Finite Group Databasebeta

← #11662450

Subgroups — #11662450

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×C3×AGL(1,ℤ/8)C211yesmaximal, characteristic, complemented
2#87268C211yesmaximal, characteristic, complemented
3C2×#5805C211yesmaximal, characteristic, complemented
4C2×#5745C211yesmaximal, characteristic, complemented
5#84072C211yesmaximal, characteristic, complemented
6C2×#5215C211yesmaximal, characteristic, complemented
7#83951C218yesmaximal, complemented
8#83950C218yesmaximal, complemented
9#83863C214yesmaximal, complemented
10#83836C214yesmaximal, complemented
11#83752C211yesmaximal, characteristic, complemented
12#60454—31nomaximal, Sylow
13C22×C3×#41C2211yescharacteristic
14C23×C3×D4C2211yescharacteristic
15C22×#235C2211yescharacteristic
16C2×C3×AGL(1,ℤ/8)—28no—
17C2×C3×AGL(1,ℤ/8)C2214yes—
18C2×C3×AGL(1,ℤ/8)C2214yescomplemented
19C22×C3×SD16C2212yescomplemented
20C22×C3×D8C2212yescomplemented
21C22×C3×M16C2211yescharacteristic
22C2×#953—22no—
23#5833C2214yescomplemented
24#5832—22no—
25#5830—21no—
26#5828C2214yescomplemented
27#5825—21no—
28#5816C2212yescomplemented
29#5805—22no—
30#5805—22no—
31#5805C2214yes—
32#5805C2214yescomplemented
33#5805C2212yescomplemented
34#5804—22no—
35C2×#940—21no—
36C2×#938C2214yescomplemented
37C2×#936—21no—
38#5791—22no—
39C2×#926C2211yescharacteristic
40C2×#926C2211yescharacteristic
41C22×#208C2211yescharacteristic
42#5745—28no—
43C2×#907C2214yescomplemented
44C2×D3×M16C2214yescomplemented
45C2×#901C2214yescomplemented
46#5233C2214yescomplemented
47#5232C2214yescomplemented
48#5225C2212yescomplemented
49#5215C2214yescomplemented
50#5215C2214yescomplemented
51#5215C2212yescomplemented
52C2×#832C2214yescomplemented
53#5174C22116yescomplemented
54#5171C22116yescomplemented
55#5159C22116yescomplemented
56#5157C22116yescomplemented
57#5121C2214yescomplemented
58#5121C2214yescomplemented
59#5106C2214yescomplemented
60C22×AGL(1,ℤ/8)D311yescharacteristic, complemented
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#2896—31no—
70C2×#418—31no—
71C25×C3—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×D4—24no—
81C22×C3×D4—21no—
82C22×C3×D4—21no—
83C22×C3×D4—21no—
84C22×C3×D4—24no—
85C22×C3×D4C2311yescharacteristic
86C22×C3×D4C2312yes—
87C22×C3×D4C2312yes—
88C22×C3×D4C2312yes—
89C23×C3×C4—21no—
90C23×C3×C4C2311yescharacteristic
91C22×#67—24no—
92C22×#67D412yes—
93C2×#235—24no—
94C2×#235—24no—
95C2×#235D412yes—
96C2×#235D412yes—
97C2×#235C2312yes—
98C2×#235C2312yes—
99C22×D12D411yescharacteristic
100C22×C4×D3C2312yes—
101C22×Dic6D411yescharacteristic
102C3×AGL(1,ℤ/8)—216no—
103C3×AGL(1,ℤ/8)—216no—
104C2×C3×SD16—28no—
105C2×C3×SD16C2318yes—
106C2×C3×D8—28no—
107C2×C3×D8C2318yes—
108C2×C3×M16D418yescomplemented
109C2×C3×M16C2312yes—
110C2×C3×M16C2312yes—
111C22×C3×C8C2312yes—
112#953—24no—
113#953—28no—
114#953—28no—
115C2×#217—21no—
116C2×#217—21no—
117C2×#217—24no—
118#947—22no—
119#946C2318yescomplemented
120#944—22no—
121#940—24no—
122#940—24no—
123#940—22no—
124#939—28no—
125#938—28no—
126#938C23116yes—
127#938C2318yescomplemented
128#937—24no—
129#936—24no—
130#936—24no—
131#936—22no—
132#935—24no—
133#930—24no—
134#929—24no—
135C4×#67—28no—
136#926—21no—
137#926—21no—
138#926—21no—
139#926—21no—
140#926C2312yes—
141#926C2312yes—
142#926C2312yes—
143#926C2312yes—
144C2×C4⋊Dic3C2312yes—
145C2×C4⋊Dic3C2312yes—
146#924—24no—
147C2×C4×Dic3—21no—
148C2×C4×Dic3—21no—
149C2×#208—24no—
150C2×#208C2312yes—
151C2×#208C2312yes—
152C22×C3⋊C8C2312yes—
153#907—216no—
154D3×M16—216no—
155#901—216no—
156C2×ΓL(1,𝔽25)C2318yescomplemented
157C2×D3×C8C2318yescomplemented
158#835C2318yescomplemented
159#832C23116yes—
160#832C2318yescomplemented
161#818C2318yescomplemented
162#817C2318yescomplemented
163C22×#41—31no—
164C22×#41C2×D311yescharacteristic
165C23×D4C2×D311yescharacteristic
166C2×AGL(1,ℤ/8)—28no—
167C2×AGL(1,ℤ/8)C2×D314yes—
168C2×AGL(1,ℤ/8)C2×D314yescomplemented
169C22×SD16C2×D312yescomplemented
170C22×D8C2×D312yescomplemented
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—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×#117—31no—
196#469—316no—
197#468—316no—
198#466—316no—
199#465—316no—
200#428—34no—
201#428—34no—
202#427—34no—
203#419—32no—
204#418—34no—
205#418—34no—
206#418—32no—
207#417—34no—
208#416—34no—
209C2×#102—34no—
210C24×C3—22no—
211C24×C3—22no—
212C24×C3—22no—
213C24×C3—21no—
214C24×C3—44no—
215C24×C3—84no—
216C24×C3C2×D411yescharacteristic
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—24no—
232C2×C3×D4—24no—
233C2×C3×D4—24no—
234C2×C3×D4—22no—
235C2×C3×D4—22no—
236C2×C3×D4—21no—
237C2×C3×D4—22no—
238C2×C3×D4—24no—
239C2×C3×D4—24no—
240C2×C3×D4—24no—
241C2×C3×D4—22no—
242C2×C3×D4—44no—
243C2×C3×D4—44no—
244C2×C3×D4—44no—
245C2×C3×D4—44no—
246C2×C3×D4C2414yes—
247C2×C3×D4C2412yes—
248C22×C3×C4—21no—
249C22×C3×C4—22no—
250C22×C3×C4—22no—
251C22×C3×C4—44no—
252C22×C3×C4C2×D411yescharacteristic
253C22×C3×C4C2×D411yescharacteristic
254C22×C3×C4C2×D411yescharacteristic
255C22×C3×C4C2×D414yes—
256C22×C3×C4#4114yes—
257C22×C3×C4C2411yescharacteristic
258C22×C3×C4C2411yescharacteristic
259C22×C3×C4C2411yescharacteristic
260C2×#67—28no—
261C2×#67—28no—
262C2×#67—216no—
263C2×#67—88no—
264C2×#67C2×D414yes—
265C2×#67C2×D414yes—
266C22×Dic3—22no—
267C22×Dic3—22no—
268C22×Dic3C2×D412yes—
269#235—28no—
270#235—28no—
271#235—24no—
272#235—24no—
273#235—24no—
274#235—24no—
275C2×D12—24no—
276C2×D12C2×D412yes—
277C2×D12C2×D412yes—
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×SD16—216no—
286C3×D8—216no—
287C3×M16C2×D418yes—
288C3×M16C2×D418yescomplemented
289C2×C3×C8C2×D418yescomplemented
290C2×C3×C8C2414yes—
291#217—22no—
292#217—22no—
293#217—22no—
294#217—22no—
295#217—28no—
296#217—216no—
297(C2×D3)⋊C4—28no—
298C4⋊Dic3—22no—
299C4⋊Dic3—22no—
300C4⋊Dic3C2414yes—
301C4⋊Dic3C2414yes—
302Dic3⋊C4—28no—
303C4×Dic3—22no—
304C4×Dic3—22no—
305C4×Dic3—22no—
306C4×Dic3—22no—
307#208—24no—
308#208—24no—
309C2×C3⋊C8—24no—
310C2×C3⋊C8C2414yes—
311ΓL(1,𝔽25)—216no—
312D3×C8—216no—
313C25—21no—
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—21no—
330C22×D4—21no—
331C22×D4—21no—
332C22×D4—24no—
333C22×D4—24no—
334C22×D4—32no—
335C22×D4—31no—
336C22×D4—64no—
337C22×D4C22×D312yes—
338C22×D4C22×D311yescharacteristic
339C22×D4C22×D312yes—
340C22×D4C22×D312yes—
341C23×C4—21no—
342C23×C4—32no—
343C23×C4C22×D311yescharacteristic
344AGL(1,ℤ/8)—216no—
345AGL(1,ℤ/8)—216no—
346C2×SD16—28no—
347C2×SD16C22×D318yes—
348C2×D8—28no—
349C2×D8C22×D318yes—
350#131—616no—
351#131—616no—
352C2×M16—32no—
353C2×M16—32no—
354C2×M16—38no—
355C2×M16—64no—
356C2×M16—616no—
357C2×M16#6718yescomplemented
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—316no—
376#121—38no—
377#121—64no—
378#121—68no—
379#121—68no—
380#120—64no—
381#120—68no—
382#120—68no—
383#120—64no—
384C4×D4—68no—
385#117—32no—
386#117—32no—
387#117—32no—
388#117—32no—
389#117—61no—
390#117—61no—
391#117—61no—
392#117—61no—
393C2×C4⋊C4—32no—
394C2×C4⋊C4—32no—
395C2×C22⋊C4—64no—
396C2×C22⋊C4—61no—
397C2×C22⋊C4—61no—
398C2×C42—61no—
399C2×C42—61no—
400#107—38no—
401#106—38no—
402#103—38no—
403#102—316no—
404#102—38no—
405C23×C3—22no—
406C23×C3—24no—
407C23×C3—42no—
408C23×C3—48no—
409C23×C3—42no—
410C23×C3—44no—
411C23×C3—44no—
412C23×C3—44no—
413C23×C3—82no—
414C23×C3—84no—
415C23×C3—84no—
416C23×C3—816no—
417C23×C3#12114yes—
418C23×C3C22×D411yescharacteristic
419C23×C3C22×D411yescharacteristic
420C23×C3C22×D411yescharacteristic
421C22×D3—24no—
422C22×D3—44no—
423C22×D3C22×D414yes—
424C3×Q8—22no—
425C3×Q8—22no—
426C3×Q8—42no—
427C3×D4—22no—
428C3×D4—24no—
429C3×D4—24no—
430C3×D4—22no—
431C3×D4—44no—
432C3×D4—44no—
433C3×D4—44no—
434C3×D4—44no—
435C3×D4—44no—
436C3×D4—42no—
437C2×C3×C4—22no—
438C2×C3×C4—42no—
439C2×C3×C4—44no—
440C2×C3×C4—44no—
441C2×C3×C4#12114yes—
442C2×C3×C4#12114yes—
443C2×C3×C4#12114yes—
444C2×C3×C4C22×D411yescharacteristic
445C2×C3×C4C22×D411yescharacteristic
446C2×C3×C4C22×D411yescharacteristic
447C2×C3×C4C22×D412yes—
448C2×C3×C4C22×D412yes—
449C2×C3×C4C22×D412yes—
450C2×C3×C4C2×#4112yes—
451C2×C3×C4C2×#4112yes—
452C2×C3×C4C2×#4112yes—
453C2×C3×C4C2511yescharacteristic
454#67—28no—
455#67—28no—
456#67—88no—
457#67—816no—
458C2×Dic3—24no—
459C2×Dic3—24no—
460C2×Dic3—24no—
461C2×Dic3—44no—
462C2×Dic3—82no—
463C2×Dic3C22×D414yes—
464D12—24no—
465D12—24no—
466C4×D3—28no—
467C4×D3—24no—
468C4×D3—24no—
469Dic6—24no—
470Dic6—24no—
471C3×C8C22×D418yes—
472C3⋊C8—24no—
473C24—22no—
474C24—21no—
475C24—22no—
476C24—22no—
477C24—32no—
478C24—44no—
479C24—84no—
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—22no—
505C2×D4—21no—
506C2×D4—24no—
507C2×D4—24no—
508C2×D4—24no—
509C2×D4—22no—
510C2×D4—22no—
511C2×D4—24no—
512C2×D4—24no—
513C2×D4—24no—
514C2×D4—34no—
515C2×D4—34no—
516C2×D4—32no—
517C2×D4—32no—
518C2×D4—44no—
519C2×D4—44no—
520C2×D4—44no—
521C2×D4—44no—
522C2×D4—68no—
523C2×D4—68no—
524C2×D4—616no—
525C2×D4—64no—
526C2×D4—248no—
527C2×D4C23×D312yes—
528C2×D4C23×D314yes—
529C22×C4—21no—
530C22×C4—22no—
531C22×C4—22no—
532C22×C4—32no—
533C22×C4—34no—
534C22×C4—34no—
535C22×C4—44no—
536C22×C4—64no—
537C22×C4—64no—
538C22×C4—62no—
539C22×C4—62no—
540C22×C4D3×D411yescharacteristic
541C22×C4D3×D411yescharacteristic
542C22×C4D3×D411yescharacteristic
543C22×C4#23714yes—
544C22×C4C2×#6714yes—
545C22×C4C23×D311yescharacteristic
546C22×C4C23×D311yescharacteristic
547C22×C4C23×D311yescharacteristic
548SD16—216no—
549D8—216no—
550M16—64no—
551M16—64no—
552M16—616no—
553M16C2×#6718yes—
554M16C2×#6718yescomplemented
555C2×C8—34no—
556C2×C8—64no—
557C2×C8—616no—
558C2×C8C2×#6718yescomplemented
559C2×C8C23×D314yes—
560C4⋊C4—34no—
561C4⋊C4—34no—
562C4⋊C4—68no—
563C4⋊C4—62no—
564C4⋊C4—62no—
565C22⋊C4—68no—
566C22⋊C4—616no—
567C22⋊C4—68no—
568C22⋊C4—62no—
569C22⋊C4—62no—
570C22⋊C4—62no—
571C22⋊C4—62no—
572C42—62no—
573C42—62no—
574C42—62no—
575C42—62no—
576C22×C3—22no—
577C22×C3—22no—
578C22×C3—22no—
579C22×C3—44no—
580C22×C3—48no—
581C22×C3—44no—
582C22×C3—42no—
583C22×C3—44no—
584C22×C3—88no—
585C22×C3—84no—
586C22×C3—88no—
587C22×C3—88no—
588C22×C3C2×#12112yes—
589C22×C3C2×#12112yes—
590C22×C3C2×#12112yes—
591C22×C3C23×D411yescharacteristic
592C2×D3—28no—
593C2×D3—24no—
594C2×D3—84no—
595C3×C4—42no—
596C3×C4C2×#12112yes—
597C3×C4C2×#12112yes—
598C3×C4C2×#12112yes—
599C3×C4C23×D411yescharacteristic
600C3×C4C22×#4111yescharacteristic
601Dic3—24no—
602Dic3—44no—
603Dic3—44no—
604C23—22no—
605C23—24no—
606C23—34no—
607C23—42no—
608C23—48no—
609C23—42no—
610C23—44no—
611C23—44no—
612C23—44no—
613C23—64no—
614C23—84no—
615C23—84no—
616C23—82no—
617C23—816no—
618C23—124no—
619C23#93814yes—
620C23C2×D3×D411yescharacteristic
621C23C2×D3×D411yescharacteristic
622C23C2×D3×D411yescharacteristic
623Q8—22no—
624Q8—22no—
625Q8—42no—
626Q8—64no—
627Q8—64no—
628D4—24no—
629D4—22no—
630D4—22no—
631D4—24no—
632D4—44no—
633D4—44no—
634D4—44no—
635D4—44no—
636D4—44no—
637D4—42no—
638D4—68no—
639D4—68no—
640D4—64no—
641D4—64no—
642D4—2416no—
643D4—248no—
644C2×C4—22no—
645C2×C4—34no—
646C2×C4—42no—
647C2×C4—44no—
648C2×C4—44no—
649C2×C4—64no—
650C2×C4—64no—
651C2×C4—64no—
652C2×C4—68no—
653C2×C4—64no—
654C2×C4—64no—
655C2×C4—124no—
656C2×C4—242no—
657C2×C4#93814yes—
658C2×C4#93814yes—
659C2×C4#93814yes—
660C2×C4C2×D3×D411yescharacteristic
661C2×C4C2×D3×D411yescharacteristic
662C2×C4C2×D3×D411yescharacteristic
663C2×C4C2×#23712yes—
664C2×C4C2×#23712yes—
665C2×C4C2×#23712yes—
666C2×C4C22×#6712yes—
667C2×C4C22×#6712yes—
668C2×C4C22×#6712yes—
669C2×C4C24×D311yescharacteristic
670C8—64no—
671C8C22×#6718yes—
672C2×C3—22no—
673C2×C3—22no—
674C2×C3—22no—
675C2×C3—82no—
676C2×C3—84no—
677C2×C3C22×#12111yescharacteristic
678C2×C3#358512yes—
679D3—44no—
680C22—22no—
681C22—22no—
682C22—22no—
683C22—44no—
684C22—48no—
685C22—44no—
686C22—42no—
687C22—44no—
688C22—68no—
689C22—64no—
690C22—88no—
691C22—88no—
692C22—88no—
693C22—84no—
694C22—244no—
695C22C2×#93812yes—
696C22C2×#93812yes—
697C22C2×#93812yes—
698C22C22×D3×D411yescharacteristic
699C4—42no—
700C4—64no—
701C4—124no—
702C4—124no—
703C4C2×#93812yes—
704C4C2×#93812yes—
705C4C2×#93812yes—
706C4C22×#23711yescharacteristic
707C4C23×#6711yescharacteristic
708C3#6045411yescharacteristic, complemented, Sylow
709C2—22no—
710C2—22no—
711C2—22no—
712C2—84no—
713C2—82no—
714C2—124no—
715C2#8719212yes—
716C2C22×#93811yescharacteristic
subgroupG/H
centerC22C22×D3×D4
commutator, cosocleC2×C3×C4C25
FrattiniC2×C4C24×D3
FittingC22×C3×AGL(1,ℤ/8)C2
socleC22×C3C23×D4
hypercenterC22×AGL(1,ℤ/8)D3
hypocenterC3#60454
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ C22×AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. C22×C3×AGL(1,ℤ/8)C2semidirect
2. #87268C2semidirect
3. C2×#5805C2semidirect
4. C2×#5745C2semidirect
5. #84072C2semidirect
6. C2×#5215C2semidirect
7. #83951C2semidirect
8. #83950C2semidirect
9. #83863C2semidirect
10. #83836C2semidirect
11. #83752C2semidirect
18. C2×C3×AGL(1,ℤ/8)C22semidirect
19. C22×C3×SD16C22semidirect
20. C22×C3×D8C22semidirect
23. #5833C22semidirect
26. #5828C22semidirect
28. #5816C22semidirect
32. #5805C22semidirect
33. #5805C22semidirect
36. C2×#938C22semidirect
43. C2×#907C22semidirect
44. C2×D3×M16C22semidirect
45. C2×#901C22semidirect
46. #5233C22semidirect
47. #5232C22semidirect
48. #5225C22semidirect
49. #5215C22semidirect
50. #5215C22semidirect
51. #5215C22semidirect
52. C2×#832C22semidirect
53. #5174C22semidirect
54. #5171C22semidirect
55. #5159C22semidirect
56. #5157C22semidirect
57. #5121C22semidirect
58. #5121C22semidirect
59. #5106C22semidirect
60. C22×AGL(1,ℤ/8)D3semidirect
108. C2×C3×M16D4semidirect
119. #946C23semidirect
127. #938C23semidirect
156. C2×ΓL(1,𝔽25)C23semidirect
157. C2×D3×C8C23semidirect
158. #835C23semidirect
160. #832C23semidirect
161. #818C23semidirect
162. #817C23semidirect
168. C2×AGL(1,ℤ/8)C2×D3semidirect
169. C22×SD16C2×D3semidirect
170. C22×D8C2×D3semidirect
288. C3×M16C2×D4semidirect
289. C2×C3×C8C2×D4semidirect
357. C2×M16#67semidirect
554. M16C2×#67semidirect
558. C2×C8C2×#67semidirect
708. C3#60454semidirect

Maximal subgroups

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

subgroupindex
C22×C3×AGL(1,ℤ/8)2
#872682
C2×#58052
C2×#57452
#840722
C2×#52152
#839512
#839502
#838632
#838362
#837522
#604543