Finite Group Databasebeta

← #11920560

Subgroups — #11920560

Order 1152.

275 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 275 conjugacy classes of subgroups — too many to draw. Only its 49 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
1C4×D3×SL(2,𝔽3)C211yesmaximal, characteristic
2C3⋊(SL(2,𝔽3)⋊C8)C211yesmaximal, characteristic, complemented
3C3×SL(2,𝔽3)⋊C8C211yesmaximal, characteristic, complemented
4#73055—31nomaximal
5#68236—31nomaximal
6C2×D3×SL(2,𝔽3)C411yescharacteristic
7C3×C4×SL(2,𝔽3)C2211yescharacteristic
8C2×#3711—41nomaximal
9Dic3×SL(2,𝔽3)C411yescharacteristic
10C4×D3×Q8D311yescharacteristic
11C2×C4×SL(2,𝔽3)—31no—
12#4837—31no—
13#4685—31no—
14SL(2,𝔽3)⋊C8—31no—
15SL(2,𝔽3)⋊C8—61no—
16SL(2,𝔽3)⋊C8D311yescharacteristic, complemented
17C3×Q8⋊C8—31no—
18#4535—31no—
19#4484—31no—
20#4456—31no—
21C2×C3×C4×D3—41no—
22C2×C3×SL(2,𝔽3)C2×C411yescharacteristic
23D3×SL(2,𝔽3)—21no—
24C2×C3⋊(C3⋊C8)—41no—
25C2×C3×C3⋊C8—41no—
26#3711—44no—
27#1487—91noSylow
28C2×D3×Q8Dic311yescharacteristic
29C22×SL(2,𝔽3)—31no—
30C3×C4×Q8C2×D311yescharacteristic
31Q8×Dic3Dic311yescharacteristic
32C2×#208—121no—
33C2×ΓL(1,𝔽25)—61no—
34D3×C4⋊C4—31no—
35#887—31no—
36C42×D3—31no—
37C4×Dic6—31no—
38C4×SL(2,𝔽3)—21no—
39C4×SL(2,𝔽3)—31no—
40C4×SL(2,𝔽3)C2×D311yescharacteristic
41C3×C4⋊C8—31no—
42C3×C4×C8—31no—
43(C2×D3)⋊C8—61no—
44Dic3⋊C8—61no—
45Dic3⋊C8—61no—
46C4⋊(C3⋊C8)—31no—
47C4×C3⋊C8—31no—
48C22×C3×D3—41no—
49C2×C32×C4—41no—
50C2×C3×Dic3—41no—
51C3×C4×D3—42no—
52C3×C4×D3—42no—
53C3×SL(2,𝔽3)M1611yescharacteristic, complemented
54C3⋊(C3⋊C8)—42no—
55C3×C3⋊C8—42no—
56C2×C4×Q8—31no—
57#431—91no—
58#423—91no—
59Q8⋊C8—31no—
60Q8⋊C8—91no—
61Q8⋊C8—91no—
62Q8⋊C8—91no—
63C2×C3×Q8C2×Dic311yescharacteristic
64C22×C3×C4—121no—
65D3×Q8—21no—
66D3×Q8—61no—
67C2×C4×D3—31no—
68C2×C4×D3—31no—
69C2×C4×D3S411yescharacteristic
70C2×Dic6—31no—
71C2×SL(2,𝔽3)—21no—
72C2×SL(2,𝔽3)—61no—
73C2×SL(2,𝔽3)C4×D311yescharacteristic
74C2×C3×C8—61no—
75C2×C3×C8—61no—
76C3×C4⋊C4—31no—
77C3×C42—31no—
78C4⋊Dic3—31no—
79Dic3⋊C4—61no—
80C4×Dic3—31no—
81C4×Dic3—31no—
82C4×Dic3—31no—
83#208—124no—
84C2×C3⋊C8—41no—
85C2×C3⋊C8—61no—
86C2×C3⋊C8—61no—
87C2×C3⋊C8—121no—
88C2×C3⋊C8—241no—
89ΓL(1,𝔽25)—122no—
90C22×C32—41no—
91C2×C3×D3—42no—
92C2×C3×D3—81no—
93C2×C3×D3—81no—
94C32×C4—41no—
95C32×C4—41no—
96C3×Dic3—42no—
97C22×Q8—31no—
98C2×M16—181no—
99C4×Q8—31no—
100C4×Q8—91no—
101C4×Q8—91no—
102C4×Q8D3211yescharacteristic
103C2×C4⋊C4—91no—
104C2×C42—91no—
105C4⋊C8—31no—
106C4⋊C8—91no—
107C4⋊C8—181no—
108C4⋊C8—181no—
109C22⋊C8—181no—
110C4×C8—31no—
111C4×C8—91no—
112C23×C3—121no—
113C22×D3SL(2,ℤ/4)11yescharacteristic
114C3×Q8—31no—
115C3×Q8#20811yescharacteristic, complemented
116C2×C3×C4—31no—
117C2×C3×C4—31no—
118C2×C3×C4—41no—
119C2×C3×C4—81no—
120C2×C3×C4—122no—
121C2×C3×C4—122no—
122C2×C3×C4—121no—
123C2×C3×C4C2×S411yescharacteristic
124C2×Dic3—31no—
125C2×Dic3—31no—
126C2×Dic3SL(2,ℤ/4)11yescharacteristic
127C4×D3—61no—
128C4×D3—61no—
129C4×D3—121no—
130C4×D3#22612yes—
131C4×D3GL(2,𝔽3)12yes—
132Dic6—32no—
133Dic6—61no—
134SL(2,𝔽3)—21no—
135SL(2,𝔽3)ΓL(1,𝔽25)11yescharacteristic, complemented
136C3×C8—121no—
137C3×C8—121no—
138C3⋊C8—42no—
139C3⋊C8—122no—
140C3⋊C8—121no—
141C3⋊C8—121no—
142C3⋊C8—242no—
143C2×C32—41no—
144C2×C32—41no—
145C2×C32—41no—
146C3×D3—82no—
147C2×Q8—61no—
148C2×Q8—91no—
149C2×Q8—181no—
150C2×Q8D3×Dic311yescharacteristic
151C22×C4—31no—
152C22×C4—91no—
153C22×C4—91no—
154M16—362no—
155C2×C8—61no—
156C2×C8—61no—
157C2×C8—181no—
158C2×C8—181no—
159C4⋊C4—31no—
160C4⋊C4—91no—
161C4⋊C4—181no—
162C42—31no—
163C42—91no—
164C42—91no—
165C42—91no—
166C22×C3—41no—
167C22×C3—81no—
168C22×C3—122no—
169C22×C3—241no—
170C22×C3—241no—
171C22×C3C2×SL(2,ℤ/4)11yescharacteristic
172C2×D3—21no—
173C2×D3—21no—
174C2×D3#85912yes—
175C3×C4—31no—
176C3×C4—31no—
177C3×C4—41no—
178C3×C4—41no—
179C3×C4—61no—
180C3×C4—81no—
181C3×C4—81no—
182C3×C4—122no—
183C3×C4C2×#22611yescharacteristic
184C3×C4C2×GL(2,𝔽3)11yescharacteristic
185Dic3—61no—
186Dic3—61no—
187Dic3#85912yes—
188C32—41noSylow
189C23—31no—
190Q8—31no—
191Q8—92no—
192Q8—181no—
193Q8#371111yescharacteristic, complemented
194C2×C4—31no—
195C2×C4—31no—
196C2×C4—32no—
197C2×C4—32no—
198C2×C4—31no—
199C2×C4—91no—
200C2×C4—91no—
201C2×C4—181no—
202C2×C4—181no—
203C2×C4—361no—
204C2×C4D3×S411yescharacteristic
205C8—121no—
206C8—121no—
207C8—361no—
208C8—361no—
209C2×C3—41no—
210C2×C3—41no—
211C2×C3—41no—
212C2×C3—81no—
213C2×C3—81no—
214C2×C3—81no—
215C2×C3—242no—
216C2×C3#540911yescharacteristic
217C2×C3C2×#85911yescharacteristic
218C2×C3#542311yescharacteristic
219D3—22no—
220C22—32no—
221C22—61no—
222C22—61no—
223C22D3×SL(2,ℤ/4)11yescharacteristic
224C4—31no—
225C4—31no—
226C4—32no—
227C4—61no—
228C4—181no—
229C4—181no—
230C4D3×#22611yescharacteristic
231C4D3×GL(2,𝔽3)11yescharacteristic
232C3—41no—
233C3—81no—
234C3#7305511yescharacteristic, complemented
235C2—62no—
236C2#1059318211yescharacteristic
237C2D3×#85911yescharacteristic
238C2#1059322811yescharacteristic
subgroupG/H
center, Frattini, hypercenterC2×C4D3×S4
commutatorC2×C3×SL(2,𝔽3)C2×C4
FittingC3×C4×Q8C2×D3
socleC22×C3C2×SL(2,ℤ/4)
prime radicalC2×D3×SL(2,𝔽3)C4
cosocleC3×C4×SL(2,𝔽3)C22
hypocenterC3×SL(2,𝔽3)M16
LCSG ▷ C2×C3×SL(2,𝔽3) ▷ C3×SL(2,𝔽3)
UCSC1 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
2. C3⋊(SL(2,𝔽3)⋊C8)C2semidirect
3. C3×SL(2,𝔽3)⋊C8C2semidirect
16. SL(2,𝔽3)⋊C8D3semidirect
53. C3×SL(2,𝔽3)M16semidirect
115. C3×Q8#208semidirect
135. SL(2,𝔽3)ΓL(1,𝔽25)semidirect
193. Q8#3711semidirect
234. C3#73055semidirect

Maximal subgroups

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

subgroupindex
C4×D3×SL(2,𝔽3)2
C3⋊(SL(2,𝔽3)⋊C8)2
C3×SL(2,𝔽3)⋊C82
#730553
#682363
C2×#37114