Finite Group Databasebeta

← C2×C3⋊D3×#226 #13180752

Subgroups — C2×C3⋊D3×#226 #13180752

Order 1728.

1022 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 1022 conjugacy classes of subgroups — too many to draw. Only its 97 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⋊D3×SL(2,𝔽3)C211yesmaximal, characteristic
2C2×C3⋊(C3⋊#226)C211yesmaximal, characteristic, complemented
3C2×C32×#226C211yesmaximal, characteristic, complemented
4C3⋊D3×#226C214yesmaximal, direct product factor
5C2×D3×#226—34nomaximal
6C2×Q16×C3⋊D3—31nomaximal
7C2×C32×SL(2,𝔽3)C2211yescharacteristic
8C2×Dic3×C3⋊D3—41nomaximal
9C3⋊D3×SL(2,𝔽3)C2212yes—
10C3⋊(C3⋊#226)C2212yescomplemented
11C32×#226C2212yescomplemented
12C2×Q8×C3⋊D3—31no—
13C2×Q8×C3⋊D3D311yescharacteristic
14C2×D3×SL(2,𝔽3)—34no—
15C2×C3⋊#226—34no—
16C2×C3⋊#226—64no—
17C2×C3×#226D314yescomplemented
18D3×#226—316no—
19C2×C32×Q16—31no—
20C2×#3756—31no—
21C2×#3756—31no—
22Q16×C3⋊D3—34no—
23Q16×C3⋊D3—34no—
24C2×#3746—31no—
25C2×C8×C3⋊D3—31no—
26C22×C3×C3⋊D3—41no—
27C2×C3⋊(C3⋊Dic3)—41no—
28C2×C32×Dic3—41no—
29C32×SL(2,𝔽3)C2311yescharacteristic
30Dic3×C3⋊D3—44no—
31C22×#226—91no—
32C2×D3×Q16—94no—
33C2×C32×Q8—31no—
34C2×C32×Q8C2×D311yescharacteristic
35Q8×C3⋊D3—32no—
36Q8×C3⋊D3—34no—
37Q8×C3⋊D3—62no—
38Q8×C3⋊D3C2×D312yes—
39C2×C4×C3⋊D3—31no—
40C2×C4×C3⋊D3—61no—
41C2×#634—31no—
42C2×#634—31no—
43C2×#634—61no—
44C2×C3×SL(2,𝔽3)—24no—
45C2×C3×SL(2,𝔽3)C2×D314yes—
46C2×D3×Dic3—124no—
47D3×SL(2,𝔽3)—38no—
48C3⋊#226—38no—
49C3⋊#226—68no—
50C3×#226C2×D318yescomplemented
51C32×Q16—32no—
52C32×Q16—32no—
53C2×C32×C8—31no—
54#3756—34no—
55#3756—32no—
56#3756—32no—
57C2×C3⋊(C3⋊C8)—31no—
58#3746—34no—
59C8×C3⋊D3—34no—
60C22×C33—41no—
61C2×C3×C3⋊D3—44no—
62C2×C3×C3⋊D3—42no—
63C3⋊(C3⋊Dic3)—42no—
64C32×Dic3—42no—
65C2×D3×Q8—34no—
66C2×D3×Q8—94no—
67C22×SL(2,𝔽3)—91no—
68C2×#226—91no—
69C2×#226—94no—
70C2×#226—184no—
71C2×#226C3⋊D311yescharacteristic, direct product factor
72C2×C3×Q16—34no—
73C2×#216—94no—
74C2×#216—94no—
75D3×Q16—916no—
76D3×Q16—916no—
77C2×Dic12—94no—
78C2×D3×C8—94no—
79C22×C3⋊D3S411yescharacteristic
80C22×C3×D3—124no—
81C32×Q8—31no—
82C32×Q8—32no—
83C32×Q8—61no—
84C32×Q8C22×D311yescharacteristic
85C2×C32×C4—31no—
86C2×C32×C4—61no—
87C2×C3⋊Dic3—31no—
88C2×C3⋊Dic3—61no—
89C2×C3⋊Dic3—124no—
90C2×C3⋊Dic3—244no—
91C4×C3⋊D3—32no—
92C4×C3⋊D3—32no—
93C4×C3⋊D3—64no—
94#634—32no—
95#634—32no—
96#634—61no—
97#634—64no—
98#634—61no—
99C2×C3×Dic3—44no—
100C3×SL(2,𝔽3)—24no—
101C3×SL(2,𝔽3)C22×D314yes—
102D3×Dic3—1216no—
103C32×C8—32no—
104C3⋊(C3⋊C8)—32no—
105C22×Q16—271noSylow
106C2×C33—42no—
107C2×C33—41no—
108C3×C3⋊D3—44no—
109C2×C3×Q8—34no—
110C2×C3×Q8D3214yes—
111C22×Dic3—361no—
112D3×Q8—38no—
113D3×Q8—98no—
114D3×Q8—916no—
115D3×Q8—188no—
116C2×C4×D3—94no—
117C2×C4×D3—184no—
118C2×Dic6—94no—
119C2×Dic6—94no—
120C2×Dic6—184no—
121C2×SL(2,𝔽3)—24no—
122C2×SL(2,𝔽3)—92no—
123C2×SL(2,𝔽3)C2×C3⋊D311yescharacteristic
124#226—92no—
125#226—188no—
126#226C2×C3⋊D312yesdirect product factor
127C3×Q16—38no—
128C3×Q16—38no—
129C2×C3×C8—34no—
130#216—916no—
131#216—98no—
132#216—98no—
133C2×C3⋊C8—94no—
134Dic12—916no—
135D3×C8—916no—
136C22×C32—44no—
137C22×C32—84no—
138C22×C32C2×S411yescharacteristic
139C2×C3⋊D3#22614yesdirect product factor
140C2×C3⋊D3C2×S412yes—
141C2×C3×D3—128no—
142C2×C3×D3—1216no—
143C32×C4—31no—
144C32×C4—31no—
145C32×C4—62no—
146C3⋊Dic3—32no—
147C3⋊Dic3—62no—
148C3⋊Dic3—128no—
149C3⋊Dic3—248no—
150C3×Dic3—48no—
151C22×Q8—91no—
152C22×Q8—271no—
153C2×Q16—31no—
154C2×Q16—271no—
155C2×Q16—274no—
156C2×Q16—274no—
157C2×Q16—271no—
158C2×Q16—271no—
159C22×C8—271no—
160C33—41noSylow
161C23×C3—361no—
162C22×D3—34no—
163C3×Q8—34no—
164C3×Q8—38no—
165C3×Q8—64no—
166C3×Q8C2×D3214yes—
167C2×C3×C4—34no—
168C2×C3×C4—64no—
169C2×Dic3—41no—
170C2×Dic3—94no—
171C2×Dic3—184no—
172C2×Dic3—364no—
173C2×Dic3—361no—
174C2×Dic3—724no—
175C4×D3—98no—
176C4×D3—98no—
177C4×D3—1816no—
178Dic6—98no—
179Dic6—98no—
180Dic6—184no—
181Dic6—1816no—
182Dic6—184no—
183SL(2,𝔽3)—24no—
184SL(2,𝔽3)C22×C3⋊D311yescharacteristic
185C3×C8—38no—
186C3⋊C8—98no—
187C2×C32—44no—
188C2×C32—48no—
189C2×C32—84no—
190C2×C32—88no—
191C2×C32C2×#22612yescomplemented
192C2×C32C22×S411yescharacteristic
193C3⋊D3C2×#22614yesdirect product factor
194C3×D3—1216no—
195C2×Q8—31no—
196C2×Q8—92no—
197C2×Q8—271no—
198C2×Q8—271no—
199C2×Q8—272no—
200C2×Q8—274no—
201C2×Q8—541no—
202C2×Q8—542no—
203C2×Q8D3×C3⋊D311yescharacteristic
204C22×C4—271no—
205C22×C4—541no—
206Q16—32no—
207Q16—32no—
208Q16—274no—
209Q16—272no—
210Q16—274no—
211Q16—272no—
212C2×C8—31no—
213C2×C8—274no—
214C2×C8—271no—
215C22×C3—41no—
216C22×C3—84no—
217C22×C3—362no—
218C22×C3—364no—
219C22×C3D3×S414yes—
220C2×D3—38no—
221C2×D3—316no—
222C3×C4—34no—
223C3×C4—34no—
224C3×C4—68no—
225Dic3—42no—
226Dic3—98no—
227Dic3—188no—
228Dic3—362no—
229Dic3—728no—
230C32—44no—
231C32—84no—
232C32C22×#22611yescharacteristic, complemented
233C23—91no—
234Q8—31no—
235Q8—32no—
236Q8—61no—
237Q8—272no—
238Q8—272no—
239Q8—541no—
240Q8—541no—
241Q8—544no—
242Q8C2×D3×C3⋊D311yescharacteristic
243C2×C4—31no—
244C2×C4—61no—
245C2×C4—272no—
246C2×C4—272no—
247C2×C4—271no—
248C2×C4—544no—
249C2×C4—541no—
250C8—32no—
251C8—272no—
252C2×C3—42no—
253C2×C3—41no—
254C2×C3—88no—
255C2×C3—84no—
256C2×C3—364no—
257C2×C3D3×#22618yescomplemented
258C2×C3C2×D3×S414yes—
259D3—316no—
260C22—94no—
261C22—92no—
262C22C3⋊D3×S411yescharacteristic
263C4—31no—
264C4—31no—
265C4—62no—
266C4—272no—
267C4—542no—
268C3—41no—
269C3—84no—
270C3C2×D3×#22614yescomplemented
271C2—94no—
272C2C3⋊D3×#22612yesdirect product factor
273C2C2×C3⋊D3×S411yescharacteristic
subgroupG/H
center, hypercenterC22C3⋊D3×S4
commutator, cosocle, hypocenterC32×SL(2,𝔽3)C23
FrattiniC2C2×C3⋊D3×S4
FittingC2×C32×Q8C2×D3
socleC22×C32C2×S4
prime radicalC2×C3⋊D3×SL(2,𝔽3)C2
LCSG ▷ C32×SL(2,𝔽3)
UCSC1 ◁ C22

Complements

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

NH ≅ G/Ntype
2. C2×C3⋊(C3⋊#226)C2semidirect
3. C2×C32×#226C2semidirect
4. C3⋊D3×#226C2direct (N × H)
10. C3⋊(C3⋊#226)C22semidirect
11. C32×#226C22semidirect
17. C2×C3×#226D3semidirect
50. C3×#226C2×D3semidirect
71. C2×#226C3⋊D3direct (N × H)
126. #226C2×C3⋊D3direct (N × H)
139. C2×C3⋊D3#226direct (N × H)
191. C2×C32C2×#226semidirect
193. C3⋊D3C2×#226direct (N × H)
232. C32C22×#226semidirect
257. C2×C3D3×#226semidirect
270. C3C2×D3×#226semidirect
272. C2C3⋊D3×#226direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3⋊D3×SL(2,𝔽3)2
C2×C3⋊(C3⋊#226)2
C2×C32×#2262
C3⋊D3×#2262
C2×D3×#2263
C2×Q16×C3⋊D33
C2×Dic3×C3⋊D34