Finite Group Databasebeta

← #10822307

Subgroups — #10822307

Order 768.

2178 conjugacy classes of subgroups.

The subgroup lattice has 732 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5805C211yesmaximal, characteristic, complemented
2C22×#932C211yesmaximal, characteristic, complemented
3C22×#904C211yesmaximal, characteristic, complemented
4C22×C3×#102C211yesmaximal, characteristic, complemented
5#82186C211yesmaximal, characteristic, complemented
6C2×#4966C211yesmaximal, characteristic, complemented
7#71965C211yesmaximal, characteristic, complemented
8#70215C212yesmaximal, complemented
9#70214C212yesmaximal, complemented
10#70210C212yesmaximal, complemented
11#70209C212yesmaximal, complemented
12#23426—31nomaximal, Sylow
13C23×C3×D4C2211yescharacteristic
14C22×#235C2211yescharacteristic
15C22×#235C2211yescharacteristic
16C23×C3×C8C2211yescharacteristic
17C22×C3×C4⋊C4C2211yescharacteristic
18C2×#953—22no—
19#5805—24no—
20#5805—24no—
21C2×#940C2212yes—
22C2×#938—22no—
23C2×#936C2212yes—
24C22×#214C2212yes—
25C2×#932—24no—
26C2×#932—24no—
27C2×#932—24no—
28C22×#213C2212yes—
29C2×#926C2211yescharacteristic
30C22×#208C2211yescharacteristic
31C22×Dic12C2211yescharacteristic, complemented
32C2×#904—24no—
33C2×#904—24no—
34C2×#904—24no—
35C22×D24C2211yescharacteristic, complemented
36C22×#204C2211yescharacteristic, complemented
37C22×#204C2211yescharacteristic, complemented
38C2×C3×#102—24no—
39C2×C3×#102C2212yes—
40C2×C3×#102C2212yes—
41#5213—22no—
42#4988C2212yescomplemented
43#4987—22no—
44#4973C2212yescomplemented
45C2×#810C2212yescomplemented
46#4966—24no—
47C2×#809C2212yescomplemented
48#4550C2212yescomplemented
49#4550C2212yescomplemented
50C22×AGL(1,ℤ/8)—31no—
51C22×#135—31no—
52C2×#535—31no—
53C2×#417—31no—
54C22×#102D311yescharacteristic, complemented
55#2311—31no—
56#2015—32no—
57#2014—32no—
58#2012—32no—
59#2011—32no—
60#1729—31no—
61C25×C3—21no—
62C22×C3×D4—22no—
63C22×C3×D4—22no—
64C22×C3×D4—22no—
65C22×C3×D4—21no—
66C22×C3×D4—22no—
67C22×C3×D4—22no—
68C22×C3×D4D414yes—
69C22×C3×D4C2312yes—
70C23×C3×C4—21no—
71C23×C3×C4C2311yescharacteristic
72C22×#67—21no—
73C22×#67—21no—
74C22×#67—24no—
75C2×#235—22no—
76C2×#235—22no—
77C2×#235—22no—
78C2×#235—22no—
79C2×#235—22no—
80C2×#235—22no—
81C2×#235—22no—
82C2×#235—22no—
83C2×#235—22no—
84C2×#235—22no—
85C2×#235D414yes—
86C2×#235D414yescomplemented
87C22×D12C2311yescharacteristic
88C22×D12C2311yescharacteristic
89C22×C4×D3—21no—
90C22×C4×D3—21no—
91C22×Dic6C2311yescharacteristic
92C22×Dic6C2311yescharacteristic
93C22×C3×C8—22no—
94C22×C3×C8—22no—
95C22×C3×C8D414yescomplemented
96C22×C3×C8C2311yescharacteristic
97C22×C3×C8C2311yescharacteristic
98C2×C3×C4⋊C4—22no—
99C2×C3×C4⋊C4—22no—
100C2×C3×C4⋊C4—21no—
101C2×C3×C4⋊C4C2312yes—
102#953—216no—
103C2×#217—24no—
104C2×#217D412yes—
105#940—24no—
106#940—24no—
107#938—216no—
108#936—24no—
109#936—24no—
110C2×#214—24no—
111C2×#214—24no—
112C2×#214—24no—
113#932—232no—
114C2×#213—24no—
115C2×#213—24no—
116C2×#213—24no—
117C2×(C2×D3)⋊C4—24no—
118#926—22no—
119#926—22no—
120C2×C4⋊Dic3D412yes—
121C2×C4×Dic3C2312yes—
122C2×#208—22no—
123C2×#208—22no—
124C2×#208D414yescomplemented
125C22×C3⋊C8C2312yes—
126C2×Dic12—22no—
127C2×Dic12—22no—
128C2×Dic12—22no—
129#904—232no—
130C2×D24—22no—
131C2×D24—22no—
132C2×D24—22no—
133C2×#204—22no—
134C2×#204—22no—
135C2×#204—22no—
136C2×#204—22no—
137C2×#204—22no—
138C2×#204—22no—
139C3×#102—24no—
140C3×#102—24no—
141C3⋊#95—24no—
142#810—24no—
143#809—24no—
144C22×#41—31no—
145C22×#41—31no—
146C23×D4C2×D311yescharacteristic
147C2×AGL(1,ℤ/8)—64no—
148C2×AGL(1,ℤ/8)—64no—
149C2×AGL(1,ℤ/8)—64no—
150C2×#135—64no—
151C2×#135—64no—
152C2×#135—64no—
153C22×Q16—31no—
154C22×SD16—32no—
155C22×SD16—31no—
156C22×SD16—31no—
157C22×D8—32no—
158C22×D8—31no—
159C22×M16—31no—
160C23×C8C2×D311yescharacteristic
161#535—64no—
162#535—64no—
163C2×#127—32no—
164C2×#124—32no—
165C2×#121—62no—
166C2×#120—62no—
167C2×#117—31no—
168C22×C4⋊C4C2×D311yescharacteristic
169#417—64no—
170C2×#103—32no—
171C2×#102—24no—
172C2×#102—32no—
173C2×#102C2×D312yes—
174C2×#102C2×D312yes—
175#391—32no—
176#390—32no—
177#386—62no—
178#380—62no—
179#336—32no—
180#336—32no—
181C24×C3—22no—
182C24×C3—24no—
183C24×C3—42no—
184C24×C3—42no—
185C24×C3—82no—
186C24×C3—82no—
187C24×C3C2×D411yescharacteristic
188C23×D3—21no—
189C23×D3—21no—
190C2×C3×D4—28no—
191C2×C3×D4—24no—
192C2×C3×D4—28no—
193C2×C3×D4—22no—
194C2×C3×D4—22no—
195C2×C3×D4—22no—
196C2×C3×D4—22no—
197C2×C3×D4—22no—
198C2×C3×D4—44no—
199C2×C3×D4—44no—
200C2×C3×D4—42no—
201C2×C3×D4—42no—
202C2×C3×D4C2×D414yes—
203C22×C3×C4—22no—
204C22×C3×C4—42no—
205C22×C3×C4—82no—
206C22×C3×C4C2×D412yes—
207C22×C3×C4C2×D412yes—
208C22×C3×C4C2×D412yes—
209C22×C3×C4C2×D411yescharacteristic
210C22×C3×C4C2×D412yes—
211C22×C3×C4C2×D412yes—
212C22×C3×C4#4112yes—
213C22×C3×C4C2411yescharacteristic
214C2×#67—24no—
215C2×#67—24no—
216C2×#67—216no—
217C2×#67—42no—
218C2×#67—42no—
219C2×#67—42no—
220C2×#67—42no—
221C2×#67—84no—
222C2×#67—84no—
223C22×Dic3—21no—
224C22×Dic3—21no—
225C22×Dic3C2×D414yes—
226#235—28no—
227#235—28no—
228#235—28no—
229#235—28no—
230#235—44no—
231#235—44no—
232#235—44no—
233#235—44no—
234C2×D12—21no—
235C2×D12—21no—
236C2×D12—21no—
237C2×D12—21no—
238C2×D12—21no—
239C2×D12—21no—
240C2×D12—21no—
241C2×D12—21no—
242C2×D12—21no—
243C2×D12—21no—
244C2×D12C2×D412yes—
245C2×D12C2×D412yes—
246C2×C4×D3—24no—
247C2×C4×D3—24no—
248C2×C4×D3—42no—
249C2×C4×D3—42no—
250C2×C4×D3—42no—
251C2×C4×D3—42no—
252C2×Dic6—21no—
253C2×Dic6—21no—
254C2×Dic6—21no—
255C2×Dic6—21no—
256C2×Dic6—21no—
257C2×Dic6—21no—
258C2×Dic6—21no—
259C2×Dic6—21no—
260C2×Dic6—21no—
261C2×Dic6—21no—
262C2×Dic6C2×D412yes—
263C2×Dic6C2×D412yes—
264C2×C3×C8—28no—
265C2×C3×C8—21no—
266C2×C3×C8—21no—
267C2×C3×C8—21no—
268C2×C3×C8—21no—
269C2×C3×C8C2×D412yes—
270C2×C3×C8C2×D412yes—
271C3×C4⋊C4—22no—
272C3×C4⋊C4—22no—
273C3×C4⋊C4—42no—
274#217—216no—
275#217—24no—
276#214—216no—
277#213—216no—
278(C2×D3)⋊C4—84no—
279C4⋊Dic3—24no—
280C4×Dic3—22no—
281C4×Dic3—22no—
282#208—28no—
283C2×C3⋊C8—22no—
284C2×C3⋊C8—22no—
285C2×C3⋊C8C2×D414yes—
286Dic12—28no—
287D24—28no—
288#204—28no—
289#204—28no—
290C25—21no—
291C2×#41—34no—
292C2×#41—34no—
293C2×#41—62no—
294C2×#41—62no—
295C2×#41—62no—
296C2×#41—62no—
297C2×#41—62no—
298C2×#41—62no—
299C2×#41—62no—
300C2×#41—62no—
301C2×#41—62no—
302C2×#41—62no—
303C22×Q8—31no—
304C22×Q8—31no—
305C22×D4—22no—
306C22×D4—22no—
307C22×D4—22no—
308C22×D4—21no—
309C22×D4—22no—
310C22×D4—22no—
311C22×D4—31no—
312C22×D4—31no—
313C22×D4—64no—
314C22×D4—61no—
315C22×D4—61no—
316C22×D4D1214yes—
317C22×D4C22×D312yes—
318C23×C4—21no—
319C23×C4—61no—
320C23×C4—61no—
321C23×C4C22×D311yescharacteristic
322AGL(1,ℤ/8)—632no—
323#135—632no—
324C2×Q16—62no—
325C2×Q16—62no—
326C2×Q16—62no—
327C2×SD16—62no—
328C2×SD16—62no—
329C2×SD16—62no—
330C2×SD16—62no—
331C2×SD16—64no—
332C2×SD16—64no—
333C2×SD16—64no—
334C2×SD16—62no—
335C2×SD16—62no—
336C2×D8—62no—
337C2×D8—62no—
338C2×D8—64no—
339C2×D8—64no—
340C2×D8—64no—
341C2×D8—62no—
342C2×M16—34no—
343C2×M16—62no—
344C2×M16—62no—
345C22×C8—22no—
346C22×C8—22no—
347C22×C8—32no—
348C22×C8#6714yescomplemented
349C22×C8C22×D311yescharacteristic
350C22×C8C22×D311yescharacteristic
351#127—64no—
352#127—64no—
353#124—64no—
354#124—64no—
355#121—616no—
356#120—616no—
357#117—62no—
358#117—62no—
359C2×C4⋊C4—22no—
360C2×C4⋊C4—22no—
361C2×C4⋊C4—21no—
362C2×C4⋊C4—32no—
363C2×C4⋊C4C22×D312yes—
364C2×C22⋊C4—32no—
365C2×C22⋊C4—64no—
366C2×C22⋊C4—64no—
367C2×C42—32no—
368#103—64no—
369#102—24no—
370#102—24no—
371#102—64no—
372#95—64no—
373C23×C3—22no—
374C23×C3—22no—
375C23×C3—24no—
376C23×C3—42no—
377C23×C3—42no—
378C23×C3—44no—
379C23×C3—44no—
380C23×C3—48no—
381C23×C3—88no—
382C23×C3—84no—
383C23×C3—84no—
384C23×C3—88no—
385C23×C3#12012yes—
386C23×C3#12112yes—
387C23×C3#12712yes—
388C23×C3C22×D411yescharacteristic
389C22×D3—22no—
390C22×D3—22no—
391C22×D3—41no—
392C22×D3—41no—
393C22×D3—41no—
394C22×D3—41no—
395C22×D3—42no—
396C22×D3—42no—
397C22×D3—42no—
398C22×D3—42no—
399C3×D4—24no—
400C3×D4—24no—
401C3×D4—42no—
402C3×D4—42no—
403C3×D4—48no—
404C2×C3×C4—42no—
405C2×C3×C4—44no—
406C2×C3×C4—84no—
407C2×C3×C4#12014yes—
408C2×C3×C4#12012yes—
409C2×C3×C4#12014yes—
410C2×C3×C4#12114yes—
411C2×C3×C4#12112yes—
412C2×C3×C4#12114yes—
413C2×C3×C4#12712yes—
414C2×C3×C4C22×D411yescharacteristic
415C2×C3×C4C22×D411yescharacteristic
416C2×C3×C4C22×D411yescharacteristic
417C2×C3×C4C22×D411yescharacteristic
418C2×C3×C4C22×D411yescharacteristic
419C2×C3×C4C2×#4111yescharacteristic
420#67—48no—
421#67—48no—
422#67—816no—
423C2×Dic3—24no—
424C2×Dic3—22no—
425C2×Dic3—22no—
426C2×Dic3—24no—
427C2×Dic3—24no—
428C2×Dic3—41no—
429C2×Dic3—41no—
430C2×Dic3—41no—
431C2×Dic3—41no—
432D12—22no—
433D12—22no—
434D12—22no—
435D12—22no—
436D12—41no—
437D12—41no—
438D12—41no—
439D12—41no—
440C4×D3—48no—
441C4×D3—48no—
442Dic6—22no—
443Dic6—22no—
444Dic6—22no—
445Dic6—22no—
446Dic6—41no—
447Dic6—41no—
448Dic6—41no—
449Dic6—41no—
450C3×C8—22no—
451C3×C8—22no—
452C3⋊C8—24no—
453C24—22no—
454C24—24no—
455C24—42no—
456C24—42no—
457C24—61no—
458C24—61no—
459C24—82no—
460C24—82no—
461C24D3×D411yescharacteristic
462#41—68no—
463#41—68no—
464#41—68no—
465#41—68no—
466#41—124no—
467#41—124no—
468#41—124no—
469#41—124no—
470C2×Q8—32no—
471C2×Q8—32no—
472C2×Q8—61no—
473C2×Q8—61no—
474C2×Q8—61no—
475C2×Q8—61no—
476C2×Q8—61no—
477C2×Q8—61no—
478C2×Q8—61no—
479C2×Q8—61no—
480C2×Q8—61no—
481C2×Q8—61no—
482C2×D4—28no—
483C2×D4—24no—
484C2×D4—28no—
485C2×D4—22no—
486C2×D4—22no—
487C2×D4—22no—
488C2×D4—22no—
489C2×D4—22no—
490C2×D4—32no—
491C2×D4—32no—
492C2×D4—42no—
493C2×D4—44no—
494C2×D4—44no—
495C2×D4—42no—
496C2×D4—64no—
497C2×D4—64no—
498C2×D4—616no—
499C2×D4—61no—
500C2×D4—61no—
501C2×D4—61no—
502C2×D4—61no—
503C2×D4—61no—
504C2×D4—61no—
505C2×D4—61no—
506C2×D4—61no—
507C2×D4—61no—
508C2×D4—61no—
509C2×D4—122no—
510C2×D4—122no—
511C2×D4—122no—
512C2×D4—122no—
513C2×D4—244no—
514C2×D4—244no—
515C2×D4C2×D1214yes—
516C22×C4—22no—
517C22×C4—34no—
518C22×C4—42no—
519C22×C4—61no—
520C22×C4—61no—
521C22×C4—64no—
522C22×C4—64no—
523C22×C4—82no—
524C22×C4—122no—
525C22×C4—122no—
526C22×C4—122no—
527C22×C4—122no—
528C22×C4C2×D1212yes—
529C22×C4#23512yes—
530C22×C4D3×D411yescharacteristic
531C22×C4D3×D412yes—
532C22×C4D3×D412yes—
533C22×C4D3×D412yes—
534C22×C4C2×#6712yes—
535C22×C4C23×D311yescharacteristic
536Q16—68no—
537SD16—68no—
538SD16—68no—
539SD16—616no—
540D8—68no—
541D8—616no—
542M16—68no—
543C2×C8—28no—
544C2×C8—21no—
545C2×C8—21no—
546C2×C8—21no—
547C2×C8—21no—
548C2×C8—34no—
549C2×C8—62no—
550C2×C8—62no—
551C2×C8C2×#6712yes—
552C2×C8C2×#6712yes—
553C4⋊C4—22no—
554C4⋊C4—22no—
555C4⋊C4—42no—
556C4⋊C4—64no—
557C22⋊C4—616no—
558C22⋊C4—64no—
559C22⋊C4—244no—
560C42—62no—
561C42—62no—
562C22×C3—22no—
563C22×C3—22no—
564C22×C3—24no—
565C22×C3—24no—
566C22×C3—44no—
567C22×C3—48no—
568C22×C3—816no—
569C22×C3—82no—
570C22×C3—82no—
571C22×C3—84no—
572C22×C3#39214yes—
573C22×C3C2×#12011yescharacteristic
574C22×C3C2×#12111yescharacteristic
575C22×C3C2×#12711yescharacteristic
576C22×C3#57612yes—
577C22×C3#57712yes—
578C2×D3—42no—
579C2×D3—42no—
580C2×D3—44no—
581C2×D3—44no—
582C2×D3—81no—
583C2×D3—81no—
584C2×D3—81no—
585C2×D3—81no—
586C2×D3—82no—
587C2×D3—82no—
588C3×C4—82no—
589C3×C4#39214yes—
590C3×C4C2×#12011yescharacteristic
591C3×C4C2×#12011yescharacteristic
592C3×C4C2×#12111yescharacteristic
593C3×C4C2×#12111yescharacteristic
594Dic3—42no—
595Dic3—42no—
596Dic3—44no—
597C23—22no—
598C23—22no—
599C23—24no—
600C23—48no—
601C23—42no—
602C23—44no—
603C23—44no—
604C23—42no—
605C23—62no—
606C23—62no—
607C23—88no—
608C23—84no—
609C23—84no—
610C23—88no—
611C23—121no—
612C23—121no—
613C23—122no—
614C23—122no—
615C23—122no—
616C23—122no—
617C23—121no—
618C23—121no—
619C23#88212yes—
620C23#88412yes—
621C23#94012yes—
622C23C2×D3×D411yescharacteristic
623Q8—62no—
624Q8—62no—
625Q8—62no—
626Q8—62no—
627Q8—121no—
628Q8—121no—
629Q8—121no—
630Q8—121no—
631D4—24no—
632D4—24no—
633D4—48no—
634D4—42no—
635D4—42no—
636D4—62no—
637D4—62no—
638D4—62no—
639D4—62no—
640D4—128no—
641D4—128no—
642D4—121no—
643D4—121no—
644D4—121no—
645D4—121no—
646D4—2416no—
647C2×C4—44no—
648C2×C4—42no—
649C2×C4—64no—
650C2×C4—62no—
651C2×C4—62no—
652C2×C4—64no—
653C2×C4—64no—
654C2×C4—84no—
655C2×C4—121no—
656C2×C4—121no—
657C2×C4—121no—
658C2×C4—121no—
659C2×C4—128no—
660C2×C4—128no—
661C2×C4#88212yes—
662C2×C4#88214yes—
663C2×C4#88412yes—
664C2×C4#88414yes—
665C2×C4#93014yes—
666C2×C4#93714yes—
667C2×C4#94012yes—
668C2×C4C22×D1211yescharacteristic
669C2×C4C2×#23511yescharacteristic
670C2×C4C2×D3×D411yescharacteristic
671C2×C4C2×D3×D411yescharacteristic
672C2×C4C2×D3×D411yescharacteristic
673C2×C4C22×#6711yescharacteristic
674C8—22no—
675C8—22no—
676C8—64no—
677C2×C3—24no—
678C2×C3—84no—
679C2×C3C2×#39211yescharacteristic
680C2×C3#301212yes—
681C2×C3#304512yes—
682C2×C3#315411yescharacteristic
683C2×C3#315511yescharacteristic
684D3—82no—
685D3—82no—
686C22—22no—
687C22—22no—
688C22—24no—
689C22—24no—
690C22—48no—
691C22—44no—
692C22—816no—
693C22—82no—
694C22—82no—
695C22—84no—
696C22—124no—
697C22—124no—
698C22—122no—
699C22—122no—
700C22—241no—
701C22—241no—
702C22—241no—
703C22—241no—
704C22—242no—
705C22—242no—
706C22#496014yes—
707C22C2×#88211yescharacteristic
708C22C2×#88411yescharacteristic
709C22#575712yes—
710C22#576212yes—
711C22C2×#94011yescharacteristic
712C4—82no—
713C4—124no—
714C4—122no—
715C4—122no—
716C4#496014yes—
717C4C2×#88211yescharacteristic
718C4C2×#88411yescharacteristic
719C4C2×#93011yescharacteristic
720C4C2×#93711yescharacteristic
721C3#2342611yescharacteristic, complemented, Sylow
722C2—24no—
723C2—84no—
724C2—242no—
725C2—242no—
726C2#7874712yes—
727C2#7876712yes—
728C2C2×#496011yescharacteristic
729C2#8383011yescharacteristic
730C2#8385411yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC22×C3×#102C2
socleC23×C3C22×D4
hypercenterC22×#102D3
hypocenterC3#23426
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×#102

Complements

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

NH ≅ G/Ntype
1. C2×#5805C2semidirect
2. C22×#932C2semidirect
3. C22×#904C2semidirect
4. C22×C3×#102C2semidirect
5. #82186C2semidirect
6. C2×#4966C2semidirect
7. #71965C2semidirect
8. #70215C2semidirect
9. #70214C2semidirect
10. #70210C2semidirect
11. #70209C2semidirect
31. C22×Dic12C22semidirect
35. C22×D24C22semidirect
36. C22×#204C22semidirect
37. C22×#204C22semidirect
42. #4988C22semidirect
44. #4973C22semidirect
45. C2×#810C22semidirect
47. C2×#809C22semidirect
48. #4550C22semidirect
49. #4550C22semidirect
54. C22×#102D3semidirect
86. C2×#235D4semidirect
95. C22×C3×C8D4semidirect
124. C2×#208D4semidirect
348. C22×C8#67semidirect
721. C3#23426semidirect

Maximal subgroups

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

subgroupindex
C2×#58052
C22×#9322
C22×#9042
C22×C3×#1022
#821862
C2×#49662
#719652
#702152
#702142
#702102
#702092
#234263