Finite Group Databasebeta

← #13164658

Subgroups — #13164658

Order 1728.

968 conjugacy classes of subgroups; only the 41 normal subgroups are shown.

normal subgroups only

This group has 968 conjugacy classes of subgroups — too many to draw. Only its 41 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
1#11722587C211yesmaximal, characteristic, complemented
2C22×Q8×C3⋊D3C311yesmaximal, characteristic, complemented
3D3×C22⋊SL(2,𝔽3)—31nomaximal
4C2×(C3⋊D3)⋊A4—41nomaximal
5#88623—44nomaximal
6C22×C32×Q8C2×C311yescharacteristic, complemented
7C2×Q8×C3⋊D3—34no—
8C2×Q8×C3⋊D3—34no—
9C22×C4×C3⋊D3—31no—
10C22×#634—31no—
11C3×C22⋊SL(2,𝔽3)—21no—
12C3×C22⋊SL(2,𝔽3)D311yescharacteristic, complemented
13C2×C22⋊H(3,𝔽3)—41no—
14(C3⋊D3)⋊A4—42no—
15Q8⋊H(3,𝔽3)—44no—
16C22×D3×Q8—31no—
17C22×D3×Q8—91no—
18C2×C22⋊SL(2,𝔽3)—91no—
19C23×C3⋊D3A411yescharacteristic
20C2×D3×A4—121no—
21C2×C32×Q8—34no—
22C22×C32×C4—31no—
23C22×C3⋊Dic3—31no—
24Q8×C3⋊D3—34no—
25Q8×C3⋊D3—312no—
26Q8×C3⋊D3—34no—
27Q8×C3⋊D3A414yescomplemented
28C2×C4×C3⋊D3—36no—
29C2×C4×C3⋊D3—36no—
30C2×#634—312no—
31D3×SL(2,𝔽3)—124no—
32C2×(C3⋊D3)⋊C3—161no—
33C22⋊H(3,𝔽3)—41no—
34C22×C3×Q8—31no—
35C22×C3×Q8C3×D311yescharacteristic
36C2×D3×Q8—94no—
37C2×D3×Q8—912no—
38C2×D3×Q8—912no—
39C2×D3×Q8—94no—
40C22×C4×D3—91no—
41C22×C4×D3—93no—
42C22×Dic6—93no—
43C22×Dic6—91no—
44C22⋊SL(2,𝔽3)—31no—
45C22⋊SL(2,𝔽3)—61no—
46C23×C32C2×A411yescharacteristic
47C22×C3⋊D3—31no—
48C22×C3⋊D3—32no—
49C22×C3⋊D3—31no—
50C22×C3⋊D3SL(2,𝔽3)12yescomplemented
51C2×C3×A4—41no—
52C2×C3×A4—81no—
53D3×A4—122no—
54C32×Q8—34no—
55C32×Q8C2×A414yescomplemented
56C2×C32×C4—36no—
57C2×C3⋊Dic3—36no—
58C4×C3⋊D3—312no—
59C4×C3⋊D3—34no—
60#634—312no—
61#634—34no—
62C3×SL(2,𝔽3)—44no—
63C3×SL(2,𝔽3)—84no—
64C23×Q8—91noSylow
65C2×H(3,𝔽3)—161no—
66(C3⋊D3)⋊C3—162no—
67C23×D3—31no—
68C23×D3—91no—
69C22×A4—361no—
70C2×C3×Q8—312no—
71C2×C3×Q8—34no—
72C22×C3×C4—33no—
73C22×C3×C4—31no—
74C22×Dic3—93no—
75C22×Dic3—91no—
76D3×Q8—34no—
77D3×Q8—94no—
78D3×Q8—94no—
79D3×Q8—912no—
80D3×Q8—936no—
81D3×Q8—912no—
82D3×Q8—912no—
83D3×Q8—94no—
84C2×C4×D3—96no—
85C2×C4×D3—918no—
86C2×C4×D3—918no—
87C2×C4×D3—96no—
88C2×Dic6—936no—
89C2×Dic6—912no—
90C2×SL(2,𝔽3)—364no—
91C22×C32—31no—
92C22×C32—31no—
93C22×C32C2×SL(2,𝔽3)11yescharacteristic, complemented
94C2×C3⋊D3—32no—
95C2×C3⋊D3—31no—
96C2×C3⋊D3—32no—
97C2×C3⋊D3—32no—
98C2×C3⋊D3—32no—
99C2×C3⋊D3C22⋊A411yescharacteristic
100C2×C3×D3—481no—
101C3×A4—41no—
102C3×A4—81no—
103C32×C4—34no—
104C3⋊Dic3—34no—
105C22×Q8—271no—
106C22×Q8—274no—
107C22×Q8—274no—
108C22×Q8(C3⋊D3)⋊C311yescharacteristic, complemented
109C23×C4—271no—
110H(3,𝔽3)—161noSylow
111C23×C3—31no—
112C23×C3D3×A411yescharacteristic
113C22×D3—32no—
114C22×D3—92no—
115C22×D3—91no—
116C22×D3—93no—
117C22×D3—93no—
118C22×D3—96no—
119C22×D3—91no—
120C22×D3—92no—
121C2×A4—121no—
122C2×A4—241no—
123C2×A4—362no—
124C3×Q8—312no—
125C3×Q8—34no—
126C3×Q8—34no—
127C3×Q8D3×A414yes—
128C2×C3×C4—318no—
129C2×C3×C4—36no—
130C2×Dic3—918no—
131C2×Dic3—96no—
132C4×D3—94no—
133C4×D3—912no—
134C4×D3—936no—
135C4×D3—912no—
136Dic6—912no—
137Dic6—936no—
138Dic6—912no—
139Dic6—94no—
140SL(2,𝔽3)—124no—
141SL(2,𝔽3)—244no—
142C2×C32—31no—
143C2×C32—31no—
144C2×C32—161no—
145C2×C32—321no—
146C2×C32C2×C22⋊A411yescharacteristic
147C3⋊D3—32no—
148C3⋊D3C22⋊SL(2,𝔽3)12yescomplemented
149C3×D3—482no—
150C24—91no—
151C2×Q8—34no—
152C2×Q8—94no—
153C2×Q8—2712no—
154C2×Q8—274no—
155C2×Q8—2712no—
156C2×Q8—274no—
157C22×C4—31no—
158C22×C4—276no—
159C22×C4—276no—
160C22×C4—271no—
161C22×C3—31no—
162C22×C3—33no—
163C22×C3—33no—
164C22×C3—31no—
165C22×C3—31no—
166C22×C3—1441no—
167C22×C3D3×SL(2,𝔽3)11yescharacteristic
168C2×D3—31no—
169C2×D3—91no—
170C2×D3—92no—
171C2×D3—92no—
172C2×D3—93no—
173C2×D3—96no—
174C2×D3—96no—
175C2×D3—96no—
176C2×D3—96no—
177C2×D3—92no—
178C2×D3—91no—
179C2×D3—92no—
180A4—121no—
181A4—241no—
182C3×C4—312no—
183C3×C4—34no—
184Dic3—912no—
185Dic3—94no—
186C32—161no—
187C32—321no—
188C32C2×C22⋊SL(2,𝔽3)11yescharacteristic, complemented
189C23—92no—
190C23—271no—
191C23—271no—
192C23—272no—
193C23(C3⋊D3)⋊A411yescharacteristic
194Q8—34no—
195Q8—274no—
196Q8—2712no—
197Q8(C3⋊D3)⋊A414yescomplemented
198C2×C4—36no—
199C2×C4—274no—
200C2×C4—2712no—
201C2×C4—276no—
202C2×C3—33no—
203C2×C3—33no—
204C2×C3—31no—
205C2×C3—31no—
206C2×C3—31no—
207C2×C3—481no—
208C2×C3—961no—
209C2×C3—1442no—
210C2×C3D3×C22⋊A411yescharacteristic
211D3—32no—
212D3—92no—
213D3—96no—
214D3—92no—
215C22—31no—
216C22—31no—
217C22—91no—
218C22—272no—
219C22—272no—
220C22—272no—
221C22—272no—
222C22—271no—
223C22#8862311yescharacteristic, complemented
224C4—34no—
225C4—274no—
226C3—31no—
227C3—481no—
228C3—961no—
229C3D3×C22⋊SL(2,𝔽3)11yescharacteristic
230C2—31no—
231C2—31no—
232C2—92no—
233C2—272no—
234C2(C3⋊D3)⋊(C22⋊A4)11yescharacteristic
subgroupG/H
center, hypercenterC2(C3⋊D3)⋊(C22⋊A4)
commutator, Fitting, cosocle, hypocenterC22×C32×Q8C2×C3
FrattiniC2×C3D3×C22⋊A4
socleC23×C3D3×A4
LCSG ▷ C22×C32×Q8
UCSC1 ◁ C2

Complements

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

NH ≅ G/Ntype
1. #11722587C2semidirect
2. C22×Q8×C3⋊D3C3semidirect
6. C22×C32×Q8C2×C3semidirect
12. C3×C22⋊SL(2,𝔽3)D3semidirect
27. Q8×C3⋊D3A4semidirect
50. C22×C3⋊D3SL(2,𝔽3)semidirect
55. C32×Q8C2×A4semidirect
93. C22×C32C2×SL(2,𝔽3)semidirect
108. C22×Q8(C3⋊D3)⋊C3semidirect
148. C3⋊D3C22⋊SL(2,𝔽3)semidirect
188. C32C2×C22⋊SL(2,𝔽3)semidirect
197. Q8(C3⋊D3)⋊A4semidirect
223. C22#88623semidirect

Maximal subgroups

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

subgroupindex
#117225872
C22×Q8×C3⋊D33
D3×C22⋊SL(2,𝔽3)3
C2×(C3⋊D3)⋊A44
#886234