Finite Group Databasebeta

← #13143997

Subgroups — #13143997

Order 1728.

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

normal subgroups only

This group has 550 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#11721864C211yesmaximal, characteristic
2#11721756C211yesmaximal, characteristic, complemented
3H(3,𝔽3)×#103C211yesmaximal, characteristic, complemented
4#11720583C211yesmaximal, characteristic, complemented
5#11720554C211yesmaximal, characteristic
6#11720548C211yesmaximal, characteristic, complemented
7#11720531C211yesmaximal, characteristic
8C3×#4794—34nomaximal
9C2×Q8×H(3,𝔽3)C2211yescharacteristic
10C2×C4⋊#272C2211yescharacteristic
11C2×#6229—21no—
12C2×C8×H(3,𝔽3)C2211yescharacteristic
13C4⋊C4×H(3,𝔽3)C2211yescharacteristic
14#88542—21no—
15#88542—21no—
16C4⋊#1084C2211yescharacteristic
17#88540—21no—
18C4×#1084C2211yescharacteristic
19C2×#6180C2211yescharacteristic
20C3×#947—34no—
21C3×#888—34no—
22C32×#103D314yescomplemented
23C3×#835—34no—
24C3×#821—34no—
25C3×#815—34no—
26C3×#809—34no—
27C22×#272—21no—
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×#1084D411yescharacteristic
35C2×#1084D411yescharacteristic
36C4⋊#272—21no—
37C4⋊#272—21no—
38#6229—42no—
39C8×H(3,𝔽3)—21no—
40#6180—21no—
41C3×#489—91no—
42#4794—34no—
43C2×C32×Q8C2×D314yes—
44C2×C3×D12—34no—
45C2×C3×Dic6—64no—
46C2×C32×C8C2×D314yes—
47C32×C4⋊C4C2×D314yes—
48C3×(C2×D3)⋊C4—64no—
49C3×(C2×D3)⋊C4—64no—
50C3×C4⋊Dic3—34no—
51C3×Dic3⋊C4—64no—
52C3×C4×Dic3—34no—
53C2×C3×C3⋊C8—34no—
54C22×H(3,𝔽3)C2×D411yescharacteristic
55C2×#272—41no—
56C2×#272—41no—
57C2×#272—41no—
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—21no—
64#1084—41no—
65C3×C4⋊Q8—91no—
66C3×#124—91no—
67#947—34no—
68#888—34no—
69C3×#103—34no—
70C3×#103—91no—
71C3×#103C3⋊D311yescharacteristic
72C3×#102—91no—
73C3×#102—91no—
74C3×#97—91no—
75#835—34no—
76#821—34no—
77#815—34no—
78#809—34no—
79C22×C3×D3—64no—
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—34no—
87C2×C3×Dic3—64no—
88C3×D12—64no—
89C3×D12—64no—
90C3×Dic6—128no—
91C32×C8—24no—
92C3×C3⋊C8—64no—
93#489—91noSylow
94C2×H(3,𝔽3)#12411yescharacteristic
95C2×H(3,𝔽3)AGL(1,ℤ/8)11yescharacteristic
96C2×H(3,𝔽3)#13711yescharacteristic
97#272—81no—
98C2×C3×Q8—34no—
99C2×C3×Q8—181no—
100C2×C3×Q8C2×C3⋊D311yescharacteristic
101C2×C3×D4—91no—
102C2×D12—34no—
103C2×Dic6—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—181no—
110C3×C4⋊C4C2×C3⋊D311yescharacteristic
111C3×C22⋊C4—181no—
112C3×C22⋊C4—181no—
113C3×C42—91no—
114(C2×D3)⋊C4—64no—
115(C2×D3)⋊C4—64no—
116C4⋊Dic3—34no—
117Dic3⋊C4—64no—
118C4×Dic3—34no—
119C2×C3⋊C8—34no—
120C22×C32D3×D414yes—
121C2×C3×D3—124no—
122C2×C3×D3—124no—
123C2×C3×D3—124no—
124C32×C4—44no—
125C32×C4—44no—
126C32×C4#23514yes—
127C32×C4#23714yes—
128C3×Dic3—64no—
129C3×Dic3—64no—
130C3×Dic3—124no—
131C4⋊Q8—91no—
132#124—91no—
133#103—91no—
134#103#27211yescharacteristic, complemented
135#102—91no—
136#102—91no—
137#97—91no—
138H(3,𝔽3)#48911yescharacteristic, complemented, Sylow
139C23×C3—181no—
140C22×D3—64no—
141C3×Q8—21no—
142C3×Q8—21no—
143C3×Q8—64no—
144C3×Q8—64no—
145C3×Q8—362no—
146C3×D4—181no—
147C3×D4—181no—
148C2×C3×C4—21no—
149C2×C3×C4—21no—
150C2×C3×C4—34no—
151C2×C3×C4—64no—
152C2×C3×C4—64no—
153C2×C3×C4—91no—
154C2×C3×C4—91no—
155C2×C3×C4—181no—
156C2×C3×C4C22×C3⋊D311yescharacteristic
157C2×Dic3—34no—
158C2×Dic3—34no—
159C2×Dic3—64no—
160D12—64no—
161D12—64no—
162Dic6—128no—
163C3×C8—21no—
164C3×C8—64no—
165C3×C8—181no—
166C3⋊C8—64no—
167C2×C32#88514yes—
168C2×C32#91414yes—
169C2×C32#91814yes—
170C3×D3—244no—
171C2×Q8—181no—
172C2×Q8C2×#27211yescharacteristic
173C2×D4—91no—
174C2×C8—91no—
175C2×C8C2×#27211yescharacteristic
176C4⋊C4—91no—
177C4⋊C4—181no—
178C4⋊C4C2×#27211yescharacteristic
179C22⋊C4—181no—
180C22⋊C4—181no—
181C42—91no—
182C22×C3—34no—
183C22×C3—361no—
184C22×C3—361no—
185C22×C3—361no—
186C22×C3D4×C3⋊D311yescharacteristic
187C2×D3—124no—
188C2×D3—124no—
189C2×D3—124no—
190C3×C4—34no—
191C3×C4—34no—
192C3×C4—41no—
193C3×C4—41no—
194C3×C4—124no—
195C3×C4—124no—
196C3×C4—181no—
197C3×C4—181no—
198C3×C4—361no—
199C3×C4#382811yescharacteristic
200C3×C4#383011yescharacteristic
201Dic3—64no—
202Dic3—64no—
203Dic3—124no—
204C32#479414yescomplemented
205C23—181no—
206Q8—21no—
207Q8—21no—
208Q8—362no—
209D4—181no—
210D4—181no—
211C2×C4—21no—
212C2×C4—21no—
213C2×C4—91no—
214C2×C4—91no—
215C2×C4—181no—
216C2×C4C22×#27211yescharacteristic
217C8—21no—
218C8—181no—
219C2×C3—34no—
220C2×C3—34no—
221C2×C3—34no—
222C2×C3—721no—
223C2×C3#6409411yescharacteristic
224C2×C3#6412311yescharacteristic
225C2×C3#6412711yescharacteristic
226D3—244no—
227C22—361no—
228C22—361no—
229C22—361no—
230C22D4×#27211yescharacteristic
231C4—41no—
232C4—41no—
233C4—181no—
234C4—181no—
235C4—361no—
236C4#8874011yescharacteristic
237C4#8874411yescharacteristic
238C3—34no—
239C3#1059249311yescharacteristic
240C2—721no—
241C2#1172174111yescharacteristic
242C2#1172180011yescharacteristic
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 radicalC2×C4⋊#272C22
hypercenterC3×#103C3⋊D3
hypocenterH(3,𝔽3)#489
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
2. #11721756C2semidirect
3. H(3,𝔽3)×#103C2semidirect
4. #11720583C2semidirect
6. #11720548C2semidirect
22. C32×#103D3semidirect
134. #103#272semidirect
138. H(3,𝔽3)#489semidirect
204. C32#4794semidirect

Maximal subgroups

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

subgroupindex
#117218642
#117217562
H(3,𝔽3)×#1032
#117205832
#117205542
#117205482
#117205312
C3×#47943