Finite Group Databasebeta

← #6542057

Subgroups — #6542057

Order 512.

1338 conjugacy classes of subgroups.

The subgroup lattice has 749 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36556C211yesmaximal, characteristic
2#34328C211yesmaximal, characteristic
3#30097C212yesmaximal
4#29867C211yesmaximal, characteristic
5#29577C212yesmaximal
6#29399C212yesmaximal
7#29274C211yesmaximal, characteristic
8#28688C211yesmaximal, characteristic
9#27368C212yesmaximal
10#27366C212yesmaximal
11#26870C212yesmaximal
12#26868C212yesmaximal
13#26281C211yesmaximal, characteristic
14#26277C211yesmaximal, characteristic
15#25840C211yesmaximal, characteristic
16#25836C211yesmaximal, characteristic
17#23836C212yesmaximal
18#23779C211yesmaximal, characteristic
19#23157C211yesmaximal, characteristic
20#23156C211yesmaximal, characteristic
21#23063C211yesmaximal, characteristic
22#23062C211yesmaximal, characteristic
23#22828C211yesmaximal, characteristic
24C2×#557C2211yescharacteristic
25C2×#500C2211yescharacteristic
26C2×#498C2211yescharacteristic
27C2×#477C2211yescharacteristic
28C2×#475C2211yescharacteristic
29#2864C2212yes—
30#2862C2211yescharacteristic
31#2588—22no—
32#2567C2211yescharacteristic
33#2546C2211yescharacteristic
34#2520—22no—
35#2511—21no—
36#2509C2212yes—
37#2508C2212yes—
38#2446—21no—
39#2371—22no—
40#2370C2212yes—
41#2369C2211yescharacteristic
42#2365—21no—
43#2365—21no—
44#2365C2211yescharacteristic
45#2365C2211yescharacteristic
46#2356—21no—
47#2356C2212yes—
48#2355—21no—
49#2355C2211yescharacteristic
50#2355C2211yescharacteristic
51#2336—22no—
52#2327—21no—
53C4×C4⋊Q8C2211yescharacteristic
54C4×#125C2212yes—
55C4×#125C2212yes—
56C4×#125C2211yescharacteristic
57#2295C2212yes—
58#2295C2211yescharacteristic
59#2276—21no—
60C42×Q8C2211yescharacteristic
61#2101C2212yes—
62#2101C2211yescharacteristic
63#2101C2212yes—
64#2101C2212yes—
65#2097C2211yescharacteristic
66#2088C2211yescharacteristic
67#2086C2211yescharacteristic
68#2085C2212yes—
69#2085C2212yes—
70#2085C2212yes—
71#2085C2212yes—
72#2085C2212yes—
73#2085C2212yes—
74#2074C2212yes—
75#2074C2212yes—
76#2068C2211yescharacteristic
77#2066C2211yescharacteristic
78#2059C2211yescharacteristic
79#2059C2212yes—
80#2059C2212yes—
81#2059C2212yes—
82#2059C2212yes—
83#2059C2212yes—
84#2026C2211yescharacteristic
85#1990C2211yescharacteristic
86#1990C2211yescharacteristic
87#1988C2211yescharacteristic
88#1988C2211yescharacteristic
89#1988C2211yescharacteristic
90#1987C2211yescharacteristic
91#1984C2211yescharacteristic
92#1984C2211yescharacteristic
93#1949C2212yes—
94#1949C2212yes—
95#1948C2211yescharacteristic
96#1947C2211yescharacteristic
97#1942C2212yes—
98#1941C2212yes—
99#1933C2212yes—
100#1933C2212yes—
101#1928C2212yes—
102#1927C2212yes—
103#1921C2212yes—
104#1921C2212yes—
105#1921C2212yes—
106#1920C2212yes—
107#1920C2212yes—
108#1919C2212yes—
109#1919C2212yes—
110#1919C2212yes—
111#1919C2212yes—
112#1918C2212yes—
113#1918C2212yes—
114#1918C2212yes—
115#1916C2212yes—
116#1916C2212yes—
117#1865C2212yes—
118#1865C2212yes—
119#1863C2212yes—
120#1863C2212yes—
121#1839C2212yes—
122#1831C2211yescharacteristic
123C4×#107C2212yes—
124C4×#107C2212yes—
125C4×#107C2211yescharacteristic
126C4×#106C2212yes—
127C4×#106C2212yes—
128C4×#106C2211yescharacteristic
129C8×C4⋊C4C2211yescharacteristic
130C8×C4⋊C4C2211yescharacteristic
131C4×C4⋊C8C2211yescharacteristic
132C4×#103C2211yescharacteristic
133C4×#103C2211yescharacteristic
134#1744C2212yes—
135#557—24no—
136#557—24no—
137C2×C4⋊Q8—21no—
138C2×C4⋊Q8—21no—
139C2×C4⋊Q8C2311yescharacteristic
140C2×C4⋊Q8C2311yescharacteristic
141C2×C4⋊Q8C2311yescharacteristic
142C2×#125—22no—
143C2×#125C2311yescharacteristic
144C2×#125C2312yes—
145C2×C4×Q8—21no—
146C2×C4×Q8—21no—
147C2×C4×Q8—21no—
148C2×C4×Q8—21no—
149C2×C4×Q8D411yescharacteristic
150C2×C4×Q8D411yescharacteristic
151C2×C4×Q8C2312yes—
152C2×C4×Q8C2311yescharacteristic
153C2×C4×Q8C2311yescharacteristic
154#500—24no—
155#498—24no—
156#477—24no—
157#475—24no—
158C2×#107C2312yes—
159C2×#107C2311yescharacteristic
160C2×#107C2312yes—
161C2×#107C2312yes—
162C2×#107C2312yes—
163C2×#107C2311yescharacteristic
164C2×#106C2312yes—
165C2×#106C2311yescharacteristic
166C2×#106C2312yes—
167C2×#106C2312yes—
168C2×#106C2312yes—
169C2×#106C2311yescharacteristic
170C2×C4⋊C8C2311yescharacteristic
171C2×C4⋊C8C2311yescharacteristic
172C2×C4⋊C8C2311yescharacteristic
173C2×C4⋊C8C2311yescharacteristic
174C2×#103C2312yes—
175C2×#103C2312yes—
176C2×#103C2311yescharacteristic
177C2×#103C2311yescharacteristic
178C2×#103C2311yescharacteristic
179C2×#103C2311yescharacteristic
180C2×C4×C8C2311yescharacteristic
181C2×C4×C8C2311yescharacteristic
182C2×C4×C8C2312yes—
183C2×C4×C8C2312yes—
184#400—22no—
185#400—22no—
186#400—22no—
187#398—22no—
188#398—22no—
189#395—22no—
190#391—21no—
191#391—21no—
192#391—21no—
193#391—21no—
194#391—21no—
195#391—21no—
196#391—21no—
197#391—21no—
198#391—21no—
199#391—21no—
200#391—22no—
201#391C2311yescharacteristic
202#391C2311yescharacteristic
203#391C2312yes—
204#389—22no—
205#389—22no—
206#389—21no—
207#389—21no—
208#389C2311yescharacteristic
209#389C2312yes—
210#389C2312yes—
211#389C2312yes—
212#389C2311yescharacteristic
213#389C2311yescharacteristic
214#389C2312yes—
215#389C2312yes—
216#389C2311yescharacteristic
217#389C2312yes—
218#389C2312yes—
219#389C2312yes—
220#387—22no—
221#387—22no—
222#387—22no—
223#387—22no—
224#387—22no—
225#387—22no—
226#387—21no—
227#387—22no—
228#387—22no—
229#387—22no—
230#387—21no—
231#387—22no—
232#387—22no—
233#387—22no—
234#387—21no—
235#387—22no—
236#387—22no—
237#387—22no—
238#387—21no—
239#387—22no—
240#387—22no—
241#387—22no—
242#387—22no—
243#387—22no—
244#384—21no—
245#384C2311yescharacteristic
246#383—22no—
247#382—21no—
248#382—22no—
249#382C2312yes—
250#382C2312yes—
251#382C2311yescharacteristic
252#382C2312yes—
253C4×C4⋊C4—21no—
254C4×C4⋊C4—21no—
255C4×C4⋊C4—21no—
256C4×C4⋊C4—22no—
257C4×C4⋊C4—22no—
258C4×C4⋊C4—21no—
259C4×C4⋊C4—22no—
260C4×C4⋊C4—21no—
261C4×C4⋊C4—22no—
262C4×C4⋊C4—21no—
263C4×C4⋊C4—22no—
264C4×C4⋊C4—21no—
265C4×C4⋊C4—22no—
266C4×C4⋊C4D412yes—
267C4×C4⋊C4D411yescharacteristic
268C4×C4⋊C4D411yescharacteristic
269C4×C4⋊C4D412yes—
270C4×C4⋊C4C2311yescharacteristic
271C4×C4⋊C4C2312yes—
272C4×C4⋊C4C2312yes—
273C4×C4⋊C4C2312yes—
274C4×C4⋊C4C2312yes—
275C4×C4⋊C4C2311yescharacteristic
276C4×C4⋊C4C2312yes—
277C4×C4⋊C4C2311yescharacteristic
278C4×C4⋊C4C2312yes—
279C4×C4⋊C4C2311yescharacteristic
280C4×C4⋊C4C2312yes—
281#376—21no—
282#376—22no—
283#376—22no—
284C43C2312yes—
285C43C2311yescharacteristic
286#340C2311yescharacteristic
287#340C2311yescharacteristic
288#340C2311yescharacteristic
289#340C2311yescharacteristic
290#340C2312yes—
291#340C2312yes—
292#340C2312yes—
293#340C2312yes—
294#340C2311yescharacteristic
295#340C2311yescharacteristic
296#340C2312yes—
297#340C2312yes—
298#340C2312yes—
299#340C2312yes—
300#340C2312yes—
301#340C2312yes—
302#340C2312yes—
303#340C2312yes—
304#340C2311yescharacteristic
305#340C2311yescharacteristic
306#340C2312yes—
307#340C2312yes—
308#336C2312yes—
309#336C2311yescharacteristic
310#336C2312yes—
311#336C2312yes—
312#336C2312yes—
313#336C2312yes—
314#336C2312yes—
315#336C2311yescharacteristic
316#336C2312yes—
317#336C2312yes—
318C22×Q8—21no—
319C22×Q8—21no—
320C22×Q8—21no—
321C22×Q8C2×D411yescharacteristic
322C22×Q8C2411yescharacteristic
323C22×C8C2411yescharacteristic
324C22×C8C2412yes—
325C22×C8C2411yescharacteristic
326C22×C8C2412yes—
327C22×C8C2411yescharacteristic
328C22×C8C2411yescharacteristic
329C4⋊Q8—28no—
330C4⋊Q8—22no—
331C4⋊Q8—22no—
332C4⋊Q8—22no—
333C4⋊Q8—22no—
334C4⋊Q8—22no—
335C4⋊Q8—44no—
336C4⋊Q8#4114yes—
337#125—216no—
338#125—24no—
339#125—24no—
340#125—22no—
341#125—22no—
342C4×Q8—28no—
343C4×Q8—24no—
344C4×Q8—24no—
345C4×Q8—24no—
346C4×Q8—22no—
347C4×Q8—22no—
348C4×Q8—22no—
349C4×Q8—42no—
350C4×Q8—44no—
351C4×Q8—44no—
352C4×Q8—44no—
353C4×Q8#4114yes—
354C2×C4⋊C4—22no—
355C2×C4⋊C4—21no—
356C2×C4⋊C4—22no—
357C2×C4⋊C4—21no—
358C2×C4⋊C4—22no—
359C2×C4⋊C4—21no—
360C2×C4⋊C4—21no—
361C2×C4⋊C4—21no—
362C2×C4⋊C4—22no—
363C2×C4⋊C4—22no—
364C2×C4⋊C4—22no—
365C2×C4⋊C4—22no—
366C2×C4⋊C4—22no—
367C2×C4⋊C4—21no—
368C2×C4⋊C4—21no—
369C2×C4⋊C4—21no—
370C2×C4⋊C4—21no—
371C2×C4⋊C4—22no—
372C2×C4⋊C4—21no—
373C2×C4⋊C4—21no—
374C2×C4⋊C4—21no—
375C2×C4⋊C4—22no—
376C2×C4⋊C4—22no—
377C2×C4⋊C4C2×D411yescharacteristic
378C2×C4⋊C4C2×D411yescharacteristic
379C2×C4⋊C4C2×D412yes—
380C2×C4⋊C4C2×D412yes—
381C2×C4⋊C4C2×D412yes—
382C2×C4⋊C4C2×D411yescharacteristic
383C2×C4⋊C4C2412yes—
384C2×C4⋊C4C2412yes—
385C2×C4⋊C4C2411yescharacteristic
386C2×C4⋊C4C2412yes—
387C2×C4⋊C4C2411yescharacteristic
388C2×C4⋊C4C2411yescharacteristic
389C2×C4⋊C4C2411yescharacteristic
390C2×C4⋊C4C2411yescharacteristic
391C2×C4⋊C4C2412yes—
392C2×C4⋊C4C2411yescharacteristic
393C2×C4⋊C4C2411yescharacteristic
394C2×C42—22no—
395C2×C42—21no—
396C2×C42—22no—
397C2×C42—22no—
398C2×C42—21no—
399C2×C42—22no—
400C2×C42—21no—
401C2×C42—21no—
402C2×C42—22no—
403C2×C42—22no—
404C2×C42—22no—
405C2×C42C2×D411yescharacteristic
406C2×C42C2×D412yes—
407C2×C42C2×D411yescharacteristic
408C2×C42C2×D412yes—
409C2×C42C2×D412yes—
410C2×C42C2×D412yes—
411C2×C42C2×D411yescharacteristic
412C2×C42C2×D411yescharacteristic
413C2×C42C2412yes—
414C2×C42C2411yescharacteristic
415C2×C42C2412yes—
416C2×C42C2411yescharacteristic
417C2×C42C2411yescharacteristic
418#107—22no—
419#107—24no—
420#107—24no—
421#107—24no—
422#107—22no—
423#107—24no—
424#106—22no—
425#106—24no—
426#106—24no—
427#106—24no—
428#106—24no—
429#106—22no—
430C4⋊C8—22no—
431C4⋊C8—22no—
432C4⋊C8—22no—
433C4⋊C8—22no—
434#103—24no—
435#103—24no—
436#103—22no—
437#103—22no—
438#103—22no—
439#103—22no—
440C4×C8—24no—
441C4×C8—24no—
442C4×C8—22no—
443C4×C8—22no—
444#95—21no—
445#95—21no—
446#95—22no—
447#95—21no—
448#95—21no—
449#95—21no—
450#95—22no—
451#95—22no—
452#95—22no—
453#95—22no—
454#95—22no—
455#95—21no—
456#95—21no—
457#95—22no—
458#95—22no—
459#95—22no—
460#95—21no—
461#95—21no—
462#95—21no—
463#95—22no—
464#95—22no—
465#95—22no—
466#95—22no—
467#95—22no—
468#95—21no—
469#95—22no—
470#95—21no—
471#95—22no—
472#95—22no—
473#95—21no—
474#95—22no—
475#95—22no—
476#95—22no—
477#95—22no—
478#95—22no—
479#95C2×D412yes—
480#95C2×D412yes—
481#95C2×D412yes—
482C2×Q8—22no—
483C2×Q8—24no—
484C2×Q8—22no—
485C2×Q8—22no—
486C2×Q8—21no—
487C2×Q8—21no—
488C2×Q8—21no—
489C2×Q8—21no—
490C2×Q8—21no—
491C2×Q8—42no—
492C2×Q8—42no—
493C2×Q8—42no—
494C2×Q8—42no—
495C2×Q8—42no—
496C2×Q8—42no—
497C2×Q8—81no—
498C2×Q8—81no—
499C2×Q8C2×#4112yes—
500C22×C4—21no—
501C22×C4—21no—
502C22×C4—21no—
503C22×C4—21no—
504C22×C4—22no—
505C22×C4—21no—
506C22×C4—22no—
507C22×C4—21no—
508C22×C4—21no—
509C22×C4—21no—
510C22×C4—22no—
511C22×C4—22no—
512C22×C4C22×D412yes—
513C22×C4C22×D411yescharacteristic
514C22×C4C22×D411yescharacteristic
515C22×C4C22×D411yescharacteristic
516C22×C4C22×D411yescharacteristic
517C22×C4C22×D412yes—
518C22×C4C22×D412yes—
519C22×C4C22×D411yescharacteristic
520C22×C4C22×D412yes—
521C22×C4C22×D411yescharacteristic
522C22×C4C2511yescharacteristic
523C2×C8—21no—
524C2×C8—21no—
525C2×C8—21no—
526C2×C8—21no—
527C2×C8—21no—
528C2×C8—21no—
529C2×C8—22no—
530C2×C8—22no—
531C2×C8—22no—
532C2×C8—22no—
533C2×C8—21no—
534C2×C8—21no—
535C2×C8—21no—
536C2×C8—21no—
537C2×C8—22no—
538C2×C8—22no—
539C2×C82−1+412yes—
540C2×C82−1+412yes—
541C4⋊C4—28no—
542C4⋊C4—28no—
543C4⋊C4—24no—
544C4⋊C4—24no—
545C4⋊C4—28no—
546C4⋊C4—24no—
547C4⋊C4—24no—
548C4⋊C4—22no—
549C4⋊C4—24no—
550C4⋊C4—24no—
551C4⋊C4—22no—
552C4⋊C4—21no—
553C4⋊C4—22no—
554C4⋊C4—21no—
555C4⋊C4—21no—
556C4⋊C4—22no—
557C4⋊C4—22no—
558C4⋊C4—22no—
559C4⋊C4—22no—
560C4⋊C4—21no—
561C4⋊C4—21no—
562C4⋊C4—21no—
563C4⋊C4—22no—
564C4⋊C4—21no—
565C4⋊C4—21no—
566C4⋊C4—21no—
567C4⋊C4—22no—
568C4⋊C4—21no—
569C4⋊C4—21no—
570C4⋊C4—42no—
571C4⋊C4—42no—
572C4⋊C4—44no—
573C4⋊C4—44no—
574C4⋊C4—44no—
575C4⋊C4—42no—
576C4⋊C4—42no—
577C4⋊C4—44no—
578C4⋊C4—44no—
579C4⋊C4—42no—
580C4⋊C4—81no—
581C4⋊C4—82no—
582C4⋊C4—81no—
583C4⋊C4—81no—
584C4⋊C4—81no—
585C4⋊C4—82no—
586C4⋊C4—82no—
587C4⋊C4#13514yes—
588C4⋊C4C2×#4114yes—
589C4⋊C4C2×#4112yes—
590C4⋊C42−1+412yes—
591C4⋊C42−1+412yes—
592C42—28no—
593C42—24no—
594C42—22no—
595C42—24no—
596C42—22no—
597C42—22no—
598C42—22no—
599C42—22no—
600C42—22no—
601C42—21no—
602C42—21no—
603C42—21no—
604C42—21no—
605C42—42no—
606C42—41no—
607C42—41no—
608C42—42no—
609C42—42no—
610C42—44no—
611C42—42no—
612C42—44no—
613C42—44no—
614C42—42no—
615C42—42no—
616C42—82no—
617C42—82no—
618C42—82no—
619C42C2×#4112yes—
620C42C2×#4112yes—
621C23C23×D411yescharacteristic
622Q8—22no—
623Q8—22no—
624Q8—44no—
625Q8—48no—
626Q8—41no—
627Q8—41no—
628Q8—84no—
629Q8—162no—
630C2×C4—22no—
631C2×C4—21no—
632C2×C4—21no—
633C2×C4—22no—
634C2×C4—21no—
635C2×C4—22no—
636C2×C4—21no—
637C2×C4—21no—
638C2×C4—22no—
639C2×C4—21no—
640C2×C4—22no—
641C2×C4—22no—
642C2×C4—22no—
643C2×C4—22no—
644C2×C4—22no—
645C2×C4—24no—
646C2×C4—21no—
647C2×C4—21no—
648C2×C4—22no—
649C2×C4—22no—
650C2×C4—21no—
651C2×C4—22no—
652C2×C4—21no—
653C2×C4—21no—
654C2×C4—21no—
655C2×C4—22no—
656C2×C4—21no—
657C2×C4—21no—
658C2×C4—22no—
659C2×C4—22no—
660C2×C4—41no—
661C2×C4—41no—
662C2×C4—41no—
663C2×C4—41no—
664C2×C4—41no—
665C2×C4—41no—
666C2×C4—41no—
667C2×C4—42no—
668C2×C4—41no—
669C2×C4—42no—
670C2×C4—41no—
671C2×C4—42no—
672C2×C4—42no—
673C2×C4—41no—
674C2×C4—41no—
675C2×C4—41no—
676C2×C4—42no—
677C2×C4—42no—
678C2×C4—42no—
679C2×C4—41no—
680C2×C4—42no—
681C2×C4—41no—
682C2×C4—41no—
683C2×C4—41no—
684C2×C4—41no—
685C2×C4—41no—
686C2×C4—41no—
687C2×C4—42no—
688C2×C4—41no—
689C2×C4—42no—
690C2×C4—42no—
691C2×C4#52512yes—
692C2×C4#52512yes—
693C2×C4C2×#13514yes—
694C2×C4C2×#13512yes—
695C2×C4#57512yes—
696C2×C4C22×#4111yescharacteristic
697C2×C4C22×#4111yescharacteristic
698C2×C4C2×2−1+411yescharacteristic
699C2×C4C2×2−1+411yescharacteristic
700C2×C4C2×2−1+411yescharacteristic
701C2×C4C2×2−1+411yescharacteristic
702C8—22no—
703C8—22no—
704C8—42no—
705C8—42no—
706C8—41no—
707C8—41no—
708C22C2×#52511yescharacteristic
709C22#348611yescharacteristic
710C22#349911yescharacteristic
711C22C22×#13511yescharacteristic
712C22C2×#57511yescharacteristic
713C22#358411yescharacteristic
714C22#358611yescharacteristic
715C4—24no—
716C4—22no—
717C4—41no—
718C4—41no—
719C4—42no—
720C4—42no—
721C4—44no—
722C4—41no—
723C4—42no—
724C4—41no—
725C4—42no—
726C4—42no—
727C4—42no—
728C4—42no—
729C4—81no—
730C4—81no—
731C4—81no—
732C4—82no—
733C4—81no—
734C4—82no—
735C4—82no—
736C4#310214yes—
737C4#350911yescharacteristic
738C4#350911yescharacteristic
739C4#352311yescharacteristic
740C4#352311yescharacteristic
741C2#3574911yescharacteristic
742C2C2×#310211yescharacteristic
743C2#6059511yescharacteristic
744C2#6182411yescharacteristic
745C2#6184511yescharacteristic
746C2#6247511yescharacteristic
747C2#6250711yescharacteristic
subgroupG/H
center, socle, prime radicalC23C23×D4
commutator, Frattini, cosocleC22×C4C25
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C42×Q8 ◁ 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
#365562
#343282
#300972
#298672
#295772
#293992
#292742
#286882
#273682
#273662
#268702
#268682
#262812
#262772
#258402
#258362
#238362
#237792
#231572
#231562
#230632
#230622
#228282