Finite Group Databasebeta

← #767090

Subgroups — #767090

Order 512.

2386 conjugacy classes of subgroups.

The subgroup lattice has 744 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2320C211yesmaximal, characteristic, complemented
2C2×#2319C211yesmaximal, characteristic
3#16078C214yesmaximal, complemented
4#15959C214yesmaximal, complemented
5#15455C214yesmaximal, complemented
6#15385C214yesmaximal, complemented
7#15234C211yesmaximal, characteristic, complemented
8#15217C211yesmaximal, characteristic
9#15184C211yesmaximal, characteristic, complemented
10#15134C211yesmaximal, characteristic
11#14049C211yesmaximal, characteristic, complemented
12#14048C211yesmaximal, characteristic, complemented
13#14027C211yesmaximal, characteristic
14#14026C211yesmaximal, characteristic
15#14003C211yesmaximal, characteristic, complemented
16#14002C211yesmaximal, characteristic
17#13977C211yesmaximal, characteristic
18#13976C211yesmaximal, characteristic, complemented
19#13854C211yesmaximal, characteristic, complemented
20C22×#124C2211yescharacteristic
21C22×C4×Q8C2211yescharacteristic
22C22×C4×D4C2211yescharacteristic
23C2×#396C2214yes—
24#2320—28no—
25#2319—28no—
26C2×#391C2211yescharacteristic
27C2×#391C2211yescharacteristic
28C2×#391C2211yescharacteristic
29C2×#391C2211yescharacteristic
30C2×#391C2211yescharacteristic
31C2×#390C2211yescharacteristic
32C2×#390C2211yescharacteristic
33C2×#390C2211yescharacteristic
34C2×#390C2211yescharacteristic
35C2×#390C2211yescharacteristic
36C2×#389C2211yescharacteristic
37C2×#389C2211yescharacteristic
38C2×#389C2211yescharacteristic
39C2×#389C2211yescharacteristic
40C2×#388C2212yes—
41C2×#388C2212yes—
42C2×#387C2212yes—
43C2×#387C2212yes—
44C2×#387C2214yes—
45C2×#385C2214yes—
46C2×#384C2211yescharacteristic
47C2×#384C2211yescharacteristic
48C2×#383C2212yes—
49C2×#382C2212yes—
50C2×#382C2211yescharacteristic
51C2×#381C2212yes—
52C2×#380C2212yes—
53C2×C4×C4⋊C4C2211yescharacteristic
54C2×C4×C4⋊C4C2211yescharacteristic
55C2×C4×C4⋊C4C2211yescharacteristic
56C2×C4×C4⋊C4C2211yescharacteristic
57C2×#376C2211yescharacteristic
58#1445C2214yes—
59#1444C2214yes—
60#1443C2214yes—
61#1442C2212yes—
62#1442C2214yes—
63#1442C2212yes—
64#1442C2212yes—
65#1442C2212yes—
66#1441C2214yes—
67#1441C2214yes—
68#1441C2214yes—
69#1441C2214yes—
70#1440C2214yes—
71#1440C2214yes—
72#1440C2214yes—
73#1439C2214yes—
74#1439C2214yes—
75#1439C2214yes—
76#1439C2214yes—
77#1437C2214yes—
78#1437C2214yes—
79#1436C2214yes—
80#1436C2214yes—
81C4⋊#95C2211yescharacteristic
82C4⋊#95C2211yescharacteristic
83C4⋊#95C2211yescharacteristic
84C4⋊#95C2211yescharacteristic
85C4⋊#95C2211yescharacteristic
86C4⋊#95C2211yescharacteristic
87#1434C2214yes—
88C22⋊#95C2214yes—
89C4×#95C2211yescharacteristic
90C4×#95C2211yescharacteristic
91C4×#95C2211yescharacteristic
92C4×#95C2211yescharacteristic
93C23×Q8C2311yescharacteristic
94C23×D4C2311yescharacteristic
95C24×C4C2312yes—
96C2×#124—24no—
97C2×#124—24no—
98C2×#124—42no—
99C2×C4×Q8—24no—
100C2×C4×Q8—24no—
101C2×C4×Q8—24no—
102C2×C4×D4—24no—
103C2×C4×D4—24no—
104C2×C4×D4—24no—
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2311yescharacteristic
107C22×C4⋊C4C2311yescharacteristic
108C22×C4⋊C4C2311yescharacteristic
109C22×C4⋊C4C2311yescharacteristic
110C22×C4⋊C4C2312yes—
111C22×C4⋊C4C2314yes—
112C22×C4⋊C4C2311yescharacteristic
113C22×C4⋊C4C2311yescharacteristic
114C22×C4⋊C4C2312yes—
115C22×C4⋊C4C2314yes—
116C22×C4⋊C4C2312yes—
117C22×C4⋊C4C2312yes—
118C22×C4⋊C4C2312yes—
119C22×C4⋊C4C2312yes—
120C22×C22⋊C4C2312yes—
121C22×C22⋊C4C2314yes—
122C22×C22⋊C4C2314yes—
123C22×C22⋊C4C2312yes—
124C22×C22⋊C4C2314yes—
125C22×C22⋊C4C2312yes—
126C22×C22⋊C4C2312yes—
127C22×C22⋊C4C2314yes—
128C22×C22⋊C4C2312yes—
129C22×C42C2312yes—
130C22×C42C2311yescharacteristic
131C22×C42C2311yescharacteristic
132C22×C42C2311yescharacteristic
133C22×C42C2311yescharacteristic
134C22×C42C2311yescharacteristic
135C22×C42C2311yescharacteristic
136C22×C42C2311yescharacteristic
137C22×C42C2312yes—
138C22×C42C2312yes—
139#396—48no—
140#391—24no—
141#391—24no—
142#391—24no—
143#391—24no—
144#391—42no—
145#390—24no—
146#390—24no—
147#390—24no—
148#390—24no—
149#390—42no—
150#389—24no—
151#389—24no—
152#389—24no—
153#389—24no—
154#388—28no—
155#388—28no—
156#387—28no—
157#387—28no—
158#387—48no—
159#385—48no—
160#384—24no—
161#384—24no—
162#383—28no—
163#382—28no—
164#382—42no—
165#381—28no—
166#380—28no—
167C4×C4⋊C4—24no—
168C4×C4⋊C4—24no—
169C4×C4⋊C4—42no—
170C4×C4⋊C4—42no—
171#376D418yes—
172C2×#95C2314yes—
173C2×#95C2312yes—
174C2×#95C2312yes—
175C2×#95C2312yes—
176C2×#95C2312yes—
177C2×#95C2312yes—
178C2×#95C2314yes—
179C2×#95C2312yes—
180C2×#95C2314yes—
181C2×#95C2312yes—
182C2×#95C2312yes—
183C2×#95C2312yes—
184C2×#95C2312yes—
185C2×#95C2312yes—
186C2×#95C2312yes—
187C2×#95C2312yes—
188C2×#95C2312yes—
189C2×#95C2312yes—
190C2×#95C2312yes—
191C2×#95C2312yes—
192C2×#95C2312yes—
193C2×#95C2314yes—
194C2×#95C2314yes—
195C2×#95C2314yes—
196C2×#95C2314yes—
197C2×#95C2314yes—
198C2×#95C2314yes—
199C2×#95C2314yes—
200C2×#95C2312yes—
201C2×#95C2312yes—
202C2×#95C2311yescharacteristic
203C2×#95C2311yescharacteristic
204C2×#95C2311yescharacteristic
205C2×#95C2311yescharacteristic
206C25C2412yes—
207C22×Q8—22no—
208C22×Q8—22no—
209C22×Q8—22no—
210C22×Q8—22no—
211C22×Q8—22no—
212C22×Q8—22no—
213C22×Q8#4114yes—
214C22×D4—22no—
215C22×D4—22no—
216C22×D4—22no—
217C22×D4—22no—
218C22×D4—22no—
219C22×D4—22no—
220C22×D4#4114yes—
221C23×C4—24no—
222C23×C4—24no—
223C23×C4—24no—
224C23×C4—44no—
225C23×C4—44no—
226C23×C4C2412yes—
227C23×C4C2412yes—
228C23×C4C2412yes—
229C23×C4C2412yes—
230C23×C4C2412yes—
231C23×C4C2412yes—
232C23×C4C2411yescharacteristic
233C23×C4C2411yescharacteristic
234C23×C4C2411yescharacteristic
235C23×C4C2411yescharacteristic
236C23×C4C2411yescharacteristic
237C23×C4C2411yescharacteristic
238C23×C4C2412yes—
239C23×C4C2412yes—
240C23×C4C2412yes—
241C23×C4C2412yes—
242C23×C4C2412yes—
243C23×C4C2411yescharacteristic
244#124—416no—
245C4×Q8—232no—
246C4×D4—232no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—28no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—28no—
266C2×C4⋊C4—24no—
267C2×C4⋊C4—24no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—24no—
271C2×C4⋊C4—24no—
272C2×C4⋊C4—24no—
273C2×C4⋊C4—24no—
274C2×C4⋊C4—28no—
275C2×C4⋊C4—44no—
276C2×C4⋊C4—42no—
277C2×C4⋊C4—42no—
278C2×C4⋊C4—42no—
279C2×C4⋊C4—42no—
280C2×C4⋊C4—44no—
281C2×C4⋊C4—44no—
282C2×C4⋊C4—44no—
283C2×C4⋊C4—44no—
284C2×C4⋊C4—44no—
285C2×C4⋊C4—44no—
286C2×C4⋊C4#4114yes—
287C2×C4⋊C4#4114yes—
288C2×C22⋊C4—24no—
289C2×C22⋊C4—24no—
290C2×C22⋊C4—24no—
291C2×C22⋊C4—28no—
292C2×C22⋊C4—28no—
293C2×C22⋊C4—28no—
294C2×C22⋊C4—28no—
295C2×C22⋊C4—28no—
296C2×C22⋊C4—24no—
297C2×C22⋊C4—24no—
298C2×C22⋊C4—24no—
299C2×C22⋊C4—24no—
300C2×C22⋊C4—28no—
301C2×C22⋊C4—44no—
302C2×C22⋊C4—44no—
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×C42—24no—
312C2×C42—22no—
313C2×C42—22no—
314C2×C42—24no—
315C2×C42—24no—
316C2×C42—22no—
317C2×C42—22no—
318C2×C42—22no—
319C2×C42—22no—
320C2×C42—22no—
321C2×C42—22no—
322C2×C42—22no—
323C2×C42—22no—
324C2×C42—22no—
325C2×C42—22no—
326C2×C42—22no—
327C2×C42—22no—
328C2×C42—22no—
329C2×C42—22no—
330C2×C42—22no—
331C2×C42—22no—
332C2×C42—24no—
333C2×C42—24no—
334C2×C42—44no—
335C2×C42—44no—
336C2×C42C2×D414yes—
337C2×C42C2×D414yes—
338C2×C42C2×D414yes—
339#95—24no—
340#95—24no—
341#95—24no—
342#95—24no—
343#95—24no—
344#95—24no—
345#95—24no—
346#95—24no—
347#95—24no—
348#95—24no—
349#95—28no—
350#95—28no—
351#95—24no—
352#95—24no—
353#95—28no—
354#95—24no—
355#95—24no—
356#95—42no—
357#95—44no—
358#95—42no—
359#95—42no—
360#95—42no—
361#95—44no—
362#95—42no—
363#95—42no—
364#95—44no—
365#95—44no—
366#95—44no—
367#95—44no—
368#95—44no—
369#95C2×D414yes—
370#95C2×D414yes—
371#95C2×D414yes—
372#95C2×D414yes—
373C24—22no—
374C24—22no—
375C24—22no—
376C24—22no—
377C24—24no—
378C24—22no—
379C24—24no—
380C24—22no—
381C24—24no—
382C24—24no—
383C24C2×#4114yes—
384C24C2511yescharacteristic
385C2×Q8—28no—
386C2×Q8—28no—
387C2×Q8—28no—
388C2×Q8—44no—
389C2×Q8—44no—
390C2×Q8—44no—
391C2×Q8—44no—
392C2×D4—28no—
393C2×D4—28no—
394C2×D4—28no—
395C2×D4—44no—
396C2×D4—44no—
397C2×D4—44no—
398C2×D4—44no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—21no—
405C22×C4—21no—
406C22×C4—21no—
407C22×C4—21no—
408C22×C4—22no—
409C22×C4—21no—
410C22×C4—21no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—21no—
423C22×C4—21no—
424C22×C4—22no—
425C22×C4—24no—
426C22×C4—22no—
427C22×C4—24no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—21no—
433C22×C4—21no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—24no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—24no—
449C22×C4—216no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
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—22no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—24no—
465C22×C4—24no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—24no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—24no—
474C22×C4—22no—
475C22×C4—22no—
476C22×C4—22no—
477C22×C4—22no—
478C22×C4—22no—
479C22×C4—24no—
480C22×C4—24no—
481C22×C4—24no—
482C22×C4—24no—
483C22×C4—22no—
484C22×C4—84no—
485C22×C4—84no—
486C22×C4—88no—
487C22×C4—88no—
488C22×C4C22×D412yes—
489C22×C4C22×D412yes—
490C22×C4C22×D412yes—
491C22×C4C22×D412yes—
492C22×C4C22×D412yes—
493C22×C4C22×D412yes—
494C22×C4C22×D412yes—
495C22×C4C2×#4114yes—
496C22×C4C2×#4114yes—
497C22×C4C2×#4114yes—
498C22×C4C2×#4112yes—
499C22×C4C2×#4112yes—
500C22×C4C2×#4112yes—
501C22×C4C2×#4112yes—
502C22×C4H(5,𝔽2)14yes—
503C22×C4H(5,𝔽2)12yes—
504C22×C42−1+414yes—
505C22×C42−1+412yes—
506C4⋊C4—216no—
507C4⋊C4—28no—
508C4⋊C4—28no—
509C4⋊C4—416no—
510C4⋊C4—48no—
511C4⋊C4—48no—
512C4⋊C4—44no—
513C4⋊C4—44no—
514C4⋊C4—44no—
515C4⋊C4—44no—
516C4⋊C4—84no—
517C4⋊C4—84no—
518C4⋊C4—84no—
519C4⋊C4—84no—
520C4⋊C4—88no—
521C22⋊C4—216no—
522C22⋊C4—416no—
523C22⋊C4—416no—
524C22⋊C4—84no—
525C22⋊C4—84no—
526C22⋊C4—84no—
527C22⋊C4—84no—
528C22⋊C4—88no—
529C22⋊C4—88no—
530C42—216no—
531C42—28no—
532C42—28no—
533C42—28no—
534C42—44no—
535C42—44no—
536C42—44no—
537C42—44no—
538C42—84no—
539C42—84no—
540C23—24no—
541C23—24no—
542C23—24no—
543C23—28no—
544C23—28no—
545C23—42no—
546C23—42no—
547C23—42no—
548C23—42no—
549C23—42no—
550C23—42no—
551C23—42no—
552C23—42no—
553C23—44no—
554C23—44no—
555C23—44no—
556C23—44no—
557C23—44no—
558C23—44no—
559C23—44no—
560C23—44no—
561C23—44no—
562C23—44no—
563C23C23×D411yescharacteristic
564C23C22×#4111yescharacteristic
565C23C22×#4111yescharacteristic
566C23C22×#4111yescharacteristic
567C23C22×#4111yescharacteristic
568C23C2×H(5,𝔽2)11yescharacteristic
569C23C2×H(5,𝔽2)12yes—
570C23C2×2−1+411yescharacteristic
571C23C2×2−1+412yes—
572C23#58512yes—
573C23#58512yes—
574Q8—416no—
575D4—416no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—24no—
580C2×C4—24no—
581C2×C4—22no—
582C2×C4—24no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—24no—
586C2×C4—24no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—24no—
600C2×C4—28no—
601C2×C4—24no—
602C2×C4—28no—
603C2×C4—24no—
604C2×C4—24no—
605C2×C4—24no—
606C2×C4—24no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—44no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—44no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—42no—
637C2×C4—42no—
638C2×C4—42no—
639C2×C4—42no—
640C2×C4—42no—
641C2×C4—42no—
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—44no—
652C2×C4—44no—
653C2×C4—42no—
654C2×C4—44no—
655C2×C4—42no—
656C2×C4—42no—
657C2×C4—42no—
658C2×C4—42no—
659C2×C4—42no—
660C2×C4—44no—
661C2×C4—42no—
662C2×C4—42no—
663C2×C4—44no—
664C2×C4—42no—
665C2×C4—42no—
666C2×C4—42no—
667C2×C4—42no—
668C2×C4—44no—
669C2×C4—44no—
670C2×C4—44no—
671C2×C4—816no—
672C2×C4#52514yes—
673C2×C4#52514yes—
674C2×C4#53714yes—
675C2×C4#55114yes—
676C2×C4#55214yes—
677C2×C4#55214yes—
678C2×C4#55514yes—
679C22—44no—
680C22—44no—
681C22—48no—
682C22—44no—
683C22—48no—
684C22—82no—
685C22—84no—
686C22—82no—
687C22—82no—
688C22—82no—
689C22—84no—
690C22C2×#52511yescharacteristic
691C22C2×#52511yescharacteristic
692C22C2×#53711yescharacteristic
693C22C2×#55111yescharacteristic
694C22C2×#55211yescharacteristic
695C22C2×#55211yescharacteristic
696C22C2×#55511yescharacteristic
697C22#349811yescharacteristic
698C22#349911yescharacteristic
699C22#350011yescharacteristic
700C22#350011yescharacteristic
701C22#351312yes—
702C22#351312yes—
703C22#351412yes—
704C22#351512yes—
705C22#351612yes—
706C22#351612yes—
707C22#351712yes—
708C22#351812yes—
709C22#353612yes—
710C22#353912yes—
711C22#355012yes—
712C22#355712yes—
713C4—42no—
714C4—42no—
715C4—44no—
716C4—44no—
717C4—44no—
718C4—42no—
719C4—42no—
720C4—42no—
721C4—44no—
722C4—44no—
723C4—42no—
724C4—82no—
725C4—82no—
726C4—82no—
727C4—82no—
728C4—82no—
729C4—82no—
730C4#260918yes—
731C2—84no—
732C2C2×#260911yescharacteristic
733C2#3553411yescharacteristic
734C2#3553611yescharacteristic
735C2#3562211yescharacteristic
736C2#3566911yescharacteristic
737C2#3729911yescharacteristic
738C2#3742011yescharacteristic
739C2#4500812yes—
740C2#4502112yes—
741C2#4538212yes—
742C2#4540612yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC23×D4C23
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×#2320C2semidirect
3. #16078C2semidirect
4. #15959C2semidirect
5. #15455C2semidirect
6. #15385C2semidirect
7. #15234C2semidirect
9. #15184C2semidirect
11. #14049C2semidirect
12. #14048C2semidirect
15. #14003C2semidirect
18. #13976C2semidirect
19. #13854C2semidirect

Maximal subgroups

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

subgroupindex
C2×#23202
C2×#23192
#160782
#159592
#154552
#153852
#152342
#152172
#151842
#151342
#140492
#140482
#140272
#140262
#140032
#140022
#139772
#139762
#138542