Finite Group Databasebeta

← #11662455

Subgroups — #11662455

Order 768.

2870 conjugacy classes of subgroups.

The subgroup lattice has 718 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#908C211yesmaximal, characteristic, complemented
2C2×#5745C211yesmaximal, characteristic, complemented
3#85735C211yesmaximal, characteristic, complemented
4C2×#5525C211yesmaximal, characteristic, complemented
5C3×#2911C211yesmaximal, characteristic, complemented
6#82213C211yesmaximal, characteristic, complemented
7C2×#5001C211yesmaximal, characteristic, complemented
8#79213C218yesmaximal, complemented
9#79211C218yesmaximal, complemented
10#79183C214yesmaximal, complemented
11#79182C214yesmaximal, complemented
12#60454—31nomaximal, Sylow
13C22×#235C2211yescharacteristic
14C22×#235C2211yescharacteristic
15C23×D12C2211yescharacteristic
16C22×C3×M16C2211yescharacteristic
17C2×C3×#117C2211yescharacteristic
18C2×C3×#117C2211yescharacteristic
19C22×#208C2211yescharacteristic
20C2×#908—28no—
21C2×#908C2214yes—
22C2×#908C2214yescomplemented
23#5745—28no—
24C2×#907C2214yescomplemented
25C2×D3×M16C2214yescomplemented
26C22×D24C2212yescomplemented
27C22×#204C2212yescomplemented
28C2×#901C2214yescomplemented
29#5526—21no—
30#5526C2212yescomplemented
31#5525—22no—
32#5525—22no—
33#5525—21no—
34#5525C2214yes—
35#5525C2214yescomplemented
36#5525C2212yescomplemented
37#5524—22no—
38#5524C2214yescomplemented
39#5510C2214yescomplemented
40C2×#895C2214yescomplemented
41#5493—22no—
42C2×#882—22no—
43#5477—22no—
44C2×#875—21no—
45C2×#874—21no—
46C3×#428C2214yescomplemented
47C3×#428C2214yescomplemented
48C3×#427C2214yescomplemented
49#5004C2212yescomplemented
50#5002C2214yescomplemented
51#5001C2214yescomplemented
52#5001C2214yescomplemented
53#5001C2212yescomplemented
54#4966C2214yescomplemented
55C2×#809C2214yescomplemented
56#4886C22116yescomplemented
57#4883C22116yescomplemented
58#4865C22116yescomplemented
59#4864C22116yescomplemented
60C22×AGL(1,ℤ/8)—31no—
61C2×#568—31no—
62#3449—31no—
63C2×#535—31no—
64#3062—38no—
65#3058—34no—
66#3057—34no—
67#3055—38no—
68#2911D311yescharacteristic, complemented
69#2896—31no—
70C2×#418—31no—
71C24×D3—21no—
72C23×C3×C4C2311yescharacteristic
73C22×#67—21no—
74C22×#67D412yes—
75C2×#235—24no—
76C2×#235—24no—
77C2×#235—21no—
78C2×#235—21no—
79C2×#235—24no—
80C2×#235—24no—
81C2×#235D412yes—
82C2×#235D412yes—
83C2×#235C2312yes—
84C2×#235C2312yes—
85C2×#235C2312yes—
86C2×#235C2312yes—
87C22×D12—24no—
88C22×D12—24no—
89C22×D12—21no—
90C22×D12—21no—
91C22×D12—21no—
92C22×D12—24no—
93C22×D12D411yescharacteristic
94C22×D12C2312yes—
95C22×D12C2312yes—
96C22×D12C2312yes—
97C22×D12C2311yescharacteristic
98C22×C4×D3—21no—
99C22×C4×D3C2312yes—
100C22×Dic6D411yescharacteristic
101C22×Dic6C2311yescharacteristic
102C2×C3×M16D418yescomplemented
103C2×C3×M16C2312yes—
104C2×C3×M16C2312yes—
105C22×C3×C8C2312yes—
106C3×#117—21no—
107C3×#117—21no—
108C3×#117—21no—
109C3×#117—21no—
110C3×#117C2312yes—
111C3×#117C2312yes—
112C3×#117C2312yes—
113C3×#117C2312yes—
114C2×C3×C4⋊C4C2312yes—
115C2×C3×C4⋊C4C2312yes—
116C2×C3×C22⋊C4—21no—
117C2×C3×C22⋊C4—21no—
118C2×C3×C42—21no—
119C2×C3×C42—21no—
120#930—28no—
121C2×(C2×D3)⋊C4—24no—
122C2×#208—24no—
123C2×#208C2312yes—
124C2×#208C2312yes—
125C22×C3⋊C8C2312yes—
126#908—216no—
127#908—216no—
128#907—216no—
129D3×M16—216no—
130C2×D24—28no—
131C2×D24C2318yes—
132C2×#204—28no—
133C2×#204C2318yes—
134#901—216no—
135C2×ΓL(1,𝔽25)C2318yescomplemented
136C2×D3×C8C2318yescomplemented
137#897—24no—
138#897C2318yescomplemented
139#895—24no—
140#895—28no—
141#895C23116yes—
142#895C2318yescomplemented
143#886—24no—
144#886—24no—
145#882—24no—
146#882—28no—
147#882—28no—
148#882—24no—
149#875—22no—
150#875—24no—
151#875—24no—
152#875—22no—
153#874—24no—
154#874—24no—
155#874—22no—
156C4×D12—28no—
157#869—22no—
158C3×#107C2318yescomplemented
159C3×#106C2318yescomplemented
160#810C2318yescomplemented
161#809C23116yes—
162#809C2318yescomplemented
163C22×#41—31no—
164C22×#41—31no—
165C23×D4—31no—
166C2×AGL(1,ℤ/8)—34no—
167C2×AGL(1,ℤ/8)—34no—
168C2×AGL(1,ℤ/8)—68no—
169C22×SD16—32no—
170C22×D8—32no—
171#568—68no—
172C2×#131—34no—
173C2×#131—34no—
174C22×M16—31no—
175C22×M16—34no—
176C22×M16C2×D311yescharacteristic
177#537—34no—
178#536—32no—
179#536—61no—
180#535—34no—
181#535—34no—
182#535—32no—
183#535—61no—
184#535—62no—
185#535—62no—
186#534—62no—
187#532—62no—
188C2×#127—61no—
189C2×#124—61no—
190#525—34no—
191#525—62no—
192C2×#121—34no—
193C2×#120—62no—
194C2×#117C2×D311yescharacteristic
195C2×#117C2×D311yescharacteristic
196#469—316no—
197#468—316no—
198#466—316no—
199#465—316no—
200#428C2×D314yescomplemented
201#428C2×D314yescomplemented
202#427C2×D314yescomplemented
203#419—32no—
204#418—34no—
205#418—34no—
206#418—32no—
207#417—34no—
208#416—34no—
209C2×#102—34no—
210C24×C3C2×D411yescharacteristic
211C23×D3—22no—
212C23×D3—21no—
213C23×D3—22no—
214C23×D3—22no—
215C23×D3—44no—
216C23×D3—84no—
217C23×D3C2×D412yes—
218C22×C3×C4—22no—
219C22×C3×C4—22no—
220C22×C3×C4C2×D414yes—
221C22×C3×C4C2×D411yescharacteristic
222C22×C3×C4C2×D411yescharacteristic
223C22×C3×C4C2×D411yescharacteristic
224C22×C3×C4#4114yes—
225C22×C3×C4C2411yescharacteristic
226C22×C3×C4C2411yescharacteristic
227C22×C3×C4C2411yescharacteristic
228C2×#67—28no—
229C2×#67—22no—
230C2×#67—22no—
231C2×#67—44no—
232C2×#67C2×D414yes—
233C2×#67C2×D414yes—
234C22×Dic3—21no—
235C22×Dic3C2×D412yes—
236#235—28no—
237#235—28no—
238#235—24no—
239#235—24no—
240#235—24no—
241#235—24no—
242#235—24no—
243#235—24no—
244#235—24no—
245#235—24no—
246#235—44no—
247#235—44no—
248C2×D12—24no—
249C2×D12—24no—
250C2×D12—22no—
251C2×D12—24no—
252C2×D12—24no—
253C2×D12—24no—
254C2×D12—24no—
255C2×D12—24no—
256C2×D12—22no—
257C2×D12—22no—
258C2×D12—22no—
259C2×D12—21no—
260C2×D12—22no—
261C2×D12—28no—
262C2×D12—216no—
263C2×D12—44no—
264C2×D12—44no—
265C2×D12—44no—
266C2×D12—88no—
267C2×D12C2×D412yes—
268C2×D12C2×D412yes—
269C2×D12C2414yes—
270C2×D12C2412yes—
271C2×C4×D3—24no—
272C2×C4×D3—24no—
273C2×C4×D3—22no—
274C2×C4×D3—22no—
275C2×C4×D3—44no—
276C2×C4×D3C2×D414yes—
277C2×C4×D3C2414yes—
278C2×Dic6—24no—
279C2×Dic6—22no—
280C2×Dic6—22no—
281C2×Dic6—21no—
282C2×Dic6C2×D412yes—
283C2×Dic6C2×D412yes—
284C2×Dic6C2412yes—
285C3×M16C2×D418yes—
286C3×M16C2×D418yescomplemented
287C2×C3×C8C2×D418yescomplemented
288C2×C3×C8C2414yes—
289C3×C4⋊C4—22no—
290C3×C4⋊C4—22no—
291C3×C4⋊C4C2414yes—
292C3×C4⋊C4C2414yes—
293C3×C22⋊C4—22no—
294C3×C22⋊C4—22no—
295C3×C22⋊C4—22no—
296C3×C22⋊C4—22no—
297C3×C42—22no—
298C3×C42—22no—
299C3×C42—22no—
300C3×C42—22no—
301(C2×D3)⋊C4—28no—
302(C2×D3)⋊C4—28no—
303(C2×D3)⋊C4—216no—
304C4⋊Dic3—28no—
305#208—24no—
306#208—24no—
307C2×C3⋊C8—24no—
308C2×C3⋊C8C2414yes—
309D24—216no—
310#204—216no—
311ΓL(1,𝔽25)—216no—
312D3×C8—216no—
313C25—61no—
314C2×#41—32no—
315C2×#41—32no—
316C2×#41—32no—
317C2×#41—32no—
318C2×#41—32no—
319C2×#41—32no—
320C2×#41—64no—
321C2×#41—61no—
322C2×#41—61no—
323C2×#41—64no—
324C2×#41—64no—
325C2×#41—64no—
326C22×Q8—31no—
327C22×Q8—31no—
328C22×D4—32no—
329C22×D4—31no—
330C22×D4—32no—
331C22×D4—31no—
332C22×D4—32no—
333C22×D4—32no—
334C22×D4—64no—
335C22×D4—61no—
336C22×D4—64no—
337C22×D4—61no—
338C22×D4—61no—
339C22×D4—61no—
340C22×D4—64no—
341C23×C4—32no—
342C23×C4—61no—
343C23×C4C22×D311yescharacteristic
344AGL(1,ℤ/8)—616no—
345AGL(1,ℤ/8)—616no—
346C2×SD16—38no—
347C2×SD16—68no—
348C2×D8—38no—
349C2×D8—68no—
350#131—616no—
351#131—616no—
352C2×M16—32no—
353C2×M16—32no—
354C2×M16—38no—
355C2×M16—64no—
356C2×M16—616no—
357C2×M16D1218yescomplemented
358C2×M16C22×D312yes—
359C2×M16C22×D312yes—
360C22×C8—32no—
361C22×C8—38no—
362C22×C8C22×D312yes—
363C4⋊Q8—62no—
364#127—64no—
365#127—64no—
366#127—62no—
367#124—62no—
368#124—64no—
369#124—64no—
370#124—62no—
371#123—64no—
372#123—64no—
373C22⋊Q8—38no—
374C22⋊Q8—64no—
375#121—316no—
376#121—38no—
377#121—68no—
378#121—64no—
379#121—68no—
380#120—68no—
381#120—68no—
382#120—64no—
383#120—64no—
384C4×D4—68no—
385#117—21no—
386#117—21no—
387#117—21no—
388#117—21no—
389#117C22×D312yes—
390#117C22×D312yes—
391#117C22×D312yes—
392#117C22×D312yes—
393C2×C4⋊C4C22×D312yes—
394C2×C4⋊C4C22×D312yes—
395C2×C22⋊C4—21no—
396C2×C22⋊C4—21no—
397C2×C22⋊C4—64no—
398C2×C42—21no—
399C2×C42—21no—
400#107C22×D318yescomplemented
401#106C22×D318yescomplemented
402#103—38no—
403#102—316no—
404#102—38no—
405C23×C3—42no—
406C23×C3#12114yes—
407C23×C3C22×D411yescharacteristic
408C23×C3C22×D411yescharacteristic
409C23×C3C22×D411yescharacteristic
410C22×D3—24no—
411C22×D3—22no—
412C22×D3—24no—
413C22×D3—44no—
414C22×D3—48no—
415C22×D3—42no—
416C22×D3—44no—
417C22×D3—44no—
418C22×D3—44no—
419C22×D3—82no—
420C22×D3—84no—
421C22×D3—84no—
422C22×D3—816no—
423C22×D3C22×D414yes—
424C2×C3×C4—24no—
425C2×C3×C4—24no—
426C2×C3×C4—44no—
427C2×C3×C4—82no—
428C2×C3×C4#12114yes—
429C2×C3×C4#12114yes—
430C2×C3×C4#12114yes—
431C2×C3×C4C22×D412yes—
432C2×C3×C4C22×D412yes—
433C2×C3×C4C22×D412yes—
434C2×C3×C4C22×D411yescharacteristic
435C2×C3×C4C22×D411yescharacteristic
436C2×C3×C4C22×D411yescharacteristic
437C2×C3×C4C2×#4112yes—
438C2×C3×C4C2×#4112yes—
439C2×C3×C4C2×#4112yes—
440C2×C3×C4C2511yescharacteristic
441#67—28no—
442#67—28no—
443#67—44no—
444#67—44no—
445C2×Dic3—24no—
446C2×Dic3—22no—
447C2×Dic3—42no—
448C2×Dic3C22×D414yes—
449D12—24no—
450D12—24no—
451D12—24no—
452D12—22no—
453D12—24no—
454D12—22no—
455D12—44no—
456D12—44no—
457D12—44no—
458D12—42no—
459D12—88no—
460D12—816no—
461C4×D3—28no—
462C4×D3—24no—
463C4×D3—24no—
464C4×D3—44no—
465C4×D3—44no—
466Dic6—24no—
467Dic6—24no—
468Dic6—22no—
469Dic6—22no—
470Dic6—42no—
471C3×C8C22×D418yes—
472C3⋊C8—24no—
473C24—32no—
474C24—62no—
475C24—61no—
476C24—62no—
477C24—62no—
478C24—124no—
479C24—244no—
480C24D3×D411yescharacteristic
481#41—68no—
482#41—68no—
483#41—64no—
484#41—64no—
485#41—64no—
486#41—64no—
487#41—64no—
488#41—64no—
489#41—64no—
490#41—64no—
491#41—124no—
492#41—124no—
493C2×Q8—32no—
494C2×Q8—32no—
495C2×Q8—32no—
496C2×Q8—64no—
497C2×Q8—61no—
498C2×Q8—62no—
499C2×Q8—62no—
500C2×D4—34no—
501C2×D4—34no—
502C2×D4—32no—
503C2×D4—32no—
504C2×D4—32no—
505C2×D4—34no—
506C2×D4—68no—
507C2×D4—616no—
508C2×D4—68no—
509C2×D4—62no—
510C2×D4—62no—
511C2×D4—64no—
512C2×D4—62no—
513C2×D4—64no—
514C2×D4—64no—
515C2×D4—64no—
516C2×D4—62no—
517C2×D4—62no—
518C2×D4—62no—
519C2×D4—61no—
520C2×D4—64no—
521C2×D4—64no—
522C2×D4—64no—
523C2×D4—62no—
524C2×D4—124no—
525C2×D4—124no—
526C2×D4—124no—
527C2×D4—124no—
528C2×D4—248no—
529C22×C4—22no—
530C22×C4—22no—
531C22×C4—32no—
532C22×C4—34no—
533C22×C4—34no—
534C22×C4—64no—
535C22×C4—64no—
536C22×C4—61no—
537C22×C4—62no—
538C22×C4—62no—
539C22×C4—124no—
540C22×C4C2×D1214yes—
541C22×C4D3×D411yescharacteristic
542C22×C4D3×D411yescharacteristic
543C22×C4D3×D411yescharacteristic
544C22×C4#23914yes—
545C22×C4C23×D311yescharacteristic
546C22×C4C23×D311yescharacteristic
547C22×C4C23×D311yescharacteristic
548SD16—616no—
549D8—616no—
550M16—64no—
551M16—64no—
552M16—616no—
553M16C2×D1218yes—
554M16C2×D1218yescomplemented
555C2×C8—34no—
556C2×C8—64no—
557C2×C8—616no—
558C2×C8C2×D1218yescomplemented
559C2×C8C23×D314yes—
560C4⋊C4—22no—
561C4⋊C4—22no—
562C4⋊C4—68no—
563C4⋊C4C23×D314yes—
564C4⋊C4C23×D314yes—
565C22⋊C4—22no—
566C22⋊C4—22no—
567C22⋊C4—22no—
568C22⋊C4—22no—
569C22⋊C4—616no—
570C22⋊C4—68no—
571C22⋊C4—68no—
572C42—22no—
573C42—22no—
574C42—22no—
575C42—22no—
576C22×C3—22no—
577C22×C3—22no—
578C22×C3—22no—
579C22×C3—44no—
580C22×C3C2×#12112yes—
581C22×C3C2×#12112yes—
582C22×C3C2×#12112yes—
583C22×C3C23×D411yescharacteristic
584C2×D3—28no—
585C2×D3—24no—
586C2×D3—48no—
587C2×D3—44no—
588C2×D3—42no—
589C2×D3—44no—
590C2×D3—84no—
591C2×D3—88no—
592C2×D3—84no—
593C2×D3—88no—
594C2×D3—88no—
595C3×C4—44no—
596C3×C4—44no—
597C3×C4C2×#12112yes—
598C3×C4C2×#12112yes—
599C3×C4C2×#12112yes—
600C3×C4C23×D411yescharacteristic
601C3×C4C22×#4111yescharacteristic
602Dic3—24no—
603Dic3—42no—
604C23—34no—
605C23—42no—
606C23—64no—
607C23—62no—
608C23—64no—
609C23—124no—
610C23—128no—
611C23—122no—
612C23—124no—
613C23—124no—
614C23—124no—
615C23—244no—
616C23—244no—
617C23—242no—
618C23—2416no—
619C23#89514yes—
620C23C2×D3×D411yescharacteristic
621C23C2×D3×D411yescharacteristic
622C23C2×D3×D411yescharacteristic
623Q8—64no—
624Q8—64no—
625Q8—62no—
626Q8—62no—
627Q8—122no—
628D4—68no—
629D4—68no—
630D4—62no—
631D4—64no—
632D4—64no—
633D4—62no—
634D4—64no—
635D4—64no—
636D4—124no—
637D4—124no—
638D4—124no—
639D4—124no—
640D4—124no—
641D4—122no—
642D4—2416no—
643D4—248no—
644C2×C4—24no—
645C2×C4—24no—
646C2×C4—34no—
647C2×C4—44no—
648C2×C4—64no—
649C2×C4—64no—
650C2×C4—64no—
651C2×C4—68no—
652C2×C4—62no—
653C2×C4—82no—
654C2×C4—122no—
655C2×C4—124no—
656C2×C4—124no—
657C2×C4#89514yes—
658C2×C4#89514yes—
659C2×C4#89514yes—
660C2×C4C22×D1212yes—
661C2×C4C22×D1212yes—
662C2×C4C22×D1212yes—
663C2×C4C2×D3×D411yescharacteristic
664C2×C4C2×D3×D411yescharacteristic
665C2×C4C2×D3×D411yescharacteristic
666C2×C4C2×#23912yes—
667C2×C4C2×#23912yes—
668C2×C4C2×#23912yes—
669C2×C4C24×D311yescharacteristic
670C8—64no—
671C8C22×D1218yes—
672C2×C3—22no—
673C2×C3—22no—
674C2×C3—22no—
675C2×C3C22×#12111yescharacteristic
676C2×C3#358512yes—
677D3—44no—
678D3—82no—
679D3—84no—
680C22—22no—
681C22—22no—
682C22—22no—
683C22—44no—
684C22—68no—
685C22—64no—
686C22—128no—
687C22—124no—
688C22—122no—
689C22—124no—
690C22—244no—
691C22—248no—
692C22—248no—
693C22—248no—
694C22—244no—
695C22C2×#89512yes—
696C22C2×#89512yes—
697C22C2×#89512yes—
698C22C22×D3×D411yescharacteristic
699C4—44no—
700C4—44no—
701C4—64no—
702C4—122no—
703C4C2×#89512yes—
704C4C2×#89512yes—
705C4C2×#89512yes—
706C4C23×D1211yescharacteristic
707C4C22×#23911yescharacteristic
708C3#6045411yescharacteristic, complemented, Sylow
709C2—22no—
710C2—22no—
711C2—22no—
712C2—124no—
713C2—244no—
714C2—242no—
715C2C22×#89511yescharacteristic
716C2#8720012yes—
subgroupG/H
centerC22C22×D3×D4
commutator, cosocleC2×C3×C4C25
FrattiniC2×C4C24×D3
FittingC3×#2911C2
socleC22×C3C23×D4
hypercenter#2911D3
hypocenterC3#60454
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2911

Complements

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

NH ≅ G/Ntype
1. C22×#908C2semidirect
2. C2×#5745C2semidirect
3. #85735C2semidirect
4. C2×#5525C2semidirect
5. C3×#2911C2semidirect
6. #82213C2semidirect
7. C2×#5001C2semidirect
8. #79213C2semidirect
9. #79211C2semidirect
10. #79183C2semidirect
11. #79182C2semidirect
22. C2×#908C22semidirect
24. C2×#907C22semidirect
25. C2×D3×M16C22semidirect
26. C22×D24C22semidirect
27. C22×#204C22semidirect
28. C2×#901C22semidirect
30. #5526C22semidirect
35. #5525C22semidirect
36. #5525C22semidirect
38. #5524C22semidirect
39. #5510C22semidirect
40. C2×#895C22semidirect
46. C3×#428C22semidirect
47. C3×#428C22semidirect
48. C3×#427C22semidirect
49. #5004C22semidirect
50. #5002C22semidirect
51. #5001C22semidirect
52. #5001C22semidirect
53. #5001C22semidirect
54. #4966C22semidirect
55. C2×#809C22semidirect
56. #4886C22semidirect
57. #4883C22semidirect
58. #4865C22semidirect
59. #4864C22semidirect
68. #2911D3semidirect
102. C2×C3×M16D4semidirect
135. C2×ΓL(1,𝔽25)C23semidirect
136. C2×D3×C8C23semidirect
138. #897C23semidirect
142. #895C23semidirect
158. C3×#107C23semidirect
159. C3×#106C23semidirect
160. #810C23semidirect
162. #809C23semidirect
200. #428C2×D3semidirect
201. #428C2×D3semidirect
202. #427C2×D3semidirect
286. C3×M16C2×D4semidirect
287. C2×C3×C8C2×D4semidirect
357. C2×M16D12semidirect
400. #107C22×D3semidirect
401. #106C22×D3semidirect
554. M16C2×D12semidirect
558. C2×C8C2×D12semidirect
708. C3#60454semidirect

Maximal subgroups

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

subgroupindex
C22×#9082
C2×#57452
#857352
C2×#55252
C3×#29112
#822132
C2×#50012
#792132
#792112
#791832
#791822
#604543