Finite Group Databasebeta

← C2×#11724181 #13180755

Subgroups — C2×#11724181 #13180755

Order 1728.

1022 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 1022 conjugacy classes of subgroups — too many to draw. Only its 89 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, complemented
2C2×C3⋊(C3⋊#226)C211yesmaximal, characteristic, complemented
3C2×C32×GL(2,𝔽3)C211yesmaximal, characteristic, complemented
4#11724181C214yesmaximal, direct product factor
5C2×#64201—34nomaximal
6C2×#64124—31nomaximal
7C2×C32×SL(2,𝔽3)C2211yescharacteristic
8C2×#6290—41nomaximal
9C3⋊D3×SL(2,𝔽3)C2212yescomplemented
10C3⋊(C3⋊#226)C2212yescomplemented
11C32×GL(2,𝔽3)C2212yescomplemented
12C2×Q8×C3⋊D3D311yescharacteristic, complemented
13C2×#3830—31no—
14C2×D3×SL(2,𝔽3)—34no—
15C2×C3⋊#226—34no—
16C2×C3⋊#226—64no—
17C2×C3×GL(2,𝔽3)D314yescomplemented
18#64201—316no—
19C2×C32×SD16—31no—
20C2×#3756—31no—
21C2×#3754—31no—
22#64124—34no—
23#64124—34no—
24C2×#3746—31no—
25C2×#3743—31no—
26C22×C3×C3⋊D3—41no—
27C22×C32×D3—41no—
28C2×C3⋊(C3⋊Dic3)—41no—
29C32×SL(2,𝔽3)C2311yescharacteristic
30#6290—44no—
31C2×#983—91no—
32C2×#915—94no—
33C2×C32×Q8C2×D311yescharacteristic
34C2×C32×D4—31no—
35C2×#638—61no—
36C22×C3⋊Dic3—61no—
37Q8×C3⋊D3—32no—
38Q8×C3⋊D3C2×D312yescomplemented
39#3830—34no—
40#3830—62no—
41C2×C4×C3⋊D3—31no—
42C2×#634—31no—
43C2×#634—31no—
44C2×C3×SL(2,𝔽3)—24no—
45C2×C3×SL(2,𝔽3)C2×D314yes—
46C2×#625—124no—
47D3×SL(2,𝔽3)—38no—
48C3⋊#226—38no—
49C3⋊#226—68no—
50C3×GL(2,𝔽3)C2×D318yescomplemented
51C32×SD16—32no—
52C32×SD16—32no—
53C2×C32×C8—31no—
54#3756—32no—
55#3756—32no—
56#3754—34no—
57C2×C3⋊(C3⋊C8)—31no—
58#3746—34no—
59#3743—34no—
60C22×C33—41no—
61C2×C3×C3⋊D3—42no—
62C2×C3×C3⋊D3—82no—
63C2×C32×D3—42no—
64C2×C32×D3—82no—
65C3⋊(C3⋊Dic3)—42no—
66C2×D3×Q8—34no—
67C2×#237—94no—
68C22×SL(2,𝔽3)—91no—
69#983—94no—
70C2×GL(2,𝔽3)C3⋊D311yescharacteristic, complemented
71C2×#226—91no—
72C2×#226—184no—
73C2×C3×SD16—34no—
74C2×#216—94no—
75C2×#214—94no—
76#915—916no—
77#915—916no—
78C2×Dic12—94no—
79C2×ΓL(1,𝔽25)—94no—
80C23×C32—61no—
81C22×C3⋊D3S411yescharacteristic
82C22×C3×D3—44no—
83C22×C3×D3—124no—
84C32×Q8—31no—
85C32×Q8C22×D311yescharacteristic
86C32×D4—32no—
87C32×D4—61no—
88C2×C32×C4—31no—
89#638—64no—
90C2×C3⋊Dic3—31no—
91C2×C3⋊Dic3—64no—
92C2×C3⋊Dic3—61no—
93C2×C3⋊Dic3—124no—
94C2×C3⋊Dic3—244no—
95C4×C3⋊D3—32no—
96C4×C3⋊D3—32no—
97#634—32no—
98#634—32no—
99#634—61no—
100#634—61no—
101C3×SL(2,𝔽3)—24no—
102C3×SL(2,𝔽3)C22×D314yes—
103#625—1216no—
104C32×C8—32no—
105C3⋊(C3⋊C8)—32no—
106C2×#137—271noSylow
107C2×C33—42no—
108C2×C33—41no—
109C3×C3⋊D3—82no—
110C32×D3—82no—
111C2×C3×Q8D3214yes—
112C2×C3×D4—34no—
113C2×#67—184no—
114C2×#67—361no—
115C22×Dic3—184no—
116D3×Q8—38no—
117D3×Q8—98no—
118#237—916no—
119#237—188no—
120C2×C4×D3—94no—
121C2×Dic6—94no—
122C2×Dic6—94no—
123C2×SL(2,𝔽3)—24no—
124C2×SL(2,𝔽3)—92no—
125C2×SL(2,𝔽3)C2×C3⋊D311yescharacteristic
126GL(2,𝔽3)C2×C3⋊D312yescomplemented
127#226—92no—
128#226—188no—
129C3×SD16—38no—
130C3×SD16—38no—
131C2×C3×C8—34no—
132#216—98no—
133#216—98no—
134#214—916no—
135C2×C3⋊C8—94no—
136Dic12—916no—
137ΓL(1,𝔽25)—916no—
138C22×C32—44no—
139C22×C32—62no—
140C22×C32—84no—
141C22×C32—122no—
142C22×C32C2×S411yescharacteristic
143C2×C3⋊D3—22no—
144C2×C3⋊D3C2×S412yes—
145C2×C3×D3—48no—
146C2×C3×D3—88no—
147C2×C3×D3—128no—
148C2×C3×D3—248no—
149C32×C4—31no—
150C32×C4—31no—
151C3⋊Dic3—32no—
152C3⋊Dic3—62no—
153C3⋊Dic3—128no—
154C3⋊Dic3—248no—
155C2×#41—271no—
156C22×Q8—91no—
157#137—274no—
158#137—274no—
159C2×Q16—271no—
160C2×Q16—271no—
161C2×SD16—31no—
162C2×SD16—271no—
163C2×M16—271no—
164C33—41noSylow
165C23×C3—64no—
166C23×C3—361no—
167C22×D3—34no—
168C22×D3—41no—
169C3×Q8—34no—
170C3×Q8C2×D3214yes—
171C3×D4—38no—
172C3×D4—64no—
173C2×C3×C4—34no—
174#67—1816no—
175#67—364no—
176C2×Dic3—94no—
177C2×Dic3—1816no—
178C2×Dic3—184no—
179C2×Dic3—361no—
180C2×Dic3—724no—
181C4×D3—98no—
182C4×D3—98no—
183Dic6—98no—
184Dic6—98no—
185Dic6—184no—
186Dic6—184no—
187SL(2,𝔽3)—24no—
188SL(2,𝔽3)C22×C3⋊D311yescharacteristic
189C3×C8—38no—
190C3⋊C8—98no—
191C2×C32—44no—
192C2×C32—48no—
193C2×C32—84no—
194C2×C32—88no—
195C2×C32—122no—
196C2×C32#98312yescomplemented
197C2×C32C22×S411yescharacteristic
198C3⋊D3—22no—
199C3×D3—88no—
200C3×D3—248no—
201#41—274no—
202#41—542no—
203C2×Q8—92no—
204C2×Q8—271no—
205C2×Q8—271no—
206C2×Q8—272no—
207C2×Q8D3×C3⋊D311yescharacteristic
208C2×D4—31no—
209C2×D4—541no—
210C22×C4—271no—
211C22×C4—541no—
212Q16—274no—
213Q16—272no—
214Q16—272no—
215SD16—32no—
216SD16—32no—
217SD16—274no—
218M16—274no—
219C2×C8—31no—
220C2×C8—271no—
221C22×C3—41no—
222C22×C3—68no—
223C22×C3—84no—
224C22×C3—128no—
225C22×C3—362no—
226C22×C3—722no—
227C22×C3D3×S414yes—
228C2×D3—38no—
229C2×D3—42no—
230C2×D3—68no—
231C2×D3—82no—
232C3×C4—34no—
233C3×C4—34no—
234Dic3—98no—
235Dic3—188no—
236Dic3—362no—
237Dic3—728no—
238C32—44no—
239C32—84no—
240C32C2×#98311yescharacteristic, complemented
241C23—61no—
242C23—91no—
243Q8—31no—
244Q8—272no—
245Q8—272no—
246Q8—541no—
247Q8—541no—
248Q8C2×D3×C3⋊D311yescharacteristic
249D4—32no—
250D4—61no—
251D4—544no—
252C2×C4—31no—
253C2×C4—272no—
254C2×C4—272no—
255C2×C4—271no—
256C2×C4—544no—
257C2×C4—541no—
258C8—32no—
259C8—272no—
260C2×C3—42no—
261C2×C3—41no—
262C2×C3—88no—
263C2×C3—84no—
264C2×C3—128no—
265C2×C3—722no—
266C2×C3#6420118yescomplemented
267C2×C3C2×D3×S414yes—
268D3—68no—
269D3—82no—
270C22—62no—
271C22—92no—
272C22—122no—
273C22—182no—
274C22C3⋊D3×S411yescharacteristic
275C4—31no—
276C4—31no—
277C4—272no—
278C4—542no—
279C3—41no—
280C3—84no—
281C3C2×#6420114yescomplemented
282C2—122no—
283C2—182no—
284C2#1172418112yesdirect product factor
285C2C2×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
LCSG ▷ C32×SL(2,𝔽3)
UCSC1 ◁ C22

Complements

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

NH ≅ G/Ntype
1. C2×C3⋊D3×SL(2,𝔽3)C2semidirect
2. C2×C3⋊(C3⋊#226)C2semidirect
3. C2×C32×GL(2,𝔽3)C2semidirect
4. #11724181C2direct (N × H)
9. C3⋊D3×SL(2,𝔽3)C22semidirect
10. C3⋊(C3⋊#226)C22semidirect
11. C32×GL(2,𝔽3)C22semidirect
12. C2×Q8×C3⋊D3D3semidirect
17. C2×C3×GL(2,𝔽3)D3semidirect
38. Q8×C3⋊D3C2×D3semidirect
50. C3×GL(2,𝔽3)C2×D3semidirect
70. C2×GL(2,𝔽3)C3⋊D3semidirect
126. GL(2,𝔽3)C2×C3⋊D3semidirect
196. C2×C32#983semidirect
240. C32C2×#983semidirect
266. C2×C3#64201semidirect
281. C3C2×#64201semidirect
284. C2#11724181direct (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×GL(2,𝔽3)2
#117241812
C2×#642013
C2×#641243
C2×#62904