Finite Group Databasebeta

← #678382

Subgroups — #678382

Order 512.

4470 conjugacy classes of subgroups.

The subgroup lattice has 719 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#554C211yesmaximal, characteristic, complemented
2C2×#2563C211yesmaximal, characteristic, complemented
3C2×#2537C211yesmaximal, characteristic
4C2×C4×#124C211yesmaximal, characteristic, complemented
5#15275C218yesmaximal, complemented
6#15236C212yesmaximal, complemented
7#15216C214yesmaximal, complemented
8#15215C212yesmaximal, complemented
9#14714C211yesmaximal, characteristic, complemented
10#14137C212yesmaximal, complemented
11#13970C214yesmaximal, complemented
12#13926C211yesmaximal, characteristic, complemented
13#13881C212yesmaximal, complemented
14C4×#390C211yesmaximal, characteristic, complemented
15C2×#554—28no—
16C2×#554—28no—
17C2×#554—28no—
18C22×C4⋊Q8C2211yescharacteristic
19C22×#125C2212yes—
20C22×#124C2211yescharacteristic
21C22×C22⋊Q8C2212yes—
22C22×C22⋊Q8C2214yes—
23C22×#121C2212yes—
24C22×#121C2212yes—
25C22×C4×Q8C2211yescharacteristic
26C22×C4×D4C2211yescharacteristic
27C22×C4×D4C2212yes—
28C22×C4×D4C2211yescharacteristic
29C23×C4⋊C4C2212yes—
30#2563—28no—
31#2537—28no—
32C2×#394C2214yes—
33C2×#393C2214yes—
34C4×#124—28no—
35C2×#391C2211yescharacteristic
36C2×#391C2212yes—
37C2×#391C2214yes—
38C2×#390C2211yescharacteristic
39C2×#390C2212yes—
40C2×#390C2212yes—
41C2×#388C2214yes—
42C2×#386C2218yes—
43C2×#384C2211yescharacteristic
44C2×#382C2211yescharacteristic
45C2×#382C2212yes—
46C2×#380C2212yes—
47C2×#380C2212yes—
48C2×#380C2214yes—
49C2×C4×C4⋊C4C2211yescharacteristic
50C2×C4×C22⋊C4C2212yes—
51C2×C4×C22⋊C4C2214yes—
52C2×C4×C22⋊C4C2212yes—
53C2×C43C2211yescharacteristic
54#1442C2218yes—
55#1442C2218yes—
56#1441C2214yes—
57#1440C2218yes—
58#1439C2218yes—
59#1439C2218yes—
60#1438C2218yes—
61#1437C2214yes—
62C4⋊#95C2212yes—
63C4⋊#95C2212yes—
64C4⋊#95C2212yes—
65C4⋊#95C2214yes—
66C4⋊#95C2212yes—
67C4⋊#95C2214yes—
68C22⋊#95C2218yes—
69C23×Q8C2311yescharacteristic
70C23×Q8C2312yes—
71C23×D4C2311yescharacteristic
72C23×D4C2314yes—
73C23×D4C2311yescharacteristic
74C24×C4C2312yes—
75C24×C4C2312yes—
76C24×C4C2314yes—
77#554—2128no—
78C2×C4⋊Q8—24no—
79C2×C4⋊Q8—24no—
80C2×C4⋊Q8D418yescomplemented
81C2×#125—28no—
82C2×#125—28no—
83C2×#125—28no—
84C2×#124—24no—
85C2×#124—24no—
86C2×#124—24no—
87C2×C22⋊Q8—216no—
88C2×C22⋊Q8—28no—
89C2×C22⋊Q8—28no—
90C2×C22⋊Q8—216no—
91C2×C22⋊Q8—28no—
92C2×C22⋊Q8—216no—
93C2×#121—28no—
94C2×#121—28no—
95C2×#121—216no—
96C2×#121—28no—
97C2×#121—44no—
98C2×C4×Q8—24no—
99C2×C4×Q8—24no—
100C2×C4×Q8D418yescomplemented
101C2×C4×D4—28no—
102C2×C4×D4—28no—
103C2×C4×D4—28no—
104C2×C4×D4—24no—
105C2×C4×D4—24no—
106C2×C4×D4—24no—
107C2×C4×D4—24no—
108C2×C4×D4—24no—
109C2×C4×D4D418yes—
110C22×C4⋊C4—28no—
111C22×C4⋊C4—28no—
112C22×C4⋊C4—28no—
113C22×C4⋊C4C2314yes—
114C22×C4⋊C4C2311yescharacteristic
115C22×C4⋊C4C2312yes—
116C22×C4⋊C4C2312yes—
117C22×C4⋊C4C2314yes—
118C22×C4⋊C4C2312yes—
119C22×C4⋊C4C2311yescharacteristic
120C22×C4⋊C4C2312yes—
121C22×C4⋊C4C2311yescharacteristic
122C22×C4⋊C4C2312yes—
123C22×C4⋊C4C2312yes—
124C22×C4⋊C4C2312yes—
125C22×C22⋊C4C2312yes—
126C22×C22⋊C4C2314yes—
127C22×C22⋊C4C2314yes—
128C22×C22⋊C4C2314yes—
129C22×C22⋊C4C2314yes—
130C22×C22⋊C4C2314yes—
131C22×C22⋊C4C2314yes—
132C22×C22⋊C4C2314yes—
133C22×C22⋊C4C2312yes—
134C22×C22⋊C4C2318yes—
135C22×C42C2312yes—
136C22×C42C2311yescharacteristic
137C22×C42C2312yes—
138C22×C42C2312yes—
139C22×C42C2312yes—
140C22×C42C2311yescharacteristic
141C22×C42C2311yescharacteristic
142#394—216no—
143#393—216no—
144#391—24no—
145#391—48no—
146#391—44no—
147#390—28no—
148#390—28no—
149#390—24no—
150#388—216no—
151#386—416no—
152#384—24no—
153#382—28no—
154#382—24no—
155#380—28no—
156#380—216no—
157#380—28no—
158C4×C4⋊C4—24no—
159C4×C22⋊C4—216no—
160C4×C22⋊C4—28no—
161C4×C22⋊C4—44no—
162C2×#95C2312yes—
163C2×#95C2318yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2318yes—
167C2×#95C2314yes—
168C2×#95C2314yes—
169C2×#95C2318yes—
170C2×#95C2318yes—
171C2×#95C2314yes—
172C2×#95C2318yes—
173C43—24no—
174C25C2412yes—
175C25C2412yes—
176C22×Q8—22no—
177C22×Q8—22no—
178C22×Q8—24no—
179C22×Q8—24no—
180C22×Q8—22no—
181C22×Q8—22no—
182C22×Q8—22no—
183C22×Q8—22no—
184C22×Q8—44no—
185C22×Q8—44no—
186C22×Q8C2×D414yes—
187C22×Q8C2×D418yes—
188C22×D4—28no—
189C22×D4—28no—
190C22×D4—216no—
191C22×D4—22no—
192C22×D4—22no—
193C22×D4—22no—
194C22×D4—22no—
195C22×D4—22no—
196C22×D4—22no—
197C22×D4—22no—
198C22×D4—24no—
199C22×D4—22no—
200C22×D4—44no—
201C22×D4—48no—
202C22×D4C2×D414yes—
203C22×D4#4114yes—
204C22×D4#4118yes—
205C23×C4—28no—
206C23×C4—24no—
207C23×C4—24no—
208C23×C4—24no—
209C23×C4—24no—
210C23×C4—24no—
211C23×C4—24no—
212C23×C4—28no—
213C23×C4—28no—
214C23×C4—24no—
215C23×C4—24no—
216C23×C4—24no—
217C23×C4—28no—
218C23×C4—28no—
219C23×C4—48no—
220C23×C4—48no—
221C23×C4C2×D418yes—
222C23×C4C2414yes—
223C23×C4C2412yes—
224C23×C4C2414yes—
225C23×C4C2412yes—
226C23×C4C2412yes—
227C23×C4C2414yes—
228C23×C4C2412yes—
229C23×C4C2412yes—
230C23×C4C2412yes—
231C23×C4C2411yescharacteristic
232C23×C4C2411yescharacteristic
233C23×C4C2411yescharacteristic
234C4⋊Q8—232no—
235#125—264no—
236#124—232no—
237C22⋊Q8—264no—
238C22⋊Q8—2128no—
239#121—432no—
240#121—432no—
241C4×Q8—232no—
242C4×D4—264no—
243C4×D4—232no—
244C4×D4—416no—
245C2×C4⋊C4—264no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—28no—
249C2×C4⋊C4—28no—
250C2×C4⋊C4—28no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—28no—
260C2×C4⋊C4—28no—
261C2×C4⋊C4—28no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—24no—
266C2×C4⋊C4—24no—
267C2×C4⋊C4—24no—
268C2×C4⋊C4—28no—
269C2×C4⋊C4—24no—
270C2×C4⋊C4—24no—
271C2×C4⋊C4—24no—
272C2×C4⋊C4—24no—
273C2×C4⋊C4—24no—
274C2×C4⋊C4—44no—
275C2×C4⋊C4C2×D414yes—
276C2×C4⋊C4C2×D418yes—
277C2×C4⋊C4C2×D418yes—
278C2×C4⋊C4C2×D414yes—
279C2×C4⋊C4C2×D418yes—
280C2×C22⋊C4—28no—
281C2×C22⋊C4—28no—
282C2×C22⋊C4—28no—
283C2×C22⋊C4—28no—
284C2×C22⋊C4—24no—
285C2×C22⋊C4—24no—
286C2×C22⋊C4—28no—
287C2×C22⋊C4—28no—
288C2×C22⋊C4—28no—
289C2×C22⋊C4—28no—
290C2×C22⋊C4—216no—
291C2×C22⋊C4—28no—
292C2×C22⋊C4—28no—
293C2×C22⋊C4—24no—
294C2×C22⋊C4—24no—
295C2×C22⋊C4—24no—
296C2×C22⋊C4—28no—
297C2×C22⋊C4—28no—
298C2×C22⋊C4—28no—
299C2×C22⋊C4—28no—
300C2×C22⋊C4—28no—
301C2×C22⋊C4—216no—
302C2×C22⋊C4—44no—
303C2×C22⋊C4—44no—
304C2×C22⋊C4—44no—
305C2×C22⋊C4—44no—
306C2×C22⋊C4—48no—
307C2×C22⋊C4—48no—
308C2×C22⋊C4C2×D418yes—
309C2×C42—24no—
310C2×C42—24no—
311C2×C42—22no—
312C2×C42—22no—
313C2×C42—24no—
314C2×C42—24no—
315C2×C42—22no—
316C2×C42—24no—
317C2×C42—24no—
318C2×C42—24no—
319C2×C42—22no—
320C2×C42—24no—
321C2×C42—24no—
322C2×C42—22no—
323C2×C42—24no—
324C2×C42—42no—
325C2×C42C2×D418yes—
326C2×C42C2×D414yes—
327C2×C42C2×D414yes—
328C2×C42C2×D414yes—
329C2×C42#4114yes—
330#95—28no—
331#95—28no—
332#95—24no—
333#95—28no—
334#95—216no—
335#95—28no—
336#95—216no—
337#95—44no—
338#95—48no—
339#95—48no—
340#95—48no—
341C24—22no—
342C24—22no—
343C24—22no—
344C24—22no—
345C24—24no—
346C24—22no—
347C24—22no—
348C24—24no—
349C24—24no—
350C24—22no—
351C24—22no—
352C24—22no—
353C24—22no—
354C24—24no—
355C24—24no—
356C24—22no—
357C24—24no—
358C24—22no—
359C24—24no—
360C24C22×D414yes—
361C24C2×#4114yes—
362C24C2×#4118yes—
363C24C2511yescharacteristic
364C2×Q8—28no—
365C2×Q8—216no—
366C2×Q8—28no—
367C2×Q8—28no—
368C2×Q8—416no—
369C2×Q8—44no—
370C2×Q8—44no—
371C2×Q8—44no—
372C2×Q8—44no—
373C2×Q8—88no—
374C2×Q8—84no—
375C2×Q8—84no—
376C2×D4—28no—
377C2×D4—28no—
378C2×D4—28no—
379C2×D4—28no—
380C2×D4—216no—
381C2×D4—216no—
382C2×D4—28no—
383C2×D4—416no—
384C2×D4—432no—
385C2×D4—416no—
386C2×D4—44no—
387C2×D4—44no—
388C2×D4—44no—
389C2×D4—44no—
390C2×D4—88no—
391C2×D4—816no—
392C2×D4#124116yescomplemented
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—24no—
404C22×C4—24no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—28no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—24no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—24no—
426C22×C4—28no—
427C22×C4—232no—
428C22×C4—216no—
429C22×C4—216no—
430C22×C4—216no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—24no—
434C22×C4—22no—
435C22×C4—24no—
436C22×C4—24no—
437C22×C4—24no—
438C22×C4—24no—
439C22×C4—28no—
440C22×C4—24no—
441C22×C4—28no—
442C22×C4—24no—
443C22×C4—24no—
444C22×C4—24no—
445C22×C4—24no—
446C22×C4—24no—
447C22×C4—24no—
448C22×C4—48no—
449C22×C4—48no—
450C22×C4—48no—
451C22×C4—48no—
452C22×C4—48no—
453C22×C4—48no—
454C22×C4—48no—
455C22×C4—416no—
456C22×C4—416no—
457C22×C4—816no—
458C22×C4—816no—
459C22×C4—88no—
460C22×C4—88no—
461C22×C4C22×D414yes—
462C22×C4C22×D414yes—
463C22×C4C22×D418yes—
464C22×C4C22×D414yes—
465C22×C4C22×D412yes—
466C22×C4C22×D412yes—
467C22×C4C22×D412yes—
468C22×C4C22×D412yes—
469C22×C4C22×D412yes—
470C22×C4C22×D412yes—
471C22×C4C22×D412yes—
472C22×C4C22×D412yes—
473C22×C4C22×D412yes—
474C22×C4C2×#4114yes—
475C22×C4C2×#4112yes—
476C22×C4C2×#4112yes—
477C22×C4C2×#4114yes—
478C22×C4H(5,𝔽2)14yes—
479C22×C4H(5,𝔽2)12yes—
480C22×C42−1+414yes—
481C22×C42−1+412yes—
482C4⋊C4—232no—
483C4⋊C4—232no—
484C4⋊C4—28no—
485C4⋊C4—216no—
486C4⋊C4—28no—
487C4⋊C4—216no—
488C4⋊C4—216no—
489C4⋊C4—44no—
490C4⋊C4—48no—
491C4⋊C4—48no—
492C4⋊C4—48no—
493C4⋊C4—84no—
494C22⋊C4—232no—
495C22⋊C4—232no—
496C22⋊C4—216no—
497C22⋊C4—416no—
498C22⋊C4—416no—
499C22⋊C4—416no—
500C22⋊C4—416no—
501C22⋊C4—416no—
502C22⋊C4—48no—
503C22⋊C4—816no—
504C42—216no—
505C42—28no—
506C42—48no—
507C42—44no—
508C42—48no—
509C42—48no—
510C42#120116yes—
511C23—24no—
512C23—24no—
513C23—24no—
514C23—24no—
515C23—28no—
516C23—28no—
517C23—28no—
518C23—28no—
519C23—24no—
520C23—24no—
521C23—28no—
522C23—28no—
523C23—24no—
524C23—24no—
525C23—24no—
526C23—28no—
527C23—28no—
528C23—42no—
529C23—42no—
530C23—42no—
531C23—42no—
532C23—44no—
533C23—44no—
534C23—44no—
535C23—44no—
536C23—44no—
537C23—44no—
538C23—42no—
539C23—42no—
540C23—42no—
541C23—42no—
542C23—42no—
543C23—44no—
544C23—44no—
545C23—42no—
546C23—44no—
547C23—44no—
548C23—44no—
549C23—42no—
550C23—42no—
551C23—44no—
552C23—44no—
553C23—44no—
554C23—44no—
555C23—44no—
556C23C2×#12418yes—
557C23C23×D411yescharacteristic
558C23C23×D411yescharacteristic
559C23C23×D411yescharacteristic
560C23C22×#4111yescharacteristic
561C23C22×#4111yescharacteristic
562C23C22×#4112yes—
563C23C2×H(5,𝔽2)11yescharacteristic
564C23C2×H(5,𝔽2)12yes—
565C23C2×2−1+412yes—
566C23C2×2−1+411yescharacteristic
567C23#58512yes—
568Q8—416no—
569Q8—816no—
570D4—232no—
571D4—416no—
572D4—832no—
573C2×C4—24no—
574C2×C4—24no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—24no—
578C2×C4—24no—
579C2×C4—24no—
580C2×C4—28no—
581C2×C4—24no—
582C2×C4—24no—
583C2×C4—24no—
584C2×C4—24no—
585C2×C4—24no—
586C2×C4—24no—
587C2×C4—24no—
588C2×C4—28no—
589C2×C4—216no—
590C2×C4—24no—
591C2×C4—24no—
592C2×C4—24no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—48no—
607C2×C4—44no—
608C2×C4—42no—
609C2×C4—44no—
610C2×C4—42no—
611C2×C4—44no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—432no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—44no—
619C2×C4—42no—
620C2×C4—44no—
621C2×C4—44no—
622C2×C4—44no—
623C2×C4—44no—
624C2×C4—44no—
625C2×C4—44no—
626C2×C4—48no—
627C2×C4—44no—
628C2×C4—44no—
629C2×C4—44no—
630C2×C4—44no—
631C2×C4—48no—
632C2×C4—44no—
633C2×C4—832no—
634C2×C4—816no—
635C2×C4C2×#12014yes—
636C2×C4C2×#12014yes—
637C2×C4C2×#12014yes—
638C2×C4#52514yes—
639C2×C4C2×#12414yes—
640C2×C4#53614yes—
641C2×C4#53714yes—
642C2×C4#54614yes—
643C2×C4#54714yes—
644C2×C4#54814yes—
645C2×C4#54818yes—
646C2×C4#54818yes—
647C2×C4#55014yes—
648C22—28no—
649C22—28no—
650C22—28no—
651C22—48no—
652C22—48no—
653C22—44no—
654C22—44no—
655C22—48no—
656C22—44no—
657C22—44no—
658C22—48no—
659C22—48no—
660C22—44no—
661C22—48no—
662C22—44no—
663C22—44no—
664C22—84no—
665C22—82no—
666C22—82no—
667C22—82no—
668C22—82no—
669C22—84no—
670C22C22×#12011yescharacteristic
671C22C2×#52511yescharacteristic
672C22C22×#12411yescharacteristic
673C22C2×#53611yescharacteristic
674C22C2×#53711yescharacteristic
675C22C2×#54611yescharacteristic
676C22C2×#54711yescharacteristic
677C22C2×#54812yes—
678C22C2×#54811yescharacteristic
679C22C2×#54812yes—
680C22C2×#55011yescharacteristic
681C22#347512yes—
682C22#348112yes—
683C22#348812yes—
684C22#349612yes—
685C22#349811yescharacteristic
686C22#350011yescharacteristic
687C22#351112yes—
688C22#351512yes—
689C22#351512yes—
690C22#351912yes—
691C22#354412yes—
692C22#354712yes—
693C4—24no—
694C4—44no—
695C4—44no—
696C4—44no—
697C4—44no—
698C4—48no—
699C4—44no—
700C4—84no—
701C4—82no—
702C4—84no—
703C4#240718yes—
704C4#243618yes—
705C2—48no—
706C2—84no—
707C2C2×#243611yescharacteristic
708C2C2×#240711yescharacteristic
709C2#3551311yescharacteristic
710C2#3616411yescharacteristic
711C2#3628911yescharacteristic
712C2#3649212yes—
713C2#3895511yescharacteristic
714C2#3912612yes—
715C2#3948911yescharacteristic
716C2#5965212yes—
717C2#5966012yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC2×#2537C2
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×#554C2semidirect
2. C2×#2563C2semidirect
4. C2×C4×#124C2semidirect
5. #15275C2semidirect
6. #15236C2semidirect
7. #15216C2semidirect
8. #15215C2semidirect
9. #14714C2semidirect
10. #14137C2semidirect
11. #13970C2semidirect
12. #13926C2semidirect
13. #13881C2semidirect
14. C4×#390C2semidirect
80. C2×C4⋊Q8D4semidirect
100. C2×C4×Q8D4semidirect
392. C2×D4#124semidirect

Maximal subgroups

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

subgroupindex
C22×#5542
C2×#25632
C2×#25372
C2×C4×#1242
#152752
#152362
#152162
#152152
#147142
#141372
#139702
#139262
#138812
C4×#3902