Finite Group Databasebeta

← #13150985

Subgroups — #13150985

Order 1728.

1096 conjugacy classes of subgroups.

The subgroup lattice has 606 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×D4×D9C211yesmaximal, characteristic, complemented
2C2×#88634C211yesmaximal, characteristic, complemented
3#11721838C211yesmaximal, characteristic, complemented
4C9×#936C211yesmaximal, characteristic, complemented
5#11721276C212yesmaximal, complemented
6#11721271C212yesmaximal, complemented
7#11721269C212yesmaximal, complemented
8D9×#217C212yesmaximal, complemented
9C4×#6192C211yesmaximal, characteristic, complemented
10Dic3×D36C211yesmaximal, characteristic, complemented
11#11720944C211yesmaximal, characteristic, complemented
12#10595229—31nomaximal
13#10589778—31nomaximal
14C23×C3×D9C2212yes—
15C2×C3×D4×C9C2211yescharacteristic
16C2×#6233C2212yescomplemented
17C2×C4×C3⋊D9C2211yescharacteristic, complemented
18C2×C3×#611C2212yes—
19C3×D4×D9—24no—
20C2×C3×D36C2211yescharacteristic
21C2×C3×C4×D9C2211yescharacteristic
22C2×C9×Dic6C2211yescharacteristic, complemented
23C2×#6192C2211yescharacteristic
24C2×#6192C2212yescomplemented
25C2×Dic3×D9C2212yescomplemented
26C2×Dic3×D9C2212yescomplemented
27#88634—24no—
28#88551C2212yescomplemented
29C4⋊(C3⋊Dic9)C2211yescharacteristic, complemented
30C9×#217C2212yescomplemented
31C9×#217C2212yescomplemented
32C4×C9×Dic3C2211yescharacteristic, complemented
33#88445C2211yescharacteristic, complemented
34(C2×D9)⋊Dic3C2211yescharacteristic
35(C2×D9)⋊Dic3C2212yes—
36(C2×D9)⋊Dic3C2212yescomplemented
37Dic3⋊Dic9C2211yescharacteristic, complemented
38#88442C2212yescomplemented
39C2×C3×D3×D4—31no—
40C2×#3796—31no—
41#64147—31no—
42C3×#936D311yescharacteristic
43#63977—32no—
44#63971—32no—
45#63969—32no—
46D3×#217—32no—
47C4×#626—31no—
48Dic3×D12—31no—
49#63856—31no—
50C2×#3701—31no—
51C2×D4×D9D311yescharacteristic, complemented
52C9×#124—31no—
53#63508—31no—
54#63500—31no—
55#63500—61no—
56#63446—32no—
57#63445—32no—
58#63444—32no—
59C22⋊C4×D9—32no—
60C4×D36—31no—
61C4×D36—31no—
62C23×C3×C9C2312yes—
63C22×C3⋊D9C2311yescharacteristic
64C22×C3×D9—24no—
65C22×C3×D9—24no—
66C22×C3×D9—42no—
67C22×C3×D9C2312yes—
68C22×C3×D9C2311yescharacteristic
69C3×D4×C9—22no—
70C2×C3×C4×C9C2311yescharacteristic
71#6233—42no—
72C2×C3⋊Dic9C2311yescharacteristic
73C2×C3⋊Dic9C2312yes—
74C4×C3⋊D9—22no—
75C3×#611—24no—
76C2×C9×Dic3C2311yescharacteristic
77C2×C9×Dic3C2312yes—
78C2×C9×Dic3C2311yescharacteristic
79C2×C3×Dic9C2311yescharacteristic
80C3×D36D414yescomplemented
81C3×C4×D9—22no—
82C9×Dic6—22no—
83#6192—24no—
84#6192—22no—
85Dic3×D9—24no—
86Dic3×D9—42no—
87#5806—91no—
88#5676—91no—
89C23×C3×D3—32no—
90C2×C32×D4C2×D311yescharacteristic
91C2×#638—32no—
92C2×C4×C3⋊D3—31no—
93C2×C3×#67—32no—
94C3×D3×D4—64no—
95C2×C3×D12—31no—
96C2×C3×C4×D3—31no—
97C2×C3×Dic6C2×D311yescharacteristic
98C2×#626—32no—
99C2×#626—31no—
100C2×D3×Dic3—32no—
101C2×D3×Dic3—32no—
102#3796—64no—
103C23×D9C2×D312yes—
104#3757—32no—
105C4⋊(C3⋊Dic3)—31no—
106C3×#217C2×D312yes—
107C3×#217C2×D312yes—
108C3×C4×Dic3C2×D311yescharacteristic
109#3724—32no—
110Dic3⋊Dic3—31no—
111#3722—31no—
112(C2×D3)⋊Dic3—32no—
113(C2×D3)⋊Dic3—31no—
114(C2×D3)⋊Dic3—32no—
115C2×Q8×C9—31no—
116C2×D4×C9—21no—
117C2×D4×C9C2×D311yescharacteristic
118C2×#611—32no—
119C2×#611—62no—
120C2×#611C2×D312yes—
121#3701—64no—
122D4×D9—24no—
123C2×D36—31no—
124C2×D36—32no—
125C2×D36C2×D311yescharacteristic
126C2×C4×D9—32no—
127C2×C4×D9—31no—
128C2×C4×D9—32no—
129C2×C4×D9—61no—
130C2×C4×D9C2×D311yescharacteristic
131C9×C22⋊C4—32no—
132C9×C22⋊C4—32no—
133C42×C9—31no—
134#3676—32no—
135#3676—62no—
136(C2×D9)⋊C4—32no—
137(C2×D9)⋊C4—31no—
138(C2×D9)⋊C4—32no—
139(C2×D9)⋊C4—31no—
140C4⋊Dic9—31no—
141C4⋊Dic9—31no—
142C4⋊Dic9—61no—
143Dic9⋊C4—32no—
144C22×C3×C9—22no—
145C22×C3×C9—22no—
146C22×C3×C9—22no—
147C22×C3×C9C2411yescharacteristic
148C2×C3⋊D9—21no—
149C2×C3⋊D9—21no—
150C2×C3⋊D9—21no—
151C2×C3×D9—21no—
152C2×C3×D9—22no—
153C2×C3×D9—22no—
154C2×C3×D9—21no—
155C2×C3×D9—48no—
156C2×C3×D9C2×D414yes—
157C2×C3×D9#4112yescomplemented
158C3×C4×C9C2×D412yes—
159C3⋊Dic9—21no—
160C3⋊Dic9—22no—
161C9×Dic3—22no—
162C9×Dic3—21no—
163C9×Dic3—21no—
164C3×Dic9—21no—
165C22×C3×D4—91no—
166C2×#239—91no—
167C2×#237—91no—
168C2×D3×D4—31no—
169C3×#124—31no—
170#953—92no—
171C2×#217—92no—
172#946—91no—
173#939—92no—
174#938—91no—
175#938—91no—
176#938—181no—
177#936D911yescharacteristic, complemented
178#935—92no—
179D4×Dic3—91no—
180C4×#67—91no—
181#924—91no—
182#884—92no—
183#883—92no—
184#882—92no—
185D3×C22⋊C4—92no—
186C4×D12—91no—
187C4×D12—91no—
188C23×C32C22×D312yes—
189C22×C3⋊D3—31no—
190C22×C3×D3—31no—
191C22×C3×D3—32no—
192C22×C3×D3—64no—
193C22×C3×D3—64no—
194C22×C3×D3—122no—
195C32×D4—22no—
196C2×C32×C4C22×D311yescharacteristic
197#638—122no—
198C2×C3⋊Dic3—32no—
199C2×C3⋊Dic3—31no—
200C4×C3⋊D3—62no—
201C3×#67—64no—
202C2×C3×Dic3—31no—
203C2×C3×Dic3C22×D311yescharacteristic
204C2×C3×Dic3C22×D311yescharacteristic
205C2×C3×Dic3C22×D312yes—
206C3×D12—34no—
207C3×C4×D3—62no—
208C3×Dic6—22no—
209#626—64no—
210#626—62no—
211D3×Dic3—64no—
212D3×Dic3—122no—
213C23×C9—22no—
214C23×C9C22×D312yes—
215C22×D9—24no—
216C22×D9—24no—
217C22×D9—31no—
218C22×D9—42no—
219C22×D9—61no—
220C22×D9C22×D311yescharacteristic
221C22×D9C22×D312yes—
222Q8×C9—62no—
223D4×C9—22no—
224D4×C9—42no—
225C2×C4×C9—21no—
226C2×C4×C9—32no—
227C2×C4×C9—31no—
228C2×C4×C9—31no—
229C2×C4×C9C22×D311yescharacteristic
230#611—24no—
231#611—122no—
232#611—124no—
233C2×Dic9—32no—
234C2×Dic9—31no—
235C2×Dic9—62no—
236C2×Dic9—61no—
237C2×Dic9C22×D311yescharacteristic
238D36—62no—
239D36—64no—
240D36#6714yescomplemented
241C4×D9—22no—
242C4×D9—64no—
243C4×D9—62no—
244C4×D9—122no—
245C4×D9—122no—
246#546—271noSylow
247C2×C3×C9—42no—
248C2×C3×C9C22×D411yescharacteristic
249C2×C3×C9C2×#4111yescharacteristic
250C2×C3×C9H(5,𝔽2)11yescharacteristic
251C3⋊D9—41no—
252C3×D9—22no—
253C3×D9—24no—
254C24×C3—92no—
255C23×D3—32no—
256C2×C3×Q8—31no—
257C2×C3×D4—21no—
258C2×C3×D4—92no—
259C2×C3×D4—91no—
260C2×C3×D4—184no—
261C2×C3×D4C2×D911yescharacteristic
262C2×C3×D4D3211yescharacteristic
263C22×C3×C4—91no—
264C2×#67—32no—
265C2×#67—92no—
266C2×#67—92no—
267C2×#67—91no—
268C2×#67—92no—
269C2×#67—182no—
270C22×Dic3—92no—
271C22×Dic3—92no—
272#239—184no—
273#237—184no—
274D3×D4—64no—
275C2×D12—31no—
276C2×D12—91no—
277C2×D12—92no—
278C2×C4×D3—31no—
279C2×C4×D3—92no—
280C2×C4×D3—91no—
281C2×C4×D3—91no—
282C2×C4×D3—92no—
283C2×C4×D3—181no—
284C2×Dic6C2×D911yescharacteristic, complemented
285C3×C22⋊C4—32no—
286C3×C22⋊C4—32no—
287C3×C42—31no—
288#217—92no—
289#217—92no—
290#217—92no—
291#217—91no—
292#217—92no—
293#217—182no—
294#217C2×D912yescomplemented
295#217C2×D912yescomplemented
296(C2×D3)⋊C4—91no—
297(C2×D3)⋊C4—91no—
298(C2×D3)⋊C4—92no—
299(C2×D3)⋊C4—92no—
300(C2×D3)⋊C4—91no—
301C4⋊Dic3—91no—
302C4⋊Dic3—91no—
303C4⋊Dic3—91no—
304C4⋊Dic3—181no—
305Dic3⋊C4—92no—
306Dic3⋊C4—91no—
307Dic3⋊C4—92no—
308C4×Dic3C2×D911yescharacteristic, complemented
309C22×C32—22no—
310C22×C32—22no—
311C22×C32—22no—
312C22×C32C23×D311yescharacteristic
313C2×C3⋊D3—61no—
314C2×C3⋊D3—61no—
315C2×C3⋊D3—61no—
316C2×C3×D3—34no—
317C2×C3×D3—32no—
318C2×C3×D3—61no—
319C2×C3×D3—62no—
320C2×C3×D3—62no—
321C2×C3×D3—61no—
322C2×C3×D3—128no—
323C32×C4D3×D412yes—
324C3⋊Dic3—61no—
325C3⋊Dic3—62no—
326C3×Dic3—22no—
327C3×Dic3—21no—
328C3×Dic3—21no—
329C3×Dic3—61no—
330C22×C9—22no—
331C22×C9—22no—
332C22×C9—22no—
333C22×C9—21no—
334C22×C9—42no—
335C22×C9—42no—
336C22×C9—42no—
337C22×C9C23×D311yescharacteristic
338C2×D9—22no—
339C2×D9—22no—
340C2×D9—21no—
341C2×D9—21no—
342C2×D9—48no—
343C2×D9—62no—
344C2×D9—61no—
345C2×D9—61no—
346C2×D9—61no—
347C2×D9—121no—
348C2×D9—121no—
349C2×D9#23712yescomplemented
350C2×D9C2×#6714yes—
351C4×C9—22no—
352C4×C9—61no—
353C4×C9—61no—
354C4×C9—62no—
355C4×C9C2×#6712yes—
356Dic9—21no—
357Dic9—62no—
358Dic9—61no—
359Dic9—121no—
360Dic9—122no—
361C2×#41—271no—
362C22×D4—91no—
363#124—31no—
364#123—272no—
365C22⋊Q8—271no—
366#121—272no—
367#121—271no—
368#120—272no—
369C4×D4—271no—
370C4×D4—271no—
371C2×C22⋊C4—272no—
372C3×C9#54611yescharacteristic, complemented, Sylow
373C23×C3—22no—
374C23×C3—91no—
375C23×C3—92no—
376C23×C3—184no—
377C23×C3—184no—
378C23×C3—362no—
379C23×C3C22×D912yes—
380C23×C3C2×D3212yes—
381C22×D3—31no—
382C22×D3—32no—
383C22×D3—64no—
384C22×D3—64no—
385C22×D3—91no—
386C22×D3—91no—
387C22×D3—122no—
388C22×D3—181no—
389C3×Q8—62no—
390C3×D4—22no—
391C3×D4—22no—
392C3×D4—42no—
393C3×D4—94no—
394C3×D4—184no—
395C2×C3×C4—21no—
396C2×C3×C4—32no—
397C2×C3×C4—31no—
398C2×C3×C4—31no—
399C2×C3×C4—91no—
400C2×C3×C4—182no—
401C2×C3×C4C22×D911yescharacteristic
402C2×C3×C4C2×D3211yescharacteristic
403#67—64no—
404#67—184no—
405#67—182no—
406#67—362no—
407#67—362no—
408#67—364no—
409C2×Dic3—31no—
410C2×Dic3—92no—
411C2×Dic3—91no—
412C2×Dic3—92no—
413C2×Dic3—91no—
414C2×Dic3—182no—
415C2×Dic3—181no—
416C2×Dic3—184no—
417C2×Dic3—362no—
418C2×Dic3C22×D911yescharacteristic
419C2×Dic3C22×D911yescharacteristic
420C2×Dic3C22×D912yes—
421D12—34no—
422D12—182no—
423D12—184no—
424C4×D3—62no—
425C4×D3—184no—
426C4×D3—182no—
427C4×D3—182no—
428C4×D3—362no—
429C4×D3—362no—
430Dic6—22no—
431C2×C32—42no—
432C2×C32C2×D3×D411yescharacteristic
433C2×C32D3×#4111yescharacteristic
434C2×C32#100911yescharacteristic
435C3⋊D3—121no—
436C3×D3—64no—
437C3×D3—62no—
438C2×C9—21no—
439C2×C9—21no—
440C2×C9—21no—
441C2×C9—42no—
442C2×C9—82no—
443C2×C9C2×#23711yescharacteristic
444C2×C9#100411yescharacteristic
445C2×C9C22×#6711yescharacteristic
446D9—22no—
447D9—24no—
448D9—122no—
449D9—121no—
450C24—92no—
451#41—544no—
452C2×Q8—31no—
453C2×D4—92no—
454C2×D4—91no—
455C2×D4—184no—
456C2×D4—271no—
457C2×D4—272no—
458C2×D4—272no—
459C2×D4D3×D911yescharacteristic
460C22×C4—91no—
461C22×C4—272no—
462C22×C4—272no—
463C22×C4—271no—
464C4⋊C4—271no—
465C4⋊C4—272no—
466C4⋊C4—271no—
467C22⋊C4—32no—
468C22⋊C4—32no—
469C22⋊C4—271no—
470C22⋊C4—271no—
471C22⋊C4—272no—
472C22⋊C4—272no—
473C22⋊C4—272no—
474C42—31no—
475C22×C3—22no—
476C22×C3—22no—
477C22×C3—22no—
478C22×C3—22no—
479C22×C3—22no—
480C22×C3—22no—
481C22×C3—21no—
482C22×C3—42no—
483C22×C3—42no—
484C22×C3—42no—
485C22×C3—94no—
486C22×C3—92no—
487C22×C3—181no—
488C22×C3—182no—
489C22×C3—182no—
490C22×C3—181no—
491C22×C3—368no—
492C22×C3C23×D911yescharacteristic
493C22×C3C22×D3211yescharacteristic
494C2×D3—32no—
495C2×D3—34no—
496C2×D3—62no—
497C2×D3—62no—
498C2×D3—61no—
499C2×D3—61no—
500C2×D3—128no—
501C2×D3—182no—
502C2×D3—181no—
503C2×D3—181no—
504C2×D3—181no—
505C2×D3—181no—
506C2×D3—181no—
507C2×D3—181no—
508C2×D3—361no—
509C2×D3—361no—
510C3×C4—22no—
511C3×C4—61no—
512C3×C4—61no—
513C3×C4—62no—
514C3×C4—181no—
515C3×C4D4×D912yes—
516C3×C4D3×#6712yes—
517Dic3—22no—
518Dic3—21no—
519Dic3—21no—
520Dic3—61no—
521Dic3—182no—
522Dic3—181no—
523Dic3—181no—
524Dic3—182no—
525Dic3—361no—
526Dic3—362no—
527C32#567611yescharacteristic
528C9—21no—
529C9#580611yescharacteristic, complemented
530C23—91no—
531C23—92no—
532C23—184no—
533C23—184no—
534C23—271no—
535C23—362no—
536C23C2×D3×D912yes—
537Q8—62no—
538D4—22no—
539D4—94no—
540D4—184no—
541D4—544no—
542D4—542no—
543D4—1082no—
544C2×C4—31no—
545C2×C4—31no—
546C2×C4—32no—
547C2×C4—91no—
548C2×C4—182no—
549C2×C4—272no—
550C2×C4—271no—
551C2×C4—544no—
552C2×C4—542no—
553C2×C4—1082no—
554C2×C4C2×D3×D911yescharacteristic
555C2×C3—21no—
556C2×C3—21no—
557C2×C3—21no—
558C2×C3—42no—
559C2×C3—42no—
560C2×C3—82no—
561C2×C3—182no—
562C2×C3—184no—
563C2×C3C2×D4×D911yescharacteristic
564C2×C3#41×D911yescharacteristic
565C2×C3#6371511yescharacteristic
566C2×C3D3×#23711yescharacteristic
567C2×C3#6431411yescharacteristic
568C2×C3C2×D3×#6711yescharacteristic
569D3—62no—
570D3—64no—
571D3—362no—
572D3—361no—
573D3—361no—
574C22—22no—
575C22—22no—
576C22—22no—
577C22—92no—
578C22—94no—
579C22—182no—
580C22—182no—
581C22—181no—
582C22—181no—
583C22—368no—
584C22—541no—
585C22—541no—
586C22—541no—
587C22C22×D3×D911yescharacteristic
588C4—61no—
589C4—61no—
590C4—62no—
591C4—181no—
592C4—542no—
593C4—541no—
594C4D9×#6712yes—
595C3—21no—
596C3#1058977811yescharacteristic, complemented
597C3#1059522911yescharacteristic
598C2—42no—
599C2—182no—
600C2—184no—
601C2—1081no—
602C2D9×#23711yescharacteristic
603C2#1172250111yescharacteristic
604C2C2×D9×#6711yescharacteristic
subgroupG/H
centerC22C22×D3×D9
commutator, cosocleC22×C3×C9C24
FrattiniC22×C3C22×D32
FittingC2×C3×D4×C9C22
socleC22×C32C23×D3
hypercenterC2×D4D3×D9
hypocenterC3×C9#546
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×C3×D4×D9C2semidirect
2. C2×#88634C2semidirect
3. #11721838C2semidirect
4. C9×#936C2semidirect
5. #11721276C2semidirect
6. #11721271C2semidirect
7. #11721269C2semidirect
8. D9×#217C2semidirect
9. C4×#6192C2semidirect
10. Dic3×D36C2semidirect
11. #11720944C2semidirect
16. C2×#6233C22semidirect
17. C2×C4×C3⋊D9C22semidirect
22. C2×C9×Dic6C22semidirect
24. C2×#6192C22semidirect
25. C2×Dic3×D9C22semidirect
26. C2×Dic3×D9C22semidirect
28. #88551C22semidirect
29. C4⋊(C3⋊Dic9)C22semidirect
30. C9×#217C22semidirect
31. C9×#217C22semidirect
32. C4×C9×Dic3C22semidirect
33. #88445C22semidirect
36. (C2×D9)⋊Dic3C22semidirect
37. Dic3⋊Dic9C22semidirect
38. #88442C22semidirect
51. C2×D4×D9D3semidirect
80. C3×D36D4semidirect
157. C2×C3×D9#41semidirect
177. #936D9semidirect
240. D36#67semidirect
284. C2×Dic6C2×D9semidirect
294. #217C2×D9semidirect
295. #217C2×D9semidirect
308. C4×Dic3C2×D9semidirect
349. C2×D9#237semidirect
372. C3×C9#546semidirect
529. C9#5806semidirect
596. C3#10589778semidirect

Maximal subgroups

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

subgroupindex
C2×C3×D4×D92
C2×#886342
#117218382
C9×#9362
#117212762
#117212712
#117212692
D9×#2172
C4×#61922
Dic3×D362
#117209442
#105952293
#105897783