Finite Group Databasebeta

← C9×#5419 #13146783

Subgroups — C9×#5419 #13146783

Order 1728.

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

normal subgroups only

This group has 282 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
1C22×C9×SL(2,𝔽3)C211yesmaximal, characteristic
2C2×C9×#226C211yesmaximal, characteristic, complemented
3C9×#859C211yesmaximal, characteristic, complemented
4C3×#5419C311yesmaximal, characteristic
5C9×#451—31nomaximal
6C2×C9×SL(2,𝔽3)—21no—
7C2×C9×SL(2,𝔽3)C2211yescharacteristic
8C9×#226—21no—
9C9×#217—41nomaximal
10C22×C3×SL(2,𝔽3)C2×C311yescharacteristic
11C2×C3×#226C2×C311yescharacteristic
12C3×#859C2×C311yescharacteristic
13C22×Q8×C9D311yescharacteristic
14C22×Q8⋊C9—21no—
15C2×C9×Q16—31no—
16C2×C9×Q16—31no—
17C9×C22⋊Q8—31no—
18C9×#103—31no—
19C9×#103—31no—
20C9×C22⋊C8—31no—
21C23×C3×C9—41no—
22C2×C9×Dic3—41no—
23C2×C9×Dic3—41no—
24C9×SL(2,𝔽3)D411yescharacteristic
25#5419C911yescharacteristic, direct product factor
26C3×#451—31no—
27C2×C3×SL(2,𝔽3)—21no—
28C2×C3×SL(2,𝔽3)C22×C311yescharacteristic
29C3×#226—21no—
30C3×#217—41no—
31C2×Q8×C9—21no—
32C2×Q8×C9—31no—
33C2×Q8×C9—31no—
34C2×Q8×C9—61no—
35C2×Q8×C9C2×D311yescharacteristic
36C22×C4×C9—31no—
37C2×Q8⋊C9—22no—
38C2×Q8⋊C9—21no—
39C9×Q16—61no—
40C9×Q16—62no—
41C9×Q16—61no—
42C2×C8×C9—31no—
43C2×C8×C9—31no—
44C9×C4⋊C4—31no—
45C9×C4⋊C4—61no—
46C9×C22⋊C4—61no—
47C22×C3×C9—41no—
48C22×C3×C9—42no—
49C22×C3×C9—81no—
50C22×C3×C9—81no—
51C9×Dic3—81no—
52C9×Dic3—81no—
53C22×C3×Q8C3×D311yescharacteristic
54C22×SL(2,𝔽3)—21no—
55C22×SL(2,𝔽3)C2×C911yescharacteristic
56C2×#226C2×C911yescharacteristic, complemented
57C2×C3×Q16—31no—
58C2×C3×Q16—31no—
59C3×C22⋊Q8—31no—
60#859C2×C911yescharacteristic, complemented
61C3×#103—31no—
62C3×#103—31no—
63C3×C22⋊C8—31no—
64C23×C32—41no—
65C2×C3×Dic3—41no—
66C2×C3×Dic3—41no—
67C3×SL(2,𝔽3)C3×D411yescharacteristic
68C23×C9—81no—
69C23×C9S411yescharacteristic
70Q8×C9—31no—
71Q8×C9—32no—
72Q8×C9—61no—
73Q8×C9—61no—
74Q8×C9—61no—
75Q8×C9#6711yescharacteristic
76C2×C4×C9—31no—
77C2×C4×C9—31no—
78C2×C4×C9—61no—
79C2×C4×C9—61no—
80C2×C4×C9—61no—
81C2×C4×C9—61no—
82Q8⋊C9—21no—
83C8×C9—61no—
84C8×C9—61no—
85#451—31noSylow
86C2×C3×C9—41no—
87C2×C3×C9—41no—
88C2×C3×C9—41no—
89C2×C3×C9—82no—
90C2×C3×Q8—21no—
91C2×C3×Q8—31no—
92C2×C3×Q8—31no—
93C2×C3×Q8—61no—
94C2×C3×Q8C2×C3×D311yescharacteristic
95C22×C3×C4—31no—
96C2×SL(2,𝔽3)—22no—
97C2×SL(2,𝔽3)—21no—
98C2×SL(2,𝔽3)—21no—
99C2×SL(2,𝔽3)C22×C911yescharacteristic
100#226—21no—
101C3×Q16—61no—
102C3×Q16—62no—
103C3×Q16—61no—
104C2×C3×C8—31no—
105C2×C3×C8—31no—
106C3×C4⋊C4—31no—
107C3×C4⋊C4—61no—
108C3×C22⋊C4—61no—
109#217—41no—
110C22×C32—41no—
111C22×C32—42no—
112C22×C32—81no—
113C22×C32—81no—
114C3×Dic3—81no—
115C3×Dic3—81no—
116C22×C9—21no—
117C22×C9—21no—
118C22×C9—82no—
119C22×C9—82no—
120C22×C9—81no—
121C22×C9—82no—
122C22×C9#22612yescomplemented
123C22×C9C2×S411yescharacteristic
124C4×C9—31no—
125C4×C9—31no—
126C4×C9—61no—
127C4×C9—121no—
128C4×C9—121no—
129C22×Q8D3×C911yescharacteristic
130C2×Q16—31no—
131C2×Q16—31no—
132C22⋊Q8—31no—
133#103—31no—
134#103—31no—
135C22⋊C8—31no—
136C3×C9—41noSylow
137C23×C3—41no—
138C23×C3—81no—
139C23×C3C3×S411yescharacteristic
140C3×Q8—31no—
141C3×Q8—32no—
142C3×Q8—61no—
143C3×Q8—61no—
144C3×Q8—61no—
145C3×Q8C3×#6711yescharacteristic
146C2×C3×C4—31no—
147C2×C3×C4—31no—
148C2×C3×C4—61no—
149C2×C3×C4—61no—
150C2×C3×C4—61no—
151C2×C3×C4—61no—
152C2×Dic3—41no—
153C2×Dic3—41no—
154SL(2,𝔽3)—21no—
155SL(2,𝔽3)D4×C911yescharacteristic
156C3×C8—61no—
157C3×C8—61no—
158C2×C32—41no—
159C2×C32—41no—
160C2×C32—41no—
161C2×C32—82no—
162C2×C9—22no—
163C2×C9—84no—
164C2×C9—81no—
165C2×C9—81no—
166C2×C9—81no—
167C2×C9C2×#22611yescharacteristic
168C2×C9#98311yescharacteristic
169C2×C9GL(2,ℤ/4)11yescharacteristic
170C2×Q8—21no—
171C2×Q8—31no—
172C2×Q8—31no—
173C2×Q8—61no—
174C2×Q8C2×D3×C911yescharacteristic
175C22×C4—31no—
176Q16—61no—
177Q16—62no—
178Q16—61no—
179C2×C8—31no—
180C2×C8—31no—
181C4⋊C4—31no—
182C4⋊C4—61no—
183C22⋊C4—61no—
184C22×C3—21no—
185C22×C3—21no—
186C22×C3—42no—
187C22×C3—41no—
188C22×C3—81no—
189C22×C3—81no—
190C22×C3—82no—
191C22×C3—82no—
192C22×C3—82no—
193C22×C3—81no—
194C22×C3C3×#22612yes—
195C22×C3C2×C3×S411yescharacteristic
196C3×C4—31no—
197C3×C4—31no—
198C3×C4—61no—
199C3×C4—121no—
200C3×C4—121no—
201Dic3—81no—
202Dic3—81no—
203C32—41no—
204C9—81no—
205C9#541911yescharacteristic, direct product factor
206C23C9×S411yescharacteristic
207Q8—31no—
208Q8—32no—
209Q8—61no—
210Q8—61no—
211Q8—61no—
212Q8C9×#6711yescharacteristic
213C2×C4—31no—
214C2×C4—31no—
215C2×C4—61no—
216C2×C4—61no—
217C2×C4—61no—
218C2×C4—61no—
219C8—61no—
220C8—61no—
221C2×C3—22no—
222C2×C3—41no—
223C2×C3—41no—
224C2×C3—41no—
225C2×C3—82no—
226C2×C3—84no—
227C2×C3—81no—
228C2×C3—81no—
229C2×C3—81no—
230C2×C3C2×C3×#22611yescharacteristic
231C2×C3C3×#98311yescharacteristic
232C2×C3C3×GL(2,ℤ/4)11yescharacteristic
233C22—21no—
234C22—21no—
235C22C9×#22612yescomplemented
236C22C2×C9×S411yescharacteristic
237C4—31no—
238C4—31no—
239C4—61no—
240C4—121no—
241C4—121no—
242C3—41no—
243C3—81no—
244C3C3×#541911yescharacteristic
245C2—22no—
246C2C2×C9×#22611yescharacteristic
247C2C9×#98311yescharacteristic
248C2C9×GL(2,ℤ/4)11yescharacteristic
subgroupG/H
centerC22×C9C2×S4
commutatorC2×SL(2,𝔽3)C22×C9
Frattini, socleC22×C3C2×C3×S4
FittingC22×Q8×C9D3
prime radicalC22×C3×SL(2,𝔽3)C2×C3
cosocleC2×C3×SL(2,𝔽3)C22×C3
hypercenterC23×C9S4
hypocenterSL(2,𝔽3)D4×C9
LCSG ▷ C2×SL(2,𝔽3) ▷ SL(2,𝔽3)
UCSC1 ◁ C22×C9 ◁ C23×C9

Complements

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

NH ≅ G/Ntype
2. C2×C9×#226C2semidirect
3. C9×#859C2semidirect
25. #5419C9direct (N × H)
56. C2×#226C2×C9semidirect
60. #859C2×C9semidirect
122. C22×C9#226semidirect
205. C9#5419direct (N × H)
235. C22C9×#226semidirect

Maximal subgroups

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

subgroupindex
C22×C9×SL(2,𝔽3)2
C2×C9×#2262
C9×#8592
C3×#54193
C9×#4513
C9×#2174