Finite Group Databasebeta

← #83586⋊C3 #11921306

Subgroups — #83586⋊C3 #11921306

Order 1152.

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

normal subgroups only

This group has 255 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×C4⋊(#41⋊C3)C211yesmaximal, characteristic
2SL(2,𝔽3)×Dic6C211yesmaximal, characteristic, complemented
3C3⋊C8×SL(2,𝔽3)C211yesmaximal, characteristic, complemented
4#83586C311yesmaximal, characteristic, complemented
5#3362⋊C3—31nomaximal
6C2×C3×#41⋊C3—21no—
7C3×C4×SL(2,𝔽3)C2211yescharacteristic
8Dic3×SL(2,𝔽3)—21no—
9C3×#810—41nomaximal
10C3×#550C2×C311yescharacteristic
11Q8×Dic6C2×C311yescharacteristic
12Q8×SL(2,𝔽3)—31no—
13C4⋊(#41⋊C3)—21no—
14C4⋊(#41⋊C3)D311yescharacteristic
15#5075—31no—
16#5070—31no—
17Q8×C3⋊C8C2×C311yescharacteristic
18C4×#214—31no—
19#5008—31no—
20C8×SL(2,𝔽3)—31no—
21C2×C3×Dic6—41no—
22C3×#41⋊C3—41no—
23C2×C3×SL(2,𝔽3)D411yescharacteristic
24C32×C4⋊C4—41no—
25C2×C3×C3⋊C8—41no—
26#3362—31noSylow
27C2×C3×#41—21no—
28C2×#41⋊C3—22no—
29C2×#41⋊C3—21no—
30C3×#127—31no—
31C3×#121—61no—
32C3×C4×Q8C22×C311yescharacteristic
33C3×C4×D4—31no—
34Q8×Dic3—21no—
35C2×#214—31no—
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×#103—121no—
43#832—32no—
44#832—31no—
45#810A411yescharacteristic
46#808—31no—
47C4⋊(C3⋊C8)—31no—
48C4×C3⋊C8—31no—
49C2×C32×C4—41no—
50C2×C32×C4—81no—
51C2×C3×Dic3—81no—
52C3×Dic6—42no—
53C3×Dic6—81no—
54C3×SL(2,𝔽3)SD1611yescharacteristic, complemented
55C3×C3⋊C8—81no—
56Q82—31no—
57#550C3×D311yescharacteristic
58#486—91no—
59#476—91no—
60#460—91no—
61C8×Q8—31no—
62C4×SD16—91no—
63C3×#41—41no—
64C3×#41—121no—
65C2×C3×Q8—121no—
66C2×C3×Q8C3×D411yescharacteristic
67C2×C3×D4—31no—
68C2×C3×D4—61no—
69C2×C3×D4—61no—
70C22×C3×C4—61no—
71C2×Dic6—61no—
72C2×Dic6C2×A411yescharacteristic
73#41⋊C3—42no—
74#41⋊C3—41no—
75C2×SL(2,𝔽3)—21no—
76C2×SL(2,𝔽3)#6711yescharacteristic
77C2×C3×C8—121no—
78C3×C4⋊C4—31no—
79C3×C4⋊C4—41no—
80C3×C4⋊C4—81no—
81C3×C4⋊C4C2×A411yescharacteristic
82C3×C22⋊C4—61no—
83C3×C42—31no—
84#214—62no—
85C4⋊Dic3—31no—
86C4⋊Dic3—61no—
87Dic3⋊C4—61no—
88C4×Dic3—61no—
89C2×C3⋊C8—31no—
90C2×C3⋊C8C2×A411yescharacteristic
91C22×C32—41no—
92C32×C4—41no—
93C32×C4—41no—
94C32×C4—161no—
95C3×Dic3—82no—
96C2×#41—21no—
97C2×SD16—91no—
98C4⋊Q8—91no—
99C4⋊Q8—181no—
100#127—31no—
101#121—61no—
102C4×Q8—61no—
103C4×Q8—91no—
104C4×Q8C2×C3×D311yescharacteristic
105C4×D4—31no—
106#106—91no—
107C4⋊C8—91no—
108#103—31no—
109#102—91no—
110#102—92no—
111C4×C8—91no—
112C23×C3—61no—
113C3×Q8—31no—
114C3×Q8—122no—
115C3×Q8—241no—
116C3×Q8C3×SD1611yescharacteristic
117C3×D4—61no—
118C3×D4—61no—
119C3×D4—121no—
120C3×D4—121no—
121C3×D4—241no—
122C2×C3×C4—21no—
123C2×C3×C4—31no—
124C2×C3×C4—31no—
125C2×C3×C4—41no—
126C2×C3×C4—81no—
127C2×C3×C4—82no—
128C2×C3×C4—81no—
129C2×C3×C4—121no—
130C2×C3×C4—121no—
131C2×C3×C4—241no—
132C2×C3×C4C22×A411yescharacteristic
133C2×Dic3—21no—
134C2×Dic3—61no—
135Dic6—21no—
136Dic6—241no—
137Dic6#41⋊C312yescomplemented
138SL(2,𝔽3)—21no—
139SL(2,𝔽3)#21411yescharacteristic, complemented
140C3×C8—241no—
141C3⋊C8—21no—
142C3⋊C8—61no—
143C2×C32—41no—
144C2×C32—41no—
145C2×C32—41no—
146#41—41no—
147#41—121no—
148C2×Q8—31no—
149C2×Q8—181no—
150C2×Q8C3×#6711yescharacteristic
151C2×D4—31no—
152C2×D4—61no—
153C2×D4—61no—
154C22×C4—61no—
155SD16—182no—
156C2×C8—31no—
157C2×C8—91no—
158C4⋊C4—31no—
159C4⋊C4—91no—
160C4⋊C4—181no—
161C4⋊C4—181no—
162C4⋊C4D3×A411yescharacteristic
163C22⋊C4—61no—
164C42—31no—
165C42—181no—
166C22×C3—41no—
167C22×C3—81no—
168C22×C3—121no—
169C22×C3—121no—
170C22×C3—121no—
171C22×C3D4×A411yescharacteristic
172C3×C4—31no—
173C3×C4—31no—
174C3×C4—41no—
175C3×C4—41no—
176C3×C4—41no—
177C3×C4—61no—
178C3×C4—81no—
179C3×C4—81no—
180C3×C4—161no—
181C3×C4—162no—
182C3×C4—242no—
183C3×C4C2×#41⋊C311yescharacteristic
184C3×C42−1+4⋊C311yescharacteristic
185Dic3—22no—
186Dic3—121no—
187C32—41noSylow
188C23—61no—
189Q8—31no—
190Q8—32no—
191Q8—61no—
192Q8—721no—
193Q8C3×#21411yescharacteristic
194D4—61no—
195D4—61no—
196D4—121no—
197D4—121no—
198D4—241no—
199C2×C4—21no—
200C2×C4—31no—
201C2×C4—31no—
202C2×C4—61no—
203C2×C4—121no—
204C2×C4—121no—
205C2×C4—181no—
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—241no—
216C2×C3#548⋊C311yescharacteristic
217C2×C3A4×SD1611yescharacteristic
218C2×C3#576⋊C311yescharacteristic
219C22—121no—
220C22—121no—
221C22—121no—
222C22A4×#6711yescharacteristic
223C4—31no—
224C4—31no—
225C4—41no—
226C4—62no—
227C4—61no—
228C4—361no—
229C4D3×#41⋊C311yescharacteristic
230C4#1010⋊C311yescharacteristic
231C3—41no—
232C3—81no—
233C3#3362⋊C311yescharacteristic, complemented
234C2—241no—
235C2A4×#21411yescharacteristic
236C2#5831⋊C311yescharacteristic
237C2#5835⋊C311yescharacteristic
subgroupG/H
centerC22A4×#67
commutator, cosocleC3×C4×Q8C22×C3
FrattiniC2×C4C2×D3×A4
FittingC3×#550C2×C3
socleC22×C3D4×A4
prime radicalC3×C4⋊(#41⋊C3)C2
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
2. SL(2,𝔽3)×Dic6C2semidirect
3. C3⋊C8×SL(2,𝔽3)C2semidirect
4. #83586C3semidirect
54. C3×SL(2,𝔽3)SD16semidirect
137. Dic6#41⋊C3semidirect
139. SL(2,𝔽3)#214semidirect
233. C3#3362⋊C3semidirect

Maximal subgroups

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

subgroupindex
C3×C4⋊(#41⋊C3)2
SL(2,𝔽3)×Dic62
C3⋊C8×SL(2,𝔽3)2
#835863
#3362⋊C33
C3×#8104