Finite Group Databasebeta

← #6971525

Subgroups — #6971525

Order 512.

1788 conjugacy classes of subgroups.

The subgroup lattice has 670 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36469C211yesmaximal, characteristic
2D4×C4⋊Q8C211yesmaximal, characteristic, complemented
3#32738C212yesmaximal, complemented
4#31738C214yesmaximal, complemented
5#31078C214yesmaximal, complemented
6#30203C211yesmaximal, characteristic, complemented
7#30165C212yesmaximal, complemented
8#29638C212yesmaximal, complemented
9#29346C211yesmaximal, characteristic, complemented
10#29208C211yesmaximal, characteristic, complemented
11#26460C212yesmaximal, complemented
12#26440C212yesmaximal, complemented
13#26296C212yesmaximal, complemented
14#26230C212yesmaximal
15#24298C212yesmaximal, complemented
16#23835C211yesmaximal, characteristic, complemented
17#22008C211yesmaximal, characteristic, complemented
18C2×#559—21no—
19C2×D4×Q8—21no—
20C22×C4⋊Q8C2212yes—
21C2×#480C2214yes—
22C2×#474C2214yes—
23#2868C2211yescharacteristic
24#2858C2211yescharacteristic
25#2857C2212yes—
26#2854C2212yes—
27#2602—22no—
28#2592—21no—
29#2557—22no—
30#2551—21no—
31#2551C2212yes—
32#2547C2211yescharacteristic
33#2546—21no—
34#2544—21no—
35#2541C2211yescharacteristic
36#2491C2212yes—
37#2440—22no—
38#2440C2214yes—
39#2434—22no—
40#2422C2212yes—
41#2367—21no—
42#2366C2211yescharacteristic
43#2365C2211yescharacteristic
44D4×C4⋊C4—21no—
45D4×C4⋊C4C2212yes—
46C4×C4⋊Q8C2211yescharacteristic
47C4×C4⋊Q8C2211yescharacteristic
48C4×#121C2212yes—
49C42×D4C2211yescharacteristic
50#2086C2214yes—
51#2083C2212yes—
52#2083C2211yescharacteristic
53#2083C2211yescharacteristic
54#2075C2214yes—
55#2075C2214yes—
56#2071C2214yes—
57#2071C2212yes—
58#2067C2214yes—
59#2057C2212yes—
60#2057C2212yes—
61#2057C2211yescharacteristic
62#2057C2211yescharacteristic
63#2057C2212yes—
64#2057C2214yes—
65#2033C2214yes—
66#2033C2214yes—
67#2023C2214yes—
68#2023C2212yes—
69#2023C2212yes—
70#2023C2212yes—
71#1990C2212yes—
72#1990C2212yes—
73#1988C2212yes—
74#1988C2214yes—
75#1982C2212yes—
76#1957C2212yes—
77#1954C2212yes—
78#1947C2212yes—
79#1947C2212yes—
80#1939C2212yes—
81#1934C2214yes—
82#1923C2214yes—
83#1918C2214yes—
84#1874C2214yes—
85#1862C2214yes—
86#1852C2214yes—
87#1845C2212yes—
88#1823C2214yes—
89#1817C2212yes—
90#1797C2212yes—
91C4×#107C2212yes—
92C4×#107C2212yes—
93C4×C4⋊C8C2211yescharacteristic
94#1744C2211yescharacteristic
95C23×Q8—22no—
96#559—48no—
97D4×Q8—48no—
98C2×C4⋊Q8—22no—
99C2×C4⋊Q8—21no—
100C2×C4⋊Q8—22no—
101C2×C4⋊Q8—22no—
102C2×C4⋊Q8—21no—
103C2×C4⋊Q8—48no—
104C2×C4⋊Q8C2311yescharacteristic
105C2×C4⋊Q8C2311yescharacteristic
106C2×C4⋊Q8C2311yescharacteristic
107C2×C4⋊Q8C2314yes—
108C2×C4⋊Q8C2312yes—
109C2×#127—21no—
110C2×#127C2311yescharacteristic
111C2×#124—21no—
112C2×#124—22no—
113C2×#124—21no—
114C2×#124—21no—
115C2×#123—22no—
116C2×#123—22no—
117C2×C22⋊Q8—22no—
118C2×C22⋊Q8—24no—
119C2×#121C2312yes—
120C2×#121C2312yes—
121C2×C4×Q8—21no—
122C2×C4×Q8—21no—
123C2×C4×Q8—21no—
124C2×C4×D4—22no—
125C2×C4×D4—22no—
126C2×C4×D4—21no—
127C2×C4×D4D411yescharacteristic
128C2×C4×D4D411yescharacteristic
129C2×C4×D4C2312yes—
130C2×C4×D4C2311yescharacteristic
131C22×C4⋊C4—22no—
132C22×C4⋊C4C2314yes—
133C22×C42C2312yes—
134#480—48no—
135#474—48no—
136C2×#107C2312yes—
137C2×#107C2312yes—
138C2×#107C2312yes—
139C2×#107C2314yes—
140C2×#107C2312yes—
141C2×#107C2314yes—
142C2×C4⋊C8C2312yes—
143C2×C4⋊C8C2314yes—
144C2×#102C2314yes—
145C2×#102C2314yes—
146C2×#102C2314yes—
147C2×#102C2314yes—
148C2×C22⋊C8C2314yes—
149C2×C22⋊C8C2314yes—
150C2×C4×C8C2312yes—
151C2×C4×C8C2312yes—
152#398—24no—
153#397—22no—
154#397—22no—
155#395—24no—
156#395—24no—
157#394—22no—
158#394—22no—
159#393—22no—
160#392—22no—
161#391—21no—
162#391—21no—
163#391—21no—
164#391—21no—
165#391—22no—
166#390—21no—
167#390—21no—
168#390—21no—
169#390—21no—
170#390C2312yes—
171#390C2312yes—
172#390C2312yes—
173#389—22no—
174#389—22no—
175#389—21no—
176#389—21no—
177#389—22no—
178#389C2311yescharacteristic
179#389C2311yescharacteristic
180#389C2312yes—
181#389C2312yes—
182#389C2314yes—
183#388—22no—
184#388—22no—
185#388—22no—
186#388—22no—
187#387—22no—
188#386—22no—
189#386—22no—
190#386—22no—
191#385—24no—
192#385—24no—
193#385—22no—
194#384—21no—
195#384—21no—
196#384C2311yescharacteristic
197#384C2311yescharacteristic
198#384C2312yes—
199#384C2312yes—
200#380—22no—
201#380C2312yes—
202#380C2314yes—
203#380C2312yes—
204C4×C4⋊C4—21no—
205C4×C4⋊C4—21no—
206C4×C4⋊C4D411yescharacteristic
207C4×C4⋊C4D411yescharacteristic
208C4×C4⋊C4C2312yes—
209C4×C4⋊C4C2311yescharacteristic
210C4×C4⋊C4C2311yescharacteristic
211C4×C4⋊C4C2312yes—
212C4×C4⋊C4C2311yescharacteristic
213C4×C22⋊C4—22no—
214C4×C22⋊C4D412yes—
215C4×C22⋊C4D412yes—
216C4×C22⋊C4C2312yes—
217C43C2311yescharacteristic
218#340C2312yes—
219#340C2312yes—
220#340C2314yes—
221#340C2314yes—
222#340C2314yes—
223#340C2314yes—
224#340C2314yes—
225#340C2314yes—
226#340C2312yes—
227#340C2312yes—
228#336C2311yescharacteristic
229#336C2312yes—
230#336C2312yes—
231#336C2311yescharacteristic
232#336C2314yes—
233C22×Q8—21no—
234C22×Q8—21no—
235C22×Q8—22no—
236C22×Q8—21no—
237C22×Q8—24no—
238C22×Q8—48no—
239C22×Q8—48no—
240C22×Q8—84no—
241C22×D4—22no—
242C22×D4—22no—
243C22×D4—21no—
244C22×D4—21no—
245C22×D4C2×D411yescharacteristic
246C22×D4C2412yes—
247C23×C4—24no—
248C23×C4—22no—
249C23×C4C2×D412yes—
250C23×C4C2×D412yes—
251C23×C4C2412yes—
252C22×C8C2412yes—
253C22×C8C2414yes—
254C22×C8C2412yes—
255C4⋊Q8—24no—
256C4⋊Q8—28no—
257C4⋊Q8—22no—
258C4⋊Q8—24no—
259C4⋊Q8—44no—
260C4⋊Q8—44no—
261C4⋊Q8—41no—
262C4⋊Q8—48no—
263C4⋊Q8—48no—
264C4⋊Q8—48no—
265C4⋊Q8—44no—
266C4⋊Q8—82no—
267#127—24no—
268#127—44no—
269#124—44no—
270#124—48no—
271#124—44no—
272#124—44no—
273#123—48no—
274#123—48no—
275C22⋊Q8—48no—
276C22⋊Q8—416no—
277#121—24no—
278#121—42no—
279#121—44no—
280C4×Q8—44no—
281C4×Q8—44no—
282C4×Q8—44no—
283C4×D4—24no—
284C4×D4—24no—
285C4×D4—44no—
286C4×D4—48no—
287C4×D4—48no—
288C4×D4—44no—
289C4×D4D818yes—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—28no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—24no—
294C2×C4⋊C4—24no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—24no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—44no—
311C2×C4⋊C4—48no—
312C2×C4⋊C4C2×D411yescharacteristic
313C2×C4⋊C4C2×D411yescharacteristic
314C2×C4⋊C4C2×D411yescharacteristic
315C2×C4⋊C4C2414yes—
316C2×C4⋊C4C2411yescharacteristic
317C2×C4⋊C4C2411yescharacteristic
318C2×C4⋊C4C2412yes—
319C2×C4⋊C4C2411yescharacteristic
320C2×C4⋊C4C2412yes—
321C2×C4⋊C4C2412yes—
322C2×C4⋊C4C2411yescharacteristic
323C2×C22⋊C4—24no—
324C2×C22⋊C4—22no—
325C2×C22⋊C4—22no—
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—22no—
335C2×C22⋊C4C2×D412yes—
336C2×C22⋊C4C2×D412yes—
337C2×C22⋊C4C2×D412yes—
338C2×C42—22no—
339C2×C42—21no—
340C2×C42—22no—
341C2×C42—21no—
342C2×C42—44no—
343C2×C42C2×D411yescharacteristic
344C2×C42C2×D412yes—
345C2×C42C2×D412yes—
346C2×C42C2×D411yescharacteristic
347C2×C42C2×D411yescharacteristic
348C2×C42C2×D411yescharacteristic
349C2×C42C2412yes—
350C2×C42C2411yescharacteristic
351C2×C42C2411yescharacteristic
352C2×C42C2411yescharacteristic
353#107—24no—
354#107—24no—
355#107—24no—
356#107—42no—
357#107—44no—
358#107—44no—
359C4⋊C8—24no—
360C4⋊C8—44no—
361#102—28no—
362#102—44no—
363#102—44no—
364#102—44no—
365C22⋊C8—44no—
366C22⋊C8—44no—
367C4×C8—24no—
368C4×C8—42no—
369#95—21no—
370#95—22no—
371#95—22no—
372#95—21no—
373#95—22no—
374#95—21no—
375#95—22no—
376#95—21no—
377#95—24no—
378#95—24no—
379#95—22no—
380#95—22no—
381#95C2×D412yes—
382#95C2×D412yes—
383#95C2×D412yes—
384C24—22no—
385C24C22×D412yes—
386C2×Q8—24no—
387C2×Q8—48no—
388C2×Q8—42no—
389C2×Q8—42no—
390C2×Q8—44no—
391C2×Q8—44no—
392C2×Q8—42no—
393C2×Q8—42no—
394C2×Q8—42no—
395C2×Q8—42no—
396C2×Q8—48no—
397C2×Q8—44no—
398C2×Q8—42no—
399C2×Q8—81no—
400C2×Q8—84no—
401C2×Q8—832no—
402C2×D4—22no—
403C2×D4—22no—
404C2×D4—24no—
405C2×D4—22no—
406C2×D4—24no—
407C2×D4—22no—
408C2×D4—44no—
409C2×D4—44no—
410C2×D4—44no—
411C2×D4—42no—
412C2×D4—42no—
413C2×D4—42no—
414C2×D4—42no—
415C2×D4—81no—
416C2×D4—81no—
417C2×D4—82no—
418C2×D4—82no—
419C2×D4—82no—
420C2×D4C2×D814yes—
421C22×C4—21no—
422C22×C4—22no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—21no—
430C22×C4—22no—
431C22×C4—21no—
432C22×C4—21no—
433C22×C4—24no—
434C22×C4—42no—
435C22×C4—42no—
436C22×C4—42no—
437C22×C4—42no—
438C22×C4—48no—
439C22×C4—48no—
440C22×C4—44no—
441C22×C4—82no—
442C22×C4—82no—
443C22×C4—84no—
444C22×C4C2×D818yes—
445C22×C4C22×D412yes—
446C22×C4C22×D412yes—
447C22×C4C22×D411yescharacteristic
448C22×C4C22×D412yes—
449C22×C4C22×D411yescharacteristic
450C22×C4C22×D411yescharacteristic
451C22×C4C22×D411yescharacteristic
452C22×C4C22×D411yescharacteristic
453C22×C4C22×D411yescharacteristic
454C22×C4C2511yescharacteristic
455C2×C8—22no—
456C2×C8—24no—
457C2×C8—22no—
458C2×C8—22no—
459C2×C8—22no—
460C2×C8—24no—
461C2×C8—22no—
462C2×C8—24no—
463C2×C8—22no—
464C4⋊C4—24no—
465C4⋊C4—22no—
466C4⋊C4—22no—
467C4⋊C4—22no—
468C4⋊C4—21no—
469C4⋊C4—24no—
470C4⋊C4—24no—
471C4⋊C4—22no—
472C4⋊C4—22no—
473C4⋊C4—22no—
474C4⋊C4—24no—
475C4⋊C4—21no—
476C4⋊C4—22no—
477C4⋊C4—21no—
478C4⋊C4—48no—
479C4⋊C4—48no—
480C4⋊C4—42no—
481C4⋊C4—42no—
482C4⋊C4—44no—
483C4⋊C4—42no—
484C4⋊C4—42no—
485C4⋊C4—44no—
486C4⋊C4—44no—
487C4⋊C4—42no—
488C4⋊C4—44no—
489C4⋊C4—42no—
490C4⋊C4—42no—
491C4⋊C4—48no—
492C4⋊C4—44no—
493C4⋊C4—81no—
494C4⋊C4—82no—
495C4⋊C4—82no—
496C4⋊C4C2×D814yes—
497C4⋊C4H(5,𝔽2)14yes—
498C4⋊C42−1+412yes—
499C22⋊C4—24no—
500C22⋊C4—24no—
501C22⋊C4—44no—
502C22⋊C4—48no—
503C22⋊C4—44no—
504C22⋊C4—44no—
505C22⋊C4—44no—
506C22⋊C4—44no—
507C22⋊C4—44no—
508C22⋊C4—44no—
509C22⋊C4—44no—
510C22⋊C4—82no—
511C22⋊C4—82no—
512C22⋊C4—82no—
513C22⋊C4C2×D818yes—
514C42—24no—
515C42—22no—
516C42—21no—
517C42—21no—
518C42—42no—
519C42—42no—
520C42—41no—
521C42—41no—
522C42—44no—
523C42—42no—
524C42—44no—
525C42—42no—
526C42C2×D814yes—
527C42#13714yes—
528C422−1+414yes—
529C23—22no—
530C23—22no—
531C23—22no—
532C23—22no—
533C23—22no—
534C23—22no—
535C23—42no—
536C23—42no—
537C23—42no—
538C23—42no—
539C23—42no—
540C23—42no—
541C23—42no—
542C23C22×D814yes—
543C23C23×D411yescharacteristic
544Q8—48no—
545Q8—84no—
546Q8—816no—
547Q8—88no—
548Q8—84no—
549D4—28no—
550D4—44no—
551D4—44no—
552D4—88no—
553D4—84no—
554D4—162no—
555D4—164no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—22no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—22no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—24no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—48no—
577C2×C4—48no—
578C2×C4—42no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—42no—
582C2×C4—44no—
583C2×C4—44no—
584C2×C4—44no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—44no—
593C2×C4—41no—
594C2×C4—42no—
595C2×C4—41no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—41no—
599C2×C4—41no—
600C2×C4—44no—
601C2×C4—42no—
602C2×C4—816no—
603C2×C4—164no—
604C2×C4#53612yes—
605C2×C4#53612yes—
606C2×C4C22×D814yes—
607C2×C4C22×D812yes—
608C2×C4C22×D812yes—
609C2×C4C22×D812yes—
610C2×C4C2×#13712yes—
611C2×C4C2×#13712yes—
612C2×C4C2×#13712yes—
613C2×C4C2×H(5,𝔽2)12yes—
614C2×C4C2×2−1+411yescharacteristic
615C2×C4C2×2−1+412yes—
616C2×C4#58511yescharacteristic
617C8—42no—
618C8—44no—
619C8—42no—
620C22—24no—
621C22—24no—
622C22—24no—
623C22—42no—
624C22—42no—
625C22—42no—
626C22—42no—
627C22—82no—
628C22—82no—
629C22—82no—
630C22—82no—
631C22—82no—
632C22—82no—
633C22—82no—
634C22C2×#53611yescharacteristic
635C22#348411yescharacteristic
636C22#348911yescharacteristic
637C22C23×D811yescharacteristic
638C22C22×#13711yescharacteristic
639C22#358511yescharacteristic
640C22#358611yescharacteristic
641C4—22no—
642C4—24no—
643C4—22no—
644C4—22no—
645C4—22no—
646C4—22no—
647C4—44no—
648C4—42no—
649C4—42no—
650C4—42no—
651C4—84no—
652C4—81no—
653C4—81no—
654C4—82no—
655C4—81no—
656C4—82no—
657C4#322614yes—
658C4#355012yes—
659C4#355412yes—
660C2—44no—
661C2—162no—
662C2#4248111yescharacteristic
663C2C2×#322611yescharacteristic
664C2#6088511yescharacteristic
665C2#6155911yescharacteristic
666C2#6158411yescharacteristic
667C2#6209511yescharacteristic
668C2#6211211yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#2541C22
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. D4×C4⋊Q8C2semidirect
3. #32738C2semidirect
4. #31738C2semidirect
5. #31078C2semidirect
6. #30203C2semidirect
7. #30165C2semidirect
8. #29638C2semidirect
9. #29346C2semidirect
10. #29208C2semidirect
11. #26460C2semidirect
12. #26440C2semidirect
13. #26296C2semidirect
15. #24298C2semidirect
16. #23835C2semidirect
17. #22008C2semidirect

Maximal subgroups

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

subgroupindex
#364692
D4×C4⋊Q82
#327382
#317382
#310782
#302032
#301652
#296382
#293462
#292082
#264602
#264402
#262962
#262302
#242982
#238352
#220082