Finite Group Databasebeta

← C3×#10594374 #13173936

Subgroups — C3×#10594374 #13173936

Order 1728.

1414 conjugacy classes of subgroups.

The subgroup lattice has 494 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#3797C212yesmaximal
2C3×C4⋊C4×C3⋊D3C211yesmaximal, characteristic
3C32×#888C212yesmaximal, complemented
4C3×#63918C212yesmaximal, complemented
5C3×#63900C211yesmaximal, characteristic
6C3×#63883C212yesmaximal
7C3×C4×#624C211yesmaximal, characteristic
8C3×#63839C214yesmaximal, complemented
9C3×D3×C4⋊Q8—32nomaximal
10#10594374C311yesmaximal, characteristic, direct product factor
11C2×C3×C4×C3⋊D3C2212yes—
12C2×C3×C4×C3⋊D3C2211yescharacteristic
13C2×C32×Dic6C2214yes—
14C2×C3×#627C2214yes—
15C2×C3×#624C2212yes—
16C2×C3×#624C2211yescharacteristic
17C2×C3×#624C2211yescharacteristic
18C3×#3797—28no—
19C33×C4⋊C4C2211yescharacteristic
20C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
21C3×#3750C2212yes—
22C32×C4⋊Dic3C2212yes—
23C32×Dic3⋊C4C2214yes—
24C32×C4×Dic3C2212yes—
25C3×#3724C2212yes—
26C3×Dic3⋊Dic3C2214yes—
27C3×Dic32C2212yes—
28C2×C3×D3×Q8—34no—
29C2×#3797C2×C312yes—
30C32×C4⋊Q8—32no—
31C4⋊C4×C3⋊D3C2×C311yescharacteristic
32C3×#944—34no—
33C3×D3×C4⋊C4—32no—
34C3×D3×C4⋊C4—32no—
35C3×D3×C4⋊C4—34no—
36C3×D3×C4⋊C4—61no—
37C3×#888—22no—
38C3×#888—34no—
39C3×#888—32no—
40C3×#888D312yescomplemented
41C3×#888C2×C312yescomplemented
42C3×C42×D3—32no—
43C3×#869—32no—
44#63918C2×C312yescomplemented
45#63900C2×C311yescharacteristic
46#63883C2×C312yes—
47C4×#624C2×C311yescharacteristic
48#63839C2×C314yescomplemented
49C22×C3×C3⋊D3C2311yescharacteristic
50C2×C33×C4C2312yes—
51C2×C33×C4C2311yescharacteristic
52C2×C3×C3⋊Dic3C2312yes—
53C2×C3×C3⋊Dic3C2311yescharacteristic
54C3×C4×C3⋊D3—24no—
55C3×C4×C3⋊D3Q814yescomplemented
56C2×C32×Dic3C2314yes—
57C2×C32×Dic3C2312yes—
58C2×C32×Dic3C2312yes—
59C32×Dic6—28no—
60C3×#627—28no—
61C3×#624—24no—
62C3×#624D414yes—
63C3×#624Q814yescomplemented
64C2×C3×C4⋊Q8—91no—
65D3×C4⋊Q8—32no—
66C2×C32×Q8—34no—
67C2×C4×C3⋊D3C22×C312yes—
68C2×C4×C3⋊D3C22×C311yescharacteristic
69C3×D3×Q8—616no—
70C2×C3×C4×D3—32no—
71C2×C3×C4×D3—32no—
72C2×C3×C4×D3—32no—
73C2×C3×C4×D3—34no—
74C2×C3×C4×D3—34no—
75C2×C3×C4×D3—61no—
76C2×C3×C4×D3—62no—
77C2×C3×Dic6—24no—
78C2×C3×Dic6—34no—
79C2×C3×Dic6—34no—
80C2×C3×Dic6C2×D314yes—
81C2×C3×Dic6C22×C314yes—
82C2×#627C22×C314yes—
83C2×#624C22×C312yes—
84C2×#624C22×C311yescharacteristic
85C2×#624C22×C311yescharacteristic
86#3797—28no—
87C32×C4⋊C4—21no—
88C32×C4⋊C4—22no—
89C32×C4⋊C4—34no—
90C32×C4⋊C4—32no—
91C32×C4⋊C4—41no—
92C32×C4⋊C4C2×D312yes—
93C32×C4⋊C4C22×C311yescharacteristic
94C32×C42—32no—
95C4⋊(C3⋊Dic3)C22×C311yescharacteristic
96#3750C22×C312yes—
97C3×C4⋊Dic3—22no—
98C3×C4⋊Dic3—32no—
99C3×C4⋊Dic3—34no—
100C3×C4⋊Dic3—61no—
101C3×C4⋊Dic3C2×D312yes—
102C3×C4⋊Dic3C22×C312yes—
103C3×Dic3⋊C4—24no—
104C3×Dic3⋊C4—34no—
105C3×Dic3⋊C4—34no—
106C3×Dic3⋊C4—34no—
107C3×Dic3⋊C4—62no—
108C3×Dic3⋊C4C2×D314yes—
109C3×Dic3⋊C4C22×C314yes—
110C3×C4×Dic3—22no—
111C3×C4×Dic3—32no—
112C3×C4×Dic3—32no—
113C3×C4×Dic3C2×D312yes—
114C3×C4×Dic3C22×C312yes—
115#3724C22×C312yes—
116Dic3⋊Dic3C22×C314yes—
117Dic32C22×C312yes—
118C22×C33C2411yescharacteristic
119C2×C3×C3⋊D3C2×D412yes—
120C2×C3×C3⋊D3C2×Q812yes—
121C2×C3×C3⋊D3C2×Q812yes—
122C33×C4—22no—
123C33×C4C2×Q812yescomplemented
124C3×C3⋊Dic3—22no—
125C3×C3⋊Dic3C2×Q812yescomplemented
126C32×Dic3—24no—
127C32×Dic3C2×D414yes—
128C32×Dic3C2×Q814yescomplemented
129C22×C3×Q8—92no—
130C2×D3×Q8—34no—
131C3×C4⋊Q8—32no—
132C3×C4⋊Q8—32no—
133C3×C4⋊Q8—62no—
134C3×C4⋊Q8—92no—
135C3×C4⋊Q8—94no—
136C2×C3×C4⋊C4—92no—
137C2×C3×C4⋊C4—91no—
138C2×C3×C4⋊C4—91no—
139C2×C3×C42—91no—
140#944—34no—
141D3×C4⋊C4—32no—
142D3×C4⋊C4—32no—
143D3×C4⋊C4—34no—
144D3×C4⋊C4—61no—
145#888—34no—
146#888—32no—
147#888C3×D312yescomplemented
148C42×D3—32no—
149#869—32no—
150C22×C3⋊D3C23×C311yescharacteristic
151C22×C3×D3—32no—
152C22×C3×D3—61no—
153C32×Q8—68no—
154C2×C32×C4—21no—
155C2×C32×C4—22no—
156C2×C32×C4—22no—
157C2×C32×C4—24no—
158C2×C32×C4—32no—
159C2×C32×C4—32no—
160C2×C32×C4—34no—
161C2×C32×C4—41no—
162C2×C32×C4—42no—
163C2×C32×C4C22×D314yes—
164C2×C32×C4C22×D312yes—
165C2×C32×C4C23×C312yes—
166C2×C32×C4C23×C311yescharacteristic
167C2×C3⋊Dic3C23×C312yes—
168C2×C3⋊Dic3C23×C311yescharacteristic
169C4×C3⋊D3—24no—
170C4×C3⋊D3C3×Q814yescomplemented
171C2×C3×Dic3—22no—
172C2×C3×Dic3—22no—
173C2×C3×Dic3—24no—
174C2×C3×Dic3—32no—
175C2×C3×Dic3—34no—
176C2×C3×Dic3—61no—
177C2×C3×Dic3—62no—
178C2×C3×Dic3C22×D314yes—
179C2×C3×Dic3C22×D312yes—
180C2×C3×Dic3C22×D312yes—
181C2×C3×Dic3C23×C314yes—
182C2×C3×Dic3C23×C312yes—
183C2×C3×Dic3C23×C312yes—
184C3×C4×D3—38no—
185C3×C4×D3—38no—
186C3×C4×D3—38no—
187C3×C4×D3—64no—
188C3×C4×D3—68no—
189C3×C4×D3—68no—
190C3×C4×D3—124no—
191C3×Dic6—28no—
192C3×Dic6—28no—
193C3×Dic6—48no—
194C3×Dic6—68no—
195C3×Dic6—68no—
196#627—28no—
197#624—24no—
198#624C3×D414yes—
199#624C3×Q814yescomplemented
200C2×C4⋊Q8—91noSylow
201C2×C33C22×D411yescharacteristic
202C2×C33C22×Q811yescharacteristic
203C2×C33C22×Q811yescharacteristic
204C3×C3⋊D3C4⋊Q814yescomplemented
205C2×C3×Q8—34no—
206C2×C3×Q8—34no—
207C2×C3×Q8—64no—
208C2×C3×Q8—94no—
209C2×C3×Q8—188no—
210C22×C3×C4—91no—
211C22×C3×C4—91no—
212C22×C3×C4—92no—
213C22×C3×C4—91no—
214C22×C3×C4—92no—
215D3×Q8—616no—
216C2×C4×D3—32no—
217C2×C4×D3—32no—
218C2×C4×D3—32no—
219C2×C4×D3—34no—
220C2×C4×D3—34no—
221C2×C4×D3—61no—
222C2×C4×D3—62no—
223C2×Dic6—34no—
224C2×Dic6—34no—
225C2×Dic6C2×C3×D314yes—
226C3×C4⋊C4—21no—
227C3×C4⋊C4—22no—
228C3×C4⋊C4—34no—
229C3×C4⋊C4—32no—
230C3×C4⋊C4—34no—
231C3×C4⋊C4—32no—
232C3×C4⋊C4—41no—
233C3×C4⋊C4—64no—
234C3×C4⋊C4—62no—
235C3×C4⋊C4—92no—
236C3×C4⋊C4—94no—
237C3×C4⋊C4—92no—
238C3×C4⋊C4—91no—
239C3×C4⋊C4D3211yescharacteristic
240C3×C4⋊C4C2×C3×D312yes—
241C3×C42—32no—
242C3×C42—32no—
243C3×C42—62no—
244C3×C42—92no—
245C4⋊Dic3—32no—
246C4⋊Dic3—34no—
247C4⋊Dic3—61no—
248C4⋊Dic3C2×C3×D312yes—
249Dic3⋊C4—34no—
250Dic3⋊C4—34no—
251Dic3⋊C4—34no—
252Dic3⋊C4—62no—
253Dic3⋊C4C2×C3×D314yes—
254C4×Dic3—32no—
255C4×Dic3—32no—
256C4×Dic3C2×C3×D312yes—
257C22×C32—21no—
258C22×C32—22no—
259C22×C32—41no—
260C22×C32C23×D312yes—
261C22×C32C24×C311yescharacteristic
262C2×C3⋊D3C2×C3×D412yes—
263C2×C3⋊D3C2×C3×Q812yes—
264C2×C3⋊D3C2×C3×Q812yes—
265C2×C3×D3—34no—
266C2×C3×D3—34no—
267C2×C3×D3—34no—
268C2×C3×D3—62no—
269C2×C3×D3—62no—
270C2×C3×D3—62no—
271C32×C4—24no—
272C32×C4—22no—
273C32×C4—22no—
274C32×C4—24no—
275C32×C4—34no—
276C32×C4—34no—
277C32×C4—42no—
278C32×C4—44no—
279C32×C4—42no—
280C32×C4—64no—
281C32×C4—82no—
282C32×C4D3×Q814yescomplemented
283C32×C4C2×C3×Q812yescomplemented
284C3⋊Dic3—22no—
285C3⋊Dic3C2×C3×Q812yescomplemented
286C3×Dic3—24no—
287C3×Dic3—24no—
288C3×Dic3—24no—
289C3×Dic3—24no—
290C3×Dic3—34no—
291C3×Dic3—44no—
292C3×Dic3—62no—
293C3×Dic3—64no—
294C3×Dic3—122no—
295C3×Dic3D3×D414yes—
296C3×Dic3D3×Q814yescomplemented
297C3×Dic3C2×C3×D414yes—
298C3×Dic3C2×C3×Q814yescomplemented
299C22×Q8—92no—
300C4⋊Q8—32no—
301C4⋊Q8—92no—
302C4⋊Q8—94no—
303C2×C4⋊C4—92no—
304C2×C4⋊C4—91no—
305C2×C4⋊C4—91no—
306C2×C42—91no—
307C33C2×C4⋊Q811yescharacteristic, complemented, Sylow
308C23×C3—91no—
309C22×D3—32no—
310C22×D3—61no—
311C3×Q8—68no—
312C3×Q8—68no—
313C3×Q8—128no—
314C3×Q8—188no—
315C2×C3×C4—22no—
316C2×C3×C4—21no—
317C2×C3×C4—24no—
318C2×C3×C4—22no—
319C2×C3×C4—32no—
320C2×C3×C4—32no—
321C2×C3×C4—32no—
322C2×C3×C4—32no—
323C2×C3×C4—34no—
324C2×C3×C4—34no—
325C2×C3×C4—41no—
326C2×C3×C4—42no—
327C2×C3×C4—62no—
328C2×C3×C4—62no—
329C2×C3×C4—64no—
330C2×C3×C4—94no—
331C2×C3×C4—94no—
332C2×C3×C4—91no—
333C2×C3×C4—94no—
334C2×C3×C4—92no—
335C2×C3×C4—184no—
336C2×C3×C4—184no—
337C2×C3×C4C2×D3212yes—
338C2×C3×C4C2×D3211yescharacteristic
339C2×C3×C4C22×C3×D314yes—
340C2×C3×C4C22×C3×D312yes—
341C2×Dic3—32no—
342C2×Dic3—34no—
343C2×Dic3—61no—
344C2×Dic3—62no—
345C2×Dic3C22×C3×D314yes—
346C2×Dic3C22×C3×D312yes—
347C2×Dic3C22×C3×D312yes—
348C4×D3—38no—
349C4×D3—38no—
350C4×D3—38no—
351C4×D3—64no—
352C4×D3—68no—
353C4×D3—68no—
354C4×D3—124no—
355Dic6—28no—
356Dic6—68no—
357Dic6—68no—
358C2×C32—21no—
359C2×C32—21no—
360C2×C32—21no—
361C2×C32—22no—
362C2×C32—22no—
363C2×C32—22no—
364C2×C32—41no—
365C2×C32—41no—
366C2×C32—41no—
367C2×C32C2×D3×D412yes—
368C2×C32C2×D3×Q812yes—
369C2×C32C2×D3×Q812yes—
370C2×C32C22×C3×D411yescharacteristic
371C2×C32C22×C3×Q811yescharacteristic
372C2×C32C22×C3×Q811yescharacteristic
373C3⋊D3C3×C4⋊Q814yescomplemented
374C3×D3—38no—
375C3×D3—64no—
376C2×Q8—34no—
377C2×Q8—94no—
378C2×Q8—188no—
379C22×C4—91no—
380C22×C4—91no—
381C22×C4—92no—
382C22×C4—91no—
383C22×C4—92no—
384C4⋊C4—34no—
385C4⋊C4—32no—
386C4⋊C4—92no—
387C4⋊C4—94no—
388C4⋊C4—92no—
389C4⋊C4—91no—
390C4⋊C4C3×D3211yescharacteristic
391C42—32no—
392C42—92no—
393C22×C3—21no—
394C22×C3—22no—
395C22×C3—41no—
396C22×C3—92no—
397C22×C3—92no—
398C22×C3—92no—
399C22×C3C22×D3211yescharacteristic
400C22×C3C23×C3×D312yes—
401C2×D3—34no—
402C2×D3—34no—
403C2×D3—34no—
404C2×D3—62no—
405C2×D3—62no—
406C2×D3—62no—
407C3×C4—22no—
408C3×C4—24no—
409C3×C4—22no—
410C3×C4—24no—
411C3×C4—34no—
412C3×C4—34no—
413C3×C4—34no—
414C3×C4—34no—
415C3×C4—42no—
416C3×C4—44no—
417C3×C4—42no—
418C3×C4—64no—
419C3×C4—64no—
420C3×C4—64no—
421C3×C4—64no—
422C3×C4—82no—
423C3×C4—92no—
424C3×C4—124no—
425C3×C4—182no—
426C3×C4#379712yescomplemented
427C3×C4C3×D3×Q814yescomplemented
428Dic3—24no—
429Dic3—34no—
430Dic3—62no—
431Dic3—64no—
432Dic3—122no—
433Dic3C3×D3×D414yes—
434Dic3C3×D3×Q814yescomplemented
435C32—21no—
436C32—22no—
437C32—41no—
438C32D3×C4⋊Q812yescomplemented
439C32C2×C3×C4⋊Q811yescharacteristic, complemented
440C23—91no—
441Q8—68no—
442Q8—188no—
443C2×C4—32no—
444C2×C4—32no—
445C2×C4—34no—
446C2×C4—94no—
447C2×C4—94no—
448C2×C4—91no—
449C2×C4—94no—
450C2×C4—92no—
451C2×C4—184no—
452C2×C4—184no—
453C2×C4C2×C3×D3212yes—
454C2×C4C2×C3×D3211yescharacteristic
455C2×C3—22no—
456C2×C3—21no—
457C2×C3—21no—
458C2×C3—22no—
459C2×C3—21no—
460C2×C3—22no—
461C2×C3—41no—
462C2×C3—41no—
463C2×C3—41no—
464C2×C3—94no—
465C2×C3C2×#379711yescharacteristic
466C2×C3D32×D411yescharacteristic
467C2×C3D32×Q811yescharacteristic
468C2×C3C2×C3×D3×D412yes—
469C2×C3C2×C3×D3×Q812yes—
470C2×C3C2×C3×D3×Q812yes—
471D3—38no—
472D3—64no—
473C22—92no—
474C22—92no—
475C22—92no—
476C22C22×C3×D3211yescharacteristic
477C4—22no—
478C4—34no—
479C4—34no—
480C4—64no—
481C4—92no—
482C4—182no—
483C4C3×#379712yescomplemented
484C3—21no—
485C3—22no—
486C3—41no—
487C3#1059437411yescharacteristic, direct product factor
488C3C3×D3×C4⋊Q812yescomplemented
489C2—94no—
490C2C2×C3×#379711yescharacteristic
491C2C3×D32×D411yescharacteristic
492C2C3×D32×Q811yescharacteristic
subgroupG/H
centerC22×C3C22×D32
commutator, cosocleC22×C32C24×C3
FrattiniC22C22×C3×D32
FittingC33×C4⋊C4C22
socleC22×C33C24
prime radicalC22×C3×C3⋊D3C23
hypercenterC3×C4⋊C4D32
hypocenterC32C2×C3×C4⋊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
3. C32×#888C2semidirect
4. C3×#63918C2semidirect
8. C3×#63839C2semidirect
10. #10594374C3direct (N × H)
40. C3×#888D3semidirect
41. C3×#888C2×C3semidirect
44. #63918C2×C3semidirect
48. #63839C2×C3semidirect
55. C3×C4×C3⋊D3Q8semidirect
63. C3×#624Q8semidirect
123. C33×C4C2×Q8semidirect
125. C3×C3⋊Dic3C2×Q8semidirect
128. C32×Dic3C2×Q8semidirect
147. #888C3×D3semidirect
170. C4×C3⋊D3C3×Q8semidirect
199. #624C3×Q8semidirect
204. C3×C3⋊D3C4⋊Q8semidirect
282. C32×C4D3×Q8semidirect
283. C32×C4C2×C3×Q8semidirect
285. C3⋊Dic3C2×C3×Q8semidirect
296. C3×Dic3D3×Q8semidirect
298. C3×Dic3C2×C3×Q8semidirect
307. C33C2×C4⋊Q8semidirect
373. C3⋊D3C3×C4⋊Q8semidirect
426. C3×C4#3797semidirect
427. C3×C4C3×D3×Q8semidirect
434. Dic3C3×D3×Q8semidirect
438. C32D3×C4⋊Q8semidirect
439. C32C2×C3×C4⋊Q8semidirect
483. C4C3×#3797semidirect
487. C3#10594374direct (N × H)
488. C3C3×D3×C4⋊Q8semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#37972
C3×C4⋊C4×C3⋊D32
C32×#8882
C3×#639182
C3×#639002
C3×#638832
C3×C4×#6242
C3×#638392
C3×D3×C4⋊Q83
#105943743