Finite Group Databasebeta

← #10822136

Subgroups — #10822136

Order 768.

1942 conjugacy classes of subgroups.

The subgroup lattice has 748 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5816C211yesmaximal, characteristic, complemented
2C2×#5816C211yesmaximal, characteristic, complemented
3C2×#5748C211yesmaximal, characteristic
4C22×C3×#101C211yesmaximal, characteristic, complemented
5C2×#5139C211yesmaximal, characteristic, complemented
6#83768C211yesmaximal, characteristic, complemented
7#71982C211yesmaximal, characteristic, complemented
8#70119C214yesmaximal, complemented
9#70116C214yesmaximal, complemented
10#23403—31nomaximal, Sylow
11C23×C3×Q8C2211yescharacteristic
12C22×#235C2211yescharacteristic
13C22×#235C2211yescharacteristic
14C22×C3×M16C2211yescharacteristic
15C22×C3×M16C2211yescharacteristic
16#5816—22no—
17#5816—22no—
18#5816—24no—
19#5816—22no—
20#5816—24no—
21#5816C2214yes—
22C2×#947—21no—
23C2×#947C2212yescomplemented
24C2×#946—22no—
25C2×#946—22no—
26C2×#944—21no—
27C2×#944C2212yescomplemented
28C2×#929—22no—
29C2×#929—22no—
30C2×#926C2211yescharacteristic
31C2×#926C2211yescharacteristic
32#5748—24no—
33#5748—24no—
34C2×#909—21no—
35C2×#909C2212yes—
36C2×#908—21no—
37C2×#908C2212yes—
38C2×#904—22no—
39C2×C3×#101—24no—
40C2×C3×#101C2214yescomplemented
41#5139—24no—
42C2×#825C2214yes—
43#5133—22no—
44#5131C2212yescomplemented
45#5120C2212yescomplemented
46#5113—22no—
47#4555C2214yescomplemented
48C2×#575—31no—
49C2×#536—31no—
50C2×#536—31no—
51C2×#421—31no—
52#2896—31no—
53C22×#101D311yescharacteristic, complemented
54#2005—34no—
55#2003—34no—
56#1733—31no—
57C22×C3×Q8—22no—
58C22×C3×Q8—21no—
59C22×C3×Q8—21no—
60C22×C3×Q8—21no—
61C22×C3×Q8—22no—
62C22×C3×Q8—21no—
63C22×C3×Q8—22no—
64C22×C3×Q8—22no—
65C22×C3×Q8D412yes—
66C22×C3×Q8C2312yes—
67C23×C3×C4D412yes—
68C23×C3×C4C2311yescharacteristic
69C22×#67—21no—
70C22×#67—21no—
71C2×#235—21no—
72C2×#235—21no—
73C2×#235—22no—
74C2×#235—21no—
75C2×#235—22no—
76C2×#235—21no—
77C2×#235—21no—
78C2×#235—22no—
79C2×#235—22no—
80C2×#235—22no—
81C2×#235—21no—
82C2×#235—22no—
83C2×#235—22no—
84C2×#235D412yes—
85C2×#235D414yes—
86C2×#235C2312yes—
87C22×D12D411yescharacteristic
88C22×D12C2311yescharacteristic
89C22×C4×D3—21no—
90C22×C4×D3—21no—
91C22×Dic6D411yescharacteristic
92C22×Dic6C2311yescharacteristic
93C2×C3×M16—21no—
94C2×C3×M16—22no—
95C2×C3×M16—21no—
96C2×C3×M16—22no—
97C2×C3×M16—22no—
98C2×C3×M16D414yes—
99C2×C3×M16C2312yes—
100C2×C3×M16C2312yes—
101C22×C3×C8—21no—
102C22×C3×C8—21no—
103C2×#217—21no—
104C2×#217—21no—
105#947—24no—
106#947—24no—
107#947—24no—
108#947—24no—
109#946—216no—
110#946—28no—
111#946—28no—
112#944—24no—
113#944—24no—
114#944—24no—
115#944—24no—
116#929—28no—
117#929—216no—
118#929—28no—
119C2×(C2×D3)⋊C4—24no—
120C2×(C2×D3)⋊C4—24no—
121#926—21no—
122#926—21no—
123#926—22no—
124#926D412yes—
125#926D414yes—
126#926C2312yes—
127C2×C4⋊Dic3—21no—
128C2×C4⋊Dic3D412yes—
129C2×Dic3⋊C4—24no—
130C2×Dic3⋊C4—24no—
131C2×C4×Dic3—21no—
132C2×C4×Dic3C2312yes—
133#909—28no—
134#909—24no—
135#909—24no—
136#908—28no—
137#908—24no—
138#908—24no—
139C2×Dic12—22no—
140#904—216no—
141C2×D24—22no—
142C2×#204—22no—
143C2×#204—22no—
144C3×#101—28no—
145#825—28no—
146#821—24no—
147#816—24no—
148C22×#41—31no—
149C22×#41—31no—
150C23×Q8C2×D311yescharacteristic
151#575—64no—
152#575—64no—
153C2×#137—32no—
154C2×#137—61no—
155C2×AGL(1,ℤ/8)—32no—
156C2×AGL(1,ℤ/8)—61no—
157C2×#135—62no—
158C22×M16C2×D311yescharacteristic
159C22×M16C2×D311yescharacteristic
160#536—34no—
161#536—64no—
162#536—62no—
163#536—64no—
164#536—62no—
165#536—62no—
166C2×C4⋊Q8—32no—
167C2×C4⋊Q8—61no—
168C2×#124—32no—
169C2×#124—61no—
170C2×#123—62no—
171C2×#123—62no—
172C2×C22⋊Q8—62no—
173C2×C22⋊Q8—62no—
174C2×#117—31no—
175C2×#117—31no—
176#421—64no—
177C2×#104—34no—
178#419—32no—
179#418—32no—
180#417—62no—
181#416—62no—
182C2×#101—24no—
183C2×#101C2×D314yescomplemented
184#339—34no—
185C24×C3C2×D411yescharacteristic
186C23×D3—21no—
187C23×D3—21no—
188C2×C3×Q8—22no—
189C2×C3×Q8—22no—
190C2×C3×Q8—24no—
191C2×C3×Q8—24no—
192C2×C3×Q8—22no—
193C2×C3×Q8—22no—
194C2×C3×Q8—22no—
195C2×C3×Q8—22no—
196C2×C3×Q8—22no—
197C2×C3×Q8—22no—
198C2×C3×Q8—42no—
199C2×C3×Q8—42no—
200C2×C3×Q8—42no—
201C2×C3×Q8—42no—
202C2×C3×Q8—42no—
203C2×C3×Q8—42no—
204C2×C3×Q8—42no—
205C2×C3×Q8—42no—
206C22×C3×C4—22no—
207C22×C3×C4—22no—
208C22×C3×C4—24no—
209C22×C3×C4—24no—
210C22×C3×C4C2×D414yes—
211C22×C3×C4C2×D411yescharacteristic
212C22×C3×C4C2×D411yescharacteristic
213C22×C3×C4C2×D411yescharacteristic
214C22×C3×C4C2×D412yes—
215C22×C3×C4C2×D411yescharacteristic
216C22×C3×C4C2×D411yescharacteristic
217C22×C3×C4C2×D412yes—
218C22×C3×C4C2×D412yes—
219C22×C3×C4#4112yes—
220C22×C3×C4C2411yescharacteristic
221C2×#67—22no—
222C2×#67—24no—
223C2×#67—22no—
224C2×#67—41no—
225C2×#67—41no—
226C2×#67—42no—
227C2×#67—42no—
228C2×#67—42no—
229C22×Dic3—22no—
230C22×Dic3—21no—
231C22×Dic3—22no—
232C22×Dic3—21no—
233#235—24no—
234#235—24no—
235#235—24no—
236#235—24no—
237#235—24no—
238#235—24no—
239#235—22no—
240#235—22no—
241#235—22no—
242#235—22no—
243#235—42no—
244#235—42no—
245#235—42no—
246#235—42no—
247#235—42no—
248#235—42no—
249#235—42no—
250#235—42no—
251C2×D12—22no—
252C2×D12—22no—
253C2×D12—21no—
254C2×D12—21no—
255C2×D12—21no—
256C2×D12—21no—
257C2×D12—21no—
258C2×D12—21no—
259C2×D12C2×D412yes—
260C2×D12C2×D412yes—
261C2×C4×D3—22no—
262C2×C4×D3—24no—
263C2×C4×D3—22no—
264C2×C4×D3—41no—
265C2×C4×D3—41no—
266C2×C4×D3—42no—
267C2×C4×D3—42no—
268C2×C4×D3—42no—
269C2×Dic6—22no—
270C2×Dic6—22no—
271C2×Dic6—21no—
272C2×Dic6—21no—
273C2×Dic6—21no—
274C2×Dic6—21no—
275C2×Dic6—21no—
276C2×Dic6—21no—
277C2×Dic6C2×D412yes—
278C2×Dic6C2×D412yes—
279C3×M16—24no—
280C3×M16—22no—
281C3×M16—22no—
282C3×M16—22no—
283C3×M16—42no—
284C3×M16C2×D414yes—
285C2×C3×C8—24no—
286C2×C3×C8—22no—
287C2×C3×C8—22no—
288C2×C3×C8—42no—
289#217—22no—
290#217—24no—
291#217—22no—
292(C2×D3)⋊C4—216no—
293(C2×D3)⋊C4—216no—
294C4⋊Dic3—24no—
295C4⋊Dic3—22no—
296C4⋊Dic3C2×D414yes—
297Dic3⋊C4—216no—
298Dic3⋊C4—216no—
299C4×Dic3—22no—
300C4×Dic3—22no—
301C4×Dic3—22no—
302C4×Dic3C2×D414yes—
303Dic12—28no—
304D24—28no—
305#204—28no—
306#204—28no—
307C2×#41—32no—
308C2×#41—34no—
309C2×#41—32no—
310C2×#41—61no—
311C2×#41—62no—
312C2×#41—61no—
313C2×#41—62no—
314C2×#41—62no—
315C2×#41—61no—
316C2×#41—61no—
317C2×#41—62no—
318C2×#41—61no—
319C2×#41—61no—
320C2×#41—62no—
321C2×#41—62no—
322C2×#41—62no—
323C22×Q8—22no—
324C22×Q8—21no—
325C22×Q8—21no—
326C22×Q8—21no—
327C22×Q8—22no—
328C22×Q8—21no—
329C22×Q8—22no—
330C22×Q8—22no—
331C22×Q8—31no—
332C22×Q8—31no—
333C22×Q8D1212yes—
334C22×Q8C22×D312yes—
335C22×D4—31no—
336C22×D4—31no—
337C22×D4—61no—
338C22×D4—61no—
339C23×C4—61no—
340C23×C4—61no—
341C23×C4D1212yes—
342C23×C4C22×D311yescharacteristic
343#137—64no—
344#137—68no—
345#137—64no—
346AGL(1,ℤ/8)—64no—
347AGL(1,ℤ/8)—68no—
348AGL(1,ℤ/8)—64no—
349#135—616no—
350C2×Q16—62no—
351C2×SD16—62no—
352C2×SD16—62no—
353C2×D8—62no—
354C2×M16—21no—
355C2×M16—22no—
356C2×M16—21no—
357C2×M16—22no—
358C2×M16—22no—
359C2×M16#6714yes—
360C2×M16C22×D312yes—
361C2×M16C22×D312yes—
362C22×C8—21no—
363C22×C8—21no—
364C4⋊Q8—64no—
365C4⋊Q8—64no—
366C4⋊Q8—64no—
367C4⋊Q8—64no—
368#124—64no—
369#124—64no—
370#124—64no—
371#124—64no—
372#123—616no—
373#123—68no—
374#123—68no—
375C22⋊Q8—68no—
376C22⋊Q8—68no—
377C22⋊Q8—616no—
378#117—34no—
379#117—32no—
380#117—32no—
381#117—61no—
382#117—62no—
383#117—61no—
384C2×C4⋊C4—32no—
385C2×C4⋊C4—64no—
386C2×C4⋊C4—61no—
387C2×C4⋊C4—64no—
388C2×C22⋊C4—64no—
389C2×C22⋊C4—61no—
390C2×C22⋊C4—61no—
391C2×C22⋊C4—64no—
392C2×C42—32no—
393C2×C42—61no—
394#104—68no—
395#103—64no—
396#102—64no—
397#101—28no—
398C23×C3—42no—
399C23×C3#12011yescharacteristic
400C23×C3#12011yescharacteristic
401C23×C3#12012yes—
402C23×C3#12112yes—
403C23×C3C22×D411yescharacteristic
404C22×D3—22no—
405C22×D3—22no—
406C22×D3—44no—
407C22×D3—44no—
408C22×D3—41no—
409C22×D3—41no—
410C22×D3—42no—
411C3×Q8—42no—
412C3×Q8—42no—
413C3×Q8—42no—
414C3×Q8—42no—
415C3×Q8—44no—
416C3×Q8—44no—
417C2×C3×C4—24no—
418C2×C3×C4—24no—
419C2×C3×C4—24no—
420C2×C3×C4—44no—
421C2×C3×C4—44no—
422C2×C3×C4#12011yescharacteristic
423C2×C3×C4#12011yescharacteristic
424C2×C3×C4#12012yes—
425C2×C3×C4#12012yes—
426C2×C3×C4#12012yes—
427C2×C3×C4#12112yes—
428C2×C3×C4#12112yes—
429C2×C3×C4#12112yes—
430C2×C3×C4#12112yes—
431C2×C3×C4#12112yes—
432C2×C3×C4#12712yes—
433C2×C3×C4#12712yes—
434C2×C3×C4C22×D411yescharacteristic
435C2×C3×C4C22×D411yescharacteristic
436C2×C3×C4C22×D411yescharacteristic
437C2×C3×C4C22×D411yescharacteristic
438C2×C3×C4C22×D411yescharacteristic
439C2×C3×C4C2×#4111yescharacteristic
440#67—42no—
441#67—42no—
442#67—44no—
443#67—44no—
444#67—44no—
445C2×Dic3—24no—
446C2×Dic3—24no—
447C2×Dic3—22no—
448C2×Dic3—22no—
449C2×Dic3—44no—
450C2×Dic3—44no—
451C2×Dic3—41no—
452C2×Dic3—41no—
453C2×Dic3—42no—
454D12—22no—
455D12—22no—
456D12—22no—
457D12—22no—
458D12—41no—
459D12—41no—
460D12—42no—
461C4×D3—42no—
462C4×D3—42no—
463C4×D3—44no—
464C4×D3—44no—
465C4×D3—44no—
466Dic6—22no—
467Dic6—22no—
468Dic6—22no—
469Dic6—22no—
470Dic6—41no—
471Dic6—41no—
472Dic6—42no—
473C3×C8—24no—
474C3×C8—42no—
475C24—61no—
476C24—61no—
477C24C2×D1211yescharacteristic
478#41—64no—
479#41—64no—
480#41—64no—
481#41—64no—
482#41—64no—
483#41—64no—
484#41—62no—
485#41—62no—
486#41—62no—
487#41—62no—
488#41—122no—
489#41—122no—
490#41—122no—
491#41—122no—
492#41—122no—
493#41—122no—
494#41—122no—
495#41—122no—
496C2×Q8—24no—
497C2×Q8—22no—
498C2×Q8—22no—
499C2×Q8—24no—
500C2×Q8—22no—
501C2×Q8—22no—
502C2×Q8—22no—
503C2×Q8—22no—
504C2×Q8—22no—
505C2×Q8—22no—
506C2×Q8—32no—
507C2×Q8—32no—
508C2×Q8—42no—
509C2×Q8—42no—
510C2×Q8—42no—
511C2×Q8—42no—
512C2×Q8—42no—
513C2×Q8—42no—
514C2×Q8—42no—
515C2×Q8—42no—
516C2×Q8—61no—
517C2×Q8—61no—
518C2×Q8—62no—
519C2×Q8—62no—
520C2×Q8—61no—
521C2×Q8—61no—
522C2×Q8—61no—
523C2×Q8—61no—
524C2×D4—32no—
525C2×D4—32no—
526C2×D4—64no—
527C2×D4—62no—
528C2×D4—62no—
529C2×D4—61no—
530C2×D4—62no—
531C2×D4—62no—
532C2×D4—61no—
533C2×D4—61no—
534C2×D4—61no—
535C2×D4—61no—
536C2×D4—61no—
537C2×D4—122no—
538C2×D4—121no—
539C2×D4—121no—
540C2×D4—122no—
541C2×D4—122no—
542C22×C4—22no—
543C22×C4—24no—
544C22×C4—22no—
545C22×C4—24no—
546C22×C4—62no—
547C22×C4—61no—
548C22×C4—62no—
549C22×C4—61no—
550C22×C4—64no—
551C22×C4—62no—
552C22×C4—62no—
553C22×C4—121no—
554C22×C4—121no—
555C22×C4—122no—
556C22×C4—122no—
557C22×C4—122no—
558C22×C4C2×D1214yes—
559C22×C4C2×D1211yescharacteristic
560C22×C4#23512yes—
561C22×C4D3×D411yescharacteristic
562C22×C4D3×D411yescharacteristic
563C22×C4D3×D412yes—
564C22×C4D3×D411yescharacteristic
565C22×C4D3×D411yescharacteristic
566C22×C4D3×D412yes—
567C22×C4C2×#6712yes—
568C22×C4C23×D311yescharacteristic
569Q16—68no—
570SD16—68no—
571SD16—68no—
572D8—68no—
573M16—24no—
574M16—22no—
575M16—22no—
576M16—22no—
577M16—42no—
578M16C2×#6714yes—
579C2×C8—24no—
580C2×C8—22no—
581C2×C8—22no—
582C2×C8—42no—
583C4⋊C4—34no—
584C4⋊C4—616no—
585C4⋊C4—616no—
586C4⋊C4—62no—
587C4⋊C4—64no—
588C22⋊C4—616no—
589C22⋊C4—616no—
590C22⋊C4—64no—
591C22⋊C4—62no—
592C22⋊C4—62no—
593C42—34no—
594C42—62no—
595C42—62no—
596C42—62no—
597C22×C3—22no—
598C22×C3—22no—
599C22×C3—22no—
600C22×C3—42no—
601C22×C3—42no—
602C22×C3#39212yes—
603C22×C3#39212yes—
604C22×C3C2×#12011yescharacteristic
605C22×C3C2×#12011yescharacteristic
606C22×C3C2×#12111yescharacteristic
607C2×D3—44no—
608C2×D3—42no—
609C2×D3—42no—
610C2×D3—44no—
611C2×D3—82no—
612C2×D3—84no—
613C2×D3—82no—
614C3×C4—44no—
615C3×C4#39212yes—
616C3×C4#39212yes—
617C3×C4C2×#12011yescharacteristic
618C3×C4C2×#12111yescharacteristic
619C3×C4C2×#12111yescharacteristic
620C3×C4C2×#12711yescharacteristic
621Dic3—44no—
622Dic3—42no—
623Dic3—42no—
624Dic3—44no—
625C23—42no—
626C23—62no—
627C23—62no—
628C23—124no—
629C23—124no—
630C23—122no—
631C23—121no—
632C23—121no—
633C23#88211yescharacteristic
634C23#88211yescharacteristic
635C23#88212yes—
636C23#93012yes—
637C23C22×D1211yescharacteristic
638Q8—44no—
639Q8—44no—
640Q8—42no—
641Q8—42no—
642Q8—42no—
643Q8—42no—
644Q8—62no—
645Q8—62no—
646Q8—62no—
647Q8—62no—
648Q8—122no—
649Q8—121no—
650Q8—121no—
651D4—62no—
652D4—62no—
653D4—62no—
654D4—62no—
655D4—124no—
656D4—124no—
657D4—124no—
658D4—122no—
659D4—122no—
660D4—122no—
661D4—121no—
662D4—121no—
663C2×C4—24no—
664C2×C4—24no—
665C2×C4—24no—
666C2×C4—44no—
667C2×C4—44no—
668C2×C4—64no—
669C2×C4—62no—
670C2×C4—64no—
671C2×C4—62no—
672C2×C4—124no—
673C2×C4—124no—
674C2×C4—122no—
675C2×C4—121no—
676C2×C4—121no—
677C2×C4—124no—
678C2×C4—124no—
679C2×C4—124no—
680C2×C4—122no—
681C2×C4—122no—
682C2×C4#88212yes—
683C2×C4#88211yescharacteristic
684C2×C4#88211yescharacteristic
685C2×C4#88412yes—
686C2×C4#88412yes—
687C2×C4#88412yes—
688C2×C4#88412yes—
689C2×C4#93012yes—
690C2×C4#93712yes—
691C2×C4#93712yes—
692C2×C4#94012yes—
693C2×C4#94012yes—
694C2×C4C2×#23511yescharacteristic
695C2×C4C2×D3×D411yescharacteristic
696C2×C4C2×D3×D411yescharacteristic
697C2×C4C2×D3×D411yescharacteristic
698C2×C4C2×D3×D411yescharacteristic
699C2×C4C22×#6711yescharacteristic
700C8—24no—
701C8—42no—
702C2×C3—21no—
703C2×C3—21no—
704C2×C3—22no—
705C2×C3—22no—
706C2×C3C2×#39211yescharacteristic
707C2×C3#302012yes—
708D3—82no—
709D3—82no—
710C22—22no—
711C22—22no—
712C22—22no—
713C22—42no—
714C22—42no—
715C22—124no—
716C22—124no—
717C22—122no—
718C22—122no—
719C22—244no—
720C22—242no—
721C22—242no—
722C22#496012yes—
723C22#496012yes—
724C22C2×#88211yescharacteristic
725C22C2×#88211yescharacteristic
726C22C2×#93011yescharacteristic
727C4—44no—
728C4—124no—
729C4—124no—
730C4—122no—
731C4—122no—
732C4#496012yes—
733C4#496012yes—
734C4C2×#88411yescharacteristic
735C4C2×#88411yescharacteristic
736C4C2×#93711yescharacteristic
737C4C2×#94011yescharacteristic
738C3#2340311yescharacteristic, complemented, Sylow
739C2—21no—
740C2—22no—
741C2—21no—
742C2—22no—
743C2—242no—
744C2—242no—
745C2#7870912yes—
746C2C2×#496011yescharacteristic
subgroupG/H
centerC22C2×#882
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC22×C3×#101C2
socleC22×C3C2×#120
prime radicalC2×#5748C2
hypercenterC22×#101D3
hypocenterC3#23403
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ C22×#101

Complements

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

NH ≅ G/Ntype
1. C2×#5816C2semidirect
2. C2×#5816C2semidirect
4. C22×C3×#101C2semidirect
5. C2×#5139C2semidirect
6. #83768C2semidirect
7. #71982C2semidirect
8. #70119C2semidirect
9. #70116C2semidirect
23. C2×#947C22semidirect
27. C2×#944C22semidirect
40. C2×C3×#101C22semidirect
44. #5131C22semidirect
45. #5120C22semidirect
47. #4555C22semidirect
53. C22×#101D3semidirect
183. C2×#101C2×D3semidirect
738. C3#23403semidirect

Maximal subgroups

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

subgroupindex
C2×#58162
C2×#57482
C22×C3×#1012
C2×#51392
#837682
#719822
#701192
#701162
#234033