Finite Group Databasebeta

← #6375632

Subgroups — #6375632

Order 512.

2124 conjugacy classes of subgroups.

The subgroup lattice has 723 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3095C211yesmaximal, characteristic
2C2×#2323C211yesmaximal, characteristic
3#34052C211yesmaximal, characteristic
4#22094C214yesmaximal, complemented
5#22092C214yesmaximal, complemented
6#22089C214yesmaximal, complemented
7#22087C214yesmaximal, complemented
8#21159C212yesmaximal, complemented
9#21094C212yesmaximal, complemented
10#21033C212yesmaximal, complemented
11#20911C211yesmaximal, characteristic
12#20874C212yesmaximal, complemented
13#20846C211yesmaximal, characteristic
14#20702C211yesmaximal, characteristic
15#20377C211yesmaximal, characteristic
16C22×C22⋊Q8C2211yescharacteristic
17C22×C4×Q8C2211yescharacteristic
18C22×#117C2211yescharacteristic
19C22×#117C2211yescharacteristic
20C22×#117C2211yescharacteristic
21#3095C4116yescomplemented
22C2×#484C2212yes—
23C2×#485C2214yes—
24C2×#483C2212yes—
25C2×#428C2212yes—
26C2×#428C2212yes—
27C2×#419C2212yes—
28C2×#419C2212yes—
29C2×#407C2211yescharacteristic
30C2×#407C2211yescharacteristic
31C2×C4×M16C2212yes—
32#2323—22no—
33#2323—22no—
34#2323C2214yes—
35#2323C2214yes—
36#2317—21no—
37#2317—21no—
38#2317C2212yes—
39#2317C2212yes—
40#2316C2212yes—
41#2316C2212yes—
42#2315—21no—
43#2315—21no—
44#2296—22no—
45#2296C2212yes—
46#2296C2212yes—
47C2×#391C2211yescharacteristic
48C2×#389—21no—
49C2×#389C2211yescharacteristic
50C2×#388—21no—
51C2×#388—21no—
52C2×#387—21no—
53#2285—22no—
54C2×#380C2211yescharacteristic
55C2×C4×C4⋊C4C2211yescharacteristic
56#1944C2214yescomplemented
57#1943C2214yescomplemented
58#1942C2214yescomplemented
59#1941C2214yescomplemented
60#1935C2214yescomplemented
61#1935C2214yescomplemented
62#1933C2214yescomplemented
63#1933C2214yescomplemented
64#1921C2214yescomplemented
65#1920C2214yescomplemented
66#1919C2214yescomplemented
67#1918C2214yescomplemented
68#1873C2214yes—
69#1872C2212yes—
70#1872C2212yes—
71#1865C2214yescomplemented
72#1865C2214yescomplemented
73#1863C2214yescomplemented
74#1863C2214yescomplemented
75#1854C2214yes—
76#1853C2212yes—
77#1852C2212yes—
78#1825C2214yes—
79#1824C2212yes—
80#1823C2212yes—
81#1750C2214yes—
82#1750C2214yes—
83#1736C2212yes—
84#1736C2212yes—
85#1736C2212yes—
86#1736C2212yes—
87#1728C2212yes—
88C23×Q8C2311yescharacteristic
89C24×C4C2311yescharacteristic
90C22×M16C2312yes—
91C22×M16C2312yes—
92C2×C22⋊Q8—21no—
93C2×C22⋊Q8—21no—
94C2×C22⋊Q8—24no—
95C2×C22⋊Q8C2×C418yes—
96C2×C22⋊Q8C2312yes—
97C2×C22⋊Q8C2312yes—
98C2×C4×Q8—22no—
99C2×C4×Q8—44no—
100C2×C4×Q8C2314yes—
101C2×#117—21no—
102C2×#117—21no—
103C2×#117—24no—
104C2×#117—21no—
105C2×#117—21no—
106C2×#117—44no—
107C2×#117—44no—
108C2×#117C2×C418yes—
109C2×#117D412yes—
110C2×#117D412yes—
111C2×#117C2312yes—
112C2×#117C2312yes—
113C2×#117C2312yes—
114C2×#117C2312yes—
115C2×#117C2312yes—
116C2×#117C2312yes—
117C22×C4⋊C4—21no—
118C22×C4⋊C4—21no—
119C22×C4⋊C4—21no—
120C22×C4⋊C4D411yescharacteristic
121C22×C4⋊C4D411yescharacteristic
122C22×C4⋊C4C2311yescharacteristic
123C22×C4⋊C4C2311yescharacteristic
124C22×C4⋊C4C2311yescharacteristic
125C22×C4⋊C4C2311yescharacteristic
126C22×C4⋊C4C2311yescharacteristic
127C22×C4⋊C4C2311yescharacteristic
128C22×C4⋊C4C2311yescharacteristic
129C22×C22⋊C4—21no—
130C22×C22⋊C4—21no—
131C22×C22⋊C4—21no—
132C22×C22⋊C4—21no—
133C22×C22⋊C4D411yescharacteristic
134C22×C22⋊C4D411yescharacteristic
135C22×C42—21no—
136C22×C42—21no—
137C22×C42—21no—
138C22×C42—21no—
139C22×C42C2311yescharacteristic
140C22×C42C2311yescharacteristic
141#485C2×C4132yescomplemented
142#484C2×C4116yescomplemented
143#483C2×C4116yescomplemented
144#428—44no—
145#428C2×C4116yescomplemented
146C2×#107C2314yes—
147C2×#107C2314yes—
148C2×#106C2314yes—
149C2×#106C2314yes—
150#419—44no—
151#419C2×C4116yescomplemented
152C2×#103C2314yes—
153C2×#103C2314yes—
154C2×#103C2314yes—
155C2×#103C2314yes—
156#407—24no—
157#407C2×C418yes—
158C2×C22⋊C8C2312yes—
159C2×C22⋊C8C2312yes—
160C2×C22⋊C8C2312yes—
161C2×C22⋊C8C2312yes—
162C4×M16—44no—
163C2×#97C2314yes—
164C2×C4×C8C2314yes—
165#391—22no—
166#391C2314yes—
167#389—22no—
168#389—22no—
169#389—22no—
170#389—22no—
171#389—24no—
172#389—24no—
173#389—22no—
174#389C2312yes—
175#389C2312yes—
176#389C2312yes—
177#389C2312yes—
178#389C2312yes—
179#389C2312yes—
180#389C2312yes—
181#389C2312yes—
182#389C2314yes—
183#388—24no—
184#388—24no—
185#388—24no—
186#388—24no—
187#387—24no—
188#387—24no—
189#385—22no—
190#385—22no—
191#385—22no—
192#385—22no—
193#385—22no—
194#385—22no—
195#385—22no—
196#385—22no—
197#384—22no—
198#384—22no—
199#382—22no—
200#382—22no—
201#380—21no—
202#380—21no—
203#380—22no—
204#380—22no—
205#380—22no—
206#380—22no—
207#380C2312yes—
208#380C2312yes—
209C4×C4⋊C4—22no—
210C4×C4⋊C4—22no—
211C4×C4⋊C4—22no—
212C4×C4⋊C4—22no—
213C4×C4⋊C4—22no—
214C4×C4⋊C4C2314yes—
215C2×#95—21no—
216C2×#95—21no—
217C2×#95—21no—
218C2×#95—21no—
219C2×#95—21no—
220#340C2314yes—
221#340C2314yes—
222#340C2314yes—
223#340C2314yes—
224#340C2314yes—
225#340C2314yes—
226#340C2314yes—
227#340C2314yes—
228#336C2314yes—
229#336C2314yes—
230C25C2×D411yescharacteristic
231C22×Q8—21no—
232C22×Q8—22no—
233C22×Q8—24no—
234C22×Q8—24no—
235C22×Q8C22×C414yes—
236C22×Q8C2412yes—
237C23×C4—24no—
238C23×C4—24no—
239C23×C4—21no—
240C23×C4—21no—
241C23×C4—21no—
242C23×C4—21no—
243C23×C4—21no—
244C23×C4—21no—
245C23×C4—21no—
246C23×C4—21no—
247C23×C4C22×C414yes—
248C23×C4C2×D411yescharacteristic
249C23×C4C2×D411yescharacteristic
250C23×C4C2×D411yescharacteristic
251C23×C4C2×D411yescharacteristic
252C23×C4C2×D411yescharacteristic
253C23×C4C2×D411yescharacteristic
254C23×C4C2×D411yescharacteristic
255C23×C4#4114yes—
256C23×C4C2411yescharacteristic
257C23×C4C2411yescharacteristic
258C23×C4C2411yescharacteristic
259C2×M16—24no—
260C2×M16—24no—
261C2×M16—24no—
262C2×M16—24no—
263C2×M16—24no—
264C2×M16C22×C418yes—
265C22×C8C2414yes—
266C22×C8C2414yes—
267C22⋊Q8—28no—
268C22⋊Q8—28no—
269C22⋊Q8—24no—
270C22⋊Q8—24no—
271C22⋊Q8C22×C418yes—
272C22⋊Q8C22×C418yes—
273C4×Q8—44no—
274C4×Q8—44no—
275C4×Q8—48no—
276#117—28no—
277#117—28no—
278#117—24no—
279#117—24no—
280#117—24no—
281#117—24no—
282#117—44no—
283#117—44no—
284#117—42no—
285#117—42no—
286#117—42no—
287#117—42no—
288#117—42no—
289#117—42no—
290#117—44no—
291#117—44no—
292#117C22×C418yes—
293#117C22×C418yes—
294C2×C4⋊C4—28no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—24no—
304C2×C4⋊C4—24no—
305C2×C4⋊C4—24no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—24no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—22no—
318C2×C4⋊C4—22no—
319C2×C4⋊C4—42no—
320C2×C4⋊C4—42no—
321C2×C4⋊C4—42no—
322C2×C4⋊C4—42no—
323C2×C4⋊C4—44no—
324C2×C4⋊C4—82no—
325C2×C4⋊C4C22×C414yes—
326C2×C4⋊C4C22×C414yes—
327C2×C4⋊C4C22×C414yes—
328C2×C4⋊C4C2×D412yes—
329C2×C4⋊C4C2×D412yes—
330C2×C4⋊C4C2×D412yes—
331C2×C4⋊C4C2×D412yes—
332C2×C4⋊C4C2412yes—
333C2×C4⋊C4C2412yes—
334C2×C4⋊C4C2412yes—
335C2×C4⋊C4C2412yes—
336C2×C4⋊C4C2412yes—
337C2×C4⋊C4C2412yes—
338C2×C4⋊C4C2412yes—
339C2×C22⋊C4—28no—
340C2×C22⋊C4—28no—
341C2×C22⋊C4—22no—
342C2×C22⋊C4—22no—
343C2×C22⋊C4—22no—
344C2×C22⋊C4—22no—
345C2×C22⋊C4—22no—
346C2×C22⋊C4—22no—
347C2×C22⋊C4—22no—
348C2×C22⋊C4—22no—
349C2×C22⋊C4—42no—
350C2×C22⋊C4—42no—
351C2×C22⋊C4—44no—
352C2×C22⋊C4—82no—
353C2×C22⋊C4C2×D412yes—
354C2×C22⋊C4C2×D412yes—
355C2×C22⋊C4C2×D412yes—
356C2×C22⋊C4C2×D412yes—
357C2×C42—28no—
358C2×C42—22no—
359C2×C42—22no—
360C2×C42—24no—
361C2×C42—24no—
362C2×C42—22no—
363C2×C42—22no—
364C2×C42—24no—
365C2×C42—24no—
366C2×C42—44no—
367C2×C42—44no—
368C2×C42—82no—
369C2×C42C2×D412yes—
370C2×C42C2×D412yes—
371C2×C42C2412yes—
372C2×C42C2412yes—
373#107—44no—
374#107C22×C4116yes—
375#106—44no—
376#106C22×C4116yes—
377#103—44no—
378#103—44no—
379#103C22×C4116yes—
380#103C22×C4116yes—
381C22⋊C8—24no—
382C22⋊C8—24no—
383C22⋊C8C22×C418yes—
384C22⋊C8C22×C418yes—
385#97—44no—
386C4×C8—44no—
387#95—24no—
388#95—22no—
389#95—22no—
390#95—24no—
391#95—24no—
392#95—24no—
393#95—22no—
394#95—22no—
395#95—22no—
396#95—22no—
397#95—22no—
398#95—22no—
399#95—22no—
400#95—22no—
401C24—22no—
402C24—22no—
403C24—41no—
404C24—41no—
405C24—41no—
406C24—41no—
407C24C4×D414yes—
408C24C22×D411yescharacteristic
409C24C22×D411yescharacteristic
410C24C22×D411yescharacteristic
411C2×Q8—24no—
412C2×Q8—22no—
413C2×Q8—216no—
414C2×Q8—24no—
415C2×Q8—24no—
416C2×Q8—42no—
417C2×Q8—42no—
418C2×Q8—48no—
419C2×Q8C23×C414yes—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—216no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—216no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—24no—
433C22×C4—24no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—24no—
441C22×C4—24no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—41no—
449C22×C4—41no—
450C22×C4—41no—
451C22×C4—41no—
452C22×C4—41no—
453C22×C4—41no—
454C22×C4—41no—
455C22×C4—41no—
456C22×C4—42no—
457C22×C4—42no—
458C22×C4—42no—
459C22×C4—42no—
460C22×C4—42no—
461C22×C4—42no—
462C22×C4—42no—
463C22×C4—42no—
464C22×C4—42no—
465C22×C4—42no—
466C22×C4—42no—
467C22×C4—42no—
468C22×C4—81no—
469C22×C4—81no—
470C22×C4—81no—
471C22×C4—81no—
472C22×C4—81no—
473C22×C4—81no—
474C22×C4—81no—
475C22×C4—81no—
476C22×C4C4×D414yes—
477C22×C4C4×D414yes—
478C22×C4C4×D414yes—
479C22×C4C23×C412yes—
480C22×C4C23×C412yes—
481C22×C4C23×C412yes—
482C22×C4C22×D412yes—
483C22×C4C22×D412yes—
484C22×C4C22×D412yes—
485C22×C4C22×D412yes—
486C22×C4C22×D411yescharacteristic
487C22×C4C22×D411yescharacteristic
488C22×C4C22×D411yescharacteristic
489C22×C4C2×#4112yes—
490C22×C4C2×#4112yes—
491C22×C4C2×#4112yes—
492C22×C4H(5,𝔽2)14yes—
493C22×C42−1+414yes—
494C22×C4C2511yescharacteristic
495M16—216no—
496M16—48no—
497C2×C8—24no—
498C2×C8—24no—
499C2×C8—24no—
500C2×C8—24no—
501C2×C8—24no—
502C2×C8C23×C418yes—
503C4⋊C4—28no—
504C4⋊C4—28no—
505C4⋊C4—24no—
506C4⋊C4—24no—
507C4⋊C4—24no—
508C4⋊C4—22no—
509C4⋊C4—22no—
510C4⋊C4—22no—
511C4⋊C4—22no—
512C4⋊C4—22no—
513C4⋊C4—22no—
514C4⋊C4—24no—
515C4⋊C4—22no—
516C4⋊C4—24no—
517C4⋊C4—42no—
518C4⋊C4—42no—
519C4⋊C4—42no—
520C4⋊C4—42no—
521C4⋊C4—42no—
522C4⋊C4—42no—
523C4⋊C4—48no—
524C4⋊C4—42no—
525C4⋊C4—42no—
526C4⋊C4—82no—
527C4⋊C4—82no—
528C4⋊C4—82no—
529C4⋊C4—82no—
530C4⋊C4—82no—
531C4⋊C4—82no—
532C4⋊C4—82no—
533C4⋊C4—82no—
534C4⋊C4—84no—
535C4⋊C4C23×C414yes—
536C4⋊C4C23×C414yes—
537C4⋊C4C23×C414yes—
538C22⋊C4—28no—
539C22⋊C4—28no—
540C22⋊C4—28no—
541C22⋊C4—28no—
542C22⋊C4—42no—
543C22⋊C4—42no—
544C22⋊C4—42no—
545C22⋊C4—42no—
546C22⋊C4—44no—
547C22⋊C4—44no—
548C22⋊C4—82no—
549C22⋊C4—82no—
550C42—28no—
551C42—28no—
552C42—24no—
553C42—24no—
554C42—22no—
555C42—22no—
556C42—22no—
557C42—22no—
558C42—44no—
559C42—48no—
560C42—44no—
561C42—84no—
562C23—21no—
563C23—21no—
564C23—21no—
565C23—21no—
566C23—21no—
567C23—21no—
568C23—21no—
569C23—21no—
570C23—21no—
571C23—21no—
572C23—21no—
573C23—21no—
574C23—21no—
575C23—21no—
576C23—21no—
577C23—21no—
578C23—28no—
579C23—44no—
580C23—44no—
581C23—82no—
582C23—82no—
583C23—82no—
584C23—82no—
585C23C2×C4×D412yes—
586C23C2×C4×D412yes—
587C23C2×C4×D412yes—
588C23#53412yes—
589C23#53612yes—
590C23C23×D411yescharacteristic
591Q8—28no—
592Q8—48no—
593Q8—44no—
594C2×C4—28no—
595C2×C4—21no—
596C2×C4—21no—
597C2×C4—21no—
598C2×C4—21no—
599C2×C4—28no—
600C2×C4—28no—
601C2×C4—24no—
602C2×C4—24no—
603C2×C4—24no—
604C2×C4—24no—
605C2×C4—22no—
606C2×C4—22no—
607C2×C4—22no—
608C2×C4—22no—
609C2×C4—22no—
610C2×C4—22no—
611C2×C4—22no—
612C2×C4—22no—
613C2×C4—24no—
614C2×C4—24no—
615C2×C4—24no—
616C2×C4—24no—
617C2×C4—22no—
618C2×C4—22no—
619C2×C4—22no—
620C2×C4—22no—
621C2×C4—48no—
622C2×C4—44no—
623C2×C4—44no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—48no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—42no—
637C2×C4—42no—
638C2×C4—42no—
639C2×C4—42no—
640C2×C4—42no—
641C2×C4—42no—
642C2×C4—42no—
643C2×C4—42no—
644C2×C4—42no—
645C2×C4—82no—
646C2×C4—82no—
647C2×C4—84no—
648C2×C4—84no—
649C2×C4—84no—
650C2×C4—84no—
651C2×C4—84no—
652C2×C4—84no—
653C2×C4C2×C4×D412yes—
654C2×C4C2×C4×D412yes—
655C2×C4C2×C4×D412yes—
656C2×C4#52018yes—
657C2×C4#53412yes—
658C2×C4#53612yes—
659C2×C4#53918yes—
660C2×C4C24×C411yescharacteristic
661C2×C4C22×#4111yescharacteristic
662C2×C4C2×H(5,𝔽2)12yes—
663C2×C4C2×2−1+412yes—
664C8—28no—
665C8—44no—
666C22—22no—
667C22—22no—
668C22—22no—
669C22—22no—
670C22—22no—
671C22—22no—
672C22—22no—
673C22—22no—
674C22—22no—
675C22—22no—
676C22—22no—
677C22—22no—
678C22—41no—
679C22—41no—
680C22—41no—
681C22—41no—
682C22—41no—
683C22—41no—
684C22—41no—
685C22—41no—
686C22—88no—
687C22#232214yes—
688C22C22×C4×D411yescharacteristic
689C22C2×#53411yescharacteristic
690C22C2×#53611yescharacteristic
691C22#358511yescharacteristic
692C22#358611yescharacteristic
693C22#358611yescharacteristic
694C22#358611yescharacteristic
695C4—22no—
696C4—22no—
697C4—44no—
698C4—44no—
699C4—44no—
700C4—44no—
701C4—42no—
702C4—42no—
703C4—42no—
704C4—42no—
705C4—82no—
706C4—82no—
707C4—82no—
708C4—82no—
709C4#232214yes—
710C4C2×#52012yes—
711C4C2×#53912yes—
712C2—24no—
713C2—42no—
714C2—42no—
715C2C2×#232211yescharacteristic
716C2#6016911yescharacteristic
717C2#6017011yescharacteristic
718C2#6075411yescharacteristic
719C2#6075511yescharacteristic
720C2#6086911yescharacteristic
721C2#6087011yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutatorC2×C4C24×C4
Frattini, cosocleC22×C4C25
prime radicalC25C2×D4
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #22094C2semidirect
5. #22092C2semidirect
6. #22089C2semidirect
7. #22087C2semidirect
8. #21159C2semidirect
9. #21094C2semidirect
10. #21033C2semidirect
12. #20874C2semidirect
21. #3095C4semidirect
56. #1944C22semidirect
57. #1943C22semidirect
58. #1942C22semidirect
59. #1941C22semidirect
60. #1935C22semidirect
61. #1935C22semidirect
62. #1933C22semidirect
63. #1933C22semidirect
64. #1921C22semidirect
65. #1920C22semidirect
66. #1919C22semidirect
67. #1918C22semidirect
71. #1865C22semidirect
72. #1865C22semidirect
73. #1863C22semidirect
74. #1863C22semidirect
141. #485C2×C4semidirect
142. #484C2×C4semidirect
143. #483C2×C4semidirect
145. #428C2×C4semidirect
151. #419C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#30952
C2×#23232
#340522
#220942
#220922
#220892
#220872
#211592
#210942
#210332
#209112
#208742
#208462
#207022
#203772