Finite Group Databasebeta

← #13149327

Subgroups — #13149327

Order 1728.

1078 conjugacy classes of subgroups.

The subgroup lattice has 632 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#88646C211yesmaximal, characteristic, complemented
2C2×#88640C211yesmaximal, characteristic, complemented
3#11721748C211yesmaximal, characteristic, complemented
4C9×#895C211yesmaximal, characteristic, complemented
5C3×#63457C211yesmaximal, characteristic, complemented
6#11721050C212yesmaximal, complemented
7#11721049C212yesmaximal, complemented
8#11721044C212yesmaximal, complemented
9#11721043C212yesmaximal, complemented
10#11721037C211yesmaximal, characteristic, complemented
11#11721036C211yesmaximal, characteristic, complemented
12#10594634—31nomaximal
13#10589709—31nomaximal
14C22×D3×D9C2212yes—
15C2×C4×C3⋊D9C2211yescharacteristic
16C2×C9×D12C2211yescharacteristic
17C2×C9×D12C2212yescomplemented
18C2×C4×D3×C9C2211yescharacteristic, complemented
19C2×C3×D36C2211yescharacteristic
20C2×C3×D36C2212yescomplemented
21C2×C3×C4×D9C2211yescharacteristic, complemented
22C2×#6195C2211yescharacteristic
23C2×#6194C2212yes—
24C2×#6194C2211yescharacteristic, complemented
25C2×#6194C2212yescomplemented
26C2×#6194C2212yescomplemented
27C2×#6192C2211yescharacteristic
28C2×#6189C2211yescharacteristic, complemented
29#88646—24no—
30#88640—24no—
31C3×C9×C4⋊C4C2211yescharacteristic, complemented
32#88536C2212yes—
33C4⋊(C3⋊Dic9)C2211yescharacteristic, complemented
34C4×C3⋊Dic9C2212yescomplemented
35C9×(C2×D3)⋊C4C2212yescomplemented
36C3×(C2×D9)⋊C4C2212yescomplemented
37(C2×D3)⋊Dic9C2212yescomplemented
38(C2×D9)⋊Dic3C2212yescomplemented
39C2×#3802—31no—
40C2×#3798—31no—
41#64101—31no—
42C3×#895—31no—
43C3×#895D311yescharacteristic
44#63922—32no—
45#63922—32no—
46#63915—32no—
47#63914—32no—
48#63908—31no—
49#63908—31no—
50C2×D4×D9—31no—
51C2×#3697—31no—
52C9×#121—31no—
53#63502—32no—
54#63500—31no—
55#63499—32no—
56#63492—31no—
57#63457D311yescharacteristic, complemented
58#63454—31no—
59#63454—61no—
60#63447—32no—
61#63444—32no—
62C22×C3⋊D9C2311yescharacteristic
63C22×D3×C9C2312yes—
64C22×D3×C9C2311yescharacteristic
65C22×C3×D9C2312yes—
66C22×C3×D9C2311yescharacteristic
67C2×D3×D9—24no—
68C2×D3×D9—42no—
69C2×D3×D9—42no—
70C2×C3×C4×C9C2312yes—
71C2×C3×C4×C9C2311yescharacteristic
72C2×C3⋊Dic9C2312yes—
73C2×C3⋊Dic9C2311yescharacteristic
74C4×C3⋊D9D414yescomplemented
75C2×C9×Dic3C2311yescharacteristic
76C2×C3×Dic9C2311yescharacteristic
77C9×D12—22no—
78C9×D12—42no—
79C4×D3×C9—22no—
80C3×D36—22no—
81C3×D36—42no—
82C3×C4×D9—22no—
83#6195—22no—
84#6194—22no—
85#6194—24no—
86#6194—42no—
87#6194—42no—
88#6192—22no—
89#6189—22no—
90#5607—91no—
91#5607—91no—
92C22×D32—32no—
93C2×C4×C3⋊D3—31no—
94C2×C3×D12—31no—
95C2×C3×D12—32no—
96C2×C3×D12C2×D311yescharacteristic
97C2×C3×D12C2×D312yes—
98C2×C3×C4×D3—31no—
99C2×C3×C4×D3C2×D311yescharacteristic
100C2×#627—31no—
101C2×#626—31no—
102C2×#626—31no—
103C2×#625—31no—
104C2×#625—32no—
105C2×#625—32no—
106C2×#625—32no—
107#3802—64no—
108#3798—64no—
109C23×D9—32no—
110C32×C4⋊C4C2×D311yescharacteristic
111#3752—32no—
112C4⋊(C3⋊Dic3)—31no—
113C4×C3⋊Dic3—32no—
114C3×(C2×D3)⋊C4—32no—
115C3×(C2×D3)⋊C4C2×D312yes—
116(C2×D3)⋊Dic3—32no—
117(C2×D3)⋊Dic3—32no—
118C2×D4×C9—31no—
119C2×D4×C9—32no—
120C22×C4×C9—31no—
121C2×#611—31no—
122C2×#611—32no—
123C2×#611—32no—
124C2×#611—32no—
125C2×#611—31no—
126D4×D9—64no—
127#3697—64no—
128C2×D36—31no—
129C2×D36C2×D311yescharacteristic
130C2×D36C2×D312yescomplemented
131C2×C4×D9—31no—
132C2×C4×D9—61no—
133C2×C4×D9C2×D311yescharacteristic, complemented
134C2×Dic18—31no—
135C9×C4⋊C4—21no—
136C9×C4⋊C4C2×D311yescharacteristic, complemented
137C9×C22⋊C4—32no—
138#3676—32no—
139(C2×D9)⋊C4—32no—
140(C2×D9)⋊C4—32no—
141(C2×D9)⋊C4—62no—
142(C2×D9)⋊C4C2×D312yescomplemented
143C4⋊Dic9—31no—
144C4⋊Dic9—61no—
145C4×Dic9—32no—
146C4×Dic9—62no—
147C22×C3×C9C2411yescharacteristic
148C2×C3⋊D9—21no—
149C2×C3⋊D9—21no—
150C2×C3⋊D9C2×D412yes—
151C2×D3×C9—21no—
152C2×D3×C9—22no—
153C2×D3×C9—21no—
154C2×D3×C9—22no—
155C2×D3×C9—21no—
156C2×D3×C9—22no—
157C2×C3×D9—21no—
158C2×C3×D9—22no—
159C2×C3×D9—21no—
160C2×C3×D9—22no—
161C2×C3×D9—21no—
162C2×C3×D9—22no—
163D3×D9—48no—
164C3×C4×C9—22no—
165C3×C4×C9C2×D412yes—
166C3⋊Dic9—22no—
167C3⋊Dic9C2×D412yes—
168C9×Dic3—21no—
169C3×Dic9—21no—
170C2×D3×D4—91no—
171C2×D3×D4—91no—
172C2×#235—91no—
173C2×#235—91no—
174C3×#121—31no—
175C3×#121—91no—
176#940—92no—
177#940—92no—
178#938—91no—
179#938—91no—
180#937—92no—
181#937—92no—
182#930—91no—
183#930—91no—
184#895—31no—
185#895D911yescharacteristic, complemented
186#892—91no—
187#892—91no—
188#892—181no—
189#885—92no—
190#885—92no—
191#882—92no—
192#882—92no—
193C22×C3⋊D3—31no—
194C22×C3×D3—32no—
195C22×C3×D3—31no—
196C22×C3×D3C22×D312yes—
197C22×C3×D3C22×D311yescharacteristic
198C2×D32—64no—
199C2×D32—122no—
200C2×D32—122no—
201C2×C32×C4C22×D312yes—
202C2×C32×C4C22×D311yescharacteristic
203C2×C3⋊Dic3—31no—
204C2×C3⋊Dic3—32no—
205C4×C3⋊D3—34no—
206C2×C3×Dic3—31no—
207C2×C3×Dic3C22×D311yescharacteristic
208C3×D12—22no—
209C3×D12—42no—
210C3×D12—62no—
211C3×D12—122no—
212C3×C4×D3—22no—
213C3×C4×D3—62no—
214#627—62no—
215#626—62no—
216#626—62no—
217#625—62no—
218#625—64no—
219#625—122no—
220#625—122no—
221C23×C9—32no—
222C23×C9—31no—
223C22×D9—31no—
224C22×D9—61no—
225C22×D9—64no—
226C22×D9—122no—
227C22×D9—122no—
228C22×D9C22×D312yes—
229C22×D9C22×D311yescharacteristic
230D4×C9—62no—
231D4×C9—122no—
232C2×C4×C9—21no—
233C2×C4×C9—22no—
234C2×C4×C9—31no—
235C2×C4×C9—62no—
236C2×C4×C9C22×D312yes—
237C2×C4×C9C22×D311yescharacteristic
238#611—62no—
239#611—62no—
240#611—64no—
241#611—122no—
242#611—122no—
243C2×Dic9—31no—
244C2×Dic9—32no—
245C2×Dic9—61no—
246C2×Dic9—62no—
247C2×Dic9C22×D311yescharacteristic
248D36—22no—
249D36—42no—
250D36—62no—
251C4×D9—22no—
252C4×D9—34no—
253C4×D9—64no—
254Dic18—62no—
255#535—271noSylow
256C2×C3×C9C22×D411yescharacteristic
257C2×C3×C9H(5,𝔽2)11yescharacteristic
258C2×C3×C9H(5,𝔽2)11yescharacteristic
259C3⋊D9—22no—
260D3×C9—42no—
261D3×C9—41no—
262C3×D9—42no—
263C3×D9—41no—
264C23×D3—92no—
265C23×D3—92no—
266C2×C3×D4—31no—
267C2×C3×D4—32no—
268C2×C3×D4—91no—
269C2×C3×D4—92no—
270C22×C3×C4—31no—
271C22×C3×C4—91no—
272C2×#67—91no—
273C2×#67—92no—
274C2×#67—91no—
275C2×#67—92no—
276C2×#67—91no—
277C2×#67—92no—
278C2×#67—92no—
279C2×#67—92no—
280C2×#67—92no—
281C2×#67—91no—
282D3×D4—184no—
283D3×D4—184no—
284#235—184no—
285#235—184no—
286C2×D12—31no—
287C2×D12—32no—
288C2×D12—91no—
289C2×D12—91no—
290C2×D12C2×D911yescharacteristic
291C2×D12C2×D912yescomplemented
292C2×C4×D3—31no—
293C2×C4×D3—91no—
294C2×C4×D3—91no—
295C2×C4×D3—181no—
296C2×C4×D3C2×D911yescharacteristic, complemented
297C2×Dic6—91no—
298C2×Dic6—91no—
299C3×C4⋊C4—21no—
300C3×C4⋊C4C2×D911yescharacteristic, complemented
301C3×C4⋊C4D3211yescharacteristic
302C3×C22⋊C4—32no—
303C3×C22⋊C4—92no—
304#217—92no—
305#217—92no—
306(C2×D3)⋊C4—32no—
307(C2×D3)⋊C4—92no—
308(C2×D3)⋊C4—92no—
309(C2×D3)⋊C4—92no—
310(C2×D3)⋊C4—92no—
311(C2×D3)⋊C4—182no—
312(C2×D3)⋊C4C2×D912yescomplemented
313C4⋊Dic3—91no—
314C4⋊Dic3—91no—
315C4⋊Dic3—181no—
316C4×Dic3—92no—
317C4×Dic3—92no—
318C4×Dic3—182no—
319C22×C32C23×D311yescharacteristic
320C2×C3⋊D3—32no—
321C2×C3⋊D3—61no—
322C2×C3⋊D3—61no—
323C2×C3×D3—21no—
324C2×C3×D3—22no—
325C2×C3×D3—21no—
326C2×C3×D3—22no—
327C2×C3×D3—21no—
328C2×C3×D3—22no—
329C2×C3×D3—61no—
330C2×C3×D3—62no—
331C2×C3×D3—61no—
332C2×C3×D3—62no—
333C2×C3×D3—61no—
334C2×C3×D3—62no—
335D32—128no—
336C32×C4—22no—
337C32×C4D3×D412yes—
338C3⋊Dic3—32no—
339C3⋊Dic3—62no—
340C3×Dic3—21no—
341C3×Dic3—61no—
342C22×C9—21no—
343C22×C9—61no—
344C22×C9—62no—
345C22×C9—61no—
346C22×C9—62no—
347C22×C9—61no—
348C22×C9—62no—
349C22×C9C23×D311yescharacteristic
350C2×D9—21no—
351C2×D9—22no—
352C2×D9—21no—
353C2×D9—22no—
354C2×D9—21no—
355C2×D9—22no—
356C2×D9—32no—
357C2×D9—62no—
358C2×D9—61no—
359C2×D9—61no—
360C2×D9—121no—
361C2×D9—121no—
362C2×D9—128no—
363C4×C9—22no—
364C4×C9—22no—
365C4×C9—42no—
366C4×C9—61no—
367C4×C9D3×D412yes—
368Dic9—21no—
369Dic9—32no—
370Dic9—62no—
371Dic9—62no—
372Dic9—122no—
373C2×#41—271no—
374C22×D4—271no—
375#127—272no—
376#124—272no—
377#121—31no—
378#121—91no—
379#121—271no—
380#121—271no—
381#120—272no—
382#120—272no—
383#117—271no—
384C3×C9#53511yescharacteristic, complemented, Sylow
385C23×C3—32no—
386C23×C3—31no—
387C23×C3—92no—
388C23×C3—91no—
389C22×D3—32no—
390C22×D3—31no—
391C22×D3—91no—
392C22×D3—91no—
393C22×D3—181no—
394C22×D3—184no—
395C22×D3—184no—
396C22×D3—362no—
397C22×D3—362no—
398C22×D3—362no—
399C22×D3—362no—
400C22×D3C22×D912yes—
401C22×D3C22×D911yescharacteristic
402C3×D4—62no—
403C3×D4—122no—
404C3×D4—182no—
405C3×D4—362no—
406C2×C3×C4—21no—
407C2×C3×C4—22no—
408C2×C3×C4—31no—
409C2×C3×C4—62no—
410C2×C3×C4—91no—
411C2×C3×C4—182no—
412C2×C3×C4C22×D912yes—
413C2×C3×C4C22×D911yescharacteristic
414C2×C3×C4C2×D3212yes—
415C2×C3×C4C2×D3211yescharacteristic
416#67—182no—
417#67—182no—
418#67—182no—
419#67—184no—
420#67—182no—
421#67—184no—
422#67—362no—
423#67—362no—
424#67—362no—
425#67—362no—
426C2×Dic3—31no—
427C2×Dic3—91no—
428C2×Dic3—91no—
429C2×Dic3—92no—
430C2×Dic3—92no—
431C2×Dic3—181no—
432C2×Dic3—182no—
433C2×Dic3C22×D911yescharacteristic
434D12—22no—
435D12—42no—
436D12—62no—
437D12—122no—
438D12—182no—
439D12—182no—
440C4×D3—22no—
441C4×D3—62no—
442C4×D3—94no—
443C4×D3—94no—
444C4×D3—184no—
445Dic6—182no—
446Dic6—182no—
447C2×C32C2×D3×D411yescharacteristic
448C2×C32#100411yescharacteristic
449C2×C32#100911yescharacteristic
450C3⋊D3—62no—
451C3×D3—42no—
452C3×D3—41no—
453C3×D3—122no—
454C3×D3—121no—
455C2×C9—21no—
456C2×C9—21no—
457C2×C9—21no—
458C2×C9—122no—
459C2×C9—121no—
460C2×C9C2×D3×D411yescharacteristic
461C2×C9#100411yescharacteristic
462C2×C9#100911yescharacteristic
463D9—41no—
464D9—42no—
465D9—62no—
466D9—122no—
467C24—272no—
468#41—544no—
469C2×Q8—271no—
470C2×D4—31no—
471C2×D4—32no—
472C2×D4—91no—
473C2×D4—92no—
474C2×D4—271no—
475C2×D4—271no—
476C2×D4—272no—
477C2×D4—271no—
478C2×D4—272no—
479C2×D4—272no—
480C2×D4—544no—
481C22×C4—31no—
482C22×C4—91no—
483C22×C4—271no—
484C4⋊C4—271no—
485C4⋊C4D3×D911yescharacteristic, complemented
486C22⋊C4—32no—
487C22⋊C4—92no—
488C22⋊C4—272no—
489C22⋊C4—272no—
490C22⋊C4—272no—
491C42—272no—
492C22×C3—21no—
493C22×C3—61no—
494C22×C3—62no—
495C22×C3—61no—
496C22×C3—62no—
497C22×C3—61no—
498C22×C3—62no—
499C22×C3—181no—
500C22×C3—182no—
501C22×C3—181no—
502C22×C3—182no—
503C22×C3—181no—
504C22×C3—182no—
505C22×C3C23×D911yescharacteristic
506C22×C3C22×D3211yescharacteristic
507C2×D3—21no—
508C2×D3—22no—
509C2×D3—21no—
510C2×D3—22no—
511C2×D3—21no—
512C2×D3—22no—
513C2×D3—61no—
514C2×D3—62no—
515C2×D3—61no—
516C2×D3—62no—
517C2×D3—61no—
518C2×D3—62no—
519C2×D3—92no—
520C2×D3—92no—
521C2×D3—182no—
522C2×D3—181no—
523C2×D3—181no—
524C2×D3—181no—
525C2×D3—181no—
526C2×D3—361no—
527C2×D3—361no—
528C2×D3—368no—
529C2×D3—368no—
530C3×C4—22no—
531C3×C4—22no—
532C3×C4—22no—
533C3×C4—42no—
534C3×C4—61no—
535C3×C4—181no—
536C3×C4D4×D912yes—
537C3×C4#380212yes—
538Dic3—21no—
539Dic3—61no—
540Dic3—92no—
541Dic3—92no—
542Dic3—182no—
543Dic3—182no—
544Dic3—182no—
545Dic3—362no—
546C32#560711yescharacteristic
547C9—21no—
548C9#560711yescharacteristic, complemented
549C23—32no—
550C23—31no—
551C23—92no—
552C23—91no—
553C23—271no—
554C23—544no—
555C23—1082no—
556C23—1082no—
557Q8—542no—
558D4—62no—
559D4—122no—
560D4—182no—
561D4—362no—
562D4—542no—
563D4—542no—
564D4—544no—
565D4—542no—
566D4—1082no—
567D4—1082no—
568C2×C4—31no—
569C2×C4—62no—
570C2×C4—91no—
571C2×C4—182no—
572C2×C4—271no—
573C2×C4—274no—
574C2×C4—272no—
575C2×C4C2×D3×D912yes—
576C2×C4C2×D3×D911yescharacteristic
577C2×C3—21no—
578C2×C3—21no—
579C2×C3—21no—
580C2×C3—122no—
581C2×C3—121no—
582C2×C3—362no—
583C2×C3—361no—
584C2×C3C2×D4×D911yescharacteristic
585C2×C3#6371011yescharacteristic
586C2×C3#6371511yescharacteristic
587C2×C3C2×#380211yescharacteristic
588C2×C3#6431311yescharacteristic
589C2×C3#6432011yescharacteristic
590D3—42no—
591D3—41no—
592D3—122no—
593D3—121no—
594D3—182no—
595D3—182no—
596D3—362no—
597C22—61no—
598C22—62no—
599C22—61no—
600C22—62no—
601C22—61no—
602C22—62no—
603C22—181no—
604C22—182no—
605C22—181no—
606C22—182no—
607C22—181no—
608C22—182no—
609C22—272no—
610C22—541no—
611C22—541no—
612C22—1088no—
613C22C22×D3×D911yescharacteristic
614C4—22no—
615C4—61no—
616C4—181no—
617C4—272no—
618C4—542no—
619C4#8864612yes—
620C3—21no—
621C3#1058970911yescharacteristic, complemented
622C3#1059463411yescharacteristic
623C2—122no—
624C2—121no—
625C2—362no—
626C2—361no—
627C2—542no—
628C2C2×#8864611yescharacteristic
629C2#1172249911yescharacteristic
630C2#1172251511yescharacteristic
subgroupG/H
centerC22C22×D3×D9
commutator, cosocleC22×C3×C9C24
FrattiniC22×C3C22×D32
FittingC3×C9×C4⋊C4C22
socleC22×C32C23×D3
hypercenterC4⋊C4D3×D9
hypocenterC3×C9#535
LCSG ▷ C22×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
1. C2×#88646C2semidirect
2. C2×#88640C2semidirect
3. #11721748C2semidirect
4. C9×#895C2semidirect
5. C3×#63457C2semidirect
6. #11721050C2semidirect
7. #11721049C2semidirect
8. #11721044C2semidirect
9. #11721043C2semidirect
10. #11721037C2semidirect
11. #11721036C2semidirect
17. C2×C9×D12C22semidirect
18. C2×C4×D3×C9C22semidirect
20. C2×C3×D36C22semidirect
21. C2×C3×C4×D9C22semidirect
24. C2×#6194C22semidirect
25. C2×#6194C22semidirect
26. C2×#6194C22semidirect
28. C2×#6189C22semidirect
31. C3×C9×C4⋊C4C22semidirect
33. C4⋊(C3⋊Dic9)C22semidirect
34. C4×C3⋊Dic9C22semidirect
35. C9×(C2×D3)⋊C4C22semidirect
36. C3×(C2×D9)⋊C4C22semidirect
37. (C2×D3)⋊Dic9C22semidirect
38. (C2×D9)⋊Dic3C22semidirect
57. #63457D3semidirect
74. C4×C3⋊D9D4semidirect
130. C2×D36C2×D3semidirect
133. C2×C4×D9C2×D3semidirect
136. C9×C4⋊C4C2×D3semidirect
142. (C2×D9)⋊C4C2×D3semidirect
185. #895D9semidirect
291. C2×D12C2×D9semidirect
296. C2×C4×D3C2×D9semidirect
300. C3×C4⋊C4C2×D9semidirect
312. (C2×D3)⋊C4C2×D9semidirect
384. C3×C9#535semidirect
485. C4⋊C4D3×D9semidirect
548. C9#5607semidirect
621. C3#10589709semidirect

Maximal subgroups

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

subgroupindex
C2×#886462
C2×#886402
#117217482
C9×#8952
C3×#634572
#117210502
#117210492
#117210442
#117210432
#117210372
#117210362
#105946343
#105897093