Finite Group Databasebeta

← #7545540

Subgroups — #7545540

Order 512.

1295 conjugacy classes of subgroups.

The subgroup lattice has 765 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#59666C211yesmaximal, characteristic
2#36598C211yesmaximal, characteristic
3#32557C212yesmaximal
4#32548C212yesmaximal
5#29225C211yesmaximal, characteristic
6#28717C211yesmaximal, characteristic
7#27745C212yesmaximal
8#27737C211yesmaximal, characteristic
9#27730C211yesmaximal, characteristic
10#26513C211yesmaximal, characteristic
11#26510C211yesmaximal, characteristic
12#26123C212yesmaximal
13#25323C211yesmaximal, characteristic
14#25323C211yesmaximal, characteristic
15#25009C212yesmaximal
16#25001C212yesmaximal
17#24691C212yesmaximal
18#23248C211yesmaximal, characteristic
19#23222C211yesmaximal, characteristic
20#23214C211yesmaximal, characteristic
21#22777C212yesmaximal
22C4×#479C211yesmaximal, characteristic
23#21506C211yesmaximal, characteristic
24C2×#487C2211yescharacteristic
25C2×#487C2211yescharacteristic
26C2×#479C2211yescharacteristic
27C2×#462C2211yescharacteristic
28C2×#441C2212yes—
29#2868C2211yescharacteristic
30#2862—21no—
31#2802—22no—
32#2582C2212yes—
33#2582C2212yes—
34#2581C2212yes—
35#2571C2212yes—
36#2571C2211yescharacteristic
37#2570—22no—
38#2568C2211yescharacteristic
39#2568C2211yescharacteristic
40#2565C2212yes—
41#2552C2211yescharacteristic
42#2550—21no—
43#2520—22no—
44#2511—22no—
45#2508—21no—
46#2507—21no—
47#2446—21no—
48#2446—21no—
49#2370—21no—
50#2365C2211yescharacteristic
51#2365C2211yescharacteristic
52#2365C2211yescharacteristic
53#2365C2211yescharacteristic
54#2355C2212yes—
55#2355C2212yes—
56#2351C2212yes—
57C4×C4⋊Q8—21no—
58C4×C4⋊Q8C2211yescharacteristic
59C4×C4⋊Q8C2211yescharacteristic
60C4×#125C2211yescharacteristic
61C42×Q8C2211yescharacteristic
62#2101C2212yes—
63#2101C2212yes—
64#2101C2212yes—
65#2100C2212yes—
66#2100C2212yes—
67#2100C2212yes—
68#2088C2212yes—
69#2087C2212yes—
70#2074C2212yes—
71#2074C2212yes—
72#2074C2212yes—
73#2072C2212yes—
74#2072C2212yes—
75#2072C2212yes—
76#2056C2212yes—
77#2056C2212yes—
78#1992C2212yes—
79#1990C2211yescharacteristic
80#1990C2211yescharacteristic
81#1989C2211yescharacteristic
82#1988C2211yescharacteristic
83#1988C2211yescharacteristic
84#1988C2211yescharacteristic
85#1988C2211yescharacteristic
86#1985C2212yes—
87#1984C2212yes—
88#1981C2211yescharacteristic
89#1949C2211yescharacteristic
90#1949C2212yes—
91#1948C2211yescharacteristic
92#1948C2212yes—
93#1940C2212yes—
94#1939C2211yescharacteristic
95#1939C2211yescharacteristic
96#1935C2212yes—
97#1935C2212yes—
98#1933C2212yes—
99#1933C2212yes—
100#1928C2212yes—
101#1927C2212yes—
102#1927C2212yes—
103#1920C2211yescharacteristic
104#1920C2212yes—
105#1919C2212yes—
106#1918C2211yescharacteristic
107#1878C2212yes—
108#1878C2212yes—
109#1878C2212yes—
110#1865C2212yes—
111#1865C2211yescharacteristic
112#1863C2212yes—
113#1863C2211yescharacteristic
114#1836C2212yes—
115#1836C2212yes—
116#1835C2212yes—
117#1805C2212yes—
118#1805C2212yes—
119#1805C2211yescharacteristic
120#1803C2211yescharacteristic
121#1803C2212yes—
122#1803C2212yes—
123C4×#107C2211yescharacteristic
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⋊Q8—21no—
137C2×C4⋊Q8—21no—
138C2×C4⋊Q8D411yescharacteristic
139C2×C4⋊Q8C2311yescharacteristic
140C2×C4⋊Q8C2311yescharacteristic
141C2×C4⋊Q8C2311yescharacteristic
142C2×#125—21no—
143C2×#125C2311yescharacteristic
144C2×C4×Q8—21no—
145C2×C4×Q8—21no—
146C2×C4×Q8C2312yes—
147C2×C4×Q8C2311yescharacteristic
148C2×C4×Q8C2311yescharacteristic
149C2×C4×Q8C2312yes—
150C2×C4×Q8C2312yes—
151C2×C4×Q8C2311yescharacteristic
152C2×C4×Q8C2312yes—
153#487—24no—
154#487—24no—
155#479—24no—
156#462—24no—
157#441—28no—
158C2×#107C2311yescharacteristic
159C2×#107C2311yescharacteristic
160C2×#106C2311yescharacteristic
161C2×#106C2312yes—
162C2×#106C2311yescharacteristic
163C2×C4⋊C8C2311yescharacteristic
164C2×C4⋊C8C2311yescharacteristic
165C2×C4⋊C8C2311yescharacteristic
166C2×C4⋊C8C2311yescharacteristic
167C2×C4⋊C8C2312yes—
168C2×C4⋊C8C2312yes—
169C2×#103C2311yescharacteristic
170C2×#103C2311yescharacteristic
171C2×#103C2312yes—
172C2×#103C2312yes—
173C2×#103C2312yes—
174C2×#103C2312yes—
175C2×#103C2312yes—
176C2×#103C2312yes—
177C2×#103C2312yes—
178C2×#103C2311yescharacteristic
179C2×#103C2311yescharacteristic
180C2×#103C2312yes—
181C2×#97C2312yes—
182C2×#97C2312yes—
183C2×C4×C8C2311yescharacteristic
184C2×C4×C8C2311yescharacteristic
185C2×C4×C8C2312yes—
186#400—22no—
187#400—22no—
188#400—22no—
189#400—22no—
190#400—22no—
191#400—22no—
192#398—22no—
193#398—22no—
194#398—22no—
195#398—22no—
196#398—22no—
197#398—22no—
198#398—22no—
199#395—22no—
200#391—22no—
201#391—22no—
202#391—22no—
203#391—21no—
204#391—21no—
205#391—21no—
206#391—21no—
207#391—21no—
208#391—21no—
209#391D412yes—
210#391C2312yes—
211#391C2312yes—
212#391C2312yes—
213#391C2312yes—
214#391C2312yes—
215#389—22no—
216#389—22no—
217#389—21no—
218#389—22no—
219#389—22no—
220#389—22no—
221#389—21no—
222#389—21no—
223#389—21no—
224#389—21no—
225#389—21no—
226#389D412yes—
227#389D411yescharacteristic
228#389D411yescharacteristic
229#389C2311yescharacteristic
230#389C2311yescharacteristic
231#389C2312yes—
232#389C2312yes—
233#389C2312yes—
234#389C2312yes—
235#387—22no—
236#387—22no—
237#387—22no—
238#387—22no—
239#387—21no—
240#387—21no—
241#387—22no—
242#387—22no—
243#387—22no—
244#387—22no—
245#387—22no—
246#387—22no—
247#387—22no—
248#387—22no—
249#384—21no—
250#384—21no—
251#384D411yescharacteristic
252#384C2311yescharacteristic
253#384C2312yes—
254#384C2312yes—
255#384C2312yes—
256#382—22no—
257#382—22no—
258#382—21no—
259#382—22no—
260#382C2311yescharacteristic
261#382C2312yes—
262#382C2312yes—
263#382C2312yes—
264C4×C4⋊C4—22no—
265C4×C4⋊C4—22no—
266C4×C4⋊C4—21no—
267C4×C4⋊C4—21no—
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⋊C4—22no—
276C4×C4⋊C4C2311yescharacteristic
277C4×C4⋊C4C2311yescharacteristic
278C4×C4⋊C4C2311yescharacteristic
279C4×C4⋊C4C2312yes—
280C4×C4⋊C4C2312yes—
281C4×C4⋊C4C2311yescharacteristic
282C4×C4⋊C4C2311yescharacteristic
283C4×C4⋊C4C2312yes—
284C4×C4⋊C4C2311yescharacteristic
285C4×C4⋊C4C2311yescharacteristic
286C43—21no—
287C43C2311yescharacteristic
288#340C2312yes—
289#340C2312yes—
290#340C2312yes—
291#340C2312yes—
292#340C2312yes—
293#340C2312yes—
294#340C2312yes—
295#340C2311yescharacteristic
296#340C2311yescharacteristic
297#340C2312yes—
298#340C2312yes—
299#340C2311yescharacteristic
300#340C2311yescharacteristic
301#340C2312yes—
302#340C2311yescharacteristic
303#340C2311yescharacteristic
304#340C2312yes—
305#340C2312yes—
306#340C2312yes—
307#340C2312yes—
308#340C2311yescharacteristic
309#340C2311yescharacteristic
310#336C2311yescharacteristic
311#336C2311yescharacteristic
312#336C2312yes—
313#336C2312yes—
314#336C2312yes—
315#336C2312yes—
316#336C2312yes—
317C22×Q8—21no—
318C22×Q8—21no—
319C22×Q8C2×D412yes—
320C22×Q8C2412yes—
321C22×Q8C2411yescharacteristic
322C2×Q16—24no—
323C2×Q16—24no—
324C2×Q16—42no—
325C22×C8C2412yes—
326C22×C8C2411yescharacteristic
327C22×C8C2411yescharacteristic
328C22×C8C2412yes—
329C22×C8C2411yescharacteristic
330C22×C8C2411yescharacteristic
331C4⋊Q8—22no—
332C4⋊Q8—22no—
333C4⋊Q8—22no—
334C4⋊Q8—22no—
335C4⋊Q8—22no—
336C4⋊Q8—42no—
337C4⋊Q8—44no—
338C4⋊Q8—82no—
339C4⋊Q8#4114yes—
340#125—22no—
341#125—22no—
342#125—44no—
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—82no—
357C4×Q8—82no—
358C4×Q8#4114yes—
359C2×C4⋊C4—21no—
360C2×C4⋊C4—21no—
361C2×C4⋊C4—21no—
362C2×C4⋊C4—21no—
363C2×C4⋊C4—21no—
364C2×C4⋊C4—22no—
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—22no—
372C2×C4⋊C4—21no—
373C2×C4⋊C4—21no—
374C2×C4⋊C4—21no—
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—21no—
383C2×C4⋊C4—22no—
384C2×C4⋊C4—22no—
385C2×C4⋊C4C2×D411yescharacteristic
386C2×C4⋊C4C2×D411yescharacteristic
387C2×C4⋊C4C2×D411yescharacteristic
388C2×C4⋊C4C2×D411yescharacteristic
389C2×C4⋊C4C2×D412yes—
390C2×C4⋊C4C2×D412yes—
391C2×C4⋊C4C2×D412yes—
392C2×C4⋊C4C2×D412yes—
393C2×C4⋊C4C2×D412yes—
394C2×C4⋊C4C2411yescharacteristic
395C2×C4⋊C4C2411yescharacteristic
396C2×C4⋊C4C2411yescharacteristic
397C2×C4⋊C4C2412yes—
398C2×C4⋊C4C2411yescharacteristic
399C2×C4⋊C4C2411yescharacteristic
400C2×C4⋊C4C2411yescharacteristic
401C2×C4⋊C4C2412yes—
402C2×C4⋊C4C2412yes—
403C2×C4⋊C4C2411yescharacteristic
404C2×C42—21no—
405C2×C42—21no—
406C2×C42—21no—
407C2×C42—21no—
408C2×C42—22no—
409C2×C42—21no—
410C2×C42—21no—
411C2×C42—21no—
412C2×C42—21no—
413C2×C42—22no—
414C2×C42—22no—
415C2×C42—22no—
416C2×C42—22no—
417C2×C42C2×D411yescharacteristic
418C2×C42C2×D411yescharacteristic
419C2×C42C2×D411yescharacteristic
420C2×C42C2×D411yescharacteristic
421C2×C42C2411yescharacteristic
422C2×C42C2412yes—
423C2×C42C2412yes—
424C2×C42C2411yescharacteristic
425C2×C42C2411yescharacteristic
426#107—22no—
427#107—41no—
428#106—24no—
429#106—22no—
430#106—41no—
431C4⋊C8—22no—
432C4⋊C8—22no—
433C4⋊C8—22no—
434C4⋊C8—22no—
435C4⋊C8—42no—
436C4⋊C8—42no—
437#103—24no—
438#103—24no—
439#103—22no—
440#103—22no—
441#103—24no—
442#103—24no—
443#103—24no—
444#103—24no—
445#103—24no—
446#103—42no—
447#103—41no—
448#103—41no—
449#97—24no—
450#97—42no—
451C4×C8—22no—
452C4×C8—22no—
453C4×C8—42no—
454#95—21no—
455#95—21no—
456#95—21no—
457#95—21no—
458#95—21no—
459#95—22no—
460#95—22no—
461#95—21no—
462#95—22no—
463#95—22no—
464#95—22no—
465#95—22no—
466#95—22no—
467#95—22no—
468#95—21no—
469#95—22no—
470#95—22no—
471#95—21no—
472#95—21no—
473#95—22no—
474#95—22no—
475#95—22no—
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#95—22no—
485#95C2×D412yes—
486#95C2×D412yes—
487#95C2×D412yes—
488#95C2×D412yes—
489C2×Q8—24no—
490C2×Q8—21no—
491C2×Q8—22no—
492C2×Q8—22no—
493C2×Q8—21no—
494C2×Q8—21no—
495C2×Q8—22no—
496C2×Q8—21no—
497C2×Q8—22no—
498C2×Q8—22no—
499C2×Q8—21no—
500C2×Q8—42no—
501C2×Q8—42no—
502C2×Q8—42no—
503C2×Q8—42no—
504C2×Q8—42no—
505C2×Q8—42no—
506C2×Q8—42no—
507C2×Q8—42no—
508C2×Q8C2×#4114yes—
509C2×Q8C2×#4112yes—
510C22×C4—21no—
511C22×C4—21no—
512C22×C4—22no—
513C22×C4—21no—
514C22×C4—21no—
515C22×C4—21no—
516C22×C4—21no—
517C22×C4—21no—
518C22×C4—22no—
519C22×C4—21no—
520C22×C4—22no—
521C22×C4—22no—
522C22×C4C22×D412yes—
523C22×C4C22×D412yes—
524C22×C4C22×D412yes—
525C22×C4C22×D412yes—
526C22×C4C22×D411yescharacteristic
527C22×C4C22×D411yescharacteristic
528C22×C4C22×D411yescharacteristic
529C22×C4C22×D411yescharacteristic
530C22×C4C22×D411yescharacteristic
531C22×C4C22×D411yescharacteristic
532C22×C4C2511yescharacteristic
533Q16—48no—
534C2×C8—21no—
535C2×C8—21no—
536C2×C8—21no—
537C2×C8—21no—
538C2×C8—22no—
539C2×C8—22no—
540C2×C8—21no—
541C2×C8—21no—
542C2×C8—22no—
543C2×C8—22no—
544C2×C8—21no—
545C2×C8—21no—
546C2×C8—21no—
547C2×C8—22no—
548C2×C8—21no—
549C2×C8—21no—
550C2×C8—21no—
551C2×C8—22no—
552C4⋊C4—24no—
553C4⋊C4—24no—
554C4⋊C4—24no—
555C4⋊C4—22no—
556C4⋊C4—22no—
557C4⋊C4—22no—
558C4⋊C4—22no—
559C4⋊C4—21no—
560C4⋊C4—21no—
561C4⋊C4—22no—
562C4⋊C4—22no—
563C4⋊C4—22no—
564C4⋊C4—21no—
565C4⋊C4—21no—
566C4⋊C4—21no—
567C4⋊C4—22no—
568C4⋊C4—21no—
569C4⋊C4—21no—
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—42no—
579C4⋊C4—42no—
580C4⋊C4—42no—
581C4⋊C4—42no—
582C4⋊C4—42no—
583C4⋊C4—42no—
584C4⋊C4—44no—
585C4⋊C4—44no—
586C4⋊C4—44no—
587C4⋊C4—44no—
588C4⋊C4—44no—
589C4⋊C4—42no—
590C4⋊C4—42no—
591C4⋊C4—42no—
592C4⋊C4—42no—
593C4⋊C4—42no—
594C4⋊C4—42no—
595C4⋊C4—42no—
596C4⋊C4—81no—
597C4⋊C4—81no—
598C4⋊C4—82no—
599C4⋊C4—82no—
600C4⋊C4—82no—
601C4⋊C4—82no—
602C4⋊C4—82no—
603C4⋊C4—82no—
604C4⋊C4—82no—
605C4⋊C4—82no—
606C4⋊C4C2×#4112yes—
607C42—24no—
608C42—22no—
609C42—22no—
610C42—22no—
611C42—22no—
612C42—22no—
613C42—22no—
614C42—22no—
615C42—21no—
616C42—21no—
617C42—41no—
618C42—42no—
619C42—42no—
620C42—44no—
621C42—42no—
622C42—42no—
623C42—42no—
624C42—42no—
625C42—42no—
626C42—82no—
627C42—82no—
628C42—82no—
629C42—82no—
630C42C2×#4112yes—
631C42C2×#4112yes—
632C422−1+412yes—
633C422−1+412yes—
634C23C23×D411yescharacteristic
635Q8—22no—
636Q8—24no—
637Q8—22no—
638Q8—24no—
639Q8—42no—
640Q8—41no—
641Q8—41no—
642Q8—42no—
643Q8—84no—
644Q8—84no—
645Q8—84no—
646C2×C4—21no—
647C2×C4—21no—
648C2×C4—21no—
649C2×C4—21no—
650C2×C4—21no—
651C2×C4—21no—
652C2×C4—21no—
653C2×C4—21no—
654C2×C4—21no—
655C2×C4—22no—
656C2×C4—22no—
657C2×C4—22no—
658C2×C4—22no—
659C2×C4—24no—
660C2×C4—22no—
661C2×C4—22no—
662C2×C4—21no—
663C2×C4—22no—
664C2×C4—22no—
665C2×C4—22no—
666C2×C4—22no—
667C2×C4—21no—
668C2×C4—22no—
669C2×C4—22no—
670C2×C4—22no—
671C2×C4—22no—
672C2×C4—41no—
673C2×C4—42no—
674C2×C4—41no—
675C2×C4—42no—
676C2×C4—41no—
677C2×C4—41no—
678C2×C4—41no—
679C2×C4—41no—
680C2×C4—41no—
681C2×C4—42no—
682C2×C4—41no—
683C2×C4—42no—
684C2×C4—41no—
685C2×C4—41no—
686C2×C4—41no—
687C2×C4—42no—
688C2×C4—42no—
689C2×C4—41no—
690C2×C4—41no—
691C2×C4—41no—
692C2×C4—41no—
693C2×C4—41no—
694C2×C4—42no—
695C2×C4—42no—
696C2×C4—42no—
697C2×C4—41no—
698C2×C4—41no—
699C2×C4—41no—
700C2×C4—42no—
701C2×C4—42no—
702C2×C4—41no—
703C2×C4—41no—
704C2×C4—41no—
705C2×C4#54812yes—
706C2×C4#54812yes—
707C2×C4#54812yes—
708C2×C4#54812yes—
709C2×C4#57512yes—
710C2×C4#57712yes—
711C2×C4#57812yes—
712C2×C4C22×#4111yescharacteristic
713C2×C4C22×#4111yescharacteristic
714C2×C4C2×2−1+411yescharacteristic
715C2×C4C2×2−1+411yescharacteristic
716C2×C4#58511yescharacteristic
717C2×C4#58511yescharacteristic
718C8—41no—
719C8—41no—
720C8—42no—
721C8—42no—
722C8—41no—
723C8—41no—
724C22C2×#54811yescharacteristic
725C22C2×#54811yescharacteristic
726C22#348511yescharacteristic
727C22C2×#57511yescharacteristic
728C22C2×#57711yescharacteristic
729C22C2×#57811yescharacteristic
730C22#358611yescharacteristic
731C4—22no—
732C4—22no—
733C4—22no—
734C4—22no—
735C4—42no—
736C4—42no—
737C4—41no—
738C4—42no—
739C4—42no—
740C4—42no—
741C4—42no—
742C4—44no—
743C4—81no—
744C4—81no—
745C4—81no—
746C4—82no—
747C4—82no—
748C4—82no—
749C4—81no—
750C4—81no—
751C4—81no—
752C4#337114yes—
753C4#347511yescharacteristic
754C4#350711yescharacteristic
755C4#351511yescharacteristic
756C4#355511yescharacteristic
757C2#5907911yescharacteristic
758C2C2×#337111yescharacteristic
759C2#6132011yescharacteristic
760C2#6133011yescharacteristic
761C2#6134211yescharacteristic
762C2#6173011yescharacteristic
763C2#6181211yescharacteristic
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
#596662
#365982
#325572
#325482
#292252
#287172
#277452
#277372
#277302
#265132
#265102
#261232
#253232
#250092
#250012
#246912
#232482
#232222
#232142
#227772
C4×#4792
#215062