Finite Group Databasebeta

← #79925⋊C3 #11921220

Subgroups — #79925⋊C3 #11921220

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
1SL(2,𝔽3)×D12C211yesmaximal, characteristic
2#5570⋊C3C211yesmaximal, characteristic, complemented
3C3×C8×SL(2,𝔽3)C211yesmaximal, characteristic, complemented
4#79925C311yesmaximal, characteristic, complemented
5#3359⋊C3—31nomaximal
6C2×D3×SL(2,𝔽3)—21no—
7C3×C4×SL(2,𝔽3)C2211yescharacteristic
8Dic3×SL(2,𝔽3)—21no—
9C3×#821—41nomaximal
10Q8×D12C2×C311yescharacteristic
11#5570C2×C311yescharacteristic, complemented
12#557⋊C3—31no—
13D4×SL(2,𝔽3)—31no—
14C3×C8×Q8C2×C311yescharacteristic, complemented
15#4851—31no—
16#4841—31no—
17C4×#204—31no—
18#4681—31no—
19C8×SL(2,𝔽3)—21no—
20C8×SL(2,𝔽3)D311yescharacteristic, complemented
21C2×C3×D12—41no—
22C2×C3×SL(2,𝔽3)D411yescharacteristic
23D3×SL(2,𝔽3)—41no—
24C2×C32×C8—41no—
25C3×C4⋊Dic3—41no—
26#3359—31noSylow
27C2×D3×Q8—21no—
28C22×SL(2,𝔽3)—61no—
29C3×C4×Q8C22×C311yescharacteristic
30Q8×Dic3—21no—
31C2×#204—31no—
32#897—61no—
33#890—61no—
34C4×D12—31no—
35#869—31no—
36#869—31no—
37C4×Dic6—31no—
38C4×SL(2,𝔽3)—21no—
39C4×SL(2,𝔽3)—61no—
40C4×SL(2,𝔽3)C2×D311yescharacteristic
41C3×C4⋊C8—31no—
42C3×#102—121no—
43C3×C4×C8—31no—
44#821A411yescharacteristic
45#817—31no—
46#816—32no—
47#816—31no—
48C22×C3×D3—81no—
49C2×C32×C4—41no—
50C2×C3×Dic3—81no—
51C3×D12—42no—
52C3×D12—81no—
53C3×SL(2,𝔽3)SD1611yescharacteristic, complemented
54C32×C8—81no—
55#557—31no—
56D4×Q8—31no—
57#487—91no—
58#475—91no—
59#461—91no—
60C8×Q8C3×D311yescharacteristic, complemented
61C4×SD16—91no—
62C2×C3×Q8C3×D411yescharacteristic
63C2×C3×D4—121no—
64D3×Q8—41no—
65D3×Q8—121no—
66C2×D12C2×A411yescharacteristic
67C2×C4×D3—61no—
68C2×Dic6—31no—
69C2×Dic6—61no—
70C2×SL(2,𝔽3)—21no—
71C2×SL(2,𝔽3)—121no—
72C2×SL(2,𝔽3)D1211yescharacteristic
73C2×C3×C8—31no—
74C2×C3×C8—41no—
75C2×C3×C8—81no—
76C2×C3×C8C2×A411yescharacteristic
77C3×C4⋊C4—31no—
78C3×C4⋊C4—121no—
79C3×C42—31no—
80(C2×D3)⋊C4—61no—
81C4⋊Dic3—31no—
82C4⋊Dic3—61no—
83C4⋊Dic3—61no—
84C4⋊Dic3—61no—
85C4⋊Dic3C2×A411yescharacteristic
86Dic3⋊C4—61no—
87C4×Dic3—61no—
88#204—62no—
89C22×C32—41no—
90C2×C3×D3—82no—
91C2×C3×D3—161no—
92C2×C3×D3—161no—
93C32×C4—41no—
94C32×C4—41no—
95C3×Dic3—161no—
96C22×Q8—61no—
97C2×SD16—91no—
98C4⋊Q8—91no—
99C4⋊Q8—91no—
100#125—181no—
101C22⋊Q8—181no—
102C4×Q8—61no—
103C4×Q8—91no—
104C4×Q8C2×C3×D311yescharacteristic
105C4×D4—91no—
106#106—91no—
107C4⋊C8—31no—
108#103—92no—
109#103—91no—
110#102—31no—
111C4×C8—31no—
112C23×C3—241no—
113C22×D3—21no—
114C3×Q8—31no—
115C3×Q8C3×SD1611yescharacteristic
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—241no—
123C2×C3×C4C22×A411yescharacteristic
124C2×Dic3—21no—
125C2×Dic3—61no—
126C2×Dic3—61no—
127D12—21no—
128D12#41⋊C312yes—
129C4×D3—121no—
130C4×D3—121no—
131Dic6—61no—
132Dic6—61no—
133Dic6—121no—
134Dic6—121no—
135SL(2,𝔽3)—21no—
136SL(2,𝔽3)#20411yescharacteristic, complemented
137C3×C8—21no—
138C3×C8—61no—
139C3×C8—81no—
140C3×C8—82no—
141C2×C32—41no—
142C2×C32—41no—
143C2×C32—41no—
144C3×D3—162no—
145C2×Q8—91no—
146C2×Q8—121no—
147C2×Q8—181no—
148C2×Q8—361no—
149C2×Q8C3×D1211yescharacteristic
150C2×D4—31no—
151C22×C4—181no—
152SD16—182no—
153C2×C8—31no—
154C2×C8D3×A411yescharacteristic
155C4⋊C4—31no—
156C4⋊C4—31no—
157C4⋊C4—91no—
158C4⋊C4—181no—
159C4⋊C4—181no—
160C4⋊C4—181no—
161C4⋊C4—181no—
162C22⋊C4—181no—
163C42—31no—
164C42—181no—
165C22×C3—41no—
166C22×C3—81no—
167C22×C3—242no—
168C22×C3—481no—
169C22×C3—481no—
170C22×C3D4×A411yescharacteristic
171C2×D3—22no—
172C2×D3—41no—
173C2×D3—41no—
174C3×C4—31no—
175C3×C4—31no—
176C3×C4—41no—
177C3×C4—41no—
178C3×C4—61no—
179C3×C4—81no—
180C3×C4—81no—
181C3×C4—481no—
182C3×C4C2×#41⋊C311yescharacteristic
183C3×C42−1+4⋊C311yescharacteristic
184Dic3—41no—
185Dic3—121no—
186Dic3—121no—
187C32—41noSylow
188C23—61no—
189Q8—31no—
190Q8—181no—
191Q8—181no—
192Q8—361no—
193Q8—361no—
194Q8C3×#20411yescharacteristic
195D4—32no—
196D4—61no—
197C2×C4—31no—
198C2×C4—31no—
199C2×C4—61no—
200C2×C4—181no—
201C2×C4—181no—
202C2×C4—361no—
203C2×C4—361no—
204C2×C4C2×D3×A411yescharacteristic
205C8—21no—
206C8—61no—
207C2×C3—41no—
208C2×C3—41no—
209C2×C3—41no—
210C2×C3—81no—
211C2×C3—81no—
212C2×C3—81no—
213C2×C3—482no—
214C2×C3#548⋊C311yescharacteristic
215C2×C3A4×SD1611yescharacteristic
216C2×C3#578⋊C311yescharacteristic
217D3—42no—
218C22—62no—
219C22—121no—
220C22—121no—
221C22A4×D1211yescharacteristic
222C4—31no—
223C4—31no—
224C4—61no—
225C4—121no—
226C4—361no—
227C4—361no—
228C4#1007⋊C311yescharacteristic
229C4D3×#41⋊C311yescharacteristic
230C3—41no—
231C3—81no—
232C3#3359⋊C311yescharacteristic, complemented
233C2—122no—
234C2#5576⋊C311yescharacteristic
235C2A4×#20411yescharacteristic
236C2#5753⋊C311yescharacteristic
subgroupG/H
centerC22A4×D12
commutator, cosocleC3×C4×Q8C22×C3
FrattiniC2×C4C2×D3×A4
FittingC3×C8×Q8C2×C3
socleC22×C3D4×A4
prime radicalSL(2,𝔽3)×D12C2
hypercenterC2×C8D3×A4
hypocenterC3×Q8C3×SD16
LCSG ▷ C3×C4×Q8 ▷ C2×C3×Q8 ▷ C3×Q8
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×C8

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. #79925C3semidirect
11. #5570C2×C3semidirect
14. C3×C8×Q8C2×C3semidirect
20. C8×SL(2,𝔽3)D3semidirect
53. C3×SL(2,𝔽3)SD16semidirect
60. C8×Q8C3×D3semidirect
136. SL(2,𝔽3)#204semidirect
232. C3#3359⋊C3semidirect

Maximal subgroups

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

subgroupindex
SL(2,𝔽3)×D122
#5570⋊C32
C3×C8×SL(2,𝔽3)2
#799253
#3359⋊C33
C3×#8214