Finite Group Databasebeta

← #6581598

Subgroups — #6581598

Order 512.

1372 conjugacy classes of subgroups.

The subgroup lattice has 781 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36447C211yesmaximal, characteristic
2#36313C211yesmaximal, characteristic
3#33278C211yesmaximal, characteristic
4#33073C211yesmaximal, characteristic
5#33033C211yesmaximal, characteristic
6#29013C212yesmaximal
7#28033C212yesmaximal
8#27749C211yesmaximal, characteristic
9#27730C211yesmaximal, characteristic
10#27551C212yesmaximal
11#27371C212yesmaximal
12#26757C212yesmaximal
13#26527C211yesmaximal, characteristic
14#26506C211yesmaximal, characteristic
15#26167C212yesmaximal
16#24818C211yesmaximal, characteristic
17#24732C211yesmaximal, characteristic
18#22998C212yesmaximal
19#22998C211yesmaximal, characteristic
20#22964C211yesmaximal, characteristic
21#22769C212yesmaximal
22C8×#125C211yesmaximal, characteristic
23C42×Q16C211yesmaximal, characteristic
24C2×#475C2211yescharacteristic
25C2×#462C2211yescharacteristic
26C2×#462C2212yes—
27C2×C4×Q16C2211yescharacteristic
28C2×C4×Q16C2211yescharacteristic
29C2×C4×Q16C2211yescharacteristic
30C2×C4×Q16C2212yes—
31#2863—21no—
32#2862C2211yescharacteristic
33#2859—21no—
34#2859C2211yescharacteristic
35#2571—21no—
36#2571C2211yescharacteristic
37#2568C2211yescharacteristic
38#2552—21no—
39#2552C2211yescharacteristic
40#2546C2211yescharacteristic
41#2528—22no—
42#2524—22no—
43#2514—21no—
44#2514C2212yes—
45#2508—22no—
46#2508C2212yes—
47#2506—22no—
48#2506C2212yes—
49#2453—21no—
50#2446C2212yes—
51#2365C2211yescharacteristic
52#2365C2212yes—
53#2365C2211yescharacteristic
54#2356C2212yes—
55#2351C2212yes—
56Q8×C4⋊C4C2211yescharacteristic
57Q8×C4⋊C4C2212yes—
58Q8×C4⋊C4C2211yescharacteristic
59C4×#125—21no—
60C4×#125C2211yescharacteristic
61#2295—21no—
62C42×Q8C2211yescharacteristic
63C42×Q8C2211yescharacteristic
64#2101C2212yes—
65#2099C2212yes—
66#2097C2212yes—
67#2086C2212yes—
68#2085C2212yes—
69#2074C2212yes—
70#2072C2212yes—
71#2068C2212yes—
72#2066C2212yes—
73#2066C2212yes—
74#2043C2212yes—
75#2043C2212yes—
76#1990C2211yescharacteristic
77#1990C2211yescharacteristic
78#1988C2211yescharacteristic
79#1981C2211yescharacteristic
80#1956C2212yes—
81#1956C2211yescharacteristic
82#1956C2211yescharacteristic
83#1948C2211yescharacteristic
84#1947C2211yescharacteristic
85#1947C2212yes—
86#1941C2212yes—
87#1939C2211yescharacteristic
88#1939C2211yescharacteristic
89#1933C2212yes—
90#1933C2212yes—
91#1933C2212yes—
92#1933C2212yes—
93#1933C2212yes—
94#1933C2212yes—
95#1933C2212yes—
96#1927C2212yes—
97#1927C2212yes—
98#1920C2212yes—
99#1920C2212yes—
100#1918C2212yes—
101#1918C2212yes—
102#1916C2212yes—
103#1916C2212yes—
104#1878C2212yes—
105#1878C2212yes—
106#1878C2212yes—
107#1878C2212yes—
108#1878C2212yes—
109#1865C2212yes—
110#1865C2212yes—
111#1863C2212yes—
112#1863C2212yes—
113#1863C2212yes—
114#1835C2212yes—
115#1831C2211yescharacteristic
116#1805C2212yes—
117#1805C2212yes—
118#1803C2211yescharacteristic
119#1803C2211yescharacteristic
120C4×#107C2211yescharacteristic
121C4×#107C2211yescharacteristic
122C4×#107C2211yescharacteristic
123C8×C4⋊C4C2211yescharacteristic
124C8×C4⋊C4C2212yes—
125C8×C4⋊C4C2211yescharacteristic
126C8×C4⋊C4C2212yes—
127C4×#103C2211yescharacteristic
128C4×#103C2211yescharacteristic
129C4×#103C2211yescharacteristic
130C4×#103C2211yescharacteristic
131C4×#103C2211yescharacteristic
132C4×#103C2211yescharacteristic
133C4×#103C2212yes—
134C42×C8C2211yescharacteristic
135C22×Q16C2312yes—
136C22×Q16C2311yescharacteristic
137C2×C4⋊Q8C2311yescharacteristic
138C2×C4⋊Q8C2312yes—
139C2×C4⋊Q8C2311yescharacteristic
140C2×#125—21no—
141C2×#125—22no—
142C2×#125—21no—
143C2×#125D411yescharacteristic
144C2×C4×Q8—21no—
145C2×C4×Q8C2311yescharacteristic
146C2×C4×Q8C2312yes—
147C2×C4×Q8C2312yes—
148C2×C4×Q8C2312yes—
149C2×C4×Q8C2311yescharacteristic
150C2×C4×Q8C2311yescharacteristic
151C2×C4×Q8C2311yescharacteristic
152C2×C4×Q8C2311yescharacteristic
153C2×C4×Q8C2311yescharacteristic
154#475—24no—
155#462—28no—
156#462—24no—
157C4×Q16—24no—
158C4×Q16—24no—
159C4×Q16—24no—
160C4×Q16—28no—
161C2×#107C2311yescharacteristic
162C2×#107C2311yescharacteristic
163C2×#107C2312yes—
164C2×#107C2311yescharacteristic
165C2×#106C2312yes—
166C2×#106C2312yes—
167C2×C4⋊C8C2311yescharacteristic
168C2×C4⋊C8C2311yescharacteristic
169C2×C4⋊C8C2312yes—
170C2×C4⋊C8C2312yes—
171C2×#103C2312yes—
172C2×#103C2312yes—
173C2×#103C2311yescharacteristic
174C2×#103C2311yescharacteristic
175C2×#103C2312yes—
176C2×#103C2312yes—
177C2×#103C2311yescharacteristic
178C2×#103C2311yescharacteristic
179C2×#103C2312yes—
180C2×#103C2311yescharacteristic
181C2×#103C2312yes—
182C2×#103C2312yes—
183C2×#103C2312yes—
184C2×#103C2312yes—
185C2×#103C2312yes—
186C2×#103C2311yescharacteristic
187C2×C4×C8C2312yes—
188C2×C4×C8C2311yescharacteristic
189C2×C4×C8C2311yescharacteristic
190C2×C4×C8C2311yescharacteristic
191C2×C4×C8C2311yescharacteristic
192C2×C4×C8C2311yescharacteristic
193C2×C4×C8C2311yescharacteristic
194C2×C4×C8C2312yes—
195#401—22no—
196#400—22no—
197#400—22no—
198#400—22no—
199#400—22no—
200#400—22no—
201#400—22no—
202#398—22no—
203#398—22no—
204#398—22no—
205#398—22no—
206#398—22no—
207#391—21no—
208#391—21no—
209#391—21no—
210#391—21no—
211#391—21no—
212#391C2312yes—
213#391C2312yes—
214#391C2312yes—
215#391C2312yes—
216#391C2312yes—
217#389—21no—
218#389—21no—
219#389—22no—
220#389—22no—
221#389—21no—
222#389—21no—
223#389—22no—
224#389—21no—
225#389—21no—
226#389—21no—
227#389—22no—
228#389—22no—
229#389—22no—
230#389—22no—
231#389D411yescharacteristic
232#389D411yescharacteristic
233#389C2312yes—
234#389C2312yes—
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#387D412yes—
252#387D412yes—
253#384—21no—
254#384—21no—
255#384—22no—
256#384C2311yescharacteristic
257#384C2311yescharacteristic
258#384C2312yes—
259#383—22no—
260#383—22no—
261#382—21no—
262#382—21no—
263#382—21no—
264#382—21no—
265#382—22no—
266#382—22no—
267#382D411yescharacteristic
268#382C2312yes—
269#382C2312yes—
270C4×C4⋊C4—21no—
271C4×C4⋊C4—22no—
272C4×C4⋊C4—21no—
273C4×C4⋊C4—21no—
274C4×C4⋊C4—22no—
275C4×C4⋊C4—21no—
276C4×C4⋊C4—21no—
277C4×C4⋊C4—21no—
278C4×C4⋊C4—21no—
279C4×C4⋊C4—21no—
280C4×C4⋊C4—21no—
281C4×C4⋊C4—22no—
282C4×C4⋊C4C2311yescharacteristic
283C4×C4⋊C4C2311yescharacteristic
284C4×C4⋊C4C2312yes—
285C4×C4⋊C4C2311yescharacteristic
286C4×C4⋊C4C2311yescharacteristic
287C4×C4⋊C4C2311yescharacteristic
288C4×C4⋊C4C2311yescharacteristic
289C4×C4⋊C4C2311yescharacteristic
290C4×C4⋊C4C2312yes—
291C4×C4⋊C4C2311yescharacteristic
292C4×C4⋊C4C2312yes—
293#376—22no—
294C43—21no—
295C43—21no—
296C43C2311yescharacteristic
297#340C2312yes—
298#340C2312yes—
299#340C2311yescharacteristic
300#340C2312yes—
301#340C2312yes—
302#340C2311yescharacteristic
303#340C2311yescharacteristic
304#340C2312yes—
305#340C2312yes—
306#340C2312yes—
307#340C2312yes—
308#340C2311yescharacteristic
309#340C2312yes—
310#340C2312yes—
311#340C2311yescharacteristic
312#340C2311yescharacteristic
313#336C2312yes—
314#336C2312yes—
315#336C2312yes—
316#336C2312yes—
317#336C2312yes—
318#336C2312yes—
319C22×Q8—21no—
320C22×Q8C2412yes—
321C22×Q8C2411yescharacteristic
322C22×Q8C2411yescharacteristic
323C2×Q16—22no—
324C2×Q16—22no—
325C2×Q16—24no—
326C2×Q16—24no—
327C2×Q16—24no—
328C2×Q16#4114yes—
329C22×C8C2412yes—
330C22×C8C2411yescharacteristic
331C22×C8C2412yes—
332C22×C8C2411yescharacteristic
333C22×C8C2411yescharacteristic
334C22×C8C2411yescharacteristic
335C4⋊Q8—24no—
336C4⋊Q8—22no—
337C4⋊Q8—22no—
338C4⋊Q8—22no—
339C4⋊Q8—22no—
340C4⋊Q8—42no—
341#125—24no—
342#125—44no—
343#125—82no—
344#125—84no—
345C4×Q8—22no—
346C4×Q8—22no—
347C4×Q8—22no—
348C4×Q8—24no—
349C4×Q8—22no—
350C4×Q8—24no—
351C4×Q8—22no—
352C4×Q8—24no—
353C4×Q8—22no—
354C4×Q8—22no—
355C4×Q8—41no—
356C4×Q8—41no—
357C4×Q8—41no—
358C4×Q8—41no—
359C4×Q8—42no—
360C4×Q8—42no—
361C4×Q8—42no—
362C4×Q8—44no—
363C4×Q8#4114yes—
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—22no—
372C2×C4⋊C4—22no—
373C2×C4⋊C4—22no—
374C2×C4⋊C4—22no—
375C2×C4⋊C4—22no—
376C2×C4⋊C4—21no—
377C2×C4⋊C4—22no—
378C2×C4⋊C4—22no—
379C2×C4⋊C4—21no—
380C2×C4⋊C4—21no—
381C2×C4⋊C4—21no—
382C2×C4⋊C4—22no—
383C2×C4⋊C4—21no—
384C2×C4⋊C4—22no—
385C2×C4⋊C4—22no—
386C2×C4⋊C4—21no—
387C2×C4⋊C4—21no—
388C2×C4⋊C4—22no—
389C2×C4⋊C4—22no—
390C2×C4⋊C4—22no—
391C2×C4⋊C4C2×D412yes—
392C2×C4⋊C4C2×D412yes—
393C2×C4⋊C4C2×D411yescharacteristic
394C2×C4⋊C4C2×D412yes—
395C2×C4⋊C4C2×D411yescharacteristic
396C2×C4⋊C4C2×D412yes—
397C2×C4⋊C4C2×D411yescharacteristic
398C2×C4⋊C4C2×D411yescharacteristic
399C2×C4⋊C4C2411yescharacteristic
400C2×C4⋊C4C2411yescharacteristic
401C2×C4⋊C4C2411yescharacteristic
402C2×C4⋊C4C2411yescharacteristic
403C2×C4⋊C4C2412yes—
404C2×C4⋊C4C2412yes—
405C2×C4⋊C4C2412yes—
406C2×C4⋊C4C2411yescharacteristic
407C2×C4⋊C4C2411yescharacteristic
408C2×C42—21no—
409C2×C42—21no—
410C2×C42—22no—
411C2×C42—21no—
412C2×C42—21no—
413C2×C42—21no—
414C2×C42—21no—
415C2×C42—21no—
416C2×C42—22no—
417C2×C42—22no—
418C2×C42—22no—
419C2×C42—21no—
420C2×C42—21no—
421C2×C42—21no—
422C2×C42—21no—
423C2×C42—21no—
424C2×C42C2×D411yescharacteristic
425C2×C42C2×D411yescharacteristic
426C2×C42C2×D411yescharacteristic
427C2×C42C2×D411yescharacteristic
428C2×C42C2411yescharacteristic
429C2×C42C2411yescharacteristic
430C2×C42C2412yes—
431C2×C42C2412yes—
432C2×C42C2411yescharacteristic
433#107—22no—
434#107—22no—
435#107—22no—
436#107—24no—
437#106—24no—
438#106—42no—
439C4⋊C8—24no—
440C4⋊C8—22no—
441C4⋊C8—22no—
442C4⋊C8—42no—
443#103—22no—
444#103—22no—
445#103—22no—
446#103—24no—
447#103—24no—
448#103—22no—
449#103—24no—
450#103—24no—
451#103—22no—
452#103—24no—
453#103—42no—
454#103—42no—
455#103—42no—
456#103—42no—
457#103—42no—
458#103#4114yes—
459C4×C8—22no—
460C4×C8—22no—
461C4×C8—24no—
462C4×C8—22no—
463C4×C8—22no—
464C4×C8—41no—
465C4×C8—41no—
466C4×C8—42no—
467#95—22no—
468#95—22no—
469#95—22no—
470#95—21no—
471#95—21no—
472#95—21no—
473#95—21no—
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#95—22no—
485#95—22no—
486#95—22no—
487#95—22no—
488#95—22no—
489#95—22no—
490#95—22no—
491#95—22no—
492#95—22no—
493#95—22no—
494#95—22no—
495#95C2×D412yes—
496#95C2×D412yes—
497#95C2×D412yes—
498#95C2×D412yes—
499#95C2×D412yes—
500#95C2×D412yes—
501C2×Q8—21no—
502C2×Q8—21no—
503C2×Q8—21no—
504C2×Q8—21no—
505C2×Q8—21no—
506C2×Q8—21no—
507C2×Q8—22no—
508C2×Q8—22no—
509C2×Q8—22no—
510C2×Q8—21no—
511C2×Q8—22no—
512C2×Q8—21no—
513C2×Q8—22no—
514C2×Q8—22no—
515C2×Q8—42no—
516C2×Q8—42no—
517C2×Q8—42no—
518C2×Q8C2×#4112yes—
519C2×Q8C2×#4112yes—
520C2×Q8C2×#4112yes—
521C2×Q8C2×#4112yes—
522C22×C4—21no—
523C22×C4—21no—
524C22×C4—21no—
525C22×C4—21no—
526C22×C4—21no—
527C22×C4—22no—
528C22×C4—21no—
529C22×C4—21no—
530C22×C4—22no—
531C22×C4—22no—
532C22×C4—21no—
533C22×C4—22no—
534C22×C4C22×D412yes—
535C22×C4C22×D411yescharacteristic
536C22×C4C22×D412yes—
537C22×C4C22×D411yescharacteristic
538C22×C4C22×D412yes—
539C22×C4C22×D411yescharacteristic
540C22×C4C22×D412yes—
541C22×C4C22×D411yescharacteristic
542C22×C4C22×D411yescharacteristic
543C22×C4C22×D411yescharacteristic
544C22×C4C2511yescharacteristic
545Q16—28no—
546Q16—48no—
547C2×C8—21no—
548C2×C8—21no—
549C2×C8—21no—
550C2×C8—21no—
551C2×C8—21no—
552C2×C8—21no—
553C2×C8—22no—
554C2×C8—22no—
555C2×C8—22no—
556C2×C8—22no—
557C2×C8—21no—
558C2×C8—21no—
559C2×C8—21no—
560C2×C8—22no—
561C2×C8—21no—
562C2×C8—22no—
563C2×C8C2×#4112yes—
564C2×C8C2×#4112yes—
565C4⋊C4—24no—
566C4⋊C4—24no—
567C4⋊C4—24no—
568C4⋊C4—22no—
569C4⋊C4—24no—
570C4⋊C4—24no—
571C4⋊C4—22no—
572C4⋊C4—22no—
573C4⋊C4—21no—
574C4⋊C4—21no—
575C4⋊C4—22no—
576C4⋊C4—22no—
577C4⋊C4—21no—
578C4⋊C4—21no—
579C4⋊C4—21no—
580C4⋊C4—21no—
581C4⋊C4—22no—
582C4⋊C4—21no—
583C4⋊C4—22no—
584C4⋊C4—21no—
585C4⋊C4—21no—
586C4⋊C4—22no—
587C4⋊C4—22no—
588C4⋊C4—21no—
589C4⋊C4—42no—
590C4⋊C4—42no—
591C4⋊C4—44no—
592C4⋊C4—44no—
593C4⋊C4—42no—
594C4⋊C4—44no—
595C4⋊C4—42no—
596C4⋊C4—44no—
597C4⋊C4—44no—
598C4⋊C4—42no—
599C4⋊C4—42no—
600C4⋊C4—42no—
601C4⋊C4—42no—
602C4⋊C4—44no—
603C4⋊C4—44no—
604C4⋊C4—44no—
605C4⋊C4—42no—
606C4⋊C4—44no—
607C4⋊C4—42no—
608C4⋊C4—81no—
609C4⋊C4—81no—
610C4⋊C4—82no—
611C4⋊C4—82no—
612C4⋊C4—82no—
613C4⋊C4—82no—
614C4⋊C4—82no—
615C4⋊C4#13514yes—
616C4⋊C4C2×#4112yes—
617C4⋊C4C2×#4112yes—
618C42—24no—
619C42—22no—
620C42—22no—
621C42—22no—
622C42—22no—
623C42—21no—
624C42—21no—
625C42—22no—
626C42—22no—
627C42—21no—
628C42—21no—
629C42—22no—
630C42—22no—
631C42—42no—
632C42—42no—
633C42—42no—
634C42—42no—
635C42—44no—
636C42—44no—
637C42—44no—
638C42—42no—
639C42—42no—
640C42—44no—
641C42—42no—
642C42—42no—
643C42—42no—
644C42—42no—
645C42—42no—
646C42C2×#4112yes—
647C422−1+412yes—
648C23C23×D411yescharacteristic
649Q8—22no—
650Q8—22no—
651Q8—22no—
652Q8—22no—
653Q8—22no—
654Q8—41no—
655Q8—41no—
656Q8—42no—
657Q8—42no—
658Q8—42no—
659Q8—42no—
660Q8—84no—
661Q8#52914yes—
662C2×C4—22no—
663C2×C4—21no—
664C2×C4—21no—
665C2×C4—22no—
666C2×C4—22no—
667C2×C4—22no—
668C2×C4—21no—
669C2×C4—21no—
670C2×C4—21no—
671C2×C4—21no—
672C2×C4—22no—
673C2×C4—22no—
674C2×C4—22no—
675C2×C4—22no—
676C2×C4—21no—
677C2×C4—21no—
678C2×C4—22no—
679C2×C4—22no—
680C2×C4—22no—
681C2×C4—21no—
682C2×C4—21no—
683C2×C4—22no—
684C2×C4—21no—
685C2×C4—22no—
686C2×C4—22no—
687C2×C4—21no—
688C2×C4—22no—
689C2×C4—22no—
690C2×C4—22no—
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—42no—
700C2×C4—42no—
701C2×C4—41no—
702C2×C4—41no—
703C2×C4—41no—
704C2×C4—42no—
705C2×C4—42no—
706C2×C4—42no—
707C2×C4—41no—
708C2×C4—41no—
709C2×C4—41no—
710C2×C4—42no—
711C2×C4—42no—
712C2×C4—41no—
713C2×C4—41no—
714C2×C4—42no—
715C2×C4—42no—
716C2×C4—41no—
717C2×C4—41no—
718C2×C4—41no—
719C2×C4—42no—
720C2×C4—42no—
721C2×C4#54812yes—
722C2×C4#54812yes—
723C2×C4C2×#13514yes—
724C2×C4C2×#13512yes—
725C2×C4#57712yes—
726C2×C4#57812yes—
727C2×C4#57812yes—
728C2×C4C22×#4111yescharacteristic
729C2×C4C22×#4111yescharacteristic
730C2×C4C22×#4111yescharacteristic
731C2×C4C2×2−1+411yescharacteristic
732C2×C4#58511yescharacteristic
733C2×C4#58511yescharacteristic
734C8—22no—
735C8—22no—
736C8—42no—
737C8—41no—
738C8—42no—
739C8—41no—
740C22C2×#54811yescharacteristic
741C22D4×#4111yescharacteristic
742C22D4×#4111yescharacteristic
743C22C22×#13511yescharacteristic
744C22C2×#57711yescharacteristic
745C22C2×#57811yescharacteristic
746C22C2×#57811yescharacteristic
747C4—24no—
748C4—22no—
749C4—22no—
750C4—22no—
751C4—24no—
752C4—42no—
753C4—42no—
754C4—42no—
755C4—42no—
756C4—41no—
757C4—41no—
758C4—42no—
759C4—42no—
760C4—82no—
761C4—82no—
762C4—82no—
763C4—81no—
764C4—81no—
765C4—81no—
766C4—81no—
767C4—82no—
768C4#336414yes—
769C4C2×#52911yescharacteristic
770C4#351211yescharacteristic
771C4#351511yescharacteristic
772C4#351511yescharacteristic
773C2#3611211yescharacteristic
774C2C2×#336411yescharacteristic
775C2#6133011yescharacteristic
776C2#6143311yescharacteristic
777C2#6143411yescharacteristic
778C2#6148411yescharacteristic
779C2#6149911yescharacteristic
subgroupG/H
center, socle, prime radicalC23C23×D4
commutator, Frattini, cosocleC22×C4C25
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×#125 ◁ 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
#364472
#363132
#332782
#330732
#330332
#290132
#280332
#277492
#277302
#275512
#273712
#267572
#265272
#265062
#261672
#248182
#247322
#229982
#229642
#227692
C8×#1252
C42×Q162