Finite Group Databasebeta

← #13165961

Subgroups — #13165961

Order 1728.

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

normal subgroups only

This group has 584 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
2C2×C3⋊(C3⋊#226)C211yesmaximal, characteristic, complemented
3C32×#859C211yesmaximal, characteristic, complemented
4#10593211—34nomaximal
5#10592508—31nomaximal
6C2×C32×SL(2,𝔽3)C2211yescharacteristic
7C3⋊D3×SL(2,𝔽3)—21no—
8C3⋊(C3⋊#226)—21no—
9#88803—41nomaximal
10C2×Q8×C3⋊D3D311yescharacteristic
11C2×D3×SL(2,𝔽3)—34no—
12C2×C3⋊#226—34no—
13C2×C3⋊#226—64no—
14C2×#3756—31no—
15C2×#3746—31no—
16#64106—31no—
17C3×#859D314yescomplemented
18C32×#103—31no—
19#63647—31no—
20#63637—31no—
21C22×C3×C3⋊D3—41no—
22C2×C3⋊(C3⋊Dic3)—41no—
23C2×C32×Dic3—41no—
24C32×SL(2,𝔽3)D411yescharacteristic
25#5419—91no—
26#4809—94no—
27C2×C32×Q8C2×D311yescharacteristic
28Q8×C3⋊D3—21no—
29Q8×C3⋊D3—61no—
30C2×C4×C3⋊D3—31no—
31C2×#634—31no—
32C2×#634—31no—
33C2×C3×SL(2,𝔽3)—24no—
34C2×C3×SL(2,𝔽3)C2×D314yes—
35D3×SL(2,𝔽3)—64no—
36C3⋊#226—68no—
37C3⋊#226—64no—
38C2×C32×C8—31no—
39C32×C4⋊C4—31no—
40#3756—61no—
41#3756—61no—
42#3752—61no—
43C4⋊(C3⋊Dic3)—61no—
44C2×C3⋊(C3⋊C8)—31no—
45#3746—62no—
46(C2×D3)⋊Dic3—124no—
47C22×C33—41no—
48C2×C3×C3⋊D3—42no—
49C2×C3×C3⋊D3—81no—
50C2×C3×C3⋊D3—81no—
51C3⋊(C3⋊Dic3)—81no—
52C32×Dic3—81no—
53C2×D3×Q8—34no—
54C22×SL(2,𝔽3)—91no—
55C2×#226—91no—
56C2×#226—184no—
57C2×#216—94no—
58C2×Dic12—94no—
59#897—94no—
60#859C3⋊D311yescharacteristic, complemented
61C3×#103—34no—
62(C2×D3)⋊C8—94no—
63#810—94no—
64C22×C3⋊D3S411yescharacteristic
65C22×C3×D3—124no—
66C32×Q8—31no—
67C32×Q8#6711yescharacteristic
68C2×C32×C4—31no—
69C2×C32×C4—61no—
70C2×C3⋊Dic3—31no—
71C2×C3⋊Dic3—61no—
72C2×C3⋊Dic3—124no—
73C2×C3⋊Dic3—244no—
74C4×C3⋊D3—61no—
75C4×C3⋊D3—61no—
76#634—32no—
77#634—61no—
78#634—61no—
79#634—61no—
80C2×C3×Dic3—44no—
81C3×SL(2,𝔽3)—24no—
82C3×SL(2,𝔽3)D1214yes—
83C32×C8—61no—
84C3⋊(C3⋊C8)—61no—
85#451—271noSylow
86C2×C33—41no—
87C2×C33—41no—
88C2×C33—41no—
89C3×C3⋊D3—82no—
90C2×C3×Q8D3214yes—
91D3×Q8—64no—
92D3×Q8—184no—
93C2×C4×D3—94no—
94C2×Dic6—94no—
95C2×Dic6—94no—
96C2×SL(2,𝔽3)—24no—
97C2×SL(2,𝔽3)—181no—
98C2×SL(2,𝔽3)C2×C3⋊D311yescharacteristic
99#226—188no—
100#226—181no—
101C2×C3×C8—34no—
102C3×C4⋊C4—34no—
103#217—361no—
104#216—184no—
105#216—184no—
106(C2×D3)⋊C4—184no—
107C4⋊Dic3—184no—
108C2×C3⋊C8—94no—
109Dic12—188no—
110C22×C32—44no—
111C22×C32—84no—
112C22×C32C2×S411yescharacteristic
113C2×C3⋊D3—21no—
114C2×C3⋊D3—21no—
115C2×C3⋊D3#22612yescomplemented
116C2×C3×D3—128no—
117C2×C3×D3—244no—
118C2×C3×D3—244no—
119C32×C4—31no—
120C32×C4—31no—
121C32×C4—121no—
122C3⋊Dic3—61no—
123C3⋊Dic3—121no—
124C3⋊Dic3—244no—
125C3⋊Dic3—248no—
126C3×Dic3—84no—
127C22×Q8—91no—
128C2×Q16—271no—
129C2×Q16—271no—
130C22⋊Q8—271no—
131#103—31no—
132#103—271no—
133C22⋊C8—271no—
134C33—41noSylow
135C23×C3—361no—
136C22×D3—34no—
137C3×Q8—34no—
138C3×Q8#62614yes—
139C2×C3×C4—34no—
140C2×C3×C4—64no—
141C2×Dic3—41no—
142C2×Dic3—94no—
143C2×Dic3—184no—
144C2×Dic3—361no—
145C2×Dic3—724no—
146C4×D3—184no—
147C4×D3—184no—
148Dic6—98no—
149Dic6—184no—
150Dic6—184no—
151Dic6—184no—
152SL(2,𝔽3)—24no—
153SL(2,𝔽3)C4⋊(C3⋊D3)11yescharacteristic
154C3×C8—64no—
155C3⋊C8—184no—
156C2×C32—44no—
157C2×C32—44no—
158C2×C32—44no—
159C2×C32—84no—
160C2×C32—84no—
161C2×C32—84no—
162C2×C32C2×#22611yescharacteristic
163C2×C32#98311yescharacteristic
164C2×C32GL(2,ℤ/4)11yescharacteristic
165C3⋊D3—22no—
166C3×D3—248no—
167C2×Q8—181no—
168C2×Q8—271no—
169C2×Q8—271no—
170C2×Q8—541no—
171C2×Q8D3×C3⋊D311yescharacteristic
172C22×C4—271no—
173Q16—542no—
174Q16—541no—
175Q16—541no—
176C2×C8—31no—
177C2×C8—271no—
178C4⋊C4—31no—
179C4⋊C4—541no—
180C22⋊C4—541no—
181C22×C3—41no—
182C22×C3—84no—
183C22×C3—362no—
184C22×C3—721no—
185C22×C3—721no—
186C22×C3D3×S414yes—
187C2×D3—38no—
188C2×D3—64no—
189C2×D3—64no—
190C3×C4—34no—
191C3×C4—34no—
192C3×C4—124no—
193Dic3—81no—
194Dic3—184no—
195Dic3—364no—
196Dic3—721no—
197Dic3—728no—
198C32—44no—
199C32—84no—
200C32#541911yescharacteristic, complemented
201C23—91no—
202Q8—31no—
203Q8—272no—
204Q8—541no—
205Q8—541no—
206Q8—541no—
207Q8#629211yescharacteristic
208C2×C4—31no—
209C2×C4—61no—
210C2×C4—271no—
211C2×C4—541no—
212C2×C4—541no—
213C2×C4—541no—
214C8—61no—
215C8—541no—
216C2×C3—41no—
217C2×C3—41no—
218C2×C3—41no—
219C2×C3—84no—
220C2×C3—84no—
221C2×C3—84no—
222C2×C3—722no—
223C2×C3D3×#22614yes—
224C2×C3#6420114yes—
225C2×C3#6421014yes—
226D3—68no—
227C22—92no—
228C22—181no—
229C22—181no—
230C22C3⋊D3×S411yescharacteristic
231C4—31no—
232C4—31no—
233C4—121no—
234C4—541no—
235C4—1081no—
236C3—41no—
237C3—84no—
238C3#1059321114yescomplemented
239C2—182no—
240C2C3⋊D3×#22611yescharacteristic
241C2#1172418111yescharacteristic
242C2#1172419011yescharacteristic
subgroupG/H
center, Frattini, hypercenterC22C3⋊D3×S4
commutator, cosocleC2×C32×SL(2,𝔽3)C22
FittingC2×C32×Q8C2×D3
socleC22×C32C2×S4
prime radicalC2×C3⋊D3×SL(2,𝔽3)C2
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
2. C2×C3⋊(C3⋊#226)C2semidirect
3. C32×#859C2semidirect
17. C3×#859D3semidirect
60. #859C3⋊D3semidirect
115. C2×C3⋊D3#226semidirect
200. C32#5419semidirect
238. C3#10593211semidirect

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⋊#226)2
C32×#8592
#105932113
#105925083
#888034