Finite Group Databasebeta

← #13144793

Subgroups — #13144793

Order 1728.

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

normal subgroups only

This group has 572 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
1H(3,𝔽3)×#137C211yesmaximal, characteristic, complemented
2#11721870C211yesmaximal, characteristic
3#11721869C211yesmaximal, characteristic, complemented
4C2×#88549C211yesmaximal, characteristic, complemented
5#11720584C211yesmaximal, characteristic, complemented
6#11720560C211yesmaximal, characteristic, complemented
7#11720559C211yesmaximal, characteristic, complemented
8C3×#5203—34nomaximal
9#41×H(3,𝔽3)C2211yescharacteristic
10C2×Q8×H(3,𝔽3)C2211yescharacteristic
11C2×C4⋊#272C2211yescharacteristic
12Q16×H(3,𝔽3)—21no—
13SD16×H(3,𝔽3)—21no—
14M16×H(3,𝔽3)C2211yescharacteristic
15#88549—21no—
16#88549—21no—
17#88549—21no—
18#88545—21no—
19#88538—21no—
20#88538C2211yescharacteristic
21#88538C2211yescharacteristic
22C2×#6180—21no—
23C2×#6180C2211yescharacteristic
24C32×#137D314yescomplemented
25C3×#949—34no—
26C3×#948—34no—
27C2×C3×#215—34no—
28C3×#836—34no—
29C3×#823—34no—
30C3×#822—34no—
31C22×#272—21no—
32Q8×H(3,𝔽3)—21no—
33Q8×H(3,𝔽3)—21no—
34Q8×H(3,𝔽3)D411yescharacteristic
35D4×H(3,𝔽3)—21no—
36D4×H(3,𝔽3)D411yescharacteristic
37C2×C4×H(3,𝔽3)—21no—
38C2×C4×H(3,𝔽3)—21no—
39C2×C4×H(3,𝔽3)C2311yescharacteristic
40C4⋊#272—21no—
41C4⋊#272—21no—
42C8×H(3,𝔽3)—21no—
43#6180—21no—
44#6180—21no—
45#6180D411yescharacteristic
46#6180D411yescharacteristic
47C3×#471—91no—
48#5203—34no—
49C32×#41C2×D314yes—
50C2×C32×Q8C2×D314yes—
51C2×C3×D12—34no—
52C32×Q16—24no—
53C32×SD16—24no—
54C32×M16C2×D314yes—
55C3×#215—64no—
56C3×#215—64no—
57C3×#215—64no—
58C3×#213—64no—
59C3×#208—34no—
60C3×#208—34no—
61C3×#208—64no—
62C2×C3×C3⋊C8—34no—
63C2×C3×C3⋊C8—64no—
64C22×H(3,𝔽3)—21no—
65C22×H(3,𝔽3)#4111yescharacteristic
66C2×#272—41no—
67C2×#272—81no—
68C4×H(3,𝔽3)—21no—
69C4×H(3,𝔽3)—41no—
70C4×H(3,𝔽3)C2×D411yescharacteristic
71C4×H(3,𝔽3)C2×D411yescharacteristic
72C3×#137—34no—
73C3×#137C3⋊D311yescharacteristic
74C3×AGL(1,ℤ/8)—91no—
75C2×C3×SD16—91no—
76C3×#131—91no—
77#949—34no—
78#948—34no—
79C2×#215—34no—
80C3×#108—91no—
81C3×#101—91no—
82C3×#100—91no—
83#836—34no—
84#823—34no—
85#822—34no—
86C22×C3×D3—64no—
87C32×Q8—24no—
88C32×Q8—24no—
89C32×Q8#6714yes—
90C32×D4—24no—
91C32×D4#6714yes—
92C2×C32×C4—24no—
93C2×C32×C4—24no—
94C2×C32×C4C22×D314yes—
95C3×D12—64no—
96C3×D12—64no—
97C32×C8—24no—
98C3×C3⋊C8—34no—
99C3×C3⋊C8—34no—
100C3×C3⋊C8—64no—
101C3×C3⋊C8—64no—
102#471—91noSylow
103C2×H(3,𝔽3)—21no—
104C2×H(3,𝔽3)—41no—
105C2×H(3,𝔽3)#12111yescharacteristic
106#272—81no—
107C3×#41—34no—
108C3×#41C2×C3⋊D311yescharacteristic
109C2×C3×Q8—34no—
110C2×C3×Q8C2×C3⋊D311yescharacteristic
111C2×C3×D4—91no—
112C2×D12—34no—
113C3×Q16—21no—
114C3×Q16—64no—
115C3×SD16—21no—
116C3×SD16—64no—
117C3×SD16—181no—
118C3×SD16—181no—
119C3×SD16—181no—
120C3×D8—181no—
121C3×M16—34no—
122C3×M16—91no—
123C3×M16—91no—
124C3×M16—181no—
125C3×M16C2×C3⋊D311yescharacteristic
126C2×C3×C8—91no—
127C2×C3×C8—181no—
128#215—64no—
129#215—64no—
130#215—64no—
131#213—64no—
132#208—34no—
133#208—34no—
134#208—64no—
135C2×C3⋊C8—34no—
136C2×C3⋊C8—64no—
137C22×C32—24no—
138C22×C32#23714yes—
139C2×C3×D3—124no—
140C2×C3×D3—244no—
141C32×C4—24no—
142C32×C4—44no—
143C32×C4D3×D414yes—
144C32×C4C2×#6714yes—
145#137#27211yescharacteristic, complemented
146AGL(1,ℤ/8)—91no—
147C2×SD16—91no—
148#131—91no—
149#108—91no—
150#101—91no—
151#100—91no—
152H(3,𝔽3)#47111yescharacteristic, complemented, Sylow
153C23×C3—181no—
154C22×D3—64no—
155C3×Q8—21no—
156C3×Q8—21no—
157C3×Q8—34no—
158C3×Q8—64no—
159C3×Q8—64no—
160C3×Q8#63811yescharacteristic
161C3×D4—21no—
162C3×D4—34no—
163C3×D4—64no—
164C3×D4—181no—
165C3×D4—181no—
166C3×D4#63811yescharacteristic
167C2×C3×C4—21no—
168C2×C3×C4—21no—
169C2×C3×C4—34no—
170C2×C3×C4—64no—
171C2×C3×C4—64no—
172C2×C3×C4C22×C3⋊D311yescharacteristic
173D12—64no—
174D12—64no—
175C3×C8—21no—
176C3×C8—64no—
177C3×C8—91no—
178C3×C8—91no—
179C3×C8—181no—
180C3×C8—181no—
181C3⋊C8—34no—
182C3⋊C8—34no—
183C3⋊C8—64no—
184C3⋊C8—64no—
185C2×C32—24no—
186C2×C32—44no—
187C2×C32#93914yes—
188C3×D3—244no—
189#41C2×#27211yescharacteristic
190C2×Q8C2×#27211yescharacteristic
191C2×D4—91no—
192Q16—21no—
193SD16—21no—
194SD16—181no—
195SD16—181no—
196SD16—181no—
197D8—181no—
198M16—91no—
199M16—91no—
200M16—181no—
201M16C2×#27211yescharacteristic
202C2×C8—91no—
203C2×C8—181no—
204C22×C3—21no—
205C22×C3—34no—
206C22×C3—64no—
207C22×C3—361no—
208C22×C3—721no—
209C22×C3#383011yescharacteristic
210C2×D3—124no—
211C2×D3—244no—
212C3×C4—21no—
213C3×C4—34no—
214C3×C4—34no—
215C3×C4—41no—
216C3×C4—64no—
217C3×C4—124no—
218C3×C4D4×C3⋊D311yescharacteristic
219C3×C4C2×#63811yescharacteristic
220C32#520314yescomplemented
221C23—181no—
222Q8—21no—
223Q8—21no—
224Q8#623511yescharacteristic
225D4—21no—
226D4—181no—
227D4—181no—
228D4#623511yescharacteristic
229C2×C4—21no—
230C2×C4—21no—
231C2×C4C22×#27211yescharacteristic
232C8—21no—
233C8—91no—
234C8—91no—
235C8—181no—
236C8—181no—
237C2×C3—21no—
238C2×C3—34no—
239C2×C3—41no—
240C2×C3—64no—
241C2×C3—124no—
242C2×C3—721no—
243C2×C3#6414811yescharacteristic
244D3—244no—
245C22—21no—
246C22—361no—
247C22—721no—
248C22#8874411yescharacteristic
249C4—21no—
250C4—41no—
251C4D4×#27211yescharacteristic
252C4C2×#623511yescharacteristic
253C3—34no—
254C3#1059290211yescharacteristic
255C2—21no—
256C2—41no—
257C2—721no—
258C2#1172184911yescharacteristic
subgroupG/H
center, socleC2×C3#64148
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C22×C3⋊D3
FittingH(3,𝔽3)×#137C2
prime radical#11721870C2
hypercenterC3×#137C3⋊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×#137

Complements

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

NH ≅ G/Ntype
1. H(3,𝔽3)×#137C2semidirect
3. #11721869C2semidirect
4. C2×#88549C2semidirect
5. #11720584C2semidirect
6. #11720560C2semidirect
7. #11720559C2semidirect
24. C32×#137D3semidirect
145. #137#272semidirect
152. H(3,𝔽3)#471semidirect
220. C32#5203semidirect

Maximal subgroups

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

subgroupindex
H(3,𝔽3)×#1372
#117218702
#117218692
C2×#885492
#117205842
#117205602
#117205592
C3×#52033