Finite Group Databasebeta

← C32×#4827 #13162263

Subgroups — C32×#4827 #13162263

Order 1728.

1052 conjugacy classes of subgroups.

The subgroup lattice has 426 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C32×#1010C211yesmaximal, characteristic, complemented
2C32×#949C211yesmaximal, characteristic, complemented
3C32×#908C211yesmaximal, characteristic, complemented
4C32×#875C211yesmaximal, characteristic, complemented
5C33×#104C211yesmaximal, characteristic, complemented
6C32×#824C211yesmaximal, characteristic, complemented
7C32×#805C211yesmaximal, characteristic, complemented
8C32×#454—31nomaximal
9C3×#4827C314yesmaximal, direct product factor
10C33×#41C2211yescharacteristic
11C32×D3×Q8—21no—
12C32×#237—21no—
13C32×#237—22no—
14C32×#235—21no—
15C32×#235C2211yescharacteristic
16C2×C32×D12C2211yescharacteristic
17C2×C32×Dic6—21no—
18C2×C32×Dic6C2211yescharacteristic
19C33×M16C2211yescharacteristic
20C33×C42C2211yescharacteristic
21C32×#215—21no—
22C32×#213—21no—
23C32×(C2×D3)⋊C4—22no—
24C32×#208C2211yescharacteristic
25C32×D24—21no—
26C32×#204—21no—
27C32×2−1+4—31no—
28C3×#1010C2×C314yescomplemented
29C32×AGL(1,ℤ/8)—31no—
30C32×AGL(1,ℤ/8)—31no—
31C32×#124—31no—
32C3×#949C2×C314yescomplemented
33C3×#908C2×C314yescomplemented
34C3×#875C2×C314yescomplemented
35C32×#104—24no—
36C32×#104—31no—
37C32×#104D311yescharacteristic, complemented
38C32×#104C2×C314yescomplemented
39C32×#101—31no—
40C3×#824C2×C314yescomplemented
41C3×#805C2×C314yescomplemented
42C22×C32×D3—21no—
43C33×Q8D411yescharacteristic
44C33×D4—21no—
45C33×D4D411yescharacteristic
46C2×C33×C4—21no—
47C2×C33×C4—21no—
48C2×C33×C4C2311yescharacteristic
49C32×#67—22no—
50C32×#67—21no—
51C2×C32×Dic3—22no—
52C2×C32×Dic3D412yes—
53C32×D12—21no—
54C32×D12—21no—
55C32×D12D411yescharacteristic
56C32×C4×D3—22no—
57C32×C4×D3—21no—
58C32×Dic6—22no—
59C32×Dic6—21no—
60C32×Dic6—21no—
61C32×Dic6D411yescharacteristic
62C33×C8—21no—
63C32×C3⋊C8—21no—
64C3×#454—34no—
65#4827C3211yescharacteristic, direct product factor
66C32×#41—24no—
67C32×#41—31no—
68C32×#41—61no—
69C32×#41—62no—
70C32×#41—61no—
71C32×#41C2×D311yescharacteristic
72C32×#41C22×C314yes—
73C2×C32×Q8—31no—
74C2×C32×Q8—61no—
75C2×C32×Q8—61no—
76C2×C32×D4—31no—
77C3×D3×Q8—24no—
78C3×#237—24no—
79C3×#237—28no—
80C3×#235—24no—
81C3×#235C22×C314yes—
82C2×C3×D12C22×C314yes—
83C2×C3×Dic6—24no—
84C2×C3×Dic6C22×C314yes—
85C32×SD16—61no—
86C32×SD16—61no—
87C32×D8—61no—
88C32×D8—61no—
89C32×M16—24no—
90C32×M16—31no—
91C32×M16C2×D311yescharacteristic
92C32×M16C22×C314yes—
93C32×C22⋊C4—62no—
94C32×C42—24no—
95C32×C42C2×D311yescharacteristic
96C32×C42C22×C314yes—
97C3×#215—24no—
98C3×#213—24no—
99C3×(C2×D3)⋊C4—28no—
100C3×#208C22×C314yes—
101C3×D24—24no—
102C3×#204—24no—
103C22×C33—21no—
104C22×C33C2×D411yescharacteristic
105C2×C32×D3—21no—
106C2×C32×D3—42no—
107C2×C32×D3—41no—
108C33×C4—21no—
109C33×C4—41no—
110C33×C4C2×D411yescharacteristic
111C33×C4C2×D411yescharacteristic
112C32×Dic3—22no—
113C32×Dic3—21no—
114C3×2−1+4—34no—
115#1010C2×C3211yescharacteristic, complemented
116C3×AGL(1,ℤ/8)—34no—
117C3×AGL(1,ℤ/8)—34no—
118C3×#124—34no—
119#949C2×C3211yescharacteristic, complemented
120#908C2×C3211yescharacteristic, complemented
121#875C2×C3211yescharacteristic, complemented
122C3×#104—24no—
123C3×#104—34no—
124C3×#104C3×D314yescomplemented
125C3×#104C2×C3211yescharacteristic, complemented
126C3×#101—34no—
127#824C2×C3211yescharacteristic, complemented
128#805C2×C3211yescharacteristic, complemented
129C23×C32—61no—
130C22×C3×D3—24no—
131C32×Q8—24no—
132C32×Q8—31no—
133C32×Q8—62no—
134C32×Q8—61no—
135C32×Q8—61no—
136C32×Q8D1211yescharacteristic
137C32×Q8C3×D414yes—
138C32×D4—21no—
139C32×D4—24no—
140C32×D4—24no—
141C32×D4—31no—
142C32×D4—44no—
143C32×D4—62no—
144C32×D4—61no—
145C32×D4—61no—
146C32×D4—61no—
147C32×D4D1211yescharacteristic
148C32×D4C3×D414yes—
149C2×C32×C4—21no—
150C2×C32×C4—21no—
151C2×C32×C4—24no—
152C2×C32×C4—24no—
153C2×C32×C4—24no—
154C2×C32×C4—32no—
155C2×C32×C4—44no—
156C2×C32×C4—44no—
157C2×C32×C4—62no—
158C2×C32×C4—62no—
159C2×C32×C4—61no—
160C2×C32×C4C22×D311yescharacteristic
161C2×C32×C4C23×C314yes—
162C3×#67—28no—
163C3×#67—24no—
164C2×C3×Dic3—28no—
165C2×C3×Dic3C3×D418yes—
166C3×D12—24no—
167C3×D12—24no—
168C3×D12C3×D414yes—
169C3×C4×D3—28no—
170C3×C4×D3—24no—
171C3×Dic6—28no—
172C3×Dic6—24no—
173C3×Dic6—24no—
174C3×Dic6C3×D414yes—
175C32×C8—21no—
176C32×C8—24no—
177C32×C8—44no—
178C32×C8—61no—
179C3×C3⋊C8—24no—
180#454—31noSylow
181C2×C33—21no—
182C2×C33—41no—
183C2×C33#12011yescharacteristic
184C32×D3—41no—
185C32×D3—81no—
186C3×#41—24no—
187C3×#41—34no—
188C3×#41—64no—
189C3×#41—68no—
190C3×#41—64no—
191C3×#41C2×C3×D314yes—
192C3×#41C22×C3211yescharacteristic
193C2×C3×Q8—34no—
194C2×C3×Q8—64no—
195C2×C3×Q8—64no—
196C2×C3×D4—34no—
197D3×Q8—21no—
198#237—21no—
199#237—22no—
200#235—21no—
201#235C22×C3211yescharacteristic
202C2×D12C22×C3211yescharacteristic
203C2×Dic6—21no—
204C2×Dic6C22×C3211yescharacteristic
205C3×SD16—64no—
206C3×SD16—64no—
207C3×D8—64no—
208C3×D8—64no—
209C3×M16—24no—
210C3×M16—34no—
211C3×M16C2×C3×D314yes—
212C3×M16C22×C3211yescharacteristic
213C3×C22⋊C4—68no—
214C3×C42—24no—
215C3×C42C2×C3×D314yes—
216C3×C42C22×C3211yescharacteristic
217#215—21no—
218#213—21no—
219(C2×D3)⋊C4—22no—
220#208C22×C3211yescharacteristic
221D24—21no—
222#204—21no—
223C22×C32—21no—
224C22×C32—24no—
225C22×C32—24no—
226C22×C32—44no—
227C22×C32—61no—
228C22×C32—122no—
229C22×C32—121no—
230C22×C32D3×D411yescharacteristic
231C22×C32C2×C3×D414yes—
232C2×C3×D3—24no—
233C2×C3×D3—48no—
234C2×C3×D3—44no—
235C32×C4—21no—
236C32×C4—24no—
237C32×C4—24no—
238C32×C4—24no—
239C32×C4—44no—
240C32×C4—41no—
241C32×C4—44no—
242C32×C4—48no—
243C32×C4—62no—
244C32×C4—61no—
245C32×C4C2×D1211yescharacteristic
246C32×C4D3×D411yescharacteristic
247C32×C4C2×C3×D414yes—
248C32×C4C2×C3×D414yes—
249C3×Dic3—28no—
250C3×Dic3—24no—
2512−1+4—31no—
252AGL(1,ℤ/8)—31no—
253AGL(1,ℤ/8)—31no—
254#124—31no—
255#104—31no—
256#104C32×D311yescharacteristic, complemented
257#101—31no—
258C33#45411yescharacteristic, complemented, Sylow
259C23×C3—64no—
260C22×D3—21no—
261C3×Q8—24no—
262C3×Q8—34no—
263C3×Q8—68no—
264C3×Q8—64no—
265C3×Q8—64no—
266C3×Q8C3×D1214yes—
267C3×Q8C32×D411yescharacteristic
268C3×D4—24no—
269C3×D4—21no—
270C3×D4—24no—
271C3×D4—34no—
272C3×D4—44no—
273C3×D4—68no—
274C3×D4—64no—
275C3×D4—64no—
276C3×D4—64no—
277C3×D4C3×D1214yes—
278C3×D4C32×D411yescharacteristic
279C2×C3×C4—24no—
280C2×C3×C4—21no—
281C2×C3×C4—24no—
282C2×C3×C4—21no—
283C2×C3×C4—24no—
284C2×C3×C4—38no—
285C2×C3×C4—44no—
286C2×C3×C4—44no—
287C2×C3×C4—68no—
288C2×C3×C4—68no—
289C2×C3×C4—64no—
290C2×C3×C4C22×C3×D314yes—
291C2×C3×C4C23×C3211yescharacteristic
292#67—22no—
293#67—21no—
294C2×Dic3—22no—
295C2×Dic3C32×D412yes—
296D12—21no—
297D12—21no—
298D12C32×D411yescharacteristic
299C4×D3—22no—
300C4×D3—21no—
301Dic6—22no—
302Dic6—21no—
303Dic6—21no—
304Dic6C32×D411yescharacteristic
305C3×C8—24no—
306C3×C8—21no—
307C3×C8—44no—
308C3×C8—64no—
309C3⋊C8—21no—
310C2×C32—21no—
311C2×C32—24no—
312C2×C32—24no—
313C2×C32—44no—
314C2×C32—41no—
315C2×C32—44no—
316C2×C32—84no—
317C2×C32—121no—
318C2×C32—241no—
319C2×C32#88211yescharacteristic
320C2×C32C3×#12014yes—
321C3×D3—44no—
322C3×D3—84no—
323#41—31no—
324#41—61no—
325#41—62no—
326#41—61no—
327#41C2×C32×D311yescharacteristic
328C2×Q8—31no—
329C2×Q8—61no—
330C2×Q8—61no—
331C2×D4—31no—
332SD16—61no—
333SD16—61no—
334D8—61no—
335D8—61no—
336M16—31no—
337M16C2×C32×D311yescharacteristic
338C22⋊C4—62no—
339C42C2×C32×D311yescharacteristic
340C22×C3—24no—
341C22×C3—21no—
342C22×C3—24no—
343C22×C3—44no—
344C22×C3—64no—
345C22×C3—128no—
346C22×C3—124no—
347C22×C3C3×D3×D414yes—
348C22×C3C2×C32×D411yescharacteristic
349C2×D3—21no—
350C2×D3—42no—
351C2×D3—41no—
352C3×C4—24no—
353C3×C4—21no—
354C3×C4—24no—
355C3×C4—24no—
356C3×C4—44no—
357C3×C4—41no—
358C3×C4—44no—
359C3×C4—48no—
360C3×C4—68no—
361C3×C4—64no—
362C3×C4C2×C3×D1214yes—
363C3×C4C3×D3×D414yes—
364C3×C4C2×C32×D411yescharacteristic
365C3×C4C2×C32×D411yescharacteristic
366Dic3—22no—
367Dic3—21no—
368C32—24no—
369C32#482711yescharacteristic, direct product factor
370C32C3×#45414yescomplemented
371C23—61no—
372Q8—31no—
373Q8—62no—
374Q8—61no—
375Q8—61no—
376Q8C32×D1211yescharacteristic
377D4—21no—
378D4—31no—
379D4—62no—
380D4—61no—
381D4—61no—
382D4—61no—
383D4C32×D1211yescharacteristic
384C2×C4—21no—
385C2×C4—21no—
386C2×C4—32no—
387C2×C4—62no—
388C2×C4—62no—
389C2×C4—61no—
390C2×C4C22×C32×D311yescharacteristic
391C8—21no—
392C8—61no—
393C2×C3—21no—
394C2×C3—24no—
395C2×C3—24no—
396C2×C3—44no—
397C2×C3—41no—
398C2×C3—44no—
399C2×C3—84no—
400C2×C3—124no—
401C2×C3—244no—
402C2×C3C3×#88214yes—
403C2×C3C32×#12011yescharacteristic
404D3—41no—
405D3—81no—
406C22—21no—
407C22—61no—
408C22—122no—
409C22—121no—
410C22C32×D3×D411yescharacteristic
411C4—21no—
412C4—41no—
413C4—62no—
414C4—61no—
415C4C2×C32×D1211yescharacteristic
416C4C32×D3×D411yescharacteristic
417C3—24no—
418C3C3×#482714yesdirect product factor
419C3C32×#45411yescharacteristic, complemented
420C2—21no—
421C2—41no—
422C2—121no—
423C2—241no—
424C2C32×#88211yescharacteristic
subgroupG/H
centerC2×C32#882
commutator, cosocleC2×C3×C4C23×C32
FrattiniC2×C4C22×C32×D3
FittingC33×#104C2
socleC2×C33#120
hypercenterC32×#104D3
hypocenterC3C32×#454
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C32 ◁ C2×C32×C4 ◁ C32×#104

Complements

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

NH ≅ G/Ntype
1. C32×#1010C2semidirect
2. C32×#949C2semidirect
3. C32×#908C2semidirect
4. C32×#875C2semidirect
5. C33×#104C2semidirect
6. C32×#824C2semidirect
7. C32×#805C2semidirect
9. C3×#4827C3direct (N × H)
28. C3×#1010C2×C3semidirect
32. C3×#949C2×C3semidirect
33. C3×#908C2×C3semidirect
34. C3×#875C2×C3semidirect
37. C32×#104D3semidirect
38. C32×#104C2×C3semidirect
40. C3×#824C2×C3semidirect
41. C3×#805C2×C3semidirect
65. #4827C32direct (N × H)
115. #1010C2×C32semidirect
119. #949C2×C32semidirect
120. #908C2×C32semidirect
121. #875C2×C32semidirect
124. C3×#104C3×D3semidirect
125. C3×#104C2×C32semidirect
127. #824C2×C32semidirect
128. #805C2×C32semidirect
256. #104C32×D3semidirect
258. C33#454semidirect
369. C32#4827direct (N × H)
370. C32C3×#454semidirect
418. C3C3×#4827direct (N × H)
419. C3C32×#454semidirect

Maximal subgroups

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

subgroupindex
C32×#10102
C32×#9492
C32×#9082
C32×#8752
C33×#1042
C32×#8242
C32×#8052
C32×#4543
C3×#48273