Finite Group Databasebeta

← Dic3×C22⋊SL(2,𝔽3) #11923432

Subgroups — Dic3×C22⋊SL(2,𝔽3) #11923432

Order 1152.

678 conjugacy classes of subgroups; only the 56 normal subgroups are shown.

normal subgroups only

This group has 678 conjugacy classes of subgroups — too many to draw. Only its 56 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
1C2×C3×C22⋊SL(2,𝔽3)C211yesmaximal, characteristic
2C22×Q8×Dic3C311yesmaximal, characteristic, complemented
3C4×C22⋊SL(2,𝔽3)—31nomaximal
4C3×C22⋊SL(2,𝔽3)C411yescharacteristic, complemented
5C2×Dic3×A4—41nomaximal
6Dic3×SL(2,𝔽3)—44nomaximal
7C23×C3×Q8C2×C311yescharacteristic
8C2×C22⋊SL(2,𝔽3)—21no—
9C2×C22⋊SL(2,𝔽3)D311yescharacteristic
10C2×Q8×Dic3—34no—
11C2×Q8×Dic3—34no—
12C22×C4⋊Dic3—31no—
13C22×C4×Dic3—31no—
14C22×C3×A4—41no—
15C2×C3×SL(2,𝔽3)—44no—
16Dic3×A4—42no—
17C22×C4×Q8—31noSylow
18C22×C3×Q8—34no—
19C22×C3×Q8—34no—
20C22×C3×Q8—31no—
21C22×C3×Q8C3×C411yescharacteristic, complemented
22C23×C3×C4—31no—
23C23×Dic3—31no—
24C23×Dic3A411yescharacteristic
25C22⋊SL(2,𝔽3)—21no—
26C22⋊SL(2,𝔽3)Dic311yescharacteristic, direct product factor
27C2×C4×A4—121no—
28Q8×Dic3—312no—
29Q8×Dic3—34no—
30Q8×Dic3—34no—
31Q8×Dic3A414yescomplemented
32C2×C4⋊Dic3—312no—
33C2×C4×Dic3—36no—
34C2×C4×Dic3—36no—
35C4×SL(2,𝔽3)—124no—
36C2×C3×A4—41no—
37C2×C3×A4—41no—
38C2×C3×A4—41no—
39C2×C3×Dic3—161no—
40C3×SL(2,𝔽3)—44no—
41C23×Q8C3×D311yescharacteristic
42C2×C4×Q8—94no—
43C2×C4×Q8—94no—
44C22×C4⋊C4—91no—
45C22×C42—91no—
46C24×C3C2×A411yescharacteristic
47C22×A4—41no—
48C22×A4—81no—
49C2×C3×Q8—34no—
50C2×C3×Q8—312no—
51C2×C3×Q8—312no—
52C2×C3×Q8—34no—
53C2×C3×Q8—34no—
54C2×C3×Q8C2×A414yes—
55C22×C3×C4—36no—
56C22×C3×C4—36no—
57C22×C3×C4—31no—
58C22×C3×C4—31no—
59C22×Dic3—31no—
60C22×Dic3—31no—
61C22×Dic3—32no—
62C22×Dic3—36no—
63C22×Dic3—63no—
64C22×Dic3—61no—
65C22×Dic3SL(2,𝔽3)12yescomplemented
66C2×SL(2,𝔽3)—44no—
67C2×SL(2,𝔽3)—84no—
68C4×A4—122no—
69C4⋊Dic3—312no—
70C4⋊Dic3—34no—
71C4×Dic3—312no—
72C4×Dic3—34no—
73C22×C32—161no—
74C3×A4—41no—
75C3×Dic3—162no—
76C22×Q8—34no—
77C22×Q8—34no—
78C22×Q8—31no—
79C22×Q8C3×Dic311yescharacteristic, complemented
80C23×C4—31no—
81C23×C4—31no—
82C23×C4—91no—
83C4×Q8—34no—
84C4×Q8—94no—
85C4×Q8—912no—
86C4×Q8—94no—
87C2×C4⋊C4—912no—
88C2×C42—96no—
89C2×C42—96no—
90C23×C3—31no—
91C23×C3—31no—
92C23×C3—31no—
93C23×C3—31no—
94C23×C3C4×A411yescharacteristic
95C23×C3C2×SL(2,𝔽3)11yescharacteristic
96C23×C3#41⋊C311yescharacteristic
97C2×A4—41no—
98C2×A4—41no—
99C2×A4—41no—
100C2×A4—81no—
101C2×A4—81no—
102C2×A4—81no—
103C3×Q8—34no—
104C3×Q8—312no—
105C3×Q8—34no—
106C3×Q8C4×A414yescomplemented
107C2×C3×C4—36no—
108C2×C3×C4—312no—
109C2×C3×C4—36no—
110C2×C3×C4—34no—
111C2×C3×C4—481no—
112C2×Dic3—32no—
113C2×Dic3—31no—
114C2×Dic3—32no—
115C2×Dic3—32no—
116C2×Dic3—32no—
117C2×Dic3—34no—
118C2×Dic3—66no—
119C2×Dic3—66no—
120C2×Dic3C22⋊A411yescharacteristic
121SL(2,𝔽3)—44no—
122SL(2,𝔽3)—84no—
123C2×C32—161no—
124C2×C32—161no—
125C2×C32—161no—
126C24D3×A411yescharacteristic
127C2×Q8—34no—
128C2×Q8—312no—
129C2×Q8—312no—
130C2×Q8—34no—
131C2×Q8—34no—
132C2×Q8D3×A414yes—
133C22×C4—36no—
134C22×C4—36no—
135C22×C4—31no—
136C22×C4—31no—
137C22×C4—32no—
138C22×C4—91no—
139C22×C4—91no—
140C22×C4—92no—
141C22×C4—96no—
142C22×C4—183no—
143C22×C4—181no—
144C4⋊C4—94no—
145C4⋊C4—912no—
146C42—94no—
147C42—912no—
148C22×C3—31no—
149C22×C3—31no—
150C22×C3—31no—
151C22×C3—32no—
152C22×C3—31no—
153C22×C3—31no—
154C22×C3—31no—
155C22×C3—31no—
156C22×C3—31no—
157C22×C3—31no—
158C22×C3—161no—
159C22×C3—321no—
160C22×C3C4×SL(2,𝔽3)11yescharacteristic, complemented
161C22×C3C2×C22⋊A411yescharacteristic
162A4—41no—
163A4—81no—
164C3×C4—34no—
165C3×C4—34no—
166C3×C4—482no—
167Dic3—32no—
168Dic3—64no—
169Dic3C22⋊SL(2,𝔽3)12yesdirect product factor
170C32—161noSylow
171C23—31no—
172C23—31no—
173C23—31no—
174C23—31no—
175C23C3⋊(#41⋊C3)11yescharacteristic
176C23D3×SL(2,𝔽3)11yescharacteristic
177C23Dic3×A411yescharacteristic
178Q8—34no—
179Q8—34no—
180Q8—312no—
181Q8Dic3×A414yescomplemented
182C2×C4—36no—
183C2×C4—312no—
184C2×C4—36no—
185C2×C4—34no—
186C2×C4—31no—
187C2×C4—92no—
188C2×C4—92no—
189C2×C4—92no—
190C2×C4—91no—
191C2×C4—92no—
192C2×C4—94no—
193C2×C4—186no—
194C2×C4—186no—
195C2×C3—31no—
196C2×C3—31no—
197C2×C3—31no—
198C2×C3—31no—
199C2×C3—161no—
200C2×C3—161no—
201C2×C3—161no—
202C2×C3—321no—
203C2×C3—321no—
204C2×C3—321no—
205C2×C3C4×C22⋊A411yescharacteristic
206C2×C3C2×C22⋊SL(2,𝔽3)11yescharacteristic
207C2×C3#594811yescharacteristic
208C22—31no—
209C22—31no—
210C22—31no—
211C22—32no—
212C22—31no—
213C22—31no—
214C22—31no—
215C22—31no—
216C22—31no—
217C22—31no—
218C22Dic3×SL(2,𝔽3)11yescharacteristic, complemented
219C22D3×C22⋊A411yescharacteristic
220C4—32no—
221C4—34no—
222C4—34no—
223C4—92no—
224C4—184no—
225C3—161no—
226C3—321no—
227C3C4×C22⋊SL(2,𝔽3)11yescharacteristic, complemented
228C2—31no—
229C2—31no—
230C2—31no—
231C2—31no—
232C2C3⋊#594811yescharacteristic
233C2D3×C22⋊SL(2,𝔽3)11yescharacteristic
234C2Dic3×C22⋊A411yescharacteristic
subgroupG/H
center, Frattini, hypercenterC22D3×C22⋊A4
commutator, hypocenterC22×C3×Q8C3×C4
Fitting, cosocleC23×C3×Q8C2×C3
socleC24×C3C2×A4
prime radicalC2×C3×C22⋊SL(2,𝔽3)C2
LCSG ▷ C22×C3×Q8
UCSC1 ◁ C22

Complements

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

NH ≅ G/Ntype
2. C22×Q8×Dic3C3semidirect
4. C3×C22⋊SL(2,𝔽3)C4semidirect
21. C22×C3×Q8C3×C4semidirect
26. C22⋊SL(2,𝔽3)Dic3direct (N × H)
31. Q8×Dic3A4semidirect
65. C22×Dic3SL(2,𝔽3)semidirect
79. C22×Q8C3×Dic3semidirect
106. C3×Q8C4×A4semidirect
160. C22×C3C4×SL(2,𝔽3)semidirect
169. Dic3C22⋊SL(2,𝔽3)direct (N × H)
181. Q8Dic3×A4semidirect
218. C22Dic3×SL(2,𝔽3)semidirect
227. C3C4×C22⋊SL(2,𝔽3)semidirect

Maximal subgroups

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

subgroupindex
C2×C3×C22⋊SL(2,𝔽3)2
C22×Q8×Dic33
C4×C22⋊SL(2,𝔽3)3
C2×Dic3×A44
Dic3×SL(2,𝔽3)4