Finite Group Databasebeta

← #6434809

Subgroups — #6434809

Order 512.

1354 conjugacy classes of subgroups.

The subgroup lattice has 779 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35265C211yesmaximal, characteristic
2#34217C211yesmaximal, characteristic
3#30104C212yesmaximal, complemented
4#30104C212yesmaximal, complemented
5#30103C212yesmaximal, complemented
6#30103C212yesmaximal, complemented
7#30099C212yesmaximal, complemented
8#30098C212yesmaximal, complemented
9#30097C212yesmaximal, complemented
10#30097C212yesmaximal, complemented
11#23749C211yesmaximal, characteristic
12#23749C211yesmaximal, characteristic
13#23747C211yesmaximal, characteristic
14#23744C211yesmaximal, characteristic
15#23661C211yesmaximal, characteristic
16#23660C211yesmaximal, characteristic
17#23553C211yesmaximal, characteristic
18C2×#2059C211yesmaximal, characteristic
19#21038C211yesmaximal, characteristic
20#21037C211yesmaximal, characteristic
21#21035C211yesmaximal, characteristic
22#21035C211yesmaximal, characteristic
23#20774C211yesmaximal, characteristic
24C22×#125C2211yescharacteristic
25C2×#428C2211yescharacteristic
26C2×#428C2211yescharacteristic
27C22×#107C2211yescharacteristic
28C22×#106C2211yescharacteristic
29#2864C2212yes—
30#2859C2212yes—
31#2647C2212yes—
32#2647C2212yes—
33#2509—21no—
34#2508—21no—
35#2507—21no—
36#2506—21no—
37#2505—22no—
38#2473—22no—
39#2370—21no—
40#2370—21no—
41#2370—21no—
42#2370—21no—
43#2345—22no—
44#2345—22no—
45#2330C2212yes—
46#2330C2212yes—
47#2317C2211yescharacteristic
48#2317C2211yescharacteristic
49#2317C2211yescharacteristic
50#2316C2211yescharacteristic
51#2316C2211yescharacteristic
52#2316C2211yescharacteristic
53C4×#125C2212yes—
54C4×#125C2212yes—
55#2296C2211yescharacteristic
56C2×#389C2211yescharacteristic
57#2285C2211yescharacteristic
58#2274C2211yescharacteristic
59#2274C2211yescharacteristic
60C4×#117C2211yescharacteristic
61#2101C2211yescharacteristic
62#2101C2211yescharacteristic
63#2101C2212yes—
64#2101C2212yes—
65#2101C2212yes—
66#2101C2212yes—
67#2101C2212yes—
68#2101C2212yes—
69#2101C2211yescharacteristic
70#2101C2211yescharacteristic
71#2100C2212yes—
72#2100C2212yes—
73#2100C2211yescharacteristic
74#2100C2212yes—
75#2100C2211yescharacteristic
76#2099C2212yes—
77#2099C2212yes—
78#2099C2211yescharacteristic
79#2099C2212yes—
80#2099C2211yescharacteristic
81#2085C2212yes—
82#2085C2212yes—
83#2085C2212yes—
84#2085C2212yes—
85#2084C2212yes—
86#2084C2212yes—
87#2083C2212yes—
88#2083C2212yes—
89#2059C2212yes—
90#2059C2212yes—
91#2059C2212yes—
92#2059C2212yes—
93#2059C2212yes—
94#2059C2212yes—
95#2059C2212yes—
96#2059C2212yes—
97#1921C2212yes—
98#1921C2212yes—
99#1921C2212yes—
100#1921C2212yes—
101#1920C2212yes—
102#1920C2212yes—
103#1920C2212yes—
104#1920C2212yes—
105#1919C2212yes—
106#1919C2212yes—
107#1919C2212yes—
108#1919C2212yes—
109#1918C2212yes—
110#1918C2212yes—
111#1918C2212yes—
112#1918C2212yes—
113#1840C2212yes—
114#1840C2212yes—
115#1839C2212yes—
116#1839C2212yes—
117#1736C2211yescharacteristic
118#1736C2211yescharacteristic
119#1736C2211yescharacteristic
120#1736C2211yescharacteristic
121#1736C2211yescharacteristic
122#1736C2211yescharacteristic
123#1735C2211yescharacteristic
124#1735C2211yescharacteristic
125#1735C2211yescharacteristic
126#1735C2211yescharacteristic
127C2×#340C2211yescharacteristic
128C2×#340C2211yescharacteristic
129#1729C2211yescharacteristic
130#1729C2211yescharacteristic
131#1729C2211yescharacteristic
132#1729C2211yescharacteristic
133#1728C2211yescharacteristic
134C2×#336C2211yescharacteristic
135C22×M16C2311yescharacteristic
136C22×M16C2311yescharacteristic
137C23×C8C2311yescharacteristic
138C23×C8C2311yescharacteristic
139C2×#125—24no—
140C2×#125—24no—
141C2×#125—21no—
142C2×#125—21no—
143C2×#125—22no—
144C2×#125C2312yes—
145C2×#125C2312yes—
146C2×#117C2311yescharacteristic
147C2×#117C2311yescharacteristic
148C2×#117C2311yescharacteristic
149C2×#117C2311yescharacteristic
150C2×#117C2311yescharacteristic
151C2×#117C2311yescharacteristic
152C22×C4⋊C4—21no—
153C22×C4⋊C4—21no—
154C22×C4⋊C4—21no—
155C22×C4⋊C4C2311yescharacteristic
156C22×C4⋊C4C2311yescharacteristic
157C22×C4⋊C4C2311yescharacteristic
158C22×C4⋊C4C2311yescharacteristic
159C22×C42C2311yescharacteristic
160#428—24no—
161#428—24no—
162C2×#107—24no—
163C2×#107C2312yes—
164C2×#107C2312yes—
165C2×#107C2312yes—
166C2×#107C2312yes—
167C2×#106—24no—
168C2×#106C2312yes—
169C2×#106C2312yes—
170C2×#106C2312yes—
171C2×#106C2312yes—
172#400—22no—
173#400—22no—
174#400—22no—
175#400—22no—
176#398—22no—
177#398—22no—
178#398—22no—
179#398—22no—
180#389—21no—
181#389—21no—
182#389—21no—
183#389—21no—
184#389—21no—
185#389—21no—
186#389—21no—
187#389—21no—
188#389—21no—
189#389—21no—
190#389—22no—
191#389—22no—
192#389—21no—
193#389—21no—
194#389—22no—
195#389—22no—
196#389—22no—
197#389—22no—
198#389C2312yes—
199#389C2312yes—
200#389C2312yes—
201#389C2312yes—
202#389C2311yescharacteristic
203#389C2312yes—
204#389C2311yescharacteristic
205#389C2312yes—
206#389C2311yescharacteristic
207#389C2311yescharacteristic
208#389C2312yes—
209#389C2312yes—
210#389C2312yes—
211#389C2312yes—
212#387—22no—
213#387—22no—
214#387—22no—
215#387—22no—
216#387—22no—
217#387—22no—
218#387—22no—
219#387—22no—
220#387—22no—
221#387—22no—
222#387—22no—
223#387—22no—
224#385—21no—
225#385—21no—
226#385—22no—
227#385—22no—
228#385—22no—
229#384—21no—
230#384C2312yes—
231#384C2312yes—
232#384C2311yescharacteristic
233#384C2311yescharacteristic
234#382—21no—
235#382—22no—
236#382C2311yescharacteristic
237#382C2311yescharacteristic
238#382C2312yes—
239#382C2312yes—
240#380—21no—
241#380—21no—
242#380—22no—
243#380—22no—
244#380—22no—
245C4×C4⋊C4—21no—
246C4×C4⋊C4—21no—
247C4×C4⋊C4—21no—
248C4×C4⋊C4—21no—
249C4×C4⋊C4—22no—
250C4×C4⋊C4—22no—
251C4×C4⋊C4—22no—
252C4×C4⋊C4—22no—
253C4×C4⋊C4—21no—
254C4×C4⋊C4—21no—
255C4×C4⋊C4—21no—
256C4×C4⋊C4—21no—
257C4×C4⋊C4D411yescharacteristic
258C4×C4⋊C4D411yescharacteristic
259C4×C4⋊C4D411yescharacteristic
260C4×C4⋊C4D411yescharacteristic
261C4×C4⋊C4C2312yes—
262C4×C4⋊C4C2312yes—
263C4×C4⋊C4C2312yes—
264C4×C4⋊C4C2312yes—
265C4×C22⋊C4—22no—
266C4×C22⋊C4—22no—
267C4×C22⋊C4D412yes—
268C4×C22⋊C4D412yes—
269#376C2312yes—
270C2×#95—21no—
271C43C2312yes—
272#340C2312yes—
273#340C2312yes—
274#340C2312yes—
275#340C2312yes—
276#340C2312yes—
277#340C2312yes—
278#340C2312yes—
279#340C2312yes—
280#340C2311yescharacteristic
281#340C2311yescharacteristic
282#340C2312yes—
283#340C2312yes—
284#340C2311yescharacteristic
285#340C2311yescharacteristic
286#340C2312yes—
287#340C2312yes—
288#340C2311yescharacteristic
289#340C2311yescharacteristic
290#340C2311yescharacteristic
291#340C2311yescharacteristic
292#340C2311yescharacteristic
293#340C2311yescharacteristic
294#340C2312yes—
295#340C2311yescharacteristic
296#340C2312yes—
297#340C2312yes—
298#340C2311yescharacteristic
299#340C2311yescharacteristic
300#340C2311yescharacteristic
301#340C2312yes—
302#340C2311yescharacteristic
303#340C2311yescharacteristic
304#336C2311yescharacteristic
305#336C2311yescharacteristic
306#336C2312yes—
307#336C2311yescharacteristic
308#336C2311yescharacteristic
309#336C2312yes—
310#336C2312yes—
311#336C2312yes—
312#336C2312yes—
313#336C2312yes—
314#336C2312yes—
315#336C2312yes—
316#336C2312yes—
317#336C2312yes—
318C23×C4—21no—
319C23×C4—21no—
320C23×C4—21no—
321C23×C4—21no—
322C23×C4C2×D411yescharacteristic
323C23×C4C2×D411yescharacteristic
324C23×C4C2411yescharacteristic
325C2×M16—22no—
326C2×M16—22no—
327C2×M16—22no—
328C2×M16—22no—
329C2×M16—22no—
330C2×M16—22no—
331C22×C8—22no—
332C22×C8—22no—
333C22×C8—22no—
334C22×C8—22no—
335C22×C8—22no—
336C22×C8—22no—
337C22×C8C2411yescharacteristic
338C22×C8C2411yescharacteristic
339C22×C8C2411yescharacteristic
340C22×C8C2411yescharacteristic
341C22×C8C2412yes—
342C22×C8C2412yes—
343#125—28no—
344#125—28no—
345#125—24no—
346#125—24no—
347#125—24no—
348#125—24no—
349#125—84no—
350#117—22no—
351#117—22no—
352#117—22no—
353#117—22no—
354#117—22no—
355#117—22no—
356#117—22no—
357#117—22no—
358#117—22no—
359#117—22no—
360#117—22no—
361#117—22no—
362C2×C4⋊C4—28no—
363C2×C4⋊C4—28no—
364C2×C4⋊C4—22no—
365C2×C4⋊C4—22no—
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—22no—
373C2×C4⋊C4—22no—
374C2×C4⋊C4—22no—
375C2×C4⋊C4—22no—
376C2×C4⋊C4—22no—
377C2×C4⋊C4—21no—
378C2×C4⋊C4—21no—
379C2×C4⋊C4—21no—
380C2×C4⋊C4—21no—
381C2×C4⋊C4—22no—
382C2×C4⋊C4—22no—
383C2×C4⋊C4—22no—
384C2×C4⋊C4—22no—
385C2×C4⋊C4—22no—
386C2×C4⋊C4—21no—
387C2×C4⋊C4—21no—
388C2×C4⋊C4—21no—
389C2×C4⋊C4—21no—
390C2×C4⋊C4—22no—
391C2×C4⋊C4—22no—
392C2×C4⋊C4—22no—
393C2×C4⋊C4—22no—
394C2×C4⋊C4—21no—
395C2×C4⋊C4—21no—
396C2×C4⋊C4—22no—
397C2×C4⋊C4—22no—
398C2×C4⋊C4—22no—
399C2×C4⋊C4—22no—
400C2×C4⋊C4—82no—
401C2×C4⋊C4C2×D411yescharacteristic
402C2×C4⋊C4C2×D411yescharacteristic
403C2×C4⋊C4C2×D411yescharacteristic
404C2×C4⋊C4C2×D411yescharacteristic
405C2×C4⋊C4#4114yes—
406C2×C4⋊C4#4114yes—
407C2×C4⋊C4C2412yes—
408C2×C4⋊C4C2412yes—
409C2×C4⋊C4C2411yescharacteristic
410C2×C4⋊C4C2411yescharacteristic
411C2×C4⋊C4C2411yescharacteristic
412C2×C4⋊C4C2411yescharacteristic
413C2×C4⋊C4C2411yescharacteristic
414C2×C4⋊C4C2411yescharacteristic
415C2×C4⋊C4C2411yescharacteristic
416C2×C4⋊C4C2411yescharacteristic
417C2×C4⋊C4C2412yes—
418C2×C4⋊C4C2412yes—
419C2×C22⋊C4—21no—
420C2×C22⋊C4—21no—
421C2×C22⋊C4—21no—
422C2×C22⋊C4—21no—
423C2×C22⋊C4C2×D412yes—
424C2×C22⋊C4C2×D412yes—
425C2×C42—22no—
426C2×C42—22no—
427C2×C42—21no—
428C2×C42—21no—
429C2×C42—21no—
430C2×C42—21no—
431C2×C42—22no—
432C2×C42—22no—
433C2×C42—22no—
434C2×C42—22no—
435C2×C42C2×D411yescharacteristic
436C2×C42C2×D411yescharacteristic
437C2×C42C2×D412yes—
438C2×C42C2×D412yes—
439C2×C42C2×D412yes—
440C2×C42C2×D412yes—
441C2×C42C2412yes—
442C2×C42C2412yes—
443C2×C42C2411yescharacteristic
444C2×C42C2411yescharacteristic
445#107—24no—
446#107—24no—
447#107—24no—
448#107—24no—
449#106—24no—
450#106—24no—
451#106—24no—
452#106—24no—
453#95—22no—
454#95—22no—
455#95—22no—
456#95—22no—
457#95—22no—
458#95—22no—
459#95—21no—
460#95—21no—
461#95—21no—
462#95—21no—
463#95—22no—
464#95—22no—
465#95—22no—
466#95—22no—
467#95—21no—
468#95—21no—
469#95—21no—
470#95—21no—
471#95—22no—
472#95—22no—
473#95—22no—
474#95—22no—
475#95—22no—
476#95—22no—
477#95—22no—
478#95—22no—
479#95C2×D412yes—
480#95C2×D412yes—
481#95C2×D412yes—
482#95C2×D412yes—
483C24C22×D411yescharacteristic
484C22×C4—22no—
485C22×C4—22no—
486C22×C4—22no—
487C22×C4—22no—
488C22×C4—24no—
489C22×C4—24no—
490C22×C4—21no—
491C22×C4—21no—
492C22×C4—21no—
493C22×C4—21no—
494C22×C4—21no—
495C22×C4—21no—
496C22×C4—22no—
497C22×C4—22no—
498C22×C4—21no—
499C22×C4—21no—
500C22×C4—41no—
501C22×C4—41no—
502C22×C4—41no—
503C22×C4—41no—
504C22×C4—42no—
505C22×C4—42no—
506C22×C4—81no—
507C22×C4—81no—
508C22×C4—81no—
509C22×C4—81no—
510C22×C4—81no—
511C22×C4—81no—
512C22×C4—81no—
513C22×C4—81no—
514C22×C4C22×D412yes—
515C22×C4C22×D411yescharacteristic
516C22×C4C22×D412yes—
517C22×C4C22×D412yes—
518C22×C4C22×D411yescharacteristic
519C22×C4C22×D412yes—
520C22×C4C22×D411yescharacteristic
521C22×C4C22×D411yescharacteristic
522C22×C4C22×D411yescharacteristic
523C22×C4C2×#4112yes—
524C22×C4C2×#4112yes—
525C22×C42−1+412yes—
526C22×C42−1+412yes—
527C22×C42−1+412yes—
528C22×C42−1+412yes—
529C22×C4C2511yescharacteristic
530M16—44no—
531M16—44no—
532C2×C8—21no—
533C2×C8—21no—
534C2×C8—22no—
535C2×C8—22no—
536C2×C8—21no—
537C2×C8—21no—
538C2×C8—21no—
539C2×C8—21no—
540C2×C8—21no—
541C2×C8—21no—
542C2×C8—21no—
543C2×C8—21no—
544C2×C8—21no—
545C2×C8—21no—
546C2×C8—22no—
547C2×C8—22no—
548C2×C8—22no—
549C2×C8—22no—
550C2×C8—44no—
551C2×C8—44no—
552C4⋊C4—28no—
553C4⋊C4—28no—
554C4⋊C4—28no—
555C4⋊C4—28no—
556C4⋊C4—24no—
557C4⋊C4—24no—
558C4⋊C4—22no—
559C4⋊C4—22no—
560C4⋊C4—22no—
561C4⋊C4—22no—
562C4⋊C4—22no—
563C4⋊C4—22no—
564C4⋊C4—22no—
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—21no—
579C4⋊C4—21no—
580C4⋊C4—21no—
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—82no—
590C4⋊C4—82no—
591C4⋊C4—82no—
592C4⋊C4—82no—
593C4⋊C4—82no—
594C4⋊C4—82no—
595C4⋊C4—81no—
596C4⋊C4—81no—
597C4⋊C4—81no—
598C4⋊C4—81no—
599C4⋊C4—82no—
600C4⋊C4—81no—
601C4⋊C4—82no—
602C4⋊C4—81no—
603C4⋊C4—82no—
604C4⋊C4—82no—
605C4⋊C4—82no—
606C4⋊C4—82no—
607C4⋊C4—82no—
608C4⋊C4C2×#4114yes—
609C4⋊C4C2×#4114yes—
610C22⋊C4—24no—
611C22⋊C4—24no—
612C22⋊C4—42no—
613C22⋊C4—42no—
614C22⋊C4—42no—
615C22⋊C4—42no—
616C42—22no—
617C42—22no—
618C42—21no—
619C42—21no—
620C42—21no—
621C42—21no—
622C42—22no—
623C42—22no—
624C42—22no—
625C42—22no—
626C42—42no—
627C42—42no—
628C42—42no—
629C42—42no—
630C42—42no—
631C42—42no—
632C42—42no—
633C42—42no—
634C42—82no—
635C42—82no—
636C42—82no—
637C42—82no—
638C23—21no—
639C23—21no—
640C23—21no—
641C23—21no—
642C23#52512yes—
643C23#53612yes—
644C23#53612yes—
645C23C23×D411yescharacteristic
646C2×C4—21no—
647C2×C4—21no—
648C2×C4—22no—
649C2×C4—22no—
650C2×C4—22no—
651C2×C4—22no—
652C2×C4—21no—
653C2×C4—21no—
654C2×C4—21no—
655C2×C4—22no—
656C2×C4—22no—
657C2×C4—21no—
658C2×C4—21no—
659C2×C4—22no—
660C2×C4—22no—
661C2×C4—21no—
662C2×C4—21no—
663C2×C4—21no—
664C2×C4—21no—
665C2×C4—21no—
666C2×C4—24no—
667C2×C4—24no—
668C2×C4—22no—
669C2×C4—22no—
670C2×C4—22no—
671C2×C4—22no—
672C2×C4—42no—
673C2×C4—42no—
674C2×C4—42no—
675C2×C4—42no—
676C2×C4—41no—
677C2×C4—41no—
678C2×C4—41no—
679C2×C4—41no—
680C2×C4—41no—
681C2×C4—41no—
682C2×C4—41no—
683C2×C4—41no—
684C2×C4—42no—
685C2×C4—42no—
686C2×C4—41no—
687C2×C4—41no—
688C2×C4—41no—
689C2×C4—42no—
690C2×C4—42no—
691C2×C4—41no—
692C2×C4—41no—
693C2×C4—41no—
694C2×C4—41no—
695C2×C4—41no—
696C2×C4—41no—
697C2×C4—41no—
698C2×C4—42no—
699C2×C4—41no—
700C2×C4—41no—
701C2×C4—41no—
702C2×C4—41no—
703C2×C4—41no—
704C2×C4—41no—
705C2×C4—42no—
706C2×C4—82no—
707C2×C4—82no—
708C2×C4—84no—
709C2×C4—84no—
710C2×C4—84no—
711C2×C4—162no—
712C2×C4#52512yes—
713C2×C4#53612yes—
714C2×C4#53612yes—
715C2×C4#54114yes—
716C2×C4#54114yes—
717C2×C4#54114yes—
718C2×C4#54114yes—
719C2×C4C22×#4111yescharacteristic
720C2×C4C22×#4111yescharacteristic
721C2×C4C2×2−1+411yescharacteristic
722C2×C4C2×2−1+411yescharacteristic
723C2×C4C2×2−1+411yescharacteristic
724C2×C4C2×2−1+411yescharacteristic
725C8—41no—
726C8—41no—
727C8—41no—
728C8—41no—
729C8—42no—
730C8—42no—
731C22—22no—
732C22—22no—
733C22—22no—
734C22—22no—
735C22—22no—
736C22—22no—
737C22#263414yes—
738C22C2×#52511yescharacteristic
739C22C2×#53611yescharacteristic
740C22C2×#53611yescharacteristic
741C22#358411yescharacteristic
742C22#358411yescharacteristic
743C22#358511yescharacteristic
744C22#358611yescharacteristic
745C4—41no—
746C4—41no—
747C4—44no—
748C4—41no—
749C4—41no—
750C4—44no—
751C4—42no—
752C4—42no—
753C4—42no—
754C4—42no—
755C4—81no—
756C4—82no—
757C4—81no—
758C4—81no—
759C4—81no—
760C4—81no—
761C4—81no—
762C4—81no—
763C4—82no—
764C4—81no—
765C4#263414yes—
766C4C2×#54111yescharacteristic
767C4C2×#54111yescharacteristic
768C4C2×#54111yescharacteristic
769C4C2×#54111yescharacteristic
770C2—24no—
771C2C2×#263411yescharacteristic
772C2#6058011yescharacteristic
773C2#6058111yescharacteristic
774C2#6088211yescharacteristic
775C2#6088711yescharacteristic
776C2#6089011yescharacteristic
777C2#6089111yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
3. #30104C2semidirect
4. #30104C2semidirect
5. #30103C2semidirect
6. #30103C2semidirect
7. #30099C2semidirect
8. #30098C2semidirect
9. #30097C2semidirect
10. #30097C2semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#352652
#342172
#301042
#301032
#300992
#300982
#300972
#237492
#237472
#237442
#236612
#236602
#235532
C2×#20592
#210382
#210372
#210352
#207742