Finite Group Databasebeta

← #13148424

Subgroups — #13148424

Order 1728.

1170 conjugacy classes of subgroups.

The subgroup lattice has 636 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#88668C211yesmaximal, characteristic, complemented
2C22×#6192C211yesmaximal, characteristic, complemented
3C22⋊C4×C3⋊D9C211yesmaximal, characteristic, complemented
4C9×#886C211yesmaximal, characteristic, complemented
5C3×#63444C211yesmaximal, characteristic, complemented
6#11721285C211yesmaximal, characteristic, complemented
7#11721259C211yesmaximal, characteristic, complemented
8#11721049C212yesmaximal, complemented
9#11721041C212yesmaximal, complemented
10#11721007C212yesmaximal, complemented
11#11720942C212yesmaximal, complemented
12#10594296—31nomaximal
13#10589755—31nomaximal
14C23×C3⋊D9C2211yescharacteristic
15C23×C3×D9C2211yescharacteristic, complemented
16C22×D3×D9C2211yescharacteristic, complemented
17C2×C4×C3⋊D9C2212yescomplemented
18C2×C9×#67C2211yescharacteristic
19C2×C3×#611C2211yescharacteristic
20C22×C9×Dic3C2211yescharacteristic, complemented
21C2×C3×D36C2212yescomplemented
22#88668—24no—
23C2×#6195C2211yescharacteristic, complemented
24C2×#6194C2212yescomplemented
25C2×#6192—24no—
26C2×#6192C2212yes—
27C2×#6192C2212yescomplemented
28C2×#6192C2211yescharacteristic, complemented
29C2×(C3⋊D9)⋊C4C2211yescharacteristic, complemented
30C3×C9×C22⋊C4C2211yescharacteristic, complemented
31#88551C2211yescharacteristic, complemented
32#88536C2212yescomplemented
33C9×(C2×D3)⋊C4C2212yescomplemented
34C3×(C2×D9)⋊C4C2212yescomplemented
35C9×C4⋊Dic3C2212yescomplemented
36(C2×D9)⋊Dic3C2212yescomplemented
37(C2×D9)⋊Dic3C2212yescomplemented
38#88442C2212yescomplemented
39C2×#3811—31no—
40C22×#626—31no—
41C22⋊C4×C3⋊D3—31no—
42C3×#886D311yescharacteristic
43C3×#882—31no—
44#63976—31no—
45#63961—31no—
46#63922—32no—
47#63916—32no—
48#63912—32no—
49#63854—32no—
50C2×D4×D9—31no—
51C22×D36—31no—
52C9×#123—31no—
53#63499—31no—
54#63491—31no—
55#63457—32no—
56#63456—32no—
57#63446—32no—
58#63444—32no—
59#63444D311yescharacteristic, complemented
60C22⋊C4×D9—31no—
61C22⋊C4×D9—61no—
62C23×C3×C9C2311yescharacteristic
63C22×C3⋊D9—42no—
64C22×C3⋊D9D414yescomplemented
65C22×C3⋊D9C2311yescharacteristic
66C22×C3⋊D9C2311yescharacteristic
67C22×D3×C9C2311yescharacteristic
68C22×C3×D9—22no—
69C22×C3×D9—42no—
70C22×C3×D9C2311yescharacteristic
71C22×C3×D9C2312yes—
72C2×D3×D9—22no—
73C2×D3×D9—42no—
74C2×C3×C4×C9C2312yes—
75C2×C3⋊Dic9C2312yes—
76C4×C3⋊D9—42no—
77C9×#67—22no—
78C3×#611—22no—
79C2×C9×Dic3—22no—
80C2×C9×Dic3C2311yescharacteristic
81C2×C9×Dic3C2312yes—
82C2×C3×Dic9C2311yescharacteristic
83C3×D36—42no—
84#6195—22no—
85#6194—42no—
86#6192—24no—
87#6192—24no—
88#6192—22no—
89(C3⋊D9)⋊C4—22no—
90#5653—91no—
91#5589—91no—
92C23×C3⋊D3—31no—
93C23×C3×D3—31no—
94C22×D32—31no—
95C2×C4×C3⋊D3—32no—
96C2×C3×#67—31no—
97C2×C3×#67C2×D311yescharacteristic
98C22×C3×Dic3C2×D311yescharacteristic
99C2×C3×D12—32no—
100#3811—64no—
101C2×#626—31no—
102C2×#626—32no—
103C2×#626—32no—
104C2×#626—31no—
105C2×#626—64no—
106C2×#625—32no—
107C2×#624—31no—
108C23×D9—31no—
109C23×D9—31no—
110C23×D9—61no—
111C23×D9C2×D311yescharacteristic, complemented
112C32×C22⋊C4C2×D311yescharacteristic
113#3757—31no—
114#3752—32no—
115C3×(C2×D3)⋊C4—32no—
116C3×(C2×D3)⋊C4C2×D312yes—
117C3×C4⋊Dic3C2×D312yes—
118#3724—32no—
119(C2×D3)⋊Dic3—32no—
120(C2×D3)⋊Dic3—32no—
121C2×D4×C9—31no—
122C22×C4×C9—31no—
123C2×#611—32no—
124C2×#611—31no—
125C2×#611C2×D311yescharacteristic
126D4×D9—64no—
127C2×D36—32no—
128C2×D36—32no—
129C2×D36—31no—
130C2×D36—64no—
131C2×D36C2×D312yescomplemented
132C2×C4×D9—31no—
133C2×C4×D9—32no—
134C2×C4×D9—62no—
135C9×C4⋊C4—32no—
136C9×C22⋊C4—21no—
137C9×C22⋊C4—32no—
138C9×C22⋊C4C2×D311yescharacteristic, complemented
139#3676—31no—
140#3676—61no—
141(C2×D9)⋊C4—32no—
142(C2×D9)⋊C4—32no—
143(C2×D9)⋊C4—32no—
144(C2×D9)⋊C4—62no—
145(C2×D9)⋊C4C2×D312yescomplemented
146Dic9⋊C4—32no—
147C22×C3×C9—22no—
148C22×C3×C9C2×D412yes—
149C22×C3×C9C2411yescharacteristic
150C2×C3⋊D9—22no—
151C2×C3⋊D9—22no—
152C2×C3⋊D9—44no—
153C2×C3⋊D9C2×D412yes—
154C2×C3⋊D9C2×D412yes—
155C2×D3×C9—21no—
156C2×D3×C9—22no—
157C2×C3×D9—22no—
158C2×C3×D9—21no—
159C2×C3×D9—22no—
160C2×C3×D9—22no—
161C2×C3×D9—22no—
162C2×C3×D9—44no—
163D3×D9—44no—
164C3×C4×C9—22no—
165C3⋊Dic9—22no—
166C9×Dic3—21no—
167C9×Dic3—22no—
168C3×Dic9—21no—
169C22×#67—91no—
170C2×D3×D4—91no—
171C2×D3×D4—91no—
172C22×D12—91no—
173C3×#123—31no—
174C3×#120—91no—
175#953—91no—
176#938—92no—
177#937—91no—
178#937—92no—
179#935—91no—
180#929—91no—
181#895—92no—
182#894—92no—
183#886D911yescharacteristic, complemented
184#884—92no—
185#884—92no—
186#883—92no—
187#882—31no—
188#882—92no—
189D3×C22⋊C4—91no—
190D3×C22⋊C4—91no—
191D3×C22⋊C4—181no—
192C23×C32C22×D311yescharacteristic
193C22×C3⋊D3—31no—
194C22×C3⋊D3—31no—
195C22×C3⋊D3—34no—
196C22×C3⋊D3—122no—
197C22×C3×D3—31no—
198C22×C3×D3—32no—
199C22×C3×D3—62no—
200C22×C3×D3—122no—
201C22×C3×D3C22×D311yescharacteristic
202C2×D32—62no—
203C2×D32—122no—
204C2×C32×C4C22×D312yes—
205C2×C3⋊Dic3—32no—
206C4×C3⋊D3—122no—
207C3×#67—22no—
208C3×#67—62no—
209C2×C3×Dic3—22no—
210C2×C3×Dic3—31no—
211C2×C3×Dic3C22×D311yescharacteristic
212C2×C3×Dic3C22×D312yes—
213C3×D12—122no—
214#626—64no—
215#626—64no—
216#626—62no—
217#626—62no—
218#625—122no—
219#624—62no—
220C23×C9—21no—
221C23×C9—31no—
222C23×C9C22×D311yescharacteristic
223C22×D9—22no—
224C22×D9—31no—
225C22×D9—31no—
226C22×D9—34no—
227C22×D9—42no—
228C22×D9—61no—
229C22×D9—61no—
230C22×D9—64no—
231C22×D9—62no—
232C22×D9—122no—
233C22×D9—122no—
234C22×D9—124no—
235C22×D9C22×D311yescharacteristic
236C22×D9C22×D312yes—
237D4×C9—62no—
238C2×C4×C9—22no—
239C2×C4×C9—32no—
240C2×C4×C9—31no—
241C2×C4×C9—62no—
242C2×C4×C9C22×D312yes—
243#611—22no—
244#611—62no—
245#611—122no—
246C2×Dic9—32no—
247C2×Dic9—62no—
248C2×Dic9C22×D311yescharacteristic
249D36—42no—
250D36—64no—
251D36—64no—
252D36—62no—
253C4×D9—62no—
254C4×D9—122no—
255C4×D9—124no—
256#534—271noSylow
257C2×C3×C9—22no—
258C2×C3×C9C22×D411yescharacteristic
259C2×C3×C9H(5,𝔽2)12yes—
260C3⋊D9—22no—
261C3⋊D9—22no—
262D3×C9—41no—
263C3×D9—41no—
264C3×D9—42no—
265C24×C3—91no—
266C23×D3—31no—
267C23×D3—91no—
268C23×D3—91no—
269C23×D3—91no—
270C23×D3—91no—
271C23×D3—181no—
272C2×C3×D4—31no—
273C2×C3×D4—91no—
274C2×C3×D4—92no—
275C22×C3×C4—31no—
276C2×#67—31no—
277C2×#67—91no—
278C2×#67—92no—
279C2×#67—92no—
280C2×#67—92no—
281C2×#67—92no—
282C2×#67—91no—
283C2×#67—184no—
284C2×#67C2×D911yescharacteristic
285C22×Dic3C2×D911yescharacteristic, complemented
286D3×D4—184no—
287D3×D4—184no—
288C2×D12—32no—
289C2×D12—92no—
290C2×D12—92no—
291C2×D12—91no—
292C2×D12—91no—
293C2×D12—184no—
294C2×C4×D3—91no—
295C2×C4×D3—91no—
296C2×C4×D3—92no—
297C2×C4×D3—92no—
298C2×C4×D3—182no—
299C3×C4⋊C4—32no—
300C3×C22⋊C4—21no—
301C3×C22⋊C4—32no—
302C3×C22⋊C4—92no—
303C3×C22⋊C4C2×D911yescharacteristic, complemented
304C3×C22⋊C4D3211yescharacteristic
305#217—91no—
306#217—91no—
307#217—92no—
308#217—92no—
309#217—181no—
310(C2×D3)⋊C4—32no—
311(C2×D3)⋊C4—92no—
312(C2×D3)⋊C4—92no—
313(C2×D3)⋊C4—92no—
314(C2×D3)⋊C4—92no—
315(C2×D3)⋊C4—182no—
316(C2×D3)⋊C4C2×D912yescomplemented
317C4⋊Dic3C2×D912yescomplemented
318Dic3⋊C4—92no—
319Dic3⋊C4—92no—
320C22×C32—22no—
321C22×C32D3×D412yes—
322C22×C32C23×D311yescharacteristic
323C2×C3⋊D3—32no—
324C2×C3⋊D3—32no—
325C2×C3⋊D3—62no—
326C2×C3⋊D3—62no—
327C2×C3⋊D3—124no—
328C2×C3×D3—21no—
329C2×C3×D3—22no—
330C2×C3×D3—62no—
331C2×C3×D3—61no—
332C2×C3×D3—62no—
333C2×C3×D3—62no—
334C2×C3×D3—62no—
335C2×C3×D3—124no—
336D32—124no—
337C32×C4—22no—
338C3⋊Dic3—62no—
339C3×Dic3—21no—
340C3×Dic3—22no—
341C3×Dic3—61no—
342C22×C9—22no—
343C22×C9—21no—
344C22×C9—22no—
345C22×C9—42no—
346C22×C9—61no—
347C22×C9—62no—
348C22×C9D3×D412yes—
349C22×C9C23×D311yescharacteristic
350C2×D9—22no—
351C2×D9—21no—
352C2×D9—22no—
353C2×D9—22no—
354C2×D9—22no—
355C2×D9—32no—
356C2×D9—32no—
357C2×D9—44no—
358C2×D9—62no—
359C2×D9—62no—
360C2×D9—64no—
361C2×D9—62no—
362C2×D9—62no—
363C2×D9—122no—
364C2×D9—124no—
365C2×D9—124no—
366C2×D9—128no—
367C4×C9—22no—
368C4×C9—42no—
369C4×C9—61no—
370C4×C9—62no—
371Dic9—21no—
372Dic9—62no—
373Dic9—122no—
374C22×D4—271no—
375C22×D4—271no—
376#123—31no—
377#123—272no—
378#123—271no—
379#121—272no—
380#121—272no—
381#120—91no—
382#120—272no—
383#120—271no—
384C2×C22⋊C4—271no—
385C3×C9#53411yescharacteristic, complemented, Sylow
386C23×C3—21no—
387C23×C3—31no—
388C23×C3—91no—
389C23×C3—92no—
390C23×C3—182no—
391C23×C3—362no—
392C23×C3C22×D911yescharacteristic
393C23×C3C2×D3211yescharacteristic
394C22×D3—32no—
395C22×D3—31no—
396C22×D3—62no—
397C22×D3—91no—
398C22×D3—91no—
399C22×D3—91no—
400C22×D3—91no—
401C22×D3—94no—
402C22×D3—94no—
403C22×D3—122no—
404C22×D3—181no—
405C22×D3—181no—
406C22×D3—184no—
407C22×D3—182no—
408C22×D3—182no—
409C22×D3—362no—
410C22×D3—362no—
411C22×D3—362no—
412C22×D3—362no—
413C22×D3—364no—
414C22×D3C22×D911yescharacteristic
415C3×D4—62no—
416C3×D4—182no—
417C3×D4—362no—
418C2×C3×C4—22no—
419C2×C3×C4—32no—
420C2×C3×C4—31no—
421C2×C3×C4—62no—
422C2×C3×C4—91no—
423C2×C3×C4C22×D912yes—
424C2×C3×C4C2×D3212yes—
425#67—22no—
426#67—62no—
427#67—182no—
428#67—184no—
429#67—184no—
430#67—182no—
431#67—362no—
432#67—362no—
433C2×Dic3—22no—
434C2×Dic3—31no—
435C2×Dic3—92no—
436C2×Dic3—92no—
437C2×Dic3—182no—
438C2×Dic3C22×D912yes—
439C2×Dic3C22×D911yescharacteristic
440D12—122no—
441D12—184no—
442D12—184no—
443D12—182no—
444D12—182no—
445C4×D3—182no—
446C4×D3—182no—
447C4×D3—362no—
448C4×D3—362no—
449C4×D3—364no—
450C2×C32—22no—
451C2×C32C2×D3×D411yescharacteristic
452C2×C32#100912yes—
453C3⋊D3—62no—
454C3⋊D3—62no—
455C3×D3—41no—
456C3×D3—121no—
457C3×D3—122no—
458C2×C9—22no—
459C2×C9—21no—
460C2×C9—22no—
461C2×C9—42no—
462C2×C9—121no—
463C2×C9C2×D3×D411yescharacteristic
464C2×C9#100412yes—
465D9—42no—
466D9—41no—
467D9—64no—
468D9—62no—
469D9—62no—
470D9—122no—
471C24—91no—
472C24—271no—
473C24—271no—
474C2×D4—31no—
475C2×D4—91no—
476C2×D4—92no—
477C2×D4—272no—
478C2×D4—272no—
479C2×D4—271no—
480C2×D4—271no—
481C2×D4—272no—
482C2×D4—544no—
483C2×D4—544no—
484C22×C4—31no—
485C22×C4—271no—
486C22×C4—272no—
487C4⋊C4—32no—
488C4⋊C4—272no—
489C22⋊C4—32no—
490C22⋊C4—92no—
491C22⋊C4—272no—
492C22⋊C4—272no—
493C22⋊C4—271no—
494C22⋊C4—272no—
495C22⋊C4D3×D911yescharacteristic, complemented
496C22×C3—22no—
497C22×C3—22no—
498C22×C3—21no—
499C22×C3—22no—
500C22×C3—42no—
501C22×C3—61no—
502C22×C3—62no—
503C22×C3—182no—
504C22×C3—181no—
505C22×C3—182no—
506C22×C3—182no—
507C22×C3—182no—
508C22×C3—364no—
509C22×C3D4×D912yes—
510C22×C3C23×D911yescharacteristic
511C22×C3#380212yes—
512C22×C3C22×D3211yescharacteristic
513C2×D3—21no—
514C2×D3—22no—
515C2×D3—62no—
516C2×D3—61no—
517C2×D3—62no—
518C2×D3—62no—
519C2×D3—62no—
520C2×D3—92no—
521C2×D3—92no—
522C2×D3—92no—
523C2×D3—92no—
524C2×D3—124no—
525C2×D3—182no—
526C2×D3—182no—
527C2×D3—184no—
528C2×D3—182no—
529C2×D3—182no—
530C2×D3—182no—
531C2×D3—182no—
532C2×D3—362no—
533C2×D3—364no—
534C2×D3—364no—
535C2×D3—364no—
536C2×D3—364no—
537C2×D3—368no—
538C3×C4—22no—
539C3×C4—22no—
540C3×C4—42no—
541C3×C4—61no—
542C3×C4—62no—
543C3×C4—181no—
544Dic3—21no—
545Dic3—22no—
546Dic3—61no—
547Dic3—182no—
548Dic3—182no—
549Dic3—362no—
550C32#565311yescharacteristic
551C9—21no—
552C9#558911yescharacteristic, complemented
553C23—31no—
554C23—92no—
555C23—91no—
556C23—182no—
557C23—271no—
558C23—271no—
559C23—274no—
560C23—362no—
561C23—542no—
562C23—1082no—
563C23—1082no—
564C23C2×D3×D911yescharacteristic
565D4—62no—
566D4—182no—
567D4—362no—
568D4—544no—
569D4—544no—
570D4—542no—
571D4—542no—
572D4—1082no—
573C2×C4—32no—
574C2×C4—31no—
575C2×C4—62no—
576C2×C4—91no—
577C2×C4—272no—
578C2×C4—542no—
579C2×C4—1082no—
580C2×C4C2×D3×D912yes—
581C2×C3—22no—
582C2×C3—22no—
583C2×C3—22no—
584C2×C3—21no—
585C2×C3—42no—
586C2×C3—121no—
587C2×C3—361no—
588C2×C3—362no—
589C2×C3C2×D4×D911yescharacteristic
590C2×C3#6371512yes—
591C2×C3C2×#380211yescharacteristic
592C2×C3#6431412yes—
593D3—41no—
594D3—122no—
595D3—121no—
596D3—184no—
597D3—182no—
598D3—182no—
599D3—182no—
600D3—182no—
601D3—362no—
602C22—22no—
603C22—62no—
604C22—61no—
605C22—182no—
606C22—182no—
607C22—182no—
608C22—181no—
609C22—182no—
610C22—272no—
611C22—272no—
612C22—364no—
613C22—542no—
614C22—542no—
615C22—1084no—
616C22—1084no—
617C22#8864612yes—
618C22C22×D3×D911yescharacteristic
619C4—22no—
620C4—62no—
621C4—61no—
622C4—181no—
623C4—542no—
624C3—21no—
625C3#1058975511yescharacteristic, complemented
626C3#1059429611yescharacteristic
627C2—22no—
628C2—121no—
629C2—362no—
630C2—361no—
631C2—542no—
632C2—542no—
633C2C2×#8864611yescharacteristic
634C2#1172250112yes—
subgroupG/H
centerC22C22×D3×D9
commutator, cosocleC22×C3×C9C24
FrattiniC22×C3C22×D32
FittingC3×C9×C22⋊C4C22
socleC22×C32C23×D3
hypercenterC22⋊C4D3×D9
hypocenterC3×C9#534
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×#88668C2semidirect
2. C22×#6192C2semidirect
3. C22⋊C4×C3⋊D9C2semidirect
4. C9×#886C2semidirect
5. C3×#63444C2semidirect
6. #11721285C2semidirect
7. #11721259C2semidirect
8. #11721049C2semidirect
9. #11721041C2semidirect
10. #11721007C2semidirect
11. #11720942C2semidirect
15. C23×C3×D9C22semidirect
16. C22×D3×D9C22semidirect
17. C2×C4×C3⋊D9C22semidirect
20. C22×C9×Dic3C22semidirect
21. C2×C3×D36C22semidirect
23. C2×#6195C22semidirect
24. C2×#6194C22semidirect
27. C2×#6192C22semidirect
28. C2×#6192C22semidirect
29. C2×(C3⋊D9)⋊C4C22semidirect
30. C3×C9×C22⋊C4C22semidirect
31. #88551C22semidirect
32. #88536C22semidirect
33. C9×(C2×D3)⋊C4C22semidirect
34. C3×(C2×D9)⋊C4C22semidirect
35. C9×C4⋊Dic3C22semidirect
36. (C2×D9)⋊Dic3C22semidirect
37. (C2×D9)⋊Dic3C22semidirect
38. #88442C22semidirect
59. #63444D3semidirect
64. C22×C3⋊D9D4semidirect
111. C23×D9C2×D3semidirect
131. C2×D36C2×D3semidirect
138. C9×C22⋊C4C2×D3semidirect
145. (C2×D9)⋊C4C2×D3semidirect
183. #886D9semidirect
285. C22×Dic3C2×D9semidirect
303. C3×C22⋊C4C2×D9semidirect
316. (C2×D3)⋊C4C2×D9semidirect
317. C4⋊Dic3C2×D9semidirect
385. C3×C9#534semidirect
495. C22⋊C4D3×D9semidirect
552. C9#5589semidirect
625. C3#10589755semidirect

Maximal subgroups

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

subgroupindex
C2×#886682
C22×#61922
C22⋊C4×C3⋊D92
C9×#8862
C3×#634442
#117212852
#117212592
#117210492
#117210412
#117210072
#117209422
#105942963
#105897553