Finite Group Databasebeta

← #636602

Subgroups — #636602

Order 512.

4338 conjugacy classes of subgroups.

The subgroup lattice has 678 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#533C211yesmaximal, characteristic, complemented
2C2×#2538C212yesmaximal, complemented
3C2×C4×#121C211yesmaximal, characteristic
4#16604C214yesmaximal, complemented
5#16265C214yesmaximal, complemented
6#15202C212yesmaximal, complemented
7#15178C212yesmaximal, complemented
8#15177C214yesmaximal, complemented
9#15157C212yesmaximal, complemented
10#15125C212yesmaximal, complemented
11#14686C211yesmaximal, characteristic, complemented
12#14264C212yesmaximal, complemented
13#14257C212yesmaximal
14#13926C212yesmaximal, complemented
15C2×#533—28no—
16C2×#533—216no—
17C22×C4⋊Q8C2211yescharacteristic
18C22×C4⋊Q8C2211yescharacteristic
19C22×#125C2212yes—
20C22×C22⋊Q8C2212yes—
21C22×C22⋊Q8C2212yes—
22C22×C22⋊Q8C2214yes—
23C22×#121C2211yescharacteristic
24C22×C4×Q8C2212yes—
25C22×C4×Q8C2212yes—
26C22×C4×D4C2211yescharacteristic
27C22×C4×D4C2212yes—
28C22×#117C2212yes—
29C23×C42C2211yescharacteristic
30#2538—216no—
31C2×#396C2218yes—
32C2×#393C2214yes—
33C4×#121—28no—
34C2×#391C2214yes—
35C2×#390C2211yescharacteristic
36C2×#390C2214yes—
37C2×#389C2211yescharacteristic
38C2×#389C2214yes—
39C2×#388C2212yes—
40C2×#386C2212yes—
41C2×#384C2212yes—
42C2×#380C2214yes—
43C2×#380C2212yes—
44C2×#380C2211yescharacteristic
45C2×#380C2212yes—
46C2×#380C2212yes—
47C2×C4×C4⋊C4C2211yescharacteristic
48C2×C4×C22⋊C4C2212yes—
49#1446C2214yes—
50#1444C2218yes—
51#1444C2214yes—
52#1444C2214yes—
53#1442C2214yes—
54#1441C2218yes—
55#1439C2214yes—
56#1439C2218yes—
57#1439C2214yes—
58#1439C2218yes—
59#1439C2214yes—
60#1438C2218yes—
61C4⋊#95C2214yes—
62C4⋊#95C2212yes—
63C4⋊#95C2212yes—
64C4⋊#95C2214yes—
65C4⋊#95C2212yes—
66C4⋊#95C2214yes—
67C23×Q8C2312yes—
68C23×Q8C2312yes—
69C23×D4C2311yescharacteristic
70C23×D4C2312yes—
71C24×C4C2312yes—
72C24×C4C2311yescharacteristic
73C24×C4C2311yescharacteristic
74C24×C4C2311yescharacteristic
75#533—2128no—
76C2×C4⋊Q8—28no—
77C2×C4⋊Q8—28no—
78C2×C4⋊Q8—24no—
79C2×C4⋊Q8—24no—
80C2×#125—216no—
81C2×#125—28no—
82C2×C22⋊Q8—28no—
83C2×C22⋊Q8—28no—
84C2×C22⋊Q8—28no—
85C2×C22⋊Q8—216no—
86C2×C22⋊Q8—28no—
87C2×C22⋊Q8—216no—
88C2×C22⋊Q8—48no—
89C2×C22⋊Q8—48no—
90C2×#121—24no—
91C2×#121—24no—
92C2×#121Q818yescomplemented
93C2×C4×Q8—28no—
94C2×C4×Q8—28no—
95C2×C4×Q8—28no—
96C2×C4×Q8—216no—
97C2×C4×Q8D4116yescomplemented
98C2×C4×D4—28no—
99C2×C4×D4—28no—
100C2×C4×D4—24no—
101C2×C4×D4—48no—
102C2×#117—28no—
103C2×#117—216no—
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2311yescharacteristic
107C22×C4⋊C4C2314yes—
108C22×C4⋊C4C2312yes—
109C22×C4⋊C4C2314yes—
110C22×C4⋊C4C2311yescharacteristic
111C22×C4⋊C4C2314yes—
112C22×C4⋊C4C2314yes—
113C22×C4⋊C4C2312yes—
114C22×C4⋊C4C2311yescharacteristic
115C22×C4⋊C4C2314yes—
116C22×C4⋊C4C2318yes—
117C22×C4⋊C4C2312yes—
118C22×C22⋊C4C2312yes—
119C22×C22⋊C4C2312yes—
120C22×C22⋊C4C2314yes—
121C22×C22⋊C4C2312yes—
122C22×C22⋊C4C2318yes—
123C22×C22⋊C4C2318yes—
124C22×C22⋊C4C2312yes—
125C22×C22⋊C4C2312yes—
126C22×C22⋊C4C2314yes—
127C22×C42—28no—
128C22×C42D418yes—
129C22×C42C2312yes—
130C22×C42C2311yescharacteristic
131C22×C42C2311yescharacteristic
132C22×C42C2311yescharacteristic
133C22×C42C2314yes—
134C22×C42C2311yescharacteristic
135#396—232no—
136#393—48no—
137#391—48no—
138#390—216no—
139#390—24no—
140#389—216no—
141#389—24no—
142#388—28no—
143#386—28no—
144#384—28no—
145#380—24no—
146#380—28no—
147#380—216no—
148#380—44no—
149#380—44no—
150C4×C4⋊C4—24no—
151C4×C22⋊C4—28no—
152C2×#95C2314yes—
153C2×#95C2318yes—
154C2×#95C2314yes—
155C2×#95C2312yes—
156C2×#95C2312yes—
157C2×#95C2312yes—
158C2×#95C2318yes—
159C2×#95C2318yes—
160C2×#95C2314yes—
161C2×#95C2318yes—
162C2×#95C2318yes—
163C25C2412yes—
164C25C2411yescharacteristic
165C22×Q8—28no—
166C22×Q8—24no—
167C22×Q8—24no—
168C22×Q8—24no—
169C22×Q8—24no—
170C22×Q8—24no—
171C22×Q8—24no—
172C22×Q8—24no—
173C22×Q8—24no—
174C22×Q8—24no—
175C22×Q8—44no—
176C22×Q8C2×D418yes—
177C22×D4—24no—
178C22×D4—24no—
179C22×D4—22no—
180C22×D4—24no—
181C22×D4—22no—
182C22×D4—24no—
183C22×D4—44no—
184C22×D4—44no—
185C22×D4C2×Q814yes—
186C22×D4C2×Q818yes—
187C23×C4—24no—
188C23×C4—24no—
189C23×C4—24no—
190C23×C4—24no—
191C23×C4—24no—
192C23×C4—22no—
193C23×C4—22no—
194C23×C4—22no—
195C23×C4—22no—
196C23×C4—24no—
197C23×C4—28no—
198C23×C4—28no—
199C23×C4—24no—
200C23×C4—24no—
201C23×C4—42no—
202C23×C4C2×D414yes—
203C23×C4C2×D414yes—
204C23×C4C2×D414yes—
205C23×C4C2×Q814yes—
206C23×C4#4114yes—
207C23×C4C2418yes—
208C23×C4C2412yes—
209C23×C4C2412yes—
210C23×C4C2414yes—
211C23×C4C2414yes—
212C23×C4C2411yescharacteristic
213C23×C4C2412yes—
214C23×C4C2411yescharacteristic
215C23×C4C2411yescharacteristic
216C23×C4C2411yescharacteristic
217C23×C4C2411yescharacteristic
218C23×C4C2411yescharacteristic
219C4⋊Q8—232no—
220C4⋊Q8—232no—
221#125—264no—
222C22⋊Q8—264no—
223C22⋊Q8—264no—
224C22⋊Q8—464no—
225#121—232no—
226C4×Q8—264no—
227C4×Q8—264no—
228C4×D4—432no—
229C4×D4—416no—
230#117—264no—
231C2×C4⋊C4—28no—
232C2×C4⋊C4—28no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—28no—
236C2×C4⋊C4—28no—
237C2×C4⋊C4—28no—
238C2×C4⋊C4—28no—
239C2×C4⋊C4—28no—
240C2×C4⋊C4—28no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—28no—
250C2×C4⋊C4—216no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—28no—
253C2×C4⋊C4—28no—
254C2×C4⋊C4—28no—
255C2×C4⋊C4—28no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—28no—
258C2×C4⋊C4—28no—
259C2×C4⋊C4—44no—
260C2×C4⋊C4—48no—
261C2×C4⋊C4—48no—
262C2×C4⋊C4—44no—
263C2×C4⋊C4—44no—
264C2×C4⋊C4C2×D4116yes—
265C2×C4⋊C4C2×D418yes—
266C2×C4⋊C4C2×Q814yes—
267C2×C22⋊C4—28no—
268C2×C22⋊C4—28no—
269C2×C22⋊C4—28no—
270C2×C22⋊C4—24no—
271C2×C22⋊C4—24no—
272C2×C22⋊C4—24no—
273C2×C22⋊C4—24no—
274C2×C22⋊C4—24no—
275C2×C22⋊C4—28no—
276C2×C22⋊C4—216no—
277C2×C22⋊C4—24no—
278C2×C22⋊C4—24no—
279C2×C22⋊C4—24no—
280C2×C22⋊C4—28no—
281C2×C22⋊C4—216no—
282C2×C22⋊C4—44no—
283C2×C22⋊C4—44no—
284C2×C22⋊C4—48no—
285C2×C22⋊C4—48no—
286C2×C22⋊C4—48no—
287C2×C22⋊C4—44no—
288C2×C22⋊C4—44no—
289C2×C22⋊C4—48no—
290C2×C22⋊C4C2×Q818yes—
291C2×C42—28no—
292C2×C42—28no—
293C2×C42—24no—
294C2×C42—24no—
295C2×C42—24no—
296C2×C42—232no—
297C2×C42—24no—
298C2×C42—28no—
299C2×C42C2×D4116yes—
300C2×C42C2×D414yes—
301C2×C42C2×D414yes—
302C2×C42C2×D414yes—
303C2×C42C2×D414yes—
304C2×C42C2×D418yes—
305#95—24no—
306#95—24no—
307#95—24no—
308#95—216no—
309#95—28no—
310#95—28no—
311#95—48no—
312#95—44no—
313#95—48no—
314#95—48no—
315#95—48no—
316C24—22no—
317C24—21no—
318C24—21no—
319C24—22no—
320C24—22no—
321C24—21no—
322C24—21no—
323C24—22no—
324C24—24no—
325C24—22no—
326C24—22no—
327C24—22no—
328C24—22no—
329C24—24no—
330C24—22no—
331C24—22no—
332C24—24no—
333C24—24no—
334C24C22×D412yes—
335C24C22×Q814yes—
336C24C22×Q812yes—
337C24C2×#4112yes—
338C24C2511yescharacteristic
339C2×Q8—216no—
340C2×Q8—216no—
341C2×Q8—216no—
342C2×Q8—216no—
343C2×Q8—416no—
344C2×Q8—416no—
345C2×Q8—48no—
346C2×Q8—48no—
347C2×Q8—48no—
348C2×Q8—48no—
349C2×Q8—48no—
350C2×Q8—48no—
351C2×D4—216no—
352C2×D4—28no—
353C2×D4—216no—
354C2×D4—416no—
355C2×D4—48no—
356C2×D4—48no—
357C2×D4—416no—
358C2×D4—44no—
359C2×D4—44no—
360C2×D4—48no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—21no—
364C22×C4—22no—
365C22×C4—21no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—28no—
369C22×C4—28no—
370C22×C4—216no—
371C22×C4—216no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—24no—
375C22×C4—24no—
376C22×C4—24no—
377C22×C4—24no—
378C22×C4—22no—
379C22×C4—21no—
380C22×C4—24no—
381C22×C4—24no—
382C22×C4—24no—
383C22×C4—22no—
384C22×C4—24no—
385C22×C4—24no—
386C22×C4—21no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—24no—
390C22×C4—24no—
391C22×C4—24no—
392C22×C4—28no—
393C22×C4—28no—
394C22×C4—28no—
395C22×C4—28no—
396C22×C4—28no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—24no—
400C22×C4—28no—
401C22×C4—28no—
402C22×C4—28no—
403C22×C4—22no—
404C22×C4—24no—
405C22×C4—28no—
406C22×C4—28no—
407C22×C4—24no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—24no—
411C22×C4—28no—
412C22×C4—22no—
413C22×C4—28no—
414C22×C4—44no—
415C22×C4—48no—
416C22×C4—48no—
417C22×C4—48no—
418C22×C4—48no—
419C22×C4—48no—
420C22×C4—48no—
421C22×C4—416no—
422C22×C4—48no—
423C22×C4—416no—
424C22×C4C22⋊Q8116yescomplemented
425C22×C4C22×D418yes—
426C22×C4C22×D412yes—
427C22×C4C22×D418yes—
428C22×C4C22×D412yes—
429C22×C4C22×D414yes—
430C22×C4C22×D412yes—
431C22×C4C22×D412yes—
432C22×C4C22×D412yes—
433C22×C4C22×D412yes—
434C22×C4C22×D414yes—
435C22×C4C22×D414yes—
436C22×C4C22×Q814yes—
437C22×C4C22×Q812yes—
438C22×C4C22×Q812yes—
439C22×C4C2×#4112yes—
440C22×C4C2×#4112yes—
441C22×C4H(5,𝔽2)12yes—
442C22×C4H(5,𝔽2)14yes—
443C22×C4H(5,𝔽2)12yes—
444C22×C42−1+412yes—
445C22×C42−1+414yes—
446C22×C42−1+412yes—
447C4⋊C4—216no—
448C4⋊C4—216no—
449C4⋊C4—232no—
450C4⋊C4—216no—
451C4⋊C4—232no—
452C4⋊C4—216no—
453C4⋊C4—28no—
454C4⋊C4—48no—
455C4⋊C4—432no—
456C4⋊C4—44no—
457C4⋊C4—416no—
458C4⋊C4—44no—
459C4⋊C4—416no—
460C4⋊C4—88no—
461C22⋊C4—232no—
462C22⋊C4—216no—
463C22⋊C4—48no—
464C22⋊C4—432no—
465C22⋊C4—48no—
466C22⋊C4—48no—
467C22⋊C4—48no—
468C22⋊C4—816no—
469C22⋊C4—88no—
470C42—28no—
471C42—28no—
472C42—28no—
473C42—232no—
474C42—48no—
475C42#120116yes—
476C23—22no—
477C23—22no—
478C23—22no—
479C23—22no—
480C23—22no—
481C23—22no—
482C23—24no—
483C23—24no—
484C23—24no—
485C23—22no—
486C23—22no—
487C23—24no—
488C23—22no—
489C23—22no—
490C23—24no—
491C23—28no—
492C23—28no—
493C23—24no—
494C23—24no—
495C23—42no—
496C23—42no—
497C23—42no—
498C23—42no—
499C23—42no—
500C23—42no—
501C23—42no—
502C23—42no—
503C23—42no—
504C23—44no—
505C23—42no—
506C23—42no—
507C23—42no—
508C23—44no—
509C23—42no—
510C23—42no—
511C23—44no—
512C23—44no—
513C23—44no—
514C23—44no—
515C23—42no—
516C23—42no—
517C23—44no—
518C23—44no—
519C23—44no—
520C23—44no—
521C23C2×C22⋊Q814yes—
522C23C23×D411yescharacteristic
523C23C23×D412yes—
524C23C23×Q811yescharacteristic
525C23C22×#4111yescharacteristic
526C23C2×H(5,𝔽2)12yes—
527C23C2×H(5,𝔽2)11yescharacteristic
528C23C2×H(5,𝔽2)11yescharacteristic
529C23C2×2−1+412yes—
530C23C2×2−1+411yescharacteristic
531C23C2×2−1+411yescharacteristic
532C23#58512yes—
533Q8—432no—
534Q8—432no—
535D4—432no—
536D4—416no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—24no—
544C2×C4—24no—
545C2×C4—24no—
546C2×C4—24no—
547C2×C4—24no—
548C2×C4—24no—
549C2×C4—232no—
550C2×C4—22no—
551C2×C4—28no—
552C2×C4—28no—
553C2×C4—24no—
554C2×C4—22no—
555C2×C4—28no—
556C2×C4—22no—
557C2×C4—28no—
558C2×C4—28no—
559C2×C4—28no—
560C2×C4—28no—
561C2×C4—416no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—44no—
565C2×C4—44no—
566C2×C4—42no—
567C2×C4—44no—
568C2×C4—44no—
569C2×C4—44no—
570C2×C4—42no—
571C2×C4—44no—
572C2×C4—44no—
573C2×C4—44no—
574C2×C4—44no—
575C2×C4—416no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—48no—
579C2×C4—42no—
580C2×C4—44no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—44no—
584C2×C4—42no—
585C2×C4—48no—
586C2×C4—48no—
587C2×C4—48no—
588C2×C4—48no—
589C2×C4—44no—
590C2×C4—48no—
591C2×C4—44no—
592C2×C4—48no—
593C2×C4—816no—
594C2×C4C2×#12014yes—
595C2×C4C2×#12014yes—
596C2×C4C2×#12014yes—
597C2×C4C2×C22⋊Q814yes—
598C2×C4C2×C22⋊Q814yes—
599C2×C4#53514yes—
600C2×C4#53518yes—
601C2×C4#53614yes—
602C2×C4#53714yes—
603C2×C4#53714yes—
604C2×C4#53718yes—
605C2×C4#54818yes—
606C2×C4D4×Q818yes—
607C22—24no—
608C22—24no—
609C22—24no—
610C22—42no—
611C22—42no—
612C22—44no—
613C22—44no—
614C22—44no—
615C22—44no—
616C22—42no—
617C22—42no—
618C22—44no—
619C22—44no—
620C22—48no—
621C22—48no—
622C22—44no—
623C22—82no—
624C22—82no—
625C22—84no—
626C22—84no—
627C22—82no—
628C22—82no—
629C22C22×#12011yescharacteristic
630C22C22×C22⋊Q811yescharacteristic
631C22C2×#53511yescharacteristic
632C22C2×#53512yes—
633C22C2×#53611yescharacteristic
634C22C2×#53711yescharacteristic
635C22C2×#53712yes—
636C22C2×#53711yescharacteristic
637C22#344911yescharacteristic
638C22C2×#54812yes—
639C22C2×D4×Q812yes—
640C22#348812yes—
641C22#348912yes—
642C22#350112yes—
643C22#350311yescharacteristic
644C22#350311yescharacteristic
645C22#350612yes—
646C22#350811yescharacteristic
647C22#350911yescharacteristic
648C22#354812yes—
649C22#355012yes—
650C22#355412yes—
651C22#355512yes—
652C4—24no—
653C4—24no—
654C4—24no—
655C4—42no—
656C4—48no—
657C4—44no—
658C4—48no—
659C4—82no—
660C4—82no—
661C4—88no—
662C4#240918yes—
663C4#261518yes—
664C2—44no—
665C2—84no—
666C2C2×#261511yescharacteristic
667C2C2×#240911yescharacteristic
668C2#3517211yescharacteristic
669C2#3525111yescharacteristic
670C2#3528011yescharacteristic
671C2#3762211yescharacteristic
672C2#3764011yescharacteristic
673C2#3959612yes—
674C2#3959812yes—
675C2#3972512yes—
676C2#3972712yes—
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×#533C2semidirect
2. C2×#2538C2semidirect
4. #16604C2semidirect
5. #16265C2semidirect
6. #15202C2semidirect
7. #15178C2semidirect
8. #15177C2semidirect
9. #15157C2semidirect
10. #15125C2semidirect
11. #14686C2semidirect
12. #14264C2semidirect
14. #13926C2semidirect
92. C2×#121Q8semidirect
97. C2×C4×Q8D4semidirect
424. C22×C4C22⋊Q8semidirect

Maximal subgroups

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

subgroupindex
C22×#5332
C2×#25382
C2×C4×#1212
#166042
#162652
#152022
#151782
#151772
#151572
#151252
#146862
#142642
#142572
#139262