Finite Group Databasebeta

← #629998

Subgroups — #629998

Order 512.

4040 conjugacy classes of subgroups.

The subgroup lattice has 756 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#544C211yesmaximal, characteristic, complemented
2C2×#2850C211yesmaximal, characteristic
3C2×#2639C211yesmaximal, characteristic
4C2×#2618C211yesmaximal, characteristic
5#15882C212yesmaximal, complemented
6#15881C212yesmaximal, complemented
7#15880C214yesmaximal, complemented
8#15783C212yesmaximal, complemented
9#15761C212yesmaximal, complemented
10#15322C214yesmaximal, complemented
11#14759C214yesmaximal, complemented
12#14702C211yesmaximal, characteristic, complemented
13#14281C211yesmaximal, characteristic, complemented
14#14135C214yesmaximal, complemented
15C2×#1436C211yesmaximal, characteristic, complemented
16C2×#544—28no—
17C2×#544—28no—
18C2×#544—28no—
19C22×C4⋊Q8C2212yes—
20C22×C4⋊Q8C2212yes—
21C22×#125C2212yes—
22C22×#125C2212yes—
23C22×#124C2211yescharacteristic
24C22×#123C2212yes—
25C22×C22⋊Q8C2212yes—
26C22×C22⋊Q8C2211yescharacteristic
27C22×C22⋊Q8C2211yescharacteristic
28C22×#121C2211yescharacteristic
29C22×#117C2212yes—
30C23×C4⋊C4C2211yescharacteristic
31C23×C42C2211yescharacteristic
32#2850—28no—
33#2639—28no—
34#2618—28no—
35C2×#400C2212yes—
36C2×#398C2212yes—
37C2×#397C2212yes—
38C2×#396C2214yes—
39C2×#394C2212yes—
40C2×#394C2212yes—
41C2×#393C2212yes—
42C2×#393C2214yes—
43C2×#391C2211yescharacteristic
44C2×#391C2212yes—
45C2×#390C2211yescharacteristic
46C2×#388C2214yes—
47C2×#386C2214yes—
48C2×#386C2214yes—
49C2×#386C2212yes—
50C2×#382C2211yescharacteristic
51C2×#380C2211yescharacteristic
52C22×#95C2211yescharacteristic
53C22×#95C2212yes—
54C22×#95C2212yes—
55#1446C2212yes—
56#1446C2214yes—
57#1445C2212yes—
58#1442C2214yes—
59#1442C2214yes—
60#1442C2214yes—
61#1441C2214yes—
62#1441C2214yes—
63#1441C2214yes—
64#1439C2214yes—
65#1439C2214yes—
66#1439C2214yes—
67#1439C2214yes—
68#1439C2214yes—
69#1437C2214yes—
70#1437C2214yes—
71#1436C2214yes—
72#1436C2214yes—
73#1436C2214yes—
74#1436C2212yes—
75#1436C2212yes—
76#1436C2212yes—
77#1436C2212yes—
78C4⋊#95C2212yes—
79C22⋊#95C2214yes—
80C22⋊#95C2214yes—
81C23×Q8C2312yes—
82C23×Q8C2311yescharacteristic
83C23×Q8C2311yescharacteristic
84C23×D4C2312yes—
85C23×D4C2311yescharacteristic
86C24×C4C2312yes—
87C24×C4C2311yescharacteristic
88C24×C4C2312yes—
89C24×C4C2311yescharacteristic
90C24×C4C2311yescharacteristic
91#544—2128no—
92C2×C4⋊Q8—28no—
93C2×C4⋊Q8—28no—
94C2×C4⋊Q8—28no—
95C2×C4⋊Q8—28no—
96C2×C4⋊Q8—28no—
97C2×C4⋊Q8—28no—
98C2×#125—28no—
99C2×#125—28no—
100C2×#125—28no—
101C2×#125—28no—
102C2×#125—28no—
103C2×#125—28no—
104C2×#124—24no—
105C2×#124—44no—
106C2×#123—28no—
107C2×#123—216no—
108C2×C22⋊Q8—28no—
109C2×C22⋊Q8—24no—
110C2×C22⋊Q8—24no—
111C2×C22⋊Q8—24no—
112C2×C22⋊Q8—24no—
113C2×C22⋊Q8—28no—
114C2×C22⋊Q8—28no—
115C2×C22⋊Q8D418yes—
116C2×C22⋊Q8D418yes—
117C2×#121—24no—
118C2×#121—24no—
119C2×#121—24no—
120C2×#117—28no—
121C2×#117—28no—
122C2×#117—28no—
123C22×C4⋊C4—24no—
124C22×C4⋊C4—24no—
125C22×C4⋊C4D418yes—
126C22×C4⋊C4C2314yes—
127C22×C4⋊C4C2311yescharacteristic
128C22×C4⋊C4C2312yes—
129C22×C4⋊C4C2312yes—
130C22×C4⋊C4C2312yes—
131C22×C4⋊C4C2312yes—
132C22×C4⋊C4C2312yes—
133C22×C4⋊C4C2312yes—
134C22×C4⋊C4C2312yes—
135C22×C4⋊C4C2312yes—
136C22×C4⋊C4C2311yescharacteristic
137C22×C4⋊C4C2311yescharacteristic
138C22×C4⋊C4C2312yes—
139C22×C4⋊C4C2312yes—
140C22×C22⋊C4C2312yes—
141C22×C22⋊C4C2314yes—
142C22×C22⋊C4C2312yes—
143C22×C22⋊C4C2314yes—
144C22×C22⋊C4C2314yes—
145C22×C22⋊C4C2312yes—
146C22×C22⋊C4C2312yes—
147C22×C22⋊C4C2314yes—
148C22×C22⋊C4C2314yes—
149C22×C22⋊C4C2314yes—
150C22×C42—24no—
151C22×C42—44no—
152C22×C42C2311yescharacteristic
153C22×C42C2312yes—
154C22×C42C2312yes—
155C22×C42C2312yes—
156C22×C42C2311yescharacteristic
157#400—28no—
158#398—28no—
159#397—28no—
160#396—216no—
161#394—28no—
162#394—28no—
163#393—28no—
164#393—48no—
165#391—24no—
166#391—44no—
167#390—24no—
168#388—216no—
169#386—28no—
170#386—216no—
171#386—216no—
172#382—24no—
173#380—24no—
174C2×#95—24no—
175C2×#95—28no—
176C2×#95—44no—
177C2×#95C2312yes—
178C2×#95C2314yes—
179C2×#95C2314yes—
180C2×#95C2314yes—
181C2×#95C2312yes—
182C2×#95C2312yes—
183C2×#95C2314yes—
184C2×#95C2314yes—
185C2×#95C2314yes—
186C2×#95C2314yes—
187C2×#95C2314yes—
188C2×#95C2314yes—
189C2×#95C2314yes—
190C2×#95C2314yes—
191C2×#95C2314yes—
192C2×#95C2314yes—
193C2×#95C2314yes—
194C2×#95C2314yes—
195C2×#95C2314yes—
196C2×#95C2314yes—
197C25C2412yes—
198C25C2411yescharacteristic
199C22×Q8—24no—
200C22×Q8—22no—
201C22×Q8—22no—
202C22×Q8—22no—
203C22×Q8—22no—
204C22×Q8—24no—
205C22×Q8—24no—
206C22×Q8—42no—
207C22×Q8—42no—
208C22×Q8—42no—
209C22×Q8—42no—
210C22×Q8—44no—
211C22×Q8—44no—
212C22×Q8C2×D414yes—
213C22×Q8C2×D414yes—
214C22×D4—24no—
215C22×D4—24no—
216C22×D4—24no—
217C22×D4—28no—
218C22×D4—28no—
219C22×D4—22no—
220C22×D4—22no—
221C22×D4—22no—
222C22×D4—42no—
223C22×D4—42no—
224C23×C4—22no—
225C23×C4—22no—
226C23×C4—24no—
227C23×C4—24no—
228C23×C4—22no—
229C23×C4—22no—
230C23×C4—24no—
231C23×C4—24no—
232C23×C4—24no—
233C23×C4—24no—
234C23×C4—24no—
235C23×C4—24no—
236C23×C4—24no—
237C23×C4—42no—
238C23×C4—42no—
239C23×C4—42no—
240C23×C4—42no—
241C23×C4—42no—
242C23×C4—42no—
243C23×C4—42no—
244C23×C4—42no—
245C23×C4—42no—
246C23×C4—42no—
247C23×C4C2×D414yes—
248C23×C4C2×D414yes—
249C23×C4C2×D414yes—
250C23×C4C2×D414yes—
251C23×C4C2×D414yes—
252C23×C4#4118yes—
253C23×C4C2414yes—
254C23×C4C2414yes—
255C23×C4C2412yes—
256C23×C4C2412yes—
257C23×C4C2412yes—
258C23×C4C2412yes—
259C23×C4C2412yes—
260C23×C4C2412yes—
261C23×C4C2412yes—
262C23×C4C2412yes—
263C23×C4C2411yescharacteristic
264C23×C4C2411yescharacteristic
265C23×C4C2412yes—
266C4⋊Q8—264no—
267C4⋊Q8—264no—
268#125—264no—
269#125—264no—
270#124—416no—
271#123—432no—
272C22⋊Q8—264no—
273C22⋊Q8—232no—
274C22⋊Q8—232no—
275#121—416no—
276#117—264no—
277C2×C4⋊C4—232no—
278C2×C4⋊C4—24no—
279C2×C4⋊C4—24no—
280C2×C4⋊C4—24no—
281C2×C4⋊C4—24no—
282C2×C4⋊C4—24no—
283C2×C4⋊C4—24no—
284C2×C4⋊C4—24no—
285C2×C4⋊C4—24no—
286C2×C4⋊C4—24no—
287C2×C4⋊C4—24no—
288C2×C4⋊C4—24no—
289C2×C4⋊C4—24no—
290C2×C4⋊C4—24no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—24no—
296C2×C4⋊C4—24no—
297C2×C4⋊C4—24no—
298C2×C4⋊C4—24no—
299C2×C4⋊C4—24no—
300C2×C4⋊C4—24no—
301C2×C4⋊C4—24no—
302C2×C4⋊C4—24no—
303C2×C4⋊C4—24no—
304C2×C4⋊C4—24no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—28no—
309C2×C4⋊C4—28no—
310C2×C4⋊C4—28no—
311C2×C4⋊C4—24no—
312C2×C4⋊C4—42no—
313C2×C4⋊C4—42no—
314C2×C4⋊C4C2×D418yes—
315C2×C4⋊C4C2×D418yes—
316C2×C4⋊C4C2×D414yes—
317C2×C4⋊C4C2×D414yes—
318C2×C4⋊C4C2×D418yes—
319C2×C4⋊C4C2×D418yes—
320C2×C22⋊C4—24no—
321C2×C22⋊C4—24no—
322C2×C22⋊C4—24no—
323C2×C22⋊C4—24no—
324C2×C22⋊C4—24no—
325C2×C22⋊C4—28no—
326C2×C22⋊C4—24no—
327C2×C22⋊C4—24no—
328C2×C22⋊C4—24no—
329C2×C22⋊C4—28no—
330C2×C22⋊C4—28no—
331C2×C22⋊C4—28no—
332C2×C22⋊C4—28no—
333C2×C22⋊C4—28no—
334C2×C22⋊C4—28no—
335C2×C22⋊C4—28no—
336C2×C22⋊C4—28no—
337C2×C22⋊C4—28no—
338C2×C22⋊C4—28no—
339C2×C22⋊C4—44no—
340C2×C22⋊C4—44no—
341C2×C22⋊C4—44no—
342C2×C22⋊C4—44no—
343C2×C22⋊C4—44no—
344C2×C22⋊C4—44no—
345C2×C22⋊C4—44no—
346C2×C22⋊C4—44no—
347C2×C22⋊C4C2×D418yes—
348C2×C22⋊C4C2×D418yes—
349C2×C42—24no—
350C2×C42—24no—
351C2×C42—24no—
352C2×C42—24no—
353C2×C42—24no—
354C2×C42—24no—
355C2×C42—22no—
356C2×C42—24no—
357C2×C42—24no—
358C2×C42—22no—
359C2×C42—24no—
360C2×C42—416no—
361C2×C42—42no—
362C2×C42—42no—
363#95—24no—
364#95—24no—
365#95—24no—
366#95—28no—
367#95—28no—
368#95—28no—
369#95—28no—
370#95—28no—
371#95—28no—
372#95—28no—
373#95—28no—
374#95—28no—
375#95—28no—
376#95—44no—
377#95—44no—
378#95—44no—
379#95—44no—
380#95—44no—
381#95—44no—
382#95—44no—
383C24—22no—
384C24—22no—
385C24—22no—
386C24—22no—
387C24—22no—
388C24—22no—
389C24—22no—
390C24—24no—
391C24—24no—
392C24—24no—
393C24—24no—
394C24—24no—
395C24—24no—
396C24C22×D412yes—
397C24C22×D412yes—
398C24C22×D412yes—
399C24C2×#4114yes—
400C242−1+414yes—
401C24C2511yescharacteristic
402C2×Q8—216no—
403C2×Q8—28no—
404C2×Q8—28no—
405C2×Q8—44no—
406C2×Q8—44no—
407C2×Q8—48no—
408C2×Q8—82no—
409C2×Q8—82no—
410C2×Q8—84no—
411C2×Q8—84no—
412C2×Q8—84no—
413C2×Q8—84no—
414C2×Q8—84no—
415C2×Q8—88no—
416C2×Q8—88no—
417C2×D4—44no—
418C2×D4—44no—
419C2×D4—48no—
420C2×D4—48no—
421C2×D4—48no—
422C2×D4—416no—
423C2×D4—416no—
424C2×D4—44no—
425C2×D4—84no—
426C2×D4—84no—
427C22×C4—28no—
428C22×C4—28no—
429C22×C4—216no—
430C22×C4—216no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—24no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—216no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—24no—
454C22×C4—24no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—24no—
458C22×C4—22no—
459C22×C4—24no—
460C22×C4—24no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—24no—
465C22×C4—22no—
466C22×C4—24no—
467C22×C4—24no—
468C22×C4—22no—
469C22×C4—24no—
470C22×C4—24no—
471C22×C4—28no—
472C22×C4—24no—
473C22×C4—22no—
474C22×C4—22no—
475C22×C4—24no—
476C22×C4—22no—
477C22×C4—24no—
478C22×C4—24no—
479C22×C4—24no—
480C22×C4—24no—
481C22×C4—24no—
482C22×C4—24no—
483C22×C4—24no—
484C22×C4—24no—
485C22×C4—24no—
486C22×C4—24no—
487C22×C4—24no—
488C22×C4—24no—
489C22×C4—24no—
490C22×C4—24no—
491C22×C4—44no—
492C22×C4—44no—
493C22×C4—48no—
494C22×C4—48no—
495C22×C4—48no—
496C22×C4—48no—
497C22×C4—48no—
498C22×C4—48no—
499C22×C4—48no—
500C22×C4—48no—
501C22×C4—48no—
502C22×C4—48no—
503C22×C4—48no—
504C22×C4—48no—
505C22×C4—82no—
506C22×C4—82no—
507C22×C4—84no—
508C22×C4—84no—
509C22×C4—84no—
510C22×C4—84no—
511C22×C4—84no—
512C22×C4#120116yes—
513C22×C4#124116yescomplemented
514C22×C4C22×D414yes—
515C22×C4C22×D414yes—
516C22×C4C22×D414yes—
517C22×C4C22×D414yes—
518C22×C4C22×D414yes—
519C22×C4C22×D412yes—
520C22×C4C22×D412yes—
521C22×C4C22×D412yes—
522C22×C4C22×D412yes—
523C22×C4C22×D412yes—
524C22×C4C22×D412yes—
525C22×C4C22×D414yes—
526C22×C4C2×#4118yes—
527C22×C4H(5,𝔽2)14yes—
528C22×C42−1+414yes—
529C4⋊C4—216no—
530C4⋊C4—216no—
531C4⋊C4—216no—
532C4⋊C4—216no—
533C4⋊C4—216no—
534C4⋊C4—216no—
535C4⋊C4—216no—
536C4⋊C4—28no—
537C4⋊C4—28no—
538C4⋊C4—216no—
539C4⋊C4—216no—
540C4⋊C4—44no—
541C4⋊C4—416no—
542C4⋊C4—84no—
543C22⋊C4—216no—
544C22⋊C4—216no—
545C22⋊C4—416no—
546C22⋊C4—416no—
547C22⋊C4—48no—
548C22⋊C4—48no—
549C22⋊C4—416no—
550C22⋊C4—88no—
551C22⋊C4—88no—
552C22⋊C4—88no—
553C42—28no—
554C42—216no—
555C42—216no—
556C42—48no—
557C42—44no—
558C23—24no—
559C23—24no—
560C23—22no—
561C23—22no—
562C23—24no—
563C23—24no—
564C23—24no—
565C23—24no—
566C23—24no—
567C23—24no—
568C23—22no—
569C23—22no—
570C23—24no—
571C23—24no—
572C23—24no—
573C23—24no—
574C23—42no—
575C23—42no—
576C23—42no—
577C23—44no—
578C23—44no—
579C23—44no—
580C23—44no—
581C23—42no—
582C23—42no—
583C23—44no—
584C23—44no—
585C23—44no—
586C23—44no—
587C23—44no—
588C23—44no—
589C23—42no—
590C23—42no—
591C23—44no—
592C23—44no—
593C23—44no—
594C23—44no—
595C23C2×#12014yes—
596C23C2×#12414yes—
597C23#53614yes—
598C23#54818yes—
599C23#54818yes—
600C23C23×D411yescharacteristic
601C23C23×D411yescharacteristic
602C23C23×D411yescharacteristic
603C23C22×#4112yes—
604C23C2×H(5,𝔽2)12yes—
605C23C2×2−1+412yes—
606C23C2×2−1+412yes—
607C23#58512yes—
608C23#58512yes—
609Q8—88no—
610Q8—88no—
611Q8—816no—
612D4—88no—
613D4—816no—
614C2×C4—24no—
615C2×C4—24no—
616C2×C4—232no—
617C2×C4—28no—
618C2×C4—28no—
619C2×C4—22no—
620C2×C4—22no—
621C2×C4—22no—
622C2×C4—22no—
623C2×C4—24no—
624C2×C4—24no—
625C2×C4—24no—
626C2×C4—24no—
627C2×C4—24no—
628C2×C4—24no—
629C2×C4—24no—
630C2×C4—24no—
631C2×C4—24no—
632C2×C4—24no—
633C2×C4—24no—
634C2×C4—24no—
635C2×C4—24no—
636C2×C4—28no—
637C2×C4—24no—
638C2×C4—24no—
639C2×C4—28no—
640C2×C4—28no—
641C2×C4—24no—
642C2×C4—24no—
643C2×C4—24no—
644C2×C4—24no—
645C2×C4—416no—
646C2×C4—432no—
647C2×C4—42no—
648C2×C4—42no—
649C2×C4—42no—
650C2×C4—42no—
651C2×C4—42no—
652C2×C4—42no—
653C2×C4—42no—
654C2×C4—42no—
655C2×C4—42no—
656C2×C4—44no—
657C2×C4—42no—
658C2×C4—44no—
659C2×C4—44no—
660C2×C4—44no—
661C2×C4—44no—
662C2×C4—44no—
663C2×C4—44no—
664C2×C4—42no—
665C2×C4—42no—
666C2×C4—42no—
667C2×C4—42no—
668C2×C4—42no—
669C2×C4—44no—
670C2×C4—42no—
671C2×C4—42no—
672C2×C4—44no—
673C2×C4—44no—
674C2×C4—44no—
675C2×C4—44no—
676C2×C4—44no—
677C2×C4—44no—
678C2×C4—44no—
679C2×C4—44no—
680C2×C4—44no—
681C2×C4—44no—
682C2×C4—44no—
683C2×C4—44no—
684C2×C4—44no—
685C2×C4—44no—
686C2×C4—44no—
687C2×C4—88no—
688C2×C4—816no—
689C2×C4C2×#12014yes—
690C2×C4C2×#12014yes—
691C2×C4C2×#12418yes—
692C22—24no—
693C22—24no—
694C22—28no—
695C22—28no—
696C22—28no—
697C22—28no—
698C22—28no—
699C22—28no—
700C22—82no—
701C22—82no—
702C22—84no—
703C22—84no—
704C22—84no—
705C22—84no—
706C22—82no—
707C22—84no—
708C22—84no—
709C22#243618yes—
710C22C22×#12011yescharacteristic
711C22C22×#12411yescharacteristic
712C22C2×#53611yescharacteristic
713C22#346012yes—
714C22#346112yes—
715C22C2×#54812yes—
716C22C2×#54812yes—
717C22#348411yescharacteristic
718C22#348511yescharacteristic
719C22#348912yes—
720C22#349212yes—
721C22#349312yes—
722C22#349612yes—
723C22#349612yes—
724C22#349712yes—
725C22#349911yescharacteristic
726C22#350011yescharacteristic
727C22#353512yes—
728C22#353812yes—
729C22#356312yes—
730C22#356412yes—
731C4—24no—
732C4—24no—
733C4—28no—
734C4—44no—
735C4—44no—
736C4—44no—
737C4—44no—
738C4—44no—
739C4—44no—
740C4—44no—
741C4—84no—
742C4—82no—
743C4—84no—
744C2—28no—
745C2—162no—
746C2C2×#243611yescharacteristic
747C2#3515812yes—
748C2#3534711yescharacteristic
749C2#3566011yescharacteristic
750C2#3674212yes—
751C2#3775312yes—
752C2#3775512yes—
753C2#3942112yes—
754C2#3942312yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC22×#121C22
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C22×#544C2semidirect
5. #15882C2semidirect
6. #15881C2semidirect
7. #15880C2semidirect
8. #15783C2semidirect
9. #15761C2semidirect
10. #15322C2semidirect
11. #14759C2semidirect
12. #14702C2semidirect
13. #14281C2semidirect
14. #14135C2semidirect
15. C2×#1436C2semidirect
513. C22×C4#124semidirect

Maximal subgroups

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

subgroupindex
C22×#5442
C2×#28502
C2×#26392
C2×#26182
#158822
#158812
#158802
#157832
#157612
#153222
#147592
#147022
#142812
#141352
C2×#14362