Finite Group Databasebeta

← C3×#10591122 #13158158

Subgroups — C3×#10591122 #13158158

Order 1728.

738 conjugacy classes of subgroups.

The subgroup lattice has 460 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#64135C211yesmaximal, characteristic
2C32×#926C211yesmaximal, characteristic
3C22×C32×C3⋊C8C211yesmaximal, characteristic
4C3×#63575C212yesmaximal, complemented
5C3×#63573C212yesmaximal, complemented
6C3×#5106—31nomaximal
7C3×#4998—31nomaximal
8#10591122C311yesmaximal, characteristic, direct product factor
9C22×C33×C4C2211yescharacteristic
10C3×#3757—21no—
11C3×C4⋊(C3⋊Dic3)C2212yes—
12C3×C4×C3⋊Dic3—21no—
13C32×#217—21no—
14C32×C4⋊Dic3C2212yes—
15C32×C4×Dic3—21no—
16C2×C32×C3⋊C8C414yescomplemented
17C2×C32×C3⋊C8C2211yescharacteristic
18C2×C32×C3⋊C8C2211yescharacteristic
19C22×C32×C8—31no—
20C32×#117—31no—
21#64135C2×C311yescharacteristic
22C3×#926—21no—
23C3×#926—31no—
24C3×#926—31no—
25C3×#926—61no—
26C3×#926D311yescharacteristic
27C3×#926C2×C311yescharacteristic
28C22×C3×C3⋊C8—21no—
29C22×C3×C3⋊C8D311yescharacteristic
30C22×C3×C3⋊C8C2×C311yescharacteristic
31C3×#818—32no—
32C3×#817—32no—
33C3×#808—32no—
34C3×#807—32no—
35#63575C2×C312yescomplemented
36#63573C2×C312yescomplemented
37C23×C33D411yescharacteristic
38C2×C33×C4C2×C412yes—
39C2×C33×C4D411yescharacteristic
40C2×C33×C4Q812yes—
41C2×C33×C4C2311yescharacteristic
42C2×C3×C3⋊Dic3—22no—
43C2×C32×Dic3—22no—
44C32×C3⋊C8C2×C412yes—
45C32×C3⋊C8C2×C412yes—
46C3×#427—91no—
47#5106—31no—
48#4998—31no—
49C22×C32×C4—21no—
50C22×C32×C4—21no—
51C22×C32×C4—21no—
52C22×C32×C4—41no—
53C22×C32×C4C2×D311yescharacteristic
54C22×C32×C4C2×D311yescharacteristic
55C22×C32×C4C22×C311yescharacteristic
56C2×C32×C8—34no—
57C2×C32×C8—31no—
58C2×C32×C8—31no—
59C32×C4⋊C4—32no—
60C32×C22⋊C4—61no—
61C32×C42—61no—
62#3757—21no—
63C4⋊(C3⋊Dic3)C22×C312yes—
64C4×C3⋊Dic3—21no—
65C3×#217—22no—
66C3×#217—21no—
67C3×#217—21no—
68C3×#217—62no—
69C3×#217—61no—
70C3×#217—61no—
71C3×C4⋊Dic3—22no—
72C3×C4⋊Dic3—32no—
73C3×C4⋊Dic3—32no—
74C3×C4⋊Dic3—62no—
75C3×C4⋊Dic3C2×D312yes—
76C3×C4⋊Dic3C22×C312yes—
77C3×C4×Dic3—22no—
78C3×C4×Dic3—21no—
79C3×C4×Dic3—21no—
80C3×C4×Dic3—62no—
81C3×C4×Dic3—61no—
82C3×C4×Dic3—61no—
83C2×C3×C3⋊C8—24no—
84C2×C3×C3⋊C8—21no—
85C2×C3×C3⋊C8—21no—
86C2×C3×C3⋊C8Dic314yescomplemented
87C2×C3×C3⋊C8C3×C414yescomplemented
88C2×C3×C3⋊C8C2×D311yescharacteristic
89C2×C3×C3⋊C8C2×D311yescharacteristic
90C2×C3×C3⋊C8C22×C311yescharacteristic
91C2×C3×C3⋊C8C22×C311yescharacteristic
92C22×C33—22no—
93C22×C33C4⋊C412yes—
94C22×C33C2×D411yescharacteristic
95C33×C4C4⋊C412yes—
96C33×C4C22×C411yescharacteristic
97C33×C4C2×Q811yescharacteristic
98C3×C3⋊Dic3—42no—
99C32×Dic3—42no—
100C22×C3×C8—31no—
101C22×C3×C8—31no—
102C22×C3×C8—61no—
103C3×#117—31no—
104C3×#117—31no—
105C3×#117—61no—
106C3×#117—91no—
107#926—31no—
108#926—31no—
109#926—61no—
110#926C3×D311yescharacteristic
111C22×C3⋊C8C3×D311yescharacteristic
112C3×#107—92no—
113C3×#106—92no—
114#818—32no—
115#817—32no—
116#808—32no—
117#807—32no—
118C23×C32—21no—
119C23×C32—21no—
120C23×C32—21no—
121C23×C32—41no—
122C23×C32D1211yescharacteristic
123C23×C32#6711yescharacteristic
124C23×C32C3×D411yescharacteristic
125C2×C32×C4—22no—
126C2×C32×C4—21no—
127C2×C32×C4—22no—
128C2×C32×C4—22no—
129C2×C32×C4—22no—
130C2×C32×C4—22no—
131C2×C32×C4—21no—
132C2×C32×C4—22no—
133C2×C32×C4—21no—
134C2×C32×C4—21no—
135C2×C32×C4—21no—
136C2×C32×C4—21no—
137C2×C32×C4—42no—
138C2×C32×C4—41no—
139C2×C32×C4—42no—
140C2×C32×C4—41no—
141C2×C32×C4—62no—
142C2×C32×C4Dic612yes—
143C2×C32×C4Dic612yes—
144C2×C32×C4C4×D312yes—
145C2×C32×C4D1211yescharacteristic
146C2×C32×C4C2×Dic312yes—
147C2×C32×C4#6711yescharacteristic
148C2×C32×C4C2×C3×C412yes—
149C2×C32×C4C3×D411yescharacteristic
150C2×C32×C4C3×Q812yes—
151C2×C32×C4C22×D311yescharacteristic
152C2×C32×C4C22×D311yescharacteristic
153C2×C32×C4C23×C311yescharacteristic
154C2×C3⋊Dic3—22no—
155C2×C3×Dic3—24no—
156C2×C3×Dic3—22no—
157C2×C3×Dic3—22no—
158C2×C3×Dic3—64no—
159C2×C3×Dic3—62no—
160C2×C3×Dic3—62no—
161C32×C8—32no—
162C32×C8—32no—
163C3×C3⋊C8—22no—
164C3×C3⋊C8—22no—
165C3×C3⋊C8C2×Dic312yes—
166C3×C3⋊C8C2×Dic312yes—
167C3×C3⋊C8C2×C3×C412yes—
168C3×C3⋊C8C2×C3×C412yes—
169#427—91noSylow
170C2×C33—22no—
171C2×C33C2×C4⋊C411yescharacteristic
172C2×C33#13512yes—
173C22×C3×C4—21no—
174C22×C3×C4—21no—
175C22×C3×C4—21no—
176C22×C3×C4—41no—
177C22×C3×C4D3211yescharacteristic
178C22×C3×C4C2×C3×D311yescharacteristic
179C22×C3×C4C2×C3×D311yescharacteristic
180C2×C3×C8—34no—
181C2×C3×C8—34no—
182C2×C3×C8—31no—
183C2×C3×C8—31no—
184C2×C3×C8—31no—
185C2×C3×C8—31no—
186C2×C3×C8—64no—
187C2×C3×C8—61no—
188C2×C3×C8—61no—
189C3×C4⋊C4—32no—
190C3×C4⋊C4—32no—
191C3×C4⋊C4—62no—
192C3×C4⋊C4—92no—
193C3×C22⋊C4—62no—
194C3×C22⋊C4—61no—
195C3×C22⋊C4—61no—
196C3×C22⋊C4—181no—
197C3×C42—62no—
198C3×C42—61no—
199C3×C42—61no—
200C3×C42—181no—
201#217—21no—
202#217—62no—
203#217—61no—
204#217—61no—
205C4⋊Dic3—32no—
206C4⋊Dic3—32no—
207C4⋊Dic3—62no—
208C4⋊Dic3C2×C3×D312yes—
209C4×Dic3—21no—
210C4×Dic3—62no—
211C4×Dic3—61no—
212C4×Dic3—61no—
213C2×C3⋊C8C3×Dic314yescomplemented
214C2×C3⋊C8C2×C3×D311yescharacteristic
215C2×C3⋊C8C2×C3×D311yescharacteristic
216C22×C32—22no—
217C22×C32—24no—
218C22×C32—22no—
219C22×C32—22no—
220C22×C32—22no—
221C22×C32—22no—
222C22×C32—22no—
223C22×C32—21no—
224C22×C32—21no—
225C22×C32—21no—
226C22×C32—42no—
227C22×C32—41no—
228C22×C32—42no—
229C22×C32—42no—
230C22×C32—44no—
231C22×C32Dic3⋊C412yes—
232C22×C32C4⋊Dic312yes—
233C22×C32C3×C4⋊C412yes—
234C22×C32C2×D1211yescharacteristic
235C22×C32C2×#6711yescharacteristic
236C22×C32C2×C3×D411yescharacteristic
237C32×C4—22no—
238C32×C4—22no—
239C32×C4—22no—
240C32×C4—21no—
241C32×C4—21no—
242C32×C4—21no—
243C32×C4—21no—
244C32×C4—21no—
245C32×C4—21no—
246C32×C4—42no—
247C32×C4—41no—
248C32×C4—41no—
249C32×C4—122no—
250C32×C4Dic3⋊C412yes—
251C32×C4C4⋊Dic312yes—
252C32×C4C3×C4⋊C412yes—
253C32×C4C2×Dic611yescharacteristic
254C32×C4C2×Dic611yescharacteristic
255C32×C4C2×C4×D311yescharacteristic
256C32×C4C22×Dic311yescharacteristic
257C32×C4C22×C3×C411yescharacteristic
258C32×C4C2×C3×Q811yescharacteristic
259C3⋊Dic3—42no—
260C3×Dic3—44no—
261C3×Dic3—42no—
262C3×Dic3—42no—
263C3×Dic3—124no—
264C3×Dic3—122no—
265C3×Dic3—122no—
266C22×C8—31no—
267#117—31no—
268#117—91no—
269#107—92no—
270#106—92no—
271C33#42711yescharacteristic, complemented, Sylow
272C23×C3—21no—
273C23×C3—21no—
274C23×C3—21no—
275C23×C3—41no—
276C23×C3#62611yescharacteristic
277C23×C3C3×D1211yescharacteristic
278C23×C3C3×#6711yescharacteristic
279C2×C3×C4—22no—
280C2×C3×C4—22no—
281C2×C3×C4—22no—
282C2×C3×C4—21no—
283C2×C3×C4—21no—
284C2×C3×C4—21no—
285C2×C3×C4—22no—
286C2×C3×C4—22no—
287C2×C3×C4—22no—
288C2×C3×C4—21no—
289C2×C3×C4—21no—
290C2×C3×C4—21no—
291C2×C3×C4—42no—
292C2×C3×C4—41no—
293C2×C3×C4—42no—
294C2×C3×C4—41no—
295C2×C3×C4—64no—
296C2×C3×C4—62no—
297C2×C3×C4—62no—
298C2×C3×C4—182no—
299C2×C3×C4D3×Dic312yes—
300C2×C3×C4#62611yescharacteristic
301C2×C3×C4#62712yes—
302C2×C3×C4C3×Dic612yes—
303C2×C3×C4C3×Dic612yes—
304C2×C3×C4C3×C4×D312yes—
305C2×C3×C4C3×D1211yescharacteristic
306C2×C3×C4C2×C3×Dic312yes—
307C2×C3×C4C3×#6711yescharacteristic
308C2×C3×C4C2×D3211yescharacteristic
309C2×C3×C4C22×C3×D311yescharacteristic
310C2×C3×C4C22×C3×D311yescharacteristic
311C2×Dic3—22no—
312C2×Dic3—64no—
313C2×Dic3—62no—
314C2×Dic3—62no—
315C3×C8—32no—
316C3×C8—32no—
317C3×C8—32no—
318C3×C8—32no—
319C3×C8—62no—
320C3×C8—62no—
321C3⋊C8C2×C3×Dic312yes—
322C3⋊C8C2×C3×Dic312yes—
323C2×C32—22no—
324C2×C32—24no—
325C2×C32—22no—
326C2×C32—22no—
327C2×C32—22no—
328C2×C32—21no—
329C2×C32—21no—
330C2×C32—22no—
331C2×C32—22no—
332C2×C32—21no—
333C2×C32—41no—
334C2×C32—42no—
335C2×C32—42no—
336C2×C32—42no—
337C2×C32—44no—
338C2×C32#90412yes—
339C2×C32C2×Dic3⋊C411yescharacteristic
340C2×C32C2×C4⋊Dic311yescharacteristic
341C2×C32#95012yes—
342C2×C32C2×C3×C4⋊C411yescharacteristic
343C2×C32C3×#13512yes—
344C22×C4C3×D3211yescharacteristic
345C2×C8—34no—
346C2×C8—31no—
347C2×C8—31no—
348C4⋊C4—32no—
349C4⋊C4—92no—
350C22⋊C4—61no—
351C22⋊C4—181no—
352C42—61no—
353C42—181no—
354C22×C3—24no—
355C22×C3—22no—
356C22×C3—22no—
357C22×C3—22no—
358C22×C3—21no—
359C22×C3—21no—
360C22×C3—21no—
361C22×C3—22no—
362C22×C3—22no—
363C22×C3—22no—
364C22×C3—42no—
365C22×C3—42no—
366C22×C3—44no—
367C22×C3—41no—
368C22×C3—42no—
369C22×C3Dic3⋊Dic312yes—
370C22×C3C3×Dic3⋊C412yes—
371C22×C3C3×C4⋊Dic312yes—
372C22×C3C2×#62611yescharacteristic
373C22×C3C2×C3×D1211yescharacteristic
374C22×C3C2×C3×#6711yescharacteristic
375C3×C4—21no—
376C3×C4—21no—
377C3×C4—21no—
378C3×C4—21no—
379C3×C4—21no—
380C3×C4—21no—
381C3×C4—22no—
382C3×C4—22no—
383C3×C4—22no—
384C3×C4—41no—
385C3×C4—41no—
386C3×C4—42no—
387C3×C4—124no—
388C3×C4—122no—
389C3×C4—122no—
390C3×C4—362no—
391C3×C4Dic3⋊Dic312yes—
392C3×C4C3×Dic3⋊C412yes—
393C3×C4C3×C4⋊Dic312yes—
394C3×C4C2×D3×Dic311yescharacteristic
395C3×C4C2×#62711yescharacteristic
396C3×C4C2×C3×Dic611yescharacteristic
397C3×C4C2×C3×Dic611yescharacteristic
398C3×C4C2×C3×C4×D311yescharacteristic
399C3×C4C22×C3×Dic311yescharacteristic
400Dic3—42no—
401Dic3—124no—
402Dic3—122no—
403Dic3—122no—
404C32—21no—
405C32—21no—
406C32—21no—
407C32—41no—
408C32#499811yescharacteristic, complemented
409C32#510611yescharacteristic, complemented
410C32C3×#42711yescharacteristic, complemented
411C23C3×#62611yescharacteristic
412C2×C4—62no—
413C2×C4—182no—
414C2×C4C3×D3×Dic312yes—
415C2×C4C3×#62611yescharacteristic
416C2×C4C3×#62712yes—
417C2×C4C2×C3×D3211yescharacteristic
418C8—32no—
419C8—32no—
420C2×C3—24no—
421C2×C3—22no—
422C2×C3—22no—
423C2×C3—22no—
424C2×C3—21no—
425C2×C3—21no—
426C2×C3—21no—
427C2×C3—22no—
428C2×C3—22no—
429C2×C3—22no—
430C2×C3—42no—
431C2×C3—42no—
432C2×C3—44no—
433C2×C3—41no—
434C2×C3—42no—
435C2×C3#6382912yes—
436C2×C3C2×Dic3⋊Dic311yescharacteristic
437C2×C3C3×#90412yes—
438C2×C3C2×C3×Dic3⋊C411yescharacteristic
439C2×C3C2×C3×C4⋊Dic311yescharacteristic
440C2×C3C3×#95012yes—
441C22—22no—
442C22C3×Dic3⋊Dic312yes—
443C22C2×C3×#62611yescharacteristic
444C4—122no—
445C4—362no—
446C4C3×Dic3⋊Dic312yes—
447C4C2×C3×D3×Dic311yescharacteristic
448C4C2×C3×#62711yescharacteristic
449C3—21no—
450C3—21no—
451C3—21no—
452C3—41no—
453C3#1059112211yescharacteristic, direct product factor
454C3C3×#499811yescharacteristic, complemented
455C3C3×#510611yescharacteristic, complemented
456C2—22no—
457C2C3×#6382912yes—
458C2C2×C3×Dic3⋊Dic311yescharacteristic
subgroupG/H
centerC2×C3×C4#626
commutatorC32×C4C22×C3×C4
FrattiniC2×C4C2×C3×D32
FittingC22×C33×C4C22
socleC22×C33C2×D4
prime radicalC23×C33D4
cosocleC2×C32×C4C23×C3
hypercenterC22×C3×C4D32
hypocenterC32C3×#427
LCSG ▷ C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2×C3×C4 ◁ C22×C3×C4

Complements

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

NH ≅ G/Ntype
4. C3×#63575C2semidirect
5. C3×#63573C2semidirect
8. #10591122C3direct (N × H)
16. C2×C32×C3⋊C8C4semidirect
35. #63575C2×C3semidirect
36. #63573C2×C3semidirect
86. C2×C3×C3⋊C8Dic3semidirect
87. C2×C3×C3⋊C8C3×C4semidirect
213. C2×C3⋊C8C3×Dic3semidirect
271. C33#427semidirect
408. C32#4998semidirect
409. C32#5106semidirect
410. C32C3×#427semidirect
453. C3#10591122direct (N × H)
454. C3C3×#4998semidirect
455. C3C3×#5106semidirect

Maximal subgroups

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

subgroupindex
C3×#641352
C32×#9262
C22×C32×C3⋊C82
C3×#635752
C3×#635732
C3×#51063
C3×#49983
#105911223