Finite Group Databasebeta

← #13150181

Subgroups — #13150181

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
1C22×C3×D36C211yesmaximal, characteristic, complemented
2C2×#88634C211yesmaximal, characteristic, complemented
3C4×#6233C211yesmaximal, characteristic, complemented
4C9×#924C211yesmaximal, characteristic, complemented
5#11721276C212yesmaximal, complemented
6D9×#217C212yesmaximal, complemented
7#11721007C211yesmaximal, characteristic, complemented
8#11720998C211yesmaximal, characteristic, complemented
9#11720983C212yesmaximal, complemented
10#11720981C212yesmaximal, complemented
11#11720951C211yesmaximal, characteristic, complemented
12#10594889—31nomaximal
13#10589732—31nomaximal
14C23×C3×D9C2212yes—
15C22×C3×C4×C9C2211yescharacteristic
16C2×#6233C2211yescharacteristic
17C2×C4×C3⋊D9C2211yescharacteristic, complemented
18C2×C3×D36—24no—
19C2×C3×D36C2212yes—
20C2×C3×D36C2211yescharacteristic
21C2×C3×D36C2211yescharacteristic
22C2×C9×Dic6C2211yescharacteristic, complemented
23C2×#6192C2212yescomplemented
24C2×#6192C2212yescomplemented
25C2×Dic3×D9C2212yescomplemented
26C2×Dic3×D9C2212yescomplemented
27#88634—24no—
28#88551C2211yescharacteristic
29#88536C2211yescharacteristic, complemented
30#88534C2211yescharacteristic, complemented
31C4×C3⋊Dic9C2211yescharacteristic, complemented
32C9×#217C2212yes—
33C9×C4⋊Dic3C2211yescharacteristic, complemented
34C9×Dic3⋊C4C2212yescomplemented
35(C2×D9)⋊Dic3C2212yescomplemented
36(C2×D9)⋊Dic3C2212yescomplemented
37(C2×D9)⋊Dic3C2212yescomplemented
38(C2×D9)⋊Dic3C2212yescomplemented
39C22×C3×D12—31no—
40C2×#3796—31no—
41C4×#638—31no—
42C3×#924D311yescharacteristic
43#63977—32no—
44D3×#217—32no—
45#63916—31no—
46#63898—31no—
47#63886—32no—
48#63884—32no—
49#63863—31no—
50C2×#3701—31no—
51C22×D36D311yescharacteristic, complemented
52C9×C22⋊Q8—31no—
53#63509—31no—
54C4×#611—31no—
55C4×#611—61no—
56#63457—32no—
57#63457—31no—
58#63456—32no—
59#63455—31no—
60#63444—32no—
61C22⋊C4×D9—32no—
62C23×C3×C9C2311yescharacteristic
63C22×C3⋊D9C2311yescharacteristic
64C22×C3×D9—24no—
65C22×C3×D9—24no—
66C22×C3×D9—42no—
67C22×C3×D9C2312yes—
68C22×C3×D9C2312yes—
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—
77C2×C9×Dic3C2312yes—
78C2×C9×Dic3C2312yes—
79C3×D36—24no—
80C3×D36—22no—
81C3×D36D414yescomplemented
82C9×Dic6—22no—
83#6192—24no—
84#6192—42no—
85Dic3×D9—24no—
86Dic3×D9—42no—
87#5806—91no—
88#5630—91no—
89C23×C3×D3—32no—
90C22×C32×C4C2×D311yescharacteristic
91C2×#638—31no—
92C2×C4×C3⋊D3—31no—
93C2×C3×D12—31no—
94C2×C3×D12—31no—
95C2×C3×D12—32no—
96C2×C3×D12—64no—
97C2×C3×Dic6C2×D311yescharacteristic
98C2×#626—32no—
99C2×#626—32no—
100C2×D3×Dic3—32no—
101C2×D3×Dic3—32no—
102#3796—64no—
103C23×D9C2×D312yes—
104#3757—31no—
105#3752—31no—
106#3750—31no—
107C4×C3⋊Dic3—31no—
108C3×#217C2×D312yes—
109C3×C4⋊Dic3C2×D311yescharacteristic
110C3×Dic3⋊C4C2×D312yes—
111(C2×D3)⋊Dic3—32no—
112(C2×D3)⋊Dic3—32no—
113(C2×D3)⋊Dic3—32no—
114(C2×D3)⋊Dic3—32no—
115C2×Q8×C9—31no—
116C22×C4×C9—21no—
117C22×C4×C9C2×D311yescharacteristic
118C2×#611—31no—
119C2×#611—61no—
120#3701—64no—
121C2×D36—24no—
122C2×D36—32no—
123C2×D36—32no—
124C2×D36C2×D311yescharacteristic
125C2×D36C2×D312yes—
126C2×D36C2×D311yescharacteristic
127C2×C4×D9—32no—
128C2×C4×D9—31no—
129C2×C4×D9—32no—
130C2×C4×D9—61no—
131C9×C4⋊C4—32no—
132C9×C4⋊C4—31no—
133C9×C22⋊C4—32no—
134#3676—31no—
135#3676—61no—
136(C2×D9)⋊C4—32no—
137(C2×D9)⋊C4—32no—
138(C2×D9)⋊C4—32no—
139(C2×D9)⋊C4—32no—
140(C2×D9)⋊C4—31no—
141(C2×D9)⋊C4—61no—
142Dic9⋊C4—31no—
143Dic9⋊C4—61no—
144C4×Dic9—31no—
145C4×Dic9—61no—
146C22×C3×C9—21no—
147C22×C3×C9—21no—
148C22×C3×C9#4112yescomplemented
149C22×C3×C9C2411yescharacteristic
150C2×C3⋊D9—21no—
151C2×C3⋊D9—21no—
152C2×C3⋊D9—21no—
153C2×C3×D9—22no—
154C2×C3×D9—22no—
155C2×C3×D9—22no—
156C2×C3×D9—22no—
157C2×C3×D9—22no—
158C2×C3×D9—48no—
159C2×C3×D9C2×D414yes—
160C3×C4×C9—21no—
161C3×C4×C9C2×D412yes—
162C3⋊Dic9—21no—
163C3⋊Dic9—21no—
164C3⋊Dic9—21no—
165C9×Dic3—22no—
166C9×Dic3—22no—
167C22×C3×D4—91no—
168C2×#239—91no—
169C2×#237—91no—
170C22×D12—31no—
171C3×C22⋊Q8—31no—
172#953—92no—
173C2×#217—92no—
174#947—91no—
175#939—92no—
176#938—91no—
177#936—91no—
178#935—92no—
179D4×Dic3—91no—
180C4×#67—91no—
181C4×#67—91no—
182C4×#67—181no—
183#924D911yescharacteristic, complemented
184#895—92no—
185#895—91no—
186#894—92no—
187#893—91no—
188#882—92no—
189D3×C22⋊C4—92no—
190C23×C32C22×D311yescharacteristic
191C22×C3⋊D3—31no—
192C22×C3×D3—32no—
193C22×C3×D3—32no—
194C22×C3×D3—64no—
195C22×C3×D3—64no—
196C22×C3×D3—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—
205C2×C3×Dic3C22×D312yes—
206C2×C3×Dic3C22×D312yes—
207C3×D12—34no—
208C3×D12—64no—
209C3×D12—62no—
210C3×Dic6—22no—
211#626—64no—
212#626—122no—
213D3×Dic3—64no—
214D3×Dic3—122no—
215C23×C9—21no—
216C23×C9C22×D311yescharacteristic
217C22×D9—24no—
218C22×D9—24no—
219C22×D9—31no—
220C22×D9—42no—
221C22×D9—61no—
222C22×D9C22×D312yes—
223C22×D9C22×D312yes—
224Q8×C9—62no—
225C2×C4×C9—22no—
226C2×C4×C9—21no—
227C2×C4×C9—21no—
228C2×C4×C9—32no—
229C2×C4×C9—32no—
230C2×C4×C9—42no—
231C2×C4×C9C22×D311yescharacteristic
232C2×C4×C9C22×D311yescharacteristic
233#611—64no—
234#611—62no—
235C2×Dic9—31no—
236C2×Dic9—31no—
237C2×Dic9—31no—
238C2×Dic9—61no—
239C2×Dic9—61no—
240C2×Dic9—61no—
241D36—24no—
242D36—22no—
243D36—64no—
244D36—122no—
245D36#6714yescomplemented
246C4×D9—64no—
247C4×D9—62no—
248C4×D9—122no—
249C4×D9—122no—
250#546—271noSylow
251C2×C3×C9—22no—
252C2×C3×C9C22×D411yescharacteristic
253C2×C3×C9C2×#4111yescharacteristic
254C2×C3×C9H(5,𝔽2)11yescharacteristic
255C3⋊D9—41no—
256C3×D9—24no—
257C3×D9—42no—
258C24×C3—92no—
259C23×D3—32no—
260C2×C3×Q8—31no—
261C2×C3×D4—91no—
262C2×C3×D4—92no—
263C2×C3×D4—91no—
264C2×C3×D4—184no—
265C22×C3×C4—21no—
266C22×C3×C4C2×D911yescharacteristic
267C22×C3×C4D3211yescharacteristic
268C2×#67—91no—
269C2×#67—91no—
270C2×#67—92no—
271C2×#67—92no—
272C2×#67—181no—
273C22×Dic3—92no—
274C22×Dic3—92no—
275#239—184no—
276#237—184no—
277C2×D12—31no—
278C2×D12—32no—
279C2×D12—31no—
280C2×D12—64no—
281C2×D12—92no—
282C2×D12—92no—
283C2×C4×D3—92no—
284C2×C4×D3—91no—
285C2×C4×D3—91no—
286C2×C4×D3—92no—
287C2×C4×D3—181no—
288C2×Dic6C2×D911yescharacteristic, complemented
289C3×C4⋊C4—31no—
290C3×C4⋊C4—32no—
291C3×C22⋊C4—32no—
292#217—91no—
293#217—91no—
294#217—92no—
295#217—92no—
296#217—92no—
297#217—92no—
298#217—181no—
299#217C2×D912yes—
300(C2×D3)⋊C4—92no—
301(C2×D3)⋊C4—92no—
302(C2×D3)⋊C4—92no—
303(C2×D3)⋊C4—92no—
304(C2×D3)⋊C4—91no—
305(C2×D3)⋊C4—91no—
306(C2×D3)⋊C4—181no—
307C4⋊Dic3C2×D911yescharacteristic, complemented
308Dic3⋊C4—91no—
309Dic3⋊C4—91no—
310Dic3⋊C4—181no—
311Dic3⋊C4C2×D912yescomplemented
312C4×Dic3—91no—
313C4×Dic3—91no—
314C4×Dic3—181no—
315C22×C32—21no—
316C22×C32—21no—
317C22×C32#23912yes—
318C22×C32C23×D311yescharacteristic
319C2×C3⋊D3—61no—
320C2×C3⋊D3—61no—
321C2×C3⋊D3—61no—
322C2×C3×D3—34no—
323C2×C3×D3—62no—
324C2×C3×D3—62no—
325C2×C3×D3—62no—
326C2×C3×D3—62no—
327C2×C3×D3—62no—
328C2×C3×D3—128no—
329C32×C4—21no—
330C32×C4D3×D412yes—
331C3⋊Dic3—61no—
332C3⋊Dic3—61no—
333C3⋊Dic3—61no—
334C3×Dic3—22no—
335C3×Dic3—22no—
336C22×C9—21no—
337C22×C9—21no—
338C22×C9—21no—
339C22×C9—22no—
340C22×C9—41no—
341C22×C9—41no—
342C22×C9#23712yescomplemented
343C22×C9C23×D311yescharacteristic
344C2×D9—22no—
345C2×D9—22no—
346C2×D9—22no—
347C2×D9—22no—
348C2×D9—22no—
349C2×D9—48no—
350C2×D9—62no—
351C2×D9—61no—
352C2×D9—61no—
353C2×D9—61no—
354C2×D9—121no—
355C2×D9—121no—
356C2×D9C2×#6714yes—
357C4×C9—21no—
358C4×C9—22no—
359C4×C9—41no—
360C4×C9—62no—
361C4×C9—62no—
362C4×C9C2×#6712yes—
363Dic9—62no—
364Dic9—61no—
365Dic9—61no—
366Dic9—61no—
367Dic9—121no—
368Dic9—121no—
369C2×#41—271no—
370C22×D4—91no—
371#124—271no—
372#123—272no—
373C22⋊Q8—31no—
374#121—271no—
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—92no—
383C23×C3—92no—
384C23×C3—184no—
385C23×C3—184no—
386C23×C3—362no—
387C23×C3C22×D911yescharacteristic
388C23×C3C2×D3211yescharacteristic
389C22×D3—32no—
390C22×D3—32no—
391C22×D3—64no—
392C22×D3—64no—
393C22×D3—91no—
394C22×D3—91no—
395C22×D3—122no—
396C22×D3—181no—
397C3×Q8—62no—
398C3×D4—94no—
399C3×D4—184no—
400C3×D4—182no—
401C2×C3×C4—22no—
402C2×C3×C4—22no—
403C2×C3×C4—21no—
404C2×C3×C4—21no—
405C2×C3×C4—32no—
406C2×C3×C4—32no—
407C2×C3×C4—42no—
408C2×C3×C4C22×D911yescharacteristic
409C2×C3×C4C22×D911yescharacteristic
410C2×C3×C4C2×D3211yescharacteristic
411C2×C3×C4C2×D3211yescharacteristic
412#67—184no—
413#67—184no—
414#67—182no—
415#67—182no—
416#67—362no—
417C2×Dic3—91no—
418C2×Dic3—91no—
419C2×Dic3—91no—
420C2×Dic3—91no—
421C2×Dic3—91no—
422C2×Dic3—91no—
423C2×Dic3—181no—
424C2×Dic3—181no—
425C2×Dic3—181no—
426C2×Dic3—184no—
427C2×Dic3—362no—
428C2×Dic3C22×D912yes—
429C2×Dic3C22×D912yes—
430D12—34no—
431D12—64no—
432D12—62no—
433D12—184no—
434D12—362no—
435C4×D3—184no—
436C4×D3—182no—
437C4×D3—182no—
438C4×D3—362no—
439C4×D3—362no—
440Dic6—22no—
441C2×C32—22no—
442C2×C32C2×D3×D411yescharacteristic
443C2×C32C2×#23911yescharacteristic
444C2×C32#100911yescharacteristic
445C3⋊D3—121no—
446C3×D3—64no—
447C3×D3—122no—
448C2×C9—22no—
449C2×C9—21no—
450C2×C9—21no—
451C2×C9—21no—
452C2×C9—42no—
453C2×C9C2×#23711yescharacteristic
454C2×C9#100411yescharacteristic
455C2×C9C22×#6711yescharacteristic
456D9—24no—
457D9—42no—
458D9—122no—
459D9—121no—
460C24—92no—
461#41—544no—
462C2×Q8—31no—
463C2×D4—91no—
464C2×D4—92no—
465C2×D4—91no—
466C2×D4—184no—
467C2×D4—272no—
468C2×D4—272no—
469C2×D4—271no—
470C22×C4—272no—
471C22×C4—272no—
472C22×C4—271no—
473C22×C4D3×D911yescharacteristic
474C4⋊C4—31no—
475C4⋊C4—32no—
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—21no—
486C22×C3—21no—
487C22×C3—21no—
488C22×C3—21no—
489C22×C3—22no—
490C22×C3—21no—
491C22×C3—41no—
492C22×C3—41no—
493C22×C3—94no—
494C22×C3—182no—
495C22×C3—182no—
496C22×C3—182no—
497C22×C3—182no—
498C22×C3—182no—
499C22×C3—368no—
500C22×C3#370112yescomplemented
501C22×C3C23×D911yescharacteristic
502C22×C3#379612yes—
503C22×C3C22×D3211yescharacteristic
504C2×D3—34no—
505C2×D3—62no—
506C2×D3—62no—
507C2×D3—62no—
508C2×D3—62no—
509C2×D3—62no—
510C2×D3—128no—
511C2×D3—182no—
512C2×D3—181no—
513C2×D3—181no—
514C2×D3—181no—
515C2×D3—181no—
516C2×D3—181no—
517C2×D3—181no—
518C2×D3—361no—
519C2×D3—361no—
520C3×C4—21no—
521C3×C4—21no—
522C3×C4—22no—
523C3×C4—41no—
524C3×C4—62no—
525C3×C4—62no—
526C3×C4D4×D912yes—
527C3×C4D3×#6712yes—
528Dic3—22no—
529Dic3—22no—
530Dic3—182no—
531Dic3—181no—
532Dic3—181no—
533Dic3—181no—
534Dic3—181no—
535Dic3—181no—
536Dic3—181no—
537Dic3—361no—
538Dic3—361no—
539C32#563011yescharacteristic
540C9—21no—
541C9#580611yescharacteristic, complemented
542C23—92no—
543C23—92no—
544C23—184no—
545C23—184no—
546C23—271no—
547C23—362no—
548C23C2×D3×D911yescharacteristic
549Q8—62no—
550D4—94no—
551D4—184no—
552D4—182no—
553D4—544no—
554D4—542no—
555D4—1082no—
556C2×C4—22no—
557C2×C4—32no—
558C2×C4—32no—
559C2×C4—271no—
560C2×C4—271no—
561C2×C4—271no—
562C2×C4—544no—
563C2×C4—542no—
564C2×C4—1082no—
565C2×C4C2×D3×D911yescharacteristic
566C2×C4C2×D3×D911yescharacteristic
567C2×C3—22no—
568C2×C3—22no—
569C2×C3—21no—
570C2×C3—21no—
571C2×C3—21no—
572C2×C3—42no—
573C2×C3—184no—
574C2×C3—362no—
575C2×C3C2×D4×D911yescharacteristic
576C2×C3C2×#370111yescharacteristic
577C2×C3#6371511yescharacteristic
578C2×C3C2×#379611yescharacteristic
579C2×C3#6430711yescharacteristic
580C2×C3C2×D3×#6711yescharacteristic
581D3—64no—
582D3—122no—
583D3—362no—
584D3—361no—
585D3—361no—
586C22—21no—
587C22—21no—
588C22—94no—
589C22—182no—
590C22—182no—
591C22—182no—
592C22—182no—
593C22—182no—
594C22—368no—
595C22—541no—
596C22—541no—
597C22—541no—
598C22#8863412yescomplemented
599C22C22×D3×D911yescharacteristic
600C4—21no—
601C4—62no—
602C4—62no—
603C4—541no—
604C4—541no—
605C4—541no—
606C4D9×#6712yes—
607C3—21no—
608C3#1058973211yescharacteristic, complemented
609C3#1059488911yescharacteristic
610C2—22no—
611C2—184no—
612C2—362no—
613C2—1081no—
614C2C2×#8863411yescharacteristic
615C2#1172247711yescharacteristic
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. C22×C3×D36C2semidirect
2. C2×#88634C2semidirect
3. C4×#6233C2semidirect
4. C9×#924C2semidirect
5. #11721276C2semidirect
6. D9×#217C2semidirect
7. #11721007C2semidirect
8. #11720998C2semidirect
9. #11720983C2semidirect
10. #11720981C2semidirect
11. #11720951C2semidirect
17. C2×C4×C3⋊D9C22semidirect
22. C2×C9×Dic6C22semidirect
23. C2×#6192C22semidirect
24. C2×#6192C22semidirect
25. C2×Dic3×D9C22semidirect
26. C2×Dic3×D9C22semidirect
29. #88536C22semidirect
30. #88534C22semidirect
31. C4×C3⋊Dic9C22semidirect
33. C9×C4⋊Dic3C22semidirect
34. C9×Dic3⋊C4C22semidirect
35. (C2×D9)⋊Dic3C22semidirect
36. (C2×D9)⋊Dic3C22semidirect
37. (C2×D9)⋊Dic3C22semidirect
38. (C2×D9)⋊Dic3C22semidirect
51. C22×D36D3semidirect
81. C3×D36D4semidirect
148. C22×C3×C9#41semidirect
183. #924D9semidirect
245. D36#67semidirect
288. C2×Dic6C2×D9semidirect
307. C4⋊Dic3C2×D9semidirect
311. Dic3⋊C4C2×D9semidirect
342. C22×C9#237semidirect
380. C3×C9#546semidirect
500. C22×C3#3701semidirect
541. C9#5806semidirect
598. C22#88634semidirect
608. C3#10589732semidirect

Maximal subgroups

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

subgroupindex
C22×C3×D362
C2×#886342
C4×#62332
C9×#9242
#117212762
D9×#2172
#117210072
#117209982
#117209832
#117209812
#117209512
#105948893
#105897323