Finite Group Databasebeta

← #13140887

Subgroups — #13140887

Order 1728.

412 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 412 conjugacy classes of subgroups — too many to draw. Only its 67 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#11721864C211yesmaximal, characteristic
2#11721095C211yesmaximal, characteristic, complemented
3#63662⋊C3C212yesmaximal, complemented
4#11720324C212yesmaximal
5#11720302C211yesmaximal, characteristic, complemented
6#10591614—32nomaximal
7C2×Q8×H(3,𝔽3)C2211yescharacteristic
8C2×C4⋊#272C2211yescharacteristic
9#88542—22no—
10C4×#1084C2211yescharacteristic
11C4⋊((C3⋊Dic3)⋊C3)C2212yes—
12C2×(C3⋊(C3⋊C8))⋊C3C2212yes—
13#88448—22no—
14C3×#947—32no—
15C3×#947—61no—
16#63869—32no—
17#63662D312yescomplemented
18C3×#835—32no—
19#63563—32no—
20#63562—32no—
21Dic3×C3⋊C8—32no—
22C22×#272—21no—
23Q8×H(3,𝔽3)—22no—
24C2×C4×H(3,𝔽3)—21no—
25C2×C4×H(3,𝔽3)C2311yescharacteristic
26C2×#1084D412yes—
27C4⋊#272—22no—
28C2×(C3⋊Dic3)⋊C3—22no—
29(C3⋊(C3⋊C8))⋊C3—22no—
30#5056—91no—
31#4800—92no—
32C2×C32×Q8—21no—
33C2×C32×Q8C2×D312yes—
34C2×C3×D12—32no—
35C2×C3×D12—61no—
36C4⋊(C3⋊Dic3)C2×D312yes—
37C2×C3⋊(C3⋊C8)C2×D312yes—
38C3×(C2×D3)⋊C4—62no—
39C3×(C2×D3)⋊C4—62no—
40C3×(C2×D3)⋊C4—64no—
41C3×C4⋊Dic3—32no—
42C3×C4×Dic3—32no—
43C3×C4×Dic3—61no—
44C2×C3×C3⋊C8—32no—
45#3724—62no—
46Dic3⋊Dic3—62no—
47C22×H(3,𝔽3)C2×D411yescharacteristic
48C2×#272—41no—
49C2×#272—42no—
50C4×H(3,𝔽3)—41no—
51C4×H(3,𝔽3)#4112yes—
52#1084—22no—
53(C3⋊Dic3)⋊C3—42no—
54C3×#124—91no—
55#947—32no—
56#947—61no—
57#890—92no—
58#870—91no—
59C3×#103—92no—
60#835—32no—
61#835—92no—
62#832—92no—
63#821—92no—
64C8×Dic3—92no—
65#809—92no—
66C4×C3⋊C8—91no—
67C22×C3×D3—62no—
68C22×C3×D3—62no—
69C32×Q8—22no—
70C32×Q8—22no—
71C32×Q8—42no—
72C2×C32×C4—22no—
73C2×C32×C4—22no—
74C2×C32×C4—21no—
75C2×C32×C4C22×D312yes—
76C2×C3⋊Dic3—22no—
77C2×C3×Dic3—32no—
78C2×C3×Dic3—32no—
79C2×C3×Dic3—62no—
80C2×C3×Dic3—62no—
81C3×D12—62no—
82C3×D12—62no—
83C3×D12—122no—
84C3⋊(C3⋊C8)—22no—
85C3×C3⋊C8—62no—
86#488—271noSylow
87C2×H(3,𝔽3)#12411yescharacteristic
88C2×H(3,𝔽3)#13512yes—
89#272—81no—
90C2×C3×Q8—32no—
91C2×C3×Q8—61no—
92C2×C3×Q8D3211yescharacteristic
93C2×C3×D4—91no—
94C2×D12—32no—
95C2×D12—61no—
96C2×C3×C8—92no—
97C3×C4⋊C4—92no—
98C3×C22⋊C4—182no—
99C3×C42—91no—
100(C2×D3)⋊C4—62no—
101(C2×D3)⋊C4—62no—
102(C2×D3)⋊C4—64no—
103C4⋊Dic3—32no—
104C4⋊Dic3—92no—
105C4⋊Dic3—182no—
106Dic3⋊C4—182no—
107Dic3⋊C4—182no—
108C4×Dic3—32no—
109C4×Dic3—61no—
110C2×C3⋊C8—32no—
111C2×C3⋊C8—92no—
112C22×C32—21no—
113C22×C32D3×D412yes—
114C2×C3×D3—122no—
115C2×C3×D3—124no—
116C2×C3×D3—122no—
117C2×C3×D3—122no—
118C2×C3×D3—122no—
119C32×C4—22no—
120C32×C4—42no—
121C32×C4—42no—
122C32×C4#23512yes—
123C32×C4#23712yes—
124C3⋊Dic3—42no—
125C3×Dic3—62no—
126C3×Dic3—62no—
127C3×Dic3—122no—
128C3×Dic3—122no—
129#125—271no—
130#124—91no—
131#103—92no—
132#102—272no—
133C4×C8—271no—
134H(3,𝔽3)#48811yescharacteristic, complemented, Sylow
135C23×C3—181no—
136C22×D3—62no—
137C22×D3—62no—
138C3×Q8—22no—
139C3×Q8—62no—
140C3×Q8—62no—
141C3×Q8—122no—
142C3×D4—182no—
143C2×C3×C4—21no—
144C2×C3×C4—32no—
145C2×C3×C4—62no—
146C2×C3×C4—62no—
147C2×C3×C4—61no—
148C2×C3×C4—92no—
149C2×C3×C4—182no—
150C2×C3×C4C2×D3211yescharacteristic
151C2×Dic3—32no—
152C2×Dic3—32no—
153C2×Dic3—62no—
154C2×Dic3—62no—
155C2×Dic3—182no—
156D12—62no—
157D12—62no—
158D12—122no—
159C3×C8—182no—
160C3⋊C8—62no—
161C3⋊C8—182no—
162C2×C32—22no—
163C2×C32—21no—
164C2×C32#88512yes—
165C2×C32#91612yes—
166C2×C32#91912yes—
167C3×D3—242no—
168C3×D3—242no—
169C2×Q8((C3⋊D3)⋊C3)⋊C211yescharacteristic
170C2×D4—91no—
171C2×C8—92no—
172C4⋊C4—92no—
173C4⋊C4—542no—
174C22⋊C4—182no—
175C42—91no—
176C22×C3—32no—
177C22×C3—61no—
178C22×C3—362no—
179C22×C3—361no—
180C22×C3#381111yescharacteristic
181C2×D3—122no—
182C2×D3—122no—
183C2×D3—122no—
184C2×D3—122no—
185C2×D3—124no—
186C3×C4—32no—
187C3×C4—32no—
188C3×C4—41no—
189C3×C4—62no—
190C3×C4—122no—
191C3×C4—122no—
192C3×C4—182no—
193C3×C4—362no—
194C3×C4#380412yes—
195Dic3—62no—
196Dic3—62no—
197Dic3—122no—
198Dic3—122no—
199Dic3—362no—
200C32—21no—
201C32#480012yescomplemented
202C23—181no—
203Q8—22no—
204D4—182no—
205C2×C4—21no—
206C2×C4—92no—
207C2×C4—182no—
208C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
209C8—182no—
210C2×C3—32no—
211C2×C3—32no—
212C2×C3—32no—
213C2×C3—62no—
214C2×C3—61no—
215C2×C3—721no—
216C2×C3#6395012yes—
217C2×C3#6395311yescharacteristic
218D3—242no—
219D3—242no—
220C22—362no—
221C22—361no—
222C22#8867711yescharacteristic
223C4—41no—
224C4—182no—
225C4—362no—
226C4#8867112yes—
227C3—32no—
228C3—61no—
229C3#1059153911yescharacteristic
230C2—721no—
231C2#1172123012yes—
232C2#1172129411yescharacteristic
subgroupG/H
centerC22#88677
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C2×D32
FittingC2×Q8×H(3,𝔽3)C22
socleC22×C3#3811
prime radicalC2×C4⋊#272C22
hypercenterC2×Q8((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#488
LCSG ▷ C2×C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
2. #11721095C2semidirect
3. #63662⋊C3C2semidirect
5. #11720302C2semidirect
17. #63662D3semidirect
134. H(3,𝔽3)#488semidirect
201. C32#4800semidirect

Maximal subgroups

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

subgroupindex
#117218642
#117210952
#63662⋊C32
#117203242
#117203022
#105916143