Finite Group Databasebeta

← #83703⋊C3 #11921311

Subgroups — #83703⋊C3 #11921311

Order 1152.

259 conjugacy classes of subgroups; only the 47 normal subgroups are shown.

normal subgroups only

This group has 259 conjugacy classes of subgroups — too many to draw. Only its 47 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
1C3×Q8×SL(2,𝔽3)C211yesmaximal, characteristic, complemented
2C4⋊(C3⋊(#41⋊C3))C211yesmaximal, characteristic
3C3⋊C8×SL(2,𝔽3)C211yesmaximal, characteristic, complemented
4#83703C311yesmaximal, characteristic, complemented
5#3362⋊C3—31nomaximal
6C2×C3⋊(#41⋊C3)—21no—
7C3×C4×SL(2,𝔽3)—21no—
8C3×C4×SL(2,𝔽3)C2211yescharacteristic
9C3×#835—41nomaximal
10C3×Q82C2×C311yescharacteristic
11#5577C2×C311yescharacteristic
12Q8×SL(2,𝔽3)—21no—
13Q8×SL(2,𝔽3)D311yescharacteristic, complemented
14C4⋊(#41⋊C3)—31no—
15#5088—31no—
16#5084—31no—
17#5027—31no—
18C4×#215—31no—
19Q8×C3⋊C8C2×C311yescharacteristic
20C8×SL(2,𝔽3)—31no—
21C2×C32×Q8—41no—
22C2×C3×SL(2,𝔽3)D411yescharacteristic
23C3⋊(#41⋊C3)—41no—
24C3×C4⋊Dic3—41no—
25C2×C3×C3⋊C8—41no—
26#3362—31noSylow
27C2×#239—21no—
28C2×#41⋊C3—61no—
29C3×C4⋊Q8—31no—
30C3×C4⋊Q8—61no—
31C3×C4×Q8—21no—
32C3×C4×Q8—31no—
33C3×C4×Q8C22×C311yescharacteristic
34C2×#215—31no—
35#895—61no—
36#874—31no—
37C4×D12—31no—
38C4×SL(2,𝔽3)—22no—
39C4×SL(2,𝔽3)—21no—
40C4×SL(2,𝔽3)—21no—
41C4×SL(2,𝔽3)C2×D311yescharacteristic
42C3×#103—121no—
43#835A411yescharacteristic
44#809—32no—
45#809—31no—
46#808—31no—
47C4⋊(C3⋊C8)—31no—
48C4×C3⋊C8—31no—
49C32×Q8—42no—
50C32×Q8—81no—
51C2×C32×C4—41no—
52C2×C32×C4—81no—
53C2×C3×Dic3—81no—
54C3×SL(2,𝔽3)SD1611yescharacteristic, complemented
55C3×C3⋊C8—81no—
56Q82C3×D311yescharacteristic
57#550—31no—
58#486—91no—
59#476—91no—
60#460—91no—
61C8×Q8—31no—
62C4×SD16—91no—
63C2×C3×Q8—41no—
64C2×C3×Q8—61no—
65C2×C3×Q8—81no—
66C2×C3×Q8C3×D411yescharacteristic
67C2×C3×Q8C2×A411yescharacteristic
68#239—41no—
69#239—121no—
70C2×D12—31no—
71C2×D12—61no—
72C2×D12—61no—
73C2×C4×D3—61no—
74#41⋊C3—121no—
75C2×SL(2,𝔽3)—21no—
76C2×SL(2,𝔽3)#6711yescharacteristic
77C2×C3×C8—121no—
78C3×C4⋊C4—31no—
79C3×C4⋊C4—31no—
80C3×C4⋊C4—61no—
81C3×C4⋊C4—61no—
82C3×C4⋊C4—121no—
83C3×C42—31no—
84C3×C42—61no—
85#215—62no—
86(C2×D3)⋊C4—61no—
87C4⋊Dic3C2×A411yescharacteristic
88C2×C3⋊C8—31no—
89C2×C3⋊C8C2×A411yescharacteristic
90C22×C32—41no—
91C32×C4—41no—
92C32×C4—41no—
93C32×C4—82no—
94C3×Dic3—161no—
95C2×#41—61no—
96C2×SD16—91no—
97C4⋊Q8—31no—
98C4⋊Q8—61no—
99#127—91no—
100#121—181no—
101C4×Q8—21no—
102C4×Q8—31no—
103C4×Q8C2×C3×D311yescharacteristic
104C4×D4—91no—
105#106—91no—
106C4⋊C8—91no—
107#103—31no—
108#102—91no—
109#102—92no—
110C4×C8—91no—
111C22×D3—61no—
112C3×Q8—21no—
113C3×Q8—31no—
114C3×Q8—42no—
115C3×Q8—81no—
116C3×Q8—82no—
117C3×Q8—82no—
118C3×Q8—241no—
119C3×Q8C3×SD1611yescharacteristic
120C3×Q8#41⋊C312yescomplemented
121C2×C3×C4—21no—
122C2×C3×C4—31no—
123C2×C3×C4—31no—
124C2×C3×C4—41no—
125C2×C3×C4—61no—
126C2×C3×C4—81no—
127C2×C3×C4—82no—
128C2×C3×C4—81no—
129C2×C3×C4—241no—
130C2×C3×C4C22×A411yescharacteristic
131C2×Dic3—21no—
132D12—61no—
133D12—61no—
134D12—121no—
135D12—121no—
136D12—241no—
137C4×D3—121no—
138C4×D3—121no—
139SL(2,𝔽3)—21no—
140SL(2,𝔽3)#21511yescharacteristic, complemented
141C3×C8—241no—
142C3⋊C8—21no—
143C3⋊C8—61no—
144C2×C32—41no—
145C2×C32—41no—
146C2×C32—41no—
147#41—121no—
148#41—361no—
149C2×Q8—61no—
150C2×Q8C3×#6711yescharacteristic
151C2×Q8D3×A411yescharacteristic
152C2×D4—91no—
153C2×D4—181no—
154C2×D4—181no—
155C22×C4—181no—
156SD16—182no—
157C2×C8—31no—
158C2×C8—91no—
159C4⋊C4—31no—
160C4⋊C4—31no—
161C4⋊C4—31no—
162C4⋊C4—61no—
163C4⋊C4—61no—
164C22⋊C4—181no—
165C42—31no—
166C42—61no—
167C22×C3—41no—
168C22×C3—81no—
169C22×C3D4×A411yescharacteristic
170C2×D3—121no—
171C2×D3—121no—
172C2×D3—121no—
173C3×C4—22no—
174C3×C4—31no—
175C3×C4—31no—
176C3×C4—41no—
177C3×C4—41no—
178C3×C4—61no—
179C3×C4—82no—
180C3×C4—84no—
181C3×C4—81no—
182C3×C4—81no—
183C3×C4—121no—
184C3×C4—481no—
185C3×C4C2×#41⋊C311yescharacteristic
186C3×C42−1+4⋊C311yescharacteristic
187Dic3—41no—
188C32—41noSylow
189C23—181no—
190Q8—21no—
191Q8—31no—
192Q8—241no—
193Q8C3×#21511yescharacteristic
194Q8C3⋊(#41⋊C3)12yescomplemented
195D4—181no—
196D4—181no—
197D4—361no—
198D4—361no—
199D4—721no—
200C2×C4—21no—
201C2×C4—31no—
202C2×C4—31no—
203C2×C4—61no—
204C2×C4—61no—
205C2×C4—361no—
206C2×C4—361no—
207C2×C4C2×D3×A411yescharacteristic
208C8—61no—
209C8—181no—
210C2×C3—41no—
211C2×C3—41no—
212C2×C3—41no—
213C2×C3—81no—
214C2×C3—81no—
215C2×C3—81no—
216C2×C3#548⋊C311yescharacteristic
217C2×C3A4×SD1611yescharacteristic
218C2×C3#576⋊C311yescharacteristic
219D3—241no—
220C22—361no—
221C22—361no—
222C22—361no—
223C22A4×#6711yescharacteristic
224C4—22no—
225C4—31no—
226C4—31no—
227C4—61no—
228C4—121no—
229C4—121no—
230C4C2×C3⋊(#41⋊C3)11yescharacteristic
231C4#1007⋊C311yescharacteristic
232C3—41no—
233C3—81no—
234C3#3362⋊C311yescharacteristic, complemented
235C2—721no—
236C2A4×#21511yescharacteristic
237C2#5817⋊C311yescharacteristic
238C2#5835⋊C311yescharacteristic
subgroupG/H
centerC22A4×#67
commutator, cosocleC3×C4×Q8C22×C3
FrattiniC2×C4C2×D3×A4
FittingC3×Q82C2×C3
socleC22×C3D4×A4
prime radicalC4⋊(C3⋊(#41⋊C3))C2
hypercenterC2×Q8D3×A4
hypocenterC3×Q8C3×SD16
LCSG ▷ C3×C4×Q8 ▷ C2×C3×Q8 ▷ C3×Q8
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
1. C3×Q8×SL(2,𝔽3)C2semidirect
3. C3⋊C8×SL(2,𝔽3)C2semidirect
4. #83703C3semidirect
13. Q8×SL(2,𝔽3)D3semidirect
54. C3×SL(2,𝔽3)SD16semidirect
120. C3×Q8#41⋊C3semidirect
140. SL(2,𝔽3)#215semidirect
194. Q8C3⋊(#41⋊C3)semidirect
234. C3#3362⋊C3semidirect

Maximal subgroups

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

subgroupindex
C3×Q8×SL(2,𝔽3)2
C4⋊(C3⋊(#41⋊C3))2
C3⋊C8×SL(2,𝔽3)2
#837033
#3362⋊C33
C3×#8354