Finite Group Databasebeta

← C2×C32×#916 #13179138

Subgroups — C2×C32×#916 #13179138

Order 1728.

1868 conjugacy classes of subgroups.

The subgroup lattice has 466 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C32×#239C211yesmaximal, characteristic, complemented
2C2×C32×#237C211yesmaximal, characteristic, complemented
3C2×C33×SD16C211yesmaximal, characteristic, complemented
4C2×C32×#216C211yesmaximal, characteristic, complemented
5C2×C32×#213C211yesmaximal, characteristic, complemented
6C32×#916C218yesmaximal, direct product factor
7C2×C32×#204C211yesmaximal, characteristic, complemented
8C2×C32×D3×C8C211yesmaximal, characteristic, complemented
9C2×C32×#135—31nomaximal
10C2×C3×#916C314yesmaximal, direct product factor
11C2×C33×Q8C2211yescharacteristic
12C2×C33×D4C2211yescharacteristic
13C2×C32×#67—21no—
14C22×C32×Dic3—21no—
15C32×#239—22no—
16C32×#239C2214yescomplemented
17C32×#237—22no—
18C32×#237C2214yescomplemented
19C2×C32×D12—21no—
20C2×C32×D12C2211yescharacteristic
21C2×C32×C4×D3—21no—
22C2×C32×C4×D3C2211yescharacteristic
23C2×C32×Dic6C2211yescharacteristic
24C33×SD16C2214yescomplemented
25C2×C33×C8C2211yescharacteristic
26C32×#216C2214yescomplemented
27C32×#213C2214yescomplemented
28C2×C32×C3⋊C8C2211yescharacteristic
29C32×#204C2214yescomplemented
30C32×D3×C8C2214yescomplemented
31C2×C32×#41—31no—
32C2×C32×#41—31no—
33C2×C3×#239C2×C314yescomplemented
34C2×C3×#237C2×C314yescomplemented
35C32×#135—38no—
36C2×C32×Q16—31no—
37C2×C32×SD16—24no—
38C2×C32×SD16—31no—
39C2×C32×SD16D311yescharacteristic, complemented
40C2×C32×SD16C2×C314yescomplemented
41C2×C32×D8—31no—
42C22×C32×C8—31no—
43C2×C3×#216C2×C314yescomplemented
44C2×C3×#213C2×C314yescomplemented
45C3×#916C2×C3132yesdirect product factor
46C2×C3×#204C2×C314yescomplemented
47C2×C3×D3×C8C2×C314yescomplemented
48C23×C33—21no—
49C22×C32×D3—21no—
50C22×C32×D3D411yescharacteristic
51C33×Q8—21no—
52C33×Q8C2312yes—
53C33×D4—21no—
54C33×D4C2312yes—
55C2×C33×C4—21no—
56C2×C33×C4C2311yescharacteristic
57C32×#67—24no—
58C2×C32×Dic3—21no—
59C2×C32×Dic3—24no—
60C2×C32×Dic3D411yescharacteristic
61C32×D12—24no—
62C32×D12—21no—
63C32×D12C2312yes—
64C32×C4×D3—24no—
65C32×C4×D3D412yes—
66C32×C4×D3C2312yes—
67C32×Dic6—21no—
68C32×Dic6C2312yes—
69C33×C8C2312yes—
70C32×C3⋊C8C2312yes—
71C2×C3×#135—34no—
72C2×#916C3211yescharacteristic, direct product factor
73C32×#41—34no—
74C32×#41—34no—
75C32×#41—62no—
76C32×#41—62no—
77C2×C32×Q8—24no—
78C2×C32×Q8—31no—
79C2×C32×Q8C2×D311yescharacteristic
80C2×C32×Q8C22×C314yes—
81C2×C32×D4—24no—
82C2×C32×D4—31no—
83C2×C32×D4—61no—
84C2×C32×D4—61no—
85C2×C32×D4C2×D311yescharacteristic
86C2×C32×D4C22×C314yes—
87C22×C32×C4—31no—
88C22×C32×C4—61no—
89C22×C32×C4—61no—
90C2×C3×#67—24no—
91C22×C3×Dic3—24no—
92C3×#239—28no—
93C3×#239C22×C3116yescomplemented
94C3×#237—28no—
95C3×#237C22×C3116yescomplemented
96C2×C3×D12—24no—
97C2×C3×D12C22×C314yes—
98C2×C3×C4×D3—24no—
99C2×C3×C4×D3C22×C314yes—
100C2×C3×Dic6C22×C314yes—
101C32×Q16—34no—
102C32×SD16—216no—
103C32×SD16—34no—
104C32×SD16C2×D314yescomplemented
105C32×SD16C22×C3116yescomplemented
106C32×D8—34no—
107C2×C32×C8—24no—
108C2×C32×C8—31no—
109C2×C32×C8—34no—
110C2×C32×C8C2×D311yescharacteristic
111C2×C32×C8C22×C314yes—
112C3×#216C22×C3116yescomplemented
113C3×#213C22×C3116yescomplemented
114C2×C3×C3⋊C8C22×C314yes—
115C3×#204C22×C3116yescomplemented
116C3×D3×C8C22×C3116yescomplemented
117C22×C33—22no—
118C22×C33—42no—
119C22×C33C2×D411yescharacteristic
120C2×C32×D3—22no—
121C2×C32×D3—22no—
122C2×C32×D3—42no—
123C2×C32×D3C2×D412yes—
124C33×C4—22no—
125C33×C4C2×D411yescharacteristic
126C33×C4C2411yescharacteristic
127C32×Dic3—22no—
128C32×Dic3C2×D412yes—
129C2×C3×#41—34no—
130C2×C3×#41—34no—
131C2×#239C2×C3211yescharacteristic, complemented
132C2×#237C2×C3211yescharacteristic, complemented
133C3×#135—332no—
134C2×C3×Q16—34no—
135C2×C3×SD16—24no—
136C2×C3×SD16—34no—
137C2×C3×SD16C3×D314yescomplemented
138C2×C3×SD16C2×C3211yescharacteristic, complemented
139C2×C3×D8—34no—
140C22×C3×C8—34no—
141C2×#216C2×C3211yescharacteristic, complemented
142C2×#213C2×C3211yescharacteristic, complemented
143#916C2×C3218yesdirect product factor
144C2×#204C2×C3211yescharacteristic, complemented
145C2×D3×C8C2×C3211yescharacteristic, complemented
146C23×C32—21no—
147C23×C32—24no—
148C23×C32—31no—
149C23×C32—44no—
150C23×C32—61no—
151C22×C3×D3—24no—
152C22×C3×D3C3×D414yes—
153C32×Q8—21no—
154C32×Q8—28no—
155C32×Q8—24no—
156C32×Q8—32no—
157C32×Q8—44no—
158C32×Q8—61no—
159C32×Q8C22×D312yes—
160C32×Q8C23×C318yes—
161C32×D4—21no—
162C32×D4—28no—
163C32×D4—24no—
164C32×D4—32no—
165C32×D4—44no—
166C32×D4—64no—
167C32×D4—64no—
168C32×D4—61no—
169C32×D4C22×D312yes—
170C32×D4C23×C318yes—
171C2×C32×C4—21no—
172C2×C32×C4—24no—
173C2×C32×C4—24no—
174C2×C32×C4—31no—
175C2×C32×C4—32no—
176C2×C32×C4—32no—
177C2×C32×C4—44no—
178C2×C32×C4—64no—
179C2×C32×C4—64no—
180C2×C32×C4—61no—
181C2×C32×C4C22×D311yescharacteristic
182C2×C32×C4C23×C314yes—
183C3×#67—216no—
184C2×C3×Dic3—24no—
185C2×C3×Dic3—216no—
186C2×C3×Dic3C3×D414yes—
187C3×D12—216no—
188C3×D12—24no—
189C3×D12C23×C318yes—
190C3×C4×D3—216no—
191C3×C4×D3C3×D418yes—
192C3×C4×D3C23×C318yes—
193C3×Dic6—24no—
194C3×Dic6C23×C318yes—
195C32×C8—28no—
196C32×C8—32no—
197C32×C8C22×D312yes—
198C32×C8C23×C318yes—
199C3×C3⋊C8C23×C318yes—
200C2×#135—31noSylow
201C2×C33—42no—
202C2×C33#13512yescomplemented
203C2×C33C22×D411yescharacteristic
204C32×D3—22no—
205C32×D3—42no—
206C3×#41—316no—
207C3×#41—316no—
208C3×#41—68no—
209C3×#41—68no—
210C2×C3×Q8—24no—
211C2×C3×Q8—34no—
212C2×C3×Q8C2×C3×D314yes—
213C2×C3×Q8C22×C3211yescharacteristic
214C2×C3×D4—24no—
215C2×C3×D4—34no—
216C2×C3×D4—64no—
217C2×C3×D4—64no—
218C2×C3×D4C2×C3×D314yes—
219C2×C3×D4C22×C3211yescharacteristic
220C22×C3×C4—34no—
221C22×C3×C4—64no—
222C22×C3×C4—64no—
223C2×#67—21no—
224C22×Dic3—21no—
225#239—22no—
226#239C22×C3214yescomplemented
227#237—22no—
228#237C22×C3214yescomplemented
229C2×D12—21no—
230C2×D12C22×C3211yescharacteristic
231C2×C4×D3—21no—
232C2×C4×D3C22×C3211yescharacteristic
233C2×Dic6C22×C3211yescharacteristic
234C3×Q16—316no—
235C3×SD16—216no—
236C3×SD16—316no—
237C3×SD16C2×C3×D3116yescomplemented
238C3×SD16C22×C3214yescomplemented
239C3×D8—316no—
240C2×C3×C8—24no—
241C2×C3×C8—34no—
242C2×C3×C8—316no—
243C2×C3×C8C2×C3×D314yes—
244C2×C3×C8C22×C3211yescharacteristic
245#216C22×C3214yescomplemented
246#213C22×C3214yescomplemented
247C2×C3⋊C8C22×C3211yescharacteristic
248#204C22×C3214yescomplemented
249D3×C8C22×C3214yescomplemented
250C22×C32—22no—
251C22×C32—24no—
252C22×C32—28no—
253C22×C32—32no—
254C22×C32—48no—
255C22×C32—42no—
256C22×C32—48no—
257C22×C32—62no—
258C22×C32—62no—
259C22×C32—88no—
260C22×C32—122no—
261C22×C32D3×D411yescharacteristic
262C22×C32C2×C3×D414yes—
263C2×C3×D3—28no—
264C2×C3×D3—28no—
265C2×C3×D3—48no—
266C2×C3×D3C2×C3×D418yes—
267C32×C4—22no—
268C32×C4—24no—
269C32×C4—24no—
270C32×C4—28no—
271C32×C4—32no—
272C32×C4—48no—
273C32×C4—62no—
274C32×C4D3×D411yescharacteristic
275C32×C4C2×C3×D414yes—
276C32×C4C23×D311yescharacteristic
277C32×C4C24×C314yes—
278C3×Dic3—28no—
279C3×Dic3C2×C3×D418yes—
280C2×#41—31no—
281C2×#41—31no—
282#135—38no—
283C2×Q16—31no—
284C2×SD16—31no—
285C2×SD16C32×D311yescharacteristic, complemented
286C2×D8—31no—
287C22×C8—31no—
288C33C2×#13511yescharacteristic, complemented, Sylow
289C23×C3—24no—
290C23×C3—21no—
291C23×C3—34no—
292C23×C3—44no—
293C23×C3—64no—
294C22×D3—21no—
295C22×D3C32×D411yescharacteristic
296C3×Q8—24no—
297C3×Q8—21no—
298C3×Q8—28no—
299C3×Q8—38no—
300C3×Q8—44no—
301C3×Q8—64no—
302C3×Q8C22×C3×D318yes—
303C3×Q8C23×C3212yes—
304C3×D4—24no—
305C3×D4—21no—
306C3×D4—28no—
307C3×D4—38no—
308C3×D4—44no—
309C3×D4—616no—
310C3×D4—616no—
311C3×D4—64no—
312C3×D4C22×C3×D318yes—
313C3×D4C23×C3212yes—
314C2×C3×C4—24no—
315C2×C3×C4—21no—
316C2×C3×C4—24no—
317C2×C3×C4—34no—
318C2×C3×C4—38no—
319C2×C3×C4—38no—
320C2×C3×C4—44no—
321C2×C3×C4—616no—
322C2×C3×C4—616no—
323C2×C3×C4—64no—
324C2×C3×C4C22×C3×D314yes—
325C2×C3×C4C23×C3211yescharacteristic
326#67—24no—
327C2×Dic3—21no—
328C2×Dic3—24no—
329C2×Dic3C32×D411yescharacteristic
330D12—24no—
331D12—21no—
332D12C23×C3212yes—
333C4×D3—24no—
334C4×D3C32×D412yes—
335C4×D3C23×C3212yes—
336Dic6—21no—
337Dic6C23×C3212yes—
338C3×C8—28no—
339C3×C8—38no—
340C3×C8C22×C3×D318yes—
341C3×C8C23×C3212yes—
342C3⋊C8C23×C3212yes—
343C2×C32—28no—
344C2×C32—24no—
345C2×C32—42no—
346C2×C32—48no—
347C2×C32—62no—
348C2×C32—88no—
349C2×C32—122no—
350C2×C32#91612yesdirect product factor
351C2×C32C3×#13518yescomplemented
352C2×C32C2×D3×D411yescharacteristic
353C2×C32C22×C3×D414yes—
354C3×D3—28no—
355C3×D3—48no—
356#41—34no—
357#41—34no—
358#41—62no—
359#41—62no—
360C2×Q8—31no—
361C2×Q8C2×C32×D311yescharacteristic
362C2×D4—31no—
363C2×D4—61no—
364C2×D4—61no—
365C2×D4C2×C32×D311yescharacteristic
366C22×C4—31no—
367C22×C4—61no—
368C22×C4—61no—
369Q16—34no—
370SD16—34no—
371SD16C2×C32×D314yescomplemented
372D8—34no—
373C2×C8—31no—
374C2×C8—34no—
375C2×C8C2×C32×D311yescharacteristic
376C22×C3—28no—
377C22×C3—22no—
378C22×C3—24no—
379C22×C3—38no—
380C22×C3—48no—
381C22×C3—48no—
382C22×C3—42no—
383C22×C3—68no—
384C22×C3—68no—
385C22×C3—88no—
386C22×C3—128no—
387C22×C3C3×D3×D414yes—
388C22×C3C2×C32×D411yescharacteristic
389C2×D3—22no—
390C2×D3—22no—
391C2×D3—42no—
392C2×D3C2×C32×D412yes—
393C3×C4—28no—
394C3×C4—22no—
395C3×C4—24no—
396C3×C4—24no—
397C3×C4—38no—
398C3×C4—48no—
399C3×C4—68no—
400C3×C4C3×D3×D414yes—
401C3×C4C2×C32×D411yescharacteristic
402C3×C4C23×C3×D314yes—
403C3×C4C24×C3211yescharacteristic
404Dic3—22no—
405Dic3C2×C32×D412yes—
406C32—24no—
407C32C2×#91611yescharacteristic, direct product factor
408C32C2×C3×#13514yescomplemented
409C23—21no—
410C23—31no—
411C23—61no—
412Q8—21no—
413Q8—32no—
414Q8—61no—
415Q8C22×C32×D312yes—
416D4—21no—
417D4—32no—
418D4—64no—
419D4—64no—
420D4—61no—
421D4C22×C32×D312yes—
422C2×C4—21no—
423C2×C4—31no—
424C2×C4—32no—
425C2×C4—32no—
426C2×C4—64no—
427C2×C4—64no—
428C2×C4—61no—
429C2×C4C22×C32×D311yescharacteristic
430C8—32no—
431C8C22×C32×D312yes—
432C2×C3—28no—
433C2×C3—24no—
434C2×C3—48no—
435C2×C3—42no—
436C2×C3—68no—
437C2×C3—88no—
438C2×C3—128no—
439C2×C3C3×#91618yesdirect product factor
440C2×C3C32×#13512yescomplemented
441C2×C3C2×C3×D3×D414yes—
442C2×C3C22×C32×D411yescharacteristic
443D3—22no—
444D3—42no—
445C22—22no—
446C22—32no—
447C22—42no—
448C22—62no—
449C22—62no—
450C22—122no—
451C22C32×D3×D411yescharacteristic
452C4—22no—
453C4—32no—
454C4—62no—
455C4C32×D3×D411yescharacteristic
456C4C23×C32×D311yescharacteristic
457C3—24no—
458C3C2×C3×#91614yesdirect product factor
459C3C2×C32×#13511yescharacteristic, complemented
460C2—42no—
461C2—62no—
462C2—122no—
463C2C32×#91612yesdirect product factor
464C2C2×C32×D3×D411yescharacteristic
subgroupG/H
centerC22×C32D3×D4
commutator, cosocleC3×C4C24×C32
FrattiniC4C23×C32×D3
FittingC2×C33×SD16C2
socleC22×C33C2×D4
hypercenterC2×C32×SD16D3
hypocenterC3C2×C32×#135
LCSG ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22×C32 ◁ C2×C32×C4 ◁ C2×C32×SD16

Complements

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

NH ≅ G/Ntype
1. C2×C32×#239C2semidirect
2. C2×C32×#237C2semidirect
3. C2×C33×SD16C2semidirect
4. C2×C32×#216C2semidirect
5. C2×C32×#213C2semidirect
6. C32×#916C2direct (N × H)
7. C2×C32×#204C2semidirect
8. C2×C32×D3×C8C2semidirect
10. C2×C3×#916C3direct (N × H)
16. C32×#239C22semidirect
18. C32×#237C22semidirect
24. C33×SD16C22semidirect
26. C32×#216C22semidirect
27. C32×#213C22semidirect
29. C32×#204C22semidirect
30. C32×D3×C8C22semidirect
33. C2×C3×#239C2×C3semidirect
34. C2×C3×#237C2×C3semidirect
39. C2×C32×SD16D3semidirect
40. C2×C32×SD16C2×C3semidirect
43. C2×C3×#216C2×C3semidirect
44. C2×C3×#213C2×C3semidirect
45. C3×#916C2×C3direct (N × H)
46. C2×C3×#204C2×C3semidirect
47. C2×C3×D3×C8C2×C3semidirect
72. C2×#916C32direct (N × H)
93. C3×#239C22×C3semidirect
95. C3×#237C22×C3semidirect
104. C32×SD16C2×D3semidirect
105. C32×SD16C22×C3semidirect
112. C3×#216C22×C3semidirect
113. C3×#213C22×C3semidirect
115. C3×#204C22×C3semidirect
116. C3×D3×C8C22×C3semidirect
131. C2×#239C2×C32semidirect
132. C2×#237C2×C32semidirect
137. C2×C3×SD16C3×D3semidirect
138. C2×C3×SD16C2×C32semidirect
141. C2×#216C2×C32semidirect
142. C2×#213C2×C32semidirect
143. #916C2×C32direct (N × H)
144. C2×#204C2×C32semidirect
145. C2×D3×C8C2×C32semidirect
202. C2×C33#135semidirect
226. #239C22×C32semidirect
228. #237C22×C32semidirect
237. C3×SD16C2×C3×D3semidirect
238. C3×SD16C22×C32semidirect
245. #216C22×C32semidirect
246. #213C22×C32semidirect
248. #204C22×C32semidirect
249. D3×C8C22×C32semidirect
285. C2×SD16C32×D3semidirect
288. C33C2×#135semidirect
350. C2×C32#916direct (N × H)
351. C2×C32C3×#135semidirect
371. SD16C2×C32×D3semidirect
407. C32C2×#916direct (N × H)
408. C32C2×C3×#135semidirect
439. C2×C3C3×#916direct (N × H)
440. C2×C3C32×#135semidirect
458. C3C2×C3×#916direct (N × H)
459. C3C2×C32×#135semidirect
463. C2C32×#916direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C32×#2392
C2×C32×#2372
C2×C33×SD162
C2×C32×#2162
C2×C32×#2132
C32×#9162
C2×C32×#2042
C2×C32×D3×C82
C2×C32×#1353
C2×C3×#9163