Finite Group Databasebeta

← #6864804

Subgroups — #6864804

Order 512.

1235 conjugacy classes of subgroups.

The subgroup lattice has 755 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36535C211yesmaximal, characteristic
2#34951C211yesmaximal, characteristic
3#33069C211yesmaximal, characteristic
4#32402C212yesmaximal
5#32250C212yesmaximal
6#31600C212yesmaximal
7#30828C212yesmaximal
8#30080C212yesmaximal
9#29583C212yesmaximal
10#29399C212yesmaximal
11#29255C211yesmaximal, characteristic
12#29193C211yesmaximal, characteristic
13#28835C211yesmaximal, characteristic
14#28800C211yesmaximal, characteristic
15#27777C211yesmaximal, characteristic
16#27757C211yesmaximal, characteristic
17#26319C211yesmaximal, characteristic
18#26319C211yesmaximal, characteristic
19#25063C211yesmaximal, characteristic
20#24997C211yesmaximal, characteristic
21#22832C211yesmaximal, characteristic
22#22225C212yesmaximal
23C4×#475C211yesmaximal, characteristic
24C2×#491C2212yes—
25C2×#475C2211yescharacteristic
26#2864C2212yes—
27#2859C2211yescharacteristic
28#2590—22no—
29#2588—22no—
30#2585—22no—
31#2582C2211yescharacteristic
32#2581—21no—
33#2571C2211yescharacteristic
34#2570C2212yes—
35#2568C2211yescharacteristic
36#2565—21no—
37#2552C2211yescharacteristic
38#2552C2211yescharacteristic
39#2550C2212yes—
40#2546C2211yescharacteristic
41#2511C2212yes—
42#2509—21no—
43#2453—21no—
44#2446C2212yes—
45#2402—22no—
46#2370—21no—
47#2355—21no—
48#2355C2211yescharacteristic
49#2355C2211yescharacteristic
50C4×C4⋊Q8C2211yescharacteristic
51C4×C4⋊Q8C2211yescharacteristic
52C4×C4⋊Q8C2212yes—
53C4×#125—21no—
54C4×#125C2212yes—
55#2295—21no—
56#2295C2212yes—
57C42×Q8C2211yescharacteristic
58C42×Q8C2211yescharacteristic
59#2101C2212yes—
60#2097C2212yes—
61#2097C2212yes—
62#2097C2212yes—
63#2096C2212yes—
64#2085C2212yes—
65#2085C2212yes—
66#2084C2212yes—
67#2083C2212yes—
68#2074C2212yes—
69#2074C2212yes—
70#2074C2212yes—
71#2074C2212yes—
72#2068C2212yes—
73#2068C2212yes—
74#2059C2212yes—
75#2043C2212yes—
76#2043C2212yes—
77#2026C2212yes—
78#2024C2212yes—
79#1992C2212yes—
80#1988C2211yescharacteristic
81#1984C2211yescharacteristic
82#1984C2211yescharacteristic
83#1981C2211yescharacteristic
84#1981C2211yescharacteristic
85#1960C2212yes—
86#1956C2211yescharacteristic
87#1956C2211yescharacteristic
88#1948C2211yescharacteristic
89#1948C2211yescharacteristic
90#1948C2211yescharacteristic
91#1947C2211yescharacteristic
92#1947C2211yescharacteristic
93#1944C2212yes—
94#1943C2212yes—
95#1935C2212yes—
96#1935C2212yes—
97#1928C2212yes—
98#1928C2212yes—
99#1928C2212yes—
100#1927C2212yes—
101#1927C2212yes—
102#1927C2212yes—
103#1921C2212yes—
104#1921C2212yes—
105#1919C2211yescharacteristic
106#1919C2211yescharacteristic
107#1919C2212yes—
108#1918C2212yes—
109#1917C2212yes—
110#1917C2212yes—
111#1865C2212yes—
112#1865C2212yes—
113#1865C2211yescharacteristic
114#1865C2211yescharacteristic
115#1863C2212yes—
116#1863C2212yes—
117#1863C2212yes—
118#1863C2212yes—
119#1847C2212yes—
120#1845C2211yescharacteristic
121#1839C2212yes—
122#1834C2211yescharacteristic
123#1831C2211yescharacteristic
124#1803C2211yescharacteristic
125#1803C2211yescharacteristic
126#1803C2211yescharacteristic
127C4×#106C2211yescharacteristic
128C4×#106C2211yescharacteristic
129#1770C2211yescharacteristic
130#1770C2211yescharacteristic
131C4×C4⋊C8C2211yescharacteristic
132C4×#103C2211yescharacteristic
133C4×#103C2211yescharacteristic
134#1745C2212yes—
135C2×C4⋊Q8C2311yescharacteristic
136C2×C4⋊Q8C2312yes—
137C2×C4⋊Q8C2311yescharacteristic
138C2×#125—21no—
139C2×#125—21no—
140C2×#125C2312yes—
141C2×C4×Q8—21no—
142C2×C4×Q8D411yescharacteristic
143C2×C4×Q8D411yescharacteristic
144C2×C4×Q8C2311yescharacteristic
145C2×C4×Q8C2311yescharacteristic
146C2×C4×Q8C2311yescharacteristic
147C2×C4×Q8C2312yes—
148C2×C4×Q8C2311yescharacteristic
149#491—28no—
150#475—24no—
151C2×#107C2312yes—
152C2×#107C2312yes—
153C2×#107C2312yes—
154C2×#107C2312yes—
155C2×#106C2311yescharacteristic
156C2×#106C2311yescharacteristic
157C2×#106C2311yescharacteristic
158C2×#106C2312yes—
159C2×#106C2312yes—
160C2×#106C2311yescharacteristic
161C2×C4⋊C8C2311yescharacteristic
162C2×C4⋊C8C2311yescharacteristic
163C2×C4⋊C8C2311yescharacteristic
164C2×C4⋊C8C2311yescharacteristic
165C2×#103C2311yescharacteristic
166C2×#103C2312yes—
167C2×#103C2312yes—
168C2×#103C2312yes—
169C2×#103C2311yescharacteristic
170C2×#103C2312yes—
171C2×#103C2311yescharacteristic
172C2×#103C2312yes—
173C2×#103C2312yes—
174C2×#103C2312yes—
175C2×#103C2312yes—
176C2×#103C2312yes—
177C2×#103C2312yes—
178C2×#103C2311yescharacteristic
179C2×#97C2312yes—
180C2×#97C2312yes—
181C2×C4×C8C2311yescharacteristic
182C2×C4×C8C2311yescharacteristic
183#400—22no—
184#400—22no—
185#400—22no—
186#400—22no—
187#400—22no—
188#400—22no—
189#398—22no—
190#398—22no—
191#398—22no—
192#398—22no—
193#398—22no—
194#398—22no—
195#398—22no—
196#395—22no—
197#391—21no—
198#391—21no—
199#391—21no—
200#391—21no—
201#391—21no—
202#391—22no—
203#391C2312yes—
204#391C2312yes—
205#391C2312yes—
206#391C2312yes—
207#391C2312yes—
208#389—21no—
209#389—21no—
210#389—21no—
211#389—21no—
212#389—21no—
213#389—21no—
214#389—21no—
215#389—22no—
216#389—22no—
217#389—21no—
218#389—21no—
219#389—21no—
220#389—21no—
221#389—22no—
222#389C2311yescharacteristic
223#389C2312yes—
224#389C2311yescharacteristic
225#389C2312yes—
226#389C2312yes—
227#389C2312yes—
228#389C2312yes—
229#389C2312yes—
230#387—22no—
231#387—22no—
232#387—22no—
233#387—22no—
234#387—22no—
235#387—22no—
236#387—22no—
237#387—22no—
238#387—22no—
239#387—22no—
240#387—22no—
241#387—22no—
242#387—22no—
243#387—22no—
244#387—22no—
245#387—22no—
246#384—21no—
247#384C2311yescharacteristic
248#384C2311yescharacteristic
249#384C2312yes—
250#384C2311yescharacteristic
251#384C2311yescharacteristic
252#382—21no—
253#382—21no—
254#382—21no—
255#382C2312yes—
256#382C2312yes—
257#382C2312yes—
258C4×C4⋊C4—21no—
259C4×C4⋊C4—21no—
260C4×C4⋊C4—22no—
261C4×C4⋊C4—21no—
262C4×C4⋊C4—22no—
263C4×C4⋊C4—22no—
264C4×C4⋊C4—22no—
265C4×C4⋊C4—21no—
266C4×C4⋊C4—21no—
267C4×C4⋊C4—22no—
268C4×C4⋊C4—21no—
269C4×C4⋊C4—21no—
270C4×C4⋊C4—21no—
271C4×C4⋊C4D411yescharacteristic
272C4×C4⋊C4D412yes—
273C4×C4⋊C4D412yes—
274C4×C4⋊C4D411yescharacteristic
275C4×C4⋊C4C2312yes—
276C4×C4⋊C4C2312yes—
277C4×C4⋊C4C2311yescharacteristic
278C4×C4⋊C4C2311yescharacteristic
279C4×C4⋊C4C2311yescharacteristic
280C4×C4⋊C4C2311yescharacteristic
281C4×C4⋊C4C2311yescharacteristic
282C4×C4⋊C4C2312yes—
283C4×C4⋊C4C2311yescharacteristic
284#376—22no—
285C43—21no—
286C43C2312yes—
287C43C2311yescharacteristic
288#340C2311yescharacteristic
289#340C2312yes—
290#340C2311yescharacteristic
291#340C2312yes—
292#340C2312yes—
293#340C2312yes—
294#340C2312yes—
295#340C2311yescharacteristic
296#340C2311yescharacteristic
297#340C2312yes—
298#340C2312yes—
299#340C2312yes—
300#340C2311yescharacteristic
301#340C2311yescharacteristic
302#340C2311yescharacteristic
303#340C2311yescharacteristic
304#336C2312yes—
305#336C2312yes—
306#336C2312yes—
307#336C2312yes—
308#336C2312yes—
309#336C2312yes—
310#336C2312yes—
311#336C2312yes—
312#336C2311yescharacteristic
313#336C2311yescharacteristic
314C22×Q8—21no—
315C22×Q8C2×D411yescharacteristic
316C22×Q8C2412yes—
317C22×Q8C2411yescharacteristic
318C22×C8C2411yescharacteristic
319C22×C8C2411yescharacteristic
320C22×C8C2412yes—
321C22×C8C2412yes—
322C22×C8C2411yescharacteristic
323C22×C8C2411yescharacteristic
324C4⋊Q8—22no—
325C4⋊Q8—24no—
326C4⋊Q8—22no—
327C4⋊Q8—22no—
328C4⋊Q8—41no—
329C4⋊Q8—42no—
330#125—24no—
331#125—42no—
332#125—44no—
333#125—44no—
334C4×Q8—24no—
335C4×Q8—22no—
336C4×Q8—22no—
337C4×Q8—22no—
338C4×Q8—22no—
339C4×Q8—22no—
340C4×Q8—42no—
341C4×Q8—41no—
342C4×Q8—41no—
343C4×Q8—42no—
344C4×Q8—42no—
345C4×Q8—41no—
346C4×Q8—44no—
347C2×C4⋊C4—21no—
348C2×C4⋊C4—21no—
349C2×C4⋊C4—21no—
350C2×C4⋊C4—21no—
351C2×C4⋊C4—21no—
352C2×C4⋊C4—21no—
353C2×C4⋊C4—22no—
354C2×C4⋊C4—21no—
355C2×C4⋊C4—21no—
356C2×C4⋊C4—22no—
357C2×C4⋊C4—21no—
358C2×C4⋊C4—22no—
359C2×C4⋊C4—22no—
360C2×C4⋊C4—22no—
361C2×C4⋊C4—22no—
362C2×C4⋊C4—22no—
363C2×C4⋊C4—21no—
364C2×C4⋊C4—22no—
365C2×C4⋊C4—22no—
366C2×C4⋊C4—21no—
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—22no—
376C2×C4⋊C4C2×D411yescharacteristic
377C2×C4⋊C4C2×D412yes—
378C2×C4⋊C4C2×D411yescharacteristic
379C2×C4⋊C4C2×D411yescharacteristic
380C2×C4⋊C4C2×D412yes—
381C2×C4⋊C4C2×D412yes—
382C2×C4⋊C4C2411yescharacteristic
383C2×C4⋊C4C2411yescharacteristic
384C2×C4⋊C4C2411yescharacteristic
385C2×C4⋊C4C2412yes—
386C2×C4⋊C4C2412yes—
387C2×C4⋊C4C2411yescharacteristic
388C2×C4⋊C4C2411yescharacteristic
389C2×C4⋊C4C2411yescharacteristic
390C2×C4⋊C4C2412yes—
391C2×C4⋊C4C2411yescharacteristic
392C2×C42—21no—
393C2×C42—21no—
394C2×C42—21no—
395C2×C42—21no—
396C2×C42—21no—
397C2×C42—22no—
398C2×C42—21no—
399C2×C42—21no—
400C2×C42—21no—
401C2×C42—22no—
402C2×C42—22no—
403C2×C42C2×D411yescharacteristic
404C2×C42C2×D412yes—
405C2×C42C2×D412yes—
406C2×C42C2×D412yes—
407C2×C42C2×D411yescharacteristic
408C2×C42C2×D412yes—
409C2×C42C2×D411yescharacteristic
410C2×C42C2×D411yescharacteristic
411C2×C42C2411yescharacteristic
412C2×C42C2411yescharacteristic
413C2×C42C2412yes—
414C2×C42C2412yes—
415C2×C42C2411yescharacteristic
416#107—42no—
417#107—42no—
418#107—42no—
419#107—42no—
420#106—22no—
421#106—22no—
422#106—41no—
423#106—42no—
424#106—41no—
425#106—42no—
426C4⋊C8—22no—
427C4⋊C8—22no—
428C4⋊C8—22no—
429C4⋊C8—41no—
430#103—22no—
431#103—24no—
432#103—24no—
433#103—24no—
434#103—42no—
435#103—41no—
436#103—42no—
437#103—42no—
438#103—42no—
439#103—42no—
440#103—41no—
441#103—42no—
442#103—42no—
443#103#4114yes—
444#97—24no—
445#97—42no—
446C4×C8—22no—
447C4×C8—22no—
448#95—22no—
449#95—22no—
450#95—22no—
451#95—22no—
452#95—21no—
453#95—21no—
454#95—21no—
455#95—21no—
456#95—22no—
457#95—22no—
458#95—22no—
459#95—22no—
460#95—22no—
461#95—22no—
462#95—22no—
463#95—22no—
464#95—21no—
465#95—22no—
466#95—22no—
467#95—22no—
468#95—22no—
469#95—22no—
470#95—22no—
471#95—22no—
472#95—21no—
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—22no—
484C2×Q8—22no—
485C2×Q8—21no—
486C2×Q8—22no—
487C2×Q8—21no—
488C2×Q8—21no—
489C2×Q8—22no—
490C2×Q8—22no—
491C2×Q8—21no—
492C2×Q8—21no—
493C2×Q8—22no—
494C2×Q8—22no—
495C2×Q8—22no—
496C2×Q8—42no—
497C2×Q8—42no—
498C2×Q8—81no—
499C2×Q8C2×#4112yes—
500C22×C4—21no—
501C22×C4—21no—
502C22×C4—21no—
503C22×C4—21no—
504C22×C4—21no—
505C22×C4—22no—
506C22×C4—21no—
507C22×C4—22no—
508C22×C4—21no—
509C22×C4—21no—
510C22×C4—22no—
511C22×C4—22no—
512C22×C4C22×D412yes—
513C22×C4C22×D412yes—
514C22×C4C22×D412yes—
515C22×C4C22×D411yescharacteristic
516C22×C4C22×D412yes—
517C22×C4C22×D411yescharacteristic
518C22×C4C22×D411yescharacteristic
519C22×C4C22×D411yescharacteristic
520C22×C4C22×D411yescharacteristic
521C22×C4C22×D411yescharacteristic
522C22×C4C2511yescharacteristic
523C2×C8—21no—
524C2×C8—21no—
525C2×C8—22no—
526C2×C8—21no—
527C2×C8—21no—
528C2×C8—21no—
529C2×C8—21no—
530C2×C8—22no—
531C2×C8—22no—
532C2×C8—21no—
533C2×C8—22no—
534C2×C8—21no—
535C2×C8—21no—
536C2×C8—21no—
537C2×C8—22no—
538C2×C8—22no—
539C2×C8—21no—
540C2×C8C2×#4112yes—
541C4⋊C4—22no—
542C4⋊C4—24no—
543C4⋊C4—22no—
544C4⋊C4—24no—
545C4⋊C4—22no—
546C4⋊C4—21no—
547C4⋊C4—22no—
548C4⋊C4—21no—
549C4⋊C4—21no—
550C4⋊C4—21no—
551C4⋊C4—21no—
552C4⋊C4—21no—
553C4⋊C4—21no—
554C4⋊C4—22no—
555C4⋊C4—21no—
556C4⋊C4—21no—
557C4⋊C4—22no—
558C4⋊C4—22no—
559C4⋊C4—21no—
560C4⋊C4—21no—
561C4⋊C4—22no—
562C4⋊C4—22no—
563C4⋊C4—21no—
564C4⋊C4—21no—
565C4⋊C4—42no—
566C4⋊C4—42no—
567C4⋊C4—44no—
568C4⋊C4—44no—
569C4⋊C4—44no—
570C4⋊C4—44no—
571C4⋊C4—42no—
572C4⋊C4—42no—
573C4⋊C4—44no—
574C4⋊C4—44no—
575C4⋊C4—44no—
576C4⋊C4—44no—
577C4⋊C4—44no—
578C4⋊C4—42no—
579C4⋊C4—42no—
580C4⋊C4—42no—
581C4⋊C4—42no—
582C4⋊C4—42no—
583C4⋊C4—42no—
584C4⋊C4—42no—
585C4⋊C4—42no—
586C4⋊C4—42no—
587C4⋊C4—81no—
588C4⋊C4—81no—
589C4⋊C4—81no—
590C4⋊C4—82no—
591C4⋊C4—82no—
592C4⋊C4—82no—
593C4⋊C4—81no—
594C4⋊C4—82no—
595C4⋊C4C2×#4112yes—
596C42—24no—
597C42—22no—
598C42—22no—
599C42—21no—
600C42—22no—
601C42—22no—
602C42—22no—
603C42—21no—
604C42—21no—
605C42—21no—
606C42—41no—
607C42—41no—
608C42—42no—
609C42—42no—
610C42—42no—
611C42—42no—
612C42—42no—
613C42—42no—
614C42—44no—
615C42—42no—
616C42—42no—
617C42—42no—
618C42—42no—
619C42—81no—
620C42—82no—
621C42—82no—
622C42#13714yes—
623C422−1+412yes—
624C422−1+412yes—
625C23C23×D411yescharacteristic
626Q8—22no—
627Q8—22no—
628Q8—44no—
629Q8—41no—
630Q8—41no—
631Q8—42no—
632Q8—42no—
633Q8—42no—
634Q8—42no—
635Q8—84no—
636C2×C4—21no—
637C2×C4—21no—
638C2×C4—21no—
639C2×C4—21no—
640C2×C4—21no—
641C2×C4—21no—
642C2×C4—21no—
643C2×C4—22no—
644C2×C4—22no—
645C2×C4—22no—
646C2×C4—21no—
647C2×C4—22no—
648C2×C4—22no—
649C2×C4—22no—
650C2×C4—22no—
651C2×C4—21no—
652C2×C4—22no—
653C2×C4—22no—
654C2×C4—22no—
655C2×C4—21no—
656C2×C4—22no—
657C2×C4—22no—
658C2×C4—21no—
659C2×C4—21no—
660C2×C4—22no—
661C2×C4—22no—
662C2×C4—41no—
663C2×C4—41no—
664C2×C4—41no—
665C2×C4—41no—
666C2×C4—42no—
667C2×C4—41no—
668C2×C4—41no—
669C2×C4—42no—
670C2×C4—41no—
671C2×C4—41no—
672C2×C4—41no—
673C2×C4—41no—
674C2×C4—42no—
675C2×C4—41no—
676C2×C4—41no—
677C2×C4—42no—
678C2×C4—41no—
679C2×C4—42no—
680C2×C4—41no—
681C2×C4—42no—
682C2×C4—41no—
683C2×C4—42no—
684C2×C4—42no—
685C2×C4—41no—
686C2×C4—41no—
687C2×C4—42no—
688C2×C4—41no—
689C2×C4—41no—
690C2×C4—41no—
691C2×C4—41no—
692C2×C4—41no—
693C2×C4—42no—
694C2×C4—42no—
695C2×C4—42no—
696C2×C4#53612yes—
697C2×C4#53612yes—
698C2×C4C2×#13712yes—
699C2×C4C2×#13712yes—
700C2×C4C2×#13712yes—
701C2×C4#57512yes—
702C2×C4C22×#4111yescharacteristic
703C2×C4C2×2−1+411yescharacteristic
704C2×C4C2×2−1+411yescharacteristic
705C2×C4#58511yescharacteristic
706C2×C4#58511yescharacteristic
707C2×C4#58511yescharacteristic
708C8—22no—
709C8—42no—
710C8—41no—
711C8—42no—
712C8—41no—
713C8—41no—
714C22C2×#53611yescharacteristic
715C22D4×#4111yescharacteristic
716C22#348711yescharacteristic
717C22C22×#13711yescharacteristic
718C22C2×#57511yescharacteristic
719C22#358411yescharacteristic
720C22#358611yescharacteristic
721C4—22no—
722C4—22no—
723C4—22no—
724C4—41no—
725C4—42no—
726C4—42no—
727C4—42no—
728C4—41no—
729C4—42no—
730C4—42no—
731C4—42no—
732C4—81no—
733C4—82no—
734C4—82no—
735C4—81no—
736C4—81no—
737C4—81no—
738C4—82no—
739C4—82no—
740C4—81no—
741C4—81no—
742C4#323714yes—
743C4#350711yescharacteristic
744C4#351511yescharacteristic
745C4#355711yescharacteristic
746C4#356511yescharacteristic
747C2#3985911yescharacteristic
748C2C2×#323711yescharacteristic
749C2#6089111yescharacteristic
750C2#6143811yescharacteristic
751C2#6146711yescharacteristic
752C2#6189311yescharacteristic
753C2#6190811yescharacteristic
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
#365352
#349512
#330692
#324022
#322502
#316002
#308282
#300802
#295832
#293992
#292552
#291932
#288352
#288002
#277772
#277572
#263192
#250632
#249972
#228322
#222252
C4×#4752