Finite Group Databasebeta

← #6389036

Subgroups — #6389036

Order 512.

2088 conjugacy classes of subgroups.

The subgroup lattice has 659 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3035C211yesmaximal, characteristic
2#34393C211yesmaximal, characteristic, complemented
3#34193C211yesmaximal, characteristic, complemented
4#23099C212yesmaximal, complemented
5#23094C212yesmaximal, complemented
6#23004C212yesmaximal, complemented
7#22997C212yesmaximal, complemented
8#22577C212yesmaximal, complemented
9#22568C212yesmaximal, complemented
10#22449C212yesmaximal, complemented
11#22441C212yesmaximal, complemented
12#22161C214yesmaximal, complemented
13#21959C211yesmaximal, characteristic, complemented
14#21805C211yesmaximal, characteristic, complemented
15#21511C214yesmaximal, complemented
16#20737C211yesmaximal, characteristic, complemented
17#20545C211yesmaximal, characteristic, complemented
18C22×#135C2212yes—
19C2×#532C2211yescharacteristic
20C2×C4×#41C2211yescharacteristic
21#3035C4116yescomplemented
22C2×#464C2212yes—
23C2×#462C2211yescharacteristic
24C2×#460C2211yescharacteristic
25C2×#463C2212yes—
26C2×#461C2211yescharacteristic
27C2×#459C2211yescharacteristic
28C2×#441C2214yes—
29C2×C4×SD16C2214yes—
30C2×#431C2211yescharacteristic
31C22×C4⋊C8C2211yescharacteristic
32C2×#416C2212yes—
33#2377—22no—
34#2366C2212yes—
35#2365C2212yes—
36#2357—21no—
37#2354—22no—
38#2354C2212yes—
39Q8×C4⋊C4C2212yes—
40#2349—21no—
41#2347—22no—
42#2347C2212yes—
43#2341C2212yes—
44#2341C2212yes—
45#2339C2212yes—
46#2326C2214yes—
47#2324—22no—
48#2319—22no—
49#2319C2212yes—
50#2318—22no—
51#2285—21no—
52#2285C2211yescharacteristic
53#2285C2211yescharacteristic
54#2283—21no—
55#2283C2211yescharacteristic
56#2276C2212yes—
57#1983C2214yes—
58#1972C2214yes—
59#1968C2214yes—
60#1957C2212yes—
61#1956C2212yes—
62#1948C2214yes—
63#1948C2212yes—
64#1948C2212yes—
65#1935C2214yes—
66#1934C2214yes—
67#1927C2214yes—
68#1926C2214yes—
69#1880C2214yes—
70#1880C2214yes—
71#1879C2214yes—
72#1878C2214yes—
73#1865C2214yes—
74#1864C2214yes—
75#1825C2211yescharacteristic
76#1825C2211yescharacteristic
77#1807C2212yes—
78#1807C2212yes—
79#1803C2212yes—
80#1802C2212yes—
81#1788C2211yescharacteristic
82#1788C2211yescharacteristic
83#1788C2212yes—
84#1788C2212yes—
85#1770C2212yes—
86C8×C4⋊C4C2212yes—
87#1768C2214yes—
88#1762C2214yes—
89C4×#103C2212yes—
90C4×#103C2214yes—
91C4×#102C2212yes—
92#1752C2212yes—
93C8×C22⋊C4C2212yes—
94C22×#41C2311yescharacteristic
95C22×#41C2312yes—
96C2×#135—48no—
97C2×#135C2×C4116yescomplemented
98C22×Q16C2312yes—
99C22×SD16C2312yes—
100C22×SD16C2312yes—
101C22×D8C2312yes—
102C23×C8C2312yes—
103#532—24no—
104#532C2×C418yes—
105C2×C4⋊Q8C2311yescharacteristic
106C2×#127C2311yescharacteristic
107C2×#124C2312yes—
108C2×#121—21no—
109C2×#121—21no—
110C4×#41—24no—
111C4×#41C2×C418yescomplemented
112C2×C4×Q8C2314yes—
113C2×C4×Q8C2311yescharacteristic
114C2×C4×Q8C2311yescharacteristic
115C2×C4×Q8C2314yes—
116C2×C4×D4—21no—
117C2×C4×D4—21no—
118C2×C4×D4—22no—
119C2×C4×D4—21no—
120C2×C4×D4—21no—
121C2×C4×D4C2311yescharacteristic
122C2×C4×D4C2311yescharacteristic
123C2×C4×D4C2314yes—
124C2×#117—21no—
125C2×#117—21no—
126C2×#117—22no—
127C2×#117—21no—
128C2×#117C2311yescharacteristic
129C22×C42—21no—
130C22×C42C2311yescharacteristic
131#464C2×C4116yescomplemented
132#463C2×C4116yescomplemented
133#462C2×C418yescomplemented
134#461C2×C418yescomplemented
135#460C2×C418yes—
136#459C2×C418yes—
137#441—48no—
138C4×SD16—48no—
139#431—24no—
140C2×#106C2314yes—
141C2×#106C2314yes—
142C2×C4⋊C8C2×C418yescomplemented
143C2×C4⋊C8C2312yes—
144C2×C4⋊C8C2311yescharacteristic
145C2×C4⋊C8C2311yescharacteristic
146C2×C4⋊C8C2312yes—
147#416C2×C4116yescomplemented
148C2×#103C2312yes—
149C2×#103C2312yes—
150C2×#103C2314yes—
151C2×#103C2314yes—
152C2×#103C2312yes—
153C2×#103C2312yes—
154C2×#102C2312yes—
155C2×#102C2312yes—
156C2×#102C2314yes—
157C2×#102C2314yes—
158C2×#102C2312yes—
159C2×#102C2312yes—
160C2×C22⋊C8C2311yescharacteristic
161C2×C22⋊C8C2311yescharacteristic
162C2×#97C2314yes—
163C2×C4×C8C2312yes—
164C2×C4×C8C2314yes—
165#391C2314yes—
166#390—22no—
167#390—24no—
168#390—24no—
169#390C2314yes—
170#390C2314yes—
171#389—22no—
172#389—24no—
173#389C2314yes—
174#388—24no—
175#388—24no—
176#388—24no—
177#388—24no—
178#388—24no—
179#387—24no—
180#386—24no—
181#386—24no—
182#385—24no—
183#385—24no—
184#384—22no—
185#384C2312yes—
186#384C2312yes—
187#383—22no—
188#383D412yes—
189#382—21no—
190#382—21no—
191#382D411yescharacteristic
192#382D411yescharacteristic
193#381—22no—
194#381—22no—
195#381D412yes—
196#380—21no—
197#380—21no—
198#380—22no—
199#380—22no—
200#380—22no—
201#380—22no—
202#380D412yes—
203#380C2311yescharacteristic
204#380C2311yescharacteristic
205#380C2312yes—
206C4×C4⋊C4—22no—
207C4×C4⋊C4—22no—
208C4×C4⋊C4—21no—
209C4×C4⋊C4—21no—
210C4×C4⋊C4—22no—
211C4×C4⋊C4—22no—
212C4×C4⋊C4—24no—
213C4×C4⋊C4C2312yes—
214C4×C4⋊C4C2314yes—
215C4×C4⋊C4C2312yes—
216C4×C4⋊C4C2312yes—
217C4×C22⋊C4—22no—
218C4×C22⋊C4—22no—
219C4×C22⋊C4C2312yes—
220#376—22no—
221#376C2312yes—
222C2×#95—22no—
223#340C2312yes—
224#340C2312yes—
225#340C2312yes—
226#340C2312yes—
227#340C2314yes—
228#340C2314yes—
229#336C2314yes—
230#336C2314yes—
231C2×#41—22no—
232C2×#41—24no—
233C2×#41—24no—
234C2×#41—42no—
235C2×#41—44no—
236C2×#41—44no—
237C2×#41C22×C414yes—
238C2×#41C22×C418yes—
239C22×Q8C2412yes—
240C22×Q8C2411yescharacteristic
241C22×D4—21no—
242C22×D4—22no—
243C22×D4—21no—
244C22×D4—21no—
245C22×D4C2412yes—
246C22×D4C2411yescharacteristic
247C23×C4—21no—
248C23×C4—21no—
249C23×C4—22no—
250C23×C4—21no—
251C23×C4—24no—
252C23×C4C2×D411yescharacteristic
253C23×C4C2×D411yescharacteristic
254C23×C4C2411yescharacteristic
255#135—432no—
256C2×Q16—28no—
257C2×Q16C22×C418yes—
258C2×SD16—28no—
259C2×SD16—44no—
260C2×SD16C22×C418yes—
261C2×SD16C22×C418yes—
262C2×D8—44no—
263C2×D8C22×C418yes—
264C22×C8—44no—
265C22×C8C22×C418yes—
266C22×C8C2412yes—
267C22×C8C2412yes—
268C22×C8C2412yes—
269C22×C8C2412yes—
270C4⋊Q8—22no—
271C4⋊Q8C22×C414yes—
272#127—22no—
273#127C22×C414yes—
274#124—24no—
275#124C22×C418yes—
276#121—28no—
277#121—28no—
278C4×Q8—22no—
279C4×Q8—22no—
280C4×Q8—28no—
281C4×Q8—44no—
282C4×Q8—44no—
283C4×Q8—44no—
284C4×Q8C22×C414yes—
285C4×Q8C22×C414yes—
286C4×D4—28no—
287C4×D4—28no—
288C4×D4—28no—
289C4×D4—28no—
290C4×D4—22no—
291C4×D4—22no—
292C4×D4—44no—
293C4×D4—44no—
294C4×D4—84no—
295C4×D4C22×C414yes—
296C4×D4C22×C414yes—
297#117—22no—
298#117—28no—
299#117—82no—
300#117—84no—
301#117—82no—
302#117C22×C414yes—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—24no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—24no—
311C2×C4⋊C4—24no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4C2×D412yes—
315C2×C4⋊C4C2×D412yes—
316C2×C4⋊C4C2412yes—
317C2×C4⋊C4C2412yes—
318C2×C4⋊C4C2414yes—
319C2×C4⋊C4C2411yescharacteristic
320C2×C4⋊C4C2411yescharacteristic
321C2×C22⋊C4—21no—
322C2×C22⋊C4—21no—
323C2×C22⋊C4—21no—
324C2×C22⋊C4—22no—
325C2×C22⋊C4—22no—
326C2×C22⋊C4—21no—
327C2×C22⋊C4—22no—
328C2×C22⋊C4—21no—
329C2×C22⋊C4—24no—
330C2×C22⋊C4—24no—
331C2×C22⋊C4—24no—
332C2×C22⋊C4—24no—
333C2×C22⋊C4—24no—
334C2×C22⋊C4C2×D414yes—
335C2×C42—21no—
336C2×C42—21no—
337C2×C42—21no—
338C2×C42—28no—
339C2×C42—22no—
340C2×C42—24no—
341C2×C42—24no—
342C2×C42—22no—
343C2×C42—24no—
344C2×C42C22×C414yes—
345C2×C42C2×D411yescharacteristic
346C2×C42C2×D411yescharacteristic
347C2×C42#4114yes—
348C2×C42C2414yes—
349C2×C42C2411yescharacteristic
350C2×C42C2411yescharacteristic
351#106—44no—
352#106—44no—
353C4⋊C8—24no—
354C4⋊C8C22×C414yes—
355C4⋊C8C22×C414yes—
356C4⋊C8C22×C418yes—
357#103—24no—
358#103—24no—
359#103—44no—
360#103—44no—
361#103C22×C418yes—
362#103C22×C418yes—
363#102—44no—
364#102—42no—
365#102—42no—
366#102—44no—
367#102C22×C418yes—
368#102C22×C418yes—
369C22⋊C8—22no—
370C22⋊C8—22no—
371#97—44no—
372C4×C8—24no—
373C4×C8—44no—
374#95—21no—
375#95—24no—
376#95—22no—
377#95—24no—
378#95—21no—
379#95—22no—
380#95—24no—
381#95—24no—
382#95—22no—
383#95C2×D412yes—
384#95C2×D412yes—
385#95C2×D412yes—
386#95C2×D414yes—
387#95C2×D412yes—
388#95C2×D414yes—
389C24—21no—
390C24—22no—
391C24C22×D411yescharacteristic
392#41—28no—
393#41—44no—
394#41—48no—
395#41—48no—
396#41—48no—
397#41—88no—
398C2×Q8—21no—
399C2×Q8—22no—
400C2×Q8—22no—
401C2×Q8—24no—
402C2×Q8—24no—
403C2×Q8C23×C414yes—
404C2×Q8C23×C412yes—
405C2×Q8H(5,𝔽2)14yes—
406C2×D4—24no—
407C2×D4—28no—
408C2×D4—24no—
409C2×D4—24no—
410C2×D4—21no—
411C2×D4—22no—
412C2×D4—24no—
413C2×D4—22no—
414C2×D4—22no—
415C2×D4—24no—
416C2×D4—44no—
417C2×D4—42no—
418C2×D4—42no—
419C2×D4—82no—
420C2×D4—84no—
421C2×D4C23×C414yes—
422C2×D4C23×C412yes—
423C22×C4—21no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—22no—
431C22×C4—24no—
432C22×C4—24no—
433C22×C4—24no—
434C22×C4—28no—
435C22×C4—22no—
436C22×C4—24no—
437C22×C4—22no—
438C22×C4—42no—
439C22×C4—44no—
440C22×C4—42no—
441C22×C4—42no—
442C22×C4—82no—
443C22×C4—84no—
444C22×C4—84no—
445C22×C4—84no—
446C22×C4—84no—
447C22×C4C4×D414yes—
448C22×C4C4×D414yes—
449C22×C4C23×C412yes—
450C22×C4C22×D414yes—
451C22×C4C22×D414yes—
452C22×C4C22×D411yescharacteristic
453C22×C4C22×D411yescharacteristic
454C22×C4C22×D411yescharacteristic
455C22×C4C22×D411yescharacteristic
456C22×C4C22×D411yescharacteristic
457C22×C4C2×#4112yes—
458C22×C4C2511yescharacteristic
459Q16—216no—
460SD16—216no—
461SD16—48no—
462D8—48no—
463C2×C8—24no—
464C2×C8—24no—
465C2×C8—22no—
466C2×C8—22no—
467C2×C8—22no—
468C2×C8—22no—
469C2×C8—22no—
470C2×C8—22no—
471C2×C8—48no—
472C2×C8C23×C414yes—
473C2×C8C23×C414yes—
474C4⋊C4—24no—
475C4⋊C4—24no—
476C4⋊C4—24no—
477C4⋊C4—24no—
478C4⋊C4—24no—
479C4⋊C4—24no—
480C4⋊C4—21no—
481C4⋊C4—21no—
482C4⋊C4—24no—
483C4⋊C4—22no—
484C4⋊C4—22no—
485C4⋊C4—24no—
486C4⋊C4—22no—
487C4⋊C4—22no—
488C4⋊C4—48no—
489C4⋊C4—44no—
490C4⋊C4—82no—
491C4⋊C4—82no—
492C4⋊C4—84no—
493C4⋊C4—82no—
494C4⋊C4—84no—
495C4⋊C4C23×C412yes—
496C4⋊C4C23×C412yes—
497C22⋊C4—24no—
498C22⋊C4—24no—
499C22⋊C4—24no—
500C22⋊C4—28no—
501C22⋊C4—28no—
502C22⋊C4—44no—
503C22⋊C4—84no—
504C22⋊C4—84no—
505C22⋊C4—84no—
506C22⋊C4—84no—
507C22⋊C4—84no—
508C22⋊C4—81no—
509C22⋊C4—81no—
510C22⋊C4—82no—
511C42—24no—
512C42—24no—
513C42—24no—
514C42—24no—
515C42—24no—
516C42—48no—
517C42—44no—
518C42—82no—
519C42—84no—
520C42—84no—
521C42C4×D414yes—
522C42C4×D414yes—
523C42C23×C412yes—
524C42C23×C412yes—
525C42C2×#4112yes—
526C42C2×#4112yes—
527C23—22no—
528C23—22no—
529C23—21no—
530C23—21no—
531C23—22no—
532C23—24no—
533C23—42no—
534C23—44no—
535C23—44no—
536C23—42no—
537C23—41no—
538C23—42no—
539C23—41no—
540C23—42no—
541C23C2×C4×D412yes—
542C23C23×D411yescharacteristic
543Q8—22no—
544Q8—24no—
545Q8—24no—
546Q8—24no—
547Q8—44no—
548Q8#51814yes—
549D4—22no—
550D4—24no—
551D4—22no—
552D4—24no—
553D4—48no—
554D4—48no—
555D4—42no—
556D4—44no—
557D4—84no—
558D4—88no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—24no—
563C2×C4—22no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—28no—
569C2×C4—24no—
570C2×C4—24no—
571C2×C4—24no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—24no—
575C2×C4—24no—
576C2×C4—24no—
577C2×C4—24no—
578C2×C4—24no—
579C2×C4—44no—
580C2×C4—48no—
581C2×C4—42no—
582C2×C4—44no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—48no—
586C2×C4—42no—
587C2×C4—44no—
588C2×C4—42no—
589C2×C4—44no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—44no—
595C2×C4—84no—
596C2×C4—88no—
597C2×C4—168no—
598C2×C4C2×C4×D412yes—
599C2×C4C2×C4×D412yes—
600C2×C4C2×C4×D412yes—
601C2×C4C2×C4×D412yes—
602C2×C4C2×C4×D412yes—
603C2×C4#57712yes—
604C2×C4#57712yes—
605C2×C4C24×C411yescharacteristic
606C2×C4C22×#4111yescharacteristic
607C2×C4C2×H(5,𝔽2)12yes—
608C2×C4#58512yes—
609C8—24no—
610C8—24no—
611C8—42no—
612C8—42no—
613C22—22no—
614C22—22no—
615C22—22no—
616C22—42no—
617C22—42no—
618C22—42no—
619C22—44no—
620C22—42no—
621C22—44no—
622C22—42no—
623C22—44no—
624C22—84no—
625C22—82no—
626C22—82no—
627C22—84no—
628C22C22×C4×D411yescharacteristic
629C22#348812yes—
630C22C2×#57711yescharacteristic
631C22C2×#57711yescharacteristic
632C22#358412yes—
633C4—22no—
634C4—22no—
635C4—22no—
636C4—22no—
637C4—24no—
638C4—42no—
639C4—42no—
640C4—44no—
641C4—44no—
642C4—44no—
643C4—82no—
644C4—82no—
645C4—84no—
646C4#295214yes—
647C4C2×#51811yescharacteristic
648C4#342811yescharacteristic
649C4#349412yes—
650C2—42no—
651C2—82no—
652C2—84no—
653C2#3420411yescharacteristic
654C2#6016111yescharacteristic
655C2C2×#295211yescharacteristic
656C2#6207112yes—
657C2#6207312yes—
subgroupG/H
center, socleC23C23×D4
commutatorC2×C4C24×C4
Frattini, cosocleC22×C4C25
prime radicalC2×#3035C2
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. #34393C2semidirect
3. #34193C2semidirect
4. #23099C2semidirect
5. #23094C2semidirect
6. #23004C2semidirect
7. #22997C2semidirect
8. #22577C2semidirect
9. #22568C2semidirect
10. #22449C2semidirect
11. #22441C2semidirect
12. #22161C2semidirect
13. #21959C2semidirect
14. #21805C2semidirect
15. #21511C2semidirect
16. #20737C2semidirect
17. #20545C2semidirect
21. #3035C4semidirect
97. C2×#135C2×C4semidirect
111. C4×#41C2×C4semidirect
131. #464C2×C4semidirect
132. #463C2×C4semidirect
133. #462C2×C4semidirect
134. #461C2×C4semidirect
142. C2×C4⋊C8C2×C4semidirect
147. #416C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#30352
#343932
#341932
#230992
#230942
#230042
#229972
#225772
#225682
#224492
#224412
#221612
#219592
#218052
#215112
#207372
#205452