Finite Group Databasebeta

← #13135905

Subgroups — #13135905

Order 1728.

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

normal subgroups only

This group has 440 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
1H(3,𝔽3)×C22⋊Q8C211yesmaximal, characteristic
2#11720574C211yesmaximal, characteristic
3#11720573C211yesmaximal, characteristic
4C3×#4538—34nomaximal
5C2×Q8×H(3,𝔽3)C411yescharacteristic, complemented
6C22×C4×H(3,𝔽3)C2211yescharacteristic
7C22×#1084—21no—
8C4⋊C4×H(3,𝔽3)—21no—
9C4⋊C4×H(3,𝔽3)C411yescharacteristic, complemented
10C22⋊C4×H(3,𝔽3)—21no—
11C2×#6180—21no—
12C32×C22⋊Q8D314yes—
13C3×C3⋊#95—34no—
14C3×#830—34no—
15C23×H(3,𝔽3)D411yescharacteristic
16Q8×H(3,𝔽3)—41no—
17C2×C4×H(3,𝔽3)—21no—
18C2×C4×H(3,𝔽3)—21no—
19C2×C4×H(3,𝔽3)—41no—
20C2×C4×H(3,𝔽3)C2×C411yescharacteristic
21C2×C4×H(3,𝔽3)D411yescharacteristic
22C2×#1084—22no—
23C2×#1084—41no—
24C2×#1084—41no—
25#6180—41no—
26C3×#328—91no—
27#4538—34no—
28C2×C32×Q8Dic314yescomplemented
29C22×C32×C4C2×D314yes—
30C22×C3×Dic3—64no—
31C32×C4⋊C4—24no—
32C32×C4⋊C4Dic314yescomplemented
33C32×C22⋊C4—24no—
34C2×C3×C3⋊C8—64no—
35C22×H(3,𝔽3)—21no—
36C22×H(3,𝔽3)—21no—
37C22×H(3,𝔽3)C22⋊C411yescharacteristic
38C22×H(3,𝔽3)SD1611yescharacteristic
39C22×H(3,𝔽3)Q1611yescharacteristic
40C4×H(3,𝔽3)—21no—
41C4×H(3,𝔽3)—21no—
42C4×H(3,𝔽3)—41no—
43C4×H(3,𝔽3)—41no—
44#1084—42no—
45C3×C22⋊Q8—34no—
46C3×C22⋊Q8C3⋊D311yescharacteristic
47C3×C22⋊C8—91no—
48C3×#95—91no—
49C3⋊#95—34no—
50#830—34no—
51C23×C32#6714yes—
52C32×Q8—44no—
53C2×C32×C4—24no—
54C2×C32×C4—24no—
55C2×C32×C4—44no—
56C2×C32×C4C2×Dic314yes—
57C2×C32×C4#6714yes—
58C2×C3×Dic3—68no—
59C2×C3×Dic3—124no—
60C2×C3×Dic3—124no—
61C3×C3⋊C8—124no—
62#328—91noSylow
63C2×H(3,𝔽3)—21no—
64C2×H(3,𝔽3)—21no—
65C2×H(3,𝔽3)#9911yescharacteristic
66C2×H(3,𝔽3)#10311yescharacteristic
67C2×H(3,𝔽3)#10411yescharacteristic
68C2×C3×Q8—34no—
69C2×C3×Q8C3⋊Dic311yescharacteristic
70C22×C3×C4—34no—
71C22×C3×C4—181no—
72C22×C3×C4C2×C3⋊D311yescharacteristic
73C22×Dic3—64no—
74C2×C3×C8—181no—
75C3×C4⋊C4—21no—
76C3×C4⋊C4—34no—
77C3×C4⋊C4—64no—
78C3×C4⋊C4C3⋊Dic311yescharacteristic
79C3×C22⋊C4—21no—
80C3×C22⋊C4—64no—
81C2×C3⋊C8—64no—
82C22×C32—24no—
83C22×C32—24no—
84C22×C32#21514yes—
85C22×C32#21614yes—
86C22×C32#21714yes—
87C32×C4—24no—
88C32×C4—24no—
89C32×C4—44no—
90C32×C4—44no—
91C3×Dic3—128no—
92C22⋊Q8#27211yescharacteristic
93C22⋊C8—91no—
94#95—91no—
95H(3,𝔽3)#32811yescharacteristic, complemented, Sylow
96C23×C3—34no—
97C23×C3#63811yescharacteristic
98C3×Q8—41no—
99C3×Q8—124no—
100C2×C3×C4—21no—
101C2×C3×C4—21no—
102C2×C3×C4—34no—
103C2×C3×C4—34no—
104C2×C3×C4—41no—
105C2×C3×C4—64no—
106C2×C3×C4—64no—
107C2×C3×C4—124no—
108C2×C3×C4—182no—
109C2×C3×C4—361no—
110C2×C3×C4—361no—
111C2×C3×C4C2×C3⋊Dic311yescharacteristic
112C2×C3×C4#63811yescharacteristic
113C2×Dic3—68no—
114C2×Dic3—124no—
115C2×Dic3—124no—
116C3×C8—361no—
117C3⋊C8—124no—
118C2×C32—24no—
119C2×C32—24no—
120C2×C32#83414yes—
121C2×C32#83514yes—
122C2×C32#83714yes—
123C2×Q8#108411yescharacteristic, complemented
124C22×C4—181no—
125C22×C4C2×#27211yescharacteristic
126C2×C8—181no—
127C4⋊C4—21no—
128C4⋊C4#108411yescharacteristic, complemented
129C22⋊C4—21no—
130C22×C3—21no—
131C22×C3—21no—
132C22×C3—34no—
133C22×C3—34no—
134C22×C3—34no—
135C22×C3—64no—
136C22×C3—64no—
137C22×C3#375511yescharacteristic
138C22×C3#375611yescharacteristic
139C22×C3#375711yescharacteristic
140C3×C4—21no—
141C3×C4—21no—
142C3×C4—41no—
143C3×C4—41no—
144C3×C4—64no—
145C3×C4—64no—
146C3×C4—124no—
147C3×C4—124no—
148C3×C4—362no—
149Dic3—128no—
150C32#453814yescomplemented
151C23#623511yescharacteristic
152Q8—41no—
153C2×C4—21no—
154C2×C4—21no—
155C2×C4—41no—
156C2×C4—182no—
157C2×C4—361no—
158C2×C4—361no—
159C2×C4C2×#108411yescharacteristic
160C2×C4#623511yescharacteristic
161C8—361no—
162C2×C3—21no—
163C2×C3—21no—
164C2×C3—34no—
165C2×C3—34no—
166C2×C3—34no—
167C2×C3—64no—
168C2×C3—64no—
169C2×C3#6366111yescharacteristic
170C2×C3#6366211yescharacteristic
171C2×C3#6366411yescharacteristic
172C22—21no—
173C22—21no—
174C22#8854911yescharacteristic
175C22#8855011yescharacteristic
176C22#8855211yescharacteristic
177C4—21no—
178C4—21no—
179C4—41no—
180C4—41no—
181C4—362no—
182C3—34no—
183C3#1058940411yescharacteristic
184C2—21no—
185C2—21no—
186C2#1172058011yescharacteristic
187C2#1172058311yescharacteristic
188C2#1172058611yescharacteristic
subgroupG/H
center, socleC22×C3#3757
commutatorC2×C4×H(3,𝔽3)C2×C4
FrattiniC22×C3×C4C2×C3⋊D3
FittingH(3,𝔽3)×C22⋊Q8C2
prime radicalC23×H(3,𝔽3)D4
cosocleC22×C4×H(3,𝔽3)C22
hypercenterC3×C22⋊Q8C3⋊D3
hypocenterH(3,𝔽3)#328
LCSG ▷ C2×C4×H(3,𝔽3) ▷ C22×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
5. C2×Q8×H(3,𝔽3)C4semidirect
9. C4⋊C4×H(3,𝔽3)C4semidirect
28. C2×C32×Q8Dic3semidirect
32. C32×C4⋊C4Dic3semidirect
95. H(3,𝔽3)#328semidirect
123. C2×Q8#1084semidirect
128. C4⋊C4#1084semidirect
150. C32#4538semidirect

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
#117205742
#117205732
C3×#45383