Finite Group Databasebeta

← #13148815

Subgroups — #13148815

Order 1728.

998 conjugacy classes of subgroups.

The subgroup lattice has 548 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#88641C212yesmaximal, complemented
2#11721744C211yesmaximal, characteristic, complemented
3C9×#893C211yesmaximal, characteristic, complemented
4C3×#63457C211yesmaximal, characteristic, complemented
5#11721043C211yesmaximal, characteristic, complemented
6#11721042C212yesmaximal, complemented
7#11721036C211yesmaximal, characteristic, complemented
8Dic3×D36C211yesmaximal, characteristic, complemented
9#11720982C212yesmaximal, complemented
10#11720976C212yesmaximal, complemented
11C4×Dic3×D9C211yesmaximal, characteristic, complemented
12#10594367—31nomaximal
13#10589650—31nomaximal
14C2×#6226C2212yes—
15C2×C9×D12C2211yescharacteristic
16C2×C4×D3×C9C2212yes—
17C2×C3×D36C2212yes—
18C2×C3×D36C2211yescharacteristic
19C2×C3×C4×D9C2211yescharacteristic
20C2×#6194C2212yes—
21C2×#6194C2212yes—
22C2×#6194C2212yes—
23C2×Dic3×D9C2211yescharacteristic
24C2×Dic3×D9C2212yes—
25C2×Dic3×D9C2211yescharacteristic
26#88641—28no—
27C3×C9×C4⋊C4C2211yescharacteristic
28C4⋊(C3⋊Dic9)C2211yescharacteristic
29#88534C2212yes—
30C4×C3⋊Dic9C2211yescharacteristic
31C9×(C2×D3)⋊C4C2212yes—
32C3×(C2×D9)⋊C4C2212yes—
33C4×C9×Dic3C2211yescharacteristic
34(C2×D3)⋊Dic9C2212yes—
35(C2×D9)⋊Dic3C2212yes—
36Dic3×Dic9C2211yescharacteristic
37Dic3×Dic9C2211yescharacteristic
38C2×#3795—32no—
39#64097—31no—
40C3×#895—31no—
41C3×#893D311yescharacteristic
42#63914—31no—
43#63913—32no—
44#63908—31no—
45Dic3×D12—31no—
46#63885—32no—
47#63879—32no—
48C4×D3×Dic3—31no—
49C2×#3697—32no—
50C4×D4×C9—31no—
51#63502—31no—
52#63500—31no—
53C4×#611—32no—
54#63457D311yescharacteristic, complemented
55#63450—31no—
56#63450—61no—
57#63447—32no—
58#63446—32no—
59C4×D36—31no—
60C42×D9—31no—
61C22×D3×C9C2312yes—
62C22×C3×D9C2312yes—
63C22×C3×D9C2311yescharacteristic
64C2×C3×C4×C9C2312yes—
65C2×C3×C4×C9C2311yescharacteristic
66C2×C3⋊Dic9C2312yes—
67C2×C3⋊Dic9C2311yescharacteristic
68C2×C3⋊Dic9C2311yescharacteristic
69#6226—24no—
70C2×C9×Dic3C2311yescharacteristic
71C2×C9×Dic3C2311yescharacteristic
72C2×C3×Dic9C2311yescharacteristic
73C9×D12—22no—
74C4×D3×C9—24no—
75C3×D36—24no—
76C3×D36—22no—
77C3×C4×D9—22no—
78#6194—24no—
79#6194—24no—
80#6194—24no—
81Dic3×D9—22no—
82Dic3×D9—24no—
83Dic3×D9D414yescomplemented
84#5596—91no—
85#5548—91no—
86C2×#634—32no—
87C2×C3×D12—32no—
88C2×C3×D12—31no—
89C2×C3×D12C2×D311yescharacteristic
90C2×C3×C4×D3—31no—
91C2×C3×C4×D3C2×D312yes—
92C2×#625—32no—
93C2×#625—32no—
94C2×#625—32no—
95C2×D3×Dic3—31no—
96C2×D3×Dic3—32no—
97C2×D3×Dic3—31no—
98#3795—68no—
99C32×C4⋊C4C2×D311yescharacteristic
100C4⋊(C3⋊Dic3)—31no—
101#3750—32no—
102C4×C3⋊Dic3—31no—
103C3×(C2×D3)⋊C4—32no—
104C3×(C2×D3)⋊C4C2×D312yes—
105C3×C4×Dic3C2×D311yescharacteristic
106(C2×D3)⋊Dic3—32no—
107(C2×D3)⋊Dic3—32no—
108Dic32—31no—
109Dic32—31no—
110C2×D4×C9—31no—
111C22×C4×C9—32no—
112C2×#611—32no—
113C2×#611—32no—
114C2×#611—32no—
115#3697—68no—
116C2×D36C2×D311yescharacteristic
117C2×D36C2×D312yes—
118C2×C4×D9—31no—
119C2×C4×D9—32no—
120C2×C4×D9—31no—
121C2×C4×D9C2×D311yescharacteristic
122C2×Dic18—32no—
123C2×Dic18—62no—
124C9×C4⋊C4—21no—
125C9×C4⋊C4C2×D311yescharacteristic
126C9×C22⋊C4—32no—
127C42×C9—31no—
128#3676—32no—
129(C2×D9)⋊C4—32no—
130(C2×D9)⋊C4C2×D312yes—
131C4⋊Dic9—31no—
132C4⋊Dic9—61no—
133Dic9⋊C4—32no—
134Dic9⋊C4—62no—
135C4×Dic9—31no—
136C4×Dic9—31no—
137C4×Dic9—31no—
138C4×Dic9—61no—
139C22×C3×C9C2411yescharacteristic
140C2×D3×C9—22no—
141C2×D3×C9—22no—
142C2×D3×C9—22no—
143C2×C3×D9—21no—
144C2×C3×D9—21no—
145C2×C3×D9—22no—
146C2×C3×D9—22no—
147C2×C3×D9—22no—
148C2×C3×D9C2×D412yes—
149C3×C4×C9—22no—
150C3×C4×C9#4112yescomplemented
151C3⋊Dic9—22no—
152C3⋊Dic9C2×D412yes—
153C3⋊Dic9#4112yescomplemented
154C9×Dic3—21no—
155C9×Dic3C2×D412yes—
156C3×Dic9—21no—
157C2×#237—92no—
158C2×#235—92no—
159C3×#121—91no—
160C3×C4×D4—31no—
161#940—91no—
162#940—91no—
163#939—92no—
164#938—91no—
165D4×Dic3—91no—
166#930—91no—
167C4×#67—92no—
168C2×C4×Dic3—91no—
169#895—31no—
170#893D911yescharacteristic, complemented
171#888—91no—
172#888—91no—
173#888—181no—
174#885—92no—
175#885—92no—
176#884—92no—
177#67⋊C4—92no—
178C4×D12—91no—
179C42×D3—91no—
180C22×C3×D3—31no—
181C22×C3×D3—32no—
182C22×C3×D3C22×D312yes—
183C2×C32×C4C22×D312yes—
184C2×C32×C4C22×D311yescharacteristic
185C2×C3⋊Dic3—31no—
186C2×C3⋊Dic3—31no—
187C2×C3⋊Dic3—32no—
188#634—64no—
189C2×C3×Dic3—31no—
190C2×C3×Dic3C22×D311yescharacteristic
191C2×C3×Dic3C22×D311yescharacteristic
192C3×D12—22no—
193C3×D12—64no—
194C3×D12—62no—
195C3×C4×D3—24no—
196C3×C4×D3—62no—
197#625—64no—
198#625—64no—
199#625—64no—
200D3×Dic3—34no—
201D3×Dic3—62no—
202D3×Dic3—64no—
203C23×C9—32no—
204C22×D9C22×D312yes—
205C22×D9C22×D311yescharacteristic
206D4×C9—62no—
207C2×C4×C9—21no—
208C2×C4×C9—22no—
209C2×C4×C9—31no—
210C2×C4×C9—31no—
211C2×C4×C9—64no—
212C2×C4×C9C22×D312yes—
213C2×C4×C9C22×D311yescharacteristic
214#611—64no—
215#611—64no—
216#611—64no—
217C2×Dic9—31no—
218C2×Dic9—31no—
219C2×Dic9—32no—
220C2×Dic9—61no—
221C2×Dic9—61no—
222C2×Dic9—62no—
223C2×Dic9C22×D311yescharacteristic
224D36—24no—
225D36—22no—
226C4×D9—22no—
227C4×D9—34no—
228C4×D9—62no—
229C4×D9—64no—
230Dic18—64no—
231Dic18—124no—
232#532—271noSylow
233C2×C3×C9C22×D411yescharacteristic
234C2×C3×C9C2×#4111yescharacteristic
235C2×C3×C9C2×#4111yescharacteristic
236D3×C9—42no—
237C3×D9—22no—
238C3×D9—42no—
239C2×C3×D4—31no—
240C2×C3×D4—91no—
241C2×C3×D4—92no—
242C22×C3×C4—32no—
243C22×C3×C4—91no—
244C2×#67—92no—
245C2×#67—92no—
246C2×#67—92no—
247C2×#67—92no—
248C2×#67—92no—
249C2×#67—92no—
250C22×Dic3—91no—
251C22×Dic3—92no—
252C22×Dic3—91no—
253#237—188no—
254#235—188no—
255C2×D12—31no—
256C2×D12—32no—
257C2×D12C2×D911yescharacteristic
258C2×C4×D3—31no—
259C2×C4×D3—91no—
260C2×C4×D3—91no—
261C2×C4×D3—92no—
262C2×C4×D3C2×D912yes—
263C2×Dic6—92no—
264C2×Dic6—92no—
265C2×Dic6—182no—
266C3×C4⋊C4—21no—
267C3×C4⋊C4C2×D911yescharacteristic
268C3×C4⋊C4D3211yescharacteristic
269C3×C22⋊C4—32no—
270C3×C22⋊C4—92no—
271C3×C42—31no—
272#217—92no—
273#217—92no—
274(C2×D3)⋊C4—32no—
275(C2×D3)⋊C4—92no—
276(C2×D3)⋊C4—92no—
277(C2×D3)⋊C4C2×D912yes—
278C4⋊Dic3—91no—
279C4⋊Dic3—91no—
280C4⋊Dic3—181no—
281Dic3⋊C4—92no—
282Dic3⋊C4—92no—
283Dic3⋊C4—182no—
284C4×Dic3—91no—
285C4×Dic3—91no—
286C4×Dic3—91no—
287C4×Dic3—91no—
288C4×Dic3—91no—
289C4×Dic3—91no—
290C4×Dic3—181no—
291C4×Dic3C2×D911yescharacteristic
292C22×C32C23×D311yescharacteristic
293C2×C3×D3—22no—
294C2×C3×D3—22no—
295C2×C3×D3—22no—
296C2×C3×D3—32no—
297C2×C3×D3—61no—
298C2×C3×D3—61no—
299C2×C3×D3—62no—
300C2×C3×D3—62no—
301C2×C3×D3—62no—
302C32×C4—22no—
303C32×C4#23512yes—
304C3⋊Dic3—32no—
305C3⋊Dic3—32no—
306C3⋊Dic3—62no—
307C3×Dic3—21no—
308C3×Dic3—61no—
309C3×Dic3D3×D412yes—
310C22×C9—21no—
311C22×C9—62no—
312C22×C9—62no—
313C22×C9—62no—
314C22×C9C23×D311yescharacteristic
315C2×D9—21no—
316C2×D9—21no—
317C2×D9—22no—
318C2×D9—22no—
319C2×D9—22no—
320C2×D9D3×D412yes—
321C4×C9—22no—
322C4×C9—22no—
323C4×C9—32no—
324C4×C9—42no—
325C4×C9—61no—
326C4×C9#23712yescomplemented
327Dic9—21no—
328Dic9—32no—
329Dic9—32no—
330Dic9—62no—
331Dic9—62no—
332Dic9—62no—
333Dic9—122no—
334C2×#41—272no—
335C4⋊Q8—271no—
336#127—271no—
337#124—272no—
338#121—91no—
339#121—272no—
340#121—271no—
341C4×D4—31no—
342C4×D4—271no—
343C4×D4—272no—
344C2×C42—271no—
345C3×C9#53211yescharacteristic, complemented, Sylow
346C23×C3—32no—
347C23×C3—91no—
348C23×C3—92no—
349C22×D3—31no—
350C22×D3—32no—
351C22×D3C22×D912yes—
352C3×D4—62no—
353C3×D4—184no—
354C3×D4—182no—
355C2×C3×C4—21no—
356C2×C3×C4—22no—
357C2×C3×C4—31no—
358C2×C3×C4—31no—
359C2×C3×C4—64no—
360C2×C3×C4—91no—
361C2×C3×C4—182no—
362C2×C3×C4C22×D912yes—
363C2×C3×C4C22×D911yescharacteristic
364C2×C3×C4C2×D3212yes—
365C2×C3×C4C2×D3211yescharacteristic
366#67—184no—
367#67—184no—
368#67—184no—
369#67—184no—
370#67—184no—
371#67—184no—
372C2×Dic3—31no—
373C2×Dic3—91no—
374C2×Dic3—91no—
375C2×Dic3—91no—
376C2×Dic3—91no—
377C2×Dic3—94no—
378C2×Dic3—92no—
379C2×Dic3—92no—
380C2×Dic3—181no—
381C2×Dic3—181no—
382C2×Dic3—182no—
383C2×Dic3—182no—
384C2×Dic3—184no—
385C2×Dic3C22×D911yescharacteristic
386C2×Dic3C22×D911yescharacteristic
387D12—22no—
388D12—64no—
389D12—62no—
390C4×D3—24no—
391C4×D3—62no—
392C4×D3—94no—
393C4×D3—182no—
394C4×D3—184no—
395Dic6—184no—
396Dic6—184no—
397Dic6—364no—
398C2×C32C2×#23511yescharacteristic
399C2×C32C2×D3×D411yescharacteristic
400C2×C32D3×#4111yescharacteristic
401C3×D3—42no—
402C3×D3—62no—
403C3×D3—122no—
404C2×C9—21no—
405C2×C9—21no—
406C2×C9—21no—
407C2×C9—122no—
408C2×C9C2×D3×D411yescharacteristic
409C2×C9C2×#23711yescharacteristic
410C2×C9D3×#4111yescharacteristic
411D9—22no—
412D9—42no—
413#41—548no—
414C2×Q8—272no—
415C2×D4—31no—
416C2×D4—91no—
417C2×D4—92no—
418C2×D4—272no—
419C2×D4—272no—
420C2×D4—272no—
421C22×C4—32no—
422C22×C4—91no—
423C22×C4—271no—
424C22×C4—271no—
425C22×C4—272no—
426C4⋊C4—272no—
427C4⋊C4—271no—
428C4⋊C4D3×D911yescharacteristic
429C22⋊C4—32no—
430C22⋊C4—92no—
431C22⋊C4—272no—
432C22⋊C4—272no—
433C42—31no—
434C42—271no—
435C42—271no—
436C42—271no—
437C22×C3—21no—
438C22×C3—62no—
439C22×C3—62no—
440C22×C3—62no—
441C22×C3—92no—
442C22×C3—181no—
443C22×C3—181no—
444C22×C3—182no—
445C22×C3—182no—
446C22×C3—182no—
447C22×C3C23×D911yescharacteristic
448C22×C3C22×D3211yescharacteristic
449C2×D3—22no—
450C2×D3—22no—
451C2×D3—22no—
452C2×D3—32no—
453C2×D3—61no—
454C2×D3—61no—
455C2×D3—62no—
456C2×D3—62no—
457C2×D3—62no—
458C3×C4—22no—
459C3×C4—22no—
460C3×C4—22no—
461C3×C4—32no—
462C3×C4—42no—
463C3×C4—61no—
464C3×C4—181no—
465C3×C4#369712yescomplemented
466C3×C4#379512yes—
467Dic3—21no—
468Dic3—61no—
469Dic3—92no—
470Dic3—92no—
471Dic3—92no—
472Dic3—92no—
473Dic3—182no—
474Dic3—182no—
475Dic3—182no—
476Dic3—182no—
477Dic3—362no—
478Dic3D4×D912yes—
479C32#554811yescharacteristic
480C9—21no—
481C9#559611yescharacteristic, complemented
482C23—32no—
483C23—91no—
484C23—92no—
485Q8—544no—
486D4—62no—
487D4—184no—
488D4—182no—
489D4—544no—
490D4—544no—
491D4—544no—
492C2×C4—31no—
493C2×C4—31no—
494C2×C4—64no—
495C2×C4—91no—
496C2×C4—182no—
497C2×C4—274no—
498C2×C4—271no—
499C2×C4—271no—
500C2×C4—272no—
501C2×C4—544no—
502C2×C4—542no—
503C2×C4C2×D3×D912yes—
504C2×C4C2×D3×D911yescharacteristic
505C2×C3—21no—
506C2×C3—21no—
507C2×C3—21no—
508C2×C3—122no—
509C2×C3—182no—
510C2×C3—362no—
511C2×C3C2×#369711yescharacteristic
512C2×C3C2×D4×D911yescharacteristic
513C2×C3#41×D911yescharacteristic
514C2×C3C2×#379511yescharacteristic
515C2×C3D32×D411yescharacteristic
516C2×C3#6431911yescharacteristic
517D3—42no—
518D3—62no—
519D3—122no—
520C22—62no—
521C22—62no—
522C22—62no—
523C22—92no—
524C22—181no—
525C22—181no—
526C22—182no—
527C22—182no—
528C22—182no—
529C22C22×D3×D911yescharacteristic
530C4—22no—
531C4—32no—
532C4—61no—
533C4—181no—
534C4—272no—
535C4—272no—
536C4—542no—
537C4#8864112yescomplemented
538C3—21no—
539C3#1058965011yescharacteristic, complemented
540C3#1059436711yescharacteristic
541C2—122no—
542C2—182no—
543C2—362no—
544C2C2×#8864111yescharacteristic
545C2D3×D4×D911yescharacteristic
546C2#1172251411yescharacteristic
subgroupG/H
centerC22C22×D3×D9
commutator, cosocleC22×C3×C9C24
FrattiniC22×C3C22×D32
FittingC3×C9×C4⋊C4C22
socleC22×C32C23×D3
hypercenterC4⋊C4D3×D9
hypocenterC3×C9#532
LCSG ▷ C22×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
1. C2×#88641C2semidirect
2. #11721744C2semidirect
3. C9×#893C2semidirect
4. C3×#63457C2semidirect
5. #11721043C2semidirect
6. #11721042C2semidirect
7. #11721036C2semidirect
8. Dic3×D36C2semidirect
9. #11720982C2semidirect
10. #11720976C2semidirect
11. C4×Dic3×D9C2semidirect
54. #63457D3semidirect
83. Dic3×D9D4semidirect
150. C3×C4×C9#41semidirect
153. C3⋊Dic9#41semidirect
170. #893D9semidirect
326. C4×C9#237semidirect
345. C3×C9#532semidirect
465. C3×C4#3697semidirect
481. C9#5596semidirect
537. C4#88641semidirect
539. C3#10589650semidirect

Maximal subgroups

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

subgroupindex
C2×#886412
#117217442
C9×#8932
C3×#634572
#117210432
#117210422
#117210362
Dic3×D362
#117209822
#117209762
C4×Dic3×D92
#105943673
#105896503