Finite Group Databasebeta

← #6520634

Subgroups — #6520634

Order 512.

1308 conjugacy classes of subgroups.

The subgroup lattice has 713 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#37561C211yesmaximal, characteristic
2#35192C211yesmaximal, characteristic
3#29325C212yesmaximal, complemented
4#29322C212yesmaximal, complemented
5#29217C212yesmaximal, complemented
6#28840C212yesmaximal, complemented
7#28819C212yesmaximal, complemented
8#28806C212yesmaximal, complemented
9#28801C212yesmaximal, complemented
10#28787C212yesmaximal
11#28782C212yesmaximal
12#28696C212yesmaximal, complemented
13#26079C212yesmaximal
14#26072C212yesmaximal
15#21986C211yesmaximal, characteristic
16#21964C211yesmaximal, characteristic
17#21364C211yesmaximal, characteristic
18#21172C211yesmaximal, characteristic
19#20551C211yesmaximal, characteristic
20C2×#501C2212yes—
21C2×#498C2212yes—
22C2×#432C2211yescharacteristic
23C2×#428C2211yescharacteristic
24C2×#428C2211yescharacteristic
25C2×#423C2211yescharacteristic
26#2864—21no—
27#2862—21no—
28#2845—21no—
29#2844—21no—
30#2666—22no—
31#2665—22no—
32#2660C2212yes—
33#2659—21no—
34#2647C2212yes—
35#2646—21no—
36#2605—22no—
37#2571C2211yescharacteristic
38#2571C2211yescharacteristic
39#2570C2212yes—
40#2551C2211yescharacteristic
41#2551C2211yescharacteristic
42#2550C2212yes—
43#2442C2212yes—
44#2440C2212yes—
45#2439—21no—
46#2439—21no—
47#2439C2212yes—
48#2439C2212yes—
49#2438—22no—
50#2331C2212yes—
51#2331C2212yes—
52#2285C2211yescharacteristic
53#2285C2211yescharacteristic
54#2283C2211yescharacteristic
55C2×#382C2211yescharacteristic
56C4×#117C2211yescharacteristic
57#2085C2212yes—
58#2085C2212yes—
59#2085C2212yes—
60#2085C2212yes—
61#2085C2212yes—
62#2085C2212yes—
63#2084C2212yes—
64#2084C2212yes—
65#2059C2212yes—
66#2059C2212yes—
67#2058C2212yes—
68#2058C2212yes—
69#2057C2212yes—
70#2057C2212yes—
71#2057C2212yes—
72#2057C2212yes—
73#1993C2212yes—
74#1993C2212yes—
75#1991C2212yes—
76#1990C2212yes—
77#1949C2212yes—
78#1949C2212yes—
79#1949C2212yes—
80#1949C2212yes—
81#1949C2212yes—
82#1949C2212yes—
83#1948C2211yescharacteristic
84#1948C2212yes—
85#1948C2211yescharacteristic
86#1948C2212yes—
87#1947C2211yescharacteristic
88#1947C2211yescharacteristic
89#1947C2212yes—
90#1947C2212yes—
91#1944C2212yes—
92#1943C2212yes—
93#1943C2212yes—
94#1943C2212yes—
95#1942C2212yes—
96#1942C2212yes—
97#1942C2212yes—
98#1941C2212yes—
99#1854C2212yes—
100#1854C2212yes—
101#1853C2212yes—
102#1852C2212yes—
103#1847C2212yes—
104#1847C2212yes—
105#1846C2212yes—
106#1836C2212yes—
107#1809C2212yes—
108#1809C2212yes—
109#1776C2212yes—
110#1776C2212yes—
111#1771C2212yes—
112#1771C2212yes—
113#1768C2212yes—
114#1768C2212yes—
115#1753C2212yes—
116#1753C2212yes—
117#1736C2211yescharacteristic
118#1736C2211yescharacteristic
119C22×M16C2311yescharacteristic
120C22×M16C2311yescharacteristic
121C2×C4⋊Q8—21no—
122C2×C4⋊Q8—21no—
123C2×C4⋊Q8C2312yes—
124C2×C4⋊Q8C2312yes—
125C2×C4⋊Q8C2312yes—
126C2×#125C2312yes—
127C2×C22⋊Q8—22no—
128C2×#117—21no—
129C2×#117—21no—
130C2×#117C2311yescharacteristic
131C2×#117C2311yescharacteristic
132C22×C4⋊C4C2311yescharacteristic
133C22×C4⋊C4C2311yescharacteristic
134C22×C42C2311yescharacteristic
135#501—44no—
136#498—44no—
137#432Q818yescomplemented
138#428—42no—
139#428—42no—
140C2×#107C2312yes—
141C2×#107C2312yes—
142C2×#107C2312yes—
143C2×#107C2312yes—
144C2×#107C2312yes—
145C2×#107C2312yes—
146C2×#107C2312yes—
147C2×#107C2312yes—
148C2×#106C2312yes—
149C2×#106C2312yes—
150C2×#106C2312yes—
151C2×#106C2312yes—
152C2×#106C2312yes—
153C2×#106C2312yes—
154C2×#106C2312yes—
155C2×#106C2312yes—
156#423Q818yescomplemented
157C2×C4⋊C8C2312yes—
158C2×C4⋊C8C2312yes—
159C2×C4⋊C8C2311yescharacteristic
160C2×C4⋊C8C2311yescharacteristic
161C2×C22⋊C8C2312yes—
162C2×#97C2312yes—
163C2×#97C2312yes—
164C2×#97C2312yes—
165C2×C4×C8C2312yes—
166C2×C4×C8C2312yes—
167#400—22no—
168#400—22no—
169#400—22no—
170#400—22no—
171#400—22no—
172#400—22no—
173#398—22no—
174#398—22no—
175#398—22no—
176#398—22no—
177#398—22no—
178#398—22no—
179#398—22no—
180#398—22no—
181#396—22no—
182#396—22no—
183#395—22no—
184#395—22no—
185#395—22no—
186#395—22no—
187#395—22no—
188#395—22no—
189#393—22no—
190#393—22no—
191#391—22no—
192#391—22no—
193#391—22no—
194#391—22no—
195#391—22no—
196#391—22no—
197#391—22no—
198#389—21no—
199#389—21no—
200#389—21no—
201#389—21no—
202#389—22no—
203#389C2312yes—
204#389C2312yes—
205#389C2312yes—
206#389C2312yes—
207#389C2312yes—
208#389C2312yes—
209#389C2312yes—
210#389C2312yes—
211#387—22no—
212#387—22no—
213#387—22no—
214#387—22no—
215#387—22no—
216#387—22no—
217#387—22no—
218#387—22no—
219#385—22no—
220#385—22no—
221#384C2312yes—
222#384C2311yescharacteristic
223#384C2311yescharacteristic
224#384C2311yescharacteristic
225#384C2311yescharacteristic
226#383D412yes—
227#382—21no—
228#382—21no—
229#382—21no—
230#382—21no—
231#382—21no—
232#382—21no—
233#382D411yescharacteristic
234#382D411yescharacteristic
235#382C2312yes—
236#382C2312yes—
237#382C2312yes—
238#382C2312yes—
239#382C2312yes—
240#381D412yes—
241#380—21no—
242#380—21no—
243#380—22no—
244#380D411yescharacteristic
245#380D411yescharacteristic
246#380C2312yes—
247#380C2312yes—
248C4×C4⋊C4—21no—
249C4×C4⋊C4—21no—
250C4×C4⋊C4C2312yes—
251C4×C4⋊C4C2311yescharacteristic
252C4×C4⋊C4C2311yescharacteristic
253C4×C4⋊C4C2312yes—
254C4×C4⋊C4C2311yescharacteristic
255C4×C4⋊C4C2311yescharacteristic
256C4×C4⋊C4C2311yescharacteristic
257C4×C4⋊C4C2311yescharacteristic
258C4×C22⋊C4—21no—
259C4×C22⋊C4—21no—
260C4×C22⋊C4C2312yes—
261#376—21no—
262#376C2312yes—
263C2×#95—21no—
264C2×#95—21no—
265C43—21no—
266#340C2312yes—
267#340C2312yes—
268#340C2312yes—
269#340C2312yes—
270#340C2312yes—
271#340C2312yes—
272#340C2312yes—
273#340C2312yes—
274#340C2312yes—
275#340C2312yes—
276#340C2312yes—
277#340C2312yes—
278#340C2312yes—
279#340C2312yes—
280#340C2312yes—
281#340C2312yes—
282#336C2312yes—
283#336C2312yes—
284#336C2312yes—
285#336C2312yes—
286C22×Q8—22no—
287C22×Q8—22no—
288C22×Q8—22no—
289C23×C4—21no—
290C23×C4—21no—
291C23×C4C2×D411yescharacteristic
292C23×C4C2×D411yescharacteristic
293C23×C4C2411yescharacteristic
294C2×M16—41no—
295C2×M16—41no—
296C2×M16—41no—
297C2×M16—41no—
298C2×M16C2×Q814yes—
299C2×M16C2×Q814yes—
300C22×C8C2412yes—
301C22×C8C2412yes—
302C22×C8C2412yes—
303C22×C8C2412yes—
304C4⋊Q8—24no—
305C4⋊Q8—24no—
306C4⋊Q8—42no—
307C4⋊Q8—42no—
308C4⋊Q8—42no—
309C4⋊Q8—42no—
310C4⋊Q8—82no—
311C4⋊Q8—82no—
312#125—42no—
313#125—42no—
314C22⋊Q8—84no—
315#117—22no—
316#117—22no—
317#117—41no—
318#117—41no—
319#117—82no—
320#117—82no—
321C2×C4⋊C4—21no—
322C2×C4⋊C4—21no—
323C2×C4⋊C4—22no—
324C2×C4⋊C4—22no—
325C2×C4⋊C4—21no—
326C2×C4⋊C4—21no—
327C2×C4⋊C4—21no—
328C2×C4⋊C4—21no—
329C2×C4⋊C4—22no—
330C2×C4⋊C4—22no—
331C2×C4⋊C4—22no—
332C2×C4⋊C4—22no—
333C2×C4⋊C4—22no—
334C2×C4⋊C4—22no—
335C2×C4⋊C4—22no—
336C2×C4⋊C4—22no—
337C2×C4⋊C4—21no—
338C2×C4⋊C4—21no—
339C2×C4⋊C4—21no—
340C2×C4⋊C4—21no—
341C2×C4⋊C4—22no—
342C2×C4⋊C4—22no—
343C2×C4⋊C4—22no—
344C2×C4⋊C4—22no—
345C2×C4⋊C4—22no—
346C2×C4⋊C4—41no—
347C2×C4⋊C4—41no—
348C2×C4⋊C4—41no—
349C2×C4⋊C4—41no—
350C2×C4⋊C4C2×D411yescharacteristic
351C2×C4⋊C4C2×D411yescharacteristic
352C2×C4⋊C4C2×D411yescharacteristic
353C2×C4⋊C4C2×D411yescharacteristic
354C2×C4⋊C4C2412yes—
355C2×C4⋊C4C2412yes—
356C2×C4⋊C4C2411yescharacteristic
357C2×C4⋊C4C2411yescharacteristic
358C2×C4⋊C4C2412yes—
359C2×C4⋊C4C2412yes—
360C2×C4⋊C4C2412yes—
361C2×C4⋊C4C2412yes—
362C2×C4⋊C4C2411yescharacteristic
363C2×C4⋊C4C2411yescharacteristic
364C2×C22⋊C4—22no—
365C2×C22⋊C4—21no—
366C2×C22⋊C4—21no—
367C2×C22⋊C4—22no—
368C2×C22⋊C4C2×D412yes—
369C2×C22⋊C4C2×D412yes—
370C2×C42—21no—
371C2×C42—21no—
372C2×C42—21no—
373C2×C42—21no—
374C2×C42—22no—
375C2×C42—22no—
376C2×C42—21no—
377C2×C42—21no—
378C2×C42C2×D411yescharacteristic
379C2×C42C2×D411yescharacteristic
380C2×C42C2×Q814yes—
381C2×C42C2×Q814yes—
382C2×C42C2412yes—
383C2×C42C2412yes—
384C2×C42C2411yescharacteristic
385C2×C42C2411yescharacteristic
386#107—24no—
387#107—24no—
388#107—42no—
389#107—42no—
390#107—42no—
391#107—42no—
392#107—42no—
393#107—42no—
394#106—24no—
395#106—24no—
396#106—42no—
397#106—42no—
398#106—42no—
399#106—42no—
400#106—42no—
401#106—42no—
402C4⋊C8C2×Q818yes—
403C4⋊C8C2×Q818yes—
404C4⋊C8C2×Q814yes—
405C4⋊C8C2×Q814yes—
406C22⋊C8C2×Q818yes—
407#97—42no—
408#97—42no—
409#97C2×Q818yes—
410C4×C8—42no—
411C4×C8—42no—
412#95—22no—
413#95—22no—
414#95—22no—
415#95—22no—
416#95—21no—
417#95—21no—
418#95—21no—
419#95—21no—
420#95—22no—
421#95—22no—
422#95—22no—
423#95—22no—
424#95—22no—
425#95—22no—
426#95—22no—
427#95—22no—
428#95—22no—
429#95—22no—
430#95—22no—
431#95—22no—
432#95—22no—
433#95—22no—
434#95—22no—
435#95—22no—
436#95—22no—
437#95—22no—
438#95C2×D412yes—
439#95C2×D412yes—
440#95C2×D412yes—
441#95C2×D412yes—
442#95C2×D412yes—
443#95C2×D412yes—
444#95C2×D412yes—
445#95C2×D412yes—
446C24C22×D411yescharacteristic
447C2×Q8—44no—
448C2×Q8—44no—
449C2×Q8—82no—
450C2×Q8—82no—
451C2×Q8—82no—
452C2×Q8—82no—
453C2×Q8—82no—
454C2×Q8—82no—
455C2×Q8—82no—
456C22×C4—21no—
457C22×C4—21no—
458C22×C4—21no—
459C22×C4—21no—
460C22×C4—21no—
461C22×C4—21no—
462C22×C4—21no—
463C22×C4—21no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—41no—
471C22×C4—41no—
472C22×C4—41no—
473C22×C4—41no—
474C22×C4—42no—
475C22×C4—42no—
476C22×C4—81no—
477C22×C4—81no—
478C22×C4—81no—
479C22×C4—81no—
480C22×C4C4⋊Q814yes—
481C22×C4C4⋊Q814yes—
482C22×C4C22×D412yes—
483C22×C4C22×D412yes—
484C22×C4C22×D412yes—
485C22×C4C22×D412yes—
486C22×C4C22×D411yescharacteristic
487C22×C4C22×D411yescharacteristic
488C22×C4C22×D411yescharacteristic
489C22×C4C22×D411yescharacteristic
490C22×C4C22×D411yescharacteristic
491C22×C4C22×Q812yes—
492C22×C4C22×Q812yes—
493C22×C4C2511yescharacteristic
494M16—44no—
495M16—44no—
496C2×C8—22no—
497C2×C8—22no—
498C2×C8—22no—
499C2×C8—22no—
500C2×C8—22no—
501C2×C8—22no—
502C2×C8—22no—
503C2×C8—22no—
504C2×C8C22×Q814yes—
505C2×C8C22×Q814yes—
506C2×C8C22×Q814yes—
507C2×C8C22×Q814yes—
508C4⋊C4—21no—
509C4⋊C4—21no—
510C4⋊C4—22no—
511C4⋊C4—22no—
512C4⋊C4—22no—
513C4⋊C4—21no—
514C4⋊C4—22no—
515C4⋊C4—22no—
516C4⋊C4—21no—
517C4⋊C4—22no—
518C4⋊C4—21no—
519C4⋊C4—22no—
520C4⋊C4—22no—
521C4⋊C4—22no—
522C4⋊C4—21no—
523C4⋊C4—22no—
524C4⋊C4—22no—
525C4⋊C4—22no—
526C4⋊C4—41no—
527C4⋊C4—41no—
528C4⋊C4—41no—
529C4⋊C4—41no—
530C4⋊C4—44no—
531C4⋊C4—44no—
532C4⋊C4—42no—
533C4⋊C4—42no—
534C4⋊C4—81no—
535C4⋊C4—81no—
536C4⋊C4—81no—
537C4⋊C4—81no—
538C4⋊C4—82no—
539C4⋊C4—82no—
540C4⋊C4—82no—
541C4⋊C4—82no—
542C4⋊C4—82no—
543C4⋊C4—81no—
544C4⋊C4—81no—
545C4⋊C4—81no—
546C4⋊C4—81no—
547C4⋊C4—82no—
548C4⋊C4—82no—
549C4⋊C4—82no—
550C4⋊C4—82no—
551C4⋊C4—82no—
552C4⋊C4—82no—
553C4⋊C4—82no—
554C4⋊C4—82no—
555C4⋊C4H(5,𝔽2)12yes—
556C4⋊C42−1+412yes—
557C22⋊C4—42no—
558C22⋊C4—42no—
559C22⋊C4—42no—
560C22⋊C4—42no—
561C22⋊C4—44no—
562C22⋊C4—82no—
563C42—22no—
564C42—22no—
565C42—22no—
566C42—22no—
567C42—42no—
568C42—42no—
569C42—42no—
570C42—42no—
571C42—81no—
572C42—81no—
573C42—82no—
574C42—82no—
575C42C4⋊Q814yes—
576C42C4⋊Q814yes—
577C42C22×Q812yes—
578C42C22×Q812yes—
579C42C22×Q812yes—
580C42C22×Q812yes—
581C23—21no—
582C23—21no—
583C23—21no—
584C23—21no—
585C23—21no—
586C23—21no—
587C23C2×C4⋊Q812yes—
588C23C23×D411yescharacteristic
589Q8—164no—
590Q8—164no—
591Q8—164no—
592C2×C4—21no—
593C2×C4—21no—
594C2×C4—21no—
595C2×C4—21no—
596C2×C4—21no—
597C2×C4—21no—
598C2×C4—21no—
599C2×C4—21no—
600C2×C4—21no—
601C2×C4—21no—
602C2×C4—22no—
603C2×C4—22no—
604C2×C4—22no—
605C2×C4—22no—
606C2×C4—22no—
607C2×C4—22no—
608C2×C4—22no—
609C2×C4—22no—
610C2×C4—22no—
611C2×C4—22no—
612C2×C4—22no—
613C2×C4—22no—
614C2×C4—22no—
615C2×C4—22no—
616C2×C4—22no—
617C2×C4—22no—
618C2×C4—22no—
619C2×C4—22no—
620C2×C4—44no—
621C2×C4—44no—
622C2×C4—41no—
623C2×C4—41no—
624C2×C4—41no—
625C2×C4—41no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—41no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—41no—
636C2×C4—41no—
637C2×C4—41no—
638C2×C4—42no—
639C2×C4—42no—
640C2×C4—42no—
641C2×C4—42no—
642C2×C4—41no—
643C2×C4—42no—
644C2×C4—42no—
645C2×C4—42no—
646C2×C4—42no—
647C2×C4—41no—
648C2×C4—42no—
649C2×C4—42no—
650C2×C4—162no—
651C2×C4—162no—
652C2×C4C2×C4⋊Q812yes—
653C2×C4C2×C4⋊Q812yes—
654C2×C4C2×C4⋊Q812yes—
655C2×C4C2×C4⋊Q812yes—
656C2×C4C2×C4⋊Q812yes—
657C2×C4C23×Q811yescharacteristic
658C2×C4C23×Q811yescharacteristic
659C2×C4C2×H(5,𝔽2)11yescharacteristic
660C2×C4C2×2−1+411yescharacteristic
661C2×C4#58511yescharacteristic
662C2×C4#58511yescharacteristic
663C8—24no—
664C8—24no—
665C8—24no—
666C8—24no—
667C22—22no—
668C22—22no—
669C22—22no—
670C22—41no—
671C22—41no—
672C22—41no—
673C22—41no—
674C22C22×C4⋊Q811yescharacteristic
675C22#348811yescharacteristic
676C22#348911yescharacteristic
677C22#358511yescharacteristic
678C22#358511yescharacteristic
679C22#358611yescharacteristic
680C22#358611yescharacteristic
681C4—22no—
682C4—22no—
683C4—22no—
684C4—22no—
685C4—22no—
686C4—42no—
687C4—42no—
688C4—42no—
689C4—42no—
690C4—42no—
691C4—42no—
692C4—82no—
693C4—82no—
694C4—81no—
695C4—82no—
696C4—82no—
697C4—81no—
698C4—82no—
699C4—82no—
700C4#350111yescharacteristic
701C4#350211yescharacteristic
702C4#350311yescharacteristic
703C4#350411yescharacteristic
704C2—42no—
705C2#3556611yescharacteristic
706C2#6066411yescharacteristic
707C2#6066411yescharacteristic
708C2#6208711yescharacteristic
709C2#6209211yescharacteristic
710C2#6223511yescharacteristic
711C2#6224011yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ #2283 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #29325C2semidirect
4. #29322C2semidirect
5. #29217C2semidirect
6. #28840C2semidirect
7. #28819C2semidirect
8. #28806C2semidirect
9. #28801C2semidirect
12. #28696C2semidirect
137. #432Q8semidirect
156. #423Q8semidirect

Maximal subgroups

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

subgroupindex
#375612
#351922
#293252
#293222
#292172
#288402
#288192
#288062
#288012
#287872
#287822
#286962
#260792
#260722
#219862
#219642
#213642
#211722
#205512