Finite Group Databasebeta

← #13151279

Subgroups — #13151279

Order 1728.

1084 conjugacy classes of subgroups.

The subgroup lattice has 612 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C9×#237C211yesmaximal, characteristic, complemented
2C2×C4×D3×D9C211yesmaximal, characteristic, complemented
3#11721838C211yesmaximal, characteristic, complemented
4C3×#63500C211yesmaximal, characteristic, complemented
5#11721292C212yesmaximal, complemented
6#11721280C212yesmaximal, complemented
7Dic3×#611C212yesmaximal, complemented
8#11721240C212yesmaximal, complemented
9#11720944C211yesmaximal, characteristic, complemented
10#11720929C211yesmaximal, characteristic, complemented
11#11720906C211yesmaximal, characteristic, complemented
12#10595301—31nomaximal
13#10589930—31nomaximal
14C22×D3×D9C2211yescharacteristic
15C2×C3×D4×C9C2211yescharacteristic, complemented
16C2×#6233C2212yes—
17C2×C4×C3⋊D9C2211yescharacteristic
18C2×C9×#67C2212yes—
19C2×C3×#611C2212yes—
20C22×C9×Dic3C2212yescomplemented
21C9×#237—24no—
22C2×C4×D3×C9C2211yescharacteristic
23C2×C3×C4×D9C2211yescharacteristic
24C2×C9×Dic6C2211yescharacteristic, complemented
25C2×D3×Dic9C2211yescharacteristic
26C2×(C3⋊D9)⋊C4C2211yescharacteristic
27C2×Dic3×D9C2211yescharacteristic
28C4×D3×D9—24no—
29#88551C2212yescomplemented
30C4⋊(C3⋊Dic9)C2211yescharacteristic, complemented
31C3×#3676C2212yescomplemented
32C3×C4⋊Dic9C2211yescharacteristic, complemented
33(C2×D3)⋊Dic9C2212yes—
34#88445C2212yes—
35(C2×D9)⋊Dic3C2212yes—
36#88442C2212yescomplemented
37Dic9⋊Dic3C2212yescomplemented
38Dic3×Dic9C2212yescomplemented
39C2×C3×#237D311yescharacteristic
40C2×C4×D32—31no—
41#64147—31no—
42C3×#938—31no—
43#63983—32no—
44Dic3×#67—32no—
45#63969—32no—
46#63952—32no—
47#63871—31no—
48#63856—31no—
49#63841—31no—
50C2×C9×#41—31no—
51C22×C4×D9—31no—
52#63508—31no—
53#63508—31no—
54#63500—31no—
55#63500—61no—
56#63500D311yescharacteristic, complemented
57#63499—32no—
58#63497—32no—
59C4×#611—32no—
60C4×#611—32no—
61#63488—31no—
62C23×C3×C9C2312yes—
63C22×C3⋊D9C2311yescharacteristic
64C22×D3×C9C2311yescharacteristic
65C22×C3×D9C2311yescharacteristic
66C2×D3×D9—22no—
67C2×D3×D9—22no—
68C2×D3×D9—22no—
69C3×D4×C9—22no—
70C2×C3×C4×C9C2311yescharacteristic
71#6233—24no—
72C2×C3⋊Dic9C2311yescharacteristic
73C2×C3⋊Dic9C2312yes—
74C4×C3⋊D9—22no—
75C9×#67—24no—
76C3×#611—24no—
77C2×C9×Dic3—24no—
78C2×C9×Dic3C2312yes—
79C2×C9×Dic3C2311yescharacteristic
80C2×C3×Dic9C2311yescharacteristic
81C2×C3×Dic9C2312yes—
82C4×D3×C9D414yescomplemented
83C3×C4×D9—22no—
84C9×Dic6—22no—
85D3×Dic9—22no—
86(C3⋊D9)⋊C4—22no—
87Dic3×D9—22no—
88#5828—91no—
89#5606—91no—
90C22×D32—31no—
91C2×C32×D4C2×D311yescharacteristic
92C2×#638—32no—
93C2×C4×C3⋊D3—31no—
94C2×C3×#67—32no—
95C2×C3×#67C2×D312yes—
96C22×C3×Dic3C2×D312yes—
97C3×#237—24no—
98C2×C3×C4×D3—31no—
99C2×C3×C4×D3C2×D311yescharacteristic
100C2×C3×Dic6C2×D311yescharacteristic
101C2×#624—31no—
102C2×D3×Dic3—31no—
103C2×D3×Dic3—31no—
104C4×D32—64no—
105C23×D9—31no—
106#3757—32no—
107C4⋊(C3⋊Dic3)—31no—
108C3×#217—32no—
109C3×C4⋊Dic3—31no—
110#3724—32no—
111Dic3⋊Dic3—32no—
112#3722—32no—
113(C2×D3)⋊Dic3—32no—
114(C2×D3)⋊Dic3—32no—
115Dic32—32no—
116C9×#41—64no—
117C2×Q8×C9—31no—
118C2×D4×C9—21no—
119C2×D4×C9—32no—
120C2×D4×C9C2×D311yescharacteristic, complemented
121C22×C4×C9—31no—
122C22×C4×C9—32no—
123C2×#611—32no—
124C2×#611—62no—
125C2×#611C2×D312yes—
126C22×Dic9—31no—
127C2×C4×D9—31no—
128C2×C4×D9—31no—
129C2×C4×D9—31no—
130C2×C4×D9—64no—
131C2×C4×D9—61no—
132C2×C4×D9C2×D311yescharacteristic
133#3676—32no—
134#3676—32no—
135#3676—62no—
136#3676C2×D312yescomplemented
137(C2×D9)⋊C4—32no—
138(C2×D9)⋊C4—32no—
139C4⋊Dic9—31no—
140C4⋊Dic9—61no—
141C4⋊Dic9C2×D311yescharacteristic, complemented
142Dic9⋊C4—32no—
143Dic9⋊C4—32no—
144C4×Dic9—32no—
145C22×C3×C9—22no—
146C22×C3×C9—22no—
147C22×C3×C9—22no—
148C22×C3×C9C2411yescharacteristic
149C2×C3⋊D9—21no—
150C2×C3⋊D9—21no—
151C2×C3⋊D9#4112yescomplemented
152C2×D3×C9—21no—
153C2×D3×C9—21no—
154C2×D3×C9C2×D412yes—
155C2×C3×D9—21no—
156C2×C3×D9—21no—
157C2×C3×D9#4112yescomplemented
158D3×D9—28no—
159C3×C4×C9C2×D412yes—
160C3⋊Dic9—21no—
161C3⋊Dic9—22no—
162C9×Dic3—22no—
163C9×Dic3C2×D412yes—
164C3×Dic9—21no—
165C3×Dic9—22no—
166C2×C3×#41—31no—
167C2×#237D911yescharacteristic, complemented
168C22×C4×D3—91no—
169C22×C4×D3—91no—
170C3×#121—91no—
171#946—91no—
172#946—91no—
173#938—31no—
174#938—91no—
175#938—91no—
176#938—181no—
177#937—92no—
178#937—92no—
179#935—92no—
180D4×Dic3—92no—
181C4×#67—92no—
182C4×#67—92no—
183#926—91no—
184#924—91no—
185#897—91no—
186#892—91no—
187#883—92no—
188#67⋊C4—92no—
189C23×C32C22×D312yes—
190C22×C3⋊D3—31no—
191C22×C3×D3—31no—
192C22×C3×D3C22×D311yescharacteristic
193C2×D32—62no—
194C2×D32—62no—
195C2×D32—62no—
196C32×D4—22no—
197C2×C32×C4C22×D311yescharacteristic
198#638—64no—
199C2×C3⋊Dic3—32no—
200C2×C3⋊Dic3—31no—
201C4×C3⋊D3—62no—
202C3×#67—24no—
203C3×#67—64no—
204C2×C3×Dic3—24no—
205C2×C3×Dic3—32no—
206C2×C3×Dic3—31no—
207C2×C3×Dic3C22×D312yes—
208C2×C3×Dic3C22×D311yescharacteristic
209C3×C4×D3—62no—
210C3×C4×D3#6714yes—
211C3×Dic6—22no—
212#624—62no—
213D3×Dic3—62no—
214D3×Dic3—62no—
215C23×C9—22no—
216C23×C9—31no—
217C23×C9C22×D312yes—
218C22×D9—31no—
219C22×D9—61no—
220C22×D9—62no—
221C22×D9—62no—
222C22×D9—62no—
223C22×D9C22×D311yescharacteristic
224Q8×C9—62no—
225D4×C9—22no—
226D4×C9—42no—
227D4×C9—64no—
228C2×C4×C9—21no—
229C2×C4×C9—31no—
230C2×C4×C9—34no—
231C2×C4×C9—32no—
232C2×C4×C9—64no—
233C2×C4×C9C22×D311yescharacteristic
234#611—24no—
235#611—64no—
236#611—124no—
237C2×Dic9—32no—
238C2×Dic9—31no—
239C2×Dic9—62no—
240C2×Dic9—61no—
241C2×Dic9—62no—
242C2×Dic9C22×D312yes—
243C2×Dic9C22×D311yescharacteristic
244C4×D9—22no—
245C4×D9—62no—
246C4×D9—62no—
247C4×D9—62no—
248C4×D9—122no—
249#525—271noSylow
250C2×C3×C9—42no—
251C2×C3×C9C22×D411yescharacteristic
252C2×C3×C9C2×#4111yescharacteristic
253C2×C3×C9C2×#4111yescharacteristic
254C3⋊D9—22no—
255D3×C9—22no—
256C3×D9—22no—
257C23×D3—91no—
258C23×D3—91no—
259C3×#41—64no—
260C2×C3×Q8—31no—
261C2×C3×D4—21no—
262C2×C3×D4—32no—
263C2×C3×D4—92no—
264C2×C3×D4C2×D911yescharacteristic, complemented
265C2×C3×D4D3211yescharacteristic
266C22×C3×C4—31no—
267C22×C3×C4—32no—
268C22×C3×C4—91no—
269C2×#67—32no—
270C2×#67—92no—
271C2×#67—92no—
272C2×#67—182no—
273C2×#67C2×D912yes—
274C22×Dic3—91no—
275C22×Dic3—91no—
276C22×Dic3C2×D912yescomplemented
277#237—24no—
278C2×C4×D3—31no—
279C2×C4×D3—91no—
280C2×C4×D3—91no—
281C2×C4×D3—91no—
282C2×C4×D3—91no—
283C2×C4×D3—91no—
284C2×C4×D3—91no—
285C2×C4×D3—181no—
286C2×C4×D3—184no—
287C2×C4×D3—184no—
288C2×C4×D3C2×D911yescharacteristic
289C2×Dic6C2×D911yescharacteristic, complemented
290C3×C4⋊C4—91no—
291C3×C22⋊C4—92no—
292#217—32no—
293#217—92no—
294#217—92no—
295#217—92no—
296#217—92no—
297#217—182no—
298(C2×D3)⋊C4—92no—
299(C2×D3)⋊C4—92no—
300(C2×D3)⋊C4—92no—
301(C2×D3)⋊C4—92no—
302C4⋊Dic3—31no—
303C4⋊Dic3—91no—
304C4⋊Dic3—91no—
305C4⋊Dic3—92no—
306C4⋊Dic3—181no—
307Dic3⋊C4—92no—
308Dic3⋊C4—92no—
309Dic3⋊C4—92no—
310C4×Dic3—92no—
311C4×Dic3—92no—
312C22×C32—22no—
313C22×C32—22no—
314C22×C32—22no—
315C22×C32C23×D311yescharacteristic
316C2×C3⋊D3—32no—
317C2×C3⋊D3—61no—
318C2×C3⋊D3—61no—
319C2×C3×D3—21no—
320C2×C3×D3—21no—
321C2×C3×D3—32no—
322C2×C3×D3—61no—
323C2×C3×D3—61no—
324C2×C3×D3C2×#6712yes—
325D32—68no—
326C32×C4C2×#6712yes—
327C3⋊Dic3—62no—
328C3⋊Dic3—61no—
329C3×Dic3—22no—
330C3×Dic3—62no—
331C3×Dic3—61no—
332C3×Dic3C2×#6712yes—
333C22×C9—22no—
334C22×C9—22no—
335C22×C9—22no—
336C22×C9—21no—
337C22×C9—32no—
338C22×C9—42no—
339C22×C9—42no—
340C22×C9—42no—
341C22×C9—61no—
342C22×C9—61no—
343C22×C9C23×D311yescharacteristic
344C2×D9—21no—
345C2×D9—21no—
346C2×D9—32no—
347C2×D9—62no—
348C2×D9—68no—
349C2×D9—61no—
350C2×D9—61no—
351C2×D9—121no—
352C2×D9—121no—
353C2×D9#23712yescomplemented
354C4×C9—22no—
355C4×C9—32no—
356C4×C9—62no—
357C4×C9D3×D412yes—
358Dic9—21no—
359Dic9—22no—
360Dic9—61no—
361Dic9—62no—
362Dic9—121no—
363Dic9—122no—
364C2×#41—31no—
365C23×C4—271no—
366#123—272no—
367C22⋊Q8—271no—
368C22⋊Q8—271no—
369#121—91no—
370#121—271no—
371#120—272no—
372C4×D4—272no—
373C4×D4—272no—
374#117—271no—
375C3×C9#52511yescharacteristic, complemented, Sylow
376C23×C3—22no—
377C23×C3—31no—
378C23×C3—91no—
379C23×C3C22×D912yes—
380C23×C3C2×D3212yes—
381C22×D3—31no—
382C22×D3—91no—
383C22×D3—91no—
384C22×D3—181no—
385C22×D3—182no—
386C22×D3—182no—
387C22×D3—182no—
388C22×D3—182no—
389C22×D3—182no—
390C22×D3—182no—
391C22×D3C22×D911yescharacteristic
392C3×Q8—62no—
393C3×D4—22no—
394C3×D4—22no—
395C3×D4—42no—
396C3×D4—64no—
397C3×D4—184no—
398C2×C3×C4—21no—
399C2×C3×C4—31no—
400C2×C3×C4—34no—
401C2×C3×C4—32no—
402C2×C3×C4—64no—
403C2×C3×C4—92no—
404C2×C3×C4—91no—
405C2×C3×C4—182no—
406C2×C3×C4C22×D911yescharacteristic
407C2×C3×C4C2×D3211yescharacteristic
408#67—24no—
409#67—64no—
410#67—184no—
411#67—184no—
412#67—364no—
413C2×Dic3—24no—
414C2×Dic3—32no—
415C2×Dic3—31no—
416C2×Dic3—92no—
417C2×Dic3—91no—
418C2×Dic3—92no—
419C2×Dic3—91no—
420C2×Dic3—182no—
421C2×Dic3—181no—
422C2×Dic3—182no—
423C2×Dic3—182no—
424C2×Dic3C22×D912yes—
425C2×Dic3C22×D911yescharacteristic
426C4×D3—62no—
427C4×D3—182no—
428C4×D3—182no—
429C4×D3—182no—
430C4×D3—182no—
431C4×D3—182no—
432C4×D3—182no—
433C4×D3—362no—
434C4×D3#61114yescomplemented
435Dic6—22no—
436C2×C32—42no—
437C2×C32D3×#4111yescharacteristic
438C2×C32D3×#4111yescharacteristic
439C2×C32C22×#6711yescharacteristic
440C3⋊D3—62no—
441C3×D3—22no—
442C3×D3—62no—
443C2×C9—21no—
444C2×C9—21no—
445C2×C9—21no—
446C2×C9—42no—
447C2×C9—62no—
448C2×C9—82no—
449C2×C9C2×D3×D411yescharacteristic
450C2×C9C2×#23711yescharacteristic
451C2×C9D3×#4111yescharacteristic
452D9—22no—
453D9—62no—
454D9—122no—
455C24—271no—
456#41—64no—
457C2×Q8—31no—
458C2×D4—32no—
459C2×D4—92no—
460C2×D4—272no—
461C2×D4D3×D911yescharacteristic, complemented
462C22×C4—31no—
463C22×C4—32no—
464C22×C4—91no—
465C22×C4—271no—
466C22×C4—271no—
467C22×C4—271no—
468C22×C4—271no—
469C22×C4—544no—
470C4⋊C4—91no—
471C4⋊C4—272no—
472C4⋊C4—272no—
473C4⋊C4—271no—
474C22⋊C4—92no—
475C22⋊C4—272no—
476C22⋊C4—272no—
477C22⋊C4—272no—
478C22⋊C4—272no—
479C42—272no—
480C22×C3—22no—
481C22×C3—22no—
482C22×C3—22no—
483C22×C3—22no—
484C22×C3—22no—
485C22×C3—22no—
486C22×C3—21no—
487C22×C3—32no—
488C22×C3—42no—
489C22×C3—42no—
490C22×C3—42no—
491C22×C3—61no—
492C22×C3—61no—
493C22×C3—92no—
494C22×C3—181no—
495C22×C3—181no—
496C22×C3C23×D911yescharacteristic
497C22×C3C22×D3211yescharacteristic
498C2×D3—21no—
499C2×D3—21no—
500C2×D3—32no—
501C2×D3—61no—
502C2×D3—61no—
503C2×D3—92no—
504C2×D3—92no—
505C2×D3—182no—
506C2×D3—188no—
507C2×D3—188no—
508C2×D3—181no—
509C2×D3—181no—
510C2×D3—181no—
511C2×D3—181no—
512C2×D3—361no—
513C2×D3—361no—
514C2×D3C2×#61112yes—
515C3×C4—22no—
516C3×C4—32no—
517C3×C4—62no—
518C3×C4—182no—
519C3×C4—181no—
520C3×C4C2×#61112yes—
521C3×C4D3×#6712yes—
522Dic3—22no—
523Dic3—62no—
524Dic3—61no—
525Dic3—182no—
526Dic3—181no—
527Dic3—182no—
528Dic3—181no—
529Dic3—361no—
530Dic3—362no—
531Dic3C2×#61112yes—
532C32#582811yescharacteristic
533C9—21no—
534C9#560611yescharacteristic, complemented
535C23—31no—
536C23—91no—
537C23—271no—
538C23—542no—
539C23—542no—
540C23—542no—
541C23C2×D3×D912yes—
542Q8—62no—
543D4—22no—
544D4—64no—
545D4—184no—
546D4—544no—
547C2×C4—31no—
548C2×C4—34no—
549C2×C4—32no—
550C2×C4—64no—
551C2×C4—92no—
552C2×C4—91no—
553C2×C4—182no—
554C2×C4—272no—
555C2×C4—271no—
556C2×C4—542no—
557C2×C4—542no—
558C2×C4—542no—
559C2×C4—542no—
560C2×C4C2×D3×D911yescharacteristic
561C2×C3—21no—
562C2×C3—21no—
563C2×C3—21no—
564C2×C3—42no—
565C2×C3—42no—
566C2×C3—62no—
567C2×C3—82no—
568C2×C3—182no—
569C2×C3#41×D911yescharacteristic
570C2×C3#41×D911yescharacteristic
571C2×C3C22×#61111yescharacteristic
572C2×C3D3×#23711yescharacteristic
573C2×C3#6431211yescharacteristic
574C2×C3C2×D3×#6711yescharacteristic
575D3—22no—
576D3—62no—
577D3—182no—
578D3—182no—
579D3—362no—
580C22—22no—
581C22—22no—
582C22—22no—
583C22—32no—
584C22—61no—
585C22—61no—
586C22—92no—
587C22—181no—
588C22—181no—
589C22—272no—
590C22—548no—
591C22—541no—
592C22—541no—
593C22C22×D3×D911yescharacteristic
594C4—32no—
595C4—62no—
596C4—182no—
597C4—181no—
598C4—542no—
599C4—541no—
600C4D3×#61112yes—
601C3—21no—
602C3#1058993011yescharacteristic, complemented
603C3#1059530111yescharacteristic
604C2—42no—
605C2—62no—
606C2—182no—
607C2—542no—
608C2D9×#23711yescharacteristic
609C2#1172249811yescharacteristic
610C2C2×D3×#61111yescharacteristic
subgroupG/H
centerC22C22×D3×D9
commutator, cosocleC22×C3×C9C24
FrattiniC22×C3C22×D32
FittingC2×C3×D4×C9C22
socleC22×C32C23×D3
hypercenterC2×D4D3×D9
hypocenterC3×C9#525
LCSG ▷ C22×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
1. C2×C9×#237C2semidirect
2. C2×C4×D3×D9C2semidirect
3. #11721838C2semidirect
4. C3×#63500C2semidirect
5. #11721292C2semidirect
6. #11721280C2semidirect
7. Dic3×#611C2semidirect
8. #11721240C2semidirect
9. #11720944C2semidirect
10. #11720929C2semidirect
11. #11720906C2semidirect
15. C2×C3×D4×C9C22semidirect
20. C22×C9×Dic3C22semidirect
24. C2×C9×Dic6C22semidirect
29. #88551C22semidirect
30. C4⋊(C3⋊Dic9)C22semidirect
31. C3×#3676C22semidirect
32. C3×C4⋊Dic9C22semidirect
36. #88442C22semidirect
37. Dic9⋊Dic3C22semidirect
38. Dic3×Dic9C22semidirect
56. #63500D3semidirect
82. C4×D3×C9D4semidirect
120. C2×D4×C9C2×D3semidirect
136. #3676C2×D3semidirect
141. C4⋊Dic9C2×D3semidirect
151. C2×C3⋊D9#41semidirect
157. C2×C3×D9#41semidirect
167. C2×#237D9semidirect
264. C2×C3×D4C2×D9semidirect
276. C22×Dic3C2×D9semidirect
289. C2×Dic6C2×D9semidirect
353. C2×D9#237semidirect
375. C3×C9#525semidirect
434. C4×D3#611semidirect
461. C2×D4D3×D9semidirect
534. C9#5606semidirect
602. C3#10589930semidirect

Maximal subgroups

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

subgroupindex
C2×C9×#2372
C2×C4×D3×D92
#117218382
C3×#635002
#117212922
#117212802
Dic3×#6112
#117212402
#117209442
#117209292
#117209062
#105953013
#105899303