Finite Group Databasebeta

← #13152134

Subgroups — #13152134

Order 1728.

231 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 231 conjugacy classes of subgroups — too many to draw. Only its 53 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
1#11721335C211yesmaximal, characteristic, complemented
2#11721333C211yesmaximal, characteristic, complemented
3#11721332C211yesmaximal, characteristic, complemented
4C3×#560⋊C3C311yesmaximal, characteristic
5C9×#560C311yesmaximal, characteristic, complemented
6#560⋊C9C312yesmaximal, complemented
7C22×C9⋊A4—41nomaximal
8C9⋊(C42⋊C3)C2211yescharacteristic, complemented
9C3×#127⋊C3C2×C311yescharacteristic
10C3×#126⋊C3C2×C311yescharacteristic
11C3×#126⋊C3C2×C311yescharacteristic
12C9×#127C2×C311yescharacteristic, complemented
13C9×#126C2×C311yescharacteristic, complemented
14C9×#126C2×C311yescharacteristic, complemented
15C9×#123—31no—
16C9×#121—31no—
17C9×#121—31no—
18C9×#120—31no—
19#127⋊C9C2×C312yescomplemented
20#126⋊C9C2×C312yescomplemented
21#126⋊C9C2×C312yescomplemented
22C2×C9⋊A4—41no—
23C2×C9⋊A4—41no—
24C2×C9⋊A4—41no—
25C3×#560C3211yescharacteristic
26#560⋊C3—31no—
27C22×C3×A4—41no—
28C24×C9A411yescharacteristic
29C22×C22⋊C9—42no—
30C3×C42⋊C3C22×C311yescharacteristic
31C2×D4×C9—31no—
32C2×D4×C9—31no—
33C2×D4×C9—31no—
34C2×D4×C9—31no—
35C22×C4×C9—31no—
36C9×C4⋊C4—31no—
37C9×C4⋊C4—31no—
38C9×C22⋊C4—31no—
39C9×C22⋊C4—31no—
40C9×C22⋊C4—31no—
41C9×C22⋊C4—31no—
42C42×C9C22×C311yescharacteristic, complemented
43C42⋊C9C22×C312yescomplemented
44C22×3−1+2—161no—
45C9⋊A4—41no—
46C3×#127C2×C3211yescharacteristic
47C3×#126C2×C3211yescharacteristic
48C3×#126C2×C3211yescharacteristic
49C3×#123—31no—
50C3×#121—31no—
51C3×#121—31no—
52C3×#120—31no—
53#127⋊C3—31no—
54#126⋊C3—31no—
55#126⋊C3—31no—
56C2×C3×A4—41no—
57C2×C3×A4—41no—
58C2×C3×A4—41no—
59C23×C9—31no—
60C23×C9—121no—
61C23×C9C2×A411yescharacteristic
62C23×C9C2×A411yescharacteristic
63C23×C9C2×A411yescharacteristic
64C2×C22⋊C9—42no—
65C2×C22⋊C9—42no—
66C2×C22⋊C9—42no—
67D4×C9—121no—
68D4×C9—121no—
69D4×C9—121no—
70D4×C9—121no—
71C2×C4×C9—31no—
72C2×C4×C9—31no—
73C2×C4×C9—31no—
74C2×C4×C9—121no—
75#5603−1+211yescharacteristic, complemented, Sylow
76C2×3−1+2—161no—
77C2×3−1+2—161no—
78C2×3−1+2—161no—
79C24×C3C3×A411yescharacteristic
80C22×A4—121no—
81C2×C3×D4—31no—
82C2×C3×D4—31no—
83C2×C3×D4—31no—
84C2×C3×D4—31no—
85C22×C3×C4—31no—
86C3×C4⋊C4—31no—
87C3×C4⋊C4—31no—
88C3×C22⋊C4—31no—
89C3×C22⋊C4—31no—
90C3×C22⋊C4—31no—
91C3×C22⋊C4—31no—
92C3×C42C22×C3211yescharacteristic
93C42⋊C3—31no—
94C22×C32—161no—
95C3×A4—41no—
96C22×C9—41no—
97C22×C9—61no—
98C22×C9—61no—
99C22×C9—61no—
100C22×C9—61no—
101C22×C9—61no—
102C22×C9—61no—
103C22×C9—121no—
104C22×C9—162no—
105C22×C9C22×A411yescharacteristic
106C22⋊C9—42no—
107C4×C9—61no—
108C4×C9—61no—
109C4×C9—61no—
110#127C2×3−1+211yescharacteristic, complemented
111#126C2×3−1+211yescharacteristic, complemented
112#126C2×3−1+211yescharacteristic, complemented
113#123—31no—
114#121—31no—
115#121—31no—
116#120—31no—
1173−1+2—161noSylow
118C23×C3—31no—
119C23×C3—121no—
120C23×C3C2×C3×A411yescharacteristic
121C23×C3C2×C3×A411yescharacteristic
122C23×C3C2×C3×A411yescharacteristic
123C2×A4—121no—
124C2×A4—121no—
125C2×A4—121no—
126C3×D4—121no—
127C3×D4—121no—
128C3×D4—121no—
129C3×D4—121no—
130C2×C3×C4—31no—
131C2×C3×C4—31no—
132C2×C3×C4—31no—
133C2×C3×C4—121no—
134C2×C32—161no—
135C2×C32—161no—
136C2×C32—161no—
137C2×C9—31no—
138C2×C9—41no—
139C2×C9—41no—
140C2×C9—41no—
141C2×C9—121no—
142C2×C9—162no—
143C2×C9—162no—
144C2×C9—162no—
145C24C9⋊A411yescharacteristic
146C2×D4—31no—
147C2×D4—31no—
148C2×D4—31no—
149C2×D4—31no—
150C22×C4—31no—
151C4⋊C4—31no—
152C4⋊C4—31no—
153C22⋊C4—31no—
154C22⋊C4—31no—
155C22⋊C4—31no—
156C22⋊C4—31no—
157C42C22×3−1+211yescharacteristic, complemented
158C22×C3—41no—
159C22×C3—61no—
160C22×C3—61no—
161C22×C3—61no—
162C22×C3—61no—
163C22×C3—61no—
164C22×C3—61no—
165C22×C3—121no—
166C22×C3—481no—
167C22×C3C22×C3×A411yescharacteristic
168A4—121no—
169C3×C4—61no—
170C3×C4—61no—
171C3×C4—61no—
172C32—161no—
173C9—162no—
174C9#560⋊C311yescharacteristic, complemented
175C23—31no—
176C23—121no—
177C23C2×C9⋊A411yescharacteristic
178C23C2×C9⋊A411yescharacteristic
179C23C2×C9⋊A411yescharacteristic
180D4—121no—
181D4—121no—
182D4—121no—
183D4—121no—
184C2×C4—31no—
185C2×C4—31no—
186C2×C4—31no—
187C2×C4—121no—
188C2×C3—31no—
189C2×C3—41no—
190C2×C3—41no—
191C2×C3—41no—
192C2×C3—121no—
193C2×C3—481no—
194C2×C3—481no—
195C2×C3—481no—
196C22—41no—
197C22—61no—
198C22—61no—
199C22—61no—
200C22—61no—
201C22—61no—
202C22—61no—
203C22—121no—
204C22C22×C9⋊A411yescharacteristic
205C4—61no—
206C4—61no—
207C4—61no—
208C3—481no—
209C3C3×#560⋊C311yescharacteristic
210C2—31no—
211C2—41no—
212C2—41no—
213C2—41no—
214C2—121no—
subgroupG/H
centerC3C3×#560⋊C3
commutator, cosocleC3×C42C22×C32
Frattini, socleC22×C3C22×C3×A4
FittingC9×#560C3
prime radicalC3×#560⋊C3C3
hypercenterC9#560⋊C3
hypocenterC42C22×3−1+2
LCSG ▷ C3×C42 ▷ C42
UCSC1 ◁ C3 ◁ C9

Complements

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

NH ≅ G/Ntype
1. #11721335C2semidirect
2. #11721333C2semidirect
3. #11721332C2semidirect
5. C9×#560C3semidirect
6. #560⋊C9C3semidirect
8. C9⋊(C42⋊C3)C22semidirect
12. C9×#127C2×C3semidirect
13. C9×#126C2×C3semidirect
14. C9×#126C2×C3semidirect
19. #127⋊C9C2×C3semidirect
20. #126⋊C9C2×C3semidirect
21. #126⋊C9C2×C3semidirect
42. C42×C9C22×C3semidirect
43. C42⋊C9C22×C3semidirect
75. #5603−1+2semidirect
110. #127C2×3−1+2semidirect
111. #126C2×3−1+2semidirect
112. #126C2×3−1+2semidirect
157. C42C22×3−1+2semidirect
174. C9#560⋊C3semidirect

Maximal subgroups

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

subgroupindex
#117213352
#117213332
#117213322
C3×#560⋊C33
C9×#5603
#560⋊C93
C22×C9⋊A44