Finite Group Databasebeta

← C3×#10591375 #13158213

Subgroups — C3×#10591375 #13158213

Order 1728.

1274 conjugacy classes of subgroups.

The subgroup lattice has 536 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×#624C211yesmaximal, characteristic
2C2×C3×#3752C211yesmaximal, characteristic
3C2×C3×Dic3⋊Dic3C212yesmaximal, complemented
4C2×C3×#3722C211yesmaximal, characteristic
5C32×C3⋊#95C212yesmaximal, complemented
6C3×#4668—32nomaximal
7#10591375C311yesmaximal, characteristic, direct product factor
8C23×C3×C3⋊D3C2211yescharacteristic
9C22×C33×C4C2211yescharacteristic
10C22×C3×C3⋊Dic3C2211yescharacteristic
11C22×C32×Dic3C2212yes—
12C22×C32×Dic3C2212yes—
13C2×C3×#624—22no—
14C2×C3×#624—24no—
15C3×#3752—22no—
16C3×Dic3⋊Dic3—24no—
17C3×#3722C414yescomplemented
18C22×C3×C4×D3—32no—
19C22×#624C2×C311yescharacteristic
20C2×#3752C2×C311yescharacteristic
21C2×C3×(C2×D3)⋊C4—32no—
22C2×C3×(C2×D3)⋊C4—32no—
23C2×C3×(C2×D3)⋊C4—61no—
24C2×C3×C4⋊Dic3—32no—
25C2×C3×Dic3⋊C4—32no—
26C2×Dic3⋊Dic3C2×C312yescomplemented
27C2×#3722C2×C311yescharacteristic
28C32×#95—32no—
29C3×C3⋊#95—22no—
30C3×C3⋊#95D312yescomplemented
31C3×C3⋊#95C2×C312yescomplemented
32C23×C33C2311yescharacteristic
33C22×C3×C3⋊D3—22no—
34C22×C3×C3⋊D3—22no—
35C22×C3×C3⋊D3C2×C412yes—
36C22×C3×C3⋊D3D412yes—
37C22×C3×C3⋊D3Q812yes—
38C2×C33×C4—21no—
39C2×C33×C4—22no—
40C2×C3×C3⋊Dic3—21no—
41C2×C3×C3⋊Dic3—22no—
42C2×C32×Dic3—22no—
43C2×C32×Dic3—22no—
44C2×C32×Dic3—22no—
45C2×C32×Dic3—22no—
46C2×C32×Dic3C2×C414yescomplemented
47C2×C32×Dic3D414yes—
48C3×#624—28no—
49C3×#385—91no—
50#4668—32no—
51C23×C3⋊D3C22×C311yescharacteristic
52C23×C3×D3—32no—
53C23×C3×D3—61no—
54C22×C32×C4—21no—
55C22×C32×C4—22no—
56C22×C32×C4—32no—
57C22×C32×C4—32no—
58C22×C32×C4—41no—
59C22×C32×C4C2×D312yes—
60C22×C32×C4C22×C311yescharacteristic
61C22×C3⋊Dic3C22×C311yescharacteristic
62C22×C3×Dic3—22no—
63C22×C3×Dic3—22no—
64C22×C3×Dic3—32no—
65C22×C3×Dic3—61no—
66C22×C3×Dic3C2×D312yes—
67C22×C3×Dic3C2×D312yes—
68C22×C3×Dic3C22×C312yes—
69C22×C3×Dic3C22×C312yes—
70C2×C3×C4×D3—64no—
71C2×C3×C4×D3—64no—
72C2×C3×C4×D3—64no—
73C2×#624—22no—
74C2×#624—24no—
75#3752—22no—
76C3×(C2×D3)⋊C4—38no—
77C3×(C2×D3)⋊C4—64no—
78C3×(C2×D3)⋊C4—64no—
79C3×C4⋊Dic3—64no—
80C3×Dic3⋊C4—64no—
81Dic3⋊Dic3—24no—
82#3722C3×C414yescomplemented
83C22×C33C22×C411yescharacteristic
84C22×C33C2×D411yescharacteristic
85C22×C33C2×D412yes—
86C22×C33C2×Q811yescharacteristic
87C22×C33#4112yes—
88C2×C3×C3⋊D3—24no—
89C2×C3×C3⋊D3—22no—
90C2×C3×C3⋊D3—24no—
91C2×C3×C3⋊D3—22no—
92C2×C3×C3⋊D3C4⋊C414yescomplemented
93C33×C4—41no—
94C3×C3⋊Dic3—41no—
95C32×Dic3—24no—
96C32×Dic3—24no—
97C23×C3×C4—91no—
98C22×C4×D3—32no—
99C2×C3×C4⋊C4—92no—
100C2×C3×C22⋊C4—91no—
101C2×C3×C22⋊C4—91no—
102C2×(C2×D3)⋊C4—32no—
103C2×(C2×D3)⋊C4—32no—
104C2×(C2×D3)⋊C4—61no—
105C2×C4⋊Dic3—32no—
106C2×Dic3⋊C4—32no—
107C3×#95—32no—
108C3×#95—32no—
109C3×#95—62no—
110C3⋊#95C3×D312yescomplemented
111C23×C32—21no—
112C23×C32—22no—
113C23×C32—41no—
114C23×C32C22×D312yes—
115C23×C32C23×C311yescharacteristic
116C22×C3⋊D3—22no—
117C22×C3⋊D3—22no—
118C22×C3⋊D3C2×C3×C412yes—
119C22×C3⋊D3C3×D412yes—
120C22×C3⋊D3C3×Q812yes—
121C22×C3×D3—34no—
122C22×C3×D3—34no—
123C22×C3×D3—34no—
124C22×C3×D3—62no—
125C22×C3×D3—62no—
126C22×C3×D3—62no—
127C22×C3×D3—62no—
128C22×C3×D3—62no—
129C22×C3×D3—62no—
130C22×C3×D3—62no—
131C22×C3×D3—122no—
132C22×C3×D3—122no—
133C2×C32×C4—22no—
134C2×C32×C4—21no—
135C2×C32×C4—22no—
136C2×C32×C4—24no—
137C2×C32×C4—22no—
138C2×C32×C4—22no—
139C2×C32×C4—22no—
140C2×C32×C4—34no—
141C2×C32×C4—34no—
142C2×C32×C4—42no—
143C2×C32×C4—42no—
144C2×C32×C4—42no—
145C2×C32×C4—42no—
146C2×C32×C4—44no—
147C2×C32×C4—62no—
148C2×C32×C4—62no—
149C2×C32×C4—62no—
150C2×C32×C4—62no—
151C2×C3⋊Dic3—21no—
152C2×C3⋊Dic3—22no—
153C2×C3×Dic3—22no—
154C2×C3×Dic3—22no—
155C2×C3×Dic3—22no—
156C2×C3×Dic3—22no—
157C2×C3×Dic3—22no—
158C2×C3×Dic3—22no—
159C2×C3×Dic3—22no—
160C2×C3×Dic3—22no—
161C2×C3×Dic3—24no—
162C2×C3×Dic3—24no—
163C2×C3×Dic3—42no—
164C2×C3×Dic3—42no—
165C2×C3×Dic3—42no—
166C2×C3×Dic3—42no—
167C2×C3×Dic3—62no—
168C2×C3×Dic3—62no—
169C2×C3×Dic3—62no—
170C2×C3×Dic3—62no—
171C2×C3×Dic3—122no—
172C2×C3×Dic3C4×D314yescomplemented
173C2×C3×Dic3D1214yes—
174C2×C3×Dic3C2×C3×C414yescomplemented
175C2×C3×Dic3C3×D414yes—
176C3×C4×D3—616no—
177#624—28no—
178#385—91noSylow
179C2×C33C2×C4⋊C411yescharacteristic
180C2×C33C4×D412yes—
181C2×C33#12011yescharacteristic
182C2×C33C22⋊Q812yes—
183C2×C33#12311yescharacteristic
184C3×C3⋊D3—24no—
185C24×C3—91no—
186C23×D3—32no—
187C23×D3—61no—
188C22×C3×C4—22no—
189C22×C3×C4—21no—
190C22×C3×C4—32no—
191C22×C3×C4—32no—
192C22×C3×C4—32no—
193C22×C3×C4—32no—
194C22×C3×C4—41no—
195C22×C3×C4—62no—
196C22×C3×C4—62no—
197C22×C3×C4—91no—
198C22×C3×C4—184no—
199C22×C3×C4—182no—
200C22×C3×C4D3211yescharacteristic
201C22×C3×C4C2×C3×D312yes—
202C22×Dic3—32no—
203C22×Dic3—61no—
204C22×Dic3C2×C3×D312yes—
205C22×Dic3C2×C3×D312yes—
206C2×C4×D3—64no—
207C2×C4×D3—64no—
208C2×C4×D3—64no—
209C3×C4⋊C4—184no—
210C3×C22⋊C4—94no—
211C3×C22⋊C4—182no—
212(C2×D3)⋊C4—38no—
213(C2×D3)⋊C4—64no—
214(C2×D3)⋊C4—64no—
215C4⋊Dic3—64no—
216Dic3⋊C4—64no—
217C22×C32—21no—
218C22×C32—21no—
219C22×C32—22no—
220C22×C32—22no—
221C22×C32—21no—
222C22×C32—22no—
223C22×C32—22no—
224C22×C32—22no—
225C22×C32—22no—
226C22×C32—22no—
227C22×C32—22no—
228C22×C32—22no—
229C22×C32—41no—
230C22×C32—41no—
231C22×C32—42no—
232C22×C32—41no—
233C22×C32—42no—
234C22×C32C2×C4×D312yes—
235C22×C32C2×D1212yes—
236C22×C32#23512yes—
237C22×C32D3×D412yes—
238C22×C32D3×D412yes—
239C22×C32#23712yes—
240C22×C32D3×Q812yes—
241C22×C32C22×C3×C411yescharacteristic
242C22×C32C2×C3×D412yes—
243C22×C32C2×C3×D411yescharacteristic
244C22×C32C2×C3×Q811yescharacteristic
245C22×C32C3×#4112yes—
246C2×C3⋊D3—24no—
247C2×C3⋊D3—22no—
248C2×C3⋊D3—24no—
249C2×C3⋊D3—22no—
250C2×C3⋊D3C3×C4⋊C414yescomplemented
251C2×C3×D3—38no—
252C2×C3×D3—64no—
253C2×C3×D3—64no—
254C2×C3×D3—64no—
255C2×C3×D3—64no—
256C2×C3×D3—64no—
257C2×C3×D3—64no—
258C2×C3×D3—64no—
259C2×C3×D3—124no—
260C2×C3×D3—124no—
261C2×C3×D3—122no—
262C2×C3×D3—122no—
263C32×C4—44no—
264C32×C4—42no—
265C32×C4—44no—
266C32×C4—42no—
267C32×C4—41no—
268C32×C4—64no—
269C32×C4—64no—
270C3⋊Dic3—41no—
271C3×Dic3—24no—
272C3×Dic3—24no—
273C3×Dic3—24no—
274C3×Dic3—24no—
275C3×Dic3—44no—
276C3×Dic3—44no—
277C3×Dic3—122no—
278C3×Dic3—122no—
279C23×C4—91no—
280C2×C4⋊C4—92no—
281C2×C22⋊C4—91no—
282C2×C22⋊C4—91no—
283#95—32no—
284C33#38511yescharacteristic, complemented, Sylow
285C23×C3—22no—
286C23×C3—21no—
287C23×C3—41no—
288C23×C3—92no—
289C23×C3—92no—
290C23×C3—92no—
291C23×C3—182no—
292C23×C3—182no—
293C23×C3C2×D3211yescharacteristic
294C23×C3C22×C3×D312yes—
295C22×D3—34no—
296C22×D3—34no—
297C22×D3—34no—
298C22×D3—62no—
299C22×D3—62no—
300C22×D3—62no—
301C22×D3—62no—
302C22×D3—62no—
303C22×D3—62no—
304C22×D3—62no—
305C22×D3—122no—
306C22×D3—122no—
307C2×C3×C4—22no—
308C2×C3×C4—24no—
309C2×C3×C4—22no—
310C2×C3×C4—21no—
311C2×C3×C4—22no—
312C2×C3×C4—22no—
313C2×C3×C4—22no—
314C2×C3×C4—34no—
315C2×C3×C4—34no—
316C2×C3×C4—34no—
317C2×C3×C4—34no—
318C2×C3×C4—42no—
319C2×C3×C4—42no—
320C2×C3×C4—42no—
321C2×C3×C4—44no—
322C2×C3×C4—42no—
323C2×C3×C4—64no—
324C2×C3×C4—64no—
325C2×C3×C4—62no—
326C2×C3×C4—62no—
327C2×C3×C4—62no—
328C2×C3×C4—62no—
329C2×C3×C4—62no—
330C2×C3×C4—62no—
331C2×C3×C4—62no—
332C2×C3×C4—62no—
333C2×C3×C4—122no—
334C2×C3×C4—122no—
335C2×C3×C4—122no—
336C2×C3×C4—122no—
337C2×C3×C4—188no—
338C2×C3×C4—182no—
339C2×C3×C4—181no—
340C2×Dic3—22no—
341C2×Dic3—22no—
342C2×Dic3—22no—
343C2×Dic3—22no—
344C2×Dic3—62no—
345C2×Dic3—62no—
346C2×Dic3—62no—
347C2×Dic3—62no—
348C2×Dic3—122no—
349C2×Dic3C3×C4×D314yescomplemented
350C2×Dic3C3×D1214yes—
351C4×D3—616no—
352C2×C32—22no—
353C2×C32—22no—
354C2×C32—22no—
355C2×C32—22no—
356C2×C32—22no—
357C2×C32—22no—
358C2×C32—22no—
359C2×C32—21no—
360C2×C32—22no—
361C2×C32—21no—
362C2×C32—21no—
363C2×C32—22no—
364C2×C32—41no—
365C2×C32—42no—
366C2×C32—41no—
367C2×C32—42no—
368C2×C32—41no—
369C2×C32C4×D1212yes—
370C2×C32#67⋊C412yes—
371C2×C32#88212yes—
372C2×C32#88312yes—
373C2×C32D3×C4⋊C412yes—
374C2×C32#89612yes—
375C2×C32#89712yes—
376C2×C32C2×C3×C4⋊C411yescharacteristic
377C2×C32C3×C4×D412yes—
378C2×C32C3×#12011yescharacteristic
379C2×C32C3×C22⋊Q812yes—
380C2×C32C3×#12311yescharacteristic
381C3⋊D3—24no—
382C3×D3—68no—
383C3×D3—124no—
384C24—91no—
385C22×C4—32no—
386C22×C4—32no—
387C22×C4—91no—
388C22×C4—184no—
389C22×C4—182no—
390C22×C4C3×D3211yescharacteristic
391C4⋊C4—184no—
392C22⋊C4—94no—
393C22⋊C4—182no—
394C22×C3—21no—
395C22×C3—22no—
396C22×C3—22no—
397C22×C3—21no—
398C22×C3—22no—
399C22×C3—22no—
400C22×C3—22no—
401C22×C3—22no—
402C22×C3—22no—
403C22×C3—22no—
404C22×C3—21no—
405C22×C3—22no—
406C22×C3—41no—
407C22×C3—41no—
408C22×C3—42no—
409C22×C3—42no—
410C22×C3—41no—
411C22×C3—94no—
412C22×C3—184no—
413C22×C3—184no—
414C22×C3—182no—
415C22×C3—182no—
416C22×C3#379711yescharacteristic
417C22×C3C4×D3211yescharacteristic
418C22×C3D3×D1212yes—
419C22×C3#380412yes—
420C22×C3#381111yescharacteristic
421C22×C3C2×C3×C4×D312yes—
422C22×C3C2×C3×D1212yes—
423C22×C3C3×#23512yes—
424C22×C3C3×D3×D412yes—
425C22×C3C3×D3×D412yes—
426C22×C3C3×#23712yes—
427C22×C3C3×D3×Q812yes—
428C2×D3—38no—
429C2×D3—64no—
430C2×D3—64no—
431C2×D3—64no—
432C2×D3—64no—
433C2×D3—64no—
434C2×D3—64no—
435C2×D3—64no—
436C2×D3—122no—
437C2×D3—124no—
438C2×D3—124no—
439C2×D3—122no—
440C3×C4—42no—
441C3×C4—44no—
442C3×C4—44no—
443C3×C4—42no—
444C3×C4—41no—
445C3×C4—64no—
446C3×C4—64no—
447C3×C4—64no—
448C3×C4—64no—
449C3×C4—124no—
450C3×C4—124no—
451C3×C4—361no—
452Dic3—24no—
453Dic3—24no—
454Dic3—122no—
455Dic3—122no—
456C32—22no—
457C32—21no—
458C32—41no—
459C32#466812yescomplemented
460C32C3×#38511yescharacteristic, complemented
461C23—92no—
462C23—92no—
463C23—92no—
464C23—182no—
465C23—182no—
466C23C2×C3×D3211yescharacteristic
467C2×C4—22no—
468C2×C4—21no—
469C2×C4—34no—
470C2×C4—34no—
471C2×C4—62no—
472C2×C4—62no—
473C2×C4—62no—
474C2×C4—62no—
475C2×C4—188no—
476C2×C4—182no—
477C2×C4—181no—
478C2×C3—21no—
479C2×C3—22no—
480C2×C3—22no—
481C2×C3—22no—
482C2×C3—22no—
483C2×C3—21no—
484C2×C3—22no—
485C2×C3—22no—
486C2×C3—22no—
487C2×C3—22no—
488C2×C3—21no—
489C2×C3—22no—
490C2×C3—41no—
491C2×C3—42no—
492C2×C3—41no—
493C2×C3—42no—
494C2×C3—41no—
495C2×C3—184no—
496C2×C3#6388012yes—
497C2×C3#6388311yescharacteristic
498C2×C3#6389512yes—
499C2×C3#6392311yescharacteristic
500C2×C3#6397511yescharacteristic
501C2×C3C3×C4×D1212yes—
502C2×C3C3×#67⋊C412yes—
503C2×C3C3×#88212yes—
504C2×C3C3×#88312yes—
505C2×C3C3×D3×C4⋊C412yes—
506C2×C3C3×#89612yes—
507C2×C3C3×#89712yes—
508D3—68no—
509D3—124no—
510C22—94no—
511C22—182no—
512C22—184no—
513C22—184no—
514C22—182no—
515C22C3×#379711yescharacteristic
516C22C3×C4×D3211yescharacteristic
517C22C3×D3×D1212yes—
518C22C3×#380412yes—
519C22C3×#381111yescharacteristic
520C4—41no—
521C4—64no—
522C4—64no—
523C4—361no—
524C3—21no—
525C3—22no—
526C3—41no—
527C3#1059137511yescharacteristic, direct product factor
528C3C3×#466812yescomplemented
529C2—184no—
530C2C3×#6388012yes—
531C2C3×#6388311yescharacteristic
532C2C3×#6389512yes—
533C2C3×#6392311yescharacteristic
534C2C3×#6397511yescharacteristic
subgroupG/H
centerC23×C3C2×D32
commutatorC22×C32C22×C3×C4
FrattiniC23C2×C3×D32
FittingC22×C33×C4C22
socleC23×C33C23
prime radicalC23×C3×C3⋊D3C22
cosocleC23×C32C23×C3
hypercenterC22×C3×C4D32
hypocenterC32C3×#385
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C23×C3 ◁ C22×C3×C4

Complements

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

NH ≅ G/Ntype
3. C2×C3×Dic3⋊Dic3C2semidirect
5. C32×C3⋊#95C2semidirect
7. #10591375C3direct (N × H)
17. C3×#3722C4semidirect
26. C2×Dic3⋊Dic3C2×C3semidirect
30. C3×C3⋊#95D3semidirect
31. C3×C3⋊#95C2×C3semidirect
46. C2×C32×Dic3C2×C4semidirect
82. #3722C3×C4semidirect
92. C2×C3×C3⋊D3C4⋊C4semidirect
110. C3⋊#95C3×D3semidirect
172. C2×C3×Dic3C4×D3semidirect
174. C2×C3×Dic3C2×C3×C4semidirect
250. C2×C3⋊D3C3×C4⋊C4semidirect
284. C33#385semidirect
349. C2×Dic3C3×C4×D3semidirect
459. C32#4668semidirect
460. C32C3×#385semidirect
527. C3#10591375direct (N × H)
528. C3C3×#4668semidirect

Maximal subgroups

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

subgroupindex
C22×C3×#6242
C2×C3×#37522
C2×C3×Dic3⋊Dic32
C2×C3×#37222
C32×C3⋊#952
C3×#46683
#105913753