Finite Group Databasebeta

← C2×C3×C4×#638 #13179539

Subgroups — C2×C3×C4×#638 #13179539

Order 1728.

3500 conjugacy classes of subgroups.

The subgroup lattice has 496 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×C33×C4C211yesmaximal, characteristic, complemented
2C22×C3×#638C211yesmaximal, characteristic
3C22×C3×C4×C3⋊D3C211yesmaximal, characteristic, complemented
4C2×C3×#3757C211yesmaximal, characteristic, complemented
5C3×C4×#638C218yesmaximal, direct product factor
6C2×C3×#3752C211yesmaximal, characteristic, complemented
7C2×C3×#3750C211yesmaximal, characteristic, complemented
8C2×C3×C4×C3⋊Dic3C211yesmaximal, characteristic, complemented
9C2×C4×#638C311yesmaximal, characteristic, direct product factor
10C2×C3×C4×#67—34nomaximal
11C24×C33C2211yescharacteristic
12C23×C3×C3⋊D3C2211yescharacteristic
13C22×C33×C4—24no—
14C22×C33×C4C2211yescharacteristic
15C22×C33×C4C2211yescharacteristic
16C22×C33×C4C2214yescomplemented
17C2×C3×#638C418yesdirect product factor
18C2×C3×#638C2214yes—
19C22×C3×C3⋊Dic3C2211yescharacteristic
20C22×C3×C3⋊Dic3C2211yescharacteristic
21C22×C3×C3⋊Dic3C2211yescharacteristic
22C2×C3×C4×C3⋊D3—24no—
23C2×C3×C4×C3⋊D3C2214yescomplemented
24C3×#3757C2214yescomplemented
25C3×#3752C2214yescomplemented
26C3×#3750C2214yescomplemented
27C3×C4×C3⋊Dic3C2214yescomplemented
28C23×C32×C4—24no—
29C23×C32×C4D314yescomplemented
30C23×C32×C4C2×C311yescharacteristic, complemented
31C22×#638C2×C311yescharacteristic
32C22×C4×C3⋊D3C2×C311yescharacteristic, complemented
33C22×C3×#67—34no—
34C22×C3×C4×D3—34no—
35C2×#3757C2×C311yescharacteristic, complemented
36C4×#638C2×C318yesdirect product factor
37C2×#3752C2×C311yescharacteristic, complemented
38C2×#3750C2×C311yescharacteristic, complemented
39C2×C4×C3⋊Dic3C2×C311yescharacteristic, complemented
40C2×C3×#217—34no—
41C3×C4×#67—332no—
42C2×C3×(C2×D3)⋊C4—34no—
43C2×C3×Dic3⋊C4—34no—
44C2×C3×C4×Dic3—34no—
45C23×C33—22no—
46C23×C33—22no—
47C23×C33C2×C414yescomplemented
48C23×C33C2312yes—
49C23×C33C2311yescharacteristic
50C22×C3×C3⋊D3—22no—
51C22×C3×C3⋊D3—22no—
52C22×C3×C3⋊D3C2×C414yescomplemented
53C22×C3×C3⋊D3C2312yes—
54C2×C33×C4—22no—
55C2×C33×C4—28no—
56C2×C33×C4D414yescomplemented
57C2×C33×C4C2312yes—
58C2×C33×C4C2312yes—
59C3×#638C2×C4116yesdirect product factor
60C2×C3×C3⋊Dic3—22no—
61C2×C3×C3⋊Dic3—22no—
62C2×C3×C3⋊Dic3C2×C414yescomplemented
63C2×C3×C3⋊Dic3C2312yes—
64C2×C3×C3⋊Dic3C2312yes—
65C2×C3×C3⋊Dic3C2312yes—
66C3×C4×C3⋊D3—28no—
67C2×C3×C4×D4—91no—
68C2×C4×#67—34no—
69C24×C32—24no—
70C24×C32C2×D314yes—
71C24×C32C22×C311yescharacteristic
72C23×C3⋊D3C22×C311yescharacteristic
73C23×C3×D3—34no—
74C22×C32×C4—232no—
75C22×C32×C4—24no—
76C22×C32×C4—24no—
77C22×C32×C4—216no—
78C22×C32×C4—216no—
79C22×C32×C4—24no—
80C22×C32×C4C2×D314yes—
81C22×C32×C4C2×D314yes—
82C22×C32×C4C2×D3116yescomplemented
83C22×C32×C4C22×C311yescharacteristic
84C22×C32×C4C22×C311yescharacteristic
85C22×C32×C4C22×C314yescomplemented
86C2×#638C3×C418yesdirect product factor
87C2×#638C22×C314yes—
88C22×C3⋊Dic3C22×C311yescharacteristic
89C22×C3⋊Dic3C22×C311yescharacteristic
90C22×C3⋊Dic3C22×C311yescharacteristic
91C2×C4×C3⋊D3—24no—
92C2×C4×C3⋊D3C22×C314yescomplemented
93C2×C3×#67—332no—
94C2×C3×#67—316no—
95C22×C3×Dic3—34no—
96C22×C3×Dic3—34no—
97C22×C3×Dic3—34no—
98C2×C3×C4×D3—316no—
99C2×C3×C4×D3—616no—
100#3757C22×C314yescomplemented
101#3752C22×C314yescomplemented
102#3750C22×C314yescomplemented
103C4×C3⋊Dic3C22×C314yescomplemented
104C3×#217—316no—
105C3×(C2×D3)⋊C4—316no—
106C3×Dic3⋊C4—316no—
107C3×C4×Dic3—316no—
108C22×C33—28no—
109C22×C33—22no—
110C22×C33—22no—
111C22×C33C22×C412yes—
112C22×C33C22×C414yescomplemented
113C22×C33C2×D412yes—
114C22×C33#4112yes—
115C22×C33C2411yescharacteristic
116C2×C3×C3⋊D3—28no—
117C2×C3×C3⋊D3—22no—
118C2×C3×C3⋊D3—22no—
119C2×C3×C3⋊D3C22×C414yescomplemented
120C33×C4—22no—
121C33×C4C2×D414yescomplemented
122C3×C3⋊Dic3—22no—
123C3×C3⋊Dic3—22no—
124C3×C3⋊Dic3C22×C414yescomplemented
125C22×C3×D4—91no—
126C23×C3×C4—24no—
127C23×C3×C4—91no—
128C23×C3×C4C3×D314yescomplemented
129C23×C3×C4C3⋊D311yescharacteristic, complemented
130C22×#67—34no—
131C22×C4×D3—34no—
132C3×C4×D4—98no—
133C2×C3×C4⋊C4—91no—
134C2×C3×C22⋊C4—91no—
135C2×C3×C22⋊C4—91no—
136C2×C3×C42—91no—
137C2×#217—34no—
138C4×#67—332no—
139C2×(C2×D3)⋊C4—34no—
140C2×Dic3⋊C4—34no—
141C2×C4×Dic3—34no—
142C23×C32—216no—
143C23×C32—216no—
144C23×C32—28no—
145C23×C32—28no—
146C23×C32—22no—
147C23×C32—22no—
148C23×C32—216no—
149C23×C32—28no—
150C23×C32—24no—
151C23×C32C4×D3116yescomplemented
152C23×C32C2×C3×C414yescomplemented
153C23×C32C22×D318yes—
154C23×C32C22×D314yes—
155C23×C32C23×C312yes—
156C23×C32C23×C311yescharacteristic
157C22×C3⋊D3—22no—
158C22×C3⋊D3—22no—
159C22×C3⋊D3C2×C3×C414yescomplemented
160C22×C3⋊D3C23×C312yes—
161C22×C3×D3—316no—
162C22×C3×D3—38no—
163C22×C3×D3—68no—
164C22×C3×D3—68no—
165C2×C32×C4—216no—
166C2×C32×C4—264no—
167C2×C32×C4—28no—
168C2×C32×C4—22no—
169C2×C32×C4—232no—
170C2×C32×C4—28no—
171C2×C32×C4—216no—
172C2×C32×C4—28no—
173C2×C32×C4—28no—
174C2×C32×C4#67116yescomplemented
175C2×C32×C4C3×D414yescomplemented
176C2×C32×C4C22×D318yes—
177C2×C32×C4C22×D318yes—
178C2×C32×C4C23×C312yes—
179C2×C32×C4C23×C312yes—
180#638C2×C3×C4116yesdirect product factor
181C2×C3⋊Dic3—22no—
182C2×C3⋊Dic3—22no—
183C2×C3⋊Dic3C2×C3×C414yescomplemented
184C2×C3⋊Dic3C23×C312yes—
185C2×C3⋊Dic3C23×C312yes—
186C2×C3⋊Dic3C23×C312yes—
187C4×C3⋊D3—28no—
188C3×#67—364no—
189C2×C3×Dic3—316no—
190C2×C3×Dic3—38no—
191C2×C3×Dic3—38no—
192C2×C3×Dic3—38no—
193C2×C3×Dic3—68no—
194C2×C3×Dic3—68no—
195C3×C4×D3—632no—
196C2×C4×D4—91noSylow
197C2×C33—24no—
198C2×C33C4×D414yescomplemented
199C2×C33C23×C411yescharacteristic
200C2×C33C22×D411yescharacteristic
201C2×C33C2×#4111yescharacteristic
202C3×C3⋊D3—24no—
203C24×C3—24no—
204C24×C3—91no—
205C24×C3C2×C3×D314yes—
206C24×C3C2×C3⋊D311yescharacteristic
207C23×D3—34no—
208C2×C3×D4—98no—
209C2×C3×D4—94no—
210C22×C3×C4—232no—
211C22×C3×C4—216no—
212C22×C3×C4—24no—
213C22×C3×C4—24no—
214C22×C3×C4—24no—
215C22×C3×C4—216no—
216C22×C3×C4—91no—
217C22×C3×C4—91no—
218C22×C3×C4—91no—
219C22×C3×C4—94no—
220C22×C3×C4—184no—
221C22×C3×C4C2×C3×D314yes—
222C22×C3×C4C2×C3×D314yes—
223C22×C3×C4C2×C3×D3116yescomplemented
224C22×C3×C4C2×C3⋊D311yescharacteristic
225C22×C3×C4C2×C3⋊D311yescharacteristic
226C22×C3×C4C2×C3⋊D314yescomplemented
227C2×#67—332no—
228C2×#67—316no—
229C22×Dic3—34no—
230C22×Dic3—34no—
231C22×Dic3—34no—
232C2×C4×D3—316no—
233C2×C4×D3—616no—
234C3×C4⋊C4—94no—
235C3×C22⋊C4—94no—
236C3×C22⋊C4—94no—
237C3×C42—94no—
238#217—316no—
239(C2×D3)⋊C4—316no—
240Dic3⋊C4—316no—
241C4×Dic3—316no—
242C22×C32—216no—
243C22×C32—264no—
244C22×C32—216no—
245C22×C32—232no—
246C22×C32—28no—
247C22×C32—28no—
248C22×C32—28no—
249C22×C32—22no—
250C22×C32—22no—
251C22×C32—24no—
252C22×C32—28no—
253C22×C32—216no—
254C22×C32—28no—
255C22×C32—28no—
256C22×C32C2×C4×D318yes—
257C22×C32C2×C4×D3116yescomplemented
258C22×C32#23518yes—
259C22×C32C2×#6718yes—
260C22×C32C22×C3×C412yes—
261C22×C32C22×C3×C414yescomplemented
262C22×C32C2×C3×D412yes—
263C22×C32C3×#4112yes—
264C22×C32C23×D314yes—
265C22×C32C24×C311yescharacteristic
266C2×C3⋊D3—28no—
267C2×C3⋊D3—22no—
268C2×C3⋊D3—22no—
269C2×C3⋊D3C22×C3×C414yescomplemented
270C2×C3×D3—316no—
271C2×C3×D3—632no—
272C2×C3×D3—68no—
273C2×C3×D3—68no—
274C32×C4—216no—
275C32×C4—28no—
276C32×C4—22no—
277C32×C4—216no—
278C32×C4C2×#67116yescomplemented
279C32×C4C2×C3×D414yescomplemented
280C3⋊Dic3—22no—
281C3⋊Dic3—22no—
282C3⋊Dic3C22×C3×C414yescomplemented
283C3×Dic3—316no—
284C3×Dic3—68no—
285C3×Dic3—68no—
286C22×D4—91no—
287C23×C4—91no—
288C23×C4C3×C3⋊D311yescharacteristic, complemented
289C4×D4—98no—
290C2×C4⋊C4—91no—
291C2×C22⋊C4—91no—
292C2×C22⋊C4—91no—
293C2×C42—91no—
294C33C2×C4×D411yescharacteristic, complemented, Sylow
295C23×C3—216no—
296C23×C3—216no—
297C23×C3—28no—
298C23×C3—28no—
299C23×C3—22no—
300C23×C3—22no—
301C23×C3—216no—
302C23×C3—28no—
303C23×C3—24no—
304C23×C3—92no—
305C23×C3—94no—
306C23×C3—182no—
307C23×C3—182no—
308C23×C3C3×C4×D3116yescomplemented
309C23×C3C4×C3⋊D314yescomplemented
310C23×C3C22×C3×D318yes—
311C23×C3C22×C3×D314yes—
312C23×C3C22×C3⋊D312yes—
313C23×C3C22×C3⋊D311yescharacteristic
314C22×D3—38no—
315C22×D3—316no—
316C22×D3—68no—
317C22×D3—68no—
318C3×D4—916no—
319C2×C3×C4—216no—
320C2×C3×C4—264no—
321C2×C3×C4—28no—
322C2×C3×C4—22no—
323C2×C3×C4—232no—
324C2×C3×C4—28no—
325C2×C3×C4—216no—
326C2×C3×C4—28no—
327C2×C3×C4—28no—
328C2×C3×C4—92no—
329C2×C3×C4—94no—
330C2×C3×C4—92no—
331C2×C3×C4—92no—
332C2×C3×C4—182no—
333C2×C3×C4—182no—
334C2×C3×C4—188no—
335C2×C3×C4C3×#67116yescomplemented
336C2×C3×C4#63814yesdirect product factor
337C2×C3×C4C22×C3×D318yes—
338C2×C3×C4C22×C3×D318yes—
339C2×C3×C4C22×C3⋊D312yes—
340C2×C3×C4C22×C3⋊D312yes—
341#67—364no—
342C2×Dic3—38no—
343C2×Dic3—316no—
344C2×Dic3—38no—
345C2×Dic3—38no—
346C2×Dic3—68no—
347C2×Dic3—68no—
348C4×D3—632no—
349C2×C32—232no—
350C2×C32—216no—
351C2×C32—24no—
352C2×C32—216no—
353C2×C32—24no—
354C2×C32—24no—
355C2×C32—24no—
356C2×C32C4×#67116yescomplemented
357C2×C32C3×C4×D414yescomplemented
358C2×C32C22×C4×D314yes—
359C2×C32C2×#23514yes—
360C2×C32C22×#6714yes—
361C2×C32C23×C3×C411yescharacteristic
362C2×C32C22×C3×D411yescharacteristic
363C2×C32C2×C3×#4111yescharacteristic
364C3⋊D3—24no—
365C3×D3—616no—
366C24—91no—
367C24C2×C3×C3⋊D311yescharacteristic
368C2×D4—98no—
369C2×D4—94no—
370C22×C4—24no—
371C22×C4—91no—
372C22×C4—91no—
373C22×C4—91no—
374C22×C4—94no—
375C22×C4—184no—
376C22×C4C2×C3×C3⋊D311yescharacteristic
377C22×C4C2×C3×C3⋊D311yescharacteristic
378C22×C4C2×C3×C3⋊D314yescomplemented
379C4⋊C4—94no—
380C22⋊C4—94no—
381C22⋊C4—94no—
382C42—94no—
383C22×C3—216no—
384C22×C3—264no—
385C22×C3—216no—
386C22×C3—232no—
387C22×C3—28no—
388C22×C3—28no—
389C22×C3—22no—
390C22×C3—22no—
391C22×C3—28no—
392C22×C3—24no—
393C22×C3—28no—
394C22×C3—216no—
395C22×C3—28no—
396C22×C3—28no—
397C22×C3—94no—
398C22×C3—182no—
399C22×C3—182no—
400C22×C3—188no—
401C22×C3C2×C3×C4×D318yes—
402C22×C3C2×C3×C4×D3116yescomplemented
403C22×C3C3×#23518yes—
404C22×C3C2×C3×#6718yes—
405C22×C3C2×C4×C3⋊D312yes—
406C22×C3C2×C4×C3⋊D314yescomplemented
407C22×C3#382812yes—
408C22×C3C2×#63812yes—
409C22×C3C23×C3×D314yes—
410C22×C3C23×C3⋊D311yescharacteristic
411C2×D3—316no—
412C2×D3—632no—
413C2×D3—68no—
414C2×D3—68no—
415C3×C4—216no—
416C3×C4—28no—
417C3×C4—22no—
418C3×C4—216no—
419C3×C4—94no—
420C3×C4—182no—
421C3×C4—182no—
422C3×C4C2×C3×#67116yescomplemented
423C3×C4C2×#63814yesdirect product factor
424Dic3—316no—
425Dic3—68no—
426Dic3—68no—
427C32—24no—
428C32C2×C4×#6714yescomplemented
429C32C2×C3×C4×D411yescharacteristic, complemented
430C23—22no—
431C23—22no—
432C23—92no—
433C23—94no—
434C23—182no—
435C23—182no—
436C23C3×C4×C3⋊D314yescomplemented
437C23C22×C3×C3⋊D312yes—
438C23C22×C3×C3⋊D311yescharacteristic
439D4—916no—
440C2×C4—22no—
441C2×C4—28no—
442C2×C4—92no—
443C2×C4—94no—
444C2×C4—92no—
445C2×C4—92no—
446C2×C4—182no—
447C2×C4—182no—
448C2×C4—188no—
449C2×C4C3×#63814yesdirect product factor
450C2×C4C22×C3×C3⋊D312yes—
451C2×C4C22×C3×C3⋊D312yes—
452C2×C3—232no—
453C2×C3—216no—
454C2×C3—24no—
455C2×C3—216no—
456C2×C3—24no—
457C2×C3—24no—
458C2×C3—24no—
459C2×C3—184no—
460C2×C3C3×C4×#67116yescomplemented
461C2×C3C4×#63814yesdirect product factor
462C2×C3C22×C3×C4×D314yes—
463C2×C3C2×C3×#23514yes—
464C2×C3C22×C3×#6714yes—
465C2×C3C22×C4×C3⋊D311yescharacteristic
466C2×C3C2×#382811yescharacteristic
467C2×C3C22×#63811yescharacteristic
468D3—616no—
469C22—28no—
470C22—22no—
471C22—22no—
472C22—94no—
473C22—188no—
474C22—182no—
475C22—182no—
476C22C2×C3×C4×C3⋊D312yes—
477C22C2×C3×C4×C3⋊D314yescomplemented
478C22C3×#382812yes—
479C22C2×C3×#63812yes—
480C22C23×C3×C3⋊D311yescharacteristic
481C4—22no—
482C4—94no—
483C4—182no—
484C4—182no—
485C4C2×C3×#63814yesdirect product factor
486C3—24no—
487C3C2×C3×C4×#6714yescomplemented
488C3C2×C4×#63811yescharacteristic, direct product factor
489C2—24no—
490C2—184no—
491C2C3×C4×#63814yesdirect product factor
492C2C22×C3×C4×C3⋊D311yescharacteristic
493C2C2×C3×#382811yescharacteristic
494C2C22×C3×#63811yescharacteristic
subgroupG/H
centerC22×C3×C4C2×C3⋊D3
commutatorC2×C32C23×C3×C4
FrattiniC22C23×C3×C3⋊D3
FittingC23×C33×C4C2
socleC23×C33C23
prime radicalC22×C3×#638C2
cosocleC22×C32C24×C3
hypercenterC23×C3×C4C3⋊D3
hypocenterC32C2×C3×C4×D4
LCSG ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C3×C4 ◁ C23×C3×C4

Complements

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

NH ≅ G/Ntype
1. C23×C33×C4C2semidirect
3. C22×C3×C4×C3⋊D3C2semidirect
4. C2×C3×#3757C2semidirect
5. C3×C4×#638C2direct (N × H)
6. C2×C3×#3752C2semidirect
7. C2×C3×#3750C2semidirect
8. C2×C3×C4×C3⋊Dic3C2semidirect
9. C2×C4×#638C3direct (N × H)
16. C22×C33×C4C22semidirect
17. C2×C3×#638C4direct (N × H)
23. C2×C3×C4×C3⋊D3C22semidirect
24. C3×#3757C22semidirect
25. C3×#3752C22semidirect
26. C3×#3750C22semidirect
27. C3×C4×C3⋊Dic3C22semidirect
29. C23×C32×C4D3semidirect
30. C23×C32×C4C2×C3semidirect
32. C22×C4×C3⋊D3C2×C3semidirect
35. C2×#3757C2×C3semidirect
36. C4×#638C2×C3direct (N × H)
37. C2×#3752C2×C3semidirect
38. C2×#3750C2×C3semidirect
39. C2×C4×C3⋊Dic3C2×C3semidirect
47. C23×C33C2×C4semidirect
52. C22×C3×C3⋊D3C2×C4semidirect
56. C2×C33×C4D4semidirect
59. C3×#638C2×C4direct (N × H)
62. C2×C3×C3⋊Dic3C2×C4semidirect
82. C22×C32×C4C2×D3semidirect
85. C22×C32×C4C22×C3semidirect
86. C2×#638C3×C4direct (N × H)
92. C2×C4×C3⋊D3C22×C3semidirect
100. #3757C22×C3semidirect
101. #3752C22×C3semidirect
102. #3750C22×C3semidirect
103. C4×C3⋊Dic3C22×C3semidirect
112. C22×C33C22×C4semidirect
119. C2×C3×C3⋊D3C22×C4semidirect
121. C33×C4C2×D4semidirect
124. C3×C3⋊Dic3C22×C4semidirect
128. C23×C3×C4C3×D3semidirect
129. C23×C3×C4C3⋊D3semidirect
151. C23×C32C4×D3semidirect
152. C23×C32C2×C3×C4semidirect
159. C22×C3⋊D3C2×C3×C4semidirect
174. C2×C32×C4#67semidirect
175. C2×C32×C4C3×D4semidirect
180. #638C2×C3×C4direct (N × H)
183. C2×C3⋊Dic3C2×C3×C4semidirect
198. C2×C33C4×D4semidirect
223. C22×C3×C4C2×C3×D3semidirect
226. C22×C3×C4C2×C3⋊D3semidirect
257. C22×C32C2×C4×D3semidirect
261. C22×C32C22×C3×C4semidirect
269. C2×C3⋊D3C22×C3×C4semidirect
278. C32×C4C2×#67semidirect
279. C32×C4C2×C3×D4semidirect
282. C3⋊Dic3C22×C3×C4semidirect
288. C23×C4C3×C3⋊D3semidirect
294. C33C2×C4×D4semidirect
308. C23×C3C3×C4×D3semidirect
309. C23×C3C4×C3⋊D3semidirect
335. C2×C3×C4C3×#67semidirect
336. C2×C3×C4#638direct (N × H)
356. C2×C32C4×#67semidirect
357. C2×C32C3×C4×D4semidirect
378. C22×C4C2×C3×C3⋊D3semidirect
402. C22×C3C2×C3×C4×D3semidirect
406. C22×C3C2×C4×C3⋊D3semidirect
422. C3×C4C2×C3×#67semidirect
423. C3×C4C2×#638direct (N × H)
428. C32C2×C4×#67semidirect
429. C32C2×C3×C4×D4semidirect
436. C23C3×C4×C3⋊D3semidirect
449. C2×C4C3×#638direct (N × H)
460. C2×C3C3×C4×#67semidirect
461. C2×C3C4×#638direct (N × H)
477. C22C2×C3×C4×C3⋊D3semidirect
485. C4C2×C3×#638direct (N × H)
487. C3C2×C3×C4×#67semidirect
488. C3C2×C4×#638direct (N × H)
491. C2C3×C4×#638direct (N × H)

Maximal subgroups

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

subgroupindex
C23×C33×C42
C22×C3×#6382
C22×C3×C4×C3⋊D32
C2×C3×#37572
C3×C4×#6382
C2×C3×#37522
C2×C3×#37502
C2×C3×C4×C3⋊Dic32
C2×C4×#6383
C2×C3×C4×#673