Finite Group Databasebeta

← #13150550

Subgroups — #13150550

Order 1728.

1096 conjugacy classes of subgroups.

The subgroup lattice has 618 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#3697C211yesmaximal, characteristic, complemented
2C2×#88646C211yesmaximal, characteristic, complemented
3C4×#6233C211yesmaximal, characteristic, complemented
4C9×#930C211yesmaximal, characteristic, complemented
5#11721292C212yesmaximal, complemented
6#11721270C212yesmaximal, complemented
7D9×(C2×D3)⋊C4C212yesmaximal, complemented
8#11721031C212yesmaximal, complemented
9#11720998C211yesmaximal, characteristic, complemented
10#11720967C211yesmaximal, characteristic, complemented
11#11720936C211yesmaximal, characteristic, complemented
12#10595070—31nomaximal
13#10589714—31nomaximal
14C22×D3×D9C2212yes—
15C22×C3×C4×C9C2211yescharacteristic, complemented
16C2×#6233C2211yescharacteristic
17C2×C4×C3⋊D9C2211yescharacteristic
18C2×C9×#67C2212yescomplemented
19C2×C3×#611C2212yescomplemented
20C2×C9×D12C2211yescharacteristic
21C3×#3697—24no—
22C2×C3×D36C2211yescharacteristic
23C2×C3×C4×D9C2212yescomplemented
24C2×C3×Dic18C2211yescharacteristic, complemented
25C2×#6194C2212yes—
26C2×Dic3×D9C2212yescomplemented
27#88646—24no—
28#88551C2211yescharacteristic, complemented
29#88536C2211yescharacteristic
30#88534C2211yescharacteristic, complemented
31C4×C3⋊Dic9C2211yescharacteristic, complemented
32C9×(C2×D3)⋊C4C2212yescomplemented
33C9×C4⋊Dic3C2211yescharacteristic, complemented
34(C2×D3)⋊Dic9C2212yescomplemented
35(C2×D3)⋊Dic9C2212yescomplemented
36#88445C2212yes—
37(C2×D9)⋊Dic3C2212yescomplemented
38Dic3⋊Dic9C2212yescomplemented
39C2×C3×#235—31no—
40C2×#3802—31no—
41C4×#638—31no—
42C3×#930D311yescharacteristic
43#63983—32no—
44#63970—32no—
45D3×(C2×D3)⋊C4—32no—
46#63905—32no—
47#63898—31no—
48#63871—31no—
49#63863—31no—
50C2×D4×D9—31no—
51C2×#3697D311yescharacteristic, complemented
52C9×#121—31no—
53#63500—32no—
54#63499—32no—
55#63498—31no—
56C4×#611—31no—
57C4×#611—61no—
58#63459—31no—
59#63455—31no—
60#63445—32no—
61C22⋊C4×D9—32no—
62C23×C3×C9C2311yescharacteristic
63C22×C3⋊D9C2311yescharacteristic
64C22×D3×C9C2312yes—
65C22×C3×D9C2312yes—
66C2×D3×D9—24no—
67C2×D3×D9—24no—
68C2×D3×D9—42no—
69C2×C3×C4×C9—22no—
70C2×C3×C4×C9C2311yescharacteristic
71C2×C3×C4×C9C2311yescharacteristic
72#6233—22no—
73C2×C3⋊Dic9C2311yescharacteristic
74C2×C3⋊Dic9C2311yescharacteristic
75C2×C3⋊Dic9C2311yescharacteristic
76C4×C3⋊D9—22no—
77C9×#67—42no—
78C3×#611—24no—
79C2×C9×Dic3C2312yes—
80C2×C3×Dic9C2312yes—
81C9×D12—22no—
82C3×D36D414yescomplemented
83C3×C4×D9—24no—
84C3×Dic18—22no—
85#6194—24no—
86Dic3×D9—42no—
87#5829—91no—
88#5612—91no—
89C22×D32—32no—
90C22×C32×C4C2×D311yescharacteristic
91C2×#638—31no—
92C2×C4×C3⋊D3—31no—
93C2×C3×#67—32no—
94C2×C3×#67C2×D312yes—
95C3×#235—64no—
96C2×C3×D12—31no—
97C2×C3×D12C2×D311yescharacteristic
98C2×C3×C4×D3—32no—
99C2×C3×Dic6—31no—
100C2×#625—32no—
101C2×D3×Dic3—32no—
102#3802—64no—
103C23×D9—32no—
104#3757—31no—
105#3752—31no—
106#3750—31no—
107C4×C3⋊Dic3—31no—
108C3×(C2×D3)⋊C4C2×D312yes—
109C3×C4⋊Dic3C2×D311yescharacteristic
110Dic3⋊Dic3—32no—
111#3722—32no—
112(C2×D3)⋊Dic3—32no—
113(C2×D3)⋊Dic3—32no—
114(C2×D3)⋊Dic3—32no—
115C2×D4×C9—32no—
116C2×D4×C9—31no—
117C22×C4×C9—21no—
118C22×C4×C9C2×D311yescharacteristic, complemented
119C2×#611—31no—
120C2×#611—32no—
121C2×#611—61no—
122C2×#611C2×D312yescomplemented
123D4×D9—64no—
124#3697—24no—
125C2×D36C2×D311yescharacteristic
126C2×C4×D9—31no—
127C2×C4×D9—32no—
128C2×C4×D9—61no—
129C2×C4×D9C2×D312yescomplemented
130C2×Dic18C2×D311yescharacteristic, complemented
131C9×C4⋊C4—31no—
132C9×C22⋊C4—32no—
133#3676—31no—
134#3676—32no—
135#3676—32no—
136#3676—61no—
137(C2×D9)⋊C4—32no—
138(C2×D9)⋊C4—31no—
139(C2×D9)⋊C4—32no—
140(C2×D9)⋊C4—61no—
141C4⋊Dic9—32no—
142Dic9⋊C4—31no—
143Dic9⋊C4—61no—
144C4×Dic9—31no—
145C4×Dic9—61no—
146C22×C3×C9—21no—
147C22×C3×C9—21no—
148C22×C3×C9—21no—
149C22×C3×C9C2411yescharacteristic
150C2×C3⋊D9—21no—
151C2×C3⋊D9—21no—
152C2×C3⋊D9#4112yescomplemented
153C2×D3×C9—22no—
154C2×D3×C9—22no—
155C2×D3×C9—22no—
156C2×C3×D9—22no—
157C2×C3×D9—22no—
158C2×C3×D9C2×D414yes—
159D3×D9—48no—
160C3×C4×C9—21no—
161C3×C4×C9C2×D412yes—
162C3⋊Dic9—21no—
163C3⋊Dic9—21no—
164C3⋊Dic9—21no—
165C9×Dic3—22no—
166C3×Dic9—22no—
167C2×C3×#41—91no—
168C2×D3×D4—91no—
169C2×D3×D4—91no—
170C2×#235—31no—
171C3×#121—31no—
172#947—91no—
173#946—91no—
174#939—92no—
175#938—92no—
176#937—92no—
177#937—92no—
178#936—91no—
179D4×Dic3—91no—
180#930D911yescharacteristic, complemented
181#929—92no—
182C4×#67—91no—
183C4×#67—91no—
184C4×#67—181no—
185C2×(C2×D3)⋊C4—92no—
186#897—91no—
187#893—91no—
188#883—92no—
189D3×C22⋊C4—92no—
190C23×C32C22×D311yescharacteristic
191C22×C3⋊D3—31no—
192C22×C3×D3—32no—
193C22×C3×D3C22×D312yes—
194C2×D32—64no—
195C2×D32—64no—
196C2×D32—122no—
197C2×C32×C4—22no—
198C2×C32×C4C22×D311yescharacteristic
199C2×C32×C4C22×D311yescharacteristic
200#638—62no—
201C2×C3⋊Dic3—31no—
202C2×C3⋊Dic3—31no—
203C2×C3⋊Dic3—31no—
204C4×C3⋊D3—62no—
205C3×#67—42no—
206C3×#67—64no—
207C2×C3×Dic3—32no—
208C2×C3×Dic3C22×D312yes—
209C3×D12—22no—
210C3×D12—34no—
211C3×C4×D3—64no—
212C3×Dic6—62no—
213#625—64no—
214D3×Dic3—122no—
215C23×C9—21no—
216C23×C9—32no—
217C23×C9C22×D311yescharacteristic
218C22×D9—31no—
219C22×D9—61no—
220C22×D9—64no—
221C22×D9—64no—
222C22×D9—122no—
223C22×D9C22×D312yes—
224D4×C9—62no—
225D4×C9—122no—
226C2×C4×C9—22no—
227C2×C4×C9—21no—
228C2×C4×C9—21no—
229C2×C4×C9—32no—
230C2×C4×C9—42no—
231C2×C4×C9C22×D311yescharacteristic
232C2×C4×C9C22×D311yescharacteristic
233#611—24no—
234#611—62no—
235#611—64no—
236#611—64no—
237C2×Dic9—31no—
238C2×Dic9—31no—
239C2×Dic9—31no—
240C2×Dic9—61no—
241C2×Dic9—61no—
242C2×Dic9—61no—
243C2×Dic9C22×D312yes—
244D36#6714yescomplemented
245C4×D9—24no—
246C4×D9—62no—
247C4×D9—122no—
248C4×D9—122no—
249Dic18—22no—
250#546—271noSylow
251C2×C3×C9—41no—
252C2×C3×C9C22×D411yescharacteristic
253C2×C3×C9C2×#4111yescharacteristic
254C2×C3×C9H(5,𝔽2)11yescharacteristic
255C3⋊D9—22no—
256D3×C9—42no—
257C3×D9—24no—
258C23×D3—92no—
259C23×D3—92no—
260C3×#41—184no—
261C2×C3×Q8—91no—
262C2×C3×D4—32no—
263C2×C3×D4—31no—
264C2×C3×D4—92no—
265C2×C3×D4—91no—
266C22×C3×C4—21no—
267C22×C3×C4—92no—
268C22×C3×C4C2×D911yescharacteristic, complemented
269C22×C3×C4D3211yescharacteristic
270C2×#67—32no—
271C2×#67—92no—
272C2×#67—91no—
273C2×#67—91no—
274C2×#67—92no—
275C2×#67—181no—
276C2×#67C2×D912yescomplemented
277C22×Dic3—92no—
278D3×D4—184no—
279D3×D4—184no—
280#235—64no—
281C2×D12—31no—
282C2×D12C2×D911yescharacteristic
283C2×C4×D3—32no—
284C2×C4×D3—91no—
285C2×C4×D3—91no—
286C2×C4×D3—92no—
287C2×C4×D3—181no—
288C2×Dic6—31no—
289C3×C4⋊C4—31no—
290C3×C22⋊C4—32no—
291#217—92no—
292#217—92no—
293#217—91no—
294#217—91no—
295#217—92no—
296#217—181no—
297(C2×D3)⋊C4—92no—
298(C2×D3)⋊C4—92no—
299(C2×D3)⋊C4—92no—
300(C2×D3)⋊C4—92no—
301(C2×D3)⋊C4—91no—
302(C2×D3)⋊C4—91no—
303(C2×D3)⋊C4—92no—
304(C2×D3)⋊C4—181no—
305(C2×D3)⋊C4C2×D912yescomplemented
306C4⋊Dic3—92no—
307C4⋊Dic3C2×D911yescharacteristic, complemented
308Dic3⋊C4—91no—
309Dic3⋊C4—91no—
310Dic3⋊C4—92no—
311Dic3⋊C4—181no—
312C4×Dic3—91no—
313C4×Dic3—91no—
314C4×Dic3—181no—
315C22×C32—21no—
316C22×C32—21no—
317C22×C32—21no—
318C22×C32C23×D311yescharacteristic
319C2×C3⋊D3—32no—
320C2×C3⋊D3—61no—
321C2×C3⋊D3—61no—
322C2×C3×D3—22no—
323C2×C3×D3—22no—
324C2×C3×D3—22no—
325C2×C3×D3—34no—
326C2×C3×D3—62no—
327C2×C3×D3—62no—
328D32—128no—
329C32×C4—21no—
330C32×C4D3×D412yes—
331C3⋊Dic3—61no—
332C3⋊Dic3—61no—
333C3⋊Dic3—61no—
334C3×Dic3—22no—
335C3×Dic3—62no—
336C22×C9—22no—
337C22×C9—21no—
338C22×C9—21no—
339C22×C9—21no—
340C22×C9—21no—
341C22×C9—41no—
342C22×C9—41no—
343C22×C9—62no—
344C22×C9—62no—
345C22×C9—62no—
346C22×C9C23×D311yescharacteristic
347C2×D9—22no—
348C2×D9—22no—
349C2×D9—32no—
350C2×D9—62no—
351C2×D9—61no—
352C2×D9—61no—
353C2×D9—121no—
354C2×D9—121no—
355C2×D9—128no—
356C2×D9C2×#6714yes—
357C4×C9—21no—
358C4×C9—22no—
359C4×C9—41no—
360C4×C9—62no—
361C4×C9C2×#6712yes—
362Dic9—22no—
363Dic9—62no—
364Dic9—61no—
365Dic9—61no—
366Dic9—61no—
367Dic9—121no—
368Dic9—121no—
369C2×#41—91no—
370C22×D4—271no—
371#124—271no—
372#123—272no—
373C22⋊Q8—271no—
374#121—31no—
375#121—272no—
376#120—272no—
377C4×D4—271no—
378C4×D4—271no—
379C2×C22⋊C4—272no—
380C3×C9#54611yescharacteristic, complemented, Sylow
381C23×C3—21no—
382C23×C3—32no—
383C23×C3—92no—
384C23×C3C22×D911yescharacteristic
385C23×C3C2×D3211yescharacteristic
386C22×D3—32no—
387C22×D3—91no—
388C22×D3—91no—
389C22×D3—181no—
390C22×D3—184no—
391C22×D3—184no—
392C22×D3—184no—
393C22×D3—184no—
394C22×D3—362no—
395C22×D3—362no—
396C22×D3C22×D912yes—
397C3×Q8—182no—
398C3×D4—62no—
399C3×D4—94no—
400C3×D4—122no—
401C3×D4—184no—
402C2×C3×C4—22no—
403C2×C3×C4—22no—
404C2×C3×C4—21no—
405C2×C3×C4—21no—
406C2×C3×C4—32no—
407C2×C3×C4—42no—
408C2×C3×C4—92no—
409C2×C3×C4—184no—
410C2×C3×C4C22×D911yescharacteristic
411C2×C3×C4C22×D911yescharacteristic
412C2×C3×C4C2×D3211yescharacteristic
413C2×C3×C4C2×D3211yescharacteristic
414#67—42no—
415#67—64no—
416#67—184no—
417#67—184no—
418#67—182no—
419#67—182no—
420#67—184no—
421C2×Dic3—32no—
422C2×Dic3—91no—
423C2×Dic3—91no—
424C2×Dic3—91no—
425C2×Dic3—91no—
426C2×Dic3—91no—
427C2×Dic3—91no—
428C2×Dic3—181no—
429C2×Dic3—181no—
430C2×Dic3—181no—
431C2×Dic3—362no—
432C2×Dic3C22×D912yes—
433D12—22no—
434D12—34no—
435C4×D3—64no—
436C4×D3—182no—
437C4×D3—182no—
438C4×D3—362no—
439C4×D3—362no—
440Dic6—62no—
441C2×C32—41no—
442C2×C32C2×D3×D411yescharacteristic
443C2×C32#100411yescharacteristic
444C2×C32D3×#4111yescharacteristic
445C3⋊D3—62no—
446C3×D3—42no—
447C3×D3—64no—
448C2×C9—21no—
449C2×C9—21no—
450C2×C9—21no—
451C2×C9—42no—
452C2×C9—41no—
453C2×C9—122no—
454C2×C9D3×#4111yescharacteristic
455C2×C9#100911yescharacteristic
456C2×C9C22×#6711yescharacteristic
457D9—24no—
458D9—62no—
459D9—122no—
460C24—272no—
461#41—184no—
462C2×Q8—91no—
463C2×D4—32no—
464C2×D4—31no—
465C2×D4—92no—
466C2×D4—91no—
467C2×D4—272no—
468C2×D4—271no—
469C2×D4—544no—
470C22×C4—92no—
471C22×C4—272no—
472C22×C4—271no—
473C22×C4D3×D911yescharacteristic, complemented
474C4⋊C4—31no—
475C4⋊C4—272no—
476C4⋊C4—271no—
477C22⋊C4—32no—
478C22⋊C4—272no—
479C22⋊C4—272no—
480C22⋊C4—272no—
481C22⋊C4—272no—
482C22⋊C4—271no—
483C22⋊C4—271no—
484C42—271no—
485C22×C3—22no—
486C22×C3—21no—
487C22×C3—21no—
488C22×C3—21no—
489C22×C3—21no—
490C22×C3—21no—
491C22×C3—21no—
492C22×C3—21no—
493C22×C3—41no—
494C22×C3—41no—
495C22×C3—62no—
496C22×C3—62no—
497C22×C3—62no—
498C22×C3—94no—
499C22×C3—182no—
500C22×C3—182no—
501C22×C3C23×D911yescharacteristic
502C22×C3C22×D3211yescharacteristic
503C2×D3—22no—
504C2×D3—22no—
505C2×D3—22no—
506C2×D3—34no—
507C2×D3—62no—
508C2×D3—62no—
509C2×D3—92no—
510C2×D3—92no—
511C2×D3—182no—
512C2×D3—181no—
513C2×D3—181no—
514C2×D3—181no—
515C2×D3—181no—
516C2×D3—361no—
517C2×D3—361no—
518C2×D3—368no—
519C2×D3—368no—
520C3×C4—21no—
521C3×C4—21no—
522C3×C4—22no—
523C3×C4—41no—
524C3×C4—62no—
525C3×C4—182no—
526C3×C4D4×D912yes—
527C3×C4D3×#6712yes—
528Dic3—22no—
529Dic3—62no—
530Dic3—182no—
531Dic3—181no—
532Dic3—181no—
533Dic3—181no—
534Dic3—181no—
535Dic3—181no—
536Dic3—181no—
537Dic3—361no—
538Dic3—361no—
539C32#561211yescharacteristic
540C9—21no—
541C9#582911yescharacteristic, complemented
542C23—32no—
543C23—92no—
544C23—271no—
545C23—544no—
546C23—544no—
547C23—1082no—
548C23C2×D3×D911yescharacteristic
549Q8—182no—
550D4—62no—
551D4—94no—
552D4—122no—
553D4—184no—
554D4—542no—
555D4—544no—
556C2×C4—22no—
557C2×C4—32no—
558C2×C4—92no—
559C2×C4—184no—
560C2×C4—271no—
561C2×C4—271no—
562C2×C4—271no—
563C2×C4—542no—
564C2×C4—1082no—
565C2×C4C2×D3×D911yescharacteristic
566C2×C4C2×D3×D911yescharacteristic
567C2×C3—21no—
568C2×C3—21no—
569C2×C3—21no—
570C2×C3—42no—
571C2×C3—41no—
572C2×C3—41no—
573C2×C3—122no—
574C2×C3—184no—
575C2×C3C2×D4×D911yescharacteristic
576C2×C3#6371011yescharacteristic
577C2×C3#41×D911yescharacteristic
578C2×C3#6430611yescharacteristic
579C2×C3#6430711yescharacteristic
580C2×C3C2×D3×#6711yescharacteristic
581D3—42no—
582D3—64no—
583D3—182no—
584D3—182no—
585D3—362no—
586C22—21no—
587C22—21no—
588C22—21no—
589C22—62no—
590C22—62no—
591C22—62no—
592C22—94no—
593C22—182no—
594C22—182no—
595C22—272no—
596C22—541no—
597C22—541no—
598C22—1088no—
599C22C22×D3×D911yescharacteristic
600C4—21no—
601C4—62no—
602C4—182no—
603C4—541no—
604C4—541no—
605C4—541no—
606C4D9×#6712yes—
607C3—21no—
608C3#1058971411yescharacteristic, complemented
609C3#1059507011yescharacteristic
610C2—41no—
611C2—122no—
612C2—184no—
613C2—542no—
614C2#1172247511yescharacteristic
615C2#1172247611yescharacteristic
616C2C2×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#546
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×C3×#3697C2semidirect
2. C2×#88646C2semidirect
3. C4×#6233C2semidirect
4. C9×#930C2semidirect
5. #11721292C2semidirect
6. #11721270C2semidirect
7. D9×(C2×D3)⋊C4C2semidirect
8. #11721031C2semidirect
9. #11720998C2semidirect
10. #11720967C2semidirect
11. #11720936C2semidirect
15. C22×C3×C4×C9C22semidirect
18. C2×C9×#67C22semidirect
19. C2×C3×#611C22semidirect
23. C2×C3×C4×D9C22semidirect
24. C2×C3×Dic18C22semidirect
26. C2×Dic3×D9C22semidirect
28. #88551C22semidirect
30. #88534C22semidirect
31. C4×C3⋊Dic9C22semidirect
32. C9×(C2×D3)⋊C4C22semidirect
33. C9×C4⋊Dic3C22semidirect
34. (C2×D3)⋊Dic9C22semidirect
35. (C2×D3)⋊Dic9C22semidirect
37. (C2×D9)⋊Dic3C22semidirect
38. Dic3⋊Dic9C22semidirect
51. C2×#3697D3semidirect
82. C3×D36D4semidirect
118. C22×C4×C9C2×D3semidirect
122. C2×#611C2×D3semidirect
129. C2×C4×D9C2×D3semidirect
130. C2×Dic18C2×D3semidirect
152. C2×C3⋊D9#41semidirect
180. #930D9semidirect
244. D36#67semidirect
268. C22×C3×C4C2×D9semidirect
276. C2×#67C2×D9semidirect
305. (C2×D3)⋊C4C2×D9semidirect
307. C4⋊Dic3C2×D9semidirect
380. C3×C9#546semidirect
473. C22×C4D3×D9semidirect
541. C9#5829semidirect
608. C3#10589714semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#36972
C2×#886462
C4×#62332
C9×#9302
#117212922
#117212702
D9×(C2×D3)⋊C42
#117210312
#117209982
#117209672
#117209362
#105950703
#105897143