Finite Group Databasebeta

← #13143806

Subgroups — #13143806

Order 1728.

550 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 550 conjugacy classes of subgroups — too many to draw. Only its 109 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#11721721C211yesmaximal, characteristic
2#11721715C211yesmaximal, characteristic, complemented
3H(3,𝔽3)×#97C211yesmaximal, characteristic, complemented
4#11720554C212yesmaximal
5#11720549C212yesmaximal, complemented
6C3×#4714—34nomaximal
7C2×C4⋊#272C2211yescharacteristic
8C2×#6229—21no—
9C2×#6229C2211yescharacteristic
10C2×C8×H(3,𝔽3)C2212yes—
11C42×H(3,𝔽3)C2211yescharacteristic
12#88542—21no—
13#88542—21no—
14C4⋊#1084—21no—
15C4⋊#1084C2212yes—
16C3×#875—34no—
17C3×#869—34no—
18C32×#97D314yescomplemented
19C3×#821—38no—
20C3×#816—38no—
21C22×#272—21no—
22C2×C4×H(3,𝔽3)D411yescharacteristic
23C2×C4×H(3,𝔽3)D411yescharacteristic
24C2×C4×H(3,𝔽3)C2311yescharacteristic
25C2×#1084—22no—
26C2×#1084—21no—
27C4⋊#272—22no—
28#6229—22no—
29#6229—42no—
30C8×H(3,𝔽3)—22no—
31C3×#489—91no—
32#4714—34no—
33C2×C3×D12—34no—
34C2×C3×Dic6—34no—
35C2×C3×Dic6—64no—
36C2×C32×C8C2×D318yes—
37C32×C42C2×D314yes—
38C3×(C2×D3)⋊C4—64no—
39C3×(C2×D3)⋊C4—64no—
40C3×C4⋊Dic3—38no—
41C3×C4⋊Dic3—64no—
42C22×H(3,𝔽3)C2×D411yescharacteristic
43C2×#272—41no—
44C2×#272—41no—
45C2×#272—41no—
46C4×H(3,𝔽3)—21no—
47C4×H(3,𝔽3)—21no—
48C4×H(3,𝔽3)#4112yes—
49#1084—41no—
50#1084—42no—
51C3×C4⋊Q8—91no—
52C3×#124—91no—
53#875—34no—
54#869—34no—
55C3×#103—92no—
56C3×#102—92no—
57C3×#97—34no—
58C3×#97C3⋊D311yescharacteristic
59#821—38no—
60#816—38no—
61C22×C3×D3—64no—
62C2×C32×C4D1214yes—
63C2×C32×C4D1214yes—
64C2×C32×C4C22×D314yes—
65C2×C3×Dic3—68no—
66C2×C3×Dic3—64no—
67C3×D12—68no—
68C3×Dic6—68no—
69C3×Dic6—128no—
70C32×C8—28no—
71#489—91noSylow
72C2×H(3,𝔽3)#12411yescharacteristic
73C2×H(3,𝔽3)AGL(1,ℤ/8)11yescharacteristic
74C2×H(3,𝔽3)#13711yescharacteristic
75#272—81no—
76C2×C3×Q8—91no—
77C2×C3×Q8—181no—
78C2×C3×D4—91no—
79C2×D12—34no—
80C2×Dic6—34no—
81C2×Dic6—64no—
82C2×C3×C8—38no—
83C2×C3×C8C2×C3⋊D312yes—
84C3×C4⋊C4—92no—
85C3×C4⋊C4—181no—
86C3×C22⋊C4—181no—
87C3×C22⋊C4—181no—
88C3×C42—34no—
89C3×C42C2×C3⋊D311yescharacteristic
90(C2×D3)⋊C4—64no—
91(C2×D3)⋊C4—64no—
92C4⋊Dic3—38no—
93C4⋊Dic3—64no—
94C22×C32C2×D1214yes—
95C2×C3×D3—124no—
96C2×C3×D3—124no—
97C2×C3×D3—124no—
98C32×C4—24no—
99C32×C4—24no—
100C32×C4#23518yes—
101C3×Dic3—124no—
102C3×Dic3—128no—
103C4⋊Q8—91no—
104#124—91no—
105#103—92no—
106#102—92no—
107#97#27211yescharacteristic, complemented
108H(3,𝔽3)#48911yescharacteristic, complemented, Sylow
109C23×C3—181no—
110C22×D3—64no—
111C3×Q8—182no—
112C3×Q8—362no—
113C3×D4—182no—
114C2×C3×C4—34no—
115C2×C3×C4—34no—
116C2×C3×C4—34no—
117C2×C3×C4—181no—
118C2×C3×C4—182no—
119C2×C3×C4C4⋊(C3⋊D3)11yescharacteristic
120C2×C3×C4C4⋊(C3⋊D3)11yescharacteristic
121C2×C3×C4C22×C3⋊D311yescharacteristic
122C2×Dic3—68no—
123C2×Dic3—64no—
124D12—68no—
125Dic6—68no—
126Dic6—128no—
127C3×C8—22no—
128C3×C8—68no—
129C2×C32#87514yes—
130C2×C32#90814yes—
131C2×C32#90914yes—
132C3×D3—244no—
133C2×Q8—91no—
134C2×Q8—181no—
135C2×D4—91no—
136C2×C8C2×#27212yes—
137C4⋊C4—92no—
138C4⋊C4—181no—
139C22⋊C4—181no—
140C22⋊C4—181no—
141C42C2×#27211yescharacteristic
142C22×C3—34no—
143C22×C3—361no—
144C22×C3—361no—
145C22×C3—361no—
146C22×C3C2×C4⋊(C3⋊D3)11yescharacteristic
147C2×D3—124no—
148C2×D3—124no—
149C2×D3—124no—
150C3×C4—21no—
151C3×C4—21no—
152C3×C4—38no—
153C3×C4—64no—
154C3×C4—64no—
155C3×C4—361no—
156C3×C4—362no—
157C3×C4#382812yes—
158Dic3—124no—
159Dic3—128no—
160C32#471414yescomplemented
161C23—181no—
162Q8—182no—
163Q8—362no—
164D4—182no—
165C2×C4—181no—
166C2×C4—182no—
167C2×C4C4⋊#27211yescharacteristic
168C2×C4C4⋊#27211yescharacteristic
169C2×C4C22×#27211yescharacteristic
170C8—22no—
171C2×C3—34no—
172C2×C3—34no—
173C2×C3—34no—
174C2×C3—721no—
175C2×C3#6408411yescharacteristic
176C2×C3#6411711yescharacteristic
177C2×C3#6411811yescharacteristic
178D3—244no—
179C22—361no—
180C22—361no—
181C22—361no—
182C22C2×C4⋊#27211yescharacteristic
183C4—21no—
184C4—21no—
185C4—362no—
186C4—361no—
187C4#8874012yes—
188C3—34no—
189C3#1059241311yescharacteristic
190C2—721no—
191C2#1172172111yescharacteristic
192C2#1172178711yescharacteristic
193C2#1172178811yescharacteristic
subgroupG/H
center, socleC22×C3C2×C4⋊(C3⋊D3)
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C22×C3⋊D3
FittingH(3,𝔽3)×#97C2
prime radicalC2×C4⋊#272C22
hypercenterC3×#97C3⋊D3
hypocenterH(3,𝔽3)#489
LCSG ▷ C2×C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22×C3 ◁ C3×C42 ◁ C3×#97

Complements

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

NH ≅ G/Ntype
2. #11721715C2semidirect
3. H(3,𝔽3)×#97C2semidirect
5. #11720549C2semidirect
18. C32×#97D3semidirect
107. #97#272semidirect
108. H(3,𝔽3)#489semidirect
160. C32#4714semidirect

Maximal subgroups

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

subgroupindex
#117217212
#117217152
H(3,𝔽3)×#972
#117205542
#117205492
C3×#47143