Finite Group Databasebeta

← #7545545

Subgroups — #7545545

Order 512.

1296 conjugacy classes of subgroups.

The subgroup lattice has 769 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#59871C211yesmaximal, characteristic
2#36286C211yesmaximal, characteristic
3#32557C212yesmaximal
4#32553C212yesmaximal
5#29224C211yesmaximal, characteristic
6#28719C211yesmaximal, characteristic
7#27746C212yesmaximal
8#27737C211yesmaximal, characteristic
9#27729C211yesmaximal, characteristic
10#26510C211yesmaximal, characteristic
11#26508C211yesmaximal, characteristic
12#26121C212yesmaximal
13#25324C211yesmaximal, characteristic
14#25323C211yesmaximal, characteristic
15#25009C212yesmaximal
16#24997C212yesmaximal
17#24698C212yesmaximal
18#23248C211yesmaximal, characteristic
19#23221C211yesmaximal, characteristic
20#23214C211yesmaximal, characteristic
21#22779C212yesmaximal
22C4×#475C211yesmaximal, characteristic
23#21506C211yesmaximal, characteristic
24C2×#487C2211yescharacteristic
25C2×#487C2211yescharacteristic
26C2×#475C2211yescharacteristic
27C2×#462C2211yescharacteristic
28C2×#441C2212yes—
29#2863—21no—
30#2862C2211yescharacteristic
31#2806—22no—
32#2605—22no—
33#2588—22no—
34#2584—21no—
35#2584—21no—
36#2582C2212yes—
37#2582C2212yes—
38#2581C2212yes—
39#2570C2212yes—
40#2568C2211yescharacteristic
41#2567C2211yescharacteristic
42#2567C2212yes—
43#2552—21no—
44#2552C2211yescharacteristic
45#2552C2211yescharacteristic
46#2511—21no—
47#2507—22no—
48#2451—21no—
49#2369—21no—
50#2369C2211yescharacteristic
51#2365C2211yescharacteristic
52#2365C2211yescharacteristic
53#2365C2211yescharacteristic
54#2356C2212yes—
55#2355C2212yes—
56Q8×C4⋊C4C2212yes—
57C4×C4⋊Q8C2211yescharacteristic
58C4×C4⋊Q8C2211yescharacteristic
59C4×C4⋊Q8C2211yescharacteristic
60#2295—21no—
61C42×Q8C2211yescharacteristic
62#2101C2212yes—
63#2101C2212yes—
64#2101C2212yes—
65#2100C2212yes—
66#2100C2212yes—
67#2099C2212yes—
68#2088C2212yes—
69#2088C2212yes—
70#2074C2212yes—
71#2074C2212yes—
72#2074C2212yes—
73#2072C2212yes—
74#2072C2212yes—
75#2072C2212yes—
76#2056C2212yes—
77#2056C2212yes—
78#1993C2212yes—
79#1990C2211yescharacteristic
80#1990C2211yescharacteristic
81#1988C2211yescharacteristic
82#1988C2211yescharacteristic
83#1988C2211yescharacteristic
84#1987C2211yescharacteristic
85#1987C2211yescharacteristic
86#1985C2212yes—
87#1984C2212yes—
88#1981C2211yescharacteristic
89#1948C2211yescharacteristic
90#1948C2211yescharacteristic
91#1948C2212yes—
92#1948C2212yes—
93#1940C2212yes—
94#1939C2211yescharacteristic
95#1939C2211yescharacteristic
96#1935C2212yes—
97#1935C2212yes—
98#1933C2212yes—
99#1933C2212yes—
100#1927C2212yes—
101#1927C2212yes—
102#1927C2212yes—
103#1920C2212yes—
104#1919C2211yescharacteristic
105#1919C2211yescharacteristic
106#1919C2212yes—
107#1878C2212yes—
108#1878C2212yes—
109#1878C2212yes—
110#1865C2212yes—
111#1865C2212yes—
112#1865C2211yescharacteristic
113#1865C2211yescharacteristic
114#1836C2212yes—
115#1836C2212yes—
116#1834C2212yes—
117#1805C2212yes—
118#1805C2212yes—
119#1805C2212yes—
120#1803C2211yescharacteristic
121#1803C2212yes—
122#1803C2211yescharacteristic
123C4×#106C2211yescharacteristic
124C4×#106C2211yescharacteristic
125#1771C2212yes—
126#1770C2211yescharacteristic
127C8×C4⋊C4C2211yescharacteristic
128#1767C2211yescharacteristic
129C4×C4⋊C8C2211yescharacteristic
130#1763C2212yes—
131#1763C2212yes—
132C4×#103C2211yescharacteristic
133C4×#103C2211yescharacteristic
134C4×#103C2212yes—
135C22×Q16C2312yes—
136C2×C4⋊Q8D411yescharacteristic
137C2×C4⋊Q8C2311yescharacteristic
138C2×C4⋊Q8C2311yescharacteristic
139C2×C4⋊Q8C2311yescharacteristic
140C2×C4⋊Q8C2311yescharacteristic
141C2×#125—21no—
142C2×#125—21no—
143C2×C4×Q8—21no—
144C2×C4×Q8—21no—
145C2×C4×Q8C2312yes—
146C2×C4×Q8C2311yescharacteristic
147C2×C4×Q8C2311yescharacteristic
148C2×C4×Q8C2312yes—
149C2×C4×Q8C2312yes—
150C2×C4×Q8C2311yescharacteristic
151C2×C4×Q8C2312yes—
152#487—24no—
153#487—24no—
154#475—24no—
155#462—24no—
156#441—28no—
157C2×#106C2311yescharacteristic
158C2×#106C2311yescharacteristic
159C2×#106C2312yes—
160C2×#106C2311yescharacteristic
161C2×#106C2311yescharacteristic
162C2×C4⋊C8C2311yescharacteristic
163C2×C4⋊C8C2311yescharacteristic
164C2×C4⋊C8C2311yescharacteristic
165C2×C4⋊C8C2311yescharacteristic
166C2×C4⋊C8C2312yes—
167C2×C4⋊C8C2312yes—
168C2×#103C2311yescharacteristic
169C2×#103C2311yescharacteristic
170C2×#103C2312yes—
171C2×#103C2312yes—
172C2×#103C2312yes—
173C2×#103C2312yes—
174C2×#103C2312yes—
175C2×#103C2312yes—
176C2×#103C2312yes—
177C2×#103C2311yescharacteristic
178C2×#103C2311yescharacteristic
179C2×#103C2312yes—
180C2×#97C2312yes—
181C2×#97C2312yes—
182C2×C4×C8C2311yescharacteristic
183C2×C4×C8C2311yescharacteristic
184C2×C4×C8C2312yes—
185#400—22no—
186#400—22no—
187#400—22no—
188#398—22no—
189#398—22no—
190#398—22no—
191#398—22no—
192#398—22no—
193#398—22no—
194#398—22no—
195#395—22no—
196#395—22no—
197#395—22no—
198#395—22no—
199#391—22no—
200#391—22no—
201#391—22no—
202#391—21no—
203#391—21no—
204#391—21no—
205#391—21no—
206#391—21no—
207#391—21no—
208#391—21no—
209#391—21no—
210#391—22no—
211#391D412yes—
212#391C2312yes—
213#391C2312yes—
214#391C2312yes—
215#391C2312yes—
216#391C2312yes—
217#389—21no—
218#389—22no—
219#389—21no—
220#389—22no—
221#389—22no—
222#389—22no—
223#389—21no—
224#389—21no—
225#389—21no—
226#389—21no—
227#389—21no—
228#389—21no—
229#389—21no—
230#389—21no—
231#389—22no—
232#389—22no—
233#389D411yescharacteristic
234#389D411yescharacteristic
235#389D412yes—
236#389C2311yescharacteristic
237#389C2311yescharacteristic
238#389C2312yes—
239#389C2312yes—
240#389C2312yes—
241#389C2312yes—
242#387—22no—
243#387—22no—
244#387—22no—
245#387—22no—
246#387—22no—
247#387—22no—
248#387—22no—
249#387—22no—
250#387—22no—
251#387—22no—
252#387—22no—
253#384—22no—
254#384D411yescharacteristic
255#384C2311yescharacteristic
256#384C2311yescharacteristic
257#384C2312yes—
258#384C2312yes—
259#384C2312yes—
260#383—22no—
261#382C2312yes—
262#382C2312yes—
263#382C2312yes—
264C4×C4⋊C4—21no—
265C4×C4⋊C4—21no—
266C4×C4⋊C4—22no—
267C4×C4⋊C4—22no—
268C4×C4⋊C4—21no—
269C4×C4⋊C4—21no—
270C4×C4⋊C4—21no—
271C4×C4⋊C4—21no—
272C4×C4⋊C4—21no—
273C4×C4⋊C4—21no—
274C4×C4⋊C4—21no—
275C4×C4⋊C4C2311yescharacteristic
276C4×C4⋊C4C2311yescharacteristic
277C4×C4⋊C4C2311yescharacteristic
278C4×C4⋊C4C2311yescharacteristic
279C4×C4⋊C4C2312yes—
280C4×C4⋊C4C2312yes—
281C4×C4⋊C4C2311yescharacteristic
282C4×C4⋊C4C2312yes—
283C4×C4⋊C4C2311yescharacteristic
284C4×C4⋊C4C2311yescharacteristic
285#376—22no—
286C43—21no—
287C43C2311yescharacteristic
288#340C2312yes—
289#340C2312yes—
290#340C2311yescharacteristic
291#340C2312yes—
292#340C2312yes—
293#340C2311yescharacteristic
294#340C2311yescharacteristic
295#340C2311yescharacteristic
296#340C2312yes—
297#340C2312yes—
298#340C2312yes—
299#340C2312yes—
300#340C2312yes—
301#340C2311yescharacteristic
302#340C2312yes—
303#340C2312yes—
304#340C2311yescharacteristic
305#340C2312yes—
306#340C2312yes—
307#340C2312yes—
308#340C2311yescharacteristic
309#340C2311yescharacteristic
310#336C2312yes—
311#336C2312yes—
312#336C2312yes—
313#336C2311yescharacteristic
314#336C2311yescharacteristic
315#336C2312yes—
316#336C2312yes—
317C22×Q8—21no—
318C22×Q8—21no—
319C22×Q8—21no—
320C22×Q8C2×D412yes—
321C22×Q8C2412yes—
322C22×Q8C2411yescharacteristic
323C2×Q16—24no—
324C2×Q16—24no—
325C2×Q16—42no—
326C22×C8C2412yes—
327C22×C8C2411yescharacteristic
328C22×C8C2411yescharacteristic
329C22×C8C2412yes—
330C22×C8C2411yescharacteristic
331C22×C8C2411yescharacteristic
332C4⋊Q8—24no—
333C4⋊Q8—22no—
334C4⋊Q8—22no—
335C4⋊Q8—22no—
336C4⋊Q8—22no—
337C4⋊Q8—22no—
338C4⋊Q8—22no—
339C4⋊Q8—22no—
340C4⋊Q8#4114yes—
341#125—82no—
342#125—82no—
343C4×Q8—22no—
344C4×Q8—24no—
345C4×Q8—22no—
346C4×Q8—24no—
347C4×Q8—22no—
348C4×Q8—42no—
349C4×Q8—42no—
350C4×Q8—41no—
351C4×Q8—41no—
352C4×Q8—42no—
353C4×Q8—42no—
354C4×Q8—42no—
355C4×Q8—42no—
356C4×Q8—44no—
357C4×Q8—82no—
358C4×Q8#4114yes—
359C2×C4⋊C4—21no—
360C2×C4⋊C4—21no—
361C2×C4⋊C4—21no—
362C2×C4⋊C4—22no—
363C2×C4⋊C4—21no—
364C2×C4⋊C4—21no—
365C2×C4⋊C4—22no—
366C2×C4⋊C4—21no—
367C2×C4⋊C4—21no—
368C2×C4⋊C4—21no—
369C2×C4⋊C4—21no—
370C2×C4⋊C4—21no—
371C2×C4⋊C4—21no—
372C2×C4⋊C4—21no—
373C2×C4⋊C4—21no—
374C2×C4⋊C4—22no—
375C2×C4⋊C4—22no—
376C2×C4⋊C4—22no—
377C2×C4⋊C4—22no—
378C2×C4⋊C4—22no—
379C2×C4⋊C4—22no—
380C2×C4⋊C4—22no—
381C2×C4⋊C4—21no—
382C2×C4⋊C4—22no—
383C2×C4⋊C4—22no—
384C2×C4⋊C4—22no—
385C2×C4⋊C4—22no—
386C2×C4⋊C4C2×D411yescharacteristic
387C2×C4⋊C4C2×D411yescharacteristic
388C2×C4⋊C4C2×D412yes—
389C2×C4⋊C4C2×D411yescharacteristic
390C2×C4⋊C4C2×D412yes—
391C2×C4⋊C4C2×D412yes—
392C2×C4⋊C4C2×D412yes—
393C2×C4⋊C4C2×D412yes—
394C2×C4⋊C4C2×D411yescharacteristic
395C2×C4⋊C4C2411yescharacteristic
396C2×C4⋊C4C2411yescharacteristic
397C2×C4⋊C4C2411yescharacteristic
398C2×C4⋊C4C2412yes—
399C2×C4⋊C4C2411yescharacteristic
400C2×C4⋊C4C2412yes—
401C2×C4⋊C4C2412yes—
402C2×C4⋊C4C2411yescharacteristic
403C2×C4⋊C4C2411yescharacteristic
404C2×C4⋊C4C2411yescharacteristic
405C2×C42—21no—
406C2×C42—21no—
407C2×C42—21no—
408C2×C42—21no—
409C2×C42—22no—
410C2×C42—21no—
411C2×C42—21no—
412C2×C42—22no—
413C2×C42—22no—
414C2×C42—22no—
415C2×C42—21no—
416C2×C42—21no—
417C2×C42—22no—
418C2×C42C2×D411yescharacteristic
419C2×C42C2×D411yescharacteristic
420C2×C42C2×D411yescharacteristic
421C2×C42C2×D411yescharacteristic
422C2×C42C2411yescharacteristic
423C2×C42C2412yes—
424C2×C42C2412yes—
425C2×C42C2411yescharacteristic
426C2×C42C2411yescharacteristic
427#106—24no—
428#106—22no—
429#106—22no—
430#106—41no—
431#106—41no—
432C4⋊C8—22no—
433C4⋊C8—22no—
434C4⋊C8—22no—
435C4⋊C8—22no—
436C4⋊C8—42no—
437C4⋊C8—42no—
438#103—24no—
439#103—24no—
440#103—22no—
441#103—22no—
442#103—24no—
443#103—24no—
444#103—24no—
445#103—24no—
446#103—24no—
447#103—42no—
448#103—41no—
449#103—41no—
450#97—24no—
451#97—42no—
452C4×C8—22no—
453C4×C8—22no—
454C4×C8—42no—
455#95—22no—
456#95—22no—
457#95—22no—
458#95—21no—
459#95—21no—
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—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#95C2×D412yes—
485#95C2×D412yes—
486#95C2×D412yes—
487#95C2×D412yes—
488C2×Q8—24no—
489C2×Q8—21no—
490C2×Q8—22no—
491C2×Q8—22no—
492C2×Q8—21no—
493C2×Q8—21no—
494C2×Q8—22no—
495C2×Q8—21no—
496C2×Q8—22no—
497C2×Q8—22no—
498C2×Q8—21no—
499C2×Q8—42no—
500C2×Q8—42no—
501C2×Q8—42no—
502C2×Q8—42no—
503C2×Q8—42no—
504C2×Q8—42no—
505C2×Q8—42no—
506C2×Q8—81no—
507C2×Q8—81no—
508C2×Q8—81no—
509C2×Q8—81no—
510C2×Q8C2×#4114yes—
511C2×Q8C2×#4112yes—
512C22×C4—21no—
513C22×C4—21no—
514C22×C4—22no—
515C22×C4—22no—
516C22×C4—22no—
517C22×C4—21no—
518C22×C4—21no—
519C22×C4—21no—
520C22×C4—21no—
521C22×C4—21no—
522C22×C4—21no—
523C22×C4—22no—
524C22×C4C22×D411yescharacteristic
525C22×C4C22×D412yes—
526C22×C4C22×D412yes—
527C22×C4C22×D412yes—
528C22×C4C22×D411yescharacteristic
529C22×C4C22×D411yescharacteristic
530C22×C4C22×D411yescharacteristic
531C22×C4C22×D411yescharacteristic
532C22×C4C22×D412yes—
533C22×C4C22×D411yescharacteristic
534C22×C4C2511yescharacteristic
535Q16—48no—
536C2×C8—21no—
537C2×C8—21no—
538C2×C8—21no—
539C2×C8—21no—
540C2×C8—22no—
541C2×C8—22no—
542C2×C8—21no—
543C2×C8—21no—
544C2×C8—22no—
545C2×C8—22no—
546C2×C8—21no—
547C2×C8—21no—
548C2×C8—21no—
549C2×C8—22no—
550C2×C8—21no—
551C2×C8—21no—
552C2×C8—21no—
553C2×C8—22no—
554C4⋊C4—24no—
555C4⋊C4—24no—
556C4⋊C4—24no—
557C4⋊C4—24no—
558C4⋊C4—22no—
559C4⋊C4—22no—
560C4⋊C4—22no—
561C4⋊C4—21no—
562C4⋊C4—21no—
563C4⋊C4—22no—
564C4⋊C4—22no—
565C4⋊C4—22no—
566C4⋊C4—21no—
567C4⋊C4—21no—
568C4⋊C4—21no—
569C4⋊C4—22no—
570C4⋊C4—22no—
571C4⋊C4—22no—
572C4⋊C4—21no—
573C4⋊C4—21no—
574C4⋊C4—21no—
575C4⋊C4—21no—
576C4⋊C4—21no—
577C4⋊C4—21no—
578C4⋊C4—21no—
579C4⋊C4—21no—
580C4⋊C4—42no—
581C4⋊C4—42no—
582C4⋊C4—42no—
583C4⋊C4—42no—
584C4⋊C4—42no—
585C4⋊C4—42no—
586C4⋊C4—44no—
587C4⋊C4—42no—
588C4⋊C4—42no—
589C4⋊C4—42no—
590C4⋊C4—42no—
591C4⋊C4—44no—
592C4⋊C4—42no—
593C4⋊C4—81no—
594C4⋊C4—82no—
595C4⋊C4—82no—
596C4⋊C4—82no—
597C4⋊C4—82no—
598C4⋊C4—82no—
599C4⋊C4—82no—
600C4⋊C4—82no—
601C4⋊C4—81no—
602C4⋊C4—81no—
603C4⋊C4—81no—
604C4⋊C4—82no—
605C4⋊C4—82no—
606C4⋊C4—82no—
607C4⋊C4—82no—
608C4⋊C4#13714yes—
609C4⋊C4C2×#4112yes—
610C42—24no—
611C42—22no—
612C42—22no—
613C42—22no—
614C42—22no—
615C42—22no—
616C42—22no—
617C42—21no—
618C42—21no—
619C42—41no—
620C42—41no—
621C42—42no—
622C42—44no—
623C42—42no—
624C42—42no—
625C42—44no—
626C42—42no—
627C42—42no—
628C42—81no—
629C42—81no—
630C42—82no—
631C42—82no—
632C42—82no—
633C42C2×#4112yes—
634C42C2×#4112yes—
635C422−1+412yes—
636C422−1+412yes—
637C23C23×D411yescharacteristic
638Q8—22no—
639Q8—24no—
640Q8—22no—
641Q8—24no—
642Q8—42no—
643Q8—41no—
644Q8—41no—
645Q8—42no—
646Q8—84no—
647Q8—84no—
648Q8—84no—
649Q8—162no—
650C2×C4—21no—
651C2×C4—21no—
652C2×C4—22no—
653C2×C4—21no—
654C2×C4—21no—
655C2×C4—22no—
656C2×C4—21no—
657C2×C4—21no—
658C2×C4—21no—
659C2×C4—22no—
660C2×C4—21no—
661C2×C4—22no—
662C2×C4—22no—
663C2×C4—22no—
664C2×C4—22no—
665C2×C4—24no—
666C2×C4—21no—
667C2×C4—22no—
668C2×C4—21no—
669C2×C4—22no—
670C2×C4—22no—
671C2×C4—21no—
672C2×C4—22no—
673C2×C4—21no—
674C2×C4—22no—
675C2×C4—22no—
676C2×C4—41no—
677C2×C4—41no—
678C2×C4—42no—
679C2×C4—41no—
680C2×C4—42no—
681C2×C4—41no—
682C2×C4—41no—
683C2×C4—41no—
684C2×C4—42no—
685C2×C4—41no—
686C2×C4—41no—
687C2×C4—42no—
688C2×C4—41no—
689C2×C4—42no—
690C2×C4—41no—
691C2×C4—41no—
692C2×C4—42no—
693C2×C4—41no—
694C2×C4—42no—
695C2×C4—41no—
696C2×C4—41no—
697C2×C4—42no—
698C2×C4—41no—
699C2×C4—41no—
700C2×C4—41no—
701C2×C4—41no—
702C2×C4—42no—
703C2×C4—41no—
704C2×C4—41no—
705C2×C4—41no—
706C2×C4—41no—
707C2×C4—42no—
708C2×C4—42no—
709C2×C4#54812yes—
710C2×C4#54812yes—
711C2×C4#54812yes—
712C2×C4#54812yes—
713C2×C4C2×#13714yes—
714C2×C4C2×#13712yes—
715C2×C4#57712yes—
716C2×C4C22×#4111yescharacteristic
717C2×C4C22×#4111yescharacteristic
718C2×C4C2×2−1+411yescharacteristic
719C2×C4C2×2−1+411yescharacteristic
720C2×C4#58511yescharacteristic
721C2×C4#58511yescharacteristic
722C8—41no—
723C8—41no—
724C8—42no—
725C8—42no—
726C8—41no—
727C8—41no—
728C22C2×#54811yescharacteristic
729C22C2×#54811yescharacteristic
730C22#348511yescharacteristic
731C22C22×#13711yescharacteristic
732C22C2×#57711yescharacteristic
733C22#358411yescharacteristic
734C22#358611yescharacteristic
735C4—24no—
736C4—22no—
737C4—22no—
738C4—22no—
739C4—42no—
740C4—41no—
741C4—41no—
742C4—42no—
743C4—42no—
744C4—42no—
745C4—42no—
746C4—44no—
747C4—81no—
748C4—81no—
749C4—81no—
750C4—82no—
751C4—82no—
752C4—81no—
753C4—81no—
754C4—81no—
755C4—82no—
756C4#336914yes—
757C4#347511yescharacteristic
758C4#350711yescharacteristic
759C4#351511yescharacteristic
760C4#355511yescharacteristic
761C2#5907911yescharacteristic
762C2C2×#336911yescharacteristic
763C2#6132211yescharacteristic
764C2#6132711yescharacteristic
765C2#6134411yescharacteristic
766C2#6174211yescharacteristic
767C2#6180911yescharacteristic
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
#598712
#362862
#325572
#325532
#292242
#287192
#277462
#277372
#277292
#265102
#265082
#261212
#253242
#253232
#250092
#249972
#246982
#232482
#232212
#232142
#227792
C4×#4752
#215062