Finite Group Databasebeta

← #83568⋊C3 #11921300

Subgroups — #83568⋊C3 #11921300

Order 1152.

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

normal subgroups only

This group has 275 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×D4×SL(2,𝔽3)C211yesmaximal, characteristic
2#5570⋊C3C211yesmaximal, characteristic, complemented
3C3⋊C8×SL(2,𝔽3)C211yesmaximal, characteristic, complemented
4#83568C311yesmaximal, characteristic, complemented
5#3359⋊C3—31nomaximal
6C22×C3×SL(2,𝔽3)—21no—
7C3×C4×SL(2,𝔽3)C2211yescharacteristic
8Dic3×SL(2,𝔽3)—21no—
9C3×#832—41nomaximal
10C3×D4×Q8C2×C311yescharacteristic
11#5570C2×C311yescharacteristic, complemented
12#557⋊C3—31no—
13D4×SL(2,𝔽3)—21no—
14D4×SL(2,𝔽3)D311yescharacteristic
15#5094—31no—
16#5093—31no—
17Q8×C3⋊C8C2×C311yescharacteristic, complemented
18#5019—31no—
19C4×#214—31no—
20C8×SL(2,𝔽3)—31no—
21C2×C32×D4—41no—
22C2×C3×SL(2,𝔽3)—41no—
23C2×C3×SL(2,𝔽3)D411yescharacteristic
24C3×C4⋊Dic3—41no—
25C2×C3×C3⋊C8—41no—
26#3359—31noSylow
27C22×C3×Q8—21no—
28C22×SL(2,𝔽3)—22no—
29C22×SL(2,𝔽3)—21no—
30C3×C4⋊Q8—31no—
31C3×C22⋊Q8—61no—
32C3×C4×Q8C22×C311yescharacteristic
33C3×C4×D4—31no—
34Q8×Dic3—21no—
35C2×#214—31no—
36#890—61no—
37#869—31no—
38C4×Dic6—31no—
39C4×SL(2,𝔽3)—21no—
40C4×SL(2,𝔽3)—61no—
41C4×SL(2,𝔽3)C2×D311yescharacteristic
42C3×#102—121no—
43#832A411yescharacteristic
44#810—32no—
45#810—31no—
46#808—31no—
47C4⋊(C3⋊C8)—31no—
48C4×C3⋊C8—31no—
49C23×C32—81no—
50C32×D4—42no—
51C32×D4—81no—
52C2×C32×C4—41no—
53C2×C3×Dic3—81no—
54C3×SL(2,𝔽3)SD1611yescharacteristic, complemented
55C3×C3⋊C8—81no—
56#557—31no—
57D4×Q8C3×D311yescharacteristic
58#487—91no—
59#475—91no—
60#461—91no—
61C8×Q8—31no—
62C4×SD16—91no—
63C2×C3×Q8—41no—
64C2×C3×Q8—61no—
65C2×C3×Q8—121no—
66C2×C3×Q8C3×D411yescharacteristic
67C2×C3×D4—41no—
68C2×C3×D4—81no—
69C2×C3×D4C2×A411yescharacteristic
70C22×C3×C4—61no—
71C2×Dic6—31no—
72C2×SL(2,𝔽3)—21no—
73C2×SL(2,𝔽3)—42no—
74C2×SL(2,𝔽3)—41no—
75C2×SL(2,𝔽3)#6711yescharacteristic
76C2×C3×C8—121no—
77C3×C4⋊C4—31no—
78C3×C4⋊C4—31no—
79C3×C4⋊C4—61no—
80C3×C4⋊C4—121no—
81C3×C22⋊C4—61no—
82C3×C42—31no—
83#214—62no—
84C4⋊Dic3—61no—
85C4⋊Dic3—61no—
86C4⋊Dic3C2×A411yescharacteristic
87Dic3⋊C4—61no—
88C4×Dic3—61no—
89C2×C3⋊C8—31no—
90C2×C3⋊C8C2×A411yescharacteristic
91C22×C32—41no—
92C22×C32—82no—
93C22×C32—161no—
94C22×C32—161no—
95C32×C4—41no—
96C32×C4—41no—
97C3×Dic3—161no—
98C22×Q8—21no—
99C2×SD16—91no—
100C4⋊Q8—31no—
101C4⋊Q8—91no—
102#125—181no—
103C22⋊Q8—61no—
104C4×Q8—61no—
105C4×Q8—91no—
106C4×Q8C2×C3×D311yescharacteristic
107C4×D4—31no—
108#106—91no—
109C4⋊C8—91no—
110#103—91no—
111#103—92no—
112#102—31no—
113C4×C8—91no—
114C23×C3—21no—
115C23×C3—81no—
116C23×C3—82no—
117C3×Q8—31no—
118C3×Q8—121no—
119C3×Q8—121no—
120C3×Q8C3×SD1611yescharacteristic
121C3×D4—21no—
122C3×D4—42no—
123C3×D4—81no—
124C3×D4—82no—
125C3×D4—82no—
126C3×D4#41⋊C312yes—
127C2×C3×C4—31no—
128C2×C3×C4—31no—
129C2×C3×C4—41no—
130C2×C3×C4—61no—
131C2×C3×C4—81no—
132C2×C3×C4—121no—
133C2×C3×C4—121no—
134C2×C3×C4—241no—
135C2×C3×C4C22×A411yescharacteristic
136C2×Dic3—21no—
137C2×Dic3—61no—
138Dic6—61no—
139Dic6—61no—
140SL(2,𝔽3)—21no—
141SL(2,𝔽3)#21411yescharacteristic, complemented
142C3×C8—241no—
143C3⋊C8—21no—
144C3⋊C8—61no—
145C2×C32—41no—
146C2×C32—41no—
147C2×C32—41no—
148C2×C32—162no—
149C2×Q8—41no—
150C2×Q8—61no—
151C2×Q8—91no—
152C2×Q8—121no—
153C2×Q8C3×#6711yescharacteristic
154C2×D4D3×A411yescharacteristic
155C22×C4—61no—
156SD16—182no—
157C2×C8—31no—
158C2×C8—91no—
159C4⋊C4—31no—
160C4⋊C4—31no—
161C4⋊C4—31no—
162C4⋊C4—61no—
163C4⋊C4—181no—
164C4⋊C4—181no—
165C4⋊C4—181no—
166C22⋊C4—61no—
167C42—31no—
168C42—181no—
169C22×C3—22no—
170C22×C3—41no—
171C22×C3—41no—
172C22×C3—41no—
173C22×C3—82no—
174C22×C3—84no—
175C22×C3—81no—
176C22×C3—161no—
177C22×C3—161no—
178C22×C3—162no—
179C22×C3—162no—
180C22×C3D4×A411yescharacteristic
181C3×C4—31no—
182C3×C4—31no—
183C3×C4—41no—
184C3×C4—41no—
185C3×C4—61no—
186C3×C4—81no—
187C3×C4—81no—
188C3×C4—121no—
189C3×C4—481no—
190C3×C4C2×#41⋊C311yescharacteristic
191C3×C42−1+4⋊C311yescharacteristic
192Dic3—41no—
193Dic3—121no—
194C32—41noSylow
195C23—21no—
196Q8—31no—
197Q8—121no—
198Q8—121no—
199Q8—181no—
200Q8—181no—
201Q8C3×#21411yescharacteristic
202D4—21no—
203D4C3⋊(#41⋊C3)12yes—
204C2×C4—31no—
205C2×C4—31no—
206C2×C4—61no—
207C2×C4—61no—
208C2×C4—121no—
209C2×C4—121no—
210C2×C4—181no—
211C2×C4C2×D3×A411yescharacteristic
212C8—61no—
213C8—181no—
214C2×C3—42no—
215C2×C3—41no—
216C2×C3—41no—
217C2×C3—41no—
218C2×C3—81no—
219C2×C3—81no—
220C2×C3—81no—
221C2×C3—162no—
222C2×C3—164no—
223C2×C3#548⋊C311yescharacteristic
224C2×C3A4×SD1611yescharacteristic
225C2×C3#578⋊C311yescharacteristic
226C22—22no—
227C22—41no—
228C22—41no—
229C22A4×#6711yescharacteristic
230C4—31no—
231C4—31no—
232C4—61no—
233C4—121no—
234C4—121no—
235C4—361no—
236C4C2×C3⋊(#41⋊C3)11yescharacteristic
237C4#1007⋊C311yescharacteristic
238C3—41no—
239C3—81no—
240C3#3359⋊C311yescharacteristic, complemented
241C2—42no—
242C2A4×#21411yescharacteristic
243C2#5817⋊C311yescharacteristic
244C2#5838⋊C311yescharacteristic
subgroupG/H
centerC22A4×#67
commutator, cosocleC3×C4×Q8C22×C3
FrattiniC2×C4C2×D3×A4
FittingC3×D4×Q8C2×C3
socleC22×C3D4×A4
prime radicalC3×D4×SL(2,𝔽3)C2
hypercenterC2×D4D3×A4
hypocenterC3×Q8C3×SD16
LCSG ▷ C3×C4×Q8 ▷ C2×C3×Q8 ▷ C3×Q8
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×D4

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. #83568C3semidirect
11. #5570C2×C3semidirect
17. Q8×C3⋊C8C2×C3semidirect
54. C3×SL(2,𝔽3)SD16semidirect
141. SL(2,𝔽3)#214semidirect
240. C3#3359⋊C3semidirect

Maximal subgroups

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

subgroupindex
C3×D4×SL(2,𝔽3)2
#5570⋊C32
C3⋊C8×SL(2,𝔽3)2
#835683
#3359⋊C33
C3×#8324