Finite Group Databasebeta

← #617362

Subgroups — #617362

Order 512.

6208 conjugacy classes of subgroups.

The subgroup lattice has 735 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×C22⋊Q8C211yesmaximal, characteristic, complemented
2C2×#2849C212yesmaximal, complemented
3C2×#2539C211yesmaximal, characteristic
4C2×#2501C211yesmaximal, characteristic, complemented
5C2×#2614C212yesmaximal, complemented
6C2×#2618C211yesmaximal, characteristic, complemented
7#15453C214yesmaximal, complemented
8#15430C214yesmaximal, complemented
9#15325C218yesmaximal, complemented
10#15069C212yesmaximal, complemented
11#14911C212yesmaximal, complemented
12#13904C211yesmaximal, characteristic, complemented
13#13903C211yesmaximal, characteristic, complemented
14#13776C211yesmaximal, characteristic, complemented
15C24×Q8C2211yescharacteristic
16C25×C4C2211yescharacteristic
17C22×#123C2212yes—
18C22×#123C2212yes—
19C22×#123C2212yes—
20C22×#123C2212yes—
21C22×C22⋊Q8—28no—
22C22×C22⋊Q8—28no—
23C22×C22⋊Q8—28no—
24C22×C22⋊Q8C2214yes—
25C22×C22⋊Q8C2214yes—
26C22×#121C2211yescharacteristic
27C22×#120C2212yes—
28C23×C4⋊C4C2211yescharacteristic
29C23×C4⋊C4C2211yescharacteristic
30C23×C4⋊C4C2211yescharacteristic
31C23×C22⋊C4C2211yescharacteristic
32C23×C22⋊C4C2211yescharacteristic
33#2849—216no—
34#2618—28no—
35#2614—216no—
36#2539—28no—
37#2501—28no—
38C2×#399C2212yes—
39C2×#399C2214yescomplemented
40C2×#398C2212yescomplemented
41C2×#396C2214yescomplemented
42C2×#395C2218yescomplemented
43C2×#394C2212yescomplemented
44C2×#393C2218yescomplemented
45C2×#392C2212yes—
46C2×#390C2214yes—
47C2×#390C2212yescomplemented
48C2×#386C2218yes—
49C2×#386C2212yes—
50C2×#385C2218yes—
51C2×#384C2212yescomplemented
52C2×#381C2212yes—
53C2×#381C2212yes—
54C2×#381C2212yes—
55C2×#381C2212yes—
56C2×#380C2211yescharacteristic
57C2×#379C2212yes—
58C2×C4×C22⋊C4C2211yescharacteristic
59C2×C4×C22⋊C4C2211yescharacteristic
60#1445C2214yescomplemented
61#1444C2218yescomplemented
62#1439C2218yes—
63#1439C2214yes—
64#1439C2218yescomplemented
65#1438C2218yes—
66#1437C2214yes—
67#1436C2214yescomplemented
68#1436C2214yescomplemented
69C4⋊#95C2214yescomplemented
70C22⋊#95C2212yes—
71C26C2311yescharacteristic
72C23×Q8—24no—
73C23×Q8—24no—
74C23×Q8—24no—
75C23×Q8—28no—
76C23×Q8—44no—
77C23×Q8C2314yes—
78C23×D4C2312yes—
79C23×D4C2314yes—
80C23×D4C2311yescharacteristic
81C24×C4—24no—
82C24×C4—24no—
83C24×C4—24no—
84C24×C4—28no—
85C24×C4—44no—
86C24×C4C2311yescharacteristic
87C24×C4C2311yescharacteristic
88C24×C4C2311yescharacteristic
89C24×C4C2314yes—
90C24×C4C2311yescharacteristic
91C24×C4C2311yescharacteristic
92C24×C4C2311yescharacteristic
93C2×#123—28no—
94C2×#123—28no—
95C2×#123—28no—
96C2×#123—28no—
97C2×#123—28no—
98C2×#123—28no—
99C2×#123—28no—
100C2×#123—28no—
101C2×#123—28no—
102C2×#123—44no—
103C2×#123—44no—
104C2×#123—44no—
105C2×C22⋊Q8—2128no—
106C2×C22⋊Q8—216no—
107C2×C22⋊Q8—216no—
108C2×C22⋊Q8—216no—
109C2×C22⋊Q8—216no—
110C2×C22⋊Q8—216no—
111C2×C22⋊Q8—216no—
112C2×#121—24no—
113C2×#121—24no—
114C2×#121—24no—
115C2×#120—28no—
116C2×#120—28no—
117C2×#120—44no—
118C22×C4⋊C4—24no—
119C22×C4⋊C4—24no—
120C22×C4⋊C4—24no—
121C22×C4⋊C4—24no—
122C22×C4⋊C4—24no—
123C22×C4⋊C4—24no—
124C22×C4⋊C4—24no—
125C22×C4⋊C4D418yescomplemented
126C22×C4⋊C4D418yescomplemented
127C22×C4⋊C4C2314yes—
128C22×C4⋊C4C2311yescharacteristic
129C22×C4⋊C4C2314yes—
130C22×C4⋊C4C2314yes—
131C22×C4⋊C4C2314yes—
132C22×C4⋊C4C2314yes—
133C22×C4⋊C4C2312yes—
134C22×C4⋊C4C2312yes—
135C22×C4⋊C4C2312yes—
136C22×C4⋊C4C2312yes—
137C22×C4⋊C4C2312yes—
138C22×C22⋊C4—24no—
139C22×C22⋊C4—24no—
140C22×C22⋊C4—24no—
141C22×C22⋊C4—24no—
142C22×C22⋊C4—24no—
143C22×C22⋊C4D418yes—
144C22×C22⋊C4C2314yes—
145C22×C22⋊C4C2312yes—
146C22×C22⋊C4C2314yes—
147C22×C22⋊C4C2312yes—
148C22×C22⋊C4C2314yes—
149C22×C22⋊C4C2314yes—
150C22×C22⋊C4C2314yes—
151C22×C22⋊C4C2314yes—
152C22×C22⋊C4C2318yes—
153C22×C22⋊C4C2312yes—
154C22×C22⋊C4C2312yes—
155C22×C22⋊C4C2312yes—
156C22×C22⋊C4C2312yes—
157C22×C22⋊C4C2314yes—
158C22×C42C2312yes—
159C22×C42C2312yes—
160#399—216no—
161#399—28no—
162#398—28no—
163#396—216no—
164#395—232no—
165#394—28no—
166#393—232no—
167#392—28no—
168#390—28no—
169#390—216no—
170#386—28no—
171#386—232no—
172#385—232no—
173#384—28no—
174#381—28no—
175#381—28no—
176#381—28no—
177#381D4116yescomplemented
178#380—24no—
179#379—28no—
180C4×C22⋊C4—24no—
181C4×C22⋊C4—24no—
182C2×#95C2312yes—
183C2×#95C2318yes—
184C2×#95C2318yes—
185C2×#95C2312yes—
186C2×#95C2314yes—
187C2×#95C2312yes—
188C2×#95C2318yes—
189C2×#95C2318yes—
190C2×#95C2318yes—
191C25—22no—
192C25—22no—
193C25—28no—
194C25—24no—
195C25—24no—
196C25—44no—
197C25C2×D414yes—
198C25C2412yes—
199C25C2412yes—
200C25C2411yescharacteristic
201C22×Q8—232no—
202C22×Q8—28no—
203C22×Q8—28no—
204C22×Q8—28no—
205C22×Q8—216no—
206C22×Q8—48no—
207C22×Q8—432no—
208C22×Q8—432no—
209C22×Q8—416no—
210C22×Q8—416no—
211C22×D4—24no—
212C22×D4—24no—
213C22×D4—24no—
214C22×D4—24no—
215C22×D4—24no—
216C22×D4—24no—
217C22×D4—24no—
218C22×D4—28no—
219C22×D4—216no—
220C22×D4—28no—
221C22×D4—22no—
222C22×D4—22no—
223C22×D4—22no—
224C22×D4—44no—
225C22×D4—44no—
226C22×D4—48no—
227C23×C4—22no—
228C23×C4—22no—
229C23×C4—24no—
230C23×C4—232no—
231C23×C4—22no—
232C23×C4—22no—
233C23×C4—22no—
234C23×C4—22no—
235C23×C4—22no—
236C23×C4—22no—
237C23×C4—22no—
238C23×C4—22no—
239C23×C4—28no—
240C23×C4—24no—
241C23×C4—24no—
242C23×C4—24no—
243C23×C4—24no—
244C23×C4—24no—
245C23×C4—24no—
246C23×C4—28no—
247C23×C4—28no—
248C23×C4—216no—
249C23×C4—48no—
250C23×C4—44no—
251C23×C4—44no—
252C23×C4—432no—
253C23×C4—432no—
254C23×C4—416no—
255C23×C4—416no—
256C23×C4C2×D414yes—
257C23×C4C2×D414yes—
258C23×C4C2×D414yes—
259C23×C4C2×D414yes—
260C23×C4C2×D418yes—
261C23×C4C2×D414yes—
262C23×C4C2×D414yes—
263C23×C4C2×D414yes—
264C23×C4C2×D414yes—
265C23×C4C2414yes—
266C23×C4C2412yes—
267C23×C4C2414yes—
268C23×C4C2412yes—
269C23×C4C2412yes—
270C23×C4C2412yes—
271C23×C4C2414yes—
272C23×C4C2412yes—
273C23×C4C2411yescharacteristic
274C23×C4C2411yescharacteristic
275C23×C4C2412yes—
276#123—432no—
277#123—432no—
278#123—432no—
279#123—432no—
280C22⋊Q8—2128no—
281C22⋊Q8—2128no—
282#121—416no—
283#120—432no—
284C2×C4⋊C4—232no—
285C2×C4⋊C4—232no—
286C2×C4⋊C4—232no—
287C2×C4⋊C4—28no—
288C2×C4⋊C4—28no—
289C2×C4⋊C4—28no—
290C2×C4⋊C4—24no—
291C2×C4⋊C4—24no—
292C2×C4⋊C4—24no—
293C2×C4⋊C4—24no—
294C2×C4⋊C4—24no—
295C2×C4⋊C4—24no—
296C2×C4⋊C4—24no—
297C2×C4⋊C4—24no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—28no—
302C2×C4⋊C4—28no—
303C2×C4⋊C4—28no—
304C2×C4⋊C4—28no—
305C2×C4⋊C4—28no—
306C2×C4⋊C4—28no—
307C2×C4⋊C4—28no—
308C2×C4⋊C4—28no—
309C2×C4⋊C4—28no—
310C2×C4⋊C4—24no—
311C2×C4⋊C4—44no—
312C2×C4⋊C4—44no—
313C2×C4⋊C4—44no—
314C2×C4⋊C4—44no—
315C2×C4⋊C4C2×D418yes—
316C2×C4⋊C4C2×D418yes—
317C2×C4⋊C4C2×D418yes—
318C2×C4⋊C4C2×D418yes—
319C2×C22⋊C4—232no—
320C2×C22⋊C4—28no—
321C2×C22⋊C4—24no—
322C2×C22⋊C4—24no—
323C2×C22⋊C4—24no—
324C2×C22⋊C4—24no—
325C2×C22⋊C4—232no—
326C2×C22⋊C4—28no—
327C2×C22⋊C4—28no—
328C2×C22⋊C4—24no—
329C2×C22⋊C4—24no—
330C2×C22⋊C4—24no—
331C2×C22⋊C4—24no—
332C2×C22⋊C4—28no—
333C2×C22⋊C4—24no—
334C2×C22⋊C4—24no—
335C2×C22⋊C4—24no—
336C2×C22⋊C4—216no—
337C2×C22⋊C4—28no—
338C2×C22⋊C4—24no—
339C2×C22⋊C4—24no—
340C2×C22⋊C4—24no—
341C2×C22⋊C4—28no—
342C2×C22⋊C4—28no—
343C2×C22⋊C4—28no—
344C2×C22⋊C4—28no—
345C2×C22⋊C4—28no—
346C2×C22⋊C4—28no—
347C2×C22⋊C4—28no—
348C2×C22⋊C4—28no—
349C2×C22⋊C4—28no—
350C2×C22⋊C4—216no—
351C2×C22⋊C4—28no—
352C2×C22⋊C4—48no—
353C2×C22⋊C4—44no—
354C2×C22⋊C4—44no—
355C2×C22⋊C4—44no—
356C2×C22⋊C4—44no—
357C2×C22⋊C4—48no—
358C2×C22⋊C4C2×D418yes—
359C2×C22⋊C4C2×D418yes—
360C2×C22⋊C4C2×D418yes—
361C2×C22⋊C4C2×D418yes—
362C2×C42—24no—
363C2×C42—24no—
364C2×C42—44no—
365C2×C42—44no—
366#95—24no—
367#95—24no—
368#95—24no—
369#95—216no—
370#95—216no—
371#95—216no—
372#95—28no—
373#95—48no—
374#95C2×D4132yes—
375C24—24no—
376C24—24no—
377C24—216no—
378C24—216no—
379C24—28no—
380C24—216no—
381C24—22no—
382C24—22no—
383C24—24no—
384C24—24no—
385C24—24no—
386C24—28no—
387C24—216no—
388C24—22no—
389C24—22no—
390C24—22no—
391C24—22no—
392C24—22no—
393C24—22no—
394C24—22no—
395C24—28no—
396C24—28no—
397C24—48no—
398C24—432no—
399C24—432no—
400C24—48no—
401C24—48no—
402C24—44no—
403C24—44no—
404C24—48no—
405C24C22×D414yes—
406C24C22×D412yes—
407C24C22×D412yes—
408C24C22×D412yes—
409C24C22×D414yes—
410C24H(5,𝔽2)14yes—
411C242−1+418yes—
412C24C2511yescharacteristic
413C2×Q8—232no—
414C2×Q8—432no—
415C2×Q8—432no—
416C2×Q8—416no—
417C2×Q8—4128no—
418C2×Q8—416no—
419C2×D4—416no—
420C2×D4—432no—
421C2×D4—48no—
422C2×D4—48no—
423C2×D4—48no—
424C2×D4—48no—
425C2×D4—416no—
426C2×D4—48no—
427C2×D4—48no—
428C2×D4—48no—
429C2×D4—48no—
430C2×D4—44no—
431C2×D4—84no—
432C2×D4—84no—
433C2×D4—88no—
434C2×D4—816no—
435C22×C4—28no—
436C22×C4—28no—
437C22×C4—216no—
438C22×C4—216no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—24no—
442C22×C4—24no—
443C22×C4—24no—
444C22×C4—232no—
445C22×C4—28no—
446C22×C4—28no—
447C22×C4—28no—
448C22×C4—28no—
449C22×C4—216no—
450C22×C4—216no—
451C22×C4—216no—
452C22×C4—24no—
453C22×C4—24no—
454C22×C4—24no—
455C22×C4—24no—
456C22×C4—24no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—24no—
466C22×C4—24no—
467C22×C4—24no—
468C22×C4—24no—
469C22×C4—24no—
470C22×C4—28no—
471C22×C4—28no—
472C22×C4—28no—
473C22×C4—28no—
474C22×C4—28no—
475C22×C4—24no—
476C22×C4—24no—
477C22×C4—24no—
478C22×C4—22no—
479C22×C4—22no—
480C22×C4—22no—
481C22×C4—22no—
482C22×C4—22no—
483C22×C4—22no—
484C22×C4—22no—
485C22×C4—28no—
486C22×C4—28no—
487C22×C4—48no—
488C22×C4—48no—
489C22×C4—48no—
490C22×C4—48no—
491C22×C4—432no—
492C22×C4—432no—
493C22×C4—416no—
494C22×C4—4128no—
495C22×C4—48no—
496C22×C4—48no—
497C22×C4—44no—
498C22×C4—44no—
499C22×C4—44no—
500C22×C4—44no—
501C22×C4—44no—
502C22×C4—44no—
503C22×C4—48no—
504C22×C4—48no—
505C22×C4—48no—
506C22×C4—416no—
507C22×C4#120116yes—
508C22×C4#127116yes—
509C22×C4C22×D418yes—
510C22×C4C22×D418yes—
511C22×C4C22×D414yes—
512C22×C4C22×D414yes—
513C22×C4C22×D414yes—
514C22×C4C22×D414yes—
515C22×C4C22×D412yes—
516C22×C4C22×D412yes—
517C22×C4C22×D412yes—
518C22×C4C22×D412yes—
519C22×C4C22×D412yes—
520C22×C4C22×D412yes—
521C22×C4C22×D418yes—
522C22×C4C22×D414yes—
523C4⋊C4—232no—
524C4⋊C4—216no—
525C4⋊C4—216no—
526C4⋊C4—216no—
527C4⋊C4—216no—
528C4⋊C4—416no—
529C4⋊C4—416no—
530C4⋊C4—416no—
531C4⋊C4—44no—
532C4⋊C4—416no—
533C4⋊C4—84no—
534C22⋊C4—232no—
535C22⋊C4—216no—
536C22⋊C4—216no—
537C22⋊C4—216no—
538C22⋊C4—216no—
539C22⋊C4—416no—
540C22⋊C4—416no—
541C22⋊C4—416no—
542C22⋊C4—432no—
543C22⋊C4—416no—
544C22⋊C4—48no—
545C22⋊C4—48no—
546C22⋊C4—416no—
547C22⋊C4—88no—
548C42—48no—
549C42—84no—
550C23—24no—
551C23—24no—
552C23—24no—
553C23—24no—
554C23—24no—
555C23—24no—
556C23—216no—
557C23—216no—
558C23—232no—
559C23—24no—
560C23—24no—
561C23—28no—
562C23—28no—
563C23—28no—
564C23—28no—
565C23—216no—
566C23—216no—
567C23—464no—
568C23—416no—
569C23—464no—
570C23—432no—
571C23—432no—
572C23—42no—
573C23—42no—
574C23—44no—
575C23—416no—
576C23—44no—
577C23—44no—
578C23—48no—
579C23—48no—
580C23—42no—
581C23—42no—
582C23—42no—
583C23—44no—
584C23—44no—
585C23—42no—
586C23—42no—
587C23—42no—
588C23—42no—
589C23—44no—
590C23—44no—
591C23—44no—
592C23—44no—
593C23—48no—
594C23—48no—
595C23—48no—
596C23—48no—
597C23C2×#12014yes—
598C23C2×#12714yes—
599C23#53514yes—
600C23#53618yes—
601C23D4218yes—
602C23D4218yes—
603C23C23×D411yescharacteristic
604C23C23×D411yescharacteristic
605C23C23×D411yescharacteristic
606C23C23×D412yes—
607C23C2×H(5,𝔽2)12yes—
608C23C2×2−1+414yes—
609C23#58514yes—
610Q8—464no—
611D4—88no—
612D4—832no—
613D4—816no—
614C2×C4—24no—
615C2×C4—24no—
616C2×C4—24no—
617C2×C4—24no—
618C2×C4—232no—
619C2×C4—28no—
620C2×C4—28no—
621C2×C4—24no—
622C2×C4—24no—
623C2×C4—24no—
624C2×C4—24no—
625C2×C4—24no—
626C2×C4—24no—
627C2×C4—24no—
628C2×C4—24no—
629C2×C4—28no—
630C2×C4—28no—
631C2×C4—28no—
632C2×C4—28no—
633C2×C4—28no—
634C2×C4—28no—
635C2×C4—216no—
636C2×C4—216no—
637C2×C4—416no—
638C2×C4—464no—
639C2×C4—416no—
640C2×C4—42no—
641C2×C4—42no—
642C2×C4—42no—
643C2×C4—42no—
644C2×C4—42no—
645C2×C4—42no—
646C2×C4—44no—
647C2×C4—44no—
648C2×C4—44no—
649C2×C4—416no—
650C2×C4—416no—
651C2×C4—416no—
652C2×C4—44no—
653C2×C4—44no—
654C2×C4—44no—
655C2×C4—44no—
656C2×C4—44no—
657C2×C4—44no—
658C2×C4—48no—
659C2×C4—48no—
660C2×C4—42no—
661C2×C4—44no—
662C2×C4—42no—
663C2×C4—42no—
664C2×C4—42no—
665C2×C4—42no—
666C2×C4—42no—
667C2×C4—42no—
668C2×C4—44no—
669C2×C4—44no—
670C2×C4—48no—
671C2×C4—48no—
672C2×C4C2×#12014yes—
673C2×C4C2×#12014yes—
674C2×C4C2×#12718yes—
675C22—216no—
676C22—24no—
677C22—216no—
678C22—28no—
679C22—24no—
680C22—28no—
681C22—216no—
682C22—28no—
683C22—464no—
684C22—48no—
685C22—44no—
686C22—416no—
687C22—48no—
688C22—44no—
689C22—48no—
690C22—84no—
691C22—84no—
692C22—82no—
693C22—84no—
694C22—84no—
695C22—84no—
696C22—82no—
697C22—82no—
698C22—84no—
699C22#243318yes—
700C22C22×#12011yescharacteristic
701C22C22×#12711yescharacteristic
702C22C2×#53511yescharacteristic
703C22C2×#53612yes—
704C22#344912yes—
705C22C2×D4212yes—
706C22C2×D4212yes—
707C22#348512yes—
708C22#348512yes—
709C22#348612yes—
710C22#348612yes—
711C22#348918yes—
712C22#353518yes—
713C4—24no—
714C4—24no—
715C4—28no—
716C4—48no—
717C4—44no—
718C4—48no—
719C4—44no—
720C4—44no—
721C4—44no—
722C4—84no—
723C4—82no—
724C2—28no—
725C2—48no—
726C2—162no—
727C2C2×#243311yescharacteristic
728C2#3508811yescharacteristic
729C2#3524211yescharacteristic
730C2#3563412yes—
731C2#3711812yes—
732C2#3764714yes—
733C2#3765314yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC2×#2539C2
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C23×C22⋊Q8C2semidirect
2. C2×#2849C2semidirect
4. C2×#2501C2semidirect
5. C2×#2614C2semidirect
6. C2×#2618C2semidirect
7. #15453C2semidirect
8. #15430C2semidirect
9. #15325C2semidirect
10. #15069C2semidirect
11. #14911C2semidirect
12. #13904C2semidirect
13. #13903C2semidirect
14. #13776C2semidirect
39. C2×#399C22semidirect
40. C2×#398C22semidirect
41. C2×#396C22semidirect
42. C2×#395C22semidirect
43. C2×#394C22semidirect
44. C2×#393C22semidirect
47. C2×#390C22semidirect
51. C2×#384C22semidirect
60. #1445C22semidirect
61. #1444C22semidirect
64. #1439C22semidirect
67. #1436C22semidirect
68. #1436C22semidirect
69. C4⋊#95C22semidirect
125. C22×C4⋊C4D4semidirect
126. C22×C4⋊C4D4semidirect
177. #381D4semidirect

Maximal subgroups

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

subgroupindex
C23×C22⋊Q82
C2×#28492
C2×#25392
C2×#25012
C2×#26142
C2×#26182
#154532
#154302
#153252
#150692
#149112
#139042
#139032
#137762