Finite Group Databasebeta

← C2×C3×#63822 #13174419

Subgroups — C2×C3×#63822 #13174419

Order 1728.

1522 conjugacy classes of subgroups.

The subgroup lattice has 482 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C32×#235C212yesmaximal, complemented
2C22×C3×C3⋊(C3⋊C8)C211yesmaximal, characteristic, complemented
3C2×C3×#3718C211yesmaximal, characteristic, complemented
4C2×C3×#3715C212yesmaximal, complemented
5C3×#63822C218yesmaximal, direct product factor
6C2×C3×#3713C211yesmaximal, characteristic, complemented
7C2×C3×#950—32nomaximal
8C2×#63822C311yesmaximal, characteristic, direct product factor
9C22×C33×C4C2211yescharacteristic
10C2×C32×#67—22no—
11C32×#235—24no—
12C32×#235C2218yescomplemented
13C2×C32×D12C2212yes—
14C2×C32×C4×D3—22no—
15C2×C32×Dic6C2212yes—
16C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
17C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
18C2×C3×C3⋊(C3⋊C8)C2214yescomplemented
19C3×#3718C2214yescomplemented
20C3×#3715C2218yescomplemented
21C3×#3713C2214yescomplemented
22C2×C32×#41—32no—
23C2×C3×#235—22no—
24C2×C3×#235D312yescomplemented
25C2×C3×#235C2×C312yescomplemented
26C22×C3⋊(C3⋊C8)C2×C311yescharacteristic, complemented
27C3×#950—316no—
28C2×C3×#216—32no—
29C2×C3×#215—32no—
30C2×C3×#214—32no—
31C2×C3×#213—32no—
32C22×C3×C3⋊C8—32no—
33C22×C3×C3⋊C8—61no—
34C2×#3718C2×C311yescharacteristic, complemented
35C2×#3715C2×C312yescomplemented
36#63822C2×C318yesdirect product factor
37C2×#3713C2×C311yescharacteristic, complemented
38C23×C33D411yescharacteristic
39C22×C32×D3—22no—
40C2×C33×C4D411yescharacteristic
41C2×C33×C4D412yes—
42C2×C33×C4C2311yescharacteristic
43C2×C33×C4C2312yes—
44C32×#67—28no—
45C2×C32×Dic3—22no—
46C32×D12—22no—
47C32×D12C2314yes—
48C32×C4×D3—28no—
49C32×Dic6—22no—
50C32×Dic6C2314yes—
51C3×C3⋊(C3⋊C8)C2312yes—
52C3×C3⋊(C3⋊C8)C2312yes—
53C2×C3×#135—91no—
54C2×#950—32no—
55C32×#41—38no—
56C32×#41—64no—
57C2×C32×Q8—32no—
58C2×C32×D4—32no—
59C2×C32×D4—62no—
60C22×C32×C4—21no—
61C22×C32×C4—22no—
62C22×C32×C4—41no—
63C22×C32×C4—62no—
64C22×C32×C4C2×D312yes—
65C22×C32×C4C22×C311yescharacteristic
66C2×C3×#67—24no—
67C2×C3×#67—22no—
68C2×C3×#67—22no—
69C3×#235—28no—
70C3×#235—28no—
71C3×#235—24no—
72C3×#235—24no—
73C3×#235C2×D318yescomplemented
74C3×#235C22×C318yescomplemented
75C2×C3×D12—22no—
76C2×C3×D12C2×D312yes—
77C2×C3×D12C22×C312yes—
78C2×C3×C4×D3—24no—
79C2×C3×C4×D3—22no—
80C2×C3×C4×D3—22no—
81C2×C3×Dic6—22no—
82C2×C3×Dic6C2×D312yes—
83C2×C3×Dic6C22×C312yes—
84C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
85C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
86C2×C3⋊(C3⋊C8)C22×C314yescomplemented
87C3×#216—38no—
88C3×#215—38no—
89C3×#214—38no—
90C3×#213—38no—
91C2×C3×C3⋊C8—32no—
92C2×C3×C3⋊C8—32no—
93C2×C3×C3⋊C8—38no—
94C2×C3×C3⋊C8—61no—
95C2×C3×C3⋊C8—61no—
96C2×C3×C3⋊C8—64no—
97#3718C22×C314yescomplemented
98#3715C22×C318yescomplemented
99#3713C22×C314yescomplemented
100C22×C33—22no—
101C22×C33C2×D412yes—
102C22×C33C2×D411yescharacteristic
103C2×C32×D3—24no—
104C2×C32×D3—44no—
105C33×C4C2×D411yescharacteristic
106C33×C4C2×D412yes—
107C33×C4C2411yescharacteristic
108C32×Dic3—24no—
109C2×C3×#41—32no—
110C2×C3×#41—32no—
111C2×C3×#41—62no—
112C2×#235C3×D312yescomplemented
113C3×#135—98no—
114C2×C3×Q16—91no—
115C2×C3×SD16—92no—
116C2×C3×D8—91no—
117C22×C3×C8—91no—
118#950—316no—
119C2×#216—32no—
120C2×#215—32no—
121C2×#214—32no—
122C2×#213—32no—
123C22×C3⋊C8—32no—
124C22×C3⋊C8—61no—
125C23×C32—21no—
126C23×C32—22no—
127C23×C32—41no—
128C23×C32—62no—
129C23×C32#6712yes—
130C23×C32C3×D411yescharacteristic
131C22×C3×D3—24no—
132C22×C3×D3—22no—
133C22×C3×D3—22no—
134C32×Q8—34no—
135C32×Q8—62no—
136C32×D4—34no—
137C32×D4—68no—
138C32×D4—62no—
139C2×C32×C4—22no—
140C2×C32×C4—21no—
141C2×C32×C4—22no—
142C2×C32×C4—24no—
143C2×C32×C4—21no—
144C2×C32×C4—22no—
145C2×C32×C4—22no—
146C2×C32×C4—24no—
147C2×C32×C4—41no—
148C2×C32×C4—42no—
149C2×C32×C4—41no—
150C2×C32×C4—42no—
151C2×C32×C4—62no—
152C2×C32×C4—68no—
153C2×C32×C4#6714yes—
154C2×C32×C4#6712yes—
155C2×C32×C4C3×D411yescharacteristic
156C2×C32×C4C3×D412yes—
157C2×C32×C4C22×D312yes—
158C2×C32×C4C22×D314yes—
159C2×C32×C4C23×C311yescharacteristic
160C2×C32×C4C23×C312yes—
161C3×#67—216no—
162C3×#67—28no—
163C3×#67—28no—
164C2×C3×Dic3—24no—
165C2×C3×Dic3—22no—
166C2×C3×Dic3—22no—
167C3×D12—24no—
168C3×D12—22no—
169C3×D12—22no—
170C3×D12—24no—
171C3×D12C22×D314yes—
172C3×D12C23×C314yes—
173C3×C4×D3—216no—
174C3×C4×D3—28no—
175C3×C4×D3—28no—
176C3×Dic6—24no—
177C3×Dic6—22no—
178C3×Dic6—22no—
179C3×Dic6—24no—
180C3×Dic6C22×D314yes—
181C3×Dic6C23×C314yes—
182C3⋊(C3⋊C8)C23×C312yes—
183C3⋊(C3⋊C8)C23×C312yes—
184C3×C3⋊C8—34no—
185C3×C3⋊C8—34no—
186C3×C3⋊C8—62no—
187C3×C3⋊C8—62no—
188C2×#135—91noSylow
189C2×C33—22no—
190C2×C33#13512yescomplemented
191C2×C33C22×D411yescharacteristic
192C32×D3—44no—
193C3×#41—38no—
194C3×#41—38no—
195C3×#41—68no—
196C3×#41—68no—
197C3×#41—64no—
198C3×#41—64no—
199C2×C3×Q8—32no—
200C2×C3×Q8—32no—
201C2×C3×Q8—62no—
202C2×C3×D4—32no—
203C2×C3×D4—32no—
204C2×C3×D4—62no—
205C2×C3×D4—64no—
206C2×C3×D4—62no—
207C2×C3×D4—62no—
208C22×C3×C4—21no—
209C22×C3×C4—22no—
210C22×C3×C4—41no—
211C22×C3×C4—64no—
212C22×C3×C4—62no—
213C22×C3×C4—62no—
214C22×C3×C4D3211yescharacteristic
215C22×C3×C4C2×C3×D312yes—
216C2×#67—22no—
217#235—24no—
218#235C2×C3×D318yescomplemented
219C2×D12C2×C3×D312yes—
220C2×C4×D3—22no—
221C2×Dic6C2×C3×D312yes—
222C3×Q16—94no—
223C3×SD16—98no—
224C3×D8—94no—
225C2×C3×C8—91no—
226C2×C3×C8—91no—
227C2×C3×C8—94no—
228#216—38no—
229#215—38no—
230#214—38no—
231#213—38no—
232C2×C3⋊C8—32no—
233C2×C3⋊C8—32no—
234C2×C3⋊C8—38no—
235C2×C3⋊C8—61no—
236C2×C3⋊C8—61no—
237C2×C3⋊C8—64no—
238C22×C32—22no—
239C22×C32—24no—
240C22×C32—24no—
241C22×C32—22no—
242C22×C32—21no—
243C22×C32—22no—
244C22×C32—24no—
245C22×C32—41no—
246C22×C32—42no—
247C22×C32—44no—
248C22×C32—44no—
249C22×C32—64no—
250C22×C32—124no—
251C22×C32C2×#6714yes—
252C22×C32C2×#6712yes—
253C22×C32C2×C3×D412yes—
254C22×C32C2×C3×D411yescharacteristic
255C2×C3×D3—28no—
256C2×C3×D3—24no—
257C2×C3×D3—24no—
258C2×C3×D3—48no—
259C2×C3×D3—44no—
260C2×C3×D3—44no—
261C32×C4—21no—
262C32×C4—22no—
263C32×C4—22no—
264C32×C4—21no—
265C32×C4—22no—
266C32×C4—24no—
267C32×C4—42no—
268C32×C4—41no—
269C32×C4—41no—
270C32×C4—64no—
271C32×C4C2×#6712yes—
272C32×C4C2×#6714yes—
273C32×C4C2×C3×D411yescharacteristic
274C32×C4C2×C3×D412yes—
275C32×C4C23×D312yes—
276C32×C4C24×C311yescharacteristic
277C3×Dic3—28no—
278C3×Dic3—24no—
279C3×Dic3—24no—
280C2×#41—32no—
281#135—98no—
282C2×Q16—91no—
283C2×SD16—92no—
284C2×D8—91no—
285C22×C8—91no—
286C33C2×#13511yescharacteristic, complemented, Sylow
287C23×C3—21no—
288C23×C3—22no—
289C23×C3—41no—
290C23×C3—64no—
291C23×C3—62no—
292C23×C3—62no—
293C23×C3#62511yescharacteristic
294C23×C3C3×#6712yes—
295C22×D3—22no—
296C3×Q8—34no—
297C3×Q8—34no—
298C3×Q8—64no—
299C3×Q8—64no—
300C3×Q8—62no—
301C3×Q8—62no—
302C3×D4—34no—
303C3×D4—34no—
304C3×D4—64no—
305C3×D4—64no—
306C3×D4—616no—
307C3×D4—68no—
308C3×D4—68no—
309C3×D4—62no—
310C3×D4—62no—
311C2×C3×C4—22no—
312C2×C3×C4—24no—
313C2×C3×C4—21no—
314C2×C3×C4—22no—
315C2×C3×C4—22no—
316C2×C3×C4—24no—
317C2×C3×C4—21no—
318C2×C3×C4—22no—
319C2×C3×C4—41no—
320C2×C3×C4—42no—
321C2×C3×C4—41no—
322C2×C3×C4—42no—
323C2×C3×C4—616no—
324C2×C3×C4—64no—
325C2×C3×C4—62no—
326C2×C3×C4—62no—
327C2×C3×C4—68no—
328C2×C3×C4—68no—
329C2×C3×C4#62512yes—
330C2×C3×C4#62511yescharacteristic
331C2×C3×C4C3×#6712yes—
332C2×C3×C4C3×#6714yes—
333C2×C3×C4C2×D3211yescharacteristic
334C2×C3×C4C2×D3212yes—
335C2×C3×C4C22×C3×D312yes—
336C2×C3×C4C22×C3×D314yes—
337#67—28no—
338C2×Dic3—22no—
339D12—22no—
340D12C22×C3×D314yes—
341C4×D3—28no—
342Dic6—22no—
343Dic6C22×C3×D314yes—
344C3×C8—92no—
345C3×C8—92no—
346C3⋊C8—34no—
347C3⋊C8—34no—
348C3⋊C8—62no—
349C3⋊C8—62no—
350C2×C32—22no—
351C2×C32—24no—
352C2×C32—24no—
353C2×C32—21no—
354C2×C32—24no—
355C2×C32—22no—
356C2×C32—22no—
357C2×C32—42no—
358C2×C32—41no—
359C2×C32—44no—
360C2×C32—44no—
361C2×C32—124no—
362C2×C32#95014yescomplemented
363C2×C32C3×#13512yescomplemented
364C2×C32C22×#6712yes—
365C2×C32C22×C3×D411yescharacteristic
366C3×D3—48no—
367C3×D3—44no—
368C3×D3—44no—
369#41—38no—
370#41—64no—
371C2×Q8—32no—
372C2×D4—32no—
373C2×D4—62no—
374C22×C4—62no—
375C22×C4C3×D3211yescharacteristic
376Q16—94no—
377SD16—98no—
378D8—94no—
379C2×C8—91no—
380C2×C8—91no—
381C2×C8—94no—
382C22×C3—24no—
383C22×C3—24no—
384C22×C3—22no—
385C22×C3—22no—
386C22×C3—22no—
387C22×C3—21no—
388C22×C3—24no—
389C22×C3—44no—
390C22×C3—44no—
391C22×C3—42no—
392C22×C3—41no—
393C22×C3—68no—
394C22×C3—64no—
395C22×C3—64no—
396C22×C3—128no—
397C22×C3—124no—
398C22×C3—124no—
399C22×C3C2×#62512yes—
400C22×C3C2×#62511yescharacteristic
401C22×C3C2×C3×#6714yes—
402C22×C3C2×C3×#6712yes—
403C2×D3—24no—
404C2×D3—44no—
405C3×C4—21no—
406C3×C4—22no—
407C3×C4—24no—
408C3×C4—22no—
409C3×C4—22no—
410C3×C4—21no—
411C3×C4—41no—
412C3×C4—42no—
413C3×C4—41no—
414C3×C4—68no—
415C3×C4—64no—
416C3×C4—64no—
417C3×C4C2×#62512yes—
418C3×C4C2×#62511yescharacteristic
419C3×C4C2×C3×#6712yes—
420C3×C4C2×C3×#6714yes—
421C3×C4C22×D3211yescharacteristic
422C3×C4C23×C3×D312yes—
423Dic3—24no—
424C32—22no—
425C32—21no—
426C32—41no—
427C32C2×#95012yescomplemented
428C32C2×C3×#13511yescharacteristic, complemented
429C23—62no—
430C23C3×#62511yescharacteristic
431Q8—34no—
432Q8—62no—
433D4—34no—
434D4—68no—
435D4—62no—
436C2×C4—62no—
437C2×C4—68no—
438C2×C4C3×#62511yescharacteristic
439C2×C4C3×#62512yes—
440C2×C4C2×C3×D3211yescharacteristic
441C2×C4C2×C3×D3212yes—
442C8—92no—
443C8—92no—
444C2×C3—24no—
445C2×C3—24no—
446C2×C3—22no—
447C2×C3—22no—
448C2×C3—21no—
449C2×C3—22no—
450C2×C3—24no—
451C2×C3—44no—
452C2×C3—44no—
453C2×C3—41no—
454C2×C3—42no—
455C2×C3—128no—
456C2×C3—124no—
457C2×C3—124no—
458C2×C3#6382212yesdirect product factor
459C2×C3C3×#95014yescomplemented
460C2×C3C22×#62511yescharacteristic
461C2×C3C22×C3×#6712yes—
462D3—44no—
463C22—22no—
464C22—64no—
465C22—124no—
466C22C2×C3×#62512yes—
467C22C2×C3×#62511yescharacteristic
468C4—64no—
469C4C2×C3×#62512yes—
470C4C2×C3×#62511yescharacteristic
471C4C22×C3×D3211yescharacteristic
472C3—21no—
473C3—22no—
474C3—41no—
475C3C2×#6382211yescharacteristic, direct product factor
476C3C2×C3×#95012yescomplemented
477C2—22no—
478C2—124no—
479C2C3×#6382212yesdirect product factor
480C2C22×C3×#62511yescharacteristic
subgroupG/H
centerC2×C3×C4#625
commutator, cosocleC32×C4C24×C3
FrattiniC4C22×C3×D32
FittingC22×C33×C4C22
socleC22×C33C2×D4
hypercenterC22×C3×C4D32
hypocenterC32C2×C3×#135
LCSG ▷ C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2×C3×C4 ◁ C22×C3×C4

Complements

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

NH ≅ G/Ntype
1. C2×C32×#235C2semidirect
2. C22×C3×C3⋊(C3⋊C8)C2semidirect
3. C2×C3×#3718C2semidirect
4. C2×C3×#3715C2semidirect
5. C3×#63822C2direct (N × H)
6. C2×C3×#3713C2semidirect
8. C2×#63822C3direct (N × H)
12. C32×#235C22semidirect
18. C2×C3×C3⋊(C3⋊C8)C22semidirect
19. C3×#3718C22semidirect
20. C3×#3715C22semidirect
21. C3×#3713C22semidirect
24. C2×C3×#235D3semidirect
25. C2×C3×#235C2×C3semidirect
26. C22×C3⋊(C3⋊C8)C2×C3semidirect
34. C2×#3718C2×C3semidirect
35. C2×#3715C2×C3semidirect
36. #63822C2×C3direct (N × H)
37. C2×#3713C2×C3semidirect
73. C3×#235C2×D3semidirect
74. C3×#235C22×C3semidirect
86. C2×C3⋊(C3⋊C8)C22×C3semidirect
97. #3718C22×C3semidirect
98. #3715C22×C3semidirect
99. #3713C22×C3semidirect
112. C2×#235C3×D3semidirect
190. C2×C33#135semidirect
218. #235C2×C3×D3semidirect
286. C33C2×#135semidirect
362. C2×C32#950semidirect
363. C2×C32C3×#135semidirect
427. C32C2×#950semidirect
428. C32C2×C3×#135semidirect
458. C2×C3#63822direct (N × H)
459. C2×C3C3×#950semidirect
475. C3C2×#63822direct (N × H)
476. C3C2×C3×#950semidirect
479. C2C3×#63822direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C32×#2352
C22×C3×C3⋊(C3⋊C8)2
C2×C3×#37182
C2×C3×#37152
C3×#638222
C2×C3×#37132
C2×C3×#9503
C2×#638223