Finite Group Databasebeta

← C32×#5923 #13181365

Subgroups — C32×#5923 #13181365

Order 1728.

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

normal subgroups only

This group has 968 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
1C2×C32×#41⋊C3C211yesmaximal, characteristic, complemented
2C32×#986C211yesmaximal, characteristic, complemented
3C32×#985C212yesmaximal, complemented
4C32×#984C211yesmaximal, characteristic, complemented
5C32×#983C212yesmaximal, complemented
6C3×#5923C314yesmaximal, direct product factor
7C32×#575—31nomaximal
8C32×#235—41nomaximal
9C32×#41⋊C3C2211yescharacteristic
10C32×#41⋊C3C2211yescharacteristic, complemented
11C2×C32×SL(2,𝔽3)C2211yescharacteristic, complemented
12C32×GL(2,𝔽3)C2212yescomplemented
13C32×#226C2212yescomplemented
14C2×C32×#41—31no—
15C2×C32×#41D311yescharacteristic, complemented
16C2×C3×#41⋊C3—24no—
17C2×C3×#41⋊C3C2×C314yescomplemented
18C3×#986C2×C314yescomplemented
19C3×#985C2×C318yescomplemented
20C3×#984C2×C314yescomplemented
21C3×#983C2×C318yescomplemented
22C32×#137—31no—
23C32×#137—31no—
24C32×#137—32no—
25C32×AGL(1,ℤ/8)—32no—
26C32×AGL(1,ℤ/8)—31no—
27C32×AGL(1,ℤ/8)—31no—
28C32×#135—32no—
29C32×#135—32no—
30C2×C32×M16—31no—
31C2×C33×C4—41no—
32C32×#67—42no—
33C32×D12—41no—
34C32×C4×D3—42no—
35C32×Dic6—41no—
36C32×SL(2,𝔽3)C2311yescharacteristic
37#5923C3211yescharacteristic, direct product factor
38C3×#575—34no—
39C32×#41—31no—
40C32×#41—31no—
41C32×#41—32no—
42C32×#41—32no—
43C32×#41—61no—
44C32×#41—61no—
45C32×#41C2×D311yescharacteristic
46C32×#41C2×D311yescharacteristic
47C2×C32×Q8—31no—
48C2×C32×Q8C2×D311yescharacteristic
49C2×C32×D4—31no—
50C2×C32×D4—31no—
51C2×C32×D4—61no—
52C22×C32×C4—31no—
53C22×C32×C4—61no—
54C3×#235—44no—
55C3×#41⋊C3—24no—
56C3×#41⋊C3—24no—
57C3×#41⋊C3C22×C314yes—
58C3×#41⋊C3C22×C314yescomplemented
59C2×C3×SL(2,𝔽3)—24no—
60C2×C3×SL(2,𝔽3)C22×C314yescomplemented
61C3×GL(2,𝔽3)C22×C318yescomplemented
62C3×#226C22×C318yescomplemented
63C32×Q16—32no—
64C32×Q16—32no—
65C32×SD16—32no—
66C32×SD16—32no—
67C32×SD16—32no—
68C32×SD16—32no—
69C32×D8—32no—
70C32×D8—32no—
71C32×M16—32no—
72C32×M16—31no—
73C32×M16—31no—
74C2×C32×C8—32no—
75C22×C33—41no—
76C2×C32×D3—42no—
77C33×C4—41no—
78C33×C4—41no—
79C32×Dic3—42no—
80C2×C3×#41—34no—
81C2×C3×#41C3×D314yescomplemented
82C2×#41⋊C3—24no—
83C2×#41⋊C3C2×C3211yescharacteristic, complemented
84#986C2×C3211yescharacteristic, complemented
85#985C2×C3212yescomplemented
86#984C2×C3211yescharacteristic, complemented
87#983C2×C3212yescomplemented
88C3×#137—34no—
89C3×#137—34no—
90C3×#137—38no—
91C3×AGL(1,ℤ/8)—38no—
92C3×AGL(1,ℤ/8)—34no—
93C3×AGL(1,ℤ/8)—34no—
94C3×#135—38no—
95C3×#135—38no—
96C2×C3×M16—34no—
97C23×C32—31no—
98C23×C32—61no—
99C32×Q8—31no—
100C32×Q8—32no—
101C32×Q8—61no—
102C32×Q8C22×D311yescharacteristic
103C32×D4—31no—
104C32×D4—31no—
105C32×D4—32no—
106C32×D4—61no—
107C32×D4—62no—
108C32×D4—62no—
109C32×D4—61no—
110C2×C32×C4—31no—
111C2×C32×C4—31no—
112C2×C32×C4—31no—
113C2×C32×C4—31no—
114C2×C32×C4—31no—
115C2×C32×C4—44no—
116C2×C32×C4—62no—
117C2×C32×C4—61no—
118C2×C32×C4—62no—
119C2×C32×C4—84no—
120C2×C32×C4S411yescharacteristic
121C3×#67—48no—
122C3×D12—44no—
123C3×C4×D3—48no—
124C3×Dic6—44no—
125C3×SL(2,𝔽3)—24no—
126C3×SL(2,𝔽3)C23×C314yes—
127C32×C8—32no—
128C32×C8—32no—
129#575—31noSylow
130C2×C33—41no—
131C2×C33—81no—
132C32×D3—82no—
133C3×#41—34no—
134C3×#41—34no—
135C3×#41—38no—
136C3×#41—38no—
137C3×#41—64no—
138C3×#41—64no—
139C3×#41C2×C3×D314yes—
140C3×#41C2×C3×D314yes—
141C2×C3×Q8—34no—
142C2×C3×Q8C2×C3×D314yes—
143C2×C3×D4—34no—
144C2×C3×D4—34no—
145C2×C3×D4—64no—
146C22×C3×C4—34no—
147C22×C3×C4—64no—
148#235—41no—
149#41⋊C3—24no—
150#41⋊C3—24no—
151#41⋊C3C22×C3211yescharacteristic
152#41⋊C3C22×C3211yescharacteristic, complemented
153C2×SL(2,𝔽3)—24no—
154C2×SL(2,𝔽3)C22×C3211yescharacteristic, complemented
155GL(2,𝔽3)C22×C3212yescomplemented
156#226C22×C3212yescomplemented
157C3×Q16—38no—
158C3×Q16—38no—
159C3×SD16—38no—
160C3×SD16—38no—
161C3×SD16—38no—
162C3×SD16—38no—
163C3×D8—38no—
164C3×D8—38no—
165C3×M16—38no—
166C3×M16—34no—
167C3×M16—34no—
168C2×C3×C8—38no—
169C22×C32—31no—
170C22×C32—31no—
171C22×C32—44no—
172C22×C32—62no—
173C22×C32—84no—
174C22×C32—121no—
175C22×C32—122no—
176C22×C32C2×S411yescharacteristic
177C2×C3×D3—48no—
178C32×C4—31no—
179C32×C4—31no—
180C32×C4—44no—
181C32×C4—44no—
182C32×C4—62no—
183C32×C4—84no—
184C32×C4—84no—
185C32×C4C2×S411yescharacteristic
186C32×C4C2×S411yescharacteristic
187C3×Dic3—48no—
188C2×#41—31no—
189C2×#41C32×D311yescharacteristic, complemented
190#137—31no—
191#137—31no—
192#137—32no—
193AGL(1,ℤ/8)—32no—
194AGL(1,ℤ/8)—31no—
195AGL(1,ℤ/8)—31no—
196#135—32no—
197#135—32no—
198C2×M16—31no—
199C33—41noSylow
200C23×C3—34no—
201C23×C3—64no—
202C3×Q8—34no—
203C3×Q8—38no—
204C3×Q8—64no—
205C3×Q8C22×C3×D314yes—
206C3×D4—34no—
207C3×D4—34no—
208C3×D4—38no—
209C3×D4—64no—
210C3×D4—68no—
211C3×D4—68no—
212C3×D4—64no—
213C2×C3×C4—34no—
214C2×C3×C4—34no—
215C2×C3×C4—34no—
216C2×C3×C4—34no—
217C2×C3×C4—34no—
218C2×C3×C4—41no—
219C2×C3×C4—68no—
220C2×C3×C4—64no—
221C2×C3×C4—68no—
222C2×C3×C4—84no—
223C2×C3×C4C3×S414yes—
224#67—42no—
225D12—41no—
226C4×D3—42no—
227Dic6—41no—
228SL(2,𝔽3)—24no—
229SL(2,𝔽3)C23×C3211yescharacteristic
230C3×C8—38no—
231C3×C8—38no—
232C2×C32—21no—
233C2×C32—44no—
234C2×C32—61no—
235C2×C32—61no—
236C2×C32—84no—
237C2×C32—88no—
238C2×C32—84no—
239C2×C32—122no—
240C2×C32C22×S411yescharacteristic
241C3×D3—88no—
242#41—31no—
243#41—31no—
244#41—32no—
245#41—32no—
246#41—61no—
247#41—61no—
248#41C2×C32×D311yescharacteristic
249#41C2×C32×D311yescharacteristic
250C2×Q8—31no—
251C2×Q8C2×C32×D311yescharacteristic
252C2×D4—31no—
253C2×D4—31no—
254C2×D4—61no—
255C22×C4—31no—
256C22×C4—61no—
257Q16—32no—
258Q16—32no—
259SD16—32no—
260SD16—32no—
261SD16—32no—
262SD16—32no—
263D8—32no—
264D8—32no—
265M16—32no—
266M16—31no—
267M16—31no—
268C2×C8—32no—
269C22×C3—34no—
270C22×C3—34no—
271C22×C3—41no—
272C22×C3—68no—
273C22×C3—84no—
274C22×C3—124no—
275C22×C3—128no—
276C22×C3C2×C3×S414yes—
277C2×D3—42no—
278C3×C4—34no—
279C3×C4—34no—
280C3×C4—41no—
281C3×C4—41no—
282C3×C4—68no—
283C3×C4—84no—
284C3×C4—84no—
285C3×C4C2×C3×S414yes—
286C3×C4C2×C3×S414yes—
287Dic3—42no—
288C32—44no—
289C32—84no—
290C32#592311yescharacteristic, direct product factor
291C23—31no—
292C23—61no—
293Q8—31no—
294Q8—32no—
295Q8—61no—
296Q8C22×C32×D311yescharacteristic
297D4—31no—
298D4—31no—
299D4—32no—
300D4—61no—
301D4—62no—
302D4—62no—
303D4—61no—
304C2×C4—31no—
305C2×C4—31no—
306C2×C4—31no—
307C2×C4—31no—
308C2×C4—31no—
309C2×C4—62no—
310C2×C4—61no—
311C2×C4—62no—
312C2×C4C32×S411yescharacteristic
313C8—32no—
314C8—32no—
315C2×C3—24no—
316C2×C3—41no—
317C2×C3—64no—
318C2×C3—64no—
319C2×C3—81no—
320C2×C3—88no—
321C2×C3—84no—
322C2×C3—128no—
323C2×C3C22×C3×S414yes—
324D3—82no—
325C22—31no—
326C22—31no—
327C22—62no—
328C22—121no—
329C22—122no—
330C22C2×C32×S411yescharacteristic
331C4—31no—
332C4—31no—
333C4—62no—
334C4C2×C32×S411yescharacteristic
335C4C2×C32×S411yescharacteristic
336C3—41no—
337C3—84no—
338C3C3×#592314yesdirect product factor
339C2—21no—
340C2—61no—
341C2—61no—
342C2—122no—
343C2C22×C32×S411yescharacteristic
subgroupG/H
centerC32×C4C2×S4
commutator, cosocle, hypocenterSL(2,𝔽3)C23×C32
FrattiniC2C22×C32×S4
FittingC2×C32×#41D3
socleC2×C32C22×S4
hypercenterC2×C32×C4S4
LCSG ▷ SL(2,𝔽3)
UCSC1 ◁ C32×C4 ◁ C2×C32×C4

Complements

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

NH ≅ G/Ntype
1. C2×C32×#41⋊C3C2semidirect
2. C32×#986C2semidirect
3. C32×#985C2semidirect
4. C32×#984C2semidirect
5. C32×#983C2semidirect
6. C3×#5923C3direct (N × H)
10. C32×#41⋊C3C22semidirect
11. C2×C32×SL(2,𝔽3)C22semidirect
12. C32×GL(2,𝔽3)C22semidirect
13. C32×#226C22semidirect
15. C2×C32×#41D3semidirect
17. C2×C3×#41⋊C3C2×C3semidirect
18. C3×#986C2×C3semidirect
19. C3×#985C2×C3semidirect
20. C3×#984C2×C3semidirect
21. C3×#983C2×C3semidirect
37. #5923C32direct (N × H)
58. C3×#41⋊C3C22×C3semidirect
60. C2×C3×SL(2,𝔽3)C22×C3semidirect
61. C3×GL(2,𝔽3)C22×C3semidirect
62. C3×#226C22×C3semidirect
81. C2×C3×#41C3×D3semidirect
83. C2×#41⋊C3C2×C32semidirect
84. #986C2×C32semidirect
85. #985C2×C32semidirect
86. #984C2×C32semidirect
87. #983C2×C32semidirect
152. #41⋊C3C22×C32semidirect
154. C2×SL(2,𝔽3)C22×C32semidirect
155. GL(2,𝔽3)C22×C32semidirect
156. #226C22×C32semidirect
189. C2×#41C32×D3semidirect
290. C32#5923direct (N × H)
338. C3C3×#5923direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C32×#41⋊C32
C32×#9862
C32×#9852
C32×#9842
C32×#9832
C3×#59233
C32×#5753
C32×#2354