Finite Group Databasebeta

← #13147108

Subgroups — #13147108

Order 1728.

280 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 280 conjugacy classes of subgroups — too many to draw. Only its 51 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×D9×SL(2,𝔽3)C211yesmaximal, characteristic
2C4×C9×SL(2,𝔽3)C211yesmaximal, characteristic, complemented
3SL(2,𝔽3)×Dic9C211yesmaximal, characteristic, complemented
4#5572⋊C3—31nomaximal
5#10589674C311yesmaximal, characteristic, complemented
6C2×C9×SL(2,𝔽3)C2211yescharacteristic
7D9×SL(2,𝔽3)C412yescomplemented
8C3×(C2×D9)⋊C4—41nomaximal
9C2×D3×SL(2,𝔽3)—31no—
10C3×C4×SL(2,𝔽3)D311yescharacteristic
11Dic3×SL(2,𝔽3)—31no—
12C2×Q8×D9C2×C311yescharacteristic
13C4×Q8×C9C2×C311yescharacteristic, complemented
14Q8×Dic9C2×C311yescharacteristic, complemented
15#63454—31no—
16#63449—31no—
17#63434—31no—
18C4×Dic18—31no—
19C4×Q8⋊C9—21no—
20C22×C3×D9—41no—
21C2×C3×C4×C9—41no—
22C2×C3×Dic9—41no—
23C9×SL(2,𝔽3)C2×C411yescharacteristic
24#5572—31no—
25#519⋊C3—91no—
26C2×C3×SL(2,𝔽3)C2×D311yescharacteristic
27D3×SL(2,𝔽3)—32no—
28C3×(C2×D3)⋊C4—121no—
29C2×Q8×C9C22×C311yescharacteristic
30Q8×D9—32no—
31Q8×D9C3×C412yescomplemented
32C2×C4×D9—31no—
33C2×Dic18—31no—
34C2×Q8⋊C9—21no—
35C9×C4⋊C4—31no—
36C42×C9—31no—
37(C2×D9)⋊C4—31no—
38(C2×D9)⋊C4A411yescharacteristic
39C4⋊Dic9—31no—
40Dic9⋊C4—31no—
41Dic9⋊C4—31no—
42C4×Dic9—32no—
43C4×Dic9—31no—
44C22×C3×C9—41no—
45C2×C3×D9—42no—
46C2×C3×D9—81no—
47C2×C3×D9—81no—
48C3×C4×C9—81no—
49C3×Dic9—81no—
50C2×D3×Q8—31no—
51C22×SL(2,𝔽3)—91no—
52C3×C4×Q8C3×D311yescharacteristic
53Q8×Dic3—31no—
54#892—91no—
55#887—91no—
56#872—91no—
57C4×Dic6—91no—
58C4×SL(2,𝔽3)—21no—
59C4×SL(2,𝔽3)—91no—
60C4×SL(2,𝔽3)D911yescharacteristic, complemented
61C22×C3×D3—121no—
62C2×C32×C4—41no—
63C2×C3×Dic3—121no—
64C3×SL(2,𝔽3)C4×D311yescharacteristic
65C22×D9C2×A411yescharacteristic
66Q8×C9—31no—
67Q8×C9C2×C3×C411yescharacteristic
68C2×C4×C9—31no—
69C2×C4×C9—31no—
70C2×C4×C9—81no—
71C2×C4×C9C2×A411yescharacteristic
72C2×Dic9—31no—
73C2×Dic9—31no—
74C2×Dic9C2×A411yescharacteristic
75C4×D9—32no—
76C4×D9—32no—
77Dic18—32no—
78Dic18—32no—
79Q8⋊C9—21no—
80#519—91noSylow
81C2×C3×C9—41no—
82C2×C3×C9—41no—
83C2×C3×C9—41no—
84C3×D9—82no—
85C2×C3×Q8C2×C3×D311yescharacteristic
86D3×Q8—32no—
87D3×Q8—92no—
88C2×C4×D3—91no—
89C2×Dic6—91no—
90C2×SL(2,𝔽3)—21no—
91C2×SL(2,𝔽3)—92no—
92C2×SL(2,𝔽3)C2×D911yescharacteristic
93C3×C4⋊C4—31no—
94C3×C22⋊C4—361no—
95C3×C42—31no—
96(C2×D3)⋊C4—31no—
97(C2×D3)⋊C4—91no—
98C4⋊Dic3—91no—
99Dic3⋊C4—91no—
100Dic3⋊C4—91no—
101C4×Dic3—91no—
102C4×Dic3—92no—
103C22×C32—41no—
104C2×C3×D3—122no—
105C2×C3×D3—241no—
106C2×C3×D3—241no—
107C32×C4—81no—
108C3×Dic3—241no—
109C22×C9—81no—
110C22×C9C22×A411yescharacteristic
111C2×D9—21no—
112C2×D9—21no—
113C2×D9C4×A412yes—
114C4×C9—21no—
115C4×C9—31no—
116C4×C9—31no—
117C4×C9—61no—
118C4×C9—82no—
119Dic9—21no—
120Dic9—32no—
121Dic9—61no—
122C22×Q8—91no—
123C4×Q8—91no—
124C4×Q8—271no—
125C4×Q8—271no—
126C4×Q8C3×D911yescharacteristic, complemented
127#117—271no—
128#117—271no—
129C3×C9—41noSylow
130C23×C3—361no—
131C22×D3—31no—
132C3×Q8—31no—
133C3×Q8C3×C4×D311yescharacteristic
134C2×C3×C4—31no—
135C2×C3×C4—31no—
136C2×C3×C4—41no—
137C2×C3×C4—81no—
138C2×C3×C4—361no—
139C2×C3×C4D3×A411yescharacteristic
140C2×Dic3—31no—
141C2×Dic3—91no—
142C2×Dic3—91no—
143C4×D3—92no—
144C4×D3—92no—
145Dic6—92no—
146Dic6—92no—
147SL(2,𝔽3)—21no—
148SL(2,𝔽3)C4×D911yescharacteristic
149C2×C32—41no—
150C2×C32—41no—
151C2×C32—41no—
152C3×D3—242no—
153C2×C9—81no—
154C2×C9—81no—
155C2×C9—81no—
156C2×C9C2×C4×A411yescharacteristic
157C2×C92−1+4⋊C311yescharacteristic
158C2×C92−1+4⋊C311yescharacteristic
159D9—22no—
160C2×Q8—92no—
161C2×Q8—271no—
162C2×Q8—272no—
163C2×Q8C2×C3×D911yescharacteristic
164C22×C4—271no—
165C4⋊C4—31no—
166C4⋊C4—271no—
167C4⋊C4—271no—
168C4⋊C4—271no—
169C22⋊C4—91no—
170C22⋊C4—271no—
171C42—31no—
172C42—271no—
173C42—272no—
174C22×C3—41no—
175C22×C3—81no—
176C22×C3—362no—
177C22×C3—721no—
178C22×C3—721no—
179C22×C3C2×D3×A411yescharacteristic
180C2×D3—32no—
181C2×D3—61no—
182C2×D3—61no—
183C3×C4—21no—
184C3×C4—31no—
185C3×C4—31no—
186C3×C4—61no—
187C3×C4—81no—
188C3×C4—82no—
189C3×C4—721no—
190Dic3—61no—
191Dic3—92no—
192Dic3—181no—
193C32—41no—
194C9—81no—
195C9#519⋊C311yescharacteristic, complemented
196C23—91no—
197Q8—31no—
198Q8—272no—
199Q8—272no—
200Q8C3×C4×D911yescharacteristic
201C2×C4—31no—
202C2×C4—31no—
203C2×C4—91no—
204C2×C4—272no—
205C2×C4—272no—
206C2×C4—271no—
207C2×C4—271no—
208C2×C4A4×D911yescharacteristic
209C2×C3—41no—
210C2×C3—41no—
211C2×C3—41no—
212C2×C3—81no—
213C2×C3—81no—
214C2×C3—81no—
215C2×C3—722no—
216C2×C3C4×D3×A411yescharacteristic
217C2×C3#1007⋊C311yescharacteristic
218C2×C3#1010⋊C311yescharacteristic
219D3—62no—
220C22—92no—
221C22—181no—
222C22—181no—
223C22C2×A4×D911yescharacteristic
224C4—21no—
225C4—31no—
226C4—31no—
227C4—61no—
228C4—181no—
229C4—272no—
230C4—541no—
231C3—41no—
232C3—81no—
233C3#5572⋊C311yescharacteristic
234C2—182no—
235C2C4×A4×D911yescharacteristic
236C2#1172241011yescharacteristic
237C2#1172241611yescharacteristic
subgroupG/H
centerC22C2×A4×D9
commutator, hypocenterQ8×C9C2×C3×C4
Frattini, socleC22×C3C2×D3×A4
FittingC4×Q8×C9C2×C3
prime radicalC2×D9×SL(2,𝔽3)C2
cosocleC2×Q8×C9C22×C3
hypercenterC2×C4A4×D9
LCSG ▷ Q8×C9
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
2. C4×C9×SL(2,𝔽3)C2semidirect
3. SL(2,𝔽3)×Dic9C2semidirect
5. #10589674C3semidirect
7. D9×SL(2,𝔽3)C4semidirect
13. C4×Q8×C9C2×C3semidirect
14. Q8×Dic9C2×C3semidirect
31. Q8×D9C3×C4semidirect
60. C4×SL(2,𝔽3)D9semidirect
126. C4×Q8C3×D9semidirect
195. C9#519⋊C3semidirect

Maximal subgroups

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

subgroupindex
C2×D9×SL(2,𝔽3)2
C4×C9×SL(2,𝔽3)2
SL(2,𝔽3)×Dic92
#5572⋊C33
#105896743
C3×(C2×D9)⋊C44