Finite Group Databasebeta

← #11920563

Subgroups — #11920563

Order 1152.

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

normal subgroups only

This group has 267 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)×D12C211yesmaximal, characteristic
2C3×#5386C211yesmaximal, characteristic, complemented
3C3⋊(SL(2,𝔽3)⋊C8)C211yesmaximal, characteristic, complemented
4#73087—31nomaximal
5#71127—31nomaximal
6C2×D3×SL(2,𝔽3)—21no—
7C3×C4×SL(2,𝔽3)C2211yescharacteristic
8C2×#3715—41nomaximal
9C3×#859—21no—
10Q8×D12D311yescharacteristic
11D4×SL(2,𝔽3)—31no—
12#5386D311yescharacteristic, complemented
13C3×#475—31no—
14#5083—31no—
15#5080—31no—
16SL(2,𝔽3)⋊C8—31no—
17SL(2,𝔽3)⋊C8—61no—
18#4535—31no—
19#4483—31no—
20#4482—31no—
21C2×C3×D12—41no—
22C2×C3×Dic6—41no—
23C2×C3×SL(2,𝔽3)D411yescharacteristic
24D3×SL(2,𝔽3)—41no—
25C2×C3⋊(C3⋊C8)—41no—
26#3715—44no—
27#1651—91noSylow
28C2×D3×Q8—21no—
29C22×SL(2,𝔽3)—61no—
30C3×C4⋊Q8—31no—
31C3×C4×Q8C2×D311yescharacteristic
32C2×#215—61no—
33C2×#214—121no—
34#897—61no—
35C4×D12—31no—
36#869—31no—
37C4×SL(2,𝔽3)—21no—
38C4×SL(2,𝔽3)C2×D311yescharacteristic
39#859—21no—
40C3×#106—61no—
41C3×C4⋊C8—31no—
42C3×#103—61no—
43#835—61no—
44#808—62no—
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—
56D4×Q8—31no—
57#498—91no—
58#475—31no—
59#461—91no—
60#332—91no—
61Q8⋊C8—91no—
62D4⋊C8—91no—
63C2×C3×Q8—61no—
64C2×C3×Q8#6711yescharacteristic
65C2×C3×D4—121no—
66D3×Q8—41no—
67D3×Q8—121no—
68C2×D12S411yescharacteristic
69C2×C4×D3—61no—
70C2×Dic6—41no—
71C2×Dic6—61no—
72C2×SL(2,𝔽3)—21no—
73C2×SL(2,𝔽3)—121no—
74C2×SL(2,𝔽3)#6711yescharacteristic
75C2×C3×C8—61no—
76C3×C4⋊C4—31no—
77C3×C4⋊C4—61no—
78C3×C4⋊C4—61no—
79C3×C42—31no—
80#215—122no—
81#214—124no—
82(C2×D3)⋊C4—61no—
83C4⋊Dic3—32no—
84C4⋊Dic3—31no—
85C2×C3⋊C8—61no—
86C2×C3⋊C8—61no—
87C2×C3⋊C8—121no—
88C2×C3⋊C8—241no—
89C22×C32—41no—
90C2×C3×D3—82no—
91C2×C3×D3—161no—
92C2×C3×D3—161no—
93C32×C4—41no—
94C32×C4—41no—
95C3×Dic3—82no—
96C22×Q8—61no—
97C2×SD16—181no—
98C4⋊Q8—31no—
99C4⋊Q8—91no—
100C22⋊Q8—181no—
101C4×Q8D3211yescharacteristic
102C4×D4—91no—
103#106—61no—
104#106—182no—
105C4⋊C8—31no—
106C4⋊C8—91no—
107#103—61no—
108#103—181no—
109C4×C8—91no—
110C23×C3—241no—
111C22×D3—21no—
112C3×Q8—31no—
113C3×Q8—121no—
114C3×Q8—121no—
115C3×Q8#21411yescharacteristic, complemented
116C3×D4—122no—
117C3×D4—241no—
118C2×C3×C4—31no—
119C2×C3×C4—31no—
120C2×C3×C4—41no—
121C2×C3×C4—81no—
122C2×C3×C4—121no—
123C2×C3×C4C2×S411yescharacteristic
124C2×Dic3—61no—
125C2×Dic3—81no—
126D12—21no—
127D12GL(2,𝔽3)12yes—
128C4×D3—121no—
129C4×D3—121no—
130Dic6—42no—
131Dic6—81no—
132Dic6—121no—
133Dic6—121no—
134SL(2,𝔽3)—21no—
135SL(2,𝔽3)#21511yescharacteristic, complemented
136C3×C8—121no—
137C3⋊C8—62no—
138C3⋊C8—122no—
139C3⋊C8—121no—
140C3⋊C8—242no—
141C2×C32—41no—
142C2×C32—41no—
143C2×C32—41no—
144C3×D3—162no—
145C2×Q8—61no—
146C2×Q8—121no—
147C2×Q8—181no—
148C2×Q8—361no—
149C2×Q8#62511yescharacteristic
150C2×D4—31no—
151C22×C4—181no—
152SD16—362no—
153C2×C8—61no—
154C2×C8—181no—
155C2×C8—181no—
156C4⋊C4—31no—
157C4⋊C4—61no—
158C4⋊C4—61no—
159C4⋊C4—92no—
160C4⋊C4—91no—
161C22⋊C4—181no—
162C42—31no—
163C22×C3—41no—
164C22×C3—81no—
165C22×C3—242no—
166C22×C3—481no—
167C22×C3—481no—
168C22×C3GL(2,ℤ/4)11yescharacteristic
169C2×D3—22no—
170C2×D3—41no—
171C2×D3—41no—
172C3×C4—31no—
173C3×C4—31no—
174C3×C4—41no—
175C3×C4—41no—
176C3×C4—61no—
177C3×C4—81no—
178C3×C4—81no—
179C3×C4—241no—
180C3×C4C2×GL(2,𝔽3)11yescharacteristic
181C3×C4#98311yescharacteristic
182Dic3—82no—
183Dic3—121no—
184C32—41noSylow
185C23—61no—
186Q8—31no—
187Q8—121no—
188Q8—121no—
189Q8—361no—
190Q8—361no—
191Q8#371511yescharacteristic, complemented
192D4—32no—
193D4—61no—
194C2×C4—31no—
195C2×C4—31no—
196C2×C4—121no—
197C2×C4—181no—
198C2×C4—361no—
199C2×C4—361no—
200C2×C4D3×S411yescharacteristic
201C8—121no—
202C8—182no—
203C8—361no—
204C2×C3—41no—
205C2×C3—41no—
206C2×C3—41no—
207C2×C3—81no—
208C2×C3—81no—
209C2×C3—81no—
210C2×C3—482no—
211C2×C3#541411yescharacteristic
212C2×C3#542111yescharacteristic
213C2×C3#543011yescharacteristic
214D3—42no—
215C22—62no—
216C22—121no—
217C22—121no—
218C22#6420911yescharacteristic
219C4—31no—
220C4—31no—
221C4—61no—
222C4—241no—
223C4—361no—
224C4#6420111yescharacteristic
225C4D3×GL(2,𝔽3)11yescharacteristic
226C3—41no—
227C3—81no—
228C3#7308711yescharacteristic, complemented
229C2—122no—
230C2#1059318511yescharacteristic
231C2#1059321711yescharacteristic
232C2#1059323111yescharacteristic
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 radicalSL(2,𝔽3)×D12C2
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
2. C3×#5386C2semidirect
3. C3⋊(SL(2,𝔽3)⋊C8)C2semidirect
12. #5386D3semidirect
54. C3×SL(2,𝔽3)SD16semidirect
115. C3×Q8#214semidirect
135. SL(2,𝔽3)#215semidirect
191. Q8#3715semidirect
228. C3#73087semidirect

Maximal subgroups

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

subgroupindex
SL(2,𝔽3)×D122
C3×#53862
C3⋊(SL(2,𝔽3)⋊C8)2
#730873
#711273
C2×#37154