Finite Group Databasebeta

← #4380485

Subgroups — #4380485

Order 512.

3434 conjugacy classes of subgroups.

The subgroup lattice has 698 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#544C211yesmaximal, characteristic
2C2×#2822C211yesmaximal, characteristic, complemented
3C2×#2611C211yesmaximal, characteristic
4C2×#2660C211yesmaximal, characteristic, complemented
5#16683C212yesmaximal, complemented
6#16507C212yesmaximal, complemented
7#15881C212yesmaximal, complemented
8#15850C212yesmaximal, complemented
9#15670C212yesmaximal
10#15430C212yesmaximal
11#15318C214yesmaximal, complemented
12#15279C214yesmaximal
13#14906C212yesmaximal
14#14786C212yesmaximal
15#14274C211yesmaximal, characteristic, complemented
16#14170C211yesmaximal, characteristic, complemented
17#13957C211yesmaximal, characteristic
18C2×#544—28no—
19C2×#544—28no—
20C2×#544—28no—
21C22×C4⋊Q8C2211yescharacteristic
22C22×C4⋊Q8C2212yes—
23C22×C4⋊Q8C2211yescharacteristic
24C22×#125C2212yes—
25C22×#125C2211yescharacteristic
26C22×#125C2211yescharacteristic
27C22×#125C2211yescharacteristic
28C22×C22⋊Q8C2212yes—
29C22×C22⋊Q8C2212yes—
30C22×#117C2211yescharacteristic
31C22×#117C2211yescharacteristic
32C23×C4⋊C4C2211yescharacteristic
33#2822—28no—
34#2660—28no—
35#2611—28no—
36C2×#400C2212yes—
37C2×#400C2212yes—
38C2×#400C2211yescharacteristic
39C2×#399C2211yescharacteristic
40C2×#398C2211yescharacteristic
41C2×#398C2212yes—
42C2×#398C2212yes—
43C2×#398C2212yes—
44C2×#398C2212yes—
45C2×#398C2214yes—
46C2×#397C2211yescharacteristic
47C2×#395C2214yes—
48C2×#395C2214yes—
49C2×#391C2214yes—
50C2×#389C2212yes—
51C2×#388C2212yes—
52C2×#387C2212yes—
53C2×#385C2212yes—
54C2×#385C2214yes—
55C2×#384C2212yes—
56C2×#382C2212yes—
57C2×#382C2211yescharacteristic
58C2×#376C2211yescharacteristic
59C2×#376C2211yescharacteristic
60#1446C2214yes—
61#1445C2212yes—
62#1445C2212yes—
63#1445C2212yes—
64#1445C2212yes—
65#1445C2212yes—
66#1445C2212yes—
67#1444C2214yes—
68#1444C2214yes—
69#1444C2214yes—
70#1442C2214yes—
71#1442C2214yes—
72#1442C2214yes—
73#1441C2214yes—
74#1441C2214yes—
75#1441C2212yes—
76#1440C2214yes—
77#1439C2214yes—
78#1439C2212yes—
79#1437C2212yes—
80#1436C2214yes—
81#1436C2212yes—
82#1436C2212yes—
83#1436C2212yes—
84#1436C2214yes—
85#1436C2214yes—
86#1434C2212yes—
87C22⋊#95C2214yes—
88C22⋊#95C2212yes—
89C23×Q8C2312yes—
90C23×Q8C2312yes—
91C24×C4C2311yescharacteristic
92C24×C4C2312yes—
93#544—2128no—
94C2×C4⋊Q8—28no—
95C2×C4⋊Q8—28no—
96C2×C4⋊Q8—24no—
97C2×C4⋊Q8—28no—
98C2×C4⋊Q8—24no—
99C2×C4⋊Q8—24no—
100C2×C4⋊Q8—24no—
101C2×C4⋊Q8—24no—
102C2×C4⋊Q8D418yes—
103C2×#125—28no—
104C2×#125—28no—
105C2×#125—24no—
106C2×#125—28no—
107C2×#125—24no—
108C2×#125—24no—
109C2×#125—24no—
110C2×#125—24no—
111C2×#125—24no—
112C2×#125—24no—
113C2×#125—24no—
114C2×#125D418yes—
115C2×C22⋊Q8—28no—
116C2×C22⋊Q8—28no—
117C2×C22⋊Q8—28no—
118C2×C22⋊Q8—28no—
119C2×C22⋊Q8—28no—
120C2×C22⋊Q8—28no—
121C2×#117—24no—
122C2×#117—24no—
123C2×#117—24no—
124C2×#117—24no—
125C2×#117—24no—
126C2×#117D418yes—
127C22×C4⋊C4—24no—
128C22×C4⋊C4—24no—
129C22×C4⋊C4—24no—
130C22×C4⋊C4C2312yes—
131C22×C4⋊C4C2311yescharacteristic
132C22×C4⋊C4C2312yes—
133C22×C4⋊C4C2311yescharacteristic
134C22×C4⋊C4C2312yes—
135C22×C4⋊C4C2312yes—
136C22×C4⋊C4C2312yes—
137C22×C4⋊C4C2312yes—
138C22×C4⋊C4C2312yes—
139C22×C4⋊C4C2312yes—
140C22×C4⋊C4C2311yescharacteristic
141C22×C4⋊C4C2311yescharacteristic
142C22×C4⋊C4C2311yescharacteristic
143C22×C4⋊C4C2312yes—
144C22×C4⋊C4C2311yescharacteristic
145C22×C4⋊C4C2312yes—
146C22×C4⋊C4C2311yescharacteristic
147C22×C4⋊C4C2311yescharacteristic
148C22×C4⋊C4C2312yes—
149C22×C4⋊C4C2314yes—
150C22×C4⋊C4C2314yes—
151C22×C4⋊C4C2314yes—
152C22×C4⋊C4C2312yes—
153C22×C22⋊C4C2312yes—
154C22×C22⋊C4C2312yes—
155C22×C22⋊C4C2312yes—
156C22×C22⋊C4C2314yes—
157C22×C22⋊C4C2311yescharacteristic
158C22×C22⋊C4C2311yescharacteristic
159C22×C42C2312yes—
160C22×C42C2312yes—
161C22×C42C2312yes—
162C22×C42C2311yescharacteristic
163C22×C42C2311yescharacteristic
164#400—28no—
165#400—28no—
166#400—24no—
167#399—24no—
168#398—28no—
169#398—28no—
170#398—216no—
171#398—24no—
172#398—28no—
173#398—28no—
174#397—24no—
175#395—216no—
176#395—48no—
177#391—48no—
178#389—28no—
179#388—28no—
180#387—28no—
181#385—28no—
182#385—48no—
183#384—28no—
184#382—28no—
185#382—24no—
186#376—24no—
187#376—24no—
188C2×#95C2312yes—
189C2×#95C2314yes—
190C2×#95C2314yes—
191C2×#95C2314yes—
192C2×#95C2314yes—
193C2×#95C2311yescharacteristic
194C2×#95C2312yes—
195C2×#95C2314yes—
196C2×#95C2314yes—
197C2×#95C2312yes—
198C2×#95C2311yescharacteristic
199C2×#95C2314yes—
200C2×#95C2312yes—
201C2×#95C2312yes—
202C2×#95C2312yes—
203C2×#95C2312yes—
204C2×#95C2312yes—
205C2×#95C2314yes—
206C2×#95C2314yes—
207C2×#95C2314yes—
208C2×#95C2314yes—
209C2×#95C2314yes—
210C2×#95C2312yes—
211C2×#95C2314yes—
212C2×#95C2314yes—
213C2×#95C2314yes—
214C2×#95C2314yes—
215C25C2411yescharacteristic
216C22×Q8—24no—
217C22×Q8—24no—
218C22×Q8—24no—
219C22×Q8—24no—
220C22×Q8—24no—
221C22×Q8—44no—
222C22×Q8—44no—
223C22×Q8—44no—
224C22×Q8—44no—
225C22×Q8C2×D418yes—
226C23×C4—24no—
227C23×C4—22no—
228C23×C4—22no—
229C23×C4—24no—
230C23×C4—24no—
231C23×C4—24no—
232C23×C4—24no—
233C23×C4—44no—
234C23×C4—44no—
235C23×C4C2×D414yes—
236C23×C4C2412yes—
237C23×C4C2412yes—
238C23×C4C2414yes—
239C23×C4C2412yes—
240C23×C4C2412yes—
241C23×C4C2411yescharacteristic
242C23×C4C2412yes—
243C23×C4C2411yescharacteristic
244C23×C4C2412yes—
245C23×C4C2412yes—
246C23×C4C2412yes—
247C23×C4C2414yes—
248C23×C4C2411yescharacteristic
249C23×C4C2411yescharacteristic
250C23×C4C2411yescharacteristic
251C23×C4C2411yescharacteristic
252C4⋊Q8—264no—
253C4⋊Q8—232no—
254C4⋊Q8—232no—
255#125—264no—
256#125—232no—
257#125—232no—
258#125—416no—
259C22⋊Q8—264no—
260C22⋊Q8—264no—
261#117—232no—
262#117—232no—
263C2×C4⋊C4—232no—
264C2×C4⋊C4—24no—
265C2×C4⋊C4—24no—
266C2×C4⋊C4—24no—
267C2×C4⋊C4—22no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—22no—
273C2×C4⋊C4—22no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—22no—
279C2×C4⋊C4—24no—
280C2×C4⋊C4—24no—
281C2×C4⋊C4—24no—
282C2×C4⋊C4—24no—
283C2×C4⋊C4—24no—
284C2×C4⋊C4—24no—
285C2×C4⋊C4—24no—
286C2×C4⋊C4—24no—
287C2×C4⋊C4—24no—
288C2×C4⋊C4—24no—
289C2×C4⋊C4—24no—
290C2×C4⋊C4—24no—
291C2×C4⋊C4—24no—
292C2×C4⋊C4—24no—
293C2×C4⋊C4—24no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—24no—
297C2×C4⋊C4—24no—
298C2×C4⋊C4—24no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—24no—
301C2×C4⋊C4—24no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—24no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—24no—
307C2×C4⋊C4—24no—
308C2×C4⋊C4—24no—
309C2×C4⋊C4—28no—
310C2×C4⋊C4—24no—
311C2×C4⋊C4—28no—
312C2×C4⋊C4—28no—
313C2×C4⋊C4—28no—
314C2×C4⋊C4—42no—
315C2×C4⋊C4—44no—
316C2×C4⋊C4—44no—
317C2×C4⋊C4—44no—
318C2×C4⋊C4—44no—
319C2×C4⋊C4—44no—
320C2×C4⋊C4—44no—
321C2×C4⋊C4—44no—
322C2×C4⋊C4C2×D414yes—
323C2×C4⋊C4C2×D414yes—
324C2×C4⋊C4C2×D414yes—
325C2×C4⋊C4C2×D414yes—
326C2×C4⋊C4C2×D414yes—
327C2×C4⋊C4C2×D414yes—
328C2×C4⋊C4C2×D418yes—
329C2×C4⋊C4C2×D418yes—
330C2×C4⋊C4C2×D418yes—
331C2×C22⋊C4—24no—
332C2×C22⋊C4—24no—
333C2×C22⋊C4—24no—
334C2×C22⋊C4—22no—
335C2×C22⋊C4—22no—
336C2×C22⋊C4—22no—
337C2×C22⋊C4—22no—
338C2×C22⋊C4—24no—
339C2×C22⋊C4—24no—
340C2×C22⋊C4—28no—
341C2×C22⋊C4—28no—
342C2×C22⋊C4—24no—
343C2×C22⋊C4—28no—
344C2×C22⋊C4—42no—
345C2×C22⋊C4—44no—
346C2×C22⋊C4C2×D414yes—
347C2×C22⋊C4C2×D414yes—
348C2×C42—24no—
349C2×C42—24no—
350C2×C42—24no—
351C2×C42—22no—
352C2×C42—22no—
353C2×C42—24no—
354C2×C42—28no—
355C2×C42—24no—
356C2×C42—24no—
357C2×C42—42no—
358C2×C42C2×D414yes—
359C2×C42C2×D414yes—
360C2×C42C2×D414yes—
361C2×C42C2×D414yes—
362#95—24no—
363#95—24no—
364#95—22no—
365#95—24no—
366#95—22no—
367#95—28no—
368#95—28no—
369#95—24no—
370#95—24no—
371#95—24no—
372#95—24no—
373#95—24no—
374#95—24no—
375#95—28no—
376#95—28no—
377#95—28no—
378#95—28no—
379#95—28no—
380#95—28no—
381#95—28no—
382#95—28no—
383#95—28no—
384#95—28no—
385#95—44no—
386#95—44no—
387#95—44no—
388#95—44no—
389C24—21no—
390C24—21no—
391C24—22no—
392C24—24no—
393C24—22no—
394C24—22no—
395C24C22×D412yes—
396C242−1+414yes—
397C24C2511yescharacteristic
398C2×Q8—216no—
399C2×Q8—216no—
400C2×Q8—416no—
401C2×Q8—88no—
402C2×Q8—84no—
403C2×Q8—84no—
404C2×Q8—88no—
405C2×Q8—88no—
406C2×Q8—84no—
407C2×Q8—84no—
408C22×C4—216no—
409C22×C4—216no—
410C22×C4—28no—
411C22×C4—21no—
412C22×C4—21no—
413C22×C4—22no—
414C22×C4—24no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—28no—
420C22×C4—28no—
421C22×C4—24no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—21no—
425C22×C4—22no—
426C22×C4—21no—
427C22×C4—22no—
428C22×C4—21no—
429C22×C4—22no—
430C22×C4—21no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—24no—
436C22×C4—24no—
437C22×C4—24no—
438C22×C4—24no—
439C22×C4—22no—
440C22×C4—24no—
441C22×C4—24no—
442C22×C4—24no—
443C22×C4—24no—
444C22×C4—24no—
445C22×C4—24no—
446C22×C4—24no—
447C22×C4—24no—
448C22×C4—24no—
449C22×C4—24no—
450C22×C4—22no—
451C22×C4—24no—
452C22×C4—22no—
453C22×C4—24no—
454C22×C4—216no—
455C22×C4—22no—
456C22×C4—24no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—24no—
464C22×C4—24no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—22no—
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—22no—
477C22×C4—22no—
478C22×C4—22no—
479C22×C4—88no—
480C22×C4—84no—
481C22×C4—84no—
482C22×C4—88no—
483C22×C4C22×D414yes—
484C22×C4C22×D412yes—
485C22×C4C22×D414yes—
486C22×C4C22×D412yes—
487C22×C4C22×D414yes—
488C22×C4C22×D414yes—
489C22×C4C22×D412yes—
490C22×C4C22×D412yes—
491C22×C4C22×D412yes—
492C22×C4C22×D412yes—
493C22×C4C22×D412yes—
494C22×C4C22×D412yes—
495C22×C4C22×D412yes—
496C22×C4C22×D412yes—
497C22×C4C22×D412yes—
498C22×C4C22×D412yes—
499C22×C4H(5,𝔽2)14yes—
500C22×C42−1+418yes—
501C22×C42−1+414yes—
502C4⋊C4—216no—
503C4⋊C4—216no—
504C4⋊C4—216no—
505C4⋊C4—216no—
506C4⋊C4—28no—
507C4⋊C4—28no—
508C4⋊C4—28no—
509C4⋊C4—28no—
510C4⋊C4—28no—
511C4⋊C4—28no—
512C4⋊C4—216no—
513C4⋊C4—216no—
514C4⋊C4—216no—
515C4⋊C4—48no—
516C4⋊C4—48no—
517C4⋊C4—44no—
518C4⋊C4—44no—
519C4⋊C4—48no—
520C4⋊C4—48no—
521C4⋊C4—84no—
522C4⋊C4—88no—
523C4⋊C4—88no—
524C4⋊C4—88no—
525C22⋊C4—216no—
526C22⋊C4—28no—
527C22⋊C4—28no—
528C22⋊C4—48no—
529C22⋊C4—416no—
530C22⋊C4—48no—
531C42—216no—
532C42—216no—
533C42—28no—
534C42—48no—
535C42#120116yes—
536C23—22no—
537C23—22no—
538C23—24no—
539C23—24no—
540C23—24no—
541C23—24no—
542C23—28no—
543C23—28no—
544C23—42no—
545C23—42no—
546C23—41no—
547C23—42no—
548C23—41no—
549C23—44no—
550C23—44no—
551C23#53614yes—
552C23C23×D411yescharacteristic
553C23C23×D411yescharacteristic
554C23C23×D411yescharacteristic
555C23C2×H(5,𝔽2)12yes—
556C23C2×2−1+412yes—
557C23C2×2−1+412yes—
558C23C2×2−1+414yes—
559C23#58512yes—
560Q8—816no—
561Q8—816no—
562C2×C4—232no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—24no—
570C2×C4—24no—
571C2×C4—24no—
572C2×C4—24no—
573C2×C4—24no—
574C2×C4—22no—
575C2×C4—24no—
576C2×C4—24no—
577C2×C4—22no—
578C2×C4—24no—
579C2×C4—22no—
580C2×C4—24no—
581C2×C4—22no—
582C2×C4—24no—
583C2×C4—24no—
584C2×C4—28no—
585C2×C4—24no—
586C2×C4—24no—
587C2×C4—28no—
588C2×C4—28no—
589C2×C4—24no—
590C2×C4—28no—
591C2×C4—24no—
592C2×C4—24no—
593C2×C4—28no—
594C2×C4—28no—
595C2×C4—28no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—44no—
601C2×C4—44no—
602C2×C4—44no—
603C2×C4—44no—
604C2×C4—44no—
605C2×C4—44no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—41no—
610C2×C4—42no—
611C2×C4—44no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—41no—
615C2×C4—42no—
616C2×C4—44no—
617C2×C4—44no—
618C2×C4—44no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—44no—
625C2×C4—44no—
626C2×C4—44no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—44no—
633C2×C4—42no—
634C2×C4—44no—
635C2×C4—42no—
636C2×C4—42no—
637C2×C4—44no—
638C2×C4—44no—
639C2×C4—44no—
640C2×C4—44no—
641C2×C4—44no—
642C2×C4—44no—
643C2×C4—816no—
644C2×C4C2×#12014yes—
645C2×C4C2×#12014yes—
646C2×C4C2×#12014yes—
647C2×C4#53614yes—
648C22—24no—
649C22—28no—
650C22—42no—
651C22—44no—
652C22—44no—
653C22—48no—
654C22—42no—
655C22—44no—
656C22C22×#12011yescharacteristic
657C22C2×#53611yescharacteristic
658C22C2×#53611yescharacteristic
659C22#348612yes—
660C22#348612yes—
661C22#348612yes—
662C22#348812yes—
663C22#349612yes—
664C22#349811yescharacteristic
665C22#349912yes—
666C22#349912yes—
667C22#350011yescharacteristic
668C22#353814yes—
669C22#354414yes—
670C22#356314yes—
671C22#356314yes—
672C4—24no—
673C4—24no—
674C4—24no—
675C4—24no—
676C4—44no—
677C4—42no—
678C4—48no—
679C4—44no—
680C4—44no—
681C4—42no—
682C4—44no—
683C4—44no—
684C4—84no—
685C4—82no—
686C4—82no—
687C4—82no—
688C2—44no—
689C2#3524211yescharacteristic
690C2#3551311yescharacteristic
691C2#3566611yescharacteristic
692C2#4193912yes—
693C2#4296412yes—
694C2#4779914yes—
695C2#5154312yes—
696C2#5154412yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC25C24
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. C2×#2822C2semidirect
4. C2×#2660C2semidirect
5. #16683C2semidirect
6. #16507C2semidirect
7. #15881C2semidirect
8. #15850C2semidirect
11. #15318C2semidirect
15. #14274C2semidirect
16. #14170C2semidirect

Maximal subgroups

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

subgroupindex
C22×#5442
C2×#28222
C2×#26112
C2×#26602
#166832
#165072
#158812
#158502
#156702
#154302
#153182
#152792
#149062
#147862
#142742
#141702
#139572