Finite Group Databasebeta

← #13150455

Subgroups — #13150455

Order 1728.

1064 conjugacy classes of subgroups.

The subgroup lattice has 604 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×C4×D9C211yesmaximal, characteristic, complemented
2C2×#88638C211yesmaximal, characteristic, complemented
3#11721813C211yesmaximal, characteristic, complemented
4C9×#930C211yesmaximal, characteristic, complemented
5#11721285C212yesmaximal, complemented
6#67×Dic9C212yesmaximal, complemented
7#11721036C211yesmaximal, characteristic, complemented
8#11721031C212yesmaximal, complemented
9#11721012C212yesmaximal, complemented
10#11720945C211yesmaximal, characteristic, complemented
11#11720907C211yesmaximal, characteristic, complemented
12#10594892—31nomaximal
13#10589708—31nomaximal
14C23×C3×D9C2211yescharacteristic
15C22×C3×C4×C9C2211yescharacteristic
16C2×#6226C2211yescharacteristic, complemented
17C2×C9×#67C2212yes—
18C22×C3×Dic9C2211yescharacteristic
19C2×C9×D12C2211yescharacteristic, complemented
20C2×C3×C4×D9—24no—
21C2×C3×C4×D9C2211yescharacteristic
22C2×C3×C4×D9C2211yescharacteristic
23C2×C3×C4×D9C2211yescharacteristic
24C2×C3×C4×D9C2211yescharacteristic
25C2×#6194C2212yes—
26C2×#6194C2212yes—
27C2×D3×Dic9C2212yescomplemented
28#88638—24no—
29#88551C2212yes—
30C4⋊(C3⋊Dic9)C2211yescharacteristic, complemented
31#88534C2212yescomplemented
32C9×(C2×D3)⋊C4C2212yescomplemented
33C9×C4⋊Dic3C2211yescharacteristic, complemented
34(C2×D3)⋊Dic9C2212yescomplemented
35(C2×D9)⋊Dic3C2212yes—
36(C2×D9)⋊Dic3C2212yes—
37Dic3⋊Dic9C2212yescomplemented
38Dic3×Dic9C2212yescomplemented
39C22×C3×C4×D3—31no—
40C2×#3795—31no—
41#64133—31no—
42C3×#930D311yescharacteristic
43#63976—32no—
44Dic3×#67—32no—
45#63908—31no—
46#63905—32no—
47#63879—32no—
48#63857—31no—
49#63841—31no—
50C2×#3699—31no—
51C22×C4×D9D311yescharacteristic, complemented
52C9×#121—31no—
53#63500—31no—
54#63499—32no—
55D4×Dic9—32no—
56#63486—31no—
57#63486—61no—
58#63459—31no—
59#63454—31no—
60#63445—32no—
61#611⋊C4—32no—
62C23×C3×C9C2311yescharacteristic
63C22×D3×C9C2312yes—
64C22×C3×D9—22no—
65C22×C3×D9—22no—
66C22×C3×D9—22no—
67C22×C3×D9C2311yescharacteristic
68C22×C3×D9C2311yescharacteristic
69C2×C3×C4×C9—22no—
70C2×C3×C4×C9C2311yescharacteristic
71C2×C3×C4×C9C2311yescharacteristic
72C2×C3⋊Dic9C2312yes—
73C2×C3⋊Dic9C2312yes—
74#6226—22no—
75C9×#67—24no—
76C2×C9×Dic3C2312yes—
77C2×C3×Dic9—22no—
78C2×C3×Dic9C2311yescharacteristic
79C2×C3×Dic9C2311yescharacteristic
80C9×D12—22no—
81C3×C4×D9—22no—
82C3×C4×D9—22no—
83C3×C4×D9—22no—
84C3×C4×D9D414yescomplemented
85#6194—24no—
86#6194—24no—
87D3×Dic9—24no—
88#5791—91no—
89#5606—91no—
90C23×C3×D3—31no—
91C22×C32×C4C2×D311yescharacteristic
92C2×#634—31no—
93C2×C3×#67C2×D312yes—
94C22×C3×Dic3—31no—
95C2×C3×D12C2×D311yescharacteristic
96C2×C3×C4×D3—31no—
97C2×C3×C4×D3—31no—
98C2×C3×C4×D3—31no—
99C2×C3×C4×D3—31no—
100C2×C3×C4×D3—64no—
101C2×#625—32no—
102C2×#625—32no—
103C2×D3×Dic3—32no—
104#3795—64no—
105C23×D9C2×D311yescharacteristic
106#3757—32no—
107C4⋊(C3⋊Dic3)—31no—
108#3750—32no—
109C3×(C2×D3)⋊C4C2×D312yes—
110C3×C4⋊Dic3C2×D311yescharacteristic
111Dic3⋊Dic3—32no—
112(C2×D3)⋊Dic3—32no—
113(C2×D3)⋊Dic3—32no—
114(C2×D3)⋊Dic3—32no—
115Dic32—32no—
116C2×D4×C9—32no—
117C2×D4×C9—31no—
118C22×C4×C9—21no—
119C22×C4×C9C2×D311yescharacteristic
120C2×#611—32no—
121C2×#611—32no—
122C22×Dic9—32no—
123C22×Dic9C2×D311yescharacteristic
124#3699—64no—
125C2×C4×D9—24no—
126C2×C4×D9C2×D311yescharacteristic
127C2×C4×D9C2×D311yescharacteristic
128C2×C4×D9C2×D311yescharacteristic
129C2×C4×D9C2×D311yescharacteristic
130C2×Dic18—31no—
131C2×Dic18—61no—
132C9×C4⋊C4—31no—
133C9×C22⋊C4—32no—
134#3676—32no—
135#3676—32no—
136#3676—62no—
137(C2×D9)⋊C4—32no—
138(C2×D9)⋊C4—32no—
139C4⋊Dic9—31no—
140C4⋊Dic9—32no—
141C4⋊Dic9—61no—
142Dic9⋊C4—32no—
143Dic9⋊C4—62no—
144C4×Dic9—32no—
145C22×C3×C9—21no—
146C22×C3×C9—21no—
147C22×C3×C9#4112yescomplemented
148C22×C3×C9C2411yescharacteristic
149C2×D3×C9—22no—
150C2×D3×C9—22no—
151C2×D3×C9—22no—
152C2×C3×D9—28no—
153C2×C3×D9—21no—
154C2×C3×D9—21no—
155C2×C3×D9—21no—
156C2×C3×D9—21no—
157C2×C3×D9C2×D412yes—
158C2×C3×D9#4112yescomplemented
159C3×C4×C9—21no—
160C3×C4×C9C2×D412yes—
161C3⋊Dic9—22no—
162C3⋊Dic9—22no—
163C9×Dic3—22no—
164C3×Dic9—21no—
165C3×Dic9C2×D412yes—
166C23×C3×C4—91no—
167C2×#237—91no—
168C2×#235—91no—
169C22×C4×D3—31no—
170C3×#121—31no—
171#953—92no—
172#938—91no—
173#937—92no—
174D4×Dic3—92no—
175#930—91no—
176#930D911yescharacteristic, complemented
177#929—92no—
178C4×#67—92no—
179C4×#67—92no—
180#926—91no—
181#924—91no—
182#924—91no—
183#924—91no—
184#924—181no—
185#897—91no—
186#892—91no—
187#883—92no—
188#67⋊C4—92no—
189C23×C32C22×D311yescharacteristic
190C22×C3×D3—31no—
191C22×C3×D3—31no—
192C22×C3×D3—62no—
193C22×C3×D3—62no—
194C22×C3×D3—62no—
195C22×C3×D3C22×D312yes—
196C2×C32×C4—22no—
197C2×C32×C4C22×D311yescharacteristic
198C2×C32×C4C22×D311yescharacteristic
199C2×C3⋊Dic3—32no—
200C2×C3⋊Dic3—32no—
201#634—62no—
202C3×#67—24no—
203C2×C3×Dic3—31no—
204C2×C3×Dic3—31no—
205C2×C3×Dic3—62no—
206C2×C3×Dic3C22×D312yes—
207C3×D12—22no—
208C3×C4×D3—34no—
209C3×C4×D3—62no—
210C3×C4×D3—62no—
211C3×C4×D3—62no—
212#625—64no—
213#625—64no—
214D3×Dic3—64no—
215C23×C9—21no—
216C23×C9—32no—
217C23×C9C22×D311yescharacteristic
218C22×D9—22no—
219C22×D9—22no—
220C22×D9—22no—
221C22×D9C22×D311yescharacteristic
222C22×D9C22×D311yescharacteristic
223D4×C9—64no—
224D4×C9—62no—
225C2×C4×C9—22no—
226C2×C4×C9—21no—
227C2×C4×C9—21no—
228C2×C4×C9—32no—
229C2×C4×C9—42no—
230C2×C4×C9C22×D311yescharacteristic
231C2×C4×C9C22×D311yescharacteristic
232#611—64no—
233#611—64no—
234C2×Dic9—22no—
235C2×Dic9—32no—
236C2×Dic9—32no—
237C2×Dic9—62no—
238C2×Dic9—62no—
239C2×Dic9—64no—
240C2×Dic9C22×D311yescharacteristic
241C2×Dic9C22×D311yescharacteristic
242C4×D9—22no—
243C4×D9—22no—
244C4×D9—22no—
245C4×D9#6714yescomplemented
246Dic18—62no—
247Dic18—122no—
248#525—271noSylow
249C2×C3×C9—22no—
250C2×C3×C9C22×D411yescharacteristic
251C2×C3×C9C2×#4111yescharacteristic
252C2×C3×C9C2×#4111yescharacteristic
253D3×C9—42no—
254C3×D9—22no—
255C3×D9—22no—
256C24×C3—91no—
257C23×D3—31no—
258C2×C3×D4—32no—
259C2×C3×D4—31no—
260C22×C3×C4—21no—
261C22×C3×C4—91no—
262C22×C3×C4—91no—
263C22×C3×C4—91no—
264C22×C3×C4—91no—
265C22×C3×C4—91no—
266C22×C3×C4—184no—
267C22×C3×C4C2×D911yescharacteristic
268C22×C3×C4D3211yescharacteristic
269C2×#67—92no—
270C2×#67—92no—
271C2×#67—92no—
272C2×#67—92no—
273C2×#67C2×D912yes—
274C22×Dic3—31no—
275C22×Dic3—92no—
276#237—184no—
277#235—184no—
278C2×D12C2×D911yescharacteristic, complemented
279C2×C4×D3—31no—
280C2×C4×D3—31no—
281C2×C4×D3—31no—
282C2×C4×D3—31no—
283C2×C4×D3—64no—
284C2×C4×D3—92no—
285C2×Dic6—91no—
286C2×Dic6—91no—
287C2×Dic6—181no—
288C3×C4⋊C4—31no—
289C3×C22⋊C4—32no—
290#217—92no—
291#217—92no—
292#217—92no—
293#217—92no—
294#217—92no—
295#217—182no—
296(C2×D3)⋊C4—92no—
297(C2×D3)⋊C4—92no—
298(C2×D3)⋊C4—92no—
299(C2×D3)⋊C4C2×D912yescomplemented
300C4⋊Dic3—92no—
301C4⋊Dic3—91no—
302C4⋊Dic3—91no—
303C4⋊Dic3—181no—
304C4⋊Dic3C2×D911yescharacteristic, complemented
305Dic3⋊C4—92no—
306Dic3⋊C4—92no—
307Dic3⋊C4—92no—
308Dic3⋊C4—182no—
309C4×Dic3—92no—
310C4×Dic3—92no—
311C22×C32—21no—
312C22×C32—21no—
313C22×C32#23712yes—
314C22×C32C23×D311yescharacteristic
315C2×C3×D3—22no—
316C2×C3×D3—22no—
317C2×C3×D3—22no—
318C2×C3×D3—32no—
319C2×C3×D3—32no—
320C2×C3×D3—68no—
321C2×C3×D3—61no—
322C2×C3×D3—61no—
323C2×C3×D3—61no—
324C2×C3×D3—61no—
325C32×C4—21no—
326C32×C4D3×D412yes—
327C3⋊Dic3—62no—
328C3⋊Dic3—62no—
329C3×Dic3—22no—
330C3×Dic3—32no—
331C3×Dic3—61no—
332C22×C9—21no—
333C22×C9—21no—
334C22×C9—21no—
335C22×C9—22no—
336C22×C9—41no—
337C22×C9—41no—
338C22×C9—62no—
339C22×C9—62no—
340C22×C9—62no—
341C22×C9#23512yescomplemented
342C22×C9C23×D311yescharacteristic
343C2×D9—28no—
344C2×D9—21no—
345C2×D9—21no—
346C2×D9—21no—
347C2×D9—21no—
348C2×D9#23512yescomplemented
349C2×D9C2×#6712yes—
350C4×C9—21no—
351C4×C9—22no—
352C4×C9—41no—
353C4×C9—62no—
354C4×C9C2×#6712yes—
355Dic9—21no—
356Dic9—62no—
357Dic9—62no—
358Dic9—122no—
359Dic9—122no—
360Dic9C2×#6712yes—
361C2×#41—271no—
362C23×C4—91no—
363#123—272no—
364C22⋊Q8—271no—
365C22⋊Q8—271no—
366#121—31no—
367#121—271no—
368#120—272no—
369C4×D4—272no—
370C4×D4—272no—
371#117—271no—
372C3×C9#52511yescharacteristic, complemented, Sylow
373C23×C3—21no—
374C23×C3—32no—
375C23×C3—91no—
376C23×C3—91no—
377C23×C3—182no—
378C23×C3—182no—
379C23×C3—182no—
380C23×C3C22×D911yescharacteristic
381C23×C3C2×D3211yescharacteristic
382C22×D3—31no—
383C22×D3—31no—
384C22×D3—62no—
385C22×D3—62no—
386C22×D3—62no—
387C22×D3C22×D912yes—
388C3×D4—64no—
389C3×D4—62no—
390C2×C3×C4—22no—
391C2×C3×C4—22no—
392C2×C3×C4—21no—
393C2×C3×C4—21no—
394C2×C3×C4—32no—
395C2×C3×C4—42no—
396C2×C3×C4—91no—
397C2×C3×C4—94no—
398C2×C3×C4—91no—
399C2×C3×C4—182no—
400C2×C3×C4—182no—
401C2×C3×C4—182no—
402C2×C3×C4—182no—
403C2×C3×C4C22×D911yescharacteristic
404C2×C3×C4C22×D911yescharacteristic
405C2×C3×C4C2×D3211yescharacteristic
406C2×C3×C4C2×D3211yescharacteristic
407#67—24no—
408#67—184no—
409#67—184no—
410#67—184no—
411#67—184no—
412C2×Dic3—31no—
413C2×Dic3—31no—
414C2×Dic3—62no—
415C2×Dic3—92no—
416C2×Dic3—92no—
417C2×Dic3—92no—
418C2×Dic3—92no—
419C2×Dic3—182no—
420C2×Dic3—182no—
421C2×Dic3—184no—
422C2×Dic3C22×D912yes—
423D12—22no—
424C4×D3—34no—
425C4×D3—62no—
426C4×D3—62no—
427C4×D3—62no—
428C4×D3—184no—
429Dic6—182no—
430Dic6—182no—
431Dic6—362no—
432C2×C32—22no—
433C2×C32C2×D3×D411yescharacteristic
434C2×C32C2×#23711yescharacteristic
435C2×C32D3×#4111yescharacteristic
436C3×D3—42no—
437C3×D3—62no—
438C3×D3—62no—
439C2×C9—22no—
440C2×C9—21no—
441C2×C9—21no—
442C2×C9—21no—
443C2×C9—42no—
444C2×C9—122no—
445C2×C9C2×#23511yescharacteristic
446C2×C9C2×#23511yescharacteristic
447C2×C9C22×#6711yescharacteristic
448D9—22no—
449D9—22no—
450C24—91no—
451#41—544no—
452C2×Q8—271no—
453C2×D4—32no—
454C2×D4—31no—
455C2×D4—272no—
456C2×D4—272no—
457C22×C4—91no—
458C22×C4—91no—
459C22×C4—91no—
460C22×C4—91no—
461C22×C4—91no—
462C22×C4—184no—
463C22×C4—272no—
464C22×C4D3×D911yescharacteristic
465C4⋊C4—31no—
466C4⋊C4—272no—
467C4⋊C4—271no—
468C4⋊C4—272no—
469C22⋊C4—32no—
470C22⋊C4—272no—
471C22⋊C4—272no—
472C22⋊C4—272no—
473C22⋊C4—272no—
474C42—272no—
475C22×C3—21no—
476C22×C3—21no—
477C22×C3—21no—
478C22×C3—21no—
479C22×C3—22no—
480C22×C3—21no—
481C22×C3—41no—
482C22×C3—41no—
483C22×C3—62no—
484C22×C3—62no—
485C22×C3—62no—
486C22×C3—92no—
487C22×C3—92no—
488C22×C3—188no—
489C22×C3—181no—
490C22×C3—181no—
491C22×C3—181no—
492C22×C3—181no—
493C22×C3#369912yescomplemented
494C22×C3C23×D911yescharacteristic
495C22×C3#379512yes—
496C22×C3C22×D3211yescharacteristic
497C2×D3—22no—
498C2×D3—22no—
499C2×D3—22no—
500C2×D3—32no—
501C2×D3—32no—
502C2×D3—68no—
503C2×D3—61no—
504C2×D3—61no—
505C2×D3—61no—
506C2×D3—61no—
507C3×C4—21no—
508C3×C4—21no—
509C3×C4—22no—
510C3×C4—41no—
511C3×C4—62no—
512C3×C4—92no—
513C3×C4—181no—
514C3×C4D4×D912yes—
515C3×C4D3×#6712yes—
516Dic3—22no—
517Dic3—32no—
518Dic3—61no—
519Dic3—182no—
520Dic3—182no—
521Dic3—182no—
522Dic3—182no—
523Dic3—362no—
524Dic3—362no—
525C32#560611yescharacteristic
526C9—21no—
527C9#579111yescharacteristic, complemented
528C23—32no—
529C23—91no—
530C23—91no—
531C23—182no—
532C23—182no—
533C23—182no—
534C23C2×D3×D911yescharacteristic
535Q8—542no—
536D4—64no—
537D4—62no—
538D4—544no—
539D4—544no—
540C2×C4—22no—
541C2×C4—32no—
542C2×C4—91no—
543C2×C4—94no—
544C2×C4—91no—
545C2×C4—182no—
546C2×C4—182no—
547C2×C4—182no—
548C2×C4—182no—
549C2×C4—272no—
550C2×C4—272no—
551C2×C4—544no—
552C2×C4C2×D3×D911yescharacteristic
553C2×C4C2×D3×D911yescharacteristic
554C2×C3—22no—
555C2×C3—22no—
556C2×C3—21no—
557C2×C3—21no—
558C2×C3—21no—
559C2×C3—42no—
560C2×C3—122no—
561C2×C3—182no—
562C2×C3—182no—
563C2×C3C2×D4×D911yescharacteristic
564C2×C3C2×#369911yescharacteristic
565C2×C3#41×D911yescharacteristic
566C2×C3C2×#379511yescharacteristic
567C2×C3D3×#23511yescharacteristic
568C2×C3C2×D3×#6711yescharacteristic
569D3—42no—
570D3—62no—
571D3—62no—
572C22—21no—
573C22—21no—
574C22—62no—
575C22—62no—
576C22—62no—
577C22—92no—
578C22—92no—
579C22—188no—
580C22—181no—
581C22—181no—
582C22—181no—
583C22—181no—
584C22#8863812yescomplemented
585C22C22×D3×D911yescharacteristic
586C4—21no—
587C4—62no—
588C4—92no—
589C4—181no—
590C4—542no—
591C4—542no—
592C4D9×#6712yes—
593C3—21no—
594C3#1058970811yescharacteristic, complemented
595C3#1059489211yescharacteristic
596C2—22no—
597C2—122no—
598C2—182no—
599C2—182no—
600C2C2×#8863811yescharacteristic
601C2D9×#23511yescharacteristic
602C2C2×D9×#6711yescharacteristic
subgroupG/H
centerC22C22×D3×D9
commutator, cosocleC22×C3×C9C24
FrattiniC22×C3C22×D32
FittingC22×C3×C4×C9C22
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. C22×C3×C4×D9C2semidirect
2. C2×#88638C2semidirect
3. #11721813C2semidirect
4. C9×#930C2semidirect
5. #11721285C2semidirect
6. #67×Dic9C2semidirect
7. #11721036C2semidirect
8. #11721031C2semidirect
9. #11721012C2semidirect
10. #11720945C2semidirect
11. #11720907C2semidirect
16. C2×#6226C22semidirect
19. C2×C9×D12C22semidirect
27. C2×D3×Dic9C22semidirect
30. C4⋊(C3⋊Dic9)C22semidirect
31. #88534C22semidirect
32. C9×(C2×D3)⋊C4C22semidirect
33. C9×C4⋊Dic3C22semidirect
34. (C2×D3)⋊Dic9C22semidirect
37. Dic3⋊Dic9C22semidirect
38. Dic3×Dic9C22semidirect
51. C22×C4×D9D3semidirect
84. C3×C4×D9D4semidirect
147. C22×C3×C9#41semidirect
158. C2×C3×D9#41semidirect
176. #930D9semidirect
245. C4×D9#67semidirect
278. C2×D12C2×D9semidirect
299. (C2×D3)⋊C4C2×D9semidirect
304. C4⋊Dic3C2×D9semidirect
341. C22×C9#235semidirect
348. C2×D9#235semidirect
372. C3×C9#525semidirect
493. C22×C3#3699semidirect
527. C9#5791semidirect
584. C22#88638semidirect
594. C3#10589708semidirect

Maximal subgroups

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

subgroupindex
C22×C3×C4×D92
C2×#886382
#117218132
C9×#9302
#117212852
#67×Dic92
#117210362
#117210312
#117210122
#117209452
#117209072
#105948923
#105897083