Finite Group Databasebeta

← #644176

Subgroups — #644176

Order 512.

3022 conjugacy classes of subgroups.

The subgroup lattice has 789 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2851C211yesmaximal, characteristic, complemented
2C2×C4×#121C211yesmaximal, characteristic
3#17016C214yesmaximal, complemented
4#16985C212yesmaximal, complemented
5#16802C212yesmaximal, complemented
6#15233C212yesmaximal, complemented
7#15216C212yesmaximal, complemented
8#15200C211yesmaximal, characteristic, complemented
9#15181C211yesmaximal, characteristic, complemented
10#15156C211yesmaximal, characteristic, complemented
11#15129C211yesmaximal, characteristic, complemented
12#14826C212yesmaximal, complemented
13#14747C212yesmaximal
14#14259C214yesmaximal, complemented
15#13872C212yesmaximal, complemented
16#13872C212yesmaximal, complemented
17C2×C4⋊#95C211yesmaximal, characteristic, complemented
18C22×#125C2211yescharacteristic
19C22×#121C2211yescharacteristic
20C22×C4×D4C2212yes—
21C22×C4×D4C2211yescharacteristic
22C23×C4⋊C4C2212yes—
23C23×C42C2211yescharacteristic
24#2851—28no—
25C2×#400C2212yes—
26C2×#394C2212yes—
27C4×#121—28no—
28C2×#390C2211yescharacteristic
29C2×#389C2211yescharacteristic
30C2×#389C2211yescharacteristic
31C2×#389C2211yescharacteristic
32C2×#388C2212yes—
33C2×#387C2212yes—
34C2×#387C2212yes—
35C2×#386C2214yes—
36C2×#386C2212yes—
37C2×#386C2214yes—
38C2×#385C2212yes—
39C2×#385C2212yes—
40C2×#382C2211yescharacteristic
41C2×#380C2211yescharacteristic
42C2×#380C2211yescharacteristic
43C2×#380C2211yescharacteristic
44C2×#380C2212yes—
45C2×#380C2212yes—
46C2×#380C2212yes—
47C2×#380C2212yes—
48C2×C4×C4⋊C4C2211yescharacteristic
49C2×C4×C22⋊C4C2212yes—
50C22×#95C2212yes—
51C22×#95C2212yes—
52#1445C2212yes—
53#1443C2214yes—
54#1443C2214yes—
55#1443C2214yes—
56#1442C2212yes—
57#1442C2212yes—
58#1442C2214yes—
59#1442C2214yes—
60#1442C2214yes—
61#1441C2212yes—
62#1441C2214yes—
63#1441C2214yes—
64#1441C2212yes—
65#1440C2214yes—
66#1440C2214yes—
67#1440C2214yes—
68#1439C2214yes—
69#1439C2214yes—
70#1439C2214yes—
71#1439C2214yes—
72#1439C2214yes—
73#1438C2212yes—
74C4⋊#95C2211yescharacteristic
75C4⋊#95C2212yes—
76C4⋊#95C2212yes—
77C4⋊#95C2212yes—
78C4⋊#95C2212yes—
79C4⋊#95C2211yescharacteristic
80C4⋊#95C2212yes—
81C4⋊#95C2212yes—
82C4⋊#95C2212yes—
83C22⋊#95C2214yes—
84C22⋊#95C2214yes—
85C23×D4C2312yes—
86C23×D4C2311yescharacteristic
87C24×C4C2312yes—
88C24×C4C2311yescharacteristic
89C24×C4C2311yescharacteristic
90C24×C4C2314yes—
91C24×C4C2311yescharacteristic
92C2×#125—24no—
93C2×#125—44no—
94C2×#121—28no—
95C2×#121—24no—
96C2×C4×D4—28no—
97C2×C4×D4—28no—
98C2×C4×D4—28no—
99C2×C4×D4—24no—
100C2×C4×D4—44no—
101C22×C4⋊C4—28no—
102C22×C4⋊C4—44no—
103C22×C4⋊C4—44no—
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2311yescharacteristic
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2312yes—
108C22×C4⋊C4C2312yes—
109C22×C4⋊C4C2311yescharacteristic
110C22×C4⋊C4C2311yescharacteristic
111C22×C4⋊C4C2311yescharacteristic
112C22×C4⋊C4C2311yescharacteristic
113C22×C4⋊C4C2311yescharacteristic
114C22×C4⋊C4C2314yes—
115C22×C4⋊C4C2311yescharacteristic
116C22×C4⋊C4C2312yes—
117C22×C4⋊C4C2312yes—
118C22×C4⋊C4C2312yes—
119C22×C22⋊C4C2312yes—
120C22×C22⋊C4C2312yes—
121C22×C22⋊C4C2312yes—
122C22×C22⋊C4C2312yes—
123C22×C22⋊C4C2312yes—
124C22×C22⋊C4C2314yes—
125C22×C22⋊C4C2312yes—
126C22×C22⋊C4C2314yes—
127C22×C22⋊C4C2312yes—
128C22×C22⋊C4C2314yes—
129C22×C22⋊C4C2314yes—
130C22×C42—44no—
131C22×C42D418yes—
132C22×C42C2312yes—
133C22×C42C2311yescharacteristic
134C22×C42C2311yescharacteristic
135C22×C42C2311yescharacteristic
136C22×C42C2311yescharacteristic
137#400—44no—
138#394—44no—
139#390—24no—
140#389—24no—
141#389—24no—
142#389—24no—
143#388—28no—
144#387—28no—
145#387—28no—
146#386—216no—
147#386—28no—
148#386—48no—
149#385—28no—
150#385—28no—
151#382—24no—
152#380—24no—
153#380—24no—
154#380—24no—
155#380—44no—
156#380—44no—
157#380—44no—
158#380—44no—
159C4×C4⋊C4—24no—
160C4×C22⋊C4—28no—
161C2×#95—44no—
162C2×#95—44no—
163C2×#95C2312yes—
164C2×#95C2312yes—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2312yes—
168C2×#95C2312yes—
169C2×#95C2312yes—
170C2×#95C2312yes—
171C2×#95C2312yes—
172C2×#95C2312yes—
173C2×#95C2312yes—
174C2×#95C2312yes—
175C2×#95C2312yes—
176C2×#95C2314yes—
177C2×#95C2314yes—
178C2×#95C2314yes—
179C2×#95C2314yes—
180C2×#95C2314yes—
181C2×#95C2314yes—
182C2×#95C2314yes—
183C2×#95C2314yes—
184C2×#95C2312yes—
185C2×#95C2314yes—
186C2×#95C2314yes—
187C2×#95C2312yes—
188C2×#95C2314yes—
189C2×#95C2314yes—
190C25C2412yes—
191C25C2411yescharacteristic
192C22×D4—24no—
193C22×D4—24no—
194C22×D4—24no—
195C22×D4—24no—
196C22×D4—24no—
197C22×D4—24no—
198C22×D4—24no—
199C22×D4—22no—
200C22×D4—22no—
201C22×D4—22no—
202C22×D4—42no—
203C22×D4#4118yes—
204C23×C4—22no—
205C23×C4—24no—
206C23×C4—22no—
207C23×C4—24no—
208C23×C4—22no—
209C23×C4—22no—
210C23×C4—22no—
211C23×C4—22no—
212C23×C4—28no—
213C23×C4—28no—
214C23×C4—28no—
215C23×C4—24no—
216C23×C4—24no—
217C23×C4—24no—
218C23×C4—24no—
219C23×C4—24no—
220C23×C4—24no—
221C23×C4—24no—
222C23×C4—42no—
223C23×C4—42no—
224C23×C4—42no—
225C23×C4—44no—
226C23×C4—44no—
227C23×C4—44no—
228C23×C4—44no—
229C23×C4C2×D414yes—
230C23×C4C2×D414yes—
231C23×C4C2×D414yes—
232C23×C4#4118yes—
233C23×C4C2412yes—
234C23×C4C2412yes—
235C23×C4C2412yes—
236C23×C4C2412yes—
237C23×C4C2412yes—
238C23×C4C2412yes—
239C23×C4C2414yes—
240C23×C4C2414yes—
241C23×C4C2411yescharacteristic
242C23×C4C2411yescharacteristic
243C23×C4C2411yescharacteristic
244C23×C4C2412yes—
245C23×C4C2411yescharacteristic
246C23×C4C2411yescharacteristic
247C23×C4C2411yescharacteristic
248#125—416no—
249#121—232no—
250C4×D4—416no—
251C4×D4—432no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—24no—
264C2×C4⋊C4—24no—
265C2×C4⋊C4—28no—
266C2×C4⋊C4—24no—
267C2×C4⋊C4—24no—
268C2×C4⋊C4—24no—
269C2×C4⋊C4—24no—
270C2×C4⋊C4—24no—
271C2×C4⋊C4—24no—
272C2×C4⋊C4—24no—
273C2×C4⋊C4—24no—
274C2×C4⋊C4—24no—
275C2×C4⋊C4—24no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—24no—
279C2×C4⋊C4—22no—
280C2×C4⋊C4—24no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—44no—
285C2×C4⋊C4—44no—
286C2×C4⋊C4—44no—
287C2×C4⋊C4—42no—
288C2×C4⋊C4—44no—
289C2×C4⋊C4—44no—
290C2×C4⋊C4—432no—
291C2×C22⋊C4—24no—
292C2×C22⋊C4—24no—
293C2×C22⋊C4—24no—
294C2×C22⋊C4—24no—
295C2×C22⋊C4—24no—
296C2×C22⋊C4—24no—
297C2×C22⋊C4—24no—
298C2×C22⋊C4—24no—
299C2×C22⋊C4—28no—
300C2×C22⋊C4—24no—
301C2×C22⋊C4—24no—
302C2×C22⋊C4—24no—
303C2×C22⋊C4—24no—
304C2×C22⋊C4—24no—
305C2×C22⋊C4—24no—
306C2×C22⋊C4—24no—
307C2×C22⋊C4—28no—
308C2×C22⋊C4—28no—
309C2×C22⋊C4—42no—
310C2×C22⋊C4—42no—
311C2×C22⋊C4—42no—
312C2×C22⋊C4—42no—
313C2×C22⋊C4—44no—
314C2×C22⋊C4—44no—
315C2×C22⋊C4—44no—
316C2×C22⋊C4—44no—
317C2×C22⋊C4—44no—
318C2×C22⋊C4—44no—
319C2×C22⋊C4—44no—
320C2×C22⋊C4—44no—
321C2×C22⋊C4—44no—
322C2×C22⋊C4—44no—
323C2×C42—24no—
324C2×C42—24no—
325C2×C42—24no—
326C2×C42—24no—
327C2×C42—24no—
328C2×C42—416no—
329C2×C42—42no—
330C2×C42—42no—
331C2×C42C2×D414yes—
332C2×C42C2×D414yes—
333C2×C42C2×D414yes—
334C2×C42C2×D414yes—
335#95—24no—
336#95—24no—
337#95—24no—
338#95—24no—
339#95—24no—
340#95—24no—
341#95—24no—
342#95—24no—
343#95—24no—
344#95—28no—
345#95—28no—
346#95—24no—
347#95—24no—
348#95—44no—
349#95—42no—
350#95—44no—
351#95—42no—
352#95—44no—
353#95—44no—
354#95—44no—
355#95—44no—
356#95—44no—
357#95—44no—
358#95—44no—
359#95—44no—
360#95—44no—
361#95—44no—
362C24—22no—
363C24—21no—
364C24—21no—
365C24—21no—
366C24—21no—
367C24—22no—
368C24—22no—
369C24—22no—
370C24—22no—
371C24—22no—
372C24—22no—
373C24—22no—
374C24—22no—
375C24—22no—
376C24—22no—
377C24—22no—
378C24—22no—
379C24—22no—
380C24—22no—
381C24—22no—
382C24C22×D412yes—
383C24C2×#4114yes—
384C24C2×#4114yes—
385C24C2×#4114yes—
386C24H(5,𝔽2)12yes—
387C242−1+412yes—
388C24C2511yescharacteristic
389C2×D4—28no—
390C2×D4—216no—
391C2×D4—216no—
392C2×D4—216no—
393C2×D4—48no—
394C2×D4—48no—
395C2×D4—44no—
396C2×D4—48no—
397C2×D4—48no—
398C2×D4—48no—
399C2×D4—48no—
400C2×D4—44no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—21no—
405C22×C4—21no—
406C22×C4—28no—
407C22×C4—28no—
408C22×C4—216no—
409C22×C4—216no—
410C22×C4—28no—
411C22×C4—28no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—21no—
418C22×C4—21no—
419C22×C4—21no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—28no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—24no—
438C22×C4—24no—
439C22×C4—24no—
440C22×C4—24no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—24no—
446C22×C4—24no—
447C22×C4—24no—
448C22×C4—24no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—24no—
452C22×C4—24no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—24no—
462C22×C4—24no—
463C22×C4—24no—
464C22×C4—24no—
465C22×C4—24no—
466C22×C4—24no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—22no—
474C22×C4—22no—
475C22×C4—22no—
476C22×C4—22no—
477C22×C4—24no—
478C22×C4—24no—
479C22×C4—24no—
480C22×C4—24no—
481C22×C4—44no—
482C22×C4—44no—
483C22×C4—44no—
484C22×C4—48no—
485C22×C4—48no—
486C22×C4—48no—
487C22×C4—48no—
488C22×C4—416no—
489C22×C4—416no—
490C22×C4—416no—
491C22×C4—48no—
492C22×C4—48no—
493C22×C4—48no—
494C22×C4—48no—
495C22×C4—48no—
496C22×C4—48no—
497C22×C4—48no—
498C22×C4—88no—
499C22×C4—88no—
500C22×C4—88no—
501C22×C4—88no—
502C22×C4#124116yes—
503C22×C4C22×D412yes—
504C22×C4C22×D412yes—
505C22×C4C22×D412yes—
506C22×C4C22×D412yes—
507C22×C4C22×D412yes—
508C22×C4C22×D412yes—
509C22×C4C2×#4114yes—
510C22×C4C2×#4114yes—
511C22×C4C2×#4114yes—
512C22×C4H(5,𝔽2)12yes—
513C22×C4H(5,𝔽2)14yes—
514C22×C4H(5,𝔽2)12yes—
515C22×C42−1+412yes—
516C22×C42−1+412yes—
517C4⋊C4—28no—
518C4⋊C4—48no—
519C4⋊C4—416no—
520C4⋊C4—44no—
521C4⋊C4—48no—
522C4⋊C4—48no—
523C4⋊C4—48no—
524C4⋊C4—44no—
525C4⋊C4—44no—
526C4⋊C4—44no—
527C4⋊C4—48no—
528C4⋊C4—44no—
529C4⋊C4—44no—
530C4⋊C4—48no—
531C4⋊C4—48no—
532C22⋊C4—216no—
533C22⋊C4—48no—
534C22⋊C4—48no—
535C22⋊C4—48no—
536C22⋊C4—48no—
537C22⋊C4—48no—
538C22⋊C4—48no—
539C22⋊C4—88no—
540C22⋊C4—88no—
541C22⋊C4—88no—
542C22⋊C4—88no—
543C42—28no—
544C42—28no—
545C42—44no—
546C42—44no—
547C42—48no—
548C23—22no—
549C23—22no—
550C23—22no—
551C23—22no—
552C23—24no—
553C23—24no—
554C23—24no—
555C23—24no—
556C23—24no—
557C23—24no—
558C23—24no—
559C23—24no—
560C23—24no—
561C23—24no—
562C23—22no—
563C23—22no—
564C23—22no—
565C23—22no—
566C23—24no—
567C23—24no—
568C23—24no—
569C23—24no—
570C23—24no—
571C23—24no—
572C23—24no—
573C23—42no—
574C23—42no—
575C23—42no—
576C23—42no—
577C23—42no—
578C23—42no—
579C23—42no—
580C23—42no—
581C23—42no—
582C23—42no—
583C23—42no—
584C23—42no—
585C23—42no—
586C23—42no—
587C23—42no—
588C23—42no—
589C23—42no—
590C23—42no—
591C23—42no—
592C23—42no—
593C23—42no—
594C23—42no—
595C23—42no—
596C23—42no—
597C23—42no—
598C23—42no—
599C23—42no—
600C23—42no—
601C23#52514yes—
602C23C2×#12414yes—
603C23#53714yes—
604C23#55218yes—
605C23#55318yes—
606C23C23×D411yescharacteristic
607C23C22×#4112yes—
608C23C22×#4112yes—
609C23C2×H(5,𝔽2)12yes—
610C23C2×H(5,𝔽2)11yescharacteristic
611C23C2×H(5,𝔽2)11yescharacteristic
612C23C2×H(5,𝔽2)11yescharacteristic
613C23C2×2−1+411yescharacteristic
614C23C2×2−1+411yescharacteristic
615C23C2×2−1+411yescharacteristic
616C23#58512yes—
617D4—416no—
618D4—432no—
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—232no—
628C2×C4—22no—
629C2×C4—24no—
630C2×C4—22no—
631C2×C4—22no—
632C2×C4—22no—
633C2×C4—22no—
634C2×C4—22no—
635C2×C4—22no—
636C2×C4—22no—
637C2×C4—22no—
638C2×C4—24no—
639C2×C4—24no—
640C2×C4—416no—
641C2×C4—416no—
642C2×C4—42no—
643C2×C4—42no—
644C2×C4—42no—
645C2×C4—42no—
646C2×C4—42no—
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—42no—
657C2×C4—42no—
658C2×C4—42no—
659C2×C4—42no—
660C2×C4—42no—
661C2×C4—42no—
662C2×C4—42no—
663C2×C4—42no—
664C2×C4—42no—
665C2×C4—42no—
666C2×C4—42no—
667C2×C4—42no—
668C2×C4—42no—
669C2×C4—42no—
670C2×C4—44no—
671C2×C4—44no—
672C2×C4—42no—
673C2×C4—42no—
674C2×C4—42no—
675C2×C4—42no—
676C2×C4—44no—
677C2×C4—44no—
678C2×C4—42no—
679C2×C4—42no—
680C2×C4—44no—
681C2×C4—44no—
682C2×C4—42no—
683C2×C4—42no—
684C2×C4—44no—
685C2×C4—44no—
686C2×C4—44no—
687C2×C4—44no—
688C2×C4—44no—
689C2×C4—44no—
690C2×C4—44no—
691C2×C4—44no—
692C2×C4—44no—
693C2×C4—44no—
694C2×C4—44no—
695C2×C4—44no—
696C2×C4—42no—
697C2×C4—44no—
698C2×C4—44no—
699C2×C4—42no—
700C2×C4—42no—
701C2×C4—42no—
702C2×C4—832no—
703C2×C4—816no—
704C2×C4C2×#12414yes—
705C2×C4C2×#12414yes—
706C2×C4#53414yes—
707C2×C4#53714yes—
708C2×C4#53714yes—
709C2×C4#53818yes—
710C2×C4#53918yes—
711C22—24no—
712C22—24no—
713C22—28no—
714C22—28no—
715C22—24no—
716C22—28no—
717C22—28no—
718C22—24no—
719C22—24no—
720C22—24no—
721C22—44no—
722C22—44no—
723C22—44no—
724C22—44no—
725C22—44no—
726C22—44no—
727C22—44no—
728C22—82no—
729C22—82no—
730C22—82no—
731C22—82no—
732C22—82no—
733C22—82no—
734C22—82no—
735C22—82no—
736C22#256218yes—
737C22C2×#52511yescharacteristic
738C22C22×#12411yescharacteristic
739C22C2×#53411yescharacteristic
740C22C2×#53711yescharacteristic
741C22C2×#53711yescharacteristic
742C22C2×#53711yescharacteristic
743C22#344911yescharacteristic
744C22C2×#53812yes—
745C22C2×#53912yes—
746C22C2×#55212yes—
747C22C2×#55312yes—
748C22#350612yes—
749C22#350712yes—
750C22#351412yes—
751C22#351512yes—
752C22#351912yes—
753C22#352012yes—
754C22#354112yes—
755C22#354212yes—
756C22#354311yescharacteristic
757C22#354411yescharacteristic
758C22#355211yescharacteristic
759C22#355211yescharacteristic
760C4—24no—
761C4—24no—
762C4—42no—
763C4—42no—
764C4—42no—
765C4—44no—
766C4—82no—
767C4—82no—
768C4—84no—
769C4—84no—
770C4—82no—
771C4—82no—
772C4—82no—
773C4—82no—
774C4#263918yes—
775C2—28no—
776C2—84no—
777C2C2×#263911yescharacteristic
778C2C2×#256211yescharacteristic
779C2#3507711yescharacteristic
780C2#3513011yescharacteristic
781C2#3518011yescharacteristic
782C2#3744411yescharacteristic
783C2#3762011yescharacteristic
784C2#3897312yes—
785C2#3897412yes—
786C2#3918912yes—
787C2#3919012yes—
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. C2×#2851C2semidirect
3. #17016C2semidirect
4. #16985C2semidirect
5. #16802C2semidirect
6. #15233C2semidirect
7. #15216C2semidirect
8. #15200C2semidirect
9. #15181C2semidirect
10. #15156C2semidirect
11. #15129C2semidirect
12. #14826C2semidirect
14. #14259C2semidirect
15. #13872C2semidirect
16. #13872C2semidirect
17. C2×C4⋊#95C2semidirect

Maximal subgroups

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

subgroupindex
C2×#28512
C2×C4×#1212
#170162
#169852
#168022
#152332
#152162
#152002
#151812
#151562
#151292
#148262
#147472
#142592
#138722
C2×C4⋊#952