Finite Group Databasebeta

← C32×#5061 #13162497

Subgroups — C32×#5061 #13162497

Order 1728.

1216 conjugacy classes of subgroups; only the 234 normal subgroups are shown.

normal subgroups only

This group has 1216 conjugacy classes of subgroups — too many to draw. Only its 234 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×#1004C211yesmaximal, characteristic
2C33×#124C211yesmaximal, characteristic, complemented
3C32×#942C212yesmaximal, complemented
4C32×#833C211yesmaximal, characteristic, complemented
5C32×#805C212yesmaximal, complemented
6C32×#455—31nomaximal
7C3×#5061C314yesmaximal, direct product factor
8C2×C33×Q8C2211yescharacteristic, complemented
9C2×C33×D4C2211yescharacteristic
10C2×C32×#67—22no—
11C32×#237—22no—
12C32×D3×D4—22no—
13C32×#235C2212yes—
14C33×C22⋊C4—22no—
15C33×C42C2211yescharacteristic, complemented
16C32×#216—22no—
17C32×#215—22no—
18C32×#208C2212yescomplemented
19C32×H(5,𝔽2)—31no—
20C3×#1004C2×C314yes—
21C32×#137—32no—
22C32×#124—24no—
23C32×#124D311yescharacteristic, complemented
24C32×#124C2×C314yescomplemented
25C3×#942C2×C318yescomplemented
26C32×#104—32no—
27C32×#100—31no—
28C3×#833C2×C314yescomplemented
29C3×#805C2×C318yescomplemented
30C23×C33D412yes—
31C22×C32×D3—22no—
32C33×Q8—22no—
33C33×D4—22no—
34C2×C33×C4—21no—
35C2×C33×C4—21no—
36C2×C33×C4C2311yescharacteristic
37C32×#67—24no—
38C32×#67—22no—
39C2×C32×Dic3—22no—
40C32×D12D412yes—
41C32×C4×D3—22no—
42C32×Dic6D412yes—
43C32×C3⋊C8—22no—
44C3×#455—34no—
45#5061C3211yescharacteristic, direct product factor
46C32×#41—32no—
47C32×#41—62no—
48C2×C32×Q8—24no—
49C2×C32×Q8C2×D311yescharacteristic, complemented
50C2×C32×Q8C22×C314yescomplemented
51C2×C32×D4—24no—
52C2×C32×D4—62no—
53C2×C32×D4—62no—
54C2×C32×D4C2×D311yescharacteristic
55C2×C32×D4C22×C314yes—
56C2×C3×#67—28no—
57C3×#237—28no—
58C3×D3×D4—28no—
59C3×#235C22×C318yes—
60C32×Q16—62no—
61C32×SD16—62no—
62C32×M16—32no—
63C32×C22⋊C4—216no—
64C32×C22⋊C4—22no—
65C32×C22⋊C4—28no—
66C32×C42—24no—
67C32×C42C2×D311yescharacteristic, complemented
68C32×C42C22×C314yescomplemented
69C3×#216—28no—
70C3×#215—28no—
71C3×#208C22×C318yescomplemented
72C22×C33—22no—
73C22×C33—42no—
74C22×C33C2×D411yescharacteristic
75C2×C32×D3—22no—
76C2×C32×D3—82no—
77C33×C4—41no—
78C33×C4—41no—
79C33×C4C2×D412yes—
80C32×Dic3—22no—
81C3×H(5,𝔽2)—34no—
82#1004C2×C3211yescharacteristic
83C3×#137—38no—
84C3×#124—24no—
85C3×#124C3×D314yescomplemented
86C3×#124C2×C3211yescharacteristic, complemented
87#942C2×C3212yescomplemented
88C3×#104—38no—
89C3×#100—34no—
90#833C2×C3211yescharacteristic, complemented
91#805C2×C3212yescomplemented
92C23×C32—28no—
93C23×C32—62no—
94C23×C32#6712yes—
95C23×C32C3×D418yes—
96C22×C3×D3—28no—
97C32×Q8—22no—
98C32×Q8—28no—
99C32×Q8—32no—
100C32×Q8—48no—
101C32×D4—22no—
102C32×D4—28no—
103C32×D4—32no—
104C32×D4—48no—
105C32×D4—64no—
106C32×D4—62no—
107C2×C32×C4—28no—
108C2×C32×C4—28no—
109C2×C32×C4—21no—
110C2×C32×C4—21no—
111C2×C32×C4—24no—
112C2×C32×C4—24no—
113C2×C32×C4—24no—
114C2×C32×C4—62no—
115C2×C32×C4—62no—
116C2×C32×C4C22×D311yescharacteristic
117C2×C32×C4C23×C314yes—
118C3×#67—216no—
119C3×#67—28no—
120C2×C3×Dic3—28no—
121C3×D12C3×D418yes—
122C3×C4×D3—28no—
123C3×Dic6C3×D418yes—
124C32×C8—62no—
125C3×C3⋊C8—28no—
126#455—31noSylow
127C2×C33—21no—
128C2×C33—42no—
129C2×C33#12011yescharacteristic
130C32×D3—42no—
131C3×#41—38no—
132C3×#41—68no—
133C2×C3×Q8—24no—
134C2×C3×Q8C2×C3×D314yescomplemented
135C2×C3×Q8C22×C3211yescharacteristic, complemented
136C2×C3×D4—24no—
137C2×C3×D4—68no—
138C2×C3×D4—68no—
139C2×C3×D4C2×C3×D314yes—
140C2×C3×D4C22×C3211yescharacteristic
141C2×#67—22no—
142#237—22no—
143D3×D4—22no—
144#235C22×C3212yes—
145C3×Q16—68no—
146C3×SD16—68no—
147C3×M16—38no—
148C3×C22⋊C4—216no—
149C3×C22⋊C4—28no—
150C3×C22⋊C4—22no—
151C3×C42—24no—
152C3×C42C2×C3×D314yescomplemented
153C3×C42C22×C3211yescharacteristic, complemented
154#216—22no—
155#215—22no—
156#208C22×C3212yescomplemented
157C22×C32—22no—
158C22×C32—24no—
159C22×C32—28no—
160C22×C32—48no—
161C22×C32—42no—
162C22×C32—48no—
163C22×C32—416no—
164C22×C32—62no—
165C22×C32—242no—
166C22×C32C2×#6711yescharacteristic
167C22×C32C2×C3×D414yes—
168C2×C3×D3—28no—
169C2×C3×D3—88no—
170C32×C4—28no—
171C32×C4—48no—
172C32×C4—41no—
173C32×C4—44no—
174C32×C4—48no—
175C32×C4—41no—
176C32×C4—44no—
177C32×C4—62no—
178C32×C4D3×D412yes—
179C32×C4C2×C3×D418yes—
180C3×Dic3—28no—
181H(5,𝔽2)—31no—
182#137—32no—
183#124C32×D311yescharacteristic, complemented
184#104—32no—
185#100—31no—
186C33#45511yescharacteristic, complemented, Sylow
187C23×C3—28no—
188C23×C3—68no—
189C23×C3C3×#6718yes—
190C23×C3C32×D412yes—
191C22×D3—22no—
192C3×Q8—28no—
193C3×Q8—22no—
194C3×Q8—38no—
195C3×Q8—48no—
196C3×D4—28no—
197C3×D4—22no—
198C3×D4—38no—
199C3×D4—48no—
200C3×D4—616no—
201C3×D4—68no—
202C2×C3×C4—28no—
203C2×C3×C4—28no—
204C2×C3×C4—24no—
205C2×C3×C4—24no—
206C2×C3×C4—21no—
207C2×C3×C4—21no—
208C2×C3×C4—24no—
209C2×C3×C4—68no—
210C2×C3×C4—68no—
211C2×C3×C4C22×C3×D314yes—
212C2×C3×C4C23×C3211yescharacteristic
213#67—24no—
214#67—22no—
215C2×Dic3—22no—
216D12C32×D412yes—
217C4×D3—22no—
218Dic6C32×D412yes—
219C3×C8—68no—
220C3⋊C8—22no—
221C2×C32—28no—
222C2×C32—21no—
223C2×C32—24no—
224C2×C32—24no—
225C2×C32—42no—
226C2×C32—48no—
227C2×C32—88no—
228C2×C32—122no—
229C2×C32#93711yescharacteristic
230C2×C32C3×#12014yes—
231C3×D3—48no—
232#41—32no—
233#41—62no—
234C2×Q8C2×C32×D311yescharacteristic, complemented
235C2×D4—62no—
236C2×D4—62no—
237C2×D4C2×C32×D311yescharacteristic
238Q16—62no—
239SD16—62no—
240M16—32no—
241C22⋊C4—22no—
242C42C2×C32×D311yescharacteristic, complemented
243C22×C3—28no—
244C22×C3—22no—
245C22×C3—24no—
246C22×C3—48no—
247C22×C3—48no—
248C22×C3—42no—
249C22×C3—416no—
250C22×C3—68no—
251C22×C3—248no—
252C22×C3C2×C3×#6714yes—
253C22×C3C2×C32×D411yescharacteristic
254C2×D3—22no—
255C2×D3—82no—
256C3×C4—28no—
257C3×C4—48no—
258C3×C4—48no—
259C3×C4—41no—
260C3×C4—44no—
261C3×C4—44no—
262C3×C4—41no—
263C3×C4—68no—
264C3×C4C3×D3×D418yes—
265C3×C4C2×C32×D412yes—
266Dic3—22no—
267C32—24no—
268C32#506111yescharacteristic, direct product factor
269C32C3×#45514yescomplemented
270C23—62no—
271C23C32×#6712yes—
272Q8—22no—
273Q8—32no—
274D4—22no—
275D4—32no—
276D4—64no—
277D4—62no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—62no—
281C2×C4—62no—
282C2×C4C22×C32×D311yescharacteristic
283C8—62no—
284C2×C3—28no—
285C2×C3—21no—
286C2×C3—24no—
287C2×C3—24no—
288C2×C3—42no—
289C2×C3—48no—
290C2×C3—88no—
291C2×C3—128no—
292C2×C3C3×#93714yes—
293C2×C3C32×#12011yescharacteristic
294D3—42no—
295C22—22no—
296C22—42no—
297C22—62no—
298C22—242no—
299C22C2×C32×#6711yescharacteristic
300C4—41no—
301C4—41no—
302C4—62no—
303C4C32×D3×D412yes—
304C3—24no—
305C3C3×#506114yesdirect product factor
306C3C32×#45511yescharacteristic, complemented
307C2—21no—
308C2—42no—
309C2—122no—
310C2C32×#93711yescharacteristic
subgroupG/H
centerC2×C32#937
commutator, cosocleC2×C3×C4C23×C32
FrattiniC2×C4C22×C32×D3
FittingC33×#124C2
socleC2×C33#120
prime radicalC32×#1004C2
hypercenterC32×#124D3
hypocenterC3C32×#455
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C32 ◁ C2×C32×C4 ◁ C32×#124

Complements

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

NH ≅ G/Ntype
2. C33×#124C2semidirect
3. C32×#942C2semidirect
4. C32×#833C2semidirect
5. C32×#805C2semidirect
7. C3×#5061C3direct (N × H)
8. C2×C33×Q8C22semidirect
15. C33×C42C22semidirect
18. C32×#208C22semidirect
23. C32×#124D3semidirect
24. C32×#124C2×C3semidirect
25. C3×#942C2×C3semidirect
28. C3×#833C2×C3semidirect
29. C3×#805C2×C3semidirect
45. #5061C32direct (N × H)
49. C2×C32×Q8C2×D3semidirect
50. C2×C32×Q8C22×C3semidirect
67. C32×C42C2×D3semidirect
68. C32×C42C22×C3semidirect
71. C3×#208C22×C3semidirect
85. C3×#124C3×D3semidirect
86. C3×#124C2×C32semidirect
87. #942C2×C32semidirect
90. #833C2×C32semidirect
91. #805C2×C32semidirect
134. C2×C3×Q8C2×C3×D3semidirect
135. C2×C3×Q8C22×C32semidirect
152. C3×C42C2×C3×D3semidirect
153. C3×C42C22×C32semidirect
156. #208C22×C32semidirect
183. #124C32×D3semidirect
186. C33#455semidirect
234. C2×Q8C2×C32×D3semidirect
242. C42C2×C32×D3semidirect
268. C32#5061direct (N × H)
269. C32C3×#455semidirect
305. C3C3×#5061direct (N × H)
306. C3C32×#455semidirect

Maximal subgroups

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

subgroupindex
C32×#10042
C33×#1242
C32×#9422
C32×#8332
C32×#8052
C32×#4553
C3×#50613