Finite Group Databasebeta

← C3×#10590491 #13157458

Subgroups — C3×#10590491 #13157458

Order 1728.

520 conjugacy classes of subgroups.

The subgroup lattice has 482 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C32×#890C211yesmaximal, characteristic
2C3×Dic3×Dic6C211yesmaximal, characteristic
3C3×#63635C211yesmaximal, characteristic
4C32×#810C211yesmaximal, characteristic
5C3×#63575C211yesmaximal, characteristic
6C3×Dic3⋊(C3⋊C8)C211yesmaximal, characteristic
7C3×#63565C211yesmaximal, characteristic
8C3×#5063—31nomaximal
9C3×#4796—31nomaximal
10#10590491C311yesmaximal, characteristic, direct product factor
11C2×C32×Dic6C2211yescharacteristic
12C33×C4⋊C4C2211yescharacteristic
13C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
14C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
15C32×C4⋊Dic3—21no—
16C32×C4⋊Dic3C2211yescharacteristic
17C32×Dic3⋊C4—21no—
18C32×C4×Dic3C2211yescharacteristic
19C2×C32×C3⋊C8C2211yescharacteristic
20C3×Dic3⋊Dic3—21no—
21C3×Dic32—21no—
22C32×#125—31no—
23C3×Q8×Dic3—31no—
24C3×#890—21no—
25C3×#890D311yescharacteristic
26C3×#890C2×C311yescharacteristic
27C3×C4×Dic6—31no—
28Dic3×Dic6C2×C311yescharacteristic
29C32×#103—31no—
30#63635C2×C311yescharacteristic
31C3×#835—31no—
32C3×#818—31no—
33C3×Dic3⋊C8—31no—
34C3×#810—21no—
35C3×#810—31no—
36C3×#810D311yescharacteristic
37C3×#810C2×C311yescharacteristic
38C3×#808—31no—
39C3×#808—31no—
40C3×#808—61no—
41C3×#807—31no—
42C3×C4⋊(C3⋊C8)—31no—
43#63575C2×C311yescharacteristic
44Dic3⋊(C3⋊C8)C2×C311yescharacteristic
45#63565C2×C311yescharacteristic
46C2×C33×C4—21no—
47C2×C33×C4C2311yescharacteristic
48C2×C3×C3⋊Dic3—21no—
49C2×C32×Dic3—21no—
50C2×C32×Dic3—21no—
51C2×C32×Dic3D411yescharacteristic
52C2×C32×Dic3D411yescharacteristic
53C32×Dic6—21no—
54C32×Dic6Q812yes—
55C3×C3⋊(C3⋊C8)—21no—
56C32×C3⋊C8—21no—
57C3×#479—91no—
58#5063—31no—
59#4796—31no—
60C2×C32×Q8—31no—
61C2×C3×Dic6—21no—
62C2×C3×Dic6C2×D311yescharacteristic
63C2×C3×Dic6C22×C311yescharacteristic
64C2×C32×C8—31no—
65C32×C4⋊C4—21no—
66C32×C4⋊C4—21no—
67C32×C4⋊C4—21no—
68C32×C4⋊C4—31no—
69C32×C4⋊C4—41no—
70C32×C4⋊C4—61no—
71C32×C4⋊C4—61no—
72C32×C4⋊C4C2×D311yescharacteristic
73C32×C4⋊C4C2×D311yescharacteristic
74C32×C4⋊C4C22×C311yescharacteristic
75C32×C42—31no—
76C4⋊(C3⋊Dic3)C22×C311yescharacteristic
77C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
78C3×C4⋊Dic3—22no—
79C3×C4⋊Dic3—21no—
80C3×C4⋊Dic3—21no—
81C3×C4⋊Dic3—21no—
82C3×C4⋊Dic3—31no—
83C3×C4⋊Dic3—31no—
84C3×C4⋊Dic3—61no—
85C3×C4⋊Dic3—61no—
86C3×C4⋊Dic3C2×D311yescharacteristic
87C3×C4⋊Dic3C22×C311yescharacteristic
88C3×Dic3⋊C4—22no—
89C3×Dic3⋊C4—21no—
90C3×Dic3⋊C4—21no—
91C3×Dic3⋊C4—61no—
92C3×C4×Dic3—21no—
93C3×C4×Dic3—61no—
94C3×C4×Dic3—61no—
95C3×C4×Dic3C2×D311yescharacteristic
96C3×C4×Dic3C22×C311yescharacteristic
97C2×C3×C3⋊C8—21no—
98C2×C3×C3⋊C8—31no—
99C2×C3×C3⋊C8—31no—
100C2×C3×C3⋊C8—61no—
101C2×C3×C3⋊C8C2×D311yescharacteristic
102C2×C3×C3⋊C8C22×C311yescharacteristic
103Dic3⋊Dic3—21no—
104Dic32—21no—
105C22×C33C2×D411yescharacteristic
106C33×C4—41no—
107C33×C4C2×Q811yescharacteristic
108C33×C4#4111yescharacteristic
109C3×C3⋊Dic3—41no—
110C32×Dic3—22no—
111C32×Dic3—21no—
112C32×Dic3—21no—
113C32×Dic3—41no—
114C3×#125—31no—
115C3×#125—31no—
116C3×#125—61no—
117C3×C4×Q8—91no—
118Q8×Dic3—31no—
119#890C3×D311yescharacteristic
120C4×Dic6—31no—
121C3×#107—91no—
122C3×#106—91no—
123C3×C4⋊C8—91no—
124C3×#103—31no—
125C3×#103—31no—
126C3×#103—61no—
127C3×#103—91no—
128#835—31no—
129#818—31no—
130Dic3⋊C8—31no—
131#810—31no—
132#810C3×D311yescharacteristic
133#808—31no—
134#808—31no—
135#808—61no—
136#807—31no—
137C4⋊(C3⋊C8)—31no—
138C32×Q8—32no—
139C32×Q8—61no—
140C2×C32×C4—21no—
141C2×C32×C4—22no—
142C2×C32×C4—21no—
143C2×C32×C4—21no—
144C2×C32×C4—21no—
145C2×C32×C4—21no—
146C2×C32×C4—21no—
147C2×C32×C4—31no—
148C2×C32×C4—31no—
149C2×C32×C4—41no—
150C2×C32×C4—41no—
151C2×C32×C4—41no—
152C2×C32×C4—42no—
153C2×C32×C4—61no—
154C2×C32×C4—61no—
155C2×C32×C4C22×D311yescharacteristic
156C2×C32×C4C22×D311yescharacteristic
157C2×C32×C4C23×C311yescharacteristic
158C2×C3⋊Dic3—21no—
159C2×C3×Dic3—21no—
160C2×C3×Dic3—21no—
161C2×C3×Dic3—22no—
162C2×C3×Dic3—21no—
163C2×C3×Dic3—21no—
164C2×C3×Dic3—21no—
165C2×C3×Dic3—21no—
166C2×C3×Dic3—41no—
167C2×C3×Dic3—61no—
168C2×C3×Dic3—61no—
169C2×C3×Dic3—121no—
170C2×C3×Dic3#6711yescharacteristic
171C2×C3×Dic3#6711yescharacteristic
172C2×C3×Dic3C3×D411yescharacteristic
173C2×C3×Dic3C3×D411yescharacteristic
174C3×Dic6—21no—
175C3×Dic6—21no—
176C3×Dic6—22no—
177C3×Dic6—41no—
178C3×Dic6Dic612yes—
179C3×Dic6C3×Q812yes—
180C32×C8—61no—
181C3⋊(C3⋊C8)—21no—
182C3×C3⋊C8—21no—
183C3×C3⋊C8—21no—
184C3×C3⋊C8—41no—
185C3×C3⋊C8—62no—
186C3×C3⋊C8—61no—
187C3×C3⋊C8—61no—
188#479—91noSylow
189C2×C33C22⋊Q811yescharacteristic
190C2×C33#13511yescharacteristic
191C2×C33#13711yescharacteristic
192C2×C3×Q8—31no—
193C2×C3×Q8—31no—
194C2×C3×Q8—61no—
195C2×Dic6C2×C3×D311yescharacteristic
196C2×C3×C8—31no—
197C2×C3×C8—31no—
198C2×C3×C8—61no—
199C2×C3×C8—91no—
200C3×C4⋊C4—21no—
201C3×C4⋊C4—21no—
202C3×C4⋊C4—21no—
203C3×C4⋊C4—31no—
204C3×C4⋊C4—31no—
205C3×C4⋊C4—41no—
206C3×C4⋊C4—61no—
207C3×C4⋊C4—61no—
208C3×C4⋊C4—61no—
209C3×C4⋊C4—61no—
210C3×C4⋊C4—61no—
211C3×C4⋊C4—62no—
212C3×C4⋊C4—62no—
213C3×C4⋊C4—91no—
214C3×C4⋊C4—181no—
215C3×C4⋊C4D3211yescharacteristic
216C3×C4⋊C4C2×C3×D311yescharacteristic
217C3×C4⋊C4C2×C3×D311yescharacteristic
218C3×C42—31no—
219C3×C42—31no—
220C3×C42—61no—
221C3×C42—181no—
222C4⋊Dic3—21no—
223C4⋊Dic3—31no—
224C4⋊Dic3—31no—
225C4⋊Dic3—61no—
226C4⋊Dic3—61no—
227C4⋊Dic3C2×C3×D311yescharacteristic
228Dic3⋊C4—21no—
229Dic3⋊C4—61no—
230C4×Dic3—61no—
231C4×Dic3—61no—
232C4×Dic3C2×C3×D311yescharacteristic
233C2×C3⋊C8—31no—
234C2×C3⋊C8—31no—
235C2×C3⋊C8—61no—
236C2×C3⋊C8C2×C3×D311yescharacteristic
237C22×C32—21no—
238C22×C32—21no—
239C22×C32—21no—
240C22×C32—41no—
241C22×C32D3×D411yescharacteristic
242C22×C32C2×#6711yescharacteristic
243C22×C32C2×C3×D411yescharacteristic
244C32×C4—21no—
245C32×C4—21no—
246C32×C4—21no—
247C32×C4—21no—
248C32×C4—21no—
249C32×C4—21no—
250C32×C4—41no—
251C32×C4—41no—
252C32×C4—41no—
253C32×C4—42no—
254C32×C4—41no—
255C32×C4—41no—
256C32×C4—61no—
257C32×C4—61no—
258C32×C4—62no—
259C32×C4—81no—
260C32×C4—81no—
261C32×C4—82no—
262C32×C4—121no—
263C32×C4C2×Dic611yescharacteristic
264C32×C4#23711yescharacteristic
265C32×C4D3×Q811yescharacteristic
266C32×C4#23911yescharacteristic
267C32×C4C2×C3×Q811yescharacteristic
268C32×C4C3×#4111yescharacteristic
269C3⋊Dic3—41no—
270C3×Dic3—22no—
271C3×Dic3—21no—
272C3×Dic3—21no—
273C3×Dic3—22no—
274C3×Dic3—21no—
275C3×Dic3—21no—
276C3×Dic3—42no—
277C3×Dic3—41no—
278C3×Dic3—41no—
279C3×Dic3—42no—
280C3×Dic3—41no—
281C3×Dic3—41no—
282C3×Dic3—121no—
283C3×Dic3—121no—
284C3×Dic3—241no—
285#125—31no—
286C4×Q8—91no—
287#107—91no—
288#106—91no—
289C4⋊C8—91no—
290#103—31no—
291#103—91no—
292C33#47911yescharacteristic, complemented, Sylow
293C3×Q8—32no—
294C3×Q8—32no—
295C3×Q8—62no—
296C3×Q8—61no—
297C3×Q8—61no—
298C3×Q8—121no—
299C2×C3×C4—22no—
300C2×C3×C4—21no—
301C2×C3×C4—21no—
302C2×C3×C4—21no—
303C2×C3×C4—21no—
304C2×C3×C4—21no—
305C2×C3×C4—21no—
306C2×C3×C4—31no—
307C2×C3×C4—31no—
308C2×C3×C4—31no—
309C2×C3×C4—31no—
310C2×C3×C4—41no—
311C2×C3×C4—41no—
312C2×C3×C4—42no—
313C2×C3×C4—41no—
314C2×C3×C4—61no—
315C2×C3×C4—61no—
316C2×C3×C4—61no—
317C2×C3×C4—61no—
318C2×C3×C4—62no—
319C2×C3×C4—61no—
320C2×C3×C4—61no—
321C2×C3×C4—121no—
322C2×C3×C4—181no—
323C2×C3×C4C2×D3211yescharacteristic
324C2×C3×C4C22×C3×D311yescharacteristic
325C2×C3×C4C22×C3×D311yescharacteristic
326C2×Dic3—21no—
327C2×Dic3—21no—
328C2×Dic3—61no—
329C2×Dic3—61no—
330C2×Dic3—121no—
331C2×Dic3C3×#6711yescharacteristic
332C2×Dic3C3×#6711yescharacteristic
333Dic6—21no—
334Dic6C3×Dic612yes—
335C3×C8—61no—
336C3×C8—61no—
337C3×C8—121no—
338C3×C8—181no—
339C3⋊C8—21no—
340C3⋊C8—62no—
341C3⋊C8—61no—
342C3⋊C8—61no—
343C2×C32—21no—
344C2×C32—21no—
345C2×C32—21no—
346C2×C32—21no—
347C2×C32—21no—
348C2×C32—21no—
349C2×C32—21no—
350C2×C32—21no—
351C2×C32—21no—
352C2×C32—41no—
353C2×C32—41no—
354C2×C32—41no—
355C2×C32#87811yescharacteristic
356C2×C32#91511yescharacteristic
357C2×C32#91911yescharacteristic
358C2×C32#94611yescharacteristic
359C2×C32#95011yescharacteristic
360C2×C32#95111yescharacteristic
361C2×C32C3×C22⋊Q811yescharacteristic
362C2×C32C3×#13511yescharacteristic
363C2×C32C3×#13711yescharacteristic
364C2×Q8—31no—
365C2×C8—31no—
366C2×C8—91no—
367C4⋊C4—31no—
368C4⋊C4—61no—
369C4⋊C4—61no—
370C4⋊C4—91no—
371C4⋊C4—181no—
372C4⋊C4C3×D3211yescharacteristic
373C42—31no—
374C42—181no—
375C22×C3—21no—
376C22×C3—21no—
377C22×C3—21no—
378C22×C3—41no—
379C22×C3D3×#6711yescharacteristic
380C22×C3C3×D3×D411yescharacteristic
381C22×C3C2×C3×#6711yescharacteristic
382C3×C4—21no—
383C3×C4—21no—
384C3×C4—21no—
385C3×C4—21no—
386C3×C4—21no—
387C3×C4—21no—
388C3×C4—42no—
389C3×C4—41no—
390C3×C4—41no—
391C3×C4—41no—
392C3×C4—41no—
393C3×C4—41no—
394C3×C4—61no—
395C3×C4—61no—
396C3×C4—62no—
397C3×C4—62no—
398C3×C4—61no—
399C3×C4—61no—
400C3×C4—81no—
401C3×C4—81no—
402C3×C4—82no—
403C3×C4—121no—
404C3×C4—121no—
405C3×C4—122no—
406C3×C4—122no—
407C3×C4—121no—
408C3×C4—121no—
409C3×C4—361no—
410C3×C4D3×Dic611yescharacteristic
411C3×C4#379611yescharacteristic
412C3×C4C2×C3×Dic611yescharacteristic
413C3×C4C3×#23711yescharacteristic
414C3×C4C3×D3×Q811yescharacteristic
415C3×C4C3×#23911yescharacteristic
416Dic3—22no—
417Dic3—21no—
418Dic3—21no—
419Dic3—41no—
420Dic3—121no—
421Dic3—121no—
422Dic3—241no—
423C32—21no—
424C32—21no—
425C32—21no—
426C32—41no—
427C32#479611yescharacteristic, complemented
428C32#506311yescharacteristic, complemented
429C32C3×#47911yescharacteristic, complemented
430Q8—32no—
431Q8—61no—
432C2×C4—21no—
433C2×C4—31no—
434C2×C4—31no—
435C2×C4—61no—
436C2×C4—61no—
437C2×C4—181no—
438C2×C4C2×C3×D3211yescharacteristic
439C8—61no—
440C8—181no—
441C2×C3—21no—
442C2×C3—21no—
443C2×C3—21no—
444C2×C3—21no—
445C2×C3—21no—
446C2×C3—21no—
447C2×C3—21no—
448C2×C3—21no—
449C2×C3—21no—
450C2×C3—41no—
451C2×C3—41no—
452C2×C3—41no—
453C2×C3#6388711yescharacteristic
454C2×C3#6392911yescharacteristic
455C2×C3#6394911yescharacteristic
456C2×C3C3×#87811yescharacteristic
457C2×C3C3×#91511yescharacteristic
458C2×C3C3×#91911yescharacteristic
459C2×C3C3×#94611yescharacteristic
460C2×C3C3×#95011yescharacteristic
461C2×C3C3×#95111yescharacteristic
462C22C3×D3×#6711yescharacteristic
463C4—41no—
464C4—61no—
465C4—62no—
466C4—61no—
467C4—121no—
468C4—361no—
469C4C3×D3×Dic611yescharacteristic
470C4C3×#379611yescharacteristic
471C3—21no—
472C3—21no—
473C3—21no—
474C3—41no—
475C3#1059049111yescharacteristic, direct product factor
476C3C3×#479611yescharacteristic, complemented
477C3C3×#506311yescharacteristic, complemented
478C2C3×#6388711yescharacteristic
479C2C3×#6392911yescharacteristic
480C2C3×#6394911yescharacteristic
subgroupG/H
centerC22×C3D3×#67
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C4C2×C3×D32
FittingC33×C4⋊C4C22
socle, prime radicalC22×C33C2×D4
hypercenterC3×C4⋊C4D32
hypocenterC32C3×#479
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C2×C3×C4 ◁ C3×C4⋊C4

Complements

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

NH ≅ G/Ntype
10. #10590491C3direct (N × H)
292. C33#479semidirect
427. C32#4796semidirect
428. C32#5063semidirect
429. C32C3×#479semidirect
475. C3#10590491direct (N × H)
476. C3C3×#4796semidirect
477. C3C3×#5063semidirect

Maximal subgroups

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

subgroupindex
C32×#8902
C3×Dic3×Dic62
C3×#636352
C32×#8102
C3×#635752
C3×Dic3⋊(C3⋊C8)2
C3×#635652
C3×#50633
C3×#47963
#105904913