Finite Group Databasebeta

← C3×#10593625 #13172993

Subgroups — C3×#10593625 #13172993

Order 1728.

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

normal subgroups only

This group has 408 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#10593625C311yesmaximal, characteristic, direct product factor
4C3×#5391—33nomaximal
5C3×#5021—31nomaximal
6C2×C32×SL(2,𝔽3)C2211yescharacteristic
7C3×C4⋊(C3⋊Dic3)—41nomaximal
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×#807—32no—
18C3×C4⋊(C3⋊C8)—31no—
19C2×C33×C4—41no—
20C2×C3×C3⋊Dic3—42no—
21C32×SL(2,𝔽3)Q811yescharacteristic, complemented
22#5391—33no—
23C3×#477—91no—
24#5021—31no—
25C2×C32×Q8C2×D311yescharacteristic
26C2×C3×Dic6—61no—
27C2×C3×SL(2,𝔽3)—21no—
28C2×C3×SL(2,𝔽3)—23no—
29C2×C3×SL(2,𝔽3)C2×D313yes—
30C2×C3×SL(2,𝔽3)C22×C311yescharacteristic
31C32×C4⋊C4—31no—
32C32×C42—31no—
33C4⋊(C3⋊Dic3)—41no—
34C3×C4⋊Dic3—32no—
35C3×C4⋊Dic3—61no—
36C3×C4⋊Dic3—123no—
37C2×C3×C3⋊C8—32no—
38C22×C33—41no—
39C33×C4—42no—
40C3×C3⋊Dic3—82no—
41C3×C4⋊Q8—91no—
42C3×C4×Q8—21no—
43C3×C4×Q8C3×D311yescharacteristic
44C3×C4×Q8C3⋊D311yescharacteristic
45#869—31no—
46C4×SL(2,𝔽3)—23no—
47C4×SL(2,𝔽3)C3×D313yes—
48#859—36no—
49C3×#107—92no—
50C3×C4⋊C8—91no—
51C3×#103—92no—
52#835—32no—
53#807—32no—
54C4⋊(C3⋊C8)—31no—
55C32×Q8—31no—
56C32×Q8Dic611yescharacteristic
57C2×C32×C4—31no—
58C2×C32×C4—31no—
59C2×C32×C4—41no—
60C2×C32×C4—43no—
61C2×C32×C4—83no—
62C2×C32×C4—81no—
63C2×C32×C4S411yescharacteristic
64C2×C3⋊Dic3—42no—
65C2×C3×Dic3—62no—
66C2×C3×Dic3—126no—
67C3×Dic6—122no—
68C3×SL(2,𝔽3)—21no—
69C3×SL(2,𝔽3)—23no—
70C3×SL(2,𝔽3)Dic613yescomplemented
71C3×SL(2,𝔽3)C3×Q811yescharacteristic, complemented
72C3×C3⋊C8—62no—
73#477—91noSylow
74C2×C33—41no—
75C2×C33—41no—
76C2×C33—41no—
77C2×C3×Q8—21no—
78C2×C3×Q8—181no—
79C2×C3×Q8C2×C3×D311yescharacteristic
80C2×C3×Q8C2×C3⋊D311yescharacteristic
81C2×Dic6—61no—
82C2×SL(2,𝔽3)—23no—
83C2×SL(2,𝔽3)C2×C3×D313yes—
84C2×C3×C8—92no—
85C3×C4⋊C4—31no—
86C3×C4⋊C4—31no—
87C3×C4⋊C4—61no—
88C3×C4⋊C4—92no—
89C3×C4⋊C4—181no—
90C3×C42—31no—
91C3×C42—31no—
92C3×C42—61no—
93C4⋊Dic3—32no—
94C4⋊Dic3—61no—
95C4⋊Dic3—123no—
96C2×C3⋊C8—32no—
97C22×C32—41no—
98C22×C32—43no—
99C22×C32—83no—
100C22×C32—81no—
101C22×C32C2×S411yescharacteristic
102C32×C4—31no—
103C32×C4—31no—
104C32×C4—42no—
105C32×C4—46no—
106C32×C4—61no—
107C32×C4—86no—
108C32×C4—82no—
109C32×C4#22612yes—
110C3⋊Dic3—82no—
111C3×Dic3—122no—
112C3×Dic3—246no—
113C4⋊Q8—91no—
114C4×Q8C3×C3⋊D311yescharacteristic
115#107—92no—
116C4⋊C8—91no—
117#103—92no—
118C33—41noSylow
119C3×Q8—21no—
120C3×Q8—31no—
121C3×Q8—31no—
122C3×Q8—61no—
123C3×Q8—362no—
124C3×Q8C3×Dic611yescharacteristic
125C3×Q8#63411yescharacteristic
126C2×C3×C4—21no—
127C2×C3×C4—31no—
128C2×C3×C4—31no—
129C2×C3×C4—31no—
130C2×C3×C4—31no—
131C2×C3×C4—43no—
132C2×C3×C4—61no—
133C2×C3×C4—61no—
134C2×C3×C4—83no—
135C2×C3×C4—182no—
136C2×C3×C4C3×S411yescharacteristic
137C2×C3×C4C3⋊S411yescharacteristic
138C2×Dic3—62no—
139C2×Dic3—126no—
140Dic6—122no—
141SL(2,𝔽3)—23no—
142SL(2,𝔽3)C3×Dic613yescomplemented
143C3×C8—182no—
144C3⋊C8—62no—
145C2×C32—41no—
146C2×C32—41no—
147C2×C32—41no—
148C2×C32—43no—
149C2×C32—43no—
150C2×C32—43no—
151C2×C32—83no—
152C2×C32—83no—
153C2×C32—83no—
154C2×C32—81no—
155C2×C32—81no—
156C2×C32—81no—
157C2×C32A4⋊Q811yescharacteristic
158C2×C32C2×#22611yescharacteristic
159C2×C32#98611yescharacteristic
160C2×Q8—181no—
161C2×Q8C2×C3×C3⋊D311yescharacteristic
162C2×C8—92no—
163C4⋊C4—31no—
164C4⋊C4—92no—
165C4⋊C4—181no—
166C42—31no—
167C22×C3—21no—
168C22×C3—43no—
169C22×C3—83no—
170C22×C3C2×C3×S411yescharacteristic
171C22×C3C2×C3⋊S411yescharacteristic
172C3×C4—22no—
173C3×C4—31no—
174C3×C4—31no—
175C3×C4—31no—
176C3×C4—31no—
177C3×C4—46no—
178C3×C4—61no—
179C3×C4—61no—
180C3×C4—61no—
181C3×C4—61no—
182C3×C4—86no—
183C3×C4—121no—
184C3×C4—362no—
185C3×C4C3×#22612yes—
186C3×C4C3⋊#22612yes—
187Dic3—122no—
188Dic3—246no—
189C32—41no—
190C32—43no—
191C32—83no—
192C32—81no—
193C32#539111yescharacteristic, complemented
194Q8—31no—
195Q8—362no—
196Q8C3×#63411yescharacteristic
197C2×C4—31no—
198C2×C4—31no—
199C2×C4—182no—
200C2×C4C3×C3⋊S411yescharacteristic
201C8—182no—
202C2×C3—21no—
203C2×C3—21no—
204C2×C3—21no—
205C2×C3—43no—
206C2×C3—43no—
207C2×C3—43no—
208C2×C3—83no—
209C2×C3—83no—
210C2×C3—83no—
211C2×C3C3×A4⋊Q811yescharacteristic
212C2×C3C2×C3×#22611yescharacteristic
213C2×C3C3×#98611yescharacteristic
214C2×C3#6425911yescharacteristic
215C2×C3C2×C3⋊#22611yescharacteristic
216C2×C3#6426711yescharacteristic
217C22C2×C3×C3⋊S411yescharacteristic
218C4—31no—
219C4—31no—
220C4—61no—
221C4—362no—
222C4C3×C3⋊#22612yes—
223C3—21no—
224C3—43no—
225C3—83no—
226C3C3×#539111yescharacteristic, complemented
227C3#1059362511yescharacteristic, direct product factor
228C2C3×#6425911yescharacteristic
229C2C2×C3×C3⋊#22611yescharacteristic
230C2C3×#6426711yescharacteristic
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. #10593625C3direct (N × H)
21. C32×SL(2,𝔽3)Q8semidirect
70. C3×SL(2,𝔽3)Dic6semidirect
71. C3×SL(2,𝔽3)C3×Q8semidirect
142. SL(2,𝔽3)C3×Dic6semidirect
193. C32#5391semidirect
226. C3C3×#5391semidirect
227. C3#10593625direct (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
#105936253
C3×#53913
C3×#50213
C3×C4⋊(C3⋊Dic3)4