Finite Group Databasebeta

← #11662457

Subgroups — #11662457

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
1#87268C211yesmaximal, characteristic, complemented
2C22×#949C211yesmaximal, characteristic, complemented
3C2×#5745C211yesmaximal, characteristic, complemented
4C2×#5525C211yesmaximal, characteristic, complemented
5C3×#2896C211yesmaximal, characteristic, complemented
6C2×#5131C211yesmaximal, characteristic, complemented
7#82207C211yesmaximal, characteristic, complemented
8#78963C214yesmaximal, complemented
9#78932C214yesmaximal, complemented
10#78888C218yesmaximal, complemented
11#78840C218yesmaximal, complemented
12#60454—31nomaximal, Sylow
13C22×C3×#41C2211yescharacteristic
14C22×#235C2211yescharacteristic
15C23×D12C2211yescharacteristic
16C22×C3×M16C2211yescharacteristic
17C2×C3×#117C2211yescharacteristic
18#5833C2214yescomplemented
19#5832—22no—
20#5830—21no—
21#5828C2214yescomplemented
22#5825—21no—
23C2×#949—28no—
24C2×#949C2214yes—
25C2×#949C2214yescomplemented
26#5816C2212yescomplemented
27C22×#215C2212yescomplemented
28#5805C2212yescomplemented
29#5804—22no—
30C22×#213C2212yescomplemented
31#5791—22no—
32C2×#926C2211yescharacteristic
33C22×#208C2211yescharacteristic
34#5745—28no—
35C2×#907C2214yescomplemented
36C2×D3×M16C2214yescomplemented
37C2×#901C2214yescomplemented
38#5525—22no—
39#5525—22no—
40#5525C2214yes—
41#5525C2214yescomplemented
42C2×#895C2214yescomplemented
43C2×#882—22no—
44C2×#875—21no—
45C2×#874—21no—
46C3×#419C2212yescomplemented
47C3×#418C2212yescomplemented
48C3×#417C2214yescomplemented
49C3×#416C2214yescomplemented
50#5131C2214yescomplemented
51#5131C2214yescomplemented
52C2×#821C2214yescomplemented
53#4998C2214yescomplemented
54#4995C2214yescomplemented
55#4964C2214yescomplemented
56#4812C22116yescomplemented
57#4807C22116yescomplemented
58#4782C22116yescomplemented
59#4775C22116yescomplemented
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#2911—31no—
69#2896D311yescharacteristic, complemented
70C2×#418—31no—
71C24×D3—21no—
72C2×C3×#41—21no—
73C2×C3×#41—21no—
74C2×C3×#41—24no—
75C2×C3×#41—24no—
76C2×C3×#41C2312yes—
77C2×C3×#41C2312yes—
78C22×C3×Q8C2311yescharacteristic
79C22×C3×D4—21no—
80C22×C3×D4C2311yescharacteristic
81C23×C3×C4—21no—
82C23×C3×C4C2311yescharacteristic
83C22×#67D412yes—
84C2×#235—24no—
85C2×#235—24no—
86C2×#235D412yes—
87C2×#235D412yes—
88C2×#235C2312yes—
89C2×#235C2312yes—
90C22×D12—24no—
91C22×D12—24no—
92C22×D12—21no—
93C22×D12—21no—
94C22×D12—21no—
95C22×D12—24no—
96C22×D12D411yescharacteristic
97C22×D12C2312yes—
98C22×D12C2312yes—
99C22×D12C2312yes—
100C22×C4×D3C2312yes—
101C22×Dic6D411yescharacteristic
102C2×C3×M16—24no—
103C2×C3×M16C2312yes—
104C2×C3×M16C2312yes—
105C22×C3×C8C2312yes—
106C3×#117—21no—
107C3×#117—21no—
108C3×#117C2312yes—
109C3×#117C2312yes—
110C2×C3×C4⋊C4C2312yes—
111C2×C3×C22⋊C4—21no—
112C2×C3×C42—21no—
113#953—24no—
114C2×#217—21no—
115#949—216no—
116#949—216no—
117#947—22no—
118#946C2318yescomplemented
119#944—22no—
120C2×#215—28no—
121C2×#215C2318yes—
122#940—22no—
123#939—28no—
124#938C2318yescomplemented
125#937—24no—
126#936—22no—
127#935—24no—
128C2×#213—28no—
129C2×#213C2318yes—
130#930—24no—
131#929—24no—
132C4×#67—28no—
133C2×(C2×D3)⋊C4—24no—
134#926—21no—
135#926—21no—
136#926C2312yes—
137#926C2312yes—
138C2×C4⋊Dic3C2312yes—
139#924—24no—
140C2×C4×Dic3—21no—
141C2×#208D418yescomplemented
142C2×#208C2312yes—
143C2×#208C2312yes—
144C22×C3⋊C8C2312yes—
145#907—216no—
146D3×M16—216no—
147#901—216no—
148C2×ΓL(1,𝔽25)C2318yescomplemented
149C2×D3×C8C2318yescomplemented
150#895—28no—
151#895C23116yes—
152#882—28no—
153#882—28no—
154#875—24no—
155#875—24no—
156#874—24no—
157#874—24no—
158C3×#103C2318yescomplemented
159C3×#102C2318yescomplemented
160#821C23116yes—
161#808C2318yescomplemented
162#807C2318yescomplemented
163C22×#41—31no—
164C22×#41C2×D311yescharacteristic
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—32no—
181#535—34no—
182#535—34no—
183#535—62no—
184#535—62no—
185#535—61no—
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×#117—31no—
195C2×#117C2×D311yescharacteristic
196#469—316no—
197#468—316no—
198#466—316no—
199#465—316no—
200#428—34no—
201#428—34no—
202#427—34no—
203#419C2×D312yescomplemented
204#418—34no—
205#418—34no—
206#418C2×D312yescomplemented
207#417C2×D314yescomplemented
208#416C2×D314yescomplemented
209C2×#102—34no—
210C24×C3—21no—
211C24×C3C2×D411yescharacteristic
212C23×D3—22no—
213C23×D3—22no—
214C23×D3—22no—
215C23×D3—44no—
216C23×D3—84no—
217C23×D3C2×D412yes—
218C3×#41—24no—
219C3×#41—24no—
220C3×#41—24no—
221C3×#41—24no—
222C3×#41—44no—
223C3×#41—44no—
224C2×C3×Q8—21no—
225C2×C3×Q8—22no—
226C2×C3×Q8—22no—
227C2×C3×Q8C2412yes—
228C2×C3×D4—22no—
229C2×C3×D4—22no—
230C2×C3×D4—22no—
231C2×C3×D4—21no—
232C2×C3×D4—22no—
233C2×C3×D4—44no—
234C2×C3×D4C2412yes—
235C22×C3×C4—22no—
236C22×C3×C4—21no—
237C22×C3×C4—22no—
238C22×C3×C4—22no—
239C22×C3×C4—44no—
240C22×C3×C4C2×D411yescharacteristic
241C22×C3×C4C2×D411yescharacteristic
242C22×C3×C4C2×D411yescharacteristic
243C22×C3×C4C2×D414yes—
244C22×C3×C4#4114yes—
245C22×C3×C4C2411yescharacteristic
246C22×C3×C4C2411yescharacteristic
247C22×C3×C4C2411yescharacteristic
248C2×#67—28no—
249C2×#67—28no—
250C2×#67C2×D414yes—
251C2×#67C2×D414yes—
252C22×Dic3—22no—
253C22×Dic3C2×D412yes—
254#235—28no—
255#235—28no—
256#235—24no—
257#235—24no—
258#235—24no—
259#235—24no—
260C2×D12—24no—
261C2×D12—24no—
262C2×D12—24no—
263C2×D12—24no—
264C2×D12—24no—
265C2×D12—24no—
266C2×D12—24no—
267C2×D12—22no—
268C2×D12—22no—
269C2×D12—22no—
270C2×D12—216no—
271C2×D12—44no—
272C2×D12—44no—
273C2×D12—44no—
274C2×D12—88no—
275C2×D12C2×D412yes—
276C2×D12C2×D412yes—
277C2×D12C2414yes—
278C2×C4×D3—24no—
279C2×C4×D3—24no—
280C2×C4×D3C2×D414yes—
281C2×C4×D3C2414yes—
282C2×Dic6—24no—
283C2×Dic6C2×D412yes—
284C2×Dic6C2×D412yes—
285C3×M16—24no—
286C3×M16—24no—
287C2×C3×C8—24no—
288C2×C3×C8C2414yes—
289C3×C4⋊C4—22no—
290C3×C4⋊C4C2414yes—
291C3×C22⋊C4—22no—
292C3×C22⋊C4—22no—
293C3×C42—22no—
294C3×C42—22no—
295#217—22no—
296#217—22no—
297#217—28no—
298#215—216no—
299#213—216no—
300(C2×D3)⋊C4—28no—
301(C2×D3)⋊C4—216no—
302C4⋊Dic3—22no—
303C4⋊Dic3C2414yes—
304Dic3⋊C4—28no—
305C4×Dic3—22no—
306C4×Dic3—22no—
307#208C2×D418yes—
308#208C2×D418yescomplemented
309C2×C3⋊C8C2×D418yescomplemented
310C2×C3⋊C8C2414yes—
311ΓL(1,𝔽25)—216no—
312D3×C8—216no—
313C25—61no—
314C2×#41—21no—
315C2×#41—21no—
316C2×#41—24no—
317C2×#41—24no—
318C2×#41—32no—
319C2×#41—32no—
320C2×#41—32no—
321C2×#41—32no—
322C2×#41—64no—
323C2×#41—64no—
324C2×#41C22×D312yes—
325C2×#41C22×D312yes—
326C22×Q8—31no—
327C22×Q8C22×D311yescharacteristic
328C22×D4—21no—
329C22×D4—32no—
330C22×D4—31no—
331C22×D4—32no—
332C22×D4—32no—
333C22×D4—32no—
334C22×D4—64no—
335C22×D4—64no—
336C22×D4—64no—
337C22×D4—61no—
338C22×D4—61no—
339C22×D4—61no—
340C22×D4C22×D311yescharacteristic
341C23×C4—21no—
342C23×C4—32no—
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—24no—
353C2×M16—38no—
354C2×M16—32no—
355C2×M16—32no—
356C2×M16—38no—
357C2×M16—616no—
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—64no—
368#124—64no—
369#124—62no—
370#124—62no—
371#123—64no—
372#123—64no—
373C22⋊Q8—38no—
374C22⋊Q8—64no—
375#121—38no—
376#121—316no—
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—32no—
388#117—32no—
389#117—61no—
390#117—61no—
391#117C22×D312yes—
392#117C22×D312yes—
393C2×C4⋊C4—32no—
394C2×C4⋊C4C22×D312yes—
395C2×C22⋊C4—21no—
396C2×C22⋊C4—64no—
397C2×C22⋊C4—61no—
398C2×C42—21no—
399C2×C42—61no—
400#107—38no—
401#106—38no—
402#103C22×D318yescomplemented
403#102—316no—
404#102C22×D318yescomplemented
405C23×C3—22no—
406C23×C3—42no—
407C23×C3—42no—
408C23×C3—82no—
409C23×C3#12114yes—
410C23×C3C22×D411yescharacteristic
411C23×C3C22×D411yescharacteristic
412C23×C3C22×D411yescharacteristic
413C22×D3—24no—
414C22×D3—24no—
415C22×D3—44no—
416C22×D3—48no—
417C22×D3—44no—
418C22×D3—44no—
419C22×D3—44no—
420C22×D3—84no—
421C22×D3—84no—
422C22×D3—816no—
423C22×D3C22×D414yes—
424C3×Q8—22no—
425C3×Q8—22no—
426C3×Q8—42no—
427C3×D4—22no—
428C3×D4—22no—
429C3×D4—44no—
430C3×D4—44no—
431C3×D4—42no—
432C2×C3×C4—22no—
433C2×C3×C4—24no—
434C2×C3×C4—44no—
435C2×C3×C4—42no—
436C2×C3×C4—44no—
437C2×C3×C4—44no—
438C2×C3×C4#12114yes—
439C2×C3×C4#12114yes—
440C2×C3×C4#12114yes—
441C2×C3×C4C22×D411yescharacteristic
442C2×C3×C4C22×D411yescharacteristic
443C2×C3×C4C22×D411yescharacteristic
444C2×C3×C4C22×D412yes—
445C2×C3×C4C22×D412yes—
446C2×C3×C4C22×D412yes—
447C2×C3×C4C2×#4112yes—
448C2×C3×C4C2×#4112yes—
449C2×C3×C4C2×#4112yes—
450C2×C3×C4C2511yescharacteristic
451#67—28no—
452#67—28no—
453#67—88no—
454C2×Dic3—24no—
455C2×Dic3—24no—
456C2×Dic3—82no—
457C2×Dic3C22×D414yes—
458D12—24no—
459D12—24no—
460D12—24no—
461D12—24no—
462D12—44no—
463D12—44no—
464D12—44no—
465D12—816no—
466C4×D3—28no—
467C4×D3—24no—
468C4×D3—24no—
469Dic6—24no—
470Dic6—24no—
471C3×C8—24no—
472C3⋊C8C22×D418yes—
473C24—21no—
474C24—32no—
475C24—62no—
476C24—62no—
477C24—62no—
478C24—124no—
479C24—244no—
480C24D3×D411yescharacteristic
481#41—24no—
482#41—24no—
483#41—24no—
484#41—24no—
485#41—44no—
486#41—44no—
487#41—68no—
488#41—68no—
489#41—64no—
490#41—64no—
491#41—64no—
492#41—64no—
493C2×Q8—21no—
494C2×Q8—22no—
495C2×Q8—22no—
496C2×Q8—32no—
497C2×Q8—32no—
498C2×Q8—64no—
499C2×Q8C23×D312yes—
500C2×D4—22no—
501C2×D4—22no—
502C2×D4—22no—
503C2×D4—22no—
504C2×D4—21no—
505C2×D4—34no—
506C2×D4—34no—
507C2×D4—32no—
508C2×D4—32no—
509C2×D4—34no—
510C2×D4—44no—
511C2×D4—616no—
512C2×D4—68no—
513C2×D4—68no—
514C2×D4—64no—
515C2×D4—64no—
516C2×D4—64no—
517C2×D4—64no—
518C2×D4—64no—
519C2×D4—64no—
520C2×D4—64no—
521C2×D4—62no—
522C2×D4—62no—
523C2×D4—62no—
524C2×D4—124no—
525C2×D4—124no—
526C2×D4—124no—
527C2×D4—248no—
528C2×D4C23×D312yes—
529C22×C4—22no—
530C22×C4—21no—
531C22×C4—22no—
532C22×C4—22no—
533C22×C4—32no—
534C22×C4—34no—
535C22×C4—34no—
536C22×C4—44no—
537C22×C4—64no—
538C22×C4—64no—
539C22×C4—62no—
540C22×C4#23514yes—
541C22×C4D3×D411yescharacteristic
542C22×C4D3×D411yescharacteristic
543C22×C4D3×D411yescharacteristic
544C22×C4D3×D414yes—
545C22×C4C23×D311yescharacteristic
546C22×C4C23×D311yescharacteristic
547C22×C4C23×D311yescharacteristic
548SD16—616no—
549D8—616no—
550M16—24no—
551M16—24no—
552M16—38no—
553M16—38no—
554M16—616no—
555C2×C8—24no—
556C2×C8—38no—
557C2×C8—34no—
558C2×C8—616no—
559C2×C8C23×D314yes—
560C4⋊C4—22no—
561C4⋊C4—34no—
562C4⋊C4—68no—
563C4⋊C4—62no—
564C4⋊C4C23×D314yes—
565C22⋊C4—22no—
566C22⋊C4—22no—
567C22⋊C4—616no—
568C22⋊C4—68no—
569C22⋊C4—68no—
570C22⋊C4—62no—
571C22⋊C4—62no—
572C42—22no—
573C42—22no—
574C42—62no—
575C42—62no—
576C22×C3—22no—
577C22×C3—22no—
578C22×C3—22no—
579C22×C3—44no—
580C22×C3—44no—
581C22×C3—42no—
582C22×C3—84no—
583C22×C3C2×#12112yes—
584C22×C3C2×#12112yes—
585C22×C3C2×#12112yes—
586C22×C3C23×D411yescharacteristic
587C2×D3—28no—
588C2×D3—24no—
589C2×D3—48no—
590C2×D3—44no—
591C2×D3—84no—
592C2×D3—88no—
593C2×D3—88no—
594C2×D3—88no—
595C3×C4—44no—
596C3×C4—42no—
597C3×C4C2×#12112yes—
598C3×C4C2×#12112yes—
599C3×C4C2×#12112yes—
600C3×C4C23×D411yescharacteristic
601C3×C4C22×#4111yescharacteristic
602Dic3—24no—
603Dic3—44no—
604C23—22no—
605C23—34no—
606C23—42no—
607C23—42no—
608C23—64no—
609C23—64no—
610C23—82no—
611C23—124no—
612C23—128no—
613C23—124no—
614C23—124no—
615C23—124no—
616C23—244no—
617C23—244no—
618C23—2416no—
619C23#88414yes—
620C23C2×D3×D411yescharacteristic
621C23C2×D3×D411yescharacteristic
622C23C2×D3×D411yescharacteristic
623Q8—22no—
624Q8—22no—
625Q8—42no—
626Q8—64no—
627Q8—64no—
628D4—22no—
629D4—22no—
630D4—44no—
631D4—44no—
632D4—42no—
633D4—68no—
634D4—68no—
635D4—64no—
636D4—64no—
637D4—64no—
638D4—64no—
639D4—124no—
640D4—124no—
641D4—124no—
642D4—2416no—
643D4—248no—
644C2×C4—24no—
645C2×C4—22no—
646C2×C4—34no—
647C2×C4—44no—
648C2×C4—42no—
649C2×C4—44no—
650C2×C4—44no—
651C2×C4—64no—
652C2×C4—64no—
653C2×C4—64no—
654C2×C4—68no—
655C2×C4—64no—
656C2×C4—242no—
657C2×C4#88414yes—
658C2×C4#88414yes—
659C2×C4#88414yes—
660C2×C4C2×#23512yes—
661C2×C4C2×#23512yes—
662C2×C4C2×#23512yes—
663C2×C4C2×D3×D412yes—
664C2×C4C2×D3×D412yes—
665C2×C4C2×D3×D412yes—
666C2×C4C2×D3×D411yescharacteristic
667C2×C4C2×D3×D411yescharacteristic
668C2×C4C2×D3×D411yescharacteristic
669C2×C4C24×D311yescharacteristic
670C8—24no—
671C8—38no—
672C2×C3—22no—
673C2×C3—22no—
674C2×C3—22no—
675C2×C3—82no—
676C2×C3C22×#12111yescharacteristic
677C2×C3#358512yes—
678D3—44no—
679D3—84no—
680C22—22no—
681C22—22no—
682C22—22no—
683C22—44no—
684C22—44no—
685C22—42no—
686C22—68no—
687C22—64no—
688C22—84no—
689C22—128no—
690C22—124no—
691C22—244no—
692C22—248no—
693C22—248no—
694C22—248no—
695C22C2×#88412yes—
696C22C2×#88412yes—
697C22C2×#88412yes—
698C22C22×D3×D411yescharacteristic
699C4—44no—
700C4—42no—
701C4—64no—
702C4—124no—
703C4C2×#88412yes—
704C4C2×#88412yes—
705C4C2×#88412yes—
706C4C22×#23511yescharacteristic
707C4C22×D3×D411yescharacteristic
708C3#6045411yescharacteristic, complemented, Sylow
709C2—22no—
710C2—22no—
711C2—22no—
712C2—82no—
713C2—124no—
714C2—244no—
715C2C22×#88411yescharacteristic
716C2#8720012yes—
subgroupG/H
centerC22C22×D3×D4
commutator, cosocleC2×C3×C4C25
FrattiniC2×C4C24×D3
FittingC3×#2896C2
socleC22×C3C23×D4
hypercenter#2896D3
hypocenterC3#60454
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2896

Complements

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

NH ≅ G/Ntype
1. #87268C2semidirect
2. C22×#949C2semidirect
3. C2×#5745C2semidirect
4. C2×#5525C2semidirect
5. C3×#2896C2semidirect
6. C2×#5131C2semidirect
7. #82207C2semidirect
8. #78963C2semidirect
9. #78932C2semidirect
10. #78888C2semidirect
11. #78840C2semidirect
18. #5833C22semidirect
21. #5828C22semidirect
25. C2×#949C22semidirect
26. #5816C22semidirect
27. C22×#215C22semidirect
28. #5805C22semidirect
30. C22×#213C22semidirect
35. C2×#907C22semidirect
36. C2×D3×M16C22semidirect
37. C2×#901C22semidirect
41. #5525C22semidirect
42. C2×#895C22semidirect
46. C3×#419C22semidirect
47. C3×#418C22semidirect
48. C3×#417C22semidirect
49. C3×#416C22semidirect
50. #5131C22semidirect
51. #5131C22semidirect
52. C2×#821C22semidirect
53. #4998C22semidirect
54. #4995C22semidirect
55. #4964C22semidirect
56. #4812C22semidirect
57. #4807C22semidirect
58. #4782C22semidirect
59. #4775C22semidirect
69. #2896D3semidirect
118. #946C23semidirect
124. #938C23semidirect
141. C2×#208D4semidirect
148. C2×ΓL(1,𝔽25)C23semidirect
149. C2×D3×C8C23semidirect
158. C3×#103C23semidirect
159. C3×#102C23semidirect
161. #808C23semidirect
162. #807C23semidirect
203. #419C2×D3semidirect
206. #418C2×D3semidirect
207. #417C2×D3semidirect
208. #416C2×D3semidirect
308. #208C2×D4semidirect
309. C2×C3⋊C8C2×D4semidirect
402. #103C22×D3semidirect
404. #102C22×D3semidirect
708. C3#60454semidirect

Maximal subgroups

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

subgroupindex
#872682
C22×#9492
C2×#57452
C2×#55252
C3×#28962
C2×#51312
#822072
#789632
#789322
#788882
#788402
#604543