Finite Group Databasebeta

← C3×#10594818 #13174371

Subgroups — C3×#10594818 #13174371

Order 1728.

1426 conjugacy classes of subgroups.

The subgroup lattice has 492 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C32×Dic6C211yesmaximal, characteristic
2C3×#64135C211yesmaximal, characteristic
3C32×#926C211yesmaximal, characteristic
4C3×#63955C214yesmaximal
5C3×#63843C214yesmaximal, complemented
6C3×Dic3×Dic6C214yesmaximal, complemented
7C3×#5812—31nomaximal
8C3×#5516—31nomaximal
9#10594818C311yesmaximal, characteristic, direct product factor
10C22×C33×C4C2211yescharacteristic
11C22×C32×Dic3C2212yes—
12C2×C32×Dic6C418yescomplemented
13C2×C32×Dic6C2212yes—
14C2×C32×Dic6C2212yes—
15C3×#3757C2212yes—
16C3×C4⋊(C3⋊Dic3)C2212yes—
17C3×C4×C3⋊Dic3C2212yes—
18C32×#217C2212yes—
19C32×C4⋊Dic3C2212yes—
20C32×C4×Dic3C2212yes—
21C3×Dic3⋊Dic3C2218yes—
22C3×Dic32C2218yes—
23C22×C32×Q8—31no—
24C22×C3×Dic6—21no—
25C22×C3×Dic6D311yescharacteristic
26C22×C3×Dic6C2×C311yescharacteristic
27C32×#117—31no—
28#64135C2×C311yescharacteristic
29C3×Q8×Dic3—34no—
30C3×Q8×Dic3—34no—
31C3×#926—21no—
32C3×#926—31no—
33C3×#926—31no—
34C3×#926—34no—
35C3×#926—61no—
36C3×#926D311yescharacteristic
37C3×#926C2×C311yescharacteristic
38C3×#887—34no—
39C3×#877—34no—
40C3×C4×Dic6—34no—
41#63955C2×C314yes—
42#63843C2×C314yescomplemented
43Dic3×Dic6C2×C314yescomplemented
44C23×C33C2311yescharacteristic
45C2×C33×C4C2×C414yes—
46C2×C33×C4C2311yescharacteristic
47C2×C33×C4C2311yescharacteristic
48C2×C3×C3⋊Dic3C2314yes—
49C2×C32×Dic3C2×C418yes—
50C2×C32×Dic3C2314yes—
51C2×C32×Dic3C2314yes—
52C32×Dic6C2×C418yes—
53C32×Dic6C2×C418yes—
54C3×#519—91no—
55#5812—31no—
56#5516—31no—
57C2×C32×Q8—38no—
58C2×C32×Q8—32no—
59C2×C32×Q8—32no—
60C22×C32×C4—21no—
61C22×C32×C4—21no—
62C22×C32×C4—21no—
63C22×C32×C4—32no—
64C22×C32×C4—41no—
65C22×C32×C4C2×D311yescharacteristic
66C22×C32×C4C2×D311yescharacteristic
67C22×C32×C4C22×C311yescharacteristic
68C22×C3×Dic3—22no—
69C22×C3×Dic3C2×D312yes—
70C22×C3×Dic3C22×C312yes—
71C2×C3×Dic6—28no—
72C2×C3×Dic6—22no—
73C2×C3×Dic6—22no—
74C2×C3×Dic6Dic318yescomplemented
75C2×C3×Dic6C3×C418yescomplemented
76C2×C3×Dic6C2×D312yes—
77C2×C3×Dic6C2×D312yes—
78C2×C3×Dic6C22×C312yes—
79C2×C3×Dic6C22×C312yes—
80C32×C4⋊C4—32no—
81C32×C22⋊C4—32no—
82C32×C42—32no—
83#3757C22×C312yes—
84C4⋊(C3⋊Dic3)C22×C312yes—
85C4×C3⋊Dic3C22×C312yes—
86C3×#217—22no—
87C3×#217—32no—
88C3×#217—32no—
89C3×#217—62no—
90C3×#217C2×D312yes—
91C3×#217C22×C312yes—
92C3×C4⋊Dic3—22no—
93C3×C4⋊Dic3—32no—
94C3×C4⋊Dic3—32no—
95C3×C4⋊Dic3—38no—
96C3×C4⋊Dic3—62no—
97C3×C4⋊Dic3C2×D312yes—
98C3×C4⋊Dic3C22×C312yes—
99C3×Dic3⋊C4—38no—
100C3×C4×Dic3—22no—
101C3×C4×Dic3—32no—
102C3×C4×Dic3—32no—
103C3×C4×Dic3—38no—
104C3×C4×Dic3—38no—
105C3×C4×Dic3—62no—
106C3×C4×Dic3C2×D312yes—
107C3×C4×Dic3C22×C312yes—
108Dic3⋊Dic3C22×C318yes—
109Dic32C22×C318yes—
110C22×C33—22no—
111C22×C33C22×C412yes—
112C22×C33C2411yescharacteristic
113C33×C4C22×C412yes—
114C33×C4C22×C412yes—
115C3×C3⋊Dic3—24no—
116C32×Dic3—24no—
117C32×Dic3C22×C418yes—
118C22×C3×Q8—31no—
119C22×C3×Q8—31no—
120C22×C3×Q8—61no—
121C22×Dic6C3×D311yescharacteristic
122C3×C4×Q8—94no—
123C3×C4×Q8—94no—
124C3×#117—31no—
125C3×#117—31no—
126C3×#117—61no—
127C3×#117—94no—
128C3×#117—91no—
129Q8×Dic3—34no—
130Q8×Dic3—34no—
131#926—31no—
132#926—31no—
133#926—34no—
134#926—61no—
135#926C3×D311yescharacteristic
136#887—34no—
137#877—34no—
138C4×Dic6—34no—
139C23×C32—21no—
140C23×C32—21no—
141C23×C32—21no—
142C23×C32—41no—
143C23×C32C22×D311yescharacteristic
144C23×C32C22×D311yescharacteristic
145C23×C32C23×C311yescharacteristic
146C32×Q8—38no—
147C32×Q8—38no—
148C2×C32×C4—24no—
149C2×C32×C4—24no—
150C2×C32×C4—24no—
151C2×C32×C4—21no—
152C2×C32×C4—21no—
153C2×C32×C4—21no—
154C2×C32×C4—21no—
155C2×C32×C4—21no—
156C2×C32×C4—21no—
157C2×C32×C4—38no—
158C2×C32×C4—34no—
159C2×C32×C4—34no—
160C2×C32×C4—44no—
161C2×C32×C4—41no—
162C2×C32×C4—41no—
163C2×C32×C4C4×D314yes—
164C2×C32×C4C2×Dic314yes—
165C2×C32×C4C2×C3×C414yes—
166C2×C32×C4C22×D311yescharacteristic
167C2×C32×C4C22×D311yescharacteristic
168C2×C32×C4C22×D311yescharacteristic
169C2×C32×C4C22×D311yescharacteristic
170C2×C32×C4C23×C311yescharacteristic
171C2×C32×C4C23×C311yescharacteristic
172C2×C3⋊Dic3C23×C314yes—
173C2×C3×Dic3—28no—
174C2×C3×Dic3—24no—
175C2×C3×Dic3—24no—
176C2×C3×Dic3—34no—
177C2×C3×Dic3—34no—
178C2×C3×Dic3—64no—
179C2×C3×Dic3C2×Dic318yes—
180C2×C3×Dic3C2×C3×C418yes—
181C2×C3×Dic3C22×D314yes—
182C2×C3×Dic3C22×D314yes—
183C2×C3×Dic3C23×C314yes—
184C2×C3×Dic3C23×C314yes—
185C3×Dic6—28no—
186C3×Dic6—28no—
187C3×Dic6C2×Dic318yes—
188C3×Dic6C2×Dic318yes—
189C3×Dic6C2×C3×C418yes—
190C3×Dic6C2×C3×C418yes—
191#519—91noSylow
192C2×C33—22no—
193C2×C33C23×C411yescharacteristic
194C2×C332−1+412yes—
195C2×C3×Q8—38no—
196C2×C3×Q8—38no—
197C2×C3×Q8—32no—
198C2×C3×Q8—32no—
199C2×C3×Q8—32no—
200C2×C3×Q8—32no—
201C2×C3×Q8—68no—
202C2×C3×Q8—62no—
203C2×C3×Q8—62no—
204C22×C3×C4—21no—
205C22×C3×C4—21no—
206C22×C3×C4—21no—
207C22×C3×C4—32no—
208C22×C3×C4—32no—
209C22×C3×C4—41no—
210C22×C3×C4—62no—
211C22×C3×C4D3211yescharacteristic
212C22×C3×C4C2×C3×D311yescharacteristic
213C22×C3×C4C2×C3×D311yescharacteristic
214C22×Dic3C2×C3×D312yes—
215C2×Dic6C3×Dic318yescomplemented
216C2×Dic6C2×C3×D312yes—
217C2×Dic6C2×C3×D312yes—
218C3×C4⋊C4—32no—
219C3×C4⋊C4—32no—
220C3×C4⋊C4—62no—
221C3×C4⋊C4—98no—
222C3×C4⋊C4—92no—
223C3×C22⋊C4—32no—
224C3×C22⋊C4—32no—
225C3×C22⋊C4—62no—
226C3×C22⋊C4—92no—
227C3×C42—32no—
228C3×C42—32no—
229C3×C42—62no—
230C3×C42—98no—
231C3×C42—92no—
232#217—32no—
233#217—32no—
234#217—62no—
235#217C2×C3×D312yes—
236C4⋊Dic3—32no—
237C4⋊Dic3—32no—
238C4⋊Dic3—38no—
239C4⋊Dic3—62no—
240C4⋊Dic3C2×C3×D312yes—
241Dic3⋊C4—38no—
242C4×Dic3—32no—
243C4×Dic3—32no—
244C4×Dic3—38no—
245C4×Dic3—38no—
246C4×Dic3—62no—
247C4×Dic3C2×C3×D312yes—
248C22×C32—22no—
249C22×C32—24no—
250C22×C32—22no—
251C22×C32—22no—
252C22×C32—21no—
253C22×C32—21no—
254C22×C32—21no—
255C22×C32—22no—
256C22×C32—22no—
257C22×C32—22no—
258C22×C32—42no—
259C22×C32—41no—
260C22×C32—42no—
261C22×C32—42no—
262C22×C32—44no—
263C22×C32C2×C4×D312yes—
264C22×C32C22×Dic312yes—
265C22×C32C22×C3×C412yes—
266C22×C32C23×D311yescharacteristic
267C22×C32C23×D311yescharacteristic
268C22×C32C24×C311yescharacteristic
269C32×C4—22no—
270C32×C4—22no—
271C32×C4—22no—
272C32×C4—22no—
273C32×C4—22no—
274C32×C4—22no—
275C32×C4—38no—
276C32×C4—42no—
277C32×C4—42no—
278C32×C4—64no—
279C32×C4C2×C4×D312yes—
280C32×C4C2×C4×D312yes—
281C32×C4C22×Dic312yes—
282C32×C4C22×Dic312yes—
283C32×C4C22×C3×C412yes—
284C32×C4C22×C3×C412yes—
285C3⋊Dic3—24no—
286C3×Dic3—24no—
287C3×Dic3—24no—
288C3×Dic3—28no—
289C3×Dic3—44no—
290C3×Dic3—64no—
291C3×Dic3—64no—
292C3×Dic3—124no—
293C3×Dic3C22×Dic318yes—
294C3×Dic3C22×C3×C418yes—
295C22×Q8—31no—
296C4×Q8—94no—
297C4×Q8—94no—
298#117—31no—
299#117—94no—
300#117—91no—
301C33#51911yescharacteristic, complemented, Sylow
302C23×C3—21no—
303C23×C3—21no—
304C23×C3—21no—
305C23×C3—41no—
306C23×C3C2×D3211yescharacteristic
307C23×C3C22×C3×D311yescharacteristic
308C23×C3C22×C3×D311yescharacteristic
309C3×Q8—38no—
310C3×Q8—38no—
311C3×Q8—38no—
312C3×Q8—38no—
313C3×Q8—68no—
314C3×Q8—68no—
315C2×C3×C4—24no—
316C2×C3×C4—24no—
317C2×C3×C4—24no—
318C2×C3×C4—21no—
319C2×C3×C4—21no—
320C2×C3×C4—21no—
321C2×C3×C4—21no—
322C2×C3×C4—21no—
323C2×C3×C4—21no—
324C2×C3×C4—34no—
325C2×C3×C4—34no—
326C2×C3×C4—38no—
327C2×C3×C4—38no—
328C2×C3×C4—34no—
329C2×C3×C4—34no—
330C2×C3×C4—44no—
331C2×C3×C4—41no—
332C2×C3×C4—41no—
333C2×C3×C4—64no—
334C2×C3×C4—68no—
335C2×C3×C4—64no—
336C2×C3×C4—94no—
337C2×C3×C4D3×Dic314yes—
338C2×C3×C4C3×C4×D314yes—
339C2×C3×C4C2×C3×Dic314yes—
340C2×C3×C4C2×D3211yescharacteristic
341C2×C3×C4C2×D3211yescharacteristic
342C2×C3×C4C22×C3×D311yescharacteristic
343C2×C3×C4C22×C3×D311yescharacteristic
344C2×C3×C4C22×C3×D311yescharacteristic
345C2×C3×C4C22×C3×D311yescharacteristic
346C2×Dic3—34no—
347C2×Dic3—34no—
348C2×Dic3—64no—
349C2×Dic3C2×C3×Dic318yes—
350C2×Dic3C22×C3×D314yes—
351C2×Dic3C22×C3×D314yes—
352Dic6C2×C3×Dic318yes—
353Dic6C2×C3×Dic318yes—
354C2×C32—22no—
355C2×C32—24no—
356C2×C32—22no—
357C2×C32—22no—
358C2×C32—22no—
359C2×C32—21no—
360C2×C32—21no—
361C2×C32—22no—
362C2×C32—22no—
363C2×C32—21no—
364C2×C32—41no—
365C2×C32—42no—
366C2×C32—42no—
367C2×C32—42no—
368C2×C32—44no—
369C2×C32C22×C4×D311yescharacteristic
370C2×C32#100712yes—
371C2×C32#101012yes—
372C2×C32C23×Dic311yescharacteristic
373C2×C32C23×C3×C411yescharacteristic
374C2×C32C3×2−1+412yes—
375C2×Q8—38no—
376C2×Q8—32no—
377C2×Q8—32no—
378C22×C4—32no—
379C22×C4C3×D3211yescharacteristic
380C4⋊C4—32no—
381C4⋊C4—98no—
382C4⋊C4—92no—
383C22⋊C4—32no—
384C22⋊C4—92no—
385C42—32no—
386C42—98no—
387C42—92no—
388C22×C3—24no—
389C22×C3—22no—
390C22×C3—22no—
391C22×C3—22no—
392C22×C3—21no—
393C22×C3—21no—
394C22×C3—21no—
395C22×C3—22no—
396C22×C3—22no—
397C22×C3—22no—
398C22×C3—42no—
399C22×C3—42no—
400C22×C3—44no—
401C22×C3—41no—
402C22×C3—42no—
403C22×C3C2×D3×Dic312yes—
404C22×C3C2×C3×C4×D312yes—
405C22×C3C22×C3×Dic312yes—
406C22×C3C22×D3211yescharacteristic
407C22×C3C23×C3×D311yescharacteristic
408C22×C3C23×C3×D311yescharacteristic
409C3×C4—22no—
410C3×C4—22no—
411C3×C4—22no—
412C3×C4—22no—
413C3×C4—22no—
414C3×C4—22no—
415C3×C4—38no—
416C3×C4—38no—
417C3×C4—42no—
418C3×C4—42no—
419C3×C4—68no—
420C3×C4—64no—
421C3×C4—64no—
422C3×C4—124no—
423C3×C4—184no—
424C3×C4C2×D3×Dic312yes—
425C3×C4C2×D3×Dic312yes—
426C3×C4C2×C3×C4×D312yes—
427C3×C4C2×C3×C4×D312yes—
428C3×C4C22×C3×Dic312yes—
429C3×C4C22×C3×Dic312yes—
430Dic3—24no—
431Dic3—64no—
432Dic3—64no—
433Dic3—124no—
434Dic3C22×C3×Dic318yes—
435C32—21no—
436C32—21no—
437C32—21no—
438C32—41no—
439C32#551611yescharacteristic, complemented
440C32#581211yescharacteristic, complemented
441C32C3×#51911yescharacteristic, complemented
442C23C2×C3×D3211yescharacteristic
443Q8—38no—
444Q8—38no—
445C2×C4—34no—
446C2×C4—38no—
447C2×C4—34no—
448C2×C4—94no—
449C2×C4C3×D3×Dic314yes—
450C2×C4C2×C3×D3211yescharacteristic
451C2×C4C2×C3×D3211yescharacteristic
452C2×C3—24no—
453C2×C3—22no—
454C2×C3—22no—
455C2×C3—22no—
456C2×C3—21no—
457C2×C3—21no—
458C2×C3—21no—
459C2×C3—22no—
460C2×C3—22no—
461C2×C3—22no—
462C2×C3—42no—
463C2×C3—42no—
464C2×C3—44no—
465C2×C3—41no—
466C2×C3—42no—
467C2×C3#6429712yes—
468C2×C3C22×D3×Dic311yescharacteristic
469C2×C3C22×C3×C4×D311yescharacteristic
470C2×C3C3×#100712yes—
471C2×C3C3×#101012yes—
472C2×C3C23×C3×Dic311yescharacteristic
473C22—22no—
474C22C2×C3×D3×Dic312yes—
475C22C22×C3×D3211yescharacteristic
476C4—38no—
477C4—64no—
478C4—184no—
479C4C2×C3×D3×Dic312yes—
480C4C2×C3×D3×Dic312yes—
481C3—21no—
482C3—21no—
483C3—21no—
484C3—41no—
485C3#1059481811yescharacteristic, direct product factor
486C3C3×#551611yescharacteristic, complemented
487C3C3×#581211yescharacteristic, complemented
488C2—22no—
489C2C3×#6429712yes—
490C2C22×C3×D3×Dic311yescharacteristic
subgroupG/H
centerC22×C3C22×D32
commutatorC2×C32C23×C3×C4
FrattiniC22C22×C3×D32
FittingC22×C33×C4C22
socleC22×C33C24
prime radicalC23×C33C23
cosocleC22×C32C24×C3
hypercenterC22×C3×C4D32
hypocenterC32C3×#519
LCSG ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C22×C3×C4

Complements

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

NH ≅ G/Ntype
5. C3×#63843C2semidirect
6. C3×Dic3×Dic6C2semidirect
9. #10594818C3direct (N × H)
12. C2×C32×Dic6C4semidirect
42. #63843C2×C3semidirect
43. Dic3×Dic6C2×C3semidirect
74. C2×C3×Dic6Dic3semidirect
75. C2×C3×Dic6C3×C4semidirect
215. C2×Dic6C3×Dic3semidirect
301. C33#519semidirect
439. C32#5516semidirect
440. C32#5812semidirect
441. C32C3×#519semidirect
485. C3#10594818direct (N × H)
486. C3C3×#5516semidirect
487. C3C3×#5812semidirect

Maximal subgroups

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

subgroupindex
C22×C32×Dic62
C3×#641352
C32×#9262
C3×#639552
C3×#638432
C3×Dic3×Dic62
C3×#58123
C3×#55163
#105948183