Finite Group Databasebeta

← C3⋊D3×#857 #13165921

Subgroups — C3⋊D3×#857 #13165921

Order 1728.

552 conjugacy classes of subgroups; only the 36 normal subgroups are shown.

normal subgroups only

This group has 552 conjugacy classes of subgroups — too many to draw. Only its 36 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⋊D3×C42⋊C3C211yesmaximal, characteristic, complemented
2C3⋊(C3⋊#857)C211yesmaximal, characteristic, complemented
3C32×#857C211yesmaximal, characteristic, complemented
4D3×#857—34nomaximal
5C3⋊D3×#104—31nomaximal
6C3⋊D3×S4—41nomaximal
7C32×C42⋊C3C2211yescharacteristic, complemented
8#41×C3⋊D3—31no—
9M16×C3⋊D3—31no—
10C42×C3⋊D3D311yescharacteristic, complemented
11D3×C42⋊C3—34no—
12C3⋊#857—34no—
13C3⋊#857—64no—
14C3×#857D314yescomplemented
15C32×#104—31no—
16#63664—31no—
17#63652—31no—
18#63632—31no—
19A4×C3⋊D3—41no—
20C3⋊(C3⋊S4)—41no—
21C32×S4—41no—
22C2×#857—91no—
23D3×#104—94no—
24D3×S4—124no—
25C32×#41—31no—
26#3832—31no—
27Q8×C3⋊D3—31no—
28#3830—31no—
29#3830—61no—
30D4×C3⋊D3—31no—
31D4×C3⋊D3—61no—
32#3828—31no—
33#3828—61no—
34C2×C4×C3⋊D3—31no—
35C2×C4×C3⋊D3—61no—
36C32×M16—31no—
37C32×C42C2×D311yescharacteristic, complemented
38C4×C3⋊Dic3—31no—
39#3748—31no—
40#3743—61no—
41C8×C3⋊D3—61no—
42C3×C42⋊C3—24no—
43C3×C42⋊C3C2×D314yescomplemented
44C32×A4—41no—
45D3×C3⋊D3—161no—
46D3×#41—94no—
47D3×M16—94no—
48C42×D3—34no—
49C2×C42⋊C3—91no—
50#857—91no—
51#857—184no—
52#857C3⋊D311yescharacteristic, direct product factor
53C3×#104—34no—
54#837—94no—
55#825—94no—
56#805—94no—
57C22×C3⋊D3—61no—
58C22×C3⋊D3S411yescharacteristic
59D3×A4—124no—
60C3⋊S4—124no—
61C3⋊S4—244no—
62C3×S4—44no—
63C32×Q8—31no—
64C32×D4—31no—
65C32×D4—61no—
66C2×C32×C4—31no—
67C2×C32×C4—61no—
68#638—61no—
69#638—61no—
70#638—61no—
71C2×C3⋊Dic3—31no—
72C2×C3⋊Dic3—61no—
73C4⋊(C3⋊D3)—31no—
74C4⋊(C3⋊D3)—61no—
75C4×C3⋊D3—31no—
76C4×C3⋊D3—31no—
77C4×C3⋊D3—31no—
78C4×C3⋊D3—31no—
79C4×C3⋊D3—61no—
80C4×C3⋊D3—61no—
81C4×C3⋊D3—61no—
82#634—31no—
83#634—61no—
84C32×C8—61no—
85C3⋊(C3⋊C8)—61no—
86C2×#104—271noSylow
87C3⋊(C3⋊D3)—161no—
88C3×C3⋊D3—161no—
89C32×D3—161no—
90C2×S4—361no—
91C3×#41—34no—
92#239—94no—
93D3×Q8—94no—
94#237—94no—
95#237—184no—
96D3×D4—94no—
97D3×D4—184no—
98#235—94no—
99#235—184no—
100C2×C4×D3—94no—
101C2×C4×D3—184no—
102C3×M16—34no—
103C3×C42D3214yescomplemented
104C4×Dic3—94no—
105#208—94no—
106ΓL(1,𝔽25)—184no—
107D3×C8—184no—
108C42⋊C3—24no—
109C42⋊C3C2×C3⋊D311yescharacteristic, complemented
110C22×C32—61no—
111C22×C32C2×S411yescharacteristic
112C2×C3⋊D3—31no—
113C2×C3⋊D3—31no—
114C2×C3⋊D3—61no—
115C2×C3⋊D3—121no—
116C2×C3⋊D3—121no—
117C3×A4—44no—
118C3×A4—84no—
119D32—484no—
120C32×C4—31no—
121C32×C4—31no—
122C32×C4—61no—
123C3⋊Dic3—31no—
124C3⋊Dic3—31no—
125C3⋊Dic3—61no—
126C2×#41—271no—
127C2×M16—271no—
128C2×C42—91no—
129#104—31no—
130#104—271no—
131#104—271no—
132#104—271no—
133C33—161noSylow
134C22×D3—34no—
135C22×D3—184no—
136C2×A4—361no—
137S4—41no—
138S4—361no—
139S4—724no—
140C3×Q8—34no—
141C3×D4—34no—
142C3×D4—64no—
143C2×C3×C4—34no—
144C2×C3×C4—64no—
145#67—184no—
146#67—184no—
147#67—184no—
148C2×Dic3—94no—
149C2×Dic3—184no—
150D12—94no—
151D12—184no—
152C4×D3—94no—
153C4×D3—94no—
154C4×D3—94no—
155C4×D3—94no—
156C4×D3—184no—
157C4×D3—184no—
158C4×D3—184no—
159Dic6—94no—
160Dic6—184no—
161C3×C8—64no—
162C3⋊C8—184no—
163C2×C32—31no—
164C2×C32—121no—
165C3⋊D3—31no—
166C3⋊D3—121no—
167C3⋊D3—484no—
168C3⋊D3—964no—
169C3⋊D3#85711yescharacteristic, direct product factor
170C3×D3—164no—
171C3×D3—484no—
172#41—31no—
173#41—271no—
174#41—271no—
175#41—271no—
176#41—541no—
177#41—541no—
178C2×Q8—271no—
179C2×D4—271no—
180C2×D4—541no—
181C22×C4—271no—
182C22×C4—541no—
183M16—31no—
184M16—271no—
185M16—541no—
186C2×C8—541no—
187C42—271no—
188C42D3×C3⋊D311yescharacteristic, complemented
189C22×C3—64no—
190C22×C3D3×S414yes—
191C2×D3—94no—
192C2×D3—94no—
193C2×D3—184no—
194C2×D3—364no—
195C2×D3—364no—
196C2×D3—1441no—
197A4—41no—
198A4—84no—
199C3×C4—34no—
200C3×C4—34no—
201C3×C4—64no—
202Dic3—94no—
203Dic3—94no—
204Dic3—184no—
205C32—164no—
206C32—324no—
207C32C2×#85711yescharacteristic, complemented
208C23—91no—
209C23—541no—
210Q8—31no—
211Q8—271no—
212Q8—541no—
213D4—31no—
214D4—61no—
215D4—271no—
216D4—541no—
217D4—541no—
218D4—541no—
219D4—541no—
220C2×C4—31no—
221C2×C4—61no—
222C2×C4—271no—
223C2×C4—271no—
224C2×C4—271no—
225C2×C4—271no—
226C2×C4—271no—
227C2×C4—541no—
228C2×C4—541no—
229C2×C4—541no—
230C2×C4—541no—
231C8—61no—
232C8—541no—
233C2×C3—34no—
234C2×C3—124no—
235C2×C3—1441no—
236D3—34no—
237D3—94no—
238D3—161no—
239D3—2884no—
240D3—364no—
241D3—1441no—
242C22—61no—
243C22—271no—
244C22—271no—
245C22—541no—
246C22—1081no—
247C22—1081no—
248C22C3⋊D3×S411yescharacteristic
249C4—31no—
250C4—31no—
251C4—61no—
252C4—271no—
253C4—271no—
254C4—541no—
255C3—161no—
256C3—324no—
257C3D3×#85714yescomplemented
258C2—31no—
259C2—91no—
260C2—121no—
261C2—271no—
262C2—1081no—
subgroupG/H
center, hypercenterC1G
commutator, cosocle, hypocenterC32×C42⋊C3C22
FrattiniC22C3⋊D3×S4
FittingC32×C42C2×D3
socleC22×C32C2×S4
LCSG ▷ C32×C42⋊C3
UCSC1

Complements

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

NH ≅ G/Ntype
1. C3⋊D3×C42⋊C3C2semidirect
2. C3⋊(C3⋊#857)C2semidirect
3. C32×#857C2semidirect
7. C32×C42⋊C3C22semidirect
10. C42×C3⋊D3D3semidirect
14. C3×#857D3semidirect
37. C32×C42C2×D3semidirect
43. C3×C42⋊C3C2×D3semidirect
52. #857C3⋊D3direct (N × H)
103. C3×C42D32semidirect
109. C42⋊C3C2×C3⋊D3semidirect
169. C3⋊D3#857direct (N × H)
188. C42D3×C3⋊D3semidirect
207. C32C2×#857semidirect
257. C3D3×#857semidirect

Maximal subgroups

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

subgroupindex
C3⋊D3×C42⋊C32
C3⋊(C3⋊#857)2
C32×#8572
D3×#8573
C3⋊D3×#1043
C3⋊D3×S44