Finite Group Databasebeta

← #13144727

Subgroups — #13144727

Order 1728.

660 conjugacy classes of subgroups; only the 167 normal subgroups are shown.

normal subgroups only

This group has 660 conjugacy classes of subgroups — too many to draw. Only its 167 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×SD16×H(3,𝔽3)C211yesmaximal, characteristic
2Q8×#1084C211yesmaximal, characteristic, complemented
3D4×#1084C211yesmaximal, characteristic
4#11720583C211yesmaximal, characteristic, complemented
5#11720578C211yesmaximal, characteristic
6#11720551C211yesmaximal, characteristic, complemented
7#11720548C211yesmaximal, characteristic, complemented
8C3×#5164—34nomaximal
9C2×Q8×H(3,𝔽3)C2211yescharacteristic
10C2×D4×H(3,𝔽3)C2211yescharacteristic
11C22×#1084—21no—
12SD16×H(3,𝔽3)C414yescomplemented
13C2×C8×H(3,𝔽3)C2211yescharacteristic
14#88552—21no—
15C4⋊#1084—21no—
16C4⋊#1084C2211yescharacteristic
17C4⋊#1084C2211yescharacteristic
18C4×#1084—21no—
19C4×#1084C2211yescharacteristic
20C2×#6180C2211yescharacteristic
21C2×C32×SD16D314yes—
22C3×Q8×Dic3—34no—
23C3×D4×Dic3—34no—
24C3×#835—34no—
25C3×#832—34no—
26C3×#818—34no—
27C3×#815—34no—
28C23×H(3,𝔽3)—21no—
29Q8×H(3,𝔽3)—21no—
30Q8×H(3,𝔽3)C2×C412yescomplemented
31D4×H(3,𝔽3)—21no—
32D4×H(3,𝔽3)C2×C412yes—
33C2×C4×H(3,𝔽3)—21no—
34C2×C4×H(3,𝔽3)C2311yescharacteristic
35C2×#1084—21no—
36C2×#1084—21no—
37C2×#1084—42no—
38C2×#1084D411yescharacteristic
39C2×#1084D411yescharacteristic
40C8×H(3,𝔽3)C2×C412yescomplemented
41#6180—21no—
42C3×#440—91no—
43#5164—34no—
44C2×C32×Q8C2×D314yes—
45C2×C32×D4C2×D314yes—
46C22×C3×Dic3—64no—
47C32×SD16Dic3116yescomplemented
48C2×C32×C8C2×D314yes—
49C3×#217—64no—
50C3×C4⋊Dic3—34no—
51C3×C4⋊Dic3—34no—
52C3×C4⋊Dic3—64no—
53C3×C4×Dic3—34no—
54C3×C4×Dic3—64no—
55C2×C3×C3⋊C8—34no—
56C22×H(3,𝔽3)—22no—
57C22×H(3,𝔽3)—41no—
58C22×H(3,𝔽3)—41no—
59C22×H(3,𝔽3)C2×D411yescharacteristic
60C4×H(3,𝔽3)—22no—
61C4×H(3,𝔽3)C22×C411yescharacteristic
62C4×H(3,𝔽3)#4111yescharacteristic
63#1084—21no—
64#1084—21no—
65#1084—41no—
66#1084—41no—
67C2×C3×SD16—34no—
68C2×C3×SD16C3⋊D311yescharacteristic
69C3×C4×Q8—91no—
70C3×C4×D4—91no—
71Q8×Dic3—34no—
72D4×Dic3—34no—
73C3×#107—91no—
74C3×#103—91no—
75C3×#102—91no—
76C3×#97—91no—
77#835—34no—
78#832—34no—
79#818—34no—
80#815—34no—
81C23×C32—24no—
82C32×Q8—24no—
83C32×Q8C2×Dic318yescomplemented
84C32×D4—24no—
85C32×D4C2×Dic318yes—
86C2×C32×C4—24no—
87C2×C32×C4C22×D314yes—
88C2×C3×Dic3—34no—
89C2×C3×Dic3—34no—
90C2×C3×Dic3—64no—
91C2×C3×Dic3—64no—
92C2×C3×Dic3—128no—
93C32×C8C2×Dic318yescomplemented
94C3×C3⋊C8—64no—
95#440—91noSylow
96C2×H(3,𝔽3)—42no—
97C2×H(3,𝔽3)C4×D411yescharacteristic
98C2×H(3,𝔽3)AGL(1,ℤ/8)11yescharacteristic
99C2×H(3,𝔽3)#13711yescharacteristic
100C2×C3×Q8—34no—
101C2×C3×Q8C2×C3⋊D311yescharacteristic
102C2×C3×D4—34no—
103C2×C3×D4C2×C3⋊D311yescharacteristic
104C22×C3×C4—181no—
105C22×Dic3—64no—
106C3×SD16—316no—
107C3×SD16C3⋊Dic314yes—
108C2×C3×C8—34no—
109C2×C3×C8—91no—
110C2×C3×C8C2×C3⋊D311yescharacteristic
111C3×C4⋊C4—91no—
112C3×C4⋊C4—91no—
113C3×C4⋊C4—181no—
114C3×C22⋊C4—181no—
115C3×C42—91no—
116C3×C42—181no—
117#217—64no—
118C4⋊Dic3—34no—
119C4⋊Dic3—34no—
120C4⋊Dic3—64no—
121C4×Dic3—34no—
122C4×Dic3—64no—
123C2×C3⋊C8—34no—
124C22×C32—28no—
125C22×C32—44no—
126C22×C32—44no—
127C22×C32D3×D414yes—
128C32×C4—28no—
129C32×C4#23714yes—
130C32×C4C22×Dic314yes—
131C3×Dic3—64no—
132C3×Dic3—64no—
133C3×Dic3—124no—
134C3×Dic3—124no—
135C2×SD16#27211yescharacteristic
136C4×Q8—91no—
137C4×D4—91no—
138#107—91no—
139#103—91no—
140#102—91no—
141#97—91no—
142H(3,𝔽3)#44011yescharacteristic, complemented, Sylow
143C23×C3—21no—
144C23×C3—64no—
145C3×Q8—21no—
146C3×Q8—38no—
147C3×Q8—64no—
148C3×Q8C2×C3⋊Dic312yes—
149C3×D4—21no—
150C3×D4—38no—
151C3×D4—64no—
152C3×D4C2×C3⋊Dic312yes—
153C2×C3×C4—21no—
154C2×C3×C4—34no—
155C2×C3×C4—64no—
156C2×C3×C4—91no—
157C2×C3×C4—91no—
158C2×C3×C4—181no—
159C2×C3×C4—181no—
160C2×C3×C4—362no—
161C2×C3×C4C22×C3⋊D311yescharacteristic
162C2×Dic3—34no—
163C2×Dic3—34no—
164C2×Dic3—64no—
165C2×Dic3—64no—
166C2×Dic3—128no—
167C3×C8—38no—
168C3×C8—181no—
169C3×C8C2×C3⋊Dic312yes—
170C3⋊C8—64no—
171C2×C32—48no—
172C2×C32#91414yes—
173C2×C32#91514yes—
174C2×C32D4×Dic314yes—
175C2×Q8C2×#27211yescharacteristic
176C2×D4C2×#27211yescharacteristic
177C22×C4—181no—
178SD16#108414yescomplemented
179C2×C8—91no—
180C2×C8C2×#27211yescharacteristic
181C4⋊C4—91no—
182C4⋊C4—91no—
183C4⋊C4—181no—
184C22⋊C4—181no—
185C42—91no—
186C42—181no—
187C22×C3—22no—
188C22×C3—34no—
189C22×C3—41no—
190C22×C3—41no—
191C22×C3—68no—
192C22×C3—124no—
193C22×C3—124no—
194C22×C3D4×C3⋊D311yescharacteristic
195C3×C4—22no—
196C3×C4—34no—
197C3×C4—34no—
198C3×C4—68no—
199C3×C4—181no—
200C3×C4—181no—
201C3×C4—361no—
202C3×C4—361no—
203C3×C4#383011yescharacteristic
204C3×C4C22×C3⋊Dic311yescharacteristic
205Dic3—64no—
206Dic3—64no—
207Dic3—124no—
208Dic3—124no—
209C32#516414yescomplemented
210C23—21no—
211Q8—21no—
212Q8C2×#108412yescomplemented
213D4—21no—
214D4C2×#108412yes—
215C2×C4—21no—
216C2×C4—91no—
217C2×C4—91no—
218C2×C4—181no—
219C2×C4—181no—
220C2×C4—362no—
221C2×C4C22×#27211yescharacteristic
222C8—181no—
223C8C2×#108412yescomplemented
224C2×C3—34no—
225C2×C3—34no—
226C2×C3—34no—
227C2×C3—42no—
228C2×C3—128no—
229C2×C3#6412311yescharacteristic
230C2×C3#6412411yescharacteristic
231C2×C3D4×C3⋊Dic311yescharacteristic
232C22—22no—
233C22—41no—
234C22—41no—
235C22D4×#27211yescharacteristic
236C4—22no—
237C4—181no—
238C4—181no—
239C4—361no—
240C4—361no—
241C4#8874411yescharacteristic
242C4C22×#108411yescharacteristic
243C3—34no—
244C3#1059286311yescharacteristic
245C2—42no—
246C2#1172180011yescharacteristic
247C2#1172180111yescharacteristic
248C2D4×#108411yescharacteristic
subgroupG/H
center, socleC22×C3D4×C3⋊D3
commutatorC4×H(3,𝔽3)C22×C4
FrattiniC2×C3×C4C22×C3⋊D3
FittingC2×SD16×H(3,𝔽3)C2
prime radicalC2×D4×H(3,𝔽3)C22
cosocleC2×C4×H(3,𝔽3)C23
hypercenterC2×C3×SD16C3⋊D3
hypocenterH(3,𝔽3)#440
LCSG ▷ C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22×C3 ◁ C2×C3×C4 ◁ C2×C3×SD16

Complements

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

NH ≅ G/Ntype
2. Q8×#1084C2semidirect
4. #11720583C2semidirect
6. #11720551C2semidirect
7. #11720548C2semidirect
12. SD16×H(3,𝔽3)C4semidirect
30. Q8×H(3,𝔽3)C2×C4semidirect
40. C8×H(3,𝔽3)C2×C4semidirect
47. C32×SD16Dic3semidirect
83. C32×Q8C2×Dic3semidirect
93. C32×C8C2×Dic3semidirect
142. H(3,𝔽3)#440semidirect
178. SD16#1084semidirect
209. C32#5164semidirect
212. Q8C2×#1084semidirect
223. C8C2×#1084semidirect

Maximal subgroups

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

subgroupindex
C2×SD16×H(3,𝔽3)2
Q8×#10842
D4×#10842
#117205832
#117205782
#117205512
#117205482
C3×#51643