Finite Group Databasebeta

← #6604633

Subgroups — #6604633

Order 512.

1408 conjugacy classes of subgroups.

The subgroup lattice has 778 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#38984C211yesmaximal, characteristic
2#34335C211yesmaximal, characteristic
3#33091C212yesmaximal
4#33033C211yesmaximal, characteristic
5#29525C212yesmaximal
6#29403C212yesmaximal
7#29273C211yesmaximal, characteristic
8#28653C211yesmaximal, characteristic
9#27561C212yesmaximal
10#27145C212yesmaximal
11#27000C212yesmaximal
12#26393C212yesmaximal
13#26317C211yesmaximal, characteristic
14#26283C211yesmaximal, characteristic
15#25936C211yesmaximal, characteristic
16#25837C211yesmaximal, characteristic
17#23217C211yesmaximal, characteristic
18#23163C211yesmaximal, characteristic
19#23004C211yesmaximal, characteristic
20#22964C211yesmaximal, characteristic
21#22829C211yesmaximal, characteristic
22C4×#477C211yesmaximal, characteristic
23#21127C212yesmaximal
24C2×Q82C2211yescharacteristic
25C2×#498C2211yescharacteristic
26C2×#487C2211yescharacteristic
27C2×#477C2211yescharacteristic
28C2×#475C2211yescharacteristic
29C2×#462C2211yescharacteristic
30C2×#441C2212yes—
31#2862C2211yescharacteristic
32#2767—22no—
33#2762—22no—
34#2645C2212yes—
35#2581—21no—
36#2568—21no—
37#2568C2211yescharacteristic
38#2546C2211yescharacteristic
39#2514—21no—
40#2511C2212yes—
41#2451C2212yes—
42#2446—21no—
43#2369—21no—
44#2369C2211yescharacteristic
45#2365—22no—
46#2365C2212yes—
47#2365C2211yescharacteristic
48#2365C2211yescharacteristic
49#2355—21no—
50#2355C2211yescharacteristic
51#2351C2212yes—
52Q8×C4⋊C4C2211yescharacteristic
53#2331—22no—
54#2331C2212yes—
55#2320C2212yes—
56C4×C4⋊Q8—21no—
57C4×C4⋊Q8C2211yescharacteristic
58C4×C4⋊Q8C2211yescharacteristic
59#2295—21no—
60#2295C2211yescharacteristic
61#2276C2212yes—
62C42×Q8C2211yescharacteristic
63#2101C2212yes—
64#2099C2212yes—
65#2097C2212yes—
66#2085C2212yes—
67#2074C2212yes—
68#2066C2212yes—
69#2066C2212yes—
70#2058C2212yes—
71#2058C2212yes—
72#2058C2212yes—
73#2024C2212yes—
74#2024C2212yes—
75#1990C2211yescharacteristic
76#1988C2211yescharacteristic
77#1987C2211yescharacteristic
78#1987C2211yescharacteristic
79#1987C2211yescharacteristic
80#1984C2211yescharacteristic
81#1984C2211yescharacteristic
82#1981C2211yescharacteristic
83#1956C2211yescharacteristic
84#1948C2211yescharacteristic
85#1948C2211yescharacteristic
86#1947C2211yescharacteristic
87#1944C2212yes—
88#1941C2212yes—
89#1935C2212yes—
90#1933C2212yes—
91#1928C2212yes—
92#1928C2212yes—
93#1927C2212yes—
94#1927C2212yes—
95#1927C2212yes—
96#1927C2212yes—
97#1927C2212yes—
98#1921C2212yes—
99#1921C2211yescharacteristic
100#1921C2211yescharacteristic
101#1920C2212yes—
102#1920C2212yes—
103#1920C2212yes—
104#1919C2212yes—
105#1919C2212yes—
106#1919C2212yes—
107#1917C2212yes—
108#1916C2212yes—
109#1878C2212yes—
110#1878C2212yes—
111#1878C2212yes—
112#1878C2212yes—
113#1878C2212yes—
114#1865C2212yes—
115#1865C2212yes—
116#1865C2212yes—
117#1865C2212yes—
118#1863C2212yes—
119#1863C2211yescharacteristic
120#1863C2211yescharacteristic
121#1840C2212yes—
122#1831C2211yescharacteristic
123#1803C2211yescharacteristic
124#1776C2212yes—
125C4×#107C2211yescharacteristic
126C4×#107C2211yescharacteristic
127#1770C2211yescharacteristic
128C8×C4⋊C4C2211yescharacteristic
129C4×C4⋊C8C2211yescharacteristic
130C4×#103C2211yescharacteristic
131C4×#103C2211yescharacteristic
132C4×#103C2212yes—
133C4×#103C2212yes—
134#1746C2212yes—
135C22×Q16C2312yes—
136Q82—24no—
137Q82—24no—
138C2×C4⋊Q8—21no—
139C2×C4⋊Q8—21no—
140C2×C4⋊Q8—22no—
141C2×C4⋊Q8C2311yescharacteristic
142C2×C4⋊Q8C2311yescharacteristic
143C2×C4⋊Q8C2312yes—
144C2×C4⋊Q8C2311yescharacteristic
145C2×C4⋊Q8C2311yescharacteristic
146C2×C4×Q8—21no—
147C2×C4×Q8—22no—
148C2×C4×Q8—21no—
149C2×C4×Q8—21no—
150C2×C4×Q8—21no—
151C2×C4×Q8C2312yes—
152C2×C4×Q8C2311yescharacteristic
153C2×C4×Q8C2311yescharacteristic
154C2×C4×Q8C2312yes—
155C2×C4×Q8C2312yes—
156C2×C4×Q8C2311yescharacteristic
157C2×C4×Q8C2311yescharacteristic
158#498—24no—
159#487—24no—
160#477—24no—
161#475—24no—
162#462—24no—
163#441—28no—
164C2×#107C2311yescharacteristic
165C2×#107C2311yescharacteristic
166C2×#107C2311yescharacteristic
167C2×#107C2311yescharacteristic
168C2×#106C2312yes—
169C2×#106C2312yes—
170C2×#106C2312yes—
171C2×#106C2312yes—
172C2×#106C2312yes—
173C2×C4⋊C8C2311yescharacteristic
174C2×C4⋊C8C2311yescharacteristic
175C2×C4⋊C8C2311yescharacteristic
176C2×C4⋊C8C2311yescharacteristic
177C2×#103C2312yes—
178C2×#103C2311yescharacteristic
179C2×#103C2311yescharacteristic
180C2×#103C2311yescharacteristic
181C2×#103C2312yes—
182C2×#103C2312yes—
183C2×#103C2312yes—
184C2×#103C2312yes—
185C2×#103C2312yes—
186C2×#103C2312yes—
187C2×#103C2312yes—
188C2×#103C2311yescharacteristic
189C2×#97C2312yes—
190C2×C4×C8C2311yescharacteristic
191C2×C4×C8C2311yescharacteristic
192C2×C4×C8C2312yes—
193#401—22no—
194#400—22no—
195#400—22no—
196#400—22no—
197#400—22no—
198#400—22no—
199#400—22no—
200#398—22no—
201#398—22no—
202#395—22no—
203#391—21no—
204#391—21no—
205#391—21no—
206#391—21no—
207#391—22no—
208#391—21no—
209#391—21no—
210#391—21no—
211#391—21no—
212#391—22no—
213#391—22no—
214#391—22no—
215#391—22no—
216#391D411yescharacteristic
217#391D411yescharacteristic
218#391C2312yes—
219#391C2312yes—
220#391C2312yes—
221#391C2312yes—
222#391C2312yes—
223#389—22no—
224#389—21no—
225#389—22no—
226#389—21no—
227#389—21no—
228#389—21no—
229#389—22no—
230#389D411yescharacteristic
231#389D411yescharacteristic
232#389C2312yes—
233#389C2311yescharacteristic
234#389C2311yescharacteristic
235#389C2312yes—
236#389C2312yes—
237#389C2312yes—
238#389C2312yes—
239#389C2312yes—
240#387—22no—
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#387—22no—
250#387—22no—
251#387—22no—
252#387D412yes—
253#387D412yes—
254#384—21no—
255#384C2311yescharacteristic
256#384C2311yescharacteristic
257#383—22no—
258#382—21no—
259#382—21no—
260#382—21no—
261#382—22no—
262#382—22no—
263#382C2312yes—
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⋊C4—21no—
271C4×C4⋊C4—21no—
272C4×C4⋊C4—21no—
273C4×C4⋊C4—22no—
274C4×C4⋊C4—22no—
275C4×C4⋊C4—22no—
276C4×C4⋊C4—22no—
277C4×C4⋊C4C2311yescharacteristic
278C4×C4⋊C4C2312yes—
279C4×C4⋊C4C2311yescharacteristic
280C4×C4⋊C4C2312yes—
281C4×C4⋊C4C2311yescharacteristic
282C4×C4⋊C4C2312yes—
283C4×C4⋊C4C2311yescharacteristic
284C4×C4⋊C4C2311yescharacteristic
285C4×C4⋊C4C2312yes—
286C4×C4⋊C4C2311yescharacteristic
287C4×C4⋊C4C2312yes—
288#376—22no—
289#376C2312yes—
290C43—21no—
291C43C2311yescharacteristic
292#340C2312yes—
293#340C2311yescharacteristic
294#340C2311yescharacteristic
295#340C2312yes—
296#340C2312yes—
297#340C2311yescharacteristic
298#340C2311yescharacteristic
299#340C2312yes—
300#340C2312yes—
301#340C2312yes—
302#340C2312yes—
303#340C2312yes—
304#340C2312yes—
305#340C2311yescharacteristic
306#340C2311yescharacteristic
307#340C2312yes—
308#340C2311yescharacteristic
309#340C2311yescharacteristic
310#336C2312yes—
311#336C2312yes—
312#336C2312yes—
313#336C2312yes—
314#336C2312yes—
315#336C2311yescharacteristic
316#336C2312yes—
317#336C2312yes—
318#336C2312yes—
319#336C2311yescharacteristic
320C22×Q8—22no—
321C22×Q8—21no—
322C22×Q8—21no—
323C22×Q8—21no—
324C22×Q8C2×D411yescharacteristic
325C22×Q8C2412yes—
326C22×Q8C2411yescharacteristic
327C2×Q16—24no—
328C2×Q16—24no—
329C2×Q16—24no—
330C22×C8C2412yes—
331C22×C8C2411yescharacteristic
332C22×C8C2412yes—
333C22×C8C2411yescharacteristic
334C22×C8C2411yescharacteristic
335C22×C8C2411yescharacteristic
336C4⋊Q8—216no—
337C4⋊Q8—28no—
338C4⋊Q8—24no—
339C4⋊Q8—24no—
340C4⋊Q8—22no—
341C4⋊Q8—22no—
342C4⋊Q8—22no—
343C4⋊Q8—22no—
344C4⋊Q8—22no—
345C4⋊Q8—22no—
346C4⋊Q8—41no—
347C4⋊Q8—82no—
348C4⋊Q8#4114yes—
349C4×Q8—28no—
350C4×Q8—24no—
351C4×Q8—22no—
352C4×Q8—24no—
353C4×Q8—22no—
354C4×Q8—22no—
355C4×Q8—22no—
356C4×Q8—22no—
357C4×Q8—24no—
358C4×Q8—22no—
359C4×Q8—24no—
360C4×Q8—42no—
361C4×Q8—42no—
362C4×Q8—41no—
363C4×Q8—82no—
364C4×Q8—82no—
365C4×Q8—84no—
366C4×Q8—82no—
367C4×Q8#4114yes—
368C2×C4⋊C4—21no—
369C2×C4⋊C4—22no—
370C2×C4⋊C4—22no—
371C2×C4⋊C4—21no—
372C2×C4⋊C4—21no—
373C2×C4⋊C4—22no—
374C2×C4⋊C4—21no—
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—21no—
382C2×C4⋊C4—21no—
383C2×C4⋊C4—21no—
384C2×C4⋊C4—22no—
385C2×C4⋊C4—21no—
386C2×C4⋊C4—21no—
387C2×C4⋊C4—22no—
388C2×C4⋊C4—21no—
389C2×C4⋊C4—22no—
390C2×C4⋊C4—22no—
391C2×C4⋊C4—22no—
392C2×C4⋊C4C2×D411yescharacteristic
393C2×C4⋊C4C2×D412yes—
394C2×C4⋊C4C2×D411yescharacteristic
395C2×C4⋊C4C2×D412yes—
396C2×C4⋊C4C2×D411yescharacteristic
397C2×C4⋊C4C2×D412yes—
398C2×C4⋊C4C2411yescharacteristic
399C2×C4⋊C4C2412yes—
400C2×C4⋊C4C2412yes—
401C2×C4⋊C4C2411yescharacteristic
402C2×C4⋊C4C2411yescharacteristic
403C2×C4⋊C4C2411yescharacteristic
404C2×C4⋊C4C2411yescharacteristic
405C2×C4⋊C4C2412yes—
406C2×C4⋊C4C2411yescharacteristic
407C2×C4⋊C4C2411yescharacteristic
408C2×C42—21no—
409C2×C42—22no—
410C2×C42—21no—
411C2×C42—21no—
412C2×C42—21no—
413C2×C42—21no—
414C2×C42—21no—
415C2×C42—22no—
416C2×C42—22no—
417C2×C42—21no—
418C2×C42—22no—
419C2×C42—22no—
420C2×C42—22no—
421C2×C42—22no—
422C2×C42—21no—
423C2×C42C2×D411yescharacteristic
424C2×C42C2×D411yescharacteristic
425C2×C42C2×D411yescharacteristic
426C2×C42C2×D411yescharacteristic
427C2×C42C2412yes—
428C2×C42C2412yes—
429C2×C42C2411yescharacteristic
430C2×C42C2411yescharacteristic
431C2×C42C2411yescharacteristic
432#107—22no—
433#107—22no—
434#107—22no—
435#107—22no—
436#106—24no—
437#106—24no—
438#106—24no—
439#106—24no—
440#106—42no—
441C4⋊C8—22no—
442C4⋊C8—22no—
443C4⋊C8—22no—
444C4⋊C8—41no—
445#103—24no—
446#103—22no—
447#103—22no—
448#103—22no—
449#103—24no—
450#103—24no—
451#103—24no—
452#103—24no—
453#103—24no—
454#103—42no—
455#103—42no—
456#103#4114yes—
457#97—24no—
458C4×C8—22no—
459C4×C8—22no—
460C4×C8—42no—
461#95—22no—
462#95—22no—
463#95—22no—
464#95—21no—
465#95—21no—
466#95—21no—
467#95—22no—
468#95—22no—
469#95—22no—
470#95—22no—
471#95—21no—
472#95—22no—
473#95—22no—
474#95—21no—
475#95—22no—
476#95—21no—
477#95—21no—
478#95—22no—
479#95—22no—
480#95—22no—
481#95—22no—
482#95—22no—
483#95—22no—
484#95—22no—
485#95—22no—
486#95—22no—
487#95—22no—
488#95—22no—
489#95—22no—
490#95C2×D412yes—
491#95C2×D412yes—
492#95C2×D412yes—
493#95C2×D412yes—
494#95C2×D412yes—
495#95C2×D412yes—
496#95C2×D412yes—
497C2×Q8—28no—
498C2×Q8—22no—
499C2×Q8—24no—
500C2×Q8—22no—
501C2×Q8—22no—
502C2×Q8—21no—
503C2×Q8—21no—
504C2×Q8—21no—
505C2×Q8—22no—
506C2×Q8—22no—
507C2×Q8—22no—
508C2×Q8—22no—
509C2×Q8—22no—
510C2×Q8—22no—
511C2×Q8—42no—
512C2×Q8—42no—
513C2×Q8—42no—
514C2×Q8—42no—
515C2×Q8—42no—
516C2×Q8—81no—
517C2×Q8—81no—
518C2×Q8—82no—
519C2×Q8—81no—
520C2×Q8—82no—
521C2×Q8C2×#4112yes—
522C2×Q8C2×#4112yes—
523C2×Q82−1+412yes—
524C22×C4—21no—
525C22×C4—21no—
526C22×C4—21no—
527C22×C4—21no—
528C22×C4—22no—
529C22×C4—22no—
530C22×C4—21no—
531C22×C4—21no—
532C22×C4—21no—
533C22×C4—21no—
534C22×C4—22no—
535C22×C4—22no—
536C22×C4C22×D411yescharacteristic
537C22×C4C22×D412yes—
538C22×C4C22×D411yescharacteristic
539C22×C4C22×D412yes—
540C22×C4C22×D412yes—
541C22×C4C22×D411yescharacteristic
542C22×C4C22×D412yes—
543C22×C4C22×D411yescharacteristic
544C22×C4C22×D411yescharacteristic
545C22×C4C22×D411yescharacteristic
546C22×C4C2511yescharacteristic
547Q16—48no—
548C2×C8—21no—
549C2×C8—21no—
550C2×C8—21no—
551C2×C8—21no—
552C2×C8—21no—
553C2×C8—22no—
554C2×C8—22no—
555C2×C8—21no—
556C2×C8—22no—
557C2×C8—22no—
558C2×C8—21no—
559C2×C8—21no—
560C2×C8—21no—
561C2×C8—22no—
562C2×C8—21no—
563C2×C8—22no—
564C2×C8—21no—
565C2×C8C2×#4112yes—
566C4⋊C4—24no—
567C4⋊C4—28no—
568C4⋊C4—28no—
569C4⋊C4—24no—
570C4⋊C4—24no—
571C4⋊C4—24no—
572C4⋊C4—22no—
573C4⋊C4—22no—
574C4⋊C4—24no—
575C4⋊C4—22no—
576C4⋊C4—22no—
577C4⋊C4—21no—
578C4⋊C4—21no—
579C4⋊C4—21no—
580C4⋊C4—22no—
581C4⋊C4—22no—
582C4⋊C4—21no—
583C4⋊C4—22no—
584C4⋊C4—21no—
585C4⋊C4—21no—
586C4⋊C4—21no—
587C4⋊C4—21no—
588C4⋊C4—21no—
589C4⋊C4—21no—
590C4⋊C4—22no—
591C4⋊C4—21no—
592C4⋊C4—42no—
593C4⋊C4—44no—
594C4⋊C4—42no—
595C4⋊C4—44no—
596C4⋊C4—42no—
597C4⋊C4—42no—
598C4⋊C4—42no—
599C4⋊C4—42no—
600C4⋊C4—42no—
601C4⋊C4—44no—
602C4⋊C4—42no—
603C4⋊C4—82no—
604C4⋊C4—81no—
605C4⋊C4—81no—
606C4⋊C4—81no—
607C4⋊C4—81no—
608C4⋊C4—82no—
609C4⋊C4—82no—
610C4⋊C4—82no—
611C4⋊C4—82no—
612C4⋊C4C2×#4114yes—
613C4⋊C4C2×#4112yes—
614C4⋊C4C2×#4112yes—
615C4⋊C42−1+412yes—
616C42—24no—
617C42—28no—
618C42—24no—
619C42—22no—
620C42—22no—
621C42—22no—
622C42—22no—
623C42—21no—
624C42—22no—
625C42—22no—
626C42—21no—
627C42—21no—
628C42—22no—
629C42—21no—
630C42—41no—
631C42—42no—
632C42—44no—
633C42—44no—
634C42—42no—
635C42—42no—
636C42—42no—
637C42—42no—
638C42—81no—
639C42—82no—
640C42—82no—
641C42—82no—
642C42—82no—
643C42C2×#4112yes—
644C42C2×#4112yes—
645C23C23×D411yescharacteristic
646Q8—22no—
647Q8—22no—
648Q8—22no—
649Q8—44no—
650Q8—42no—
651Q8—42no—
652Q8—42no—
653Q8—42no—
654Q8—84no—
655Q8—88no—
656Q8—84no—
657Q8—84no—
658Q8#55614yes—
659C2×C4—21no—
660C2×C4—22no—
661C2×C4—21no—
662C2×C4—22no—
663C2×C4—22no—
664C2×C4—21no—
665C2×C4—21no—
666C2×C4—21no—
667C2×C4—21no—
668C2×C4—22no—
669C2×C4—22no—
670C2×C4—22no—
671C2×C4—22no—
672C2×C4—24no—
673C2×C4—22no—
674C2×C4—22no—
675C2×C4—22no—
676C2×C4—21no—
677C2×C4—22no—
678C2×C4—21no—
679C2×C4—21no—
680C2×C4—22no—
681C2×C4—21no—
682C2×C4—21no—
683C2×C4—21no—
684C2×C4—22no—
685C2×C4—22no—
686C2×C4—22no—
687C2×C4—22no—
688C2×C4—21no—
689C2×C4—22no—
690C2×C4—22no—
691C2×C4—41no—
692C2×C4—42no—
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—42no—
702C2×C4—42no—
703C2×C4—42no—
704C2×C4—41no—
705C2×C4—42no—
706C2×C4—41no—
707C2×C4—41no—
708C2×C4—41no—
709C2×C4—42no—
710C2×C4—41no—
711C2×C4—41no—
712C2×C4—41no—
713C2×C4—41no—
714C2×C4—42no—
715C2×C4—42no—
716C2×C4—41no—
717C2×C4—41no—
718C2×C4—42no—
719C2×C4—42no—
720C2×C4#52512yes—
721C2×C4#52512yes—
722C2×C4#57712yes—
723C2×C4#57712yes—
724C2×C4#57812yes—
725C2×C4C22×#4111yescharacteristic
726C2×C4C22×#4111yescharacteristic
727C2×C4C22×#4111yescharacteristic
728C2×C4C2×2−1+411yescharacteristic
729C2×C4C2×2−1+411yescharacteristic
730C2×C4#58511yescharacteristic
731C8—22no—
732C8—42no—
733C8—41no—
734C8—42no—
735C8—41no—
736C8—41no—
737C22C2×#52511yescharacteristic
738C22D4×#4111yescharacteristic
739C22#348611yescharacteristic
740C22C2×#57711yescharacteristic
741C22C2×#57711yescharacteristic
742C22C2×#57811yescharacteristic
743C22#358611yescharacteristic
744C4—22no—
745C4—24no—
746C4—22no—
747C4—42no—
748C4—42no—
749C4—41no—
750C4—42no—
751C4—42no—
752C4—41no—
753C4—42no—
754C4—42no—
755C4—44no—
756C4—41no—
757C4—42no—
758C4—82no—
759C4—81no—
760C4—81no—
761C4—82no—
762C4—82no—
763C4—81no—
764C4—81no—
765C4#312114yes—
766C4C2×#55611yescharacteristic
767C4#348211yescharacteristic
768C4#350711yescharacteristic
769C4#351611yescharacteristic
770C2#3623311yescharacteristic
771C2C2×#312111yescharacteristic
772C2#6059111yescharacteristic
773C2#6149911yescharacteristic
774C2#6155711yescharacteristic
775C2#6185211yescharacteristic
776C2#6187111yescharacteristic
subgroupG/H
center, socle, prime radicalC23C23×D4
commutator, Frattini, cosocleC22×C4C25
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ #2295 ◁ 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
#389842
#343352
#330912
#330332
#295252
#294032
#292732
#286532
#275612
#271452
#270002
#263932
#263172
#262832
#259362
#258372
#232172
#231632
#230042
#229642
#228292
C4×#4772
#211272