Finite Group Databasebeta

← #13148022

Subgroups — #13148022

Order 1728.

1066 conjugacy classes of subgroups.

The subgroup lattice has 568 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#88663C211yesmaximal, characteristic, complemented
2C22×Dic3×D9C211yesmaximal, characteristic, complemented
3#11721723C211yesmaximal, characteristic, complemented
4C9×#886C211yesmaximal, characteristic, complemented
5C3×#63444C211yesmaximal, characteristic, complemented
6#11721285C211yesmaximal, characteristic, complemented
7#11721259C211yesmaximal, characteristic, complemented
8#11721028C212yesmaximal, complemented
9#11720999C212yesmaximal, complemented
10#11720998C212yesmaximal, complemented
11#11720982C212yesmaximal, complemented
12#10594082—31nomaximal
13#10589758—31nomaximal
14C23×C3×D9C2211yescharacteristic
15C22×C3⋊Dic9C2211yescharacteristic
16C2×C9×#67C2211yescharacteristic
17C2×C3×#611C2211yescharacteristic
18C22×C9×Dic3C2211yescharacteristic
19C2×C3×D36C2212yes—
20C2×#6194C2212yes—
21C2×#6194C2211yescharacteristic, complemented
22#88663—24no—
23C2×D3×Dic9C2211yescharacteristic, complemented
24C2×Dic3×D9—24no—
25C2×Dic3×D9C2212yes—
26C2×Dic3×D9C2212yes—
27C2×Dic3×D9C2211yescharacteristic, complemented
28C2×#6189C2211yescharacteristic, complemented
29C3×C9×C22⋊C4C2211yescharacteristic
30#88551C2211yescharacteristic
31#88534C2212yescomplemented
32C4×C3⋊Dic9C2212yescomplemented
33C9×(C2×D3)⋊C4C2212yescomplemented
34C3×(C2×D9)⋊C4C2212yes—
35C9×C4⋊Dic3C2212yescomplemented
36(C2×D9)⋊Dic3C2212yes—
37(C2×D9)⋊Dic3C2212yescomplemented
38#88442C2212yescomplemented
39C2×#3805—31no—
40C22×D3×Dic3—31no—
41#64086—31no—
42C3×#886D311yescharacteristic
43C3×#882—31no—
44#63976—31no—
45#63961—31no—
46#63902—32no—
47#63899—32no—
48#63898—32no—
49#63885—32no—
50C2×#3699—31no—
51C22×C4×D9—31no—
52C9×#123—31no—
53#63499—31no—
54#63491—31no—
55#63458—32no—
56#63455—32no—
57#63446—32no—
58#63444D311yescharacteristic, complemented
59#611⋊C4—32no—
60#63439—31no—
61#63439—61no—
62C23×C3×C9C2311yescharacteristic
63C22×D3×C9C2311yescharacteristic
64C22×C3×D9—22no—
65C22×C3×D9—24no—
66C22×C3×D9C2311yescharacteristic
67C22×C3×D9C2312yes—
68C2×C3×C4×C9C2312yes—
69C2×C3⋊Dic9D414yescomplemented
70C2×C3⋊Dic9C2312yes—
71C2×C3⋊Dic9C2311yescharacteristic
72C2×C3⋊Dic9C2311yescharacteristic
73C9×#67—22no—
74C3×#611—22no—
75C2×C9×Dic3—22no—
76C2×C9×Dic3C2311yescharacteristic
77C2×C9×Dic3C2312yes—
78C2×C3×Dic9C2311yescharacteristic
79C3×D36—24no—
80#6194—24no—
81#6194—22no—
82D3×Dic9—22no—
83Dic3×D9—24no—
84Dic3×D9—24no—
85Dic3×D9—22no—
86#6189—22no—
87#5656—91no—
88#5586—91no—
89C23×C3×D3—31no—
90C22×C3⋊Dic3—31no—
91C2×C3×#67—31no—
92C2×C3×#67C2×D311yescharacteristic
93C22×C3×Dic3C2×D311yescharacteristic
94C2×C3×D12—32no—
95C2×#627—31no—
96C2×#625—31no—
97C2×#625—32no—
98#3805—64no—
99C2×D3×Dic3—31no—
100C2×D3×Dic3—32no—
101C2×D3×Dic3—32no—
102C2×D3×Dic3—31no—
103C2×D3×Dic3—64no—
104C23×D9C2×D311yescharacteristic
105C32×C22⋊C4C2×D311yescharacteristic
106#3757—31no—
107#3750—32no—
108C4×C3⋊Dic3—32no—
109C3×(C2×D3)⋊C4—32no—
110C3×(C2×D3)⋊C4C2×D312yes—
111C3×C4⋊Dic3C2×D312yes—
112#3724—32no—
113(C2×D3)⋊Dic3—32no—
114(C2×D3)⋊Dic3—32no—
115C2×D4×C9—31no—
116C22×C4×C9—31no—
117C2×#611—32no—
118C2×#611—31no—
119C2×#611C2×D311yescharacteristic
120C22×Dic9—31no—
121C22×Dic9—31no—
122C22×Dic9—61no—
123#3699—64no—
124C2×D36C2×D312yes—
125C2×C4×D9—32no—
126C2×C4×D9—32no—
127C2×C4×D9—31no—
128C2×C4×D9—64no—
129C2×Dic18—31no—
130C9×C4⋊C4—32no—
131C9×C22⋊C4—21no—
132C9×C22⋊C4—32no—
133C9×C22⋊C4C2×D311yescharacteristic
134#3676—31no—
135#3676—61no—
136(C2×D9)⋊C4—32no—
137(C2×D9)⋊C4—32no—
138(C2×D9)⋊C4C2×D312yes—
139Dic9⋊C4—32no—
140Dic9⋊C4—32no—
141Dic9⋊C4—62no—
142C4×Dic9—32no—
143C4×Dic9—62no—
144C22×C3×C9—22no—
145C22×C3×C9C2×D412yes—
146C22×C3×C9C2411yescharacteristic
147C2×D3×C9—21no—
148C2×D3×C9—22no—
149C2×C3×D9—28no—
150C2×C3×D9—22no—
151C2×C3×D9—22no—
152C2×C3×D9—21no—
153C2×C3×D9—22no—
154C2×C3×D9#4114yescomplemented
155C3×C4×C9—22no—
156C3⋊Dic9—22no—
157C3⋊Dic9C2×D412yes—
158C3⋊Dic9C2×D412yes—
159C9×Dic3—21no—
160C9×Dic3—22no—
161C3×Dic9—21no—
162C23×Dic3—91no—
163C2×#237—91no—
164C2×#237—91no—
165C22×C4×D3—91no—
166C3×#123—31no—
167C3×#120—91no—
168#953—91no—
169#939—92no—
170#937—91no—
171#935—91no—
172D4×Dic3—92no—
173#929—91no—
174#896—92no—
175#893—92no—
176#886D911yescharacteristic, complemented
177#884—92no—
178#882—31no—
179#67⋊C4—92no—
180#67⋊C4—92no—
181#878—92no—
182#877—91no—
183#877—91no—
184#877—181no—
185C23×C32C22×D311yescharacteristic
186C22×C3×D3—32no—
187C22×C3×D3—31no—
188C22×C3×D3—64no—
189C22×C3×D3—62no—
190C22×C3×D3C22×D311yescharacteristic
191C2×C32×C4C22×D312yes—
192C2×C3⋊Dic3—34no—
193C2×C3⋊Dic3—31no—
194C2×C3⋊Dic3—31no—
195C2×C3⋊Dic3—32no—
196C3×#67—22no—
197C3×#67—62no—
198C2×C3×Dic3—22no—
199C2×C3×Dic3—31no—
200C2×C3×Dic3C22×D311yescharacteristic
201C2×C3×Dic3C22×D312yes—
202C3×D12—64no—
203#627—62no—
204#625—62no—
205#625—64no—
206D3×Dic3—64no—
207D3×Dic3—64no—
208D3×Dic3—62no—
209D3×Dic3—62no—
210C23×C9—21no—
211C23×C9—31no—
212C23×C9C22×D311yescharacteristic
213C22×D9—22no—
214C22×D9—24no—
215C22×D9C22×D311yescharacteristic
216C22×D9C22×D312yes—
217D4×C9—62no—
218C2×C4×C9—22no—
219C2×C4×C9—32no—
220C2×C4×C9—31no—
221C2×C4×C9—62no—
222C2×C4×C9C22×D312yes—
223#611—22no—
224#611—62no—
225#611—64no—
226C2×Dic9—34no—
227C2×Dic9—31no—
228C2×Dic9—31no—
229C2×Dic9—32no—
230C2×Dic9—61no—
231C2×Dic9—61no—
232C2×Dic9—64no—
233C2×Dic9—62no—
234C2×Dic9—62no—
235C2×Dic9C22×D311yescharacteristic
236D36—24no—
237C4×D9—64no—
238C4×D9—64no—
239C4×D9—62no—
240Dic18—62no—
241#525—271noSylow
242C2×C3×C9—22no—
243C2×C3×C9C22×D411yescharacteristic
244C2×C3×C9C2×#4112yes—
245D3×C9—41no—
246C3×D9—24no—
247C3×D9—41no—
248C24×C3—91no—
249C23×D3—31no—
250C2×C3×D4—31no—
251C2×C3×D4—91no—
252C2×C3×D4—92no—
253C22×C3×C4—31no—
254C2×#67—31no—
255C2×#67—92no—
256C2×#67—91no—
257C2×#67—92no—
258C2×#67—91no—
259C2×#67C2×D911yescharacteristic
260C22×Dic3—91no—
261C22×Dic3—91no—
262C22×Dic3—91no—
263C22×Dic3—92no—
264C22×Dic3—92no—
265C22×Dic3—91no—
266C22×Dic3—181no—
267C22×Dic3—184no—
268C22×Dic3C2×D911yescharacteristic
269#237—184no—
270#237—184no—
271C2×D12—32no—
272C2×C4×D3—92no—
273C2×C4×D3—92no—
274C2×C4×D3—91no—
275C2×C4×D3—91no—
276C2×C4×D3—184no—
277C2×Dic6—91no—
278C2×Dic6—91no—
279C3×C4⋊C4—32no—
280C3×C22⋊C4—21no—
281C3×C22⋊C4—32no—
282C3×C22⋊C4—92no—
283C3×C22⋊C4C2×D911yescharacteristic
284C3×C22⋊C4D3211yescharacteristic
285#217—91no—
286#217—91no—
287#217—92no—
288#217—92no—
289#217—181no—
290(C2×D3)⋊C4—32no—
291(C2×D3)⋊C4—92no—
292(C2×D3)⋊C4—92no—
293(C2×D3)⋊C4C2×D912yescomplemented
294C4⋊Dic3C2×D912yescomplemented
295Dic3⋊C4—92no—
296Dic3⋊C4—92no—
297Dic3⋊C4—92no—
298Dic3⋊C4—92no—
299Dic3⋊C4—182no—
300C4×Dic3—92no—
301C4×Dic3—92no—
302C4×Dic3—182no—
303C22×C32—22no—
304C22×C32D3×D412yes—
305C22×C32C23×D311yescharacteristic
306C2×C3×D3—21no—
307C2×C3×D3—22no—
308C2×C3×D3—34no—
309C2×C3×D3—68no—
310C2×C3×D3—62no—
311C2×C3×D3—62no—
312C2×C3×D3—61no—
313C2×C3×D3—62no—
314C32×C4—22no—
315C3⋊Dic3—32no—
316C3⋊Dic3—32no—
317C3⋊Dic3—62no—
318C3×Dic3—22no—
319C3×Dic3—21no—
320C3×Dic3—61no—
321C22×C9—22no—
322C22×C9—21no—
323C22×C9—22no—
324C22×C9—42no—
325C22×C9—61no—
326C22×C9—62no—
327C22×C9D3×D412yes—
328C22×C9C23×D311yescharacteristic
329C2×D9—28no—
330C2×D9—22no—
331C2×D9—22no—
332C2×D9—21no—
333C2×D9—22no—
334C2×D9#23714yescomplemented
335C4×C9—22no—
336C4×C9—42no—
337C4×C9—61no—
338C4×C9—62no—
339Dic9—21no—
340Dic9—32no—
341Dic9—32no—
342Dic9—62no—
343Dic9—62no—
344Dic9—62no—
345Dic9—122no—
346C2×#41—271no—
347C23×C4—271no—
348#123—31no—
349#123—271no—
350C22⋊Q8—272no—
351#121—272no—
352#120—91no—
353#120—271no—
354C4×D4—272no—
355C4×D4—272no—
356#117—271no—
357C3×C9#52511yescharacteristic, complemented, Sylow
358C23×C3—21no—
359C23×C3—31no—
360C23×C3—92no—
361C23×C3—91no—
362C23×C3—184no—
363C23×C3—182no—
364C23×C3C22×D911yescharacteristic
365C23×C3C2×D3211yescharacteristic
366C22×D3—32no—
367C22×D3—31no—
368C22×D3—64no—
369C22×D3—62no—
370C22×D3C22×D911yescharacteristic
371C3×D4—62no—
372C3×D4—184no—
373C3×D4—182no—
374C2×C3×C4—22no—
375C2×C3×C4—32no—
376C2×C3×C4—31no—
377C2×C3×C4—62no—
378C2×C3×C4—91no—
379C2×C3×C4C22×D912yes—
380C2×C3×C4C2×D3212yes—
381#67—22no—
382#67—62no—
383#67—182no—
384#67—182no—
385#67—184no—
386#67—184no—
387C2×Dic3—22no—
388C2×Dic3—31no—
389C2×Dic3—91no—
390C2×Dic3—91no—
391C2×Dic3—94no—
392C2×Dic3—91no—
393C2×Dic3—91no—
394C2×Dic3—94no—
395C2×Dic3—92no—
396C2×Dic3—92no—
397C2×Dic3—181no—
398C2×Dic3—181no—
399C2×Dic3—184no—
400C2×Dic3—182no—
401C2×Dic3—182no—
402C2×Dic3—184no—
403C2×Dic3—184no—
404C2×Dic3—182no—
405C2×Dic3C22×D912yes—
406C2×Dic3C22×D911yescharacteristic
407D12—64no—
408C4×D3—184no—
409C4×D3—184no—
410C4×D3—182no—
411C4×D3—182no—
412Dic6—182no—
413Dic6—182no—
414C2×C32—22no—
415C2×C32C2×D3×D411yescharacteristic
416C2×C32D3×#4112yes—
417C3×D3—41no—
418C3×D3—64no—
419C3×D3—121no—
420C2×C9—22no—
421C2×C9—21no—
422C2×C9—22no—
423C2×C9—42no—
424C2×C9—121no—
425C2×C9C2×D3×D411yescharacteristic
426C2×C9C2×#23712yes—
427D9—24no—
428D9—41no—
429C24—91no—
430#41—544no—
431C2×Q8—271no—
432C2×D4—31no—
433C2×D4—91no—
434C2×D4—92no—
435C2×D4—272no—
436C2×D4—271no—
437C22×C4—31no—
438C22×C4—272no—
439C22×C4—271no—
440C22×C4—272no—
441C22×C4—271no—
442C22×C4—271no—
443C22×C4—544no—
444C4⋊C4—32no—
445C4⋊C4—272no—
446C4⋊C4—272no—
447C22⋊C4—32no—
448C22⋊C4—92no—
449C22⋊C4—272no—
450C22⋊C4—272no—
451C22⋊C4—271no—
452C22⋊C4D3×D911yescharacteristic
453C42—272no—
454C22×C3—22no—
455C22×C3—22no—
456C22×C3—21no—
457C22×C3—22no—
458C22×C3—42no—
459C22×C3—61no—
460C22×C3—62no—
461C22×C3—94no—
462C22×C3—188no—
463C22×C3—182no—
464C22×C3—182no—
465C22×C3—181no—
466C22×C3—182no—
467C22×C3D4×D912yes—
468C22×C3C23×D911yescharacteristic
469C22×C3#380212yes—
470C22×C3C22×D3211yescharacteristic
471C2×D3—21no—
472C2×D3—22no—
473C2×D3—34no—
474C2×D3—68no—
475C2×D3—62no—
476C2×D3—62no—
477C2×D3—61no—
478C2×D3—62no—
479C3×C4—22no—
480C3×C4—22no—
481C3×C4—42no—
482C3×C4—61no—
483C3×C4—62no—
484C3×C4—181no—
485Dic3—21no—
486Dic3—22no—
487Dic3—61no—
488Dic3—92no—
489Dic3—92no—
490Dic3—92no—
491Dic3—92no—
492Dic3—182no—
493Dic3—182no—
494Dic3—182no—
495Dic3—182no—
496Dic3—362no—
497C32#565611yescharacteristic
498C9—21no—
499C9#558611yescharacteristic, complemented
500C23—31no—
501C23—92no—
502C23—91no—
503C23—184no—
504C23—182no—
505C23C2×D3×D911yescharacteristic
506Q8—542no—
507D4—62no—
508D4—184no—
509D4—182no—
510D4—542no—
511D4—544no—
512C2×C4—32no—
513C2×C4—31no—
514C2×C4—62no—
515C2×C4—91no—
516C2×C4—271no—
517C2×C4—271no—
518C2×C4—274no—
519C2×C4—272no—
520C2×C4—544no—
521C2×C4—544no—
522C2×C4—542no—
523C2×C4—542no—
524C2×C4C2×D3×D912yes—
525C2×C3—22no—
526C2×C3—22no—
527C2×C3—22no—
528C2×C3—21no—
529C2×C3—42no—
530C2×C3—121no—
531C2×C3—184no—
532C2×C3—361no—
533C2×C3C2×D4×D911yescharacteristic
534C2×C3#41×D912yes—
535C2×C3C2×#380211yescharacteristic
536C2×C3D3×#23712yes—
537D3—41no—
538D3—64no—
539D3—121no—
540C22—22no—
541C22—62no—
542C22—61no—
543C22—94no—
544C22—188no—
545C22—182no—
546C22—182no—
547C22—182no—
548C22—181no—
549C22#8864612yes—
550C22C22×D3×D911yescharacteristic
551C4—22no—
552C4—62no—
553C4—61no—
554C4—181no—
555C4—272no—
556C4—272no—
557C4—542no—
558C3—21no—
559C3#1058975811yescharacteristic, complemented
560C3#1059408211yescharacteristic
561C2—22no—
562C2—121no—
563C2—184no—
564C2—361no—
565C2C2×#8864611yescharacteristic
566C2D9×#23712yes—
subgroupG/H
centerC22C22×D3×D9
commutator, cosocleC22×C3×C9C24
FrattiniC22×C3C22×D32
FittingC3×C9×C22⋊C4C22
socleC22×C32C23×D3
hypercenterC22⋊C4D3×D9
hypocenterC3×C9#525
LCSG ▷ C22×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C22⋊C4

Complements

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

NH ≅ G/Ntype
1. C2×#88663C2semidirect
2. C22×Dic3×D9C2semidirect
3. #11721723C2semidirect
4. C9×#886C2semidirect
5. C3×#63444C2semidirect
6. #11721285C2semidirect
7. #11721259C2semidirect
8. #11721028C2semidirect
9. #11720999C2semidirect
10. #11720998C2semidirect
11. #11720982C2semidirect
21. C2×#6194C22semidirect
23. C2×D3×Dic9C22semidirect
27. C2×Dic3×D9C22semidirect
28. C2×#6189C22semidirect
31. #88534C22semidirect
32. C4×C3⋊Dic9C22semidirect
33. C9×(C2×D3)⋊C4C22semidirect
35. C9×C4⋊Dic3C22semidirect
37. (C2×D9)⋊Dic3C22semidirect
38. #88442C22semidirect
58. #63444D3semidirect
69. C2×C3⋊Dic9D4semidirect
154. C2×C3×D9#41semidirect
176. #886D9semidirect
293. (C2×D3)⋊C4C2×D9semidirect
294. C4⋊Dic3C2×D9semidirect
334. C2×D9#237semidirect
357. C3×C9#525semidirect
499. C9#5586semidirect
559. C3#10589758semidirect

Maximal subgroups

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

subgroupindex
C2×#886632
C22×Dic3×D92
#117217232
C9×#8862
C3×#634442
#117212852
#117212592
#117210282
#117209992
#117209982
#117209822
#105940823
#105897583