Finite Group Databasebeta

← #6552005

Subgroups — #6552005

Order 512.

1351 conjugacy classes of subgroups.

The subgroup lattice has 767 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36286C211yesmaximal, characteristic
2#34328C211yesmaximal, characteristic
3#30155C212yesmaximal
4#29830C211yesmaximal, characteristic
5#29541C212yesmaximal
6#29447C212yesmaximal
7#29287C211yesmaximal, characteristic
8#28653C211yesmaximal, characteristic
9#27512C212yesmaximal
10#27448C212yesmaximal
11#27156C212yesmaximal
12#27028C212yesmaximal
13#26317C211yesmaximal, characteristic
14#26277C211yesmaximal, characteristic
15#25931C211yesmaximal, characteristic
16#25836C211yesmaximal, characteristic
17#23785C212yesmaximal
18#23777C211yesmaximal, characteristic
19#23222C211yesmaximal, characteristic
20#23163C211yesmaximal, characteristic
21#23102C211yesmaximal, characteristic
22#23069C211yesmaximal, characteristic
23#22825C211yesmaximal, characteristic
24C2×#557C2211yescharacteristic
25C2×#498C2211yescharacteristic
26C2×#487C2211yescharacteristic
27C2×#477C2211yescharacteristic
28C2×#462C2211yescharacteristic
29#2863C2212yes—
30#2862C2211yescharacteristic
31#2605—22no—
32#2588—22no—
33#2581—21no—
34#2570—21no—
35#2568C2211yescharacteristic
36#2567C2211yescharacteristic
37#2511C2212yes—
38#2451C2212yes—
39#2371—22no—
40#2370C2212yes—
41#2369C2211yescharacteristic
42#2365—21no—
43#2365C2211yescharacteristic
44#2365C2211yescharacteristic
45#2356—21no—
46#2356—21no—
47#2356C2212yes—
48#2355—21no—
49#2355C2211yescharacteristic
50#2355C2211yescharacteristic
51#2336—22no—
52#2327—21no—
53C4×C4⋊Q8C2211yescharacteristic
54C4×C4⋊Q8C2211yescharacteristic
55C4×#125C2212yes—
56#2295C2212yes—
57#2295C2211yescharacteristic
58#2295C2212yes—
59#2276—21no—
60C42×Q8C2211yescharacteristic
61#2101C2212yes—
62#2100C2211yescharacteristic
63#2097C2212yes—
64#2095C2211yescharacteristic
65#2088C2212yes—
66#2088C2212yes—
67#2087C2211yescharacteristic
68#2087C2211yescharacteristic
69#2085C2212yes—
70#2074C2212yes—
71#2068C2211yescharacteristic
72#2068C2212yes—
73#2068C2211yescharacteristic
74#2068C2212yes—
75#2066C2212yes—
76#2066C2212yes—
77#2059C2212yes—
78#2059C2212yes—
79#2057C2211yescharacteristic
80#2056C2212yes—
81#2056C2212yes—
82#2026C2212yes—
83#2026C2212yes—
84#2024C2211yescharacteristic
85#1990C2211yescharacteristic
86#1988C2211yescharacteristic
87#1988C2211yescharacteristic
88#1987C2212yes—
89#1987C2211yescharacteristic
90#1984C2211yescharacteristic
91#1984C2211yescharacteristic
92#1981C2211yescharacteristic
93#1971C2212yes—
94#1956C2211yescharacteristic
95#1948C2212yes—
96#1948C2211yescharacteristic
97#1948C2211yescharacteristic
98#1944C2212yes—
99#1935C2212yes—
100#1935C2212yes—
101#1928C2212yes—
102#1928C2212yes—
103#1927C2212yes—
104#1927C2212yes—
105#1927C2212yes—
106#1927C2212yes—
107#1927C2212yes—
108#1920C2212yes—
109#1919C2212yes—
110#1919C2212yes—
111#1918C2212yes—
112#1918C2212yes—
113#1917C2212yes—
114#1917C2212yes—
115#1878C2212yes—
116#1878C2212yes—
117#1878C2212yes—
118#1865C2212yes—
119#1865C2212yes—
120#1863C2212yes—
121#1842C2211yescharacteristic
122#1831C2212yes—
123#1776C2212yes—
124#1776C2212yes—
125C4×#107C2211yescharacteristic
126C4×#107C2211yescharacteristic
127#1770C2211yescharacteristic
128#1770C2211yescharacteristic
129#1767C2212yes—
130C4×C4⋊C8C2211yescharacteristic
131#1763C2212yes—
132#1763C2212yes—
133C4×#103C2211yescharacteristic
134C4×#103C2211yescharacteristic
135C22×Q16C2312yes—
136#557—24no—
137#557—24no—
138C2×C4⋊Q8—21no—
139C2×C4⋊Q8C2311yescharacteristic
140C2×C4⋊Q8C2311yescharacteristic
141C2×C4⋊Q8C2311yescharacteristic
142C2×C4⋊Q8C2311yescharacteristic
143C2×#125—22no—
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×Q8C2312yes—
153C2×C4×Q8C2311yescharacteristic
154C2×C4×Q8C2311yescharacteristic
155#498—24no—
156#487—24no—
157#477—24no—
158#462—24no—
159C2×#107C2312yes—
160C2×#107C2311yescharacteristic
161C2×#107C2311yescharacteristic
162C2×#107C2311yescharacteristic
163C2×#107C2311yescharacteristic
164C2×#106C2312yes—
165C2×#106C2312yes—
166C2×#106C2312yes—
167C2×#106C2312yes—
168C2×C4⋊C8C2311yescharacteristic
169C2×C4⋊C8C2311yescharacteristic
170C2×C4⋊C8C2311yescharacteristic
171C2×C4⋊C8C2312yes—
172C2×C4⋊C8C2312yes—
173C2×C4⋊C8C2311yescharacteristic
174C2×#103C2312yes—
175C2×#103C2312yes—
176C2×#103C2312yes—
177C2×#103C2312yes—
178C2×#103C2312yes—
179C2×#103C2311yescharacteristic
180C2×#103C2311yescharacteristic
181C2×#103C2312yes—
182C2×#103C2312yes—
183C2×#103C2312yes—
184C2×#103C2311yescharacteristic
185C2×#103C2311yescharacteristic
186C2×#97C2312yes—
187C2×#97C2312yes—
188C2×C4×C8C2311yescharacteristic
189C2×C4×C8C2311yescharacteristic
190#400—22no—
191#400—22no—
192#400—22no—
193#398—22no—
194#395—22no—
195#395—22no—
196#391—21no—
197#391—21no—
198#391—21no—
199#391—21no—
200#391—21no—
201#391—21no—
202#391—21no—
203#391—21no—
204#391—22no—
205#391—22no—
206#391C2311yescharacteristic
207#391C2311yescharacteristic
208#391C2312yes—
209#391C2312yes—
210#391C2312yes—
211#391C2312yes—
212#391C2312yes—
213#391C2312yes—
214#389—21no—
215#389—21no—
216#389—22no—
217#389—21no—
218#389—21no—
219#389C2311yescharacteristic
220#389C2312yes—
221#389C2311yescharacteristic
222#389C2311yescharacteristic
223#389C2312yes—
224#389C2312yes—
225#389C2311yescharacteristic
226#389C2312yes—
227#389C2312yes—
228#387—22no—
229#387—22no—
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—21no—
243#387—22no—
244#387—21no—
245#387—22no—
246#387—22no—
247#387—22no—
248#387—22no—
249#387—22no—
250#384—21no—
251#384C2311yescharacteristic
252#384C2311yescharacteristic
253#383—22no—
254#383—22no—
255#382—21no—
256#382—21no—
257#382—21no—
258#382—21no—
259#382C2312yes—
260C4×C4⋊C4—21no—
261C4×C4⋊C4—21no—
262C4×C4⋊C4—21no—
263C4×C4⋊C4—21no—
264C4×C4⋊C4—22no—
265C4×C4⋊C4—22no—
266C4×C4⋊C4—21no—
267C4×C4⋊C4—21no—
268C4×C4⋊C4—22no—
269C4×C4⋊C4—22no—
270C4×C4⋊C4—21no—
271C4×C4⋊C4—22no—
272C4×C4⋊C4D411yescharacteristic
273C4×C4⋊C4D412yes—
274C4×C4⋊C4D412yes—
275C4×C4⋊C4D411yescharacteristic
276C4×C4⋊C4C2312yes—
277C4×C4⋊C4C2312yes—
278C4×C4⋊C4C2311yescharacteristic
279C4×C4⋊C4C2311yescharacteristic
280C4×C4⋊C4C2312yes—
281C4×C4⋊C4C2311yescharacteristic
282C4×C4⋊C4C2312yes—
283C4×C4⋊C4C2312yes—
284C4×C4⋊C4C2311yescharacteristic
285C4×C4⋊C4C2312yes—
286#376—21no—
287#376—22no—
288#376—22no—
289#376—22no—
290C43C2312yes—
291C43C2311yescharacteristic
292#340C2312yes—
293#340C2312yes—
294#340C2311yescharacteristic
295#340C2311yescharacteristic
296#340C2311yescharacteristic
297#340C2312yes—
298#340C2312yes—
299#340C2312yes—
300#340C2312yes—
301#340C2311yescharacteristic
302#340C2311yescharacteristic
303#340C2311yescharacteristic
304#340C2312yes—
305#340C2311yescharacteristic
306#340C2311yescharacteristic
307#340C2312yes—
308#340C2312yes—
309#340C2312yes—
310#336C2311yescharacteristic
311#336C2312yes—
312#336C2312yes—
313#336C2312yes—
314#336C2312yes—
315#336C2312yes—
316#336C2311yescharacteristic
317#336C2312yes—
318C22×Q8—21no—
319C22×Q8—21no—
320C22×Q8—21no—
321C22×Q8C2×D411yescharacteristic
322C22×Q8C2412yes—
323C22×Q8C2411yescharacteristic
324C2×Q16—24no—
325C2×Q16—42no—
326C2×Q16—42no—
327C22×C8C2412yes—
328C22×C8C2411yescharacteristic
329C22×C8C2411yescharacteristic
330C22×C8C2411yescharacteristic
331C22×C8C2412yes—
332C22×C8C2411yescharacteristic
333C4⋊Q8—28no—
334C4⋊Q8—22no—
335C4⋊Q8—22no—
336C4⋊Q8—22no—
337C4⋊Q8—22no—
338C4⋊Q8—22no—
339C4⋊Q8—22no—
340C4⋊Q8—41no—
341C4⋊Q8#4114yes—
342#125—216no—
343#125—24no—
344#125—24no—
345C4×Q8—28no—
346C4×Q8—24no—
347C4×Q8—24no—
348C4×Q8—24no—
349C4×Q8—22no—
350C4×Q8—22no—
351C4×Q8—22no—
352C4×Q8—24no—
353C4×Q8—42no—
354C4×Q8—42no—
355C4×Q8—44no—
356C4×Q8—44no—
357C4×Q8—44no—
358C4×Q8#4114yes—
359C2×C4⋊C4—22no—
360C2×C4⋊C4—21no—
361C2×C4⋊C4—22no—
362C2×C4⋊C4—21no—
363C2×C4⋊C4—22no—
364C2×C4⋊C4—21no—
365C2×C4⋊C4—21no—
366C2×C4⋊C4—22no—
367C2×C4⋊C4—22no—
368C2×C4⋊C4—21no—
369C2×C4⋊C4—21no—
370C2×C4⋊C4—22no—
371C2×C4⋊C4—21no—
372C2×C4⋊C4—21no—
373C2×C4⋊C4—21no—
374C2×C4⋊C4—21no—
375C2×C4⋊C4—21no—
376C2×C4⋊C4—22no—
377C2×C4⋊C4—21no—
378C2×C4⋊C4—21no—
379C2×C4⋊C4—21no—
380C2×C4⋊C4—22no—
381C2×C4⋊C4—22no—
382C2×C4⋊C4C2×D411yescharacteristic
383C2×C4⋊C4C2×D411yescharacteristic
384C2×C4⋊C4C2×D411yescharacteristic
385C2×C4⋊C4C2×D412yes—
386C2×C4⋊C4C2×D412yes—
387C2×C4⋊C4C2×D412yes—
388C2×C4⋊C4C2411yescharacteristic
389C2×C4⋊C4C2412yes—
390C2×C4⋊C4C2411yescharacteristic
391C2×C4⋊C4C2412yes—
392C2×C4⋊C4C2411yescharacteristic
393C2×C4⋊C4C2411yescharacteristic
394C2×C4⋊C4C2411yescharacteristic
395C2×C4⋊C4C2411yescharacteristic
396C2×C4⋊C4C2412yes—
397C2×C4⋊C4C2411yescharacteristic
398C2×C42—22no—
399C2×C42—21no—
400C2×C42—22no—
401C2×C42—22no—
402C2×C42—21no—
403C2×C42—21no—
404C2×C42—21no—
405C2×C42—22no—
406C2×C42—22no—
407C2×C42—22no—
408C2×C42—22no—
409C2×C42C2×D411yescharacteristic
410C2×C42C2×D412yes—
411C2×C42C2×D412yes—
412C2×C42C2×D412yes—
413C2×C42C2×D412yes—
414C2×C42C2×D411yescharacteristic
415C2×C42C2×D411yescharacteristic
416C2×C42C2×D411yescharacteristic
417C2×C42C2412yes—
418C2×C42C2412yes—
419C2×C42C2411yescharacteristic
420C2×C42C2411yescharacteristic
421C2×C42C2411yescharacteristic
422#107—22no—
423#107—24no—
424#107—22no—
425#107—41no—
426#107—41no—
427#106—24no—
428#106—24no—
429#106—42no—
430#106—42no—
431C4⋊C8—22no—
432C4⋊C8—22no—
433C4⋊C8—22no—
434C4⋊C8—41no—
435C4⋊C8—42no—
436C4⋊C8—42no—
437#103—24no—
438#103—22no—
439#103—24no—
440#103—24no—
441#103—24no—
442#103—24no—
443#103—22no—
444#103—42no—
445#103—42no—
446#103—42no—
447#103—41no—
448#103—41no—
449#97—42no—
450#97—42no—
451C4×C8—22no—
452C4×C8—22no—
453#95—21no—
454#95—21no—
455#95—21no—
456#95—22no—
457#95—21no—
458#95—21no—
459#95—22no—
460#95—21no—
461#95—22no—
462#95—22no—
463#95—22no—
464#95—22no—
465#95—22no—
466#95—22no—
467#95—22no—
468#95—22no—
469#95—22no—
470#95—21no—
471#95—21no—
472#95—22no—
473#95—22no—
474#95—22no—
475#95—22no—
476#95—22no—
477#95—21no—
478#95—22no—
479#95—21no—
480#95—22no—
481#95—21no—
482#95—22no—
483#95—22no—
484#95—22no—
485#95—22no—
486#95—22no—
487#95—22no—
488#95C2×D412yes—
489#95C2×D412yes—
490#95C2×D412yes—
491C2×Q8—22no—
492C2×Q8—24no—
493C2×Q8—22no—
494C2×Q8—22no—
495C2×Q8—21no—
496C2×Q8—22no—
497C2×Q8—22no—
498C2×Q8—21no—
499C2×Q8—21no—
500C2×Q8—22no—
501C2×Q8—22no—
502C2×Q8—21no—
503C2×Q8—21no—
504C2×Q8—22no—
505C2×Q8—22no—
506C2×Q8—42no—
507C2×Q8—42no—
508C2×Q8—42no—
509C2×Q8—42no—
510C2×Q8—42no—
511C2×Q8—81no—
512C2×Q8—81no—
513C2×Q8—81no—
514C2×Q8C2×#4112yes—
515C22×C4—21no—
516C22×C4—21no—
517C22×C4—22no—
518C22×C4—21no—
519C22×C4—21no—
520C22×C4—22no—
521C22×C4—21no—
522C22×C4—22no—
523C22×C4—21no—
524C22×C4—21no—
525C22×C4—22no—
526C22×C4—21no—
527C22×C4C22×D412yes—
528C22×C4C22×D412yes—
529C22×C4C22×D411yescharacteristic
530C22×C4C22×D411yescharacteristic
531C22×C4C22×D412yes—
532C22×C4C22×D412yes—
533C22×C4C22×D411yescharacteristic
534C22×C4C22×D411yescharacteristic
535C22×C4C22×D411yescharacteristic
536C22×C4C22×D411yescharacteristic
537C22×C4C2511yescharacteristic
538Q16—84no—
539C2×C8—21no—
540C2×C8—21no—
541C2×C8—21no—
542C2×C8—21no—
543C2×C8—21no—
544C2×C8—22no—
545C2×C8—22no—
546C2×C8—21no—
547C2×C8—21no—
548C2×C8—21no—
549C2×C8—21no—
550C2×C8—22no—
551C2×C8—22no—
552C2×C8—21no—
553C2×C8—22no—
554C2×C8—21no—
555C2×C8—22no—
556C2×C82−1+412yes—
557C4⋊C4—28no—
558C4⋊C4—28no—
559C4⋊C4—28no—
560C4⋊C4—24no—
561C4⋊C4—24no—
562C4⋊C4—22no—
563C4⋊C4—24no—
564C4⋊C4—22no—
565C4⋊C4—24no—
566C4⋊C4—22no—
567C4⋊C4—21no—
568C4⋊C4—22no—
569C4⋊C4—21no—
570C4⋊C4—22no—
571C4⋊C4—22no—
572C4⋊C4—22no—
573C4⋊C4—21no—
574C4⋊C4—21no—
575C4⋊C4—21no—
576C4⋊C4—21no—
577C4⋊C4—21no—
578C4⋊C4—21no—
579C4⋊C4—21no—
580C4⋊C4—22no—
581C4⋊C4—21no—
582C4⋊C4—21no—
583C4⋊C4—21no—
584C4⋊C4—42no—
585C4⋊C4—44no—
586C4⋊C4—42no—
587C4⋊C4—42no—
588C4⋊C4—42no—
589C4⋊C4—42no—
590C4⋊C4—42no—
591C4⋊C4—44no—
592C4⋊C4—42no—
593C4⋊C4—42no—
594C4⋊C4—42no—
595C4⋊C4—44no—
596C4⋊C4—42no—
597C4⋊C4—81no—
598C4⋊C4—81no—
599C4⋊C4—82no—
600C4⋊C4—82no—
601C4⋊C4—81no—
602C4⋊C4—82no—
603C4⋊C4C2×#4114yes—
604C4⋊C4C2×#4112yes—
605C4⋊C42−1+412yes—
606C42—28no—
607C42—24no—
608C42—24no—
609C42—22no—
610C42—22no—
611C42—22no—
612C42—21no—
613C42—22no—
614C42—21no—
615C42—22no—
616C42—21no—
617C42—21no—
618C42—41no—
619C42—42no—
620C42—41no—
621C42—42no—
622C42—42no—
623C42—44no—
624C42—44no—
625C42—44no—
626C42—42no—
627C42—42no—
628C42—42no—
629C42—82no—
630C42—82no—
631C42—82no—
632C42#13714yes—
633C42C2×#4112yes—
634C42C2×#4112yes—
635C23C23×D411yescharacteristic
636Q8—22no—
637Q8—22no—
638Q8—44no—
639Q8—48no—
640Q8—41no—
641Q8—41no—
642Q8—42no—
643Q8—42no—
644Q8—42no—
645Q8—42no—
646Q8—84no—
647Q8—162no—
648C2×C4—21no—
649C2×C4—21no—
650C2×C4—21no—
651C2×C4—21no—
652C2×C4—21no—
653C2×C4—21no—
654C2×C4—22no—
655C2×C4—22no—
656C2×C4—21no—
657C2×C4—21no—
658C2×C4—22no—
659C2×C4—22no—
660C2×C4—22no—
661C2×C4—22no—
662C2×C4—22no—
663C2×C4—22no—
664C2×C4—24no—
665C2×C4—22no—
666C2×C4—22no—
667C2×C4—21no—
668C2×C4—22no—
669C2×C4—21no—
670C2×C4—22no—
671C2×C4—22no—
672C2×C4—21no—
673C2×C4—21no—
674C2×C4—22no—
675C2×C4—22no—
676C2×C4—41no—
677C2×C4—41no—
678C2×C4—41no—
679C2×C4—41no—
680C2×C4—41no—
681C2×C4—42no—
682C2×C4—42no—
683C2×C4—41no—
684C2×C4—42no—
685C2×C4—41no—
686C2×C4—42no—
687C2×C4—41no—
688C2×C4—41no—
689C2×C4—42no—
690C2×C4—41no—
691C2×C4—41no—
692C2×C4—42no—
693C2×C4—42no—
694C2×C4—42no—
695C2×C4—41no—
696C2×C4—41no—
697C2×C4—42no—
698C2×C4—41no—
699C2×C4—41no—
700C2×C4—41no—
701C2×C4—42no—
702C2×C4—42no—
703C2×C4—41no—
704C2×C4—41no—
705C2×C4—41no—
706C2×C4—41no—
707C2×C4—41no—
708C2×C4#52512yes—
709C2×C4#52512yes—
710C2×C4C2×#13712yes—
711C2×C4C2×#13712yes—
712C2×C4C2×#13712yes—
713C2×C4#57512yes—
714C2×C4C22×#4111yescharacteristic
715C2×C4C22×#4111yescharacteristic
716C2×C4C2×2−1+411yescharacteristic
717C2×C4C2×2−1+411yescharacteristic
718C2×C4#58511yescharacteristic
719C2×C4#58511yescharacteristic
720C8—22no—
721C8—42no—
722C8—42no—
723C8—41no—
724C8—41no—
725C8—41no—
726C22C2×#52511yescharacteristic
727C22#348911yescharacteristic
728C22#349711yescharacteristic
729C22C22×#13711yescharacteristic
730C22C2×#57511yescharacteristic
731C22#358411yescharacteristic
732C22#358611yescharacteristic
733C4—22no—
734C4—22no—
735C4—22no—
736C4—42no—
737C4—44no—
738C4—42no—
739C4—42no—
740C4—42no—
741C4—41no—
742C4—42no—
743C4—41no—
744C4—42no—
745C4—42no—
746C4—82no—
747C4—82no—
748C4—81no—
749C4—81no—
750C4—81no—
751C4—81no—
752C4—81no—
753C4—82no—
754C4#311614yes—
755C4#350911yescharacteristic
756C4#351011yescharacteristic
757C4#351611yescharacteristic
758C4#351811yescharacteristic
759C2#3581711yescharacteristic
760C2C2×#311611yescharacteristic
761C2#6059511yescharacteristic
762C2#6211311yescharacteristic
763C2#6215211yescharacteristic
764C2#6236611yescharacteristic
765C2#6239911yescharacteristic
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
#362862
#343282
#301552
#298302
#295412
#294472
#292872
#286532
#275122
#274482
#271562
#270282
#263172
#262772
#259312
#258362
#237852
#237772
#232222
#231632
#231022
#230692
#228252