Finite Group Databasebeta

← C2×#11724182 #13180756

Subgroups — C2×#11724182 #13180756

Order 1728.

1086 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 1086 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⋊GL(2,𝔽3))C211yesmaximal, characteristic, complemented
3C2×C32×#226C211yesmaximal, characteristic, complemented
4#11724182C214yesmaximal, direct product factor
5C2×#64202—34nomaximal
6C2×#64127—31nomaximal
7C2×C32×SL(2,𝔽3)C2211yescharacteristic
8C2×#6292—41nomaximal
9C3⋊D3×SL(2,𝔽3)C2212yescomplemented
10C3⋊(C3⋊GL(2,𝔽3))C2212yescomplemented
11C32×#226C2212yescomplemented
12C2×#3832—31no—
13C2×Q8×C3⋊D3D311yescharacteristic, complemented
14C2×D3×SL(2,𝔽3)—34no—
15C2×C3⋊GL(2,𝔽3)—34no—
16C2×C3⋊GL(2,𝔽3)—64no—
17C2×C3×#226D314yescomplemented
18#64202—316no—
19C2×C32×Q16—31no—
20C2×#3756—31no—
21C2×#3755—31no—
22#64127—34no—
23#64127—34no—
24C2×#3744—31no—
25C2×#3743—31no—
26C22×C3⋊(C3⋊D3)—41no—
27C22×C3×C3⋊D3—41no—
28C2×C32×Dic3—41no—
29C32×SL(2,𝔽3)C2311yescharacteristic
30#6292—44no—
31C2×#983—91no—
32C2×#918—94no—
33C2×C32×Q8—31no—
34C2×C32×Q8C2×D311yescharacteristic
35#3832—34no—
36#3832—62no—
37Q8×C3⋊D3—32no—
38Q8×C3⋊D3C2×D312yescomplemented
39C2×C4⋊(C3⋊D3)—31no—
40C2×C4⋊(C3⋊D3)—61no—
41C2×C4×C3⋊D3—31no—
42C2×C4×C3⋊D3—61no—
43C2×#634—31no—
44C2×C3×SL(2,𝔽3)—24no—
45C2×C3×SL(2,𝔽3)C2×D314yes—
46C2×#626—124no—
47D3×SL(2,𝔽3)—38no—
48C3⋊GL(2,𝔽3)—38no—
49C3⋊GL(2,𝔽3)—68no—
50C3×#226C2×D318yescomplemented
51C32×Q16—32no—
52C32×Q16—32no—
53C2×C32×C8—31no—
54#3756—34no—
55#3755—32no—
56#3755—32no—
57C2×C3⋊(C3⋊C8)—31no—
58#3744—34no—
59#3743—34no—
60C22×C33—41no—
61C2×C3⋊(C3⋊D3)—42no—
62C2×C3⋊(C3⋊D3)—82no—
63C2×C3×C3⋊D3—42no—
64C2×C3×C3⋊D3—82no—
65C32×Dic3—42no—
66C2×#239—94no—
67C2×D3×Q8—34no—
68C22×SL(2,𝔽3)—91no—
69#983—94no—
70C2×GL(2,𝔽3)—91no—
71C2×GL(2,𝔽3)—184no—
72C2×#226C3⋊D311yescharacteristic, complemented
73C2×C3×Q16—34no—
74C2×#216—94no—
75C2×#215—94no—
76#918—916no—
77#918—916no—
78C2×#204—94no—
79C2×ΓL(1,𝔽25)—94no—
80C22×C3⋊D3—61no—
81C22×C3⋊D3—124no—
82C22×C3⋊D3—244no—
83C22×C3⋊D3S411yescharacteristic
84C22×C3×D3—124no—
85C32×Q8—31no—
86C32×Q8—32no—
87C32×Q8—61no—
88C32×Q8C22×D311yescharacteristic
89C2×C32×C4—31no—
90C2×C32×C4—61no—
91C2×C3⋊Dic3—31no—
92C4⋊(C3⋊D3)—32no—
93C4⋊(C3⋊D3)—61no—
94C4⋊(C3⋊D3)—64no—
95C4×C3⋊D3—32no—
96C4×C3⋊D3—32no—
97C4×C3⋊D3—64no—
98#634—32no—
99#634—61no—
100C2×C3×Dic3—44no—
101C3×SL(2,𝔽3)—24no—
102C3×SL(2,𝔽3)C22×D314yes—
103#626—1216no—
104C32×C8—32no—
105C3⋊(C3⋊C8)—32no—
106C2×#137—271noSylow
107C2×C33—42no—
108C2×C33—41no—
109C3⋊(C3⋊D3)—82no—
110C3×C3⋊D3—82no—
111C2×C3×Q8—34no—
112C2×C3×Q8D3214yes—
113C2×#67—361no—
114#239—916no—
115#239—188no—
116D3×Q8—38no—
117D3×Q8—98no—
118C2×D12—94no—
119C2×D12—184no—
120C2×C4×D3—94no—
121C2×C4×D3—184no—
122C2×Dic6—94no—
123C2×SL(2,𝔽3)—24no—
124C2×SL(2,𝔽3)—92no—
125C2×SL(2,𝔽3)C2×C3⋊D311yescharacteristic
126GL(2,𝔽3)—92no—
127GL(2,𝔽3)—188no—
128#226C2×C3⋊D312yescomplemented
129C3×Q16—38no—
130C3×Q16—38no—
131C2×C3×C8—34no—
132#216—916no—
133#215—98no—
134#215—98no—
135C2×C3⋊C8—94no—
136#204—916no—
137ΓL(1,𝔽25)—916no—
138C22×C32—44no—
139C22×C32—84no—
140C22×C32C2×S411yescharacteristic
141C2×C3⋊D3—22no—
142C2×C3⋊D3—62no—
143C2×C3⋊D3—128no—
144C2×C3⋊D3—122no—
145C2×C3⋊D3—248no—
146C2×C3⋊D3—248no—
147C2×C3⋊D3—2416no—
148C2×C3⋊D3C2×S412yes—
149C2×C3×D3—128no—
150C2×C3×D3—248no—
151C32×C4—31no—
152C32×C4—31no—
153C32×C4—62no—
154C3⋊Dic3—32no—
155C3×Dic3—48no—
156C2×#41—271no—
157C22×Q8—91no—
158#137—274no—
159#137—274no—
160C2×Q16—31no—
161C2×Q16—271no—
162C2×SD16—271no—
163C2×SD16—271no—
164C2×M16—271no—
165C33—41noSylow
166C23×C3—361no—
167C22×D3—34no—
168C22×D3—184no—
169C22×D3—361no—
170C22×D3—724no—
171C3×Q8—34no—
172C3×Q8—38no—
173C3×Q8—64no—
174C3×Q8C2×D3214yes—
175C2×C3×C4—34no—
176C2×C3×C4—64no—
177#67—364no—
178C2×Dic3—41no—
179C2×Dic3—94no—
180D12—98no—
181D12—184no—
182D12—1816no—
183C4×D3—98no—
184C4×D3—98no—
185C4×D3—1816no—
186Dic6—98no—
187Dic6—184no—
188SL(2,𝔽3)—24no—
189SL(2,𝔽3)C22×C3⋊D311yescharacteristic
190C3×C8—38no—
191C3⋊C8—98no—
192C2×C32—44no—
193C2×C32—48no—
194C2×C32—84no—
195C2×C32—88no—
196C2×C32#98312yescomplemented
197C2×C32C22×S411yescharacteristic
198C3⋊D3—22no—
199C3⋊D3—122no—
200C3⋊D3—248no—
201C3⋊D3—2416no—
202C3×D3—248no—
203#41—274no—
204#41—542no—
205C2×Q8—31no—
206C2×Q8—92no—
207C2×Q8—271no—
208C2×Q8—272no—
209C2×Q8D3×C3⋊D311yescharacteristic
210C2×D4—271no—
211C2×D4—541no—
212C22×C4—271no—
213C22×C4—541no—
214Q16—32no—
215Q16—32no—
216Q16—274no—
217SD16—274no—
218SD16—272no—
219SD16—272no—
220M16—274no—
221C2×C8—31no—
222C2×C8—271no—
223C22×C3—41no—
224C22×C3—84no—
225C22×C3—362no—
226C22×C3—722no—
227C22×C3D3×S414yes—
228C2×D3—38no—
229C2×D3—68no—
230C2×D3—188no—
231C2×D3—362no—
232C2×D3—368no—
233C2×D3—728no—
234C2×D3—722no—
235C2×D3—7216no—
236C3×C4—34no—
237C3×C4—34no—
238C3×C4—68no—
239Dic3—42no—
240Dic3—98no—
241C32—44no—
242C32—84no—
243C32C2×#98311yescharacteristic, complemented
244C23—91no—
245C23—541no—
246Q8—31no—
247Q8—32no—
248Q8—61no—
249Q8—272no—
250Q8—541no—
251Q8C2×D3×C3⋊D311yescharacteristic
252D4—272no—
253D4—541no—
254D4—544no—
255C2×C4—31no—
256C2×C4—61no—
257C2×C4—272no—
258C2×C4—272no—
259C2×C4—271no—
260C2×C4—544no—
261C8—32no—
262C8—272no—
263C2×C3—42no—
264C2×C3—41no—
265C2×C3—88no—
266C2×C3—84no—
267C2×C3—722no—
268C2×C3#6420218yescomplemented
269C2×C3C2×D3×S414yes—
270D3—68no—
271D3—368no—
272D3—722no—
273D3—7216no—
274C22—92no—
275C22—182no—
276C22—542no—
277C22—1082no—
278C22C3⋊D3×S411yescharacteristic
279C4—31no—
280C4—31no—
281C4—62no—
282C4—272no—
283C3—41no—
284C3—84no—
285C3C2×#6420214yescomplemented
286C2—182no—
287C2—1082no—
288C2#1172418212yesdirect product factor
289C2C2×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⋊GL(2,𝔽3))C2semidirect
3. C2×C32×#226C2semidirect
4. #11724182C2direct (N × H)
9. C3⋊D3×SL(2,𝔽3)C22semidirect
10. C3⋊(C3⋊GL(2,𝔽3))C22semidirect
11. C32×#226C22semidirect
13. C2×Q8×C3⋊D3D3semidirect
17. C2×C3×#226D3semidirect
38. Q8×C3⋊D3C2×D3semidirect
50. C3×#226C2×D3semidirect
72. C2×#226C3⋊D3semidirect
128. #226C2×C3⋊D3semidirect
196. C2×C32#983semidirect
243. C32C2×#983semidirect
268. C2×C3#64202semidirect
285. C3C2×#64202semidirect
288. C2#11724182direct (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⋊GL(2,𝔽3))2
C2×C32×#2262
#117241822
C2×#642023
C2×#641273
C2×#62924