Finite Group Databasebeta

← #6433199

Subgroups — #6433199

Order 512.

2414 conjugacy classes of subgroups.

The subgroup lattice has 695 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3095C211yesmaximal, characteristic, complemented
2C2×#3094C211yesmaximal, characteristic
3C2×#2850C211yesmaximal, characteristic
4C2×#2296C211yesmaximal, characteristic, complemented
5#29762C218yesmaximal, complemented
6#29761C214yesmaximal, complemented
7#29760C214yesmaximal, complemented
8#23659C212yesmaximal, complemented
9#23619C212yesmaximal, complemented
10#23618C212yesmaximal, complemented
11#23506C212yesmaximal, complemented
12#20912C211yesmaximal, characteristic
13#20911C211yesmaximal, characteristic, complemented
14#20626C211yesmaximal, characteristic, complemented
15C22×#124C2211yescharacteristic
16C22×C22⋊Q8C2211yescharacteristic
17C22×#121C2211yescharacteristic
18C22×#117C2211yescharacteristic
19C22×#117C2211yescharacteristic
20C23×C42C2211yescharacteristic
21#3095—28no—
22#3094—28no—
23C2×#484C2212yescomplemented
24C2×#481C2212yes—
25C2×#485C2214yescomplemented
26C2×#482C2214yes—
27C2×#483C2212yescomplemented
28C2×#480C2212yes—
29C2×#428C2212yes—
30C2×#428C2212yes—
31C2×#423C2212yes—
32C2×#419C2212yes—
33C2×#419C2212yes—
34C2×#418C2212yes—
35C2×#418C2212yes—
36C2×#407C2211yescharacteristic
37C2×#407C2211yescharacteristic
38#2850—21no—
39#2850—21no—
40#2850—21no—
41#2850—21no—
42#2850C2214yes—
43#2850C2214yes—
44#2296—21no—
45#2296—21no—
46#2296—22no—
47#2296C2212yes—
48#2296C2212yes—
49#2296C2214yescomplemented
50C2×#391C2211yescharacteristic
51C2×#390C2211yescharacteristic
52C2×#389C2212yescomplemented
53C2×#389C2212yescomplemented
54C2×#388—22no—
55C2×#386—22no—
56C2×#385—22no—
57C2×#382C2211yescharacteristic
58C2×C4×C4⋊C4—21no—
59C2×C4×C4⋊C4—21no—
60#2085C2218yescomplemented
61#2085C2218yescomplemented
62#2068C2218yescomplemented
63#2067C2218yescomplemented
64#2066C2218yescomplemented
65#2065C2218yescomplemented
66#2042C2218yescomplemented
67#2042C2218yescomplemented
68#1875C2214yes—
69#1874C2214yes—
70#1873C2214yescomplemented
71#1872C2214yescomplemented
72#1797C2214yescomplemented
73#1797C2214yescomplemented
74#1736C2212yes—
75#1736C2212yes—
76#1728C2212yes—
77C23×Q8C2311yescharacteristic
78C23×D4—21no—
79C23×D4C2311yescharacteristic
80C24×C4D412yes—
81C24×C4C2311yescharacteristic
82C22×M16C2312yes—
83C22×M16C2312yes—
84C2×#124—21no—
85C2×#124—21no—
86C2×#124—44no—
87C2×#124C2314yes—
88C2×C22⋊Q8—21no—
89C2×C22⋊Q8—21no—
90C2×C22⋊Q8—24no—
91C2×C22⋊Q8—24no—
92C2×C22⋊Q8C2312yes—
93C2×C22⋊Q8C2312yes—
94C2×#121—21no—
95C2×#121—21no—
96C2×#121—24no—
97C2×#121—24no—
98C2×#121C2312yes—
99C2×#121C2312yes—
100C2×#117—21no—
101C2×#117—21no—
102C2×#117—24no—
103C2×#117—21no—
104C2×#117—21no—
105C2×#117—44no—
106C2×#117D418yes—
107C2×#117C2312yes—
108C2×#117C2312yes—
109C2×#117C2312yes—
110C2×#117C2312yes—
111C22×C4⋊C4—21no—
112C22×C4⋊C4—22no—
113C22×C4⋊C4—22no—
114C22×C4⋊C4C2311yescharacteristic
115C22×C4⋊C4C2311yescharacteristic
116C22×C4⋊C4C2311yescharacteristic
117C22×C4⋊C4C2311yescharacteristic
118C22×C4⋊C4C2312yes—
119C22×C22⋊C4—21no—
120C22×C22⋊C4—21no—
121C22×C22⋊C4—21no—
122C22×C22⋊C4—21no—
123C22×C22⋊C4—22no—
124C22×C22⋊C4—22no—
125C22×C42—21no—
126C22×C42—21no—
127C22×C42—44no—
128C22×C42D411yescharacteristic
129C22×C42D411yescharacteristic
130C22×C42D411yescharacteristic
131C22×C42D411yescharacteristic
132C22×C42D412yes—
133C22×C42C2311yescharacteristic
134C22×C42C2311yescharacteristic
135C22×C42C2312yes—
136C22×C42C2312yes—
137#485—216no—
138#484—28no—
139#483—28no—
140#482—216no—
141#481—28no—
142#480—28no—
143#428—28no—
144#428—28no—
145C2×#107C2314yes—
146C2×#107C2314yes—
147C2×#106C2314yes—
148C2×#106C2314yes—
149#423—44no—
150C2×C4⋊C8C2318yes—
151#419—28no—
152#419—44no—
153#418—28no—
154#418—44no—
155C2×#103C2314yes—
156C2×#103C2314yes—
157C2×#103C2318yes—
158C2×#102C2314yes—
159C2×#102C2314yes—
160C2×#102C2318yes—
161#407—24no—
162#407—24no—
163C2×C22⋊C8C2312yes—
164C2×C22⋊C8C2312yes—
165C2×C22⋊C8C2312yes—
166C2×C22⋊C8C2312yes—
167#391—21no—
168#391—21no—
169#391C2314yes—
170#390—21no—
171#390—21no—
172#390C2314yes—
173#389—22no—
174#389—22no—
175#389—22no—
176#389—22no—
177#389C2314yes—
178#389C2314yes—
179#389C2314yes—
180#389C2314yes—
181#388—24no—
182#388—24no—
183#388—24no—
184#388—24no—
185#386—24no—
186#386—24no—
187#386—24no—
188#386—24no—
189#385—24no—
190#385—24no—
191#385—24no—
192#385—24no—
193#382—21no—
194#382—21no—
195#382C2314yes—
196C4×C4⋊C4—22no—
197C4×C4⋊C4—22no—
198C4×C4⋊C4—22no—
199C4×C4⋊C4—22no—
200C4×C4⋊C4—24no—
201C4×C4⋊C4—24no—
202C2×#95—22no—
203C2×#95—22no—
204C2×#95—22no—
205C2×#95—22no—
206#340C2318yes—
207#340C2318yes—
208#336C2318yes—
209C25—21no—
210C25C2×D411yescharacteristic
211C22×Q8—21no—
212C22×Q8—22no—
213C22×Q8—22no—
214C22×Q8—24no—
215C22×Q8—24no—
216C22×Q8C2412yes—
217C22×D4—22no—
218C22×D4—22no—
219C22×D4—22no—
220C22×D4—21no—
221C22×D4—22no—
222C22×D4—24no—
223C22×D4—24no—
224C22×D4—48no—
225C22×D4—44no—
226C22×D4C2412yes—
227C23×C4—24no—
228C23×C4—24no—
229C23×C4—21no—
230C23×C4—21no—
231C23×C4—21no—
232C23×C4—21no—
233C23×C4—22no—
234C23×C4—21no—
235C23×C4—44no—
236C23×C4—44no—
237C23×C4—44no—
238C23×C4C2×D412yes—
239C23×C4C2×D412yes—
240C23×C4C2×D411yescharacteristic
241C23×C4C2×D411yescharacteristic
242C23×C4C2×D411yescharacteristic
243C23×C4C2×D414yes—
244C23×C4C2×D412yes—
245C23×C4C2×D412yes—
246C23×C4C2×D412yes—
247C23×C4C2×D412yes—
248C23×C4#4114yes—
249C23×C4C2411yescharacteristic
250C23×C4C2411yescharacteristic
251C23×C4C2411yescharacteristic
252C2×M16—24no—
253C2×M16—24no—
254C2×M16—24no—
255C2×M16—24no—
256C2×M16—24no—
257C2×M16—24no—
258C22×C8C2414yes—
259C22×C8C2414yes—
260#124—48no—
261#124—44no—
262#124—44no—
263C22⋊Q8—24no—
264C22⋊Q8—24no—
265C22⋊Q8—24no—
266C22⋊Q8—24no—
267C22⋊Q8—44no—
268C22⋊Q8—44no—
269#121—24no—
270#121—24no—
271#121—24no—
272#121—24no—
273#121—44no—
274#121—44no—
275#117—28no—
276#117—28no—
277#117—24no—
278#117—24no—
279#117—24no—
280#117—24no—
281#117—44no—
282#117—44no—
283#117—44no—
284#117C2×D418yes—
285#117C2×D418yes—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—24no—
297C2×C4⋊C4—24no—
298C2×C4⋊C4—24no—
299C2×C4⋊C4—24no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—44no—
304C2×C4⋊C4—48no—
305C2×C4⋊C4—48no—
306C2×C4⋊C4—44no—
307C2×C4⋊C4C2×D414yes—
308C2×C4⋊C4C2×D414yes—
309C2×C4⋊C4C2414yes—
310C2×C4⋊C4C2412yes—
311C2×C4⋊C4C2412yes—
312C2×C4⋊C4C2412yes—
313C2×C4⋊C4C2412yes—
314C2×C22⋊C4—28no—
315C2×C22⋊C4—22no—
316C2×C22⋊C4—22no—
317C2×C22⋊C4—22no—
318C2×C22⋊C4—22no—
319C2×C22⋊C4—22no—
320C2×C22⋊C4—22no—
321C2×C22⋊C4—22no—
322C2×C22⋊C4—22no—
323C2×C22⋊C4—24no—
324C2×C22⋊C4—24no—
325C2×C22⋊C4—24no—
326C2×C22⋊C4—24no—
327C2×C22⋊C4—48no—
328C2×C22⋊C4—48no—
329C2×C22⋊C4—44no—
330C2×C22⋊C4—44no—
331C2×C22⋊C4—44no—
332C2×C42—28no—
333C2×C42—22no—
334C2×C42—22no—
335C2×C42—24no—
336C2×C42—24no—
337C2×C42—22no—
338C2×C42—22no—
339C2×C42—42no—
340C2×C42—42no—
341C2×C42—42no—
342C2×C42—42no—
343C2×C42—44no—
344C2×C42—44no—
345C2×C42—44no—
346C2×C42—44no—
347C2×C42C2×D411yescharacteristic
348C2×C42C2×D411yescharacteristic
349C2×C42C2×D411yescharacteristic
350C2×C42C2×D411yescharacteristic
351C2×C42C2×D412yes—
352C2×C42C2×D412yes—
353C2×C42C2×D412yes—
354C2×C42C2×D412yes—
355C2×C42C2412yes—
356C2×C42C2412yes—
357#107—28no—
358#107—28no—
359#106—28no—
360#106—28no—
361C4⋊C8—48no—
362#103—28no—
363#103—28no—
364#103—48no—
365#102—28no—
366#102—28no—
367#102—48no—
368C22⋊C8—24no—
369C22⋊C8—24no—
370C22⋊C8—24no—
371C22⋊C8—24no—
372#95—24no—
373#95—24no—
374#95—24no—
375#95—24no—
376#95—24no—
377#95—24no—
378#95—24no—
379#95—24no—
380C24—22no—
381C24—22no—
382C24—24no—
383C24—22no—
384C24—42no—
385C24—44no—
386C24—44no—
387C24—42no—
388C24—42no—
389C24—42no—
390C24#12114yes—
391C24C22×D411yescharacteristic
392C24C22×D411yescharacteristic
393C24C22×D411yescharacteristic
394C2×Q8—22no—
395C2×Q8—22no—
396C2×Q8—24no—
397C2×Q8—24no—
398C2×Q8—42no—
399C2×Q8—42no—
400C2×Q8—42no—
401C2×Q8—48no—
402C2×Q8—48no—
403C2×D4—22no—
404C2×D4—22no—
405C2×D4—24no—
406C2×D4—24no—
407C2×D4—48no—
408C2×D4—44no—
409C2×D4—44no—
410C2×D4—48no—
411C2×D4—42no—
412C2×D4—42no—
413C2×D4—42no—
414C2×D4—48no—
415C2×D4—48no—
416C2×D4—816no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—216no—
424C22×C4—216no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—24no—
430C22×C4—24no—
431C22×C4—24no—
432C22×C4—24no—
433C22×C4—24no—
434C22×C4—24no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—24no—
438C22×C4—24no—
439C22×C4—24no—
440C22×C4—22no—
441C22×C4—24no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—42no—
445C22×C4—42no—
446C22×C4—42no—
447C22×C4—42no—
448C22×C4—42no—
449C22×C4—42no—
450C22×C4—42no—
451C22×C4—42no—
452C22×C4—42no—
453C22×C4—42no—
454C22×C4—42no—
455C22×C4—44no—
456C22×C4—432no—
457C22×C4—44no—
458C22×C4—44no—
459C22×C4—44no—
460C22×C4—44no—
461C22×C4—44no—
462C22×C4—42no—
463C22×C4—42no—
464C22×C4—42no—
465C22×C4—42no—
466C22×C4—42no—
467C22×C4—44no—
468C22×C4#12114yes—
469C22×C4#12114yes—
470C22×C4#12114yes—
471C22×C4C22×D412yes—
472C22×C4C22×D412yes—
473C22×C4C22×D412yes—
474C22×C4C22×D411yescharacteristic
475C22×C4C22×D411yescharacteristic
476C22×C4C22×D412yes—
477C22×C4C22×D412yes—
478C22×C4C22×D412yes—
479C22×C4C22×D411yescharacteristic
480C22×C4C22×D412yes—
481C22×C4C2×#4112yes—
482C22×C4C2×#4112yes—
483C22×C4C2×#4112yes—
484C22×C4H(5,𝔽2)14yes—
485C22×C42−1+414yes—
486C22×C4C2511yescharacteristic
487M16—48no—
488M16—48no—
489C2×C8—24no—
490C2×C8—24no—
491C2×C8—24no—
492C2×C8—24no—
493C2×C8—24no—
494C2×C8—24no—
495C4⋊C4—24no—
496C4⋊C4—24no—
497C4⋊C4—22no—
498C4⋊C4—22no—
499C4⋊C4—22no—
500C4⋊C4—22no—
501C4⋊C4—22no—
502C4⋊C4—22no—
503C4⋊C4—24no—
504C4⋊C4—24no—
505C4⋊C4—48no—
506C4⋊C4—48no—
507C4⋊C4—48no—
508C4⋊C4—44no—
509C4⋊C4—42no—
510C4⋊C4—42no—
511C4⋊C4—44no—
512C4⋊C4—44no—
513C4⋊C4—48no—
514C4⋊C4C22×D414yes—
515C4⋊C4C22×D414yes—
516C22⋊C4—28no—
517C22⋊C4—28no—
518C22⋊C4—48no—
519C22⋊C4—48no—
520C22⋊C4—48no—
521C22⋊C4—44no—
522C22⋊C4—44no—
523C22⋊C4—44no—
524C22⋊C4—44no—
525C22⋊C4—44no—
526C22⋊C4—44no—
527C22⋊C4—48no—
528C42—28no—
529C42—28no—
530C42—22no—
531C42—22no—
532C42—22no—
533C42—22no—
534C42—24no—
535C42—24no—
536C42—42no—
537C42—42no—
538C42—42no—
539C42—42no—
540C42—44no—
541C42—44no—
542C23—21no—
543C23—21no—
544C23—22no—
545C23—22no—
546C23—22no—
547C23—28no—
548C23—21no—
549C23—21no—
550C23—28no—
551C23—22no—
552C23—22no—
553C23—44no—
554C23—48no—
555C23—48no—
556C23—42no—
557C23—42no—
558C23—42no—
559C23—44no—
560C23—44no—
561C23—84no—
562C23—88no—
563C23—88no—
564C23—84no—
565C23—84no—
566C23C2×#12112yes—
567C23C2×#12112yes—
568C23C2×#12112yes—
569C23#53412yes—
570C23#53612yes—
571C23#57514yes—
572C23C23×D411yescharacteristic
573Q8—44no—
574Q8—44no—
575Q8—84no—
576D4—44no—
577D4—44no—
578D4—88no—
579D4—88no—
580D4—84no—
581C2×C4—28no—
582C2×C4—22no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—22no—
588C2×C4—28no—
589C2×C4—28no—
590C2×C4—24no—
591C2×C4—24no—
592C2×C4—24no—
593C2×C4—24no—
594C2×C4—24no—
595C2×C4—24no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—22no—
600C2×C4—22no—
601C2×C4—22no—
602C2×C4—22no—
603C2×C4—22no—
604C2×C4—22no—
605C2×C4—48no—
606C2×C4—48no—
607C2×C4—48no—
608C2×C4—48no—
609C2×C4—48no—
610C2×C4—48no—
611C2×C4—48no—
612C2×C4—48no—
613C2×C4—44no—
614C2×C4—44no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—44no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—44no—
622C2×C4—44no—
623C2×C4—44no—
624C2×C4—44no—
625C2×C4—88no—
626C2×C4—84no—
627C2×C4—84no—
628C2×C4—84no—
629C2×C4C2×#12112yes—
630C2×C4C2×#12112yes—
631C2×C4C2×#12112yes—
632C2×C4#53412yes—
633C2×C4#53612yes—
634C2×C4#53718yes—
635C2×C4#53918yes—
636C2×C4#57514yes—
637C2×C4C23×D411yescharacteristic
638C2×C4C22×#4111yescharacteristic
639C2×C4C2×H(5,𝔽2)12yes—
640C2×C4C2×2−1+412yes—
641C8—44no—
642C8—44no—
643C22—24no—
644C22—24no—
645C22—24no—
646C22—24no—
647C22—22no—
648C22—22no—
649C22—22no—
650C22—22no—
651C22—22no—
652C22—22no—
653C22—22no—
654C22—416no—
655C22—42no—
656C22—42no—
657C22—84no—
658C22—84no—
659C22—82no—
660C22—82no—
661C22—82no—
662C22—168no—
663C22#261814yes—
664C22#304014yes—
665C22C22×#12111yescharacteristic
666C22C2×#53411yescharacteristic
667C22C2×#53611yescharacteristic
668C22C2×#57512yes—
669C22C2×#57512yes—
670C4—24no—
671C4—22no—
672C4—22no—
673C4—42no—
674C4—42no—
675C4—42no—
676C4—44no—
677C4—44no—
678C4—84no—
679C4—82no—
680C4—82no—
681C4—82no—
682C4#261814yes—
683C4C2×#53712yes—
684C4C2×#53912yes—
685C2—24no—
686C2—24no—
687C2—42no—
688C2—162no—
689C2C2×#261811yescharacteristic
690C2C2×#304011yescharacteristic
691C2#6041511yescharacteristic
692C2#6075212yes—
693C2#6086612yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#121C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C42 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#3095C2semidirect
4. C2×#2296C2semidirect
5. #29762C2semidirect
6. #29761C2semidirect
7. #29760C2semidirect
8. #23659C2semidirect
9. #23619C2semidirect
10. #23618C2semidirect
11. #23506C2semidirect
13. #20911C2semidirect
14. #20626C2semidirect
23. C2×#484C22semidirect
25. C2×#485C22semidirect
27. C2×#483C22semidirect
49. #2296C22semidirect
52. C2×#389C22semidirect
53. C2×#389C22semidirect
60. #2085C22semidirect
61. #2085C22semidirect
62. #2068C22semidirect
63. #2067C22semidirect
64. #2066C22semidirect
65. #2065C22semidirect
66. #2042C22semidirect
67. #2042C22semidirect
70. #1873C22semidirect
71. #1872C22semidirect
72. #1797C22semidirect
73. #1797C22semidirect

Maximal subgroups

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

subgroupindex
C2×#30952
C2×#30942
C2×#28502
C2×#22962
#297622
#297612
#297602
#236592
#236192
#236182
#235062
#209122
#209112
#206262