Finite Group Databasebeta

← #4420537

Subgroups — #4420537

Order 512.

3218 conjugacy classes of subgroups.

The subgroup lattice has 658 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#557C211yesmaximal, characteristic
2C2×#2822C211yesmaximal, characteristic
3C2×#2565C211yesmaximal, characteristic
4C2×#2634C211yesmaximal, characteristic
5#16683C212yesmaximal
6#16605C214yesmaximal
7#16560C212yesmaximal
8#15881C212yesmaximal
9#15850C212yesmaximal
10#15783C212yesmaximal
11#15783C212yesmaximal
12#14879C212yesmaximal
13#14878C214yesmaximal
14#14876C212yesmaximal
15#14346C211yesmaximal, characteristic
16#14289C211yesmaximal, characteristic
17#13925C211yesmaximal, characteristic
18C2×#557—28no—
19C2×#557—28no—
20C2×#557—28no—
21C22×C4⋊Q8C2212yes—
22C22×C4⋊Q8C2211yescharacteristic
23C22×#125C2212yes—
24C22×#125C2211yescharacteristic
25C22×#125C2211yescharacteristic
26C22×#125C2212yes—
27C22×#125C2211yescharacteristic
28C22×C4×Q8C2211yescharacteristic
29C22×C4×Q8C2212yes—
30C22×C4×Q8C2211yescharacteristic
31C22×C4×Q8C2211yescharacteristic
32C22×C4×Q8C2211yescharacteristic
33#2822—28no—
34#2634—28no—
35#2565—28no—
36C2×#400C2212yes—
37C2×#400C2212yes—
38C2×#400C2212yes—
39C2×#400C2212yes—
40C2×#398C2212yes—
41C2×#398C2212yes—
42C2×#398C2214yes—
43C2×#398C2212yes—
44C2×#398C2212yes—
45C2×#398C2212yes—
46C2×#395C2212yes—
47C2×#391C2211yescharacteristic
48C2×#391C2211yescharacteristic
49C2×#387C2212yes—
50C2×#387C2214yes—
51C2×#387C2212yes—
52C2×#384C2211yescharacteristic
53C2×#383C2211yescharacteristic
54C2×#382C2211yescharacteristic
55C2×#382C2211yescharacteristic
56C2×C4×C4⋊C4C2211yescharacteristic
57#1446C2212yes—
58#1446C2212yes—
59#1446C2214yes—
60#1446C2214yes—
61#1445C2212yes—
62#1445C2212yes—
63#1445C2212yes—
64#1445C2212yes—
65#1444C2214yes—
66#1444C2214yes—
67#1442C2214yes—
68#1442C2214yes—
69#1442C2214yes—
70#1442C2214yes—
71#1441C2214yes—
72#1441C2212yes—
73#1441C2212yes—
74#1441C2214yes—
75#1441C2214yes—
76#1441C2214yes—
77#1441C2214yes—
78#1436C2214yes—
79#1436C2212yes—
80#1436C2212yes—
81#1436C2214yes—
82#1436C2214yes—
83#1436C2214yes—
84#1436C2212yes—
85#1436C2212yes—
86C4⋊#95C2214yes—
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—24no—
103C2×#125—28no—
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⋊C4C2314yes—
131C22×C4⋊C4C2312yes—
132C22×C4⋊C4C2311yescharacteristic
133C22×C4⋊C4C2311yescharacteristic
134C22×C4⋊C4C2312yes—
135C22×C4⋊C4C2312yes—
136C22×C4⋊C4C2312yes—
137C22×C4⋊C4C2312yes—
138C22×C4⋊C4C2312yes—
139C22×C4⋊C4C2312yes—
140C22×C4⋊C4C2312yes—
141C22×C4⋊C4C2312yes—
142C22×C4⋊C4C2311yescharacteristic
143C22×C4⋊C4C2312yes—
144C22×C4⋊C4C2311yescharacteristic
145C22×C4⋊C4C2311yescharacteristic
146C22×C4⋊C4C2312yes—
147C22×C4⋊C4C2312yes—
148C22×C4⋊C4C2311yescharacteristic
149C22×C4⋊C4C2311yescharacteristic
150C22×C4⋊C4C2311yescharacteristic
151C22×C4⋊C4C2314yes—
152C22×C4⋊C4C2312yes—
153C22×C42C2312yes—
154C22×C42C2312yes—
155C22×C42C2311yescharacteristic
156C22×C42C2312yes—
157C22×C42C2311yescharacteristic
158C22×C42C2311yescharacteristic
159C22×C42C2311yescharacteristic
160C22×C42C2311yescharacteristic
161#400—28no—
162#400—28no—
163#400—28no—
164#400—28no—
165#398—28no—
166#398—28no—
167#398—28no—
168#398—28no—
169#398—216no—
170#398—28no—
171#395—28no—
172#391—24no—
173#391—24no—
174#387—28no—
175#387—28no—
176#387—216no—
177#384—24no—
178#383—24no—
179#382—24no—
180#382—24no—
181C4×C4⋊C4—24no—
182C2×#95C2312yes—
183C2×#95C2314yes—
184C2×#95C2314yes—
185C2×#95C2314yes—
186C2×#95C2314yes—
187C2×#95C2312yes—
188C2×#95C2312yes—
189C2×#95C2312yes—
190C2×#95C2314yes—
191C2×#95C2312yes—
192C2×#95C2312yes—
193C2×#95C2312yes—
194C2×#95C2312yes—
195C2×#95C2312yes—
196C2×#95C2312yes—
197C2×#95C2314yes—
198C2×#95C2314yes—
199C2×#95C2314yes—
200C2×#95C2314yes—
201C2×#95C2314yes—
202C2×#95C2312yes—
203C2×#95C2312yes—
204C2×#95C2314yes—
205C2×#95C2314yes—
206C2×#95C2314yes—
207C2×#95C2314yes—
208C2×#95C2314yes—
209C2×#95C2314yes—
210C2×#95C2314yes—
211C2×#95C2314yes—
212C22×Q8—22no—
213C22×Q8—22no—
214C22×Q8—24no—
215C22×Q8—24no—
216C22×Q8—22no—
217C22×Q8—22no—
218C22×Q8—24no—
219C22×Q8—28no—
220C22×Q8—24no—
221C22×Q8—28no—
222C22×Q8—42no—
223C22×Q8—44no—
224C22×Q8C2×D414yes—
225C22×Q8C2×D414yes—
226C23×C4C2412yes—
227C23×C4C2412yes—
228C23×C4C2414yes—
229C23×C4C2414yes—
230C23×C4C2412yes—
231C23×C4C2412yes—
232C23×C4C2411yescharacteristic
233C23×C4C2411yescharacteristic
234C23×C4C2412yes—
235C23×C4C2411yescharacteristic
236C23×C4C2412yes—
237C23×C4C2412yes—
238C23×C4C2411yescharacteristic
239C23×C4C2411yescharacteristic
240C23×C4C2411yescharacteristic
241C23×C4C2412yes—
242C23×C4C2411yescharacteristic
243C4⋊Q8—264no—
244C4⋊Q8—232no—
245#125—264no—
246#125—232no—
247#125—264no—
248#125—232no—
249#125—416no—
250C4×Q8—232no—
251C4×Q8—264no—
252C4×Q8—232no—
253C4×Q8—232no—
254C4×Q8—232no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—22no—
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—22no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—24no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4—24no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—24no—
272C2×C4⋊C4—24no—
273C2×C4⋊C4—24no—
274C2×C4⋊C4—24no—
275C2×C4⋊C4—24no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—22no—
279C2×C4⋊C4—22no—
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—22no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—24no—
291C2×C4⋊C4—24no—
292C2×C4⋊C4—24no—
293C2×C4⋊C4—24no—
294C2×C4⋊C4—24no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—22no—
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—28no—
304C2×C4⋊C4—24no—
305C2×C4⋊C4—24no—
306C2×C4⋊C4—24no—
307C2×C4⋊C4—28no—
308C2×C4⋊C4—28no—
309C2×C4⋊C4—24no—
310C2×C4⋊C4—28no—
311C2×C4⋊C4—42no—
312C2×C4⋊C4—42no—
313C2×C4⋊C4—42no—
314C2×C4⋊C4—44no—
315C2×C4⋊C4—44no—
316C2×C4⋊C4—44no—
317C2×C4⋊C4C2×D414yes—
318C2×C4⋊C4C2×D418yes—
319C2×C4⋊C4C2×D414yes—
320C2×C4⋊C4C2×D414yes—
321C2×C4⋊C4C2×D414yes—
322C2×C4⋊C4C2×D414yes—
323C2×C4⋊C4C2×D418yes—
324C2×C4⋊C4C2×D414yes—
325C2×C4⋊C4C2×D418yes—
326C2×C42—24no—
327C2×C42—24no—
328C2×C42—24no—
329C2×C42—24no—
330C2×C42—24no—
331C2×C42—22no—
332C2×C42—22no—
333C2×C42—24no—
334C2×C42—24no—
335C2×C42—24no—
336C2×C42—22no—
337C2×C42—22no—
338C2×C42—22no—
339C2×C42—22no—
340C2×C42—42no—
341C2×C42—42no—
342C2×C42C2×D414yes—
343C2×C42C2×D418yes—
344C2×C42C2×D414yes—
345C2×C42C2×D414yes—
346C2×C42C2×D414yes—
347C2×C42C2×D414yes—
348#95—24no—
349#95—24no—
350#95—24no—
351#95—24no—
352#95—24no—
353#95—24no—
354#95—24no—
355#95—24no—
356#95—24no—
357#95—24no—
358#95—24no—
359#95—24no—
360#95—28no—
361#95—28no—
362#95—28no—
363#95—28no—
364#95—28no—
365#95—28no—
366#95—28no—
367#95—28no—
368#95—28no—
369#95—28no—
370#95—44no—
371#95—44no—
372#95—44no—
373#95—44no—
374#95—44no—
375#95—44no—
376#95—44no—
377#95—44no—
378C24C2511yescharacteristic
379C2×Q8—216no—
380C2×Q8—216no—
381C2×Q8—216no—
382C2×Q8—28no—
383C2×Q8—28no—
384C2×Q8—216no—
385C2×Q8—28no—
386C2×Q8—44no—
387C2×Q8—48no—
388C2×Q8—48no—
389C2×Q8—416no—
390C2×Q8—416no—
391C2×Q8—48no—
392C2×Q8—48no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—21no—
396C22×C4—21no—
397C22×C4—22no—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—21no—
406C22×C4—21no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—21no—
410C22×C4—21no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—21no—
415C22×C4—22no—
416C22×C4—21no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—24no—
431C22×C4—22no—
432C22×C4—24no—
433C22×C4—22no—
434C22×C4—24no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
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—24no—
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—24no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4C22×D414yes—
464C22×C4C22×D412yes—
465C22×C4C22×D414yes—
466C22×C4C22×D412yes—
467C22×C4C22×D414yes—
468C22×C4C22×D412yes—
469C22×C4C22×D412yes—
470C22×C4C22×D412yes—
471C22×C4C22×D412yes—
472C22×C4C22×D412yes—
473C22×C4C22×D412yes—
474C22×C4C22×D412yes—
475C22×C4C22×D412yes—
476C22×C4C22×D412yes—
477C22×C4C22×D412yes—
478C22×C4C22×D412yes—
479C22×C4C22×D414yes—
480C22×C42−1+414yes—
481C22×C42−1+418yes—
482C22×C42−1+414yes—
483C4⋊C4—216no—
484C4⋊C4—216no—
485C4⋊C4—216no—
486C4⋊C4—216no—
487C4⋊C4—216no—
488C4⋊C4—28no—
489C4⋊C4—28no—
490C4⋊C4—28no—
491C4⋊C4—216no—
492C4⋊C4—28no—
493C4⋊C4—216no—
494C4⋊C4—28no—
495C4⋊C4—28no—
496C4⋊C4—216no—
497C4⋊C4—48no—
498C4⋊C4—48no—
499C4⋊C4—48no—
500C4⋊C4—48no—
501C4⋊C4—48no—
502C4⋊C4—44no—
503C4⋊C4—44no—
504C4⋊C4—416no—
505C4⋊C4—84no—
506C4⋊C4—88no—
507C42—216no—
508C42—216no—
509C42—216no—
510C42—28no—
511C42—28no—
512C42—44no—
513C42—44no—
514C42#120116yes—
515C23C23×D411yescharacteristic
516C23C23×D411yescharacteristic
517C23C23×D411yescharacteristic
518C23C2×2−1+414yes—
519C23C2×2−1+412yes—
520C23C2×2−1+412yes—
521C23#58512yes—
522C23#58512yes—
523Q8—232no—
524Q8—416no—
525Q8—432no—
526C2×C4—24no—
527C2×C4—22no—
528C2×C4—24no—
529C2×C4—24no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—24no—
534C2×C4—24no—
535C2×C4—28no—
536C2×C4—28no—
537C2×C4—24no—
538C2×C4—24no—
539C2×C4—24no—
540C2×C4—24no—
541C2×C4—24no—
542C2×C4—24no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—22no—
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—24no—
559C2×C4—24no—
560C2×C4—28no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—44no—
568C2×C4—42no—
569C2×C4—44no—
570C2×C4—42no—
571C2×C4—44no—
572C2×C4—42no—
573C2×C4—44no—
574C2×C4—42no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—44no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—44no—
583C2×C4—44no—
584C2×C4—42no—
585C2×C4—44no—
586C2×C4—44no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—44no—
594C2×C4—44no—
595C2×C4—44no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—44no—
600C2×C4—42no—
601C2×C4—44no—
602C2×C4—44no—
603C2×C4—44no—
604C2×C4—44no—
605C2×C4—44no—
606C2×C4—44no—
607C2×C4—44no—
608C2×C4—44no—
609C2×C4—44no—
610C2×C4—44no—
611C2×C4C2×#12014yes—
612C2×C4C2×#12014yes—
613C2×C4C2×#12014yes—
614C2×C4#53618yes—
615C2×C4#53614yes—
616C22C22×#12011yescharacteristic
617C22C2×#53612yes—
618C22C2×#53611yescharacteristic
619C22#344911yescharacteristic
620C22#348612yes—
621C22#348612yes—
622C22#348912yes—
623C22#348912yes—
624C22#348912yes—
625C22#349712yes—
626C22#349911yescharacteristic
627C22#350011yescharacteristic
628C22#355514yes—
629C22#355614yes—
630C22#355714yes—
631C22#356314yes—
632C4—24no—
633C4—24no—
634C4—28no—
635C4—24no—
636C4—24no—
637C4—42no—
638C4—42no—
639C4—44no—
640C4—44no—
641C4—42no—
642C4—44no—
643C4—44no—
644C4—84no—
645C4—84no—
646C4—82no—
647C4—82no—
648C4—82no—
649C2#3524211yescharacteristic
650C2#3530511yescharacteristic
651C2#3566011yescharacteristic
652C2#4217112yes—
653C2#4280114yes—
654C2#4281112yes—
655C2#5120612yes—
656C2#5121012yes—
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×#25652
C2×#26342
#166832
#166052
#165602
#158812
#158502
#157832
#148792
#148782
#148762
#143462
#142892
#139252