Finite Group Databasebeta

← C3×#10594526 #13173944

Subgroups — C3×#10594526 #13173944

Order 1728.

1364 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×C3×D3×Dic6C212yesmaximal, complemented
2C3×#64096C211yesmaximal, characteristic, complemented
3C32×#897C212yesmaximal, complemented
4C3×#63919C212yesmaximal, complemented
5C3×#63902C212yesmaximal
6C3×C4×#625C211yesmaximal, characteristic
7C3×#63899C214yesmaximal, complemented
8C3×#63873C211yesmaximal, characteristic, complemented
9C3×#5632—32nomaximal
10#10594526C311yesmaximal, characteristic, direct product factor
11C2×C3×#634C2211yescharacteristic
12C2×C32×C4×D3C2212yes—
13C2×C32×Dic6C2212yes—
14C2×C3×#627C2214yes—
15C2×C3×#625C2211yescharacteristic
16C2×C3×D3×Dic3C2214yes—
17C3×D3×Dic6—28no—
18C33×C4⋊C4C2211yescharacteristic
19C3×#3750C2212yes—
20C3×C4×C3⋊Dic3C2211yescharacteristic
21C3×C4×C3⋊Dic3C2212yes—
22C32×(C2×D3)⋊C4C2214yes—
23C32×C4⋊Dic3C2214yes—
24C3×#3724C2212yes—
25C3×#3724C2212yes—
26C3×#3724C2211yescharacteristic
27C3×(C2×D3)⋊Dic3C2212yes—
28C2×C3×D3×Q8—32no—
29C22×C3×Dic6—32no—
30C2×D3×Dic6C2×C312yescomplemented
31C32×C22⋊Q8—32no—
32#64096C2×C311yescharacteristic, complemented
33C3×#944—32no—
34C3×C4×#67—32no—
35C3×#897—22no—
36C3×#897D312yescomplemented
37C3×#897C2×C312yescomplemented
38C3×#896—34no—
39C3×#888—34no—
40C3×#887—32no—
41C3×#887—61no—
42C3×#67⋊C4—34no—
43C3×#878—34no—
44#63919C2×C312yescomplemented
45#63902C2×C312yes—
46C4×#625C2×C311yescharacteristic
47#63899C2×C314yescomplemented
48#63873C2×C311yescharacteristic, complemented
49C22×C32×D3C2312yes—
50C2×C33×C4C2312yes—
51C2×C33×C4C2311yescharacteristic
52C2×C3×C3⋊Dic3C2311yescharacteristic
53C2×C3×C3⋊Dic3C2312yes—
54C2×C3×C3⋊Dic3C2311yescharacteristic
55C3×#634D414yescomplemented
56C2×C32×Dic3C2314yes—
57C2×C32×Dic3C2312yes—
58C32×C4×D3—24no—
59C32×Dic6—24no—
60C3×#627—28no—
61C3×#625Q814yescomplemented
62C3×D3×Dic3—28no—
63C3×D4×Q8—91no—
64#5632—32no—
65C2×C32×Q8—32no—
66C22×C32×C4—32no—
67C2×#634C22×C311yescharacteristic
68C2×C3×#67—32no—
69C22×C3×Dic3—34no—
70C3×D3×Q8—68no—
71C2×C3×C4×D3—22no—
72C2×C3×C4×D3—34no—
73C2×C3×C4×D3C2×D312yes—
74C2×C3×C4×D3C22×C312yes—
75C2×C3×Dic6—22no—
76C2×C3×Dic6—32no—
77C2×C3×Dic6—34no—
78C2×C3×Dic6—34no—
79C2×C3×Dic6—68no—
80C2×C3×Dic6—61no—
81C2×C3×Dic6C2×D312yes—
82C2×C3×Dic6C22×C312yes—
83C2×#627C22×C314yes—
84C2×#625C22×C311yescharacteristic
85C2×D3×Dic3C22×C314yes—
86D3×Dic6—28no—
87C32×C4⋊C4—21no—
88C32×C4⋊C4—22no—
89C32×C4⋊C4—34no—
90C32×C4⋊C4—41no—
91C32×C4⋊C4C2×D312yes—
92C32×C4⋊C4C22×C311yescharacteristic
93C32×C22⋊C4—34no—
94#3750C22×C312yes—
95C4×C3⋊Dic3C22×C311yescharacteristic
96C4×C3⋊Dic3C22×C312yes—
97C3×#217—32no—
98C3×(C2×D3)⋊C4—24no—
99C3×(C2×D3)⋊C4—32no—
100C3×(C2×D3)⋊C4C2×D314yes—
101C3×(C2×D3)⋊C4C22×C314yes—
102C3×C4⋊Dic3—24no—
103C3×C4⋊Dic3C2×D314yes—
104C3×C4⋊Dic3C22×C314yes—
105C3×Dic3⋊C4—34no—
106C3×Dic3⋊C4—34no—
107C3×Dic3⋊C4—34no—
108C3×Dic3⋊C4—32no—
109C3×Dic3⋊C4—62no—
110C3×C4×Dic3—34no—
111C3×C4×Dic3—32no—
112C3×C4×Dic3—62no—
113C3×C4×Dic3—61no—
114#3724C22×C312yes—
115#3724C22×C312yes—
116#3724C22×C311yescharacteristic
117(C2×D3)⋊Dic3C22×C312yes—
118C22×C33C2411yescharacteristic
119C2×C32×D3—22no—
120C2×C32×D3—22no—
121C2×C32×D3C2×Q814yescomplemented
122C33×C4—22no—
123C33×C4C2×D412yes—
124C3×C3⋊Dic3—21no—
125C3×C3⋊Dic3C2×D414yes—
126C3×C3⋊Dic3C2×Q812yescomplemented
127C32×Dic3—24no—
128C32×Dic3—22no—
129C22×C3×Q8—92no—
130C2×D3×Q8—32no—
131C22×Dic6—32no—
132C3×C4⋊Q8—92no—
133C3×C4⋊Q8—91no—
134C3×C22⋊Q8—32no—
135C3×C22⋊Q8—32no—
136C3×C22⋊Q8—62no—
137C3×C22⋊Q8—94no—
138C3×C4×Q8—91no—
139C3×C4×D4—92no—
140C3×C4×D4—91no—
141#944—32no—
142C4×#67—32no—
143#897C3×D312yescomplemented
144#896—34no—
145#888—34no—
146#887—32no—
147#887—61no—
148#67⋊C4—34no—
149#878—34no—
150C23×C32—32no—
151C22×C3×D3—22no—
152C22×C3×D3C22×D312yes—
153C22×C3×D3C23×C312yes—
154C32×Q8—64no—
155C2×C32×C4—21no—
156C2×C32×C4—22no—
157C2×C32×C4—22no—
158C2×C32×C4—24no—
159C2×C32×C4—34no—
160C2×C32×C4—32no—
161C2×C32×C4—41no—
162C2×C32×C4—42no—
163C2×C32×C4—64no—
164C2×C32×C4C22×D314yes—
165C2×C32×C4C22×D312yes—
166C2×C32×C4C23×C312yes—
167C2×C32×C4C23×C311yescharacteristic
168C2×C3⋊Dic3C23×C311yescharacteristic
169C2×C3⋊Dic3C23×C312yes—
170C2×C3⋊Dic3C23×C311yescharacteristic
171#634C3×D414yescomplemented
172C3×#67—38no—
173C2×C3×Dic3—22no—
174C2×C3×Dic3—24no—
175C2×C3×Dic3—32no—
176C2×C3×Dic3—34no—
177C2×C3×Dic3—32no—
178C2×C3×Dic3—61no—
179C2×C3×Dic3—62no—
180C2×C3×Dic3—61no—
181C2×C3×Dic3—68no—
182C2×C3×Dic3C22×D314yes—
183C2×C3×Dic3C22×D312yes—
184C2×C3×Dic3C23×C312yes—
185C2×C3×Dic3C23×C314yes—
186C3×C4×D3—24no—
187C3×C4×D3—24no—
188C3×C4×D3—44no—
189C3×C4×D3—68no—
190C3×Dic6—24no—
191C3×Dic6—24no—
192C3×Dic6—38no—
193C3×Dic6—44no—
194C3×Dic6—64no—
195C3×Dic6—68no—
196C3×Dic6—68no—
197#627—28no—
198#625C3×Q814yescomplemented
199D3×Dic3—28no—
200D4×Q8—91noSylow
201C2×C33C22×D411yescharacteristic
202C2×C33C22×Q811yescharacteristic
203C2×C332−1+411yescharacteristic
204C32×D3—24no—
205C2×C3×Q8—32no—
206C2×C3×Q8—32no—
207C2×C3×Q8—62no—
208C2×C3×Q8—94no—
209C2×C3×Q8—91no—
210C2×C3×Q8—188no—
211C2×C3×D4—91no—
212C22×C3×C4—32no—
213C22×C3×C4—32no—
214C22×C3×C4—62no—
215C22×C3×C4—94no—
216C2×#67—32no—
217C22×Dic3—34no—
218D3×Q8—68no—
219C2×C4×D3—34no—
220C2×C4×D3C2×C3×D312yes—
221C2×Dic6—32no—
222C2×Dic6—34no—
223C2×Dic6—34no—
224C2×Dic6—61no—
225C2×Dic6—68no—
226C2×Dic6C2×C3×D312yes—
227C3×C4⋊C4—21no—
228C3×C4⋊C4—22no—
229C3×C4⋊C4—34no—
230C3×C4⋊C4—34no—
231C3×C4⋊C4—41no—
232C3×C4⋊C4—64no—
233C3×C4⋊C4—92no—
234C3×C4⋊C4—91no—
235C3×C4⋊C4—92no—
236C3×C4⋊C4—92no—
237C3×C4⋊C4D3211yescharacteristic
238C3×C4⋊C4C2×C3×D312yes—
239C3×C22⋊C4—34no—
240C3×C22⋊C4—34no—
241C3×C22⋊C4—64no—
242C3×C22⋊C4—92no—
243C3×C42—92no—
244C3×C42—91no—
245#217—32no—
246(C2×D3)⋊C4—32no—
247(C2×D3)⋊C4C2×C3×D314yes—
248C4⋊Dic3C2×C3×D314yes—
249Dic3⋊C4—34no—
250Dic3⋊C4—34no—
251Dic3⋊C4—34no—
252Dic3⋊C4—32no—
253Dic3⋊C4—62no—
254C4×Dic3—34no—
255C4×Dic3—32no—
256C4×Dic3—62no—
257C4×Dic3—61no—
258C22×C32—21no—
259C22×C32—22no—
260C22×C32—34no—
261C22×C32—41no—
262C22×C32—62no—
263C22×C32—62no—
264C22×C32C23×D312yes—
265C22×C32C24×C311yescharacteristic
266C2×C3×D3—22no—
267C2×C3×D3—22no—
268C2×C3×D3—22no—
269C2×C3×D3—22no—
270C2×C3×D3—24no—
271C2×C3×D3—42no—
272C2×C3×D3—42no—
273C2×C3×D3D3×Q814yescomplemented
274C2×C3×D3C2×C3×Q814yescomplemented
275C32×C4—24no—
276C32×C4—22no—
277C32×C4—22no—
278C32×C4—24no—
279C32×C4—42no—
280C32×C4—42no—
281C32×C4—44no—
282C32×C4—62no—
283C32×C4—64no—
284C32×C4—82no—
285C32×C4D3×D414yes—
286C32×C4C2×C3×D412yes—
287C3⋊Dic3—21no—
288C3⋊Dic3C2×C3×D414yes—
289C3⋊Dic3C2×C3×Q812yescomplemented
290C3×Dic3—24no—
291C3×Dic3—22no—
292C3×Dic3—24no—
293C3×Dic3—22no—
294C3×Dic3—38no—
295C3×Dic3—34no—
296C3×Dic3—44no—
297C3×Dic3—42no—
298C3×Dic3—64no—
299C3×Dic3—62no—
300C3×Dic3—62no—
301C3×Dic3—121no—
302C22×Q8—92no—
303C4⋊Q8—92no—
304C4⋊Q8—91no—
305C22⋊Q8—32no—
306C22⋊Q8—94no—
307C4×Q8—91no—
308C4×D4—92no—
309C4×D4—91no—
310C33D4×Q811yescharacteristic, complemented, Sylow
311C23×C3—32no—
312C23×C3—32no—
313C23×C3—62no—
314C22×D3C22×C3×D312yes—
315C3×Q8—64no—
316C3×Q8—64no—
317C3×Q8—94no—
318C3×Q8—124no—
319C3×Q8—188no—
320C3×D4—94no—
321C2×C3×C4—22no—
322C2×C3×C4—21no—
323C2×C3×C4—24no—
324C2×C3×C4—22no—
325C2×C3×C4—34no—
326C2×C3×C4—32no—
327C2×C3×C4—34no—
328C2×C3×C4—32no—
329C2×C3×C4—41no—
330C2×C3×C4—42no—
331C2×C3×C4—64no—
332C2×C3×C4—62no—
333C2×C3×C4—64no—
334C2×C3×C4—64no—
335C2×C3×C4—91no—
336C2×C3×C4—92no—
337C2×C3×C4—91no—
338C2×C3×C4—124no—
339C2×C3×C4—188no—
340C2×C3×C4C2×D3212yes—
341C2×C3×C4C2×D3211yescharacteristic
342C2×C3×C4C22×C3×D314yes—
343C2×C3×C4C22×C3×D312yes—
344#67—38no—
345C2×Dic3—32no—
346C2×Dic3—34no—
347C2×Dic3—32no—
348C2×Dic3—61no—
349C2×Dic3—62no—
350C2×Dic3—61no—
351C2×Dic3—68no—
352C2×Dic3C22×C3×D314yes—
353C2×Dic3C22×C3×D312yes—
354C4×D3—24no—
355C4×D3—68no—
356Dic6—24no—
357Dic6—38no—
358Dic6—64no—
359Dic6—68no—
360Dic6—68no—
361C2×C32—21no—
362C2×C32—21no—
363C2×C32—21no—
364C2×C32—22no—
365C2×C32—22no—
366C2×C32—22no—
367C2×C32—41no—
368C2×C32—41no—
369C2×C32—41no—
370C2×C32—64no—
371C2×C32C2×D3×D412yes—
372C2×C32C2×D3×Q812yes—
373C2×C32#101012yes—
374C2×C32C22×C3×D411yescharacteristic
375C2×C32C22×C3×Q811yescharacteristic
376C2×C32C3×2−1+411yescharacteristic
377C3×D3—24no—
378C3×D3—24no—
379C3×D3—44no—
380C2×Q8—32no—
381C2×Q8—94no—
382C2×Q8—91no—
383C2×Q8—188no—
384C2×D4—91no—
385C22×C4—32no—
386C22×C4—94no—
387C4⋊C4—34no—
388C4⋊C4—92no—
389C4⋊C4—91no—
390C4⋊C4—92no—
391C4⋊C4—92no—
392C4⋊C4C3×D3211yescharacteristic
393C22⋊C4—34no—
394C22⋊C4—92no—
395C42—92no—
396C42—91no—
397C22×C3—21no—
398C22×C3—22no—
399C22×C3—34no—
400C22×C3—34no—
401C22×C3—41no—
402C22×C3—64no—
403C22×C3—62no—
404C22×C3—62no—
405C22×C3—62no—
406C22×C3—62no—
407C22×C3—122no—
408C22×C3—122no—
409C22×C3C22×D3211yescharacteristic
410C22×C3C23×C3×D312yes—
411C2×D3—22no—
412C2×D3—22no—
413C2×D3C3×D3×Q814yescomplemented
414C3×C4—22no—
415C3×C4—24no—
416C3×C4—22no—
417C3×C4—24no—
418C3×C4—42no—
419C3×C4—44no—
420C3×C4—42no—
421C3×C4—62no—
422C3×C4—64no—
423C3×C4—62no—
424C3×C4—64no—
425C3×C4—82no—
426C3×C4—94no—
427C3×C4—92no—
428C3×C4—122no—
429C3×C4—124no—
430C3×C4—181no—
431C3×C4#380212yes—
432C3×C4C3×D3×D414yes—
433Dic3—24no—
434Dic3—22no—
435Dic3—38no—
436Dic3—34no—
437Dic3—64no—
438Dic3—62no—
439Dic3—62no—
440Dic3—121no—
441C32—22no—
442C32—21no—
443C32—41no—
444C32#563212yescomplemented
445C32C3×D4×Q811yescharacteristic, complemented
446C23—32no—
447Q8—64no—
448Q8—94no—
449Q8—188no—
450D4—94no—
451C2×C4—34no—
452C2×C4—32no—
453C2×C4—64no—
454C2×C4—91no—
455C2×C4—92no—
456C2×C4—91no—
457C2×C4—188no—
458C2×C4C2×C3×D3212yes—
459C2×C4C2×C3×D3211yescharacteristic
460C2×C3—21no—
461C2×C3—22no—
462C2×C3—21no—
463C2×C3—22no—
464C2×C3—21no—
465C2×C3—22no—
466C2×C3—41no—
467C2×C3—41no—
468C2×C3—41no—
469C2×C3—64no—
470C2×C3—64no—
471C2×C3—124no—
472C2×C3C2×#380211yescharacteristic
473C2×C3#6430911yescharacteristic
474C2×C3D32×Q811yescharacteristic
475C2×C3C2×C3×D3×D412yes—
476C2×C3C2×C3×D3×Q812yes—
477C2×C3C3×#101012yes—
478D3—24no—
479C22—34no—
480C22—62no—
481C22—62no—
482C22C22×C3×D3211yescharacteristic
483C4—22no—
484C4—62no—
485C4—64no—
486C4—92no—
487C4—94no—
488C4—181no—
489C4C3×#380212yes—
490C3—22no—
491C3—21no—
492C3—41no—
493C3#1059452611yescharacteristic, direct product factor
494C3C3×#563212yescomplemented
495C2—64no—
496C2C2×C3×#380211yescharacteristic
497C2C3×#6430911yescharacteristic
498C2C3×D32×Q811yescharacteristic
subgroupG/H
centerC22×C3C22×D32
commutator, cosocleC22×C32C24×C3
FrattiniC22C22×C3×D32
FittingC33×C4⋊C4C22
socleC22×C33C24
prime radicalC2×C3×#625C22
hypercenterC3×C4⋊C4D32
hypocenterC32C3×D4×Q8
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C3×C4⋊C4

Complements

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

NH ≅ G/Ntype
1. C2×C3×D3×Dic6C2semidirect
2. C3×#64096C2semidirect
3. C32×#897C2semidirect
4. C3×#63919C2semidirect
7. C3×#63899C2semidirect
8. C3×#63873C2semidirect
10. #10594526C3direct (N × H)
30. C2×D3×Dic6C2×C3semidirect
32. #64096C2×C3semidirect
36. C3×#897D3semidirect
37. C3×#897C2×C3semidirect
44. #63919C2×C3semidirect
47. #63899C2×C3semidirect
48. #63873C2×C3semidirect
55. C3×#634D4semidirect
61. C3×#625Q8semidirect
121. C2×C32×D3C2×Q8semidirect
126. C3×C3⋊Dic3C2×Q8semidirect
143. #897C3×D3semidirect
171. #634C3×D4semidirect
198. #625C3×Q8semidirect
273. C2×C3×D3D3×Q8semidirect
274. C2×C3×D3C2×C3×Q8semidirect
289. C3⋊Dic3C2×C3×Q8semidirect
310. C33D4×Q8semidirect
413. C2×D3C3×D3×Q8semidirect
444. C32#5632semidirect
445. C32C3×D4×Q8semidirect
493. C3#10594526direct (N × H)
494. C3C3×#5632semidirect

Maximal subgroups

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

subgroupindex
C2×C3×D3×Dic62
C3×#640962
C32×#8972
C3×#639192
C3×#639022
C3×C4×#6252
C3×#638992
C3×#638732
C3×#56323
#105945263