Finite Group Databasebeta

← #11920565

Subgroups — #11920565

Order 1152.

277 conjugacy classes of subgroups; only the 37 normal subgroups are shown.

normal subgroups only

This group has 277 conjugacy classes of subgroups — too many to draw. Only its 37 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
1SL(2,𝔽3)×Dic6C211yesmaximal, characteristic, complemented
2C3×C4⋊GL(2,𝔽3)C211yesmaximal, characteristic
3C3⋊(SL(2,𝔽3)⋊C8)C211yesmaximal, characteristic, complemented
4#73100—31nomaximal
5#71033—31nomaximal
6C2×C3×GL(2,𝔽3)—21no—
7C3×C4×SL(2,𝔽3)C2211yescharacteristic
8C2×#3715—41nomaximal
9Dic3×SL(2,𝔽3)—21no—
10Q8×Dic6D311yescharacteristic, complemented
11Q8×SL(2,𝔽3)—31no—
12C4⋊GL(2,𝔽3)D311yescharacteristic
13C3×#460—31no—
14#5076—31no—
15#5075—31no—
16SL(2,𝔽3)⋊C8—31no—
17SL(2,𝔽3)⋊C8—61no—
18#4535—31no—
19#4484—31no—
20#4481—31no—
21C2×C3×D12—41no—
22C2×C3×Dic6—41no—
23C2×C3×SL(2,𝔽3)D411yescharacteristic
24C3×GL(2,𝔽3)—41no—
25C2×C3⋊(C3⋊C8)—41no—
26#3715—44no—
27#1623—91noSylow
28C2×GL(2,𝔽3)—21no—
29C2×C3×SD16—61no—
30C3×#127—31no—
31C3×C4×Q8C2×D311yescharacteristic
32Q8×Dic3—21no—
33C2×#215—121no—
34C2×#214—62no—
35C2×#214—61no—
36#888—61no—
37#869—31no—
38C4×Dic6—31no—
39C4×SL(2,𝔽3)—21no—
40C4×SL(2,𝔽3)—61no—
41C4×SL(2,𝔽3)C2×D311yescharacteristic
42C3×C4⋊C8—31no—
43C3×#102—61no—
44#832—61no—
45C4⋊(C3⋊C8)—31no—
46C4×C3⋊C8—31no—
47C22×C3×D3—81no—
48C2×C32×C4—41no—
49C2×C3×Dic3—81no—
50C3×D12—42no—
51C3×D12—81no—
52C3×Dic6—42no—
53C3×Dic6—81no—
54C3×SL(2,𝔽3)SD1611yescharacteristic, complemented
55C3⋊(C3⋊C8)—42no—
56Q82—31no—
57#492—91no—
58#460—31no—
59#460—91no—
60#333—91no—
61Q8⋊C8—91no—
62Q8⋊C8—91no—
63C2×C3×Q8—121no—
64C2×C3×Q8#6711yescharacteristic
65C2×C3×D4—61no—
66C2×C3×D4—61no—
67C2×C3×D4—61no—
68C2×D12—41no—
69C2×Dic6—32no—
70C2×Dic6S411yescharacteristic
71C2×SL(2,𝔽3)—21no—
72C2×SL(2,𝔽3)#6711yescharacteristic
73GL(2,𝔽3)—41no—
74C3×SD16—121no—
75C3×SD16—121no—
76C2×C3×C8—61no—
77C3×C4⋊C4—31no—
78C3×C42—31no—
79#215—124no—
80#214—124no—
81#214—122no—
82C4⋊Dic3—31no—
83C4⋊Dic3—61no—
84Dic3⋊C4—61no—
85C4×Dic3—61no—
86C2×C3⋊C8—61no—
87C2×C3⋊C8—61no—
88C2×C3⋊C8—121no—
89C2×C3⋊C8—241no—
90C22×C32—41no—
91C2×C3×D3—82no—
92C2×C3×D3—161no—
93C2×C3×D3—161no—
94C32×C4—41no—
95C32×C4—41no—
96C3×Dic3—82no—
97C2×SD16—61no—
98C2×SD16—181no—
99C2×SD16—182no—
100C4⋊Q8—91no—
101C4⋊Q8—181no—
102#127—31no—
103C4×Q8—61no—
104C4×Q8—91no—
105C4×Q8D3211yescharacteristic
106C4⋊C8—31no—
107C4⋊C8—91no—
108#102—61no—
109#102—181no—
110C4×C8—91no—
111C23×C3—121no—
112C22×D3—81no—
113C3×Q8—31no—
114C3×Q8—122no—
115C3×Q8—241no—
116C3×Q8#21511yescharacteristic, complemented
117C3×D4—121no—
118C3×D4—121no—
119C3×D4—122no—
120C3×D4—121no—
121C3×D4—121no—
122C2×C3×C4—31no—
123C2×C3×C4—31no—
124C2×C3×C4—41no—
125C2×C3×C4—81no—
126C2×C3×C4—241no—
127C2×C3×C4C2×S411yescharacteristic
128C2×Dic3—21no—
129C2×Dic3—61no—
130D12—42no—
131D12—81no—
132Dic6—21no—
133Dic6—122no—
134Dic6GL(2,𝔽3)12yescomplemented
135SL(2,𝔽3)—21no—
136SL(2,𝔽3)#21411yescharacteristic, complemented
137C3×C8—121no—
138C3⋊C8—62no—
139C3⋊C8—122no—
140C3⋊C8—121no—
141C3⋊C8—242no—
142C2×C32—41no—
143C2×C32—41no—
144C2×C32—41no—
145C3×D3—162no—
146C2×Q8—31no—
147C2×Q8—92no—
148C2×Q8#62511yescharacteristic
149C2×D4—61no—
150C2×D4—61no—
151C2×D4—61no—
152SD16—121no—
153SD16—121no—
154SD16—364no—
155SD16—362no—
156C2×C8—61no—
157C2×C8—181no—
158C2×C8—181no—
159C4⋊C4—31no—
160C4⋊C4—91no—
161C4⋊C4—181no—
162C4⋊C4—181no—
163C42—31no—
164C42—181no—
165C22×C3—41no—
166C22×C3—81no—
167C22×C3—241no—
168C22×C3—241no—
169C22×C3—241no—
170C22×C3GL(2,ℤ/4)11yescharacteristic
171C2×D3—82no—
172C2×D3—161no—
173C2×D3—161no—
174C3×C4—31no—
175C3×C4—31no—
176C3×C4—41no—
177C3×C4—41no—
178C3×C4—61no—
179C3×C4—81no—
180C3×C4—81no—
181C3×C4—242no—
182C3×C4C2×GL(2,𝔽3)11yescharacteristic
183C3×C4#98311yescharacteristic
184Dic3—22no—
185Dic3—121no—
186C32—41noSylow
187C23—121no—
188Q8—31no—
189Q8—32no—
190Q8—61no—
191Q8—362no—
192Q8#371511yescharacteristic, complemented
193D4—121no—
194D4—121no—
195D4—122no—
196D4—121no—
197D4—121no—
198C2×C4—31no—
199C2×C4—31no—
200C2×C4—61no—
201C2×C4—181no—
202C2×C4D3×S411yescharacteristic
203C8—121no—
204C8—182no—
205C8—361no—
206C2×C3—41no—
207C2×C3—41no—
208C2×C3—41no—
209C2×C3—81no—
210C2×C3—81no—
211C2×C3—81no—
212C2×C3—481no—
213C2×C3#541711yescharacteristic
214C2×C3#542111yescharacteristic
215C2×C3#542911yescharacteristic
216D3—162no—
217C22—241no—
218C22—241no—
219C22—241no—
220C22#6420911yescharacteristic
221C4—31no—
222C4—31no—
223C4—62no—
224C4—61no—
225C4—361no—
226C4#6420111yescharacteristic
227C4D3×GL(2,𝔽3)11yescharacteristic
228C3—41no—
229C3—81no—
230C3#7310011yescharacteristic, complemented
231C2—481no—
232C2#1059318711yescharacteristic
233C2#1059321711yescharacteristic
234C2#1059323311yescharacteristic
subgroupG/H
centerC22#64209
commutator, cosocleC3×C4×SL(2,𝔽3)C22
Frattini, hypercenterC2×C4D3×S4
FittingC3×C4×Q8C2×D3
socleC22×C3GL(2,ℤ/4)
prime radicalC3×C4⋊GL(2,𝔽3)C2
hypocenterC3×SL(2,𝔽3)SD16
LCSG ▷ C3×C4×SL(2,𝔽3) ▷ C2×C3×SL(2,𝔽3) ▷ C3×SL(2,𝔽3)
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
1. SL(2,𝔽3)×Dic6C2semidirect
3. C3⋊(SL(2,𝔽3)⋊C8)C2semidirect
10. Q8×Dic6D3semidirect
54. C3×SL(2,𝔽3)SD16semidirect
116. C3×Q8#215semidirect
134. Dic6GL(2,𝔽3)semidirect
136. SL(2,𝔽3)#214semidirect
192. Q8#3715semidirect
230. C3#73100semidirect

Maximal subgroups

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

subgroupindex
SL(2,𝔽3)×Dic62
C3×C4⋊GL(2,𝔽3)2
C3⋊(SL(2,𝔽3)⋊C8)2
#731003
#710333
C2×#37154