Finite Group Databasebeta

← #13147720

Subgroups — #13147720

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×#88646C211yesmaximal, characteristic, complemented
2C2×#88639C211yesmaximal, characteristic, complemented
3C4×C4⋊(C3⋊D9)C211yesmaximal, characteristic, complemented
4C4×C9×D12C211yesmaximal, characteristic, complemented
5C3×#63437C211yesmaximal, characteristic, complemented
6#11721051C212yesmaximal, complemented
7D3×(C2×D9)⋊C4C212yesmaximal, complemented
8#11721034C212yesmaximal, complemented
9#11721021C212yesmaximal, complemented
10#11721007C211yesmaximal, characteristic, complemented
11#11720936C211yesmaximal, characteristic, complemented
12#10594005—31nomaximal
13#10589656—31nomaximal
14C22×D3×D9C2212yes—
15C2×C4⋊(C3⋊D9)C2211yescharacteristic
16C2×C4×C3⋊D9C2211yescharacteristic
17C2×C4×C3⋊D9C2211yescharacteristic, complemented
18C2×C9×D12C2211yescharacteristic
19C2×C4×D3×C9C2212yescomplemented
20C2×C3×D36C2211yescharacteristic
21C2×C3×Dic18C2211yescharacteristic, complemented
22C2×#6195C2212yescomplemented
23C2×#6194C2212yes—
24C2×D3×Dic9C2212yescomplemented
25C2×#6192C2212yescomplemented
26#88646—24no—
27#88639—24no—
28C3×C42×C9C2211yescharacteristic, complemented
29#88536C2211yescharacteristic
30#88536C2211yescharacteristic, complemented
31C4⋊(C3⋊Dic9)C2211yescharacteristic, complemented
32C9×(C2×D3)⋊C4C2212yescomplemented
33C3×(C2×D9)⋊C4C2212yescomplemented
34C3×(C2×D9)⋊C4C2212yescomplemented
35C9×C4⋊Dic3C2211yescharacteristic, complemented
36#88445C2212yes—
37(C2×D9)⋊Dic3C2212yescomplemented
38Dic3⋊Dic9C2212yescomplemented
39C2×#3802—31no—
40C2×#3796—31no—
41C4×C4⋊(C3⋊D3)—31no—
42C3×#875—31no—
43C3×C4×D12D311yescharacteristic
44#63923—32no—
45D3×(C2×D3)⋊C4—32no—
46#63916—31no—
47#63906—32no—
48#63891—32no—
49#63871—31no—
50C2×#3699—31no—
51C2×D4×D9—31no—
52C4×D4×C9—31no—
53#63492—32no—
54C2×(C2×D9)⋊C4—32no—
55#63459—31no—
56#63457—31no—
57#63448—32no—
58#63444—32no—
59#63437D311yescharacteristic, complemented
60C4×D36—31no—
61C4×D36—61no—
62C22×C3⋊D9C2311yescharacteristic
63C22×C3⋊D9C2311yescharacteristic
64C22×D3×C9C2312yes—
65C22×C3×D9C2312yes—
66C2×D3×D9—24no—
67C2×D3×D9—24no—
68C2×D3×D9—42no—
69C2×C3×C4×C9C2311yescharacteristic
70C2×C3×C4×C9C2311yescharacteristic
71C2×C3×C4×C9C2311yescharacteristic
72C2×C3⋊Dic9C2311yescharacteristic
73C2×C3⋊Dic9C2311yescharacteristic
74C4⋊(C3⋊D9)—22no—
75C4×C3⋊D9—22no—
76C4×C3⋊D9—22no—
77C2×C9×Dic3C2312yes—
78C2×C3×Dic9C2312yes—
79C9×D12D414yescomplemented
80C4×D3×C9—42no—
81C3×D36—22no—
82C3×Dic18—22no—
83#6195—24no—
84#6194—24no—
85D3×Dic9—24no—
86#6192—42no—
87#5676—91no—
88#5554—91no—
89C22×D32—32no—
90C2×C4⋊(C3⋊D3)—31no—
91C2×C4×C3⋊D3—31no—
92C2×C4×C3⋊D3—31no—
93C2×C3×D12—31no—
94C2×C3×D12C2×D311yescharacteristic
95C2×C3×C4×D3C2×D312yes—
96C2×C3×Dic6—31no—
97C2×#626—32no—
98C2×#626—32no—
99C2×#625—32no—
100C2×D3×Dic3—32no—
101#3802—64no—
102#3796—64no—
103C23×D9—32no—
104C32×C42C2×D311yescharacteristic
105#3752—31no—
106#3752—31no—
107C4⋊(C3⋊Dic3)—31no—
108C3×(C2×D3)⋊C4—32no—
109C3×(C2×D3)⋊C4—32no—
110C3×(C2×D3)⋊C4C2×D312yes—
111C3×C4⋊Dic3C2×D311yescharacteristic
112Dic3⋊Dic3—32no—
113#3722—32no—
114(C2×D3)⋊Dic3—32no—
115C2×D4×C9—31no—
116C22×C4×C9—32no—
117C2×#611—32no—
118C2×#611—32no—
119C22×Dic9—32no—
120#3699—64no—
121D4×D9—64no—
122C2×D36—31no—
123C2×D36—32no—
124C2×D36—61no—
125C2×D36C2×D311yescharacteristic
126C2×C4×D9—31no—
127C2×C4×D9—31no—
128C2×C4×D9—61no—
129C2×C4×D9—61no—
130C2×Dic18C2×D311yescharacteristic, complemented
131C9×C4⋊C4—31no—
132C9×C22⋊C4—32no—
133C42×C9—21no—
134C42×C9C2×D311yescharacteristic, complemented
135(C2×D9)⋊C4—32no—
136(C2×D9)⋊C4—31no—
137(C2×D9)⋊C4—31no—
138(C2×D9)⋊C4—32no—
139(C2×D9)⋊C4—61no—
140(C2×D9)⋊C4—61no—
141(C2×D9)⋊C4C2×D312yescomplemented
142(C2×D9)⋊C4C2×D312yescomplemented
143C4⋊Dic9—31no—
144C4⋊Dic9—32no—
145C4⋊Dic9—61no—
146C22×C3×C9C2411yescharacteristic
147C2×C3⋊D9—21no—
148C2×C3⋊D9—21no—
149C2×C3⋊D9—21no—
150C2×C3⋊D9—21no—
151C2×C3⋊D9—21no—
152C2×C3⋊D9#4112yescomplemented
153C2×D3×C9—22no—
154C2×D3×C9—22no—
155C2×D3×C9C2×D414yes—
156C2×C3×D9—22no—
157C2×C3×D9—22no—
158C2×C3×D9—22no—
159D3×D9—48no—
160C3×C4×C9—21no—
161C3×C4×C9—21no—
162C3×C4×C9C2×D412yes—
163C3⋊Dic9—21no—
164C3⋊Dic9—21no—
165C9×Dic3—22no—
166C3×Dic9—22no—
167C2×#239—91no—
168C2×#237—91no—
169C2×D3×D4—91no—
170C2×D3×D4—91no—
171C3×#124—91no—
172C3×C4×D4—31no—
173#946—91no—
174#938—91no—
175#930—92no—
176C2×(C2×D3)⋊C4—92no—
177#897—91no—
178#895—91no—
179#886—92no—
180#884—92no—
181#883—92no—
182#882—92no—
183#882—92no—
184D3×C22⋊C4—92no—
185#875—31no—
186C4×D12—91no—
187C4×D12—91no—
188C4×D12—181no—
189C4×D12D911yescharacteristic, complemented
190C22×C3⋊D3—31no—
191C22×C3⋊D3—31no—
192C22×C3×D3—32no—
193C22×C3×D3C22×D312yes—
194C2×D32—64no—
195C2×D32—64no—
196C2×D32—122no—
197C2×C32×C4C22×D311yescharacteristic
198C2×C32×C4C22×D311yescharacteristic
199C2×C32×C4C22×D311yescharacteristic
200C2×C3⋊Dic3—31no—
201C2×C3⋊Dic3—31no—
202C4⋊(C3⋊D3)—62no—
203C4×C3⋊D3—62no—
204C4×C3⋊D3—62no—
205C2×C3×Dic3—32no—
206C2×C3×Dic3C22×D312yes—
207C3×D12—62no—
208C3×D12D1214yes—
209C3×C4×D3—42no—
210C3×Dic6—62no—
211#626—64no—
212#626—122no—
213#625—64no—
214D3×Dic3—64no—
215C23×C9—32no—
216C22×D9—31no—
217C22×D9—31no—
218C22×D9—61no—
219C22×D9—61no—
220C22×D9—64no—
221C22×D9—64no—
222C22×D9—122no—
223C22×D9C22×D312yes—
224D4×C9—34no—
225C2×C4×C9—21no—
226C2×C4×C9—21no—
227C2×C4×C9—21no—
228C2×C4×C9—32no—
229C2×C4×C9—122no—
230C2×C4×C9C22×D311yescharacteristic
231C2×C4×C9C22×D311yescharacteristic
232C2×C4×C9C22×D311yescharacteristic
233#611—64no—
234#611—64no—
235C2×Dic9—31no—
236C2×Dic9—31no—
237C2×Dic9—61no—
238C2×Dic9—61no—
239C2×Dic9—64no—
240C2×Dic9C22×D312yes—
241D36—22no—
242D36—62no—
243D36—64no—
244D36—122no—
245C4×D9—62no—
246C4×D9—62no—
247C4×D9—122no—
248C4×D9—122no—
249Dic18—22no—
250#546—271noSylow
251C2×C3×C9C22×D411yescharacteristic
252C2×C3×C9C2×#4111yescharacteristic
253C2×C3×C9H(5,𝔽2)11yescharacteristic
254C3⋊D9—22no—
255C3⋊D9—41no—
256D3×C9—24no—
257C3×D9—42no—
258C23×D3—92no—
259C23×D3—92no—
260C2×C3×Q8—91no—
261C2×C3×D4—31no—
262C2×C3×D4—91no—
263C22×C3×C4—32no—
264C2×#67—92no—
265C2×#67—92no—
266C2×#67—92no—
267C2×#67—92no—
268C22×Dic3—92no—
269#239—184no—
270#237—184no—
271D3×D4—184no—
272D3×D4—184no—
273C2×D12—31no—
274C2×D12—92no—
275C2×D12—91no—
276C2×D12—91no—
277C2×D12—92no—
278C2×D12—181no—
279C2×D12C2×D911yescharacteristic
280C2×C4×D3—92no—
281C2×C4×D3—91no—
282C2×C4×D3—91no—
283C2×C4×D3—91no—
284C2×C4×D3—91no—
285C2×C4×D3—181no—
286C2×C4×D3—181no—
287C2×C4×D3C2×D912yescomplemented
288C2×Dic6—31no—
289C3×C4⋊C4—31no—
290C3×C22⋊C4—32no—
291C3×C22⋊C4—92no—
292C3×C22⋊C4—92no—
293C3×C42—21no—
294C3×C42C2×D911yescharacteristic, complemented
295C3×C42D3211yescharacteristic
296#217—92no—
297(C2×D3)⋊C4—32no—
298(C2×D3)⋊C4—32no—
299(C2×D3)⋊C4—92no—
300(C2×D3)⋊C4—92no—
301(C2×D3)⋊C4—91no—
302(C2×D3)⋊C4—91no—
303(C2×D3)⋊C4—91no—
304(C2×D3)⋊C4—91no—
305(C2×D3)⋊C4—92no—
306(C2×D3)⋊C4—181no—
307(C2×D3)⋊C4—181no—
308(C2×D3)⋊C4C2×D912yescomplemented
309C4⋊Dic3—92no—
310C4⋊Dic3—91no—
311C4⋊Dic3—91no—
312C4⋊Dic3—181no—
313C4⋊Dic3C2×D911yescharacteristic, complemented
314Dic3⋊C4—92no—
315C22×C32C23×D311yescharacteristic
316C2×C3⋊D3—32no—
317C2×C3⋊D3—61no—
318C2×C3⋊D3—61no—
319C2×C3⋊D3—61no—
320C2×C3⋊D3—61no—
321C2×C3⋊D3—61no—
322C2×C3×D3—22no—
323C2×C3×D3—22no—
324C2×C3×D3—62no—
325C2×C3×D3—62no—
326C2×C3×D3—62no—
327C2×C3×D3C2×D1214yes—
328D32—128no—
329C32×C4—21no—
330C32×C4—21no—
331C32×C4C2×D1212yes—
332C3⋊Dic3—61no—
333C3⋊Dic3—61no—
334C3×Dic3—22no—
335C3×Dic3—62no—
336C22×C9—21no—
337C22×C9—34no—
338C22×C9—62no—
339C22×C9—62no—
340C22×C9C23×D311yescharacteristic
341C2×D9—22no—
342C2×D9—22no—
343C2×D9—22no—
344C2×D9—32no—
345C2×D9—62no—
346C2×D9—61no—
347C2×D9—61no—
348C2×D9—62no—
349C2×D9—61no—
350C2×D9—61no—
351C2×D9—61no—
352C2×D9—121no—
353C2×D9—121no—
354C2×D9—121no—
355C2×D9—121no—
356C2×D9—128no—
357C4×C9—22no—
358C4×C9—21no—
359C4×C9—21no—
360C4×C9—22no—
361C4×C9—41no—
362C4×C9—62no—
363C4×C9D3×D412yes—
364Dic9—22no—
365Dic9—61no—
366Dic9—61no—
367Dic9—121no—
368Dic9—121no—
369C2×#41—271no—
370C22×D4—271no—
371#124—91no—
372#123—272no—
373C22⋊Q8—271no—
374#121—272no—
375#121—271no—
376#120—272no—
377C4×D4—31no—
378C4×D4—271no—
379C2×C22⋊C4—272no—
380C3×C9#54611yescharacteristic, complemented, Sylow
381C23×C3—32no—
382C23×C3—92no—
383C22×D3—32no—
384C22×D3—91no—
385C22×D3—91no—
386C22×D3—91no—
387C22×D3—91no—
388C22×D3—181no—
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—34no—
399C3×D4—182no—
400C2×C3×C4—21no—
401C2×C3×C4—21no—
402C2×C3×C4—21no—
403C2×C3×C4—32no—
404C2×C3×C4—92no—
405C2×C3×C4—122no—
406C2×C3×C4C22×D911yescharacteristic
407C2×C3×C4C22×D911yescharacteristic
408C2×C3×C4C22×D911yescharacteristic
409C2×C3×C4C2×D3211yescharacteristic
410C2×C3×C4C2×D3211yescharacteristic
411C2×C3×C4C2×D3211yescharacteristic
412#67—184no—
413#67—184no—
414#67—184no—
415#67—362no—
416C2×Dic3—32no—
417C2×Dic3—91no—
418C2×Dic3—91no—
419C2×Dic3—91no—
420C2×Dic3—91no—
421C2×Dic3—181no—
422C2×Dic3—181no—
423C2×Dic3—184no—
424C2×Dic3C22×D912yes—
425D12—62no—
426D12—182no—
427D12—182no—
428D12—184no—
429D12—184no—
430D12—362no—
431D12D3614yescomplemented
432C4×D3—42no—
433C4×D3—184no—
434C4×D3—182no—
435C4×D3—182no—
436C4×D3—182no—
437C4×D3—182no—
438C4×D3—362no—
439C4×D3—362no—
440Dic6—62no—
441C2×C32C22×D1211yescharacteristic
442C2×C32#100411yescharacteristic
443C2×C32D3×#4111yescharacteristic
444C3⋊D3—62no—
445C3⋊D3—121no—
446C3×D3—24no—
447C3×D3—122no—
448C2×C9—21no—
449C2×C9—21no—
450C2×C9—21no—
451C2×C9—64no—
452C2×C9C2×D3×D411yescharacteristic
453C2×C9D3×#4111yescharacteristic
454C2×C9#100911yescharacteristic
455D9—42no—
456D9—62no—
457D9—122no—
458D9—122no—
459D9—121no—
460C24—272no—
461#41—544no—
462C2×Q8—91no—
463C2×D4—31no—
464C2×D4—91no—
465C2×D4—272no—
466C2×D4—272no—
467C2×D4—272no—
468C2×D4—271no—
469C2×D4—544no—
470C22×C4—32no—
471C22×C4—272no—
472C22×C4—271no—
473C22×C4—271no—
474C4⋊C4—31no—
475C4⋊C4—272no—
476C4⋊C4—271no—
477C22⋊C4—32no—
478C22⋊C4—92no—
479C22⋊C4—92no—
480C22⋊C4—272no—
481C22⋊C4—272no—
482C22⋊C4—271no—
483C22⋊C4—271no—
484C42D3×D911yescharacteristic, complemented
485C22×C3—21no—
486C22×C3—34no—
487C22×C3—62no—
488C22×C3—62no—
489C22×C3—182no—
490C22×C3—182no—
491C22×C3—182no—
492C22×C3C23×D911yescharacteristic
493C22×C3C22×D3211yescharacteristic
494C2×D3—22no—
495C2×D3—22no—
496C2×D3—62no—
497C2×D3—62no—
498C2×D3—62no—
499C2×D3—92no—
500C2×D3—92no—
501C2×D3—182no—
502C2×D3—181no—
503C2×D3—181no—
504C2×D3—181no—
505C2×D3—181no—
506C2×D3—182no—
507C2×D3—181no—
508C2×D3—181no—
509C2×D3—181no—
510C2×D3—181no—
511C2×D3—181no—
512C2×D3—181no—
513C2×D3—361no—
514C2×D3—361no—
515C2×D3—361no—
516C2×D3—361no—
517C2×D3—368no—
518C2×D3—368no—
519C2×D3C2×D3614yes—
520C3×C4—22no—
521C3×C4—21no—
522C3×C4—21no—
523C3×C4—21no—
524C3×C4—21no—
525C3×C4—22no—
526C3×C4—41no—
527C3×C4—62no—
528C3×C4—182no—
529C3×C4C2×D3612yes—
530C3×C4D3×D1212yes—
531Dic3—22no—
532Dic3—62no—
533Dic3—181no—
534Dic3—181no—
535Dic3—181no—
536Dic3—181no—
537Dic3—361no—
538Dic3—361no—
539C32#555411yescharacteristic
540C9—21no—
541C9#567611yescharacteristic, complemented
542C23—32no—
543C23—92no—
544C23—271no—
545C23—271no—
546C23—544no—
547C23—544no—
548C23—1082no—
549Q8—182no—
550D4—34no—
551D4—182no—
552D4—542no—
553D4—544no—
554D4—544no—
555D4—1082no—
556C2×C4—32no—
557C2×C4—92no—
558C2×C4—122no—
559C2×C4—271no—
560C2×C4—271no—
561C2×C4—544no—
562C2×C4—542no—
563C2×C4—542no—
564C2×C4C2×D3×D911yescharacteristic
565C2×C4C2×D3×D911yescharacteristic
566C2×C4C2×D3×D911yescharacteristic
567C2×C3—21no—
568C2×C3—21no—
569C2×C3—21no—
570C2×C3—64no—
571C2×C3—362no—
572C2×C3C22×D3611yescharacteristic
573C2×C3#6371011yescharacteristic
574C2×C3#41×D911yescharacteristic
575C2×C3C2×D3×D1211yescharacteristic
576C2×C3#6430611yescharacteristic
577C2×C3#6430711yescharacteristic
578D3—24no—
579D3—122no—
580D3—182no—
581D3—182no—
582D3—362no—
583D3—362no—
584D3—361no—
585D3—361no—
586C22—34no—
587C22—62no—
588C22—62no—
589C22—182no—
590C22—182no—
591C22—182no—
592C22—272no—
593C22—541no—
594C22—541no—
595C22—541no—
596C22—541no—
597C22—541no—
598C22—1088no—
599C22C22×D3×D911yescharacteristic
600C4—21no—
601C4—21no—
602C4—62no—
603C4—182no—
604C4—541no—
605C4—541no—
606C4D3×D3612yes—
607C3—21no—
608C3#1058965611yescharacteristic, complemented
609C3#1059400511yescharacteristic
610C2—64no—
611C2—362no—
612C2—542no—
613C2—1081no—
614C2C2×D3×D3611yescharacteristic
615C2#1172247511yescharacteristic
616C2#1172247611yescharacteristic
subgroupG/H
centerC22C22×D3×D9
commutator, cosocleC22×C3×C9C24
FrattiniC22×C3C22×D32
FittingC3×C42×C9C22
socleC22×C32C23×D3
hypercenterC42D3×D9
hypocenterC3×C9#546
LCSG ▷ C22×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C42

Complements

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

NH ≅ G/Ntype
1. C2×#88646C2semidirect
2. C2×#88639C2semidirect
3. C4×C4⋊(C3⋊D9)C2semidirect
4. C4×C9×D12C2semidirect
5. C3×#63437C2semidirect
6. #11721051C2semidirect
7. D3×(C2×D9)⋊C4C2semidirect
8. #11721034C2semidirect
9. #11721021C2semidirect
10. #11721007C2semidirect
11. #11720936C2semidirect
17. C2×C4×C3⋊D9C22semidirect
19. C2×C4×D3×C9C22semidirect
21. C2×C3×Dic18C22semidirect
22. C2×#6195C22semidirect
24. C2×D3×Dic9C22semidirect
25. C2×#6192C22semidirect
28. C3×C42×C9C22semidirect
30. #88536C22semidirect
31. C4⋊(C3⋊Dic9)C22semidirect
32. C9×(C2×D3)⋊C4C22semidirect
33. C3×(C2×D9)⋊C4C22semidirect
34. C3×(C2×D9)⋊C4C22semidirect
35. C9×C4⋊Dic3C22semidirect
37. (C2×D9)⋊Dic3C22semidirect
38. Dic3⋊Dic9C22semidirect
59. #63437D3semidirect
79. C9×D12D4semidirect
130. C2×Dic18C2×D3semidirect
134. C42×C9C2×D3semidirect
141. (C2×D9)⋊C4C2×D3semidirect
142. (C2×D9)⋊C4C2×D3semidirect
152. C2×C3⋊D9#41semidirect
189. C4×D12D9semidirect
287. C2×C4×D3C2×D9semidirect
294. C3×C42C2×D9semidirect
308. (C2×D3)⋊C4C2×D9semidirect
313. C4⋊Dic3C2×D9semidirect
380. C3×C9#546semidirect
431. D12D36semidirect
484. C42D3×D9semidirect
541. C9#5676semidirect
608. C3#10589656semidirect

Maximal subgroups

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

subgroupindex
C2×#886462
C2×#886392
C4×C4⋊(C3⋊D9)2
C4×C9×D122
C3×#634372
#117210512
D3×(C2×D9)⋊C42
#117210342
#117210212
#117210072
#117209362
#105940053
#105896563