Finite Group Databasebeta

← C32×#5926 #13181368

Subgroups — C32×#5926 #13181368

Order 1728.

1016 conjugacy classes of subgroups; only the 162 normal subgroups are shown.

normal subgroups only

This group has 1016 conjugacy classes of subgroups — too many to draw. Only its 162 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
1C32×2−1+4⋊C3C211yesmaximal, characteristic, complemented
2C32×#986C211yesmaximal, characteristic, complemented
3C32×#985C211yesmaximal, characteristic, complemented
4C32×#983C212yesmaximal, complemented
5C2×C32×GL(2,𝔽3)C212yesmaximal, complemented
6C3×#5926C314yesmaximal, direct product factor
7C32×#577—31nomaximal
8C32×D3×D4—41nomaximal
9C32×#41⋊C3C2211yescharacteristic, complemented
10C2×C32×SL(2,𝔽3)C2212yescomplemented
11C32×GL(2,𝔽3)C2211yescharacteristic, complemented
12C32×GL(2,𝔽3)C2212yescomplemented
13C32×#226C2211yescharacteristic, complemented
14C32×2−1+4D311yescharacteristic, complemented
15C32×H(5,𝔽2)—31no—
16C3×2−1+4⋊C3—24no—
17C3×2−1+4⋊C3C2×C314yescomplemented
18C3×#986C2×C314yescomplemented
19C3×#985C2×C314yescomplemented
20C3×#983C2×C318yescomplemented
21C2×C3×GL(2,𝔽3)C2×C318yescomplemented
22C32×#137—31no—
23C32×#137—32no—
24C32×AGL(1,ℤ/8)—32no—
25C32×AGL(1,ℤ/8)—31no—
26C32×#135—32no—
27C32×#135—31no—
28C2×C32×SD16—31no—
29C2×C32×SD16—32no—
30C32×#131—31no—
31C22×C32×D3—42no—
32C33×D4—41no—
33C32×#67—42no—
34C32×D12—41no—
35C32×C4×D3—41no—
36C32×SL(2,𝔽3)C2311yescharacteristic
37#5926C3211yescharacteristic, direct product factor
38C3×#577—34no—
39C32×#41—31no—
40C32×#41—32no—
41C32×#41—32no—
42C32×#41—31no—
43C32×#41—61no—
44C32×#41C2×D311yescharacteristic, complemented
45C2×C32×Q8—31no—
46C2×C32×Q8C2×D312yescomplemented
47C2×C32×D4—32no—
48C2×C32×D4—31no—
49C2×C32×D4—62no—
50C2×C32×D4—61no—
51C3×D3×D4—44no—
52C3×#41⋊C3—24no—
53C3×#41⋊C3C22×C314yescomplemented
54C2×C3×SL(2,𝔽3)—28no—
55C2×C3×SL(2,𝔽3)C22×C318yescomplemented
56C3×GL(2,𝔽3)C22×C318yescomplemented
57C3×GL(2,𝔽3)C22×C314yescomplemented
58C3×#226C22×C314yescomplemented
59C32×Q16—32no—
60C32×Q16—31no—
61C32×SD16—32no—
62C32×SD16—32no—
63C32×SD16—32no—
64C32×SD16—31no—
65C32×SD16—31no—
66C32×SD16—32no—
67C32×D8—32no—
68C32×D8—31no—
69C32×M16—32no—
70C32×M16—31no—
71C2×C32×C8—32no—
72C2×C32×C8—31no—
73C22×C33—42no—
74C2×C32×D3—41no—
75C2×C32×D3—42no—
76C2×C32×D3—84no—
77C33×C4—41no—
78C32×Dic3—41no—
79C3×2−1+4C3×D314yescomplemented
80C3×H(5,𝔽2)—34no—
812−1+4⋊C3—24no—
822−1+4⋊C3C2×C3211yescharacteristic, complemented
83#986C2×C3211yescharacteristic, complemented
84#985C2×C3211yescharacteristic, complemented
85#983C2×C3212yescomplemented
86C2×GL(2,𝔽3)C2×C3212yescomplemented
87C3×#137—34no—
88C3×#137—38no—
89C3×AGL(1,ℤ/8)—38no—
90C3×AGL(1,ℤ/8)—34no—
91C3×#135—38no—
92C3×#135—34no—
93C2×C3×SD16—34no—
94C2×C3×SD16—38no—
95C3×#131—34no—
96C23×C32—61no—
97C23×C32—62no—
98C22×C3×D3—48no—
99C32×Q8—32no—
100C32×Q8—31no—
101C32×Q8—31no—
102C32×Q8C22×D311yescharacteristic
103C32×D4—31no—
104C32×D4—32no—
105C32×D4—32no—
106C32×D4—31no—
107C32×D4—44no—
108C32×D4—62no—
109C32×D4—61no—
110C32×D4—62no—
111C32×D4—61no—
112C32×D4—84no—
113C32×D4S411yescharacteristic
114C2×C32×C4—32no—
115C2×C32×C4—31no—
116C2×C32×C4—61no—
117C2×C32×C4—62no—
118C2×C32×C4—61no—
119C3×#67—48no—
120C3×D12—44no—
121C3×C4×D3—44no—
122C3×SL(2,𝔽3)—24no—
123C3×SL(2,𝔽3)C23×C314yes—
124C32×C8—31no—
125C32×C8—31no—
126C32×C8—32no—
127#577—31noSylow
128C2×C33—41no—
129C2×C33—82no—
130C32×D3—42no—
131C32×D3—82no—
132C3×#41—34no—
133C3×#41—38no—
134C3×#41—38no—
135C3×#41—34no—
136C3×#41—64no—
137C3×#41C2×C3×D314yescomplemented
138C2×C3×Q8—34no—
139C2×C3×Q8C2×C3×D318yescomplemented
140C2×C3×D4—38no—
141C2×C3×D4—34no—
142C2×C3×D4—68no—
143C2×C3×D4—64no—
144D3×D4—41no—
145#41⋊C3—24no—
146#41⋊C3C22×C3211yescharacteristic, complemented
147C2×SL(2,𝔽3)—28no—
148C2×SL(2,𝔽3)C22×C3212yescomplemented
149GL(2,𝔽3)C22×C3211yescharacteristic, complemented
150GL(2,𝔽3)C22×C3212yescomplemented
151#226C22×C3211yescharacteristic, complemented
152C3×Q16—38no—
153C3×Q16—34no—
154C3×SD16—38no—
155C3×SD16—38no—
156C3×SD16—38no—
157C3×SD16—34no—
158C3×SD16—34no—
159C3×SD16—38no—
160C3×D8—38no—
161C3×D8—34no—
162C3×M16—38no—
163C3×M16—34no—
164C2×C3×C8—38no—
165C2×C3×C8—34no—
166C22×C32—31no—
167C22×C32—48no—
168C22×C32—61no—
169C22×C32—62no—
170C22×C32—88no—
171C22×C32—242no—
172C22×C32—241no—
173C22×C32C2×S412yes—
174C2×C3×D3—44no—
175C2×C3×D3—48no—
176C2×C3×D3—816no—
177C32×C4—31no—
178C32×C4—32no—
179C32×C4—44no—
180C32×C4—61no—
181C32×C4—84no—
182C32×C4C2×S411yescharacteristic
183C3×Dic3—44no—
1842−1+4C32×D311yescharacteristic, complemented
185H(5,𝔽2)—31no—
186#137—31no—
187#137—32no—
188AGL(1,ℤ/8)—32no—
189AGL(1,ℤ/8)—31no—
190#135—32no—
191#135—31no—
192C2×SD16—31no—
193C2×SD16—32no—
194#131—31no—
195C33—41noSylow
196C23×C3—64no—
197C23×C3—68no—
198C22×D3—42no—
199C3×Q8—38no—
200C3×Q8—34no—
201C3×Q8—34no—
202C3×Q8C22×C3×D314yes—
203C3×D4—34no—
204C3×D4—38no—
205C3×D4—38no—
206C3×D4—34no—
207C3×D4—41no—
208C3×D4—68no—
209C3×D4—64no—
210C3×D4—68no—
211C3×D4—64no—
212C3×D4—84no—
213C3×D4C3×S414yes—
214C2×C3×C4—38no—
215C2×C3×C4—34no—
216C2×C3×C4—64no—
217C2×C3×C4—68no—
218C2×C3×C4—64no—
219#67—42no—
220D12—41no—
221C4×D3—41no—
222SL(2,𝔽3)—24no—
223SL(2,𝔽3)C23×C3211yescharacteristic
224C3×C8—34no—
225C3×C8—34no—
226C3×C8—38no—
227C2×C32—22no—
228C2×C32—44no—
229C2×C32—61no—
230C2×C32—88no—
231C2×C32—84no—
232C2×C32—121no—
233C2×C32—122no—
234C2×C32—168no—
235C2×C32C22×S411yescharacteristic
236C3×D3—48no—
237C3×D3—88no—
238#41—31no—
239#41—32no—
240#41—32no—
241#41—31no—
242#41—61no—
243#41C2×C32×D311yescharacteristic, complemented
244C2×Q8—31no—
245C2×Q8C2×C32×D312yescomplemented
246C2×D4—32no—
247C2×D4—31no—
248C2×D4—62no—
249C2×D4—61no—
250Q16—32no—
251Q16—31no—
252SD16—32no—
253SD16—32no—
254SD16—32no—
255SD16—31no—
256SD16—31no—
257SD16—32no—
258D8—32no—
259D8—31no—
260M16—32no—
261M16—31no—
262C2×C8—32no—
263C2×C8—31no—
264C22×C3—34no—
265C22×C3—42no—
266C22×C3—64no—
267C22×C3—68no—
268C22×C3—88no—
269C22×C3—248no—
270C22×C3—244no—
271C22×C3C2×C3×S418yes—
272C2×D3—41no—
273C2×D3—42no—
274C2×D3—84no—
275C3×C4—34no—
276C3×C4—38no—
277C3×C4—41no—
278C3×C4—64no—
279C3×C4—84no—
280C3×C4C2×C3×S414yes—
281Dic3—41no—
282C32—44no—
283C32—84no—
284C32#592611yescharacteristic, direct product factor
285C23—61no—
286C23—62no—
287Q8—32no—
288Q8—31no—
289Q8—31no—
290Q8C22×C32×D311yescharacteristic
291D4—31no—
292D4—32no—
293D4—32no—
294D4—31no—
295D4—62no—
296D4—61no—
297D4—62no—
298D4—61no—
299D4C32×S411yescharacteristic
300C2×C4—32no—
301C2×C4—31no—
302C2×C4—61no—
303C2×C4—62no—
304C2×C4—61no—
305C8—31no—
306C8—31no—
307C8—32no—
308C2×C3—28no—
309C2×C3—41no—
310C2×C3—64no—
311C2×C3—82no—
312C2×C3—84no—
313C2×C3—124no—
314C2×C3—128no—
315C2×C3—168no—
316C2×C3C22×C3×S414yes—
317D3—42no—
318D3—82no—
319C22—31no—
320C22—61no—
321C22—62no—
322C22—242no—
323C22—241no—
324C22C2×C32×S412yes—
325C4—31no—
326C4—32no—
327C4—61no—
328C4C2×C32×S411yescharacteristic
329C3—41no—
330C3—84no—
331C3C3×#592614yesdirect product factor
332C2—22no—
333C2—61no—
334C2—121no—
335C2—122no—
336C2C22×C32×S411yescharacteristic
subgroupG/H
center, socleC2×C32C22×S4
commutator, cosocle, hypocenterSL(2,𝔽3)C23×C32
FrattiniC2C22×C32×S4
FittingC32×2−1+4D3
hypercenterC32×D4S4
LCSG ▷ SL(2,𝔽3)
UCSC1 ◁ C2×C32 ◁ C32×D4

Complements

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

NH ≅ G/Ntype
1. C32×2−1+4⋊C3C2semidirect
2. C32×#986C2semidirect
3. C32×#985C2semidirect
4. C32×#983C2semidirect
5. C2×C32×GL(2,𝔽3)C2semidirect
6. C3×#5926C3direct (N × H)
9. C32×#41⋊C3C22semidirect
10. C2×C32×SL(2,𝔽3)C22semidirect
11. C32×GL(2,𝔽3)C22semidirect
12. C32×GL(2,𝔽3)C22semidirect
13. C32×#226C22semidirect
14. C32×2−1+4D3semidirect
17. C3×2−1+4⋊C3C2×C3semidirect
18. C3×#986C2×C3semidirect
19. C3×#985C2×C3semidirect
20. C3×#983C2×C3semidirect
21. C2×C3×GL(2,𝔽3)C2×C3semidirect
37. #5926C32direct (N × H)
44. C32×#41C2×D3semidirect
46. C2×C32×Q8C2×D3semidirect
53. C3×#41⋊C3C22×C3semidirect
55. C2×C3×SL(2,𝔽3)C22×C3semidirect
56. C3×GL(2,𝔽3)C22×C3semidirect
57. C3×GL(2,𝔽3)C22×C3semidirect
58. C3×#226C22×C3semidirect
79. C3×2−1+4C3×D3semidirect
82. 2−1+4⋊C3C2×C32semidirect
83. #986C2×C32semidirect
84. #985C2×C32semidirect
85. #983C2×C32semidirect
86. C2×GL(2,𝔽3)C2×C32semidirect
137. C3×#41C2×C3×D3semidirect
139. C2×C3×Q8C2×C3×D3semidirect
146. #41⋊C3C22×C32semidirect
148. C2×SL(2,𝔽3)C22×C32semidirect
149. GL(2,𝔽3)C22×C32semidirect
150. GL(2,𝔽3)C22×C32semidirect
151. #226C22×C32semidirect
184. 2−1+4C32×D3semidirect
243. #41C2×C32×D3semidirect
245. C2×Q8C2×C32×D3semidirect
284. C32#5926direct (N × H)
331. C3C3×#5926direct (N × H)

Maximal subgroups

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

subgroupindex
C32×2−1+4⋊C32
C32×#9862
C32×#9852
C32×#9832
C2×C32×GL(2,𝔽3)2
C3×#59263
C32×#5773
C32×D3×D44