Finite Group Databasebeta

← #13144023

Subgroups — #13144023

Order 1728.

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

normal subgroups only

This group has 605 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
1Q8×#1084C211yesmaximal, characteristic
2C2×#88550C211yesmaximal, characteristic
3#11721755C211yesmaximal, characteristic
4H(3,𝔽3)×#103C211yesmaximal, characteristic
5#11720549C211yesmaximal, characteristic
6C8×#1084C211yesmaximal, characteristic
7#11720529C211yesmaximal, characteristic
8C3×#4790—34nomaximal
9C2×Q8×H(3,𝔽3)C2211yescharacteristic
10C2×#6229C2211yescharacteristic
11C2×C8×H(3,𝔽3)C2211yescharacteristic
12C4⋊C4×H(3,𝔽3)C2211yescharacteristic
13#88550C414yescomplemented
14C4⋊#1084—21no—
15C4⋊#1084C2211yescharacteristic
16#88540—21no—
17C4×#1084—21no—
18C4×#1084—21no—
19C4×#1084C2211yescharacteristic
20C2×#6180C2211yescharacteristic
21C3×Q8×Dic3—34no—
22C2×C3×#216—34no—
23C3×#887—34no—
24C32×#103D314yes—
25C3×#816—34no—
26C3×C8×Dic3—34no—
27C3×#807—34no—
28Q8×H(3,𝔽3)—21no—
29Q8×H(3,𝔽3)C2×C412yes—
30C2×C4×H(3,𝔽3)—21no—
31C2×C4×H(3,𝔽3)—21no—
32C2×C4×H(3,𝔽3)C2311yescharacteristic
33C2×#1084—21no—
34C2×#1084—21no—
35C2×#1084D411yescharacteristic
36C2×#1084D411yescharacteristic
37#6229—21no—
38#6229C2×C412yes—
39C8×H(3,𝔽3)—21no—
40#6180C2×C412yes—
41C3×C4×Q16—91no—
42#4790—34no—
43C2×C32×Q8C2×D314yes—
44C2×C3×Dic6—34no—
45C2×C32×C8C2×D314yes—
46C32×C4⋊C4C2×D314yes—
47C3×#216—316no—
48C3×C4⋊Dic3—34no—
49C3×C4⋊Dic3—64no—
50C3×Dic3⋊C4—64no—
51C3×C4×Dic3—34no—
52C3×C4×Dic3—64no—
53C3×C4×Dic3—64no—
54C2×C3×C3⋊C8—34no—
55C22×H(3,𝔽3)C2×D411yescharacteristic
56C4×H(3,𝔽3)—22no—
57C4×H(3,𝔽3)—41no—
58C4×H(3,𝔽3)C22×C411yescharacteristic
59C4×H(3,𝔽3)#4111yescharacteristic
60#1084—22no—
61#1084—21no—
62#1084—41no—
63#1084Q1612yescomplemented
64C2×C3×Q16—91no—
65C3×C4×Q8—91no—
66C3×C4×Q8—91no—
67Q8×Dic3—34no—
68C2×#216—34no—
69#887—34no—
70C3×#107—91no—
71C3×#103—34no—
72C3×#103—91no—
73C3×#103C3⋊D311yescharacteristic
74C3×C4×C8—91no—
75#816—34no—
76C8×Dic3—34no—
77#807—34no—
78C32×Q8—24no—
79C32×Q8C4×D318yes—
80C2×C32×C4—24no—
81C2×C32×C4—24no—
82C2×C32×C4C22×D314yes—
83C2×C3×Dic3—34no—
84C2×C3×Dic3—34no—
85C2×C3×Dic3—64no—
86C2×C3×Dic3—64no—
87C3×Dic6—38no—
88C3×Dic6—64no—
89C32×C8—24no—
90C3×C3⋊C8—38no—
91C4×Q16—91noSylow
92C2×H(3,𝔽3)C4×D411yescharacteristic
93C2×H(3,𝔽3)C2×Q1611yescharacteristic
94C2×H(3,𝔽3)#13511yescharacteristic
95C2×C3×Q8—34no—
96C2×C3×Q8—91no—
97C2×C3×Q8C2×C3⋊D311yescharacteristic
98C2×Dic6—34no—
99C3×Q16—94no—
100C2×C3×C8—34no—
101C2×C3×C8—91no—
102C2×C3×C8C2×C3⋊D311yescharacteristic
103C3×C4⋊C4—34no—
104C3×C4⋊C4—91no—
105C3×C4⋊C4—181no—
106C3×C4⋊C4—181no—
107C3×C4⋊C4C2×C3⋊D311yescharacteristic
108C3×C42—91no—
109C3×C42—181no—
110C3×C42—181no—
111#216—316no—
112C4⋊Dic3—34no—
113C4⋊Dic3—64no—
114Dic3⋊C4—64no—
115C4×Dic3—34no—
116C4×Dic3—64no—
117C4×Dic3—64no—
118C2×C3⋊C8—34no—
119C22×C32D3×D414yes—
120C32×C4—28no—
121C32×C4—44no—
122C32×C4C2×C4×D314yes—
123C32×C4#23714yes—
124C3×Dic3—38no—
125C3×Dic3—68no—
126C3×Dic3—64no—
127C3×Dic3—124no—
128C2×Q16—91no—
129C4×Q8—91no—
130C4×Q8—91no—
131#107—91no—
132#103—91no—
133#103#27211yescharacteristic
134C4×C8—91no—
135H(3,𝔽3)C4×Q1611yescharacteristic, complemented, Sylow
136C3×Q8—21no—
137C3×Q8—38no—
138C3×Q8—64no—
139C3×Q8—92no—
140C3×Q8—181no—
141C3×Q8C4×C3⋊D312yes—
142C2×C3×C4—21no—
143C2×C3×C4—21no—
144C2×C3×C4—34no—
145C2×C3×C4—64no—
146C2×C3×C4—64no—
147C2×C3×C4—91no—
148C2×C3×C4—91no—
149C2×C3×C4—181no—
150C2×C3×C4—181no—
151C2×C3×C4C22×C3⋊D311yescharacteristic
152C2×Dic3—34no—
153C2×Dic3—34no—
154C2×Dic3—64no—
155C2×Dic3—64no—
156Dic6—38no—
157Dic6—64no—
158C3×C8—21no—
159C3×C8—64no—
160C3×C8—92no—
161C3⋊C8—38no—
162C2×C32#67⋊C414yes—
163C2×C32#91614yes—
164C2×C32D3×Q1614yes—
165C2×Q8—91no—
166C2×Q8C2×#27211yescharacteristic
167Q16—94no—
168C2×C8—91no—
169C2×C8C2×#27211yescharacteristic
170C4⋊C4—91no—
171C4⋊C4—181no—
172C4⋊C4—181no—
173C4⋊C4C2×#27211yescharacteristic
174C42—91no—
175C42—181no—
176C42—181no—
177C22×C3—34no—
178C22×C3D4×C3⋊D311yescharacteristic
179C3×C4—22no—
180C3×C4—34no—
181C3×C4—34no—
182C3×C4—41no—
183C3×C4—68no—
184C3×C4—92no—
185C3×C4—124no—
186C3×C4—182no—
187C3×C4—181no—
188C3×C4—361no—
189C3×C4C2×C4×C3⋊D311yescharacteristic
190C3×C4#383011yescharacteristic
191Dic3—38no—
192Dic3—68no—
193Dic3—64no—
194Dic3—124no—
195C32#479014yescomplemented
196Q8—21no—
197Q8—92no—
198Q8—181no—
199Q8C4×#27212yes—
200C2×C4—21no—
201C2×C4—21no—
202C2×C4—91no—
203C2×C4—91no—
204C2×C4—181no—
205C2×C4—181no—
206C2×C4C22×#27211yescharacteristic
207C8—21no—
208C8—92no—
209C2×C3—34no—
210C2×C3—34no—
211C2×C3—34no—
212C2×C3#6409011yescharacteristic
213C2×C3#6412511yescharacteristic
214C2×C3Q16×C3⋊D311yescharacteristic
215C22D4×#27211yescharacteristic
216C4—22no—
217C4—41no—
218C4—92no—
219C4—182no—
220C4—181no—
221C4—361no—
222C4C2×C4×#27211yescharacteristic
223C4#8874411yescharacteristic
224C3—34no—
225C3#1059248911yescharacteristic
226C2#1172173711yescharacteristic
227C2#1172180211yescharacteristic
228C2Q16×#27211yescharacteristic
subgroupG/H
center, socleC22×C3D4×C3⋊D3
commutatorC4×H(3,𝔽3)C22×C4
FrattiniC2×C3×C4C22×C3⋊D3
FittingH(3,𝔽3)×#103C2
prime radicalC22×H(3,𝔽3)C2×D4
cosocleC2×C4×H(3,𝔽3)C23
hypercenterC3×#103C3⋊D3
hypocenterH(3,𝔽3)C4×Q16
LCSG ▷ C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22×C3 ◁ C2×C3×C4 ◁ C3×#103

Complements

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

NH ≅ G/Ntype
13. #88550C4semidirect
63. #1084Q16semidirect
135. H(3,𝔽3)C4×Q16semidirect
195. C32#4790semidirect

Maximal subgroups

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

subgroupindex
Q8×#10842
C2×#885502
#117217552
H(3,𝔽3)×#1032
#117205492
C8×#10842
#117205292
C3×#47903