Finite Group Databasebeta

← #13143662

Subgroups — #13143662

Order 1728.

563 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 563 conjugacy classes of subgroups — too many to draw. Only its 102 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
13−1+2×H(5,𝔽2)C211yesmaximal, characteristic
2#63497⋊C3C213yesmaximal, complemented
3#63393⋊C3C213yesmaximal, complemented
4C3×#5243—31nomaximal
5#10588923C311yesmaximal, characteristic, complemented
6#41×3−1+2—23no—
7C2×D4×3−1+2—23no—
8C2×D4×3−1+2C2213yes—
9C22×3−1+2⋊C4C2211yescharacteristic, complemented
10#3676⋊C3—23no—
11#3676⋊C3C2213yescomplemented
12(3−1+2⋊C4)⋊C4—23no—
13C32×H(5,𝔽2)D311yescharacteristic
14C3×#935—33no—
15C9×H(5,𝔽2)—21no—
16C9×H(5,𝔽2)C2×C311yescharacteristic
17C3×#834—33no—
18#63497C2×C313yescomplemented
19#63393C2×C313yescomplemented
20C23×3−1+2—21no—
21C23×3−1+2D413yes—
22C23×3−1+2C2311yescharacteristic
23Q8×3−1+2—21no—
24D4×3−1+2—23no—
25D4×3−1+2—26no—
26C2×C4×3−1+2—23no—
27C2×C4×3−1+2D413yes—
28C2×3−1+2⋊C4—23no—
29C2×3−1+2⋊C4—23no—
30C3×#458—91no—
31#5243—31no—
32C32×#41—23no—
33C2×C32×D4—23no—
34C2×C32×D4C2×D313yes—
35C22×C3×Dic3—31no—
36C3×#217—33no—
37C3×#217—63no—
38C3×Dic3⋊C4—63no—
39C9×#41—26no—
40C9×#41—23no—
41C2×D4×C9—26no—
42C2×D4×C9—23no—
43C2×D4×C9—23no—
44C2×D4×C9C22×C313yes—
45C22×Dic9C22×C311yescharacteristic, complemented
46#3676—23no—
47#3676C22×C313yescomplemented
48Dic9⋊C4—23no—
49C22×3−1+2—23no—
50C22×3−1+2—41no—
51C22×3−1+2—43no—
52C22×3−1+2—81no—
53C22×3−1+2C2×D413yes—
54C4×3−1+2—23no—
553−1+2⋊C4—43no—
563−1+2⋊C4—41no—
57C3×H(5,𝔽2)—31no—
58C3×H(5,𝔽2)C3×D311yescharacteristic
59C3×#123—93no—
60#935—33no—
61C3×#99—93no—
62#834—33no—
63C23×C32—21no—
64C23×C32#6713yes—
65C23×C32C22×D311yescharacteristic
66C32×Q8—21no—
67C32×D4—23no—
68C32×D4—26no—
69C2×C32×C4—23no—
70C2×C32×C4#6713yes—
71C2×C3×Dic3—63no—
72C2×C3×Dic3—63no—
73C23×C9—22no—
74C23×C9—21no—
75C23×C9—23no—
76C23×C9—21no—
77C23×C9C3×D413yes—
78C23×C9C23×C311yescharacteristic
79Q8×C9—22no—
80Q8×C9—21no—
81D4×C9—26no—
82D4×C9—23no—
83D4×C9—212no—
84D4×C9—26no—
85C2×C4×C9—26no—
86C2×C4×C9—23no—
87C2×C4×C9—23no—
88C2×C4×C9C3×D413yes—
89C2×Dic9—23no—
90C2×Dic9—23no—
91#458—91noSylow
92C2×3−1+2—23no—
93C2×3−1+2—43no—
94C2×3−1+2#12011yescharacteristic
95C3×#41—23no—
96C3×#41—63no—
97C2×C3×D4—23no—
98C2×C3×D4—33no—
99C2×C3×D4—63no—
100C2×C3×D4C2×C3×D313yes—
101C22×C3×C4—91no—
102C22×Dic3—31no—
103C3×C4⋊C4—183no—
104C3×C22⋊C4—93no—
105C3×C22⋊C4—183no—
106#217—33no—
107#217—63no—
108Dic3⋊C4—63no—
109C22×C32—23no—
110C22×C32—41no—
111C22×C32—43no—
112C22×C32—81no—
113C22×C32C2×#6713yes—
114C32×C4—23no—
115C3×Dic3—123no—
116C3×Dic3—121no—
117C22×C9—26no—
118C22×C9—23no—
119C22×C9—23no—
120C22×C9—41no—
121C22×C9—43no—
122C22×C9—81no—
123C22×C9—83no—
124C22×C9—82no—
125C22×C9—81no—
126C22×C9C2×C3×D413yes—
127C4×C9—26no—
128C4×C9—23no—
129Dic9—43no—
130Dic9—41no—
131H(5,𝔽2)AGL(1,ℤ/9)11yescharacteristic
132#123—93no—
133#99—93no—
1343−1+2#45811yescharacteristic, complemented, Sylow
135C23×C3—21no—
136C23×C3—33no—
137C23×C3—31no—
138C23×C3—61no—
139C23×C3C3×#6713yes—
140C23×C3C22×C3×D311yescharacteristic
141C3×Q8—21no—
142C3×Q8—61no—
143C3×D4—23no—
144C3×D4—26no—
145C3×D4—66no—
146C3×D4—63no—
147C2×C3×C4—23no—
148C2×C3×C4—33no—
149C2×C3×C4—63no—
150C2×C3×C4—183no—
151C2×C3×C4—183no—
152C2×C3×C4C3×#6713yes—
153C2×Dic3—63no—
154C2×Dic3—63no—
155C2×C32—23no—
156C2×C32—43no—
157C2×C32#95311yescharacteristic
158C2×C9—23no—
159C2×C9—21no—
160C2×C9—43no—
161C2×C9—46no—
162C2×C9—43no—
163C2×C9C3×#12011yescharacteristic
164#41—23no—
165C2×D4—23no—
166C2×D4C2×AGL(1,ℤ/9)13yes—
167C22×C4—91no—
168C4⋊C4—183no—
169C22⋊C4—93no—
170C22⋊C4—183no—
171C22×C3—23no—
172C22×C3—33no—
173C22×C3—41no—
174C22×C3—43no—
175C22×C3—63no—
176C22×C3—81no—
177C22×C3—121no—
178C22×C3—123no—
179C22×C3—241no—
180C22×C3C2×C3×#6713yes—
181C3×C4—23no—
182C3×C4—63no—
183C3×C4—361no—
184C3×C4—363no—
185Dic3—123no—
186Dic3—121no—
187C32#524311yescharacteristic
188C9—21no—
189C9C3×#45811yescharacteristic, complemented
190C23—21no—
191C23#611⋊C313yes—
192C23C22×AGL(1,ℤ/9)11yescharacteristic
193Q8—21no—
194D4—23no—
195D4—26no—
196C2×C4—23no—
197C2×C4—183no—
198C2×C4—183no—
199C2×C4#611⋊C313yes—
200C2×C3—23no—
201C2×C3—31no—
202C2×C3—43no—
203C2×C3—63no—
204C2×C3—123no—
205C2×C3C3×#95311yescharacteristic
206C22—23no—
207C22—41no—
208C22—43no—
209C22—81no—
210C22C2×#611⋊C313yes—
211C4—23no—
212C4—363no—
213C4—361no—
214C3—31no—
215C3C3×#524311yescharacteristic
216C2—23no—
217C2—43no—
218C2#1172170411yescharacteristic
subgroupG/H
centerC2#11721704
commutator, cosocleC23×C9C23×C3
FrattiniC23×C3C22×C3×D3
Fitting3−1+2×H(5,𝔽2)C2
socleC2×C3C3×#953
prime radicalC32×H(5,𝔽2)D3
hypercenterH(5,𝔽2)AGL(1,ℤ/9)
hypocenterC9C3×#458
LCSG ▷ C23×C9 ▷ C2×C9 ▷ C9
UCSC1 ◁ C2 ◁ C23 ◁ H(5,𝔽2)

Complements

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

NH ≅ G/Ntype
2. #63497⋊C3C2semidirect
3. #63393⋊C3C2semidirect
5. #10588923C3semidirect
9. C22×3−1+2⋊C4C22semidirect
11. #3676⋊C3C22semidirect
18. #63497C2×C3semidirect
19. #63393C2×C3semidirect
45. C22×Dic9C22×C3semidirect
47. #3676C22×C3semidirect
134. 3−1+2#458semidirect
189. C9C3×#458semidirect

Maximal subgroups

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

subgroupindex
3−1+2×H(5,𝔽2)2
#63497⋊C32
#63393⋊C32
C3×#52433
#105889233