Finite Group Databasebeta

← C32×#5241 #13162678

Subgroups — C32×#5241 #13162678

Order 1728.

1672 conjugacy classes of subgroups; only the 258 normal subgroups are shown.

normal subgroups only

This group has 1672 conjugacy classes of subgroups — too many to draw. Only its 258 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
1C33×H(5,𝔽2)C211yesmaximal, characteristic, complemented
2C32×#949C212yesmaximal, complemented
3C32×#940C211yesmaximal, characteristic, complemented
4C32×#837C212yesmaximal, complemented
5C32×#833C211yesmaximal, characteristic, complemented
6C32×#453—31nomaximal
7C3×#5241C314yesmaximal, direct product factor
8C33×#41—22no—
9C33×#41C2212yes—
10C2×C33×D4—22no—
11C2×C33×D4—22no—
12C2×C33×D4C2211yescharacteristic
13C2×C32×#67—22no—
14C2×C32×D12C2211yescharacteristic, complemented
15C32×#215—22no—
16C32×#213—22no—
17C32×C4×Dic3C2211yescharacteristic, complemented
18C32×#208C2212yescomplemented
19C32×H(5,𝔽2)—24no—
20C32×H(5,𝔽2)D311yescharacteristic, complemented
21C32×H(5,𝔽2)C2×C314yescomplemented
22C32×AGL(1,ℤ/8)—32no—
23C32×#127—31no—
24C3×#949C2×C318yescomplemented
25C3×#940C2×C314yescomplemented
26C32×#104—32no—
27C32×#100—31no—
28C3×#837C2×C318yescomplemented
29C3×#833C2×C314yescomplemented
30C23×C33—22no—
31C23×C33D412yes—
32C22×C32×D3—21no—
33C33×Q8D412yescomplemented
34C33×D4—24no—
35C33×D4—22no—
36C33×D4—22no—
37C33×D4D412yescomplemented
38C2×C33×C4—22no—
39C2×C33×C4—22no—
40C2×C33×C4C2311yescharacteristic
41C32×#67—44no—
42C2×C32×Dic3—21no—
43C32×D12—22no—
44C32×C3⋊C8—22no—
45C3×#453—34no—
46#5241C3211yescharacteristic, direct product factor
47C32×#41—216no—
48C32×#41—28no—
49C32×#41—22no—
50C32×#41—28no—
51C32×#41C2×D312yes—
52C32×#41C22×C318yes—
53C2×C32×D4—216no—
54C2×C32×D4—216no—
55C2×C32×D4—24no—
56C2×C32×D4—22no—
57C2×C32×D4—28no—
58C2×C32×D4—22no—
59C2×C32×D4—28no—
60C2×C32×D4—31no—
61C2×C32×D4—62no—
62C2×C32×D4C2×D311yescharacteristic
63C2×C32×D4C22×C314yes—
64C2×C3×#67—28no—
65C2×C3×D12C22×C314yescomplemented
66C32×SD16—62no—
67C32×D8—62no—
68C32×M16—32no—
69C32×C42—31no—
70C3×#215—28no—
71C3×#213—28no—
72C3×C4×Dic3C22×C314yescomplemented
73C3×#208C22×C318yescomplemented
74C22×C33—22no—
75C22×C33—22no—
76C22×C33—42no—
77C22×C33—82no—
78C22×C33C2×D411yescharacteristic
79C2×C32×D3—42no—
80C2×C32×D3—41no—
81C33×C4—22no—
82C33×C4C2×D412yes—
83C32×Dic3—22no—
84C3×H(5,𝔽2)—24no—
85C3×H(5,𝔽2)C3×D314yescomplemented
86C3×H(5,𝔽2)C2×C3211yescharacteristic, complemented
87C3×AGL(1,ℤ/8)—38no—
88C3×#127—34no—
89#949C2×C3212yescomplemented
90#940C2×C3211yescharacteristic, complemented
91C3×#104—38no—
92C3×#100—34no—
93#837C2×C3212yescomplemented
94#833C2×C3211yescharacteristic, complemented
95C23×C32—216no—
96C23×C32—22no—
97C23×C32—28no—
98C23×C32—28no—
99C23×C32—61no—
100C23×C32#6712yes—
101C23×C32C3×D418yes—
102C22×C3×D3—24no—
103C32×Q8—28no—
104C32×Q8#6712yescomplemented
105C32×Q8C3×D418yescomplemented
106C32×D4—232no—
107C32×D4—24no—
108C32×D4—216no—
109C32×D4—216no—
110C32×D4—216no—
111C32×D4—22no—
112C32×D4—28no—
113C32×D4—22no—
114C32×D4—28no—
115C32×D4—28no—
116C32×D4—62no—
117C32×D4—124no—
118C32×D4#6712yescomplemented
119C32×D4C3×D418yescomplemented
120C2×C32×C4—216no—
121C2×C32×C4—22no—
122C2×C32×C4—28no—
123C2×C32×C4—216no—
124C2×C32×C4—22no—
125C2×C32×C4—28no—
126C2×C32×C4—24no—
127C2×C32×C4—61no—
128C2×C32×C4C22×D311yescharacteristic
129C2×C32×C4C23×C314yes—
130C3×#67—416no—
131C2×C3×Dic3—24no—
132C3×D12—28no—
133C32×C8—62no—
134C3×C3⋊C8—28no—
135#453—31noSylow
136C2×C33—21no—
137C2×C33—42no—
138C2×C33—42no—
139C2×C33#12011yescharacteristic
140C32×D3—81no—
141C3×#41—216no—
142C3×#41—28no—
143C3×#41—22no—
144C3×#41—28no—
145C3×#41C2×C3×D318yes—
146C3×#41C22×C3212yes—
147C2×C3×D4—216no—
148C2×C3×D4—216no—
149C2×C3×D4—28no—
150C2×C3×D4—22no—
151C2×C3×D4—28no—
152C2×C3×D4—22no—
153C2×C3×D4—24no—
154C2×C3×D4—34no—
155C2×C3×D4—68no—
156C2×C3×D4C2×C3×D314yes—
157C2×C3×D4C22×C3211yescharacteristic
158C2×#67—22no—
159C2×D12C22×C3211yescharacteristic, complemented
160C3×SD16—68no—
161C3×D8—68no—
162C3×M16—38no—
163C3×C42—34no—
164#215—22no—
165#213—22no—
166C4×Dic3C22×C3211yescharacteristic, complemented
167#208C22×C3212yescomplemented
168C22×C32—216no—
169C22×C32—216no—
170C22×C32—22no—
171C22×C32—22no—
172C22×C32—28no—
173C22×C32—28no—
174C22×C32—24no—
175C22×C32—42no—
176C22×C32—48no—
177C22×C32—88no—
178C22×C32—816no—
179C22×C32—82no—
180C22×C32—88no—
181C22×C32—122no—
182C22×C32—121no—
183C22×C32C2×#6711yescharacteristic
184C22×C32C2×C3×D414yes—
185C2×C3×D3—48no—
186C2×C3×D3—44no—
187C32×C4—216no—
188C32×C4—22no—
189C32×C4—28no—
190C32×C4—28no—
191C32×C4—62no—
192C32×C4C2×#6712yes—
193C32×C4C2×C3×D418yes—
194C3×Dic3—28no—
195H(5,𝔽2)C32×D311yescharacteristic, complemented
196AGL(1,ℤ/8)—32no—
197#127—31no—
198#104—32no—
199#100—31no—
200C33#45311yescharacteristic, complemented, Sylow
201C23×C3—216no—
202C23×C3—28no—
203C23×C3—22no—
204C23×C3—28no—
205C23×C3—64no—
206C23×C3C3×#6718yes—
207C23×C3C32×D412yes—
208C22×D3—21no—
209C3×Q8—28no—
210C3×Q8C3×#6718yescomplemented
211C3×Q8C32×D412yescomplemented
212C3×D4—232no—
213C3×D4—216no—
214C3×D4—24no—
215C3×D4—216no—
216C3×D4—216no—
217C3×D4—28no—
218C3×D4—22no—
219C3×D4—28no—
220C3×D4—22no—
221C3×D4—28no—
222C3×D4—68no—
223C3×D4—1216no—
224C3×D4C3×#6718yescomplemented
225C3×D4C32×D412yescomplemented
226C2×C3×C4—216no—
227C2×C3×C4—28no—
228C2×C3×C4—22no—
229C2×C3×C4—216no—
230C2×C3×C4—28no—
231C2×C3×C4—22no—
232C2×C3×C4—24no—
233C2×C3×C4—64no—
234C2×C3×C4C22×C3×D314yes—
235C2×C3×C4C23×C3211yescharacteristic
236#67—44no—
237C2×Dic3—21no—
238D12—22no—
239C3×C8—68no—
240C3⋊C8—22no—
241C2×C32—21no—
242C2×C32—24no—
243C2×C32—24no—
244C2×C32—44no—
245C2×C32—416no—
246C2×C32—416no—
247C2×C32—42no—
248C2×C32—42no—
249C2×C32—48no—
250C2×C32—48no—
251C2×C32—241no—
252C2×C32#95311yescharacteristic
253C2×C32C3×#12014yes—
254C3×D3—84no—
255#41—22no—
256#41C2×C32×D312yes—
257C2×D4—22no—
258C2×D4—22no—
259C2×D4—31no—
260C2×D4—62no—
261C2×D4C2×C32×D311yescharacteristic
262SD16—62no—
263D8—62no—
264M16—32no—
265C42—31no—
266C22×C3—216no—
267C22×C3—216no—
268C22×C3—28no—
269C22×C3—28no—
270C22×C3—22no—
271C22×C3—22no—
272C22×C3—24no—
273C22×C3—42no—
274C22×C3—48no—
275C22×C3—88no—
276C22×C3—816no—
277C22×C3—82no—
278C22×C3—88no—
279C22×C3—128no—
280C22×C3—124no—
281C22×C3C2×C3×#6714yes—
282C22×C3C2×C32×D411yescharacteristic
283C2×D3—42no—
284C2×D3—41no—
285C3×C4—216no—
286C3×C4—28no—
287C3×C4—22no—
288C3×C4—28no—
289C3×C4—68no—
290C3×C4C2×C3×#6718yes—
291C3×C4C2×C32×D412yes—
292Dic3—22no—
293C32—24no—
294C32#524111yescharacteristic, direct product factor
295C32C3×#45314yescomplemented
296C23—22no—
297C23—61no—
298C23C32×#6712yes—
299Q8C32×#6712yescomplemented
300D4—24no—
301D4—22no—
302D4—22no—
303D4—62no—
304D4—124no—
305D4C32×#6712yescomplemented
306C2×C4—22no—
307C2×C4—22no—
308C2×C4—61no—
309C2×C4C22×C32×D311yescharacteristic
310C8—62no—
311C2×C3—21no—
312C2×C3—24no—
313C2×C3—24no—
314C2×C3—44no—
315C2×C3—416no—
316C2×C3—416no—
317C2×C3—42no—
318C2×C3—42no—
319C2×C3—48no—
320C2×C3—48no—
321C2×C3—244no—
322C2×C3C3×#95314yes—
323C2×C3C32×#12011yescharacteristic
324D3—81no—
325C22—22no—
326C22—22no—
327C22—42no—
328C22—82no—
329C22—122no—
330C22—121no—
331C22C2×C32×#6711yescharacteristic
332C4—22no—
333C4—62no—
334C4C2×C32×#6712yes—
335C3—24no—
336C3C3×#524114yesdirect product factor
337C3C32×#45311yescharacteristic, complemented
338C2—21no—
339C2—42no—
340C2—42no—
341C2—241no—
342C2C32×#95311yescharacteristic
subgroupG/H
centerC2×C32#953
commutator, cosocleC2×C3×C4C23×C32
FrattiniC2×C4C22×C32×D3
FittingC33×H(5,𝔽2)C2
socleC2×C33#120
hypercenterC32×H(5,𝔽2)D3
hypocenterC3C32×#453
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C32 ◁ C2×C32×C4 ◁ C32×H(5,𝔽2)

Complements

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

NH ≅ G/Ntype
1. C33×H(5,𝔽2)C2semidirect
2. C32×#949C2semidirect
3. C32×#940C2semidirect
4. C32×#837C2semidirect
5. C32×#833C2semidirect
7. C3×#5241C3direct (N × H)
14. C2×C32×D12C22semidirect
17. C32×C4×Dic3C22semidirect
18. C32×#208C22semidirect
20. C32×H(5,𝔽2)D3semidirect
21. C32×H(5,𝔽2)C2×C3semidirect
24. C3×#949C2×C3semidirect
25. C3×#940C2×C3semidirect
28. C3×#837C2×C3semidirect
29. C3×#833C2×C3semidirect
33. C33×Q8D4semidirect
37. C33×D4D4semidirect
46. #5241C32direct (N × H)
65. C2×C3×D12C22×C3semidirect
72. C3×C4×Dic3C22×C3semidirect
73. C3×#208C22×C3semidirect
85. C3×H(5,𝔽2)C3×D3semidirect
86. C3×H(5,𝔽2)C2×C32semidirect
89. #949C2×C32semidirect
90. #940C2×C32semidirect
93. #837C2×C32semidirect
94. #833C2×C32semidirect
104. C32×Q8#67semidirect
105. C32×Q8C3×D4semidirect
118. C32×D4#67semidirect
119. C32×D4C3×D4semidirect
159. C2×D12C22×C32semidirect
166. C4×Dic3C22×C32semidirect
167. #208C22×C32semidirect
195. H(5,𝔽2)C32×D3semidirect
200. C33#453semidirect
210. C3×Q8C3×#67semidirect
211. C3×Q8C32×D4semidirect
224. C3×D4C3×#67semidirect
225. C3×D4C32×D4semidirect
294. C32#5241direct (N × H)
295. C32C3×#453semidirect
299. Q8C32×#67semidirect
305. D4C32×#67semidirect
336. C3C3×#5241direct (N × H)
337. C3C32×#453semidirect

Maximal subgroups

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

subgroupindex
C33×H(5,𝔽2)2
C32×#9492
C32×#9402
C32×#8372
C32×#8332
C32×#4533
C3×#52413