Finite Group Databasebeta

← #6490458

Subgroups — #6490458

Order 512.

1860 conjugacy classes of subgroups.

The subgroup lattice has 653 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36631C211yesmaximal, characteristic
2C2×C4×C4⋊Q8C211yesmaximal, characteristic, complemented
3#33096C214yesmaximal, complemented
4#30150C214yesmaximal, complemented
5#29393C214yesmaximal, complemented
6#29238C214yesmaximal, complemented
7#29176C212yesmaximal, complemented
8#28655C212yesmaximal, complemented
9#27973C211yesmaximal, characteristic
10#27948C211yesmaximal, characteristic, complemented
11#25903C211yesmaximal, characteristic
12#25813C211yesmaximal, characteristic, complemented
13#22248C212yesmaximal, complemented
14#22219C212yesmaximal, complemented
15#20777C211yesmaximal, characteristic, complemented
16C22×C4⋊Q8C2211yescharacteristic
17C2×#529—21no—
18C22×C4×Q8—21no—
19C2×#500C2212yes—
20C2×#486C2212yes—
21#2861—21no—
22#2860—21no—
23#2860C2212yes—
24#2859C2211yescharacteristic
25#2858C2211yescharacteristic
26#2857C2211yescharacteristic
27#2856C2211yescharacteristic
28#2854—21no—
29#2854C2212yes—
30#2850—21no—
31C4×C4⋊Q8—21no—
32C4×C4⋊Q8—21no—
33C4×C4⋊Q8—21no—
34C4×C4⋊Q8—21no—
35C4×C4⋊Q8C2211yescharacteristic
36C4×C4⋊Q8C2212yes—
37C4×C4⋊Q8C2212yes—
38C4×C4⋊Q8C2212yes—
39C4×C4⋊Q8C2212yes—
40C4×#127C2211yescharacteristic
41C4×#124—21no—
42C4×#121—21no—
43C4×#121C2212yes—
44C2×#391—21no—
45#2293—21no—
46#2293C2212yes—
47C2×#389—21no—
48C2×#389C2212yes—
49C2×#384C2211yescharacteristic
50C2×C4×C4⋊C4—21no—
51C2×C4×C4⋊C4C2212yes—
52C2×C43C2211yescharacteristic
53#2088C2218yes—
54#2084C2214yes—
55#2071C2214yes—
56#2067C2218yes—
57#2057C2214yes—
58#2023C2214yes—
59#1990C2212yes—
60#1990C2212yes—
61#1990C2212yes—
62#1957C2212yes—
63#1926C2218yes—
64#1925C2218yes—
65#1921C2218yes—
66#1921C2218yes—
67#1920C2218yes—
68#1874C2214yes—
69#1874C2214yes—
70#1864C2218yes—
71#1853C2214yes—
72#1852C2214yes—
73#1844C2212yes—
74#1839C2212yes—
75#1834C2212yes—
76#1831C2212yes—
77#1817C2212yes—
78#1797C2212yes—
79C4×C4⋊C8C2212yes—
80C4×C22⋊C8C2211yescharacteristic
81C4×C22⋊C8C2211yescharacteristic
82#1744C2212yes—
83C23×Q8—21no—
84C2×C4⋊Q8—28no—
85C2×C4⋊Q8—21no—
86C2×C4⋊Q8—21no—
87C2×C4⋊Q8C2312yes—
88C2×C4⋊Q8C2311yescharacteristic
89C2×C4⋊Q8C2312yes—
90C2×#127C2311yescharacteristic
91#529—48no—
92C2×#126—22no—
93C2×#125—21no—
94C2×#124—21no—
95C2×#124—21no—
96C2×#123—22no—
97C2×C22⋊Q8—21no—
98C2×#121—21no—
99C2×#121—21no—
100C2×#121C2312yes—
101C2×C4×Q8—22no—
102C2×C4×Q8—22no—
103C2×C4×Q8—22no—
104C2×C4×Q8—22no—
105C2×C4×Q8—21no—
106C2×C4×Q8—48no—
107C2×C4×D4—22no—
108C2×C4×D4—21no—
109C2×C4×D4—21no—
110C2×C4×D4C2312yes—
111C2×#117—22no—
112C22×C4⋊C4—21no—
113C22×C4⋊C4—21no—
114C22×C4⋊C4—22no—
115C22×C4⋊C4—21no—
116C22×C4⋊C4C2312yes—
117C22×C4⋊C4C2312yes—
118C22×C42—22no—
119C22×C42D411yescharacteristic
120C22×C42D411yescharacteristic
121C22×C42D411yescharacteristic
122C22×C42D411yescharacteristic
123C22×C42C2311yescharacteristic
124C22×C42C2311yescharacteristic
125C22×C42C2311yescharacteristic
126#500—28no—
127#486—28no—
128C2×#107C2318yes—
129C2×#106C2318yes—
130C2×C4⋊C8C2314yes—
131C2×#102C2318yes—
132C2×#102C2318yes—
133C2×C22⋊C8C2312yes—
134C2×C22⋊C8C2312yes—
135C2×C4×C8C2312yes—
136C2×C4×C8C2312yes—
137#391—22no—
138#391—22no—
139#391—22no—
140#391—22no—
141#391—21no—
142#391—21no—
143#390—21no—
144#390—21no—
145#390—21no—
146#390—21no—
147#390—21no—
148#390—21no—
149#390C2312yes—
150#390C2312yes—
151#390C2312yes—
152#390C2312yes—
153#389—22no—
154#389—22no—
155#389—22no—
156#389—22no—
157#389—22no—
158#389—22no—
159#389—21no—
160#389—21no—
161#389—21no—
162#389—21no—
163#389C2314yes—
164#389C2314yes—
165#389C2312yes—
166#389C2312yes—
167#389C2312yes—
168#389C2312yes—
169#388—22no—
170#388—22no—
171#388—22no—
172#388—22no—
173#388—22no—
174#388—22no—
175#388—22no—
176#388—22no—
177#386—22no—
178#386—22no—
179#384—21no—
180#384—21no—
181#384C2311yescharacteristic
182#384C2312yes—
183#384C2312yes—
184#384C2311yescharacteristic
185#382—21no—
186#380—21no—
187#380C2312yes—
188C4×C4⋊C4—22no—
189C4×C4⋊C4—22no—
190C4×C4⋊C4—22no—
191C4×C4⋊C4—22no—
192C4×C4⋊C4—22no—
193C4×C4⋊C4—22no—
194C4×C4⋊C4—21no—
195C4×C4⋊C4C2314yes—
196C4×C4⋊C4C2314yes—
197C4×C4⋊C4C2312yes—
198C4×C22⋊C4—22no—
199C4×C22⋊C4—22no—
200C2×#95—22no—
201C2×#95—22no—
202C43D411yescharacteristic
203C43D411yescharacteristic
204C43D411yescharacteristic
205C43D411yescharacteristic
206C43C2311yescharacteristic
207C43C2311yescharacteristic
208C43C2311yescharacteristic
209C43C2311yescharacteristic
210#340C2318yes—
211#340C2318yes—
212#340C2318yes—
213#340C2318yes—
214#336C2314yes—
215#336C2312yes—
216#336C2314yes—
217#336C2314yes—
218#336C2312yes—
219C22×Q8—28no—
220C22×Q8—22no—
221C22×Q8—22no—
222C22×Q8—21no—
223C22×Q8—44no—
224C22×Q8—44no—
225C22×D4—22no—
226C22×D4—21no—
227C22×D4—21no—
228C22×D4C2412yes—
229C23×C4—22no—
230C23×C4—22no—
231C23×C4—22no—
232C23×C4C2×D411yescharacteristic
233C23×C4C2×D411yescharacteristic
234C23×C4C2×D411yescharacteristic
235C23×C4C2×D411yescharacteristic
236C23×C4C2×D411yescharacteristic
237C23×C4C2×D411yescharacteristic
238C23×C4C2411yescharacteristic
239C22×C8C2414yes—
240C22×C8C2414yes—
241C4⋊Q8—28no—
242C4⋊Q8—28no—
243C4⋊Q8—28no—
244C4⋊Q8—24no—
245C4⋊Q8—24no—
246C4⋊Q8#4118yes—
247#127—24no—
248#126—48no—
249#125—44no—
250#124—44no—
251#124—44no—
252#123—48no—
253C22⋊Q8—44no—
254#121—28no—
255#121—44no—
256#121—44no—
257C4×Q8—48no—
258C4×Q8—48no—
259C4×Q8—48no—
260C4×Q8—44no—
261C4×Q8—48no—
262C4×D4—44no—
263C4×D4—44no—
264C4×D4—48no—
265C4×D4—44no—
266#117—48no—
267C2×C4⋊C4—28no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—28no—
272C2×C4⋊C4—22no—
273C2×C4⋊C4—22no—
274C2×C4⋊C4—24no—
275C2×C4⋊C4—24no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—22no—
279C2×C4⋊C4—22no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—44no—
289C2×C4⋊C4—44no—
290C2×C4⋊C4—48no—
291C2×C4⋊C4—44no—
292C2×C4⋊C4C2414yes—
293C2×C4⋊C4C2412yes—
294C2×C4⋊C4C2412yes—
295C2×C4⋊C4C2412yes—
296C2×C4⋊C4C2414yes—
297C2×C22⋊C4—22no—
298C2×C22⋊C4—22no—
299C2×C22⋊C4—22no—
300C2×C22⋊C4—22no—
301C2×C22⋊C4—22no—
302C2×C22⋊C4—22no—
303C2×C22⋊C4—22no—
304C2×C22⋊C4—22no—
305C2×C22⋊C4—22no—
306C2×C22⋊C4—22no—
307C2×C22⋊C4—22no—
308C2×C42—24no—
309C2×C42—24no—
310C2×C42—24no—
311C2×C42—24no—
312C2×C42—24no—
313C2×C42—24no—
314C2×C42—22no—
315C2×C42—24no—
316C2×C42—24no—
317C2×C42—42no—
318C2×C42—48no—
319C2×C42C2×D411yescharacteristic
320C2×C42C2×D411yescharacteristic
321C2×C42C2×D411yescharacteristic
322C2×C42C2×D411yescharacteristic
323C2×C42C2×D411yescharacteristic
324C2×C42C2×D411yescharacteristic
325C2×C42C2×D411yescharacteristic
326C2×C42C2×D411yescharacteristic
327C2×C42C2×D411yescharacteristic
328C2×C42C2×D411yescharacteristic
329C2×C42C2×D411yescharacteristic
330C2×C42C2×D411yescharacteristic
331C2×C42C2×D411yescharacteristic
332C2×C42C2×D411yescharacteristic
333C2×C42C2×D411yescharacteristic
334C2×C42C2×D411yescharacteristic
335C2×C42C2×D411yescharacteristic
336C2×C42C2×D411yescharacteristic
337C2×C42C2×D411yescharacteristic
338C2×C42C2×D411yescharacteristic
339C2×C42C2×D411yescharacteristic
340C2×C42C2×D411yescharacteristic
341C2×C42C2411yescharacteristic
342C2×C42C2411yescharacteristic
343C2×C42C2411yescharacteristic
344C2×C42C2411yescharacteristic
345C2×C42C2411yescharacteristic
346C2×C42C2411yescharacteristic
347#107—216no—
348#106—48no—
349C4⋊C8—44no—
350#102—216no—
351#102—48no—
352C22⋊C8—24no—
353C22⋊C8—24no—
354C4×C8—24no—
355C4×C8—24no—
356#95—24no—
357#95—24no—
358#95—24no—
359#95—24no—
360#95—22no—
361#95—22no—
362#95—22no—
363#95—22no—
364#95—22no—
365#95—22no—
366C24—22no—
367C24C22×D411yescharacteristic
368C2×Q8—28no—
369C2×Q8—28no—
370C2×Q8—44no—
371C2×Q8—432no—
372C2×Q8—42no—
373C2×Q8—44no—
374C2×Q8—44no—
375C2×Q8—44no—
376C2×Q8—42no—
377C2×Q8—42no—
378C2×D4—24no—
379C2×D4—24no—
380C2×D4—24no—
381C2×D4—44no—
382C2×D4—44no—
383C2×D4—44no—
384C2×D4—42no—
385C2×D4—42no—
386C2×D4—42no—
387C2×D4—42no—
388C2×D4—42no—
389C2×D4—42no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—28no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—24no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—24no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—41no—
413C22×C4—41no—
414C22×C4—44no—
415C22×C4—44no—
416C22×C4—44no—
417C22×C4—44no—
418C22×C4—44no—
419C22×C4—44no—
420C22×C4—44no—
421C22×C4—44no—
422C22×C4#13514yes—
423C22×C4AGL(1,ℤ/8)14yes—
424C22×C4C22×D411yescharacteristic
425C22×C4C22×D411yescharacteristic
426C22×C4C22×D411yescharacteristic
427C22×C4C22×D411yescharacteristic
428C22×C4C22×D411yescharacteristic
429C22×C4C22×D411yescharacteristic
430C22×C4C22×D411yescharacteristic
431C22×C4C22×D411yescharacteristic
432C22×C4C22×D411yescharacteristic
433C22×C4C22×D411yescharacteristic
434C22×C4C22×D411yescharacteristic
435C22×C4C22×D411yescharacteristic
436C22×C4C22×D411yescharacteristic
437C22×C42−1+414yes—
438C22×C4C2511yescharacteristic
439C2×C8—24no—
440C2×C8—24no—
441C2×C8—24no—
442C2×C8—24no—
443C2×C8—24no—
444C2×C8—24no—
445C4⋊C4—28no—
446C4⋊C4—28no—
447C4⋊C4—28no—
448C4⋊C4—24no—
449C4⋊C4—24no—
450C4⋊C4—24no—
451C4⋊C4—28no—
452C4⋊C4—24no—
453C4⋊C4—44no—
454C4⋊C4—44no—
455C4⋊C4—48no—
456C4⋊C4—44no—
457C4⋊C4—44no—
458C4⋊C4—44no—
459C4⋊C4—42no—
460C4⋊C4—44no—
461C4⋊C4—44no—
462C4⋊C4—42no—
463C4⋊C4—48no—
464C4⋊C4—44no—
465C4⋊C4—44no—
466C4⋊C4—42no—
467C4⋊C4—42no—
468C4⋊C4—42no—
469C4⋊C4C2×#4118yes—
470C22⋊C4—44no—
471C22⋊C4—44no—
472C22⋊C4—44no—
473C22⋊C4—44no—
474C22⋊C4—44no—
475C22⋊C4—44no—
476C22⋊C4—44no—
477C22⋊C4—44no—
478C22⋊C4—44no—
479C22⋊C4—44no—
480C22⋊C4—44no—
481C42—22no—
482C42—22no—
483C42—22no—
484C42—22no—
485C42—22no—
486C42—22no—
487C42—22no—
488C42—22no—
489C42—22no—
490C42—22no—
491C42—22no—
492C42—22no—
493C42—22no—
494C42—22no—
495C42—22no—
496C42—22no—
497C42—22no—
498C42—22no—
499C42—22no—
500C42—22no—
501C42—22no—
502C42—22no—
503C42—22no—
504C42—41no—
505C42—41no—
506C42—48no—
507C42—44no—
508C42—48no—
509C42AGL(1,ℤ/8)14yes—
510C42#13714yes—
511C42C2×#4114yes—
512C23—21no—
513C23—21no—
514C23—21no—
515C23—21no—
516C23—42no—
517C23—42no—
518C23—42no—
519C23—42no—
520C23—42no—
521C23—42no—
522C23—42no—
523C23#53612yes—
524C23C2×#13512yes—
525C23C2×AGL(1,ℤ/8)12yes—
526C23C23×D411yescharacteristic
527Q8—416no—
528Q8—416no—
529Q8—84no—
530D4—48no—
531D4—88no—
532D4—84no—
533D4—84no—
534C2×C4—28no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—28no—
550C2×C4—28no—
551C2×C4—28no—
552C2×C4—28no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—44no—
566C2×C4—44no—
567C2×C4—416no—
568C2×C4—44no—
569C2×C4—44no—
570C2×C4—44no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—44no—
575C2×C4—42no—
576C2×C4—44no—
577C2×C4—44no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—88no—
584C2×C4—88no—
585C2×C4#52512yes—
586C2×C4#52512yes—
587C2×C4#53612yes—
588C2×C4C2×#13512yes—
589C2×C4C2×#13512yes—
590C2×C4C2×AGL(1,ℤ/8)12yes—
591C2×C4C2×AGL(1,ℤ/8)12yes—
592C2×C4C2×AGL(1,ℤ/8)12yes—
593C2×C4C2×AGL(1,ℤ/8)12yes—
594C2×C4C2×AGL(1,ℤ/8)12yes—
595C2×C4C2×#13712yes—
596C2×C4C2×#13712yes—
597C2×C4C2×#13712yes—
598C2×C4C22×#4112yes—
599C2×C4C2×2−1+412yes—
600C2×C4#58512yes—
601C8—44no—
602C8—44no—
603C22—22no—
604C22—22no—
605C22—22no—
606C22—22no—
607C22—22no—
608C22—22no—
609C22—82no—
610C22—82no—
611C22—82no—
612C22—82no—
613C22—82no—
614C22—82no—
615C22—82no—
616C22#322814yes—
617C22C2×#52511yescharacteristic
618C22C2×#53611yescharacteristic
619C22#344911yescharacteristic
620C22C22×#13511yescharacteristic
621C22C22×AGL(1,ℤ/8)11yescharacteristic
622C22C22×AGL(1,ℤ/8)11yescharacteristic
623C22C22×#13711yescharacteristic
624C4—22no—
625C4—22no—
626C4—22no—
627C4—22no—
628C4—22no—
629C4—22no—
630C4—22no—
631C4—41no—
632C4—41no—
633C4—48no—
634C4—41no—
635C4—41no—
636C4—84no—
637C4—82no—
638C4—82no—
639C4—82no—
640C4#310814yes—
641C4#310914yes—
642C4#351614yes—
643C2—24no—
644C2—162no—
645C2#3524611yescharacteristic
646C2C2×#310811yescharacteristic
647C2C2×#310911yescharacteristic
648C2C2×#322811yescharacteristic
649C2#6086011yescharacteristic
650C2#6094611yescharacteristic
651C2#6096911yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#2856C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C43 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×C4×C4⋊Q8C2semidirect
3. #33096C2semidirect
4. #30150C2semidirect
5. #29393C2semidirect
6. #29238C2semidirect
7. #29176C2semidirect
8. #28655C2semidirect
10. #27948C2semidirect
12. #25813C2semidirect
13. #22248C2semidirect
14. #22219C2semidirect
15. #20777C2semidirect

Maximal subgroups

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

subgroupindex
#366312
C2×C4×C4⋊Q82
#330962
#301502
#293932
#292382
#291762
#286552
#279732
#279482
#259032
#258132
#222482
#222192
#207772