Finite Group Databasebeta

← #539280

Subgroups — #539280

Order 512.

3854 conjugacy classes of subgroups.

The subgroup lattice has 683 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#556C211yesmaximal, characteristic
2C2×#2365C211yesmaximal, characteristic, complemented
3C2×#2327C211yesmaximal, characteristic, complemented
4#14493C214yesmaximal, complemented
5#14448C214yesmaximal, complemented
6#14101C214yesmaximal, complemented
7#14026C212yesmaximal
8#14026C212yesmaximal
9#13944C214yesmaximal, complemented
10#13941C211yesmaximal, characteristic
11#13931C212yesmaximal, complemented
12#13925C211yesmaximal, characteristic, complemented
13#13850C211yesmaximal, characteristic, complemented
14Q8×#95C212yesmaximal, complemented
15#13785C211yesmaximal, characteristic
16C2×#556—28no—
17C2×#556—28no—
18C2×#556C4116yescomplemented
19C22×C4⋊Q8C2211yescharacteristic
20C22×#126C2214yes—
21C22×#124C2212yes—
22C22×C22⋊Q8C2212yes—
23C22×C4×Q8C2212yes—
24C22×C4×Q8C2212yes—
25C22×C4×Q8C2211yescharacteristic
26C22×C4×Q8C2212yes—
27C22×C4×D4C2211yescharacteristic
28C22×#117C2212yes—
29#2365—28no—
30#2327—28no—
31C2×#391C2212yes—
32C2×#391C2212yes—
33C2×#390C2211yescharacteristic
34C2×#390C2211yescharacteristic
35C2×#390C2212yes—
36C2×#389C2212yes—
37C2×#387C2214yes—
38C2×#387C2214yes—
39C2×#387C2214yes—
40C2×#382C2211yescharacteristic
41C2×C4×C4⋊C4C2212yes—
42C2×C4×C4⋊C4C2211yescharacteristic
43C2×C4×C4⋊C4C2211yescharacteristic
44C2×#376C2211yescharacteristic
45#1446C2212yes—
46#1446C2212yes—
47#1445C2214yes—
48#1445C2212yes—
49#1445C2212yes—
50#1445C2214yes—
51#1441C2212yes—
52#1441C2212yes—
53#1439C2212yes—
54#1439C2214yes—
55#1439C2214yes—
56#1439C2214yes—
57#1439C2212yes—
58#1439C2214yes—
59#1437C2214yes—
60#1437C2214yes—
61#1437C2214yes—
62#1437C2214yes—
63#1437C2214yes—
64#1436C2214yes—
65C4⋊#95C2214yes—
66C4⋊#95C2212yes—
67C4⋊#95C2212yes—
68C4⋊#95C2211yescharacteristic
69C4⋊#95C2212yes—
70C4⋊#95C2214yes—
71C4⋊#95C2214yes—
72C4⋊#95C2211yescharacteristic
73#1434C2214yes—
74#1434C2214yes—
75C22⋊#95C2214yes—
76C22⋊#95C2214yes—
77C4×#95C2214yes—
78C4×#95C2212yes—
79C4×#95C2212yes—
80C23×Q8C2312yes—
81C23×Q8C2311yescharacteristic
82C23×D4C2311yescharacteristic
83C24×C4C2312yes—
84#556—2128no—
85C2×C4⋊Q8C2×C418yes—
86C2×C4⋊Q8D418yes—
87C2×C4⋊Q8D418yes—
88C2×#126—216no—
89C2×#126—216no—
90C2×#126C2×C4132yes—
91C2×#124—28no—
92C2×#124—28no—
93C2×#124C2×C4116yes—
94C2×C22⋊Q8—28no—
95C2×C22⋊Q8—28no—
96C2×C22⋊Q8C2×C4116yes—
97C2×C4×Q8—28no—
98C2×C4×Q8—28no—
99C2×C4×Q8—28no—
100C2×C4×Q8—28no—
101C2×C4×Q8—28no—
102C2×C4×Q8—24no—
103C2×C4×Q8—24no—
104C2×C4×Q8—28no—
105C2×C4×Q8—44no—
106C2×C4×Q8—44no—
107C2×C4×Q8C2×C4116yes—
108C2×C4×Q8C2×C418yes—
109C2×C4×D4—24no—
110C2×C4×D4—24no—
111C2×C4×D4C2×C418yes—
112C2×#117—28no—
113C2×#117—28no—
114C2×#117C2×C4116yes—
115C22×C4⋊C4C2312yes—
116C22×C4⋊C4C2311yescharacteristic
117C22×C4⋊C4C2311yescharacteristic
118C22×C4⋊C4C2314yes—
119C22×C4⋊C4C2314yes—
120C22×C4⋊C4C2311yescharacteristic
121C22×C4⋊C4C2311yescharacteristic
122C22×C4⋊C4C2314yes—
123C22×C4⋊C4C2312yes—
124C22×C4⋊C4C2311yescharacteristic
125C22×C4⋊C4C2311yescharacteristic
126C22×C4⋊C4C2312yes—
127C22×C4⋊C4C2312yes—
128C22×C22⋊C4C2314yes—
129C22×C22⋊C4C2312yes—
130C22×C22⋊C4C2314yes—
131C22×C22⋊C4C2314yes—
132C22×C22⋊C4C2314yes—
133C22×C22⋊C4C2314yes—
134C22×C22⋊C4C2314yes—
135C22×C42C2312yes—
136C22×C42C2311yescharacteristic
137C22×C42C2314yes—
138C22×C42C2314yes—
139C22×C42C2314yes—
140C22×C42C2312yes—
141C22×C42C2311yescharacteristic
142C22×C42C2312yes—
143C22×C42C2311yescharacteristic
144#391—28no—
145#391—28no—
146#390—24no—
147#390—24no—
148#390—44no—
149#389—28no—
150#387—216no—
151#387—216no—
152#387—216no—
153#382—24no—
154C4×C4⋊C4—28no—
155C4×C4⋊C4—24no—
156C4×C4⋊C4—24no—
157#376—24no—
158C2×#95C2312yes—
159C2×#95C2312yes—
160C2×#95C2312yes—
161C2×#95C2312yes—
162C2×#95C2312yes—
163C2×#95C2312yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2314yes—
168C2×#95C2312yes—
169C2×#95C2312yes—
170C2×#95C2314yes—
171C2×#95C2314yes—
172C2×#95C2312yes—
173C2×#95C2314yes—
174C2×#95C2312yes—
175C2×#95C2312yes—
176C2×#95C2312yes—
177C2×#95C2312yes—
178C2×#95C2312yes—
179C2×#95C2314yes—
180C2×#95C2314yes—
181C2×#95C2314yes—
182C2×#95C2314yes—
183C2×#95C2314yes—
184C25C2412yes—
185C22×Q8—22no—
186C22×Q8—22no—
187C22×Q8—24no—
188C22×Q8—24no—
189C22×Q8—24no—
190C22×Q8—24no—
191C22×Q8—42no—
192C22×Q8—42no—
193C22×Q8C22×C418yes—
194C22×Q8C22×C414yes—
195C22×Q8C2×D418yes—
196C22×Q8C2×D418yes—
197C22×D4—22no—
198C22×D4—22no—
199C22×D4—22no—
200C22×D4—22no—
201C22×D4—22no—
202C22×D4—22no—
203C22×D4C22×C414yes—
204C23×C4—24no—
205C23×C4—24no—
206C23×C4—44no—
207C23×C4—44no—
208C23×C4C22×C418yes—
209C23×C4C2412yes—
210C23×C4C2414yes—
211C23×C4C2412yes—
212C23×C4C2412yes—
213C23×C4C2412yes—
214C23×C4C2412yes—
215C23×C4C2412yes—
216C23×C4C2412yes—
217C23×C4C2412yes—
218C23×C4C2412yes—
219C23×C4C2411yescharacteristic
220C23×C4C2414yes—
221C23×C4C2411yescharacteristic
222C23×C4C2411yescharacteristic
223C4⋊Q8C22⋊C4164yescomplemented
224#126—2128no—
225#124—264no—
226C22⋊Q8—264no—
227C4×Q8—264no—
228C4×Q8—232no—
229C4×Q8—432no—
230C4×Q8—432no—
231C4×D4—232no—
232#117—264no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—28no—
239C2×C4⋊C4—28no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—28no—
247C2×C4⋊C4—28no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—28no—
251C2×C4⋊C4—28no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—42no—
257C2×C4⋊C4—44no—
258C2×C4⋊C4—44no—
259C2×C4⋊C4C22×C414yes—
260C2×C4⋊C4C22×C414yes—
261C2×C4⋊C4C22×C4116yes—
262C2×C4⋊C4C22×C418yes—
263C2×C4⋊C4C22×C418yes—
264C2×C4⋊C4C22×C418yes—
265C2×C4⋊C4C2×D418yes—
266C2×C4⋊C4C2×D418yes—
267C2×C4⋊C4C2×D418yes—
268C2×C4⋊C4C2×D418yes—
269C2×C4⋊C4#4114yes—
270C2×C22⋊C4—28no—
271C2×C22⋊C4—28no—
272C2×C22⋊C4—24no—
273C2×C22⋊C4—24no—
274C2×C22⋊C4—28no—
275C2×C22⋊C4—28no—
276C2×C22⋊C4—28no—
277C2×C22⋊C4—28no—
278C2×C22⋊C4—28no—
279C2×C22⋊C4—44no—
280C2×C22⋊C4—44no—
281C2×C22⋊C4—44no—
282C2×C22⋊C4—44no—
283C2×C22⋊C4—44no—
284C2×C22⋊C4—44no—
285C2×C22⋊C4—44no—
286C2×C22⋊C4—44no—
287C2×C22⋊C4—44no—
288C2×C22⋊C4C22×C4116yes—
289C2×C22⋊C4C22×C418yes—
290C2×C22⋊C4C22×C4116yes—
291C2×C42—28no—
292C2×C42—28no—
293C2×C42—24no—
294C2×C42—24no—
295C2×C42—22no—
296C2×C42—22no—
297C2×C42—28no—
298C2×C42—24no—
299C2×C42—28no—
300C2×C42—22no—
301C2×C42—24no—
302C2×C42—28no—
303C2×C42—28no—
304C2×C42—24no—
305C2×C42—22no—
306C2×C42—24no—
307C2×C42—22no—
308C2×C42—44no—
309C2×C42—44no—
310C2×C42—44no—
311C2×C42C22×C4116yes—
312C2×C42C22×C418yes—
313C2×C42C22×C414yes—
314C2×C42C2×D414yes—
315C2×C42C2×D414yes—
316C2×C42#4114yes—
317#95—24no—
318#95—24no—
319#95—24no—
320#95—24no—
321#95—28no—
322#95—24no—
323#95—24no—
324#95—28no—
325#95—28no—
326#95—28no—
327#95—24no—
328#95—24no—
329#95—44no—
330#95—44no—
331#95—42no—
332#95—42no—
333#95—44no—
334#95—42no—
335#95—44no—
336#95—42no—
337#95—44no—
338#95—44no—
339#95—44no—
340#95—42no—
341#95—42no—
342#95—44no—
343C24—22no—
344C24—22no—
345C24—22no—
346C24—22no—
347C24—22no—
348C24—24no—
349C24—22no—
350C24—24no—
351C24—24no—
352C24—24no—
353C24C23×C414yes—
354C24C2511yescharacteristic
355C2×Q8—28no—
356C2×Q8—216no—
357C2×Q8—216no—
358C2×Q8—216no—
359C2×Q8—216no—
360C2×Q8—216no—
361C2×Q8—216no—
362C2×Q8—48no—
363C2×Q8—44no—
364C2×Q8—48no—
365C2×Q8—44no—
366C2×Q8C2×C22⋊C4132yes—
367C2×D4—28no—
368C2×D4—28no—
369C2×D4—28no—
370C2×D4—44no—
371C2×D4—44no—
372C2×D4—44no—
373C2×D4—44no—
374C22×C4—24no—
375C22×C4—24no—
376C22×C4—24no—
377C22×C4—22no—
378C22×C4—24no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
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—24no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—24no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—216no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—24no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—24no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—24no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—24no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—24no—
432C22×C4—24no—
433C22×C4—24no—
434C22×C4—24no—
435C22×C4—22no—
436C22×C4—24no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—416no—
442C22×C4—48no—
443C22×C4—416no—
444C22×C4—48no—
445C22×C4C23×C414yes—
446C22×C4C23×C414yes—
447C22×C4C23×C414yes—
448C22×C4C23×C412yes—
449C22×C4C23×C418yes—
450C22×C4C23×C412yes—
451C22×C4C23×C412yes—
452C22×C4C22×D412yes—
453C22×C4C22×D412yes—
454C22×C4C22×D412yes—
455C22×C4C22×D418yes—
456C22×C4C22×D418yes—
457C22×C4C22×D412yes—
458C22×C4C22×D412yes—
459C22×C4C22×D412yes—
460C22×C4C2×#4114yes—
461C22×C4C2×#4114yes—
462C22×C4C2×#4112yes—
463C22×C4C2×#4112yes—
464C22×C4H(5,𝔽2)12yes—
465C22×C42−1+414yes—
466C22×C42−1+412yes—
467C4⋊C4—28no—
468C4⋊C4—28no—
469C4⋊C4—28no—
470C4⋊C4—232no—
471C4⋊C4—216no—
472C4⋊C4—28no—
473C4⋊C4—416no—
474C4⋊C4—416no—
475C4⋊C4—44no—
476C4⋊C4—48no—
477C4⋊C4—44no—
478C4⋊C4C2×C22⋊C4132yes—
479C4⋊C4C2×C22⋊C4132yes—
480C22⋊C4—232no—
481C22⋊C4—216no—
482C22⋊C4—232no—
483C22⋊C4—88no—
484C22⋊C4—88no—
485C22⋊C4—88no—
486C22⋊C4—88no—
487C42—232no—
488C42—216no—
489C42—28no—
490C42—416no—
491C42—416no—
492C42—48no—
493C42—48no—
494C42—44no—
495C42C2×C22⋊C4116yes—
496C23—24no—
497C23—24no—
498C23—24no—
499C23—28no—
500C23—28no—
501C23—42no—
502C23—42no—
503C23—42no—
504C23—42no—
505C23—42no—
506C23—42no—
507C23—44no—
508C23—42no—
509C23—42no—
510C23—44no—
511C23—44no—
512C23—44no—
513C23—44no—
514C23—44no—
515C23—44no—
516C23—44no—
517C23—44no—
518C23—44no—
519C23C24×C411yescharacteristic
520C23C23×D411yescharacteristic
521C23C23×D411yescharacteristic
522C23C22×#4111yescharacteristic
523C23C22×#4111yescharacteristic
524C23C2×H(5,𝔽2)11yescharacteristic
525C23C2×2−1+411yescharacteristic
526C23C2×2−1+412yes—
527C23#58512yes—
528C23#58512yes—
529C23#58512yes—
530Q8—264no—
531Q8—416no—
532D4—416no—
533C2×C4—28no—
534C2×C4—28no—
535C2×C4—24no—
536C2×C4—24no—
537C2×C4—28no—
538C2×C4—28no—
539C2×C4—28no—
540C2×C4—28no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—24no—
545C2×C4—22no—
546C2×C4—24no—
547C2×C4—24no—
548C2×C4—24no—
549C2×C4—24no—
550C2×C4—28no—
551C2×C4—24no—
552C2×C4—24no—
553C2×C4—24no—
554C2×C4—24no—
555C2×C4—24no—
556C2×C4—24no—
557C2×C4—24no—
558C2×C4—24no—
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—28no—
562C2×C4—24no—
563C2×C4—432no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—44no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—44no—
586C2×C4—44no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—44no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—44no—
597C2×C4—42no—
598C2×C4—44no—
599C2×C4—44no—
600C2×C4—44no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—44no—
608C2×C4—44no—
609C2×C4—42no—
610C2×C4—44no—
611C2×C4—42no—
612C2×C4C22×C22⋊C4116yes—
613C2×C4C22×C22⋊C414yes—
614C2×C4C22×C22⋊C414yes—
615C2×C4C22×C22⋊C414yes—
616C2×C4C4×#4114yes—
617C2×C4#52014yes—
618C2×C4#54614yes—
619C2×C4#54614yes—
620C2×C4#54714yes—
621C2×C4#54714yes—
622C22—48no—
623C22—44no—
624C22—44no—
625C22—48no—
626C22—44no—
627C22—82no—
628C22—84no—
629C22—84no—
630C22—82no—
631C22—82no—
632C22—82no—
633C22C23×C22⋊C411yescharacteristic
634C22C2×C4×#4111yescharacteristic
635C22C2×#52011yescharacteristic
636C22#342812yes—
637C22C4×2−1+412yes—
638C22C2×#54611yescharacteristic
639C22C2×#54611yescharacteristic
640C22C2×#54711yescharacteristic
641C22C2×#54711yescharacteristic
642C22#348611yescharacteristic
643C22#348711yescharacteristic
644C22#348712yes—
645C22#348711yescharacteristic
646C22#348711yescharacteristic
647C22#349712yes—
648C22#350712yes—
649C22#351012yes—
650C22#351612yes—
651C22#352112yes—
652C22#355512yes—
653C22#355712yes—
654C22#356112yes—
655C22#356212yes—
656C4—216no—
657C4—24no—
658C4—24no—
659C4—44no—
660C4—44no—
661C4—44no—
662C4—44no—
663C4—44no—
664C4—44no—
665C4—84no—
666C4—82no—
667C4—82no—
668C4—82no—
669C4#233918yes—
670C2—84no—
671C2C2×#233911yescharacteristic
672C2#3417411yescharacteristic
673C2#3425711yescharacteristic
674C2#3432612yes—
675C2#3449912yes—
676C2#4524012yes—
677C2#4524212yes—
678C2#4526611yescharacteristic
679C2#4526811yescharacteristic
680C2#4526911yescharacteristic
681C2#4539411yescharacteristic
subgroupG/H
center, Frattini, socle, cosocleC24C25
commutatorC23C24×C4
prime radicalC23×D4C23
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2365C2semidirect
3. C2×#2327C2semidirect
4. #14493C2semidirect
5. #14448C2semidirect
6. #14101C2semidirect
9. #13944C2semidirect
11. #13931C2semidirect
12. #13925C2semidirect
13. #13850C2semidirect
14. Q8×#95C2semidirect
18. C2×#556C4semidirect
223. C4⋊Q8C22⋊C4semidirect

Maximal subgroups

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

subgroupindex
C22×#5562
C2×#23652
C2×#23272
#144932
#144482
#141012
#140262
#139442
#139412
#139312
#139252
#138502
Q8×#952
#137852