Finite Group Databasebeta

← C2×C3×#64159 #13179575

Subgroups — C2×C3×#64159 #13179575

Order 1728.

2252 conjugacy classes of subgroups.

The subgroup lattice has 490 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C33×#41C211yesmaximal, characteristic, complemented
2C2×C3×#3828C211yesmaximal, characteristic, complemented
3C3×#64159C218yesmaximal, direct product factor
4C2×C3×#3756C211yesmaximal, characteristic, complemented
5C2×C3×#3755C211yesmaximal, characteristic, complemented
6C2×C3×#3754C211yesmaximal, characteristic, complemented
7C2×C3×#3753C211yesmaximal, characteristic, complemented
8C22×C3×C3⋊(C3⋊C8)C211yesmaximal, characteristic, complemented
9C2×#64159C311yesmaximal, characteristic, direct product factor
10C2×C3×#950—34nomaximal
11C33×#41—22no—
12C33×#41C2214yescomplemented
13C2×C33×Q8C2211yescharacteristic
14C2×C33×D4—21no—
15C2×C33×D4C2211yescharacteristic
16C22×C33×C4—21no—
17C22×C33×C4C2211yescharacteristic
18C2×C3×#638—21no—
19C3×#3828—22no—
20C3×#3828C2214yescomplemented
21C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
22C2×C3×C4×C3⋊D3—21no—
23C2×C3×#634C2211yescharacteristic
24C3×#3756C2214yescomplemented
25C3×#3755C2214yescomplemented
26C3×#3754C2214yescomplemented
27C3×#3753C2214yescomplemented
28C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
29C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
30C2×C3×C3⋊(C3⋊C8)C2214yescomplemented
31C2×C32×#41—24no—
32C2×C32×#41D314yescomplemented
33C2×C32×#41C2×C311yescharacteristic, complemented
34C2×#3828C2×C311yescharacteristic, complemented
35C2×C3×#235—34no—
36#64159C2×C318yesdirect product factor
37C2×#3756C2×C311yescharacteristic, complemented
38C2×#3755C2×C311yescharacteristic, complemented
39C2×#3754C2×C311yescharacteristic, complemented
40C2×#3753C2×C311yescharacteristic, complemented
41C22×C3⋊(C3⋊C8)C2×C311yescharacteristic, complemented
42C3×#950—332no—
43C2×C3×#216—34no—
44C2×C3×#215—34no—
45C2×C3×#214—34no—
46C2×C3×#213—34no—
47C22×C3×C3⋊C8—34no—
48C23×C33—21no—
49C23×C33D411yescharacteristic
50C22×C3×C3⋊D3—21no—
51C33×Q8—21no—
52C33×Q8C2312yes—
53C33×D4—24no—
54C33×D4—21no—
55C33×D4C2312yes—
56C2×C33×C4—21no—
57C2×C33×C4—24no—
58C2×C33×C4D411yescharacteristic
59C2×C33×C4D412yes—
60C2×C33×C4C2311yescharacteristic
61C2×C33×C4C2312yes—
62C3×#638—24no—
63C2×C3×C3⋊Dic3—21no—
64C3×C4⋊(C3⋊D3)—21no—
65C3×C4⋊(C3⋊D3)C2312yes—
66C3×C4×C3⋊D3—24no—
67C3×#634—21no—
68C3×#634C2312yes—
69C3×C3⋊(C3⋊C8)C2312yes—
70C3×C3⋊(C3⋊C8)C2312yes—
71C2×C3×#135—91no—
72C2×#950—34no—
73C32×#41—216no—
74C32×#41—216no—
75C32×#41—28no—
76C32×#41—22no—
77C32×#41C2×D3116yescomplemented
78C32×#41C22×C314yescomplemented
79C2×C32×Q8—24no—
80C2×C32×Q8C2×D314yes—
81C2×C32×Q8C22×C311yescharacteristic
82C2×C32×D4—28no—
83C2×C32×D4—24no—
84C2×C32×D4—24no—
85C2×C32×D4—21no—
86C2×C32×D4C2×D314yes—
87C2×C32×D4C22×C311yescharacteristic
88C22×C32×C4—28no—
89C22×C32×C4—24no—
90C22×C32×C4—24no—
91C22×C32×C4—21no—
92C22×C32×C4C2×D314yes—
93C22×C32×C4C22×C311yescharacteristic
94C2×#638—21no—
95#3828—22no—
96#3828C22×C314yescomplemented
97C2×C4⋊(C3⋊D3)C22×C311yescharacteristic
98C2×C4×C3⋊D3—21no—
99C2×#634C22×C311yescharacteristic
100C2×C3×#67—64no—
101C3×#235—316no—
102C3×#235—68no—
103C2×C3×D12—34no—
104C2×C3×C4×D3—64no—
105C2×C3×Dic6—34no—
106#3756C22×C314yescomplemented
107#3755C22×C314yescomplemented
108#3754C22×C314yescomplemented
109#3753C22×C314yescomplemented
110C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
111C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
112C2×C3⋊(C3⋊C8)C22×C314yescomplemented
113C3×#216—316no—
114C3×#215—316no—
115C3×#214—316no—
116C3×#213—316no—
117C2×C3×C3⋊C8—34no—
118C2×C3×C3⋊C8—34no—
119C2×C3×C3⋊C8—316no—
120C22×C33—22no—
121C22×C33—22no—
122C22×C33—42no—
123C22×C33C2×D412yes—
124C22×C33C2×D411yescharacteristic
125C2×C3×C3⋊D3—22no—
126C2×C3×C3⋊D3—42no—
127C33×C4—22no—
128C33×C4C2×D411yescharacteristic
129C33×C4C2×D412yes—
130C33×C4C2411yescharacteristic
131C3×C3⋊Dic3—22no—
132C2×C3×#41—24no—
133C2×C3×#41—91no—
134C2×C3×#41C3×D314yescomplemented
135C2×C3×#41C3⋊D311yescharacteristic, complemented
136C2×#235—34no—
137C3×#135—98no—
138C2×C3×Q16—91no—
139C2×C3×SD16—91no—
140C2×C3×SD16—91no—
141C2×C3×D8—91no—
142C22×C3×C8—91no—
143#950—332no—
144C2×#216—34no—
145C2×#215—34no—
146C2×#214—34no—
147C2×#213—34no—
148C22×C3⋊C8—34no—
149C23×C32—28no—
150C23×C32—24no—
151C23×C32—21no—
152C23×C32—24no—
153C23×C32#6714yes—
154C23×C32C3×D411yescharacteristic
155C22×C3⋊D3—21no—
156C22×C3×D3—64no—
157C32×Q8—28no—
158C32×Q8—24no—
159C32×Q8—21no—
160C32×Q8—28no—
161C32×Q8C22×D318yes—
162C32×Q8C23×C312yes—
163C32×D4—232no—
164C32×D4—28no—
165C32×D4—216no—
166C32×D4—24no—
167C32×D4—24no—
168C32×D4—21no—
169C32×D4—28no—
170C32×D4C22×D318yes—
171C32×D4C23×C312yes—
172C2×C32×C4—28no—
173C2×C32×C4—232no—
174C2×C32×C4—24no—
175C2×C32×C4—21no—
176C2×C32×C4—216no—
177C2×C32×C4—24no—
178C2×C32×C4—24no—
179C2×C32×C4—28no—
180C2×C32×C4—24no—
181C2×C32×C4—28no—
182C2×C32×C4#6718yes—
183C2×C32×C4#6714yes—
184C2×C32×C4C3×D411yescharacteristic
185C2×C32×C4C3×D412yes—
186C2×C32×C4C22×D314yes—
187C2×C32×C4C22×D318yes—
188C2×C32×C4C23×C311yescharacteristic
189C2×C32×C4C23×C312yes—
190#638—24no—
191C2×C3⋊Dic3—21no—
192C4⋊(C3⋊D3)—21no—
193C4⋊(C3⋊D3)C23×C312yes—
194C4×C3⋊D3—24no—
195#634—21no—
196#634C23×C312yes—
197C3×#67—616no—
198C2×C3×Dic3—64no—
199C3×D12—38no—
200C3×D12—64no—
201C3×C4×D3—616no—
202C3×Dic6—38no—
203C3×Dic6—64no—
204C3⋊(C3⋊C8)C23×C312yes—
205C3⋊(C3⋊C8)C23×C312yes—
206C3×C3⋊C8—38no—
207C3×C3⋊C8—38no—
208C2×#135—91noSylow
209C2×C33—22no—
210C2×C33—42no—
211C2×C33#13512yescomplemented
212C2×C33C22×D411yescharacteristic
213C3×C3⋊D3—42no—
214C3×#41—216no—
215C3×#41—28no—
216C3×#41—22no—
217C3×#41—216no—
218C3×#41—94no—
219C3×#41—182no—
220C3×#41C2×C3×D3116yescomplemented
221C3×#41C2×C3⋊D314yescomplemented
222C2×C3×Q8—24no—
223C2×C3×Q8—91no—
224C2×C3×Q8C2×C3×D314yes—
225C2×C3×Q8C2×C3⋊D311yescharacteristic
226C2×C3×D4—28no—
227C2×C3×D4—24no—
228C2×C3×D4—21no—
229C2×C3×D4—24no—
230C2×C3×D4—91no—
231C2×C3×D4—181no—
232C2×C3×D4C2×C3×D314yes—
233C2×C3×D4C2×C3⋊D311yescharacteristic
234C22×C3×C4—28no—
235C22×C3×C4—24no—
236C22×C3×C4—21no—
237C22×C3×C4—24no—
238C22×C3×C4—181no—
239C22×C3×C4C2×C3×D314yes—
240C22×C3×C4C2×C3⋊D311yescharacteristic
241C2×#67—64no—
242#235—316no—
243#235—68no—
244C2×D12—34no—
245C2×C4×D3—64no—
246C2×Dic6—34no—
247C3×Q16—94no—
248C3×SD16—94no—
249C3×SD16—94no—
250C3×D8—94no—
251C2×C3×C8—91no—
252C2×C3×C8—91no—
253C2×C3×C8—94no—
254#216—316no—
255#215—316no—
256#214—316no—
257#213—316no—
258C2×C3⋊C8—34no—
259C2×C3⋊C8—34no—
260C2×C3⋊C8—316no—
261C22×C32—22no—
262C22×C32—216no—
263C22×C32—28no—
264C22×C32—22no—
265C22×C32—24no—
266C22×C32—28no—
267C22×C32—28no—
268C22×C32—48no—
269C22×C32—416no—
270C22×C32—48no—
271C22×C32—42no—
272C22×C32C2×#6718yes—
273C22×C32C2×#6714yes—
274C22×C32C2×C3×D412yes—
275C22×C32C2×C3×D411yescharacteristic
276C2×C3⋊D3—22no—
277C2×C3⋊D3—42no—
278C2×C3×D3—68no—
279C2×C3×D3—128no—
280C32×C4—216no—
281C32×C4—28no—
282C32×C4—22no—
283C32×C4—24no—
284C32×C4—28no—
285C32×C4—24no—
286C32×C4C2×#6714yes—
287C32×C4C2×#6718yes—
288C32×C4C2×C3×D411yescharacteristic
289C32×C4C2×C3×D412yes—
290C32×C4C23×D314yes—
291C32×C4C24×C311yescharacteristic
292C3⋊Dic3—22no—
293C3×Dic3—68no—
294C2×#41—91no—
295C2×#41C3×C3⋊D311yescharacteristic, complemented
296#135—98no—
297C2×Q16—91no—
298C2×SD16—91no—
299C2×SD16—91no—
300C2×D8—91no—
301C22×C8—91no—
302C33C2×#13511yescharacteristic, complemented, Sylow
303C23×C3—28no—
304C23×C3—24no—
305C23×C3—21no—
306C23×C3—24no—
307C23×C3—181no—
308C23×C3C3×#6714yes—
309C23×C3#63811yescharacteristic
310C22×D3—64no—
311C3×Q8—28no—
312C3×Q8—24no—
313C3×Q8—21no—
314C3×Q8—28no—
315C3×Q8—92no—
316C3×Q8—181no—
317C3×Q8C22×C3×D318yes—
318C3×Q8C22×C3⋊D312yes—
319C3×D4—232no—
320C3×D4—28no—
321C3×D4—216no—
322C3×D4—24no—
323C3×D4—24no—
324C3×D4—21no—
325C3×D4—28no—
326C3×D4—92no—
327C3×D4—184no—
328C3×D4—181no—
329C3×D4C22×C3×D318yes—
330C3×D4C22×C3⋊D312yes—
331C2×C3×C4—28no—
332C2×C3×C4—232no—
333C2×C3×C4—24no—
334C2×C3×C4—21no—
335C2×C3×C4—216no—
336C2×C3×C4—24no—
337C2×C3×C4—24no—
338C2×C3×C4—28no—
339C2×C3×C4—24no—
340C2×C3×C4—28no—
341C2×C3×C4—181no—
342C2×C3×C4—184no—
343C2×C3×C4C3×#6714yes—
344C2×C3×C4C3×#6718yes—
345C2×C3×C4#63812yes—
346C2×C3×C4#63811yescharacteristic
347C2×C3×C4C22×C3×D314yes—
348C2×C3×C4C22×C3×D318yes—
349C2×C3×C4C22×C3⋊D311yescharacteristic
350C2×C3×C4C22×C3⋊D312yes—
351#67—616no—
352C2×Dic3—64no—
353D12—38no—
354D12—64no—
355C4×D3—616no—
356Dic6—38no—
357Dic6—64no—
358C3×C8—92no—
359C3×C8—92no—
360C3⋊C8—38no—
361C3⋊C8—38no—
362C2×C32—22no—
363C2×C32—28no—
364C2×C32—24no—
365C2×C32—28no—
366C2×C32—48no—
367C2×C32—416no—
368C2×C32—42no—
369C2×C32—48no—
370C2×C32#95018yescomplemented
371C2×C32C3×#13512yescomplemented
372C2×C32C22×#6714yes—
373C2×C32C22×C3×D411yescharacteristic
374C3⋊D3—42no—
375C3×D3—128no—
376#41—22no—
377#41—94no—
378#41—182no—
379#41C2×C3×C3⋊D314yescomplemented
380C2×Q8—91no—
381C2×Q8C2×C3×C3⋊D311yescharacteristic
382C2×D4—21no—
383C2×D4—91no—
384C2×D4—181no—
385C2×D4C2×C3×C3⋊D311yescharacteristic
386C22×C4—21no—
387C22×C4—181no—
388C22×C4C2×C3×C3⋊D311yescharacteristic
389Q16—94no—
390SD16—94no—
391SD16—94no—
392D8—94no—
393C2×C8—91no—
394C2×C8—91no—
395C2×C8—94no—
396C22×C3—28no—
397C22×C3—22no—
398C22×C3—216no—
399C22×C3—28no—
400C22×C3—22no—
401C22×C3—24no—
402C22×C3—28no—
403C22×C3—48no—
404C22×C3—416no—
405C22×C3—48no—
406C22×C3—42no—
407C22×C3—182no—
408C22×C3—362no—
409C22×C3C2×C3×#6718yes—
410C22×C3C2×C3×#6714yes—
411C22×C3C2×#63812yes—
412C22×C3C2×#63811yescharacteristic
413C2×D3—68no—
414C2×D3—128no—
415C3×C4—216no—
416C3×C4—28no—
417C3×C4—22no—
418C3×C4—24no—
419C3×C4—28no—
420C3×C4—24no—
421C3×C4—182no—
422C3×C4C2×C3×#6714yes—
423C3×C4C2×C3×#6718yes—
424C3×C4C2×#63812yes—
425C3×C4C2×#63811yescharacteristic
426C3×C4C23×C3×D314yes—
427C3×C4C23×C3⋊D311yescharacteristic
428Dic3—68no—
429C32—24no—
430C32C2×#95014yescomplemented
431C32C2×C3×#13511yescharacteristic, complemented
432C23—21no—
433C23—181no—
434C23C3×#63811yescharacteristic
435Q8—21no—
436Q8—92no—
437Q8—181no—
438Q8C22×C3×C3⋊D312yes—
439D4—24no—
440D4—21no—
441D4—92no—
442D4—184no—
443D4—181no—
444D4C22×C3×C3⋊D312yes—
445C2×C4—21no—
446C2×C4—24no—
447C2×C4—181no—
448C2×C4—184no—
449C2×C4C3×#63811yescharacteristic
450C2×C4C3×#63812yes—
451C2×C4C22×C3×C3⋊D311yescharacteristic
452C2×C4C22×C3×C3⋊D312yes—
453C8—92no—
454C8—92no—
455C2×C3—28no—
456C2×C3—22no—
457C2×C3—24no—
458C2×C3—28no—
459C2×C3—48no—
460C2×C3—416no—
461C2×C3—48no—
462C2×C3—42no—
463C2×C3—362no—
464C2×C3C3×#95018yescomplemented
465C2×C3#6415912yesdirect product factor
466C2×C3C22×C3×#6714yes—
467C2×C3C22×#63811yescharacteristic
468D3—128no—
469C22—22no—
470C22—22no—
471C22—42no—
472C22—182no—
473C22—362no—
474C22C2×C3×#63812yes—
475C22C2×C3×#63811yescharacteristic
476C4—22no—
477C4—182no—
478C4C2×C3×#63812yes—
479C4C2×C3×#63811yescharacteristic
480C4C23×C3×C3⋊D311yescharacteristic
481C3—24no—
482C3C2×C3×#95014yescomplemented
483C3C2×#6415911yescharacteristic, direct product factor
484C2—22no—
485C2—42no—
486C2—362no—
487C2C3×#6415912yesdirect product factor
488C2C22×C3×#63811yescharacteristic
subgroupG/H
centerC2×C3×C4#638
commutator, cosocleC32×C4C24×C3
FrattiniC4C23×C3×C3⋊D3
FittingC2×C33×#41C2
socleC22×C33C2×D4
hypercenterC2×C3×#41C3⋊D3
hypocenterC32C2×C3×#135
LCSG ▷ C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2×C3×C4 ◁ C22×C3×C4 ◁ C2×C3×#41

Complements

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

NH ≅ G/Ntype
1. C2×C33×#41C2semidirect
2. C2×C3×#3828C2semidirect
3. C3×#64159C2direct (N × H)
4. C2×C3×#3756C2semidirect
5. C2×C3×#3755C2semidirect
6. C2×C3×#3754C2semidirect
7. C2×C3×#3753C2semidirect
8. C22×C3×C3⋊(C3⋊C8)C2semidirect
9. C2×#64159C3direct (N × H)
12. C33×#41C22semidirect
20. C3×#3828C22semidirect
24. C3×#3756C22semidirect
25. C3×#3755C22semidirect
26. C3×#3754C22semidirect
27. C3×#3753C22semidirect
30. C2×C3×C3⋊(C3⋊C8)C22semidirect
32. C2×C32×#41D3semidirect
33. C2×C32×#41C2×C3semidirect
34. C2×#3828C2×C3semidirect
36. #64159C2×C3direct (N × H)
37. C2×#3756C2×C3semidirect
38. C2×#3755C2×C3semidirect
39. C2×#3754C2×C3semidirect
40. C2×#3753C2×C3semidirect
41. C22×C3⋊(C3⋊C8)C2×C3semidirect
77. C32×#41C2×D3semidirect
78. C32×#41C22×C3semidirect
96. #3828C22×C3semidirect
106. #3756C22×C3semidirect
107. #3755C22×C3semidirect
108. #3754C22×C3semidirect
109. #3753C22×C3semidirect
112. C2×C3⋊(C3⋊C8)C22×C3semidirect
134. C2×C3×#41C3×D3semidirect
135. C2×C3×#41C3⋊D3semidirect
211. C2×C33#135semidirect
220. C3×#41C2×C3×D3semidirect
221. C3×#41C2×C3⋊D3semidirect
295. C2×#41C3×C3⋊D3semidirect
302. C33C2×#135semidirect
370. C2×C32#950semidirect
371. C2×C32C3×#135semidirect
379. #41C2×C3×C3⋊D3semidirect
430. C32C2×#950semidirect
431. C32C2×C3×#135semidirect
464. C2×C3C3×#950semidirect
465. C2×C3#64159direct (N × H)
482. C3C2×C3×#950semidirect
483. C3C2×#64159direct (N × H)
487. C2C3×#64159direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C33×#412
C2×C3×#38282
C3×#641592
C2×C3×#37562
C2×C3×#37552
C2×C3×#37542
C2×C3×#37532
C22×C3×C3⋊(C3⋊C8)2
C2×#641593
C2×C3×#9503