Finite Group Databasebeta

← #627071

Subgroups — #627071

Order 512.

3334 conjugacy classes of subgroups.

The subgroup lattice has 768 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2627C211yesmaximal, characteristic, complemented
2C2×#2626C211yesmaximal, characteristic
3C2×#2316C211yesmaximal, characteristic, complemented
4C22×#388C211yesmaximal, characteristic
5#15677C214yesmaximal, complemented
6#15675C214yesmaximal, complemented
7#15633C214yesmaximal, complemented
8#15620C214yesmaximal, complemented
9#14920C212yesmaximal, complemented
10#14726C212yesmaximal
11#14702C211yesmaximal, characteristic, complemented
12#14697C211yesmaximal, characteristic, complemented
13#14338C211yesmaximal, characteristic
14#13962C212yesmaximal, complemented
15#13868C212yesmaximal
16C22×C22⋊Q8C2211yescharacteristic
17C22×#117C2212yes—
18C23×C4⋊C4C2211yescharacteristic
19C23×C22⋊C4C2211yescharacteristic
20C23×C22⋊C4C2211yescharacteristic
21C23×C22⋊C4C2211yescharacteristic
22C23×C42C2211yescharacteristic
23#2627—28no—
24#2626—28no—
25C2×#400C2212yescomplemented
26C2×#399C2212yes—
27C2×#398C2212yescomplemented
28C2×#398C2212yescomplemented
29C2×#396C2214yes—
30C2×#396C2214yes—
31C2×#395C2212yescomplemented
32C2×#393C2212yes—
33#2316—28no—
34C2×#391C2212yescomplemented
35C2×#389C2214yescomplemented
36C2×#389C2214yescomplemented
37C2×#389C2212yescomplemented
38C2×#388—28no—
39C2×#388C2214yes—
40C2×#388C2214yes—
41C2×#385C2212yes—
42C2×#385C2212yes—
43C2×#385C2212yes—
44C2×#381C2212yes—
45C2×#380C2211yescharacteristic
46C2×#380C2211yescharacteristic
47C2×#380C2212yes—
48C2×#380C2211yescharacteristic
49C2×#379C2212yes—
50C2×#379C2212yes—
51C22×#95C2211yescharacteristic
52C22×#95C2211yescharacteristic
53#1445C2214yescomplemented
54#1442C2214yescomplemented
55#1441C2214yescomplemented
56#1441C2214yescomplemented
57#1441C2214yescomplemented
58#1441C2214yescomplemented
59#1439C2212yes—
60#1439C2214yes—
61#1439C2212yes—
62#1439C2214yes—
63#1439C2214yes—
64#1439C2214yes—
65#1439C2214yes—
66#1439C2214yes—
67#1437C2214yes—
68#1437C2214yes—
69#1436C2214yescomplemented
70#1436C2214yescomplemented
71C4⋊#95C2214yescomplemented
72C4⋊#95C2212yescomplemented
73C4⋊#95C2212yescomplemented
74#1434C2214yescomplemented
75C22⋊#95C2212yes—
76C22⋊#95C2212yes—
77C22⋊#95C2212yes—
78C22⋊#95C2212yes—
79C26C2311yescharacteristic
80C23×Q8C2311yescharacteristic
81C24×C4C2311yescharacteristic
82C24×C4C2311yescharacteristic
83C24×C4C2311yescharacteristic
84C24×C4C2311yescharacteristic
85C24×C4C2311yescharacteristic
86C24×C4C2311yescharacteristic
87C2×C22⋊Q8—24no—
88C2×C22⋊Q8—24no—
89C2×C22⋊Q8—24no—
90C2×#117—28no—
91C2×#117—28no—
92C2×#117—44no—
93C22×C4⋊C4—24no—
94C22×C4⋊C4—24no—
95C22×C4⋊C4D418yes—
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2311yescharacteristic
98C22×C4⋊C4C2314yes—
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2314yes—
104C22×C4⋊C4C2314yes—
105C22×C4⋊C4C2314yes—
106C22×C4⋊C4C2311yescharacteristic
107C22×C4⋊C4C2311yescharacteristic
108C22×C4⋊C4C2312yes—
109C22×C4⋊C4C2312yes—
110C22×C22⋊C4—24no—
111C22×C22⋊C4—24no—
112C22×C22⋊C4—24no—
113C22×C22⋊C4—24no—
114C22×C22⋊C4—24no—
115C22×C22⋊C4—24no—
116C22×C22⋊C4—44no—
117C22×C22⋊C4Q818yescomplemented
118C22×C22⋊C4C2312yes—
119C22×C22⋊C4C2312yes—
120C22×C22⋊C4C2312yes—
121C22×C22⋊C4C2312yes—
122C22×C22⋊C4C2312yes—
123C22×C22⋊C4C2314yes—
124C22×C22⋊C4C2312yes—
125C22×C22⋊C4C2314yes—
126C22×C22⋊C4C2314yes—
127C22×C22⋊C4C2314yes—
128C22×C22⋊C4C2314yes—
129C22×C22⋊C4C2314yes—
130C22×C42—24no—
131C22×C42—44no—
132C22×C42C2312yes—
133C22×C42C2312yes—
134C22×C42C2314yes—
135#400—28no—
136#399—28no—
137#398—28no—
138#398—28no—
139#396—216no—
140#396—48no—
141#395—28no—
142#393—28no—
143#391—44no—
144#389—216no—
145#389—216no—
146#389—44no—
147#388—216no—
148#388—216no—
149#385—28no—
150#385—28no—
151#385—44no—
152#381—28no—
153#380—24no—
154#380—24no—
155#380—28no—
156#380D418yes—
157#379—28no—
158#379—44no—
159C2×#95—24no—
160C2×#95—24no—
161C2×#95C2314yes—
162C2×#95C2312yes—
163C2×#95C2312yes—
164C2×#95C2312yes—
165C2×#95C2314yes—
166C2×#95C2312yes—
167C2×#95C2312yes—
168C2×#95C2312yes—
169C2×#95C2312yes—
170C2×#95C2314yes—
171C2×#95C2312yes—
172C2×#95C2314yes—
173C2×#95C2314yes—
174C2×#95C2314yes—
175C2×#95C2314yes—
176C2×#95C2314yes—
177C2×#95C2314yes—
178C2×#95C2314yes—
179C2×#95C2314yes—
180C2×#95C2314yes—
181C2×#95C2314yes—
182C2×#95C2312yes—
183C25—22no—
184C25—22no—
185C25—22no—
186C25—22no—
187C25—22no—
188C25—22no—
189C25—42no—
190C25—42no—
191C25—42no—
192C25—42no—
193C25C2×Q814yes—
194C25C2412yes—
195C25C2411yescharacteristic
196C22×Q8—22no—
197C22×Q8—22no—
198C22×Q8—22no—
199C22×Q8—42no—
200C22×Q8—42no—
201C23×C4—22no—
202C23×C4—22no—
203C23×C4—22no—
204C23×C4—22no—
205C23×C4—22no—
206C23×C4—22no—
207C23×C4—22no—
208C23×C4—22no—
209C23×C4—22no—
210C23×C4—24no—
211C23×C4—22no—
212C23×C4—22no—
213C23×C4—22no—
214C23×C4—22no—
215C23×C4—22no—
216C23×C4—42no—
217C23×C4—42no—
218C23×C4—42no—
219C23×C4—42no—
220C23×C4—42no—
221C23×C4—42no—
222C23×C4—42no—
223C23×C4—42no—
224C23×C4—42no—
225C23×C4C2×D414yes—
226C23×C4C2×D414yes—
227C23×C4C2×D414yes—
228C23×C4C2×D414yes—
229C23×C4C2×Q814yes—
230C23×C4C2×Q814yes—
231C23×C4#4114yes—
232C23×C4#4114yes—
233C23×C4C2412yes—
234C23×C4C2412yes—
235C23×C4C2412yes—
236C23×C4C2414yes—
237C23×C4C2412yes—
238C23×C4C2412yes—
239C23×C4C2414yes—
240C23×C4C2412yes—
241C23×C4C2412yes—
242C23×C4C2412yes—
243C23×C4C2411yescharacteristic
244C23×C4C2411yescharacteristic
245C23×C4C2412yes—
246C22⋊Q8—416no—
247#117—432no—
248C2×C4⋊C4—232no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—28no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—28no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—28no—
262C2×C4⋊C4—24no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—24no—
266C2×C4⋊C4—24no—
267C2×C4⋊C4—28no—
268C2×C4⋊C4—28no—
269C2×C4⋊C4—42no—
270C2×C4⋊C4—44no—
271C2×C4⋊C4—42no—
272C2×C4⋊C4—42no—
273C2×C4⋊C4—42no—
274C2×C4⋊C4—44no—
275C2×C4⋊C4—44no—
276C2×C4⋊C4—44no—
277C2×C4⋊C4—44no—
278C2×C4⋊C4—44no—
279C2×C4⋊C4—44no—
280C2×C4⋊C4—42no—
281C2×C4⋊C4—42no—
282C2×C4⋊C4—44no—
283C2×C4⋊C4—44no—
284C2×C4⋊C4C2×D418yes—
285C2×C4⋊C4C2×D418yes—
286C2×C4⋊C4C2×D418yes—
287C2×C22⋊C4—24no—
288C2×C22⋊C4—24no—
289C2×C22⋊C4—232no—
290C2×C22⋊C4—28no—
291C2×C22⋊C4—28no—
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—24no—
300C2×C22⋊C4—24no—
301C2×C22⋊C4—28no—
302C2×C22⋊C4—24no—
303C2×C22⋊C4—24no—
304C2×C22⋊C4—24no—
305C2×C22⋊C4—24no—
306C2×C22⋊C4—24no—
307C2×C22⋊C4—24no—
308C2×C22⋊C4—28no—
309C2×C22⋊C4—28no—
310C2×C22⋊C4—28no—
311C2×C22⋊C4—44no—
312C2×C22⋊C4—416no—
313C2×C22⋊C4—416no—
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×C22⋊C4—44no—
324C2×C22⋊C4C2×D418yes—
325C2×C22⋊C4C2×Q8116yes—
326C2×C42—28no—
327C2×C42—24no—
328C2×C42—24no—
329C2×C42—28no—
330C2×C42—24no—
331C2×C42—24no—
332C2×C42—44no—
333C2×C42—42no—
334C2×C42—42no—
335C2×C42—416no—
336#95—28no—
337#95—28no—
338#95—28no—
339#95—24no—
340#95—24no—
341#95—24no—
342#95—24no—
343#95—24no—
344#95—24no—
345#95—28no—
346#95—24no—
347#95—24no—
348#95—28no—
349#95—44no—
350#95—44no—
351#95—44no—
352#95—44no—
353#95—44no—
354#95—44no—
355#95—44no—
356#95—44no—
357#95C2×D418yes—
358C24—22no—
359C24—22no—
360C24—22no—
361C24—22no—
362C24—28no—
363C24—22no—
364C24—22no—
365C24—22no—
366C24—22no—
367C24—24no—
368C24—22no—
369C24—22no—
370C24—24no—
371C24—24no—
372C24—24no—
373C24—28no—
374C24—28no—
375C24—48no—
376C24—48no—
377C24—44no—
378C24—48no—
379C24—44no—
380C24—44no—
381C24—44no—
382C24—48no—
383C24—44no—
384C24—44no—
385C24—44no—
386C24—44no—
387C24—84no—
388C24—84no—
389C24—84no—
390C24—84no—
391C24—84no—
392C24—84no—
393C24—84no—
394C24—84no—
395C24C22×D412yes—
396C24C22×D412yes—
397C24C22×Q814yes—
398C24C22×Q812yes—
399C24C2×#4112yes—
400C24C2×#4112yes—
401C24H(5,𝔽2)12yes—
402C242−1+412yes—
403C24C2511yescharacteristic
404C2×Q8—48no—
405C2×Q8—44no—
406C2×Q8—82no—
407C2×Q8—82no—
408C2×Q8—84no—
409C22×C4—28no—
410C22×C4—28no—
411C22×C4—28no—
412C22×C4—28no—
413C22×C4—28no—
414C22×C4—28no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—28no—
424C22×C4—28no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—24no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—24no—
435C22×C4—24no—
436C22×C4—22no—
437C22×C4—24no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—24no—
443C22×C4—24no—
444C22×C4—24no—
445C22×C4—24no—
446C22×C4—22no—
447C22×C4—28no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—24no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—24no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—24no—
462C22×C4—24no—
463C22×C4—22no—
464C22×C4—24no—
465C22×C4—24no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—24no—
470C22×C4—24no—
471C22×C4—24no—
472C22×C4—24no—
473C22×C4—24no—
474C22×C4—24no—
475C22×C4—24no—
476C22×C4—24no—
477C22×C4—24no—
478C22×C4—24no—
479C22×C4—24no—
480C22×C4—44no—
481C22×C4—44no—
482C22×C4—48no—
483C22×C4—48no—
484C22×C4—48no—
485C22×C4—44no—
486C22×C4—44no—
487C22×C4—48no—
488C22×C4—48no—
489C22×C4—44no—
490C22×C4—48no—
491C22×C4—44no—
492C22×C4—44no—
493C22×C4—44no—
494C22×C4—44no—
495C22×C4—44no—
496C22×C4—44no—
497C22×C4—84no—
498C22×C4—84no—
499C22×C4—84no—
500C22×C4—84no—
501C22×C4#121116yes—
502C22×C4C22⋊Q8116yes—
503C22×C4C22×D414yes—
504C22×C4C22×D418yes—
505C22×C4C22×D412yes—
506C22×C4C22×D412yes—
507C22×C4C22×D412yes—
508C22×C4C22×D412yes—
509C22×C4C22×D414yes—
510C22×C4C22×Q814yes—
511C22×C4C22×Q814yes—
512C22×C4C2×#4112yes—
513C22×C4C2×#4112yes—
514C22×C4C2×#4114yes—
515C22×C4H(5,𝔽2)14yes—
516C4⋊C4—216no—
517C4⋊C4—216no—
518C4⋊C4—48no—
519C4⋊C4—416no—
520C4⋊C4—44no—
521C4⋊C4—48no—
522C4⋊C4—82no—
523C4⋊C4—82no—
524C4⋊C4—84no—
525C4⋊C4—84no—
526C4⋊C4—84no—
527C4⋊C4—88no—
528C4⋊C4—88no—
529C4⋊C4—88no—
530C22⋊C4—216no—
531C22⋊C4—232no—
532C22⋊C4—416no—
533C22⋊C4—48no—
534C22⋊C4—48no—
535C22⋊C4—48no—
536C22⋊C4—48no—
537C22⋊C4—48no—
538C22⋊C4—88no—
539C22⋊C4—88no—
540C22⋊C4—88no—
541C22⋊C4—88no—
542C42—48no—
543C42—48no—
544C42—416no—
545C23—24no—
546C23—24no—
547C23—22no—
548C23—22no—
549C23—22no—
550C23—22no—
551C23—24no—
552C23—24no—
553C23—24no—
554C23—24no—
555C23—24no—
556C23—24no—
557C23—24no—
558C23—24no—
559C23—24no—
560C23—24no—
561C23—24no—
562C23—28no—
563C23—24no—
564C23—24no—
565C23—28no—
566C23—416no—
567C23—416no—
568C23—42no—
569C23—416no—
570C23—42no—
571C23—44no—
572C23—42no—
573C23—42no—
574C23—42no—
575C23—42no—
576C23—42no—
577C23—42no—
578C23—44no—
579C23—44no—
580C23—44no—
581C23—44no—
582C23—44no—
583C23—44no—
584C23—44no—
585C23—44no—
586C23—44no—
587C23—816no—
588C23—816no—
589C23—88no—
590C23—88no—
591C23—88no—
592C23—88no—
593C23—816no—
594C23—816no—
595C23C2×#12114yes—
596C23C2×C22⋊Q814yes—
597C23#53714yes—
598C23#54614yes—
599C23D4×Q814yes—
600C23#55214yes—
601C23#55414yes—
602C23C23×D411yescharacteristic
603C23C23×D411yescharacteristic
604C23C23×Q811yescharacteristic
605C23C22×#4111yescharacteristic
606C23C22×#4111yescharacteristic
607C23C2×H(5,𝔽2)11yescharacteristic
608C23C2×H(5,𝔽2)12yes—
609C23C2×2−1+411yescharacteristic
610C23#58512yes—
611C23#58512yes—
612C23#58512yes—
613Q8—88no—
614C2×C4—24no—
615C2×C4—232no—
616C2×C4—24no—
617C2×C4—24no—
618C2×C4—24no—
619C2×C4—24no—
620C2×C4—24no—
621C2×C4—22no—
622C2×C4—22no—
623C2×C4—22no—
624C2×C4—22no—
625C2×C4—24no—
626C2×C4—24no—
627C2×C4—24no—
628C2×C4—24no—
629C2×C4—24no—
630C2×C4—24no—
631C2×C4—28no—
632C2×C4—24no—
633C2×C4—24no—
634C2×C4—24no—
635C2×C4—28no—
636C2×C4—28no—
637C2×C4—24no—
638C2×C4—24no—
639C2×C4—24no—
640C2×C4—24no—
641C2×C4—416no—
642C2×C4—416no—
643C2×C4—416no—
644C2×C4—42no—
645C2×C4—44no—
646C2×C4—42no—
647C2×C4—42no—
648C2×C4—42no—
649C2×C4—42no—
650C2×C4—42no—
651C2×C4—44no—
652C2×C4—44no—
653C2×C4—44no—
654C2×C4—44no—
655C2×C4—42no—
656C2×C4—42no—
657C2×C4—44no—
658C2×C4—42no—
659C2×C4—42no—
660C2×C4—42no—
661C2×C4—42no—
662C2×C4—42no—
663C2×C4—44no—
664C2×C4—44no—
665C2×C4—42no—
666C2×C4—42no—
667C2×C4—42no—
668C2×C4—42no—
669C2×C4—42no—
670C2×C4—42no—
671C2×C4—42no—
672C2×C4—42no—
673C2×C4—42no—
674C2×C4—42no—
675C2×C4—44no—
676C2×C4—44no—
677C2×C4—44no—
678C2×C4—44no—
679C2×C4—44no—
680C2×C4—44no—
681C2×C4—44no—
682C2×C4—44no—
683C2×C4—44no—
684C2×C4—44no—
685C2×C4—44no—
686C2×C4—44no—
687C2×C4—44no—
688C2×C4—44no—
689C2×C4—44no—
690C2×C4—88no—
691C2×C4—88no—
692C2×C4C2×#12114yes—
693C2×C4C2×#12114yes—
694C2×C4C2×C22⋊Q818yes—
695C22—24no—
696C22—28no—
697C22—24no—
698C22—24no—
699C22—24no—
700C22—24no—
701C22—24no—
702C22—28no—
703C22—28no—
704C22—28no—
705C22—48no—
706C22—44no—
707C22—44no—
708C22—44no—
709C22—48no—
710C22—832no—
711C22—82no—
712C22—82no—
713C22—82no—
714C22—82no—
715C22—84no—
716C22—84no—
717C22#241718yes—
718C22C22×#12111yescharacteristic
719C22C22×C22⋊Q811yescharacteristic
720C22C2×#53711yescharacteristic
721C22#345812yes—
722C22#346012yes—
723C22C2×#54611yescharacteristic
724C22C2×D4×Q811yescharacteristic
725C22C2×#55211yescharacteristic
726C22C2×#55411yescharacteristic
727C22#348411yescharacteristic
728C22#348511yescharacteristic
729C22#348711yescharacteristic
730C22#348711yescharacteristic
731C22#349312yes—
732C22#349612yes—
733C22#350012yes—
734C22#350112yes—
735C22#350512yes—
736C22#351012yes—
737C22#353412yes—
738C22#353512yes—
739C22#356012yes—
740C22#356412yes—
741C4—24no—
742C4—24no—
743C4—28no—
744C4—44no—
745C4—44no—
746C4—44no—
747C4—48no—
748C4—82no—
749C4—82no—
750C4—82no—
751C4—82no—
752C4—82no—
753C4—84no—
754C2—28no—
755C2—84no—
756C2C2×#241711yescharacteristic
757C2#3515111yescharacteristic
758C2#3517411yescharacteristic
759C2#3535811yescharacteristic
760C2#3567711yescharacteristic
761C2#3672211yescharacteristic
762C2#3697111yescharacteristic
763C2#3936412yes—
764C2#3941512yes—
765C2#3949212yes—
766C2#3954312yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC26C23
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×#2627C2semidirect
3. C2×#2316C2semidirect
5. #15677C2semidirect
6. #15675C2semidirect
7. #15633C2semidirect
8. #15620C2semidirect
9. #14920C2semidirect
11. #14702C2semidirect
12. #14697C2semidirect
14. #13962C2semidirect
25. C2×#400C22semidirect
27. C2×#398C22semidirect
28. C2×#398C22semidirect
31. C2×#395C22semidirect
34. C2×#391C22semidirect
35. C2×#389C22semidirect
36. C2×#389C22semidirect
37. C2×#389C22semidirect
53. #1445C22semidirect
54. #1442C22semidirect
55. #1441C22semidirect
56. #1441C22semidirect
57. #1441C22semidirect
58. #1441C22semidirect
69. #1436C22semidirect
70. #1436C22semidirect
71. C4⋊#95C22semidirect
72. C4⋊#95C22semidirect
73. C4⋊#95C22semidirect
74. #1434C22semidirect
117. C22×C22⋊C4Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#26272
C2×#26262
C2×#23162
C22×#3882
#156772
#156752
#156332
#156202
#149202
#147262
#147022
#146972
#143382
#139622
#138682