Finite Group Databasebeta

← #13143994

Subgroups — #13143994

Order 1728.

539 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 539 conjugacy classes of subgroups — too many to draw. Only its 101 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#11721861C211yesmaximal, characteristic
2#11721756C211yesmaximal, characteristic
3H(3,𝔽3)×#103C211yesmaximal, characteristic
4#11720583C211yesmaximal, characteristic
5#11720549C211yesmaximal, characteristic
6C8×#1084C211yesmaximal, characteristic
7#11720532C211yesmaximal, characteristic
8C3×#4792—34nomaximal
9C2×Q8×H(3,𝔽3)C2211yescharacteristic
10C2×#6229—21no—
11C2×#6229C2211yescharacteristic
12C2×C8×H(3,𝔽3)C2211yescharacteristic
13C4⋊C4×H(3,𝔽3)C2211yescharacteristic
14C4⋊#1084C2211yescharacteristic
15#88540—21no—
16#88540—21no—
17#88540—21no—
18C4×#1084C2211yescharacteristic
19C2×#6180C2211yescharacteristic
20C3×#944—34no—
21C3×#888—34no—
22C32×#103D314yes—
23C3×#835—34no—
24C3×#816—34no—
25C3×C8×Dic3—34no—
26C3×#810—34no—
27Q8×H(3,𝔽3)—21no—
28Q8×H(3,𝔽3)—21no—
29C2×C4×H(3,𝔽3)—21no—
30C2×C4×H(3,𝔽3)—21no—
31C2×C4×H(3,𝔽3)C2311yescharacteristic
32C2×#1084—21no—
33C2×#1084—21no—
34C2×#1084D411yescharacteristic
35C2×#1084D411yescharacteristic
36#6229—21no—
37#6229—21no—
38#6229—42no—
39C8×H(3,𝔽3)—21no—
40#6180—21no—
41C3×#487—91no—
42#4792—34no—
43C2×C32×Q8C2×D314yes—
44C2×C3×Dic6—34no—
45C2×C3×Dic6—64no—
46C2×C32×C8C2×D314yes—
47C32×C4⋊C4C2×D314yes—
48C3×C4⋊Dic3—34no—
49C3×Dic3⋊C4—64no—
50C3×Dic3⋊C4—64no—
51C3×Dic3⋊C4—64no—
52C3×C4×Dic3—34no—
53C2×C3×C3⋊C8—34no—
54C22×H(3,𝔽3)C2×D411yescharacteristic
55C4×H(3,𝔽3)—41no—
56C4×H(3,𝔽3)—41no—
57C4×H(3,𝔽3)#4111yescharacteristic
58C4×H(3,𝔽3)#4111yescharacteristic
59#1084—41no—
60#1084—41no—
61#1084SD1612yes—
62#1084Q1612yes—
63C3×C4⋊Q8—91no—
64C3×C4⋊Q8—91no—
65#944—34no—
66#888—34no—
67C3×#103—34no—
68C3×#103—91no—
69C3×#103—91no—
70C3×#103—91no—
71C3×#103C3⋊D311yescharacteristic
72C3×C4×C8—91no—
73#835—34no—
74#816—34no—
75C8×Dic3—34no—
76#810—34no—
77C32×Q8—24no—
78C32×Q8—24no—
79C2×C32×C4—24no—
80C2×C32×C4—24no—
81C2×C32×C4C22×D314yes—
82C2×C3×Dic3—34no—
83C2×C3×Dic3—34no—
84C2×C3×Dic3—64no—
85C2×C3×Dic3—64no—
86C3×Dic6—64no—
87C3×Dic6—64no—
88C3×Dic6—128no—
89C32×C8—24no—
90C3×C3⋊C8—64no—
91#487—91noSylow
92C2×H(3,𝔽3)#12411yescharacteristic
93C2×H(3,𝔽3)C2×SD1611yescharacteristic
94C2×H(3,𝔽3)C2×Q1611yescharacteristic
95C2×C3×Q8—34no—
96C2×C3×Q8—91no—
97C2×C3×Q8—181no—
98C2×C3×Q8C2×C3⋊D311yescharacteristic
99C2×Dic6—34no—
100C2×Dic6—64no—
101C2×C3×C8—34no—
102C2×C3×C8—91no—
103C2×C3×C8C2×C3⋊D311yescharacteristic
104C3×C4⋊C4—34no—
105C3×C4⋊C4—91no—
106C3×C4⋊C4—181no—
107C3×C4⋊C4—181no—
108C3×C4⋊C4—181no—
109C3×C4⋊C4C2×C3⋊D311yescharacteristic
110C3×C42—91no—
111C4⋊Dic3—34no—
112Dic3⋊C4—64no—
113Dic3⋊C4—64no—
114Dic3⋊C4—64no—
115C4×Dic3—34no—
116C2×C3⋊C8—34no—
117C22×C32D3×D414yes—
118C32×C4—44no—
119C32×C4—44no—
120C32×C4#23514yes—
121C32×C4#23714yes—
122C3×Dic3—38no—
123C3×Dic3—38no—
124C3×Dic3—124no—
125C3×Dic3—124no—
126C4⋊Q8—91no—
127C4⋊Q8—91no—
128#103—91no—
129#103—91no—
130#103—91no—
131#103#27211yescharacteristic
132C4×C8—91no—
133H(3,𝔽3)#48711yescharacteristic, complemented, Sylow
134C3×Q8—21no—
135C3×Q8—21no—
136C3×Q8—64no—
137C3×Q8—64no—
138C3×Q8—181no—
139C3×Q8—181no—
140C3×Q8—362no—
141C2×C3×C4—21no—
142C2×C3×C4—21no—
143C2×C3×C4—34no—
144C2×C3×C4—64no—
145C2×C3×C4—64no—
146C2×C3×C4—91no—
147C2×C3×C4—91no—
148C2×C3×C4—181no—
149C2×C3×C4—181no—
150C2×C3×C4C22×C3⋊D311yescharacteristic
151C2×Dic3—34no—
152C2×Dic3—34no—
153C2×Dic3—64no—
154C2×Dic3—64no—
155Dic6—64no—
156Dic6—64no—
157Dic6—128no—
158C3×C8—21no—
159C3×C8—64no—
160C3×C8—181no—
161C3⋊C8—64no—
162C2×C32#88514yes—
163C2×C32D3×SD1614yes—
164C2×C32D3×Q1614yes—
165C2×Q8—91no—
166C2×Q8—181no—
167C2×Q8C2×#27211yescharacteristic
168C2×C8—91no—
169C2×C8C2×#27211yescharacteristic
170C4⋊C4—91no—
171C4⋊C4—181no—
172C4⋊C4—181no—
173C4⋊C4—181no—
174C4⋊C4C2×#27211yescharacteristic
175C42—91no—
176C22×C3—34no—
177C22×C3D4×C3⋊D311yescharacteristic
178C3×C4—34no—
179C3×C4—34no—
180C3×C4—41no—
181C3×C4—41no—
182C3×C4—92no—
183C3×C4—92no—
184C3×C4—124no—
185C3×C4—124no—
186C3×C4—361no—
187C3×C4—361no—
188C3×C4#382811yescharacteristic
189C3×C4#383011yescharacteristic
190Dic3—38no—
191Dic3—38no—
192Dic3—124no—
193Dic3—124no—
194C32#479214yescomplemented
195Q8—21no—
196Q8—21no—
197Q8—181no—
198Q8—181no—
199Q8—362no—
200C2×C4—21no—
201C2×C4—21no—
202C2×C4—91no—
203C2×C4—91no—
204C2×C4—181no—
205C2×C4—181no—
206C2×C4C22×#27211yescharacteristic
207C8—21no—
208C8—181no—
209C2×C3—34no—
210C2×C3—34no—
211C2×C3—34no—
212C2×C3#6409411yescharacteristic
213C2×C3SD16×C3⋊D311yescharacteristic
214C2×C3Q16×C3⋊D311yescharacteristic
215C22D4×#27211yescharacteristic
216C4—41no—
217C4—41no—
218C4—92no—
219C4—92no—
220C4—361no—
221C4—361no—
222C4#8874011yescharacteristic
223C4#8874411yescharacteristic
224C3—34no—
225C3#1059249111yescharacteristic
226C2#1172174111yescharacteristic
227C2SD16×#27211yescharacteristic
228C2Q16×#27211yescharacteristic
subgroupG/H
center, socleC22×C3D4×C3⋊D3
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C22×C3⋊D3
FittingH(3,𝔽3)×#103C2
prime radicalC22×H(3,𝔽3)C2×D4
hypercenterC3×#103C3⋊D3
hypocenterH(3,𝔽3)#487
LCSG ▷ C2×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
133. H(3,𝔽3)#487semidirect
194. C32#4792semidirect

Maximal subgroups

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

subgroupindex
#117218612
#117217562
H(3,𝔽3)×#1032
#117205832
#117205492
C8×#10842
#117205322
C3×#47923