Finite Group Databasebeta

← #10822158

Subgroups — #10822158

Order 768.

2106 conjugacy classes of subgroups.

The subgroup lattice has 758 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
3C22×#932C211yesmaximal, characteristic, complemented
4C2×#5236C211yesmaximal, characteristic, complemented
5#84044C211yesmaximal, characteristic, complemented
6C2×#5139C211yesmaximal, characteristic, complemented
7#72098C218yesmaximal, complemented
8#71987C211yesmaximal, characteristic, complemented
9#23407—31nomaximal, Sylow
10C22×C3×#41C2211yescharacteristic
11C23×C3×D4C2211yescharacteristic
12C22×#235C2211yescharacteristic
13C2×C3×AGL(1,ℤ/8)—22no—
14C2×C3×AGL(1,ℤ/8)—24no—
15C2×C3×AGL(1,ℤ/8)—24no—
16C2×C3×AGL(1,ℤ/8)C2214yescomplemented
17C22×C3×SD16—21no—
18C22×C3×D8—21no—
19C22×C3×M16C2211yescharacteristic
20#5833C2214yescomplemented
21#5832—22no—
22#5830—21no—
23#5828—22no—
24#5825—21no—
25#5816—21no—
26#5805—21no—
27#5804—22no—
28C22×#214—21no—
29C2×#932—24no—
30C2×#932—22no—
31C2×#932—24no—
32C2×#932C2214yescomplemented
33C22×#213—21no—
34#5791—22no—
35C2×#926C2211yescharacteristic
36C2×#926C2211yescharacteristic
37C22×#208C2211yescharacteristic
38#5247—22no—
39#5236—24no—
40C2×#837C2214yescomplemented
41#5221—22no—
42#5221C2214yescomplemented
43#5139—24no—
44C2×#825C2214yescomplemented
45#4554C2214yescomplemented
46C22×AGL(1,ℤ/8)—31no—
47C22×AGL(1,ℤ/8)D311yescharacteristic, complemented
48#3449—31no—
49C2×#421—31no—
50C2×#421—31no—
51#2309—31no—
52#2006—38no—
53#1742—31no—
54C25×C3—21no—
55C2×C3×#41—21no—
56C2×C3×#41—21no—
57C2×C3×#41—22no—
58C2×C3×#41—22no—
59C2×C3×#41—22no—
60C2×C3×#41—22no—
61C2×C3×#41D412yes—
62C2×C3×#41C2312yes—
63C22×C3×Q8D411yescharacteristic
64C22×C3×D4—21no—
65C22×C3×D4—21no—
66C22×C3×D4—21no—
67C22×C3×D4—22no—
68C22×C3×D4—21no—
69C22×C3×D4—21no—
70C22×C3×D4—22no—
71C22×C3×D4—21no—
72C22×C3×D4—22no—
73C22×C3×D4D411yescharacteristic
74C22×C3×D4D414yescomplemented
75C22×C3×D4C2312yes—
76C23×C3×C4—21no—
77C23×C3×C4C2311yescharacteristic
78C22×#67—21no—
79C2×#235—21no—
80C2×#235—21no—
81C2×#235—22no—
82C2×#235—22no—
83C2×#235—22no—
84C2×#235—22no—
85C2×#235D412yes—
86C2×#235C2312yes—
87C22×D12D411yescharacteristic
88C22×C4×D3—21no—
89C22×Dic6D411yescharacteristic
90C3×AGL(1,ℤ/8)—216no—
91C3×AGL(1,ℤ/8)—28no—
92C3×AGL(1,ℤ/8)—28no—
93C2×C3×SD16—28no—
94C2×C3×SD16—24no—
95C2×C3×D8—28no—
96C2×C3×D8—24no—
97C2×C3×M16—22no—
98C2×C3×M16—21no—
99C2×C3×M16D414yescomplemented
100C2×C3×M16C2312yes—
101C22×C3×C8—21no—
102#953—24no—
103C2×#217—21no—
104C2×#217—28no—
105C2×#217D412yes—
106#947—22no—
107#946—24no—
108#944—22no—
109#940—22no—
110#939—28no—
111#938—24no—
112#937—24no—
113#936—22no—
114#935—24no—
115C2×#214—28no—
116C2×#214—24no—
117#932—216no—
118#932—28no—
119#932—28no—
120C2×#213—28no—
121C2×#213—24no—
122#930—24no—
123#929—24no—
124C4×#67—28no—
125#926—21no—
126#926—21no—
127#926—21no—
128#926—21no—
129#926—22no—
130#926—22no—
131C2×C4⋊Dic3—21no—
132C2×C4⋊Dic3C2312yes—
133#924—24no—
134C2×C4×Dic3D412yes—
135C2×C4×Dic3C2312yes—
136C2×#208—21no—
137C2×#208—22no—
138C2×#208D414yescomplemented
139C2×#208C2312yes—
140C22×C3⋊C8—21no—
141#837—28no—
142#825—28no—
143C22×#41—31no—
144C22×#41C2×D311yescharacteristic
145C23×D4C2×D311yescharacteristic
146C2×AGL(1,ℤ/8)—22no—
147C2×AGL(1,ℤ/8)—24no—
148C2×AGL(1,ℤ/8)—24no—
149C2×AGL(1,ℤ/8)—34no—
150C2×AGL(1,ℤ/8)—64no—
151C2×AGL(1,ℤ/8)—62no—
152C2×AGL(1,ℤ/8)—64no—
153C2×AGL(1,ℤ/8)C2×D314yescomplemented
154C22×SD16—21no—
155C22×SD16—61no—
156C22×D8—21no—
157C22×D8—61no—
158C22×M16—31no—
159C22×M16C2×D311yescharacteristic
160#537—34no—
161#536—61no—
162#536—61no—
163#535—61no—
164#535—61no—
165#534—62no—
166#532—62no—
167#525—62no—
168#525—62no—
169C2×#117—31no—
170C2×#117—31no—
171#421—64no—
172#421—64no—
173C2×#104—34no—
174C2×#104—34no—
175#390—34no—
176#390—62no—
177#379—62no—
178#343—34no—
179C24×C3—22no—
180C24×C3—21no—
181C24×C3—24no—
182C24×C3—41no—
183C24×C3—41no—
184C24×C3—42no—
185C24×C3—84no—
186C24×C3C2×D411yescharacteristic
187C23×D3—21no—
188C3×#41—24no—
189C3×#41—24no—
190C3×#41—22no—
191C3×#41—22no—
192C3×#41—22no—
193C3×#41—22no—
194C3×#41—42no—
195C3×#41—42no—
196C3×#41—42no—
197C3×#41—42no—
198C2×C3×Q8—21no—
199C2×C3×Q8—22no—
200C2×C3×Q8—22no—
201C2×C3×Q8C2×D412yes—
202C2×C3×D4—22no—
203C2×C3×D4—22no—
204C2×C3×D4—24no—
205C2×C3×D4—24no—
206C2×C3×D4—22no—
207C2×C3×D4—24no—
208C2×C3×D4—24no—
209C2×C3×D4—21no—
210C2×C3×D4—22no—
211C2×C3×D4—24no—
212C2×C3×D4—22no—
213C2×C3×D4—22no—
214C2×C3×D4—22no—
215C2×C3×D4—22no—
216C2×C3×D4—22no—
217C2×C3×D4—42no—
218C2×C3×D4—42no—
219C2×C3×D4—41no—
220C2×C3×D4—41no—
221C2×C3×D4—42no—
222C2×C3×D4—42no—
223C2×C3×D4—42no—
224C2×C3×D4—42no—
225C2×C3×D4—42no—
226C2×C3×D4C2×D414yes—
227C2×C3×D4C2×D412yes—
228C22×C3×C4—21no—
229C22×C3×C4—22no—
230C22×C3×C4—22no—
231C22×C3×C4—42no—
232C22×C3×C4—42no—
233C22×C3×C4C2×D411yescharacteristic
234C22×C3×C4C2×D411yescharacteristic
235C22×C3×C4C2×D411yescharacteristic
236C22×C3×C4C2×D412yes—
237C22×C3×C4C2×D412yes—
238C22×C3×C4C2×D411yescharacteristic
239C22×C3×C4C2×D411yescharacteristic
240C22×C3×C4C2×D412yes—
241C22×C3×C4#4112yes—
242C22×C3×C4C2411yescharacteristic
243C2×#67—22no—
244C2×#67—22no—
245C2×#67—28no—
246C2×#67—42no—
247C2×#67—42no—
248C22×Dic3—21no—
249C22×Dic3—22no—
250C22×Dic3C2×D414yes—
251#235—24no—
252#235—24no—
253#235—22no—
254#235—22no—
255#235—22no—
256#235—22no—
257#235—42no—
258#235—42no—
259#235—42no—
260#235—42no—
261C2×D12—22no—
262C2×D12—22no—
263C2×D12—21no—
264C2×D12C2×D412yes—
265C2×C4×D3—22no—
266C2×C4×D3—22no—
267C2×C4×D3—42no—
268C2×C4×D3—42no—
269C2×Dic6—22no—
270C2×Dic6—22no—
271C2×Dic6—21no—
272C2×Dic6C2×D412yes—
273C3×SD16—216no—
274C3×D8—216no—
275C3×M16—24no—
276C3×M16—22no—
277C3×M16C2×D414yes—
278C2×C3×C8—24no—
279C2×C3×C8—22no—
280#217—28no—
281#217—22no—
282#217—22no—
283#217—42no—
284#217—88no—
285#214—216no—
286#213—216no—
287(C2×D3)⋊C4—28no—
288C4⋊Dic3—22no—
289C4⋊Dic3—22no—
290C4⋊Dic3—42no—
291Dic3⋊C4—28no—
292C4×Dic3—22no—
293C4×Dic3—22no—
294C4×Dic3—22no—
295C4×Dic3—22no—
296#208—24no—
297#208—22no—
298#208C2×D414yes—
299C2×C3⋊C8—24no—
300C2×C3⋊C8—22no—
301C25—21no—
302C2×#41—21no—
303C2×#41—21no—
304C2×#41—22no—
305C2×#41—22no—
306C2×#41—22no—
307C2×#41—22no—
308C2×#41—32no—
309C2×#41—32no—
310C2×#41—61no—
311C2×#41—61no—
312C2×#41—62no—
313C2×#41—62no—
314C2×#41—62no—
315C2×#41—62no—
316C2×#41#6712yes—
317C2×#41C22×D312yes—
318C22×Q8—31no—
319C22×Q8#6711yescharacteristic
320C22×D4—21no—
321C22×D4—21no—
322C22×D4—21no—
323C22×D4—22no—
324C22×D4—21no—
325C22×D4—22no—
326C22×D4—21no—
327C22×D4—21no—
328C22×D4—22no—
329C22×D4—31no—
330C22×D4—61no—
331C22×D4#6711yescharacteristic
332C22×D4#6714yescomplemented
333C22×D4C22×D312yes—
334C23×C4—21no—
335C23×C4—61no—
336C23×C4C22×D311yescharacteristic
337AGL(1,ℤ/8)—216no—
338AGL(1,ℤ/8)—28no—
339AGL(1,ℤ/8)—28no—
340AGL(1,ℤ/8)—68no—
341AGL(1,ℤ/8)—616no—
342AGL(1,ℤ/8)—68no—
343C2×SD16—28no—
344C2×SD16—24no—
345C2×SD16—64no—
346C2×SD16—68no—
347C2×D8—28no—
348C2×D8—24no—
349C2×D8—64no—
350C2×D8—68no—
351C2×M16—22no—
352C2×M16—21no—
353C2×M16—34no—
354C2×M16—32no—
355C2×M16—61no—
356C2×M16—62no—
357C2×M16#6714yescomplemented
358C2×M16C22×D312yes—
359C22×C8—21no—
360C22×C8—61no—
361C4⋊Q8—62no—
362#127—62no—
363#124—62no—
364#124—62no—
365#123—64no—
366#123—64no—
367C22⋊Q8—64no—
368C22⋊Q8—64no—
369#121—64no—
370#121—68no—
371#121—64no—
372#120—64no—
373#120—64no—
374C4×D4—68no—
375#117—61no—
376#117—61no—
377#117—61no—
378#117—61no—
379#117—62no—
380#117—62no—
381C2×C4⋊C4—32no—
382C2×C4⋊C4—61no—
383C2×C22⋊C4—32no—
384C2×C22⋊C4—61no—
385C2×C22⋊C4—68no—
386C2×C42—32no—
387C2×C42—32no—
388#104—68no—
389#104—68no—
390C23×C3—22no—
391C23×C3—24no—
392C23×C3—42no—
393C23×C3—48no—
394C23×C3—42no—
395C23×C3—42no—
396C23×C3—42no—
397C23×C3—44no—
398C23×C3—44no—
399C23×C3—82no—
400C23×C3—84no—
401C23×C3—84no—
402C23×C3—88no—
403C23×C3—88no—
404C23×C3#12011yescharacteristic
405C23×C3#12011yescharacteristic
406C23×C3#12112yes—
407C23×C3#12712yes—
408C23×C3C22×D411yescharacteristic
409C22×D3—22no—
410C22×D3—42no—
411C22×D3—82no—
412C3×Q8—22no—
413C3×Q8—22no—
414C3×Q8—42no—
415C3×D4—24no—
416C3×D4—22no—
417C3×D4—24no—
418C3×D4—22no—
419C3×D4—42no—
420C3×D4—42no—
421C3×D4—44no—
422C3×D4—42no—
423C3×D4—42no—
424C3×D4—44no—
425C3×D4—44no—
426C3×D4—42no—
427C2×C3×C4—22no—
428C2×C3×C4—42no—
429C2×C3×C4—42no—
430C2×C3×C4—42no—
431C2×C3×C4—44no—
432C2×C3×C4#12012yes—
433C2×C3×C4#12012yes—
434C2×C3×C4#12011yescharacteristic
435C2×C3×C4#12011yescharacteristic
436C2×C3×C4#12012yes—
437C2×C3×C4#12012yes—
438C2×C3×C4#12112yes—
439C2×C3×C4#12112yes—
440C2×C3×C4#12112yes—
441C2×C3×C4#12112yes—
442C2×C3×C4#12112yes—
443C2×C3×C4#12712yes—
444C2×C3×C4C22×D411yescharacteristic
445C2×C3×C4C22×D411yescharacteristic
446C2×C3×C4C22×D411yescharacteristic
447C2×C3×C4C22×D411yescharacteristic
448C2×C3×C4C22×D411yescharacteristic
449C2×C3×C4C2×#4111yescharacteristic
450#67—42no—
451#67—42no—
452#67—44no—
453#67—88no—
454C2×Dic3—24no—
455C2×Dic3—22no—
456C2×Dic3—42no—
457C2×Dic3—44no—
458C2×Dic3—44no—
459C2×Dic3—42no—
460D12—22no—
461D12—22no—
462D12—42no—
463C4×D3—42no—
464C4×D3—42no—
465C4×D3—44no—
466Dic6—22no—
467Dic6—22no—
468Dic6—42no—
469C3×C8—24no—
470C3⋊C8—24no—
471C24—22no—
472C24—21no—
473C24—24no—
474C24—41no—
475C24—41no—
476C24—42no—
477C24—61no—
478C24—84no—
479C24D3×D411yescharacteristic
480#41—24no—
481#41—24no—
482#41—22no—
483#41—22no—
484#41—22no—
485#41—22no—
486#41—42no—
487#41—42no—
488#41—42no—
489#41—42no—
490#41—64no—
491#41—64no—
492#41—62no—
493#41—62no—
494#41—62no—
495#41—62no—
496#41—122no—
497#41—122no—
498#41—122no—
499#41—122no—
500C2×Q8—21no—
501C2×Q8—22no—
502C2×Q8—22no—
503C2×Q8—32no—
504C2×Q8—61no—
505C2×Q8—62no—
506C2×Q8—62no—
507C2×Q8C2×#6712yes—
508C2×D4—22no—
509C2×D4—22no—
510C2×D4—24no—
511C2×D4—24no—
512C2×D4—21no—
513C2×D4—22no—
514C2×D4—24no—
515C2×D4—22no—
516C2×D4—24no—
517C2×D4—24no—
518C2×D4—22no—
519C2×D4—22no—
520C2×D4—22no—
521C2×D4—22no—
522C2×D4—22no—
523C2×D4—32no—
524C2×D4—42no—
525C2×D4—42no—
526C2×D4—41no—
527C2×D4—41no—
528C2×D4—42no—
529C2×D4—42no—
530C2×D4—42no—
531C2×D4—42no—
532C2×D4—42no—
533C2×D4—62no—
534C2×D4—62no—
535C2×D4—68no—
536C2×D4—62no—
537C2×D4—61no—
538C2×D4—62no—
539C2×D4—122no—
540C2×D4—122no—
541C2×D4C2×#6712yes—
542C2×D4C2×#6714yes—
543C22×C4—21no—
544C22×C4—22no—
545C22×C4—22no—
546C22×C4—34no—
547C22×C4—42no—
548C22×C4—42no—
549C22×C4—61no—
550C22×C4—62no—
551C22×C4—62no—
552C22×C4—62no—
553C22×C4—122no—
554C22×C4—122no—
555C22×C4D3×D411yescharacteristic
556C22×C4D3×D411yescharacteristic
557C22×C4D3×D411yescharacteristic
558C22×C4D3×D412yes—
559C22×C4#23712yes—
560C22×C4C2×#6712yes—
561C22×C4C2×#6711yescharacteristic
562C22×C4C2×#6711yescharacteristic
563C22×C4C2×#6712yes—
564C22×C4C23×D311yescharacteristic
565SD16—216no—
566SD16—616no—
567D8—216no—
568D8—616no—
569M16—24no—
570M16—22no—
571M16—34no—
572M16—64no—
573M16—62no—
574M16C2×#6714yes—
575C2×C8—24no—
576C2×C8—22no—
577C2×C8—64no—
578C2×C8—62no—
579C4⋊C4—68no—
580C4⋊C4—62no—
581C4⋊C4—62no—
582C4⋊C4—122no—
583C22⋊C4—68no—
584C22⋊C4—68no—
585C22⋊C4—62no—
586C22⋊C4—62no—
587C22⋊C4—122no—
588C22⋊C4—248no—
589C42—62no—
590C42—62no—
591C42—62no—
592C42—62no—
593C22×C3—22no—
594C22×C3—22no—
595C22×C3—22no—
596C22×C3—42no—
597C22×C3—42no—
598C22×C3—48no—
599C22×C3—42no—
600C22×C3—44no—
601C22×C3—44no—
602C22×C3—84no—
603C22×C3—84no—
604C22×C3—88no—
605C22×C3—88no—
606C22×C3—84no—
607C22×C3#39212yes—
608C22×C3#39212yes—
609C22×C3C2×#12011yescharacteristic
610C22×C3C2×#12111yescharacteristic
611C22×C3C2×#12711yescharacteristic
612C2×D3—42no—
613C2×D3—44no—
614C2×D3—84no—
615C3×C4—42no—
616C3×C4#39212yes—
617C3×C4#39212yes—
618C3×C4C2×#12011yescharacteristic
619C3×C4C2×#12011yescharacteristic
620C3×C4C2×#12111yescharacteristic
621C3×C4C2×#12111yescharacteristic
622Dic3—42no—
623Dic3—44no—
624Dic3—82no—
625C23—22no—
626C23—24no—
627C23—42no—
628C23—48no—
629C23—42no—
630C23—44no—
631C23—44no—
632C23—42no—
633C23—42no—
634C23—62no—
635C23—82no—
636C23—84no—
637C23—84no—
638C23—88no—
639C23—88no—
640C23—122no—
641C23—242no—
642C23#93711yescharacteristic
643C23#93711yescharacteristic
644C23#93912yes—
645C23#94012yes—
646C23C2×D3×D411yescharacteristic
647Q8—22no—
648Q8—22no—
649Q8—42no—
650Q8—62no—
651Q8—62no—
652Q8—122no—
653D4—24no—
654D4—22no—
655D4—22no—
656D4—24no—
657D4—44no—
658D4—42no—
659D4—42no—
660D4—44no—
661D4—44no—
662D4—42no—
663D4—42no—
664D4—42no—
665D4—62no—
666D4—62no—
667D4—124no—
668D4—122no—
669D4—122no—
670D4—122no—
671D4—248no—
672C2×C4—22no—
673C2×C4—42no—
674C2×C4—44no—
675C2×C4—42no—
676C2×C4—42no—
677C2×C4—64no—
678C2×C4—62no—
679C2×C4—122no—
680C2×C4—124no—
681C2×C4—122no—
682C2×C4—122no—
683C2×C4—124no—
684C2×C4—124no—
685C2×C4—122no—
686C2×C4#93711yescharacteristic
687C2×C4#93711yescharacteristic
688C2×C4#93712yes—
689C2×C4#93712yes—
690C2×C4#93812yes—
691C2×C4#93812yes—
692C2×C4#93912yes—
693C2×C4#93912yes—
694C2×C4#93912yes—
695C2×C4#94012yes—
696C2×C4#95312yes—
697C2×C4#95312yes—
698C2×C4C2×D3×D411yescharacteristic
699C2×C4C2×D3×D411yescharacteristic
700C2×C4C2×#23711yescharacteristic
701C2×C4C22×#6711yescharacteristic
702C2×C4C22×#6711yescharacteristic
703C2×C4C22×#6711yescharacteristic
704C8—24no—
705C8—64no—
706C2×C3—21no—
707C2×C3—22no—
708C2×C3—21no—
709C2×C3—22no—
710C2×C3—82no—
711C2×C3—84no—
712C2×C3C2×#39211yescharacteristic
713C2×C3#302112yes—
714D3—82no—
715C22—22no—
716C22—22no—
717C22—22no—
718C22—42no—
719C22—42no—
720C22—48no—
721C22—42no—
722C22—44no—
723C22—44no—
724C22—84no—
725C22—88no—
726C22—88no—
727C22—84no—
728C22—84no—
729C22—122no—
730C22—124no—
731C22—244no—
732C22#522312yes—
733C22#522312yes—
734C22C2×#93711yescharacteristic
735C22C2×#93911yescharacteristic
736C22C2×#94011yescharacteristic
737C4—42no—
738C4—122no—
739C4—124no—
740C4—242no—
741C4#522312yes—
742C4#522312yes—
743C4C2×#93711yescharacteristic
744C4C2×#93811yescharacteristic
745C4C2×#93911yescharacteristic
746C4C2×#95311yescharacteristic
747C3#2340711yescharacteristic, complemented, Sylow
748C2—21no—
749C2—21no—
750C2—22no—
751C2—22no—
752C2—84no—
753C2—82no—
754C2—242no—
755C2#8395712yes—
756C2C2×#522311yescharacteristic
subgroupG/H
centerC22C2×#937
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC22×C3×AGL(1,ℤ/8)C2
socleC22×C3C2×#120
hypercenterC22×AGL(1,ℤ/8)D3
hypocenterC3#23407
LCSG ▷ C22×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. C22×#932C2semidirect
4. C2×#5236C2semidirect
5. #84044C2semidirect
6. C2×#5139C2semidirect
7. #72098C2semidirect
8. #71987C2semidirect
16. C2×C3×AGL(1,ℤ/8)C22semidirect
20. #5833C22semidirect
32. C2×#932C22semidirect
40. C2×#837C22semidirect
42. #5221C22semidirect
44. C2×#825C22semidirect
45. #4554C22semidirect
47. C22×AGL(1,ℤ/8)D3semidirect
74. C22×C3×D4D4semidirect
99. C2×C3×M16D4semidirect
138. C2×#208D4semidirect
153. C2×AGL(1,ℤ/8)C2×D3semidirect
332. C22×D4#67semidirect
357. C2×M16#67semidirect
747. C3#23407semidirect

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
C22×#9322
C2×#52362
#840442
C2×#51392
#720982
#719872
#234073