Finite Group Databasebeta

← #13144481

Subgroups — #13144481

Order 1728.

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

normal subgroups only

This group has 572 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
1H(3,𝔽3)×C22⋊Q8C211yesmaximal, characteristic
2C2×C4⋊#1084C211yesmaximal, characteristic
3#11720583C212yesmaximal, complemented
4#11720573C211yesmaximal, characteristic
5#11720529C212yesmaximal, complemented
6C3×#5044—34nomaximal
7C2×Q8×H(3,𝔽3)C2211yescharacteristic
8C22×C4×H(3,𝔽3)C2211yescharacteristic
9C22×#1084—21no—
10C4⋊C4×H(3,𝔽3)—21no—
11C4⋊C4×H(3,𝔽3)C2211yescharacteristic
12C22⋊C4×H(3,𝔽3)—21no—
13C4⋊#1084—21no—
14C4⋊#1084C2212yes—
15C2×#6180C2212yes—
16C32×C22⋊Q8D314yes—
17C2×C3×C4⋊Dic3—34no—
18C3×#835—38no—
19C3×#830—34no—
20C3×#807—38no—
21C23×H(3,𝔽3)D411yescharacteristic
22Q8×H(3,𝔽3)—22no—
23C2×C4×H(3,𝔽3)—22no—
24C2×C4×H(3,𝔽3)—21no—
25C2×C4×H(3,𝔽3)—21no—
26C2×C4×H(3,𝔽3)D411yescharacteristic
27C2×C4×H(3,𝔽3)C2311yescharacteristic
28C2×#1084—22no—
29C2×#1084—42no—
30#6180—22no—
31C3×#484—91no—
32#5044—34no—
33C2×C32×Q8C2×D314yes—
34C22×C32×C4C2×D314yes—
35C22×C3×Dic3—64no—
36C32×C4⋊C4—24no—
37C32×C4⋊C4C2×D314yes—
38C32×C22⋊C4—24no—
39C3×C4⋊Dic3—38no—
40C3×C4⋊Dic3—64no—
41C2×C3×C3⋊C8—38no—
42C22×H(3,𝔽3)—21no—
43C22×H(3,𝔽3)—21no—
44C22×H(3,𝔽3)Q1612yes—
45C22×H(3,𝔽3)C2×D411yescharacteristic
46C4×H(3,𝔽3)—21no—
47C4×H(3,𝔽3)—41no—
48C4×H(3,𝔽3)—41no—
49C4×H(3,𝔽3)#4112yes—
50#1084—42no—
51C3×C22⋊Q8—34no—
52C3×C22⋊Q8C3⋊D311yescharacteristic
53C2×C3×C4⋊C4—91no—
54C2×C4⋊Dic3—34no—
55C3×#107—92no—
56C3×#103—92no—
57C3×C22⋊C8—91no—
58#835—38no—
59#830—34no—
60#807—38no—
61C23×C32#6714yes—
62C32×Q8—28no—
63C2×C32×C4—28no—
64C2×C32×C4—24no—
65C2×C32×C4—24no—
66C2×C32×C4#6714yes—
67C2×C32×C4C22×D314yes—
68C2×C3×Dic3—68no—
69C2×C3×Dic3—128no—
70C3×C3⋊C8—68no—
71#484—91noSylow
72C2×H(3,𝔽3)—22no—
73C2×H(3,𝔽3)#12311yescharacteristic
74C2×H(3,𝔽3)C2×Q1611yescharacteristic
75C2×H(3,𝔽3)AGL(1,ℤ/8)11yescharacteristic
76C2×C3×Q8—34no—
77C2×C3×Q8C2×C3⋊D311yescharacteristic
78C22×C3×C4—34no—
79C22×C3×C4—181no—
80C22×C3×C4C2×C3⋊D311yescharacteristic
81C22×Dic3—64no—
82C2×C3×C8—92no—
83C3×C4⋊C4—21no—
84C3×C4⋊C4—34no—
85C3×C4⋊C4—64no—
86C3×C4⋊C4—92no—
87C3×C4⋊C4—181no—
88C3×C4⋊C4C2×C3⋊D311yescharacteristic
89C3×C22⋊C4—21no—
90C3×C22⋊C4—64no—
91C4⋊Dic3—38no—
92C4⋊Dic3—64no—
93C2×C3⋊C8—38no—
94C22×C32—24no—
95C22×C32—24no—
96C22×C32#21618yes—
97C22×C32C2×#6714yes—
98C32×C4—24no—
99C32×C4—44no—
100C32×C4—44no—
101C32×C4#23718yes—
102C3×Dic3—128no—
103C22⋊Q8#27211yescharacteristic
104C2×C4⋊C4—91no—
105#107—92no—
106#103—92no—
107C22⋊C8—91no—
108H(3,𝔽3)#48411yescharacteristic, complemented, Sylow
109C23×C3—34no—
110C23×C3#63811yescharacteristic
111C3×Q8—22no—
112C3×Q8—68no—
113C2×C3×C4—22no—
114C2×C3×C4—21no—
115C2×C3×C4—21no—
116C2×C3×C4—34no—
117C2×C3×C4—34no—
118C2×C3×C4—68no—
119C2×C3×C4—64no—
120C2×C3×C4—64no—
121C2×C3×C4—182no—
122C2×C3×C4—362no—
123C2×C3×C4#63811yescharacteristic
124C2×C3×C4C22×C3⋊D311yescharacteristic
125C2×Dic3—68no—
126C2×Dic3—128no—
127C3×C8—182no—
128C3⋊C8—68no—
129C2×C32—28no—
130C2×C32#93514yes—
131C2×C32C2×#21614yes—
132C2×C32#94914yes—
133C2×Q8C2×#27211yescharacteristic
134C22×C4—181no—
135C22×C4C2×#27211yescharacteristic
136C2×C8—92no—
137C4⋊C4—21no—
138C4⋊C4—92no—
139C4⋊C4—181no—
140C4⋊C4C2×#27211yescharacteristic
141C22⋊C4—21no—
142C22×C3—21no—
143C22×C3—21no—
144C22×C3—34no—
145C22×C3—38no—
146C22×C3—64no—
147C22×C3—64no—
148C22×C3#375612yes—
149C22×C3C2×#63811yescharacteristic
150C3×C4—21no—
151C3×C4—38no—
152C3×C4—41no—
153C3×C4—41no—
154C3×C4—64no—
155C3×C4—124no—
156C3×C4—124no—
157C3×C4—362no—
158C3×C4#383012yes—
159Dic3—128no—
160C32#504414yescomplemented
161C23#623511yescharacteristic
162Q8—22no—
163C2×C4—22no—
164C2×C4—21no—
165C2×C4—21no—
166C2×C4—182no—
167C2×C4—362no—
168C2×C4#623511yescharacteristic
169C2×C4C22×#27211yescharacteristic
170C8—182no—
171C2×C3—22no—
172C2×C3—34no—
173C2×C3—34no—
174C2×C3—34no—
175C2×C3—68no—
176C2×C3#6414411yescharacteristic
177C2×C3C2×#375611yescharacteristic
178C2×C3#6415811yescharacteristic
179C22—21no—
180C22—21no—
181C22#8855012yes—
182C22C2×#623511yescharacteristic
183C4—21no—
184C4—41no—
185C4—41no—
186C4—362no—
187C4#8874412yes—
188C3—34no—
189C3#1059274311yescharacteristic
190C2—22no—
191C2#1172184511yescharacteristic
192C2C2×#8855011yescharacteristic
193C2#1172187011yescharacteristic
subgroupG/H
center, socleC22×C3C2×#638
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C22×C3⋊D3
FittingH(3,𝔽3)×C22⋊Q8C2
prime radicalC23×H(3,𝔽3)D4
hypercenterC3×C22⋊Q8C3⋊D3
hypocenterH(3,𝔽3)#484
LCSG ▷ C2×C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22×C3 ◁ C22×C3×C4 ◁ C3×C22⋊Q8

Complements

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

NH ≅ G/Ntype
3. #11720583C2semidirect
5. #11720529C2semidirect
108. H(3,𝔽3)#484semidirect
160. C32#5044semidirect

Maximal subgroups

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

subgroupindex
H(3,𝔽3)×C22⋊Q82
C2×C4⋊#10842
#117205832
#117205732
#117205292
C3×#50443