Finite Group Databasebeta

← C3×#10596502 #13181021

Subgroups — C3×#10596502 #13181021

Order 1728.

932 conjugacy classes of subgroups; only the 170 normal subgroups are shown.

normal subgroups only

This group has 932 conjugacy classes of subgroups — too many to draw. Only its 170 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
1C3×#64306C212yesmaximal, complemented
2C2×C3×#3788C211yesmaximal, characteristic, complemented
3C3×#64237C214yesmaximal, complemented
4C3×#64226C214yesmaximal, complemented
5C3×#64224C214yesmaximal, complemented
6#10596502C311yesmaximal, characteristic, direct product factor
7C2×C3×C4×C3⋊D3C2211yescharacteristic
8C32×#235—22no—
9C3×#3811—22no—
10C3×#3805C2214yescomplemented
11C3×#3802C2212yescomplemented
12C3×C4×D32—22no—
13C3×#3798C2214yes—
14C3×#3797C2212yescomplemented
15C3×#3796—22no—
16C2×C3×#622C2212yescomplemented
17C3×#3788C2214yescomplemented
18C3×#3776C2214yescomplemented
19C3×#3775C2218yescomplemented
20C3×#3774C2214yescomplemented
21C3×D3×#41—62no—
22#64306C2×C312yescomplemented
23C2×#3788C2×C311yescharacteristic, complemented
24#64237C2×C314yescomplemented
25#64226C2×C314yescomplemented
26#64224C2×C314yescomplemented
27C22×C3×C3⋊D3D411yescharacteristic
28C2×C3×D32—22no—
29C2×C33×C4D411yescharacteristic
30C2×C3×C3⋊Dic3C2311yescharacteristic
31C3×C4×C3⋊D3D412yes—
32C3×C4×C3⋊D3C2312yes—
33C32×#67—24no—
34C32×D12—22no—
35C32×C4×D3—24no—
36C32×Dic6—22no—
37C3×#627C2314yes—
38C3×#626—24no—
39C3×#625C2314yes—
40C3×#624—22no—
41C3×D3×Dic3—24no—
42C3×#622C2314yes—
43C3×#575—91nomaximal
44C32×#41—62no—
45C2×C4×C3⋊D3C22×C311yescharacteristic
46C3×#239—62no—
47C3×D3×Q8—62no—
48C3×#237—62no—
49C3×#237—64no—
50C3×D3×D4—62no—
51C3×D3×D4—64no—
52C3×#235—22no—
53C3×#235—22no—
54C3×#235—42no—
55C3×#235—64no—
56C2×C3×C4×D3—64no—
57C2×C3×C4×D3—62no—
58#3811—22no—
59#3805C22×C314yescomplemented
60#3802C22×C312yescomplemented
61C4×D32—22no—
62#3798C22×C314yes—
63#3797C22×C312yescomplemented
64#3796—22no—
65C2×#622C22×C312yescomplemented
66#3788C22×C314yescomplemented
67#3776C22×C314yescomplemented
68#3775C22×C318yescomplemented
69#3774C22×C314yescomplemented
70C22×C33C2×D411yescharacteristic
71C2×C3×C3⋊D3—41no—
72C2×C3×C3⋊D3C2×D412yes—
73C2×C32×D3—24no—
74C3×D32—44no—
75C33×C4C2×D412yes—
76C3×C3⋊Dic3C2×D411yescharacteristic
77C3×C3⋊Dic3C2411yescharacteristic
78C32×Dic3—24no—
79C2×C3×#41—92no—
80D3×#41—62no—
81C3×#137—94no—
82C3×AGL(1,ℤ/8)—94no—
83C3×#135—94no—
84C2×C3×M16—91no—
85C22×C3⋊D3C3×D411yescharacteristic
86C22×C3×D3—62no—
87C22×C3×D3—64no—
88C2×D32—22no—
89C32×Q8—62no—
90C32×D4—62no—
91C32×D4—64no—
92C2×C32×C4—22no—
93C2×C32×C4—42no—
94C2×C32×C4—64no—
95C2×C32×C4C3×D411yescharacteristic
96C2×C3⋊Dic3C23×C311yescharacteristic
97C4×C3⋊D3C3×D412yes—
98C4×C3⋊D3C23×C312yes—
99C3×#67—24no—
100C3×#67—24no—
101C3×#67—44no—
102C3×#67—64no—
103C3×#67—64no—
104C2×C3×Dic3—62no—
105C2×C3×Dic3—64no—
106C3×D12—22no—
107C3×D12—22no—
108C3×D12—42no—
109C3×D12—64no—
110C3×C4×D3—24no—
111C3×C4×D3—24no—
112C3×C4×D3—44no—
113C3×C4×D3—62no—
114C3×C4×D3—62no—
115C3×C4×D3—62no—
116C3×C4×D3—62no—
117C3×C4×D3—64no—
118C3×C4×D3—64no—
119C3×Dic6—22no—
120C3×Dic6—22no—
121C3×Dic6—42no—
122C3×Dic6—64no—
123#627C23×C314yes—
124#626—24no—
125#625C23×C314yes—
126#624—22no—
127D3×Dic3—24no—
128#622C23×C314yes—
129#575—91noSylow
130C2×C33—21no—
131C2×C33C22×D411yescharacteristic
132C3×C3⋊D3—22no—
133C32×D3—44no—
134C3×#41—62no—
135C3×#41—62no—
136C3×#41—94no—
137C3×#41—94no—
138C3×#41—122no—
139C3×#41—182no—
140C2×C3×Q8—92no—
141C2×C3×D4—92no—
142C2×C3×D4—182no—
143C22×C3×C4—91no—
144C22×C3×C4—182no—
145#239—62no—
146D3×Q8—62no—
147#237—64no—
148#237—62no—
149D3×D4—62no—
150D3×D4—64no—
151#235—22no—
152#235—64no—
153C2×C4×D3—62no—
154C2×C4×D3—64no—
155C3×Q16—94no—
156C3×SD16—98no—
157C3×D8—94no—
158C3×M16—94no—
159C2×C3×C8—92no—
160C22×C32—22no—
161C22×C32—42no—
162C22×C32—64no—
163C22×C32C2×C3×D411yescharacteristic
164C2×C3⋊D3—41no—
165C2×C3⋊D3C2×C3×D412yes—
166C2×C3×D3—24no—
167C2×C3×D3—24no—
168C2×C3×D3—44no—
169C2×C3×D3—62no—
170C2×C3×D3—62no—
171C2×C3×D3—128no—
172C2×C3×D3—124no—
173D32—44no—
174C32×C4—22no—
175C32×C4—22no—
176C32×C4—42no—
177C32×C4—42no—
178C32×C4—64no—
179C32×C4C2×C3×D412yes—
180C3⋊Dic3C2×C3×D411yescharacteristic
181C3⋊Dic3C24×C311yescharacteristic
182C3×Dic3—24no—
183C3×Dic3—24no—
184C3×Dic3—44no—
185C3×Dic3—62no—
186C3×Dic3—62no—
187C2×#41—92no—
188#137—94no—
189AGL(1,ℤ/8)—94no—
190#135—94no—
191C2×M16—91no—
192C33#57511yescharacteristic, complemented, Sylow
193C23×C3—91no—
194C23×C3—182no—
195C22×D3—62no—
196C22×D3—64no—
197C3×Q8—62no—
198C3×Q8—62no—
199C3×Q8—94no—
200C3×Q8—122no—
201C3×D4—62no—
202C3×D4—64no—
203C3×D4—64no—
204C3×D4—62no—
205C3×D4—94no—
206C3×D4—122no—
207C3×D4—124no—
208C3×D4—184no—
209C2×C3×C4—22no—
210C2×C3×C4—42no—
211C2×C3×C4—64no—
212C2×C3×C4—64no—
213C2×C3×C4—91no—
214C2×C3×C4—92no—
215C2×C3×C4—92no—
216C2×C3×C4—124no—
217C2×C3×C4—182no—
218C2×C3×C4—184no—
219C2×C3×C4GO+(4,𝔽2)11yescharacteristic
220#67—24no—
221#67—64no—
222#67—64no—
223C2×Dic3—62no—
224C2×Dic3—64no—
225D12—22no—
226D12—64no—
227C4×D3—24no—
228C4×D3—62no—
229C4×D3—62no—
230C4×D3—62no—
231C4×D3—62no—
232C4×D3—64no—
233C4×D3—64no—
234Dic6—22no—
235Dic6—64no—
236C3×C8—94no—
237C2×C32—22no—
238C2×C32—21no—
239C2×C32—42no—
240C2×C32—42no—
241C2×C32—82no—
242C2×C32—124no—
243C2×C32C22×C3×D411yescharacteristic
244C3⋊D3—22no—
245C3×D3—44no—
246C3×D3—44no—
247C3×D3—64no—
248C3×D3—84no—
249C3×D3—122no—
250#41—62no—
251#41—94no—
252#41—94no—
253#41—182no—
254C2×Q8—92no—
255C2×D4—92no—
256C2×D4—182no—
257C22×C4—91no—
258C22×C4—182no—
259Q16—94no—
260SD16—98no—
261D8—94no—
262M16—94no—
263C2×C8—92no—
264C22×C3—22no—
265C22×C3—42no—
266C22×C3—64no—
267C22×C3—64no—
268C22×C3—92no—
269C22×C3—124no—
270C22×C3—361no—
271C22×C3—364no—
272C22×C3C2×GO+(4,𝔽2)11yescharacteristic
273C2×D3—24no—
274C2×D3—62no—
275C2×D3—62no—
276C2×D3—128no—
277C2×D3—124no—
278C3×C4—22no—
279C3×C4—22no—
280C3×C4—42no—
281C3×C4—42no—
282C3×C4—64no—
283C3×C4—64no—
284C3×C4—91no—
285C3×C4—91no—
286C3×C4—124no—
287C3×C4C2×GO+(4,𝔽2)12yes—
288Dic3—24no—
289Dic3—62no—
290Dic3—62no—
291C32—22no—
292C32—42no—
293C32C3×#57511yescharacteristic, complemented
294C23—91no—
295C23—182no—
296Q8—62no—
297Q8—94no—
298D4—64no—
299D4—62no—
300D4—94no—
301D4—184no—
302C2×C4—64no—
303C2×C4—91no—
304C2×C4—92no—
305C2×C4—92no—
306C2×C4—182no—
307C2×C4—184no—
308C2×C4C3×GO+(4,𝔽2)11yescharacteristic
309C8—94no—
310C2×C3—21no—
311C2×C3—22no—
312C2×C3—42no—
313C2×C3—42no—
314C2×C3—82no—
315C2×C3—124no—
316C2×C3—124no—
317C2×C3—182no—
318C2×C3—244no—
319C2×C3C22×GO+(4,𝔽2)11yescharacteristic
320D3—44no—
321D3—64no—
322D3—122no—
323C22—64no—
324C22—92no—
325C22—361no—
326C22—364no—
327C22C2×C3×GO+(4,𝔽2)11yescharacteristic
328C4—64no—
329C4—91no—
330C4—91no—
331C4C2×C3×GO+(4,𝔽2)12yes—
332C3—22no—
333C3—42no—
334C3#1059650211yescharacteristic, direct product factor
335C2—21no—
336C2—124no—
337C2—182no—
338C2C22×C3×GO+(4,𝔽2)11yescharacteristic
subgroupG/H
centerC2×C3C22×GO+(4,𝔽2)
commutator, cosocleC3⋊Dic3C24×C3
FrattiniC2C22×C3×GO+(4,𝔽2)
FittingC2×C33×C4D4
socleC2×C33C22×D4
hypercenterC2×C3×C4GO+(4,𝔽2)
hypocenterC32C3×#575
LCSG ▷ C3⋊Dic3 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2×C3 ◁ C2×C3×C4

Complements

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

NH ≅ G/Ntype
1. C3×#64306C2semidirect
2. C2×C3×#3788C2semidirect
3. C3×#64237C2semidirect
4. C3×#64226C2semidirect
5. C3×#64224C2semidirect
6. #10596502C3direct (N × H)
10. C3×#3805C22semidirect
11. C3×#3802C22semidirect
14. C3×#3797C22semidirect
16. C2×C3×#622C22semidirect
17. C3×#3788C22semidirect
18. C3×#3776C22semidirect
19. C3×#3775C22semidirect
20. C3×#3774C22semidirect
22. #64306C2×C3semidirect
23. C2×#3788C2×C3semidirect
24. #64237C2×C3semidirect
25. #64226C2×C3semidirect
26. #64224C2×C3semidirect
59. #3805C22×C3semidirect
60. #3802C22×C3semidirect
63. #3797C22×C3semidirect
65. C2×#622C22×C3semidirect
66. #3788C22×C3semidirect
67. #3776C22×C3semidirect
68. #3775C22×C3semidirect
69. #3774C22×C3semidirect
192. C33#575semidirect
293. C32C3×#575semidirect
334. C3#10596502direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#643062
C2×C3×#37882
C3×#642372
C3×#642262
C3×#642242
#105965023
C3×#5759