Finite Group Databasebeta

← #82460⋊C3 #11921268

Subgroups — #82460⋊C3 #11921268

Order 1152.

389 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 389 conjugacy classes of subgroups — too many to draw. Only its 83 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#82460C311yesmaximal, characteristic, complemented
4#2351⋊C3—31nomaximal
5C22×C3×SL(2,𝔽3)C2211yescharacteristic
6C3×C4×SL(2,𝔽3)C412yescomplemented
7Dic3×SL(2,𝔽3)—22no—
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
14#4984—32no—
15#4936—31no—
16C4×C4⋊Dic3—31no—
17C22×C32×C4—41no—
18C22×C3×Dic3—42no—
19C2×C3×SL(2,𝔽3)C2×C411yescharacteristic
20C2×C3×SL(2,𝔽3)D411yescharacteristic
21C2×C3×SL(2,𝔽3)Q811yescharacteristic
22#2351—31noSylow
23C22×C3×Q8C22×C311yescharacteristic
24C22×SL(2,𝔽3)—21no—
25C22×SL(2,𝔽3)C2×D311yescharacteristic
26C3×C4×Q8—32no—
27C3×C4×Q8C3×C412yescomplemented
28C2×C3×C4⋊C4—31no—
29C2×C3×C42—31no—
30Q8×Dic3—22no—
31Q8×Dic3—62no—
32C2×C4⋊Dic3—32no—
33C2×C4⋊Dic3—32no—
34C2×C4×Dic3—32no—
35C4×SL(2,𝔽3)—22no—
36C4×SL(2,𝔽3)—62no—
37C4×SL(2,𝔽3)Dic312yescomplemented
38C3×#95—121no—
39C3⋊#95—32no—
40C3⋊#95—31no—
41C3⋊#95A411yescharacteristic
42C23×C32—41no—
43C2×C32×C4—42no—
44C2×C32×C4—82no—
45C2×C3×Dic3—44no—
46C2×C3×Dic3—82no—
47C2×C3×Dic3—82no—
48C3×SL(2,𝔽3)C4⋊C411yescharacteristic, complemented
49C2×C4×Q8—32no—
50C2×C4×Q8C3×D311yescharacteristic
51#389—92no—
52#382—91no—
53C4×C4⋊C4—91no—
54C2×C3×Q8—31no—
55C2×C3×Q8—31no—
56C2×C3×Q8—31no—
57C2×C3×Q8C2×C3×C411yescharacteristic
58C2×C3×Q8C3×D411yescharacteristic
59C2×C3×Q8C3×Q811yescharacteristic
60C22×C3×C4—31no—
61C22×C3×C4—31no—
62C22×C3×C4—41no—
63C22×C3×C4—81no—
64C22×C3×C4—122no—
65C22×C3×C4C2×A411yescharacteristic
66C22×Dic3—32no—
67C22×Dic3C2×A412yes—
68C2×SL(2,𝔽3)—21no—
69C2×SL(2,𝔽3)—21no—
70C2×SL(2,𝔽3)—21no—
71C2×SL(2,𝔽3)Dic611yescharacteristic
72C2×SL(2,𝔽3)D1211yescharacteristic
73C2×SL(2,𝔽3)C2×Dic311yescharacteristic
74C3×C4⋊C4—32no—
75C3×C4⋊C4—32no—
76C3×C42—32no—
77C3×C42—32no—
78C4⋊Dic3—64no—
79C4⋊Dic3—62no—
80C4⋊Dic3—62no—
81C4×Dic3—62no—
82C4×Dic3—62no—
83C22×C32—41no—
84C22×C32—41no—
85C22×C32—41no—
86C22×C32—42no—
87C22×C32—42no—
88C32×C4—82no—
89C3×Dic3—84no—
90C22×Q8C2×C3×D311yescharacteristic
91C4×Q8—32no—
92C4×Q8—62no—
93C4×Q8—182no—
94C4×Q8C3×Dic312yescomplemented
95C2×C4⋊C4—31no—
96C2×C4⋊C4—92no—
97C2×C4⋊C4—92no—
98C2×C42—31no—
99C2×C42—92no—
100#95—31no—
101#95—92no—
102#95—91no—
103C23×C3—41no—
104C23×C3—81no—
105C23×C3C22×A411yescharacteristic
106C3×Q8—31no—
107C3×Q8—31no—
108C3×Q8—31no—
109C3×Q8—32no—
110C3×Q8C3×C4⋊C411yescharacteristic
111C2×C3×C4—22no—
112C2×C3×C4—31no—
113C2×C3×C4—31no—
114C2×C3×C4—31no—
115C2×C3×C4—31no—
116C2×C3×C4—31no—
117C2×C3×C4—31no—
118C2×C3×C4—32no—
119C2×C3×C4—42no—
120C2×C3×C4—62no—
121C2×C3×C4—82no—
122C2×C3×C4—84no—
123C2×C3×C4—82no—
124C2×C3×C4—124no—
125C2×C3×C4—242no—
126C2×C3×C4—242no—
127C2×C3×C4C4×A412yes—
128C2×Dic3—22no—
129C2×Dic3—22no—
130C2×Dic3—62no—
131C2×Dic3—62no—
132C2×Dic3—62no—
133C2×Dic3#41⋊C314yes—
134SL(2,𝔽3)—21no—
135SL(2,𝔽3)C4⋊Dic311yescharacteristic, complemented
136C2×C32—41no—
137C2×C32—41no—
138C2×C32—41no—
139C2×C32—42no—
140C2×C32—42no—
141C2×Q8—31no—
142C2×Q8—31no—
143C2×Q8—31no—
144C2×Q8C3×Dic611yescharacteristic
145C2×Q8C3×D1211yescharacteristic
146C2×Q8C2×C3×Dic311yescharacteristic
147C22×C4—31no—
148C22×C4—32no—
149C22×C4—31no—
150C22×C4—92no—
151C22×C4D3×A411yescharacteristic
152C4⋊C4—32no—
153C4⋊C4—32no—
154C4⋊C4—184no—
155C4⋊C4—182no—
156C4⋊C4—182no—
157C42—32no—
158C42—32no—
159C42—182no—
160C42—182no—
161C22×C3—41no—
162C22×C3—41no—
163C22×C3—42no—
164C22×C3—42no—
165C22×C3—41no—
166C22×C3—81no—
167C22×C3—81no—
168C22×C3—82no—
169C22×C3—82no—
170C22×C3—81no—
171C22×C3C2×C4×A411yescharacteristic
172C22×C3D4×A411yescharacteristic
173C22×C3Q8×A411yescharacteristic
174C22×C3C2×#41⋊C312yes—
175C22×C32−1+4⋊C312yes—
176C3×C4—22no—
177C3×C4—31no—
178C3×C4—31no—
179C3×C4—31no—
180C3×C4—31no—
181C3×C4—62no—
182C3×C4—82no—
183C3×C4—84no—
184C3×C4—244no—
185Dic3—24no—
186Dic3—122no—
187C32—41noSylow
188C23C2×D3×A411yescharacteristic
189Q8—31no—
190Q8—31no—
191Q8—31no—
192Q8—32no—
193Q8C3×C4⋊Dic311yescharacteristic
194C2×C4—22no—
195C2×C4—31no—
196C2×C4—31no—
197C2×C4—31no—
198C2×C4—31no—
199C2×C4—31no—
200C2×C4—31no—
201C2×C4—34no—
202C2×C4—32no—
203C2×C4—62no—
204C2×C4—62no—
205C2×C4—62no—
206C2×C4—182no—
207C2×C4—182no—
208C2×C4—182no—
209C2×C4Dic3×A412yes—
210C2×C3—42no—
211C2×C3—42no—
212C2×C3—41no—
213C2×C3—41no—
214C2×C3—41no—
215C2×C3—82no—
216C2×C3—82no—
217C2×C3—81no—
218C2×C3—81no—
219C2×C3—81no—
220C2×C3A4×C4⋊C411yescharacteristic
221C2×C3C4×#41⋊C311yescharacteristic
222C2×C3#519⋊C311yescharacteristic
223C2×C3#548⋊C312yes—
224C2×C3#557⋊C312yes—
225C22A4×Dic611yescharacteristic
226C22A4×D1211yescharacteristic
227C22#1007⋊C312yes—
228C22D3×#41⋊C312yes—
229C22C2×Dic3×A411yescharacteristic
230C4—22no—
231C4—31no—
232C4—31no—
233C4—31no—
234C4—31no—
235C4—64no—
236C4—62no—
237C4—362no—
238C3—41no—
239C3—81no—
240C3#2351⋊C311yescharacteristic, complemented
241C2#5569⋊C312yes—
242C2#5576⋊C312yes—
243C2A4×C4⋊Dic311yescharacteristic
244C2#5812⋊C311yescharacteristic
245C2Dic3×#41⋊C311yescharacteristic
subgroupG/H
center, FrattiniC23C2×D3×A4
commutatorC2×C3×Q8C2×C3×C4
FittingC2×C3×C4×Q8C2×C3
socleC23×C3C22×A4
prime radicalC22×C3×SL(2,𝔽3)C22
cosocleC22×C3×Q8C22×C3
hypercenterC22×C4D3×A4
hypocenterC3×Q8C3×C4⋊C4
LCSG ▷ C2×C3×Q8 ▷ C3×Q8
UCSC1 ◁ C23 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
3. #82460C3semidirect
6. C3×C4×SL(2,𝔽3)C4semidirect
27. C3×C4×Q8C3×C4semidirect
37. C4×SL(2,𝔽3)Dic3semidirect
48. C3×SL(2,𝔽3)C4⋊C4semidirect
94. C4×Q8C3×Dic3semidirect
135. SL(2,𝔽3)C4⋊Dic3semidirect
240. C3#2351⋊C3semidirect

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
#824603
#2351⋊C33
C3×C3⋊#954