Finite Group Databasebeta

← #13144000

Subgroups — #13144000

Order 1728.

495 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 495 conjugacy classes of subgroups — too many to draw. Only its 97 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#11721861C211yesmaximal, characteristic
2#11721758C211yesmaximal, characteristic
3H(3,𝔽3)×#103C211yesmaximal, characteristic
4#11720583C211yesmaximal, characteristic
5#11720549C211yesmaximal, characteristic
6#11720548C211yesmaximal, characteristic
7#11720532C211yesmaximal, characteristic
8C3×#4797—34nomaximal
9C2×Q8×H(3,𝔽3)C2211yescharacteristic
10C2×#6229C2211yescharacteristic
11C2×C8×H(3,𝔽3)C2211yescharacteristic
12C4⋊C4×H(3,𝔽3)C2211yescharacteristic
13C4⋊#1084—21no—
14C4⋊#1084C2211yescharacteristic
15#88540—21no—
16#88540—21no—
17#88540—21no—
18C4×#1084C2211yescharacteristic
19C2×#6180C2211yescharacteristic
20C3×#944—34no—
21C3×#890—34no—
22C32×#103D314yes—
23C3×#835—34no—
24C3×#816—34no—
25C3×#815—34no—
26C3×#810—34no—
27Q8×H(3,𝔽3)—21no—
28Q8×H(3,𝔽3)—21no—
29C2×C4×H(3,𝔽3)—21no—
30C2×C4×H(3,𝔽3)—21no—
31C2×C4×H(3,𝔽3)C2311yescharacteristic
32C2×#1084—21no—
33C2×#1084—21no—
34C2×#1084D411yescharacteristic
35C2×#1084D411yescharacteristic
36#6229—21no—
37#6229—21no—
38C8×H(3,𝔽3)—21no—
39#6180—21no—
40C3×#491—91no—
41#4797—34no—
42C2×C32×Q8C2×D314yes—
43C2×C3×Dic6—34no—
44C2×C32×C8C2×D314yes—
45C32×C4⋊C4C2×D314yes—
46C3×C4⋊Dic3—34no—
47C3×C4⋊Dic3—64no—
48C3×Dic3⋊C4—64no—
49C3×Dic3⋊C4—64no—
50C3×Dic3⋊C4—64no—
51C3×C4×Dic3—34no—
52C2×C3×C3⋊C8—34no—
53C22×H(3,𝔽3)C2×D411yescharacteristic
54C4×H(3,𝔽3)—41no—
55C4×H(3,𝔽3)—41no—
56C4×H(3,𝔽3)#4111yescharacteristic
57C4×H(3,𝔽3)#4111yescharacteristic
58#1084—21no—
59#1084—21no—
60#1084—41no—
61#1084—41no—
62C3×C4⋊Q8—91no—
63C3×#125—91no—
64#944—34no—
65#890—34no—
66C3×#103—34no—
67C3×#103—91no—
68C3×#103—91no—
69C3×#103—91no—
70C3×#103C3⋊D311yescharacteristic
71C3×#97—91no—
72#835—34no—
73#816—34no—
74#815—34no—
75#810—34no—
76C32×Q8—24no—
77C32×Q8—24no—
78C2×C32×C4—24no—
79C2×C32×C4—24no—
80C2×C32×C4C22×D314yes—
81C2×C3×Dic3—34no—
82C2×C3×Dic3—34no—
83C2×C3×Dic3—64no—
84C2×C3×Dic3—64no—
85C3×Dic6—64no—
86C3×Dic6—64no—
87C32×C8—24no—
88C3×C3⋊C8—64no—
89#491—91noSylow
90C2×H(3,𝔽3)#12411yescharacteristic
91C2×H(3,𝔽3)#13711yescharacteristic
92C2×H(3,𝔽3)#13711yescharacteristic
93C2×C3×Q8—34no—
94C2×C3×Q8—91no—
95C2×C3×Q8C2×C3⋊D311yescharacteristic
96C2×Dic6—34no—
97C2×C3×C8—34no—
98C2×C3×C8—91no—
99C2×C3×C8C2×C3⋊D311yescharacteristic
100C3×C4⋊C4—34no—
101C3×C4⋊C4—91no—
102C3×C4⋊C4—181no—
103C3×C4⋊C4—181no—
104C3×C4⋊C4—181no—
105C3×C4⋊C4—181no—
106C3×C4⋊C4C2×C3⋊D311yescharacteristic
107C3×C42—91no—
108C4⋊Dic3—34no—
109C4⋊Dic3—64no—
110Dic3⋊C4—64no—
111Dic3⋊C4—64no—
112Dic3⋊C4—64no—
113C4×Dic3—34no—
114C2×C3⋊C8—34no—
115C22×C32D3×D414yes—
116C32×C4—44no—
117C32×C4—44no—
118C32×C4#23514yes—
119C32×C4#23714yes—
120C3×Dic3—64no—
121C3×Dic3—64no—
122C3×Dic3—124no—
123C3×Dic3—124no—
124C4⋊Q8—91no—
125#125—91no—
126#103—91no—
127#103—91no—
128#103—91no—
129#103#27211yescharacteristic
130#97—91no—
131H(3,𝔽3)#49111yescharacteristic, complemented, Sylow
132C3×Q8—21no—
133C3×Q8—21no—
134C3×Q8—64no—
135C3×Q8—64no—
136C3×Q8—181no—
137C3×Q8—181no—
138C2×C3×C4—21no—
139C2×C3×C4—21no—
140C2×C3×C4—34no—
141C2×C3×C4—64no—
142C2×C3×C4—64no—
143C2×C3×C4—91no—
144C2×C3×C4—91no—
145C2×C3×C4—181no—
146C2×C3×C4—181no—
147C2×C3×C4C22×C3⋊D311yescharacteristic
148C2×Dic3—34no—
149C2×Dic3—34no—
150C2×Dic3—64no—
151C2×Dic3—64no—
152Dic6—64no—
153Dic6—64no—
154C3×C8—21no—
155C3×C8—64no—
156C3×C8—181no—
157C3⋊C8—64no—
158C2×C32#88514yes—
159C2×C32#91514yes—
160C2×C32#91814yes—
161C2×Q8—91no—
162C2×Q8C2×#27211yescharacteristic
163C2×C8—91no—
164C2×C8C2×#27211yescharacteristic
165C4⋊C4—91no—
166C4⋊C4—181no—
167C4⋊C4—181no—
168C4⋊C4—181no—
169C4⋊C4—181no—
170C4⋊C4C2×#27211yescharacteristic
171C42—91no—
172C22×C3—34no—
173C22×C3D4×C3⋊D311yescharacteristic
174C3×C4—34no—
175C3×C4—34no—
176C3×C4—41no—
177C3×C4—41no—
178C3×C4—124no—
179C3×C4—124no—
180C3×C4—181no—
181C3×C4—181no—
182C3×C4—361no—
183C3×C4—361no—
184C3×C4#382811yescharacteristic
185C3×C4#383011yescharacteristic
186Dic3—64no—
187Dic3—64no—
188Dic3—124no—
189Dic3—124no—
190C32#479714yescomplemented
191Q8—21no—
192Q8—21no—
193Q8—181no—
194Q8—181no—
195C2×C4—21no—
196C2×C4—21no—
197C2×C4—91no—
198C2×C4—91no—
199C2×C4—181no—
200C2×C4—181no—
201C2×C4C22×#27211yescharacteristic
202C8—21no—
203C8—181no—
204C2×C3—34no—
205C2×C3—34no—
206C2×C3—34no—
207C2×C3#6409411yescharacteristic
208C2×C3#6412411yescharacteristic
209C2×C3#6412711yescharacteristic
210C22D4×#27211yescharacteristic
211C4—41no—
212C4—41no—
213C4—181no—
214C4—181no—
215C4—361no—
216C4—361no—
217C4#8874011yescharacteristic
218C4#8874411yescharacteristic
219C3—34no—
220C3#1059249611yescharacteristic
221C2#1172174111yescharacteristic
222C2#1172180111yescharacteristic
223C2#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×H(3,𝔽3)C2×D4
hypercenterC3×#103C3⋊D3
hypocenterH(3,𝔽3)#491
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
131. H(3,𝔽3)#491semidirect
190. C32#4797semidirect

Maximal subgroups

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

subgroupindex
#117218612
#117217582
H(3,𝔽3)×#1032
#117205832
#117205492
#117205482
#117205322
C3×#47973