Finite Group Databasebeta

← C22×C3×C3⋊#226 #13181378

Subgroups — C22×C3×C3⋊#226 #13181378

Order 1728.

971 conjugacy classes of subgroups; only the 146 normal subgroups are shown.

normal subgroups only

This group has 971 conjugacy classes of subgroups — too many to draw. Only its 146 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×C32×SL(2,𝔽3)C211yesmaximal, characteristic
2C2×C3×C3⋊#226C216yesmaximal, direct product factor
3C22×C3⋊#226C311yesmaximal, characteristic, direct product factor
4C22×C3×#226—33nomaximal
5C22×C3×#216—31nomaximal
6C22×C3×C3⋊Dic3—41nomaximal
7C2×C32×SL(2,𝔽3)C2213yes—
8C3×C3⋊#226C2214yesdirect product factor
9C22×C32×Q8D311yescharacteristic
10C22×C3×Dic6—31no—
11C22×C3×SL(2,𝔽3)—21no—
12C22×C3×SL(2,𝔽3)—23no—
13C22×C3×SL(2,𝔽3)D313yes—
14C22×C3×SL(2,𝔽3)C2×C311yescharacteristic
15C2×C3⋊#226C2×C316yesdirect product factor
16C2×C3×#226—318no—
17C2×C3×#216—36no—
18C2×C3×#216—36no—
19C22×C3×C3⋊C8—31no—
20C23×C33—41no—
21C2×C3×C3⋊Dic3—46no—
22C32×SL(2,𝔽3)C2311yescharacteristic
23C22×#226—33no—
24C22×C3×Q16—91no—
25C22×#216—31no—
26C2×C32×Q8—33no—
27C2×C32×Q8C2×D313yes—
28C22×C32×C4—31no—
29C22×C3⋊Dic3—41no—
30C22×C3×Dic3—61no—
31C22×C3×Dic3—123no—
32C2×C3×Dic6—36no—
33C2×C3×Dic6—63no—
34C2×C3×SL(2,𝔽3)—23no—
35C2×C3×SL(2,𝔽3)—29no—
36C2×C3×SL(2,𝔽3)C2×D319yes—
37C2×C3×SL(2,𝔽3)C22×C313yes—
38C3⋊#226C22×C314yesdirect product factor
39C3×#226—312no—
40C3×#216—312no—
41C3×#216—34no—
42C2×C3×C3⋊C8—36no—
43C22×C33—44no—
44C22×C33—43no—
45C3×C3⋊Dic3—44no—
46C22×C3×Q8—21no—
47C22×C3×Q8—91no—
48C22×C3×Q8C3×D311yescharacteristic
49C22×C3×Q8C3⋊D311yescharacteristic
50C22×Dic6—31no—
51C22×SL(2,𝔽3)—23no—
52C22×SL(2,𝔽3)C3×D313yes—
53C2×#226—318no—
54C2×C3×Q16—96no—
55C2×C3×Q16—96no—
56C22×C3×C8—91no—
57C2×#216—36no—
58C2×#216—36no—
59C22×C3⋊C8—31no—
60C23×C32—41no—
61C23×C32—43no—
62C23×C32—83no—
63C23×C32—81no—
64C23×C32S411yescharacteristic
65C32×Q8—33no—
66C32×Q8—61no—
67C32×Q8C22×D311yescharacteristic
68C2×C32×C4—33no—
69C2×C32×C4—33no—
70C2×C3⋊Dic3—46no—
71C2×C3×Dic3—66no—
72C2×C3×Dic3—1218no—
73C3×Dic6—34no—
74C3×Dic6—66no—
75C3×SL(2,𝔽3)—21no—
76C3×SL(2,𝔽3)—23no—
77C3×SL(2,𝔽3)C22×D313yes—
78C3×SL(2,𝔽3)C23×C311yescharacteristic
79C3×C3⋊C8—34no—
80C22×Q16—91noSylow
81C2×C33—46no—
82C2×C33—41no—
83C2×C3×Q8—23no—
84C2×C3×Q8—33no—
85C2×C3×Q8—33no—
86C2×C3×Q8—63no—
87C2×C3×Q8—96no—
88C2×C3×Q8—183no—
89C2×C3×Q8C2×C3×D313yes—
90C2×C3×Q8C2×C3⋊D313yes—
91C22×C3×C4—31no—
92C22×C3×C4—31no—
93C22×C3×C4—61no—
94C22×C3×C4—181no—
95C22×Dic3—61no—
96C22×Dic3—123no—
97C2×Dic6—36no—
98C2×Dic6—63no—
99C2×SL(2,𝔽3)—29no—
100C2×SL(2,𝔽3)C2×C3×D319yes—
101#226—312no—
102C3×Q16—912no—
103C3×Q16—94no—
104C2×C3×C8—96no—
105#216—312no—
106#216—34no—
107C2×C3⋊C8—36no—
108C22×C32—43no—
109C22×C32—44no—
110C22×C32—49no—
111C22×C32—412no—
112C22×C32—812no—
113C22×C32—89no—
114C22×C32—83no—
115C22×C32—84no—
116C22×C32#22614yescomplemented
117C22×C32C2×S413yes—
118C32×C4—31no—
119C32×C4—33no—
120C3⋊Dic3—44no—
121C3×Dic3—64no—
122C3×Dic3—1212no—
123C22×Q8—91no—
124C22×Q8C3×C3⋊D311yescharacteristic
125C2×Q16—96no—
126C2×Q16—96no—
127C22×C8—91no—
128C33—41noSylow
129C23×C3—21no—
130C23×C3—43no—
131C23×C3—83no—
132C23×C3C3×S411yescharacteristic
133C23×C3C3⋊S411yescharacteristic
134C3×Q8—21no—
135C3×Q8—33no—
136C3×Q8—33no—
137C3×Q8—63no—
138C3×Q8—61no—
139C3×Q8—61no—
140C3×Q8—62no—
141C3×Q8—94no—
142C3×Q8—186no—
143C3×Q8C22×C3×D311yescharacteristic
144C3×Q8C22×C3⋊D311yescharacteristic
145C2×C3×C4—33no—
146C2×C3×C4—33no—
147C2×C3×C4—33no—
148C2×C3×C4—33no—
149C2×C3×C4—63no—
150C2×C3×C4—63no—
151C2×C3×C4—186no—
152C2×Dic3—66no—
153C2×Dic3—1218no—
154Dic6—34no—
155Dic6—66no—
156SL(2,𝔽3)—23no—
157SL(2,𝔽3)C22×C3×D313yes—
158C3×C8—94no—
159C3⋊C8—34no—
160C2×C32—41no—
161C2×C32—46no—
162C2×C32—418no—
163C2×C32—43no—
164C2×C32—83no—
165C2×C32—818no—
166C2×C32—81no—
167C2×C32—86no—
168C2×C32C2×#22616yescomplemented
169C2×C32C22×S411yescharacteristic
170C2×Q8—33no—
171C2×Q8—96no—
172C2×Q8—183no—
173C2×Q8C2×C3×C3⋊D313yes—
174C22×C4—31no—
175C22×C4—181no—
176Q16—912no—
177Q16—94no—
178C2×C8—96no—
179C22×C3—23no—
180C22×C3—24no—
181C22×C3—49no—
182C22×C3—412no—
183C22×C3—89no—
184C22×C3—812no—
185C22×C3C3×#22614yescomplemented
186C22×C3C3⋊#22614yesdirect product factor
187C22×C3C2×C3×S413yes—
188C22×C3C2×C3⋊S413yes—
189C3×C4—31no—
190C3×C4—33no—
191C3×C4—31no—
192C3×C4—33no—
193C3×C4—61no—
194C3×C4—63no—
195C3×C4—184no—
196Dic3—64no—
197Dic3—1212no—
198C32—41no—
199C32—43no—
200C32—83no—
201C32—81no—
202C32C22×#22611yescharacteristic, complemented
203C23C3×C3⋊S411yescharacteristic
204Q8—33no—
205Q8—61no—
206Q8—94no—
207Q8—186no—
208Q8C22×C3×C3⋊D311yescharacteristic
209C2×C4—33no—
210C2×C4—33no—
211C2×C4—186no—
212C8—94no—
213C2×C3—26no—
214C2×C3—21no—
215C2×C3—418no—
216C2×C3—43no—
217C2×C3—818no—
218C2×C3—83no—
219C2×C3C2×C3×#22616yescomplemented
220C2×C3C2×C3⋊#22616yesdirect product factor
221C2×C3C22×C3×S411yescharacteristic
222C2×C3C22×C3⋊S411yescharacteristic
223C22C3×C3⋊#22614yesdirect product factor
224C22C2×C3×C3⋊S413yes—
225C4—31no—
226C4—33no—
227C4—184no—
228C3—21no—
229C3—43no—
230C3—83no—
231C3C22×C3×#22611yescharacteristic, complemented
232C3C22×C3⋊#22611yescharacteristic, direct product factor
233C2C2×C3×C3⋊#22616yesdirect product factor
234C2C22×C3×C3⋊S411yescharacteristic
subgroupG/H
center, hypercenterC23×C3C3⋊S4
commutator, cosocle, hypocenterC3×SL(2,𝔽3)C23×C3
FrattiniC2C22×C3×C3⋊S4
FittingC22×C32×Q8D3
socleC23×C32S4
prime radicalC22×C32×SL(2,𝔽3)C2
LCSG ▷ C3×SL(2,𝔽3)
UCSC1 ◁ C23×C3

Complements

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

NH ≅ G/Ntype
2. C2×C3×C3⋊#226C2direct (N × H)
3. C22×C3⋊#226C3direct (N × H)
8. C3×C3⋊#226C22direct (N × H)
15. C2×C3⋊#226C2×C3direct (N × H)
38. C3⋊#226C22×C3direct (N × H)
116. C22×C32#226semidirect
168. C2×C32C2×#226semidirect
185. C22×C3C3×#226semidirect
186. C22×C3C3⋊#226direct (N × H)
202. C32C22×#226semidirect
219. C2×C3C2×C3×#226semidirect
220. C2×C3C2×C3⋊#226direct (N × H)
223. C22C3×C3⋊#226direct (N × H)
231. C3C22×C3×#226semidirect
232. C3C22×C3⋊#226direct (N × H)
233. C2C2×C3×C3⋊#226direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C32×SL(2,𝔽3)2
C2×C3×C3⋊#2262
C22×C3⋊#2263
C22×C3×#2263
C22×C3×#2163
C22×C3×C3⋊Dic34