Finite Group Databasebeta

← #6556243

Subgroups — #6556243

Order 512.

1736 conjugacy classes of subgroups.

The subgroup lattice has 699 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36289C211yesmaximal, characteristic
2#36194C211yesmaximal, characteristic, complemented
3#30203C211yesmaximal, characteristic, complemented
4#30200C212yesmaximal, complemented
5#29354C211yesmaximal, characteristic, complemented
6#29103C212yesmaximal, complemented
7#28726C211yesmaximal, characteristic, complemented
8#27976C212yesmaximal, complemented
9#27750C212yesmaximal, complemented
10#27728C212yesmaximal, complemented
11#27703C214yesmaximal, complemented
12#27281C214yesmaximal, complemented
13#23817C211yesmaximal, characteristic, complemented
14#23628C212yesmaximal, complemented
15#22502C212yesmaximal, complemented
16#22428C212yesmaximal
17#20567C211yesmaximal, characteristic, complemented
18C2×#554C2211yescharacteristic
19C2×D42C2211yescharacteristic
20C2×#489C2212yes—
21C2×#436C2212yes—
22C2×#418C2214yes—
23#2868C2211yescharacteristic, complemented
24#2858C2211yescharacteristic
25#2857C2212yescomplemented
26#2850C2212yes—
27#2571C2212yescomplemented
28#2563—21no—
29#2563C2211yescharacteristic
30#2548C2211yescharacteristic
31#2538—21no—
32#2504C2212yes—
33#2497—22no—
34#2492—22no—
35#2490—21no—
36#2490—21no—
37#2472—22no—
38#2472—22no—
39#2439C2214yescomplemented
40#2435C2212yes—
41#2369—21no—
42#2355—21no—
43#2354C2212yes—
44D4×C22⋊C4—21no—
45D4×C22⋊C4—21no—
46C4×C4⋊Q8C2211yescharacteristic, complemented
47C4×#124C2211yescharacteristic
48C2×#390C2212yes—
49#2285C2212yescomplemented
50C42×D4C2211yescharacteristic
51#2101C2212yescomplemented
52#2101C2214yescomplemented
53#2073C2214yes—
54#2073C2214yescomplemented
55#2072C2211yescharacteristic, complemented
56#2072C2212yescomplemented
57#2071C2211yescharacteristic
58#2071C2212yes—
59#2071C2212yescomplemented
60#2071C2212yescomplemented
61#2071C2212yescomplemented
62#2042C2212yes—
63#2042C2214yescomplemented
64#2024C2211yescharacteristic, complemented
65#2023C2211yescharacteristic
66#2023C2214yescomplemented
67#2023C2212yescomplemented
68#1993C2212yescomplemented
69#1990C2212yescomplemented
70#1987C2212yescomplemented
71#1982C2212yes—
72#1958C2214yescomplemented
73#1957C2212yes—
74#1936C2214yescomplemented
75#1934C2214yes—
76#1934C2214yescomplemented
77#1895C2214yes—
78#1894C2214yescomplemented
79#1875C2214yes—
80#1873C2214yescomplemented
81#1835C2214yescomplemented
82#1823C2214yescomplemented
83#1803C2212yescomplemented
84#1802C2212yes—
85#1786C2214yes—
86C8×C4⋊C4C2212yescomplemented
87#1767C2211yescharacteristic, complemented
88C4×#103C2212yescomplemented
89C4×#102C2212yes—
90#1753C2214yescomplemented
91C4×C22⋊C8C2212yescomplemented
92#1744C2211yescharacteristic, complemented
93#1736C2214yescomplemented
94#1728C2212yescomplemented
95C23×D4—21no—
96C23×D4C2312yes—
97C22×M16C2314yes—
98#554—28no—
99D42—28no—
100C2×C4⋊Q8—21no—
101C2×C4⋊Q8—21no—
102C2×C4⋊Q8C2311yescharacteristic
103C2×C4⋊Q8C2311yescharacteristic
104C2×C4⋊Q8C2312yes—
105C2×#127C2311yescharacteristic
106C2×#125—21no—
107C2×#124—21no—
108C2×#124—21no—
109C2×#124C2311yescharacteristic
110C2×#124C2312yes—
111C2×C22⋊Q8—22no—
112C2×C22⋊Q8—22no—
113C2×C22⋊Q8C2312yes—
114C2×#121—21no—
115C2×#121C2312yes—
116C2×#120—22no—
117C2×C4×Q8—21no—
118C2×C4×Q8C2311yescharacteristic
119C2×C4×D4—21no—
120C2×C4×D4D411yescharacteristic
121C2×C4×D4D411yescharacteristic
122C2×C4×D4C2311yescharacteristic
123C2×C4×D4C2311yescharacteristic
124C2×#117C2314yes—
125C22×C4⋊C4C2312yes—
126C22×C22⋊C4—22no—
127C22×C22⋊C4—22no—
128C22×C42C2312yes—
129#489—44no—
130#436—28no—
131C2×C4⋊C8C2312yes—
132#418—48no—
133C2×#103C2314yes—
134C2×#103C2312yes—
135C2×#103C2312yes—
136C2×#102C2314yes—
137C2×#102C2312yes—
138C2×#102C2314yes—
139C2×#102C2314yes—
140C2×#102C2314yes—
141C2×#102C2314yes—
142C2×#102C2312yes—
143C2×C22⋊C8C2314yes—
144C2×#97C2312yes—
145C2×C4×C8C2312yes—
146C2×C4×C8C2314yes—
147#400—24no—
148#397—22no—
149#397—22no—
150#396—24no—
151#394—22no—
152#394—22no—
153#394—22no—
154#393—24no—
155#393—22no—
156#391—21no—
157#391—22no—
158#391—24no—
159#391C2311yescharacteristic
160#390—22no—
161#390—21no—
162#390—21no—
163#390—21no—
164#390—21no—
165#390—22no—
166#390—22no—
167#390C2312yes—
168#390C2311yescharacteristic
169#390C2312yes—
170#390C2312yes—
171#390C2312yes—
172#389—22no—
173#389—21no—
174#389—22no—
175#389C2312yes—
176#389C2314yes—
177#388—24no—
178#388—22no—
179#388—22no—
180#388—21no—
181#388—21no—
182#387—24no—
183#386—22no—
184#386—22no—
185#386—22no—
186#386—22no—
187#384—21no—
188#384—21no—
189#384C2311yescharacteristic
190#384C2312yes—
191#384C2312yes—
192#382—22no—
193#382C2312yes—
194#382C2314yes—
195#382C2311yescharacteristic
196#380—22no—
197#380—22no—
198#380—22no—
199#380—21no—
200#380—21no—
201#380C2314yes—
202#379—22no—
203#379—22no—
204C4×C4⋊C4—22no—
205C4×C4⋊C4—21no—
206C4×C4⋊C4—22no—
207C4×C4⋊C4—22no—
208C4×C4⋊C4D411yescharacteristic
209C4×C4⋊C4D411yescharacteristic
210C4×C4⋊C4C2312yes—
211C4×C4⋊C4C2311yescharacteristic
212C4×C22⋊C4—21no—
213C4×C22⋊C4—21no—
214C4×C22⋊C4D412yes—
215C4×C22⋊C4D412yes—
216C4×C22⋊C4C2312yes—
217C43C2311yescharacteristic
218#340C2314yes—
219#340C2314yes—
220#340C2312yes—
221#340C2314yes—
222#340C2314yes—
223#340C2314yes—
224#340C2312yes—
225#340C2312yes—
226#340C2314yes—
227#340C2312yes—
228#336C2311yescharacteristic
229#336C2312yes—
230#336C2311yescharacteristic
231#336C2314yes—
232#336C2312yes—
233C25—22no—
234C22×Q8—21no—
235C22×Q8—21no—
236C22×Q8—22no—
237C22×Q8C2411yescharacteristic
238C22×D4—24no—
239C22×D4—22no—
240C22×D4—21no—
241C22×D4—22no—
242C22×D4—21no—
243C22×D4—28no—
244C22×D4—44no—
245C22×D4—42no—
246C22×D4—44no—
247C22×D4—44no—
248C22×D4—44no—
249C22×D4C2×D411yescharacteristic
250C22×D4C2412yes—
251C22×D4C2411yescharacteristic
252C23×C4—22no—
253C23×C4—21no—
254C23×C4C2×D412yes—
255C23×C4C2×D412yes—
256C23×C4C2412yes—
257C2×M16—28no—
258C2×M16—44no—
259C2×M16—44no—
260C22×C8C2412yes—
261C22×C8C2414yes—
262C22×C8C2412yes—
263C4⋊Q8—24no—
264C4⋊Q8—42no—
265C4⋊Q8—42no—
266C4⋊Q8—42no—
267C4⋊Q8—44no—
268C4⋊Q8—82no—
269#127—24no—
270#125—44no—
271#124—22no—
272#124—41no—
273#124—44no—
274#124—82no—
275#124—82no—
276C22⋊Q8—28no—
277C22⋊Q8—48no—
278C22⋊Q8—84no—
279#121—28no—
280#121—44no—
281#120—48no—
282C4×Q8—24no—
283C4×Q8—44no—
284C4×D4—24no—
285C4×D4—42no—
286C4×D4—42no—
287C4×D4—44no—
288C4×D4#4118yes—
289#117—44no—
290#117—44no—
291C2×C4⋊C4—28no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—24no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4C2×D411yescharacteristic
307C2×C4⋊C4C2×D411yescharacteristic
308C2×C4⋊C4C2×D411yescharacteristic
309C2×C4⋊C4C2411yescharacteristic
310C2×C4⋊C4C2412yes—
311C2×C4⋊C4C2412yes—
312C2×C4⋊C4C2414yes—
313C2×C4⋊C4C2411yescharacteristic
314C2×C4⋊C4C2412yes—
315C2×C22⋊C4—22no—
316C2×C22⋊C4—21no—
317C2×C22⋊C4—21no—
318C2×C22⋊C4—22no—
319C2×C22⋊C4—22no—
320C2×C22⋊C4—21no—
321C2×C22⋊C4—22no—
322C2×C22⋊C4—21no—
323C2×C22⋊C4—21no—
324C2×C22⋊C4—24no—
325C2×C22⋊C4—24no—
326C2×C22⋊C4—22no—
327C2×C22⋊C4—22no—
328C2×C22⋊C4—22no—
329C2×C22⋊C4—22no—
330C2×C22⋊C4—22no—
331C2×C22⋊C4—22no—
332C2×C22⋊C4—22no—
333C2×C22⋊C4—22no—
334C2×C22⋊C4—84no—
335C2×C22⋊C4—84no—
336C2×C22⋊C4C2×D412yes—
337C2×C22⋊C4C2×D412yes—
338C2×C22⋊C4C2×D412yes—
339C2×C42—21no—
340C2×C42—22no—
341C2×C42—24no—
342C2×C42—44no—
343C2×C42C2×D411yescharacteristic
344C2×C42C2×D411yescharacteristic
345C2×C42C2×D411yescharacteristic
346C2×C42C2×D411yescharacteristic
347C2×C42C2×D412yes—
348C2×C42C2×D412yes—
349C2×C42C2411yescharacteristic
350C2×C42C2412yes—
351C2×C42C2411yescharacteristic
352C2×C42C2411yescharacteristic
353C4⋊C8—24no—
354#103—42no—
355#103—42no—
356#103—44no—
357#102—24no—
358#102—28no—
359#102—44no—
360#102—42no—
361#102—44no—
362#102—44no—
363#102—44no—
364C22⋊C8—28no—
365#97—42no—
366C4×C8—24no—
367C4×C8—44no—
368#95—21no—
369#95—21no—
370#95—21no—
371#95—21no—
372#95—24no—
373#95—24no—
374#95—24no—
375#95—22no—
376#95—22no—
377#95—24no—
378#95—22no—
379#95—22no—
380#95C2×D412yes—
381#95C2×D412yes—
382#95C2×D412yes—
383C24—21no—
384C24—22no—
385C24—44no—
386C24—44no—
387C24—44no—
388C24—44no—
389C24—82no—
390C24—82no—
391C24—82no—
392C24C22×D412yes—
393C2×Q8—22no—
394C2×Q8—21no—
395C2×Q8—21no—
396C2×Q8—22no—
397C2×Q8—42no—
398C2×Q8—42no—
399C2×Q8—44no—
400C2×Q8—42no—
401C2×Q8—42no—
402C2×Q8—81no—
403C2×Q8—81no—
404C2×Q8—82no—
405C2×Q8—82no—
406C2×D4—22no—
407C2×D4—22no—
408C2×D4—22no—
409C2×D4—24no—
410C2×D4—22no—
411C2×D4—22no—
412C2×D4—21no—
413C2×D4—22no—
414C2×D4—24no—
415C2×D4—416no—
416C2×D4—44no—
417C2×D4—44no—
418C2×D4—44no—
419C2×D4—44no—
420C2×D4—44no—
421C2×D4—42no—
422C2×D4—42no—
423C2×D4—42no—
424C2×D4—42no—
425C2×D4—88no—
426C2×D4—82no—
427C2×D4—81no—
428C2×D4—81no—
429C2×D4—816no—
430C2×D4#12414yes—
431C2×D4H(5,𝔽2)12yes—
432C22×C4—22no—
433C22×C4—24no—
434C22×C4—22no—
435C22×C4—24no—
436C22×C4—24no—
437C22×C4—21no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—24no—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—22no—
444C22×C4—42no—
445C22×C4—42no—
446C22×C4—42no—
447C22×C4—42no—
448C22×C4—44no—
449C22×C4—42no—
450C22×C4—42no—
451C22×C4—42no—
452C22×C4—82no—
453C22×C4—82no—
454C22×C4C22×D412yes—
455C22×C4C22×D411yescharacteristic
456C22×C4C22×D412yes—
457C22×C4C22×D411yescharacteristic
458C22×C4C22×D412yes—
459C22×C4C22×D411yescharacteristic
460C22×C4C22×D411yescharacteristic
461C22×C4C22×D411yescharacteristic
462C22×C4C22×D411yescharacteristic
463C22×C4C2×#4118yes—
464C22×C4C2511yescharacteristic
465M16—88no—
466C2×C8—22no—
467C2×C8—22no—
468C2×C8—24no—
469C2×C8—24no—
470C2×C8—22no—
471C2×C8—22no—
472C2×C8—24no—
473C2×C8—22no—
474C2×C8—22no—
475C4⋊C4—22no—
476C4⋊C4—22no—
477C4⋊C4—22no—
478C4⋊C4—24no—
479C4⋊C4—22no—
480C4⋊C4—22no—
481C4⋊C4—24no—
482C4⋊C4—22no—
483C4⋊C4—22no—
484C4⋊C4—24no—
485C4⋊C4—22no—
486C4⋊C4—42no—
487C4⋊C4—44no—
488C4⋊C4—42no—
489C4⋊C4—44no—
490C4⋊C4—44no—
491C4⋊C4—42no—
492C4⋊C4—42no—
493C4⋊C4—44no—
494C4⋊C4—44no—
495C4⋊C4—44no—
496C4⋊C4—42no—
497C4⋊C4—84no—
498C4⋊C4—82no—
499C4⋊C4—82no—
500C4⋊C4#12414yes—
501C22⋊C4—42no—
502C22⋊C4—42no—
503C22⋊C4—42no—
504C22⋊C4—42no—
505C22⋊C4—44no—
506C22⋊C4—42no—
507C22⋊C4—44no—
508C22⋊C4—42no—
509C22⋊C4—44no—
510C22⋊C4—42no—
511C22⋊C4—44no—
512C22⋊C4—82no—
513C22⋊C4—82no—
514C22⋊C4—82no—
515C22⋊C4—82no—
516C22⋊C4—82no—
517C22⋊C4—82no—
518C22⋊C4—82no—
519C22⋊C4—82no—
520C22⋊C4—84no—
521C22⋊C4—84no—
522C22⋊C4#12418yes—
523C42—22no—
524C42—22no—
525C42—22no—
526C42—22no—
527C42—21no—
528C42—24no—
529C42—22no—
530C42—21no—
531C42—42no—
532C42—42no—
533C42—42no—
534C42—44no—
535C42—84no—
536C42C2×#4114yes—
537C23—22no—
538C23—22no—
539C23—22no—
540C23—22no—
541C23—22no—
542C23—22no—
543C23—22no—
544C23—41no—
545C23—41no—
546C23—41no—
547C23—41no—
548C23—41no—
549C23—41no—
550C23—42no—
551C23—42no—
552C23—42no—
553C23—42no—
554C23—42no—
555C23—42no—
556C23—42no—
557C23—88no—
558C23—88no—
559C23—88no—
560C23—88no—
561C23—164no—
562C23—164no—
563C23—164no—
564C23C2×#12414yes—
565C23C23×D411yescharacteristic
566Q8—42no—
567Q8—42no—
568Q8—84no—
569Q8—162no—
570Q8—164no—
571D4—28no—
572D4—24no—
573D4—44no—
574D4—42no—
575D4—44no—
576D4—88no—
577D4—88no—
578D4—84no—
579D4—162no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—28no—
589C2×C4—28no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—21no—
593C2×C4—21no—
594C2×C4—21no—
595C2×C4—21no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—21no—
599C2×C4—22no—
600C2×C4—21no—
601C2×C4—22no—
602C2×C4—41no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—42no—
606C2×C4—41no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—44no—
610C2×C4—41no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—44no—
614C2×C4—41no—
615C2×C4—41no—
616C2×C4—41no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—44no—
620C2×C4—41no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—84no—
624C2×C4—84no—
625C2×C4—84no—
626C2×C4—164no—
627C2×C4C2×#12412yes—
628C2×C4C2×#12414yes—
629C2×C4C2×#12412yes—
630C2×C4#57512yes—
631C2×C4#57512yes—
632C2×C4#57612yes—
633C2×C4#57712yes—
634C2×C4C22×#4112yes—
635C2×C4C2×H(5,𝔽2)11yescharacteristic
636C2×C4#58512yes—
637C2×C4#58511yescharacteristic
638C8—44no—
639C8—42no—
640C8—42no—
641C22—24no—
642C22—24no—
643C22—24no—
644C22—42no—
645C22—42no—
646C22—42no—
647C22—42no—
648C22—42no—
649C22—42no—
650C22—42no—
651C22—81no—
652C22—81no—
653C22—81no—
654C22—81no—
655C22—82no—
656C22—82no—
657C22—82no—
658C22—82no—
659C22—82no—
660C22—82no—
661C22—82no—
662C22—1616no—
663C22C22×#12411yescharacteristic
664C22#348511yescharacteristic
665C22#348811yescharacteristic
666C22C2×#57511yescharacteristic
667C22C2×#57511yescharacteristic
668C22C2×#57611yescharacteristic
669C22C2×#57711yescharacteristic
670C4—24no—
671C4—22no—
672C4—22no—
673C4—22no—
674C4—42no—
675C4—41no—
676C4—42no—
677C4—41no—
678C4—81no—
679C4—84no—
680C4—82no—
681C4—82no—
682C4—81no—
683C4—81no—
684C4—82no—
685C4#314014yes—
686C4#351312yes—
687C4#352112yes—
688C2—44no—
689C2—82no—
690C2—162no—
691C2#3584711yescharacteristic
692C2C2×#314011yescharacteristic
693C2#6062211yescharacteristic
694C2#6173911yescharacteristic
695C2#6174011yescharacteristic
696C2#6199511yescharacteristic
697C2#6199611yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×D42C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C42×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #36194C2semidirect
3. #30203C2semidirect
4. #30200C2semidirect
5. #29354C2semidirect
6. #29103C2semidirect
7. #28726C2semidirect
8. #27976C2semidirect
9. #27750C2semidirect
10. #27728C2semidirect
11. #27703C2semidirect
12. #27281C2semidirect
13. #23817C2semidirect
14. #23628C2semidirect
15. #22502C2semidirect
17. #20567C2semidirect
23. #2868C22semidirect
25. #2857C22semidirect
27. #2571C22semidirect
39. #2439C22semidirect
46. C4×C4⋊Q8C22semidirect
49. #2285C22semidirect
51. #2101C22semidirect
52. #2101C22semidirect
54. #2073C22semidirect
55. #2072C22semidirect
56. #2072C22semidirect
59. #2071C22semidirect
60. #2071C22semidirect
61. #2071C22semidirect
63. #2042C22semidirect
64. #2024C22semidirect
66. #2023C22semidirect
67. #2023C22semidirect
68. #1993C22semidirect
69. #1990C22semidirect
70. #1987C22semidirect
72. #1958C22semidirect
74. #1936C22semidirect
76. #1934C22semidirect
78. #1894C22semidirect
80. #1873C22semidirect
81. #1835C22semidirect
82. #1823C22semidirect
83. #1803C22semidirect
86. C8×C4⋊C4C22semidirect
87. #1767C22semidirect
88. C4×#103C22semidirect
90. #1753C22semidirect
91. C4×C22⋊C8C22semidirect
92. #1744C22semidirect
93. #1736C22semidirect
94. #1728C22semidirect

Maximal subgroups

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

subgroupindex
#362892
#361942
#302032
#302002
#293542
#291032
#287262
#279762
#277502
#277282
#277032
#272812
#238172
#236282
#225022
#224282
#205672