Finite Group Databasebeta

← #4418779

Subgroups — #4418779

Order 512.

3274 conjugacy classes of subgroups.

The subgroup lattice has 687 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#557C211yesmaximal, characteristic
2C2×#2822C211yesmaximal, characteristic
3C2×#2584C211yesmaximal, characteristic
4C2×#2634C211yesmaximal, characteristic
5#16683C212yesmaximal
6#16634C212yesmaximal
7#16607C212yesmaximal
8#16507C212yesmaximal
9#15847C214yesmaximal
10#15318C214yesmaximal
11#14884C212yesmaximal
12#14880C212yesmaximal
13#14879C212yesmaximal
14#14876C212yesmaximal
15#14346C211yesmaximal, characteristic
16#14289C211yesmaximal, characteristic
17#13942C211yesmaximal, characteristic
18C2×#557—28no—
19C2×#557—28no—
20C2×#557—28no—
21C22×C4⋊Q8C2211yescharacteristic
22C22×C4⋊Q8C2212yes—
23C22×#125C2211yescharacteristic
24C22×#125C2211yescharacteristic
25C22×#125C2212yes—
26C22×#125C2212yes—
27C22×#125C2211yescharacteristic
28C22×C4×Q8C2211yescharacteristic
29C22×C4×Q8C2211yescharacteristic
30C22×C4×Q8C2212yes—
31C22×C4×Q8C2211yescharacteristic
32C22×C4×Q8C2211yescharacteristic
33#2822—28no—
34#2634—28no—
35#2584—28no—
36C2×#400C2212yes—
37C2×#400C2211yescharacteristic
38C2×#400C2212yes—
39C2×#400C2212yes—
40C2×#400C2211yescharacteristic
41C2×#398C2212yes—
42C2×#398C2211yescharacteristic
43C2×#398C2212yes—
44C2×#398C2211yescharacteristic
45C2×#398C2212yes—
46C2×#398C2214yes—
47C2×#395C2212yes—
48C2×#395C2212yes—
49C2×#395C2214yes—
50C2×#391C2212yes—
51C2×#389C2212yes—
52C2×#389C2212yes—
53C2×#389C2214yes—
54C2×#387C2212yes—
55C2×#387C2212yes—
56C2×#383C2211yescharacteristic
57C2×#382C2211yescharacteristic
58C2×C4×C4⋊C4C2211yescharacteristic
59#1446C2212yes—
60#1446C2214yes—
61#1445C2212yes—
62#1445C2212yes—
63#1445C2212yes—
64#1444C2214yes—
65#1444C2214yes—
66#1444C2214yes—
67#1442C2214yes—
68#1442C2214yes—
69#1442C2214yes—
70#1442C2214yes—
71#1441C2214yes—
72#1441C2214yes—
73#1441C2214yes—
74#1441C2212yes—
75#1441C2214yes—
76#1441C2212yes—
77#1441C2212yes—
78#1441C2212yes—
79#1436C2214yes—
80#1436C2214yes—
81#1436C2214yes—
82#1436C2214yes—
83#1436C2214yes—
84C4⋊#95C2212yes—
85#1434C2212yes—
86#1434C2212yes—
87#1434C2212yes—
88#1434C2212yes—
89C23×Q8C2311yescharacteristic
90C23×Q8C2312yes—
91C23×Q8C2311yescharacteristic
92#557—2128no—
93C2×C4⋊Q8—28no—
94C2×C4⋊Q8—28no—
95C2×C4⋊Q8—24no—
96C2×C4⋊Q8—28no—
97C2×C4⋊Q8—24no—
98C2×C4⋊Q8—24no—
99C2×#125—28no—
100C2×#125—28no—
101C2×#125—24no—
102C2×#125—28no—
103C2×#125—24no—
104C2×#125—28no—
105C2×#125—28no—
106C2×#125—24no—
107C2×#125—28no—
108C2×#125—24no—
109C2×#125—24no—
110C2×#125—24no—
111C2×#125—24no—
112C2×#125—24no—
113C2×#125D418yes—
114C2×C4×Q8—24no—
115C2×C4×Q8—24no—
116C2×C4×Q8—24no—
117C2×C4×Q8—28no—
118C2×C4×Q8—28no—
119C2×C4×Q8—24no—
120C2×C4×Q8—24no—
121C2×C4×Q8—24no—
122C2×C4×Q8—24no—
123C2×C4×Q8—28no—
124C2×C4×Q8—24no—
125C2×C4×Q8—24no—
126C2×C4×Q8—24no—
127C2×C4×Q8D418yes—
128C2×C4×Q8D418yes—
129C22×C4⋊C4C2312yes—
130C22×C4⋊C4C2312yes—
131C22×C4⋊C4C2312yes—
132C22×C4⋊C4C2314yes—
133C22×C4⋊C4C2311yescharacteristic
134C22×C4⋊C4C2311yescharacteristic
135C22×C4⋊C4C2312yes—
136C22×C4⋊C4C2312yes—
137C22×C4⋊C4C2312yes—
138C22×C4⋊C4C2312yes—
139C22×C4⋊C4C2312yes—
140C22×C4⋊C4C2312yes—
141C22×C4⋊C4C2311yescharacteristic
142C22×C4⋊C4C2311yescharacteristic
143C22×C4⋊C4C2312yes—
144C22×C4⋊C4C2311yescharacteristic
145C22×C4⋊C4C2312yes—
146C22×C4⋊C4C2311yescharacteristic
147C22×C4⋊C4C2311yescharacteristic
148C22×C4⋊C4C2311yescharacteristic
149C22×C4⋊C4C2312yes—
150C22×C4⋊C4C2314yes—
151C22×C4⋊C4C2314yes—
152C22×C4⋊C4C2312yes—
153C22×C4⋊C4C2312yes—
154C22×C42C2312yes—
155C22×C42C2312yes—
156C22×C42C2311yescharacteristic
157C22×C42C2312yes—
158C22×C42C2311yescharacteristic
159C22×C42C2312yes—
160C22×C42C2311yescharacteristic
161#400—28no—
162#400—28no—
163#400—28no—
164#400—24no—
165#400—24no—
166#398—28no—
167#398—28no—
168#398—216no—
169#398—28no—
170#398—24no—
171#398—24no—
172#395—28no—
173#395—28no—
174#395—48no—
175#391—44no—
176#389—216no—
177#389—28no—
178#389—28no—
179#387—28no—
180#387—28no—
181#383—24no—
182#382—24no—
183C4×C4⋊C4—24no—
184C2×#95C2312yes—
185C2×#95C2312yes—
186C2×#95C2312yes—
187C2×#95C2312yes—
188C2×#95C2314yes—
189C2×#95C2314yes—
190C2×#95C2314yes—
191C2×#95C2314yes—
192C2×#95C2314yes—
193C2×#95C2312yes—
194C2×#95C2311yescharacteristic
195C2×#95C2311yescharacteristic
196C2×#95C2312yes—
197C2×#95C2312yes—
198C2×#95C2312yes—
199C2×#95C2314yes—
200C2×#95C2312yes—
201C2×#95C2312yes—
202C2×#95C2314yes—
203C2×#95C2314yes—
204C2×#95C2314yes—
205C2×#95C2314yes—
206C2×#95C2314yes—
207C2×#95C2314yes—
208C2×#95C2312yes—
209C2×#95C2314yes—
210C2×#95C2314yes—
211C2×#95C2314yes—
212C2×#95C2314yes—
213C2×#95C2314yes—
214C22×Q8—22no—
215C22×Q8—22no—
216C22×Q8—24no—
217C22×Q8—24no—
218C22×Q8—22no—
219C22×Q8—22no—
220C22×Q8—24no—
221C22×Q8—22no—
222C22×Q8—24no—
223C22×Q8—28no—
224C22×Q8—22no—
225C22×Q8—24no—
226C22×Q8—41no—
227C22×Q8—41no—
228C22×Q8—42no—
229C22×Q8—42no—
230C22×Q8C2×D414yes—
231C22×Q8C2×D414yes—
232C23×C4C2412yes—
233C23×C4C2412yes—
234C23×C4C2414yes—
235C23×C4C2412yes—
236C23×C4C2412yes—
237C23×C4C2411yescharacteristic
238C23×C4C2412yes—
239C23×C4C2412yes—
240C23×C4C2412yes—
241C23×C4C2411yescharacteristic
242C23×C4C2411yescharacteristic
243C23×C4C2414yes—
244C23×C4C2411yescharacteristic
245C23×C4C2411yescharacteristic
246C23×C4C2411yescharacteristic
247C23×C4C2411yescharacteristic
248C23×C4C2412yes—
249C4⋊Q8—264no—
250C4⋊Q8—232no—
251#125—264no—
252#125—232no—
253#125—264no—
254#125—232no—
255#125—416no—
256C4×Q8—232no—
257C4×Q8—264no—
258C4×Q8—232no—
259C4×Q8—232no—
260C4×Q8—232no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—24no—
264C2×C4⋊C4—24no—
265C2×C4⋊C4—24no—
266C2×C4⋊C4—24no—
267C2×C4⋊C4—24no—
268C2×C4⋊C4—24no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—22no—
273C2×C4⋊C4—24no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—24no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—24no—
278C2×C4⋊C4—24no—
279C2×C4⋊C4—24no—
280C2×C4⋊C4—24no—
281C2×C4⋊C4—24no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—22no—
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—24no—
295C2×C4⋊C4—28no—
296C2×C4⋊C4—24no—
297C2×C4⋊C4—24no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—24no—
301C2×C4⋊C4—24no—
302C2×C4⋊C4—24no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—24no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—24no—
309C2×C4⋊C4—24no—
310C2×C4⋊C4—28no—
311C2×C4⋊C4—24no—
312C2×C4⋊C4—28no—
313C2×C4⋊C4—28no—
314C2×C4⋊C4—24no—
315C2×C4⋊C4—24no—
316C2×C4⋊C4—28no—
317C2×C4⋊C4—28no—
318C2×C4⋊C4—42no—
319C2×C4⋊C4—42no—
320C2×C4⋊C4—42no—
321C2×C4⋊C4—44no—
322C2×C4⋊C4—44no—
323C2×C4⋊C4—42no—
324C2×C4⋊C4—44no—
325C2×C4⋊C4—44no—
326C2×C4⋊C4C2×D414yes—
327C2×C4⋊C4C2×D418yes—
328C2×C4⋊C4C2×D414yes—
329C2×C4⋊C4C2×D414yes—
330C2×C4⋊C4C2×D414yes—
331C2×C4⋊C4C2×D414yes—
332C2×C4⋊C4C2×D418yes—
333C2×C4⋊C4C2×D414yes—
334C2×C4⋊C4C2×D418yes—
335C2×C42—24no—
336C2×C42—24no—
337C2×C42—24no—
338C2×C42—24no—
339C2×C42—24no—
340C2×C42—22no—
341C2×C42—22no—
342C2×C42—24no—
343C2×C42—24no—
344C2×C42—24no—
345C2×C42—22no—
346C2×C42—22no—
347C2×C42—24no—
348C2×C42—42no—
349C2×C42—42no—
350C2×C42C2×D414yes—
351C2×C42C2×D418yes—
352C2×C42C2×D414yes—
353C2×C42C2×D414yes—
354C2×C42C2×D414yes—
355C2×C42C2×D414yes—
356#95—24no—
357#95—24no—
358#95—24no—
359#95—24no—
360#95—24no—
361#95—22no—
362#95—22no—
363#95—24no—
364#95—24no—
365#95—24no—
366#95—24no—
367#95—24no—
368#95—24no—
369#95—28no—
370#95—28no—
371#95—28no—
372#95—28no—
373#95—28no—
374#95—28no—
375#95—28no—
376#95—28no—
377#95—28no—
378#95—28no—
379#95—28no—
380#95—28no—
381#95—44no—
382#95—44no—
383#95—44no—
384#95—44no—
385#95—44no—
386C24C2511yescharacteristic
387C2×Q8—216no—
388C2×Q8—216no—
389C2×Q8—216no—
390C2×Q8—28no—
391C2×Q8—28no—
392C2×Q8—216no—
393C2×Q8—28no—
394C2×Q8—44no—
395C2×Q8—44no—
396C2×Q8—44no—
397C2×Q8—48no—
398C2×Q8—48no—
399C2×Q8—44no—
400C2×Q8—44no—
401C2×Q8—48no—
402C2×Q8—48no—
403C2×Q8—48no—
404C2×Q8—48no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—21no—
408C22×C4—21no—
409C22×C4—21no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—21no—
414C22×C4—21no—
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—24no—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—22no—
422C22×C4—21no—
423C22×C4—22no—
424C22×C4—21no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—21no—
429C22×C4—22no—
430C22×C4—21no—
431C22×C4—24no—
432C22×C4—21no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—24no—
436C22×C4—24no—
437C22×C4—24no—
438C22×C4—24no—
439C22×C4—24no—
440C22×C4—24no—
441C22×C4—24no—
442C22×C4—21no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—24no—
448C22×C4—21no—
449C22×C4—21no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—24no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—24no—
462C22×C4—22no—
463C22×C4—22no—
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—22no—
475C22×C4—22no—
476C22×C4—22no—
477C22×C4—22no—
478C22×C4—24no—
479C22×C4—22no—
480C22×C4—22no—
481C22×C4C22×D412yes—
482C22×C4C22×D414yes—
483C22×C4C22×D412yes—
484C22×C4C22×D414yes—
485C22×C4C22×D412yes—
486C22×C4C22×D414yes—
487C22×C4C22×D412yes—
488C22×C4C22×D412yes—
489C22×C4C22×D412yes—
490C22×C4C22×D412yes—
491C22×C4C22×D412yes—
492C22×C4C22×D412yes—
493C22×C4C22×D412yes—
494C22×C4C22×D412yes—
495C22×C4C22×D412yes—
496C22×C4C22×D414yes—
497C22×C4C22×D412yes—
498C22×C4H(5,𝔽2)12yes—
499C22×C42−1+418yes—
500C22×C42−1+418yes—
501C22×C42−1+412yes—
502C4⋊C4—216no—
503C4⋊C4—216no—
504C4⋊C4—216no—
505C4⋊C4—216no—
506C4⋊C4—216no—
507C4⋊C4—216no—
508C4⋊C4—28no—
509C4⋊C4—28no—
510C4⋊C4—28no—
511C4⋊C4—216no—
512C4⋊C4—28no—
513C4⋊C4—216no—
514C4⋊C4—28no—
515C4⋊C4—28no—
516C4⋊C4—216no—
517C4⋊C4—48no—
518C4⋊C4—44no—
519C4⋊C4—44no—
520C4⋊C4—48no—
521C4⋊C4—48no—
522C4⋊C4—416no—
523C4⋊C4—48no—
524C4⋊C4—84no—
525C4⋊C4—88no—
526C4⋊C4—88no—
527C42—216no—
528C42—216no—
529C42—216no—
530C42—28no—
531C42—28no—
532C42—84no—
533C42#120116yes—
534C23C23×D411yescharacteristic
535C23C23×D411yescharacteristic
536C23C23×D411yescharacteristic
537C23C2×H(5,𝔽2)11yescharacteristic
538C23C2×2−1+414yes—
539C23C2×2−1+411yescharacteristic
540C23C2×2−1+414yes—
541C23#58511yescharacteristic
542C23#58511yescharacteristic
543Q8—232no—
544Q8—416no—
545Q8—432no—
546C2×C4—24no—
547C2×C4—22no—
548C2×C4—24no—
549C2×C4—24no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—24no—
554C2×C4—24no—
555C2×C4—24no—
556C2×C4—28no—
557C2×C4—28no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—22no—
569C2×C4—24no—
570C2×C4—22no—
571C2×C4—24no—
572C2×C4—24no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—28no—
577C2×C4—24no—
578C2×C4—24no—
579C2×C4—24no—
580C2×C4—24no—
581C2×C4—28no—
582C2×C4—24no—
583C2×C4—24no—
584C2×C4—28no—
585C2×C4—28no—
586C2×C4—44no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—44no—
596C2×C4—41no—
597C2×C4—42no—
598C2×C4—41no—
599C2×C4—42no—
600C2×C4—41no—
601C2×C4—42no—
602C2×C4—41no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—44no—
607C2×C4—44no—
608C2×C4—44no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—44no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—44no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—44no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—44no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—44no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—44no—
631C2×C4—44no—
632C2×C4—44no—
633C2×C4—44no—
634C2×C4—44no—
635C2×C4—44no—
636C2×C4—44no—
637C2×C4—44no—
638C2×C4C2×#12014yes—
639C2×C4C2×#12014yes—
640C2×C4C2×#12014yes—
641C2×C4#53618yes—
642C2×C4#53714yes—
643C22C22×#12011yescharacteristic
644C22C2×#53612yes—
645C22C2×#53711yescharacteristic
646C22#344911yescharacteristic
647C22#348612yes—
648C22#348612yes—
649C22#348612yes—
650C22#348811yescharacteristic
651C22#348811yescharacteristic
652C22#348911yescharacteristic
653C22#348911yescharacteristic
654C22#349912yes—
655C22#349912yes—
656C22#355214yes—
657C22#355514yes—
658C22#355614yes—
659C22#356314yes—
660C4—24no—
661C4—24no—
662C4—28no—
663C4—24no—
664C4—24no—
665C4—48no—
666C4—42no—
667C4—44no—
668C4—42no—
669C4—42no—
670C4—44no—
671C4—44no—
672C4—84no—
673C4—82no—
674C4—82no—
675C4—82no—
676C4—82no—
677C2#3524211yescharacteristic
678C2#3525111yescharacteristic
679C2#3566611yescharacteristic
680C2#4231312yes—
681C2#4279112yes—
682C2#4281112yes—
683C2#4281412yes—
684C2#5069212yes—
685C2#5120612yes—
subgroupG/H
center, commutator, Frattini, socle, prime radical, cosocleC24C25
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Maximal subgroups

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

subgroupindex
C22×#5572
C2×#28222
C2×#25842
C2×#26342
#166832
#166342
#166072
#165072
#158472
#153182
#148842
#148802
#148792
#148762
#143462
#142892
#139422