Finite Group Databasebeta

← #6852322

Subgroups — #6852322

Order 512.

1627 conjugacy classes of subgroups.

The subgroup lattice has 680 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36551C211yesmaximal, characteristic
2C2×#2567C211yesmaximal, characteristic, complemented
3#32957C211yesmaximal, characteristic, complemented
4#32358C214yesmaximal, complemented
5#31534C214yesmaximal, complemented
6#29957C212yesmaximal, complemented
7#28539C212yesmaximal, complemented
8#27953C212yesmaximal, complemented
9#27824C212yesmaximal, complemented
10#27677C212yesmaximal
11#25813C211yesmaximal, characteristic, complemented
12#25455C212yesmaximal, complemented
13#24815C212yesmaximal, complemented
14#24299C211yesmaximal, characteristic
15#22629C211yesmaximal, characteristic, complemented
16#22047C212yesmaximal
17C4×#448C211yesmaximal, characteristic
18C22×C4⋊Q8C2211yescharacteristic
19C22×C4×Q8C2211yescharacteristic
20C2×#500C2212yes—
21C2×#477C2212yes—
22C2×#476C2212yes—
23C2×#448C2211yescharacteristic
24C2×C4×SD16C2212yes—
25#2857C2211yescharacteristic
26#2851C2212yes—
27#2589—22no—
28#2567—22no—
29#2567—21no—
30#2567—21no—
31#2567C2214yes—
32#2567C2212yes—
33#2567C2212yes—
34#2563C2211yescharacteristic
35#2552C2211yescharacteristic
36#2509C2212yes—
37#2497—21no—
38#2487—21no—
39#2454C2212yes—
40#2450—22no—
41C2×#395—22no—
42#2369C2211yescharacteristic
43D4×C4⋊C4—21no—
44D4×C4⋊C4C2211yescharacteristic
45#2345—21no—
46C4×#121C2212yes—
47C4×#121C2211yescharacteristic
48C2×#391C2212yes—
49C2×#389—21no—
50C2×#389—21no—
51C2×#384C2212yes—
52C2×C4×C4⋊C4C2211yescharacteristic
53#2099C2214yes—
54#2097C2214yes—
55#2093C2214yes—
56#2084C2214yes—
57#2083C2214yes—
58#2068C2214yes—
59#2066C2214yes—
60#2058C2212yes—
61#2057C2214yes—
62#2026C2214yes—
63#2023C2212yes—
64#1992C2214yes—
65#1990C2212yes—
66#1987C2212yes—
67#1973C2214yes—
68#1947C2214yes—
69#1939C2212yes—
70#1935C2214yes—
71#1928C2214yes—
72#1917C2214yes—
73#1880C2212yes—
74#1877C2214yes—
75#1874C2211yescharacteristic
76#1874C2212yes—
77#1865C2212yes—
78#1863C2214yes—
79#1853C2212yes—
80#1853C2211yescharacteristic
81#1852C2212yes—
82#1852C2212yes—
83#1852C2212yes—
84#1847C2214yes—
85#1803C2212yes—
86#1802C2214yes—
87#1797C2212yes—
88#1797C2211yescharacteristic
89#1787C2211yescharacteristic
90#1787C2212yes—
91#1770C2212yes—
92C4×#103C2212yes—
93#1749C2212yes—
94C4×C22⋊C8C2211yescharacteristic
95C23×Q8—21no—
96C23×Q8C2311yescharacteristic
97C22×SD16C2312yes—
98C2×C4⋊Q8—21no—
99C2×C4⋊Q8—21no—
100C2×C4⋊Q8—44no—
101C2×C4⋊Q8C2311yescharacteristic
102C2×C4⋊Q8C2312yes—
103C2×C4⋊Q8C2312yes—
104C2×#125C2312yes—
105C2×C22⋊Q8—21no—
106C2×#121C2311yescharacteristic
107C2×C4×Q8—21no—
108C2×C4×Q8—21no—
109C2×C4×Q8—44no—
110C2×C4×Q8C2312yes—
111C2×C4×Q8C2312yes—
112C2×C4×D4—21no—
113C2×C4×D4—22no—
114C2×C4×D4C2311yescharacteristic
115C2×C4×D4C2312yes—
116C22×C4⋊C4—21no—
117C22×C4⋊C4—21no—
118C22×C4⋊C4—22no—
119C22×C4⋊C4—22no—
120C22×C4⋊C4—21no—
121C22×C4⋊C4D411yescharacteristic
122C22×C4⋊C4D411yescharacteristic
123C22×C4⋊C4C2311yescharacteristic
124C22×C4⋊C4C2312yes—
125C22×C42—21no—
126C22×C42C2311yescharacteristic
127C22×C42C2312yes—
128#500—44no—
129#477—44no—
130#476—44no—
131#448Q818yescomplemented
132C4×SD16—44no—
133C2×#107C2314yes—
134C2×#107C2314yes—
135C2×#107C2314yes—
136C2×#106C2314yes—
137C2×#106C2312yes—
138C2×C4⋊C8C2312yes—
139C2×C4⋊C8C2314yes—
140C2×#103C2314yes—
141C2×#103C2314yes—
142C2×#103C2312yes—
143C2×#103C2312yes—
144C2×#103C2312yes—
145C2×#102C2312yes—
146C2×#102C2314yes—
147C2×C22⋊C8C2311yescharacteristic
148C2×C22⋊C8C2312yes—
149C2×C22⋊C8C2311yescharacteristic
150C2×#97C2314yes—
151C2×C4×C8C2312yes—
152#400—22no—
153#400—22no—
154#399—22no—
155#398—22no—
156#396—22no—
157#395—24no—
158#395—24no—
159#395—24no—
160#395—24no—
161#393—22no—
162#391—22no—
163#391—22no—
164#391—21no—
165#391—22no—
166#391C2314yes—
167#391C2314yes—
168#390C2312yes—
169#390C2311yescharacteristic
170#389—22no—
171#389—22no—
172#389—22no—
173#389—24no—
174#389—24no—
175#389—22no—
176#389—22no—
177#389—21no—
178#389—21no—
179#389—21no—
180#389C2312yes—
181#389C2311yescharacteristic
182#389C2312yes—
183#389C2312yes—
184#389C2312yes—
185#388—21no—
186#388—22no—
187#387—22no—
188#387—22no—
189#387—22no—
190#386—21no—
191#386—22no—
192#385—22no—
193#385—22no—
194#385—22no—
195#385—22no—
196#384—21no—
197#384—22no—
198#384—22no—
199#384C2314yes—
200#384C2314yes—
201#384C2311yescharacteristic
202#382C2312yes—
203#380—21no—
204#380—21no—
205#380—21no—
206#380—21no—
207#380—22no—
208#380C2311yescharacteristic
209#380C2312yes—
210C4×C4⋊C4—21no—
211C4×C4⋊C4D412yes—
212C4×C4⋊C4D411yescharacteristic
213C4×C4⋊C4D411yescharacteristic
214C4×C4⋊C4C2312yes—
215C4×C4⋊C4C2311yescharacteristic
216C4×C4⋊C4C2311yescharacteristic
217C4×C4⋊C4C2311yescharacteristic
218C4×C4⋊C4C2312yes—
219C4×C22⋊C4—22no—
220C4×C22⋊C4—21no—
221C2×#95—22no—
222C2×#95—22no—
223C2×#95D412yes—
224#340C2314yes—
225#340C2314yes—
226#340C2314yes—
227#340C2314yes—
228#340C2314yes—
229#340C2312yes—
230#340C2312yes—
231#336C2312yes—
232#336C2314yes—
233#336C2312yes—
234#336C2314yes—
235C22×Q8—21no—
236C22×Q8—21no—
237C22×Q8—22no—
238C22×Q8—22no—
239C22×Q8—22no—
240C22×Q8—42no—
241C22×Q8—42no—
242C22×Q8—44no—
243C22×Q8—82no—
244C22×Q8—82no—
245C22×Q8C2×Q814yes—
246C22×Q8C2412yes—
247C22×D4—21no—
248C22×D4C2411yescharacteristic
249C23×C4—21no—
250C23×C4—21no—
251C23×C4—22no—
252C23×C4—21no—
253C23×C4—22no—
254C23×C4C2×D412yes—
255C23×C4C2×D411yescharacteristic
256C23×C4C2×D411yescharacteristic
257C23×C4C2×D412yes—
258C23×C4C2411yescharacteristic
259C2×SD16—42no—
260C2×SD16—42no—
261C2×SD16C2×Q818yescomplemented
262C22×C8C2412yes—
263C22×C8C2414yes—
264C22×C8C2412yes—
265C4⋊Q8—24no—
266C4⋊Q8—24no—
267C4⋊Q8—24no—
268C4⋊Q8—44no—
269C4⋊Q8—44no—
270C4⋊Q8—44no—
271#125—44no—
272C22⋊Q8—82no—
273#121—22no—
274#121C2×Q814yes—
275C4×Q8—24no—
276C4×Q8—24no—
277C4×Q8—42no—
278C4×Q8—42no—
279C4×Q8—44no—
280C4×Q8—44no—
281C4×D4—24no—
282C4×D4—44no—
283C4×D4—44no—
284C4×D4—84no—
285C2×C4⋊C4—24no—
286C2×C4⋊C4—24no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—24no—
298C2×C4⋊C4—24no—
299C2×C4⋊C4—24no—
300C2×C4⋊C4—24no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—42no—
308C2×C4⋊C4—44no—
309C2×C4⋊C4—44no—
310C2×C4⋊C4—44no—
311C2×C4⋊C4—44no—
312C2×C4⋊C4—84no—
313C2×C4⋊C4—84no—
314C2×C4⋊C4C2×D411yescharacteristic
315C2×C4⋊C4C2×D411yescharacteristic
316C2×C4⋊C4C2×D411yescharacteristic
317C2×C4⋊C4C2×D412yes—
318C2×C4⋊C4C2×D412yes—
319C2×C4⋊C4C2×D411yescharacteristic
320C2×C4⋊C4C2412yes—
321C2×C4⋊C4C2414yes—
322C2×C4⋊C4C2412yes—
323C2×C4⋊C4C2411yescharacteristic
324C2×C4⋊C4C2411yescharacteristic
325C2×C4⋊C4C2412yes—
326C2×C4⋊C4C2411yescharacteristic
327C2×C22⋊C4—21no—
328C2×C22⋊C4—22no—
329C2×C22⋊C4—21no—
330C2×C22⋊C4—21no—
331C2×C22⋊C4—21no—
332C2×C22⋊C4—22no—
333C2×C22⋊C4—22no—
334C2×C22⋊C4—22no—
335C2×C22⋊C4—21no—
336C2×C42—24no—
337C2×C42—21no—
338C2×C42—22no—
339C2×C42—22no—
340C2×C42—22no—
341C2×C42—44no—
342C2×C42—44no—
343C2×C42C2×D411yescharacteristic
344C2×C42C2×D411yescharacteristic
345C2×C42C2×D412yes—
346C2×C42C2×D412yes—
347C2×C42C2412yes—
348C2×C42C2411yescharacteristic
349C2×C42C2412yes—
350C2×C42C2411yescharacteristic
351#107—44no—
352#107—44no—
353#107—44no—
354#106—42no—
355#106—44no—
356C4⋊C8—24no—
357C4⋊C8—44no—
358#103—44no—
359#103—42no—
360#103—42no—
361#103—44no—
362#103C2×Q818yescomplemented
363#102—42no—
364#102—44no—
365C22⋊C8—22no—
366C22⋊C8—24no—
367C22⋊C8C2×Q814yes—
368#97—44no—
369C4×C8—24no—
370#95—24no—
371#95—24no—
372#95—24no—
373#95—24no—
374#95—21no—
375#95—22no—
376#95—21no—
377#95—22no—
378#95—22no—
379#95—22no—
380#95—21no—
381#95—22no—
382#95—22no—
383#95—22no—
384#95—22no—
385#95C2×D412yes—
386#95C2×D412yes—
387#95C2×D412yes—
388#95C2×D412yes—
389C24—21no—
390C24C22×D411yescharacteristic
391C2×Q8—24no—
392C2×Q8—22no—
393C2×Q8—22no—
394C2×Q8—22no—
395C2×Q8—22no—
396C2×Q8—22no—
397C2×Q8—48no—
398C2×Q8—42no—
399C2×Q8—42no—
400C2×Q8—48no—
401C2×Q8—44no—
402C2×Q8—44no—
403C2×Q8—42no—
404C2×Q8—82no—
405C2×Q8—82no—
406C2×Q8—82no—
407C2×Q8—82no—
408C2×Q8—81no—
409C2×Q8—81no—
410C2×Q8—816no—
411C2×Q8C22×Q814yes—
412C2×D4—24no—
413C2×D4—22no—
414C2×D4—21no—
415C2×D4—22no—
416C2×D4—42no—
417C2×D4—42no—
418C2×D4C22×Q812yes—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—21no—
429C22×C4—22no—
430C22×C4—24no—
431C22×C4—21no—
432C22×C4—22no—
433C22×C4—21no—
434C22×C4—22no—
435C22×C4—24no—
436C22×C4—22no—
437C22×C4—42no—
438C22×C4—42no—
439C22×C4—42no—
440C22×C4—41no—
441C22×C4—42no—
442C22×C4—42no—
443C22×C4—42no—
444C22×C4—44no—
445C22×C4—42no—
446C22×C4—44no—
447C22×C4—42no—
448C22×C4—44no—
449C22×C4—42no—
450C22×C4—42no—
451C22×C4—42no—
452C22×C4—44no—
453C22×C4—82no—
454C22×C4—82no—
455C22×C4#13714yes—
456C22×C4C22×D412yes—
457C22×C4C22×D412yes—
458C22×C4C22×D412yes—
459C22×C4C22×D411yescharacteristic
460C22×C4C22×D411yescharacteristic
461C22×C4C22×D411yescharacteristic
462C22×C4C22×D412yes—
463C22×C4C22×D411yescharacteristic
464C22×C4C22×D411yescharacteristic
465C22×C4C22×Q812yes—
466C22×C42−1+412yes—
467C22×C4C2511yescharacteristic
468SD16—48no—
469C2×C8—22no—
470C2×C8—22no—
471C2×C8—22no—
472C2×C8—24no—
473C2×C8—22no—
474C2×C8—24no—
475C2×C8—24no—
476C2×C8—22no—
477C2×C8C22×Q814yes—
478C4⋊C4—24no—
479C4⋊C4—24no—
480C4⋊C4—22no—
481C4⋊C4—22no—
482C4⋊C4—22no—
483C4⋊C4—21no—
484C4⋊C4—22no—
485C4⋊C4—24no—
486C4⋊C4—22no—
487C4⋊C4—24no—
488C4⋊C4—22no—
489C4⋊C4—24no—
490C4⋊C4—24no—
491C4⋊C4—22no—
492C4⋊C4—42no—
493C4⋊C4—44no—
494C4⋊C4—44no—
495C4⋊C4—44no—
496C4⋊C4—44no—
497C4⋊C4—44no—
498C4⋊C4—42no—
499C4⋊C4—44no—
500C4⋊C4—42no—
501C4⋊C4—42no—
502C4⋊C4—82no—
503C4⋊C4—84no—
504C4⋊C4—84no—
505C4⋊C4—84no—
506C4⋊C4—84no—
507C4⋊C4—82no—
508C4⋊C4—81no—
509C4⋊C4—82no—
510C4⋊C4—82no—
511C4⋊C4—82no—
512C4⋊C4#13714yes—
513C4⋊C4C22×Q812yes—
514C22⋊C4—24no—
515C22⋊C4—42no—
516C22⋊C4—44no—
517C22⋊C4—42no—
518C22⋊C4—42no—
519C22⋊C4—82no—
520C22⋊C4—81no—
521C22⋊C4—82no—
522C22⋊C4—82no—
523C42—22no—
524C42—22no—
525C42—22no—
526C42—24no—
527C42—24no—
528C42—42no—
529C42—42no—
530C42—44no—
531C42—44no—
532C42—42no—
533C42—82no—
534C42H(5,𝔽2)14yes—
535C23—21no—
536C23—21no—
537C23—21no—
538C23—21no—
539C23—22no—
540C23—41no—
541C23—41no—
542C23—41no—
543C23—41no—
544C23—41no—
545C23—41no—
546C23D4×Q812yes—
547C23C2×#13712yes—
548C23#57612yes—
549C23C23×D411yescharacteristic
550Q8—24no—
551Q8—24no—
552Q8—48no—
553Q8—42no—
554Q8—42no—
555Q8—88no—
556Q8—88no—
557Q8—162no—
558D4—24no—
559D4—48no—
560D4—42no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—22no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—22no—
568C2×C4—21no—
569C2×C4—28no—
570C2×C4—24no—
571C2×C4—24no—
572C2×C4—22no—
573C2×C4—21no—
574C2×C4—22no—
575C2×C4—21no—
576C2×C4—22no—
577C2×C4—24no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—21no—
586C2×C4—48no—
587C2×C4—44no—
588C2×C4—48no—
589C2×C4—48no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—44no—
598C2×C4—42no—
599C2×C4—41no—
600C2×C4—44no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—41no—
606C2×C4—42no—
607C2×C4—41no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—44no—
612C2×C4—84no—
613C2×C4—88no—
614C2×C4—84no—
615C2×C4—164no—
616C2×C4#53512yes—
617C2×C4#53512yes—
618C2×C4D4×Q812yes—
619C2×C4C2×#13712yes—
620C2×C4C2×#13714yes—
621C2×C4C2×#13712yes—
622C2×C4C2×#13712yes—
623C2×C4C23×Q811yescharacteristic
624C2×C4C2×H(5,𝔽2)12yes—
625C2×C4C2×2−1+411yescharacteristic
626C2×C4#58512yes—
627C8—24no—
628C8—42no—
629C8—44no—
630C22—22no—
631C22—22no—
632C22—22no—
633C22—22no—
634C22—22no—
635C22—22no—
636C22—42no—
637C22—42no—
638C22—42no—
639C22—81no—
640C22—81no—
641C22—81no—
642C22—81no—
643C22#338514yes—
644C22C2×#53511yescharacteristic
645C22C2×D4×Q811yescharacteristic
646C22#348511yescharacteristic
647C22C22×#13711yescharacteristic
648C22C22×#13711yescharacteristic
649C22C2×#57611yescharacteristic
650C22#358511yescharacteristic
651C4—24no—
652C4—22no—
653C4—22no—
654C4—22no—
655C4—42no—
656C4—42no—
657C4—42no—
658C4—41no—
659C4—44no—
660C4—42no—
661C4—81no—
662C4—82no—
663C4—81no—
664C4—82no—
665C4—84no—
666C4—82no—
667C4#321814yes—
668C4#350112yes—
669C4#354112yes—
670C2—24no—
671C2—82no—
672C2#3959411yescharacteristic
673C2C2×#321811yescharacteristic
674C2#6081111yescharacteristic
675C2C2×#338511yescharacteristic
676C2Q8×#13711yescharacteristic
677C2#6171911yescharacteristic
678C2#6172611yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#121C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2567C2semidirect
3. #32957C2semidirect
4. #32358C2semidirect
5. #31534C2semidirect
6. #29957C2semidirect
7. #28539C2semidirect
8. #27953C2semidirect
9. #27824C2semidirect
11. #25813C2semidirect
12. #25455C2semidirect
13. #24815C2semidirect
15. #22629C2semidirect
131. #448Q8semidirect
261. C2×SD16C2×Q8semidirect
362. #103C2×Q8semidirect

Maximal subgroups

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

subgroupindex
#365512
C2×#25672
#329572
#323582
#315342
#299572
#285392
#279532
#278242
#276772
#258132
#254552
#248152
#242992
#226292
#220472
C4×#4482