3×C4—124no—
178Dic3—81no—
179Dic3—98no—
180Dic3—364no—
181Dic3—721no—
182Dic3—728no—
183C32—44no—
184C32—84no—
185C32#542011yescharacteristic, complemented
186C23—91no—
187Q8—31no—
188Q8—272no—
189Q8—541no—
190Q8Dic3×C3⋊D311yescharacteristic
191C2×C4—31no—
192C2×C4—61no—
193C2×C4—272no—
194C2×C4—272no—
195C2×C4—271no—
196C2×C4—541no—
197C8—61no—
198C8—541no—
199C2×C3—41no—
200C2×C3—41no—
201C2×C3—41no—
202C2×C3—84no—
203C2×C3—84no—
204C2×C3—84no—
205C2×C3—722no—
206C2×C3#6420114yes—
207C2×C3#6420214yes—
208C2×C3D3×SL(2,ℤ/4)14yes—
209D3—68no—
210C22—92no—
211C22—181no—
212C22—181no—
213C22C3⋊D3×S411yescharacteristic
214C4—31no—
215C4—31no—
216C4—121no—
217C4—272no—
218C4—1081no—
219C3—41no—
220C3—84no—
221C3#1059320914yescomplemented
222C2—182no—
223C2#1172418111yescharacteristic
224C2#1172418211yescharacteristic
225C2C3⋊D3×SL(2,ℤ/4)11yescharacteristic
subgroupG/H
center, Frattini, hypercenterC22C3⋊D3×S4
commutator, hypocenterC32×SL(2,𝔽3)C2×C4
FittingC2×C32×Q8C2×D3
socleC22×C32C2×S4
prime radicalC2×C3⋊D3×SL(2,𝔽3)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⋊#859)C2semidirect
3. C32×#859C2semidirect
7. C3⋊D3×SL(2,𝔽3)C4semidirect
15. C3×#859D3semidirect
28. Q8×C3⋊D3Dic3semidirect
54. #859C3⋊D3semidirect
185. C32#5420semidirect
221. C3#10593209semidirect

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
C3⋊(C3⋊#859)2
C32×#8592
#105932093
#105925013
#888034