Finite Group Databasebeta

← #4464129

Subgroups — #4464129

Order 512.

1862 conjugacy classes of subgroups.

The subgroup lattice has 670 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2660C211yesmaximal, characteristic
2C2×#2645C211yesmaximal, characteristic
3#17200C214yesmaximal
4#17113C214yesmaximal
5#17112C214yesmaximal
6#17099C214yesmaximal
7#15241C211yesmaximal, characteristic
8#15240C211yesmaximal, characteristic
9#15239C211yesmaximal, characteristic
10#15238C211yesmaximal, characteristic
11#15231C211yesmaximal, characteristic
12#15228C211yesmaximal, characteristic
13#15199C211yesmaximal, characteristic
14#15191C211yesmaximal, characteristic
15#15185C211yesmaximal, characteristic
16#15153C211yesmaximal, characteristic
17#15145C211yesmaximal, characteristic
18#15143C211yesmaximal, characteristic
19#13855C211yesmaximal, characteristic
20C22×C4⋊Q8C2211yescharacteristic
21C22×#125C2211yescharacteristic
22#2660—28no—
23#2645—28no—
24C2×#401C2212yes—
25C2×#400C2212yes—
26C2×#400C2212yes—
27C2×#400C2212yes—
28C2×#400C2214yes—
29C2×#398C2212yes—
30C2×#398C2212yes—
31C2×#398C2214yes—
32C2×#395C2212yes—
33C2×#395C2212yes—
34C2×#395C2214yes—
35C2×#391C2211yescharacteristic
36C2×#391C2211yescharacteristic
37C2×#391C2211yescharacteristic
38C2×#391C2211yescharacteristic
39C2×#391C2211yescharacteristic
40C2×#391C2211yescharacteristic
41C2×#391C2211yescharacteristic
42C2×#389C2211yescharacteristic
43C2×#389C2211yescharacteristic
44C2×#389C2211yescharacteristic
45C2×#384C2211yescharacteristic
46C2×#384C2211yescharacteristic
47C2×#384C2211yescharacteristic
48C2×#384C2211yescharacteristic
49C2×#384C2211yescharacteristic
50C2×#382C2211yescharacteristic
51C2×#382C2211yescharacteristic
52C2×#382C2211yescharacteristic
53C2×#376C2211yescharacteristic
54#1446C2214yes—
55#1445C2214yes—
56#1445C2214yes—
57#1445C2214yes—
58#1445C2214yes—
59#1444C2214yes—
60#1444C2214yes—
61#1444C2214yes—
62#1444C2214yes—
63#1444C2214yes—
64#1443C2212yes—
65#1443C2212yes—
66#1443C2212yes—
67#1443C2212yes—
68#1443C2214yes—
69#1443C2214yes—
70#1443C2214yes—
71#1442C2214yes—
72#1442C2214yes—
73#1442C2214yes—
74#1442C2214yes—
75#1442C2214yes—
76#1442C2214yes—
77#1441C2214yes—
78#1441C2214yes—
79C4⋊#95C2211yescharacteristic
80C4⋊#95C2211yescharacteristic
81C4⋊#95C2211yescharacteristic
82C4⋊#95C2211yescharacteristic
83C4⋊#95C2211yescharacteristic
84C4⋊#95C2211yescharacteristic
85C4⋊#95C2211yescharacteristic
86C4⋊#95C2211yescharacteristic
87C4×#95C2211yescharacteristic
88C4×#95C2211yescharacteristic
89C4×#95C2211yescharacteristic
90C4×#95C2211yescharacteristic
91C4×#95C2211yescharacteristic
92C4×#95C2211yescharacteristic
93C23×Q8C2311yescharacteristic
94C23×Q8C2312yes—
95C2×C4⋊Q8—24no—
96C2×C4⋊Q8—42no—
97C2×C4⋊Q8—42no—
98C2×#125—24no—
99C2×#125—42no—
100C2×#125—42no—
101C22×C4⋊C4C2311yescharacteristic
102C22×C4⋊C4C2311yescharacteristic
103C22×C4⋊C4C2311yescharacteristic
104C22×C4⋊C4C2311yescharacteristic
105C22×C4⋊C4C2311yescharacteristic
106C22×C4⋊C4C2311yescharacteristic
107C22×C4⋊C4C2311yescharacteristic
108C22×C4⋊C4C2312yes—
109C22×C4⋊C4C2312yes—
110C22×C4⋊C4C2312yes—
111C22×C4⋊C4C2312yes—
112C22×C4⋊C4C2312yes—
113C22×C4⋊C4C2312yes—
114C22×C4⋊C4C2314yes—
115C22×C4⋊C4C2312yes—
116C22×C4⋊C4C2312yes—
117C22×C4⋊C4C2312yes—
118C22×C4⋊C4C2312yes—
119C22×C4⋊C4C2314yes—
120C22×C4⋊C4C2314yes—
121C22×C42C2311yescharacteristic
122C22×C42C2311yescharacteristic
123C22×C42C2311yescharacteristic
124C22×C42C2311yescharacteristic
125C22×C42C2311yescharacteristic
126C22×C42C2311yescharacteristic
127C22×C42C2311yescharacteristic
128#401—28no—
129#400—28no—
130#400—28no—
131#400—28no—
132#400—48no—
133#398—28no—
134#398—28no—
135#398—48no—
136#395—28no—
137#395—28no—
138#395—48no—
139#391—24no—
140#391—24no—
141#391—24no—
142#391—42no—
143#391—42no—
144#391—42no—
145#391—42no—
146#389—24no—
147#389—24no—
148#389—24no—
149#384—24no—
150#384—24no—
151#384—42no—
152#384—42no—
153#384—42no—
154#382—24no—
155#382—24no—
156#382—42no—
157#376D418yes—
158C2×#95C2312yes—
159C2×#95C2312yes—
160C2×#95C2312yes—
161C2×#95C2312yes—
162C2×#95C2312yes—
163C2×#95C2314yes—
164C2×#95C2312yes—
165C2×#95C2314yes—
166C2×#95C2312yes—
167C2×#95C2312yes—
168C2×#95C2312yes—
169C2×#95C2312yes—
170C2×#95C2312yes—
171C2×#95C2312yes—
172C2×#95C2312yes—
173C2×#95C2312yes—
174C2×#95C2312yes—
175C2×#95C2312yes—
176C2×#95C2312yes—
177C2×#95C2312yes—
178C2×#95C2312yes—
179C2×#95C2312yes—
180C2×#95C2312yes—
181C2×#95C2312yes—
182C2×#95C2314yes—
183C2×#95C2314yes—
184C2×#95C2314yes—
185C2×#95C2314yes—
186C2×#95C2314yes—
187C2×#95C2314yes—
188C2×#95C2312yes—
189C2×#95C2312yes—
190C2×#95C2312yes—
191C2×#95C2314yes—
192C2×#95C2314yes—
193C2×#95C2314yes—
194C2×#95C2314yes—
195C2×#95C2314yes—
196C2×#95C2311yescharacteristic
197C2×#95C2311yescharacteristic
198C2×#95C2311yescharacteristic
199C2×#95C2311yescharacteristic
200C22×Q8—22no—
201C22×Q8—24no—
202C22×Q8—22no—
203C22×Q8—22no—
204C22×Q8—22no—
205C22×Q8—22no—
206C22×Q8—22no—
207C22×Q8—22no—
208C22×Q8—42no—
209C22×Q8—42no—
210C22×Q8—44no—
211C22×Q8—44no—
212C23×C4C2412yes—
213C23×C4C2412yes—
214C23×C4C2412yes—
215C23×C4C2412yes—
216C23×C4C2412yes—
217C23×C4C2412yes—
218C23×C4C2412yes—
219C23×C4C2412yes—
220C23×C4C2412yes—
221C23×C4C2411yescharacteristic
222C23×C4C2411yescharacteristic
223C23×C4C2411yescharacteristic
224C23×C4C2411yescharacteristic
225C23×C4C2411yescharacteristic
226C23×C4C2411yescharacteristic
227C23×C4C2412yes—
228C23×C4C2412yes—
229C23×C4C2412yes—
230C23×C4C2411yescharacteristic
231C4⋊Q8—416no—
232#125—416no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—22no—
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—24no—
264C2×C4⋊C4—28no—
265C2×C4⋊C4—42no—
266C2×C4⋊C4—42no—
267C2×C4⋊C4—42no—
268C2×C4⋊C4—42no—
269C2×C4⋊C4—42no—
270C2×C4⋊C4—42no—
271C2×C4⋊C4—44no—
272C2×C4⋊C4—44no—
273C2×C4⋊C4—44no—
274C2×C4⋊C4—44no—
275C2×C4⋊C4—44no—
276C2×C4⋊C4—44no—
277C2×C4⋊C4—44no—
278C2×C4⋊C4—44no—
279C2×C4⋊C4—44no—
280C2×C4⋊C4—44no—
281C2×C4⋊C4—44no—
282C2×C4⋊C4—44no—
283C2×C4⋊C4—44no—
284C2×C4⋊C4—44no—
285C2×C4⋊C4—44no—
286C2×C42—22no—
287C2×C42—22no—
288C2×C42—22no—
289C2×C42—22no—
290C2×C42—22no—
291C2×C42—22no—
292C2×C42—22no—
293C2×C42—22no—
294C2×C42—22no—
295C2×C42—22no—
296C2×C42—22no—
297C2×C42—22no—
298C2×C42—22no—
299C2×C42—22no—
300C2×C42—22no—
301C2×C42—22no—
302C2×C42—22no—
303C2×C42—22no—
304C2×C42C2×D414yes—
305C2×C42C2×D414yes—
306C2×C42C2×D414yes—
307#95—24no—
308#95—24no—
309#95—24no—
310#95—24no—
311#95—24no—
312#95—24no—
313#95—24no—
314#95—24no—
315#95—24no—
316#95—24no—
317#95—24no—
318#95—24no—
319#95—24no—
320#95—24no—
321#95—28no—
322#95—24no—
323#95—28no—
324#95—24no—
325#95—24no—
326#95—24no—
327#95—24no—
328#95—24no—
329#95—24no—
330#95—24no—
331#95—28no—
332#95—28no—
333#95—42no—
334#95—42no—
335#95—42no—
336#95—44no—
337#95—44no—
338#95—44no—
339#95—44no—
340#95—44no—
341#95—44no—
342#95—44no—
343#95—44no—
344#95—44no—
345#95C2×D414yes—
346#95C2×D414yes—
347#95C2×D414yes—
348#95C2×D414yes—
349C24C2511yescharacteristic
350C2×Q8—48no—
351C2×Q8—44no—
352C2×Q8—44no—
353C2×Q8—44no—
354C2×Q8—44no—
355C2×Q8—44no—
356C2×Q8—44no—
357C2×Q8—44no—
358C2×Q8—84no—
359C2×Q8—84no—
360C2×Q8—88no—
361C2×Q8—88no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—21no—
371C22×C4—21no—
372C22×C4—21no—
373C22×C4—21no—
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—21no—
385C22×C4—21no—
386C22×C4—21no—
387C22×C4—21no—
388C22×C4—21no—
389C22×C4—21no—
390C22×C4—21no—
391C22×C4—21no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—24no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—24no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—24no—
433C22×C4—24no—
434C22×C4—24no—
435C22×C4—24no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—24no—
450C22×C4—24no—
451C22×C4—24no—
452C22×C4—24no—
453C22×C4C22×D412yes—
454C22×C4C22×D412yes—
455C22×C4C22×D412yes—
456C22×C4C22×D412yes—
457C22×C4C22×D412yes—
458C22×C4C22×D412yes—
459C22×C4C22×D412yes—
460C22×C4H(5,𝔽2)14yes—
461C22×C4H(5,𝔽2)12yes—
462C22×C4H(5,𝔽2)12yes—
463C22×C42−1+414yes—
464C22×C42−1+414yes—
465C22×C42−1+412yes—
466C22×C42−1+412yes—
467C22×C42−1+412yes—
468C22×C42−1+412yes—
469C4⋊C4—48no—
470C4⋊C4—48no—
471C4⋊C4—48no—
472C4⋊C4—44no—
473C4⋊C4—44no—
474C4⋊C4—44no—
475C4⋊C4—44no—
476C4⋊C4—44no—
477C4⋊C4—44no—
478C4⋊C4—44no—
479C4⋊C4—84no—
480C4⋊C4—84no—
481C4⋊C4—84no—
482C4⋊C4—84no—
483C4⋊C4—84no—
484C4⋊C4—84no—
485C4⋊C4—84no—
486C4⋊C4—88no—
487C4⋊C4—88no—
488C4⋊C4—88no—
489C42—28no—
490C42—28no—
491C42—28no—
492C42—44no—
493C42—44no—
494C42—44no—
495C42—44no—
496C23C23×D411yescharacteristic
497C23C2×H(5,𝔽2)11yescharacteristic
498C23C2×H(5,𝔽2)11yescharacteristic
499C23C2×H(5,𝔽2)12yes—
500C23C2×2−1+411yescharacteristic
501C23C2×2−1+411yescharacteristic
502C23C2×2−1+411yescharacteristic
503C23C2×2−1+411yescharacteristic
504C23C2×2−1+412yes—
505C23C2×2−1+412yes—
506C23#58512yes—
507Q8—816no—
508Q8—88no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—24no—
528C2×C4—24no—
529C2×C4—24no—
530C2×C4—24no—
531C2×C4—24no—
532C2×C4—24no—
533C2×C4—24no—
534C2×C4—24no—
535C2×C4—24no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—44no—
591C2×C4—44no—
592C2×C4—44no—
593C2×C4—44no—
594C2×C4—42no—
595C2×C4—44no—
596C2×C4—44no—
597C2×C4—44no—
598C2×C4—44no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—44no—
602C2×C4—44no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—44no—
606C2×C4—44no—
607C2×C4—44no—
608C2×C4—44no—
609C2×C4#53514yes—
610C2×C4#53614yes—
611C2×C4#53614yes—
612C2×C4#56314yes—
613C2×C4#56314yes—
614C2×C4#56314yes—
615C2×C4#56314yes—
616C22C2×#53511yescharacteristic
617C22C2×#53611yescharacteristic
618C22C2×#53611yescharacteristic
619C22#349811yescharacteristic
620C22#349911yescharacteristic
621C22#349911yescharacteristic
622C22#350011yescharacteristic
623C22C2×#56311yescharacteristic
624C22C2×#56311yescharacteristic
625C22C2×#56311yescharacteristic
626C22C2×#56311yescharacteristic
627C22#354412yes—
628C22#354412yes—
629C22#354612yes—
630C22#355212yes—
631C22#355212yes—
632C22#355212yes—
633C22#355212yes—
634C22#355512yes—
635C22#355512yes—
636C22#355812yes—
637C22#356312yes—
638C22#356312yes—
639C4—44no—
640C4—42no—
641C4—42no—
642C4—42no—
643C4—44no—
644C4—44no—
645C4—42no—
646C4—42no—
647C4—42no—
648C4—82no—
649C4—82no—
650C4—82no—
651C4—82no—
652C4—82no—
653C4—82no—
654C4—82no—
655C4—82no—
656C4—82no—
657C4#282118yes—
658C2C2×#282111yescharacteristic
659C2#3730311yescharacteristic
660C2#3730311yescharacteristic
661C2#3730511yescharacteristic
662C2#3739011yescharacteristic
663C2#3739211yescharacteristic
664C2#3739211yescharacteristic
665C2#5300212yes—
666C2#5303812yes—
667C2#5304012yes—
668C2#5309112yes—
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×#26602
C2×#26452
#172002
#171132
#171122
#170992
#152412
#152402
#152392
#152382
#152312
#152282
#151992
#151912
#151852
#151532
#151452
#151432
#138552