Finite Group Databasebeta

← SL(2,𝔽3)×#627 #13166152

Subgroups — SL(2,𝔽3)×#627 #13166152

Order 1728.

396 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 396 conjugacy classes of subgroups — too many to draw. Only its 68 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)×C3⋊Dic3C211yesmaximal, characteristic
2C3×Dic3×SL(2,𝔽3)C212yesmaximal
3Q8×#627C311yesmaximal, characteristic, complemented
4SL(2,𝔽3)×Dic6—32nomaximal
5C2×C32×SL(2,𝔽3)C2211yescharacteristic
6C2×C3×#627—41nomaximal
7Q8×C3⋊Dic3C2×C311yescharacteristic
8C3×Q8×Dic3C2×C312yes—
9C3×C4×SL(2,𝔽3)—32no—
10#63919—31no—
11#63918—32no—
12C4×#627—31no—
13Dic3×SL(2,𝔽3)—22no—
14Dic3×SL(2,𝔽3)—32no—
15Dic3×SL(2,𝔽3)—61no—
16Dic3×SL(2,𝔽3)D312yes—
17C2×C3×C3⋊Dic3—41no—
18C2×C32×Dic3—42no—
19C32×SL(2,𝔽3)Q811yescharacteristic, complemented
20C3×#627—44no—
21Q8×Dic6—32no—
22Q8×SL(2,𝔽3)—91no—
23C2×C32×Q8C22×C311yescharacteristic
24C2×C3×Dic6—122no—
25C2×C3×SL(2,𝔽3)—22no—
26C2×C3×SL(2,𝔽3)—21no—
27C2×C3×SL(2,𝔽3)—41no—
28C2×C3×SL(2,𝔽3)C2×D312yes—
29C2×#627—32no—
30C2×#627A411yescharacteristic
31C4⋊(C3⋊Dic3)—31no—
32C4×C3⋊Dic3—31no—
33C3×C4⋊Dic3—32no—
34C3×C4×Dic3—32no—
35#3724—31no—
36Dic3⋊Dic3—32no—
37C22×C33—41no—
38C3×C3⋊Dic3—42no—
39C32×Dic3—44no—
40C3×C4×Q8—32no—
41Q8×Dic3—32no—
42Q8×Dic3—61no—
43Q8×Dic3C3×D312yes—
44#888—92no—
45#888—92no—
46#869—92no—
47C4×Dic6—92no—
48C4×SL(2,𝔽3)—32no—
49C4×SL(2,𝔽3)—62no—
50C4×SL(2,𝔽3)—91no—
51C32×Q8—31no—
52C32×Q8C3×Q811yescharacteristic, complemented
53C2×C32×C4—31no—
54C2×C32×C4—122no—
55C2×C3⋊Dic3—31no—
56C2×C3⋊Dic3C2×A411yescharacteristic
57C2×C3×Dic3—32no—
58C2×C3×Dic3—42no—
59C2×C3×Dic3—82no—
60C2×C3×Dic3—122no—
61C2×C3×Dic3—241no—
62C2×C3×Dic3C2×A412yes—
63C3×Dic6—128no—
64C3×SL(2,𝔽3)—22no—
65C3×SL(2,𝔽3)—21no—
66C3×SL(2,𝔽3)—41no—
67C3×SL(2,𝔽3)Dic612yescomplemented
68#627—122no—
69#627SL(2,𝔽3)14yesdirect product factor
70Q82—91noSylow
71C2×C33—41no—
72C2×C33—41no—
73C2×C33—41no—
74C2×C3×Q8—21no—
75C2×C3×Q8—361no—
76C2×C3×Q8C2×C3×D312yes—
77C2×Dic6—32no—
78C2×Dic6—94no—
79C2×SL(2,𝔽3)—22no—
80C2×SL(2,𝔽3)—41no—
81C2×SL(2,𝔽3)D3211yescharacteristic
82C3×C4⋊C4—92no—
83C3×C42—92no—
84C4⋊Dic3—32no—
85C4⋊Dic3—92no—
86C4⋊Dic3—92no—
87C4⋊Dic3—181no—
88Dic3⋊C4—92no—
89Dic3⋊C4—92no—
90C4×Dic3—32no—
91C4×Dic3—92no—
92C4×Dic3—181no—
93C22×C32—42no—
94C22×C32—81no—
95C22×C32—82no—
96C22×C32—161no—
97C22×C32C22×A411yescharacteristic
98C32×C4—31no—
99C32×C4—31no—
100C32×C4—124no—
101C3⋊Dic3—61no—
102C3⋊Dic3C2×SL(2,𝔽3)12yes—
103C3×Dic3—44no—
104C3×Dic3—62no—
105C3×Dic3—84no—
106C3×Dic3—124no—
107C3×Dic3—242no—
108C3×Dic3C2×SL(2,𝔽3)14yes—
109C4⋊Q8—272no—
110C4⋊Q8—271no—
111C4×Q8—32no—
112C4×Q8—91no—
113C4×Q8—271no—
114C33—41noSylow
115C3×Q8—21no—
116C3×Q8—32no—
117C3×Q8—61no—
118C3×Q8—364no—
119C3×Q8C3×Dic612yescomplemented
120C2×C3×C4—32no—
121C2×C3×C4—32no—
122C2×C3×C4—61no—
123C2×C3×C4—92no—
124C2×C3×C4—122no—
125C2×C3×C4—242no—
126C2×C3×C4—361no—
127C2×Dic3—32no—
128C2×Dic3—32no—
129C2×Dic3—61no—
130C2×Dic3—92no—
131C2×Dic3—181no—
132C2×Dic3D3×A412yes—
133Dic6—38no—
134Dic6—364no—
135SL(2,𝔽3)—22no—
136SL(2,𝔽3)—41no—
137SL(2,𝔽3)#62711yescharacteristic, direct product factor
138C2×C32—42no—
139C2×C32—42no—
140C2×C32—42no—
141C2×C32—81no—
142C2×C32—81no—
143C2×C32—81no—
144C2×C32—82no—
145C2×C32—82no—
146C2×C32—82no—
147C2×C32—161no—
148C2×C32—161no—
149C2×C32—161no—
150C2×C32C22×SL(2,𝔽3)11yescharacteristic
151C2×C32Q8×A411yescharacteristic
152C2×C32#99411yescharacteristic
153C2×Q8—91no—
154C2×Q8—272no—
155C2×Q8C3×D3211yescharacteristic
156C4⋊C4—92no—
157C4⋊C4—272no—
158C4⋊C4—271no—
159C4⋊C4—271no—
160C42—92no—
161C42—271no—
162C22×C3—21no—
163C22×C3—41no—
164C22×C3—82no—
165C22×C3—161no—
166C22×C3C2×D3×A412yes—
167C3×C4—32no—
168C3×C4—32no—
169C3×C4—34no—
170C3×C4—61no—
171C3×C4—61no—
172C3×C4—124no—
173C3×C4—182no—
174C3×C4—244no—
175C3×C4—362no—
176Dic3—34no—
177Dic3—62no—
178Dic3—62no—
179Dic3—182no—
180Dic3—361no—
181Dic3D3×SL(2,𝔽3)14yes—
182C32—42no—
183C32—81no—
184C32—82no—
185C32—161no—
186C32Q8×SL(2,𝔽3)11yescharacteristic, complemented
187Q8—31no—
188Q8—94no—
189Q8—1082no—
190Q8C3×#62711yescharacteristic, complemented
191C2×C4—31no—
192C2×C4—32no—
193C2×C4—91no—
194C2×C4—92no—
195C2×C4—271no—
196C2×C3—21no—
197C2×C3—21no—
198C2×C3—21no—
199C2×C3—41no—
200C2×C3—41no—
201C2×C3—41no—
202C2×C3—82no—
203C2×C3—82no—
204C2×C3—82no—
205C2×C3—161no—
206C2×C3—161no—
207C2×C3—161no—
208C2×C3A4×Dic612yes—
209C2×C3C2×D3×SL(2,𝔽3)12yes—
210C2×C3C3⋊#99412yes—
211C22D32×A411yescharacteristic
212C4—34no—
213C4—31no—
214C4—31no—
215C4—92no—
216C4—182no—
217C4—541no—
218C3—21no—
219C3—41no—
220C3—82no—
221C3—161no—
222C3SL(2,𝔽3)×Dic612yescomplemented
223C2#64320⋊C311yescharacteristic
224C2D32×SL(2,𝔽3)11yescharacteristic
225C2A4×#62711yescharacteristic
subgroupG/H
center, Frattini, hypercenterC22D32×A4
commutator, Fitting, cosocleC2×C32×Q8C22×C3
socleC22×C32C22×A4
prime radicalC2×C32×SL(2,𝔽3)C22
hypocenterC32×Q8C3×Q8
LCSG ▷ C2×C32×Q8 ▷ C32×Q8
UCSC1 ◁ C22

Complements

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

NH ≅ G/Ntype
3. Q8×#627C3semidirect
19. C32×SL(2,𝔽3)Q8semidirect
52. C32×Q8C3×Q8semidirect
67. C3×SL(2,𝔽3)Dic6semidirect
69. #627SL(2,𝔽3)direct (N × H)
119. C3×Q8C3×Dic6semidirect
137. SL(2,𝔽3)#627direct (N × H)
186. C32Q8×SL(2,𝔽3)semidirect
190. Q8C3×#627semidirect
222. C3SL(2,𝔽3)×Dic6semidirect

Maximal subgroups

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

subgroupindex
SL(2,𝔽3)×C3⋊Dic32
C3×Dic3×SL(2,𝔽3)2
Q8×#6273
SL(2,𝔽3)×Dic63
C2×C3×#6274