Finite Group Databasebeta

← C3×C3⋊#5930 #13181421

Subgroups — C3×C3⋊#5930 #13181421

Order 1728.

403 conjugacy classes of subgroups; only the 38 normal subgroups are shown.

normal subgroups only

This group has 403 conjugacy classes of subgroups — too many to draw. Only its 38 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
1C32×C22⋊SL(2,𝔽3)C211yesmaximal, characteristic
2C3⋊#5930C311yesmaximal, characteristic, direct product factor
3C3×#5930—33nomaximal
4C3×#5228—31nomaximal
5C3×C3⋊SL(2,ℤ/4)—41nomaximal
6C3×C3⋊#226—42nomaximal
7C22×C32×Q8D311yescharacteristic
8C3×C22⋊SL(2,𝔽3)—21no—
9C3×C22⋊SL(2,𝔽3)—23no—
10C3×C22⋊SL(2,𝔽3)D313yes—
11C3×C22⋊SL(2,𝔽3)C2×C311yescharacteristic
12C2×C3×#216—32no—
13C3×#924—31no—
14C3×#835—32no—
15C3×#830—31no—
16C2×C32×A4—41no—
17C32×SL(2,𝔽3)—42no—
18C32×SL(2,𝔽3)—81no—
19#5930—33no—
20C3×#451—91no—
21#5228—31no—
22C2×C32×Q8—32no—
23C2×C32×Q8—61no—
24C22×C32×C4—31no—
25C2×C3×Dic6—31no—
26C3⋊SL(2,ℤ/4)—41no—
27C3⋊#226—42no—
28C3×SL(2,ℤ/4)—123no—
29C3×#226—126no—
30C3×#217—61no—
31C3×#216—62no—
32C3×#216—62no—
33C3×C4⋊Dic3—31no—
34C3×Dic3⋊C4—61no—
35C2×C3×C3⋊C8—32no—
36C32×A4—41no—
37C3×C3⋊Dic3—161no—
38C22×C3×Q8—21no—
39C22×C3×Q8C3×D311yescharacteristic
40C22×C3×Q8C3⋊D311yescharacteristic
41C22⋊SL(2,𝔽3)—23no—
42C22⋊SL(2,𝔽3)C3×D313yes—
43C2×C3×Q16—92no—
44C3×C22⋊Q8—91no—
45C2×#216—32no—
46#924—31no—
47C3×#103—92no—
48C3×C22⋊C8—91no—
49#835—32no—
50#830—31no—
51C23×C32S411yescharacteristic
52C2×C3×A4—41no—
53C2×C3×A4—43no—
54C2×C3×A4—83no—
55C2×C3×A4—81no—
56C32×Q8—21no—
57C32×Q8—32no—
58C32×Q8—61no—
59C32×Q8S412yes—
60C2×C32×C4—31no—
61C2×C32×C4—31no—
62C2×C32×C4—62no—
63C2×C3×Dic3—61no—
64C2×C3×Dic3—61no—
65C3×Dic6—62no—
66C3×SL(2,𝔽3)—42no—
67C3×SL(2,𝔽3)—46no—
68C3×SL(2,𝔽3)—83no—
69C3×SL(2,𝔽3)—81no—
70C3×SL(2,𝔽3)—86no—
71C3×SL(2,𝔽3)—86no—
72C3×SL(2,𝔽3)—82no—
73C3×SL(2,𝔽3)—82no—
74C3×C3⋊C8—62no—
75#451—91noSylow
76C2×C33—161no—
77C2×C3×Q8—32no—
78C2×C3×Q8—32no—
79C2×C3×Q8—62no—
80C2×C3×Q8—61no—
81C2×C3×Q8—61no—
82C2×C3×Q8—62no—
83C2×C3×Q8—91no—
84C22×C3×C4—31no—
85C22×C3×C4—31no—
86C22×C3×C4—61no—
87C2×Dic6—31no—
88SL(2,ℤ/4)—123no—
89#226—126no—
90C3×Q16—182no—
91C3×Q16—182no—
92C2×C3×C8—92no—
93C3×C4⋊C4—91no—
94C3×C4⋊C4—181no—
95C3×C22⋊C4—181no—
96#217—61no—
97#216—62no—
98#216—62no—
99C4⋊Dic3—31no—
100Dic3⋊C4—61no—
101C2×C3⋊C8—32no—
102C22×C32—31no—
103C22×C32—31no—
104C22×C32#22611yescharacteristic, complemented
105C3×A4—41no—
106C3×A4—43no—
107C3×A4—83no—
108C3×A4—81no—
109C32×C4—32no—
110C32×C4—61no—
111C3⋊Dic3—161no—
112C3×Dic3—121no—
113C3×Dic3—121no—
114C3×Dic3—483no—
115C22×Q8C3×C3⋊D311yescharacteristic
116C2×Q16—92no—
117C22⋊Q8—91no—
118#103—92no—
119C22⋊C8—91no—
120C33—161noSylow
121C23×C3—21no—
122C23×C3C3×S411yescharacteristic
123C23×C3C3⋊S411yescharacteristic
124C2×A4—43no—
125C2×A4—83no—
126C3×Q8—22no—
127C3×Q8—21no—
128C3×Q8—21no—
129C3×Q8—22no—
130C3×Q8—32no—
131C3×Q8—32no—
132C3×Q8—62no—
133C3×Q8—61no—
134C3×Q8—61no—
135C3×Q8—62no—
136C3×Q8—182no—
137C3×Q8C3×S412yes—
138C3×Q8C3⋊S412yes—
139C2×C3×C4—31no—
140C2×C3×C4—31no—
141C2×C3×C4—31no—
142C2×C3×C4—31no—
143C2×C3×C4—61no—
144C2×C3×C4—61no—
145C2×C3×C4—62no—
146C2×C3×C4—62no—
147C2×C3×C4—64no—
148C2×C3×C4—181no—
149C2×C3×C4—181no—
150C2×Dic3—61no—
151C2×Dic3—61no—
152Dic6—62no—
153SL(2,𝔽3)—46no—
154SL(2,𝔽3)—83no—
155SL(2,𝔽3)—86no—
156SL(2,𝔽3)—86no—
157C3×C8—182no—
158C3⋊C8—62no—
159C2×C32—31no—
160C2×C32—31no—
161C2×C32—161no—
162C2×C32—163no—
163C2×C32—323no—
164C2×C32—321no—
165C2×C32#102011yescharacteristic
166C2×Q8—32no—
167C2×Q8—61no—
168C2×Q8—91no—
169C22×C4—31no—
170Q16—182no—
171Q16—182no—
172C2×C8—92no—
173C4⋊C4—91no—
174C4⋊C4—181no—
175C22⋊C4—181no—
176C22×C3—21no—
177C22×C3—31no—
178C22×C3—31no—
179C22×C3—31no—
180C22×C3—31no—
181C22×C3—61no—
182C22×C3—61no—
183C22×C3C3×#22611yescharacteristic, complemented
184C22×C3C3⋊#22611yescharacteristic, complemented
185A4—43no—
186A4—83no—
187C3×C4—32no—
188C3×C4—32no—
189C3×C4—62no—
190C3×C4—62no—
191C3×C4—61no—
192C3×C4—61no—
193C3×C4—361no—
194C3×C4—361no—
195Dic3—121no—
196Dic3—121no—
197Dic3—483no—
198C32—161no—
199C32—163no—
200C32—323no—
201C32—321no—
202C32#593011yescharacteristic, complemented
203C23C3×C3⋊S411yescharacteristic
204Q8—21no—
205Q8—32no—
206Q8—61no—
207Q8—182no—
208Q8C3×C3⋊S412yes—
209C2×C4—31no—
210C2×C4—31no—
211C2×C4—62no—
212C2×C4—181no—
213C2×C4—181no—
214C8—182no—
215C2×C3—21no—
216C2×C3—31no—
217C2×C3—31no—
218C2×C3—31no—
219C2×C3—31no—
220C2×C3—61no—
221C2×C3—61no—
222C2×C3—163no—
223C2×C3—323no—
224C2×C3C3×#102011yescharacteristic
225C2×C3C3⋊#102011yescharacteristic
226C22—31no—
227C22—31no—
228C22C3×C3⋊#22611yescharacteristic, complemented
229C4—32no—
230C4—61no—
231C4—361no—
232C4—361no—
233C3—21no—
234C3—163no—
235C3—323no—
236C3C3×#593011yescharacteristic, complemented
237C3C3⋊#593011yescharacteristic, direct product factor
238C2—31no—
239C2—31no—
240C2C3×C3⋊#102011yescharacteristic
subgroupG/H
center, hypercenterC2×C3C3⋊#1020
commutator, cosocle, hypocenterC3×C22⋊SL(2,𝔽3)C2×C3
FrattiniC2C3×C3⋊#1020
FittingC22×C32×Q8D3
socleC23×C32S4
prime radicalC32×C22⋊SL(2,𝔽3)C2
LCSG ▷ C3×C22⋊SL(2,𝔽3)
UCSC1 ◁ C2×C3

Complements

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

NH ≅ G/Ntype
2. C3⋊#5930C3direct (N × H)
104. C22×C32#226semidirect
183. C22×C3C3×#226semidirect
184. C22×C3C3⋊#226semidirect
202. C32#5930semidirect
228. C22C3×C3⋊#226semidirect
236. C3C3×#5930semidirect
237. C3C3⋊#5930direct (N × H)

Maximal subgroups

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

subgroupindex
C32×C22⋊SL(2,𝔽3)2
C3⋊#59303
C3×#59303
C3×#52283
C3×C3⋊SL(2,ℤ/4)4
C3×C3⋊#2264