Finite Group Databasebeta

← #13143894

Subgroups — #13143894

Order 1728.

550 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 550 conjugacy classes of subgroups — too many to draw. Only its 85 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×#88550C211yesmaximal, characteristic, complemented
2#11721859C211yesmaximal, characteristic
3#11721788C211yesmaximal, characteristic
4#11721786C211yesmaximal, characteristic
5H(3,𝔽3)×#101C211yesmaximal, characteristic, complemented
6#11720561C211yesmaximal, characteristic, complemented
7#11720559C211yesmaximal, characteristic, complemented
8C3×#4755—34nomaximal
9C2×Q8×H(3,𝔽3)C2211yescharacteristic
10#88740C2211yescharacteristic
11C2×#6229C2211yescharacteristic
12M16×H(3,𝔽3)C2211yescharacteristic
13M16×H(3,𝔽3)C2211yescharacteristic
14#88550—21no—
15#88550—21no—
16#88550—21no—
17#88549—21no—
18#88538C2211yescharacteristic
19C2×#6180C2211yescharacteristic
20#88530—21no—
21#88528—21no—
22#88527—21no—
23C8×#272—21no—
24C2×C3×#216—34no—
25C3×#942—34no—
26C3×#909—34no—
27C3×#907—34no—
28C32×#101D314yescomplemented
29C3×#824—34no—
30C3×#822—34no—
31Q8×H(3,𝔽3)—21no—
32Q8×H(3,𝔽3)—21no—
33C2×C4×H(3,𝔽3)—21no—
34C2×C4×H(3,𝔽3)C2311yescharacteristic
35#6235—21no—
36C2×#1084—21no—
37C4⋊#272D411yescharacteristic
38C4×#272—21no—
39#6229—21no—
40#6229—21no—
41#6229D411yescharacteristic
42C8×H(3,𝔽3)—21no—
43C8×H(3,𝔽3)—21no—
44#6180—21no—
45#6180D411yescharacteristic
46#6180D411yescharacteristic
47C3×#473—91no—
48#4755—34no—
49C2×C32×Q8C2×D314yes—
50C3×#235—34no—
51C2×C3×Dic6—34no—
52C32×M16C2×D314yes—
53C32×M16C2×D314yes—
54C3×#216—64no—
55C3×#216—64no—
56C3×#216—64no—
57C3×#215—64no—
58C3×#208—34no—
59C2×C3×C3⋊C8—34no—
60C3×Dic12—64no—
61C3×#204—64no—
62C3×ΓL(1,𝔽25)—64no—
63C3×D3×C8—64no—
64C22×H(3,𝔽3)#4111yescharacteristic
65C2×#272—21no—
66C4×H(3,𝔽3)—41no—
67C4×H(3,𝔽3)C2×D411yescharacteristic
68C4×H(3,𝔽3)C2×D411yescharacteristic
69#1084—21no—
70#1084—41no—
71C3×#137—91no—
72C3×#137—91no—
73C2×C3×Q16—91no—
74C3×#131—91no—
75C2×#216—34no—
76#942—34no—
77#909—34no—
78#907—34no—
79C3×#108—91no—
80C3×#101—34no—
81C3×#101—91no—
82C3×#101C3⋊D311yescharacteristic
83#824—34no—
84#822—34no—
85C32×Q8—24no—
86C32×Q8—24no—
87C2×C32×C4—24no—
88C2×C32×C4C22×D314yes—
89C3×#67—64no—
90C2×C3×Dic3—64no—
91C3×D12—34no—
92C3×C4×D3—64no—
93C3×Dic6—34no—
94C3×Dic6—64no—
95C3×Dic6—64no—
96C32×C8—24no—
97C32×C8—24no—
98C3×C3⋊C8—34no—
99C3×C3⋊C8—34no—
100C3×C3⋊C8—64no—
101#473—91noSylow
102C2×H(3,𝔽3)—21no—
103C2×H(3,𝔽3)#12111yescharacteristic
104#272—41no—
105C3×#41—91no—
106C2×C3×Q8—34no—
107C2×C3×Q8—91no—
108C2×C3×Q8C2×C3⋊D311yescharacteristic
109#235—34no—
110C2×Dic6—34no—
111C3×Q16—181no—
112C3×Q16—181no—
113C3×Q16—181no—
114C3×Q16—181no—
115C3×SD16—181no—
116C3×SD16—181no—
117C3×M16—34no—
118C3×M16—34no—
119C3×M16—91no—
120C3×M16—181no—
121C3×M16C2×C3⋊D311yescharacteristic
122C3×M16C2×C3⋊D311yescharacteristic
123C2×C3×C8—91no—
124C2×C3×C8—181no—
125#216—64no—
126#216—64no—
127#216—64no—
128#215—64no—
129#208—34no—
130C2×C3⋊C8—34no—
131Dic12—64no—
132#204—64no—
133ΓL(1,𝔽25)—64no—
134D3×C8—64no—
135C22×C32#23514yes—
136C2×C3×D3—64no—
137C32×C4—44no—
138C32×C4D3×D414yes—
139C32×C4D3×D414yes—
140C3×Dic3—64no—
141C3×Dic3—124no—
142#137—91no—
143#137—91no—
144C2×Q16—91no—
145#131—91no—
146#108—91no—
147#101—91no—
148#101#27211yescharacteristic, complemented
149H(3,𝔽3)#47311yescharacteristic, complemented, Sylow
150C3×Q8—21no—
151C3×Q8—21no—
152C3×Q8—64no—
153C3×Q8—64no—
154C3×Q8—91no—
155C3×Q8—181no—
156C3×Q8—181no—
157C3×D4—91no—
158C3×D4—181no—
159C2×C3×C4—21no—
160C2×C3×C4—34no—
161C2×C3×C4—64no—
162C2×C3×C4—181no—
163C2×C3×C4—181no—
164C2×C3×C4C22×C3⋊D311yescharacteristic
165#67—64no—
166C2×Dic3—64no—
167D12—34no—
168C4×D3—64no—
169Dic6—34no—
170Dic6—64no—
171Dic6—64no—
172C3×C8—21no—
173C3×C8—21no—
174C3×C8—64no—
175C3×C8—64no—
176C3×C8—91no—
177C3×C8—91no—
178C3×C8—181no—
179C3⋊C8—34no—
180C3⋊C8—34no—
181C3⋊C8—64no—
182C2×C32—24no—
183C2×C32#88414yes—
184C3×D3—124no—
185#41—91no—
186C2×Q8—91no—
187C2×Q8C2×#27211yescharacteristic
188Q16—181no—
189Q16—181no—
190Q16—181no—
191Q16—181no—
192SD16—181no—
193SD16—181no—
194M16—91no—
195M16—181no—
196M16C2×#27211yescharacteristic
197M16C2×#27211yescharacteristic
198C2×C8—91no—
199C2×C8—181no—
200C22×C3—34no—
201C22×C3—181no—
202C22×C3#382811yescharacteristic
203C2×D3—64no—
204C3×C4—34no—
205C3×C4—34no—
206C3×C4—41no—
207C3×C4—124no—
208C3×C4—181no—
209C3×C4—361no—
210C3×C4D4×C3⋊D311yescharacteristic
211C3×C4D4×C3⋊D311yescharacteristic
212Dic3—64no—
213Dic3—124no—
214C32#475514yescomplemented
215Q8—21no—
216Q8—21no—
217Q8—91no—
218Q8—181no—
219Q8—181no—
220D4—91no—
221D4—181no—
222C2×C4—21no—
223C2×C4—181no—
224C2×C4—181no—
225C2×C4C22×#27211yescharacteristic
226C8—21no—
227C8—21no—
228C8—91no—
229C8—91no—
230C8—181no—
231C2×C3—21no—
232C2×C3—34no—
233C2×C3—64no—
234C2×C3—361no—
235C2×C3#6409311yescharacteristic
236D3—124no—
237C22—181no—
238C22#8874011yescharacteristic
239C4—41no—
240C4—181no—
241C4—361no—
242C4D4×#27211yescharacteristic
243C4D4×#27211yescharacteristic
244C3—34no—
245C3#1059245411yescharacteristic
246C2—21no—
247C2—361no—
248C2#1172174011yescharacteristic
subgroupG/H
center, socleC2×C3#64093
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C22×C3⋊D3
FittingH(3,𝔽3)×#101C2
prime radical#88740C22
hypercenterC3×#101C3⋊D3
hypocenterH(3,𝔽3)#473
LCSG ▷ C2×C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C2×C3 ◁ C2×C3×C4 ◁ C3×#101

Complements

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

NH ≅ G/Ntype
1. C2×#88550C2semidirect
5. H(3,𝔽3)×#101C2semidirect
6. #11720561C2semidirect
7. #11720559C2semidirect
28. C32×#101D3semidirect
148. #101#272semidirect
149. H(3,𝔽3)#473semidirect
214. C32#4755semidirect

Maximal subgroups

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

subgroupindex
C2×#885502
#117218592
#117217882
#117217862
H(3,𝔽3)×#1012
#117205612
#117205592
C3×#47553