Finite Group Databasebeta

← #83688⋊C3 #11921309

Subgroups — #83688⋊C3 #11921309

Order 1152.

259 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 259 conjugacy classes of subgroups — too many to draw. Only its 45 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×#557⋊C3C211yesmaximal, characteristic, complemented
2SL(2,𝔽3)×D12C211yesmaximal, characteristic
3C3⋊C8×SL(2,𝔽3)C211yesmaximal, characteristic, complemented
4#83688C311yesmaximal, characteristic, complemented
5#3359⋊C3—31nomaximal
6C2×D3×SL(2,𝔽3)—21no—
7C3×C4×SL(2,𝔽3)—21no—
8C3×C4×SL(2,𝔽3)C2211yescharacteristic
9C3×#809—41nomaximal
10C3×#557C2×C311yescharacteristic, complemented
11Q8×D12C2×C311yescharacteristic
12#557⋊C3—21no—
13#557⋊C3D311yescharacteristic, complemented
14D4×SL(2,𝔽3)—31no—
15#5083—31no—
16#5079—31no—
17C4×#215—31no—
18Q8×C3⋊C8C2×C311yescharacteristic, complemented
19#5020—31no—
20C8×SL(2,𝔽3)—31no—
21C2×C3×D12—41no—
22C2×C3×SL(2,𝔽3)D411yescharacteristic
23D3×SL(2,𝔽3)—41no—
24C32×C4⋊C4—41no—
25C2×C3×C3⋊C8—41no—
26#3359—31noSylow
27C2×D3×Q8—21no—
28C22×SL(2,𝔽3)—61no—
29C3×C4⋊Q8—31no—
30C3×#125—61no—
31C3×C4×Q8—21no—
32C3×C4×Q8—31no—
33C3×C4×Q8C22×C311yescharacteristic
34C2×#215—31no—
35#897—61no—
36C4×D12—31no—
37#869—31no—
38C4×SL(2,𝔽3)—22no—
39C4×SL(2,𝔽3)—21no—
40C4×SL(2,𝔽3)—21no—
41C4×SL(2,𝔽3)C2×D311yescharacteristic
42C3×#102—121no—
43#835—32no—
44#835—31no—
45#809A411yescharacteristic
46#808—31no—
47C4⋊(C3⋊C8)—31no—
48C4×C3⋊C8—31no—
49C22×C3×D3—81no—
50C2×C32×C4—41no—
51C2×C32×C4—81no—
52C3×D12—42no—
53C3×D12—81no—
54C3×SL(2,𝔽3)SD1611yescharacteristic, complemented
55C3×C3⋊C8—81no—
56#557C3×D311yescharacteristic, complemented
57D4×Q8—31no—
58#487—91no—
59#475—91no—
60#461—91no—
61C8×Q8—31no—
62C4×SD16—91no—
63C2×C3×Q8—31no—
64C2×C3×Q8C3×D411yescharacteristic
65C2×C3×D4—121no—
66D3×Q8—41no—
67D3×Q8—121no—
68C2×D12C2×A411yescharacteristic
69C2×C4×D3—61no—
70C2×Dic6—61no—
71C2×SL(2,𝔽3)—21no—
72C2×SL(2,𝔽3)—121no—
73C2×SL(2,𝔽3)#6711yescharacteristic
74C2×C3×C8—121no—
75C3×C4⋊C4—31no—
76C3×C4⋊C4—41no—
77C3×C4⋊C4—61no—
78C3×C4⋊C4—61no—
79C3×C4⋊C4—61no—
80C3×C4⋊C4—81no—
81C3×C4⋊C4C2×A411yescharacteristic
82C3×C42—31no—
83C3×C42—61no—
84#215—62no—
85(C2×D3)⋊C4—61no—
86C4⋊Dic3—31no—
87C4⋊Dic3—61no—
88C2×C3⋊C8—31no—
89C2×C3⋊C8C2×A411yescharacteristic
90C22×C32—41no—
91C2×C3×D3—82no—
92C2×C3×D3—161no—
93C2×C3×D3—161no—
94C32×C4—41no—
95C32×C4—41no—
96C32×C4—161no—
97C22×Q8—61no—
98C2×SD16—91no—
99C4⋊Q8—31no—
100C4⋊Q8—91no—
101#125—61no—
102C22⋊Q8—181no—
103C4×Q8—21no—
104C4×Q8—31no—
105C4×Q8C2×C3×D311yescharacteristic
106C4×D4—91no—
107#106—91no—
108C4⋊C8—91no—
109#103—91no—
110#103—92no—
111#102—31no—
112C4×C8—91no—
113C23×C3—241no—
114C22×D3—21no—
115C3×Q8—31no—
116C3×Q8—61no—
117C3×Q8—61no—
118C3×Q8C3×SD1611yescharacteristic
119C3×D4—122no—
120C3×D4—241no—
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×C4C22×A411yescharacteristic
130C2×Dic3—61no—
131D12—21no—
132D12#41⋊C312yes—
133C4×D3—121no—
134C4×D3—121no—
135Dic6—121no—
136Dic6—121no—
137SL(2,𝔽3)—21no—
138SL(2,𝔽3)#21511yescharacteristic, complemented
139C3×C8—241no—
140C3⋊C8—21no—
141C3⋊C8—61no—
142C2×C32—41no—
143C2×C32—41no—
144C2×C32—41no—
145C3×D3—162no—
146C2×Q8—31no—
147C2×Q8—121no—
148C2×Q8—181no—
149C2×Q8—361no—
150C2×Q8C3×#6711yescharacteristic
151C2×D4—31no—
152C22×C4—181no—
153SD16—182no—
154C2×C8—31no—
155C2×C8—91no—
156C4⋊C4—31no—
157C4⋊C4—61no—
158C4⋊C4—61no—
159C4⋊C4—61no—
160C4⋊C4—91no—
161C4⋊C4—181no—
162C4⋊C4D3×A411yescharacteristic
163C22⋊C4—181no—
164C42—31no—
165C42—61no—
166C22×C3—41no—
167C22×C3—81no—
168C22×C3—242no—
169C22×C3—481no—
170C22×C3—481no—
171C22×C3D4×A411yescharacteristic
172C2×D3—22no—
173C2×D3—41no—
174C2×D3—41no—
175C3×C4—31no—
176C3×C4—31no—
177C3×C4—41no—
178C3×C4—41no—
179C3×C4—41no—
180C3×C4—61no—
181C3×C4—81no—
182C3×C4—81no—
183C3×C4—121no—
184C3×C4—161no—
185C3×C4—162no—
186C3×C4C2×#41⋊C311yescharacteristic
187C3×C42−1+4⋊C311yescharacteristic
188Dic3—121no—
189C32—41noSylow
190C23—61no—
191Q8—31no—
192Q8—61no—
193Q8—61no—
194Q8—361no—
195Q8—361no—
196Q8C3×#21511yescharacteristic
197D4—32no—
198D4—61no—
199C2×C4—21no—
200C2×C4—31no—
201C2×C4—31no—
202C2×C4—61no—
203C2×C4—181no—
204C2×C4—361no—
205C2×C4—361no—
206C2×C4C2×D3×A411yescharacteristic
207C8—61no—
208C8—181no—
209C2×C3—41no—
210C2×C3—41no—
211C2×C3—41no—
212C2×C3—81no—
213C2×C3—81no—
214C2×C3—81no—
215C2×C3—482no—
216C2×C3#548⋊C311yescharacteristic
217C2×C3A4×SD1611yescharacteristic
218C2×C3#578⋊C311yescharacteristic
219D3—42no—
220C22—62no—
221C22—121no—
222C22—121no—
223C22A4×#6711yescharacteristic
224C4—31no—
225C4—31no—
226C4—41no—
227C4—61no—
228C4—121no—
229C4—361no—
230C4D3×#41⋊C311yescharacteristic
231C4#1010⋊C311yescharacteristic
232C3—41no—
233C3—81no—
234C3#3359⋊C311yescharacteristic, complemented
235C2—122no—
236C2A4×#21511yescharacteristic
237C2#5831⋊C311yescharacteristic
238C2#5838⋊C311yescharacteristic
subgroupG/H
centerC22A4×#67
commutator, cosocleC3×C4×Q8C22×C3
FrattiniC2×C4C2×D3×A4
FittingC3×#557C2×C3
socleC22×C3D4×A4
prime radicalSL(2,𝔽3)×D12C2
hypercenterC4⋊C4D3×A4
hypocenterC3×Q8C3×SD16
LCSG ▷ C3×C4×Q8 ▷ C2×C3×Q8 ▷ C3×Q8
UCSC1 ◁ C22 ◁ C2×C4 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
1. C3×#557⋊C3C2semidirect
3. C3⋊C8×SL(2,𝔽3)C2semidirect
4. #83688C3semidirect
10. C3×#557C2×C3semidirect
13. #557⋊C3D3semidirect
18. Q8×C3⋊C8C2×C3semidirect
54. C3×SL(2,𝔽3)SD16semidirect
56. #557C3×D3semidirect
138. SL(2,𝔽3)#215semidirect
234. C3#3359⋊C3semidirect

Maximal subgroups

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

subgroupindex
C3×#557⋊C32
SL(2,𝔽3)×D122
C3⋊C8×SL(2,𝔽3)2
#836883
#3359⋊C33
C3×#8094