Finite Group Databasebeta

← #13143883

Subgroups — #13143883

Order 1728.

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

normal subgroups only

This group has 594 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#11721842C211yesmaximal, characteristic, complemented
2C2×#88545C211yesmaximal, characteristic, complemented
3#11721787C211yesmaximal, characteristic, complemented
4#11721786C211yesmaximal, characteristic, complemented
5H(3,𝔽3)×#100C211yesmaximal, characteristic, complemented
6#11720560C211yesmaximal, characteristic, complemented
7#11720559C211yesmaximal, characteristic, complemented
8C3×#4749—34nomaximal
9C2×D4×H(3,𝔽3)C2211yescharacteristic
10#88740C2211yescharacteristic
11C2×C4⋊#272C2211yescharacteristic
12M16×H(3,𝔽3)C2211yescharacteristic
13M16×H(3,𝔽3)C2211yescharacteristic
14#88546—21no—
15#88545—21no—
16#88545—21no—
17#88545—21no—
18#88538C2211yescharacteristic
19C2×#6180C2211yescharacteristic
20#88529—21no—
21#88528—21no—
22#88527—21no—
23C8×#272—21no—
24C3×#932—34no—
25C2×C3×#213—34no—
26C3×#908—34no—
27C3×#907—34no—
28C32×#100D314yescomplemented
29C3×#823—34no—
30C3×#822—34no—
31C23×H(3,𝔽3)—21no—
32C22×#272—21no—
33D4×H(3,𝔽3)—21no—
34D4×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×#472—91no—
48#4749—34no—
49C2×C32×D4C2×D314yes—
50C3×#235—34no—
51C2×C3×D12—34no—
52C32×M16C2×D314yes—
53C32×M16C2×D314yes—
54C3×#214—64no—
55C3×#213—64no—
56C3×#213—64no—
57C3×#213—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)—41no—
65C22×H(3,𝔽3)—81no—
66C22×H(3,𝔽3)#4111yescharacteristic
67C2×#272—21no—
68C2×#272—41no—
69C2×#272—81no—
70C4×H(3,𝔽3)C2×D411yescharacteristic
71C4×H(3,𝔽3)C2×D411yescharacteristic
72#1084—21no—
73C3×AGL(1,ℤ/8)—91no—
74C3×AGL(1,ℤ/8)—91no—
75C2×C3×D8—91no—
76C3×#131—91no—
77#932—34no—
78C2×#213—34no—
79#908—34no—
80#907—34no—
81C3×#108—91no—
82C3×#100—34no—
83C3×#100—91no—
84C3×#100C3⋊D311yescharacteristic
85#823—34no—
86#822—34no—
87C23×C32—24no—
88C22×C3×D3—64no—
89C32×D4—24no—
90C32×D4—24no—
91C2×C32×C4C22×D314yes—
92C3×#67—64no—
93C3×D12—34no—
94C3×D12—64no—
95C3×D12—64no—
96C3×C4×D3—64no—
97C3×Dic6—34no—
98C32×C8—24no—
99C32×C8—24no—
100C3×C3⋊C8—34no—
101C3×C3⋊C8—34no—
102C3×C3⋊C8—64no—
103#472—91noSylow
104C2×H(3,𝔽3)—21no—
105C2×H(3,𝔽3)—81no—
106C2×H(3,𝔽3)#12111yescharacteristic
107#272—41no—
108#272—81no—
109C3×#41—91no—
110C2×C3×D4—34no—
111C2×C3×D4—91no—
112C2×C3×D4C2×C3⋊D311yescharacteristic
113#235—34no—
114C2×D12—34no—
115C3×SD16—181no—
116C3×SD16—181no—
117C3×D8—181no—
118C3×D8—181no—
119C3×D8—181no—
120C3×D8—181no—
121C3×M16—34no—
122C3×M16—34no—
123C3×M16—91no—
124C3×M16—181no—
125C3×M16C2×C3⋊D311yescharacteristic
126C3×M16C2×C3⋊D311yescharacteristic
127C2×C3×C8—91no—
128C2×C3×C8—181no—
129#214—64no—
130#213—64no—
131#213—64no—
132#213—64no—
133#208—34no—
134C2×C3⋊C8—34no—
135D24—64no—
136#204—64no—
137ΓL(1,𝔽25)—64no—
138D3×C8—64no—
139C22×C32—44no—
140C22×C32—84no—
141C22×C32#23514yes—
142C2×C3×D3—64no—
143C2×C3×D3—124no—
144C2×C3×D3—244no—
145C32×C4D3×D414yes—
146C32×C4D3×D414yes—
147C3×Dic3—64no—
148AGL(1,ℤ/8)—91no—
149AGL(1,ℤ/8)—91no—
150C2×D8—91no—
151#131—91no—
152#108—91no—
153#100—91no—
154#100#27211yescharacteristic, complemented
155H(3,𝔽3)#47211yescharacteristic, complemented, Sylow
156C23×C3—21no—
157C23×C3—64no—
158C23×C3—181no—
159C22×D3—64no—
160C3×Q8—91no—
161C3×D4—21no—
162C3×D4—21no—
163C3×D4—64no—
164C3×D4—64no—
165C3×D4—91no—
166C3×D4—181no—
167C3×D4—181no—
168C3×D4—181no—
169C2×C3×C4—34no—
170C2×C3×C4—181no—
171C2×C3×C4C22×C3⋊D311yescharacteristic
172#67—64no—
173D12—34no—
174D12—64no—
175D12—64no—
176C4×D3—64no—
177Dic6—34no—
178C3×C8—21no—
179C3×C8—21no—
180C3×C8—64no—
181C3×C8—64no—
182C3×C8—91no—
183C3×C8—91no—
184C3×C8—181no—
185C3⋊C8—34no—
186C3⋊C8—34no—
187C3⋊C8—64no—
188C2×C32—24no—
189C2×C32—84no—
190C2×C32#88414yes—
191C3×D3—124no—
192C3×D3—244no—
193#41—91no—
194C2×D4—91no—
195C2×D4C2×#27211yescharacteristic
196SD16—181no—
197SD16—181no—
198D8—181no—
199D8—181no—
200D8—181no—
201D8—181no—
202M16—91no—
203M16—181no—
204M16C2×#27211yescharacteristic
205M16C2×#27211yescharacteristic
206C2×C8—91no—
207C2×C8—181no—
208C22×C3—34no—
209C22×C3—41no—
210C22×C3—81no—
211C22×C3—124no—
212C22×C3—181no—
213C22×C3—244no—
214C22×C3—361no—
215C22×C3—721no—
216C22×C3#382811yescharacteristic
217C2×D3—64no—
218C2×D3—124no—
219C2×D3—244no—
220C3×C4—34no—
221C3×C4—34no—
222C3×C4—181no—
223C3×C4D4×C3⋊D311yescharacteristic
224C3×C4D4×C3⋊D311yescharacteristic
225Dic3—64no—
226C32#474914yescomplemented
227C23—21no—
228C23—181no—
229Q8—91no—
230D4—21no—
231D4—21no—
232D4—91no—
233D4—181no—
234D4—181no—
235D4—181no—
236C2×C4—181no—
237C2×C4C22×#27211yescharacteristic
238C8—21no—
239C8—21no—
240C8—91no—
241C8—91no—
242C8—181no—
243C2×C3—21no—
244C2×C3—34no—
245C2×C3—64no—
246C2×C3—81no—
247C2×C3—244no—
248C2×C3—361no—
249C2×C3—721no—
250C2×C3#6409311yescharacteristic
251D3—124no—
252D3—244no—
253C22—41no—
254C22—81no—
255C22—181no—
256C22—361no—
257C22—721no—
258C22#8874011yescharacteristic
259C4—181no—
260C4D4×#27211yescharacteristic
261C4D4×#27211yescharacteristic
262C3—34no—
263C3#1059244811yescharacteristic
264C2—21no—
265C2—81no—
266C2—361no—
267C2—721no—
268C2#1172174011yescharacteristic
subgroupG/H
center, socleC2×C3#64093
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C22×C3⋊D3
FittingH(3,𝔽3)×#100C2
hypercenterC3×#100C3⋊D3
hypocenterH(3,𝔽3)#472
LCSG ▷ C2×C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C2×C3 ◁ C2×C3×C4 ◁ C3×#100

Complements

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

NH ≅ G/Ntype
1. #11721842C2semidirect
2. C2×#88545C2semidirect
3. #11721787C2semidirect
4. #11721786C2semidirect
5. H(3,𝔽3)×#100C2semidirect
6. #11720560C2semidirect
7. #11720559C2semidirect
28. C32×#100D3semidirect
154. #100#272semidirect
155. H(3,𝔽3)#472semidirect
226. C32#4749semidirect

Maximal subgroups

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

subgroupindex
#117218422
C2×#885452
#117217872
#117217862
H(3,𝔽3)×#1002
#117205602
#117205592
C3×#47493