Finite Group Databasebeta

← C3×#10591159 #13158338

Subgroups — C3×#10591159 #13158338

Order 1728.

1150 conjugacy classes of subgroups.

The subgroup lattice has 494 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×C4×C3⋊D3C211yesmaximal, characteristic
2C2×C3×#3712C212yesmaximal, complemented
3C32×#830C212yesmaximal, complemented
4C3×#63557C212yesmaximal, complemented
5C3×#4726—32nomaximal
6#10591159C311yesmaximal, characteristic, direct product factor
7C23×C3×C3⋊D3C411yescharacteristic
8C22×C33×C4C2211yescharacteristic
9C22×C3×C3⋊Dic3C411yescharacteristic
10C2×C3×C4×C3⋊D3—24no—
11C2×C3×C4×C3⋊D3C411yescharacteristic
12C2×C3×C4×C3⋊D3C411yescharacteristic
13C2×C3×C4×C3⋊D3C2211yescharacteristic
14C2×C3×C4×C3⋊D3C2211yescharacteristic
15C2×C32×C3⋊C8C2214yescomplemented
16C3×#3712—24no—
17C22×C4×C3⋊D3C2×C311yescharacteristic
18C22×C3×C4×D3—32no—
19C22×C3×C4×D3—61no—
20C2×C3×ΓL(1,𝔽25)—34no—
21C2×#3712C2×C312yescomplemented
22C32×C22⋊C8—32no—
23C3×#830—22no—
24C3×#830D312yescomplemented
25C3×#830C2×C312yescomplemented
26C3×(C2×D3)⋊C8—34no—
27#63557C2×C312yescomplemented
28C23×C33C2×C411yescharacteristic
29C22×C3×C3⋊D3—22no—
30C22×C3×C3⋊D3—22no—
31C22×C3×C3⋊D3—22no—
32C22×C3×C3⋊D3C2×C411yescharacteristic
33C22×C3×C3⋊D3C2×C411yescharacteristic
34C2×C33×C4—22no—
35C2×C33×C4C2×C411yescharacteristic
36C2×C33×C4C2311yescharacteristic
37C2×C3×C3⋊Dic3—22no—
38C2×C3×C3⋊Dic3C2×C411yescharacteristic
39C2×C3×C3⋊Dic3C2×C411yescharacteristic
40C3×C4×C3⋊D3—22no—
41C3×C4×C3⋊D3—22no—
42C3×C4×C3⋊D3—22no—
43C3×C4×C3⋊D3D414yes—
44C32×C3⋊C8—24no—
45C3×#407—91no—
46#4726—32no—
47C23×C3⋊D3C3×C411yescharacteristic
48C23×C3×D3—32no—
49C23×C3×D3—61no—
50C22×C32×C4—21no—
51C22×C32×C4—22no—
52C22×C32×C4—41no—
53C22×C32×C4C2×D312yes—
54C22×C32×C4C22×C311yescharacteristic
55C22×C3⋊Dic3C3×C411yescharacteristic
56C2×C4×C3⋊D3—24no—
57C2×C4×C3⋊D3C3×C411yescharacteristic
58C2×C4×C3⋊D3C3×C411yescharacteristic
59C2×C4×C3⋊D3C22×C311yescharacteristic
60C2×C4×C3⋊D3C22×C311yescharacteristic
61C22×C3×Dic3—32no—
62C22×C3×Dic3—61no—
63C2×C3×C4×D3—32no—
64C2×C3×C4×D3—32no—
65C2×C3×C4×D3—32no—
66C2×C3×C4×D3—32no—
67C2×C3×C4×D3—68no—
68C2×C3×C4×D3—61no—
69C2×C3×C4×D3—61no—
70C2×C3×C4×D3—61no—
71C2×C3×C4×D3—61no—
72C2×C3×C4×D3—68no—
73C2×C32×C8—34no—
74C2×C3×C3⋊C8—24no—
75C2×C3×C3⋊C8C2×D314yescomplemented
76C2×C3×C3⋊C8C22×C314yescomplemented
77C3×ΓL(1,𝔽25)—68no—
78#3712—24no—
79C22×C33—21no—
80C22×C33—21no—
81C22×C33M1612yescomplemented
82C22×C33C22×C411yescharacteristic
83C2×C3×C3⋊D3—28no—
84C2×C3×C3⋊D3—21no—
85C2×C3×C3⋊D3—21no—
86C2×C3×C3⋊D3—21no—
87C2×C3×C3⋊D3—21no—
88C2×C3×C3⋊D3C22⋊C412yes—
89C2×C3×C3⋊D3M1612yescomplemented
90C33×C4—21no—
91C33×C4C2×D412yes—
92C3×C3⋊Dic3—21no—
93C3×C3⋊Dic3C22⋊C412yes—
94C23×C3×C4—91no—
95C22×C4×D3—32no—
96C22×C4×D3—61no—
97C2×C3×M16—92no—
98C2×ΓL(1,𝔽25)—34no—
99C3×C22⋊C8—32no—
100C3×C22⋊C8—32no—
101C3×C22⋊C8—62no—
102C3×C22⋊C8—92no—
103#830C3×D312yescomplemented
104(C2×D3)⋊C8—34no—
105C23×C32—21no—
106C23×C32—22no—
107C23×C32—41no—
108C23×C32C4×D312yes—
109C23×C32C2×C3×C411yescharacteristic
110C22×C3⋊D3—22no—
111C22×C3⋊D3—22no—
112C22×C3⋊D3—22no—
113C22×C3⋊D3C2×C3×C411yescharacteristic
114C22×C3⋊D3C2×C3×C411yescharacteristic
115C22×C3×D3—32no—
116C22×C3×D3—32no—
117C22×C3×D3—61no—
118C22×C3×D3—61no—
119C22×C3×D3—64no—
120C22×C3×D3—64no—
121C22×C3×D3—64no—
122C22×C3×D3—64no—
123C22×C3×D3—64no—
124C22×C3×D3—64no—
125C2×C32×C4—22no—
126C2×C32×C4—24no—
127C2×C32×C4—24no—
128C2×C32×C4—21no—
129C2×C32×C4—22no—
130C2×C32×C4—21no—
131C2×C32×C4—22no—
132C2×C32×C4—41no—
133C2×C32×C4—41no—
134C2×C32×C4—44no—
135C2×C32×C4—44no—
136C2×C32×C4C4×D312yes—
137C2×C32×C4C2×C3×C411yescharacteristic
138C2×C32×C4C22×D312yes—
139C2×C32×C4C23×C311yescharacteristic
140C2×C3⋊Dic3—22no—
141C2×C3⋊Dic3C2×C3×C411yescharacteristic
142C2×C3⋊Dic3C2×C3×C411yescharacteristic
143C4×C3⋊D3—22no—
144C4×C3⋊D3—22no—
145C4×C3⋊D3—22no—
146C4×C3⋊D3C3×D414yes—
147C2×C3×Dic3—32no—
148C2×C3×Dic3—32no—
149C2×C3×Dic3—61no—
150C2×C3×Dic3—61no—
151C2×C3×Dic3—64no—
152C2×C3×Dic3—64no—
153C3×C4×D3—38no—
154C3×C4×D3—64no—
155C3×C4×D3—64no—
156C3×C4×D3—64no—
157C3×C4×D3—64no—
158C3×C4×D3—64no—
159C3×C4×D3—64no—
160C3×C4×D3—64no—
161C32×C8—64no—
162C3×C3⋊C8—24no—
163C3×C3⋊C8—24no—
164C3×C3⋊C8—44no—
165#407—91noSylow
166C2×C33—22no—
167C2×C33C2×C22⋊C411yescharacteristic
168C2×C33C2×M1611yescharacteristic
169C2×C33C2×M1611yescharacteristic
170C3×C3⋊D3—22no—
171C3×C3⋊D3—22no—
172C24×C3—91no—
173C23×D3—32no—
174C23×D3—61no—
175C22×C3×C4—21no—
176C22×C3×C4—22no—
177C22×C3×C4—41no—
178C22×C3×C4—91no—
179C22×C3×C4—91no—
180C22×C3×C4—91no—
181C22×C3×C4—91no—
182C22×C3×C4—91no—
183C22×C3×C4—184no—
184C22×C3×C4D3211yescharacteristic
185C22×C3×C4C2×C3×D312yes—
186C22×Dic3—32no—
187C22×Dic3—61no—
188C2×C4×D3—32no—
189C2×C4×D3—32no—
190C2×C4×D3—32no—
191C2×C4×D3—32no—
192C2×C4×D3—61no—
193C2×C4×D3—61no—
194C2×C4×D3—61no—
195C2×C4×D3—61no—
196C2×C4×D3—68no—
197C2×C4×D3—68no—
198C3×M16—184no—
199C2×C3×C8—34no—
200C2×C3×C8—34no—
201C2×C3×C8—64no—
202C2×C3⋊C8C2×C3×D314yescomplemented
203ΓL(1,𝔽25)—68no—
204C22×C32—21no—
205C22×C32—21no—
206C22×C32—22no—
207C22×C32—22no—
208C22×C32—22no—
209C22×C32—22no—
210C22×C32—22no—
211C22×C32—24no—
212C22×C32—21no—
213C22×C32—22no—
214C22×C32—42no—
215C22×C32—41no—
216C22×C32—42no—
217C22×C32—42no—
218C22×C32—42no—
219C22×C32—42no—
220C22×C32ΓL(1,𝔽25)14yescomplemented
221C22×C32C3×M1612yescomplemented
222C22×C32C2×C4×D312yes—
223C22×C32C22×C3×C411yescharacteristic
224C2×C3⋊D3—28no—
225C2×C3⋊D3—21no—
226C2×C3⋊D3—21no—
227C2×C3⋊D3—21no—
228C2×C3⋊D3—21no—
229C2×C3⋊D3C3×C22⋊C412yes—
230C2×C3⋊D3C3×M1612yescomplemented
231C2×C3×D3—34no—
232C2×C3×D3—34no—
233C2×C3×D3—62no—
234C2×C3×D3—62no—
235C2×C3×D3—62no—
236C2×C3×D3—616no—
237C2×C3×D3—616no—
238C2×C3×D3—62no—
239C2×C3×D3—62no—
240C2×C3×D3—62no—
241C2×C3×D3—62no—
242C2×C3×D3—62no—
243C2×C3×D3—62no—
244C2×C3×D3—62no—
245C32×C4—21no—
246C32×C4—22no—
247C32×C4—22no—
248C32×C4—22no—
249C32×C4—24no—
250C32×C4—42no—
251C32×C4—42no—
252C32×C4—42no—
253C32×C4D3×D414yes—
254C32×C4C2×C3×D412yes—
255C3⋊Dic3—21no—
256C3⋊Dic3C3×C22⋊C412yes—
257C3×Dic3—34no—
258C3×Dic3—62no—
259C3×Dic3—62no—
260C3×Dic3—62no—
261C23×C4—91no—
262C2×M16—92no—
263C22⋊C8—32no—
264C22⋊C8—92no—
265C33#40711yescharacteristic, complemented, Sylow
266C23×C3—21no—
267C23×C3—22no—
268C23×C3—41no—
269C23×C3—91no—
270C23×C3—91no—
271C23×C3—182no—
272C23×C3—182no—
273C23×C3—182no—
274C23×C3#62411yescharacteristic
275C23×C3C3×C4×D312yes—
276C22×D3—32no—
277C22×D3—32no—
278C22×D3—61no—
279C22×D3—61no—
280C22×D3—64no—
281C22×D3—64no—
282C22×D3—64no—
283C22×D3—64no—
284C22×D3—64no—
285C22×D3—64no—
286C2×C3×C4—24no—
287C2×C3×C4—22no—
288C2×C3×C4—24no—
289C2×C3×C4—21no—
290C2×C3×C4—22no—
291C2×C3×C4—22no—
292C2×C3×C4—21no—
293C2×C3×C4—44no—
294C2×C3×C4—44no—
295C2×C3×C4—41no—
296C2×C3×C4—41no—
297C2×C3×C4—91no—
298C2×C3×C4—94no—
299C2×C3×C4—91no—
300C2×C3×C4—182no—
301C2×C3×C4—182no—
302C2×C3×C4—182no—
303C2×C3×C4—182no—
304C2×C3×C4#62411yescharacteristic
305C2×C3×C4C3×C4×D312yes—
306C2×C3×C4C2×D3211yescharacteristic
307C2×C3×C4C22×C3×D312yes—
308C2×Dic3—32no—
309C2×Dic3—32no—
310C2×Dic3—61no—
311C2×Dic3—61no—
312C2×Dic3—64no—
313C2×Dic3—64no—
314C4×D3—38no—
315C4×D3—64no—
316C4×D3—64no—
317C4×D3—64no—
318C4×D3—64no—
319C4×D3—64no—
320C4×D3—64no—
321C4×D3—64no—
322C3×C8—64no—
323C3×C8—64no—
324C3×C8—124no—
325C3⋊C8—24no—
326C2×C32—22no—
327C2×C32—24no—
328C2×C32—24no—
329C2×C32—22no—
330C2×C32—22no—
331C2×C32—22no—
332C2×C32—21no—
333C2×C32—21no—
334C2×C32—21no—
335C2×C32—41no—
336C2×C32—41no—
337C2×C32—41no—
338C2×C32—44no—
339C2×C32—44no—
340C2×C32D3×C22⋊C412yes—
341C2×C32C2×ΓL(1,𝔽25)12yes—
342C2×C32D3×M1612yes—
343C2×C32C2×C3×C22⋊C411yescharacteristic
344C2×C32C2×C3×M1611yescharacteristic
345C2×C32C2×C3×M1611yescharacteristic
346C3⋊D3—22no—
347C3⋊D3—22no—
348C3×D3—64no—
349C3×D3—64no—
350C3×D3—64no—
351C3×D3—64no—
352C24—91no—
353C22×C4—91no—
354C22×C4—91no—
355C22×C4—91no—
356C22×C4—91no—
357C22×C4—91no—
358C22×C4—184no—
359C22×C4C3×D3211yescharacteristic
360M16—184no—
361C2×C8—34no—
362C22×C3—22no—
363C22×C3—22no—
364C22×C3—22no—
365C22×C3—22no—
366C22×C3—21no—
367C22×C3—21no—
368C22×C3—21no—
369C22×C3—22no—
370C22×C3—22no—
371C22×C3—24no—
372C22×C3—42no—
373C22×C3—42no—
374C22×C3—42no—
375C22×C3—42no—
376C22×C3—41no—
377C22×C3—42no—
378C22×C3—92no—
379C22×C3—92no—
380C22×C3—188no—
381C22×C3—181no—
382C22×C3—181no—
383C22×C3—181no—
384C22×C3—181no—
385C22×C3#371212yescomplemented
386C22×C3C3×ΓL(1,𝔽25)14yescomplemented
387C22×C3C2×#62411yescharacteristic
388C22×C3C2×C3×C4×D312yes—
389C2×D3—34no—
390C2×D3—34no—
391C2×D3—62no—
392C2×D3—62no—
393C2×D3—62no—
394C2×D3—616no—
395C2×D3—616no—
396C2×D3—62no—
397C2×D3—62no—
398C2×D3—62no—
399C2×D3—62no—
400C2×D3—62no—
401C2×D3—62no—
402C2×D3—62no—
403C3×C4—22no—
404C3×C4—22no—
405C3×C4—21no—
406C3×C4—24no—
407C3×C4—22no—
408C3×C4—42no—
409C3×C4—42no—
410C3×C4—42no—
411C3×C4—92no—
412C3×C4—181no—
413C3×C4#381112yes—
414C3×C4C3×D3×D414yes—
415Dic3—34no—
416Dic3—62no—
417Dic3—62no—
418Dic3—62no—
419C32—22no—
420C32—21no—
421C32—41no—
422C32#472612yescomplemented
423C32C3×#40711yescharacteristic, complemented
424C23—91no—
425C23—91no—
426C23—182no—
427C23—182no—
428C23—182no—
429C23C3×#62411yescharacteristic
430C2×C4—22no—
431C2×C4—91no—
432C2×C4—94no—
433C2×C4—91no—
434C2×C4—182no—
435C2×C4—182no—
436C2×C4—182no—
437C2×C4—182no—
438C2×C4C3×#62411yescharacteristic
439C2×C4C2×C3×D3211yescharacteristic
440C8—64no—
441C2×C3—24no—
442C2×C3—24no—
443C2×C3—22no—
444C2×C3—21no—
445C2×C3—22no—
446C2×C3—21no—
447C2×C3—22no—
448C2×C3—21no—
449C2×C3—22no—
450C2×C3—44no—
451C2×C3—44no—
452C2×C3—41no—
453C2×C3—41no—
454C2×C3—41no—
455C2×C3—182no—
456C2×C3—182no—
457C2×C3#6381811yescharacteristic
458C2×C3C2×#371211yescharacteristic
459C2×C3#6397411yescharacteristic
460C2×C3C3×D3×C22⋊C412yes—
461C2×C3C2×C3×ΓL(1,𝔽25)12yes—
462C2×C3C3×D3×M1612yes—
463D3—64no—
464D3—64no—
465D3—64no—
466D3—64no—
467C22—21no—
468C22—21no—
469C22—92no—
470C22—92no—
471C22—188no—
472C22—181no—
473C22—181no—
474C22—181no—
475C22—181no—
476C22C3×#371212yescomplemented
477C22C2×C3×#62411yescharacteristic
478C4—21no—
479C4—92no—
480C4—181no—
481C4C3×#381112yes—
482C3—21no—
483C3—22no—
484C3—41no—
485C3#1059115911yescharacteristic, direct product factor
486C3C3×#472612yescomplemented
487C2—22no—
488C2—182no—
489C2—182no—
490C2C3×#6381811yescharacteristic
491C2C2×C3×#371211yescharacteristic
492C2C3×#6397411yescharacteristic
subgroupG/H
centerC2×C3×C4C2×D32
commutatorC22×C32C22×C3×C4
FrattiniC2×C4C2×C3×D32
FittingC22×C33×C4C22
socleC22×C33C22×C4
prime radicalC23×C3×C3⋊D3C4
cosocleC2×C32×C4C23×C3
hypercenterC22×C3×C4D32
hypocenterC32C3×#407
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C2×C3×C4 ◁ C22×C3×C4

Complements

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

NH ≅ G/Ntype
2. C2×C3×#3712C2semidirect
3. C32×#830C2semidirect
4. C3×#63557C2semidirect
6. #10591159C3direct (N × H)
15. C2×C32×C3⋊C8C22semidirect
21. C2×#3712C2×C3semidirect
24. C3×#830D3semidirect
25. C3×#830C2×C3semidirect
27. #63557C2×C3semidirect
75. C2×C3×C3⋊C8C2×D3semidirect
76. C2×C3×C3⋊C8C22×C3semidirect
81. C22×C33M16semidirect
89. C2×C3×C3⋊D3M16semidirect
103. #830C3×D3semidirect
202. C2×C3⋊C8C2×C3×D3semidirect
220. C22×C32ΓL(1,𝔽25)semidirect
221. C22×C32C3×M16semidirect
230. C2×C3⋊D3C3×M16semidirect
265. C33#407semidirect
385. C22×C3#3712semidirect
386. C22×C3C3×ΓL(1,𝔽25)semidirect
422. C32#4726semidirect
423. C32C3×#407semidirect
476. C22C3×#3712semidirect
485. C3#10591159direct (N × H)
486. C3C3×#4726semidirect

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
C2×C3×#37122
C32×#8302
C3×#635572
C3×#47263
#105911593