Finite Group Databasebeta

← #13143999

Subgroups — #13143999

Order 1728.

495 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 495 conjugacy classes of subgroups — too many to draw. Only its 111 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
2#11721758C211yesmaximal, characteristic
3H(3,𝔽3)×#103C211yesmaximal, characteristic
4#11720583C211yesmaximal, characteristic
5#11720551C211yesmaximal, characteristic
6#11720547C211yesmaximal, characteristic
7#11720530C211yesmaximal, characteristic
8C3×#4796—34nomaximal
9C2×Q8×H(3,𝔽3)C2211yescharacteristic
10C2×C8×H(3,𝔽3)C2211yescharacteristic
11C4⋊C4×H(3,𝔽3)C2211yescharacteristic
12C4⋊#1084—21no—
13C4⋊#1084—21no—
14C4⋊#1084C2211yescharacteristic
15C4⋊#1084C2211yescharacteristic
16#88540—21no—
17C4×#1084—21no—
18C4×#1084C2211yescharacteristic
19C2×#6180C2211yescharacteristic
20C3×Q8×Dic3—34no—
21C3×#890—34no—
22C32×#103D314yes—
23C3×#835—34no—
24C3×#818—34no—
25C3×Dic3⋊C8—34no—
26C3×#808—34no—
27Q8×H(3,𝔽3)—21no—
28Q8×H(3,𝔽3)Q812yes—
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
36C8×H(3,𝔽3)—21no—
37#6180—21no—
38C3×#479—91no—
39#4796—34no—
40C2×C32×Q8C2×D314yes—
41C2×C32×C8C2×D314yes—
42C32×C4⋊C4C2×D314yes—
43C3×C4⋊Dic3—34no—
44C3×C4⋊Dic3—34no—
45C3×C4⋊Dic3—64no—
46C3×C4⋊Dic3—64no—
47C3×Dic3⋊C4—64no—
48C3×C4×Dic3—34no—
49C3×C4×Dic3—64no—
50C2×C3×C3⋊C8—34no—
51C22×H(3,𝔽3)C2×D411yescharacteristic
52C4×H(3,𝔽3)—22no—
53C4×H(3,𝔽3)—41no—
54C4×H(3,𝔽3)C2×Q811yescharacteristic
55C4×H(3,𝔽3)#4111yescharacteristic
56#1084—21no—
57#1084—21no—
58#1084—41no—
59#1084—41no—
60C3×#125—91no—
61C3×C4×Q8—91no—
62Q8×Dic3—34no—
63#890—34no—
64C3×#107—91no—
65C3×#106—91no—
66C3×C4⋊C8—91no—
67C3×#103—34no—
68C3×#103—91no—
69C3×#103C3⋊D311yescharacteristic
70#835—34no—
71#818—34no—
72Dic3⋊C8—34no—
73#808—34no—
74C32×Q8—24no—
75C32×Q8Dic618yes—
76C2×C32×C4—24no—
77C2×C32×C4—24no—
78C2×C32×C4C22×D314yes—
79C2×C3×Dic3—34no—
80C2×C3×Dic3—34no—
81C2×C3×Dic3—64no—
82C2×C3×Dic3—64no—
83C32×C8—24no—
84C3×C3⋊C8—64no—
85#479—91noSylow
86C2×H(3,𝔽3)C22⋊Q811yescharacteristic
87C2×H(3,𝔽3)#13511yescharacteristic
88C2×H(3,𝔽3)#13711yescharacteristic
89C2×C3×Q8—34no—
90C2×C3×Q8C2×C3⋊D311yescharacteristic
91C2×C3×C8—34no—
92C2×C3×C8—91no—
93C2×C3×C8C2×C3⋊D311yescharacteristic
94C3×C4⋊C4—34no—
95C3×C4⋊C4—91no—
96C3×C4⋊C4—91no—
97C3×C4⋊C4—181no—
98C3×C4⋊C4—181no—
99C3×C4⋊C4—181no—
100C3×C4⋊C4C2×C3⋊D311yescharacteristic
101C3×C42—91no—
102C3×C42—181no—
103C4⋊Dic3—34no—
104C4⋊Dic3—34no—
105C4⋊Dic3—64no—
106C4⋊Dic3—64no—
107Dic3⋊C4—64no—
108C4×Dic3—34no—
109C4×Dic3—64no—
110C2×C3⋊C8—34no—
111C22×C32D3×D414yes—
112C32×C4—28no—
113C32×C4—44no—
114C32×C4C2×Dic614yes—
115C32×C4#23714yes—
116C3×Dic3—64no—
117C3×Dic3—64no—
118C3×Dic3—124no—
119C3×Dic3—124no—
120#125—91no—
121C4×Q8—91no—
122#107—91no—
123#106—91no—
124C4⋊C8—91no—
125#103—91no—
126#103#27211yescharacteristic
127H(3,𝔽3)#47911yescharacteristic, complemented, Sylow
128C3×Q8—21no—
129C3×Q8—38no—
130C3×Q8—64no—
131C3×Q8#63412yes—
132C2×C3×C4—21no—
133C2×C3×C4—21no—
134C2×C3×C4—34no—
135C2×C3×C4—64no—
136C2×C3×C4—64no—
137C2×C3×C4—91no—
138C2×C3×C4—91no—
139C2×C3×C4—181no—
140C2×C3×C4—181no—
141C2×C3×C4C22×C3⋊D311yescharacteristic
142C2×Dic3—34no—
143C2×Dic3—34no—
144C2×Dic3—64no—
145C2×Dic3—64no—
146C3×C8—21no—
147C3×C8—64no—
148C3×C8—181no—
149C3⋊C8—64no—
150C2×C32#87814yes—
151C2×C32#91514yes—
152C2×C32#91914yes—
153C2×Q8C2×#27211yescharacteristic
154C2×C8—91no—
155C2×C8C2×#27211yescharacteristic
156C4⋊C4—91no—
157C4⋊C4—91no—
158C4⋊C4—181no—
159C4⋊C4—181no—
160C4⋊C4—181no—
161C4⋊C4C2×#27211yescharacteristic
162C42—91no—
163C42—181no—
164C22×C3—34no—
165C22×C3D4×C3⋊D311yescharacteristic
166C3×C4—22no—
167C3×C4—34no—
168C3×C4—34no—
169C3×C4—41no—
170C3×C4—68no—
171C3×C4—124no—
172C3×C4—181no—
173C3×C4—181no—
174C3×C4—361no—
175C3×C4—361no—
176C3×C4C2×#63411yescharacteristic
177C3×C4#383011yescharacteristic
178Dic3—64no—
179Dic3—64no—
180Dic3—124no—
181Dic3—124no—
182C32#479614yescomplemented
183Q8—21no—
184Q8#622912yes—
185C2×C4—21no—
186C2×C4—21no—
187C2×C4—91no—
188C2×C4—91no—
189C2×C4—181no—
190C2×C4—181no—
191C2×C4C22×#27211yescharacteristic
192C8—21no—
193C8—181no—
194C2×C3—34no—
195C2×C3—34no—
196C2×C3—34no—
197C2×C3#6408711yescharacteristic
198C2×C3#6412411yescharacteristic
199C2×C3#6412811yescharacteristic
200C22D4×#27211yescharacteristic
201C4—22no—
202C4—41no—
203C4—181no—
204C4—181no—
205C4—361no—
206C4—361no—
207C4C2×#622911yescharacteristic
208C4#8874411yescharacteristic
209C3—34no—
210C3#1059249511yescharacteristic
211C2#1172173411yescharacteristic
212C2#1172180111yescharacteristic
213C2#1172180811yescharacteristic
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)#479
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
127. H(3,𝔽3)#479semidirect
182. C32#4796semidirect

Maximal subgroups

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

subgroupindex
Q8×#10842
#117217582
H(3,𝔽3)×#1032
#117205832
#117205512
#117205472
#117205302
C3×#47963