Finite Group Databasebeta

← #1394092

Subgroups — #1394092

Order 512.

3094 conjugacy classes of subgroups.

The subgroup lattice has 722 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2551C211yesmaximal, characteristic, complemented
2C2×#2452C212yesmaximal
3C2×#2621C212yesmaximal, complemented
4C2×#2354C211yesmaximal, characteristic
5#16518C212yesmaximal, complemented
6#16506C212yesmaximal, complemented
7#16295C214yesmaximal, complemented
8#16264C214yesmaximal, complemented
9#15151C211yesmaximal, characteristic, complemented
10#15147C211yesmaximal, characteristic
11#15138C212yesmaximal
12#15136C211yesmaximal, characteristic
13#15133C212yesmaximal, complemented
14#14907C212yesmaximal, complemented
15#14053C212yesmaximal, complemented
16#13918C211yesmaximal, characteristic, complemented
17C4×#398C211yesmaximal, characteristic, complemented
18C22×C4⋊Q8C2212yes—
19C22×C4⋊Q8C2212yes—
20C22×C22⋊Q8C2214yes—
21C22×C4×D4C2211yescharacteristic
22C22×#117C2212yes—
23#2621—216no—
24#2551—28no—
25#2452—216no—
26C2×#398C2214yes—
27C2×#398C2212yes—
28C2×#398C2212yes—
29C2×#398C2211yescharacteristic
30C2×#397C2212yes—
31C2×#397C2214yes—
32C2×#397C2212yes—
33C2×#397C2212yes—
34C2×#396C2214yes—
35C2×#395C2214yes—
36C2×#393C2214yes—
37#2354—28no—
38C2×#391C2214yes—
39C2×#390C2212yes—
40C2×#390C2211yescharacteristic
41C2×#390C2211yescharacteristic
42C2×#390C2212yes—
43C2×#389C2212yes—
44C2×#389C2212yes—
45C2×#389C2212yes—
46C2×#389C2212yes—
47C2×#388C2212yes—
48C2×#388C2212yes—
49C2×#385C2214yes—
50C2×#384C2211yescharacteristic
51C2×#384C2211yescharacteristic
52C2×#382C2212yes—
53C2×#380C2212yes—
54C2×C4×C4⋊C4C2212yes—
55C2×C4×C4⋊C4C2211yescharacteristic
56C2×C4×C4⋊C4C2211yescharacteristic
57C2×C4×C4⋊C4C2211yescharacteristic
58C2×C4×C4⋊C4C2211yescharacteristic
59#1446C2214yes—
60#1445C2211yescharacteristic
61#1445C2212yes—
62#1445C2212yes—
63#1444C2211yescharacteristic
64#1442C2212yes—
65#1442C2214yes—
66#1441C2214yes—
67#1441C2214yes—
68#1441C2214yes—
69#1441C2211yescharacteristic
70#1441C2212yes—
71#1441C2214yes—
72#1440C2214yes—
73#1440C2214yes—
74#1440C2214yes—
75#1439C2214yes—
76#1439C2214yes—
77#1439C2212yes—
78#1436C2214yes—
79C4⋊#95C2212yes—
80C4⋊#95C2212yes—
81C4⋊#95C2212yes—
82C4⋊#95C2212yes—
83C4⋊#95C2211yescharacteristic
84#1434C2212yes—
85C4×#95C2211yescharacteristic
86C4×#95C2212yes—
87C23×Q8C2314yes—
88C23×D4C2311yescharacteristic
89C24×C4C2312yes—
90C2×C4⋊Q8—28no—
91C2×C4⋊Q8—28no—
92C2×C4⋊Q8—28no—
93C2×C4⋊Q8—28no—
94C2×C4⋊Q8—28no—
95C2×C4⋊Q8—44no—
96C2×C22⋊Q8—216no—
97C2×C22⋊Q8—216no—
98C2×C22⋊Q8—48no—
99C2×C4×D4—24no—
100C2×C4×D4—24no—
101C2×C4×D4D418yes—
102C2×#117—28no—
103C2×#117—28no—
104C2×#117—44no—
105C22×C4⋊C4C2314yes—
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2314yes—
108C22×C4⋊C4C2312yes—
109C22×C4⋊C4C2311yescharacteristic
110C22×C4⋊C4C2312yes—
111C22×C4⋊C4C2311yescharacteristic
112C22×C4⋊C4C2314yes—
113C22×C4⋊C4C2312yes—
114C22×C4⋊C4C2314yes—
115C22×C4⋊C4C2312yes—
116C22×C4⋊C4C2311yescharacteristic
117C22×C4⋊C4C2311yescharacteristic
118C22×C4⋊C4C2311yescharacteristic
119C22×C4⋊C4C2312yes—
120C22×C4⋊C4C2311yescharacteristic
121C22×C4⋊C4C2312yes—
122C22×C4⋊C4C2312yes—
123C22×C4⋊C4C2311yescharacteristic
124C22×C22⋊C4C2312yes—
125C22×C22⋊C4C2314yes—
126C22×C22⋊C4C2314yes—
127C22×C22⋊C4C2314yes—
128C22×C22⋊C4C2312yes—
129C22×C22⋊C4C2312yes—
130C22×C22⋊C4C2314yes—
131C22×C22⋊C4C2312yes—
132C22×C22⋊C4C2312yes—
133C22×C42C2312yes—
134C22×C42C2311yescharacteristic
135C22×C42C2312yes—
136C22×C42C2312yes—
137C22×C42C2312yes—
138C22×C42C2311yescharacteristic
139C22×C42C2311yescharacteristic
140#398—216no—
141#398—28no—
142#398—44no—
143#398D418yescomplemented
144#397—28no—
145#397—216no—
146#397—28no—
147#397—44no—
148#396—216no—
149#395—216no—
150#393—216no—
151#391—216no—
152#390—28no—
153#390—28no—
154#390—24no—
155#390—24no—
156#389—28no—
157#389—28no—
158#389—28no—
159#389—28no—
160#388—28no—
161#388—28no—
162#385—216no—
163#384—24no—
164#384—24no—
165#382—28no—
166#380—28no—
167C4×C4⋊C4—24no—
168C4×C4⋊C4—24no—
169C4×C4⋊C4—28no—
170C4×C4⋊C4—24no—
171C4×C4⋊C4D418yes—
172C2×#95C2312yes—
173C2×#95C2314yes—
174C2×#95C2314yes—
175C2×#95C2312yes—
176C2×#95C2314yes—
177C2×#95C2311yescharacteristic
178C2×#95C2312yes—
179C2×#95C2314yes—
180C2×#95C2314yes—
181C2×#95C2314yes—
182C2×#95C2314yes—
183C2×#95C2312yes—
184C2×#95C2312yes—
185C2×#95C2312yes—
186C2×#95C2311yescharacteristic
187C2×#95C2311yescharacteristic
188C2×#95C2314yes—
189C2×#95C2314yes—
190C2×#95C2312yes—
191C2×#95C2312yes—
192C2×#95C2312yes—
193C2×#95C2312yes—
194C2×#95C2314yes—
195C2×#95C2314yes—
196C2×#95C2312yes—
197C2×#95C2312yes—
198C2×#95C2311yescharacteristic
199C25C2412yes—
200C22×Q8—28no—
201C22×Q8—28no—
202C22×Q8—28no—
203C22×Q8—28no—
204C22×Q8—44no—
205C22×Q8—44no—
206C22×Q8—44no—
207C22×D4—22no—
208C22×D4—22no—
209C22×D4—24no—
210C22×D4—24no—
211C22×D4C2×D414yes—
212C23×C4—24no—
213C23×C4—24no—
214C23×C4—24no—
215C23×C4—24no—
216C23×C4—42no—
217C23×C4—42no—
218C23×C4C2×D418yes—
219C23×C4C2414yes—
220C23×C4C2414yes—
221C23×C4C2412yes—
222C23×C4C2411yescharacteristic
223C23×C4C2412yes—
224C23×C4C2412yes—
225C23×C4C2412yes—
226C23×C4C2411yescharacteristic
227C23×C4C2412yes—
228C23×C4C2411yescharacteristic
229C23×C4C2411yescharacteristic
230C23×C4C2412yes—
231C23×C4C2412yes—
232C23×C4C2411yescharacteristic
233C23×C4C2411yescharacteristic
234C23×C4C2411yescharacteristic
235C4⋊Q8—432no—
236C4⋊Q8—432no—
237C22⋊Q8—464no—
238C4×D4—232no—
239#117—432no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—28no—
252C2×C4⋊C4—28no—
253C2×C4⋊C4—28no—
254C2×C4⋊C4—28no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—28no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—28no—
260C2×C4⋊C4—28no—
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—24no—
269C2×C4⋊C4—28no—
270C2×C4⋊C4—24no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—24no—
273C2×C4⋊C4—24no—
274C2×C4⋊C4—24no—
275C2×C4⋊C4—24no—
276C2×C4⋊C4—24no—
277C2×C4⋊C4—24no—
278C2×C4⋊C4—41no—
279C2×C4⋊C4—42no—
280C2×C4⋊C4—42no—
281C2×C4⋊C4—44no—
282C2×C4⋊C4—44no—
283C2×C4⋊C4—44no—
284C2×C4⋊C4—42no—
285C2×C4⋊C4—42no—
286C2×C4⋊C4C2×D414yes—
287C2×C4⋊C4C2×D418yes—
288C2×C4⋊C4C2×D414yes—
289C2×C4⋊C4C2×D414yes—
290C2×C4⋊C4C2×D414yes—
291C2×C4⋊C4C2×D414yes—
292C2×C4⋊C4#4114yes—
293C2×C22⋊C4—24no—
294C2×C22⋊C4—24no—
295C2×C22⋊C4—28no—
296C2×C22⋊C4—28no—
297C2×C22⋊C4—28no—
298C2×C22⋊C4—24no—
299C2×C22⋊C4—24no—
300C2×C22⋊C4—24no—
301C2×C22⋊C4—24no—
302C2×C22⋊C4—28no—
303C2×C22⋊C4—28no—
304C2×C22⋊C4—24no—
305C2×C22⋊C4—28no—
306C2×C22⋊C4—28no—
307C2×C22⋊C4—24no—
308C2×C22⋊C4—24no—
309C2×C22⋊C4—44no—
310C2×C22⋊C4—42no—
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⋊C4C2×D418yes—
318C2×C42—24no—
319C2×C42—22no—
320C2×C42—22no—
321C2×C42—24no—
322C2×C42—24no—
323C2×C42—24no—
324C2×C42—24no—
325C2×C42—24no—
326C2×C42—24no—
327C2×C42—22no—
328C2×C42—24no—
329C2×C42—24no—
330C2×C42—24no—
331C2×C42—24no—
332C2×C42—24no—
333C2×C42—42no—
334C2×C42C2×D418yes—
335C2×C42C2×D414yes—
336C2×C42C2×D414yes—
337#95—24no—
338#95—24no—
339#95—22no—
340#95—22no—
341#95—24no—
342#95—24no—
343#95—28no—
344#95—24no—
345#95—28no—
346#95—28no—
347#95—28no—
348#95—28no—
349#95—28no—
350#95—28no—
351#95—22no—
352#95—24no—
353#95—42no—
354#95—44no—
355#95—42no—
356#95—44no—
357#95—42no—
358#95—44no—
359#95—42no—
360#95—44no—
361#95C2×D418yes—
362#95C2×D414yes—
363#95C2×D418yes—
364C24—22no—
365C24—22no—
366C24—22no—
367C24—22no—
368C24—22no—
369C24—24no—
370C24—24no—
371C24—24no—
372C24—24no—
373C24—22no—
374C24C22×D414yes—
375C24C2511yescharacteristic
376C2×Q8—416no—
377C2×Q8—416no—
378C2×Q8—416no—
379C2×Q8—416no—
380C2×Q8—88no—
381C2×Q8—88no—
382C2×Q8—88no—
383C2×D4—28no—
384C2×D4—216no—
385C2×D4—44no—
386C2×D4—44no—
387C2×D4—48no—
388C22×C4—22no—
389C22×C4—21no—
390C22×C4—21no—
391C22×C4—22no—
392C22×C4—21no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—21no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—21no—
400C22×C4—22no—
401C22×C4—21no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—21no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—21no—
410C22×C4—22no—
411C22×C4—21no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—24no—
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—21no—
431C22×C4—22no—
432C22×C4—216no—
433C22×C4—216no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—24no—
437C22×C4—21no—
438C22×C4—24no—
439C22×C4—24no—
440C22×C4—22no—
441C22×C4—24no—
442C22×C4—24no—
443C22×C4—24no—
444C22×C4—22no—
445C22×C4—21no—
446C22×C4—22no—
447C22×C4—24no—
448C22×C4—24no—
449C22×C4—24no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—24no—
455C22×C4—24no—
456C22×C4—24no—
457C22×C4—24no—
458C22×C4—22no—
459C22×C4—24no—
460C22×C4—21no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—24no—
464C22×C4—24no—
465C22×C4—48no—
466C22×C4—48no—
467C22×C4—48no—
468C22×C4—48no—
469C22×C4—48no—
470C22×C4C22×D412yes—
471C22×C4C22×D412yes—
472C22×C4C22×D414yes—
473C22×C4C22×D412yes—
474C22×C4C22×D414yes—
475C22×C4C22×D414yes—
476C22×C4C22×D414yes—
477C22×C4C22×D414yes—
478C22×C4C22×D412yes—
479C22×C4C22×D412yes—
480C22×C4C22×D412yes—
481C22×C4C22×D412yes—
482C22×C4C22×D412yes—
483C22×C4C22×D412yes—
484C22×C4C2×#4114yes—
485C22×C4C2×#4112yes—
486C22×C4H(5,𝔽2)12yes—
487C22×C4H(5,𝔽2)12yes—
488C22×C42−1+414yes—
489C22×C42−1+414yes—
490C22×C42−1+414yes—
491C4⋊C4—28no—
492C4⋊C4—28no—
493C4⋊C4—28no—
494C4⋊C4—216no—
495C4⋊C4—28no—
496C4⋊C4—44no—
497C4⋊C4—48no—
498C4⋊C4—48no—
499C4⋊C4—416no—
500C4⋊C4—416no—
501C4⋊C4—416no—
502C4⋊C4—48no—
503C4⋊C4—48no—
504C4⋊C4—44no—
505C4⋊C4—416no—
506C4⋊C4—48no—
507C4⋊C4—84no—
508C4⋊C4—84no—
509C4⋊C4#121116yes—
510C22⋊C4—216no—
511C22⋊C4—48no—
512C22⋊C4—416no—
513C22⋊C4—416no—
514C22⋊C4—416no—
515C22⋊C4—48no—
516C22⋊C4—84no—
517C22⋊C4—84no—
518C22⋊C4—88no—
519C42—28no—
520C42—28no—
521C42—216no—
522C42—48no—
523C42—48no—
524C42—44no—
525C42—48no—
526C23—24no—
527C23—28no—
528C23—24no—
529C23—28no—
530C23—24no—
531C23—42no—
532C23—42no—
533C23—44no—
534C23—42no—
535C23—42no—
536C23—42no—
537C23—42no—
538C23—44no—
539C23—44no—
540C23—42no—
541C23—44no—
542C23—42no—
543C23—44no—
544C23—44no—
545C23—44no—
546C23—44no—
547C23—44no—
548C23—44no—
549C23C23×D411yescharacteristic
550C23C23×D411yescharacteristic
551C23C23×D411yescharacteristic
552C23C22×#4111yescharacteristic
553C23C2×H(5,𝔽2)11yescharacteristic
554C23C2×H(5,𝔽2)11yescharacteristic
555C23C2×2−1+412yes—
556C23C2×2−1+412yes—
557C23C2×2−1+412yes—
558C23#58511yescharacteristic
559C23#58512yes—
560Q8—832no—
561D4—416no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—24no—
565C2×C4—22no—
566C2×C4—24no—
567C2×C4—22no—
568C2×C4—24no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—24no—
572C2×C4—22no—
573C2×C4—24no—
574C2×C4—24no—
575C2×C4—22no—
576C2×C4—24no—
577C2×C4—24no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—24no—
581C2×C4—24no—
582C2×C4—22no—
583C2×C4—24no—
584C2×C4—24no—
585C2×C4—24no—
586C2×C4—24no—
587C2×C4—24no—
588C2×C4—22no—
589C2×C4—24no—
590C2×C4—24no—
591C2×C4—24no—
592C2×C4—24no—
593C2×C4—24no—
594C2×C4—22no—
595C2×C4—24no—
596C2×C4—24no—
597C2×C4—432no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—41no—
601C2×C4—42no—
602C2×C4—41no—
603C2×C4—41no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—44no—
609C2×C4—41no—
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—44no—
619C2×C4—42no—
620C2×C4—44no—
621C2×C4—42no—
622C2×C4—41no—
623C2×C4—44no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—44no—
627C2×C4—44no—
628C2×C4—44no—
629C2×C4—44no—
630C2×C4—41no—
631C2×C4—44no—
632C2×C4—44no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—44no—
636C2×C4—44no—
637C2×C4—44no—
638C2×C4—44no—
639C2×C4—41no—
640C2×C4—42no—
641C2×C4—44no—
642C2×C4—42no—
643C2×C4—44no—
644C2×C4—42no—
645C2×C4—44no—
646C2×C4—44no—
647C2×C4—42no—
648C2×C4—42no—
649C2×C4—42no—
650C2×C4—42no—
651C2×C4C2×#12118yes—
652C2×C4C2×#12114yes—
653C2×C4#53614yes—
654C2×C4#53618yes—
655C2×C4#53614yes—
656C2×C4#54118yes—
657C2×C4D4214yes—
658C2×C4D4214yes—
659C22—44no—
660C22—44no—
661C22—48no—
662C22—48no—
663C22—44no—
664C22—82no—
665C22—84no—
666C22—82no—
667C22—82no—
668C22—82no—
669C22—84no—
670C22C22×#12111yescharacteristic
671C22C2×#53611yescharacteristic
672C22C2×#53611yescharacteristic
673C22C2×#53612yes—
674C22C2×#54112yes—
675C22C2×D4211yescharacteristic
676C22C2×D4211yescharacteristic
677C22D4×#4111yescharacteristic
678C22#348611yescharacteristic
679C22#348811yescharacteristic
680C22#348912yes—
681C22#349611yescharacteristic
682C22#349712yes—
683C22#349912yes—
684C22#350011yescharacteristic
685C22#351612yes—
686C22#351911yescharacteristic
687C22#354412yes—
688C22#354712yes—
689C22#355012yes—
690C22#355412yes—
691C22#355512yes—
692C22#356512yes—
693C4—24no—
694C4—24no—
695C4—28no—
696C4—42no—
697C4—44no—
698C4—42no—
699C4—44no—
700C4—44no—
701C4—42no—
702C4—44no—
703C4—84no—
704C4—82no—
705C4—82no—
706C4—81no—
707C4—84no—
708C4#262118yescomplemented
709C2—84no—
710C2C2×#262111yescharacteristic
711C2#3573711yescharacteristic
712C2D4×#12111yescharacteristic
713C2#3933711yescharacteristic
714C2#3934312yes—
715C2#3936311yescharacteristic
716C2#3939912yes—
717C2#4244212yes—
718C2#4258212yes—
719C2#4280911yescharacteristic
720C2#4296511yescharacteristic
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×#2551C2semidirect
3. C2×#2621C2semidirect
5. #16518C2semidirect
6. #16506C2semidirect
7. #16295C2semidirect
8. #16264C2semidirect
9. #15151C2semidirect
13. #15133C2semidirect
14. #14907C2semidirect
15. #14053C2semidirect
16. #13918C2semidirect
17. C4×#398C2semidirect
143. #398D4semidirect
708. C4#2621semidirect

Maximal subgroups

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

subgroupindex
C2×#25512
C2×#24522
C2×#26212
C2×#23542
#165182
#165062
#162952
#162642
#151512
#151472
#151382
#151362
#151332
#149072
#140532
#139182
C4×#3982