Finite Group Databasebeta

← C3×C4⋊(C3⋊#226) #13172992

Subgroups — C3×C4⋊(C3⋊#226) #13172992

Order 1728.

448 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 448 conjugacy classes of subgroups — too many to draw. Only its 74 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×C3×C3⋊#226C212yesmaximal
2C32×C4×SL(2,𝔽3)C211yesmaximal, characteristic
3C4⋊(C3⋊#226)C311yesmaximal, characteristic, direct product factor
4C3×C4⋊#226—33nomaximal
5C3×#5031—31nomaximal
6C2×C32×SL(2,𝔽3)C2211yescharacteristic
7C3×C3⋊#226—22no—
8C3×C4⋊(C3⋊Dic3)—41nomaximal
9C2×C3⋊#226C2×C312yes—
10C2×C3×#226—36no—
11C32×C4×Q8D311yescharacteristic
12C2×C3×#216—32no—
13C3×#869—31no—
14C3×C4×SL(2,𝔽3)—21no—
15C3×C4×SL(2,𝔽3)—23no—
16C3×C4×SL(2,𝔽3)D313yes—
17C3×C4×SL(2,𝔽3)C2×C311yescharacteristic
18C3×#810—32no—
19C3×C4⋊(C3⋊C8)—31no—
20C2×C33×C4—41no—
21C2×C3×C3⋊Dic3—42no—
22C32×SL(2,𝔽3)D411yescharacteristic
23C4⋊#226—33no—
24C3×#462—91no—
25#5031—31no—
26C2×C32×Q8C2×D311yescharacteristic
27C2×C3×Dic6—32no—
28C2×C3×SL(2,𝔽3)—21no—
29C2×C3×SL(2,𝔽3)—23no—
30C2×C3×SL(2,𝔽3)C2×D313yes—
31C2×C3×SL(2,𝔽3)C22×C311yescharacteristic
32C3⋊#226—22no—
33C3×#226—66no—
34C32×C4⋊C4—31no—
35C32×C42—31no—
36C4⋊(C3⋊Dic3)—41no—
37C3×#216—62no—
38C3×#216—62no—
39C3×C4⋊Dic3—61no—
40C3×C4⋊Dic3—61no—
41C3×C4⋊Dic3—123no—
42C2×C3×C3⋊C8—32no—
43C22×C33—41no—
44C33×C4—42no—
45C3×C3⋊Dic3—82no—
46C2×#226—36no—
47C2×C3×Q16—92no—
48C3×C4⋊Q8—91no—
49C3×C4×Q8—21no—
50C3×C4×Q8C3×D311yescharacteristic
51C3×C4×Q8C3⋊D311yescharacteristic
52C2×#216—32no—
53#869—31no—
54C4×SL(2,𝔽3)—23no—
55C4×SL(2,𝔽3)C3×D313yes—
56C3×C4⋊C8—91no—
57C3×#103—92no—
58#810—32no—
59C4⋊(C3⋊C8)—31no—
60C32×Q8—31no—
61C32×Q8D1211yescharacteristic
62C2×C32×C4—31no—
63C2×C32×C4—31no—
64C2×C32×C4—41no—
65C2×C32×C4—43no—
66C2×C32×C4—83no—
67C2×C32×C4—81no—
68C2×C32×C4S411yescharacteristic
69C2×C3⋊Dic3—42no—
70C2×C3×Dic3—62no—
71C2×C3×Dic3—126no—
72C3×Dic6—62no—
73C3×Dic6—62no—
74C3×SL(2,𝔽3)—21no—
75C3×SL(2,𝔽3)—23no—
76C3×SL(2,𝔽3)D1213yes—
77C3×SL(2,𝔽3)C3×D411yescharacteristic
78C3×C3⋊C8—62no—
79#462—91noSylow
80C2×C33—41no—
81C2×C33—41no—
82C2×C33—41no—
83C2×C3×Q8—21no—
84C2×C3×Q8—92no—
85C2×C3×Q8C2×C3×D311yescharacteristic
86C2×C3×Q8C2×C3⋊D311yescharacteristic
87C2×Dic6—32no—
88C2×SL(2,𝔽3)—23no—
89C2×SL(2,𝔽3)C2×C3×D313yes—
90#226—66no—
91C3×Q16—182no—
92C3×Q16—182no—
93C2×C3×C8—92no—
94C3×C4⋊C4—31no—
95C3×C4⋊C4—31no—
96C3×C4⋊C4—61no—
97C3×C4⋊C4—181no—
98C3×C4⋊C4—181no—
99C3×C42—31no—
100C3×C42—31no—
101C3×C42—61no—
102#216—62no—
103#216—62no—
104C4⋊Dic3—61no—
105C4⋊Dic3—61no—
106C4⋊Dic3—123no—
107C2×C3⋊C8—32no—
108C22×C32—41no—
109C22×C32—43no—
110C22×C32—83no—
111C22×C32—81no—
112C22×C32C2×S411yescharacteristic
113C32×C4—31no—
114C32×C4—31no—
115C32×C4—42no—
116C32×C4—46no—
117C32×C4—61no—
118C32×C4—86no—
119C32×C4—82no—
120C32×C4#22612yescomplemented
121C3⋊Dic3—82no—
122C3×Dic3—122no—
123C3×Dic3—246no—
124C2×Q16—92no—
125C4⋊Q8—91no—
126C4×Q8C3×C3⋊D311yescharacteristic
127C4⋊C8—91no—
128#103—92no—
129C33—41noSylow
130C3×Q8—21no—
131C3×Q8—31no—
132C3×Q8—31no—
133C3×Q8—61no—
134C3×Q8—182no—
135C3×Q8—182no—
136C3×Q8C3×D1211yescharacteristic
137C3×Q8C4⋊(C3⋊D3)11yescharacteristic
138C2×C3×C4—21no—
139C2×C3×C4—31no—
140C2×C3×C4—31no—
141C2×C3×C4—31no—
142C2×C3×C4—31no—
143C2×C3×C4—43no—
144C2×C3×C4—61no—
145C2×C3×C4—61no—
146C2×C3×C4—83no—
147C2×C3×C4—182no—
148C2×C3×C4C3×S411yescharacteristic
149C2×C3×C4C3⋊S411yescharacteristic
150C2×Dic3—62no—
151C2×Dic3—126no—
152Dic6—62no—
153Dic6—62no—
154SL(2,𝔽3)—23no—
155SL(2,𝔽3)C3×D1213yes—
156C3×C8—182no—
157C3⋊C8—62no—
158C2×C32—41no—
159C2×C32—41no—
160C2×C32—41no—
161C2×C32—43no—
162C2×C32—43no—
163C2×C32—43no—
164C2×C32—83no—
165C2×C32—83no—
166C2×C32—83no—
167C2×C32—81no—
168C2×C32—81no—
169C2×C32—81no—
170C2×C32C4⋊S411yescharacteristic
171C2×C32C2×#22611yescharacteristic
172C2×C32#98411yescharacteristic
173C2×Q8—92no—
174C2×Q8C2×C3×C3⋊D311yescharacteristic
175Q16—182no—
176Q16—182no—
177C2×C8—92no—
178C4⋊C4—31no—
179C4⋊C4—181no—
180C4⋊C4—181no—
181C42—31no—
182C22×C3—21no—
183C22×C3—43no—
184C22×C3—83no—
185C22×C3C2×C3×S411yescharacteristic
186C22×C3C2×C3⋊S411yescharacteristic
187C3×C4—22no—
188C3×C4—31no—
189C3×C4—31no—
190C3×C4—31no—
191C3×C4—31no—
192C3×C4—46no—
193C3×C4—61no—
194C3×C4—61no—
195C3×C4—61no—
196C3×C4—61no—
197C3×C4—86no—
198C3×C4—121no—
199C3×C4—362no—
200C3×C4C3×#22612yescomplemented
201C3×C4C3⋊#22612yescomplemented
202Dic3—122no—
203Dic3—246no—
204C32—41no—
205C32—43no—
206C32—83no—
207C32—81no—
208C32C4⋊#22611yescharacteristic, complemented
209Q8—31no—
210Q8—182no—
211Q8—182no—
212Q8C3×C4⋊(C3⋊D3)11yescharacteristic
213C2×C4—31no—
214C2×C4—31no—
215C2×C4—182no—
216C2×C4C3×C3⋊S411yescharacteristic
217C8—182no—
218C2×C3—21no—
219C2×C3—21no—
220C2×C3—21no—
221C2×C3—43no—
222C2×C3—43no—
223C2×C3—43no—
224C2×C3—83no—
225C2×C3—83no—
226C2×C3—83no—
227C2×C3C3×C4⋊S411yescharacteristic
228C2×C3C2×C3×#22611yescharacteristic
229C2×C3C3×#98411yescharacteristic
230C2×C3C4⋊(C3⋊S4)11yescharacteristic
231C2×C3C2×C3⋊#22611yescharacteristic
232C2×C3#6426511yescharacteristic
233C22C2×C3×C3⋊S411yescharacteristic
234C4—31no—
235C4—31no—
236C4—61no—
237C4—362no—
238C4C3×C3⋊#22612yescomplemented
239C3—21no—
240C3—43no—
241C3—83no—
242C3C3×C4⋊#22611yescharacteristic, complemented
243C3C4⋊(C3⋊#226)11yescharacteristic, direct product factor
244C2C3×C4⋊(C3⋊S4)11yescharacteristic
245C2C2×C3×C3⋊#22611yescharacteristic
246C2C3×#6426511yescharacteristic
subgroupG/H
centerC22×C3C2×C3⋊S4
commutator, cosocleC2×C3×SL(2,𝔽3)C22×C3
FrattiniC22C2×C3×C3⋊S4
FittingC32×C4×Q8D3
socleC22×C32C2×S4
prime radicalC2×C32×SL(2,𝔽3)C22
hypercenterC2×C3×C4C3⋊S4
hypocenterC3×SL(2,𝔽3)C3×D4
LCSG ▷ C2×C3×SL(2,𝔽3) ▷ C3×SL(2,𝔽3)
UCSC1 ◁ C22×C3 ◁ C2×C3×C4

Complements

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

NH ≅ G/Ntype
3. C4⋊(C3⋊#226)C3direct (N × H)
120. C32×C4#226semidirect
200. C3×C4C3×#226semidirect
201. C3×C4C3⋊#226semidirect
208. C32C4⋊#226semidirect
238. C4C3×C3⋊#226semidirect
242. C3C3×C4⋊#226semidirect
243. C3C4⋊(C3⋊#226)direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×C3⋊#2262
C32×C4×SL(2,𝔽3)2
C4⋊(C3⋊#226)3
C3×C4⋊#2263
C3×#50313
C3×C4⋊(C3⋊Dic3)4