Finite Group Databasebeta

← C3×#10593620 #13172988

Subgroups — C3×#10593620 #13172988

Order 1728.

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

normal subgroups only

This group has 456 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
1C32×C4×SL(2,𝔽3)C211yesmaximal, characteristic
2C3×C3⋊#859C212yesmaximal
3#10593620C311yesmaximal, characteristic, direct product factor
4C3×#5386—33nomaximal
5C3×#5020—31nomaximal
6C2×C3×#634—41nomaximal
7C2×C32×SL(2,𝔽3)C2211yescharacteristic
8C32×C4×Q8D311yescharacteristic
9C3×#869—31no—
10C3×C4×SL(2,𝔽3)—21no—
11C3×C4×SL(2,𝔽3)—23no—
12C3×C4×SL(2,𝔽3)D313yes—
13C3×C4×SL(2,𝔽3)C2×C311yescharacteristic
14C3⋊#859C2×C312yes—
15C3×#859—36no—
16C3×#835—32no—
17C3×#808—32no—
18C3×C4⋊(C3⋊C8)—31no—
19C2×C33×C4—41no—
20C2×C3×C3⋊Dic3—42no—
21C3×#634—44no—
22C32×SL(2,𝔽3)Q811yescharacteristic, complemented
23#5386—33no—
24C3×#475—91no—
25#5020—31no—
26C2×C32×Q8C2×D311yescharacteristic
27C2×#634—41no—
28C2×C3×Dic6—61no—
29C2×C3×Dic6—123no—
30C2×C3×SL(2,𝔽3)—21no—
31C2×C3×SL(2,𝔽3)—23no—
32C2×C3×SL(2,𝔽3)C2×D313yes—
33C2×C3×SL(2,𝔽3)C22×C311yescharacteristic
34C32×C4⋊C4—31no—
35C32×C42—31no—
36C3×C4⋊Dic3—32no—
37C3×C4⋊Dic3—61no—
38C2×C3×C3⋊C8—32no—
39C22×C33—41no—
40C33×C4—42no—
41C3×C3⋊Dic3—44no—
42C3×C4⋊Q8—91no—
43C3×C4×Q8—21no—
44C3×C4×Q8C3×D311yescharacteristic
45C3×C4×Q8C3⋊D311yescharacteristic
46#869—31no—
47C4×SL(2,𝔽3)—23no—
48C4×SL(2,𝔽3)C3×D313yes—
49#859—36no—
50C3×#106—92no—
51C3×C4⋊C8—91no—
52C3×#103—92no—
53#835—32no—
54#808—32no—
55C4⋊(C3⋊C8)—31no—
56C32×Q8—31no—
57C32×Q8Dic611yescharacteristic, complemented
58C2×C32×C4—31no—
59C2×C32×C4—31no—
60C2×C32×C4—41no—
61C2×C32×C4—43no—
62C2×C32×C4—83no—
63C2×C32×C4—81no—
64C2×C32×C4S411yescharacteristic
65C2×C3⋊Dic3—42no—
66#634—44no—
67C2×C3×Dic3—62no—
68C2×C3×Dic3—126no—
69C3×Dic6—1212no—
70C3×Dic6—122no—
71C3×SL(2,𝔽3)—21no—
72C3×SL(2,𝔽3)—23no—
73C3×SL(2,𝔽3)Dic613yescomplemented
74C3×SL(2,𝔽3)C3×Q811yescharacteristic, complemented
75C3×C3⋊C8—62no—
76#475—91noSylow
77C2×C33—41no—
78C2×C33—41no—
79C2×C33—41no—
80C2×C3×Q8—21no—
81C2×C3×Q8—181no—
82C2×C3×Q8C2×C3×D311yescharacteristic
83C2×C3×Q8C2×C3⋊D311yescharacteristic
84C2×Dic6—61no—
85C2×Dic6—123no—
86C2×SL(2,𝔽3)—23no—
87C2×SL(2,𝔽3)C2×C3×D313yes—
88C2×C3×C8—92no—
89C3×C4⋊C4—31no—
90C3×C4⋊C4—31no—
91C3×C4⋊C4—61no—
92C3×C4⋊C4—92no—
93C3×C4⋊C4—181no—
94C3×C42—31no—
95C3×C42—31no—
96C3×C42—61no—
97C4⋊Dic3—32no—
98C4⋊Dic3—61no—
99C2×C3⋊C8—32no—
100C22×C32—41no—
101C22×C32—43no—
102C22×C32—83no—
103C22×C32—81no—
104C22×C32C2×S411yescharacteristic
105C32×C4—31no—
106C32×C4—31no—
107C32×C4—42no—
108C32×C4—46no—
109C32×C4—61no—
110C32×C4—86no—
111C32×C4—82no—
112C32×C4GL(2,𝔽3)12yes—
113C3⋊Dic3—44no—
114C3×Dic3—1212no—
115C3×Dic3—122no—
116C4⋊Q8—91no—
117C4×Q8C3×C3⋊D311yescharacteristic
118#106—92no—
119C4⋊C8—91no—
120#103—92no—
121C33—41noSylow
122C3×Q8—21no—
123C3×Q8—31no—
124C3×Q8—31no—
125C3×Q8—61no—
126C3×Q8—362no—
127C3×Q8C3×Dic611yescharacteristic, complemented
128C3×Q8#63411yescharacteristic, complemented
129C2×C3×C4—21no—
130C2×C3×C4—31no—
131C2×C3×C4—31no—
132C2×C3×C4—31no—
133C2×C3×C4—31no—
134C2×C3×C4—43no—
135C2×C3×C4—61no—
136C2×C3×C4—61no—
137C2×C3×C4—83no—
138C2×C3×C4—182no—
139C2×C3×C4C3×S411yescharacteristic
140C2×C3×C4C3⋊S411yescharacteristic
141C2×Dic3—62no—
142C2×Dic3—126no—
143Dic6—1212no—
144Dic6—122no—
145SL(2,𝔽3)—23no—
146SL(2,𝔽3)C3×Dic613yescomplemented
147C3×C8—182no—
148C3⋊C8—62no—
149C2×C32—41no—
150C2×C32—41no—
151C2×C32—41no—
152C2×C32—43no—
153C2×C32—43no—
154C2×C32—43no—
155C2×C32—83no—
156C2×C32—83no—
157C2×C32—83no—
158C2×C32—81no—
159C2×C32—81no—
160C2×C32—81no—
161C2×C32A4⋊Q811yescharacteristic
162C2×C32C2×GL(2,𝔽3)11yescharacteristic
163C2×C32#98411yescharacteristic
164C2×Q8—181no—
165C2×Q8C2×C3×C3⋊D311yescharacteristic
166C2×C8—92no—
167C4⋊C4—31no—
168C4⋊C4—92no—
169C4⋊C4—181no—
170C42—31no—
171C22×C3—21no—
172C22×C3—43no—
173C22×C3—83no—
174C22×C3C2×C3×S411yescharacteristic
175C22×C3C2×C3⋊S411yescharacteristic
176C3×C4—22no—
177C3×C4—31no—
178C3×C4—31no—
179C3×C4—31no—
180C3×C4—31no—
181C3×C4—46no—
182C3×C4—61no—
183C3×C4—61no—
184C3×C4—61no—
185C3×C4—61no—
186C3×C4—86no—
187C3×C4—121no—
188C3×C4—362no—
189C3×C4C3×GL(2,𝔽3)12yes—
190C3×C4C3⋊GL(2,𝔽3)12yes—
191Dic3—1212no—
192Dic3—122no—
193C32—41no—
194C32—43no—
195C32—83no—
196C32—81no—
197C32#538611yescharacteristic, complemented
198Q8—31no—
199Q8—362no—
200Q8C3×#63411yescharacteristic, complemented
201C2×C4—31no—
202C2×C4—31no—
203C2×C4—182no—
204C2×C4C3×C3⋊S411yescharacteristic
205C8—182no—
206C2×C3—21no—
207C2×C3—21no—
208C2×C3—21no—
209C2×C3—43no—
210C2×C3—43no—
211C2×C3—43no—
212C2×C3—83no—
213C2×C3—83no—
214C2×C3—83no—
215C2×C3C3×A4⋊Q811yescharacteristic
216C2×C3C2×C3×GL(2,𝔽3)11yescharacteristic
217C2×C3C3×#98411yescharacteristic
218C2×C3#6425911yescharacteristic
219C2×C3C2×C3⋊GL(2,𝔽3)11yescharacteristic
220C2×C3#6426511yescharacteristic
221C22C2×C3×C3⋊S411yescharacteristic
222C4—31no—
223C4—31no—
224C4—61no—
225C4—362no—
226C4C3×C3⋊GL(2,𝔽3)12yes—
227C3—21no—
228C3—43no—
229C3—83no—
230C3C3×#538611yescharacteristic, complemented
231C3#1059362011yescharacteristic, direct product factor
232C2C3×#6425911yescharacteristic
233C2C2×C3×C3⋊GL(2,𝔽3)11yescharacteristic
234C2C3×#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×Q8
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. #10593620C3direct (N × H)
22. C32×SL(2,𝔽3)Q8semidirect
57. C32×Q8Dic6semidirect
73. C3×SL(2,𝔽3)Dic6semidirect
74. C3×SL(2,𝔽3)C3×Q8semidirect
127. C3×Q8C3×Dic6semidirect
128. C3×Q8#634semidirect
146. SL(2,𝔽3)C3×Dic6semidirect
197. C32#5386semidirect
200. Q8C3×#634semidirect
230. C3C3×#5386semidirect
231. C3#10593620direct (N × H)

Maximal subgroups

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

subgroupindex
C32×C4×SL(2,𝔽3)2
C3×C3⋊#8592
#105936203
C3×#53863
C3×#50203
C2×C3×#6344