Finite Group Databasebeta

← #695786

Subgroups — #695786

Order 512.

3074 conjugacy classes of subgroups.

The subgroup lattice has 782 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2347C211yesmaximal, characteristic, complemented
2C2×#2311C211yesmaximal, characteristic
3#16696C212yesmaximal, complemented
4#16693C212yesmaximal, complemented
5#15967C214yesmaximal, complemented
6#15345C214yesmaximal, complemented
7#15183C212yesmaximal, complemented
8#15135C212yesmaximal, complemented
9#15104C212yesmaximal, complemented
10#15084C212yesmaximal, complemented
11#14337C211yesmaximal, characteristic
12#14130C211yesmaximal, characteristic, complemented
13#13974C212yesmaximal, complemented
14#13939C211yesmaximal, characteristic, complemented
15#13880C211yesmaximal, characteristic, complemented
16#13876C212yesmaximal, complemented
17#13851C211yesmaximal, characteristic
18C23×#41C2211yescharacteristic
19C22×#117C2211yescharacteristic
20C23×C4⋊C4C2211yescharacteristic
21C2×#399C2214yes—
22#2347—28no—
23#2311—28no—
24C2×#391C2212yes—
25C2×#390C2212yes—
26C2×#389C2211yescharacteristic
27C2×#389C2211yescharacteristic
28C2×#389C2212yes—
29C2×#388C2212yes—
30C2×#388C2212yes—
31C2×#388C2214yes—
32C2×#388C2214yes—
33C2×#386C2214yes—
34C2×#385C2212yes—
35C2×#385C2214yes—
36C2×#384C2212yes—
37C2×#383C2212yes—
38C2×#380C2211yescharacteristic
39C2×#380C2211yescharacteristic
40C2×#380C2212yes—
41C2×#380C2212yes—
42C2×#380C2212yes—
43C2×#380C2212yes—
44C2×C4×C4⋊C4C2212yes—
45C2×C4×C4⋊C4C2211yescharacteristic
46C2×C4×C4⋊C4C2211yescharacteristic
47C2×C4×C22⋊C4C2211yescharacteristic
48C2×C4×C22⋊C4C2212yes—
49C2×C4×C22⋊C4C2211yescharacteristic
50C2×C4×C22⋊C4C2212yes—
51#1446C2211yescharacteristic
52#1446C2212yes—
53#1446C2211yescharacteristic
54#1445C2214yes—
55#1445C2214yes—
56#1444C2214yes—
57#1444C2212yes—
58#1444C2212yes—
59#1444C2212yes—
60#1443C2214yes—
61#1443C2214yes—
62#1442C2212yes—
63#1442C2214yes—
64#1440C2214yes—
65#1440C2214yes—
66#1440C2214yes—
67#1439C2214yes—
68#1439C2214yes—
69#1439C2214yes—
70#1438C2212yes—
71#1438C2214yes—
72#1438C2211yescharacteristic
73#1438C2212yes—
74#1438C2211yescharacteristic
75#1438C2212yes—
76#1436C2214yes—
77#1436C2214yes—
78C4⋊#95C2212yes—
79C4⋊#95C2212yes—
80C4⋊#95C2212yes—
81C4⋊#95C2212yes—
82#1434C2212yes—
83#1434C2212yes—
84C22⋊#95C2212yes—
85C22×#41—24no—
86C22×#41—28no—
87C22×#41—24no—
88C22×#41—24no—
89C22×#41—44no—
90C23×Q8C2311yescharacteristic
91C23×D4C2312yes—
92C23×D4C2311yescharacteristic
93C24×C4C2312yes—
94C24×C4C2311yescharacteristic
95C24×C4C2312yes—
96C2×#117—24no—
97C2×#117—24no—
98C2×#117D418yes—
99C22×C4⋊C4—24no—
100C22×C4⋊C4—44no—
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2311yescharacteristic
104C22×C4⋊C4C2311yescharacteristic
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2311yescharacteristic
108C22×C4⋊C4C2311yescharacteristic
109C22×C4⋊C4C2314yes—
110C22×C4⋊C4C2314yes—
111C22×C4⋊C4C2312yes—
112C22×C4⋊C4C2314yes—
113C22×C22⋊C4C2312yes—
114C22×C22⋊C4C2312yes—
115C22×C22⋊C4C2312yes—
116C22×C22⋊C4C2311yescharacteristic
117C22×C22⋊C4C2312yes—
118C22×C22⋊C4C2311yescharacteristic
119C22×C22⋊C4C2312yes—
120C22×C22⋊C4C2314yes—
121C22×C22⋊C4C2312yes—
122C22×C22⋊C4C2312yes—
123C22×C22⋊C4C2314yes—
124C22×C22⋊C4C2312yes—
125C22×C22⋊C4C2314yes—
126C22×C22⋊C4C2314yes—
127C22×C42C2312yes—
128C22×C42C2311yescharacteristic
129C22×C42C2311yescharacteristic
130C22×C42C2312yes—
131C22×C42C2312yes—
132C22×C42C2312yes—
133#399—48no—
134#391—28no—
135#390—28no—
136#389—24no—
137#389—24no—
138#389—44no—
139#388—28no—
140#388—28no—
141#388—48no—
142#388—48no—
143#386—216no—
144#385—48no—
145#385—44no—
146#384—44no—
147#383—28no—
148#380—24no—
149#380—28no—
150#380—24no—
151#380—28no—
152#380—28no—
153#380—44no—
154C4×C4⋊C4—24no—
155C4×C4⋊C4—24no—
156C4×C4⋊C4—28no—
157C4×C22⋊C4—24no—
158C4×C22⋊C4—24no—
159C4×C22⋊C4—28no—
160C4×C22⋊C4—44no—
161C2×#95C2312yes—
162C2×#95C2311yescharacteristic
163C2×#95C2311yescharacteristic
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2312yes—
168C2×#95C2314yes—
169C2×#95C2314yes—
170C2×#95C2312yes—
171C2×#95C2311yescharacteristic
172C2×#95C2311yescharacteristic
173C2×#95C2312yes—
174C2×#95C2312yes—
175C2×#95C2312yes—
176C2×#95C2312yes—
177C2×#95C2312yes—
178C2×#95C2314yes—
179C2×#95C2314yes—
180C2×#95C2314yes—
181C2×#95C2314yes—
182C2×#95C2314yes—
183C2×#95C2314yes—
184C2×#95C2314yes—
185C2×#95C2312yes—
186C2×#95C2314yes—
187C2×#95C2312yes—
188C25C2412yes—
189C25C2411yescharacteristic
190C2×#41—232no—
191C2×#41—432no—
192C2×#41—432no—
193C2×#41—416no—
194C2×#41—416no—
195C22×Q8—22no—
196C22×Q8—24no—
197C22×Q8—22no—
198C22×Q8—22no—
199C22×Q8—42no—
200C22×D4—24no—
201C22×D4—24no—
202C22×D4—24no—
203C22×D4—24no—
204C22×D4—24no—
205C22×D4—22no—
206C22×D4—24no—
207C22×D4—22no—
208C22×D4—22no—
209C22×D4—42no—
210C22×D4—42no—
211C22×D4—42no—
212C23×C4—24no—
213C23×C4—24no—
214C23×C4—24no—
215C23×C4—24no—
216C23×C4—24no—
217C23×C4—24no—
218C23×C4—24no—
219C23×C4—24no—
220C23×C4—24no—
221C23×C4—44no—
222C23×C4—42no—
223C23×C4—44no—
224C23×C4—42no—
225C23×C4C2×D414yes—
226C23×C4#4118yes—
227C23×C4#4114yes—
228C23×C4#4114yes—
229C23×C4C2412yes—
230C23×C4C2414yes—
231C23×C4C2412yes—
232C23×C4C2411yescharacteristic
233C23×C4C2411yescharacteristic
234C23×C4C2414yes—
235C23×C4C2412yes—
236C23×C4C2412yes—
237C23×C4C2412yes—
238C23×C4C2411yescharacteristic
239C23×C4C2412yes—
240C23×C4C2412yes—
241C23×C4C2411yescharacteristic
242C23×C4C2411yescharacteristic
243C23×C4C2411yescharacteristic
244#117—232no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—22no—
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—24no—
264C2×C4⋊C4—28no—
265C2×C4⋊C4—28no—
266C2×C4⋊C4—44no—
267C2×C4⋊C4—42no—
268C2×C4⋊C4—42no—
269C2×C4⋊C4—416no—
270C2×C4⋊C4—44no—
271C2×C4⋊C4—44no—
272C2×C4⋊C4—44no—
273C2×C4⋊C4—44no—
274C2×C4⋊C4—44no—
275C2×C4⋊C4—44no—
276C2×C4⋊C4—44no—
277C2×C4⋊C4C2×D414yes—
278C2×C4⋊C4C2×D414yes—
279C2×C22⋊C4—24no—
280C2×C22⋊C4—24no—
281C2×C22⋊C4—24no—
282C2×C22⋊C4—24no—
283C2×C22⋊C4—24no—
284C2×C22⋊C4—24no—
285C2×C22⋊C4—24no—
286C2×C22⋊C4—24no—
287C2×C22⋊C4—22no—
288C2×C22⋊C4—22no—
289C2×C22⋊C4—24no—
290C2×C22⋊C4—28no—
291C2×C22⋊C4—24no—
292C2×C22⋊C4—22no—
293C2×C22⋊C4—24no—
294C2×C22⋊C4—24no—
295C2×C22⋊C4—24no—
296C2×C22⋊C4—22no—
297C2×C22⋊C4—24no—
298C2×C22⋊C4—24no—
299C2×C22⋊C4—28no—
300C2×C22⋊C4—28no—
301C2×C22⋊C4—41no—
302C2×C22⋊C4—41no—
303C2×C22⋊C4—42no—
304C2×C22⋊C4—42no—
305C2×C22⋊C4—42no—
306C2×C22⋊C4—44no—
307C2×C22⋊C4—42no—
308C2×C22⋊C4—42no—
309C2×C22⋊C4—42no—
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⋊C4—44no—
319C2×C22⋊C4C2×D418yes—
320C2×C42—24no—
321C2×C42—24no—
322C2×C42—22no—
323C2×C42—22no—
324C2×C42—24no—
325C2×C42—24no—
326C2×C42—24no—
327C2×C42—24no—
328C2×C42—24no—
329C2×C42—24no—
330C2×C42—24no—
331C2×C42—44no—
332C2×C42—44no—
333C2×C42C2×D418yes—
334#95—22no—
335#95—22no—
336#95—24no—
337#95—24no—
338#95—24no—
339#95—28no—
340#95—28no—
341#95—24no—
342#95—24no—
343#95—24no—
344#95—22no—
345#95—22no—
346#95—24no—
347#95—28no—
348#95—24no—
349#95—28no—
350#95—28no—
351#95—28no—
352#95—44no—
353#95—42no—
354#95—42no—
355#95—44no—
356#95—44no—
357#95—44no—
358#95—44no—
359#95—44no—
360#95—44no—
361C24—22no—
362C24—21no—
363C24—22no—
364C24—22no—
365C24—21no—
366C24—22no—
367C24—21no—
368C24—21no—
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—
382C24—22no—
383C24C22×D412yes—
384C24C2×#4114yes—
385C24C2×#4112yes—
386C24C2×#4112yes—
387C24C2511yescharacteristic
388#41—4128no—
389C2×Q8—28no—
390C2×Q8—48no—
391C2×Q8—48no—
392C2×Q8—44no—
393C2×Q8—44no—
394C2×D4—28no—
395C2×D4—216no—
396C2×D4—48no—
397C2×D4—48no—
398C2×D4—48no—
399C2×D4—48no—
400C2×D4—44no—
401C2×D4—48no—
402C2×D4—48no—
403C2×D4—48no—
404C2×D4—48no—
405C2×D4—44no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—28no—
409C22×C4—28no—
410C22×C4—216no—
411C22×C4—216no—
412C22×C4—22no—
413C22×C4—21no—
414C22×C4—21no—
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—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—24no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—24no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—22no—
448C22×C4—216no—
449C22×C4—216no—
450C22×C4—216no—
451C22×C4—21no—
452C22×C4—21no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—24no—
456C22×C4—22no—
457C22×C4—24no—
458C22×C4—24no—
459C22×C4—24no—
460C22×C4—24no—
461C22×C4—24no—
462C22×C4—24no—
463C22×C4—21no—
464C22×C4—22no—
465C22×C4—24no—
466C22×C4—24no—
467C22×C4—24no—
468C22×C4—24no—
469C22×C4—24no—
470C22×C4—24no—
471C22×C4—24no—
472C22×C4—22no—
473C22×C4—24no—
474C22×C4—24no—
475C22×C4—21no—
476C22×C4—22no—
477C22×C4—22no—
478C22×C4—22no—
479C22×C4—24no—
480C22×C4—21no—
481C22×C4—21no—
482C22×C4—22no—
483C22×C4—24no—
484C22×C4—22no—
485C22×C4—48no—
486C22×C4—48no—
487C22×C4—48no—
488C22×C4—48no—
489C22×C4—48no—
490C22×C4—84no—
491C22×C4—84no—
492C22×C4—88no—
493C22×C4—88no—
494C22×C4#123116yes—
495C22×C4C22×D414yes—
496C22×C4C22×D414yes—
497C22×C4C22×D412yes—
498C22×C4C22×D412yes—
499C22×C4C2×#4114yes—
500C22×C4C2×#4112yes—
501C22×C4C2×#4112yes—
502C22×C4C2×#4112yes—
503C22×C4C2×#4112yes—
504C22×C4C2×#4114yes—
505C22×C4H(5,𝔽2)14yes—
506C22×C4H(5,𝔽2)12yes—
507C22×C4H(5,𝔽2)14yes—
508C22×C42−1+414yes—
509C22×C42−1+412yes—
510C4⋊C4—28no—
511C4⋊C4—28no—
512C4⋊C4—48no—
513C4⋊C4—48no—
514C4⋊C4—48no—
515C4⋊C4—44no—
516C4⋊C4—44no—
517C4⋊C4—48no—
518C4⋊C4—48no—
519C4⋊C4—88no—
520C4⋊C4—88no—
521C4⋊C4—88no—
522C22⋊C4—216no—
523C22⋊C4—216no—
524C22⋊C4—216no—
525C22⋊C4—48no—
526C22⋊C4—44no—
527C22⋊C4—44no—
528C22⋊C4—48no—
529C22⋊C4—48no—
530C22⋊C4—48no—
531C22⋊C4—416no—
532C22⋊C4—88no—
533C22⋊C4—88no—
534C22⋊C4—84no—
535C22⋊C4—88no—
536C42—216no—
537C42—44no—
538C42—44no—
539C42—48no—
540C42—84no—
541C42—84no—
542C23—22no—
543C23—22no—
544C23—22no—
545C23—22no—
546C23—24no—
547C23—24no—
548C23—22no—
549C23—22no—
550C23—24no—
551C23—22no—
552C23—22no—
553C23—22no—
554C23—24no—
555C23—22no—
556C23—24no—
557C23—24no—
558C23—24no—
559C23—24no—
560C23—24no—
561C23—24no—
562C23—24no—
563C23—42no—
564C23—42no—
565C23—42no—
566C23—42no—
567C23—42no—
568C23—42no—
569C23—42no—
570C23—42no—
571C23—42no—
572C23—42no—
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—
599C23C2×#12314yes—
600C23C23×D411yescharacteristic
601C23C22×#4111yescharacteristic
602C23C22×#4111yescharacteristic
603C23C22×#4112yes—
604C23C2×H(5,𝔽2)12yes—
605C23C2×H(5,𝔽2)11yescharacteristic
606C23C2×H(5,𝔽2)12yes—
607C23C2×2−1+411yescharacteristic
608C23C2×2−1+412yes—
609C23#58511yescharacteristic
610C23#58511yescharacteristic
611Q8—416no—
612D4—416no—
613D4—432no—
614C2×C4—22no—
615C2×C4—22no—
616C2×C4—24no—
617C2×C4—24no—
618C2×C4—232no—
619C2×C4—24no—
620C2×C4—24no—
621C2×C4—24no—
622C2×C4—24no—
623C2×C4—24no—
624C2×C4—22no—
625C2×C4—24no—
626C2×C4—22no—
627C2×C4—24no—
628C2×C4—24no—
629C2×C4—24no—
630C2×C4—24no—
631C2×C4—24no—
632C2×C4—24no—
633C2×C4—24no—
634C2×C4—24no—
635C2×C4—24no—
636C2×C4—24no—
637C2×C4—24no—
638C2×C4—42no—
639C2×C4—41no—
640C2×C4—41no—
641C2×C4—42no—
642C2×C4—42no—
643C2×C4—42no—
644C2×C4—42no—
645C2×C4—42no—
646C2×C4—42no—
647C2×C4—42no—
648C2×C4—44no—
649C2×C4—41no—
650C2×C4—41no—
651C2×C4—42no—
652C2×C4—42no—
653C2×C4—42no—
654C2×C4—42no—
655C2×C4—42no—
656C2×C4—44no—
657C2×C4—42no—
658C2×C4—42no—
659C2×C4—42no—
660C2×C4—42no—
661C2×C4—42no—
662C2×C4—432no—
663C2×C4—41no—
664C2×C4—42no—
665C2×C4—41no—
666C2×C4—44no—
667C2×C4—42no—
668C2×C4—44no—
669C2×C4—44no—
670C2×C4—41no—
671C2×C4—44no—
672C2×C4—42no—
673C2×C4—41no—
674C2×C4—42no—
675C2×C4—44no—
676C2×C4—42no—
677C2×C4—42no—
678C2×C4—42no—
679C2×C4—44no—
680C2×C4—41no—
681C2×C4—41no—
682C2×C4—44no—
683C2×C4—44no—
684C2×C4—44no—
685C2×C4—44no—
686C2×C4—42no—
687C2×C4—44no—
688C2×C4—44no—
689C2×C4—44no—
690C2×C4—44no—
691C2×C4—42no—
692C2×C4—44no—
693C2×C4—44no—
694C2×C4—44no—
695C2×C4—44no—
696C2×C4—42no—
697C2×C4—816no—
698C2×C4C2×#12314yes—
699C2×C4C2×#12314yes—
700C2×C4#52514yes—
701C2×C4#53514yes—
702C2×C4#55118yes—
703C2×C4#55118yes—
704C22—24no—
705C22—24no—
706C22—24no—
707C22—44no—
708C22—44no—
709C22—42no—
710C22—44no—
711C22—42no—
712C22—44no—
713C22—42no—
714C22—42no—
715C22—44no—
716C22—44no—
717C22—44no—
718C22—44no—
719C22—44no—
720C22—44no—
721C22—44no—
722C22—82no—
723C22—82no—
724C22—82no—
725C22—82no—
726C22—82no—
727C22—82no—
728C22—82no—
729C22—82no—
730C22C22×#12311yescharacteristic
731C22C2×#52511yescharacteristic
732C22C2×#53511yescharacteristic
733C22#346011yescharacteristic
734C22#346111yescharacteristic
735C22C2×#55112yes—
736C22C2×#55112yes—
737C22#348811yescharacteristic
738C22#348911yescharacteristic
739C22#349212yes—
740C22#349212yes—
741C22#349512yes—
742C22#349811yescharacteristic
743C22#349911yescharacteristic
744C22#350612yes—
745C22#351111yescharacteristic
746C22#351112yes—
747C22#351112yes—
748C22#351211yescharacteristic
749C22#353712yes—
750C22#354112yes—
751C22#354612yes—
752C22#355012yes—
753C4—24no—
754C4—44no—
755C4—42no—
756C4—44no—
757C4—44no—
758C4—44no—
759C4—42no—
760C4—44no—
761C4—82no—
762C4—84no—
763C4—81no—
764C4—82no—
765C4—84no—
766C4—81no—
767C4#248218yes—
768C2—44no—
769C2—84no—
770C2C2×#248211yescharacteristic
771C2#3525511yescharacteristic
772C2#3553011yescharacteristic
773C2#3568911yescharacteristic
774C2#3608911yescharacteristic
775C2#3796812yes—
776C2#3956012yes—
777C2#4291111yescharacteristic
778C2#4298611yescharacteristic
779C2#4525012yes—
780C2#4533012yes—
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×#2347C2semidirect
3. #16696C2semidirect
4. #16693C2semidirect
5. #15967C2semidirect
6. #15345C2semidirect
7. #15183C2semidirect
8. #15135C2semidirect
9. #15104C2semidirect
10. #15084C2semidirect
12. #14130C2semidirect
13. #13974C2semidirect
14. #13939C2semidirect
15. #13880C2semidirect
16. #13876C2semidirect

Maximal subgroups

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

subgroupindex
C2×#23472
C2×#23112
#166962
#166932
#159672
#153452
#151832
#151352
#151042
#150842
#143372
#141302
#139742
#139392
#138802
#138762
#138512