Finite Group Databasebeta

← C3×#10591637 #13158640

Subgroups — C3×#10591637 #13158640

Order 1728.

878 conjugacy classes of subgroups.

The subgroup lattice has 500 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C32×#239C211yesmaximal, characteristic
2C3×D3×#208C212yesmaximal, complemented
3C3×#63663C211yesmaximal, characteristic, complemented
4C32×#836C211yesmaximal, characteristic, complemented
5C3×#63558C212yesmaximal, complemented
6C3×#5237—31nomaximal
7C3×#4751—31nomaximal
8#10591637C311yesmaximal, characteristic, direct product factor
9C2×C33×Q8C2211yescharacteristic
10C32×#239—22no—
11C32×#239—22no—
12C2×C32×D12C412yes—
13C2×C32×D12C2211yescharacteristic
14C2×C32×C4×D3C412yes—
15C2×C32×C4×D3C2211yescharacteristic
16C3×#3748C2212yes—
17C32×#208C2212yes—
18C3×#3711—22no—
19C3×D3×C3⋊C8—22no—
20C2×C32×#41—31no—
21C2×C3×#239—21no—
22C2×C3×#239D311yescharacteristic
23C2×C3×#239C2×C311yescharacteristic
24C2×C3×#208—32no—
25C3×D3×M16—32no—
26D3×#208C2×C312yescomplemented
27C32×#101—31no—
28#63663C2×C311yescharacteristic, complemented
29C3×#836—21no—
30C3×#836—31no—
31C3×#836—31no—
32C3×#836—61no—
33C3×#836D311yescharacteristic, complemented
34C3×#836C2×C311yescharacteristic, complemented
35C3×#833—32no—
36C3×#823—32no—
37#63558C2×C312yescomplemented
38C22×C32×D3C2×C412yes—
39C22×C32×D3C2×C411yescharacteristic
40C33×Q8—22no—
41C2×C33×C4C2×C412yes—
42C2×C33×C4C2311yescharacteristic
43C2×C32×Dic3C2×C411yescharacteristic
44C32×D12—24no—
45C32×D12—22no—
46C32×C4×D3—22no—
47C32×C4×D3—22no—
48C32×C4×D3D412yes—
49C32×C4×D3D412yes—
50C3×C3⋊(C3⋊C8)—22no—
51C32×C3⋊C8—22no—
52C3×#413—91no—
53#5237—31no—
54#4751—31no—
55C32×#41—62no—
56C32×#41—62no—
57C2×C32×Q8—21no—
58C2×C32×Q8—21no—
59C2×C32×Q8—21no—
60C2×C32×Q8—41no—
61C2×C32×Q8C2×D311yescharacteristic
62C2×C32×Q8C2×D311yescharacteristic
63C2×C32×Q8C22×C311yescharacteristic
64C2×C32×D4—32no—
65C2×C32×D4—31no—
66C22×C32×C4—31no—
67C22×C32×C4—32no—
68C3×#239—24no—
69C3×#239—24no—
70C3×#239—22no—
71C3×#239—22no—
72C3×#239—22no—
73C3×#239—22no—
74C2×C3×D12—22no—
75C2×C3×D12—21no—
76C2×C3×D12Dic312yes—
77C2×C3×D12C3×C412yes—
78C2×C3×D12C2×D311yescharacteristic
79C2×C3×D12C22×C311yescharacteristic
80C2×C3×C4×D3—22no—
81C2×C3×C4×D3—21no—
82C2×C3×C4×D3Dic312yes—
83C2×C3×C4×D3C3×C412yes—
84C2×C3×C4×D3C2×D311yescharacteristic
85C2×C3×C4×D3C22×C311yescharacteristic
86C32×M16—32no—
87#3748C22×C312yes—
88C3×#208—22no—
89C3×#208—32no—
90C3×#208—32no—
91C3×#208—62no—
92C3×#208—62no—
93C3×#208C2×D312yes—
94C3×#208C22×C312yes—
95C2×C3×C3⋊C8—62no—
96C3×ΓL(1,𝔽25)—62no—
97C3×D3×C8—62no—
98#3711—22no—
99D3×C3⋊C8—22no—
100C22×C33C22×C411yescharacteristic
101C2×C32×D3—22no—
102C2×C32×D3—41no—
103C2×C32×D3—42no—
104C2×C32×D3C22⋊C412yes—
105C33×C4—22no—
106C33×C4C2×D412yes—
107C32×Dic3C22⋊C411yescharacteristic
108C32×Dic3C22⋊C411yescharacteristic
109C2×C3×#41—31no—
110C2×C3×#41—31no—
111C2×C3×#41—61no—
112C2×#239C3×D311yescharacteristic
113C2×C3×M16—92no—
114C2×#208—32no—
115D3×M16—32no—
116C3×#101—31no—
117C3×#101—31no—
118C3×#101—61no—
119C3×#101—91no—
120C3×#100—92no—
121#836—31no—
122#836—31no—
123#836—61no—
124#836C3×D311yescharacteristic, complemented
125#833—32no—
126#823—32no—
127C23×C32—31no—
128C23×C32—32no—
129C22×C3×D3—22no—
130C22×C3×D3—21no—
131C22×C3×D3C2×Dic312yes—
132C22×C3×D3C2×Dic311yescharacteristic
133C22×C3×D3C2×C3×C412yes—
134C22×C3×D3C2×C3×C411yescharacteristic
135C32×Q8—22no—
136C32×Q8—24no—
137C32×Q8—22no—
138C32×Q8—22no—
139C32×Q8—42no—
140C32×Q8—42no—
141C32×Q8—44no—
142C32×D4—64no—
143C32×D4—62no—
144C2×C32×C4—22no—
145C2×C32×C4—22no—
146C2×C32×C4—22no—
147C2×C32×C4—21no—
148C2×C32×C4—21no—
149C2×C32×C4—21no—
150C2×C32×C4—31no—
151C2×C32×C4—32no—
152C2×C32×C4—32no—
153C2×C32×C4—42no—
154C2×C32×C4—41no—
155C2×C32×C4—62no—
156C2×C32×C4—62no—
157C2×C32×C4C4×D312yes—
158C2×C32×C4C2×Dic312yes—
159C2×C32×C4C2×C3×C412yes—
160C2×C32×C4C22×D311yescharacteristic
161C2×C32×C4C22×D311yescharacteristic
162C2×C32×C4C23×C311yescharacteristic
163C2×C3×Dic3—21no—
164C2×C3×Dic3C2×Dic311yescharacteristic
165C2×C3×Dic3C2×C3×C411yescharacteristic
166C3×D12—28no—
167C3×D12—24no—
168C3×D12—24no—
169C3×D12—24no—
170C3×D12—22no—
171C3×D12—22no—
172C3×C4×D3—24no—
173C3×C4×D3—24no—
174C3×C4×D3—22no—
175C3×C4×D3—22no—
176C3×C4×D3—22no—
177C3×C4×D3—22no—
178C3×C4×D3—22no—
179C3×C4×D3—22no—
180C3×C4×D3#6712yes—
181C3×C4×D3#6712yes—
182C3×C4×D3C3×D412yes—
183C3×C4×D3C3×D412yes—
184C32×C8—62no—
185C3⋊(C3⋊C8)—22no—
186C3×C3⋊C8—22no—
187C3×C3⋊C8—22no—
188C3×C3⋊C8—42no—
189C3×C3⋊C8—62no—
190C3×C3⋊C8—62no—
191C3×C3⋊C8—122no—
192#413—91noSylow
193C2×C33—21no—
194C2×C33C2×C22⋊C411yescharacteristic
195C32×D3—22no—
196C32×D3—42no—
197C3×#41—62no—
198C3×#41—62no—
199C3×#41—62no—
200C3×#41—62no—
201C3×#41—64no—
202C3×#41—64no—
203C2×C3×Q8—21no—
204C2×C3×Q8—21no—
205C2×C3×Q8—21no—
206C2×C3×Q8—41no—
207C2×C3×Q8D3211yescharacteristic
208C2×C3×Q8C2×C3×D311yescharacteristic
209C2×C3×Q8C2×C3×D311yescharacteristic
210C2×C3×D4—32no—
211C2×C3×D4—32no—
212C2×C3×D4—31no—
213C2×C3×D4—31no—
214C2×C3×D4—62no—
215C2×C3×D4—61no—
216C22×C3×C4—31no—
217C22×C3×C4—31no—
218C22×C3×C4—32no—
219C22×C3×C4—32no—
220C22×C3×C4—61no—
221C22×C3×C4—62no—
222#239—22no—
223#239—22no—
224C2×D12C3×Dic312yes—
225C2×D12C2×C3×D311yescharacteristic
226C2×C4×D3C3×Dic312yes—
227C2×C4×D3C2×C3×D311yescharacteristic
228C3×M16—32no—
229C3×M16—32no—
230C3×M16—62no—
231C3×M16—92no—
232C3×M16—182no—
233C2×C3×C8—182no—
234#208—32no—
235#208—32no—
236#208—62no—
237#208—62no—
238#208C2×C3×D312yes—
239C2×C3⋊C8—62no—
240ΓL(1,𝔽25)—62no—
241D3×C8—62no—
242C22×C32—21no—
243C22×C32—21no—
244C22×C32—21no—
245C22×C32—32no—
246C22×C32—41no—
247C22×C32—62no—
248C22×C32—121no—
249C22×C32—122no—
250C22×C32C2×C4×D311yescharacteristic
251C22×C32C22×Dic311yescharacteristic
252C22×C32C22×C3×C411yescharacteristic
253C2×C3×D3—24no—
254C2×C3×D3—22no—
255C2×C3×D3—22no—
256C2×C3×D3—22no—
257C2×C3×D3—41no—
258C2×C3×D3—41no—
259C2×C3×D3—42no—
260C2×C3×D3—44no—
261C2×C3×D3—42no—
262C2×C3×D3—42no—
263C2×C3×D3#21712yes—
264C2×C3×D3C3×C22⋊C412yes—
265C32×C4—22no—
266C32×C4—24no—
267C32×C4—22no—
268C32×C4—22no—
269C32×C4—22no—
270C32×C4—22no—
271C32×C4—22no—
272C32×C4—31no—
273C32×C4—31no—
274C32×C4—42no—
275C32×C4—42no—
276C32×C4—42no—
277C32×C4—44no—
278C32×C4D3×D412yes—
279C32×C4C2×#6712yes—
280C32×C4C2×C3×D412yes—
281C3×Dic3—21no—
282C3×Dic3—21no—
283C3×Dic3#21711yescharacteristic
284C3×Dic3#21711yescharacteristic
285C3×Dic3C3×C22⋊C411yescharacteristic
286C3×Dic3C3×C22⋊C411yescharacteristic
287C2×#41—31no—
288C2×M16—92no—
289#101—31no—
290#101—91no—
291#100—92no—
292C33#41311yescharacteristic, complemented, Sylow
293C23×C3—31no—
294C23×C3—31no—
295C23×C3—32no—
296C23×C3—32no—
297C23×C3—61no—
298C23×C3—62no—
299C22×D3C2×C3×Dic312yes—
300C22×D3C2×C3×Dic311yescharacteristic
301C3×Q8—24no—
302C3×Q8—22no—
303C3×Q8—22no—
304C3×Q8—22no—
305C3×Q8—42no—
306C3×Q8—42no—
307C3×Q8—44no—
308C3×D4—64no—
309C3×D4—64no—
310C3×D4—68no—
311C3×D4—64no—
312C3×D4—62no—
313C3×D4—62no—
314C2×C3×C4—22no—
315C2×C3×C4—22no—
316C2×C3×C4—22no—
317C2×C3×C4—21no—
318C2×C3×C4—21no—
319C2×C3×C4—21no—
320C2×C3×C4—31no—
321C2×C3×C4—31no—
322C2×C3×C4—32no—
323C2×C3×C4—32no—
324C2×C3×C4—32no—
325C2×C3×C4—32no—
326C2×C3×C4—42no—
327C2×C3×C4—41no—
328C2×C3×C4—62no—
329C2×C3×C4—62no—
330C2×C3×C4—61no—
331C2×C3×C4—62no—
332C2×C3×C4—62no—
333C2×C3×C4—62no—
334C2×C3×C4—62no—
335C2×C3×C4—64no—
336C2×C3×C4—64no—
337C2×C3×C4D3×Dic312yes—
338C2×C3×C4C3×C4×D312yes—
339C2×C3×C4C2×C3×Dic312yes—
340C2×C3×C4C2×D3211yescharacteristic
341C2×C3×C4C22×C3×D311yescharacteristic
342C2×C3×C4C22×C3×D311yescharacteristic
343C2×Dic3C2×C3×Dic311yescharacteristic
344D12—24no—
345D12—22no—
346C4×D3—22no—
347C4×D3—22no—
348C4×D3C3×#6712yes—
349C4×D3C3×#6712yes—
350C3×C8—62no—
351C3×C8—62no—
352C3×C8—122no—
353C3×C8—182no—
354C3⋊C8—22no—
355C3⋊C8—62no—
356C3⋊C8—62no—
357C3⋊C8—122no—
358C2×C32—21no—
359C2×C32—22no—
360C2×C32—21no—
361C2×C32—21no—
362C2×C32—21no—
363C2×C32—21no—
364C2×C32—21no—
365C2×C32—41no—
366C2×C32—41no—
367C2×C32—41no—
368C2×C32—42no—
369C2×C32—62no—
370C2×C32—122no—
371C2×C32D3×C22⋊C411yescharacteristic
372C2×C32C2×#21711yescharacteristic
373C2×C32C2×C3×C22⋊C411yescharacteristic
374C3×D3—22no—
375C3×D3—22no—
376C3×D3—42no—
377C3×D3—44no—
378C3×D3—42no—
379C3×D3—42no—
380#41—62no—
381#41—62no—
382C2×Q8C3×D3211yescharacteristic
383C2×D4—32no—
384C2×D4—31no—
385C22×C4—31no—
386C22×C4—32no—
387M16—32no—
388M16—92no—
389M16—182no—
390C2×C8—182no—
391C22×C3—21no—
392C22×C3—21no—
393C22×C3—21no—
394C22×C3—32no—
395C22×C3—32no—
396C22×C3—41no—
397C22×C3—62no—
398C22×C3—64no—
399C22×C3—62no—
400C22×C3—62no—
401C22×C3—121no—
402C22×C3—121no—
403C22×C3—122no—
404C22×C3—122no—
405C22×C3—122no—
406C22×C3—124no—
407C22×C3C2×D3×Dic311yescharacteristic
408C22×C3C2×C3×C4×D311yescharacteristic
409C22×C3C22×C3×Dic311yescharacteristic
410C2×D3—22no—
411C2×D3—41no—
412C2×D3—42no—
413C2×D3C3×#21712yes—
414C3×C4—24no—
415C3×C4—22no—
416C3×C4—22no—
417C3×C4—22no—
418C3×C4—22no—
419C3×C4—22no—
420C3×C4—22no—
421C3×C4—31no—
422C3×C4—31no—
423C3×C4—31no—
424C3×C4—31no—
425C3×C4—42no—
426C3×C4—42no—
427C3×C4—44no—
428C3×C4—42no—
429C3×C4—61no—
430C3×C4—61no—
431C3×C4D3×#6712yes—
432C3×C4C3×D3×D412yes—
433C3×C4C2×C3×#6712yes—
434Dic3C3×#21711yescharacteristic
435Dic3C3×#21711yescharacteristic
436C32—21no—
437C32—21no—
438C32—21no—
439C32—41no—
440C32#475111yescharacteristic, complemented
441C32#523711yescharacteristic, complemented
442C32C3×#41311yescharacteristic, complemented
443C23—31no—
444C23—32no—
445Q8—22no—
446D4—64no—
447D4—62no—
448C2×C4—31no—
449C2×C4—32no—
450C2×C4—32no—
451C2×C4—62no—
452C2×C4—62no—
453C2×C4C3×D3×Dic312yes—
454C2×C4C2×C3×D3211yescharacteristic
455C8—62no—
456C8—182no—
457C2×C3—22no—
458C2×C3—21no—
459C2×C3—21no—
460C2×C3—21no—
461C2×C3—21no—
462C2×C3—21no—
463C2×C3—21no—
464C2×C3—41no—
465C2×C3—41no—
466C2×C3—42no—
467C2×C3—41no—
468C2×C3—62no—
469C2×C3—62no—
470C2×C3—122no—
471C2×C3—124no—
472C2×C3—122no—
473C2×C3—122no—
474C2×C3D3×#21711yescharacteristic
475C2×C3C3×D3×C22⋊C411yescharacteristic
476C2×C3C2×C3×#21711yescharacteristic
477D3—22no—
478D3—42no—
479C22—32no—
480C22—62no—
481C22—121no—
482C22—122no—
483C22C2×C3×D3×Dic311yescharacteristic
484C4—22no—
485C4—31no—
486C4—31no—
487C4C3×D3×#6712yes—
488C3—21no—
489C3—21no—
490C3—21no—
491C3—41no—
492C3#1059163711yescharacteristic, direct product factor
493C3C3×#475111yescharacteristic, complemented
494C3C3×#523711yescharacteristic, complemented
495C2—21no—
496C2—62no—
497C2—122no—
498C2C3×D3×#21711yescharacteristic
subgroupG/H
centerC2×C3D3×#217
commutatorC22×C32C22×C3×C4
FrattiniC2×C4C2×C3×D32
FittingC2×C33×Q8C22
socleC2×C33C2×C22⋊C4
prime radicalC2×C32×#239C2
cosocleC2×C32×C4C23×C3
hypercenterC2×C3×Q8D32
hypocenterC32C3×#413
LCSG ▷ C22×C32 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2×C3 ◁ C2×C3×C4 ◁ C2×C3×Q8

Complements

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

NH ≅ G/Ntype
2. C3×D3×#208C2semidirect
3. C3×#63663C2semidirect
4. C32×#836C2semidirect
5. C3×#63558C2semidirect
8. #10591637C3direct (N × H)
26. D3×#208C2×C3semidirect
28. #63663C2×C3semidirect
33. C3×#836D3semidirect
34. C3×#836C2×C3semidirect
37. #63558C2×C3semidirect
124. #836C3×D3semidirect
292. C33#413semidirect
440. C32#4751semidirect
441. C32#5237semidirect
442. C32C3×#413semidirect
492. C3#10591637direct (N × H)
493. C3C3×#4751semidirect
494. C3C3×#5237semidirect

Maximal subgroups

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

subgroupindex
C2×C32×#2392
C3×D3×#2082
C3×#636632
C32×#8362
C3×#635582
C3×#52373
C3×#47513
#105916373