Finite Group Databasebeta

← C2×C3×#63864 #13174396

Subgroups — C2×C3×#63864 #13174396

Order 1728.

1874 conjugacy classes of subgroups.

The subgroup lattice has 484 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×C4⋊(C3⋊D3)C211yesmaximal, characteristic
2C22×C32×Dic6C211yesmaximal, characteristic, complemented
3C2×C32×C4×Dic3C211yesmaximal, characteristic, complemented
4C2×C3×#3722C214yesmaximal, complemented
5C3×#63864C218yesmaximal, direct product factor
6C2×C3×#947—31nomaximal
7C2×C3×#875—31nomaximal
8C2×#63864C311yesmaximal, characteristic, direct product factor
9C23×C3×C3⋊D3C2212yes—
10C22×C33×C4C2211yescharacteristic
11C2×C3×C4⋊(C3⋊D3)—24no—
12C2×C3×C4⋊(C3⋊D3)C2214yes—
13C22×C32×Dic3C2212yes—
14C22×C32×Dic3C2212yes—
15C2×C32×Dic6—24no—
16C2×C32×Dic6C2214yescomplemented
17C32×C4×Dic3C2214yescomplemented
18C3×#3722C22116yescomplemented
19C22×C32×Q8—31no—
20C22×C4⋊(C3⋊D3)C2×C311yescharacteristic
21C22×C3×D12—31no—
22C22×C3×D12—31no—
23C22×C3×D12—61no—
24C22×C3×Dic6—21no—
25C22×C3×Dic6D311yescharacteristic, complemented
26C22×C3×Dic6C2×C311yescharacteristic, complemented
27C2×C32×C42—31no—
28C3×#947—38no—
29C2×C3×(C2×D3)⋊C4—34no—
30C2×C3×(C2×D3)⋊C4—34no—
31C2×C3×C4×Dic3—21no—
32C2×C3×C4×Dic3D311yescharacteristic, complemented
33C2×C3×C4×Dic3C2×C311yescharacteristic, complemented
34C3×#875—38no—
35C2×#3722C2×C314yescomplemented
36#63864C2×C318yesdirect product factor
37C23×C33C2311yescharacteristic
38C22×C3×C3⋊D3—24no—
39C22×C3×C3⋊D3—28no—
40C22×C3×C3⋊D3C2314yes—
41C2×C33×C4D414yes—
42C2×C33×C4C2312yes—
43C3×C4⋊(C3⋊D3)—28no—
44C2×C32×Dic3—24no—
45C2×C32×Dic3—24no—
46C2×C32×Dic3C2314yes—
47C2×C32×Dic3C2314yes—
48C32×Dic6—28no—
49C2×C3×#124—91no—
50C2×#947—31no—
51C2×#875—31no—
52C23×C3⋊D3C22×C312yes—
53C23×C3×D3—32no—
54C23×C3×D3—32no—
55C23×C3×D3—62no—
56C2×C32×Q8—34no—
57C2×C32×Q8—64no—
58C22×C32×C4—21no—
59C22×C32×C4—21no—
60C22×C32×C4—21no—
61C22×C32×C4—32no—
62C22×C32×C4—32no—
63C22×C32×C4—41no—
64C22×C32×C4C2×D311yescharacteristic
65C22×C32×C4C2×D311yescharacteristic
66C22×C32×C4C22×C311yescharacteristic
67C2×C4⋊(C3⋊D3)—24no—
68C2×C4⋊(C3⋊D3)C22×C314yes—
69C22×C3×Dic3—22no—
70C22×C3×Dic3—22no—
71C22×C3×Dic3C2×D312yes—
72C22×C3×Dic3C2×D312yes—
73C22×C3×Dic3C22×C312yes—
74C22×C3×Dic3C22×C312yes—
75C2×C3×D12—34no—
76C2×C3×D12—34no—
77C2×C3×D12—64no—
78C2×C3×D12—68no—
79C2×C3×D12—64no—
80C2×C3×D12—64no—
81C2×C3×Dic6—28no—
82C2×C3×Dic6—24no—
83C2×C3×Dic6—24no—
84C2×C3×Dic6—24no—
85C2×C3×Dic6C2×D314yescomplemented
86C2×C3×Dic6C22×C314yescomplemented
87C32×C42—34no—
88C3×(C2×D3)⋊C4—316no—
89C3×(C2×D3)⋊C4—316no—
90C3×C4×Dic3—24no—
91C3×C4×Dic3C2×D314yescomplemented
92C3×C4×Dic3C22×C314yescomplemented
93#3722C22×C3116yescomplemented
94C22×C33C2×D412yes—
95C22×C33#4114yes—
96C22×C33C2411yescharacteristic
97C2×C3×C3⋊D3—24no—
98C2×C3×C3⋊D3—28no—
99C2×C3×C3⋊D3—48no—
100C33×C4C2×D414yes—
101C32×Dic3—24no—
102C32×Dic3—24no—
103C22×C3×Q8—31no—
104C22×C3×Q8—31no—
105C22×C3×Q8—61no—
106C22×C3×D4—91no—
107C22×D12—31no—
108C22×D12—31no—
109C22×D12—61no—
110C22×Dic6C3×D311yescharacteristic, complemented
111C3×#124—98no—
112C2×C3×C22⋊C4—94no—
113C2×C3×C42—31no—
114C2×C3×C42—31no—
115C2×C3×C42—61no—
116#947—38no—
117C2×(C2×D3)⋊C4—34no—
118C2×(C2×D3)⋊C4—34no—
119C2×C4×Dic3C3×D311yescharacteristic, complemented
120#875—38no—
121C23×C32—21no—
122C23×C32—21no—
123C23×C32—21no—
124C23×C32—41no—
125C23×C32C22×D311yescharacteristic
126C23×C32C22×D311yescharacteristic
127C23×C32C23×C311yescharacteristic
128C22×C3⋊D3—24no—
129C22×C3⋊D3—28no—
130C22×C3⋊D3C23×C314yes—
131C22×C3×D3—34no—
132C22×C3×D3—34no—
133C22×C3×D3—64no—
134C22×C3×D3—68no—
135C22×C3×D3—64no—
136C22×C3×D3—68no—
137C22×C3×D3—68no—
138C22×C3×D3—64no—
139C22×C3×D3—128no—
140C32×Q8—68no—
141C2×C32×C4—24no—
142C2×C32×C4—24no—
143C2×C32×C4—24no—
144C2×C32×C4—22no—
145C2×C32×C4—22no—
146C2×C32×C4—22no—
147C2×C32×C4—34no—
148C2×C32×C4—34no—
149C2×C32×C4—44no—
150C2×C32×C4—42no—
151C2×C32×C4—64no—
152C2×C32×C4—64no—
153C2×C32×C4D1214yes—
154C2×C32×C4#6714yes—
155C2×C32×C4C3×D414yes—
156C2×C32×C4C22×D312yes—
157C2×C32×C4C22×D312yes—
158C2×C32×C4C23×C312yes—
159C4⋊(C3⋊D3)—28no—
160C2×C3×Dic3—28no—
161C2×C3×Dic3—28no—
162C2×C3×Dic3—24no—
163C2×C3×Dic3—24no—
164C2×C3×Dic3—24no—
165C2×C3×Dic3—24no—
166C2×C3×Dic3—24no—
167C2×C3×Dic3—24no—
168C2×C3×Dic3C22×D314yes—
169C2×C3×Dic3C22×D314yes—
170C2×C3×Dic3C23×C314yes—
171C2×C3×Dic3C23×C314yes—
172C3×D12—616no—
173C3×D12—68no—
174C3×D12—68no—
175C3×Dic6—216no—
176C3×Dic6—28no—
177C3×Dic6—28no—
178C2×#124—91noSylow
179C2×C33#12414yescomplemented
180C2×C33C22×D411yescharacteristic
181C2×C33C2×#4112yes—
182C3×C3⋊D3—44no—
183C24×C3—92no—
184C23×D3—32no—
185C23×D3—32no—
186C23×D3—62no—
187C2×C3×Q8—34no—
188C2×C3×Q8—34no—
189C2×C3×Q8—64no—
190C2×C3×Q8—68no—
191C2×C3×Q8—64no—
192C2×C3×Q8—64no—
193C2×C3×D4—94no—
194C2×C3×D4—184no—
195C22×C3×C4—21no—
196C22×C3×C4—21no—
197C22×C3×C4—21no—
198C22×C3×C4—32no—
199C22×C3×C4—32no—
200C22×C3×C4—32no—
201C22×C3×C4—32no—
202C22×C3×C4—41no—
203C22×C3×C4—62no—
204C22×C3×C4—62no—
205C22×C3×C4D3211yescharacteristic
206C22×C3×C4C2×C3×D311yescharacteristic
207C22×C3×C4C2×C3×D311yescharacteristic
208C22×Dic3C2×C3×D312yes—
209C22×Dic3C2×C3×D312yes—
210C2×D12—34no—
211C2×D12—34no—
212C2×D12—64no—
213C2×D12—68no—
214C2×D12—64no—
215C2×D12—64no—
216C2×Dic6—24no—
217C2×Dic6C2×C3×D314yescomplemented
218C3×C22⋊C4—916no—
219C3×C42—34no—
220C3×C42—34no—
221C3×C42—64no—
222(C2×D3)⋊C4—316no—
223(C2×D3)⋊C4—316no—
224C4×Dic3C2×C3×D314yescomplemented
225C22×C32—21no—
226C22×C32—21no—
227C22×C32—21no—
228C22×C32—24no—
229C22×C32—22no—
230C22×C32—22no—
231C22×C32—24no—
232C22×C32—24no—
233C22×C32—22no—
234C22×C32—42no—
235C22×C32—44no—
236C22×C32—41no—
237C22×C32C2×D1212yes—
238C22×C32#23514yes—
239C22×C32#23914yes—
240C22×C32C2×#6712yes—
241C22×C32C2×C3×D412yes—
242C22×C32C3×#4114yes—
243C22×C32C23×D311yescharacteristic
244C22×C32C23×D311yescharacteristic
245C22×C32C24×C311yescharacteristic
246C2×C3⋊D3—24no—
247C2×C3⋊D3—28no—
248C2×C3⋊D3—48no—
249C2×C3×D3—68no—
250C2×C3×D3—64no—
251C2×C3×D3—68no—
252C2×C3×D3—68no—
253C2×C3×D3—64no—
254C2×C3×D3—128no—
255C2×C3×D3—1216no—
256C2×C3×D3—128no—
257C2×C3×D3—128no—
258C32×C4—24no—
259C32×C4—24no—
260C32×C4—24no—
261C32×C4—44no—
262C32×C4—64no—
263C32×C4—64no—
264C32×C4C2×D1214yes—
265C32×C4C2×#6714yes—
266C32×C4C2×C3×D414yes—
267C3×Dic3—28no—
268C3×Dic3—28no—
269C3×Dic3—24no—
270C3×Dic3—24no—
271C3×Dic3—24no—
272C3×Dic3—24no—
273C22×Q8—31no—
274C22×D4—91no—
275#124—98no—
276C2×C22⋊C4—94no—
277C2×C42—31no—
278C33C2×#12411yescharacteristic, complemented, Sylow
279C23×C3—21no—
280C23×C3—21no—
281C23×C3—21no—
282C23×C3—41no—
283C23×C3—94no—
284C23×C3—188no—
285C23×C3—184no—
286C23×C3C2×D3211yescharacteristic
287C23×C3C22×C3×D311yescharacteristic
288C23×C3C22×C3×D311yescharacteristic
289C22×D3—34no—
290C22×D3—34no—
291C22×D3—64no—
292C22×D3—68no—
293C22×D3—64no—
294C22×D3—68no—
295C22×D3—68no—
296C22×D3—64no—
297C22×D3—128no—
298C3×Q8—616no—
299C3×Q8—68no—
300C3×Q8—68no—
301C3×D4—188no—
302C2×C3×C4—24no—
303C2×C3×C4—24no—
304C2×C3×C4—24no—
305C2×C3×C4—22no—
306C2×C3×C4—22no—
307C2×C3×C4—22no—
308C2×C3×C4—34no—
309C2×C3×C4—34no—
310C2×C3×C4—34no—
311C2×C3×C4—34no—
312C2×C3×C4—44no—
313C2×C3×C4—42no—
314C2×C3×C4—64no—
315C2×C3×C4—64no—
316C2×C3×C4—68no—
317C2×C3×C4—68no—
318C2×C3×C4—64no—
319C2×C3×C4—64no—
320C2×C3×C4—64no—
321C2×C3×C4—64no—
322C2×C3×C4#62614yes—
323C2×C3×C4C3×D1214yes—
324C2×C3×C4C3×#6714yes—
325C2×C3×C4C2×D3212yes—
326C2×C3×C4C22×C3×D312yes—
327C2×C3×C4C22×C3×D312yes—
328C2×Dic3—24no—
329C2×Dic3—24no—
330C2×Dic3C22×C3×D314yes—
331C2×Dic3C22×C3×D314yes—
332D12—616no—
333D12—68no—
334D12—68no—
335Dic6—28no—
336C2×C32—22no—
337C2×C32—24no—
338C2×C32—21no—
339C2×C32—21no—
340C2×C32—22no—
341C2×C32—22no—
342C2×C32—21no—
343C2×C32—24no—
344C2×C32—24no—
345C2×C32—44no—
346C2×C32—41no—
347C2×C32—42no—
348C2×C32#87514yescomplemented
349C2×C32#94714yescomplemented
350C2×C32C3×#12414yescomplemented
351C2×C32C22×D1211yescharacteristic
352C2×C32C2×#23512yes—
353C2×C32C2×#23912yes—
354C2×C32C22×#6711yescharacteristic
355C2×C32C22×C3×D411yescharacteristic
356C2×C32C2×C3×#4112yes—
357C3⋊D3—44no—
358C3×D3—128no—
359C3×D3—124no—
360C3×D3—124no—
361C24—92no—
362C2×Q8—34no—
363C2×Q8—64no—
364C2×D4—94no—
365C2×D4—184no—
366C22×C4—32no—
367C22×C4—32no—
368C22×C4C3×D3211yescharacteristic
369C22⋊C4—916no—
370C42—34no—
371C22×C3—21no—
372C22×C3—21no—
373C22×C3—21no—
374C22×C3—24no—
375C22×C3—24no—
376C22×C3—24no—
377C22×C3—22no—
378C22×C3—22no—
379C22×C3—22no—
380C22×C3—41no—
381C22×C3—44no—
382C22×C3—42no—
383C22×C3—188no—
384C22×C3—184no—
385C22×C3—368no—
386C22×C3#379914yes—
387C22×C3C2×#62612yes—
388C22×C3C2×C3×D1212yes—
389C22×C3C3×#23514yes—
390C22×C3C3×#23914yes—
391C22×C3C2×C3×#6712yes—
392C22×C3C22×D3211yescharacteristic
393C22×C3C23×C3×D311yescharacteristic
394C22×C3C23×C3×D311yescharacteristic
395C2×D3—68no—
396C2×D3—64no—
397C2×D3—68no—
398C2×D3—68no—
399C2×D3—64no—
400C2×D3—128no—
401C2×D3—1216no—
402C2×D3—128no—
403C2×D3—128no—
404C3×C4—24no—
405C3×C4—24no—
406C3×C4—24no—
407C3×C4—44no—
408C3×C4—68no—
409C3×C4—68no—
410C3×C4—64no—
411C3×C4—64no—
412C3×C4—64no—
413C3×C4—64no—
414C3×C4C2×#62614yes—
415C3×C4C2×C3×D1214yes—
416C3×C4C2×C3×#6714yes—
417Dic3—24no—
418Dic3—24no—
419C32—21no—
420C32—21no—
421C32—21no—
422C32—41no—
423C32C2×#87511yescharacteristic, complemented
424C32C2×#94711yescharacteristic, complemented
425C32C2×C3×#12411yescharacteristic, complemented
426C23—94no—
427C23—188no—
428C23—184no—
429C23C2×C3×D3211yescharacteristic
430Q8—68no—
431D4—188no—
432C2×C4—34no—
433C2×C4—34no—
434C2×C4—64no—
435C2×C4—64no—
436C2×C4C3×#62614yes—
437C2×C4C2×C3×D3212yes—
438C2×C3—21no—
439C2×C3—21no—
440C2×C3—21no—
441C2×C3—24no—
442C2×C3—24no—
443C2×C3—24no—
444C2×C3—22no—
445C2×C3—22no—
446C2×C3—22no—
447C2×C3—41no—
448C2×C3—44no—
449C2×C3—42no—
450C2×C3—364no—
451C2×C3#6386414yesdirect product factor
452C2×C3C3×#87514yescomplemented
453C2×C3C3×#94714yescomplemented
454C2×C3C2×#379912yes—
455C2×C3C22×#62611yescharacteristic
456C2×C3C22×C3×D1211yescharacteristic
457C2×C3C2×C3×#23512yes—
458C2×C3C2×C3×#23912yes—
459C2×C3C22×C3×#6711yescharacteristic
460D3—128no—
461D3—124no—
462D3—124no—
463C22—184no—
464C22—188no—
465C22—368no—
466C22C3×#379914yes—
467C22C2×C3×#62612yes—
468C22C22×C3×D3211yescharacteristic
469C4—64no—
470C4—64no—
471C4C2×C3×#62614yes—
472C3—21no—
473C3—21no—
474C3—21no—
475C3—41no—
476C3C2×#6386411yescharacteristic, direct product factor
477C3C2×C3×#87511yescharacteristic, complemented
478C3C2×C3×#94711yescharacteristic, complemented
479C2—364no—
480C2C3×#6386414yesdirect product factor
481C2C2×C3×#379912yes—
482C2C22×C3×#62611yescharacteristic
subgroupG/H
centerC23×C3C2×D32
commutator, cosocleC22×C32C24×C3
FrattiniC22C22×C3×D32
FittingC22×C33×C4C22
socleC23×C33C23
prime radicalC22×C3×C4⋊(C3⋊D3)C2
hypercenterC22×C3×C4D32
hypocenterC32C2×C3×#124
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C23×C3 ◁ C22×C3×C4

Complements

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

NH ≅ G/Ntype
2. C22×C32×Dic6C2semidirect
3. C2×C32×C4×Dic3C2semidirect
4. C2×C3×#3722C2semidirect
5. C3×#63864C2direct (N × H)
8. C2×#63864C3direct (N × H)
16. C2×C32×Dic6C22semidirect
17. C32×C4×Dic3C22semidirect
18. C3×#3722C22semidirect
25. C22×C3×Dic6D3semidirect
26. C22×C3×Dic6C2×C3semidirect
32. C2×C3×C4×Dic3D3semidirect
33. C2×C3×C4×Dic3C2×C3semidirect
35. C2×#3722C2×C3semidirect
36. #63864C2×C3direct (N × H)
85. C2×C3×Dic6C2×D3semidirect
86. C2×C3×Dic6C22×C3semidirect
91. C3×C4×Dic3C2×D3semidirect
92. C3×C4×Dic3C22×C3semidirect
93. #3722C22×C3semidirect
110. C22×Dic6C3×D3semidirect
119. C2×C4×Dic3C3×D3semidirect
179. C2×C33#124semidirect
217. C2×Dic6C2×C3×D3semidirect
224. C4×Dic3C2×C3×D3semidirect
278. C33C2×#124semidirect
348. C2×C32#875semidirect
349. C2×C32#947semidirect
350. C2×C32C3×#124semidirect
423. C32C2×#875semidirect
424. C32C2×#947semidirect
425. C32C2×C3×#124semidirect
451. C2×C3#63864direct (N × H)
452. C2×C3C3×#875semidirect
453. C2×C3C3×#947semidirect
476. C3C2×#63864direct (N × H)
477. C3C2×C3×#875semidirect
478. C3C2×C3×#947semidirect
480. C2C3×#63864direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×C4⋊(C3⋊D3)2
C22×C32×Dic62
C2×C32×C4×Dic32
C2×C3×#37222
C3×#638642
C2×C3×#9473
C2×C3×#8753
C2×#638643