Finite Group Databasebeta

← #13144438

Subgroups — #13144438

Order 1728.

495 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 495 conjugacy classes of subgroups — too many to draw. Only its 123 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
1C4×Q8×H(3,𝔽3)C211yesmaximal, characteristic
2#11721716C211yesmaximal, characteristic
3#11720583C211yesmaximal, characteristic
4#11720583C211yesmaximal, characteristic
5#11720530C211yesmaximal, characteristic
6#11720529C211yesmaximal, characteristic
7C4⋊#6180C211yesmaximal, characteristic
8C3×#5022—34nomaximal
9C2×Q8×H(3,𝔽3)C2211yescharacteristic
10C4⋊C4×H(3,𝔽3)—21no—
11C4⋊C4×H(3,𝔽3)C2211yescharacteristic
12C42×H(3,𝔽3)—21no—
13C42×H(3,𝔽3)C2211yescharacteristic
14C4⋊#1084C2211yescharacteristic
15C4⋊#1084C2211yescharacteristic
16#88540—21no—
17#88540—21no—
18C2×#6180C2211yescharacteristic
19C2×#6180C2211yescharacteristic
20C32×C4×Q8D314yes—
21C3×#870—34no—
22C3×#835—34no—
23C3×#835—34no—
24C3×#808—34no—
25C3×#807—34no—
26C3×C4⋊(C3⋊C8)—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)D411yescharacteristic
32C2×C4×H(3,𝔽3)D411yescharacteristic
33C2×C4×H(3,𝔽3)C2311yescharacteristic
34C2×#1084—21no—
35C2×#1084—21no—
36#6180—21no—
37#6180—21no—
38C3×#479—91no—
39#5022—34no—
40C2×C32×Q8C2×D314yes—
41C32×C4⋊C4—24no—
42C32×C4⋊C4C2×D314yes—
43C32×C42—24no—
44C32×C42C2×D314yes—
45C3×C4⋊Dic3—34no—
46C3×C4⋊Dic3—34no—
47C3×Dic3⋊C4—64no—
48C3×Dic3⋊C4—64no—
49C2×C3×C3⋊C8—34no—
50C2×C3×C3⋊C8—34no—
51C22×H(3,𝔽3)C2×D411yescharacteristic
52C4×H(3,𝔽3)—22no—
53C4×H(3,𝔽3)—21no—
54C4×H(3,𝔽3)—21no—
55C4×H(3,𝔽3)—41no—
56C4×H(3,𝔽3)C2×Q811yescharacteristic
57C4×H(3,𝔽3)#4111yescharacteristic
58#1084—41no—
59#1084—41no—
60C3×#125—91no—
61C3×C4×Q8—34no—
62C3×C4×Q8C3⋊D311yescharacteristic
63#870—34no—
64C3×#107—91no—
65C3×#106—91no—
66C3×C4⋊C8—91no—
67C3×#103—91no—
68C3×#103—91no—
69#835—34no—
70#835—34no—
71#808—34no—
72#807—34no—
73C4⋊(C3⋊C8)—34no—
74C32×Q8—24no—
75C32×Q8Dic618yes—
76C2×C32×C4—24no—
77C2×C32×C4—24no—
78C2×C32×C4#6714yes—
79C2×C32×C4#6714yes—
80C2×C32×C4C22×D314yes—
81C2×C3×Dic3—64no—
82C2×C3×Dic3—64no—
83C3×C3⋊C8—64no—
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—91no—
92C2×C3×C8—91no—
93C3×C4⋊C4—21no—
94C3×C4⋊C4—34no—
95C3×C4⋊C4—64no—
96C3×C4⋊C4—91no—
97C3×C4⋊C4—91no—
98C3×C4⋊C4—181no—
99C3×C4⋊C4—181no—
100C3×C4⋊C4C2×C3⋊D311yescharacteristic
101C3×C42—21no—
102C3×C42—34no—
103C3×C42—64no—
104C3×C42C2×C3⋊D311yescharacteristic
105C4⋊Dic3—34no—
106C4⋊Dic3—34no—
107Dic3⋊C4—64no—
108Dic3⋊C4—64no—
109C2×C3⋊C8—34no—
110C2×C3⋊C8—34no—
111C22×C32C2×#6714yes—
112C32×C4—28no—
113C32×C4—24no—
114C32×C4—24no—
115C32×C4—44no—
116C32×C4C2×Dic614yes—
117C32×C4#23514yes—
118C3×Dic3—124no—
119C3×Dic3—124no—
120#125—91no—
121C4×Q8#27211yescharacteristic
122#107—91no—
123#106—91no—
124C4⋊C8—91no—
125#103—91no—
126#103—91no—
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—34no—
136C2×C3×C4—34no—
137C2×C3×C4—64no—
138C2×C3×C4—64no—
139C2×C3×C4—181no—
140C2×C3×C4—181no—
141C2×C3×C4#63811yescharacteristic
142C2×C3×C4#63811yescharacteristic
143C2×C3×C4C22×C3⋊D311yescharacteristic
144C2×Dic3—64no—
145C2×Dic3—64no—
146C3×C8—181no—
147C3×C8—181no—
148C3⋊C8—64no—
149C3⋊C8—64no—
150C2×C32#92414yes—
151C2×C32#94214yes—
152C2×C32#95014yes—
153C2×Q8C2×#27211yescharacteristic
154C2×C8—91no—
155C2×C8—91no—
156C4⋊C4—21no—
157C4⋊C4—91no—
158C4⋊C4—91no—
159C4⋊C4—181no—
160C4⋊C4—181no—
161C4⋊C4C2×#27211yescharacteristic
162C42—21no—
163C42C2×#27211yescharacteristic
164C22×C3—34no—
165C22×C3C2×#63811yescharacteristic
166C3×C4—22no—
167C3×C4—21no—
168C3×C4—21no—
169C3×C4—34no—
170C3×C4—34no—
171C3×C4—41no—
172C3×C4—64no—
173C3×C4—64no—
174C3×C4—68no—
175C3×C4—124no—
176C3×C4—361no—
177C3×C4—361no—
178C3×C4C2×#63411yescharacteristic
179C3×C4#382811yescharacteristic
180Dic3—124no—
181Dic3—124no—
182C32#502214yescomplemented
183Q8—21no—
184Q8#622912yes—
185C2×C4—21no—
186C2×C4—21no—
187C2×C4—181no—
188C2×C4—181no—
189C2×C4#623511yescharacteristic
190C2×C4#623511yescharacteristic
191C2×C4C22×#27211yescharacteristic
192C8—181no—
193C8—181no—
194C2×C3—34no—
195C2×C3—34no—
196C2×C3—34no—
197C2×C3#6413311yescharacteristic
198C2×C3#6415111yescharacteristic
199C2×C3#6415911yescharacteristic
200C22C2×#623511yescharacteristic
201C4—22no—
202C4—21no—
203C4—21no—
204C4—41no—
205C4—361no—
206C4—361no—
207C4C2×#622911yescharacteristic
208C4#8874011yescharacteristic
209C3—34no—
210C3#1059272111yescharacteristic
211C2#1172182411yescharacteristic
212C2#1172185911yescharacteristic
213C2#1172187111yescharacteristic
subgroupG/H
center, socleC22×C3C2×#638
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C22×C3⋊D3
FittingC4×Q8×H(3,𝔽3)C2
prime radicalC22×H(3,𝔽3)C2×D4
hypercenterC3×C4×Q8C3⋊D3
hypocenterH(3,𝔽3)#479
LCSG ▷ C2×C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22×C3 ◁ C3×C42 ◁ C3×C4×Q8

Complements

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

NH ≅ G/Ntype
127. H(3,𝔽3)#479semidirect
182. C32#5022semidirect

Maximal subgroups

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

subgroupindex
C4×Q8×H(3,𝔽3)2
#117217162
#117205832
#117205302
#117205292
C4⋊#61802
C3×#50223