Finite Group Databasebeta

← #657578

Subgroups — #657578

Order 512.

6026 conjugacy classes of subgroups.

The subgroup lattice has 757 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#547C211yesmaximal, characteristic
2C2×#2501C211yesmaximal, characteristic, complemented
3C2×#2486C211yesmaximal, characteristic, complemented
4C2×#2406C214yesmaximal, complemented
5C22×#384C211yesmaximal, characteristic, complemented
6#15430C214yesmaximal, complemented
7#15384C214yesmaximal, complemented
8#15143C212yesmaximal, complemented
9#15130C212yesmaximal, complemented
10#15069C214yesmaximal, complemented
11#14698C211yesmaximal, characteristic, complemented
12#13977C212yesmaximal, complemented
13#13904C211yesmaximal, characteristic, complemented
14#13880C212yesmaximal, complemented
15C4×#386C211yesmaximal, characteristic, complemented
16C23×#41C2212yes—
17C2×#547—28no—
18C2×#547—28no—
19C2×#547—28no—
20C22×C4⋊Q8C2211yescharacteristic
21C22×#123C2212yes—
22C22×#123C2212yes—
23C22×#123C2212yes—
24C22×#123C2214yes—
25C22×C22⋊Q8C2212yes—
26C22×#121C2211yescharacteristic
27C22×#121C2211yescharacteristic
28C22×C4×D4C2211yescharacteristic
29C22×C4×D4C2211yescharacteristic
30C23×C4⋊C4C2211yescharacteristic
31C23×C4⋊C4C2211yescharacteristic
32C23×C4⋊C4C2214yes—
33C23×C4⋊C4C2211yescharacteristic
34C23×C42C2211yescharacteristic
35#2501—28no—
36#2486—28no—
37#2406—232no—
38C2×#399C2214yes—
39C2×#398C2212yes—
40C2×#397C2212yes—
41C2×#395C2218yes—
42C2×#394C2218yes—
43C2×#390C2212yes—
44C2×#390C2212yes—
45C2×#389C2212yes—
46C2×#386C2214yes—
47C2×#386C2212yes—
48C2×#385C2218yes—
49C2×#385C2218yes—
50C2×#384—28no—
51C2×#384C2214yes—
52C2×#384C2212yes—
53C2×#384C2212yes—
54C2×#384C2212yes—
55C2×#381C2214yes—
56C2×#381C2212yes—
57C2×#380C2212yes—
58C2×#380C2211yescharacteristic
59C2×C4×C22⋊C4C2212yes—
60C2×C4×C22⋊C4C2211yescharacteristic
61#1445C2214yes—
62#1444C2218yes—
63#1444C2218yes—
64#1439C2218yes—
65#1439C2218yes—
66#1437C2212yes—
67#1437C2214yes—
68#1436C2212yes—
69C4⋊#95C2212yes—
70C4⋊#95C2212yes—
71C4⋊#95C2212yes—
72C22⋊#95C2212yes—
73C4×#95C2212yes—
74C22×#41—28no—
75C22×#41—28no—
76C22×#41—216no—
77C22×#41—28no—
78C22×#41—48no—
79C23×Q8C2312yes—
80C23×D4C2312yes—
81C23×D4C2311yescharacteristic
82C23×D4C2312yes—
83C23×D4C2311yescharacteristic
84C24×C4C2312yes—
85C24×C4C2312yes—
86C24×C4C2312yes—
87C24×C4C2311yescharacteristic
88C24×C4C2312yes—
89C24×C4C2312yes—
90C24×C4C2311yescharacteristic
91#547—2128no—
92C2×C4⋊Q8—24no—
93C2×C4⋊Q8—24no—
94C2×C4⋊Q8—24no—
95C2×#123—28no—
96C2×#123—28no—
97C2×#123—216no—
98C2×#123—28no—
99C2×#123—216no—
100C2×#123—28no—
101C2×#123—28no—
102C2×#123—28no—
103C2×#123—28no—
104C2×#123—216no—
105C2×#123—48no—
106C2×C22⋊Q8—28no—
107C2×C22⋊Q8—28no—
108C2×C22⋊Q8—28no—
109C2×#121—24no—
110C2×#121—24no—
111C2×#121—24no—
112C2×#121—24no—
113C2×#121—24no—
114C2×#121D418yes—
115C2×C4×D4—24no—
116C2×C4×D4—24no—
117C2×C4×D4—24no—
118C2×C4×D4—24no—
119C2×C4×D4—24no—
120C2×C4×D4D418yes—
121C22×C4⋊C4—24no—
122C22×C4⋊C4—24no—
123C22×C4⋊C4—24no—
124C22×C4⋊C4—24no—
125C22×C4⋊C4—24no—
126C22×C4⋊C4—216no—
127C22×C4⋊C4—24no—
128C22×C4⋊C4—24no—
129C22×C4⋊C4—216no—
130C22×C4⋊C4—24no—
131C22×C4⋊C4—48no—
132C22×C4⋊C4D418yes—
133C22×C4⋊C4C2314yes—
134C22×C4⋊C4C2312yes—
135C22×C4⋊C4C2312yes—
136C22×C4⋊C4C2312yes—
137C22×C4⋊C4C2312yes—
138C22×C4⋊C4C2312yes—
139C22×C4⋊C4C2318yes—
140C22×C4⋊C4C2318yes—
141C22×C4⋊C4C2318yes—
142C22×C4⋊C4C2311yescharacteristic
143C22×C4⋊C4C2311yescharacteristic
144C22×C4⋊C4C2311yescharacteristic
145C22×C4⋊C4C2311yescharacteristic
146C22×C4⋊C4C2312yes—
147C22×C22⋊C4C2314yes—
148C22×C22⋊C4C2318yes—
149C22×C22⋊C4C2312yes—
150C22×C22⋊C4C2312yes—
151C22×C22⋊C4C2312yes—
152C22×C22⋊C4C2318yes—
153C22×C22⋊C4C2314yes—
154C22×C22⋊C4C2311yescharacteristic
155C22×C22⋊C4C2312yes—
156C22×C22⋊C4C2311yescharacteristic
157C22×C22⋊C4C2312yes—
158C22×C42—24no—
159C22×C42—28no—
160C22×C42C2312yes—
161C22×C42C2312yes—
162C22×C42C2311yescharacteristic
163C22×C42C2312yes—
164#399—216no—
165#398—28no—
166#397—28no—
167#395—232no—
168#394—232no—
169#390—28no—
170#390—28no—
171#389—28no—
172#386—216no—
173#386D4116yescomplemented
174#385—232no—
175#385—232no—
176#384—28no—
177#384—28no—
178#384—28no—
179#384—216no—
180#381—216no—
181#381—28no—
182#380—28no—
183#380—24no—
184C4×C22⋊C4—24no—
185C4×C22⋊C4—28no—
186C2×#95C2318yes—
187C2×#95C2318yes—
188C2×#95C2312yes—
189C2×#95C2314yes—
190C2×#95C2314yes—
191C2×#95C2312yes—
192C2×#95C2314yes—
193C2×#95C2314yes—
194C2×#95C2318yes—
195C2×#95C2314yes—
196C25C2412yes—
197C25C2411yescharacteristic
198C25C2411yescharacteristic
199C2×#41—264no—
200C2×#41—464no—
201C2×#41—464no—
202C2×#41—432no—
203C2×#41—432no—
204C22×Q8—24no—
205C22×Q8—24no—
206C22×Q8—24no—
207C22×Q8—28no—
208C22×Q8—44no—
209C22×D4—22no—
210C22×D4—22no—
211C22×D4—24no—
212C22×D4—24no—
213C22×D4—24no—
214C22×D4—24no—
215C22×D4—24no—
216C22×D4—28no—
217C22×D4—28no—
218C22×D4—28no—
219C22×D4—22no—
220C22×D4—22no—
221C22×D4—24no—
222C22×D4—24no—
223C22×D4—24no—
224C22×D4—44no—
225C22×D4C2×D414yes—
226C22×D4C2×D418yes—
227C22×D4C2×D414yes—
228C22×D4C2×D418yes—
229C23×C4—24no—
230C23×C4—24no—
231C23×C4—28no—
232C23×C4—24no—
233C23×C4—28no—
234C23×C4—24no—
235C23×C4—24no—
236C23×C4—24no—
237C23×C4—24no—
238C23×C4—28no—
239C23×C4—24no—
240C23×C4—28no—
241C23×C4—22no—
242C23×C4—22no—
243C23×C4—24no—
244C23×C4—24no—
245C23×C4—24no—
246C23×C4—24no—
247C23×C4—24no—
248C23×C4—24no—
249C23×C4—24no—
250C23×C4—24no—
251C23×C4—28no—
252C23×C4—24no—
253C23×C4—28no—
254C23×C4—28no—
255C23×C4—28no—
256C23×C4—44no—
257C23×C4—44no—
258C23×C4C2×D418yes—
259C23×C4C2×D418yes—
260C23×C4C2×D414yes—
261C23×C4C2×D414yes—
262C23×C4C2×D414yes—
263C23×C4C2×D414yes—
264C23×C4C2414yes—
265C23×C4C2414yes—
266C23×C4C2414yes—
267C23×C4C2411yescharacteristic
268C23×C4C2411yescharacteristic
269C23×C4C2412yes—
270C23×C4C2412yes—
271C23×C4C2412yes—
272C23×C4C2411yescharacteristic
273C23×C4C2411yescharacteristic
274C23×C4C2414yes—
275C23×C4C2411yescharacteristic
276C4⋊Q8—232no—
277#123—264no—
278#123—264no—
279#123—464no—
280#123—432no—
281C22⋊Q8—264no—
282#121—232no—
283#121—232no—
284C4×D4—232no—
285C4×D4—232no—
286C2×C4⋊C4—232no—
287C2×C4⋊C4—232no—
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—24no—
295C2×C4⋊C4—24no—
296C2×C4⋊C4—24no—
297C2×C4⋊C4—24no—
298C2×C4⋊C4—24no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—24no—
302C2×C4⋊C4—24no—
303C2×C4⋊C4—24no—
304C2×C4⋊C4—24no—
305C2×C4⋊C4—216no—
306C2×C4⋊C4—216no—
307C2×C4⋊C4—24no—
308C2×C4⋊C4—24no—
309C2×C4⋊C4—24no—
310C2×C4⋊C4—216no—
311C2×C4⋊C4—216no—
312C2×C4⋊C4—24no—
313C2×C4⋊C4—24no—
314C2×C4⋊C4—216no—
315C2×C4⋊C4—216no—
316C2×C4⋊C4—216no—
317C2×C4⋊C4—28no—
318C2×C4⋊C4—24no—
319C2×C4⋊C4—24no—
320C2×C4⋊C4—24no—
321C2×C4⋊C4—48no—
322C2×C4⋊C4—48no—
323C2×C4⋊C4—48no—
324C2×C4⋊C4—464no—
325C2×C4⋊C4—416no—
326C2×C4⋊C4C2×D414yes—
327C2×C4⋊C4C2×D414yes—
328C2×C4⋊C4C2×D414yes—
329C2×C4⋊C4C2×D418yes—
330C2×C4⋊C4C2×D414yes—
331C2×C22⋊C4—24no—
332C2×C22⋊C4—22no—
333C2×C22⋊C4—22no—
334C2×C22⋊C4—24no—
335C2×C22⋊C4—24no—
336C2×C22⋊C4—24no—
337C2×C22⋊C4—24no—
338C2×C22⋊C4—24no—
339C2×C22⋊C4—22no—
340C2×C22⋊C4—22no—
341C2×C22⋊C4—24no—
342C2×C22⋊C4—24no—
343C2×C22⋊C4—24no—
344C2×C22⋊C4—24no—
345C2×C22⋊C4—28no—
346C2×C22⋊C4—28no—
347C2×C22⋊C4—24no—
348C2×C22⋊C4—28no—
349C2×C22⋊C4—28no—
350C2×C22⋊C4—216no—
351C2×C22⋊C4—216no—
352C2×C22⋊C4—216no—
353C2×C22⋊C4—216no—
354C2×C22⋊C4—216no—
355C2×C22⋊C4—216no—
356C2×C22⋊C4—48no—
357C2×C22⋊C4C2×D414yes—
358C2×C22⋊C4C2×D414yes—
359C2×C22⋊C4C2×D418yes—
360C2×C22⋊C4C2×D418yes—
361C2×C22⋊C4C2×D4116yes—
362C2×C42—232no—
363C2×C42—22no—
364C2×C42—22no—
365C2×C42—24no—
366C2×C42—24no—
367C2×C42—28no—
368C2×C42—24no—
369C2×C42—24no—
370C2×C42—24no—
371C2×C42—24no—
372C2×C42C2×D414yes—
373C2×C42#4118yescomplemented
374#95—24no—
375#95—28no—
376#95—24no—
377#95—216no—
378#95—28no—
379#95—216no—
380#95—216no—
381#95—28no—
382#95—28no—
383#95C2×D4116yes—
384C24—22no—
385C24—22no—
386C24—24no—
387C24—21no—
388C24—21no—
389C24—22no—
390C24—22no—
391C24—24no—
392C24—22no—
393C24—24no—
394C24—22no—
395C24—22no—
396C24—24no—
397C24—24no—
398C24—28no—
399C24—24no—
400C24C22×D414yes—
401C24C22×D414yes—
402C24C22×D412yes—
403C24C22×D412yes—
404C24C22×D412yes—
405C24C22×D412yes—
406C24C22×D414yes—
407C24H(5,𝔽2)14yes—
408C24C2511yescharacteristic
409#41—4256no—
410C2×Q8—216no—
411C2×Q8—416no—
412C2×Q8—416no—
413C2×Q8—48no—
414C2×Q8—48no—
415C2×D4—28no—
416C2×D4—28no—
417C2×D4—216no—
418C2×D4—216no—
419C2×D4—216no—
420C2×D4—216no—
421C2×D4—232no—
422C2×D4—28no—
423C2×D4—216no—
424C2×D4—416no—
425C2×D4—416no—
426C2×D4—48no—
427C2×D4—48no—
428C2×D4—48no—
429C2×D4—48no—
430C2×D4—416no—
431C2×D4—44no—
432C2×D4—44no—
433C2×D4—48no—
434C2×D4#127116yes—
435C22×C4—216no—
436C22×C4—216no—
437C22×C4—232no—
438C22×C4—216no—
439C22×C4—24no—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—22no—
443C22×C4—24no—
444C22×C4—24no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—24no—
448C22×C4—28no—
449C22×C4—28no—
450C22×C4—216no—
451C22×C4—216no—
452C22×C4—232no—
453C22×C4—216no—
454C22×C4—216no—
455C22×C4—216no—
456C22×C4—232no—
457C22×C4—24no—
458C22×C4—24no—
459C22×C4—24no—
460C22×C4—24no—
461C22×C4—21no—
462C22×C4—21no—
463C22×C4—21no—
464C22×C4—21no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—24no—
473C22×C4—24no—
474C22×C4—24no—
475C22×C4—24no—
476C22×C4—24no—
477C22×C4—24no—
478C22×C4—24no—
479C22×C4—24no—
480C22×C4—24no—
481C22×C4—22no—
482C22×C4—28no—
483C22×C4—216no—
484C22×C4—28no—
485C22×C4—24no—
486C22×C4—24no—
487C22×C4—24no—
488C22×C4—28no—
489C22×C4—24no—
490C22×C4—24no—
491C22×C4—28no—
492C22×C4—28no—
493C22×C4—24no—
494C22×C4—416no—
495C22×C4—416no—
496C22×C4—416no—
497C22×C4—416no—
498C22×C4—48no—
499C22×C4—48no—
500C22×C4—48no—
501C22×C4—48no—
502C22×C4—48no—
503C22×C4—416no—
504C22×C4—44no—
505C22×C4—48no—
506C22×C4—44no—
507C22×C4—48no—
508C22×C4—48no—
509C22×C4—48no—
510C22×C4—48no—
511C22×C4—416no—
512C22×C4—416no—
513C22×C4#120116yes—
514C22×C4C22×D418yes—
515C22×C4C22×D412yes—
516C22×C4C22×D412yes—
517C22×C4C22×D414yes—
518C22×C4C22×D414yes—
519C22×C4C22×D414yes—
520C22×C4C22×D412yes—
521C22×C4C22×D412yes—
522C22×C4C22×D412yes—
523C22×C4C22×D414yes—
524C22×C4C22×D412yes—
525C22×C4C22×D418yes—
526C22×C4C22×D412yes—
527C22×C4C22×D412yes—
528C22×C4C22×D412yes—
529C22×C4C2×#4118yes—
530C22×C4C2×#4114yes—
531C22×C42−1+418yes—
532C22×C42−1+414yes—
533C4⋊C4—216no—
534C4⋊C4—216no—
535C4⋊C4—28no—
536C4⋊C4—28no—
537C4⋊C4—28no—
538C4⋊C4—216no—
539C4⋊C4—28no—
540C4⋊C4—432no—
541C4⋊C4—432no—
542C4⋊C4—432no—
543C4⋊C4—48no—
544C4⋊C4—48no—
545C4⋊C4—416no—
546C4⋊C4—48no—
547C22⋊C4—216no—
548C22⋊C4—216no—
549C22⋊C4—28no—
550C22⋊C4—216no—
551C22⋊C4—28no—
552C22⋊C4—216no—
553C22⋊C4—232no—
554C22⋊C4—432no—
555C22⋊C4—48no—
556C22⋊C4—416no—
557C22⋊C4—432no—
558C42—216no—
559C42—28no—
560C42—216no—
561C42—48no—
562C23—22no—
563C23—22no—
564C23—28no—
565C23—24no—
566C23—24no—
567C23—24no—
568C23—28no—
569C23—24no—
570C23—24no—
571C23—28no—
572C23—28no—
573C23—22no—
574C23—22no—
575C23—24no—
576C23—24no—
577C23—28no—
578C23—24no—
579C23—24no—
580C23—28no—
581C23—24no—
582C23—28no—
583C23—28no—
584C23—28no—
585C23—41no—
586C23—41no—
587C23—42no—
588C23—42no—
589C23—44no—
590C23—42no—
591C23—44no—
592C23—42no—
593C23—42no—
594C23—42no—
595C23—42no—
596C23—44no—
597C23—48no—
598C23—44no—
599C23—44no—
600C23—48no—
601C23—44no—
602C23—48no—
603C23—48no—
604C23C2×#12014yes—
605C23C2×#12714yes—
606C23C2×#12714yes—
607C23#53514yes—
608C23D4218yes—
609C23D4218yes—
610C23C23×D411yescharacteristic
611C23C23×D411yescharacteristic
612C23C23×D411yescharacteristic
613C23C23×D412yes—
614C23C22×#4112yes—
615C23C2×H(5,𝔽2)12yes—
616C23C2×2−1+412yes—
617C23C2×2−1+414yes—
618Q8—432no—
619D4—232no—
620D4—432no—
621D4—432no—
622D4—416no—
623C2×C4—264no—
624C2×C4—22no—
625C2×C4—22no—
626C2×C4—28no—
627C2×C4—22no—
628C2×C4—22no—
629C2×C4—28no—
630C2×C4—24no—
631C2×C4—24no—
632C2×C4—28no—
633C2×C4—24no—
634C2×C4—24no—
635C2×C4—28no—
636C2×C4—28no—
637C2×C4—28no—
638C2×C4—232no—
639C2×C4—22no—
640C2×C4—22no—
641C2×C4—24no—
642C2×C4—24no—
643C2×C4—24no—
644C2×C4—28no—
645C2×C4—28no—
646C2×C4—28no—
647C2×C4—28no—
648C2×C4—28no—
649C2×C4—24no—
650C2×C4—24no—
651C2×C4—28no—
652C2×C4—432no—
653C2×C4—432no—
654C2×C4—41no—
655C2×C4—42no—
656C2×C4—42no—
657C2×C4—42no—
658C2×C4—42no—
659C2×C4—42no—
660C2×C4—42no—
661C2×C4—41no—
662C2×C4—432no—
663C2×C4—44no—
664C2×C4—44no—
665C2×C4—44no—
666C2×C4—44no—
667C2×C4—42no—
668C2×C4—42no—
669C2×C4—44no—
670C2×C4—42no—
671C2×C4—44no—
672C2×C4—44no—
673C2×C4—416no—
674C2×C4—44no—
675C2×C4—44no—
676C2×C4—44no—
677C2×C4—44no—
678C2×C4—48no—
679C2×C4—48no—
680C2×C4—48no—
681C2×C4—44no—
682C2×C4—44no—
683C2×C4—48no—
684C2×C4—816no—
685C2×C4C2×#12014yes—
686C2×C4C2×#12014yes—
687C2×C4#52514yes—
688C2×C4C2×#12714yes—
689C2×C4#53614yes—
690C2×C4#54718yes—
691C2×C4#54718yes—
692C2×C4#547116yescomplemented
693C22—28no—
694C22—24no—
695C22—28no—
696C22—28no—
697C22—24no—
698C22—216no—
699C22—28no—
700C22—28no—
701C22—24no—
702C22—44no—
703C22—42no—
704C22—44no—
705C22—42no—
706C22—48no—
707C22—44no—
708C22—44no—
709C22—44no—
710C22—44no—
711C22—416no—
712C22—84no—
713C22—84no—
714C22—84no—
715C22—84no—
716C22#243318yes—
717C22C22×#12011yescharacteristic
718C22C2×#52511yescharacteristic
719C22C22×#12711yescharacteristic
720C22C2×#53511yescharacteristic
721C22C2×#53611yescharacteristic
722C22C2×D4212yes—
723C22C2×D4212yes—
724C22C2×#54714yes—
725C22C2×#54712yes—
726C22C2×#54712yes—
727C22#348612yes—
728C22#349214yes—
729C22#349912yes—
730C22#349912yes—
731C22#349914yes—
732C22#354414yes—
733C4—216no—
734C4—24no—
735C4—24no—
736C4—42no—
737C4—44no—
738C4—44no—
739C4—44no—
740C4—42no—
741C4—48no—
742C4—84no—
743C4—84no—
744C4#240618yescomplemented
745C2—28no—
746C2—44no—
747C2—84no—
748C2C2×#243311yescharacteristic
749C2C2×#240611yescharacteristic
750C2#3566611yescharacteristic
751C2#3610312yes—
752C2#3617912yes—
753C2#3786312yes—
754C2#3933712yes—
755C2#5931314yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC22×#547C2
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2501C2semidirect
3. C2×#2486C2semidirect
4. C2×#2406C2semidirect
5. C22×#384C2semidirect
6. #15430C2semidirect
7. #15384C2semidirect
8. #15143C2semidirect
9. #15130C2semidirect
10. #15069C2semidirect
11. #14698C2semidirect
12. #13977C2semidirect
13. #13904C2semidirect
14. #13880C2semidirect
15. C4×#386C2semidirect
173. #386D4semidirect
373. C2×C42#41semidirect
692. C2×C4#547semidirect
744. C4#2406semidirect

Maximal subgroups

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

subgroupindex
C22×#5472
C2×#25012
C2×#24862
C2×#24062
C22×#3842
#154302
#153842
#151432
#151302
#150692
#146982
#139772
#139042
#138802
C4×#3862