Finite Group Databasebeta

← #10593658⋊C3 #13146922

Subgroups — #10593658⋊C3 #13146922

Order 1728.

342 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 342 conjugacy classes of subgroups — too many to draw. Only its 55 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×#41⋊H(3,𝔽3)C211yesmaximal, characteristic
2(C3⋊#859)⋊C3C212yesmaximal, complemented
3#10593658C311yesmaximal, characteristic, complemented
4C3×#5233—31nomaximal
5#41⋊H(3,𝔽3)C412yescomplemented
6C2×Q8⋊H(3,𝔽3)C2211yescharacteristic
7C4×(C3⋊Dic3)⋊C3—41nomaximal
8C2×C32×#41D311yescharacteristic
9C2×C3×#41⋊C3—21no—
10C2×C3×#41⋊C3C2×C311yescharacteristic
11C3×#926—31no—
12C22×C3×C3⋊C8—31no—
13C3⋊#859C2×C312yescomplemented
14C3×#835—32no—
15C3×#832—32no—
16C2×C4×H(3,𝔽3)—41no—
17C2×(C3⋊Dic3)⋊C3—42no—
18Q8⋊H(3,𝔽3)C2×C411yescharacteristic
19#5424—91no—
20C3×#416—91no—
21#5233—31no—
22C32×#41—32no—
23C32×#41Dic312yescomplemented
24C2×C32×Q8C2×D311yescharacteristic
25C2×C32×D4—31no—
26C22×C32×C4—31no—
27C3×#41⋊C3—22no—
28C3×#41⋊C3C3×C412yescomplemented
29C2×C3×SL(2,𝔽3)—21no—
30C2×C3×SL(2,𝔽3)C22×C311yescharacteristic
31C4×C3⋊Dic3—41no—
32C3×#217—61no—
33C3×C4⋊Dic3—32no—
34C3×C4×Dic3—61no—
35C2×C3×C3⋊C8—32no—
36C2×C3×C3⋊C8—62no—
37C22×H(3,𝔽3)—41no—
38C4×H(3,𝔽3)—42no—
39(C3⋊Dic3)⋊C3—44no—
40C2×C3×#41—31no—
41C2×C3×#41C3×D311yescharacteristic
42C2×#41⋊C3—31no—
43C2×#41⋊C3—61no—
44C22×C3×C8—91no—
45C3×#117—91no—
46#926—31no—
47C22×C3⋊C8—31no—
48#859—92no—
49C3×#103—92no—
50C3×#102—92no—
51#835—32no—
52#832—32no—
53C23×C32—31no—
54C32×Q8—31no—
55C32×Q8C2×Dic311yescharacteristic
56C32×D4—32no—
57C32×D4—61no—
58C2×C32×C4—31no—
59C2×C32×C4—32no—
60C2×C32×C4—32no—
61C2×C32×C4—41no—
62C2×C32×C4—81no—
63C2×C32×C4S411yescharacteristic
64C2×C3⋊Dic3—42no—
65C2×C3×Dic3—62no—
66C3×SL(2,𝔽3)—21no—
67C3×SL(2,𝔽3)C2×C3×C411yescharacteristic
68C3×C3⋊C8—62no—
69#416—91noSylow
70C2×H(3,𝔽3)—42no—
71C2×H(3,𝔽3)—41no—
72C3×#41—32no—
73C3×#41—32no—
74C3×#41—32no—
75C3×#41—62no—
76C3×#41C3×Dic312yes—
77C2×C3×Q8—31no—
78C2×C3×Q8C2×C3×D311yescharacteristic
79C2×C3×D4—31no—
80C2×C3×D4—31no—
81C2×C3×D4—61no—
82C22×C3×C4—31no—
83C22×C3×C4—31no—
84C22×C3×C4—61no—
85#41⋊C3—32no—
86#41⋊C3—62no—
87C2×SL(2,𝔽3)—31no—
88C2×SL(2,𝔽3)—61no—
89C2×C3×C8—92no—
90C2×C3×C8—182no—
91C3×C4⋊C4—92no—
92C3×C22⋊C4—181no—
93C3×C42—181no—
94#217—61no—
95C4⋊Dic3—32no—
96C4×Dic3—61no—
97C4×Dic3—361no—
98C2×C3⋊C8—32no—
99C2×C3⋊C8—62no—
100C22×C32—32no—
101C22×C32—41no—
102C22×C32—62no—
103C22×C32—81no—
104C22×C32C2×S411yescharacteristic
105C32×C4—31no—
106C32×C4—31no—
107C32×C4—42no—
108C32×C4—82no—
109C32×C4SL(2,ℤ/4)12yes—
110C3⋊Dic3—44no—
111C3×Dic3—122no—
112C2×#41(C3⋊D3)⋊C311yescharacteristic
113C22×C8—91no—
114#117—91no—
115#103—92no—
116#102—92no—
117H(3,𝔽3)—41noSylow
118C23×C3—31no—
119C23×C3—31no—
120C23×C3—61no—
121C3×Q8—31no—
122C3×Q8—31no—
123C3×Q8—31no—
124C3×Q8—61no—
125C3×Q8C2×C3×Dic311yescharacteristic
126C3×D4—32no—
127C3×D4—32no—
128C3×D4—62no—
129C3×D4—61no—
130C3×D4—61no—
131C3×D4—62no—
132C2×C3×C4—31no—
133C2×C3×C4—32no—
134C2×C3×C4—32no—
135C2×C3×C4—31no—
136C2×C3×C4—32no—
137C2×C3×C4—32no—
138C2×C3×C4—31no—
139C2×C3×C4—62no—
140C2×C3×C4—62no—
141C2×C3×C4—61no—
142C2×C3×C4—121no—
143C2×C3×C4—182no—
144C2×C3×C4—241no—
145C2×C3×C4C3×S411yescharacteristic
146C2×Dic3—62no—
147C2×Dic3—362no—
148SL(2,𝔽3)—31no—
149SL(2,𝔽3)—61no—
150C3×C8—182no—
151C3⋊C8—62no—
152C2×C32—41no—
153C2×C32—42no—
154C2×C32—62no—
155C2×C32—81no—
156C2×C32—82no—
157C2×C32#98512yes—
158C2×C32C2×SL(2,ℤ/4)11yescharacteristic
159#41—32no—
160#41(C3⋊Dic3)⋊C312yescomplemented
161C2×Q8C2×(C3⋊D3)⋊C311yescharacteristic
162C2×D4—31no—
163C22×C4—31no—
164C2×C8—92no—
165C2×C8—182no—
166C4⋊C4—92no—
167C22⋊C4—181no—
168C42—181no—
169C22×C3—32no—
170C22×C3—32no—
171C22×C3—31no—
172C22×C3—62no—
173C22×C3—62no—
174C22×C3—62no—
175C22×C3—122no—
176C22×C3—121no—
177C22×C3—241no—
178C22×C3C2×C3×S411yescharacteristic
179C3×C4—31no—
180C3×C4—31no—
181C3×C4—31no—
182C3×C4—31no—
183C3×C4—32no—
184C3×C4—61no—
185C3×C4—61no—
186C3×C4—122no—
187C3×C4—242no—
188C3×C4—362no—
189C3×C4C3×SL(2,ℤ/4)12yes—
190Dic3—122no—
191Dic3—364no—
192C32—41no—
193C32—81no—
194C32#542411yescharacteristic, complemented
195C23—31no—
196Q8—31no—
197Q8C2×(C3⋊Dic3)⋊C311yescharacteristic
198D4—32no—
199D4—61no—
200C2×C4—31no—
201C2×C4—32no—
202C2×C4—32no—
203C2×C4—182no—
204C2×C4(C3⋊S4)⋊C311yescharacteristic
205C8—182no—
206C2×C3—32no—
207C2×C3—31no—
208C2×C3—62no—
209C2×C3—62no—
210C2×C3—122no—
211C2×C3—122no—
212C2×C3—121no—
213C2×C3—242no—
214C2×C3—241no—
215C2×C3C3×#98512yes—
216C2×C3C2×C3×SL(2,ℤ/4)11yescharacteristic
217C22—32no—
218C22—62no—
219C22C2×(C3⋊S4)⋊C311yescharacteristic
220C4—31no—
221C4—31no—
222C4—362no—
223C4(C3⋊SL(2,ℤ/4))⋊C312yes—
224C3—31no—
225C3—121no—
226C3—241no—
227C3C3×#542411yescharacteristic
228C2—62no—
229C2#64266⋊C312yes—
230C2C2×(C3⋊SL(2,ℤ/4))⋊C311yescharacteristic
subgroupG/H
center, hypercenterC2×C4(C3⋊S4)⋊C3
commutator, hypocenterC3×SL(2,𝔽3)C2×C3×C4
Frattini, socleC22×C3C2×C3×S4
FittingC2×C32×#41D3
prime radicalC2×#41⋊H(3,𝔽3)C2
cosocleC2×C3×SL(2,𝔽3)C22×C3
LCSG ▷ C3×SL(2,𝔽3)
UCSC1 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
2. (C3⋊#859)⋊C3C2semidirect
3. #10593658C3semidirect
5. #41⋊H(3,𝔽3)C4semidirect
13. C3⋊#859C2×C3semidirect
23. C32×#41Dic3semidirect
28. C3×#41⋊C3C3×C4semidirect
160. #41(C3⋊Dic3)⋊C3semidirect
194. C32#5424semidirect

Maximal subgroups

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

subgroupindex
C2×#41⋊H(3,𝔽3)2
(C3⋊#859)⋊C32
#105936583
C3×#52333
C4×(C3⋊Dic3)⋊C34