Finite Group Databasebeta

← #10593654⋊C3 #13146906

Subgroups — #10593654⋊C3 #13146906

Order 1728.

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

normal subgroups only

This group has 360 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
1C22×Q8⋊H(3,𝔽3)C211yesmaximal, characteristic
2(C3⋊#859)⋊C3C212yesmaximal, complemented
3#10593654C311yesmaximal, characteristic, complemented
4C3×#5225—31nomaximal
5C2×Q8⋊H(3,𝔽3)C412yescomplemented
6C2×Q8⋊H(3,𝔽3)C2211yescharacteristic
7#3757⋊C3—41nomaximal
8C22×C32×Q8D311yescharacteristic
9C22×C3×SL(2,𝔽3)—21no—
10C22×C3×SL(2,𝔽3)C2×C311yescharacteristic
11C3×#926—31no—
12C2×C3×#208—31no—
13C3⋊#859C2×C312yescomplemented
14C3×#835—32no—
15C3×#835—32no—
16C23×H(3,𝔽3)—41no—
17C2×(C3⋊Dic3)⋊C3—42no—
18Q8⋊H(3,𝔽3)C2×C411yescharacteristic
19#5420—91no—
20C3×#419—91no—
21#5225—31no—
22C2×C32×Q8—32no—
23C2×C32×Q8—31no—
24C2×C32×Q8Dic312yescomplemented
25C2×C32×Q8C2×D311yescharacteristic
26C22×C32×C4—31no—
27C2×C3×SL(2,𝔽3)—22no—
28C2×C3×SL(2,𝔽3)—21no—
29C2×C3×SL(2,𝔽3)C3×C412yescomplemented
30C2×C3×SL(2,𝔽3)C22×C311yescharacteristic
31#3757—41no—
32C3×#217—61no—
33C3×C4⋊Dic3—32no—
34C3×C4×Dic3—61no—
35C3×#208—62no—
36C2×C3×C3⋊C8—32no—
37C22×H(3,𝔽3)—41no—
38C22×H(3,𝔽3)—42no—
39C22×H(3,𝔽3)—82no—
40(C3⋊Dic3)⋊C3—82no—
41C22×C3×Q8—31no—
42C22×C3×Q8C3×D311yescharacteristic
43C22×SL(2,𝔽3)—31no—
44C22×SL(2,𝔽3)—61no—
45C2×C3×M16—91no—
46C3×#117—91no—
47#926—31no—
48C2×#208—31no—
49#859—92no—
50C3×#103—92no—
51C3×#103—92no—
52#835—32no—
53#835—32no—
54C23×C32—41no—
55C23×C32—81no—
56C23×C32S411yescharacteristic
57C32×Q8—32no—
58C32×Q8—31no—
59C32×Q8—61no—
60C32×Q8C2×Dic311yescharacteristic
61C2×C32×C4—32no—
62C2×C32×C4—32no—
63C2×C32×C4—31no—
64C2×C32×C4—31no—
65C2×C3⋊Dic3—42no—
66C2×C3×Dic3—62no—
67C3×SL(2,𝔽3)—21no—
68C3×SL(2,𝔽3)C2×C3×C411yescharacteristic
69C3×C3⋊C8—62no—
70#419—91noSylow
71C2×H(3,𝔽3)—42no—
72C2×H(3,𝔽3)—41no—
73C2×H(3,𝔽3)—82no—
74C2×C3×Q8—31no—
75C2×C3×Q8—32no—
76C2×C3×Q8—31no—
77C2×C3×Q8—32no—
78C2×C3×Q8—32no—
79C2×C3×Q8—31no—
80C2×C3×Q8—61no—
81C2×C3×Q8—62no—
82C2×C3×Q8C3×Dic312yes—
83C2×C3×Q8C2×C3×D311yescharacteristic
84C22×C3×C4—31no—
85C22×C3×C4—31no—
86C22×C3×C4—61no—
87C2×SL(2,𝔽3)—32no—
88C2×SL(2,𝔽3)—31no—
89C2×SL(2,𝔽3)—62no—
90C2×SL(2,𝔽3)—61no—
91C3×M16—182no—
92C2×C3×C8—92no—
93C3×C4⋊C4—92no—
94C3×C22⋊C4—181no—
95C3×C42—181no—
96#217—61no—
97#217—361no—
98C4⋊Dic3—32no—
99C4×Dic3—61no—
100#208—62no—
101C2×C3⋊C8—32no—
102C22×C32—22no—
103C22×C32—41no—
104C22×C32—42no—
105C22×C32—82no—
106C22×C32—84no—
107C22×C32—81no—
108C22×C32—82no—
109C22×C32SL(2,ℤ/4)12yes—
110C22×C32C2×S411yescharacteristic
111C32×C4—31no—
112C32×C4—31no—
113C32×C4—32no—
114C3⋊Dic3—82no—
115C3×Dic3—122no—
116C22×Q8(C3⋊D3)⋊C311yescharacteristic
117C2×M16—91no—
118#117—91no—
119#103—92no—
120#103—92no—
121H(3,𝔽3)—41noSylow
122C23×C3—31no—
123C23×C3—121no—
124C23×C3—241no—
125C23×C3C3×S411yescharacteristic
126C3×Q8—32no—
127C3×Q8—31no—
128C3×Q8—32no—
129C3×Q8—31no—
130C3×Q8—31no—
131C3×Q8—62no—
132C3×Q8—61no—
133C3×Q8—61no—
134C3×Q8—61no—
135C3×Q8—62no—
136C3×Q8C2×C3×Dic311yescharacteristic
137C2×C3×C4—31no—
138C2×C3×C4—31no—
139C2×C3×C4—32no—
140C2×C3×C4—32no—
141C2×C3×C4—31no—
142C2×C3×C4—31no—
143C2×C3×C4—32no—
144C2×C3×C4—32no—
145C2×C3×C4—62no—
146C2×C3×C4—62no—
147C2×C3×C4—61no—
148C2×C3×C4—61no—
149C2×C3×C4—182no—
150C2×Dic3—62no—
151C2×Dic3—362no—
152SL(2,𝔽3)—31no—
153SL(2,𝔽3)—61no—
154C3×C8—182no—
155C3⋊C8—62no—
156C2×C32—22no—
157C2×C32—41no—
158C2×C32—42no—
159C2×C32—82no—
160C2×C32—84no—
161C2×C32—81no—
162C2×C32—82no—
163C2×C32#98312yes—
164C2×C32C2×SL(2,ℤ/4)11yescharacteristic
165C2×Q8—31no—
166C2×Q8—32no—
167C2×Q8(C3⋊Dic3)⋊C312yescomplemented
168C2×Q8C2×(C3⋊D3)⋊C311yescharacteristic
169C22×C4—31no—
170M16—182no—
171C2×C8—92no—
172C4⋊C4—92no—
173C22⋊C4—181no—
174C42—181no—
175C22×C3—22no—
176C22×C3—32no—
177C22×C3—31no—
178C22×C3—62no—
179C22×C3—122no—
180C22×C3—121no—
181C22×C3—242no—
182C22×C3—244no—
183C22×C3—242no—
184C22×C3—241no—
185C22×C3C3×SL(2,ℤ/4)12yes—
186C22×C3C2×C3×S411yescharacteristic
187C3×C4—31no—
188C3×C4—31no—
189C3×C4—32no—
190C3×C4—31no—
191C3×C4—31no—
192C3×C4—32no—
193C3×C4—61no—
194C3×C4—61no—
195C3×C4—62no—
196C3×C4—362no—
197Dic3—122no—
198Dic3—722no—
199C32—41no—
200C32—81no—
201C32#542011yescharacteristic, complemented
202C23(C3⋊S4)⋊C311yescharacteristic
203Q8—32no—
204Q8—31no—
205Q8—61no—
206Q8C2×(C3⋊Dic3)⋊C311yescharacteristic
207C2×C4—32no—
208C2×C4—32no—
209C2×C4—31no—
210C2×C4—31no—
211C2×C4—182no—
212C8—182no—
213C2×C3—22no—
214C2×C3—32no—
215C2×C3—31no—
216C2×C3—62no—
217C2×C3—122no—
218C2×C3—121no—
219C2×C3—242no—
220C2×C3—244no—
221C2×C3—242no—
222C2×C3—241no—
223C2×C3C3×#98312yes—
224C2×C3C2×C3×SL(2,ℤ/4)11yescharacteristic
225C22—22no—
226C22(C3⋊SL(2,ℤ/4))⋊C312yes—
227C22C2×(C3⋊S4)⋊C311yescharacteristic
228C4—31no—
229C4—31no—
230C4—32no—
231C4—362no—
232C3—31no—
233C3—121no—
234C3—241no—
235C3C3×#542011yescharacteristic
236C2—22no—
237C2#64264⋊C312yes—
238C2C2×(C3⋊SL(2,ℤ/4))⋊C311yescharacteristic
subgroupG/H
centerC22C2×(C3⋊S4)⋊C3
commutator, hypocenterC3×SL(2,𝔽3)C2×C3×C4
Frattini, socleC22×C3C2×C3×S4
FittingC22×C32×Q8D3
prime radicalC22×Q8⋊H(3,𝔽3)C2
cosocleC2×C3×SL(2,𝔽3)C22×C3
hypercenterC23(C3⋊S4)⋊C3
LCSG ▷ C3×SL(2,𝔽3)
UCSC1 ◁ C22 ◁ C23

Complements

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

NH ≅ G/Ntype
2. (C3⋊#859)⋊C3C2semidirect
3. #10593654C3semidirect
5. C2×Q8⋊H(3,𝔽3)C4semidirect
13. C3⋊#859C2×C3semidirect
24. C2×C32×Q8Dic3semidirect
29. C2×C3×SL(2,𝔽3)C3×C4semidirect
167. C2×Q8(C3⋊Dic3)⋊C3semidirect
201. C32#5420semidirect

Maximal subgroups

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

subgroupindex
C22×Q8⋊H(3,𝔽3)2
(C3⋊#859)⋊C32
#105936543
C3×#52253
#3757⋊C34