Finite Group Databasebeta

← #83588⋊C3 #11921304

Subgroups — #83588⋊C3 #11921304

Order 1152.

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

normal subgroups only

This group has 249 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
2#5570⋊C3C211yesmaximal, characteristic, complemented
3C3⋊C8×SL(2,𝔽3)C211yesmaximal, characteristic, complemented
4#83588C311yesmaximal, characteristic, complemented
5#3392⋊C3—31nomaximal
6C2×C3×#41⋊C3—21no—
7C3×C4×SL(2,𝔽3)C2211yescharacteristic
8Dic3×SL(2,𝔽3)—21no—
9C3×#808—41nomaximal
10C3×#550C2×C311yescharacteristic
11#5570C2×C311yescharacteristic
12#557⋊C3—31no—
13C4⋊(#41⋊C3)—21no—
14C4⋊(#41⋊C3)D311yescharacteristic
15#5076—31no—
16#5075—31no—
17Q8×C3⋊C8C2×C311yescharacteristic
18#5019—31no—
19C4×#214—31no—
20C8×SL(2,𝔽3)—31no—
21C3×#41⋊C3—41no—
22C2×C3×SL(2,𝔽3)D411yescharacteristic
23C32×C4⋊C4—41no—
24C3×C4⋊Dic3—41no—
25C2×C3×C3⋊C8—41no—
26#3392—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—32no—
36C2×#214—31no—
37#890—61no—
38#869—31no—
39C4×Dic6—31no—
40C4×SL(2,𝔽3)—21no—
41C4×SL(2,𝔽3)—61no—
42C4×SL(2,𝔽3)C2×D311yescharacteristic
43C3×#106—121no—
44#832—31no—
45#810—31no—
46#808A411yescharacteristic
47C4⋊(C3⋊C8)—31no—
48C4×C3⋊C8—31no—
49C2×C32×C4—41no—
50C2×C32×C4—81no—
51C2×C3×Dic3—81no—
52C3×SL(2,𝔽3)SD1611yescharacteristic, complemented
53C3×C3⋊C8—42no—
54#557—31no—
55#550C3×D311yescharacteristic
56#492—91no—
57#461—91no—
58#460—91no—
59C8×Q8—31no—
60C4×SD16—91no—
61C3×#41—41no—
62C3×#41—121no—
63C2×C3×Q8C3×D411yescharacteristic
64C2×C3×D4—31no—
65C2×C3×D4—61no—
66C2×C3×D4—61no—
67C22×C3×C4—61no—
68C2×Dic6—31no—
69#41⋊C3—42no—
70#41⋊C3—41no—
71C2×SL(2,𝔽3)—21no—
72C2×SL(2,𝔽3)#6711yescharacteristic
73C2×C3×C8—121no—
74C3×C4⋊C4—31no—
75C3×C4⋊C4—41no—
76C3×C4⋊C4—81no—
77C3×C4⋊C4—121no—
78C3×C4⋊C4C2×A411yescharacteristic
79C3×C22⋊C4—61no—
80C3×C42—31no—
81#214—62no—
82#214—122no—
83C4⋊Dic3—61no—
84C4⋊Dic3—61no—
85C4⋊Dic3C2×A411yescharacteristic
86Dic3⋊C4—61no—
87C4×Dic3—61no—
88C2×C3⋊C8—31no—
89C2×C3⋊C8C2×A411yescharacteristic
90C22×C32—41no—
91C32×C4—41no—
92C32×C4—41no—
93C32×C4—161no—
94C3×Dic3—161no—
95C2×#41—21no—
96C2×SD16—91no—
97C2×SD16—92no—
98C4⋊Q8—91no—
99#127—31no—
100#125—181no—
101#121—61no—
102C4×Q8—61no—
103C4×Q8—91no—
104C4×Q8C2×C3×D311yescharacteristic
105C4×D4—31no—
106#106—31no—
107C4⋊C8—91no—
108#103—91no—
109#102—91no—
110C4×C8—91no—
111C23×C3—61no—
112C3×Q8—31no—
113C3×Q8C3×SD1611yescharacteristic
114C3×D4—61no—
115C3×D4—61no—
116C3×D4—121no—
117C3×D4—121no—
118C3×D4—241no—
119C2×C3×C4—21no—
120C2×C3×C4—31no—
121C2×C3×C4—31no—
122C2×C3×C4—41no—
123C2×C3×C4—81no—
124C2×C3×C4—82no—
125C2×C3×C4—81no—
126C2×C3×C4—121no—
127C2×C3×C4—121no—
128C2×C3×C4—241no—
129C2×C3×C4C22×A411yescharacteristic
130C2×Dic3—21no—
131C2×Dic3—61no—
132Dic6—61no—
133Dic6—61no—
134SL(2,𝔽3)—21no—
135SL(2,𝔽3)#21411yescharacteristic, complemented
136C3×C8—122no—
137C3⋊C8—61no—
138C3⋊C8#41⋊C312yescomplemented
139C2×C32—41no—
140C2×C32—41no—
141C2×C32—41no—
142#41—41no—
143#41—121no—
144C2×Q8—91no—
145C2×Q8C3×#6711yescharacteristic
146C2×D4—31no—
147C2×D4—61no—
148C2×D4—61no—
149C22×C4—61no—
150SD16—182no—
151SD16—362no—
152C2×C8—31no—
153C2×C8—91no—
154C4⋊C4—31no—
155C4⋊C4—31no—
156C4⋊C4—181no—
157C4⋊C4—181no—
158C4⋊C4—181no—
159C4⋊C4D3×A411yescharacteristic
160C22⋊C4—61no—
161C42—31no—
162C42—181no—
163C22×C3—41no—
164C22×C3—81no—
165C22×C3—121no—
166C22×C3—121no—
167C22×C3—121no—
168C22×C3D4×A411yescharacteristic
169C3×C4—31no—
170C3×C4—31no—
171C3×C4—41no—
172C3×C4—41no—
173C3×C4—41no—
174C3×C4—61no—
175C3×C4—81no—
176C3×C4—81no—
177C3×C4—161no—
178C3×C4—162no—
179C3×C4—481no—
180C3×C4C2×#41⋊C311yescharacteristic
181C3×C4#99411yescharacteristic
182Dic3—41no—
183Dic3—121no—
184C32—41noSylow
185C23—61no—
186Q8—31no—
187Q8—181no—
188Q8—181no—
189Q8C3×#21411yescharacteristic
190D4—61no—
191D4—61no—
192D4—121no—
193D4—121no—
194D4—241no—
195C2×C4—21no—
196C2×C4—31no—
197C2×C4—31no—
198C2×C4—61no—
199C2×C4—121no—
200C2×C4—121no—
201C2×C4—181no—
202C2×C4C2×D3×A411yescharacteristic
203C8—32no—
204C8—181no—
205C2×C3—41no—
206C2×C3—41no—
207C2×C3—41no—
208C2×C3—81no—
209C2×C3—81no—
210C2×C3—81no—
211C2×C3—241no—
212C2×C3C4⋊(#41⋊C3)11yescharacteristic
213C2×C3A4×SD1611yescharacteristic
214C2×C3#577⋊C311yescharacteristic
215C22—121no—
216C22—121no—
217C22—121no—
218C22A4×#6711yescharacteristic
219C4—31no—
220C4—31no—
221C4—41no—
222C4—61no—
223C4—121no—
224C4—361no—
225C4C3⋊#99411yescharacteristic
226C4D3×#41⋊C311yescharacteristic
227C3—41no—
228C3—81no—
229C3#3392⋊C311yescharacteristic, complemented
230C2—241no—
231C2A4×#21411yescharacteristic
232C2#5834⋊C311yescharacteristic
233C2#5836⋊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. #5570⋊C3C2semidirect
3. C3⋊C8×SL(2,𝔽3)C2semidirect
4. #83588C3semidirect
52. C3×SL(2,𝔽3)SD16semidirect
135. SL(2,𝔽3)#214semidirect
138. C3⋊C8#41⋊C3semidirect
229. C3#3392⋊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
#5570⋊C32
C3⋊C8×SL(2,𝔽3)2
#835883
#3392⋊C33
C3×#8084