Finite Group Databasebeta

← C2×C3×#63865 #13174636

Subgroups — C2×C3×#63865 #13174636

Order 1728.

1762 conjugacy classes of subgroups.

The subgroup lattice has 476 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×C4×C3⋊D3C211yesmaximal, characteristic
2C22×C3×#627C211yesmaximal, characteristic, complemented
3C2×C32×Dic3⋊C4C212yesmaximal, complemented
4C2×C3×#3724C211yesmaximal, characteristic, complemented
5C2×C3×#3722C212yesmaximal
6C3×#63865C218yesmaximal, direct product factor
7C2×C3×#896—32nomaximal
8C2×#63865C311yesmaximal, characteristic, direct product factor
9C23×C3×C3⋊D3C2211yescharacteristic
10C22×C33×C4C2211yescharacteristic
11C22×C3×C3⋊Dic3C2211yescharacteristic
12C2×C3×C4×C3⋊D3—24no—
13C2×C3×C4×C3⋊D3C2214yes—
14C22×C32×Dic3C2212yes—
15C22×C32×Dic3C2212yes—
16C2×C3×#627—24no—
17C2×C3×#627C2214yescomplemented
18C32×Dic3⋊C4C2218yescomplemented
19C3×#3724C2214yescomplemented
20C3×#3722C2218yes—
21C22×C4×C3⋊D3C2×C311yescharacteristic
22C22×C3×C4×D3—32no—
23C22×C3×C4×D3—61no—
24C22×C3×Dic6—32no—
25C22×#627C2×C311yescharacteristic, complemented
26C2×C32×C4⋊C4—32no—
27C2×C3×(C2×D3)⋊C4—32no—
28C2×C3×(C2×D3)⋊C4—32no—
29C2×C3×Dic3⋊C4—22no—
30C2×C3×Dic3⋊C4—32no—
31C2×C3×Dic3⋊C4D312yescomplemented
32C2×C3×Dic3⋊C4C2×C312yescomplemented
33C3×#896—316no—
34C2×#3724C2×C311yescharacteristic, complemented
35C2×#3722C2×C312yes—
36#63865C2×C318yesdirect product factor
37C23×C33C2311yescharacteristic
38C22×C3×C3⋊D3—22no—
39C22×C3×C3⋊D3—22no—
40C22×C3×C3⋊D3Q814yescomplemented
41C22×C3×C3⋊D3C2312yes—
42C2×C33×C4—22no—
43C2×C33×C4C2312yes—
44C2×C3×C3⋊Dic3D414yes—
45C2×C3×C3⋊Dic3C2312yes—
46C3×C4×C3⋊D3—28no—
47C2×C32×Dic3—24no—
48C2×C32×Dic3—24no—
49C2×C32×Dic3C2314yes—
50C2×C32×Dic3C2314yes—
51C3×#627—28no—
52C2×C3×C22⋊Q8—91no—
53C2×#896—32no—
54C23×C3⋊D3C22×C311yescharacteristic
55C23×C3×D3—32no—
56C23×C3×D3—61no—
57C22×C32×C4—21no—
58C22×C32×C4—22no—
59C22×C32×C4—32no—
60C22×C32×C4—32no—
61C22×C32×C4—41no—
62C22×C32×C4C2×D312yes—
63C22×C32×C4C22×C311yescharacteristic
64C22×C3⋊Dic3C22×C311yescharacteristic
65C2×C4×C3⋊D3—24no—
66C2×C4×C3⋊D3C22×C314yes—
67C22×C3×Dic3—22no—
68C22×C3×Dic3—22no—
69C22×C3×Dic3—32no—
70C22×C3×Dic3—61no—
71C22×C3×Dic3C2×D312yes—
72C22×C3×Dic3C2×D312yes—
73C22×C3×Dic3C22×C312yes—
74C22×C3×Dic3C22×C312yes—
75C2×C3×C4×D3—38no—
76C2×C3×C4×D3—68no—
77C2×C3×C4×D3—64no—
78C2×C3×C4×D3—68no—
79C2×C3×Dic6—38no—
80C2×C3×Dic6—68no—
81C2×#627—24no—
82C2×#627C22×C314yescomplemented
83C32×C4⋊C4—38no—
84C3×(C2×D3)⋊C4—38no—
85C3×(C2×D3)⋊C4—38no—
86C3×Dic3⋊C4—28no—
87C3×Dic3⋊C4—38no—
88C3×Dic3⋊C4C2×D318yescomplemented
89C3×Dic3⋊C4C22×C318yescomplemented
90#3724C22×C314yescomplemented
91#3722C22×C318yes—
92C22×C33C2×D412yes—
93C22×C33C2×Q812yes—
94C22×C33#4112yes—
95C22×C33C2411yescharacteristic
96C2×C3×C3⋊D3—28no—
97C2×C3×C3⋊D3—22no—
98C2×C3×C3⋊D3—22no—
99C2×C3×C3⋊D3C2×Q814yescomplemented
100C33×C4—22no—
101C3×C3⋊Dic3C2×D414yes—
102C32×Dic3—24no—
103C32×Dic3—24no—
104C22×C3×Q8—91no—
105C23×C3×C4—91no—
106C22×C4×D3—32no—
107C22×C4×D3—61no—
108C22×Dic6—32no—
109C3×C22⋊Q8—98no—
110C2×C3×C4⋊C4—32no—
111C2×C3×C4⋊C4—32no—
112C2×C3×C4⋊C4—62no—
113C2×C3×C4⋊C4—91no—
114C2×C3×C22⋊C4—92no—
115C2×(C2×D3)⋊C4—32no—
116C2×(C2×D3)⋊C4—32no—
117C2×Dic3⋊C4—32no—
118C2×Dic3⋊C4C3×D312yescomplemented
119#896—316no—
120C23×C32—21no—
121C23×C32—22no—
122C23×C32—41no—
123C23×C32C22×D312yes—
124C23×C32C23×C311yescharacteristic
125C22×C3⋊D3—22no—
126C22×C3⋊D3—22no—
127C22×C3⋊D3C3×Q814yescomplemented
128C22×C3⋊D3C23×C312yes—
129C22×C3×D3—38no—
130C22×C3×D3—34no—
131C22×C3×D3—64no—
132C22×C3×D3—62no—
133C22×C3×D3—64no—
134C22×C3×D3—64no—
135C22×C3×D3—64no—
136C22×C3×D3—64no—
137C2×C32×C4—22no—
138C2×C32×C4—24no—
139C2×C32×C4—24no—
140C2×C32×C4—22no—
141C2×C32×C4—24no—
142C2×C32×C4—34no—
143C2×C32×C4—34no—
144C2×C32×C4—42no—
145C2×C32×C4—44no—
146C2×C32×C4—44no—
147C2×C32×C4—64no—
148C2×C32×C4—64no—
149C2×C32×C4C22×D314yes—
150C2×C32×C4C23×C312yes—
151C2×C3⋊Dic3C3×D414yes—
152C2×C3⋊Dic3C23×C312yes—
153C4×C3⋊D3—28no—
154C2×C3×Dic3—28no—
155C2×C3×Dic3—24no—
156C2×C3×Dic3—24no—
157C2×C3×Dic3—24no—
158C2×C3×Dic3—24no—
159C2×C3×Dic3—24no—
160C2×C3×Dic3—24no—
161C2×C3×Dic3—38no—
162C2×C3×Dic3—34no—
163C2×C3×Dic3—44no—
164C2×C3×Dic3—64no—
165C2×C3×Dic3—62no—
166C2×C3×Dic3C22×D314yes—
167C2×C3×Dic3C22×D314yes—
168C2×C3×Dic3C23×C314yes—
169C2×C3×Dic3C23×C314yes—
170C3×C4×D3—616no—
171C3×C4×D3—616no—
172C3×Dic6—616no—
173#627—28no—
174C2×C22⋊Q8—91noSylow
175C2×C33C22⋊Q814yescomplemented
176C2×C33C22×D411yescharacteristic
177C2×C33C22×Q811yescharacteristic
178C2×C33C2×#4111yescharacteristic
179C3×C3⋊D3—24no—
180C24×C3—91no—
181C23×D3—32no—
182C23×D3—61no—
183C2×C3×Q8—94no—
184C2×C3×Q8—184no—
185C22×C3×C4—21no—
186C22×C3×C4—22no—
187C22×C3×C4—32no—
188C22×C3×C4—32no—
189C22×C3×C4—32no—
190C22×C3×C4—32no—
191C22×C3×C4—41no—
192C22×C3×C4—62no—
193C22×C3×C4—62no—
194C22×C3×C4—91no—
195C22×C3×C4—94no—
196C22×C3×C4—184no—
197C22×C3×C4D3211yescharacteristic
198C22×C3×C4C2×C3×D312yes—
199C22×Dic3—32no—
200C22×Dic3—61no—
201C22×Dic3C2×C3×D312yes—
202C22×Dic3C2×C3×D312yes—
203C2×C4×D3—38no—
204C2×C4×D3—64no—
205C2×C4×D3—68no—
206C2×C4×D3—68no—
207C2×Dic6—38no—
208C2×Dic6—68no—
209C3×C4⋊C4—38no—
210C3×C4⋊C4—38no—
211C3×C4⋊C4—68no—
212C3×C4⋊C4—94no—
213C3×C22⋊C4—98no—
214(C2×D3)⋊C4—38no—
215(C2×D3)⋊C4—38no—
216Dic3⋊C4—38no—
217Dic3⋊C4C2×C3×D318yescomplemented
218C22×C32—21no—
219C22×C32—22no—
220C22×C32—22no—
221C22×C32—22no—
222C22×C32—22no—
223C22×C32—24no—
224C22×C32—24no—
225C22×C32—24no—
226C22×C32—42no—
227C22×C32—42no—
228C22×C32—42no—
229C22×C32—41no—
230C22×C32#23514yes—
231C22×C32D3×D414yes—
232C22×C32D3×Q814yes—
233C22×C32C2×C3×D412yes—
234C22×C32C2×C3×Q812yes—
235C22×C32C3×#4112yes—
236C22×C32C23×D312yes—
237C22×C32C24×C311yescharacteristic
238C2×C3⋊D3—28no—
239C2×C3⋊D3—22no—
240C2×C3⋊D3—22no—
241C2×C3⋊D3C2×C3×Q814yescomplemented
242C2×C3×D3—38no—
243C2×C3×D3—64no—
244C2×C3×D3—64no—
245C2×C3×D3—64no—
246C2×C3×D3—616no—
247C2×C3×D3—616no—
248C2×C3×D3—64no—
249C2×C3×D3—64no—
250C32×C4—22no—
251C32×C4—24no—
252C32×C4—24no—
253C32×C4—44no—
254C32×C4—44no—
255C32×C4—64no—
256C32×C4—64no—
257C3⋊Dic3C2×C3×D414yes—
258C3×Dic3—28no—
259C3×Dic3—24no—
260C3×Dic3—24no—
261C3×Dic3—24no—
262C3×Dic3—24no—
263C3×Dic3—38no—
264C3×Dic3—44no—
265C3×Dic3—64no—
266C22×Q8—91no—
267C23×C4—91no—
268C22⋊Q8—98no—
269C2×C4⋊C4—32no—
270C2×C4⋊C4—91no—
271C2×C22⋊C4—92no—
272C33C2×C22⋊Q811yescharacteristic, complemented, Sylow
273C23×C3—22no—
274C23×C3—21no—
275C23×C3—41no—
276C23×C3—92no—
277C23×C3—94no—
278C23×C3—182no—
279C23×C3—182no—
280C23×C3C2×D3211yescharacteristic
281C23×C3C22×C3×D312yes—
282C22×D3—34no—
283C22×D3—38no—
284C22×D3—64no—
285C22×D3—62no—
286C22×D3—64no—
287C22×D3—64no—
288C22×D3—64no—
289C22×D3—64no—
290C3×Q8—188no—
291C2×C3×C4—24no—
292C2×C3×C4—22no—
293C2×C3×C4—24no—
294C2×C3×C4—22no—
295C2×C3×C4—24no—
296C2×C3×C4—34no—
297C2×C3×C4—34no—
298C2×C3×C4—34no—
299C2×C3×C4—34no—
300C2×C3×C4—44no—
301C2×C3×C4—44no—
302C2×C3×C4—42no—
303C2×C3×C4—64no—
304C2×C3×C4—64no—
305C2×C3×C4—68no—
306C2×C3×C4—64no—
307C2×C3×C4—64no—
308C2×C3×C4—64no—
309C2×C3×C4—64no—
310C2×C3×C4—92no—
311C2×C3×C4—94no—
312C2×C3×C4—124no—
313C2×C3×C4—188no—
314C2×C3×C4C2×D3212yes—
315C2×C3×C4C22×C3×D314yes—
316C2×Dic3—24no—
317C2×Dic3—24no—
318C2×Dic3—34no—
319C2×Dic3—38no—
320C2×Dic3—64no—
321C2×Dic3—62no—
322C2×Dic3C22×C3×D314yes—
323C2×Dic3C22×C3×D314yes—
324C4×D3—616no—
325C4×D3—616no—
326Dic6—616no—
327C2×C32—28no—
328C2×C32—21no—
329C2×C32—21no—
330C2×C32—21no—
331C2×C32—22no—
332C2×C32—22no—
333C2×C32—22no—
334C2×C32—24no—
335C2×C32—44no—
336C2×C32—41no—
337C2×C32—41no—
338C2×C32—41no—
339C2×C32#89618yescomplemented
340C2×C32C3×C22⋊Q814yescomplemented
341C2×C32C2×#23512yes—
342C2×C32C2×D3×D412yes—
343C2×C32C2×D3×Q812yes—
344C2×C32C22×C3×D411yescharacteristic
345C2×C32C22×C3×Q811yescharacteristic
346C2×C32C2×C3×#4111yescharacteristic
347C3⋊D3—24no—
348C3×D3—68no—
349C3×D3—68no—
350C24—91no—
351C2×Q8—94no—
352C2×Q8—184no—
353C22×C4—32no—
354C22×C4—32no—
355C22×C4—91no—
356C22×C4—94no—
357C22×C4—184no—
358C22×C4C3×D3211yescharacteristic
359C4⋊C4—38no—
360C4⋊C4—94no—
361C22⋊C4—98no—
362C22×C3—21no—
363C22×C3—22no—
364C22×C3—22no—
365C22×C3—24no—
366C22×C3—24no—
367C22×C3—22no—
368C22×C3—22no—
369C22×C3—24no—
370C22×C3—41no—
371C22×C3—42no—
372C22×C3—42no—
373C22×C3—42no—
374C22×C3—94no—
375C22×C3—182no—
376C22×C3—182no—
377C22×C3—188no—
378C22×C3#379712yes—
379C22×C3#379812yes—
380C22×C3#381112yes—
381C22×C3C3×#23514yes—
382C22×C3C3×D3×D414yes—
383C22×C3C3×D3×Q814yes—
384C22×C3C22×D3211yescharacteristic
385C22×C3C23×C3×D312yes—
386C2×D3—38no—
387C2×D3—64no—
388C2×D3—64no—
389C2×D3—616no—
390C2×D3—64no—
391C2×D3—616no—
392C2×D3—64no—
393C2×D3—64no—
394C3×C4—24no—
395C3×C4—22no—
396C3×C4—24no—
397C3×C4—44no—
398C3×C4—44no—
399C3×C4—68no—
400C3×C4—64no—
401C3×C4—64no—
402C3×C4—64no—
403C3×C4—64no—
404C3×C4—94no—
405C3×C4—124no—
406Dic3—24no—
407Dic3—24no—
408Dic3—38no—
409Dic3—64no—
410C32—22no—
411C32—21no—
412C32—41no—
413C32C2×#89612yescomplemented
414C32C2×C3×C22⋊Q811yescharacteristic, complemented
415C23—92no—
416C23—94no—
417C23—182no—
418C23—182no—
419C23C2×C3×D3211yescharacteristic
420Q8—188no—
421C2×C4—22no—
422C2×C4—34no—
423C2×C4—34no—
424C2×C4—64no—
425C2×C4—64no—
426C2×C4—92no—
427C2×C4—94no—
428C2×C4—188no—
429C2×C4C2×C3×D3212yes—
430C2×C3—21no—
431C2×C3—22no—
432C2×C3—24no—
433C2×C3—28no—
434C2×C3—21no—
435C2×C3—22no—
436C2×C3—22no—
437C2×C3—21no—
438C2×C3—41no—
439C2×C3—44no—
440C2×C3—41no—
441C2×C3—41no—
442C2×C3—184no—
443C2×C3#6386514yesdirect product factor
444C2×C3C3×#89618yescomplemented
445C2×C3C2×#379711yescharacteristic
446C2×C3C2×#379811yescharacteristic
447C2×C3C2×#381111yescharacteristic
448C2×C3C2×C3×#23512yes—
449C2×C3C2×C3×D3×D412yes—
450C2×C3C2×C3×D3×Q812yes—
451D3—68no—
452D3—68no—
453C22—94no—
454C22—182no—
455C22—188no—
456C22—182no—
457C22C3×#379712yes—
458C22C3×#379812yes—
459C22C3×#381112yes—
460C22C22×C3×D3211yescharacteristic
461C4—22no—
462C4—64no—
463C4—64no—
464C4—94no—
465C3—21no—
466C3—22no—
467C3—41no—
468C3C2×#6386511yescharacteristic, direct product factor
469C3C2×C3×#89612yescomplemented
470C2—184no—
471C2C3×#6386514yesdirect product factor
472C2C2×C3×#379711yescharacteristic
473C2C2×C3×#379811yescharacteristic
474C2C2×C3×#381111yescharacteristic
subgroupG/H
centerC23×C3C2×D32
commutator, cosocleC22×C32C24×C3
FrattiniC22C22×C3×D32
FittingC22×C33×C4C22
socleC23×C33C23
prime radicalC23×C3×C3⋊D3C22
hypercenterC22×C3×C4D32
hypocenterC32C2×C3×C22⋊Q8
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×C3×#627C2semidirect
3. C2×C32×Dic3⋊C4C2semidirect
4. C2×C3×#3724C2semidirect
6. C3×#63865C2direct (N × H)
8. C2×#63865C3direct (N × H)
17. C2×C3×#627C22semidirect
18. C32×Dic3⋊C4C22semidirect
19. C3×#3724C22semidirect
25. C22×#627C2×C3semidirect
31. C2×C3×Dic3⋊C4D3semidirect
32. C2×C3×Dic3⋊C4C2×C3semidirect
34. C2×#3724C2×C3semidirect
36. #63865C2×C3direct (N × H)
40. C22×C3×C3⋊D3Q8semidirect
82. C2×#627C22×C3semidirect
88. C3×Dic3⋊C4C2×D3semidirect
89. C3×Dic3⋊C4C22×C3semidirect
90. #3724C22×C3semidirect
99. C2×C3×C3⋊D3C2×Q8semidirect
118. C2×Dic3⋊C4C3×D3semidirect
127. C22×C3⋊D3C3×Q8semidirect
175. C2×C33C22⋊Q8semidirect
217. Dic3⋊C4C2×C3×D3semidirect
241. C2×C3⋊D3C2×C3×Q8semidirect
272. C33C2×C22⋊Q8semidirect
339. C2×C32#896semidirect
340. C2×C32C3×C22⋊Q8semidirect
413. C32C2×#896semidirect
414. C32C2×C3×C22⋊Q8semidirect
443. C2×C3#63865direct (N × H)
444. C2×C3C3×#896semidirect
468. C3C2×#63865direct (N × H)
469. C3C2×C3×#896semidirect
471. C2C3×#63865direct (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⋊D32
C22×C3×#6272
C2×C32×Dic3⋊C42
C2×C3×#37242
C2×C3×#37222
C3×#638652
C2×C3×#8963
C2×#638653