Finite Group Databasebeta

← #13150456

Subgroups — #13150456

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×C4×D3×C9C211yesmaximal, characteristic, complemented
2C2×#88641C211yesmaximal, characteristic, complemented
3#11721813C211yesmaximal, characteristic, complemented
4C3×#63492C211yesmaximal, characteristic, complemented
5#11721286C212yesmaximal, complemented
6Dic3×#611C212yesmaximal, complemented
7#11721037C211yesmaximal, characteristic, complemented
8#11721032C212yesmaximal, complemented
9#11720976C212yesmaximal, complemented
10#11720961C211yesmaximal, characteristic, complemented
11#11720906C211yesmaximal, characteristic, complemented
12#10594892—31nomaximal
13#10589893—31nomaximal
14C23×D3×C9C2211yescharacteristic
15C22×C3×C4×C9C2211yescharacteristic
16C2×#6226C2211yescharacteristic, complemented
17C2×C3×#611C2212yes—
18C22×C9×Dic3C2211yescharacteristic
19C2×C4×D3×C9—24no—
20C2×C4×D3×C9C2211yescharacteristic
21C2×C4×D3×C9C2211yescharacteristic
22C2×C4×D3×C9C2211yescharacteristic
23C2×C4×D3×C9C2211yescharacteristic
24C2×C3×D36C2211yescharacteristic, complemented
25C2×#6194C2212yes—
26C2×#6194C2212yes—
27C2×Dic3×D9C2212yescomplemented
28#88641—24no—
29#88551C2212yes—
30C4⋊(C3⋊Dic9)C2211yescharacteristic, complemented
31#88534C2212yescomplemented
32C3×(C2×D9)⋊C4C2212yescomplemented
33C3×C4⋊Dic9C2211yescharacteristic, complemented
34(C2×D3)⋊Dic9C2212yes—
35(C2×D3)⋊Dic9C2212yes—
36(C2×D9)⋊Dic3C2212yescomplemented
37Dic9⋊Dic3C2212yescomplemented
38Dic3×Dic9C2212yescomplemented
39C22×C3×C4×D3D311yescharacteristic
40C2×#3795—31no—
41#64133—31no—
42C3×#930—31no—
43#63976—32no—
44Dic3×#67—32no—
45#63908—31no—
46#63905—32no—
47#63879—32no—
48#63857—31no—
49#63841—31no—
50C23×C4×C9—31no—
51C2×#3697—31no—
52#63515—32no—
53#63492—31no—
54#63492D311yescharacteristic, complemented
55#63491—32no—
56C4×#611—32no—
57C4×#611—32no—
58#63488—31no—
59#63486—31no—
60#63486—31no—
61#63486—61no—
62C23×C3×C9C2311yescharacteristic
63C22×D3×C9—22no—
64C22×D3×C9—22no—
65C22×D3×C9—22no—
66C22×D3×C9C2311yescharacteristic
67C22×D3×C9C2311yescharacteristic
68C22×C3×D9C2312yes—
69C2×C3×C4×C9—22no—
70C2×C3×C4×C9C2311yescharacteristic
71C2×C3×C4×C9C2311yescharacteristic
72C2×C3⋊Dic9C2312yes—
73C2×C3⋊Dic9C2312yes—
74#6226—22no—
75C3×#611—24no—
76C2×C9×Dic3—22no—
77C2×C9×Dic3C2311yescharacteristic
78C2×C9×Dic3C2311yescharacteristic
79C2×C3×Dic9C2312yes—
80C4×D3×C9—22no—
81C4×D3×C9—22no—
82C4×D3×C9—22no—
83C4×D3×C9D414yescomplemented
84C3×D36—22no—
85#6194—24no—
86#6194—24no—
87Dic3×D9—24no—
88#5791—91no—
89#5606—91no—
90C23×C3×D3C2×D311yescharacteristic
91C22×C32×C4C2×D311yescharacteristic
92C2×#634—31no—
93C2×C3×#67—32no—
94C22×C3×Dic3C2×D311yescharacteristic
95C2×C3×D12—31no—
96C2×C3×C4×D3—24no—
97C2×C3×C4×D3C2×D311yescharacteristic
98C2×C3×C4×D3C2×D311yescharacteristic
99C2×C3×C4×D3C2×D311yescharacteristic
100C2×C3×C4×D3C2×D311yescharacteristic
101C2×#625—32no—
102C2×#625—32no—
103C2×D3×Dic3—32no—
104#3795—64no—
105C24×C9—31no—
106#3757—32no—
107C4⋊(C3⋊Dic3)—31no—
108#3750—32no—
109C3×(C2×D3)⋊C4—32no—
110C3×C4⋊Dic3—31no—
111Dic3⋊Dic3—32no—
112(C2×D3)⋊Dic3—32no—
113(C2×D3)⋊Dic3—32no—
114(C2×D3)⋊Dic3—32no—
115Dic32—32no—
116C22×C4×C9—21no—
117C22×C4×C9—31no—
118C22×C4×C9—31no—
119C22×C4×C9—31no—
120C22×C4×C9—31no—
121C22×C4×C9—31no—
122C22×C4×C9—64no—
123C22×C4×C9C2×D311yescharacteristic
124C2×#611—32no—
125C2×#611—32no—
126C2×#611C2×D312yes—
127#3697—64no—
128C2×D36C2×D311yescharacteristic, complemented
129C2×C4×D9—32no—
130C2×Dic18—31no—
131C2×Dic18—61no—
132#3676—32no—
133#3676—32no—
134#3676—32no—
135#3676—62no—
136(C2×D9)⋊C4—32no—
137(C2×D9)⋊C4C2×D312yescomplemented
138C4⋊Dic9—31no—
139C4⋊Dic9—61no—
140C4⋊Dic9C2×D311yescharacteristic, complemented
141Dic9⋊C4—32no—
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—28no—
150C2×D3×C9—21no—
151C2×D3×C9—21no—
152C2×D3×C9—21no—
153C2×D3×C9—21no—
154C2×D3×C9C2×D412yes—
155C2×D3×C9#4112yescomplemented
156C2×C3×D9—22no—
157C2×C3×D9—22no—
158C2×C3×D9—22no—
159C3×C4×C9—21no—
160C3×C4×C9C2×D412yes—
161C3⋊Dic9—22no—
162C3⋊Dic9—22no—
163C9×Dic3—21no—
164C9×Dic3C2×D412yes—
165C3×Dic9—22no—
166C23×C3×C4—31no—
167C2×#237—91no—
168C2×#235—91no—
169C22×C4×D3D911yescharacteristic, complemented
170C3×#121—91no—
171#953—92no—
172#938—91no—
173#937—92no—
174D4×Dic3—92no—
175#930—31no—
176#930—91no—
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—22no—
191C22×C3×D3—22no—
192C22×C3×D3—22no—
193C22×C3×D3—32no—
194C22×C3×D3C22×D311yescharacteristic
195C22×C3×D3C22×D311yescharacteristic
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—64no—
203C2×C3×Dic3—22no—
204C2×C3×Dic3—32no—
205C2×C3×Dic3C22×D311yescharacteristic
206C2×C3×Dic3C22×D311yescharacteristic
207C3×D12—62no—
208C3×C4×D3—22no—
209C3×C4×D3—22no—
210C3×C4×D3—22no—
211C3×C4×D3#6714yes—
212#625—64no—
213#625—64no—
214D3×Dic3—64no—
215C23×C9—21no—
216C23×C9—31no—
217C23×C9—31no—
218C23×C9—62no—
219C23×C9—62no—
220C23×C9—62no—
221C23×C9C22×D311yescharacteristic
222C22×D9C22×D312yes—
223C2×C4×C9—22no—
224C2×C4×C9—21no—
225C2×C4×C9—21no—
226C2×C4×C9—31no—
227C2×C4×C9—34no—
228C2×C4×C9—31no—
229C2×C4×C9—42no—
230C2×C4×C9—62no—
231C2×C4×C9—62no—
232C2×C4×C9—62no—
233C2×C4×C9—62no—
234C2×C4×C9C22×D311yescharacteristic
235C2×C4×C9C22×D311yescharacteristic
236#611—24no—
237#611—64no—
238#611—64no—
239C2×Dic9—32no—
240C2×Dic9—32no—
241C2×Dic9—62no—
242C2×Dic9—62no—
243C2×Dic9C22×D312yes—
244D36—22no—
245C4×D9—64no—
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—22no—
254D3×C9—22no—
255C3×D9—42no—
256C24×C3—31no—
257C23×D3C2×D911yescharacteristic
258C2×C3×D4—92no—
259C2×C3×D4—91no—
260C22×C3×C4—21no—
261C22×C3×C4—31no—
262C22×C3×C4—31no—
263C22×C3×C4—31no—
264C22×C3×C4—31no—
265C22×C3×C4—31no—
266C22×C3×C4—64no—
267C22×C3×C4C2×D911yescharacteristic
268C22×C3×C4D3211yescharacteristic
269C2×#67—32no—
270C2×#67—92no—
271C2×#67—92no—
272C2×#67—92no—
273C2×#67—92no—
274C22×Dic3—92no—
275C22×Dic3C2×D911yescharacteristic
276#237—184no—
277#235—184no—
278C2×D12—31no—
279C2×C4×D3—24no—
280C2×C4×D3—92no—
281C2×C4×D3C2×D911yescharacteristic
282C2×C4×D3C2×D911yescharacteristic
283C2×C4×D3C2×D911yescharacteristic
284C2×C4×D3C2×D911yescharacteristic
285C2×Dic6—91no—
286C2×Dic6—91no—
287C2×Dic6—181no—
288C3×C4⋊C4—91no—
289C3×C22⋊C4—92no—
290#217—92no—
291#217—92no—
292#217—92no—
293#217—92no—
294#217—92no—
295#217—182no—
296(C2×D3)⋊C4—32no—
297(C2×D3)⋊C4—92no—
298(C2×D3)⋊C4—92no—
299(C2×D3)⋊C4—92no—
300C4⋊Dic3—31no—
301C4⋊Dic3—91no—
302C4⋊Dic3—91no—
303C4⋊Dic3—92no—
304C4⋊Dic3—181no—
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#23512yes—
314C22×C32C23×D311yescharacteristic
315C2×C3×D3—28no—
316C2×C3×D3—21no—
317C2×C3×D3—21no—
318C2×C3×D3—21no—
319C2×C3×D3—21no—
320C2×C3×D3—62no—
321C2×C3×D3—62no—
322C2×C3×D3—62no—
323C2×C3×D3#23512yes—
324C2×C3×D3C2×#6712yes—
325C32×C4—21no—
326C32×C4C2×#6712yes—
327C3⋊Dic3—62no—
328C3⋊Dic3—62no—
329C3×Dic3—21no—
330C3×Dic3—62no—
331C3×Dic3C2×#6712yes—
332C22×C9—21no—
333C22×C9—21no—
334C22×C9—21no—
335C22×C9—22no—
336C22×C9—32no—
337C22×C9—32no—
338C22×C9—41no—
339C22×C9—41no—
340C22×C9—68no—
341C22×C9—61no—
342C22×C9—61no—
343C22×C9—61no—
344C22×C9—61no—
345C22×C9#23712yescomplemented
346C22×C9C23×D311yescharacteristic
347C2×D9—22no—
348C2×D9—22no—
349C2×D9—22no—
350C4×C9—21no—
351C4×C9—22no—
352C4×C9—32no—
353C4×C9—41no—
354C4×C9—61no—
355C4×C9D3×D412yes—
356Dic9—22no—
357Dic9—62no—
358Dic9—62no—
359Dic9—122no—
360Dic9—122no—
361C2×#41—271no—
362C23×C4—31no—
363#123—272no—
364C22⋊Q8—271no—
365C22⋊Q8—271no—
366#121—91no—
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—31no—
375C23×C3—31no—
376C23×C3—62no—
377C23×C3—62no—
378C23×C3—62no—
379C23×C3—92no—
380C23×C3C22×D911yescharacteristic
381C23×C3C2×D3211yescharacteristic
382C22×D3—22no—
383C22×D3—22no—
384C22×D3—22no—
385C22×D3—32no—
386C22×D3C22×D911yescharacteristic
387C22×D3C22×D911yescharacteristic
388C3×D4—184no—
389C3×D4—182no—
390C2×C3×C4—22no—
391C2×C3×C4—22no—
392C2×C3×C4—21no—
393C2×C3×C4—21no—
394C2×C3×C4—31no—
395C2×C3×C4—34no—
396C2×C3×C4—31no—
397C2×C3×C4—42no—
398C2×C3×C4—62no—
399C2×C3×C4—62no—
400C2×C3×C4—62no—
401C2×C3×C4—62no—
402C2×C3×C4—92no—
403C2×C3×C4C22×D911yescharacteristic
404C2×C3×C4C22×D911yescharacteristic
405C2×C3×C4C2×D3211yescharacteristic
406C2×C3×C4C2×D3211yescharacteristic
407#67—64no—
408#67—184no—
409#67—184no—
410#67—184no—
411#67—184no—
412C2×Dic3—22no—
413C2×Dic3—32no—
414C2×Dic3—92no—
415C2×Dic3—92no—
416C2×Dic3—92no—
417C2×Dic3—92no—
418C2×Dic3—182no—
419C2×Dic3—182no—
420C2×Dic3—184no—
421C2×Dic3C22×D911yescharacteristic
422C2×Dic3C22×D911yescharacteristic
423D12—62no—
424C4×D3—22no—
425C4×D3—22no—
426C4×D3—22no—
427C4×D3—184no—
428C4×D3#61114yescomplemented
429Dic6—182no—
430Dic6—182no—
431Dic6—362no—
432C2×C32—22no—
433C2×C32C2×#23511yescharacteristic
434C2×C32C2×#23511yescharacteristic
435C2×C32C22×#6711yescharacteristic
436C3×D3—22no—
437C3×D3—22no—
438C3×D3—122no—
439C2×C9—22no—
440C2×C9—21no—
441C2×C9—21no—
442C2×C9—21no—
443C2×C9—42no—
444C2×C9—62no—
445C2×C9—62no—
446C2×C9C2×D3×D411yescharacteristic
447C2×C9C2×#23711yescharacteristic
448C2×C9D3×#4111yescharacteristic
449D9—42no—
450C24—31no—
451#41—544no—
452C2×Q8—271no—
453C2×D4—92no—
454C2×D4—91no—
455C2×D4—272no—
456C2×D4—272no—
457C22×C4—31no—
458C22×C4—31no—
459C22×C4—31no—
460C22×C4—31no—
461C22×C4—31no—
462C22×C4—64no—
463C22×C4—272no—
464C22×C4D3×D911yescharacteristic
465C4⋊C4—91no—
466C4⋊C4—272no—
467C4⋊C4—271no—
468C4⋊C4—272no—
469C22⋊C4—92no—
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—32no—
482C22×C3—32no—
483C22×C3—41no—
484C22×C3—41no—
485C22×C3—68no—
486C22×C3—61no—
487C22×C3—61no—
488C22×C3—61no—
489C22×C3—61no—
490C22×C3—182no—
491C22×C3—182no—
492C22×C3—182no—
493C22×C3#369712yescomplemented
494C22×C3C23×D911yescharacteristic
495C22×C3#379512yes—
496C22×C3C22×D3211yescharacteristic
497C2×D3—28no—
498C2×D3—21no—
499C2×D3—21no—
500C2×D3—21no—
501C2×D3—21no—
502C2×D3—62no—
503C2×D3—62no—
504C2×D3—62no—
505C2×D3#369712yescomplemented
506C2×D3C2×#61112yes—
507C3×C4—21no—
508C3×C4—21no—
509C3×C4—22no—
510C3×C4—32no—
511C3×C4—41no—
512C3×C4—61no—
513C3×C4—182no—
514C3×C4C2×#61112yes—
515C3×C4D3×#6712yes—
516Dic3—21no—
517Dic3—62no—
518Dic3—182no—
519Dic3—182no—
520Dic3—182no—
521Dic3—182no—
522Dic3—362no—
523Dic3—362no—
524Dic3C2×#61112yes—
525C32#579111yescharacteristic
526C9—21no—
527C9#560611yescharacteristic, complemented
528C23—31no—
529C23—31no—
530C23—62no—
531C23—62no—
532C23—62no—
533C23—92no—
534C23C2×D3×D911yescharacteristic
535Q8—542no—
536D4—184no—
537D4—182no—
538D4—544no—
539D4—544no—
540C2×C4—22no—
541C2×C4—31no—
542C2×C4—34no—
543C2×C4—31no—
544C2×C4—62no—
545C2×C4—62no—
546C2×C4—62no—
547C2×C4—62no—
548C2×C4—92no—
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—62no—
561C2×C3—62no—
562C2×C3—362no—
563C2×C3C2×#369711yescharacteristic
564C2×C3C2×#369711yescharacteristic
565C2×C3C22×#61111yescharacteristic
566C2×C3C2×#379511yescharacteristic
567C2×C3D3×#23511yescharacteristic
568C2×C3C2×D3×#6711yescharacteristic
569D3—22no—
570D3—22no—
571D3—122no—
572C22—21no—
573C22—21no—
574C22—32no—
575C22—32no—
576C22—68no—
577C22—61no—
578C22—61no—
579C22—61no—
580C22—61no—
581C22—182no—
582C22—182no—
583C22—182no—
584C22#8864112yescomplemented
585C22C22×D3×D911yescharacteristic
586C4—21no—
587C4—32no—
588C4—61no—
589C4—182no—
590C4—542no—
591C4—542no—
592C4D3×#61112yes—
593C3—21no—
594C3#1058989311yescharacteristic, complemented
595C3#1059489211yescharacteristic
596C2—22no—
597C2—62no—
598C2—62no—
599C2—362no—
600C2C2×#8864111yescharacteristic
601C2D3×#369711yescharacteristic
602C2C2×D3×#61111yescharacteristic
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×C4×D3×C9C2semidirect
2. C2×#88641C2semidirect
3. #11721813C2semidirect
4. C3×#63492C2semidirect
5. #11721286C2semidirect
6. Dic3×#611C2semidirect
7. #11721037C2semidirect
8. #11721032C2semidirect
9. #11720976C2semidirect
10. #11720961C2semidirect
11. #11720906C2semidirect
16. C2×#6226C22semidirect
24. C2×C3×D36C22semidirect
27. C2×Dic3×D9C22semidirect
30. C4⋊(C3⋊Dic9)C22semidirect
31. #88534C22semidirect
32. C3×(C2×D9)⋊C4C22semidirect
33. C3×C4⋊Dic9C22semidirect
36. (C2×D9)⋊Dic3C22semidirect
37. Dic9⋊Dic3C22semidirect
38. Dic3×Dic9C22semidirect
54. #63492D3semidirect
83. C4×D3×C9D4semidirect
128. C2×D36C2×D3semidirect
137. (C2×D9)⋊C4C2×D3semidirect
140. C4⋊Dic9C2×D3semidirect
147. C22×C3×C9#41semidirect
155. C2×D3×C9#41semidirect
169. C22×C4×D3D9semidirect
345. C22×C9#237semidirect
372. C3×C9#525semidirect
428. C4×D3#611semidirect
493. C22×C3#3697semidirect
505. C2×D3#3697semidirect
527. C9#5606semidirect
584. C22#88641semidirect
594. C3#10589893semidirect

Maximal subgroups

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

subgroupindex
C22×C4×D3×C92
C2×#886412
#117218132
C3×#634922
#117212862
Dic3×#6112
#117210372
#117210322
#117209762
#117209612
#117209062
#105948923
#105898933