Finite Group Databasebeta

← #13160576

Subgroups — #13160576

Order 1728.

1508 conjugacy classes of subgroups.

The subgroup lattice has 516 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#88857C211yesmaximal, characteristic, complemented
2C2×C3×#3752C211yesmaximal, characteristic
3C2×#88803C211yesmaximal, characteristic
4C2×#88803C211yesmaximal, characteristic
5C2×Dic3×C3⋊Dic3C211yesmaximal, characteristic, complemented
6C32×C3⋊#95C211yesmaximal, characteristic, complemented
7#11722061C211yesmaximal, characteristic, complemented
8#10592368—31nomaximal
9#10591281—34nomaximal
10C23×C3×C3⋊D3C2211yescharacteristic
11C22×C33×C4C2211yescharacteristic
12C22×C3⋊(C3⋊Dic3)C2211yescharacteristic
13C22×C3⋊(C3⋊Dic3)C2211yescharacteristic
14C22×C3×C3⋊Dic3C2211yescharacteristic
15C22×C32×Dic3C2211yescharacteristic
16C22×C32×Dic3C2211yescharacteristic
17#88857—22no—
18C3×#3752—22no—
19#88803—22no—
20#88803C414yescomplemented
21Dic3×C3⋊Dic3—22no—
22C2×#3752—31no—
23C2×#3752—31no—
24C2×#3752D311yescharacteristic
25C2×#3750—31no—
26C2×#3750—34no—
27C2×#3750—64no—
28C2×C4×C3⋊Dic3—31no—
29C2×C3×(C2×D3)⋊C4—34no—
30C2×(C2×D3)⋊Dic3—34no—
31C2×(C2×D3)⋊Dic3—34no—
32C2×Dic32—34no—
33C32×#95—31no—
34#63658—31no—
35C3×C3⋊#95D314yescomplemented
36#63579—34no—
37C23×C33C2311yescharacteristic
38C22×C3×C3⋊D3—21no—
39C22×C3×C3⋊D3—21no—
40C22×C3×C3⋊D3—21no—
41C22×C3×C3⋊D3—21no—
42C22×C3×C3⋊D3—21no—
43C22×C3×C3⋊D3—21no—
44C22×C3×C3⋊D3C2×C412yes—
45C2×C33×C4—21no—
46C2×C33×C4—21no—
47C2×C33×C4—21no—
48C2×C3⋊(C3⋊Dic3)—21no—
49C2×C3⋊(C3⋊Dic3)—21no—
50C2×C3⋊(C3⋊Dic3)—21no—
51C2×C3⋊(C3⋊Dic3)—21no—
52C2×C3⋊(C3⋊Dic3)C2×C412yes—
53C2×C3⋊(C3⋊Dic3)D412yes—
54C2×C3×C3⋊Dic3—21no—
55C2×C3×C3⋊Dic3—21no—
56C2×C3×C3⋊Dic3D412yes—
57C2×C32×Dic3—21no—
58C2×C32×Dic3—21no—
59C2×C32×Dic3—21no—
60C2×C32×Dic3—21no—
61C2×C32×Dic3—21no—
62C2×C32×Dic3C2×C412yes—
63#4944—91no—
64#4669—94no—
65C23×C3⋊D3C2×D311yescharacteristic
66C23×C3×D3—34no—
67C22×C32×C4—24no—
68C22×C32×C4—31no—
69C22×C32×C4—31no—
70C22×C32×C4C2×D314yes—
71C22×C32×C4C2×D311yescharacteristic
72C22×C3⋊Dic3—31no—
73C22×C3⋊Dic3—31no—
74C22×C3⋊Dic3—34no—
75C22×C3⋊Dic3—34no—
76C22×C3⋊Dic3—64no—
77C22×C3⋊Dic3—64no—
78C22×C3⋊Dic3C2×D311yescharacteristic
79C22×C3×Dic3—34no—
80C22×C3×Dic3C2×D314yes—
81C22×C3×Dic3C2×D314yes—
82#3752—22no—
83#3752—34no—
84#3752—62no—
85#3750—62no—
86#3750—68no—
87#3750—616no—
88C4×C3⋊Dic3—62no—
89C3×(C2×D3)⋊C4—68no—
90(C2×D3)⋊Dic3—316no—
91(C2×D3)⋊Dic3—68no—
92Dic32—68no—
93C22×C33C22×C411yescharacteristic
94C22×C33C2×D411yescharacteristic
95C22×C33C2×D411yescharacteristic
96C22×C33#4111yescharacteristic
97C22×C33#4111yescharacteristic
98C22×C33#4111yescharacteristic
99C22×C33#4111yescharacteristic
100C2×C3×C3⋊D3—22no—
101C2×C3×C3⋊D3—22no—
102C2×C3×C3⋊D3—22no—
103C2×C3×C3⋊D3—41no—
104C2×C3×C3⋊D3—41no—
105C2×C3×C3⋊D3—41no—
106C2×C3×C3⋊D3—41no—
107C33×C4—41no—
108C3⋊(C3⋊Dic3)—22no—
109C3⋊(C3⋊Dic3)—22no—
110C3×C3⋊Dic3—22no—
111C32×Dic3—22no—
112C32×Dic3—41no—
113C2×C3×C22⋊C4—91no—
114C2×#217—91no—
115C2×#217—91no—
116C2×(C2×D3)⋊C4—34no—
117C2×(C2×D3)⋊C4—94no—
118C2×(C2×D3)⋊C4—94no—
119C2×Dic3⋊C4—94no—
120C2×Dic3⋊C4—91no—
121C2×Dic3⋊C4—184no—
122C2×C4×Dic3—94no—
123C2×C4×Dic3—91no—
124C3×#95—34no—
125C3⋊#95—94no—
126C3⋊#95—91no—
127C3⋊#95C3⋊D311yescharacteristic, complemented
128C23×C32—24no—
129C23×C32C22×D314yes—
130C23×C32C22×D311yescharacteristic
131C22×C3⋊D3—21no—
132C22×C3⋊D3—21no—
133C22×C3⋊D3—21no—
134C22×C3⋊D3—21no—
135C22×C3⋊D3—21no—
136C22×C3⋊D3—21no—
137C22×C3⋊D3C4×D312yes—
138C22×C3×D3—38no—
139C22×C3×D3—64no—
140C22×C3×D3—64no—
141C22×C3×D3—64no—
142C22×C3×D3—64no—
143C22×C3×D3—64no—
144C22×C3×D3—64no—
145C2×C32×C4—28no—
146C2×C32×C4—24no—
147C2×C32×C4—21no—
148C2×C32×C4—21no—
149C2×C32×C4—21no—
150C2×C32×C4—24no—
151C2×C32×C4—24no—
152C2×C32×C4—32no—
153C2×C32×C4—44no—
154C2×C32×C4—44no—
155C2×C32×C4—61no—
156C2×C32×C4—61no—
157C2×C32×C4—61no—
158C2×C32×C4—61no—
159C2×C32×C4—61no—
160C2×C3⋊Dic3—21no—
161C2×C3⋊Dic3—21no—
162C2×C3⋊Dic3—32no—
163C2×C3⋊Dic3—32no—
164C2×C3⋊Dic3—38no—
165C2×C3⋊Dic3—38no—
166C2×C3⋊Dic3—68no—
167C2×C3⋊Dic3—68no—
168C2×C3⋊Dic3—68no—
169C2×C3⋊Dic3—68no—
170C2×C3⋊Dic3—64no—
171C2×C3⋊Dic3—61no—
172C2×C3⋊Dic3—61no—
173C2×C3⋊Dic3—61no—
174C2×C3⋊Dic3—64no—
175C2×C3⋊Dic3—61no—
176C2×C3⋊Dic3—64no—
177C2×C3⋊Dic3—64no—
178C2×C3⋊Dic3—124no—
179C2×C3⋊Dic3—124no—
180C2×C3⋊Dic3#6712yes—
181C2×C3×Dic3—24no—
182C2×C3×Dic3—24no—
183C2×C3×Dic3—24no—
184C2×C3×Dic3—24no—
185C2×C3×Dic3—24no—
186C2×C3×Dic3—38no—
187C2×C3×Dic3—64no—
188C2×C3×Dic3—64no—
189C2×C3×Dic3C4×D318yes—
190#388—271noSylow
191C2×C33#11711yescharacteristic
192C2×C33C4×D411yescharacteristic
193C2×C33C4×D411yescharacteristic
194C2×C33#12111yescharacteristic
195C2×C33#12311yescharacteristic
196C2×C33#12411yescharacteristic
197C2×C33#12611yescharacteristic
198C3×C3⋊D3—42no—
199C24×C3—91no—
200C23×D3—34no—
201C22×C3×C4—24no—
202C22×C3×C4—34no—
203C22×C3×C4—34no—
204C22×C3×C4—91no—
205C22×C3×C4D3214yes—
206C22×C3×C4C2×C3⋊D311yescharacteristic
207C22×Dic3—34no—
208C22×Dic3—94no—
209C22×Dic3—94no—
210C22×Dic3—91no—
211C22×Dic3—91no—
212C22×Dic3—184no—
213C22×Dic3—184no—
214C22×Dic3C2×C3⋊D311yescharacteristic
215C22×Dic3C2×C3⋊D311yescharacteristic
216C3×C22⋊C4—182no—
217#217—94no—
218#217—182no—
219(C2×D3)⋊C4—68no—
220(C2×D3)⋊C4—916no—
221(C2×D3)⋊C4—188no—
222Dic3⋊C4—188no—
223Dic3⋊C4—182no—
224Dic3⋊C4—1816no—
225C4×Dic3—182no—
226C4×Dic3—188no—
227C22×C32—24no—
228C22×C32—24no—
229C22×C32—24no—
230C22×C32—24no—
231C22×C32—24no—
232C22×C32—24no—
233C22×C32—24no—
234C22×C32C2×C4×D314yes—
235C22×C32C2×C4×D311yescharacteristic
236C22×C32#23514yes—
237C22×C32#23514yes—
238C22×C32#23511yescharacteristic
239C22×C32D3×D414yes—
240C22×C32D3×D414yes—
241C22×C32D3×D411yescharacteristic
242C22×C32#23714yes—
243C22×C32#23711yescharacteristic
244C22×C32#23711yescharacteristic
245C22×C32#23711yescharacteristic
246C22×C32#23914yes—
247C22×C32C2×#6711yescharacteristic
248C2×C3⋊D3—22no—
249C2×C3⋊D3—22no—
250C2×C3⋊D3—22no—
251C2×C3⋊D3—41no—
252C2×C3⋊D3—41no—
253C2×C3⋊D3—41no—
254C2×C3⋊D3—41no—
255C2×C3×D3—68no—
256C2×C3×D3—68no—
257C2×C3×D3—68no—
258C2×C3×D3—124no—
259C2×C3×D3—124no—
260C2×C3×D3—124no—
261C2×C3×D3—124no—
262C32×C4—48no—
263C32×C4—44no—
264C32×C4—41no—
265C32×C4—62no—
266C32×C4—121no—
267C3⋊Dic3—22no—
268C3⋊Dic3—616no—
269C3⋊Dic3—62no—
270C3⋊Dic3—68no—
271C3⋊Dic3—62no—
272C3⋊Dic3—68no—
273C3⋊Dic3—128no—
274C3×Dic3—28no—
275C3×Dic3—44no—
276C3×Dic3—68no—
277C2×C4⋊C4—271no—
278C2×C22⋊C4—91no—
279C2×C22⋊C4—271no—
280C2×C22⋊C4—271no—
281C2×C42—271no—
282#95—31no—
283#95—271no—
284C33#38811yescharacteristic, complemented, Sylow
285C23×C3—24no—
286C23×C3—92no—
287C23×C3—181no—
288C23×C3—181no—
289C23×C3—181no—
290C23×C3—181no—
291C23×C3—181no—
292C23×C3—181no—
293C23×C3C2×D3214yes—
294C23×C3C22×C3⋊D311yescharacteristic
295C22×D3—38no—
296C22×D3—64no—
297C22×D3—64no—
298C22×D3—64no—
299C22×D3—64no—
300C22×D3—64no—
301C22×D3—64no—
302C2×C3×C4—28no—
303C2×C3×C4—24no—
304C2×C3×C4—24no—
305C2×C3×C4—24no—
306C2×C3×C4—21no—
307C2×C3×C4—21no—
308C2×C3×C4—21no—
309C2×C3×C4—38no—
310C2×C3×C4—44no—
311C2×C3×C4—44no—
312C2×C3×C4—64no—
313C2×C3×C4—64no—
314C2×C3×C4—64no—
315C2×C3×C4—64no—
316C2×C3×C4—64no—
317C2×C3×C4—92no—
318C2×C3×C4—181no—
319C2×C3×C4—181no—
320C2×Dic3—21no—
321C2×Dic3—21no—
322C2×Dic3—21no—
323C2×Dic3—21no—
324C2×Dic3—21no—
325C2×Dic3—38no—
326C2×Dic3—64no—
327C2×Dic3—64no—
328C2×Dic3—98no—
329C2×Dic3—98no—
330C2×Dic3—92no—
331C2×Dic3—92no—
332C2×Dic3—188no—
333C2×Dic3—188no—
334C2×Dic3—188no—
335C2×Dic3—188no—
336C2×Dic3—184no—
337C2×Dic3—184no—
338C2×Dic3—184no—
339C2×Dic3—184no—
340C2×Dic3—181no—
341C2×Dic3—181no—
342C2×Dic3—181no—
343C2×Dic3—181no—
344C2×Dic3—364no—
345C2×Dic3—364no—
346C2×Dic3C4×C3⋊D312yes—
347C2×C32—24no—
348C2×C32—24no—
349C2×C32—24no—
350C2×C32—24no—
351C2×C32—24no—
352C2×C32—24no—
353C2×C32—24no—
354C2×C32#87214yes—
355C2×C32#87711yescharacteristic
356C2×C32#87911yescharacteristic
357C2×C32#67⋊C414yes—
358C2×C32#67⋊C411yescharacteristic
359C2×C32#88414yes—
360C2×C32#88514yes—
361C2×C32#88511yescharacteristic
362C2×C32#89314yes—
363C2×C32#89414yes—
364C2×C32#89814yes—
365C2×C32C4×#6711yescharacteristic
366C2×C32#93511yescharacteristic
367C2×C32#93811yescharacteristic
368C3⋊D3—42no—
369C3×D3—128no—
370C24—91no—
371C22×C4—31no—
372C22×C4—31no—
373C22×C4—91no—
374C22×C4—271no—
375C22×C4—271no—
376C22×C4D3×C3⋊D311yescharacteristic
377C4⋊C4—542no—
378C22⋊C4—182no—
379C22⋊C4—274no—
380C22⋊C4—542no—
381C42—542no—
382C22×C3—24no—
383C22×C3—24no—
384C22×C3—24no—
385C22×C3—24no—
386C22×C3—24no—
387C22×C3—24no—
388C22×C3—24no—
389C22×C3—182no—
390C22×C3—182no—
391C22×C3—182no—
392C22×C3—361no—
393C22×C3—361no—
394C22×C3—361no—
395C22×C3—361no—
396C22×C3#379514yes—
397C22×C3#379514yes—
398C22×C3#379614yes—
399C22×C3C4×D3214yes—
400C22×C3#380414yes—
401C22×C3D3×#6714yes—
402C22×C3#381114yes—
403C22×C3C2×C4×C3⋊D311yescharacteristic
404C22×C3#382811yescharacteristic
405C22×C3#382811yescharacteristic
406C22×C3D4×C3⋊D311yescharacteristic
407C22×C3D4×C3⋊D311yescharacteristic
408C22×C3#383011yescharacteristic
409C22×C3#383211yescharacteristic
410C2×D3—68no—
411C2×D3—68no—
412C2×D3—68no—
413C2×D3—124no—
414C2×D3—124no—
415C2×D3—124no—
416C2×D3—124no—
417C3×C4—48no—
418C3×C4—41no—
419C3×C4—44no—
420C3×C4—68no—
421C3×C4—124no—
422C3×C4—182no—
423Dic3—22no—
424Dic3—41no—
425Dic3—68no—
426Dic3—1816no—
427Dic3—182no—
428Dic3—182no—
429Dic3—188no—
430Dic3—188no—
431Dic3—368no—
432C32—24no—
433C32#466914yescomplemented
434C32#494411yescharacteristic, complemented
435C23—92no—
436C23—181no—
437C23—181no—
438C23—181no—
439C23—181no—
440C23—181no—
441C23—181no—
442C23C2×D3×C3⋊D311yescharacteristic
443C2×C4—21no—
444C2×C4—21no—
445C2×C4—21no—
446C2×C4—32no—
447C2×C4—61no—
448C2×C4—61no—
449C2×C4—61no—
450C2×C4—61no—
451C2×C4—61no—
452C2×C4—92no—
453C2×C4—181no—
454C2×C4—181no—
455C2×C4—272no—
456C2×C4—272no—
457C2×C4—541no—
458C2×C4—541no—
459C2×C4—541no—
460C2×C4—541no—
461C2×C3—24no—
462C2×C3—24no—
463C2×C3—24no—
464C2×C3—24no—
465C2×C3—24no—
466C2×C3—24no—
467C2×C3—24no—
468C2×C3—362no—
469C2×C3#6385814yes—
470C2×C3#6387714yes—
471C2×C3#6387914yes—
472C2×C3#6388114yes—
473C2×C3#6388414yes—
474C2×C3#6391314yes—
475C2×C3#6395814yes—
476C2×C3#6408111yescharacteristic
477C2×C3#6409011yescharacteristic
478C2×C3#6409311yescharacteristic
479C2×C3#6409411yescharacteristic
480C2×C3#6410211yescharacteristic
481C2×C3#6410311yescharacteristic
482C2×C3#6410711yescharacteristic
483D3—128no—
484C22—182no—
485C22—182no—
486C22—182no—
487C22—361no—
488C22—361no—
489C22—361no—
490C22—361no—
491C22#8901411yescharacteristic
492C22#8901711yescharacteristic
493C22#8902111yescharacteristic
494C22C4×D3×C3⋊D311yescharacteristic
495C22#8903111yescharacteristic
496C22C3⋊D3×#6711yescharacteristic
497C22#8903911yescharacteristic
498C4—41no—
499C4—62no—
500C4—121no—
501C4—182no—
502C4—542no—
503C4—542no—
504C3—24no—
505C3#1059128114yescomplemented
506C3#1059236811yescharacteristic, complemented
507C2—362no—
508C2#1172340711yescharacteristic
509C2#1172346411yescharacteristic
510C2#1172347211yescharacteristic
511C2#1172348811yescharacteristic
512C2#1172350711yescharacteristic
513C2#1172352811yescharacteristic
514C2#1172362311yescharacteristic
subgroupG/H
center, FrattiniC23C2×D3×C3⋊D3
commutatorC22×C33C22×C4
FittingC22×C33×C4C22
socle, cosocleC23×C33C23
prime radicalC23×C3×C3⋊D3C22
hypercenterC22×C4D3×C3⋊D3
hypocenterC33#388
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C23 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
1. C2×#88857C2semidirect
5. C2×Dic3×C3⋊Dic3C2semidirect
6. C32×C3⋊#95C2semidirect
7. #11722061C2semidirect
20. #88803C4semidirect
35. C3×C3⋊#95D3semidirect
127. C3⋊#95C3⋊D3semidirect
284. C33#388semidirect
433. C32#4669semidirect
434. C32#4944semidirect
505. C3#10591281semidirect
506. C3#10592368semidirect

Maximal subgroups

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

subgroupindex
C2×#888572
C2×C3×#37522
C2×#888032
C2×Dic3×C3⋊Dic32
C32×C3⋊#952
#117220612
#105923683
#105912813