Finite Group Databasebeta

← #6382368

Subgroups — #6382368

Order 512.

2080 conjugacy classes of subgroups.

The subgroup lattice has 651 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3106C211yesmaximal, characteristic
2#34236C211yesmaximal, characteristic, complemented
3#34191C211yesmaximal, characteristic, complemented
4#23095C212yesmaximal, complemented
5#23069C212yesmaximal, complemented
6#23001C212yesmaximal, complemented
7#22963C212yesmaximal, complemented
8#22924C214yesmaximal, complemented
9#22866C214yesmaximal, complemented
10#21410C211yesmaximal, characteristic, complemented
11#21359C211yesmaximal, characteristic, complemented
12#21280C212yesmaximal, complemented
13#21134C212yesmaximal, complemented
14#21133C212yesmaximal, complemented
15#20503C212yesmaximal, complemented
16#20482C211yesmaximal, characteristic, complemented
17C4×#417C211yesmaximal, characteristic, complemented
18C2×#532C2211yescharacteristic
19C2×C4×#41C2211yescharacteristic
20#3106C4116yescomplemented
21C2×#482C2212yes—
22C2×#477C2211yescharacteristic
23C2×#476C2211yescharacteristic
24C2×#460C2211yescharacteristic
25C2×#459C2211yescharacteristic
26C2×#449C2212yes—
27C2×#442C2214yes—
28C2×C4×SD16C2212yes—
29C2×C4×D8C2212yes—
30C2×#431C2211yescharacteristic
31C2×#417C2212yes—
32C2×C4×M16C2211yescharacteristic
33#2369C2212yes—
34#2366C2212yes—
35#2361C2214yes—
36#2357—21no—
37#2355C2212yes—
38#2352—22no—
39#2349—21no—
40D4×C4⋊C4C2212yes—
41#2346C2212yes—
42#2343—22no—
43#2339—22no—
44#2339C2212yes—
45#2337—22no—
46#2320C2212yes—
47#2319C2212yes—
48#2318—22no—
49C4×#121—22no—
50#2285C2211yescharacteristic
51#2283—21no—
52#2283C2211yescharacteristic
53C2×#382—21no—
54#2275C2212yes—
55#2275C2212yes—
56C2×C4×C4⋊C4C2211yescharacteristic
57#1990C2212yes—
58#1968C2214yes—
59#1957C2212yes—
60#1948C2212yes—
61#1948C2212yes—
62#1941C2214yes—
63#1936C2214yes—
64#1935C2214yes—
65#1927C2214yes—
66#1926C2214yes—
67#1919C2214yes—
68#1918C2214yes—
69#1895C2214yes—
70#1879C2214yes—
71#1877C2214yes—
72#1864C2214yes—
73#1864C2214yes—
74#1853C2214yes—
75#1776C2214yes—
76C4×#106C2212yes—
77C4×#106C2212yes—
78#1770C2212yes—
79C8×C4⋊C4C2212yes—
80#1763C2211yescharacteristic
81#1763C2211yescharacteristic
82#1762C2211yescharacteristic
83#1762C2212yes—
84#1762C2212yes—
85#1762C2211yescharacteristic
86C4×#103C2212yes—
87C4×#102C2214yes—
88C4×#102C2214yes—
89C4×#102C2212yes—
90#1753C2214yes—
91#1746C2212yes—
92#1736C2212yes—
93#1726C2212yes—
94C22×#41—21no—
95C22×#41C2311yescharacteristic
96C22×SD16C2312yes—
97C22×D8C2312yes—
98C22×M16C2312yes—
99#532—24no—
100#532C2×C418yescomplemented
101C2×C4⋊Q8C2311yescharacteristic
102C2×#127C2311yescharacteristic
103C2×#124—21no—
104C2×#121—21no—
105C2×#121C2312yes—
106C4×#41—24no—
107C4×#41C2×C418yescomplemented
108C2×C4×Q8C2311yescharacteristic
109C2×C4×Q8C2311yescharacteristic
110C2×C4×D4—21no—
111C2×C4×D4—21no—
112C2×C4×D4—22no—
113C2×C4×D4—24no—
114C2×C4×D4—21no—
115C2×C4×D4C2314yes—
116C2×C4×D4C2314yes—
117C2×C4×D4C2312yes—
118C2×C4×D4C2311yescharacteristic
119C2×C4×D4C2311yescharacteristic
120C2×#117—21no—
121C2×#117—21no—
122C2×#117C2311yescharacteristic
123C22×C4⋊C4—22no—
124C22×C4⋊C4C2312yes—
125C22×C42—21no—
126C22×C42—21no—
127C22×C42C2311yescharacteristic
128#482C2×C4116yescomplemented
129#477C2×C418yescomplemented
130#476C2×C418yescomplemented
131#460C2×C418yescomplemented
132#459C2×C418yescomplemented
133#449C2×C4116yescomplemented
134#442—216no—
135C4×SD16C2×C4116yescomplemented
136C4×D8C2×C4116yescomplemented
137#431C2×C418yescomplemented
138C2×#107C2312yes—
139C2×#106C2314yes—
140C2×#106C2314yes—
141C2×#106C2312yes—
142C2×C4⋊C8C2311yescharacteristic
143C2×C4⋊C8C2311yescharacteristic
144#417—28no—
145C2×#103C2312yes—
146C2×#103C2312yes—
147C2×#103C2312yes—
148C2×#102C2312yes—
149C2×#102C2314yes—
150C2×#102C2314yes—
151C2×#102C2312yes—
152C2×#102C2312yes—
153C2×#102C2312yes—
154C2×#102C2312yes—
155C2×C22⋊C8C2312yes—
156C4×M16—24no—
157C2×#97C2311yescharacteristic
158C2×#97C2311yescharacteristic
159C2×#97C2314yes—
160C2×C4×C8C2312yes—
161C2×C4×C8C2314yes—
162C2×C4×C8C2312yes—
163#391—24no—
164#391C2314yes—
165#390—22no—
166#390C2314yes—
167#389—22no—
168#389C2314yes—
169#389C2314yes—
170#388—24no—
171#388—24no—
172#388—24no—
173#387—24no—
174#387—24no—
175#386—24no—
176#386—24no—
177#386—24no—
178#385—24no—
179#385—24no—
180#384C2312yes—
181#383—22no—
182#383D412yes—
183#382—22no—
184#382—21no—
185#382—22no—
186#382—22no—
187#382D411yescharacteristic
188#382D411yescharacteristic
189#381—22no—
190#381—22no—
191#381D412yes—
192#380—21no—
193#380—22no—
194#380—22no—
195#380D412yes—
196#380C2312yes—
197C4×C4⋊C4—22no—
198C4×C4⋊C4—22no—
199C4×C4⋊C4—21no—
200C4×C4⋊C4—21no—
201C4×C4⋊C4—21no—
202C4×C4⋊C4C2311yescharacteristic
203C4×C4⋊C4C2312yes—
204C4×C4⋊C4C2314yes—
205C4×C4⋊C4C2314yes—
206C4×C4⋊C4C2312yes—
207C4×C4⋊C4C2311yescharacteristic
208C4×C4⋊C4C2312yes—
209C4×C22⋊C4—22no—
210C4×C22⋊C4—24no—
211C4×C22⋊C4—22no—
212C4×C22⋊C4—24no—
213C4×C22⋊C4C2312yes—
214#376—22no—
215#376C2312yes—
216C2×#95—21no—
217C2×#95—22no—
218C2×#95—22no—
219C2×#95—22no—
220#340C2314yes—
221#340C2314yes—
222#340C2312yes—
223#340C2314yes—
224#340C2314yes—
225#340C2312yes—
226#340C2312yes—
227#340C2312yes—
228#336C2314yes—
229#336C2314yes—
230C2×#41—28no—
231C2×#41—22no—
232C2×#41—24no—
233C2×#41—42no—
234C2×#41—84no—
235C2×#41C22×C414yes—
236C22×Q8—21no—
237C22×Q8C2411yescharacteristic
238C22×D4—22no—
239C22×D4—21no—
240C22×D4—21no—
241C22×D4—21no—
242C22×D4C2412yes—
243C22×D4C2411yescharacteristic
244C23×C4—21no—
245C23×C4—21no—
246C23×C4—22no—
247C23×C4—21no—
248C23×C4—22no—
249C23×C4—24no—
250C23×C4—24no—
251C23×C4C2×D411yescharacteristic
252C23×C4C2×D411yescharacteristic
253C23×C4C2411yescharacteristic
254C2×SD16—44no—
255C2×SD16C22×C418yes—
256C2×D8—28no—
257C2×D8C22×C418yes—
258C2×M16—24no—
259C2×M16—24no—
260C2×M16—42no—
261C22×C8C2412yes—
262C22×C8C2412yes—
263C22×C8C2412yes—
264C22×C8C2412yes—
265C4⋊Q8—22no—
266C4⋊Q8C22×C414yes—
267#127—22no—
268#127C22×C414yes—
269#124—28no—
270#121—28no—
271#121—24no—
272#121C22×C418yes—
273C4×Q8—22no—
274C4×Q8—22no—
275C4×Q8C22×C414yes—
276C4×Q8C22×C414yes—
277C4×D4—28no—
278C4×D4—28no—
279C4×D4—28no—
280C4×D4—28no—
281C4×D4—28no—
282C4×D4—22no—
283C4×D4—22no—
284C4×D4—24no—
285C4×D4—44no—
286C4×D4—44no—
287C4×D4—84no—
288C4×D4—88no—
289C4×D4C22×C414yes—
290C4×D4C22×C414yes—
291C4×D4C22×C418yes—
292#117—22no—
293#117—28no—
294#117—82no—
295#117C22×C414yes—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—24no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—24no—
303C2×C4⋊C4—24no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—42no—
306C2×C4⋊C4—84no—
307C2×C4⋊C4C2×D412yes—
308C2×C4⋊C4C2×D412yes—
309C2×C4⋊C4C2412yes—
310C2×C4⋊C4C2414yes—
311C2×C4⋊C4C2411yescharacteristic
312C2×C4⋊C4C2411yescharacteristic
313C2×C4⋊C4C2412yes—
314C2×C4⋊C4C2412yes—
315C2×C22⋊C4—21no—
316C2×C22⋊C4—21no—
317C2×C22⋊C4—21no—
318C2×C22⋊C4—22no—
319C2×C22⋊C4—22no—
320C2×C22⋊C4—24no—
321C2×C22⋊C4—24no—
322C2×C22⋊C4—22no—
323C2×C22⋊C4—24no—
324C2×C22⋊C4—24no—
325C2×C22⋊C4C2×D414yes—
326C2×C42—21no—
327C2×C42—21no—
328C2×C42—21no—
329C2×C42—28no—
330C2×C42—22no—
331C2×C42—22no—
332C2×C42—24no—
333C2×C42—24no—
334C2×C42—21no—
335C2×C42—21no—
336C2×C42—82no—
337C2×C42C22×C414yes—
338C2×C42C2×D411yescharacteristic
339C2×C42C2×D411yescharacteristic
340C2×C42#4114yes—
341C2×C42C2414yes—
342C2×C42C2411yescharacteristic
343C2×C42C2411yescharacteristic
344#107C22×C418yes—
345#106—28no—
346#106—44no—
347#106C22×C418yes—
348C4⋊C8C22×C414yes—
349C4⋊C8C22×C414yes—
350#103—24no—
351#103—24no—
352#103C22×C418yes—
353#102—24no—
354#102—28no—
355#102—44no—
356#102C22×C418yes—
357#102C22×C418yes—
358#102C22×C418yes—
359#102#4118yescomplemented
360C22⋊C8C22×C418yes—
361#97—28no—
362#97—22no—
363#97—22no—
364C4×C8—24no—
365C4×C8—44no—
366C4×C8C22×C418yes—
367#95—21no—
368#95—22no—
369#95—22no—
370#95—24no—
371#95—24no—
372#95—21no—
373#95—22no—
374#95—24no—
375#95—24no—
376#95—24no—
377#95—24no—
378#95C2×D412yes—
379#95C2×D412yes—
380#95C2×D412yes—
381#95C2×D412yes—
382#95C2×D414yes—
383#95C2×D414yes—
384C24—21no—
385C24—22no—
386C24C22×D411yescharacteristic
387#41—28no—
388#41—44no—
389#41—48no—
390#41—816no—
391C2×Q8—24no—
392C2×Q8—21no—
393C2×Q8—22no—
394C2×Q8—22no—
395C2×Q8—82no—
396C2×Q8C23×C412yes—
397C2×D4—24no—
398C2×D4—28no—
399C2×D4—24no—
400C2×D4—24no—
401C2×D4—21no—
402C2×D4—24no—
403C2×D4—22no—
404C2×D4—22no—
405C2×D4—24no—
406C2×D4—44no—
407C2×D4—48no—
408C2×D4—42no—
409C2×D4—42no—
410C2×D4—82no—
411C2×D4C23×C414yes—
412C2×D4C23×C412yes—
413C2×D4C2×#4114yes—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—21no—
419C22×C4—21no—
420C22×C4—22no—
421C22×C4—21no—
422C22×C4—22no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—28no—
427C22×C4—24no—
428C22×C4—22no—
429C22×C4—42no—
430C22×C4—48no—
431C22×C4—44no—
432C22×C4—44no—
433C22×C4—48no—
434C22×C4—42no—
435C22×C4—42no—
436C22×C4—44no—
437C22×C4—82no—
438C22×C4—84no—
439C22×C4—84no—
440C22×C4—84no—
441C22×C4—84no—
442C22×C4—84no—
443C22×C4—82no—
444C22×C4C4×D414yes—
445C22×C4C4×D414yes—
446C22×C4C23×C412yes—
447C22×C4C22×D414yes—
448C22×C4C22×D414yes—
449C22×C4C22×D411yescharacteristic
450C22×C4C22×D411yescharacteristic
451C22×C4C22×D411yescharacteristic
452C22×C4C22×D411yescharacteristic
453C22×C4C22×D411yescharacteristic
454C22×C4C2×#4112yes—
455C22×C4C2511yescharacteristic
456SD16—48no—
457D8—216no—
458M16—48no—
459C2×C8—22no—
460C2×C8—24no—
461C2×C8—22no—
462C2×C8—24no—
463C2×C8—22no—
464C2×C8—22no—
465C2×C8—22no—
466C2×C8C23×C414yes—
467C2×C8C23×C414yes—
468C2×C8C2×#4114yes—
469C4⋊C4—24no—
470C4⋊C4—24no—
471C4⋊C4—24no—
472C4⋊C4—24no—
473C4⋊C4—24no—
474C4⋊C4—21no—
475C4⋊C4—21no—
476C4⋊C4—22no—
477C4⋊C4—22no—
478C4⋊C4—22no—
479C4⋊C4—24no—
480C4⋊C4—24no—
481C4⋊C4—22no—
482C4⋊C4—44no—
483C4⋊C4—44no—
484C4⋊C4—82no—
485C4⋊C4—84no—
486C4⋊C4—84no—
487C4⋊C4C23×C412yes—
488C4⋊C4C23×C412yes—
489C4⋊C4C23×C414yes—
490C4⋊C4C2×#4114yes—
491C22⋊C4—24no—
492C22⋊C4—24no—
493C22⋊C4—24no—
494C22⋊C4—28no—
495C22⋊C4—28no—
496C22⋊C4—44no—
497C22⋊C4—48no—
498C22⋊C4—48no—
499C22⋊C4—84no—
500C22⋊C4—84no—
501C22⋊C4—82no—
502C42—24no—
503C42—24no—
504C42—24no—
505C42—24no—
506C42—24no—
507C42—44no—
508C42—84no—
509C42—81no—
510C42—81no—
511C42—82no—
512C42—84no—
513C42C4×D414yes—
514C42C4×D414yes—
515C42C23×C412yes—
516C42C23×C412yes—
517C42C2×#4112yes—
518C42C2×#4112yes—
519C23—22no—
520C23—22no—
521C23—24no—
522C23—22no—
523C23—21no—
524C23—21no—
525C23—24no—
526C23—42no—
527C23—41no—
528C23—41no—
529C23—44no—
530C23—44no—
531C23—42no—
532C23—42no—
533C23C2×C4×D412yes—
534C23C23×D411yescharacteristic
535Q8—22no—
536Q8—22no—
537Q8—42no—
538Q8—84no—
539D4—24no—
540D4—22no—
541D4—24no—
542D4—24no—
543D4—48no—
544D4—416no—
545D4—48no—
546D4—44no—
547D4—84no—
548D4C4×#4114yes—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—28no—
557C2×C4—24no—
558C2×C4—24no—
559C2×C4—24no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—24no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—44no—
570C2×C4—48no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—44no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—48no—
579C2×C4—44no—
580C2×C4—42no—
581C2×C4—44no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—44no—
585C2×C4—84no—
586C2×C4—88no—
587C2×C4—88no—
588C2×C4—168no—
589C2×C4—168no—
590C2×C4C2×C4×D412yes—
591C2×C4C2×C4×D412yes—
592C2×C4C2×C4×D412yes—
593C2×C4C2×C4×D412yes—
594C2×C4C2×C4×D412yes—
595C2×C4#57712yes—
596C2×C4#57712yes—
597C2×C4C24×C411yescharacteristic
598C2×C4C22×#4112yes—
599C2×C4C22×#4111yescharacteristic
600C2×C4#58512yes—
601C8—24no—
602C8—24no—
603C8—24no—
604C8—42no—
605C22—22no—
606C22—22no—
607C22—22no—
608C22—24no—
609C22—42no—
610C22—42no—
611C22—42no—
612C22—44no—
613C22—44no—
614C22—42no—
615C22—44no—
616C22—44no—
617C22—44no—
618C22—84no—
619C22—84no—
620C22C22×C4×D411yescharacteristic
621C22D4×#4112yes—
622C22C2×#57711yescharacteristic
623C22C2×#57711yescharacteristic
624C22#358512yes—
625C4—22no—
626C4—22no—
627C4—22no—
628C4—22no—
629C4—42no—
630C4—42no—
631C4—44no—
632C4—42no—
633C4—44no—
634C4—44no—
635C4—44no—
636C4—82no—
637C4—84no—
638C4#295214yes—
639C4C2×C4×#4111yescharacteristic
640C4#342811yescharacteristic
641C4#347512yes—
642C2—42no—
643C2—44no—
644C2—84no—
645C2#3413611yescharacteristic
646C2C2×#295211yescharacteristic
647C2#6016811yescharacteristic
648C2#6155112yes—
649C2#6155412yes—
subgroupG/H
center, socleC23C23×D4
commutatorC2×C4C24×C4
Frattini, cosocleC22×C4C25
prime radicalC2×#3106C2
LCSG ▷ C2×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
2. #34236C2semidirect
3. #34191C2semidirect
4. #23095C2semidirect
5. #23069C2semidirect
6. #23001C2semidirect
7. #22963C2semidirect
8. #22924C2semidirect
9. #22866C2semidirect
10. #21410C2semidirect
11. #21359C2semidirect
12. #21280C2semidirect
13. #21134C2semidirect
14. #21133C2semidirect
15. #20503C2semidirect
16. #20482C2semidirect
17. C4×#417C2semidirect
20. #3106C4semidirect
100. #532C2×C4semidirect
107. C4×#41C2×C4semidirect
128. #482C2×C4semidirect
129. #477C2×C4semidirect
130. #476C2×C4semidirect
131. #460C2×C4semidirect
132. #459C2×C4semidirect
133. #449C2×C4semidirect
135. C4×SD16C2×C4semidirect
136. C4×D8C2×C4semidirect
137. #431C2×C4semidirect
359. #102#41semidirect

Maximal subgroups

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

subgroupindex
C2×#31062
#342362
#341912
#230952
#230692
#230012
#229632
#229242
#228662
#214102
#213592
#212802
#211342
#211332
#205032
#204822
C4×#4172