Finite Group Databasebeta

← #13161816

Subgroups — #13161816

Order 1728.

954 conjugacy classes of subgroups.

The subgroup lattice has 492 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×#634C211yesmaximal, characteristic
2C3×#64135C211yesmaximal, characteristic
3C2×C3×#3748C211yesmaximal, characteristic
4#11722076C214yesmaximal, complemented
5#10591107—31nomaximal
6#10591100—31nomaximal
7#10591099—31nomaximal
8C22×C33×C4C2211yescharacteristic
9C22×C3×C3⋊Dic3—21no—
10C2×C3×#634—21no—
11C2×C3×#634—22no—
12C2×C3×#634C414yescomplemented
13C2×C3×#634C2212yes—
14C3×#3757—21no—
15C3×C4⋊(C3⋊Dic3)C2212yes—
16C3×C4×C3⋊Dic3—21no—
17C3×#3748—22no—
18C2×C3×C3⋊(C3⋊C8)C2212yes—
19C22×#634D311yescharacteristic
20C22×C3×Dic6—31no—
21C22×C3×Dic6—31no—
22C22×C3×Dic6—61no—
23#64135D311yescharacteristic
24C2×#3748D311yescharacteristic
25C3×#926—31no—
26C3×#926—31no—
27C3×#926—61no—
28C2×C3×#208—31no—
29C2×C3×#208—31no—
30C2×C3×#208—61no—
31#63567—34no—
32#63566—34no—
33#63565—34no—
34C23×C33D411yescharacteristic
35C2×C33×C4C2×C412yes—
36C2×C33×C4D411yescharacteristic
37C2×C33×C4D412yes—
38C2×C33×C4C2311yescharacteristic
39C2×C3×C3⋊Dic3—24no—
40C2×C3×C3⋊Dic3—22no—
41C2×C3×C3⋊Dic3—22no—
42C3×#634—24no—
43C3×#634—22no—
44C3×#634C2×C414yes—
45C3×C3⋊(C3⋊C8)—22no—
46#5225—91no—
47#5120—91no—
48#5004—91no—
49C22×C32×C4—21no—
50C22×C32×C4—21no—
51C22×C32×C4—21no—
52C22×C32×C4—41no—
53C22×C32×C4C2×D311yescharacteristic
54C22×C32×C4C2×D311yescharacteristic
55C22×C32×C4C2×D311yescharacteristic
56C22×C3⋊Dic3—21no—
57C2×#634—21no—
58C2×#634—22no—
59C2×#634Dic314yescomplemented
60C2×#634C2×D312yes—
61C22×C3×Dic3—62no—
62C22×C3×Dic3—61no—
63C22×C3×Dic3—61no—
64C2×C3×Dic6—34no—
65C2×C3×Dic6—34no—
66C2×C3×Dic6—32no—
67C2×C3×Dic6—32no—
68C2×C3×Dic6—64no—
69C2×C3×Dic6—62no—
70C2×C3×Dic6—62no—
71C2×C3×Dic6—64no—
72C2×C3×Dic6—61no—
73C2×C3×Dic6—62no—
74C2×C3×Dic6—61no—
75C2×C3×Dic6—62no—
76#3757—21no—
77C4⋊(C3⋊Dic3)C2×D312yes—
78C4×C3⋊Dic3—21no—
79#3748—22no—
80C2×C3⋊(C3⋊C8)C2×D312yes—
81C3×#217—62no—
82C3×#217—61no—
83C3×#217—61no—
84C3×C4⋊Dic3—32no—
85C3×C4⋊Dic3—32no—
86C3×C4⋊Dic3—62no—
87C3×C4×Dic3—62no—
88C3×C4×Dic3—61no—
89C3×C4×Dic3—61no—
90C3×#208—64no—
91C3×#208—62no—
92C3×#208—62no—
93C2×C3×C3⋊C8—32no—
94C2×C3×C3⋊C8—32no—
95C2×C3×C3⋊C8—62no—
96C22×C33—22no—
97C22×C33C22⋊C412yes—
98C22×C33C2×D411yescharacteristic
99C33×C4C22⋊C412yes—
100C33×C4C22×C411yescharacteristic
101C33×C4C2×D411yescharacteristic
102C3×C3⋊Dic3—24no—
103C3×C3⋊Dic3—42no—
104C22×C3×Q8—91no—
105C22×Dic6—31no—
106C22×Dic6—31no—
107C22×Dic6—61no—
108C2×C3×M16—91no—
109C3×#117—91no—
110#926—31no—
111#926—31no—
112#926—61no—
113C2×#208—31no—
114C2×#208—31no—
115C2×#208—61no—
116#835—94no—
117#816—94no—
118#810—94no—
119C23×C32—21no—
120C23×C32—21no—
121C23×C32—21no—
122C23×C32—41no—
123C23×C32D1211yescharacteristic
124C23×C32#6711yescharacteristic
125C23×C32#6711yescharacteristic
126C2×C32×C4—22no—
127C2×C32×C4—22no—
128C2×C32×C4—22no—
129C2×C32×C4—21no—
130C2×C32×C4—22no—
131C2×C32×C4—22no—
132C2×C32×C4—21no—
133C2×C32×C4—21no—
134C2×C32×C4—22no—
135C2×C32×C4—21no—
136C2×C32×C4—21no—
137C2×C32×C4—21no—
138C2×C32×C4—42no—
139C2×C32×C4—42no—
140C2×C32×C4—41no—
141C2×C32×C4—41no—
142C2×C32×C4C4×D312yes—
143C2×C32×C4C4×D312yes—
144C2×C32×C4D1211yescharacteristic
145C2×C32×C4D1212yes—
146C2×C32×C4C2×Dic312yes—
147C2×C32×C4#6711yescharacteristic
148C2×C32×C4#6712yes—
149C2×C32×C4#6711yescharacteristic
150C2×C32×C4#6712yes—
151C2×C32×C4C22×D311yescharacteristic
152C2×C32×C4C22×D311yescharacteristic
153C2×C32×C4C22×D311yescharacteristic
154C2×C3⋊Dic3—24no—
155C2×C3⋊Dic3—22no—
156C2×C3⋊Dic3—22no—
157#634—24no—
158#634—22no—
159#634C2×Dic314yes—
160C2×C3×Dic3—68no—
161C2×C3×Dic3—64no—
162C2×C3×Dic3—64no—
163C2×C3×Dic3—64no—
164C2×C3×Dic3—62no—
165C2×C3×Dic3—62no—
166C2×C3×Dic3—64no—
167C2×C3×Dic3—62no—
168C2×C3×Dic3—62no—
169C3×Dic6—34no—
170C3×Dic6—34no—
171C3×Dic6—64no—
172C3×Dic6—68no—
173C3×Dic6—64no—
174C3×Dic6—64no—
175C3×Dic6—64no—
176C3×Dic6—62no—
177C3×Dic6—62no—
178C3⋊(C3⋊C8)—22no—
179C3×C3⋊C8—64no—
180C3×C3⋊C8—62no—
181C3×C3⋊C8—62no—
182#419—271noSylow
183C2×C33—22no—
184C2×C33C2×C22⋊C411yescharacteristic
185C2×C33#13712yes—
186C2×C3×Q8—94no—
187C2×C3×Q8—92no—
188C2×C3×Q8—182no—
189C2×C3×Q8—181no—
190C22×C3×C4—21no—
191C22×C3×C4—21no—
192C22×C3×C4—21no—
193C22×C3×C4—41no—
194C22×C3×C4—181no—
195C22×C3×C4D3211yescharacteristic
196C22×C3×C4D3211yescharacteristic
197C22×C3×C4D3211yescharacteristic
198C22×Dic3—62no—
199C22×Dic3—61no—
200C22×Dic3—61no—
201C2×Dic6—34no—
202C2×Dic6—34no—
203C2×Dic6—32no—
204C2×Dic6—32no—
205C2×Dic6—64no—
206C2×Dic6—62no—
207C2×Dic6—62no—
208C2×Dic6—64no—
209C2×Dic6—61no—
210C2×Dic6—62no—
211C2×Dic6—61no—
212C2×Dic6—62no—
213C3×M16—182no—
214C2×C3×C8—92no—
215C3×C4⋊C4—92no—
216C3×C22⋊C4—181no—
217C3×C42—181no—
218#217—62no—
219#217—61no—
220#217—61no—
221C4⋊Dic3—32no—
222C4⋊Dic3—32no—
223C4⋊Dic3—62no—
224C4×Dic3—62no—
225C4×Dic3—61no—
226C4×Dic3—61no—
227#208—64no—
228#208—62no—
229#208—62no—
230C2×C3⋊C8—32no—
231C2×C3⋊C8—32no—
232C2×C3⋊C8—62no—
233C22×C32—22no—
234C22×C32—24no—
235C22×C32—22no—
236C22×C32—22no—
237C22×C32—22no—
238C22×C32—22no—
239C22×C32—22no—
240C22×C32—21no—
241C22×C32—21no—
242C22×C32—21no—
243C22×C32—42no—
244C22×C32—41no—
245C22×C32—42no—
246C22×C32—42no—
247C22×C32—44no—
248C22×C32(C2×D3)⋊C412yes—
249C22×C32(C2×D3)⋊C412yes—
250C22×C32#21712yes—
251C22×C32C2×D1211yescharacteristic
252C22×C32C2×#6711yescharacteristic
253C22×C32C2×#6711yescharacteristic
254C32×C4—22no—
255C32×C4—22no—
256C32×C4—22no—
257C32×C4—21no—
258C32×C4—21no—
259C32×C4—21no—
260C32×C4—21no—
261C32×C4—21no—
262C32×C4—21no—
263C32×C4—42no—
264C32×C4—41no—
265C32×C4—41no—
266C32×C4(C2×D3)⋊C412yes—
267C32×C4(C2×D3)⋊C412yes—
268C32×C4#21712yes—
269C32×C4C2×C4×D311yescharacteristic
270C32×C4C2×C4×D311yescharacteristic
271C32×C4C2×D1211yescharacteristic
272C32×C4C22×Dic311yescharacteristic
273C32×C4C2×#6711yescharacteristic
274C32×C4C2×#6711yescharacteristic
275C3⋊Dic3—24no—
276C3⋊Dic3—42no—
277C3×Dic3—68no—
278C3×Dic3—64no—
279C3×Dic3—64no—
280C3×Dic3—124no—
281C3×Dic3—122no—
282C3×Dic3—122no—
283C22×Q8—91no—
284C2×M16—91no—
285#117—91no—
286#103—274no—
287C33#41911yescharacteristic, complemented, Sylow
288C23×C3—21no—
289C23×C3—21no—
290C23×C3—21no—
291C23×C3—41no—
292C23×C3#62511yescharacteristic
293C23×C3#62611yescharacteristic
294C23×C3#62611yescharacteristic
295C3×Q8—94no—
296C3×Q8—184no—
297C3×Q8—182no—
298C2×C3×C4—22no—
299C2×C3×C4—22no—
300C2×C3×C4—22no—
301C2×C3×C4—22no—
302C2×C3×C4—21no—
303C2×C3×C4—22no—
304C2×C3×C4—21no—
305C2×C3×C4—22no—
306C2×C3×C4—21no—
307C2×C3×C4—21no—
308C2×C3×C4—21no—
309C2×C3×C4—21no—
310C2×C3×C4—42no—
311C2×C3×C4—42no—
312C2×C3×C4—41no—
313C2×C3×C4—41no—
314C2×C3×C4—184no—
315C2×C3×C4—182no—
316C2×C3×C4—182no—
317C2×C3×C4D3×Dic312yes—
318C2×C3×C4D3×Dic312yes—
319C2×C3×C4#62412yes—
320C2×C3×C4#62511yescharacteristic
321C2×C3×C4#62512yes—
322C2×C3×C4#62611yescharacteristic
323C2×C3×C4#62612yes—
324C2×C3×C4#62611yescharacteristic
325C2×C3×C4#62612yes—
326C2×C3×C4C2×D3211yescharacteristic
327C2×C3×C4C2×D3211yescharacteristic
328C2×C3×C4C2×D3211yescharacteristic
329C2×Dic3—64no—
330C2×Dic3—68no—
331C2×Dic3—64no—
332C2×Dic3—64no—
333C2×Dic3—62no—
334C2×Dic3—62no—
335C2×Dic3—64no—
336C2×Dic3—62no—
337C2×Dic3—62no—
338Dic6—34no—
339Dic6—34no—
340Dic6—64no—
341Dic6—64no—
342Dic6—68no—
343Dic6—64no—
344Dic6—64no—
345Dic6—62no—
346Dic6—62no—
347C3×C8—182no—
348C3⋊C8—64no—
349C3⋊C8—62no—
350C3⋊C8—62no—
351C2×C32—22no—
352C2×C32—24no—
353C2×C32—22no—
354C2×C32—22no—
355C2×C32—22no—
356C2×C32—22no—
357C2×C32—22no—
358C2×C32—21no—
359C2×C32—21no—
360C2×C32—21no—
361C2×C32—41no—
362C2×C32—42no—
363C2×C32—42no—
364C2×C32—42no—
365C2×C32—44no—
366C2×C32#90912yes—
367C2×C32C2×(C2×D3)⋊C411yescharacteristic
368C2×C32C2×(C2×D3)⋊C411yescharacteristic
369C2×C32#94212yes—
370C2×C32#95112yes—
371C2×C32C2×#21711yescharacteristic
372C2×Q8—94no—
373C2×Q8—92no—
374C2×Q8—181no—
375C2×Q8—182no—
376C22×C4—181no—
377C22×C4#111311yescharacteristic
378M16—182no—
379C2×C8—92no—
380C4⋊C4—92no—
381C22⋊C4—181no—
382C42—181no—
383C22×C3—24no—
384C22×C3—22no—
385C22×C3—22no—
386C22×C3—22no—
387C22×C3—21no—
388C22×C3—21no—
389C22×C3—21no—
390C22×C3—22no—
391C22×C3—22no—
392C22×C3—22no—
393C22×C3—42no—
394C22×C3—42no—
395C22×C3—44no—
396C22×C3—41no—
397C22×C3—42no—
398C22×C3(C2×D3)⋊Dic312yes—
399C22×C3(C2×D3)⋊Dic312yes—
400C22×C3#372212yes—
401C22×C3C2×#62511yescharacteristic
402C22×C3C2×#62611yescharacteristic
403C22×C3C2×#62611yescharacteristic
404C3×C4—21no—
405C3×C4—21no—
406C3×C4—21no—
407C3×C4—21no—
408C3×C4—21no—
409C3×C4—21no—
410C3×C4—22no—
411C3×C4—22no—
412C3×C4—22no—
413C3×C4—41no—
414C3×C4—41no—
415C3×C4—42no—
416C3×C4—184no—
417C3×C4—362no—
418C3×C4(C2×D3)⋊Dic312yes—
419C3×C4(C2×D3)⋊Dic312yes—
420C3×C4#372212yes—
421C3×C4C2×D3×Dic311yescharacteristic
422C3×C4C2×D3×Dic311yescharacteristic
423C3×C4C2×#62411yescharacteristic
424C3×C4C2×#62511yescharacteristic
425C3×C4C2×#62611yescharacteristic
426C3×C4C2×#62611yescharacteristic
427Dic3—68no—
428Dic3—64no—
429Dic3—64no—
430Dic3—124no—
431Dic3—122no—
432Dic3—122no—
433C32—21no—
434C32—21no—
435C32—21no—
436C32—41no—
437C32#500411yescharacteristic, complemented
438C32#512011yescharacteristic, complemented
439C32#522511yescharacteristic, complemented
440C23#629511yescharacteristic
441Q8—94no—
442Q8—182no—
443Q8—184no—
444C2×C4—182no—
445C2×C4—184no—
446C2×C4—182no—
447C2×C4#629412yes—
448C2×C4#629512yes—
449C2×C4#629511yescharacteristic
450C2×C4C2×#111311yescharacteristic
451C8—182no—
452C2×C3—24no—
453C2×C3—22no—
454C2×C3—22no—
455C2×C3—22no—
456C2×C3—22no—
457C2×C3—22no—
458C2×C3—22no—
459C2×C3—21no—
460C2×C3—21no—
461C2×C3—21no—
462C2×C3—42no—
463C2×C3—42no—
464C2×C3—44no—
465C2×C3—42no—
466C2×C3—41no—
467C2×C3#6382712yes—
468C2×C3#6383112yes—
469C2×C3#6383312yes—
470C2×C3C2×(C2×D3)⋊Dic311yescharacteristic
471C2×C3C2×(C2×D3)⋊Dic311yescharacteristic
472C2×C3C2×#372211yescharacteristic
473C22—22no—
474C22#8881412yes—
475C22C2×#629511yescharacteristic
476C4—184no—
477C4—362no—
478C4#8881412yes—
479C4C2×#629411yescharacteristic
480C4C2×#629511yescharacteristic
481C3—21no—
482C3—21no—
483C3—21no—
484C3—41no—
485C3#1059109911yescharacteristic, complemented
486C3#1059110011yescharacteristic, complemented
487C3#1059110711yescharacteristic, complemented
488C2—22no—
489C2#1172366312yes—
490C2C2×#8881411yescharacteristic
subgroupG/H
centerC22C2×#6295
commutatorC33×C4C22×C4
FrattiniC2×C4C2×#1113
FittingC22×C33×C4C22
socleC22×C33C2×D4
prime radicalC23×C33D4
cosocleC2×C33×C4C23
hypercenterC22×C4#1113
hypocenterC33#419
LCSG ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
4. #11722076C2semidirect
12. C2×C3×#634C4semidirect
59. C2×#634Dic3semidirect
287. C33#419semidirect
437. C32#5004semidirect
438. C32#5120semidirect
439. C32#5225semidirect
485. C3#10591099semidirect
486. C3#10591100semidirect
487. C3#10591107semidirect

Maximal subgroups

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

subgroupindex
C22×C3×#6342
C3×#641352
C2×C3×#37482
#117220762
#105911073
#105911003
#105910993