Finite Group Databasebeta

← #13143959

Subgroups — #13143959

Order 1728.

671 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 671 conjugacy classes of subgroups — too many to draw. Only its 121 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
1D4×#1084C211yesmaximal, characteristic
2C2×#88546C211yesmaximal, characteristic
3#11721755C211yesmaximal, characteristic, complemented
4H(3,𝔽3)×#102C211yesmaximal, characteristic
5#11720549C211yesmaximal, characteristic, complemented
6C8×#1084C211yesmaximal, characteristic, complemented
7#11720530C211yesmaximal, characteristic, complemented
8C3×#4758—34nomaximal
9C2×D4×H(3,𝔽3)C2211yescharacteristic
10C22×#1084—21no—
11C2×#6229C2211yescharacteristic
12C2×C8×H(3,𝔽3)C2211yescharacteristic
13C4⋊C4×H(3,𝔽3)C2211yescharacteristic
14#88552—21no—
15#88546C414yescomplemented
16C4⋊#1084C2211yescharacteristic
17#88540—21no—
18C4×#1084—21no—
19C4×#1084C2211yescharacteristic
20C2×#6180C2211yescharacteristic
21C3×D4×Dic3—34no—
22C2×C3×#214—34no—
23C3×#887—34no—
24C32×#102D314yes—
25C3×#816—34no—
26C3×C8×Dic3—34no—
27C3×#808—34no—
28C23×H(3,𝔽3)—21no—
29D4×H(3,𝔽3)—21no—
30D4×H(3,𝔽3)C2×C412yes—
31C2×C4×H(3,𝔽3)—21no—
32C2×C4×H(3,𝔽3)C2311yescharacteristic
33C2×#1084—21no—
34C2×#1084—21no—
35C2×#1084—42no—
36C2×#1084D411yescharacteristic
37C2×#1084D411yescharacteristic
38#6229—21no—
39#6229C2×C412yescomplemented
40C8×H(3,𝔽3)—21no—
41#6180C2×C412yescomplemented
42C3×C4×SD16—91no—
43#4758—34no—
44C2×C32×D4C2×D314yes—
45C22×C3×Dic3—64no—
46C2×C3×Dic6—34no—
47C2×C32×C8C2×D314yes—
48C32×C4⋊C4C2×D314yes—
49C3×#217—64no—
50C3×#214—316no—
51C3×C4⋊Dic3—34no—
52C3×Dic3⋊C4—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)—41no—
61C4×H(3,𝔽3)C22×C411yescharacteristic
62C4×H(3,𝔽3)#4111yescharacteristic
63#1084—22no—
64#1084—21no—
65#1084—41no—
66#1084SD1612yescomplemented
67C2×C3×SD16—91no—
68C3×C4×Q8—91no—
69C3×C4×D4—91no—
70D4×Dic3—34no—
71C2×#214—34no—
72#887—34no—
73C3×#106—91no—
74C3×#103—91no—
75C3×#102—34no—
76C3×#102C3⋊D311yescharacteristic
77C3×C4×C8—91no—
78#816—34no—
79C8×Dic3—34no—
80#808—34no—
81C23×C32—24no—
82C32×D4—24no—
83C32×D4C4×D318yes—
84C2×C32×C4—24no—
85C2×C32×C4C22×D314yes—
86C2×C3×Dic3—34no—
87C2×C3×Dic3—34no—
88C2×C3×Dic3—64no—
89C2×C3×Dic3—64no—
90C2×C3×Dic3—128no—
91C3×Dic6—38no—
92C3×Dic6—64no—
93C32×C8—24no—
94C3×C3⋊C8—38no—
95C4×SD16—91noSylow
96C2×H(3,𝔽3)—42no—
97C2×H(3,𝔽3)C4×D411yescharacteristic
98C2×H(3,𝔽3)C2×SD1611yescharacteristic
99C2×H(3,𝔽3)#13511yescharacteristic
100C2×C3×Q8—91no—
101C2×C3×D4—34no—
102C2×C3×D4C2×C3⋊D311yescharacteristic
103C22×C3×C4—181no—
104C22×Dic3—64no—
105C2×Dic6—34no—
106C3×SD16—94no—
107C2×C3×C8—34no—
108C2×C3×C8—91no—
109C2×C3×C8C2×C3⋊D311yescharacteristic
110C3×C4⋊C4—34no—
111C3×C4⋊C4—91no—
112C3×C4⋊C4—181no—
113C3×C4⋊C4C2×C3⋊D311yescharacteristic
114C3×C22⋊C4—181no—
115C3×C42—91no—
116C3×C42—181no—
117#217—64no—
118#214—316no—
119C4⋊Dic3—34no—
120Dic3⋊C4—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—44no—
129C32×C4C2×C4×D314yes—
130C32×C4#23714yes—
131C3×Dic3—38no—
132C3×Dic3—68no—
133C3×Dic3—64no—
134C3×Dic3—124no—
135C2×SD16—91no—
136C4×Q8—91no—
137C4×D4—91no—
138#106—91no—
139#103—91no—
140#102#27211yescharacteristic
141C4×C8—91no—
142H(3,𝔽3)C4×SD1611yescharacteristic, complemented, Sylow
143C23×C3—21no—
144C23×C3—64no—
145C3×Q8—92no—
146C3×Q8—181no—
147C3×D4—21no—
148C3×D4—38no—
149C3×D4—64no—
150C3×D4C4×C3⋊D312yes—
151C2×C3×C4—21no—
152C2×C3×C4—34no—
153C2×C3×C4—64no—
154C2×C3×C4—91no—
155C2×C3×C4—91no—
156C2×C3×C4—181no—
157C2×C3×C4—181no—
158C2×C3×C4—362no—
159C2×C3×C4C22×C3⋊D311yescharacteristic
160C2×Dic3—34no—
161C2×Dic3—34no—
162C2×Dic3—64no—
163C2×Dic3—64no—
164C2×Dic3—128no—
165Dic6—38no—
166Dic6—64no—
167C3×C8—21no—
168C3×C8—64no—
169C3×C8—92no—
170C3⋊C8—38no—
171C2×C32—48no—
172C2×C32#67⋊C414yes—
173C2×C32#91214yes—
174C2×C32D3×SD1614yes—
175C2×Q8—91no—
176C2×D4C2×#27211yescharacteristic
177C22×C4—181no—
178SD16—94no—
179C2×C8—91no—
180C2×C8C2×#27211yescharacteristic
181C4⋊C4—91no—
182C4⋊C4—181no—
183C4⋊C4C2×#27211yescharacteristic
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—34no—
196C3×C4—34no—
197C3×C4—41no—
198C3×C4—92no—
199C3×C4—124no—
200C3×C4—182no—
201C3×C4—181no—
202C3×C4—361no—
203C3×C4C2×C4×C3⋊D311yescharacteristic
204C3×C4#383011yescharacteristic
205Dic3—38no—
206Dic3—68no—
207Dic3—64no—
208Dic3—124no—
209C32#475814yescomplemented
210C23—21no—
211Q8—92no—
212Q8—181no—
213D4—21no—
214D4C4×#27212yes—
215C2×C4—21no—
216C2×C4—91no—
217C2×C4—91no—
218C2×C4—181no—
219C2×C4—181no—
220C2×C4—362no—
221C2×C4C22×#27211yescharacteristic
222C8—21no—
223C8—92no—
224C2×C3—34no—
225C2×C3—34no—
226C2×C3—34no—
227C2×C3—42no—
228C2×C3—128no—
229C2×C3#6409011yescharacteristic
230C2×C3#6412111yescharacteristic
231C2×C3SD16×C3⋊D311yescharacteristic
232C22—22no—
233C22—41no—
234C22—41no—
235C22D4×#27211yescharacteristic
236C4—41no—
237C4—92no—
238C4—182no—
239C4—181no—
240C4—361no—
241C4C2×C4×#27211yescharacteristic
242C4#8874411yescharacteristic
243C3—34no—
244C3#1059245711yescharacteristic
245C2—42no—
246C2#1172173711yescharacteristic
247C2#1172179411yescharacteristic
248C2SD16×#27211yescharacteristic
subgroupG/H
center, socleC22×C3D4×C3⋊D3
commutatorC4×H(3,𝔽3)C22×C4
FrattiniC2×C3×C4C22×C3⋊D3
FittingH(3,𝔽3)×#102C2
prime radicalC2×D4×H(3,𝔽3)C22
cosocleC2×C4×H(3,𝔽3)C23
hypercenterC3×#102C3⋊D3
hypocenterH(3,𝔽3)C4×SD16
LCSG ▷ C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22×C3 ◁ C2×C3×C4 ◁ C3×#102

Complements

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

NH ≅ G/Ntype
3. #11721755C2semidirect
5. #11720549C2semidirect
6. C8×#1084C2semidirect
7. #11720530C2semidirect
15. #88546C4semidirect
39. #6229C2×C4semidirect
41. #6180C2×C4semidirect
66. #1084SD16semidirect
142. H(3,𝔽3)C4×SD16semidirect
209. C32#4758semidirect

Maximal subgroups

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

subgroupindex
D4×#10842
C2×#885462
#117217552
H(3,𝔽3)×#1022
#117205492
C8×#10842
#117205302
C3×#47583