Finite Group Databasebeta

← #6604597

Subgroups — #6604597

Order 512.

1383 conjugacy classes of subgroups.

The subgroup lattice has 770 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#38986C211yesmaximal, characteristic
2#34335C211yesmaximal, characteristic
3#33091C212yesmaximal
4#33027C211yesmaximal, characteristic
5#29526C212yesmaximal
6#29402C212yesmaximal
7#29275C211yesmaximal, characteristic
8#28656C211yesmaximal, characteristic
9#27575C212yesmaximal
10#27127C212yesmaximal
11#27016C212yesmaximal
12#26383C212yesmaximal
13#26317C211yesmaximal, characteristic
14#26279C211yesmaximal, characteristic
15#25936C211yesmaximal, characteristic
16#25841C211yesmaximal, characteristic
17#23221C211yesmaximal, characteristic
18#23158C211yesmaximal, characteristic
19#22998C211yesmaximal, characteristic
20#22971C211yesmaximal, characteristic
21#22829C211yesmaximal, characteristic
22C4×#479C211yesmaximal, characteristic
23#21135C212yesmaximal
24C2×Q82C2211yescharacteristic
25C2×#500C2211yescharacteristic
26C2×#487C2211yescharacteristic
27C2×#479C2211yescharacteristic
28C2×#477C2211yescharacteristic
29C2×#462C2211yescharacteristic
30C2×C4×Q16C2212yes—
31#2862C2211yescharacteristic
32#2789—22no—
33#2758—22no—
34#2645C2212yes—
35#2571—21no—
36#2570—21no—
37#2568C2211yescharacteristic
38#2567C2211yescharacteristic
39#2514—21no—
40#2514—21no—
41#2514C2212yes—
42#2453C2212yes—
43#2369—21no—
44#2369C2211yescharacteristic
45#2369C2211yescharacteristic
46#2365—22no—
47#2365C2212yes—
48#2365C2211yescharacteristic
49#2355—21no—
50#2355C2211yescharacteristic
51#2351C2212yes—
52Q8×C4⋊C4C2211yescharacteristic
53#2331—22no—
54#2331C2212yes—
55#2320C2212yes—
56C4×C4⋊Q8—21no—
57C4×C4⋊Q8C2211yescharacteristic
58C4×#125C2211yescharacteristic
59#2295—21no—
60#2295C2211yescharacteristic
61#2276C2212yes—
62C42×Q8C2211yescharacteristic
63#2101C2212yes—
64#2101C2212yes—
65#2097C2212yes—
66#2085C2212yes—
67#2074C2212yes—
68#2068C2212yes—
69#2068C2212yes—
70#2059C2212yes—
71#2057C2212yes—
72#2057C2212yes—
73#2024C2212yes—
74#2024C2212yes—
75#1990C2211yescharacteristic
76#1989C2211yescharacteristic
77#1987C2211yescharacteristic
78#1987C2211yescharacteristic
79#1987C2211yescharacteristic
80#1984C2211yescharacteristic
81#1984C2211yescharacteristic
82#1981C2211yescharacteristic
83#1956C2211yescharacteristic
84#1949C2211yescharacteristic
85#1948C2211yescharacteristic
86#1947C2211yescharacteristic
87#1943C2212yes—
88#1942C2212yes—
89#1935C2212yes—
90#1933C2212yes—
91#1928C2212yes—
92#1928C2212yes—
93#1928C2212yes—
94#1928C2212yes—
95#1928C2212yes—
96#1927C2212yes—
97#1927C2212yes—
98#1921C2212yes—
99#1921C2212yes—
100#1920C2212yes—
101#1920C2211yescharacteristic
102#1919C2212yes—
103#1918C2211yescharacteristic
104#1918C2212yes—
105#1918C2212yes—
106#1918C2212yes—
107#1917C2212yes—
108#1916C2212yes—
109#1878C2212yes—
110#1878C2212yes—
111#1878C2212yes—
112#1878C2212yes—
113#1878C2212yes—
114#1865C2212yes—
115#1865C2212yes—
116#1865C2211yescharacteristic
117#1863C2212yes—
118#1863C2212yes—
119#1863C2211yescharacteristic
120#1863C2212yes—
121#1840C2212yes—
122#1831C2211yescharacteristic
123#1805C2211yescharacteristic
124C4×#107C2211yescharacteristic
125C4×#107C2212yes—
126C4×#106C2211yescharacteristic
127#1770C2211yescharacteristic
128C8×C4⋊C4C2211yescharacteristic
129C4×C4⋊C8C2211yescharacteristic
130#1763C2212yes—
131#1763C2212yes—
132C4×#103C2211yescharacteristic
133C4×#103C2211yescharacteristic
134#1746C2212yes—
135C22×Q16C2312yes—
136Q82—24no—
137Q82—24no—
138C2×C4⋊Q8—21no—
139C2×C4⋊Q8—22no—
140C2×C4⋊Q8C2311yescharacteristic
141C2×C4⋊Q8C2311yescharacteristic
142C2×C4⋊Q8C2312yes—
143C2×C4⋊Q8C2311yescharacteristic
144C2×#125C2311yescharacteristic
145C2×C4×Q8—21no—
146C2×C4×Q8—22no—
147C2×C4×Q8—21no—
148C2×C4×Q8C2312yes—
149C2×C4×Q8C2311yescharacteristic
150C2×C4×Q8C2311yescharacteristic
151C2×C4×Q8C2312yes—
152C2×C4×Q8C2312yes—
153C2×C4×Q8C2311yescharacteristic
154C2×C4×Q8C2311yescharacteristic
155#500—24no—
156#487—24no—
157#479—24no—
158#477—24no—
159#462—24no—
160C4×Q16—28no—
161C2×#107C2312yes—
162C2×#107C2311yescharacteristic
163C2×#107C2312yes—
164C2×#107C2312yes—
165C2×#107C2312yes—
166C2×#107C2311yescharacteristic
167C2×#106C2311yescharacteristic
168C2×#106C2311yescharacteristic
169C2×#106C2312yes—
170C2×C4⋊C8C2311yescharacteristic
171C2×C4⋊C8C2311yescharacteristic
172C2×C4⋊C8C2311yescharacteristic
173C2×C4⋊C8C2311yescharacteristic
174C2×#103C2312yes—
175C2×#103C2311yescharacteristic
176C2×#103C2311yescharacteristic
177C2×#103C2311yescharacteristic
178C2×#103C2312yes—
179C2×#103C2312yes—
180C2×#103C2312yes—
181C2×#103C2312yes—
182C2×#103C2312yes—
183C2×#103C2312yes—
184C2×#103C2312yes—
185C2×#103C2311yescharacteristic
186C2×#97C2312yes—
187C2×C4×C8C2311yescharacteristic
188C2×C4×C8C2311yescharacteristic
189C2×C4×C8C2312yes—
190#401—22no—
191#401—22no—
192#400—22no—
193#400—22no—
194#400—22no—
195#400—22no—
196#400—22no—
197#400—22no—
198#398—22no—
199#395—22no—
200#391—21no—
201#391—21no—
202#391—21no—
203#391—22no—
204#391—21no—
205#391—21no—
206#391—21no—
207#391—22no—
208#391—22no—
209#391—22no—
210#391—22no—
211#391D411yescharacteristic
212#391D411yescharacteristic
213#391C2312yes—
214#391C2312yes—
215#391C2312yes—
216#391C2312yes—
217#391C2312yes—
218#389—22no—
219#389—21no—
220#389—21no—
221#389—22no—
222#389—22no—
223#389—21no—
224#389—21no—
225#389—22no—
226#389D411yescharacteristic
227#389D411yescharacteristic
228#389C2311yescharacteristic
229#389C2311yescharacteristic
230#389C2312yes—
231#389C2312yes—
232#389C2312yes—
233#389C2312yes—
234#389C2312yes—
235#389C2312yes—
236#387—22no—
237#387—22no—
238#387—21no—
239#387—21no—
240#387—22no—
241#387—22no—
242#387—22no—
243#387—22no—
244#387—22no—
245#387—22no—
246#387—22no—
247#387—22no—
248#387D412yes—
249#387D412yes—
250#384—21no—
251#384—21no—
252#384C2311yescharacteristic
253#383—22no—
254#383—22no—
255#382—21no—
256#382—21no—
257#382—21no—
258#382—22no—
259#382—22no—
260#382C2311yescharacteristic
261#382C2312yes—
262C4×C4⋊C4—21no—
263C4×C4⋊C4—21no—
264C4×C4⋊C4—21no—
265C4×C4⋊C4—21no—
266C4×C4⋊C4—21no—
267C4×C4⋊C4—21no—
268C4×C4⋊C4—21no—
269C4×C4⋊C4—21no—
270C4×C4⋊C4—21no—
271C4×C4⋊C4—21no—
272C4×C4⋊C4—21no—
273C4×C4⋊C4—22no—
274C4×C4⋊C4—22no—
275C4×C4⋊C4—22no—
276C4×C4⋊C4—22no—
277C4×C4⋊C4C2311yescharacteristic
278C4×C4⋊C4C2311yescharacteristic
279C4×C4⋊C4C2312yes—
280C4×C4⋊C4C2312yes—
281C4×C4⋊C4C2311yescharacteristic
282C4×C4⋊C4C2312yes—
283C4×C4⋊C4C2311yescharacteristic
284C4×C4⋊C4C2311yescharacteristic
285C4×C4⋊C4C2312yes—
286C4×C4⋊C4C2311yescharacteristic
287C4×C4⋊C4C2312yes—
288#376—22no—
289#376C2312yes—
290C43—21no—
291C43C2311yescharacteristic
292#340C2311yescharacteristic
293#340C2311yescharacteristic
294#340C2312yes—
295#340C2312yes—
296#340C2312yes—
297#340C2311yescharacteristic
298#340C2311yescharacteristic
299#340C2312yes—
300#340C2312yes—
301#340C2312yes—
302#340C2312yes—
303#340C2312yes—
304#340C2312yes—
305#340C2311yescharacteristic
306#340C2311yescharacteristic
307#340C2312yes—
308#340C2311yescharacteristic
309#340C2311yescharacteristic
310#336C2312yes—
311#336C2312yes—
312#336C2312yes—
313#336C2312yes—
314#336C2312yes—
315#336C2311yescharacteristic
316#336C2312yes—
317#336C2312yes—
318#336C2312yes—
319#336C2311yescharacteristic
320C22×Q8—22no—
321C22×Q8—21no—
322C22×Q8—21no—
323C22×Q8C2×D411yescharacteristic
324C22×Q8C2412yes—
325C22×Q8C2411yescharacteristic
326C2×Q16—24no—
327C2×Q16—24no—
328C2×Q16—24no—
329C22×C8C2412yes—
330C22×C8C2411yescharacteristic
331C22×C8C2411yescharacteristic
332C22×C8C2412yes—
333C22×C8C2411yescharacteristic
334C22×C8C2411yescharacteristic
335C4⋊Q8—216no—
336C4⋊Q8—28no—
337C4⋊Q8—24no—
338C4⋊Q8—24no—
339C4⋊Q8—22no—
340C4⋊Q8—22no—
341C4⋊Q8—22no—
342C4⋊Q8—22no—
343C4⋊Q8—22no—
344C4⋊Q8#4114yes—
345#125—22no—
346#125—41no—
347C4×Q8—28no—
348C4×Q8—24no—
349C4×Q8—22no—
350C4×Q8—24no—
351C4×Q8—22no—
352C4×Q8—22no—
353C4×Q8—22no—
354C4×Q8—22no—
355C4×Q8—24no—
356C4×Q8—22no—
357C4×Q8—24no—
358C4×Q8—42no—
359C4×Q8—42no—
360C4×Q8—41no—
361C4×Q8—82no—
362C4×Q8—84no—
363C4×Q8#4114yes—
364C2×C4⋊C4—21no—
365C2×C4⋊C4—22no—
366C2×C4⋊C4—22no—
367C2×C4⋊C4—21no—
368C2×C4⋊C4—21no—
369C2×C4⋊C4—22no—
370C2×C4⋊C4—21no—
371C2×C4⋊C4—22no—
372C2×C4⋊C4—22no—
373C2×C4⋊C4—21no—
374C2×C4⋊C4—21no—
375C2×C4⋊C4—21no—
376C2×C4⋊C4—21no—
377C2×C4⋊C4—21no—
378C2×C4⋊C4—21no—
379C2×C4⋊C4—22no—
380C2×C4⋊C4—21no—
381C2×C4⋊C4—21no—
382C2×C4⋊C4—22no—
383C2×C4⋊C4—21no—
384C2×C4⋊C4—22no—
385C2×C4⋊C4—22no—
386C2×C4⋊C4—22no—
387C2×C4⋊C4C2×D411yescharacteristic
388C2×C4⋊C4C2×D412yes—
389C2×C4⋊C4C2×D411yescharacteristic
390C2×C4⋊C4C2×D411yescharacteristic
391C2×C4⋊C4C2×D412yes—
392C2×C4⋊C4C2×D412yes—
393C2×C4⋊C4C2411yescharacteristic
394C2×C4⋊C4C2412yes—
395C2×C4⋊C4C2411yescharacteristic
396C2×C4⋊C4C2412yes—
397C2×C4⋊C4C2411yescharacteristic
398C2×C4⋊C4C2411yescharacteristic
399C2×C4⋊C4C2411yescharacteristic
400C2×C4⋊C4C2412yes—
401C2×C4⋊C4C2411yescharacteristic
402C2×C4⋊C4C2411yescharacteristic
403C2×C42—21no—
404C2×C42—22no—
405C2×C42—21no—
406C2×C42—21no—
407C2×C42—21no—
408C2×C42—22no—
409C2×C42—22no—
410C2×C42—21no—
411C2×C42—21no—
412C2×C42—21no—
413C2×C42—22no—
414C2×C42—22no—
415C2×C42—22no—
416C2×C42—21no—
417C2×C42—22no—
418C2×C42C2×D411yescharacteristic
419C2×C42C2×D411yescharacteristic
420C2×C42C2×D411yescharacteristic
421C2×C42C2×D411yescharacteristic
422C2×C42C2412yes—
423C2×C42C2412yes—
424C2×C42C2411yescharacteristic
425C2×C42C2411yescharacteristic
426C2×C42C2411yescharacteristic
427#107—22no—
428#107—24no—
429#107—24no—
430#107—24no—
431#107—24no—
432#107—22no—
433#106—22no—
434#106—22no—
435#106—42no—
436C4⋊C8—22no—
437C4⋊C8—22no—
438C4⋊C8—22no—
439C4⋊C8—41no—
440#103—24no—
441#103—22no—
442#103—22no—
443#103—22no—
444#103—24no—
445#103—24no—
446#103—24no—
447#103—24no—
448#103—24no—
449#103—42no—
450#103—42no—
451#103#4114yes—
452#97—42no—
453C4×C8—24no—
454C4×C8—22no—
455C4×C8—22no—
456#95—22no—
457#95—22no—
458#95—22no—
459#95—21no—
460#95—21no—
461#95—21no—
462#95—22no—
463#95—22no—
464#95—22no—
465#95—21no—
466#95—21no—
467#95—22no—
468#95—21no—
469#95—22no—
470#95—21no—
471#95—22no—
472#95—22no—
473#95—22no—
474#95—21no—
475#95—21no—
476#95—22no—
477#95—22no—
478#95—22no—
479#95—22no—
480#95—22no—
481#95—22no—
482#95—22no—
483#95—22no—
484#95—22no—
485#95—22no—
486#95—22no—
487#95C2×D412yes—
488#95C2×D412yes—
489#95C2×D412yes—
490#95C2×D412yes—
491#95C2×D412yes—
492#95C2×D412yes—
493#95C2×D412yes—
494C2×Q8—28no—
495C2×Q8—22no—
496C2×Q8—24no—
497C2×Q8—22no—
498C2×Q8—22no—
499C2×Q8—21no—
500C2×Q8—21no—
501C2×Q8—21no—
502C2×Q8—22no—
503C2×Q8—22no—
504C2×Q8—22no—
505C2×Q8—22no—
506C2×Q8—22no—
507C2×Q8—22no—
508C2×Q8—42no—
509C2×Q8—81no—
510C2×Q8—82no—
511C2×Q8—81no—
512C2×Q8—82no—
513C2×Q8—81no—
514C2×Q8—81no—
515C2×Q8C2×#4112yes—
516C2×Q8C2×#4112yes—
517C2×Q82−1+412yes—
518C22×C4—21no—
519C22×C4—21no—
520C22×C4—21no—
521C22×C4—21no—
522C22×C4—22no—
523C22×C4—21no—
524C22×C4—21no—
525C22×C4—22no—
526C22×C4—21no—
527C22×C4—21no—
528C22×C4—22no—
529C22×C4—22no—
530C22×C4C22×D412yes—
531C22×C4C22×D411yescharacteristic
532C22×C4C22×D411yescharacteristic
533C22×C4C22×D411yescharacteristic
534C22×C4C22×D412yes—
535C22×C4C22×D412yes—
536C22×C4C22×D411yescharacteristic
537C22×C4C22×D412yes—
538C22×C4C22×D411yescharacteristic
539C22×C4C22×D411yescharacteristic
540C22×C4C2511yescharacteristic
541Q16—48no—
542C2×C8—21no—
543C2×C8—21no—
544C2×C8—21no—
545C2×C8—21no—
546C2×C8—21no—
547C2×C8—22no—
548C2×C8—21no—
549C2×C8—22no—
550C2×C8—22no—
551C2×C8—22no—
552C2×C8—21no—
553C2×C8—21no—
554C2×C8—21no—
555C2×C8—22no—
556C2×C8—21no—
557C2×C8—22no—
558C2×C8—21no—
559C2×C8C2×#4112yes—
560C4⋊C4—24no—
561C4⋊C4—28no—
562C4⋊C4—28no—
563C4⋊C4—24no—
564C4⋊C4—24no—
565C4⋊C4—24no—
566C4⋊C4—22no—
567C4⋊C4—22no—
568C4⋊C4—22no—
569C4⋊C4—22no—
570C4⋊C4—21no—
571C4⋊C4—21no—
572C4⋊C4—21no—
573C4⋊C4—22no—
574C4⋊C4—21no—
575C4⋊C4—22no—
576C4⋊C4—21no—
577C4⋊C4—22no—
578C4⋊C4—21no—
579C4⋊C4—21no—
580C4⋊C4—21no—
581C4⋊C4—21no—
582C4⋊C4—21no—
583C4⋊C4—22no—
584C4⋊C4—21no—
585C4⋊C4—42no—
586C4⋊C4—42no—
587C4⋊C4—42no—
588C4⋊C4—44no—
589C4⋊C4—44no—
590C4⋊C4—42no—
591C4⋊C4—42no—
592C4⋊C4—44no—
593C4⋊C4—42no—
594C4⋊C4—42no—
595C4⋊C4—42no—
596C4⋊C4—42no—
597C4⋊C4—81no—
598C4⋊C4—82no—
599C4⋊C4—82no—
600C4⋊C4—82no—
601C4⋊C4—81no—
602C4⋊C4—82no—
603C4⋊C4—81no—
604C4⋊C4—82no—
605C4⋊C4C2×#4114yes—
606C4⋊C4C2×#4112yes—
607C4⋊C4C2×#4112yes—
608C4⋊C42−1+412yes—
609C42—24no—
610C42—28no—
611C42—24no—
612C42—22no—
613C42—22no—
614C42—22no—
615C42—21no—
616C42—22no—
617C42—22no—
618C42—21no—
619C42—21no—
620C42—22no—
621C42—21no—
622C42—41no—
623C42—41no—
624C42—44no—
625C42—42no—
626C42—44no—
627C42—42no—
628C42—42no—
629C42—42no—
630C42—42no—
631C42—81no—
632C42—82no—
633C42—82no—
634C42—82no—
635C42—82no—
636C42C2×#4112yes—
637C42C2×#4112yes—
638C23C23×D411yescharacteristic
639Q8—22no—
640Q8—22no—
641Q8—22no—
642Q8—44no—
643Q8—42no—
644Q8—42no—
645Q8—42no—
646Q8—42no—
647Q8—84no—
648Q8—88no—
649Q8—162no—
650Q8#55614yes—
651C2×C4—21no—
652C2×C4—21no—
653C2×C4—22no—
654C2×C4—22no—
655C2×C4—22no—
656C2×C4—21no—
657C2×C4—21no—
658C2×C4—22no—
659C2×C4—22no—
660C2×C4—21no—
661C2×C4—21no—
662C2×C4—22no—
663C2×C4—22no—
664C2×C4—22no—
665C2×C4—22no—
666C2×C4—24no—
667C2×C4—22no—
668C2×C4—22no—
669C2×C4—21no—
670C2×C4—21no—
671C2×C4—22no—
672C2×C4—22no—
673C2×C4—21no—
674C2×C4—21no—
675C2×C4—22no—
676C2×C4—21no—
677C2×C4—21no—
678C2×C4—21no—
679C2×C4—21no—
680C2×C4—22no—
681C2×C4—22no—
682C2×C4—22no—
683C2×C4—22no—
684C2×C4—41no—
685C2×C4—42no—
686C2×C4—41no—
687C2×C4—41no—
688C2×C4—42no—
689C2×C4—41no—
690C2×C4—41no—
691C2×C4—41no—
692C2×C4—41no—
693C2×C4—41no—
694C2×C4—41no—
695C2×C4—42no—
696C2×C4—42no—
697C2×C4—41no—
698C2×C4—42no—
699C2×C4—42no—
700C2×C4—42no—
701C2×C4—41no—
702C2×C4—42no—
703C2×C4—42no—
704C2×C4—41no—
705C2×C4—41no—
706C2×C4—41no—
707C2×C4—42no—
708C2×C4—41no—
709C2×C4—41no—
710C2×C4—41no—
711C2×C4—41no—
712C2×C4—42no—
713C2×C4#52512yes—
714C2×C4#52512yes—
715C2×C4#57812yes—
716C2×C4#57812yes—
717C2×C4C22×#4111yescharacteristic
718C2×C4C22×#4111yescharacteristic
719C2×C4C22×#4111yescharacteristic
720C2×C4C2×2−1+411yescharacteristic
721C2×C4C2×2−1+411yescharacteristic
722C2×C4#58511yescharacteristic
723C8—22no—
724C8—42no—
725C8—41no—
726C8—42no—
727C8—41no—
728C8—41no—
729C22C2×#52511yescharacteristic
730C22D4×#4111yescharacteristic
731C22#348611yescharacteristic
732C22C2×#57811yescharacteristic
733C22C2×#57811yescharacteristic
734C22#358411yescharacteristic
735C22#358511yescharacteristic
736C4—24no—
737C4—22no—
738C4—41no—
739C4—42no—
740C4—41no—
741C4—42no—
742C4—42no—
743C4—42no—
744C4—41no—
745C4—42no—
746C4—41no—
747C4—42no—
748C4—42no—
749C4—44no—
750C4—81no—
751C4—81no—
752C4—82no—
753C4—82no—
754C4—81no—
755C4—81no—
756C4—82no—
757C4#312714yes—
758C4C2×#55611yescharacteristic
759C4#348211yescharacteristic
760C4#350711yescharacteristic
761C4#351611yescharacteristic
762C2#3623311yescharacteristic
763C2C2×#312711yescharacteristic
764C2#6059211yescharacteristic
765C2#6147811yescharacteristic
766C2#6154211yescharacteristic
767C2#6185911yescharacteristic
768C2#6186611yescharacteristic
subgroupG/H
center, socle, prime radicalC23C23×D4
commutator, Frattini, cosocleC22×C4C25
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ #2295 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Maximal subgroups

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

subgroupindex
#389862
#343352
#330912
#330272
#295262
#294022
#292752
#286562
#275752
#271272
#270162
#263832
#263172
#262792
#259362
#258412
#232212
#231582
#229982
#229712
#228292
C4×#4792
#211352