Finite Group Databasebeta

← #4419099

Subgroups — #4419099

Order 512.

2799 conjugacy classes of subgroups.

The subgroup lattice has 739 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2498C211yesmaximal, characteristic, complemented
2C2×#2624C211yesmaximal, characteristic, complemented
3C2×#2661C211yesmaximal, characteristic
4C2×#2346C211yesmaximal, characteristic, complemented
5#17071C212yesmaximal, complemented
6#16899C212yesmaximal, complemented
7#16519C212yesmaximal, complemented
8#16518C212yesmaximal, complemented
9#15419C214yesmaximal, complemented
10#15303C214yesmaximal, complemented
11#15147C212yesmaximal, complemented
12#15081C212yesmaximal, complemented
13#14903C212yesmaximal, complemented
14#14900C212yesmaximal, complemented
15#14658C211yesmaximal, characteristic, complemented
16#14271C211yesmaximal, characteristic, complemented
17#13965C211yesmaximal, characteristic, complemented
18C22×#123C2211yescharacteristic
19C22×#121C2211yescharacteristic
20C22×#117C2211yescharacteristic
21#2661—28no—
22#2624—28no—
23#2498—28no—
24C2×#400C2211yescharacteristic
25C2×#399C2212yes—
26C2×#399C2211yescharacteristic
27C2×#398C2211yescharacteristic
28C2×#398C2211yescharacteristic
29C2×#397C2211yescharacteristic
30C2×#397C2211yescharacteristic
31C2×#397C2212yes—
32C2×#396C2214yes—
33C2×#396C2214yes—
34C2×#394C2211yescharacteristic
35C2×#394C2212yes—
36C2×#394C2214yes—
37C2×#392C2211yescharacteristic
38C2×#392C2212yes—
39#2346—28no—
40C2×#390C2212yes—
41C2×#390C2212yes—
42C2×#390C2214yes—
43C2×#389C2212yes—
44C2×#389C2212yes—
45C2×#388C2212yes—
46C2×#388C2212yes—
47C2×#388C2214yes—
48C2×#388C2214yes—
49C2×#387C2212yes—
50C2×#386C2212yes—
51C2×#386C2214yes—
52C2×#385C2214yes—
53C2×#382C2212yes—
54C2×#380C2211yescharacteristic
55C2×#380C2211yescharacteristic
56C2×#380C2212yes—
57C2×C4×C4⋊C4C2212yes—
58C2×C4×C22⋊C4C2211yescharacteristic
59C2×#376C2212yes—
60C22×#95C2211yescharacteristic
61#1445C2212yes—
62#1444C2212yes—
63#1444C2214yes—
64#1444C2212yes—
65#1441C2214yes—
66#1441C2212yes—
67#1441C2214yes—
68#1441C2212yes—
69#1441C2214yes—
70#1440C2214yes—
71#1440C2214yes—
72#1440C2212yes—
73#1439C2212yes—
74#1439C2214yes—
75#1439C2214yes—
76#1439C2212yes—
77#1439C2212yes—
78#1439C2214yes—
79#1439C2212yes—
80#1438C2214yes—
81#1438C2212yes—
82#1438C2212yes—
83C4⋊#95C2212yes—
84C4⋊#95C2214yes—
85C4⋊#95C2212yes—
86#1434C2214yes—
87C23×D4C2311yescharacteristic
88C23×D4C2312yes—
89C23×D4C2312yes—
90C23×D4C2311yescharacteristic
91C24×C4C2312yes—
92C24×C4C2311yescharacteristic
93C2×#123—24no—
94C2×#123—24no—
95C2×#123—24no—
96C2×#121—24no—
97C2×#121—24no—
98C2×#121—24no—
99C2×#117—24no—
100C2×#117—24no—
101C2×#117D418yescomplemented
102C22×C4⋊C4C2311yescharacteristic
103C22×C4⋊C4C2311yescharacteristic
104C22×C4⋊C4C2311yescharacteristic
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2311yescharacteristic
107C22×C4⋊C4C2311yescharacteristic
108C22×C4⋊C4C2314yes—
109C22×C4⋊C4C2312yes—
110C22×C4⋊C4C2314yes—
111C22×C4⋊C4C2314yes—
112C22×C4⋊C4C2312yes—
113C22×C4⋊C4C2314yes—
114C22×C4⋊C4C2312yes—
115C22×C4⋊C4C2312yes—
116C22×C22⋊C4C2312yes—
117C22×C22⋊C4C2311yescharacteristic
118C22×C22⋊C4C2311yescharacteristic
119C22×C22⋊C4C2312yes—
120C22×C22⋊C4C2311yescharacteristic
121C22×C22⋊C4C2312yes—
122C22×C22⋊C4C2312yes—
123C22×C22⋊C4C2312yes—
124C22×C22⋊C4C2312yes—
125C22×C22⋊C4C2312yes—
126C22×C22⋊C4C2312yes—
127C22×C22⋊C4C2312yes—
128C22×C22⋊C4C2314yes—
129C22×C22⋊C4C2314yes—
130C22×C22⋊C4C2314yes—
131C22×C22⋊C4C2312yes—
132C22×C22⋊C4C2312yes—
133C22×C22⋊C4C2314yes—
134C22×C22⋊C4C2314yes—
135C22×C42C2314yes—
136C22×C42C2312yes—
137C22×C42C2312yes—
138#400—24no—
139#399—28no—
140#399—24no—
141#398—24no—
142#398—24no—
143#397—28no—
144#397—24no—
145#397—24no—
146#396—48no—
147#396—48no—
148#394—216no—
149#394—28no—
150#394—24no—
151#392—28no—
152#392—24no—
153#390—28no—
154#390—28no—
155#390—48no—
156#389—28no—
157#389—28no—
158#388—216no—
159#388—28no—
160#388—28no—
161#388—48no—
162#387—28no—
163#386—28no—
164#386—48no—
165#385—48no—
166#382—28no—
167#380—24no—
168#380—24no—
169#380—28no—
170C4×C4⋊C4—28no—
171C4×C22⋊C4D418yes—
172#376—28no—
173C2×#95—24no—
174C2×#95C2312yes—
175C2×#95C2314yes—
176C2×#95C2312yes—
177C2×#95C2314yes—
178C2×#95C2312yes—
179C2×#95C2311yescharacteristic
180C2×#95C2312yes—
181C2×#95C2311yescharacteristic
182C2×#95C2312yes—
183C2×#95C2312yes—
184C2×#95C2312yes—
185C2×#95C2314yes—
186C2×#95C2314yes—
187C2×#95C2312yes—
188C2×#95C2312yes—
189C2×#95C2314yes—
190C2×#95C2314yes—
191C2×#95C2314yes—
192C2×#95C2314yes—
193C2×#95C2314yes—
194C2×#95C2314yes—
195C2×#95C2311yescharacteristic
196C2×#95C2311yescharacteristic
197C25C2411yescharacteristic
198C25C2412yes—
199C25C2411yescharacteristic
200C22×D4—22no—
201C22×D4—24no—
202C22×D4—24no—
203C22×D4—24no—
204C22×D4—22no—
205C22×D4—22no—
206C22×D4—22no—
207C22×D4—22no—
208C22×D4—24no—
209C22×D4—22no—
210C22×D4—24no—
211C22×D4—24no—
212C22×D4—44no—
213C22×D4—44no—
214C22×D4—44no—
215C22×D4—44no—
216C22×D4—44no—
217C22×D4—44no—
218C23×C4—24no—
219C23×C4—24no—
220C23×C4—44no—
221C23×C4—44no—
222C23×C4—42no—
223C23×C4—44no—
224C23×C4C2×D414yes—
225C23×C4C2×D414yes—
226C23×C4#4114yes—
227C23×C4C2411yescharacteristic
228C23×C4C2414yes—
229C23×C4C2411yescharacteristic
230C23×C4C2414yes—
231C23×C4C2412yes—
232C23×C4C2412yes—
233C23×C4C2411yescharacteristic
234C23×C4C2412yes—
235C23×C4C2412yes—
236C23×C4C2412yes—
237C23×C4C2412yes—
238C23×C4C2411yescharacteristic
239C23×C4C2411yescharacteristic
240C23×C4C2412yes—
241#123—416no—
242#121—416no—
243#117—232no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—24no—
262C2×C4⋊C4—28no—
263C2×C4⋊C4—28no—
264C2×C4⋊C4—24no—
265C2×C4⋊C4—28no—
266C2×C4⋊C4—28no—
267C2×C4⋊C4—42no—
268C2×C4⋊C4—44no—
269C2×C4⋊C4—44no—
270C2×C4⋊C4—44no—
271C2×C4⋊C4—44no—
272C2×C4⋊C4—44no—
273C2×C4⋊C4—44no—
274C2×C4⋊C4—44no—
275C2×C4⋊C4—44no—
276C2×C4⋊C4—44no—
277C2×C4⋊C4—44no—
278C2×C4⋊C4—44no—
279C2×C4⋊C4—44no—
280C2×C4⋊C4C2×D414yes—
281C2×C4⋊C4C2×D414yes—
282C2×C22⋊C4—24no—
283C2×C22⋊C4—24no—
284C2×C22⋊C4—24no—
285C2×C22⋊C4—24no—
286C2×C22⋊C4—22no—
287C2×C22⋊C4—22no—
288C2×C22⋊C4—22no—
289C2×C22⋊C4—24no—
290C2×C22⋊C4—24no—
291C2×C22⋊C4—22no—
292C2×C22⋊C4—22no—
293C2×C22⋊C4—22no—
294C2×C22⋊C4—22no—
295C2×C22⋊C4—24no—
296C2×C22⋊C4—24no—
297C2×C22⋊C4—24no—
298C2×C22⋊C4—22no—
299C2×C22⋊C4—22no—
300C2×C22⋊C4—28no—
301C2×C22⋊C4—24no—
302C2×C22⋊C4—24no—
303C2×C22⋊C4—24no—
304C2×C22⋊C4—24no—
305C2×C22⋊C4—24no—
306C2×C22⋊C4—24no—
307C2×C22⋊C4—24no—
308C2×C22⋊C4—24no—
309C2×C22⋊C4—24no—
310C2×C22⋊C4—28no—
311C2×C22⋊C4—28no—
312C2×C22⋊C4—42no—
313C2×C22⋊C4—44no—
314C2×C22⋊C4—44no—
315C2×C22⋊C4—44no—
316C2×C22⋊C4—44no—
317C2×C22⋊C4—44no—
318C2×C22⋊C4—44no—
319C2×C22⋊C4—44no—
320C2×C22⋊C4—44no—
321C2×C22⋊C4—44no—
322C2×C22⋊C4—44no—
323C2×C22⋊C4—44no—
324C2×C22⋊C4—44no—
325C2×C22⋊C4—44no—
326C2×C22⋊C4—44no—
327C2×C22⋊C4—44no—
328C2×C22⋊C4—44no—
329C2×C22⋊C4—44no—
330C2×C22⋊C4—44no—
331C2×C22⋊C4C2×D418yes—
332C2×C22⋊C4C2×D418yes—
333C2×C42—24no—
334C2×C42—24no—
335C2×C42—28no—
336C2×C42—28no—
337C2×C42—28no—
338C2×C42—44no—
339C2×C42C2×D418yes—
340C2×C42C2×D418yes—
341#95—24no—
342#95—22no—
343#95—22no—
344#95—24no—
345#95—28no—
346#95—28no—
347#95—24no—
348#95—24no—
349#95—24no—
350#95—24no—
351#95—24no—
352#95—24no—
353#95—24no—
354#95—28no—
355#95—28no—
356#95—28no—
357#95—44no—
358#95—44no—
359#95—44no—
360#95—44no—
361#95—44no—
362#95C2×D414yes—
363#95C2×D414yes—
364C24—22no—
365C24—22no—
366C24—22no—
367C24—24no—
368C24—22no—
369C24—21no—
370C24—22no—
371C24—21no—
372C24—22no—
373C24—21no—
374C24—22no—
375C24—22no—
376C24—22no—
377C24—24no—
378C24—22no—
379C24—22no—
380C24—22no—
381C24—22no—
382C24—22no—
383C24—24no—
384C24—24no—
385C24—24no—
386C24—22no—
387C24—24no—
388C24C22×D412yes—
389C24C22×D412yes—
390C24C2×#4112yes—
391C24C2511yescharacteristic
392C2×D4—48no—
393C2×D4—416no—
394C2×D4—48no—
395C2×D4—416no—
396C2×D4—48no—
397C2×D4—44no—
398C2×D4—44no—
399C2×D4—48no—
400C2×D4—84no—
401C2×D4—84no—
402C2×D4—84no—
403C2×D4—84no—
404C2×D4—88no—
405C2×D4—84no—
406C2×D4—88no—
407C2×D4—84no—
408C2×D4—84no—
409C2×D4—84no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—24no—
420C22×C4—22no—
421C22×C4—24no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—22no—
427C22×C4—24no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—24no—
432C22×C4—22no—
433C22×C4—21no—
434C22×C4—24no—
435C22×C4—21no—
436C22×C4—24no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—22no—
440C22×C4—21no—
441C22×C4—22no—
442C22×C4—21no—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—22no—
446C22×C4—21no—
447C22×C4—24no—
448C22×C4—24no—
449C22×C4—24no—
450C22×C4—24no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—24no—
455C22×C4—24no—
456C22×C4—24no—
457C22×C4—24no—
458C22×C4—24no—
459C22×C4—22no—
460C22×C4—24no—
461C22×C4—24no—
462C22×C4—24no—
463C22×C4—24no—
464C22×C4—24no—
465C22×C4—24no—
466C22×C4—24no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—24no—
470C22×C4—48no—
471C22×C4—48no—
472C22×C4—48no—
473C22×C4—48no—
474C22×C4—48no—
475C22×C4—88no—
476C22×C4—84no—
477C22×C4—88no—
478C22×C4#121116yes—
479C22×C4C22×D414yes—
480C22×C4C22×D414yes—
481C22×C4C22×D414yes—
482C22×C4C22×D412yes—
483C22×C4C22×D412yes—
484C22×C4C22×D412yes—
485C22×C4C22×D412yes—
486C22×C4C22×D414yes—
487C22×C4C2×#4112yes—
488C22×C4C2×#4112yes—
489C22×C4H(5,𝔽2)14yes—
490C22×C4H(5,𝔽2)14yes—
491C22×C42−1+414yes—
492C4⋊C4—28no—
493C4⋊C4—28no—
494C4⋊C4—48no—
495C4⋊C4—48no—
496C4⋊C4—48no—
497C4⋊C4—44no—
498C4⋊C4—44no—
499C4⋊C4—44no—
500C4⋊C4—84no—
501C4⋊C4—88no—
502C4⋊C4—88no—
503C4⋊C4—84no—
504C4⋊C4—88no—
505C4⋊C4—88no—
506C22⋊C4—216no—
507C22⋊C4—216no—
508C22⋊C4—48no—
509C22⋊C4—416no—
510C22⋊C4—44no—
511C22⋊C4—48no—
512C22⋊C4—44no—
513C22⋊C4—44no—
514C22⋊C4—48no—
515C22⋊C4—84no—
516C22⋊C4—88no—
517C22⋊C4—88no—
518C22⋊C4—84no—
519C22⋊C4—84no—
520C22⋊C4—84no—
521C22⋊C4—84no—
522C22⋊C4—84no—
523C22⋊C4—88no—
524C22⋊C4—88no—
525C42—216no—
526C42—48no—
527C42—416no—
528C23—24no—
529C23—24no—
530C23—24no—
531C23—24no—
532C23—24no—
533C23—22no—
534C23—22no—
535C23—22no—
536C23—22no—
537C23—24no—
538C23—24no—
539C23—44no—
540C23—44no—
541C23—44no—
542C23—44no—
543C23—41no—
544C23—42no—
545C23—42no—
546C23—42no—
547C23—42no—
548C23—44no—
549C23—42no—
550C23—42no—
551C23—42no—
552C23—42no—
553C23—42no—
554C23—44no—
555C23—44no—
556C23—42no—
557C23—41no—
558C23—41no—
559C23—42no—
560C23—42no—
561C23—41no—
562C23—42no—
563C23—41no—
564C23—42no—
565C23—44no—
566C23—42no—
567C23—42no—
568C23—42no—
569C23—42no—
570C23—44no—
571C23—44no—
572C23—44no—
573C23—44no—
574C23—44no—
575C23—44no—
576C23—44no—
577C23—44no—
578C23—44no—
579C23—44no—
580C23—44no—
581C23—44no—
582C23C2×#12114yes—
583C23C23×D411yescharacteristic
584C23C23×D411yescharacteristic
585C23C22×#4111yescharacteristic
586C23C2×H(5,𝔽2)12yes—
587C23C2×H(5,𝔽2)12yes—
588C23C2×2−1+412yes—
589C23#58514yes—
590C23#58512yes—
591D4—88no—
592D4—816no—
593D4—88no—
594D4—816no—
595C2×C4—24no—
596C2×C4—24no—
597C2×C4—24no—
598C2×C4—24no—
599C2×C4—24no—
600C2×C4—24no—
601C2×C4—24no—
602C2×C4—24no—
603C2×C4—28no—
604C2×C4—28no—
605C2×C4—24no—
606C2×C4—24no—
607C2×C4—24no—
608C2×C4—24no—
609C2×C4—24no—
610C2×C4—24no—
611C2×C4—24no—
612C2×C4—24no—
613C2×C4—24no—
614C2×C4—24no—
615C2×C4—28no—
616C2×C4—416no—
617C2×C4—44no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—44no—
625C2×C4—44no—
626C2×C4—44no—
627C2×C4—44no—
628C2×C4—41no—
629C2×C4—41no—
630C2×C4—42no—
631C2×C4—41no—
632C2×C4—44no—
633C2×C4—42no—
634C2×C4—44no—
635C2×C4—42no—
636C2×C4—42no—
637C2×C4—42no—
638C2×C4—44no—
639C2×C4—44no—
640C2×C4—42no—
641C2×C4—44no—
642C2×C4—42no—
643C2×C4—42no—
644C2×C4—41no—
645C2×C4—44no—
646C2×C4—44no—
647C2×C4—42no—
648C2×C4—42no—
649C2×C4—41no—
650C2×C4—41no—
651C2×C4—41no—
652C2×C4—42no—
653C2×C4—42no—
654C2×C4—44no—
655C2×C4—42no—
656C2×C4—44no—
657C2×C4—44no—
658C2×C4—44no—
659C2×C4—44no—
660C2×C4—44no—
661C2×C4—44no—
662C2×C4—44no—
663C2×C4—44no—
664C2×C4—44no—
665C2×C4—44no—
666C2×C4—816no—
667C2×C4C2×#12114yes—
668C2×C4C2×#12114yes—
669C2×C4#53718yes—
670C22—24no—
671C22—24no—
672C22—24no—
673C22—44no—
674C22—44no—
675C22—44no—
676C22—48no—
677C22—44no—
678C22—82no—
679C22—81no—
680C22—82no—
681C22—84no—
682C22—82no—
683C22—82no—
684C22—82no—
685C22—82no—
686C22—82no—
687C22—82no—
688C22—81no—
689C22—82no—
690C22—82no—
691C22—81no—
692C22—84no—
693C22—84no—
694C22—84no—
695C22—82no—
696C22—84no—
697C22C22×#12111yescharacteristic
698C22C2×#53712yes—
699C22#344912yes—
700C22#348512yes—
701C22#348812yes—
702C22#349112yes—
703C22#349811yescharacteristic
704C22#349811yescharacteristic
705C22#349911yescharacteristic
706C22#350011yescharacteristic
707C22#350612yes—
708C22#351112yes—
709C22#353314yes—
710C22#354214yes—
711C22#354714yes—
712C22#356414yes—
713C4—24no—
714C4—24no—
715C4—28no—
716C4—44no—
717C4—44no—
718C4—44no—
719C4—44no—
720C4—81no—
721C4—84no—
722C4—81no—
723C4—81no—
724C4—82no—
725C4—84no—
726C4—82no—
727C2—44no—
728C2—161no—
729C2—162no—
730C2#3508911yescharacteristic
731C2#3527311yescharacteristic
732C2#3553311yescharacteristic
733C2#4136912yes—
734C2#4195812yes—
735C2#4231412yes—
736C2#4264612yes—
737C2#4886914yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC2×#2661C2
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2498C2semidirect
2. C2×#2624C2semidirect
4. C2×#2346C2semidirect
5. #17071C2semidirect
6. #16899C2semidirect
7. #16519C2semidirect
8. #16518C2semidirect
9. #15419C2semidirect
10. #15303C2semidirect
11. #15147C2semidirect
12. #15081C2semidirect
13. #14903C2semidirect
14. #14900C2semidirect
15. #14658C2semidirect
16. #14271C2semidirect
17. #13965C2semidirect
101. C2×#117D4semidirect

Maximal subgroups

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

subgroupindex
C2×#24982
C2×#26242
C2×#26612
C2×#23462
#170712
#168992
#165192
#165182
#154192
#153032
#151472
#150812
#149032
#149002
#146582
#142712
#139652