Finite Group Databasebeta

← #13143895

Subgroups — #13143895

Order 1728.

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

normal subgroups only

This group has 572 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
1#11721859C211yesmaximal, characteristic, complemented
2C2×#88549C211yesmaximal, characteristic, complemented
3#11721787C211yesmaximal, characteristic
4#11721786C211yesmaximal, characteristic, complemented
5H(3,𝔽3)×#101C211yesmaximal, characteristic, complemented
6#11720560C211yesmaximal, characteristic, complemented
7#11720559C211yesmaximal, characteristic, complemented
8C3×#4754—34nomaximal
9C2×Q8×H(3,𝔽3)C2211yescharacteristic
10#88740C2211yescharacteristic
11C2×C4⋊#272C2211yescharacteristic
12M16×H(3,𝔽3)C2211yescharacteristic
13M16×H(3,𝔽3)C2211yescharacteristic
14#88550—21no—
15#88549—21no—
16#88549—21no—
17#88549—21no—
18#88538C2211yescharacteristic
19C2×#6180C2211yescharacteristic
20#88529—21no—
21#88528—21no—
22#88527—21no—
23C8×#272—21no—
24C3×#942—34no—
25C2×C3×#215—34no—
26C3×#908—34no—
27C3×#907—34no—
28C32×#101D314yescomplemented
29C3×#823—34no—
30C3×#822—34no—
31C22×#272—21no—
32Q8×H(3,𝔽3)—21no—
33Q8×H(3,𝔽3)—21no—
34C2×C4×H(3,𝔽3)—21no—
35C2×C4×H(3,𝔽3)C2311yescharacteristic
36#6235—21no—
37C4⋊#272—21no—
38C4⋊#272—21no—
39C4⋊#272D411yescharacteristic
40C4×#272—21no—
41#6229D411yescharacteristic
42C8×H(3,𝔽3)—21no—
43C8×H(3,𝔽3)—21no—
44#6180—21no—
45#6180D411yescharacteristic
46#6180D411yescharacteristic
47C3×#471—91no—
48#4754—34no—
49C2×C32×Q8C2×D314yes—
50C3×#235—34no—
51C2×C3×D12—34no—
52C32×M16C2×D314yes—
53C32×M16C2×D314yes—
54C3×#216—64no—
55C3×#215—64no—
56C3×#215—64no—
57C3×#215—64no—
58C3×#208—34no—
59C2×C3×C3⋊C8—34no—
60C3×D24—64no—
61C3×#204—64no—
62C3×ΓL(1,𝔽25)—64no—
63C3×D3×C8—64no—
64C22×H(3,𝔽3)#4111yescharacteristic
65C2×#272—21no—
66C2×#272—41no—
67C2×#272—81no—
68C4×H(3,𝔽3)—41no—
69C4×H(3,𝔽3)C2×D411yescharacteristic
70C4×H(3,𝔽3)C2×D411yescharacteristic
71#1084—21no—
72C3×#137—91no—
73C3×AGL(1,ℤ/8)—91no—
74C2×C3×SD16—91no—
75C3×#131—91no—
76#942—34no—
77C2×#215—34no—
78#908—34no—
79#907—34no—
80C3×#108—91no—
81C3×#101—34no—
82C3×#101C3⋊D311yescharacteristic
83C3×#100—91no—
84#823—34no—
85#822—34no—
86C22×C3×D3—64no—
87C32×Q8—24no—
88C32×Q8—24no—
89C2×C32×C4—24no—
90C2×C32×C4C22×D314yes—
91C3×#67—64no—
92C3×D12—34no—
93C3×D12—64no—
94C3×D12—64no—
95C3×C4×D3—64no—
96C3×Dic6—34no—
97C32×C8—24no—
98C32×C8—24no—
99C3×C3⋊C8—34no—
100C3×C3⋊C8—34no—
101C3×C3⋊C8—64no—
102#471—91noSylow
103C2×H(3,𝔽3)—21no—
104C2×H(3,𝔽3)#12111yescharacteristic
105#272—41no—
106#272—81no—
107C3×#41—91no—
108C2×C3×Q8—34no—
109C2×C3×Q8C2×C3⋊D311yescharacteristic
110C2×C3×D4—91no—
111#235—34no—
112C2×D12—34no—
113C3×Q16—181no—
114C3×SD16—181no—
115C3×SD16—181no—
116C3×SD16—181no—
117C3×SD16—181no—
118C3×D8—181no—
119C3×M16—34no—
120C3×M16—34no—
121C3×M16—91no—
122C3×M16—181no—
123C3×M16C2×C3⋊D311yescharacteristic
124C3×M16C2×C3⋊D311yescharacteristic
125C2×C3×C8—91no—
126C2×C3×C8—181no—
127#216—64no—
128#215—64no—
129#215—64no—
130#215—64no—
131#208—34no—
132C2×C3⋊C8—34no—
133D24—64no—
134#204—64no—
135ΓL(1,𝔽25)—64no—
136D3×C8—64no—
137C22×C32#23514yes—
138C2×C3×D3—64no—
139C2×C3×D3—124no—
140C2×C3×D3—244no—
141C32×C4—44no—
142C32×C4D3×D414yes—
143C32×C4D3×D414yes—
144C3×Dic3—64no—
145#137—91no—
146AGL(1,ℤ/8)—91no—
147C2×SD16—91no—
148#131—91no—
149#108—91no—
150#101#27211yescharacteristic, complemented
151#100—91no—
152H(3,𝔽3)#47111yescharacteristic, complemented, Sylow
153C23×C3—181no—
154C22×D3—64no—
155C3×Q8—21no—
156C3×Q8—21no—
157C3×Q8—64no—
158C3×Q8—64no—
159C3×Q8—91no—
160C3×D4—91no—
161C3×D4—181no—
162C3×D4—181no—
163C3×D4—181no—
164C2×C3×C4—21no—
165C2×C3×C4—34no—
166C2×C3×C4—64no—
167C2×C3×C4—181no—
168C2×C3×C4C22×C3⋊D311yescharacteristic
169#67—64no—
170D12—34no—
171D12—64no—
172D12—64no—
173C4×D3—64no—
174Dic6—34no—
175C3×C8—21no—
176C3×C8—21no—
177C3×C8—64no—
178C3×C8—64no—
179C3×C8—91no—
180C3×C8—91no—
181C3×C8—181no—
182C3⋊C8—34no—
183C3⋊C8—34no—
184C3⋊C8—64no—
185C2×C32—24no—
186C2×C32#88414yes—
187C3×D3—124no—
188C3×D3—244no—
189#41—91no—
190C2×Q8C2×#27211yescharacteristic
191C2×D4—91no—
192Q16—181no—
193SD16—181no—
194SD16—181no—
195SD16—181no—
196SD16—181no—
197D8—181no—
198M16—91no—
199M16—181no—
200M16C2×#27211yescharacteristic
201M16C2×#27211yescharacteristic
202C2×C8—91no—
203C2×C8—181no—
204C22×C3—34no—
205C22×C3—181no—
206C22×C3—361no—
207C22×C3—721no—
208C22×C3#382811yescharacteristic
209C2×D3—64no—
210C2×D3—124no—
211C2×D3—244no—
212C3×C4—34no—
213C3×C4—34no—
214C3×C4—41no—
215C3×C4—124no—
216C3×C4—181no—
217C3×C4D4×C3⋊D311yescharacteristic
218C3×C4D4×C3⋊D311yescharacteristic
219Dic3—64no—
220C32#475414yescomplemented
221C23—181no—
222Q8—21no—
223Q8—21no—
224Q8—91no—
225D4—91no—
226D4—181no—
227D4—181no—
228D4—181no—
229C2×C4—21no—
230C2×C4—181no—
231C2×C4C22×#27211yescharacteristic
232C8—21no—
233C8—21no—
234C8—91no—
235C8—91no—
236C8—181no—
237C2×C3—21no—
238C2×C3—34no—
239C2×C3—64no—
240C2×C3—361no—
241C2×C3—721no—
242C2×C3#6409311yescharacteristic
243D3—124no—
244D3—244no—
245C22—181no—
246C22—361no—
247C22—721no—
248C22#8874011yescharacteristic
249C4—41no—
250C4—181no—
251C4D4×#27211yescharacteristic
252C4D4×#27211yescharacteristic
253C3—34no—
254C3#1059245311yescharacteristic
255C2—21no—
256C2—361no—
257C2—721no—
258C2#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#11721787C2
hypercenterC3×#101C3⋊D3
hypocenterH(3,𝔽3)#471
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. #11721859C2semidirect
2. C2×#88549C2semidirect
4. #11721786C2semidirect
5. H(3,𝔽3)×#101C2semidirect
6. #11720560C2semidirect
7. #11720559C2semidirect
28. C32×#101D3semidirect
150. #101#272semidirect
152. H(3,𝔽3)#471semidirect
220. C32#4754semidirect

Maximal subgroups

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

subgroupindex
#117218592
C2×#885492
#117217872
#117217862
H(3,𝔽3)×#1012
#117205602
#117205592
C3×#47543