Finite Group Databasebeta

← #719804

Subgroups — #719804

Order 512.

2406 conjugacy classes of subgroups.

The subgroup lattice has 646 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2446C212yesmaximal
2C2×Q8×C4⋊C4C211yesmaximal, characteristic
3C2×#2327C211yesmaximal, characteristic
4#17063C212yesmaximal
5#17061C212yesmaximal
6#15895C214yesmaximal
7#15874C214yesmaximal
8#15221C212yesmaximal
9#15186C212yesmaximal
10#15080C212yesmaximal
11#15080C212yesmaximal
12#14277C211yesmaximal, characteristic
13#13942C211yesmaximal, characteristic
14#13942C211yesmaximal, characteristic
15#13931C212yesmaximal
16#13926C211yesmaximal, characteristic
17#13848C211yesmaximal, characteristic
18C22×C4⋊Q8C2211yescharacteristic
19C22×C4×Q8C2211yescharacteristic
20C22×C4×Q8C2212yes—
21C22×C4×Q8C2211yescharacteristic
22#2446—216no—
23C2×#400C2214yes—
24C2×#398C2214yes—
25Q8×C4⋊C4—28no—
26#2327—28no—
27C2×#391C2211yescharacteristic
28C2×#391C2211yescharacteristic
29C2×#391C2212yes—
30C2×#391C2212yes—
31C2×#391C2212yes—
32C2×#389C2214yes—
33C2×#389C2212yes—
34C2×#389C2212yes—
35C2×#387C2212yes—
36C2×#387C2214yes—
37C2×#387C2212yes—
38C2×#387C2212yes—
39C2×#387C2212yes—
40C2×#387C2214yes—
41C2×#384C2211yescharacteristic
42C2×#384C2212yes—
43C2×#382C2211yescharacteristic
44C2×#382C2211yescharacteristic
45C2×C4×C4⋊C4C2211yescharacteristic
46C2×C4×C4⋊C4C2211yescharacteristic
47C2×C4×C4⋊C4C2212yes—
48C2×C4×C4⋊C4C2212yes—
49C2×#376C2211yescharacteristic
50#1445C2214yes—
51#1445C2214yes—
52#1445C2214yes—
53#1445C2214yes—
54#1444C2212yes—
55#1444C2214yes—
56#1444C2211yescharacteristic
57#1444C2211yescharacteristic
58#1444C2212yes—
59#1443C2214yes—
60#1443C2214yes—
61#1442C2214yes—
62#1442C2212yes—
63#1442C2214yes—
64#1442C2214yes—
65#1442C2211yescharacteristic
66#1442C2211yescharacteristic
67#1442C2212yes—
68#1441C2212yes—
69#1441C2214yes—
70#1441C2212yes—
71#1441C2212yes—
72#1441C2212yes—
73#1441C2212yes—
74#1441C2212yes—
75C4⋊#95C2214yes—
76C4⋊#95C2212yes—
77C4⋊#95C2212yes—
78C4⋊#95C2212yes—
79C4⋊#95C2212yes—
80#1434C2214yes—
81#1434C2214yes—
82#1434C2214yes—
83#1434C2212yes—
84#1434C2212yes—
85C4×#95C2212yes—
86C23×Q8C2311yescharacteristic
87C23×Q8C2311yescharacteristic
88C2×C4⋊Q8—24no—
89C2×C4⋊Q8—24no—
90C2×C4⋊Q8D418yes—
91C2×C4×Q8—28no—
92C2×C4×Q8—28no—
93C2×C4×Q8—28no—
94C2×C4×Q8—24no—
95C2×C4×Q8—24no—
96C2×C4×Q8—24no—
97C2×C4×Q8—24no—
98C2×C4×Q8—24no—
99C2×C4×Q8D418yes—
100C22×C4⋊C4C2314yes—
101C22×C4⋊C4C2311yescharacteristic
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2311yescharacteristic
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2312yes—
108C22×C4⋊C4C2312yes—
109C22×C4⋊C4C2312yes—
110C22×C4⋊C4C2311yescharacteristic
111C22×C4⋊C4C2312yes—
112C22×C4⋊C4C2312yes—
113C22×C4⋊C4C2314yes—
114C22×C4⋊C4C2312yes—
115C22×C4⋊C4C2312yes—
116C22×C4⋊C4C2311yescharacteristic
117C22×C4⋊C4C2314yes—
118C22×C42C2312yes—
119C22×C42C2312yes—
120C22×C42C2314yes—
121C22×C42C2311yescharacteristic
122C22×C42C2312yes—
123C22×C42C2312yes—
124C22×C42C2311yescharacteristic
125C22×C42C2311yescharacteristic
126#400—216no—
127#398—216no—
128#391—24no—
129#391—28no—
130#391—28no—
131#391—28no—
132#391—24no—
133#389—28no—
134#389—216no—
135#389—28no—
136#387—28no—
137#387—28no—
138#387—216no—
139#387—216no—
140#387—28no—
141#387—28no—
142#384—28no—
143#384—24no—
144#382—24no—
145#382—24no—
146C4×C4⋊C4—28no—
147C4×C4⋊C4—24no—
148C4×C4⋊C4—28no—
149C4×C4⋊C4—24no—
150#376—24no—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2312yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2312yes—
159C2×#95C2312yes—
160C2×#95C2314yes—
161C2×#95C2311yescharacteristic
162C2×#95C2312yes—
163C2×#95C2312yes—
164C2×#95C2311yescharacteristic
165C2×#95C2312yes—
166C2×#95C2312yes—
167C2×#95C2312yes—
168C2×#95C2314yes—
169C2×#95C2314yes—
170C2×#95C2311yescharacteristic
171C2×#95C2311yescharacteristic
172C2×#95C2312yes—
173C2×#95C2311yescharacteristic
174C2×#95C2314yes—
175C2×#95C2312yes—
176C2×#95C2312yes—
177C2×#95C2312yes—
178C2×#95C2311yescharacteristic
179C2×#95C2312yes—
180C2×#95C2312yes—
181C2×#95C2312yes—
182C2×#95C2314yes—
183C2×#95C2314yes—
184C2×#95C2314yes—
185C2×#95C2314yes—
186C2×#95C2314yes—
187C2×#95C2314yes—
188C22×Q8—22no—
189C22×Q8—22no—
190C22×Q8—22no—
191C22×Q8—22no—
192C22×Q8—22no—
193C22×Q8—22no—
194C22×Q8—22no—
195C22×Q8—22no—
196C22×Q8—22no—
197C22×Q8—22no—
198C22×Q8C2×D414yes—
199C22×Q8C2×D414yes—
200C22×Q8C2×D414yes—
201C22×Q8#4114yes—
202C23×C4C2412yes—
203C23×C4C2414yes—
204C23×C4C2414yes—
205C23×C4C2412yes—
206C23×C4C2412yes—
207C23×C4C2411yescharacteristic
208C23×C4C2411yescharacteristic
209C23×C4C2411yescharacteristic
210C23×C4C2411yescharacteristic
211C23×C4C2412yes—
212C23×C4C2412yes—
213C23×C4C2412yes—
214C23×C4C2412yes—
215C23×C4C2412yes—
216C23×C4C2411yescharacteristic
217C23×C4C2411yescharacteristic
218C23×C4C2411yescharacteristic
219C4⋊Q8—232no—
220C4×Q8—232no—
221C4×Q8—432no—
222C4×Q8—416no—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—28no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—28no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—28no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—28no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—28no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—42no—
260C2×C4⋊C4—44no—
261C2×C4⋊C4—42no—
262C2×C4⋊C4—42no—
263C2×C4⋊C4—42no—
264C2×C4⋊C4—44no—
265C2×C4⋊C4—44no—
266C2×C4⋊C4—44no—
267C2×C4⋊C4—44no—
268C2×C4⋊C4—44no—
269C2×C4⋊C4C2×D414yes—
270C2×C4⋊C4C2×D418yes—
271C2×C4⋊C4C2×D418yes—
272C2×C4⋊C4C2×D418yes—
273C2×C42—22no—
274C2×C42—24no—
275C2×C42—24no—
276C2×C42—22no—
277C2×C42—28no—
278C2×C42—24no—
279C2×C42—24no—
280C2×C42—28no—
281C2×C42—24no—
282C2×C42—22no—
283C2×C42—24no—
284C2×C42—28no—
285C2×C42—24no—
286C2×C42—24no—
287C2×C42—24no—
288C2×C42—24no—
289C2×C42—22no—
290C2×C42—22no—
291C2×C42—24no—
292C2×C42C2×D418yes—
293C2×C42C2×D414yes—
294C2×C42C2×D414yes—
295C2×C42#4114yes—
296C2×C42#4114yes—
297#95—22no—
298#95—22no—
299#95—22no—
300#95—22no—
301#95—24no—
302#95—28no—
303#95—24no—
304#95—24no—
305#95—24no—
306#95—24no—
307#95—28no—
308#95—24no—
309#95—24no—
310#95—24no—
311#95—24no—
312#95—28no—
313#95—28no—
314#95—28no—
315#95—28no—
316#95—28no—
317#95—28no—
318#95—22no—
319#95—22no—
320#95—24no—
321#95—24no—
322#95—24no—
323#95—28no—
324#95—24no—
325#95—28no—
326#95—44no—
327#95—42no—
328#95—44no—
329#95—42no—
330#95—44no—
331#95—44no—
332#95—44no—
333#95—44no—
334C24C2511yescharacteristic
335C2×Q8—28no—
336C2×Q8—28no—
337C2×Q8—28no—
338C2×Q8—28no—
339C2×Q8—28no—
340C2×Q8—28no—
341C2×Q8—28no—
342C2×Q8—28no—
343C2×Q8—28no—
344C2×Q8—44no—
345C2×Q8—44no—
346C2×Q8—44no—
347C2×Q8—44no—
348C2×Q8#121116yes—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—21no—
358C22×C4—21no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—21no—
366C22×C4—21no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—24no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—24no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—21no—
388C22×C4—21no—
389C22×C4—22no—
390C22×C4—24no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—24no—
394C22×C4—24no—
395C22×C4—22no—
396C22×C4—24no—
397C22×C4—22no—
398C22×C4—24no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—24no—
403C22×C4—22no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—21no—
409C22×C4—21no—
410C22×C4—21no—
411C22×C4—21no—
412C22×C4—22no—
413C22×C4—24no—
414C22×C4—22no—
415C22×C4—24no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—24no—
420C22×C4—22no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—24no—
426C22×C4—22no—
427C22×C4—24no—
428C22×C4—22no—
429C22×C4—24no—
430C22×C4C22×D414yes—
431C22×C4C22×D412yes—
432C22×C4C22×D414yes—
433C22×C4C22×D412yes—
434C22×C4C22×D414yes—
435C22×C4C22×D414yes—
436C22×C4C22×D412yes—
437C22×C4C22×D412yes—
438C22×C4C22×D412yes—
439C22×C4C22×D412yes—
440C22×C4C2×#4114yes—
441C22×C4C2×#4112yes—
442C22×C4C2×#4112yes—
443C22×C4C2×#4112yes—
444C22×C4C2×#4112yes—
445C22×C4C2×#4112yes—
446C22×C4C2×#4112yes—
447C22×C4C2×#4112yes—
448C22×C42−1+414yes—
449C22×C42−1+414yes—
450C22×C42−1+412yes—
451C22×C42−1+412yes—
452C4⋊C4—28no—
453C4⋊C4—216no—
454C4⋊C4—216no—
455C4⋊C4—216no—
456C4⋊C4—216no—
457C4⋊C4—28no—
458C4⋊C4—48no—
459C4⋊C4—48no—
460C4⋊C4—44no—
461C4⋊C4—48no—
462C4⋊C4—416no—
463C4⋊C4—416no—
464C4⋊C4—48no—
465C4⋊C4—48no—
466C4⋊C4—44no—
467C4⋊C4—48no—
468C4⋊C4—84no—
469C4⋊C4—88no—
470C42—28no—
471C42—216no—
472C42—416no—
473C42—48no—
474C42—48no—
475C42—44no—
476C42—48no—
477C42#123116yes—
478C23C23×D411yescharacteristic
479C23C23×D411yescharacteristic
480C23C22×#4111yescharacteristic
481C23C22×#4111yescharacteristic
482C23C22×#4111yescharacteristic
483C23C2×2−1+411yescharacteristic
484C23C2×2−1+411yescharacteristic
485C23C2×2−1+412yes—
486C23C2×2−1+412yes—
487C23#58512yes—
488C23#58512yes—
489Q8—232no—
490Q8—416no—
491C2×C4—24no—
492C2×C4—24no—
493C2×C4—24no—
494C2×C4—24no—
495C2×C4—24no—
496C2×C4—24no—
497C2×C4—24no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—24no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—24no—
504C2×C4—24no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—24no—
508C2×C4—24no—
509C2×C4—24no—
510C2×C4—24no—
511C2×C4—24no—
512C2×C4—24no—
513C2×C4—24no—
514C2×C4—24no—
515C2×C4—24no—
516C2×C4—24no—
517C2×C4—24no—
518C2×C4—24no—
519C2×C4—24no—
520C2×C4—24no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—41no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—41no—
532C2×C4—44no—
533C2×C4—41no—
534C2×C4—42no—
535C2×C4—41no—
536C2×C4—44no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—44no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—44no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—44no—
550C2×C4—41no—
551C2×C4—44no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—44no—
555C2×C4—42no—
556C2×C4—44no—
557C2×C4—44no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—41no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—44no—
564C2×C4—42no—
565C2×C4—41no—
566C2×C4—44no—
567C2×C4—42no—
568C2×C4—41no—
569C2×C4—44no—
570C2×C4—44no—
571C2×C4—44no—
572C2×C4—44no—
573C2×C4—42no—
574C2×C4—44no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—44no—
579C2×C4—44no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4C2×#12118yes—
585C2×C4C2×#12114yes—
586C2×C4C2×#12314yes—
587C2×C4C2×#12314yes—
588C2×C4C2×#12314yes—
589C2×C4#53614yes—
590C2×C4#54714yes—
591C2×C4#54714yes—
592C2×C4#55214yes—
593C2×C4#55214yes—
594C22C22×#12111yescharacteristic
595C22C22×#12311yescharacteristic
596C22C2×#53611yescharacteristic
597C22#346112yes—
598C22C2×#54711yescharacteristic
599C22C2×#54711yescharacteristic
600C22C2×#55211yescharacteristic
601C22C2×#55211yescharacteristic
602C22#348611yescharacteristic
603C22#348612yes—
604C22#348712yes—
605C22#348711yescharacteristic
606C22#349512yes—
607C22#349911yescharacteristic
608C22#350011yescharacteristic
609C22#350712yes—
610C22#350712yes—
611C22#352312yes—
612C22#352412yes—
613C22#355512yes—
614C22#355512yes—
615C22#355512yes—
616C22#356312yes—
617C4—24no—
618C4—24no—
619C4—28no—
620C4—44no—
621C4—42no—
622C4—44no—
623C4—44no—
624C4—42no—
625C4—44no—
626C4—81no—
627C4—84no—
628C4—82no—
629C4—81no—
630C4—82no—
631C4—84no—
632C4—82no—
633C4#248618yes—
634C2C2×#248611yescharacteristic
635C2#3518211yescharacteristic
636C2#3561911yescharacteristic
637C2#3570011yescharacteristic
638C2#3601811yescharacteristic
639C2#3776512yes—
640C2#3822712yes—
641C2#4279311yescharacteristic
642C2#4287911yescharacteristic
643C2#4515412yes—
644C2#4539412yes—
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
C2×#24462
C2×Q8×C4⋊C42
C2×#23272
#170632
#170612
#158952
#158742
#152212
#151862
#150802
#142772
#139422
#139312
#139262
#138482