Finite Group Databasebeta

← C2×D32×SL(2,𝔽3) #13180896

Subgroups — C2×D32×SL(2,𝔽3) #13180896

Order 1728.

1279 conjugacy classes of subgroups; only the 160 normal subgroups are shown.

normal subgroups only

This group has 1279 conjugacy classes of subgroups — too many to draw. Only its 160 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×D3×SL(2,𝔽3)C212yesmaximal, complemented
3D32×SL(2,𝔽3)C214yesmaximal, direct product factor
4C2×D32×Q8C311yesmaximal, characteristic, complemented
5C22×D3×SL(2,𝔽3)—32nomaximal
6C22×C3×D32—41nomaximal
7C2×C32×SL(2,𝔽3)C2211yescharacteristic, complemented
8C3⋊D3×SL(2,𝔽3)C2212yescomplemented
9C3×D3×SL(2,𝔽3)C2214yescomplemented
10C2×Q8×C3⋊D3C2×C311yescharacteristic, complemented
11C2×C3×D3×Q8C2×C312yescomplemented
12C22×C3×SL(2,𝔽3)—32no—
13D32×Q8—34no—
14D32×Q8C2×C314yescomplemented
15C2×C4×D32—31no—
16C2×#3797—31no—
17C2×D3×Dic6—32no—
18C2×D3×SL(2,𝔽3)—22no—
19C2×D3×SL(2,𝔽3)—32no—
20C2×D3×SL(2,𝔽3)—38no—
21C2×D3×SL(2,𝔽3)—61no—
22C2×D3×SL(2,𝔽3)D312yesdirect product factor
23C22×C3×C3⋊D3—41no—
24C22×C32×D3—42no—
25C2×C3×D32—48no—
26C2×C3×D32—44no—
27C32×SL(2,𝔽3)C2311yescharacteristic, complemented
28C22×D3×Q8—32no—
29C23×SL(2,𝔽3)—91no—
30C23×C3×D3—122no—
31C22×D32A411yescharacteristic
32C2×C32×Q8C22×C311yescharacteristic, complemented
33Q8×C3⋊D3—32no—
34Q8×C3⋊D3C22×C312yescomplemented
35C2×C4×C3⋊D3—31no—
36C2×#634—31no—
37C3×D3×Q8—34no—
38C3×D3×Q8C22×C314yescomplemented
39C2×C3×C4×D3—32no—
40C2×C3×Dic6—32no—
41C2×C3×SL(2,𝔽3)—22no—
42C2×C3×SL(2,𝔽3)—21no—
43C2×C3×SL(2,𝔽3)—34no—
44C2×C3×SL(2,𝔽3)—41no—
45C2×C3×SL(2,𝔽3)C2×D312yescomplemented
46C2×#627—32no—
47C2×#624—31no—
48C2×D3×Dic3—32no—
49C4×D32—34no—
50C4×D32—34no—
51#3797—34no—
52#3797—34no—
53D3×Dic6—38no—
54D3×Dic6—38no—
55D3×SL(2,𝔽3)—24no—
56D3×SL(2,𝔽3)—34no—
57D3×SL(2,𝔽3)—62no—
58D3×SL(2,𝔽3)C2×D314yesdirect product factor
59C22×C33—41no—
60C2×C3×C3⋊D3—44no—
61C2×C3×C3⋊D3—42no—
62C2×C32×D3—48no—
63C2×C32×D3—44no—
64C3×D32—416no—
65C22×C3×Q8—32no—
66C2×D3×Q8—32no—
67C2×D3×Q8—38no—
68C2×D3×Q8—61no—
69C2×D3×Q8—92no—
70C2×D3×Q8—92no—
71C2×D3×Q8—98no—
72C2×D3×Q8C3×D312yescomplemented
73C22×C4×D3—92no—
74C22×Dic6—92no—
75C22×SL(2,𝔽3)—32no—
76C22×SL(2,𝔽3)—62no—
77C22×SL(2,𝔽3)—94no—
78C22×SL(2,𝔽3)—91no—
79C23×C32—122no—
80C22×C3⋊D3C2×A411yescharacteristic
81C22×C3×D3—42no—
82C22×C3×D3—82no—
83C22×C3×D3—128no—
84C22×C3×D3—1216no—
85C22×C3×D3—122no—
86C22×C3×D3—241no—
87C22×C3×D3C2×A412yes—
88C2×D32SL(2,𝔽3)18yesdirect product factor
89C2×D32C2×A414yes—
90C32×Q8—31no—
91C32×Q8C23×C311yescharacteristic, complemented
92C2×C32×C4—31no—
93C2×C3⋊Dic3—31no—
94C4×C3⋊D3—32no—
95C4×C3⋊D3—32no—
96#634—32no—
97#634—32no—
98C2×C3×Dic3—32no—
99C3×C4×D3—34no—
100C3×C4×D3—34no—
101C3×Dic6—34no—
102C3×Dic6—34no—
103C3×SL(2,𝔽3)—22no—
104C3×SL(2,𝔽3)—21no—
105C3×SL(2,𝔽3)—41no—
106C3×SL(2,𝔽3)C22×D312yescomplemented
107#627—38no—
108#624—34no—
109D3×Dic3—38no—
110C23×Q8—91noSylow
111C2×C33—42no—
112C2×C33—41no—
113C3×C3⋊D3—44no—
114C32×D3—48no—
115C24×C3—361no—
116C23×D3—32no—
117C2×C3×Q8—21no—
118C2×C3×Q8—34no—
119C2×C3×Q8—92no—
120C2×C3×Q8—94no—
121C2×C3×Q8C2×C3×D312yescomplemented
122C22×C3×C4—92no—
123C22×Dic3—92no—
124D3×Q8—34no—
125D3×Q8—34no—
126D3×Q8—62no—
127D3×Q8—98no—
128D3×Q8—98no—
129D3×Q8—98no—
130D3×Q8—98no—
131D3×Q8—94no—
132D3×Q8—182no—
133D3×Q8C2×C3×D314yescomplemented
134C2×C4×D3—32no—
135C2×C4×D3—98no—
136C2×C4×D3—98no—
137C2×C4×D3—92no—
138C2×C4×D3—92no—
139C2×C4×D3—92no—
140C2×C4×D3—181no—
141C2×Dic6—32no—
142C2×Dic6—98no—
143C2×Dic6—98no—
144C2×Dic6—94no—
145C2×Dic6—92no—
146C2×Dic6—181no—
147C2×SL(2,𝔽3)—22no—
148C2×SL(2,𝔽3)—34no—
149C2×SL(2,𝔽3)—41no—
150C2×SL(2,𝔽3)—64no—
151C2×SL(2,𝔽3)—92no—
152C2×SL(2,𝔽3)D3211yescharacteristic, direct product factor
153C22×C32—42no—
154C22×C32—81no—
155C22×C32—82no—
156C22×C32—128no—
157C22×C32—124no—
158C22×C32—161no—
159C22×C32C22×A411yescharacteristic
160C2×C3⋊D3C2×SL(2,𝔽3)14yescomplemented
161C2×C3⋊D3C22×A412yes—
162C2×C3×D3—44no—
163C2×C3×D3—48no—
164C2×C3×D3—84no—
165C2×C3×D3—88no—
166C2×C3×D3—1232no—
167C2×C3×D3—128no—
168C2×C3×D3—124no—
169C2×C3×D3—244no—
170C2×C3×D3—242no—
171C2×C3×D3C2×SL(2,𝔽3)18yescomplemented
172C2×C3×D3C22×A414yes—
173D32C2×SL(2,𝔽3)116yesdirect product factor
174C32×C4—31no—
175C32×C4—31no—
176C3⋊Dic3—32no—
177C3×Dic3—34no—
178C22×Q8—32no—
179C22×Q8—94no—
180C22×Q8—91no—
181C22×Q8—272no—
182C22×Q8—271no—
183C22×Q8—274no—
184C23×C4—271no—
185C33—41noSylow
186C23×C3—32no—
187C23×C3—122no—
188C23×C3—242no—
189C23×C3—361no—
190C23×C3—364no—
191C23×C3—368no—
192C22×D3—32no—
193C22×D3—38no—
194C22×D3—316no—
195C22×D3—61no—
196C22×D3D3×A412yes—
197C3×Q8—21no—
198C3×Q8—32no—
199C3×Q8—61no—
200C3×Q8—94no—
201C3×Q8—94no—
202C3×Q8C22×C3×D312yescomplemented
203C2×C3×C4—32no—
204C2×C3×C4—61no—
205C2×C3×C4—94no—
206C2×C3×C4—94no—
207C2×C3×C4—92no—
208C2×Dic3—32no—
209C2×Dic3—98no—
210C2×Dic3—92no—
211C2×Dic3—181no—
212C4×D3—34no—
213C4×D3—34no—
214C4×D3—98no—
215C4×D3—94no—
216C4×D3—94no—
217C4×D3—98no—
218C4×D3—182no—
219C4×D3—182no—
220Dic6—34no—
221Dic6—34no—
222Dic6—916no—
223Dic6—94no—
224Dic6—94no—
225Dic6—182no—
226Dic6—182no—
227SL(2,𝔽3)—22no—
228SL(2,𝔽3)—41no—
229SL(2,𝔽3)C2×D3211yescharacteristic, direct product factor
230C2×C32—44no—
231C2×C32—42no—
232C2×C32—81no—
233C2×C32—82no—
234C2×C32—84no—
235C2×C32—82no—
236C2×C32—128no—
237C2×C32—161no—
238C2×C32—162no—
239C2×C32C22×SL(2,𝔽3)12yescomplemented
240C2×C32C23×A411yescharacteristic
241C3⋊D3C22×SL(2,𝔽3)14yescomplemented
242C3×D3—48no—
243C3×D3—88no—
244C3×D3—128no—
245C3×D3—244no—
246C3×D3C22×SL(2,𝔽3)18yescomplemented
247C24—91no—
248C2×Q8—34no—
249C2×Q8—92no—
250C2×Q8—94no—
251C2×Q8—92no—
252C2×Q8—271no—
253C2×Q8—272no—
254C2×Q8—278no—
255C2×Q8—278no—
256C2×Q8—274no—
257C2×Q8—272no—
258C2×Q8—274no—
259C2×Q8C3×D3211yescharacteristic, complemented
260C22×C4—92no—
261C22×C4—271no—
262C22×C4—274no—
263C22×C4—274no—
264C22×C4—271no—
265C22×C4—272no—
266C22×C3—21no—
267C22×C3—38no—
268C22×C3—34no—
269C22×C3—41no—
270C22×C3—82no—
271C22×C3—128no—
272C22×C3—124no—
273C22×C3—161no—
274C22×C3—248no—
275C22×C3—244no—
276C22×C3—364no—
277C22×C3—362no—
278C22×C3—3616no—
279C22×C3C2×D3×A412yes—
280C2×D3—38no—
281C2×D3—34no—
282C2×D3—332no—
283C2×D3—62no—
284C2×D3—64no—
285C2×D3D3×SL(2,𝔽3)18yesdirect product factor
286C2×D3C2×D3×A414yes—
287C3×C4—32no—
288C3×C4—32no—
289C3×C4—61no—
290C3×C4—61no—
291C3×C4—94no—
292Dic3—34no—
293Dic3—94no—
294Dic3—182no—
295C32—42no—
296C32—81no—
297C32—82no—
298C32—161no—
299C32C23×SL(2,𝔽3)11yescharacteristic, complemented
300C23—32no—
301C23—94no—
302C23—98no—
303C23—91no—
304Q8—31no—
305Q8—94no—
306Q8—94no—
307Q8—272no—
308Q8—272no—
309Q8—278no—
310Q8C2×C3×D3211yescharacteristic, complemented
311C2×C4—31no—
312C2×C4—94no—
313C2×C4—94no—
314C2×C4—92no—
315C2×C4—272no—
316C2×C4—272no—
317C2×C4—274no—
318C2×C4—278no—
319C2×C4—271no—
320C2×C3—21no—
321C2×C3—22no—
322C2×C3—38no—
323C2×C3—41no—
324C2×C3—42no—
325C2×C3—82no—
326C2×C3—84no—
327C2×C3—128no—
328C2×C3—161no—
329C2×C3—162no—
330C2×C3—248no—
331C2×C3—364no—
332C2×C3C2×D3×SL(2,𝔽3)14yescomplemented
333C2×C3C22×D3×A412yes—
334D3—38no—
335D3—64no—
336D3C2×D3×SL(2,𝔽3)18yesdirect product factor
337C22—34no—
338C22—38no—
339C22—916no—
340C22—92no—
341C22—94no—
342C22D32×A411yescharacteristic
343C4—31no—
344C4—31no—
345C4—94no—
346C4—272no—
347C3—21no—
348C3—41no—
349C3—82no—
350C3—161no—
351C3C22×D3×SL(2,𝔽3)12yescomplemented
352C2—38no—
353C2—94no—
354C2D32×SL(2,𝔽3)12yesdirect product factor
355C2C2×D32×A411yescharacteristic
subgroupG/H
center, hypercenterC22D32×A4
commutator, cosocle, hypocenterC32×Q8C23×C3
FrattiniC2C2×D32×A4
FittingC2×C32×Q8C22×C3
socleC22×C32C22×A4
LCSG ▷ C32×Q8
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×D3×SL(2,𝔽3)C2semidirect
3. D32×SL(2,𝔽3)C2direct (N × H)
4. C2×D32×Q8C3semidirect
7. C2×C32×SL(2,𝔽3)C22semidirect
8. C3⋊D3×SL(2,𝔽3)C22semidirect
9. C3×D3×SL(2,𝔽3)C22semidirect
10. C2×Q8×C3⋊D3C2×C3semidirect
11. C2×C3×D3×Q8C2×C3semidirect
14. D32×Q8C2×C3semidirect
22. C2×D3×SL(2,𝔽3)D3direct (N × H)
27. C32×SL(2,𝔽3)C23semidirect
32. C2×C32×Q8C22×C3semidirect
34. Q8×C3⋊D3C22×C3semidirect
38. C3×D3×Q8C22×C3semidirect
45. C2×C3×SL(2,𝔽3)C2×D3semidirect
58. D3×SL(2,𝔽3)C2×D3direct (N × H)
72. C2×D3×Q8C3×D3semidirect
88. C2×D32SL(2,𝔽3)direct (N × H)
91. C32×Q8C23×C3semidirect
106. C3×SL(2,𝔽3)C22×D3semidirect
121. C2×C3×Q8C2×C3×D3semidirect
133. D3×Q8C2×C3×D3semidirect
152. C2×SL(2,𝔽3)D32direct (N × H)
160. C2×C3⋊D3C2×SL(2,𝔽3)semidirect
171. C2×C3×D3C2×SL(2,𝔽3)semidirect
173. D32C2×SL(2,𝔽3)direct (N × H)
202. C3×Q8C22×C3×D3semidirect
229. SL(2,𝔽3)C2×D32direct (N × H)
239. C2×C32C22×SL(2,𝔽3)semidirect
241. C3⋊D3C22×SL(2,𝔽3)semidirect
246. C3×D3C22×SL(2,𝔽3)semidirect
259. C2×Q8C3×D32semidirect
285. C2×D3D3×SL(2,𝔽3)direct (N × H)
299. C32C23×SL(2,𝔽3)semidirect
310. Q8C2×C3×D32semidirect
332. C2×C3C2×D3×SL(2,𝔽3)semidirect
336. D3C2×D3×SL(2,𝔽3)direct (N × H)
351. C3C22×D3×SL(2,𝔽3)semidirect
354. C2D32×SL(2,𝔽3)direct (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×D3×SL(2,𝔽3)2
D32×SL(2,𝔽3)2
C2×D32×Q83
C22×D3×SL(2,𝔽3)3
C22×C3×D324