Finite Group Databasebeta

← #4309181

Subgroups — #4309181

Order 512.

3054 conjugacy classes of subgroups.

The subgroup lattice has 781 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2628C211yesmaximal, characteristic, complemented
2C2×#2297C211yesmaximal, characteristic
3#16653C212yesmaximal, complemented
4#16630C212yesmaximal, complemented
5#16599C212yesmaximal, complemented
6#16556C212yesmaximal, complemented
7#16320C214yesmaximal, complemented
8#16319C214yesmaximal, complemented
9#15223C212yesmaximal, complemented
10#15220C212yesmaximal, complemented
11#15131C212yesmaximal, complemented
12#15125C212yesmaximal, complemented
13#14697C212yesmaximal, complemented
14#14340C211yesmaximal, characteristic, complemented
15#14337C212yesmaximal
16C23×#41C2211yescharacteristic
17C22×C4⋊Q8C2211yescharacteristic
18C22×#117C2211yescharacteristic
19C23×C42C2211yescharacteristic
20#2628—28no—
21C2×#400C2212yes—
22C2×#399C2212yes—
23C2×#396C2214yes—
24C2×#395C2212yes—
25C2×#393C2212yes—
26C2×#393C2214yes—
27#2297—28no—
28C2×#391C2211yescharacteristic
29C2×#391C2211yescharacteristic
30C2×#390C2211yescharacteristic
31C2×#390C2211yescharacteristic
32C2×#389C2212yes—
33C2×#389C2212yes—
34C2×#389C2212yes—
35C2×#386C2212yes—
36C2×#386C2212yes—
37C2×#385C2214yes—
38C2×#384C2211yescharacteristic
39C2×#382C2211yescharacteristic
40C2×#382C2211yescharacteristic
41C2×#380C2212yes—
42C2×#380C2212yes—
43C2×#380C2212yes—
44C2×#380C2212yes—
45C2×#380C2212yes—
46C2×#380C2212yes—
47C2×C4×C22⋊C4C2212yes—
48C2×C4×C22⋊C4C2212yes—
49#1446C2214yes—
50#1446C2214yes—
51#1445C2214yes—
52#1445C2214yes—
53#1443C2214yes—
54#1442C2214yes—
55#1441C2212yes—
56#1441C2212yes—
57#1441C2214yes—
58#1441C2214yes—
59#1441C2214yes—
60#1439C2212yes—
61#1439C2212yes—
62#1439C2214yes—
63#1439C2214yes—
64#1439C2214yes—
65#1439C2214yes—
66#1439C2214yes—
67#1439C2214yes—
68#1438C2214yes—
69#1438C2214yes—
70#1438C2214yes—
71#1436C2214yes—
72#1436C2214yes—
73C4⋊#95C2212yes—
74C4⋊#95C2212yes—
75C4⋊#95C2212yes—
76C22⋊#95C2214yes—
77C22×#41—24no—
78C22×#41—24no—
79C22×#41—42no—
80C22×#41—42no—
81C22×#41—42no—
82C22×#41—42no—
83C22×#41Q818yescomplemented
84C23×Q8C2311yescharacteristic
85C23×Q8C2312yes—
86C23×D4C2312yes—
87C23×D4C2311yescharacteristic
88C24×C4C2312yes—
89C24×C4C2311yescharacteristic
90C24×C4C2312yes—
91C2×C4⋊Q8—24no—
92C2×C4⋊Q8—42no—
93C2×C4⋊Q8—42no—
94C2×#117—42no—
95C2×#117—42no—
96C2×#117D418yes—
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2312yes—
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2314yes—
102C22×C4⋊C4C2314yes—
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2314yes—
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2311yescharacteristic
108C22×C4⋊C4C2311yescharacteristic
109C22×C22⋊C4C2312yes—
110C22×C22⋊C4C2312yes—
111C22×C22⋊C4C2312yes—
112C22×C22⋊C4C2312yes—
113C22×C22⋊C4C2314yes—
114C22×C22⋊C4C2314yes—
115C22×C22⋊C4C2314yes—
116C22×C22⋊C4C2314yes—
117C22×C22⋊C4C2314yes—
118C22×C22⋊C4C2314yes—
119C22×C22⋊C4C2311yescharacteristic
120C22×C22⋊C4C2311yescharacteristic
121C22×C42—24no—
122C22×C42—42no—
123C22×C42—42no—
124C22×C42C2312yes—
125C22×C42C2311yescharacteristic
126C22×C42C2311yescharacteristic
127C22×C42C2311yescharacteristic
128C22×C42C2311yescharacteristic
129#400—28no—
130#399—28no—
131#396—48no—
132#395—28no—
133#393—28no—
134#393—48no—
135#391—24no—
136#391—24no—
137#390—24no—
138#390—24no—
139#389—28no—
140#389—44no—
141#389—44no—
142#386—28no—
143#386—28no—
144#385—48no—
145#384—42no—
146#382—24no—
147#382—42no—
148#380—28no—
149#380—28no—
150#380—44no—
151#380—44no—
152#380—44no—
153#380—44no—
154C4×C22⋊C4—28no—
155C4×C22⋊C4—28no—
156C2×#95C2312yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2312yes—
162C2×#95C2314yes—
163C2×#95C2312yes—
164C2×#95C2312yes—
165C2×#95C2312yes—
166C2×#95C2312yes—
167C2×#95C2314yes—
168C2×#95C2314yes—
169C2×#95C2314yes—
170C2×#95C2314yes—
171C2×#95C2314yes—
172C2×#95C2314yes—
173C2×#95C2314yes—
174C2×#95C2314yes—
175C2×#95C2314yes—
176C2×#95C2314yes—
177C2×#95C2312yes—
178C2×#95C2312yes—
179C25C2412yes—
180C25C2411yescharacteristic
181C2×#41—232no—
182C2×#41—416no—
183C2×#41—416no—
184C2×#41—416no—
185C2×#41—416no—
186C2×#41—416no—
187C2×#41—416no—
188C22×Q8—22no—
189C22×Q8—24no—
190C22×Q8—22no—
191C22×Q8—41no—
192C22×Q8—41no—
193C22×Q8—41no—
194C22×Q8—41no—
195C22×Q8—42no—
196C22×Q8—42no—
197C22×Q8—44no—
198C22×Q8—44no—
199C22×Q8C2×Q814yes—
200C22×D4—24no—
201C22×D4—24no—
202C22×D4—22no—
203C22×D4—22no—
204C22×D4—41no—
205C22×D4—41no—
206C22×D4—41no—
207C22×D4—41no—
208C22×D4—42no—
209C22×D4—42no—
210C22×D4—42no—
211C22×D4—42no—
212C22×D4C2×Q818yes—
213C22×D4C2×Q814yes—
214C23×C4—22no—
215C23×C4—22no—
216C23×C4—22no—
217C23×C4—22no—
218C23×C4—24no—
219C23×C4—24no—
220C23×C4—24no—
221C23×C4—24no—
222C23×C4—24no—
223C23×C4—24no—
224C23×C4—24no—
225C23×C4—24no—
226C23×C4—24no—
227C23×C4—42no—
228C23×C4—42no—
229C23×C4—42no—
230C23×C4—42no—
231C23×C4C2×D414yes—
232C23×C4C2×Q818yes—
233C23×C4C2×Q814yes—
234C23×C4#4114yes—
235C23×C4C2414yes—
236C23×C4C2414yes—
237C23×C4C2414yes—
238C23×C4C2414yes—
239C23×C4C2411yescharacteristic
240C23×C4C2411yescharacteristic
241C23×C4C2411yescharacteristic
242C23×C4C2411yescharacteristic
243C23×C4C2412yes—
244C23×C4C2412yes—
245C23×C4C2412yes—
246C23×C4C2411yescharacteristic
247C23×C4C2411yescharacteristic
248C4⋊Q8—416no—
249#117—416no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—24no—
262C2×C4⋊C4—24no—
263C2×C4⋊C4—28no—
264C2×C4⋊C4—28no—
265C2×C4⋊C4—28no—
266C2×C4⋊C4—41no—
267C2×C4⋊C4—41no—
268C2×C4⋊C4—41no—
269C2×C4⋊C4—41no—
270C2×C4⋊C4—42no—
271C2×C4⋊C4—42no—
272C2×C4⋊C4—44no—
273C2×C4⋊C4—44no—
274C2×C4⋊C4—44no—
275C2×C4⋊C4—42no—
276C2×C4⋊C4—42no—
277C2×C4⋊C4—44no—
278C2×C4⋊C4—44no—
279C2×C4⋊C4—44no—
280C2×C4⋊C4—44no—
281C2×C4⋊C4—44no—
282C2×C4⋊C4C2×D414yes—
283C2×C4⋊C4C2×D414yes—
284C2×C22⋊C4—28no—
285C2×C22⋊C4—24no—
286C2×C22⋊C4—24no—
287C2×C22⋊C4—24no—
288C2×C22⋊C4—24no—
289C2×C22⋊C4—24no—
290C2×C22⋊C4—24no—
291C2×C22⋊C4—24no—
292C2×C22⋊C4—24no—
293C2×C22⋊C4—28no—
294C2×C22⋊C4—42no—
295C2×C22⋊C4—42no—
296C2×C22⋊C4—42no—
297C2×C22⋊C4—42no—
298C2×C22⋊C4—44no—
299C2×C22⋊C4—41no—
300C2×C22⋊C4—41no—
301C2×C22⋊C4—41no—
302C2×C22⋊C4—41no—
303C2×C22⋊C4—44no—
304C2×C22⋊C4—44no—
305C2×C22⋊C4—44no—
306C2×C22⋊C4—44no—
307C2×C22⋊C4—44no—
308C2×C22⋊C4—44no—
309C2×C22⋊C4—44no—
310C2×C22⋊C4—44no—
311C2×C22⋊C4—44no—
312C2×C22⋊C4—44no—
313C2×C22⋊C4—44no—
314C2×C22⋊C4—44no—
315C2×C22⋊C4—44no—
316C2×C22⋊C4—44no—
317C2×C22⋊C4—44no—
318C2×C22⋊C4C2×D414yes—
319C2×C22⋊C4C2×D414yes—
320C2×C42—22no—
321C2×C42—22no—
322C2×C42—22no—
323C2×C42—22no—
324C2×C42—24no—
325C2×C42—22no—
326C2×C42—22no—
327C2×C42—24no—
328C2×C42—24no—
329C2×C42—41no—
330C2×C42—41no—
331C2×C42—41no—
332C2×C42—41no—
333C2×C42—416no—
334C2×C42C2×D414yes—
335C2×C42C2×D414yes—
336#95—24no—
337#95—24no—
338#95—24no—
339#95—24no—
340#95—28no—
341#95—24no—
342#95—24no—
343#95—28no—
344#95—28no—
345#95—28no—
346#95—28no—
347#95—44no—
348#95—44no—
349#95—44no—
350#95—44no—
351#95—44no—
352#95—44no—
353#95—42no—
354#95—42no—
355#95—44no—
356#95—44no—
357#95—44no—
358#95—44no—
359C24—22no—
360C24—21no—
361C24—21no—
362C24—21no—
363C24—21no—
364C24—22no—
365C24—22no—
366C24—22no—
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—
381C24C22×D412yes—
382C24C22×Q814yes—
383C24C22×Q812yes—
384C24C2×#4112yes—
385C24C2511yescharacteristic
386#41—4128no—
387C2×Q8—28no—
388C2×Q8—44no—
389C2×Q8—44no—
390C2×Q8—44no—
391C2×Q8—44no—
392C2×Q8—44no—
393C2×Q8—44no—
394C2×Q8—48no—
395C2×Q8—88no—
396C2×Q8—88no—
397C2×Q8—84no—
398C2×Q8—84no—
399C2×D4—216no—
400C2×D4—28no—
401C2×D4—44no—
402C2×D4—44no—
403C2×D4—48no—
404C2×D4—48no—
405C2×D4—44no—
406C2×D4—44no—
407C2×D4—44no—
408C2×D4—44no—
409C2×D4—48no—
410C2×D4—48no—
411C2×D4—48no—
412C2×D4—48no—
413C22×C4—21no—
414C22×C4—21no—
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—28no—
418C22×C4—28no—
419C22×C4—28no—
420C22×C4—28no—
421C22×C4—28no—
422C22×C4—28no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—216no—
458C22×C4—216no—
459C22×C4—216no—
460C22×C4—24no—
461C22×C4—24no—
462C22×C4—24no—
463C22×C4—24no—
464C22×C4—24no—
465C22×C4—24no—
466C22×C4—24no—
467C22×C4—24no—
468C22×C4—24no—
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—24no—
481C22×C4—24no—
482C22×C4—24no—
483C22×C4—24no—
484C22×C4—24no—
485C22×C4—24no—
486C22×C4—24no—
487C22×C4—24no—
488C22×C4—48no—
489C22×C4—48no—
490C22×C4—48no—
491C22×C4—48no—
492C22×C4—48no—
493C22×C4—48no—
494C22×C4—48no—
495C22×C4—48no—
496C22×C4—48no—
497C22×C4—48no—
498C22×C4—48no—
499C22×C4C22⋊Q8116yes—
500C22×C4C22×D412yes—
501C22×C4C22×D412yes—
502C22×C4C22×D412yes—
503C22×C4C22×D412yes—
504C22×C4C22×D412yes—
505C22×C4C22×D412yes—
506C22×C4C22×Q814yes—
507C22×C4C22×Q812yes—
508C22×C4C22×Q812yes—
509C22×C4C2×#4112yes—
510C22×C4C2×#4112yes—
511C22×C4H(5,𝔽2)14yes—
512C22×C4H(5,𝔽2)12yes—
513C22×C4H(5,𝔽2)12yes—
514C22×C4H(5,𝔽2)12yes—
515C22×C4H(5,𝔽2)12yes—
516C22×C42−1+414yes—
517C4⋊C4—44no—
518C4⋊C4—44no—
519C4⋊C4—48no—
520C4⋊C4—48no—
521C4⋊C4—48no—
522C4⋊C4—48no—
523C4⋊C4—48no—
524C4⋊C4—88no—
525C4⋊C4—88no—
526C4⋊C4—88no—
527C4⋊C4—84no—
528C4⋊C4—84no—
529C22⋊C4—44no—
530C22⋊C4—44no—
531C22⋊C4—48no—
532C22⋊C4—48no—
533C22⋊C4—48no—
534C22⋊C4—48no—
535C22⋊C4—416no—
536C22⋊C4—88no—
537C22⋊C4—88no—
538C22⋊C4—88no—
539C22⋊C4—88no—
540C22⋊C4—88no—
541C42—28no—
542C42—28no—
543C42—44no—
544C42—44no—
545C42—48no—
546C23—22no—
547C23—22no—
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—24no—
563C23—24no—
564C23—24no—
565C23—42no—
566C23—42no—
567C23—42no—
568C23—42no—
569C23—42no—
570C23—42no—
571C23—41no—
572C23—41no—
573C23—41no—
574C23—41no—
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—42no—
602C23—42no—
603C23C2×C22⋊Q814yes—
604C23C23×D411yescharacteristic
605C23C23×Q811yescharacteristic
606C23C22×#4111yescharacteristic
607C23C2×H(5,𝔽2)11yescharacteristic
608C23C2×H(5,𝔽2)11yescharacteristic
609C23C2×H(5,𝔽2)11yescharacteristic
610C23C2×H(5,𝔽2)11yescharacteristic
611C23C2×H(5,𝔽2)12yes—
612C23C2×2−1+412yes—
613C23#58512yes—
614C23#58512yes—
615Q8—416no—
616Q8—816no—
617D4—416no—
618D4—432no—
619C2×C4—22no—
620C2×C4—22no—
621C2×C4—22no—
622C2×C4—22no—
623C2×C4—22no—
624C2×C4—22no—
625C2×C4—232no—
626C2×C4—24no—
627C2×C4—22no—
628C2×C4—22no—
629C2×C4—22no—
630C2×C4—22no—
631C2×C4—24no—
632C2×C4—24no—
633C2×C4—24no—
634C2×C4—24no—
635C2×C4—24no—
636C2×C4—24no—
637C2×C4—24no—
638C2×C4—24no—
639C2×C4—24no—
640C2×C4—24no—
641C2×C4—24no—
642C2×C4—24no—
643C2×C4—432no—
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—41no—
661C2×C4—41no—
662C2×C4—41no—
663C2×C4—41no—
664C2×C4—41no—
665C2×C4—41no—
666C2×C4—41no—
667C2×C4—41no—
668C2×C4—432no—
669C2×C4—44no—
670C2×C4—44no—
671C2×C4—44no—
672C2×C4—44no—
673C2×C4—44no—
674C2×C4—44no—
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—44no—
691C2×C4—44no—
692C2×C4—44no—
693C2×C4—44no—
694C2×C4—44no—
695C2×C4—44no—
696C2×C4—44no—
697C2×C4C2×C22⋊Q814yes—
698C2×C4C2×C22⋊Q814yes—
699C2×C4#53414yes—
700C2×C4#53514yes—
701C2×C4#53814yes—
702C2×C4#53814yes—
703C2×C4#54314yes—
704C2×C4#54314yes—
705C22—28no—
706C22—24no—
707C22—42no—
708C22—42no—
709C22—42no—
710C22—42no—
711C22—44no—
712C22—44no—
713C22—42no—
714C22—42no—
715C22—44no—
716C22—44no—
717C22—44no—
718C22—44no—
719C22—44no—
720C22—44no—
721C22—44no—
722C22—44no—
723C22—82no—
724C22—82no—
725C22—82no—
726C22—82no—
727C22—82no—
728C22—82no—
729C22—82no—
730C22—82no—
731C22C22×C22⋊Q811yescharacteristic
732C22C2×#53411yescharacteristic
733C22C2×#53511yescharacteristic
734C22C2×#53811yescharacteristic
735C22C2×#53811yescharacteristic
736C22C2×#54311yescharacteristic
737C22C2×#54311yescharacteristic
738C22#346012yes—
739C22#346112yes—
740C22#348812yes—
741C22#348912yes—
742C22#350312yes—
743C22#350412yes—
744C22#353612yes—
745C22#354012yes—
746C22#354112yes—
747C22#354212yes—
748C22#354612yes—
749C22#354712yes—
750C22#354911yescharacteristic
751C22#355011yescharacteristic
752C22#355111yescharacteristic
753C22#355111yescharacteristic
754C4—24no—
755C4—42no—
756C4—42no—
757C4—44no—
758C4—44no—
759C4—42no—
760C4—42no—
761C4—44no—
762C4—84no—
763C4—84no—
764C4—84no—
765C4—84no—
766C4#262618yes—
767C2—44no—
768C2—84no—
769C2C2×#262611yescharacteristic
770C2#3525712yes—
771C2#3727112yes—
772C2#3727311yescharacteristic
773C2#3730611yescharacteristic
774C2#3731012yes—
775C2#3737611yescharacteristic
776C2#3742611yescharacteristic
777C2#3751311yescharacteristic
778C2#3751411yescharacteristic
779C2#3751612yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC23×#41C22
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×#2628C2semidirect
3. #16653C2semidirect
4. #16630C2semidirect
5. #16599C2semidirect
6. #16556C2semidirect
7. #16320C2semidirect
8. #16319C2semidirect
9. #15223C2semidirect
10. #15220C2semidirect
11. #15131C2semidirect
12. #15125C2semidirect
13. #14697C2semidirect
14. #14340C2semidirect
83. C22×#41Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#26282
C2×#22972
#166532
#166302
#165992
#165562
#163202
#163192
#152232
#152202
#151312
#151252
#146972
#143402
#143372