Finite Group Databasebeta

← C3⋊#73167 #11921266

Subgroups — C3⋊#73167 #11921266

Order 1152.

421 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 421 conjugacy classes of subgroups — too many to draw. Only its 91 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×C4×SL(2,𝔽3)C211yesmaximal, characteristic
2C2×Dic3×SL(2,𝔽3)C212yesmaximal
3#82450C311yesmaximal, characteristic, complemented
4#73167—31nomaximal
5C22×C3×SL(2,𝔽3)C2211yescharacteristic
6C3×C4×SL(2,𝔽3)C412yescomplemented
7Dic3×SL(2,𝔽3)C414yescomplemented
8C3×C3⋊#95—41nomaximal
9C2×C3×C4×Q8C2×C311yescharacteristic
10C2×Q8×Dic3C2×C312yes—
11C2×C4×SL(2,𝔽3)—21no—
12C2×C4×SL(2,𝔽3)—32no—
13C2×C4×SL(2,𝔽3)D311yescharacteristic
14Dic3×C4⋊C4—32no—
15C4×C4⋊Dic3—31no—
16#4932—31no—
17C22×C32×C4—41no—
18C22×C3×Dic3—42no—
19C2×C3×SL(2,𝔽3)C2×C412yes—
20C2×C3×SL(2,𝔽3)C2×C411yescharacteristic
21#2276—31noSylow
22C22×C3×Q8C22×C311yescharacteristic
23C22×SL(2,𝔽3)—21no—
24C22×SL(2,𝔽3)C2×D311yescharacteristic
25C3×C4×Q8—32no—
26C3×C4×Q8C3×C412yescomplemented
27C2×C3×C4⋊C4—31no—
28C2×C3×C42—31no—
29Q8×Dic3—34no—
30Q8×Dic3C3×C414yescomplemented
31C2×C4⋊Dic3—32no—
32C2×C4×Dic3—32no—
33C2×C4×Dic3—32no—
34C4×SL(2,𝔽3)—22no—
35C4×SL(2,𝔽3)—34no—
36C4×SL(2,𝔽3)Dic312yescomplemented
37C3×#95—121no—
38C3⋊#95—31no—
39C3⋊#95—32no—
40C3⋊#95A411yescharacteristic
41C23×C32—41no—
42C2×C32×C4—42no—
43C2×C32×C4—82no—
44C2×C3×Dic3—44no—
45C2×C3×Dic3—82no—
46C2×C3×Dic3—82no—
47C3×SL(2,𝔽3)C4211yescharacteristic, complemented
48C2×C4×Q8—32no—
49C2×C4×Q8C3×D311yescharacteristic
50C4×C4⋊C4—91no—
51C4×C4⋊C4—92no—
52#376—91no—
53C2×C3×Q8—32no—
54C2×C3×Q8—31no—
55C2×C3×Q8C2×C3×C412yes—
56C2×C3×Q8C2×C3×C411yescharacteristic
57C22×C3×C4—31no—
58C22×C3×C4—31no—
59C22×C3×C4—41no—
60C22×C3×C4—81no—
61C22×C3×C4—122no—
62C22×C3×C4C2×A411yescharacteristic
63C22×Dic3—32no—
64C22×Dic3C2×A412yes—
65C2×SL(2,𝔽3)—22no—
66C2×SL(2,𝔽3)—21no—
67C2×SL(2,𝔽3)C4×D312yes—
68C2×SL(2,𝔽3)C2×Dic311yescharacteristic
69C3×C4⋊C4—32no—
70C3×C4⋊C4—32no—
71C3×C42—32no—
72C3×C42—32no—
73C4⋊Dic3—34no—
74C4⋊Dic3—34no—
75C4×Dic3—34no—
76C4×Dic3—34no—
77C4×Dic3—122no—
78C22×C32—42no—
79C22×C32—41no—
80C22×C32—41no—
81C22×C32—41no—
82C22×C32—42no—
83C32×C4—82no—
84C3×Dic3—84no—
85C22×Q8C2×C3×D311yescharacteristic
86C4×Q8—32no—
87C4×Q8—34no—
88C4×Q8—94no—
89C4×Q8C3×Dic312yescomplemented
90C2×C4⋊C4—31no—
91C2×C4⋊C4—92no—
92C2×C42—31no—
93C2×C42—92no—
94C2×C42—92no—
95#95—31no—
96#95—92no—
97#95—91no—
98C23×C3—41no—
99C23×C3—81no—
100C23×C3C22×A411yescharacteristic
101C3×Q8—32no—
102C3×Q8—31no—
103C3×Q8—32no—
104C3×Q8C3×C4211yescharacteristic
105C2×C3×C4—22no—
106C2×C3×C4—32no—
107C2×C3×C4—32no—
108C2×C3×C4—31no—
109C2×C3×C4—31no—
110C2×C3×C4—32no—
111C2×C3×C4—42no—
112C2×C3×C4—61no—
113C2×C3×C4—61no—
114C2×C3×C4—82no—
115C2×C3×C4—84no—
116C2×C3×C4—82no—
117C2×C3×C4—124no—
118C2×C3×C4—242no—
119C2×C3×C4—242no—
120C2×C3×C4C4×A412yes—
121C2×Dic3—22no—
122C2×Dic3—22no—
123C2×Dic3—34no—
124C2×Dic3—62no—
125C2×Dic3—62no—
126C2×Dic3C4×A414yes—
127SL(2,𝔽3)—21no—
128SL(2,𝔽3)C4×Dic311yescharacteristic, complemented
129C2×C32—41no—
130C2×C32—42no—
131C2×C32—41no—
132C2×C32—42no—
133C2×C32—41no—
134C2×Q8—31no—
135C2×Q8—32no—
136C2×Q8C3×C4×D312yes—
137C2×Q8C2×C3×Dic311yescharacteristic
138C22×C4—31no—
139C22×C4—32no—
140C22×C4—31no—
141C22×C4—92no—
142C22×C4D3×A411yescharacteristic
143C4⋊C4—32no—
144C4⋊C4—32no—
145C4⋊C4—94no—
146C4⋊C4—94no—
147C42—32no—
148C42—32no—
149C42—94no—
150C42—94no—
151C42—362no—
152C22×C3—42no—
153C22×C3—42no—
154C22×C3—41no—
155C22×C3—41no—
156C22×C3—41no—
157C22×C3—82no—
158C22×C3—82no—
159C22×C3—81no—
160C22×C3—81no—
161C22×C3—81no—
162C22×C3C2×C4×A412yes—
163C22×C3C2×C4×A411yescharacteristic
164C22×C3#99411yescharacteristic
165C22×C32−1+4⋊C312yes—
166C22×C32−1+4⋊C311yescharacteristic
167C3×C4—22no—
168C3×C4—31no—
169C3×C4—32no—
170C3×C4—31no—
171C3×C4—62no—
172C3×C4—82no—
173C3×C4—84no—
174C3×C4—244no—
175Dic3—24no—
176Dic3—64no—
177C32—41noSylow
178C23C2×D3×A411yescharacteristic
179Q8—32no—
180Q8—31no—
181Q8—32no—
182Q8C3×C4×Dic311yescharacteristic
183C2×C4—22no—
184C2×C4—32no—
185C2×C4—32no—
186C2×C4—31no—
187C2×C4—31no—
188C2×C4—34no—
189C2×C4—32no—
190C2×C4—62no—
191C2×C4—61no—
192C2×C4—62no—
193C2×C4—61no—
194C2×C4—94no—
195C2×C4—182no—
196C2×C4—182no—
197C2×C4Dic3×A412yes—
198C2×C3—42no—
199C2×C3—42no—
200C2×C3—41no—
201C2×C3—41no—
202C2×C3—41no—
203C2×C3—82no—
204C2×C3—82no—
205C2×C3—81no—
206C2×C3—81no—
207C2×C3—81no—
208C2×C3C42×A411yescharacteristic
209C2×C3#519⋊C312yes—
210C2×C3#519⋊C311yescharacteristic
211C2×C3#520⋊C312yes—
212C2×C3#520⋊C311yescharacteristic
213C22C4×D3×A412yes—
214C22#1007⋊C312yes—
215C22C3⋊#99411yescharacteristic
216C22#1010⋊C311yescharacteristic
217C22C2×Dic3×A411yescharacteristic
218C4—22no—
219C4—31no—
220C4—32no—
221C4—31no—
222C4—64no—
223C4—62no—
224C4—184no—
225C3—41no—
226C3—81no—
227C3#7316711yescharacteristic, complemented
228C2#5574⋊C312yes—
229C2#5572⋊C312yes—
230C2C4×Dic3×A411yescharacteristic
231C2#5812⋊C311yescharacteristic
232C2#5827⋊C311yescharacteristic
subgroupG/H
center, FrattiniC23C2×D3×A4
commutator, hypocenterC3×Q8C3×C42
FittingC2×C3×C4×Q8C2×C3
socleC23×C3C22×A4
prime radicalC22×C3×SL(2,𝔽3)C22
cosocleC22×C3×Q8C22×C3
hypercenterC22×C4D3×A4
LCSG ▷ C3×Q8
UCSC1 ◁ C23 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
3. #82450C3semidirect
6. C3×C4×SL(2,𝔽3)C4semidirect
7. Dic3×SL(2,𝔽3)C4semidirect
26. C3×C4×Q8C3×C4semidirect
30. Q8×Dic3C3×C4semidirect
36. C4×SL(2,𝔽3)Dic3semidirect
47. C3×SL(2,𝔽3)C42semidirect
89. C4×Q8C3×Dic3semidirect
128. SL(2,𝔽3)C4×Dic3semidirect
227. C3#73167semidirect

Maximal subgroups

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

subgroupindex
C2×C3×C4×SL(2,𝔽3)2
C2×Dic3×SL(2,𝔽3)2
#824503
#731673
C3×C3⋊#954