Finite Group Databasebeta

← C3×#10591501 #13158514

Subgroups — C3×#10591501 #13158514

Order 1728.

910 conjugacy classes of subgroups.

The subgroup lattice has 512 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C32×#237C211yesmaximal, characteristic
2C3×D3×#208C212yesmaximal, complemented
3C3×#63660C211yesmaximal, characteristic, complemented
4C32×#833C211yesmaximal, characteristic, complemented
5C3×#63560C212yesmaximal, complemented
6C3×#5237—31nomaximal
7C3×#4745—31nomaximal
8#10591501C311yesmaximal, characteristic, direct product factor
9C2×C33×D4C2211yescharacteristic
10C2×C32×#67C412yes—
11C22×C32×Dic3C412yes—
12C32×#237—22no—
13C32×#237—22no—
14C2×C32×C4×D3C2211yescharacteristic
15C2×C32×Dic6C2211yescharacteristic
16C3×#3748C2212yes—
17C32×#208C2212yes—
18C3×#3711—22no—
19C3×D3×C3⋊C8—22no—
20C2×C32×#41—31no—
21C2×C3×#237—21no—
22C2×C3×#237D311yescharacteristic
23C2×C3×#237C2×C311yescharacteristic
24C2×C3×#208—32no—
25C3×D3×M16—32no—
26D3×#208C2×C312yescomplemented
27C32×#100—31no—
28#63660C2×C311yescharacteristic, complemented
29C3×#836—32no—
30C3×#833—21no—
31C3×#833—31no—
32C3×#833—31no—
33C3×#833—61no—
34C3×#833D311yescharacteristic, complemented
35C3×#833C2×C311yescharacteristic, complemented
36C3×#824—32no—
37#63560C2×C312yescomplemented
38C23×C33C2×C412yes—
39C22×C32×D3C2×C411yescharacteristic
40C33×D4—22no—
41C2×C33×C4C2311yescharacteristic
42C32×#67—24no—
43C2×C32×Dic3—22no—
44C2×C32×Dic3—22no—
45C2×C32×Dic3C2×C412yes—
46C2×C32×Dic3C2×C411yescharacteristic
47C32×C4×D3D412yes—
48C32×C4×D3D412yes—
49C32×Dic6—22no—
50C3×C3⋊(C3⋊C8)—22no—
51C32×C3⋊C8—22no—
52C3×#413—91no—
53#5237—31no—
54#4745—31no—
55C32×#41—62no—
56C32×#41—62no—
57C2×C32×Q8—31no—
58C2×C32×D4—21no—
59C2×C32×D4—21no—
60C2×C32×D4—21no—
61C2×C32×D4—32no—
62C2×C32×D4—41no—
63C2×C32×D4C2×D311yescharacteristic
64C2×C32×D4C2×D311yescharacteristic
65C2×C32×D4C22×C311yescharacteristic
66C22×C32×C4—31no—
67C22×C32×C4—32no—
68C2×C3×#67—22no—
69C2×C3×#67Dic312yes—
70C2×C3×#67C3×C412yes—
71C22×C3×Dic3—22no—
72C22×C3×Dic3Dic312yes—
73C22×C3×Dic3C3×C412yes—
74C3×#237—24no—
75C3×#237—24no—
76C3×#237—22no—
77C3×#237—22no—
78C3×#237—22no—
79C3×#237—22no—
80C2×C3×C4×D3—21no—
81C2×C3×C4×D3C2×D311yescharacteristic
82C2×C3×C4×D3C22×C311yescharacteristic
83C2×C3×Dic6—21no—
84C2×C3×Dic6C2×D311yescharacteristic
85C2×C3×Dic6C22×C311yescharacteristic
86C32×M16—32no—
87#3748C22×C312yes—
88C3×#208—22no—
89C3×#208—32no—
90C3×#208—32no—
91C3×#208—62no—
92C3×#208—62no—
93C3×#208C2×D312yes—
94C3×#208C22×C312yes—
95C2×C3×C3⋊C8—62no—
96C3×ΓL(1,𝔽25)—62no—
97C3×D3×C8—62no—
98#3711—22no—
99D3×C3⋊C8—22no—
100C22×C33—22no—
101C22×C33—42no—
102C22×C33C22×C411yescharacteristic
103C2×C32×D3—41no—
104C2×C32×D3C22⋊C412yes—
105C33×C4C2×D412yes—
106C32×Dic3—22no—
107C32×Dic3C22⋊C411yescharacteristic
108C32×Dic3C22⋊C411yescharacteristic
109C2×C3×#41—31no—
110C2×C3×#41—31no—
111C2×C3×#41—61no—
112C2×#237C3×D311yescharacteristic
113C2×C3×M16—92no—
114C2×#208—32no—
115D3×M16—32no—
116C3×#101—92no—
117C3×#100—31no—
118C3×#100—31no—
119C3×#100—61no—
120C3×#100—91no—
121#836—32no—
122#833—31no—
123#833—31no—
124#833—61no—
125#833C3×D311yescharacteristic, complemented
126#824—32no—
127C23×C32—22no—
128C23×C32—22no—
129C23×C32—22no—
130C23×C32—31no—
131C23×C32—42no—
132C23×C32C4×D312yes—
133C23×C32C2×Dic312yes—
134C23×C32C2×C3×C412yes—
135C22×C3×D3—21no—
136C22×C3×D3C2×Dic311yescharacteristic
137C22×C3×D3C2×C3×C411yescharacteristic
138C32×Q8—62no—
139C32×D4—22no—
140C32×D4—24no—
141C32×D4—22no—
142C32×D4—22no—
143C32×D4—42no—
144C32×D4—42no—
145C32×D4—44no—
146C32×D4—64no—
147C2×C32×C4—21no—
148C2×C32×C4—21no—
149C2×C32×C4—21no—
150C2×C32×C4—31no—
151C2×C32×C4—32no—
152C2×C32×C4—32no—
153C2×C32×C4—32no—
154C2×C32×C4—41no—
155C2×C32×C4—62no—
156C2×C32×C4—62no—
157C2×C32×C4C22×D311yescharacteristic
158C2×C32×C4C22×D311yescharacteristic
159C2×C32×C4C23×C311yescharacteristic
160C3×#67—28no—
161C3×#67—24no—
162C3×#67—24no—
163C2×C3×Dic3—24no—
164C2×C3×Dic3—24no—
165C2×C3×Dic3—22no—
166C2×C3×Dic3—22no—
167C2×C3×Dic3—22no—
168C2×C3×Dic3—22no—
169C2×C3×Dic3—22no—
170C2×C3×Dic3—21no—
171C2×C3×Dic3C2×Dic312yes—
172C2×C3×Dic3C2×Dic311yescharacteristic
173C2×C3×Dic3C2×C3×C412yes—
174C2×C3×Dic3C2×C3×C411yescharacteristic
175C3×C4×D3—22no—
176C3×C4×D3—22no—
177C3×C4×D3#6712yes—
178C3×C4×D3#6712yes—
179C3×C4×D3C3×D412yes—
180C3×C4×D3C3×D412yes—
181C3×Dic6—24no—
182C3×Dic6—22no—
183C3×Dic6—22no—
184C32×C8—62no—
185C3⋊(C3⋊C8)—22no—
186C3×C3⋊C8—22no—
187C3×C3⋊C8—22no—
188C3×C3⋊C8—42no—
189C3×C3⋊C8—62no—
190C3×C3⋊C8—62no—
191C3×C3⋊C8—122no—
192#413—91noSylow
193C2×C33—21no—
194C2×C33—42no—
195C2×C33C2×C22⋊C411yescharacteristic
196C32×D3—22no—
197C3×#41—62no—
198C3×#41—62no—
199C3×#41—62no—
200C3×#41—62no—
201C3×#41—64no—
202C3×#41—64no—
203C2×C3×Q8—31no—
204C2×C3×Q8—31no—
205C2×C3×Q8—61no—
206C2×C3×D4—21no—
207C2×C3×D4—21no—
208C2×C3×D4—21no—
209C2×C3×D4—32no—
210C2×C3×D4—32no—
211C2×C3×D4—41no—
212C2×C3×D4—62no—
213C2×C3×D4D3211yescharacteristic
214C2×C3×D4C2×C3×D311yescharacteristic
215C2×C3×D4C2×C3×D311yescharacteristic
216C22×C3×C4—31no—
217C22×C3×C4—31no—
218C22×C3×C4—32no—
219C22×C3×C4—32no—
220C22×C3×C4—61no—
221C22×C3×C4—62no—
222C2×#67C3×Dic312yes—
223C22×Dic3C3×Dic312yes—
224#237—22no—
225#237—22no—
226C2×C4×D3C2×C3×D311yescharacteristic
227C2×Dic6C2×C3×D311yescharacteristic
228C3×M16—32no—
229C3×M16—32no—
230C3×M16—62no—
231C3×M16—92no—
232C3×M16—182no—
233C2×C3×C8—182no—
234#208—32no—
235#208—32no—
236#208—62no—
237#208—62no—
238#208C2×C3×D312yes—
239C2×C3⋊C8—62no—
240ΓL(1,𝔽25)—62no—
241D3×C8—62no—
242C22×C32—22no—
243C22×C32—24no—
244C22×C32—22no—
245C22×C32—22no—
246C22×C32—21no—
247C22×C32—21no—
248C22×C32—21no—
249C22×C32—32no—
250C22×C32—41no—
251C22×C32—42no—
252C22×C32—42no—
253C22×C32—44no—
254C22×C32—42no—
255C22×C32—42no—
256C22×C32—42no—
257C22×C32—44no—
258C22×C32—82no—
259C22×C32—82no—
260C22×C32—84no—
261C22×C32—121no—
262C22×C32C2×C4×D311yescharacteristic
263C22×C32C22×Dic311yescharacteristic
264C22×C32C22×C3×C411yescharacteristic
265C2×C3×D3—22no—
266C2×C3×D3—41no—
267C2×C3×D3—41no—
268C2×C3×D3—42no—
269C2×C3×D3#21712yes—
270C2×C3×D3C3×C22⋊C412yes—
271C32×C4—22no—
272C32×C4—22no—
273C32×C4—22no—
274C32×C4—31no—
275C32×C4—31no—
276C32×C4—42no—
277C32×C4—62no—
278C32×C4D3×D412yes—
279C32×C4C2×#6712yes—
280C32×C4C2×C3×D412yes—
281C3×Dic3—24no—
282C3×Dic3—22no—
283C3×Dic3—22no—
284C3×Dic3—21no—
285C3×Dic3—21no—
286C3×Dic3#21711yescharacteristic
287C3×Dic3#21711yescharacteristic
288C3×Dic3C3×C22⋊C411yescharacteristic
289C3×Dic3C3×C22⋊C411yescharacteristic
290C2×#41—31no—
291C2×M16—92no—
292#101—92no—
293#100—31no—
294#100—91no—
295C33#41311yescharacteristic, complemented, Sylow
296C23×C3—22no—
297C23×C3—22no—
298C23×C3—22no—
299C23×C3—31no—
300C23×C3—31no—
301C23×C3—42no—
302C23×C3—61no—
303C23×C3D3×Dic312yes—
304C23×C3C3×C4×D312yes—
305C23×C3C2×C3×Dic312yes—
306C22×D3C2×C3×Dic311yescharacteristic
307C3×Q8—64no—
308C3×Q8—62no—
309C3×Q8—62no—
310C3×D4—24no—
311C3×D4—22no—
312C3×D4—22no—
313C3×D4—22no—
314C3×D4—42no—
315C3×D4—42no—
316C3×D4—44no—
317C3×D4—64no—
318C3×D4—64no—
319C3×D4—68no—
320C2×C3×C4—21no—
321C2×C3×C4—21no—
322C2×C3×C4—21no—
323C2×C3×C4—31no—
324C2×C3×C4—31no—
325C2×C3×C4—32no—
326C2×C3×C4—32no—
327C2×C3×C4—32no—
328C2×C3×C4—32no—
329C2×C3×C4—32no—
330C2×C3×C4—32no—
331C2×C3×C4—41no—
332C2×C3×C4—62no—
333C2×C3×C4—62no—
334C2×C3×C4—61no—
335C2×C3×C4—62no—
336C2×C3×C4—62no—
337C2×C3×C4—62no—
338C2×C3×C4—62no—
339C2×C3×C4—62no—
340C2×C3×C4—64no—
341C2×C3×C4—64no—
342C2×C3×C4C2×D3211yescharacteristic
343C2×C3×C4C22×C3×D311yescharacteristic
344C2×C3×C4C22×C3×D311yescharacteristic
345#67—24no—
346C2×Dic3—22no—
347C2×Dic3—22no—
348C2×Dic3C2×C3×Dic312yes—
349C2×Dic3C2×C3×Dic311yescharacteristic
350C4×D3C3×#6712yes—
351C4×D3C3×#6712yes—
352Dic6—22no—
353C3×C8—62no—
354C3×C8—62no—
355C3×C8—122no—
356C3×C8—182no—
357C3⋊C8—22no—
358C3⋊C8—62no—
359C3⋊C8—62no—
360C3⋊C8—122no—
361C2×C32—21no—
362C2×C32—22no—
363C2×C32—21no—
364C2×C32—21no—
365C2×C32—21no—
366C2×C32—21no—
367C2×C32—21no—
368C2×C32—41no—
369C2×C32—41no—
370C2×C32—41no—
371C2×C32—42no—
372C2×C32—42no—
373C2×C32—44no—
374C2×C32—42no—
375C2×C32—42no—
376C2×C32—62no—
377C2×C32—82no—
378C2×C32—82no—
379C2×C32—84no—
380C2×C32D3×C22⋊C411yescharacteristic
381C2×C32C2×#21711yescharacteristic
382C2×C32C2×C3×C22⋊C411yescharacteristic
383C3×D3—22no—
384C3×D3—22no—
385C3×D3—42no—
386#41—62no—
387#41—62no—
388C2×Q8—31no—
389C2×D4—32no—
390C2×D4C3×D3211yescharacteristic
391C22×C4—31no—
392C22×C4—32no—
393M16—32no—
394M16—92no—
395M16—182no—
396C2×C8—182no—
397C22×C3—24no—
398C22×C3—22no—
399C22×C3—22no—
400C22×C3—22no—
401C22×C3—21no—
402C22×C3—21no—
403C22×C3—21no—
404C22×C3—32no—
405C22×C3—32no—
406C22×C3—42no—
407C22×C3—42no—
408C22×C3—44no—
409C22×C3—42no—
410C22×C3—42no—
411C22×C3—42no—
412C22×C3—44no—
413C22×C3—41no—
414C22×C3—62no—
415C22×C3—82no—
416C22×C3—82no—
417C22×C3—84no—
418C22×C3—121no—
419C22×C3—121no—
420C22×C3—122no—
421C22×C3C2×D3×Dic311yescharacteristic
422C22×C3C2×C3×C4×D311yescharacteristic
423C22×C3C22×C3×Dic311yescharacteristic
424C2×D3—41no—
425C2×D3C3×#21712yes—
426C3×C4—22no—
427C3×C4—22no—
428C3×C4—22no—
429C3×C4—31no—
430C3×C4—31no—
431C3×C4—31no—
432C3×C4—31no—
433C3×C4—42no—
434C3×C4—61no—
435C3×C4—61no—
436C3×C4—64no—
437C3×C4—62no—
438C3×C4—62no—
439C3×C4D3×#6712yes—
440C3×C4C3×D3×D412yes—
441C3×C4C2×C3×#6712yes—
442Dic3—22no—
443Dic3C3×#21711yescharacteristic
444Dic3C3×#21711yescharacteristic
445C32—21no—
446C32—21no—
447C32—21no—
448C32—41no—
449C32#474511yescharacteristic, complemented
450C32#523711yescharacteristic, complemented
451C32C3×#41311yescharacteristic, complemented
452C23—31no—
453C23C3×D3×Dic312yes—
454Q8—62no—
455D4—22no—
456D4—64no—
457C2×C4—31no—
458C2×C4—32no—
459C2×C4—32no—
460C2×C4—32no—
461C2×C4—62no—
462C2×C4—62no—
463C2×C4C2×C3×D3211yescharacteristic
464C8—62no—
465C8—182no—
466C2×C3—22no—
467C2×C3—21no—
468C2×C3—21no—
469C2×C3—21no—
470C2×C3—21no—
471C2×C3—21no—
472C2×C3—21no—
473C2×C3—41no—
474C2×C3—41no—
475C2×C3—42no—
476C2×C3—44no—
477C2×C3—42no—
478C2×C3—42no—
479C2×C3—42no—
480C2×C3—41no—
481C2×C3—62no—
482C2×C3—62no—
483C2×C3—82no—
484C2×C3—82no—
485C2×C3—84no—
486C2×C3—122no—
487C2×C3D3×#21711yescharacteristic
488C2×C3C3×D3×C22⋊C411yescharacteristic
489C2×C3C2×C3×#21711yescharacteristic
490D3—22no—
491C22—22no—
492C22—32no—
493C22—42no—
494C22—121no—
495C22C2×C3×D3×Dic311yescharacteristic
496C4—31no—
497C4—31no—
498C4—62no—
499C4C3×D3×#6712yes—
500C3—21no—
501C3—21no—
502C3—21no—
503C3—41no—
504C3#1059150111yescharacteristic, direct product factor
505C3C3×#474511yescharacteristic, complemented
506C3C3×#523711yescharacteristic, complemented
507C2—21no—
508C2—42no—
509C2—62no—
510C2C3×D3×#21711yescharacteristic
subgroupG/H
centerC2×C3D3×#217
commutatorC22×C32C22×C3×C4
FrattiniC2×C4C2×C3×D32
FittingC2×C33×D4C22
socleC2×C33C2×C22⋊C4
prime radicalC2×C32×#237C2
cosocleC2×C32×C4C23×C3
hypercenterC2×C3×D4D32
hypocenterC32C3×#413
LCSG ▷ C22×C32 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2×C3 ◁ C2×C3×C4 ◁ C2×C3×D4

Complements

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

NH ≅ G/Ntype
2. C3×D3×#208C2semidirect
3. C3×#63660C2semidirect
4. C32×#833C2semidirect
5. C3×#63560C2semidirect
8. #10591501C3direct (N × H)
26. D3×#208C2×C3semidirect
28. #63660C2×C3semidirect
34. C3×#833D3semidirect
35. C3×#833C2×C3semidirect
37. #63560C2×C3semidirect
125. #833C3×D3semidirect
295. C33#413semidirect
449. C32#4745semidirect
450. C32#5237semidirect
451. C32C3×#413semidirect
504. C3#10591501direct (N × H)
505. C3C3×#4745semidirect
506. C3C3×#5237semidirect

Maximal subgroups

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

subgroupindex
C2×C32×#2372
C3×D3×#2082
C3×#636602
C32×#8332
C3×#635602
C3×#52373
C3×#47453
#105915013