Finite Group Databasebeta

← #13152115

Subgroups — #13152115

Order 1728.

294 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 294 conjugacy classes of subgroups — too many to draw. Only its 89 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
1C2×#88683C211yesmaximal, characteristic
2C4×C9⋊SL(2,𝔽3)C212yesmaximal, complemented
3C3×#520⋊C3C311yesmaximal, characteristic
4C9×#520C311yesmaximal, characteristic, complemented
5#520⋊C9C312yesmaximal, complemented
6#88683C412yescomplemented
7C2×C9⋊SL(2,𝔽3)C2211yescharacteristic
8C4⋊C4×3−1+2—41nomaximal
9C2×C3×#41⋊C3C2×C311yescharacteristic
10C3×C4×SL(2,𝔽3)C2×C312yes—
11C2×C9×#41C2×C311yescharacteristic
12C2×#41⋊C9C2×C312yes—
13C4×Q8×C9C2×C312yes—
14C4×D4×C9—32no—
15C9×#117—31no—
16C2×C9×C4⋊C4—31no—
17C4×Q8⋊C9C2×C314yes—
18C2×C4×3−1+2—41no—
19C2×C4×3−1+2—42no—
20C9⋊SL(2,𝔽3)C2×C411yescharacteristic, complemented
21C3×#520C3211yescharacteristic
22#520⋊C3—31no—
23C3×#41⋊C3C3×C412yes—
24C2×C3×SL(2,𝔽3)C22×C311yescharacteristic
25C32×C4⋊C4—41no—
26C9×#41—32no—
27C9×#41C3×C412yescomplemented
28C2×Q8×C9C22×C311yescharacteristic
29C2×D4×C9—31no—
30C22×C4×C9—31no—
31C22×C4×C9—32no—
32#41⋊C9C3×C414yescomplemented
33C2×Q8⋊C9C22×C312yes—
34C9×C4⋊C4—31no—
35C9×C4⋊C4—32no—
36C9×C4⋊C4—42no—
37C9×C4⋊C4A411yescharacteristic
38C9×C22⋊C4—32no—
39C42×C9—32no—
40C22×3−1+2—41no—
41C4×3−1+2—42no—
42C4×3−1+2—82no—
43C2×C3×#41C2×C3211yescharacteristic
44C2×#41⋊C3—31no—
45C3×C4×Q8C2×C3212yes—
46C3×C4×D4—32no—
47C3×#117—31no—
48C2×C3×C4⋊C4—31no—
49C4×SL(2,𝔽3)—32no—
50C2×C32×C4—41no—
51C2×C32×C4—42no—
52C3×SL(2,𝔽3)C2×C3×C411yescharacteristic
53C23×C9—31no—
54Q8×C9—31no—
55Q8×C9C2×C3×C411yescharacteristic
56D4×C9—32no—
57D4×C9—32no—
58C2×C4×C9—31no—
59C2×C4×C9—32no—
60C2×C4×C9—32no—
61C2×C4×C9—32no—
62C2×C4×C9—42no—
63C2×C4×C9—44no—
64C2×C4×C9—122no—
65C2×C4×C9C2×A412yes—
66C2×C4×C9C2×A411yescharacteristic
67Q8⋊C9C2×C3×C412yes—
68#5203−1+211yescharacteristic, complemented, Sylow
69C2×3−1+2—41no—
70C2×3−1+2—41no—
71C2×3−1+2—41no—
72C3×#41—32no—
73C3×#41C32×C412yes—
74C2×C3×Q8C22×C3211yescharacteristic
75C2×C3×D4—31no—
76C22×C3×C4—31no—
77C22×C3×C4—32no—
78#41⋊C3—32no—
79C2×SL(2,𝔽3)—31no—
80C3×C4⋊C4—32no—
81C3×C4⋊C4—31no—
82C3×C4⋊C4—121no—
83C3×C4⋊C4C3×A411yescharacteristic
84C3×C22⋊C4—32no—
85C3×C42—32no—
86C22×C32—41no—
87C32×C4—42no—
88C32×C4—82no—
89C22×C9—32no—
90C22×C9—42no—
91C22×C9—61no—
92C22×C9—61no—
93C22×C9C22×A411yescharacteristic
94C4×C9—22no—
95C4×C9—31no—
96C4×C9—31no—
97C4×C9—44no—
98C4×C9—62no—
99C4×C9—84no—
100C4×C9C4×A412yes—
101C2×#41C2×3−1+211yescharacteristic
102C4×Q8C2×3−1+212yes—
103C4×D4—32no—
104#117—31no—
105C2×C4⋊C4—31no—
1063−1+2—41noSylow
107C23×C3—31no—
108C3×Q8—31no—
109C3×Q8C2×C32×C411yescharacteristic
110C3×D4—32no—
111C3×D4—32no—
112C2×C3×C4—31no—
113C2×C3×C4—32no—
114C2×C3×C4—32no—
115C2×C3×C4—32no—
116C2×C3×C4—122no—
117C2×C3×C4—121no—
118C2×C3×C4—122no—
119C2×C3×C4C2×C3×A412yes—
120C2×C3×C4C2×C3×A411yescharacteristic
121SL(2,𝔽3)—31no—
122C2×C32—41no—
123C2×C32—41no—
124C2×C32—41no—
125C2×C9—42no—
126C2×C9—42no—
127C2×C9—42no—
128C2×C9—62no—
129C2×C9C2×C4×A411yescharacteristic
130C2×C9#99411yescharacteristic
131C2×C92−1+4⋊C311yescharacteristic
132#41—32no—
133#41C4×3−1+212yescomplemented
134C2×Q8C22×3−1+211yescharacteristic
135C2×D4—31no—
136C22×C4—31no—
137C22×C4—32no—
138C4⋊C4—32no—
139C4⋊C4—31no—
140C4⋊C4C9⋊A411yescharacteristic
141C22⋊C4—32no—
142C42—32no—
143C22×C3—32no—
144C22×C3—61no—
145C22×C3—61no—
146C22×C3—121no—
147C22×C3C22×C3×A411yescharacteristic
148C3×C4—22no—
149C3×C4—31no—
150C3×C4—31no—
151C3×C4—62no—
152C3×C4—122no—
153C3×C4—242no—
154C3×C4C3×C4×A412yes—
155C32—41no—
156C9—42no—
157C9#520⋊C311yescharacteristic, complemented
158C23—31no—
159Q8—31no—
160Q8C2×C4×3−1+211yescharacteristic
161D4—32no—
162D4—32no—
163C2×C4—31no—
164C2×C4—32no—
165C2×C4—32no—
166C2×C4—32no—
167C2×C4—122no—
168C2×C4C2×C9⋊A412yes—
169C2×C4C2×C9⋊A411yescharacteristic
170C2×C3—62no—
171C2×C3—121no—
172C2×C3—121no—
173C2×C3—121no—
174C2×C3C2×C3×C4×A411yescharacteristic
175C2×C3C3×#99411yescharacteristic
176C2×C3C3×2−1+4⋊C311yescharacteristic
177C22—32no—
178C22—61no—
179C22—61no—
180C22C22×C9⋊A411yescharacteristic
181C4—22no—
182C4—31no—
183C4—31no—
184C4—62no—
185C4C4×C9⋊A412yes—
186C3—121no—
187C3C3×#520⋊C311yescharacteristic
188C2—62no—
189C2C2×C4×C9⋊A411yescharacteristic
190C2#1172255711yescharacteristic
191C2#1172255911yescharacteristic
subgroupG/H
center, Frattini, socleC22×C3C22×C3×A4
commutatorC3×Q8C2×C32×C4
FittingC9×#520C3
prime radicalC2×C3×#41⋊C3C2×C3
cosocleC2×C3×Q8C22×C32
hypercenterC9×C4⋊C4A4
hypocenterQ8C2×C4×3−1+2
LCSG ▷ C3×Q8 ▷ Q8
UCSC1 ◁ C22×C3 ◁ C9×C4⋊C4

Complements

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

NH ≅ G/Ntype
2. C4×C9⋊SL(2,𝔽3)C2semidirect
4. C9×#520C3semidirect
5. #520⋊C9C3semidirect
6. #88683C4semidirect
20. C9⋊SL(2,𝔽3)C2×C4semidirect
27. C9×#41C3×C4semidirect
32. #41⋊C9C3×C4semidirect
68. #5203−1+2semidirect
133. #41C4×3−1+2semidirect
157. C9#520⋊C3semidirect

Maximal subgroups

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

subgroupindex
C2×#886832
C4×C9⋊SL(2,𝔽3)2
C3×#520⋊C33
C9×#5203
#520⋊C93
C4⋊C4×3−1+24