Finite Group Databasebeta

← #13144792

Subgroups — #13144792

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
1H(3,𝔽3)×#137C211yesmaximal, characteristic
2#11721872C211yesmaximal, characteristic
3#11721869C211yesmaximal, characteristic
4C2×#88550C211yesmaximal, characteristic, complemented
5#11720584C211yesmaximal, characteristic, complemented
6#11720561C211yesmaximal, characteristic, complemented
7#11720559C211yesmaximal, characteristic, complemented
8C3×#5204—34nomaximal
9#41×H(3,𝔽3)C2211yescharacteristic
10C2×Q8×H(3,𝔽3)C2211yescharacteristic
11C2×#6229C2211yescharacteristic
12Q16×H(3,𝔽3)—21no—
13SD16×H(3,𝔽3)—21no—
14M16×H(3,𝔽3)C2211yescharacteristic
15#88550—21no—
16#88550—21no—
17#88550—21no—
18#88546—21no—
19#88538—21no—
20#88538C2211yescharacteristic
21#88538C2211yescharacteristic
22C2×#6180—21no—
23C2×#6180C2211yescharacteristic
24C32×#137D314yes—
25C3×#951—34no—
26C3×#948—34no—
27C2×C3×#216—34no—
28C3×#836—34no—
29C3×#824—34no—
30C3×#822—34no—
31Q8×H(3,𝔽3)—21no—
32Q8×H(3,𝔽3)—21no—
33Q8×H(3,𝔽3)D411yescharacteristic
34D4×H(3,𝔽3)—21no—
35D4×H(3,𝔽3)D411yescharacteristic
36C2×C4×H(3,𝔽3)—21no—
37C2×C4×H(3,𝔽3)—21no—
38C2×C4×H(3,𝔽3)C2311yescharacteristic
39C2×#1084—21no—
40#6229—21no—
41#6229—21no—
42C8×H(3,𝔽3)—21no—
43#6180—21no—
44#6180—21no—
45#6180D411yescharacteristic
46#6180D411yescharacteristic
47C3×#473—91no—
48#5204—34no—
49C32×#41C2×D314yes—
50C2×C32×Q8C2×D314yes—
51C2×C3×Dic6—34no—
52C32×Q16—24no—
53C32×SD16—24no—
54C32×M16C2×D314yes—
55C3×#216—64no—
56C3×#216—64no—
57C3×#216—64no—
58C3×#214—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
66C4×H(3,𝔽3)—21no—
67C4×H(3,𝔽3)—41no—
68C4×H(3,𝔽3)C2×D411yescharacteristic
69C4×H(3,𝔽3)C2×D411yescharacteristic
70#1084—41no—
71C3×#137—34no—
72C3×#137—91no—
73C3×#137C3⋊D311yescharacteristic
74C2×C3×Q16—91no—
75C3×#131—91no—
76#951—34no—
77#948—34no—
78C2×#216—34no—
79C3×#108—91no—
80C3×#101—91no—
81C3×#101—91no—
82#836—34no—
83#824—34no—
84#822—34no—
85C32×Q8—24no—
86C32×Q8—24no—
87C32×Q8#6714yes—
88C32×D4—24no—
89C32×D4#6714yes—
90C2×C32×C4—24no—
91C2×C32×C4—24no—
92C2×C32×C4C22×D314yes—
93C2×C3×Dic3—64no—
94C3×Dic6—64no—
95C3×Dic6—64no—
96C32×C8—24no—
97C3×C3⋊C8—34no—
98C3×C3⋊C8—34no—
99C3×C3⋊C8—64no—
100C3×C3⋊C8—64no—
101#473—91noSylow
102C2×H(3,𝔽3)—21no—
103C2×H(3,𝔽3)—41no—
104C2×H(3,𝔽3)#12111yescharacteristic
105C3×#41—34no—
106C3×#41C2×C3⋊D311yescharacteristic
107C2×C3×Q8—34no—
108C2×C3×Q8—91no—
109C2×C3×Q8C2×C3⋊D311yescharacteristic
110C2×Dic6—34no—
111C3×Q16—21no—
112C3×Q16—64no—
113C3×Q16—181no—
114C3×Q16—181no—
115C3×Q16—181no—
116C3×SD16—21no—
117C3×SD16—64no—
118C3×SD16—181no—
119C3×M16—34no—
120C3×M16—91no—
121C3×M16—91no—
122C3×M16—181no—
123C3×M16C2×C3⋊D311yescharacteristic
124C2×C3×C8—91no—
125C2×C3×C8—181no—
126#216—64no—
127#216—64no—
128#216—64no—
129#214—64no—
130#208—34no—
131#208—34no—
132#208—64no—
133C2×C3⋊C8—34no—
134C2×C3⋊C8—64no—
135C22×C32—24no—
136C22×C32#23714yes—
137C32×C4—24no—
138C32×C4—44no—
139C32×C4D3×D414yes—
140C32×C4C2×#6714yes—
141C3×Dic3—124no—
142#137—91no—
143#137#27211yescharacteristic
144C2×Q16—91no—
145#131—91no—
146#108—91no—
147#101—91no—
148#101—91no—
149H(3,𝔽3)#47311yescharacteristic, complemented, Sylow
150C3×Q8—21no—
151C3×Q8—21no—
152C3×Q8—34no—
153C3×Q8—64no—
154C3×Q8—64no—
155C3×Q8—181no—
156C3×Q8—181no—
157C3×Q8#63811yescharacteristic
158C3×D4—21no—
159C3×D4—34no—
160C3×D4—64no—
161C3×D4#63811yescharacteristic
162C2×C3×C4—21no—
163C2×C3×C4—21no—
164C2×C3×C4—34no—
165C2×C3×C4—64no—
166C2×C3×C4—64no—
167C2×C3×C4—181no—
168C2×C3×C4C22×C3⋊D311yescharacteristic
169C2×Dic3—64no—
170Dic6—64no—
171Dic6—64no—
172C3×C8—21no—
173C3×C8—64no—
174C3×C8—91no—
175C3×C8—91no—
176C3×C8—181no—
177C3×C8—181no—
178C3⋊C8—34no—
179C3⋊C8—34no—
180C3⋊C8—64no—
181C3⋊C8—64no—
182C2×C32—24no—
183C2×C32—44no—
184C2×C32#93914yes—
185#41C2×#27211yescharacteristic
186C2×Q8—91no—
187C2×Q8C2×#27211yescharacteristic
188Q16—21no—
189Q16—181no—
190Q16—181no—
191Q16—181no—
192SD16—21no—
193SD16—181no—
194M16—91no—
195M16—91no—
196M16—181no—
197M16C2×#27211yescharacteristic
198C2×C8—91no—
199C2×C8—181no—
200C22×C3—21no—
201C22×C3—34no—
202C22×C3—64no—
203C22×C3#383011yescharacteristic
204C3×C4—21no—
205C3×C4—34no—
206C3×C4—34no—
207C3×C4—41no—
208C3×C4—64no—
209C3×C4—124no—
210C3×C4—361no—
211C3×C4D4×C3⋊D311yescharacteristic
212C3×C4C2×#63811yescharacteristic
213Dic3—124no—
214C32#520414yescomplemented
215Q8—21no—
216Q8—21no—
217Q8—181no—
218Q8—181no—
219Q8#623511yescharacteristic
220D4—21no—
221D4#623511yescharacteristic
222C2×C4—21no—
223C2×C4—21no—
224C2×C4—181no—
225C2×C4C22×#27211yescharacteristic
226C8—21no—
227C8—91no—
228C8—91no—
229C8—181no—
230C8—181no—
231C2×C3—21no—
232C2×C3—34no—
233C2×C3—41no—
234C2×C3—64no—
235C2×C3—124no—
236C2×C3#6414811yescharacteristic
237C22—21no—
238C22#8874411yescharacteristic
239C4—21no—
240C4—41no—
241C4—361no—
242C4D4×#27211yescharacteristic
243C4C2×#623511yescharacteristic
244C3—34no—
245C3#1059290311yescharacteristic
246C2—21no—
247C2—41no—
248C2#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#41×H(3,𝔽3)C22
hypercenterC3×#137C3⋊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×#137

Complements

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

NH ≅ G/Ntype
4. C2×#88550C2semidirect
5. #11720584C2semidirect
6. #11720561C2semidirect
7. #11720559C2semidirect
149. H(3,𝔽3)#473semidirect
214. C32#5204semidirect

Maximal subgroups

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

subgroupindex
H(3,𝔽3)×#1372
#117218722
#117218692
C2×#885502
#117205842
#117205612
#117205592
C3×#52043