Finite Group Databasebeta

← #13144008

Subgroups — #13144008

Order 1728.

572 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 572 conjugacy classes of subgroups — too many to draw. Only its 99 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
1#11721863C211yesmaximal, characteristic
2C4⋊C4×#272C211yesmaximal, characteristic
3H(3,𝔽3)×#103C211yesmaximal, characteristic, complemented
4#11720583C211yesmaximal, characteristic, complemented
5#11720553C211yesmaximal, characteristic
6#11720550C211yesmaximal, characteristic, complemented
7#11720530C211yesmaximal, characteristic, complemented
8C3×#4805—34nomaximal
9C2×Q8×H(3,𝔽3)C2211yescharacteristic
10C2×C4×#272—21no—
11C2×C4×#272C2211yescharacteristic
12C2×C8×H(3,𝔽3)C2211yescharacteristic
13C4⋊C4×H(3,𝔽3)C2211yescharacteristic
14#88542—21no—
15C4⋊#1084C2211yescharacteristic
16C4⋊#1084C2211yescharacteristic
17#88540—21no—
18#88540—21no—
19C2×#6180C2211yescharacteristic
20C3×#946—34no—
21C3×D3×C4⋊C4—34no—
22C32×#103D314yescomplemented
23C3×#835—34no—
24C3×(C2×D3)⋊C8—34no—
25C3×#817—34no—
26C3×#808—34no—
27C22×#272D411yescharacteristic
28Q8×H(3,𝔽3)—21no—
29Q8×H(3,𝔽3)—21no—
30C2×C4×H(3,𝔽3)—21no—
31C2×C4×H(3,𝔽3)—21no—
32C2×C4×H(3,𝔽3)C2311yescharacteristic
33C2×#1084—21no—
34C2×#1084—21no—
35C2×#1084D411yescharacteristic
36C4×#272—21no—
37C4×#272—21no—
38C4×#272—42no—
39C8×H(3,𝔽3)—21no—
40#6180—21no—
41C3×#483—91no—
42#4805—34no—
43C2×C32×Q8C2×D314yes—
44C2×C3×C4×D3—34no—
45C2×C3×C4×D3—64no—
46C2×C32×C8C2×D314yes—
47C32×C4⋊C4C2×D314yes—
48C3×(C2×D3)⋊C4—64no—
49C3×C4⋊Dic3—34no—
50C3×C4⋊Dic3—34no—
51C3×Dic3⋊C4—64no—
52C3×Dic3⋊C4—64no—
53C2×C3×C3⋊C8—34no—
54C22×H(3,𝔽3)C2×D411yescharacteristic
55C2×#272—21no—
56C2×#272—21no—
57C2×#272SD1612yes—
58C4×H(3,𝔽3)—41no—
59C4×H(3,𝔽3)—41no—
60C4×H(3,𝔽3)#4111yescharacteristic
61C4×H(3,𝔽3)#4111yescharacteristic
62#1084—21no—
63#1084—41no—
64#1084—41no—
65C3×C22⋊Q8—91no—
66C2×C3×C4⋊C4—91no—
67#946—34no—
68D3×C4⋊C4—34no—
69C3×#106—91no—
70C3×#106—91no—
71C3×#103—34no—
72C3×#103—91no—
73C3×#103C3⋊D311yescharacteristic
74C3×C22⋊C8—91no—
75#835—34no—
76(C2×D3)⋊C8—34no—
77#817—34no—
78#808—34no—
79C22×C3×D3—34no—
80C32×Q8—24no—
81C32×Q8—24no—
82C2×C32×C4—24no—
83C2×C32×C4—24no—
84C2×C32×C4C22×D314yes—
85C2×C3×Dic3—34no—
86C2×C3×Dic3—64no—
87C2×C3×Dic3—64no—
88C3×C4×D3—64no—
89C3×C4×D3—64no—
90C3×C4×D3—128no—
91C32×C8—24no—
92C3×C3⋊C8—64no—
93#483—91noSylow
94C2×H(3,𝔽3)#12311yescharacteristic
95C2×H(3,𝔽3)C2×SD1611yescharacteristic
96C2×H(3,𝔽3)#13711yescharacteristic
97#272—22no—
98C2×C3×Q8—34no—
99C2×C3×Q8C2×C3⋊D311yescharacteristic
100C22×C3×C4—91no—
101C22×C3×C4—181no—
102C2×C4×D3—34no—
103C2×C4×D3—64no—
104C2×C3×C8—34no—
105C2×C3×C8—91no—
106C2×C3×C8C2×C3⋊D311yescharacteristic
107C3×C4⋊C4—34no—
108C3×C4⋊C4—91no—
109C3×C4⋊C4—91no—
110C3×C4⋊C4—181no—
111C3×C4⋊C4—181no—
112C3×C4⋊C4C2×C3⋊D311yescharacteristic
113C3×C22⋊C4—181no—
114(C2×D3)⋊C4—64no—
115C4⋊Dic3—34no—
116C4⋊Dic3—34no—
117Dic3⋊C4—64no—
118Dic3⋊C4—64no—
119C2×C3⋊C8—34no—
120C22×C32D3×D414yes—
121C2×C3×D3—38no—
122C2×C3×D3—64no—
123C2×C3×D3—64no—
124C32×C4—44no—
125C32×C4—44no—
126C32×C4#23514yes—
127C32×C4#23714yes—
128C3×Dic3—64no—
129C3×Dic3—124no—
130C3×Dic3—124no—
131C22⋊Q8—91no—
132C2×C4⋊C4—91no—
133#106—91no—
134#106—91no—
135#103—91no—
136#103#27211yescharacteristic, complemented
137C22⋊C8—91no—
138H(3,𝔽3)#48311yescharacteristic, complemented, Sylow
139C23×C3—91no—
140C22×D3—34no—
141C3×Q8—21no—
142C3×Q8—21no—
143C3×Q8—64no—
144C3×Q8—64no—
145C2×C3×C4—21no—
146C2×C3×C4—21no—
147C2×C3×C4—34no—
148C2×C3×C4—64no—
149C2×C3×C4—64no—
150C2×C3×C4—91no—
151C2×C3×C4—181no—
152C2×C3×C4—181no—
153C2×C3×C4—181no—
154C2×C3×C4—181no—
155C2×C3×C4—362no—
156C2×C3×C4C22×C3⋊D311yescharacteristic
157C2×Dic3—34no—
158C2×Dic3—64no—
159C2×Dic3—64no—
160C4×D3—64no—
161C4×D3—64no—
162C4×D3—128no—
163C3×C8—21no—
164C3×C8—64no—
165C3×C8—181no—
166C3⋊C8—64no—
167C2×C32#88314yes—
168C2×C32D3×SD1614yes—
169C2×C32#91814yes—
170C3×D3—68no—
171C2×Q8C2×#27211yescharacteristic
172C22×C4—91no—
173C22×C4—181no—
174C2×C8—91no—
175C2×C8C2×#27211yescharacteristic
176C4⋊C4—91no—
177C4⋊C4—91no—
178C4⋊C4—181no—
179C4⋊C4—181no—
180C4⋊C4C2×#27211yescharacteristic
181C22⋊C4—181no—
182C22×C3—34no—
183C22×C3—92no—
184C22×C3—181no—
185C22×C3—181no—
186C22×C3D4×C3⋊D311yescharacteristic
187C2×D3—38no—
188C2×D3—64no—
189C2×D3—64no—
190C3×C4—34no—
191C3×C4—34no—
192C3×C4—41no—
193C3×C4—41no—
194C3×C4—124no—
195C3×C4—124no—
196C3×C4—181no—
197C3×C4—361no—
198C3×C4—361no—
199C3×C4#382811yescharacteristic
200C3×C4#383011yescharacteristic
201Dic3—64no—
202Dic3—124no—
203Dic3—124no—
204C32#480514yescomplemented
205C23—91no—
206Q8—21no—
207Q8—21no—
208C2×C4—21no—
209C2×C4—21no—
210C2×C4—91no—
211C2×C4—181no—
212C2×C4—181no—
213C2×C4—181no—
214C2×C4—181no—
215C2×C4—362no—
216C2×C4C22×#27211yescharacteristic
217C8—21no—
218C8—181no—
219C2×C3—34no—
220C2×C3—34no—
221C2×C3—34no—
222C2×C3—182no—
223C2×C3#6409211yescharacteristic
224C2×C3SD16×C3⋊D311yescharacteristic
225C2×C3#6412711yescharacteristic
226D3—68no—
227C22—92no—
228C22—181no—
229C22—181no—
230C22D4×#27211yescharacteristic
231C4—41no—
232C4—41no—
233C4—181no—
234C4—361no—
235C4—361no—
236C4#8874011yescharacteristic
237C4#8874411yescharacteristic
238C3—34no—
239C3#1059250411yescharacteristic
240C2—182no—
241C2#1172173911yescharacteristic
242C2SD16×#27211yescharacteristic
243C2#1172180711yescharacteristic
subgroupG/H
center, socleC22×C3D4×C3⋊D3
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C22×C3⋊D3
FittingH(3,𝔽3)×#103C2
prime radicalC22×#272D4
hypercenterC3×#103C3⋊D3
hypocenterH(3,𝔽3)#483
LCSG ▷ C2×C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22×C3 ◁ C2×C3×C4 ◁ C3×#103

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
3. H(3,𝔽3)×#103C2semidirect
4. #11720583C2semidirect
6. #11720550C2semidirect
7. #11720530C2semidirect
22. C32×#103D3semidirect
136. #103#272semidirect
138. H(3,𝔽3)#483semidirect
204. C32#4805semidirect

Maximal subgroups

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

subgroupindex
#117218632
C4⋊C4×#2722
H(3,𝔽3)×#1032
#117205832
#117205532
#117205502
#117205302
C3×#48053