Finite Group Databasebeta

← C3×#10595984 #13179264

Subgroups — C3×#10595984 #13179264

Order 1728.

1652 conjugacy classes of subgroups.

The subgroup lattice has 490 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C33×C4⋊C4C211yesmaximal, characteristic, complemented
2C3×#64138C212yesmaximal
3C3×C4×#638C211yesmaximal, characteristic
4C3×#64135C212yesmaximal, complemented
5C3×#64133C211yesmaximal, characteristic, complemented
6C3×#64107C212yesmaximal, complemented
7C3×#64106C211yesmaximal, characteristic, complemented
8C3×#64101C211yesmaximal, characteristic, complemented
9C3×#64099C211yesmaximal, characteristic, complemented
10C3×#64098C212yesmaximal, complemented
11C3×#64096C211yesmaximal, characteristic, complemented
12#10595984C311yesmaximal, characteristic, direct product factor
13C3×#5513—34nomaximal
14C22×C33×C4C2211yescharacteristic
15C22×C33×C4C2212yes—
16C2×C3×#638C2211yescharacteristic
17C2×C3×C4×C3⋊D3C2211yescharacteristic
18C2×C3×#634C2211yescharacteristic
19C33×C4⋊C4C2211yescharacteristic
20C33×C4⋊C4C2212yes—
21C33×C4⋊C4C2211yescharacteristic
22C3×#3757C2212yes—
23C3×#3757C2211yescharacteristic
24C3×#3752C2212yes—
25C3×#3752C2212yes—
26C3×#3752C2211yescharacteristic
27C3×C4⋊(C3⋊Dic3)C2212yes—
28C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
29C3×C4⋊(C3⋊Dic3)C2212yes—
30C3×#3750C2212yes—
31C3×#3750C2211yescharacteristic
32C3×#3750C2212yes—
33C3×#3750C2212yes—
34C3×C4×C3⋊Dic3C2211yescharacteristic
35C3×C4×C3⋊Dic3C2212yes—
36C3×C4×C3⋊Dic3C2212yes—
37C2×C32×C4⋊C4—24no—
38C2×C32×C4⋊C4D314yescomplemented
39C2×C32×C4⋊C4C2×C311yescharacteristic, complemented
40#64138C2×C312yes—
41C4×#638C2×C311yescharacteristic
42#64135C2×C312yescomplemented
43#64133C2×C311yescharacteristic, complemented
44#64107C2×C312yescomplemented
45#64106C2×C311yescharacteristic, complemented
46#64101C2×C311yescharacteristic, complemented
47#64099C2×C311yescharacteristic, complemented
48#64098C2×C312yescomplemented
49#64096C2×C311yescharacteristic, complemented
50C3×#929—38no—
51C3×C4×#67—34no—
52C3×#926—38no—
53C3×#924—34no—
54C3×#898—38no—
55C3×#897—34no—
56C3×#892—34no—
57C3×#890—34no—
58C3×#889—38no—
59C3×#887—34no—
60C23×C33C2311yescharacteristic
61C22×C3×C3⋊D3C2311yescharacteristic
62C2×C33×C4—24no—
63C2×C33×C4—22no—
64C2×C33×C4C2312yes—
65C2×C33×C4C2311yescharacteristic
66C2×C33×C4C2312yes—
67C2×C33×C4C2311yescharacteristic
68C3×#638—22no—
69C2×C3×C3⋊Dic3C2311yescharacteristic
70C2×C3×C3⋊Dic3C2311yescharacteristic
71C2×C3×C3⋊Dic3C2311yescharacteristic
72C2×C3×C3⋊Dic3C2312yes—
73C2×C3×C3⋊Dic3C2312yes—
74C3×C4×C3⋊D3—22no—
75C3×#634—22no—
76C3×#552—91no—
77#5513—34no—
78C22×C32×C4—24no—
79C22×C32×C4—28no—
80C22×C32×C4C2×D314yes—
81C22×C32×C4C2×D318yes—
82C22×C32×C4C22×C311yescharacteristic
83C22×C32×C4C22×C312yes—
84C2×#638C22×C311yescharacteristic
85C2×C4×C3⋊D3C22×C311yescharacteristic
86C2×#634C22×C311yescharacteristic
87C2×C3×#67—34no—
88C2×C3×C4×D3—34no—
89C2×C3×Dic6—34no—
90C32×C4⋊C4—24no—
91C32×C4⋊C4—28no—
92C32×C4⋊C4—24no—
93C32×C4⋊C4C2×D314yes—
94C32×C4⋊C4C2×D318yes—
95C32×C4⋊C4C2×D314yes—
96C32×C4⋊C4C22×C311yescharacteristic
97C32×C4⋊C4C22×C312yes—
98C32×C4⋊C4C22×C311yescharacteristic
99#3757C22×C312yes—
100#3757C22×C311yescharacteristic
101#3752C22×C312yes—
102#3752C22×C312yes—
103#3752C22×C311yescharacteristic
104C4⋊(C3⋊Dic3)C22×C312yes—
105C4⋊(C3⋊Dic3)C22×C311yescharacteristic
106C4⋊(C3⋊Dic3)C22×C312yes—
107#3750C22×C312yes—
108#3750C22×C311yescharacteristic
109#3750C22×C312yes—
110#3750C22×C312yes—
111C4×C3⋊Dic3C22×C311yescharacteristic
112C4×C3⋊Dic3C22×C312yes—
113C4×C3⋊Dic3C22×C312yes—
114C3×#217—38no—
115C3×#217—34no—
116C3×(C2×D3)⋊C4—38no—
117C3×(C2×D3)⋊C4—38no—
118C3×(C2×D3)⋊C4—34no—
119C3×C4⋊Dic3—38no—
120C3×C4⋊Dic3—34no—
121C3×C4⋊Dic3—38no—
122C3×Dic3⋊C4—38no—
123C3×Dic3⋊C4—34no—
124C3×Dic3⋊C4—38no—
125C3×Dic3⋊C4—38no—
126C3×C4×Dic3—34no—
127C3×C4×Dic3—38no—
128C3×C4×Dic3—38no—
129C22×C33—21no—
130C22×C33—21no—
131C22×C33#4112yes—
132C22×C33C2411yescharacteristic
133C2×C3×C3⋊D3—21no—
134C2×C3×C3⋊D3—21no—
135C2×C3×C3⋊D3—21no—
136C33×C4—21no—
137C33×C4—22no—
138C33×C4—22no—
139C33×C4#4112yes—
140C3×C3⋊Dic3—22no—
141C3×C3⋊Dic3—21no—
142C3×C3⋊Dic3—21no—
143C3×C3⋊Dic3—21no—
144C3×C3⋊Dic3—22no—
145C3×#126—92no—
146C3×#125—92no—
147C3×#125—91no—
148C3×#123—92no—
149C3×C22⋊Q8—91no—
150C3×C22⋊Q8—91no—
151C3×C4×Q8—91no—
152C3×C4×D4—91no—
153C3×#117—92no—
154C3×#117—91no—
155C2×C3×C4⋊C4—24no—
156C2×C3×C4⋊C4C3×D314yescomplemented
157C2×C3×C4⋊C4C3⋊D311yescharacteristic, complemented
158#929—38no—
159C4×#67—34no—
160#926—38no—
161#924—34no—
162#898—38no—
163#897—34no—
164#892—34no—
165#890—34no—
166#889—38no—
167#887—34no—
168C23×C32—24no—
169C23×C32C22×D314yes—
170C23×C32C23×C311yescharacteristic
171C22×C3⋊D3C23×C311yescharacteristic
172C22×C3×D3—34no—
173C2×C32×C4—216no—
174C2×C32×C4—24no—
175C2×C32×C4—28no—
176C2×C32×C4—22no—
177C2×C32×C4—24no—
178C2×C32×C4—24no—
179C2×C32×C4—28no—
180C2×C32×C4—28no—
181C2×C32×C4—216no—
182C2×C32×C4—416no—
183C2×C32×C4C22×D318yes—
184C2×C32×C4C22×D314yes—
185C2×C32×C4C22×D318yes—
186C2×C32×C4C22×D314yes—
187C2×C32×C4C23×C312yes—
188C2×C32×C4C23×C311yescharacteristic
189C2×C32×C4C23×C312yes—
190C2×C32×C4C23×C311yescharacteristic
191#638—22no—
192C2×C3⋊Dic3C23×C311yescharacteristic
193C2×C3⋊Dic3C23×C311yescharacteristic
194C2×C3⋊Dic3C23×C311yescharacteristic
195C2×C3⋊Dic3C23×C312yes—
196C2×C3⋊Dic3C23×C312yes—
197C4×C3⋊D3—22no—
198#634—22no—
199C3×#67—68no—
200C2×C3×Dic3—34no—
201C2×C3×Dic3—34no—
202C2×C3×Dic3—34no—
203C2×C3×Dic3—38no—
204C2×C3×Dic3—38no—
205C3×C4×D3—68no—
206C3×Dic6—68no—
207#552—91noSylow
208C2×C33—22no—
209C2×C33C2×#4111yescharacteristic
210C2×C33C2×#4111yescharacteristic
211C2×C332−1+411yescharacteristic
212C3×C3⋊D3—41no—
213C2×C3×Q8—91no—
214C2×C3×D4—91no—
215C22×C3×C4—24no—
216C22×C3×C4—28no—
217C22×C3×C4—91no—
218C22×C3×C4C2×C3×D314yes—
219C22×C3×C4C2×C3×D318yes—
220C22×C3×C4C2×C3⋊D312yes—
221C22×C3×C4C2×C3⋊D311yescharacteristic
222C2×#67—34no—
223C2×C4×D3—34no—
224C2×Dic6—34no—
225C3×C4⋊C4—24no—
226C3×C4⋊C4—24no—
227C3×C4⋊C4—28no—
228C3×C4⋊C4—91no—
229C3×C4⋊C4—92no—
230C3×C4⋊C4—92no—
231C3×C4⋊C4—92no—
232C3×C4⋊C4—92no—
233C3×C4⋊C4—91no—
234C3×C4⋊C4—92no—
235C3×C4⋊C4C2×C3×D314yes—
236C3×C4⋊C4C2×C3×D318yes—
237C3×C4⋊C4C2×C3×D314yes—
238C3×C4⋊C4C2×C3⋊D311yescharacteristic
239C3×C4⋊C4C2×C3⋊D311yescharacteristic
240C3×C4⋊C4C2×C3⋊D312yes—
241C3×C22⋊C4—91no—
242C3×C22⋊C4—92no—
243C3×C22⋊C4—92no—
244C3×C22⋊C4—91no—
245C3×C22⋊C4—92no—
246C3×C42—92no—
247C3×C42—92no—
248C3×C42—91no—
249#217—38no—
250#217—34no—
251(C2×D3)⋊C4—38no—
252(C2×D3)⋊C4—38no—
253(C2×D3)⋊C4—34no—
254C4⋊Dic3—38no—
255C4⋊Dic3—34no—
256C4⋊Dic3—38no—
257Dic3⋊C4—38no—
258Dic3⋊C4—34no—
259Dic3⋊C4—38no—
260Dic3⋊C4—38no—
261C4×Dic3—34no—
262C4×Dic3—38no—
263C4×Dic3—38no—
264C22×C32—28no—
265C22×C32—28no—
266C22×C32—24no—
267C22×C32—24no—
268C22×C32—21no—
269C22×C32—21no—
270C22×C32—24no—
271C22×C32—28no—
272C22×C32#23718yes—
273C22×C32C3×#4112yes—
274C22×C32C23×D314yes—
275C22×C32C24×C311yescharacteristic
276C2×C3⋊D3—21no—
277C2×C3⋊D3—21no—
278C2×C3⋊D3—21no—
279C2×C3×D3—64no—
280C2×C3×D3—64no—
281C2×C3×D3—64no—
282C32×C4—28no—
283C32×C4—24no—
284C32×C4—21no—
285C32×C4—22no—
286C32×C4—22no—
287C32×C4—28no—
288C32×C4—28no—
289C32×C4—28no—
290C32×C4—48no—
291C32×C4—48no—
292C32×C4#23518yes—
293C32×C4C3×#4112yes—
294C3⋊Dic3—22no—
295C3⋊Dic3—21no—
296C3⋊Dic3—21no—
297C3⋊Dic3—21no—
298C3⋊Dic3—22no—
299C3×Dic3—68no—
300C3×Dic3—64no—
301C3×Dic3—64no—
302C3×Dic3—64no—
303C3×Dic3—68no—
304#126—92no—
305#125—92no—
306#125—91no—
307#123—92no—
308C22⋊Q8—91no—
309C22⋊Q8—91no—
310C4×Q8—91no—
311C4×D4—91no—
312#117—92no—
313#117—91no—
314C2×C4⋊C4C3×C3⋊D311yescharacteristic, complemented
315C33#55211yescharacteristic, complemented, Sylow
316C23×C3—24no—
317C23×C3—91no—
318C23×C3C22×C3×D314yes—
319C23×C3C22×C3⋊D311yescharacteristic
320C22×D3—34no—
321C3×Q8—182no—
322C3×D4—182no—
323C2×C3×C4—216no—
324C2×C3×C4—216no—
325C2×C3×C4—24no—
326C2×C3×C4—28no—
327C2×C3×C4—22no—
328C2×C3×C4—24no—
329C2×C3×C4—24no—
330C2×C3×C4—28no—
331C2×C3×C4—28no—
332C2×C3×C4—416no—
333C2×C3×C4—91no—
334C2×C3×C4—91no—
335C2×C3×C4—92no—
336C2×C3×C4—92no—
337C2×C3×C4—91no—
338C2×C3×C4—182no—
339C2×C3×C4C22×C3×D318yes—
340C2×C3×C4C22×C3×D314yes—
341C2×C3×C4C22×C3×D318yes—
342C2×C3×C4C22×C3×D314yes—
343C2×C3×C4C22×C3⋊D312yes—
344C2×C3×C4C22×C3⋊D311yescharacteristic
345C2×C3×C4C22×C3⋊D312yes—
346C2×C3×C4C22×C3⋊D311yescharacteristic
347#67—68no—
348C2×Dic3—34no—
349C2×Dic3—34no—
350C2×Dic3—34no—
351C2×Dic3—38no—
352C2×Dic3—38no—
353C4×D3—68no—
354Dic6—68no—
355C2×C32—216no—
356C2×C32—28no—
357C2×C32—22no—
358C2×C32—24no—
359C2×C32—24no—
360C2×C32—24no—
361C2×C32C2×#23514yes—
362C2×C32C2×#23714yes—
363C2×C32#100714yes—
364C2×C32C2×C3×#4111yescharacteristic
365C2×C32C2×C3×#4111yescharacteristic
366C2×C32C3×2−1+411yescharacteristic
367C3⋊D3—41no—
368C3×D3—124no—
369C2×Q8—91no—
370C2×D4—91no—
371C22×C4—91no—
372C22×C4C2×C3×C3⋊D312yes—
373C22×C4C2×C3×C3⋊D311yescharacteristic
374C4⋊C4—92no—
375C4⋊C4—91no—
376C4⋊C4—92no—
377C4⋊C4—91no—
378C4⋊C4—92no—
379C4⋊C4—92no—
380C4⋊C4—92no—
381C4⋊C4C2×C3×C3⋊D311yescharacteristic
382C4⋊C4C2×C3×C3⋊D311yescharacteristic
383C4⋊C4C2×C3×C3⋊D312yes—
384C22⋊C4—91no—
385C22⋊C4—92no—
386C22⋊C4—92no—
387C22⋊C4—91no—
388C22⋊C4—92no—
389C42—92no—
390C42—92no—
391C42—91no—
392C22×C3—28no—
393C22×C3—28no—
394C22×C3—24no—
395C22×C3—24no—
396C22×C3—21no—
397C22×C3—21no—
398C22×C3—24no—
399C22×C3—28no—
400C22×C3—181no—
401C22×C3—181no—
402C22×C3—181no—
403C22×C3C3×#23718yes—
404C22×C3#383012yes—
405C22×C3C23×C3×D314yes—
406C22×C3C23×C3⋊D311yescharacteristic
407C2×D3—64no—
408C2×D3—64no—
409C2×D3—64no—
410C3×C4—28no—
411C3×C4—24no—
412C3×C4—28no—
413C3×C4—28no—
414C3×C4—21no—
415C3×C4—22no—
416C3×C4—22no—
417C3×C4—28no—
418C3×C4—48no—
419C3×C4—48no—
420C3×C4—181no—
421C3×C4—182no—
422C3×C4—181no—
423C3×C4—181no—
424C3×C4—182no—
425C3×C4C3×#23518yes—
426C3×C4#382812yes—
427Dic3—68no—
428Dic3—64no—
429Dic3—64no—
430Dic3—64no—
431Dic3—68no—
432C32—24no—
433C32#551314yescomplemented
434C32C3×#55211yescharacteristic, complemented
435C23—91no—
436C23C22×C3×C3⋊D311yescharacteristic
437Q8—182no—
438D4—182no—
439C2×C4—24no—
440C2×C4—22no—
441C2×C4—91no—
442C2×C4—91no—
443C2×C4—92no—
444C2×C4—92no—
445C2×C4—91no—
446C2×C4—182no—
447C2×C4C22×C3×C3⋊D312yes—
448C2×C4C22×C3×C3⋊D311yescharacteristic
449C2×C4C22×C3×C3⋊D312yes—
450C2×C4C22×C3×C3⋊D311yescharacteristic
451C2×C3—216no—
452C2×C3—28no—
453C2×C3—22no—
454C2×C3—24no—
455C2×C3—24no—
456C2×C3—24no—
457C2×C3—361no—
458C2×C3C2×C3×#23514yes—
459C2×C3C2×C3×#23714yes—
460C2×C3C3×#100714yes—
461C2×C3C2×#382811yescharacteristic
462C2×C3C2×#383011yescharacteristic
463C2×C3#6436411yescharacteristic
464D3—124no—
465C22—21no—
466C22—21no—
467C22—181no—
468C22—181no—
469C22—181no—
470C22C3×#383012yes—
471C22C23×C3×C3⋊D311yescharacteristic
472C4—21no—
473C4—22no—
474C4—22no—
475C4—181no—
476C4—182no—
477C4—181no—
478C4—181no—
479C4—182no—
480C4C3×#382812yes—
481C3—24no—
482C3C3×#551314yescomplemented
483C3#1059598411yescharacteristic, direct product factor
484C2—22no—
485C2—361no—
486C2C2×C3×#382811yescharacteristic
487C2C2×C3×#383011yescharacteristic
488C2C3×#6436411yescharacteristic
subgroupG/H
centerC22×C3C23×C3⋊D3
commutator, cosocleC22×C32C24×C3
FrattiniC22C23×C3×C3⋊D3
FittingC2×C33×C4⋊C4C2
socleC22×C33C24
prime radicalC2×C3×#638C22
hypercenterC2×C3×C4⋊C4C3⋊D3
hypocenterC32C3×#552
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C2×C3×C4⋊C4

Complements

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

NH ≅ G/Ntype
1. C2×C33×C4⋊C4C2semidirect
4. C3×#64135C2semidirect
5. C3×#64133C2semidirect
6. C3×#64107C2semidirect
7. C3×#64106C2semidirect
8. C3×#64101C2semidirect
9. C3×#64099C2semidirect
10. C3×#64098C2semidirect
11. C3×#64096C2semidirect
12. #10595984C3direct (N × H)
38. C2×C32×C4⋊C4D3semidirect
39. C2×C32×C4⋊C4C2×C3semidirect
42. #64135C2×C3semidirect
43. #64133C2×C3semidirect
44. #64107C2×C3semidirect
45. #64106C2×C3semidirect
46. #64101C2×C3semidirect
47. #64099C2×C3semidirect
48. #64098C2×C3semidirect
49. #64096C2×C3semidirect
156. C2×C3×C4⋊C4C3×D3semidirect
157. C2×C3×C4⋊C4C3⋊D3semidirect
314. C2×C4⋊C4C3×C3⋊D3semidirect
315. C33#552semidirect
433. C32#5513semidirect
434. C32C3×#552semidirect
482. C3C3×#5513semidirect
483. C3#10595984direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C33×C4⋊C42
C3×#641382
C3×C4×#6382
C3×#641352
C3×#641332
C3×#641072
C3×#641062
C3×#641012
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