Finite Group Databasebeta

← #13143862

Subgroups — #13143862

Order 1728.

869 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 869 conjugacy classes of subgroups — too many to draw. Only its 127 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
1C22×#6229C211yesmaximal, characteristic, complemented
2#11721830C211yesmaximal, characteristic
3C2×#88528C212yesmaximal, complemented
4H(3,𝔽3)×C22⋊C8C211yesmaximal, characteristic, complemented
5#11720549C212yesmaximal, complemented
6C3×#4738—34nomaximal
7C22×C4×H(3,𝔽3)C2211yescharacteristic
8C2×#6235—21no—
9C22×#1084—21no—
10C2×C4⋊#272C2211yescharacteristic
11C2×#6229—21no—
12C2×#6229—24no—
13C2×#6229C2212yes—
14C2×C8×H(3,𝔽3)C2212yes—
15#88542—21no—
16C4⋊#1084C2211yescharacteristic
17#88528—24no—
18C22×C3×Dic6—34no—
19C3×#930—34no—
20C2×C3×#204—38no—
21C32×C22⋊C8D314yescomplemented
22C3×#816—38no—
23C23×H(3,𝔽3)D411yescharacteristic
24C22×#272—21no—
25C2×C4×H(3,𝔽3)—22no—
26C2×C4×H(3,𝔽3)D411yescharacteristic
27C2×C4×H(3,𝔽3)C2311yescharacteristic
28#6235—42no—
29C2×#1084—24no—
30C2×#1084—22no—
31C2×#1084—21no—
32C4⋊#272—22no—
33#6229—24no—
34#6229—22no—
35#6229D414yescomplemented
36C8×H(3,𝔽3)—22no—
37C3×#448—91no—
38#4738—34no—
39C22×C32×C4C2×D314yes—
40C2×C3×#67—64no—
41C22×C3×Dic3—64no—
42C2×C3×D12—34no—
43C2×C3×Dic6—38no—
44C2×C3×Dic6—616no—
45C2×C3×Dic6—64no—
46C2×C32×C8C2×D318yes—
47C3×(C2×D3)⋊C4—64no—
48C3×C4⋊Dic3—34no—
49C3×#204—616no—
50C22×H(3,𝔽3)—21no—
51C22×H(3,𝔽3)—21no—
52C22×H(3,𝔽3)SD1612yescomplemented
53C22×H(3,𝔽3)C2×D411yescharacteristic
54C2×#272—41no—
55C2×#272—41no—
56C2×#272—41no—
57C4×H(3,𝔽3)—21no—
58C4×H(3,𝔽3)C2×D412yes—
59#1084—24no—
60#1084—41no—
61C22×C3×Q8—91no—
62C22×Dic6—34no—
63C2×C3×SD16—92no—
64C3×#121—91no—
65#930—34no—
66C2×#204—38no—
67C3×#103—92no—
68C3×C22⋊C8—34no—
69C3×C22⋊C8C3⋊D311yescharacteristic
70#816—38no—
71C23×C32D1214yes—
72C22×C3×D3—64no—
73C2×C32×C4—28no—
74C2×C32×C4D1214yes—
75C2×C32×C4C22×D314yes—
76C3×#67—128no—
77C2×C3×Dic3—616no—
78C2×C3×Dic3—68no—
79C2×C3×Dic3—64no—
80C3×D12—68no—
81C3×Dic6—316no—
82C3×Dic6—616no—
83C3×Dic6—68no—
84C32×C8—28no—
85#448—91noSylow
86C2×H(3,𝔽3)—22no—
87C2×H(3,𝔽3)#12011yescharacteristic
88C2×H(3,𝔽3)C2×SD1611yescharacteristic
89C2×H(3,𝔽3)#13711yescharacteristic
90#272—81no—
91C2×C3×Q8—92no—
92C2×C3×Q8—184no—
93C2×C3×Q8—181no—
94C2×C3×D4—91no—
95C2×C3×D4—181no—
96C22×C3×C4—34no—
97C22×C3×C4—181no—
98C22×C3×C4C2×C3⋊D311yescharacteristic
99C2×#67—64no—
100C22×Dic3—64no—
101C2×D12—34no—
102C2×Dic6—38no—
103C2×Dic6—616no—
104C2×Dic6—64no—
105C3×SD16—184no—
106C2×C3×C8—38no—
107C2×C3×C8C2×C3⋊D312yes—
108C3×C4⋊C4—91no—
109C3×C22⋊C4—181no—
110(C2×D3)⋊C4—64no—
111C4⋊Dic3—34no—
112#204—616no—
113C22×C32—24no—
114C22×C32—24no—
115C22×C32#20418yescomplemented
116C22×C32C2×D1214yes—
117C2×C3×D3—124no—
118C2×C3×D3—124no—
119C2×C3×D3—124no—
120C32×C4—24no—
121C32×C4D3×D418yes—
122C3×Dic3—616no—
123C3×Dic3—124no—
124C22×Q8—91no—
125C2×SD16—92no—
126#121—91no—
127#103—92no—
128C22⋊C8#27211yescharacteristic, complemented
129H(3,𝔽3)#44811yescharacteristic, complemented, Sylow
130C23×C3—34no—
131C23×C3—181no—
132C23×C3C4⋊(C3⋊D3)11yescharacteristic
133C22×D3—64no—
134C3×Q8—94no—
135C3×Q8—184no—
136C3×Q8—182no—
137C3×D4—182no—
138C3×D4—362no—
139C2×C3×C4—22no—
140C2×C3×C4—34no—
141C2×C3×C4—34no—
142C2×C3×C4—68no—
143C2×C3×C4—184no—
144C2×C3×C4—182no—
145C2×C3×C4—181no—
146C2×C3×C4C4⋊(C3⋊D3)11yescharacteristic
147C2×C3×C4C22×C3⋊D311yescharacteristic
148#67—128no—
149C2×Dic3—68no—
150C2×Dic3—616no—
151C2×Dic3—64no—
152D12—68no—
153Dic6—316no—
154Dic6—68no—
155Dic6—616no—
156C3×C8—22no—
157C3×C8—68no—
158C2×C32—28no—
159C2×C32#88214yes—
160C2×C32C2×#20414yes—
161C2×C32#90914yes—
162C3×D3—244no—
163C2×Q8—92no—
164C2×Q8—184no—
165C2×Q8—181no—
166C2×D4—91no—
167C2×D4—181no—
168C22×C4—181no—
169C22×C4C2×#27211yescharacteristic
170SD16—184no—
171C2×C8C2×#27212yes—
172C4⋊C4—91no—
173C22⋊C4—181no—
174C22×C3—21no—
175C22×C3—21no—
176C22×C3—34no—
177C22×C3—38no—
178C22×C3—64no—
179C22×C3—64no—
180C22×C3—361no—
181C22×C3—361no—
182C22×C3—361no—
183C22×C3#374412yes—
184C22×C3C2×C4⋊(C3⋊D3)11yescharacteristic
185C2×D3—124no—
186C2×D3—124no—
187C2×D3—124no—
188C3×C4—21no—
189C3×C4—38no—
190C3×C4—64no—
191C3×C4—184no—
192C3×C4—361no—
193C3×C4D4×C3⋊D312yes—
194Dic3—616no—
195Dic3—124no—
196C32#473814yescomplemented
197C23—181no—
198C23C4⋊#27211yescharacteristic
199Q8—94no—
200Q8—184no—
201Q8—182no—
202D4—182no—
203D4—362no—
204C2×C4—22no—
205C2×C4—184no—
206C2×C4—182no—
207C2×C4—181no—
208C2×C4C4⋊#27211yescharacteristic
209C2×C4C22×#27211yescharacteristic
210C8—22no—
211C2×C3—22no—
212C2×C3—34no—
213C2×C3—34no—
214C2×C3—34no—
215C2×C3—68no—
216C2×C3—721no—
217C2×C3#6409111yescharacteristic
218C2×C3C2×#374411yescharacteristic
219C2×C3#6411811yescharacteristic
220D3—244no—
221C22—21no—
222C22—21no—
223C22—361no—
224C22—361no—
225C22—361no—
226C22#8852812yescomplemented
227C22C2×C4⋊#27211yescharacteristic
228C4—21no—
229C4—184no—
230C4—361no—
231C4D4×#27212yes—
232C3—34no—
233C3#1059243711yescharacteristic
234C2—22no—
235C2—721no—
236C2#1172173811yescharacteristic
237C2C2×#8852811yescharacteristic
238C2#1172178811yescharacteristic
subgroupG/H
center, socleC22×C3C2×C4⋊(C3⋊D3)
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C22×C3⋊D3
FittingH(3,𝔽3)×C22⋊C8C2
prime radical#11721830C2
hypercenterC3×C22⋊C8C3⋊D3
hypocenterH(3,𝔽3)#448
LCSG ▷ C2×C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22×C3 ◁ C22×C3×C4 ◁ C3×C22⋊C8

Complements

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

NH ≅ G/Ntype
1. C22×#6229C2semidirect
3. C2×#88528C2semidirect
4. H(3,𝔽3)×C22⋊C8C2semidirect
5. #11720549C2semidirect
21. C32×C22⋊C8D3semidirect
35. #6229D4semidirect
52. C22×H(3,𝔽3)SD16semidirect
115. C22×C32#204semidirect
128. C22⋊C8#272semidirect
129. H(3,𝔽3)#448semidirect
196. C32#4738semidirect
226. C22#88528semidirect

Maximal subgroups

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

subgroupindex
C22×#62292
#117218302
C2×#885282
H(3,𝔽3)×C22⋊C82
#117205492
C3×#47383