Finite Group Databasebeta

← #13147098

Subgroups — #13147098

Order 1728.

217 conjugacy classes of subgroups; only the 47 normal subgroups are shown.

normal subgroups only

This group has 217 conjugacy classes of subgroups — too many to draw. Only its 47 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
1C4×C9×SL(2,𝔽3)C211yesmaximal, characteristic
2SL(2,𝔽3)×Dic9C211yesmaximal, characteristic
3SL(2,𝔽3)×Dic9C211yesmaximal, characteristic
4#5569⋊C3—31nomaximal
5#10589671C311yesmaximal, characteristic, complemented
6C2×C9×SL(2,𝔽3)C2211yescharacteristic
7C3×Dic9⋊C4—41nomaximal
8C3×C4×SL(2,𝔽3)D311yescharacteristic
9Dic3×SL(2,𝔽3)—31no—
10Dic3×SL(2,𝔽3)—31no—
11C4×Q8×C9C2×C311yescharacteristic
12Q8×Dic9C2×C311yescharacteristic
13Q8×Dic9C2×C311yescharacteristic
14#63452—31no—
15#63450—31no—
16#63432—31no—
17C4×Dic18—31no—
18C4×Q8⋊C9—21no—
19C2×C3×C4×C9—41no—
20C2×C3×Dic9—41no—
21C2×C3×Dic9—41no—
22C9×SL(2,𝔽3)Q811yescharacteristic, complemented
23#5569—31no—
24#557⋊C3—91no—
25C2×C3×SL(2,𝔽3)C2×D311yescharacteristic
26C3×Dic3⋊C4—121no—
27C2×Q8×C9C22×C311yescharacteristic
28C2×Dic18—31no—
29C2×Q8⋊C9—21no—
30C9×C4⋊C4—31no—
31C42×C9—31no—
32C4⋊Dic9—31no—
33C4⋊Dic9—31no—
34C4⋊Dic9—31no—
35Dic9⋊C4—32no—
36Dic9⋊C4—31no—
37Dic9⋊C4A411yescharacteristic
38C4×Dic9—31no—
39C4×Dic9—31no—
40C22×C3×C9—41no—
41C3×C4×C9—81no—
42C3×Dic9—42no—
43C3×Dic9—81no—
44C3×C4×Q8C3×D311yescharacteristic
45Q8×Dic3—31no—
46Q8×Dic3—31no—
47#890—91no—
48#888—91no—
49#870—91no—
50C4×Dic6—91no—
51C4×SL(2,𝔽3)—21no—
52C4×SL(2,𝔽3)—91no—
53C4×SL(2,𝔽3)—91no—
54C4×SL(2,𝔽3)D911yescharacteristic
55C2×C32×C4—41no—
56C2×C3×Dic3—121no—
57C2×C3×Dic3—121no—
58C3×SL(2,𝔽3)Dic611yescharacteristic
59Q8×C9—31no—
60Q8×C9C3×Q811yescharacteristic
61C2×C4×C9—31no—
62C2×C4×C9—31no—
63C2×C4×C9—81no—
64C2×C4×C9C2×A411yescharacteristic
65C2×Dic9—31no—
66C2×Dic9—31no—
67C2×Dic9C2×A411yescharacteristic
68C2×Dic9C2×A411yescharacteristic
69Dic18—62no—
70Q8⋊C9—21no—
71#557—91noSylow
72C2×C3×C9—41no—
73C2×C3×C9—41no—
74C2×C3×C9—41no—
75C2×C3×Q8C2×C3×D311yescharacteristic
76C2×Dic6—91no—
77C2×SL(2,𝔽3)—21no—
78C2×SL(2,𝔽3)C2×D911yescharacteristic
79C3×C4⋊C4—31no—
80C3×C4⋊C4—361no—
81C3×C42—31no—
82C4⋊Dic3—91no—
83C4⋊Dic3—91no—
84C4⋊Dic3—91no—
85Dic3⋊C4—31no—
86Dic3⋊C4—92no—
87Dic3⋊C4—91no—
88C4×Dic3—91no—
89C4×Dic3—91no—
90C22×C32—41no—
91C32×C4—81no—
92C3×Dic3—122no—
93C3×Dic3—241no—
94C22×C9—81no—
95C22×C9C22×A411yescharacteristic
96C4×C9—21no—
97C4×C9—31no—
98C4×C9—31no—
99C4×C9—61no—
100C4×C9—82no—
101Dic9—21no—
102Dic9—61no—
103Dic9—61no—
104Dic9#41⋊C312yes—
105C4⋊Q8—271no—
106#125—271no—
107#125—271no—
108C4×Q8—91no—
109C4×Q8—91no—
110C4×Q8—271no—
111C4×Q8C3×D911yescharacteristic
112C3×C9—41noSylow
113C3×Q8—31no—
114C3×Q8C3×Dic611yescharacteristic
115C2×C3×C4—31no—
116C2×C3×C4—31no—
117C2×C3×C4—41no—
118C2×C3×C4—81no—
119C2×C3×C4—361no—
120C2×C3×C4—361no—
121C2×C3×C4D3×A411yescharacteristic
122C2×Dic3—31no—
123C2×Dic3—31no—
124C2×Dic3—91no—
125C2×Dic3—91no—
126Dic6—182no—
127SL(2,𝔽3)—21no—
128SL(2,𝔽3)Dic1811yescharacteristic, complemented
129C2×C32—41no—
130C2×C32—41no—
131C2×C32—41no—
132C2×C9—81no—
133C2×C9—81no—
134C2×C9—81no—
135C2×C9Q8×A411yescharacteristic
136C2×C9C2×#41⋊C311yescharacteristic
137C2×C92−1+4⋊C311yescharacteristic
138C2×Q8—271no—
139C2×Q8C2×C3×D911yescharacteristic
140C4⋊C4—31no—
141C4⋊C4—91no—
142C4⋊C4—271no—
143C4⋊C4—271no—
144C4⋊C4—271no—
145C4⋊C4—272no—
146C4⋊C4—271no—
147C42—31no—
148C42—271no—
149C42—271no—
150C22×C3—41no—
151C22×C3—81no—
152C22×C3C2×D3×A411yescharacteristic
153C3×C4—21no—
154C3×C4—31no—
155C3×C4—31no—
156C3×C4—61no—
157C3×C4—81no—
158C3×C4—82no—
159C3×C4—362no—
160C3×C4—721no—
161Dic3—32no—
162Dic3—61no—
163Dic3—181no—
164Dic3—181no—
165C32—41no—
166C9—81no—
167C9#557⋊C311yescharacteristic, complemented
168Q8—31no—
169Q8—542no—
170Q8C3×Dic1811yescharacteristic
171C2×C4—31no—
172C2×C4—31no—
173C2×C4—91no—
174C2×C4—91no—
175C2×C4—271no—
176C2×C4—271no—
177C2×C4A4×D911yescharacteristic
178C2×C3—41no—
179C2×C3—41no—
180C2×C3—41no—
181C2×C3—81no—
182C2×C3—81no—
183C2×C3—81no—
184C2×C3A4×Dic611yescharacteristic
185C2×C3#1007⋊C311yescharacteristic
186C2×C3D3×#41⋊C311yescharacteristic
187C22C2×A4×D911yescharacteristic
188C4—21no—
189C4—31no—
190C4—31no—
191C4—61no—
192C4—92no—
193C4—181no—
194C4—541no—
195C4—541no—
196C3—41no—
197C3—81no—
198C3#5569⋊C311yescharacteristic
199C2A4×Dic1811yescharacteristic
200C2#1172241011yescharacteristic
201C2D9×#41⋊C311yescharacteristic
subgroupG/H
centerC22C2×A4×D9
commutator, cosocleC2×Q8×C9C22×C3
Frattini, socleC22×C3C2×D3×A4
FittingC4×Q8×C9C2×C3
prime radicalC2×C3×SL(2,𝔽3)C2×D3
hypercenterC2×C4A4×D9
hypocenterQ8×C9C3×Q8
LCSG ▷ C2×Q8×C9 ▷ Q8×C9
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
5. #10589671C3semidirect
22. C9×SL(2,𝔽3)Q8semidirect
128. SL(2,𝔽3)Dic18semidirect
167. C9#557⋊C3semidirect

Maximal subgroups

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

subgroupindex
C4×C9×SL(2,𝔽3)2
SL(2,𝔽3)×Dic92
#5569⋊C33
#105896713
C3×Dic9⋊C44