Finite Group Databasebeta

← C2×C3×#63866 #13174397

Subgroups — C2×C3×#63866 #13174397

Order 1728.

1634 conjugacy classes of subgroups.

The subgroup lattice has 444 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×#634C211yesmaximal, characteristic
2C22×C32×Dic6C211yesmaximal, characteristic
3C2×C32×C4×Dic3C211yesmaximal, characteristic
4C2×C3×Dic3⋊Dic3C214yesmaximal
5C3×#63866C218yesmaximal, direct product factor
6C2×C3×#944—31nomaximal
7C2×C3×#869—31nomaximal
8C2×#63866C311yesmaximal, characteristic, direct product factor
9C22×C33×C4C2211yescharacteristic
10C22×C3×C3⋊Dic3C2212yes—
11C2×C3×#634—24no—
12C2×C3×#634C2214yes—
13C22×C32×Dic3C2212yes—
14C22×C32×Dic3C2212yes—
15C2×C32×Dic6—24no—
16C2×C32×Dic6C2214yes—
17C32×C4×Dic3C2214yes—
18C3×Dic3⋊Dic3C22116yes—
19C22×C32×Q8—31no—
20C22×#634C2×C311yescharacteristic
21C22×C3×Dic6—21no—
22C22×C3×Dic6—31no—
23C22×C3×Dic6—31no—
24C22×C3×Dic6—61no—
25C22×C3×Dic6D311yescharacteristic
26C22×C3×Dic6C2×C311yescharacteristic
27C2×C32×C42—31no—
28C3×#944—38no—
29C2×C3×C4⋊Dic3—34no—
30C2×C3×Dic3⋊C4—34no—
31C2×C3×C4×Dic3—21no—
32C2×C3×C4×Dic3D311yescharacteristic
33C2×C3×C4×Dic3C2×C311yescharacteristic
34C3×#869—38no—
35C2×Dic3⋊Dic3C2×C314yes—
36#63866C2×C318yesdirect product factor
37C23×C33C2311yescharacteristic
38C2×C33×C4D414yes—
39C2×C33×C4C2312yes—
40C2×C3×C3⋊Dic3—24no—
41C2×C3×C3⋊Dic3C2314yes—
42C3×#634—28no—
43C2×C32×Dic3—24no—
44C2×C32×Dic3Q818yescomplemented
45C2×C32×Dic3C2314yes—
46C2×C32×Dic3C2314yes—
47C32×Dic6—28no—
48C2×C3×C4⋊Q8—91no—
49C2×#944—31no—
50C2×#869—31no—
51C2×C32×Q8—34no—
52C2×C32×Q8—64no—
53C22×C32×C4—21no—
54C22×C32×C4—21no—
55C22×C32×C4—21no—
56C22×C32×C4—32no—
57C22×C32×C4—32no—
58C22×C32×C4—41no—
59C22×C32×C4C2×D311yescharacteristic
60C22×C32×C4C2×D311yescharacteristic
61C22×C32×C4C22×C311yescharacteristic
62C22×C3⋊Dic3C22×C312yes—
63C2×#634—24no—
64C2×#634C22×C314yes—
65C22×C3×Dic3—22no—
66C22×C3×Dic3—22no—
67C22×C3×Dic3—32no—
68C22×C3×Dic3—32no—
69C22×C3×Dic3—62no—
70C22×C3×Dic3C2×D312yes—
71C22×C3×Dic3C2×D312yes—
72C22×C3×Dic3C22×C312yes—
73C22×C3×Dic3C22×C312yes—
74C2×C3×Dic6—28no—
75C2×C3×Dic6—24no—
76C2×C3×Dic6—24no—
77C2×C3×Dic6—24no—
78C2×C3×Dic6—34no—
79C2×C3×Dic6—34no—
80C2×C3×Dic6—64no—
81C2×C3×Dic6—68no—
82C2×C3×Dic6—64no—
83C2×C3×Dic6—64no—
84C2×C3×Dic6C2×D314yes—
85C2×C3×Dic6C22×C314yes—
86C32×C42—34no—
87C3×C4⋊Dic3—316no—
88C3×Dic3⋊C4—316no—
89C3×C4×Dic3—24no—
90C3×C4×Dic3C2×D314yes—
91C3×C4×Dic3C22×C314yes—
92Dic3⋊Dic3C22×C3116yes—
93C22×C33C2×D412yes—
94C22×C33C2×Q814yes—
95C22×C33C2411yescharacteristic
96C33×C4C2×D414yes—
97C3×C3⋊Dic3—24no—
98C32×Dic3—24no—
99C32×Dic3C2×Q818yescomplemented
100C22×C3×Q8—31no—
101C22×C3×Q8—31no—
102C22×C3×Q8—61no—
103C22×C3×Q8—91no—
104C22×Dic6—31no—
105C22×Dic6—31no—
106C22×Dic6—61no—
107C22×Dic6C3×D311yescharacteristic
108C3×C4⋊Q8—98no—
109C2×C3×C4⋊C4—94no—
110C2×C3×C42—31no—
111C2×C3×C42—31no—
112C2×C3×C42—61no—
113#944—38no—
114C2×C4⋊Dic3—34no—
115C2×Dic3⋊C4—34no—
116C2×C4×Dic3C3×D311yescharacteristic
117#869—38no—
118C23×C32—21no—
119C23×C32—21no—
120C23×C32—21no—
121C23×C32—41no—
122C23×C32C22×D311yescharacteristic
123C23×C32C22×D311yescharacteristic
124C23×C32C23×C311yescharacteristic
125C32×Q8—68no—
126C2×C32×C4—24no—
127C2×C32×C4—24no—
128C2×C32×C4—24no—
129C2×C32×C4—22no—
130C2×C32×C4—22no—
131C2×C32×C4—22no—
132C2×C32×C4—38no—
133C2×C32×C4—34no—
134C2×C32×C4—34no—
135C2×C32×C4—44no—
136C2×C32×C4—42no—
137C2×C32×C4—64no—
138C2×C32×C4D1214yes—
139C2×C32×C4#6714yes—
140C2×C32×C4C3×D414yes—
141C2×C32×C4C22×D312yes—
142C2×C32×C4C22×D312yes—
143C2×C32×C4C23×C312yes—
144C2×C3⋊Dic3—24no—
145C2×C3⋊Dic3C23×C314yes—
146#634—28no—
147C2×C3×Dic3—28no—
148C2×C3×Dic3—24no—
149C2×C3×Dic3—24no—
150C2×C3×Dic3—28no—
151C2×C3×Dic3—24no—
152C2×C3×Dic3—24no—
153C2×C3×Dic3—34no—
154C2×C3×Dic3—34no—
155C2×C3×Dic3—64no—
156C2×C3×Dic3—68no—
157C2×C3×Dic3—64no—
158C2×C3×Dic3—64no—
159C2×C3×Dic3Dic618yescomplemented
160C2×C3×Dic3C3×Q818yescomplemented
161C2×C3×Dic3C22×D314yes—
162C2×C3×Dic3C22×D314yes—
163C2×C3×Dic3C23×C314yes—
164C2×C3×Dic3C23×C314yes—
165C3×Dic6—216no—
166C3×Dic6—28no—
167C3×Dic6—28no—
168C3×Dic6—616no—
169C3×Dic6—68no—
170C3×Dic6—68no—
171C2×C4⋊Q8—91noSylow
172C2×C33C4⋊Q814yescomplemented
173C2×C33C22×D411yescharacteristic
174C2×C33C22×Q812yes—
175C2×C3×Q8—34no—
176C2×C3×Q8—34no—
177C2×C3×Q8—64no—
178C2×C3×Q8—68no—
179C2×C3×Q8—64no—
180C2×C3×Q8—64no—
181C2×C3×Q8—94no—
182C2×C3×Q8—184no—
183C22×C3×C4—21no—
184C22×C3×C4—21no—
185C22×C3×C4—21no—
186C22×C3×C4—32no—
187C22×C3×C4—32no—
188C22×C3×C4—32no—
189C22×C3×C4—32no—
190C22×C3×C4—41no—
191C22×C3×C4—62no—
192C22×C3×C4—62no—
193C22×C3×C4—92no—
194C22×C3×C4D3211yescharacteristic
195C22×C3×C4C2×C3×D311yescharacteristic
196C22×C3×C4C2×C3×D311yescharacteristic
197C22×Dic3—32no—
198C22×Dic3—32no—
199C22×Dic3—62no—
200C22×Dic3C2×C3×D312yes—
201C22×Dic3C2×C3×D312yes—
202C2×Dic6—24no—
203C2×Dic6—34no—
204C2×Dic6—34no—
205C2×Dic6—64no—
206C2×Dic6—68no—
207C2×Dic6—64no—
208C2×Dic6—64no—
209C2×Dic6C2×C3×D314yes—
210C3×C4⋊C4—916no—
211C3×C42—34no—
212C3×C42—34no—
213C3×C42—64no—
214C4⋊Dic3—316no—
215Dic3⋊C4—316no—
216C4×Dic3C2×C3×D314yes—
217C22×C32—21no—
218C22×C32—21no—
219C22×C32—21no—
220C22×C32—24no—
221C22×C32—22no—
222C22×C32—24no—
223C22×C32—22no—
224C22×C32—24no—
225C22×C32—22no—
226C22×C32—42no—
227C22×C32—44no—
228C22×C32—41no—
229C22×C32C2×Dic614yes—
230C22×C32C2×D1212yes—
231C22×C32D3×Q814yes—
232C22×C32C2×#6712yes—
233C22×C32C2×C3×D412yes—
234C22×C32C2×C3×Q814yes—
235C22×C32C23×D311yescharacteristic
236C22×C32C23×D311yescharacteristic
237C22×C32C24×C311yescharacteristic
238C32×C4—24no—
239C32×C4—24no—
240C32×C4—24no—
241C32×C4—38no—
242C32×C4—44no—
243C32×C4—64no—
244C32×C4C2×D1214yes—
245C32×C4C2×#6714yes—
246C32×C4C2×C3×D414yes—
247C3⋊Dic3—24no—
248C3×Dic3—28no—
249C3×Dic3—24no—
250C3×Dic3—24no—
251C3×Dic3—28no—
252C3×Dic3—68no—
253C3×Dic3—64no—
254C3×Dic3—64no—
255C3×Dic3C2×Dic618yescomplemented
256C3×Dic3C2×C3×Q818yescomplemented
257C22×Q8—31no—
258C22×Q8—91no—
259C4⋊Q8—98no—
260C2×C4⋊C4—94no—
261C2×C42—31no—
262C33C2×C4⋊Q811yescharacteristic, complemented, Sylow
263C23×C3—21no—
264C23×C3—21no—
265C23×C3—21no—
266C23×C3—41no—
267C23×C3C2×D3211yescharacteristic
268C23×C3C22×C3×D311yescharacteristic
269C23×C3C22×C3×D311yescharacteristic
270C3×Q8—616no—
271C3×Q8—68no—
272C3×Q8—68no—
273C3×Q8—188no—
274C2×C3×C4—24no—
275C2×C3×C4—24no—
276C2×C3×C4—24no—
277C2×C3×C4—22no—
278C2×C3×C4—22no—
279C2×C3×C4—22no—
280C2×C3×C4—34no—
281C2×C3×C4—34no—
282C2×C3×C4—38no—
283C2×C3×C4—38no—
284C2×C3×C4—34no—
285C2×C3×C4—34no—
286C2×C3×C4—44no—
287C2×C3×C4—42no—
288C2×C3×C4—64no—
289C2×C3×C4—68no—
290C2×C3×C4—64no—
291C2×C3×C4—68no—
292C2×C3×C4—64no—
293C2×C3×C4—64no—
294C2×C3×C4—94no—
295C2×C3×C4—184no—
296C2×C3×C4#62614yes—
297C2×C3×C4C3×D1214yes—
298C2×C3×C4C3×#6714yes—
299C2×C3×C4C2×D3212yes—
300C2×C3×C4C22×C3×D312yes—
301C2×C3×C4C22×C3×D312yes—
302C2×Dic3—24no—
303C2×Dic3—34no—
304C2×Dic3—34no—
305C2×Dic3—64no—
306C2×Dic3—68no—
307C2×Dic3—64no—
308C2×Dic3—64no—
309C2×Dic3C3×Dic618yescomplemented
310C2×Dic3C22×C3×D314yes—
311C2×Dic3C22×C3×D314yes—
312Dic6—28no—
313Dic6—616no—
314Dic6—68no—
315Dic6—68no—
316C2×C32—22no—
317C2×C32—24no—
318C2×C32—21no—
319C2×C32—22no—
320C2×C32—21no—
321C2×C32—22no—
322C2×C32—21no—
323C2×C32—24no—
324C2×C32—24no—
325C2×C32—44no—
326C2×C32—41no—
327C2×C32—42no—
328C2×C32#86914yescomplemented
329C2×C32#94414yescomplemented
330C2×C32C3×C4⋊Q814yescomplemented
331C2×C32C22×Dic612yes—
332C2×C32C22×D1211yescharacteristic
333C2×C32C2×D3×Q812yes—
334C2×C32C22×#6711yescharacteristic
335C2×C32C22×C3×D411yescharacteristic
336C2×C32C22×C3×Q812yes—
337C2×Q8—34no—
338C2×Q8—64no—
339C2×Q8—94no—
340C2×Q8—184no—
341C22×C4—32no—
342C22×C4—32no—
343C22×C4—92no—
344C22×C4C3×D3211yescharacteristic
345C4⋊C4—916no—
346C42—34no—
347C22×C3—21no—
348C22×C3—21no—
349C22×C3—21no—
350C22×C3—24no—
351C22×C3—24no—
352C22×C3—24no—
353C22×C3—22no—
354C22×C3—22no—
355C22×C3—22no—
356C22×C3—41no—
357C22×C3—44no—
358C22×C3—42no—
359C22×C3D3×Dic614yes—
360C22×C3C2×#62612yes—
361C22×C3C2×C3×Dic614yes—
362C22×C3C2×C3×D1212yes—
363C22×C3C3×D3×Q814yes—
364C22×C3C2×C3×#6712yes—
365C22×C3C22×D3211yescharacteristic
366C22×C3C23×C3×D311yescharacteristic
367C22×C3C23×C3×D311yescharacteristic
368C3×C4—24no—
369C3×C4—24no—
370C3×C4—24no—
371C3×C4—38no—
372C3×C4—38no—
373C3×C4—44no—
374C3×C4—68no—
375C3×C4—68no—
376C3×C4—64no—
377C3×C4—64no—
378C3×C4—184no—
379C3×C4C2×#62614yes—
380C3×C4C2×C3×D1214yes—
381C3×C4C2×C3×#6714yes—
382Dic3—24no—
383Dic3—68no—
384Dic3—64no—
385Dic3—64no—
386Dic3C2×C3×Dic618yescomplemented
387C32—21no—
388C32—21no—
389C32—21no—
390C32—41no—
391C32C2×#86911yescharacteristic, complemented
392C32C2×#94411yescharacteristic, complemented
393C32C2×C3×C4⋊Q811yescharacteristic, complemented
394C23C2×C3×D3211yescharacteristic
395Q8—68no—
396Q8—188no—
397C2×C4—34no—
398C2×C4—38no—
399C2×C4—34no—
400C2×C4—64no—
401C2×C4—94no—
402C2×C4—184no—
403C2×C4C3×#62614yes—
404C2×C4C2×C3×D3212yes—
405C2×C3—21no—
406C2×C3—21no—
407C2×C3—21no—
408C2×C3—24no—
409C2×C3—24no—
410C2×C3—24no—
411C2×C3—22no—
412C2×C3—22no—
413C2×C3—22no—
414C2×C3—41no—
415C2×C3—44no—
416C2×C3—42no—
417C2×C3#6386614yesdirect product factor
418C2×C3C3×#86914yescomplemented
419C2×C3C3×#94414yescomplemented
420C2×C3C2×D3×Dic612yes—
421C2×C3C22×#62611yescharacteristic
422C2×C3C22×C3×Dic612yes—
423C2×C3C22×C3×D1211yescharacteristic
424C2×C3C2×C3×D3×Q812yes—
425C2×C3C22×C3×#6711yescharacteristic
426C22C3×D3×Dic614yes—
427C22C2×C3×#62612yes—
428C22C22×C3×D3211yescharacteristic
429C4—38no—
430C4—64no—
431C4—184no—
432C4C2×C3×#62614yes—
433C3—21no—
434C3—21no—
435C3—21no—
436C3—41no—
437C3C2×#6386611yescharacteristic, direct product factor
438C3C2×C3×#86911yescharacteristic, complemented
439C3C2×C3×#94411yescharacteristic, complemented
440C2C3×#6386614yesdirect product factor
441C2C2×C3×D3×Dic612yes—
442C2C22×C3×#62611yescharacteristic
subgroupG/H
centerC23×C3C2×D32
commutator, cosocleC22×C32C24×C3
FrattiniC22C22×C3×D32
FittingC22×C33×C4C22
socle, prime radicalC23×C33C23
hypercenterC22×C3×C4D32
hypocenterC32C2×C3×C4⋊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
5. C3×#63866C2direct (N × H)
8. C2×#63866C3direct (N × H)
36. #63866C2×C3direct (N × H)
44. C2×C32×Dic3Q8semidirect
99. C32×Dic3C2×Q8semidirect
159. C2×C3×Dic3Dic6semidirect
160. C2×C3×Dic3C3×Q8semidirect
172. C2×C33C4⋊Q8semidirect
255. C3×Dic3C2×Dic6semidirect
256. C3×Dic3C2×C3×Q8semidirect
262. C33C2×C4⋊Q8semidirect
309. C2×Dic3C3×Dic6semidirect
328. C2×C32#869semidirect
329. C2×C32#944semidirect
330. C2×C32C3×C4⋊Q8semidirect
386. Dic3C2×C3×Dic6semidirect
391. C32C2×#869semidirect
392. C32C2×#944semidirect
393. C32C2×C3×C4⋊Q8semidirect
417. C2×C3#63866direct (N × H)
418. C2×C3C3×#869semidirect
419. C2×C3C3×#944semidirect
437. C3C2×#63866direct (N × H)
438. C3C2×C3×#869semidirect
439. C3C2×C3×#944semidirect
440. C2C3×#63866direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×#6342
C22×C32×Dic62
C2×C32×C4×Dic32
C2×C3×Dic3⋊Dic32
C3×#638662
C2×C3×#9443
C2×C3×#8693
C2×#638663