Finite Group Databasebeta

← #6714835

Subgroups — #6714835

Order 512.

1232 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#39754C211yesmaximal, characteristic
2#36598C211yesmaximal, characteristic
3#33067C211yesmaximal, characteristic
4#32578C212yesmaximal
5#32553C212yesmaximal
6#32325C212yesmaximal
7#31469C212yesmaximal
8#30228C211yesmaximal, characteristic
9#29866C211yesmaximal, characteristic
10#27816C211yesmaximal, characteristic
11#27730C211yesmaximal, characteristic
12#27425C212yesmaximal
13#27359C212yesmaximal
14#26513C211yesmaximal, characteristic
15#26447C211yesmaximal, characteristic
16#26286C212yesmaximal
17#25000C212yesmaximal
18#24818C211yesmaximal, characteristic
19#24683C211yesmaximal, characteristic
20#23062C211yesmaximal, characteristic
21#22964C211yesmaximal, characteristic
22#22307C211yesmaximal, characteristic
23#21119C211yesmaximal, characteristic
24C2×#475C2211yescharacteristic
25C2×#475C2211yescharacteristic
26C2×C4×Q16C2211yescharacteristic
27#2868C2211yescharacteristic
28#2862—21no—
29#2859C2211yescharacteristic
30#2802—22no—
31#2774—22no—
32#2660—21no—
33#2584C2212yes—
34#2582C2212yes—
35#2581C2212yes—
36#2571—21no—
37#2571C2212yes—
38#2571C2211yescharacteristic
39#2570—21no—
40#2570—22no—
41#2568C2211yescharacteristic
42#2568C2211yescharacteristic
43#2552C2211yescharacteristic
44#2511—22no—
45#2508C2212yes—
46#2453C2212yes—
47#2446—21no—
48#2446—21no—
49#2365C2211yescharacteristic
50#2365C2211yescharacteristic
51#2355C2211yescharacteristic
52#2355C2212yes—
53#2355C2212yes—
54Q8×C4⋊C4C2211yescharacteristic
55#2330—21no—
56C4×C4⋊Q8—21no—
57C4×C4⋊Q8C2211yescharacteristic
58#2295C2211yescharacteristic
59C42×Q8C2211yescharacteristic
60#2101C2212yes—
61#2101C2212yes—
62#2101C2212yes—
63#2099C2212yes—
64#2099C2212yes—
65#2099C2212yes—
66#2099C2212yes—
67#2088C2212yes—
68#2087C2212yes—
69#2085C2212yes—
70#2084C2212yes—
71#2084C2212yes—
72#2074C2212yes—
73#2074C2212yes—
74#2072C2212yes—
75#2072C2212yes—
76#2066C2212yes—
77#2059C2212yes—
78#2056C2212yes—
79#2026C2212yes—
80#1991C2212yes—
81#1990C2211yescharacteristic
82#1990C2211yescharacteristic
83#1990C2211yescharacteristic
84#1990C2211yescharacteristic
85#1989C2212yes—
86#1988C2211yescharacteristic
87#1988C2211yescharacteristic
88#1984C2212yes—
89#1984C2212yes—
90#1949C2212yes—
91#1949C2212yes—
92#1948C2211yescharacteristic
93#1948C2211yescharacteristic
94#1941C2212yes—
95#1939C2211yescharacteristic
96#1939C2211yescharacteristic
97#1933C2212yes—
98#1933C2212yes—
99#1933C2212yes—
100#1928C2211yescharacteristic
101#1928C2211yescharacteristic
102#1927C2212yes—
103#1927C2212yes—
104#1921C2211yescharacteristic
105#1921C2212yes—
106#1921C2211yescharacteristic
107#1920C2212yes—
108#1919C2212yes—
109#1919C2212yes—
110#1919C2212yes—
111#1919C2212yes—
112#1916C2212yes—
113#1916C2212yes—
114#1878C2211yescharacteristic
115#1878C2212yes—
116#1878C2211yescharacteristic
117#1865C2212yes—
118#1865C2212yes—
119#1865C2212yes—
120#1863C2211yescharacteristic
121#1863C2211yescharacteristic
122#1835C2212yes—
123#1831C2211yescharacteristic
124#1805C2212yes—
125#1803C2211yescharacteristic
126#1803C2211yescharacteristic
127C4×#107C2211yescharacteristic
128C4×#106C2212yes—
129C8×C4⋊C4C2211yescharacteristic
130C8×C4⋊C4C2211yescharacteristic
131C4×#103C2211yescharacteristic
132C4×#103C2212yes—
133C4×#103C2211yescharacteristic
134#1744C2211yescharacteristic
135C22×Q16C2311yescharacteristic
136C2×C4⋊Q8—21no—
137C2×C4⋊Q8—21no—
138C2×C4⋊Q8D411yescharacteristic
139C2×C4⋊Q8C2311yescharacteristic
140C2×C4⋊Q8C2311yescharacteristic
141C2×#125—21no—
142C2×#125C2312yes—
143C2×C4×Q8—21no—
144C2×C4×Q8—21no—
145C2×C4×Q8C2311yescharacteristic
146C2×C4×Q8C2311yescharacteristic
147C2×C4×Q8C2311yescharacteristic
148C2×C4×Q8C2312yes—
149C2×C4×Q8C2311yescharacteristic
150C2×C4×Q8C2312yes—
151#475—24no—
152#475—24no—
153C4×Q16—24no—
154C2×#107C2311yescharacteristic
155C2×#107C2311yescharacteristic
156C2×#107C2311yescharacteristic
157C2×#106C2312yes—
158C2×#106C2312yes—
159C2×#106C2312yes—
160C2×#106C2312yes—
161C2×C4⋊C8C2311yescharacteristic
162C2×C4⋊C8C2311yescharacteristic
163C2×C4⋊C8C2312yes—
164C2×C4⋊C8C2312yes—
165C2×#103C2312yes—
166C2×#103C2312yes—
167C2×#103C2312yes—
168C2×#103C2311yescharacteristic
169C2×#103C2311yescharacteristic
170C2×#103C2311yescharacteristic
171C2×#103C2311yescharacteristic
172C2×#103C2312yes—
173C2×#103C2311yescharacteristic
174C2×#103C2311yescharacteristic
175C2×C4×C8C2311yescharacteristic
176C2×C4×C8C2311yescharacteristic
177C2×C4×C8C2312yes—
178C2×C4×C8C2312yes—
179#400—22no—
180#400—22no—
181#400—22no—
182#400—22no—
183#400—22no—
184#400—22no—
185#400—22no—
186#400—22no—
187#398—22no—
188#398—22no—
189#398—22no—
190#398—22no—
191#398—22no—
192#398—22no—
193#395—22no—
194#395—22no—
195#391—22no—
196#391—22no—
197#391—21no—
198#391—21no—
199#391—21no—
200#391—21no—
201#391—21no—
202#391—21no—
203#391D412yes—
204#391C2312yes—
205#391C2312yes—
206#391C2312yes—
207#391C2311yescharacteristic
208#391C2311yescharacteristic
209#389—22no—
210#389—22no—
211#389—21no—
212#389—22no—
213#389—21no—
214#389—22no—
215#389—22no—
216#389—21no—
217#389—21no—
218#389—22no—
219#389D411yescharacteristic
220#389D412yes—
221#389D411yescharacteristic
222#389C2312yes—
223#389C2312yes—
224#389C2312yes—
225#389C2312yes—
226#389C2312yes—
227#389C2312yes—
228#389C2311yescharacteristic
229#389C2311yescharacteristic
230#389C2312yes—
231#387—22no—
232#387—22no—
233#387—22no—
234#387—22no—
235#387—22no—
236#387—21no—
237#387—21no—
238#387—22no—
239#387—22no—
240#387—22no—
241#387—22no—
242#387—22no—
243#387—22no—
244#384—21no—
245#384—21no—
246#384—21no—
247#384—21no—
248#384D411yescharacteristic
249#384C2311yescharacteristic
250#384C2311yescharacteristic
251#384C2312yes—
252#384C2312yes—
253#382—22no—
254#382—22no—
255#382—21no—
256#382—21no—
257#382—21no—
258#382—22no—
259#382C2312yes—
260#382C2312yes—
261#382C2312yes—
262C4×C4⋊C4—22no—
263C4×C4⋊C4—22no—
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⋊C4C2311yescharacteristic
271C4×C4⋊C4C2311yescharacteristic
272C4×C4⋊C4C2312yes—
273C4×C4⋊C4C2312yes—
274C4×C4⋊C4C2312yes—
275C4×C4⋊C4C2311yescharacteristic
276C4×C4⋊C4C2311yescharacteristic
277C4×C4⋊C4C2311yescharacteristic
278C4×C4⋊C4C2311yescharacteristic
279C4×C4⋊C4C2312yes—
280#376—21no—
281C43—21no—
282C43C2311yescharacteristic
283#340C2312yes—
284#340C2312yes—
285#340C2312yes—
286#340C2312yes—
287#340C2312yes—
288#340C2312yes—
289#340C2312yes—
290#340C2312yes—
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#340C2312yes—
304#340C2311yescharacteristic
305#336C2312yes—
306#336C2312yes—
307#336C2312yes—
308#336C2312yes—
309#336C2312yes—
310#336C2311yescharacteristic
311#336C2311yescharacteristic
312#336C2311yescharacteristic
313#336C2311yescharacteristic
314#336C2312yes—
315C22×Q8—21no—
316C22×Q8—21no—
317C22×Q8C2×D412yes—
318C22×Q8C2411yescharacteristic
319C22×Q8C2411yescharacteristic
320C2×Q16—22no—
321C2×Q16—22no—
322C2×Q16—41no—
323C22×C8C2411yescharacteristic
324C22×C8C2411yescharacteristic
325C22×C8C2411yescharacteristic
326C22×C8C2412yes—
327C22×C8C2412yes—
328C22×C8C2411yescharacteristic
329C4⋊Q8—24no—
330C4⋊Q8—22no—
331C4⋊Q8—22no—
332C4⋊Q8—22no—
333C4⋊Q8—22no—
334C4⋊Q8—44no—
335C4⋊Q8—82no—
336#125—42no—
337#125—42no—
338#125—44no—
339C4×Q8—22no—
340C4×Q8—22no—
341C4×Q8—22no—
342C4×Q8—22no—
343C4×Q8—22no—
344C4×Q8—42no—
345C4×Q8—42no—
346C4×Q8—41no—
347C4×Q8—41no—
348C4×Q8—42no—
349C4×Q8—42no—
350C4×Q8—82no—
351C4×Q8—82no—
352C4×Q8#4114yes—
353C2×C4⋊C4—21no—
354C2×C4⋊C4—21no—
355C2×C4⋊C4—21no—
356C2×C4⋊C4—21no—
357C2×C4⋊C4—21no—
358C2×C4⋊C4—21no—
359C2×C4⋊C4—22no—
360C2×C4⋊C4—22no—
361C2×C4⋊C4—22no—
362C2×C4⋊C4—22no—
363C2×C4⋊C4—21no—
364C2×C4⋊C4—21no—
365C2×C4⋊C4—22no—
366C2×C4⋊C4—22no—
367C2×C4⋊C4—22no—
368C2×C4⋊C4—22no—
369C2×C4⋊C4—22no—
370C2×C4⋊C4—22no—
371C2×C4⋊C4—22no—
372C2×C4⋊C4—21no—
373C2×C4⋊C4—21no—
374C2×C4⋊C4—22no—
375C2×C4⋊C4—22no—
376C2×C4⋊C4—21no—
377C2×C4⋊C4—21no—
378C2×C4⋊C4—22no—
379C2×C4⋊C4C2×D411yescharacteristic
380C2×C4⋊C4C2×D411yescharacteristic
381C2×C4⋊C4C2×D412yes—
382C2×C4⋊C4C2×D411yescharacteristic
383C2×C4⋊C4C2×D411yescharacteristic
384C2×C4⋊C4C2×D412yes—
385C2×C4⋊C4C2×D412yes—
386C2×C4⋊C4C2×D412yes—
387C2×C4⋊C4C2×D412yes—
388C2×C4⋊C4C2412yes—
389C2×C4⋊C4C2411yescharacteristic
390C2×C4⋊C4C2411yescharacteristic
391C2×C4⋊C4C2411yescharacteristic
392C2×C4⋊C4C2412yes—
393C2×C4⋊C4C2412yes—
394C2×C4⋊C4C2411yescharacteristic
395C2×C4⋊C4C2412yes—
396C2×C4⋊C4C2411yescharacteristic
397C2×C4⋊C4C2411yescharacteristic
398C2×C42—21no—
399C2×C42—21no—
400C2×C42—21no—
401C2×C42—21no—
402C2×C42—21no—
403C2×C42—21no—
404C2×C42—21no—
405C2×C42—21no—
406C2×C42—22no—
407C2×C42—22no—
408C2×C42—22no—
409C2×C42—22no—
410C2×C42C2×D411yescharacteristic
411C2×C42C2×D411yescharacteristic
412C2×C42C2×D411yescharacteristic
413C2×C42C2×D411yescharacteristic
414C2×C42C2411yescharacteristic
415C2×C42C2412yes—
416C2×C42C2412yes—
417C2×C42C2411yescharacteristic
418C2×C42C2411yescharacteristic
419#107—22no—
420#107—22no—
421#107—41no—
422#106—24no—
423#106—24no—
424#106—24no—
425#106—42no—
426C4⋊C8—22no—
427C4⋊C8—22no—
428C4⋊C8—42no—
429C4⋊C8—42no—
430#103—22no—
431#103—24no—
432#103—24no—
433#103—24no—
434#103—22no—
435#103—22no—
436#103—22no—
437#103—41no—
438#103—42no—
439#103—41no—
440C4×C8—22no—
441C4×C8—22no—
442C4×C8—42no—
443C4×C8—42no—
444#95—21no—
445#95—21no—
446#95—21no—
447#95—21no—
448#95—21no—
449#95—21no—
450#95—21no—
451#95—22no—
452#95—22no—
453#95—21no—
454#95—21no—
455#95—22no—
456#95—22no—
457#95—22no—
458#95—22no—
459#95—22no—
460#95—22no—
461#95—22no—
462#95—21no—
463#95—22no—
464#95—22no—
465#95—22no—
466#95—22no—
467#95—22no—
468#95—22no—
469#95—22no—
470#95—22no—
471#95—22no—
472#95—22no—
473#95—22no—
474#95—22no—
475#95—22no—
476#95C2×D412yes—
477#95C2×D412yes—
478#95C2×D412yes—
479#95C2×D412yes—
480C2×Q8—24no—
481C2×Q8—21no—
482C2×Q8—21no—
483C2×Q8—21no—
484C2×Q8—21no—
485C2×Q8—21no—
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—42no—
498C2×Q8—42no—
499C2×Q8C2×#4112yes—
500C2×Q82−1+412yes—
501C22×C4—21no—
502C22×C4—21no—
503C22×C4—21no—
504C22×C4—21no—
505C22×C4—21no—
506C22×C4—21no—
507C22×C4—22no—
508C22×C4—21no—
509C22×C4—21no—
510C22×C4—22no—
511C22×C4—22no—
512C22×C4—22no—
513C22×C4C22×D412yes—
514C22×C4C22×D412yes—
515C22×C4C22×D411yescharacteristic
516C22×C4C22×D412yes—
517C22×C4C22×D412yes—
518C22×C4C22×D411yescharacteristic
519C22×C4C22×D411yescharacteristic
520C22×C4C22×D411yescharacteristic
521C22×C4C22×D411yescharacteristic
522C22×C4C22×D411yescharacteristic
523C22×C4C2511yescharacteristic
524Q16—44no—
525C2×C8—21no—
526C2×C8—21no—
527C2×C8—22no—
528C2×C8—22no—
529C2×C8—21no—
530C2×C8—21no—
531C2×C8—22no—
532C2×C8—21no—
533C2×C8—21no—
534C2×C8—22no—
535C2×C8—21no—
536C2×C8—21no—
537C2×C8—21no—
538C2×C8—21no—
539C2×C8—22no—
540C2×C8—22no—
541C2×C8—21no—
542C2×C8—21no—
543C4⋊C4—24no—
544C4⋊C4—24no—
545C4⋊C4—24no—
546C4⋊C4—22no—
547C4⋊C4—22no—
548C4⋊C4—22no—
549C4⋊C4—22no—
550C4⋊C4—21no—
551C4⋊C4—21no—
552C4⋊C4—22no—
553C4⋊C4—22no—
554C4⋊C4—21no—
555C4⋊C4—21no—
556C4⋊C4—22no—
557C4⋊C4—22no—
558C4⋊C4—21no—
559C4⋊C4—21no—
560C4⋊C4—22no—
561C4⋊C4—22no—
562C4⋊C4—21no—
563C4⋊C4—22no—
564C4⋊C4—21no—
565C4⋊C4—21no—
566C4⋊C4—22no—
567C4⋊C4—21no—
568C4⋊C4—42no—
569C4⋊C4—42no—
570C4⋊C4—42no—
571C4⋊C4—42no—
572C4⋊C4—42no—
573C4⋊C4—42no—
574C4⋊C4—42no—
575C4⋊C4—42no—
576C4⋊C4—44no—
577C4⋊C4—44no—
578C4⋊C4—42no—
579C4⋊C4—42no—
580C4⋊C4—44no—
581C4⋊C4—44no—
582C4⋊C4—81no—
583C4⋊C4—81no—
584C4⋊C4—81no—
585C4⋊C4—81no—
586C4⋊C4—82no—
587C4⋊C4—82no—
588C4⋊C4—82no—
589C4⋊C4—82no—
590C4⋊C4—82no—
591C4⋊C4—82no—
592C4⋊C4—82no—
593C4⋊C4—82no—
594C4⋊C4—82no—
595C4⋊C4—82no—
596C4⋊C4C2×#4112yes—
597C4⋊C42−1+412yes—
598C42—24no—
599C42—24no—
600C42—22no—
601C42—21no—
602C42—21no—
603C42—22no—
604C42—22no—
605C42—22no—
606C42—22no—
607C42—21no—
608C42—21no—
609C42—41no—
610C42—41no—
611C42—42no—
612C42—42no—
613C42—42no—
614C42—42no—
615C42—42no—
616C42—42no—
617C42—82no—
618C42—82no—
619C42—82no—
620C42—82no—
621C42#13514yes—
622C42C2×#4112yes—
623C422−1+412yes—
624C23C23×D411yescharacteristic
625Q8—22no—
626Q8—22no—
627Q8—22no—
628Q8—22no—
629Q8—41no—
630Q8—41no—
631Q8—41no—
632Q8—41no—
633Q8—84no—
634Q8—84no—
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—21no—
644C2×C4—22no—
645C2×C4—22no—
646C2×C4—22no—
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—22no—
659C2×C4—22no—
660C2×C4—21no—
661C2×C4—21no—
662C2×C4—22no—
663C2×C4—22no—
664C2×C4—41no—
665C2×C4—41no—
666C2×C4—41no—
667C2×C4—41no—
668C2×C4—41no—
669C2×C4—41no—
670C2×C4—41no—
671C2×C4—41no—
672C2×C4—41no—
673C2×C4—41no—
674C2×C4—41no—
675C2×C4—42no—
676C2×C4—42no—
677C2×C4—41no—
678C2×C4—41no—
679C2×C4—41no—
680C2×C4—42no—
681C2×C4—42no—
682C2×C4—41no—
683C2×C4—42no—
684C2×C4—41no—
685C2×C4—42no—
686C2×C4—41no—
687C2×C4—42no—
688C2×C4—42no—
689C2×C4—42no—
690C2×C4—42no—
691C2×C4—41no—
692C2×C4—41no—
693C2×C4—41no—
694C2×C4—42no—
695C2×C4—42no—
696C2×C4#54812yes—
697C2×C4#54812yes—
698C2×C4C2×#13512yes—
699C2×C4C2×#13512yes—
700C2×C4C2×#13512yes—
701C2×C4#57712yes—
702C2×C4C22×#4111yescharacteristic
703C2×C4C2×2−1+411yescharacteristic
704C2×C4C2×2−1+411yescharacteristic
705C2×C4C2×2−1+411yescharacteristic
706C2×C4#58511yescharacteristic
707C2×C4#58511yescharacteristic
708C8—41no—
709C8—42no—
710C8—42no—
711C8—41no—
712C8—41no—
713C8—41no—
714C22C2×#54811yescharacteristic
715C22#348511yescharacteristic
716C22#348611yescharacteristic
717C22C22×#13511yescharacteristic
718C22C2×#57711yescharacteristic
719C22#358611yescharacteristic
720C22#358611yescharacteristic
721C4—22no—
722C4—22no—
723C4—22no—
724C4—42no—
725C4—42no—
726C4—41no—
727C4—42no—
728C4—42no—
729C4—42no—
730C4—42no—
731C4—42no—
732C4—41no—
733C4—42no—
734C4—82no—
735C4—81no—
736C4—81no—
737C4—82no—
738C4—82no—
739C4—82no—
740C4—81no—
741C4—81no—
742C4#336514yes—
743C4#346111yescharacteristic
744C4#349511yescharacteristic
745C4#355511yescharacteristic
746C4#356311yescharacteristic
747C2#3775711yescharacteristic
748C2C2×#336511yescharacteristic
749C2#6133811yescharacteristic
750C2#6170111yescharacteristic
751C2#6181211yescharacteristic
752C2#6182211yescharacteristic
753C2#6185811yescharacteristic
subgroupG/H
center, socle, prime radicalC23C23×D4
commutator, Frattini, cosocleC22×C4C25
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×C4⋊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
#397542
#365982
#330672
#325782
#325532
#323252
#314692
#302282
#298662
#278162
#277302
#274252
#273592
#265132
#264472
#262862
#250002
#248182
#246832
#230622
#229642
#223072
#211192