Finite Group Databasebeta

← #6382370

Subgroups — #6382370

Order 512.

2066 conjugacy classes of subgroups.

The subgroup lattice has 650 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3106C211yesmaximal, characteristic
2#34236C211yesmaximal, characteristic, complemented
3#34193C211yesmaximal, characteristic, complemented
4#23093C212yesmaximal, complemented
5#23067C212yesmaximal, complemented
6#23003C212yesmaximal, complemented
7#22965C212yesmaximal, complemented
8#22926C214yesmaximal, complemented
9#22863C214yesmaximal, complemented
10#21410C211yesmaximal, characteristic, complemented
11#21358C211yesmaximal, characteristic, complemented
12#21282C212yesmaximal, complemented
13#21136C212yesmaximal, complemented
14#21129C212yesmaximal, complemented
15#20504C212yesmaximal, 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×#440C2214yes—
28C2×C4×SD16C2212yes—
29C2×C4×D8C2212yes—
30C2×#431C2211yescharacteristic
31C2×#417C2212yes—
32C2×C4×M16C2211yescharacteristic
33#2369C2212yes—
34#2366C2212yes—
35#2361C2214yes—
36#2357—21no—
37#2354—22no—
38#2354C2212yes—
39Q8×C4⋊C4C2212yes—
40#2349—21no—
41#2347—22no—
42#2347C2212yes—
43#2341C2212yes—
44#2339—22no—
45#2337—22no—
46#2319—22no—
47#2319C2212yes—
48#2319C2212yes—
49C4×#121—22no—
50#2285—21no—
51#2285C2211yescharacteristic
52#2283—21no—
53#2283C2211yescharacteristic
54#2276C2212yes—
55#2275C2212yes—
56C2×C4×C4⋊C4C2211yescharacteristic
57#1990C2212yes—
58#1970C2214yes—
59#1957C2212yes—
60#1947C2212yes—
61#1947C2212yes—
62#1942C2214yes—
63#1936C2214yes—
64#1936C2214yes—
65#1925C2214yes—
66#1925C2214yes—
67#1921C2214yes—
68#1919C2214yes—
69#1895C2214yes—
70#1880C2214yes—
71#1879C2214yes—
72#1863C2214yes—
73#1862C2214yes—
74#1852C2214yes—
75#1776C2214yes—
76C4×#107C2212yes—
77C4×#107C2212yes—
78#1770C2212yes—
79C8×C4⋊C4C2212yes—
80#1763C2211yescharacteristic
81#1763C2211yescharacteristic
82#1762C2212yes—
83#1762C2211yescharacteristic
84#1762C2211yescharacteristic
85#1762C2212yes—
86C4×#103C2214yes—
87C4×#103C2212yes—
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×C418yes—
101C2×C4⋊Q8C2311yescharacteristic
102C2×#127C2311yescharacteristic
103C2×#124—21no—
104C2×#121—21no—
105C2×#121C2312yes—
106C4×#41—24no—
107C4×#41C2×C418yescomplemented
108C2×C4×Q8C2314yes—
109C2×C4×Q8C2311yescharacteristic
110C2×C4×Q8C2311yescharacteristic
111C2×C4×D4—21no—
112C2×C4×D4—21no—
113C2×C4×D4—22no—
114C2×C4×D4—24no—
115C2×C4×D4—21no—
116C2×C4×D4C2314yes—
117C2×C4×D4C2311yescharacteristic
118C2×C4×D4C2312yes—
119C2×C4×D4C2311yescharacteristic
120C2×#117—21no—
121C2×#117—22no—
122C2×#117—21no—
123C2×#117C2311yescharacteristic
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#440—216no—
135C4×SD16C2×C4116yescomplemented
136C4×D8C2×C4116yescomplemented
137#431C2×C418yescomplemented
138C2×#107C2314yes—
139C2×#107C2314yes—
140C2×#107C2312yes—
141C2×#106C2312yes—
142C2×C4⋊C8C2311yescharacteristic
143C2×C4⋊C8C2311yescharacteristic
144#417—28no—
145C2×#103C2312yes—
146C2×#103C2312yes—
147C2×#103C2312yes—
148C2×#102C2312yes—
149C2×#102C2312yes—
150C2×#102C2314yes—
151C2×#102C2314yes—
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#390—22no—
165#390—24no—
166#390C2314yes—
167#390C2314yes—
168#389—22no—
169#389—24no—
170#389C2314yes—
171#389C2314yes—
172#388—24no—
173#388—24no—
174#388—24no—
175#388—24no—
176#388—24no—
177#386—24no—
178#386—24no—
179#385—24no—
180#384—22no—
181#384C2312yes—
182#383—22no—
183#383D412yes—
184#382—21no—
185#382D411yescharacteristic
186#382D411yescharacteristic
187#381—22no—
188#381—22no—
189#381D412yes—
190#380—21no—
191#380—22no—
192#380—22no—
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⋊C4—22no—
203C4×C4⋊C4—22no—
204C4×C4⋊C4—24no—
205C4×C4⋊C4C2311yescharacteristic
206C4×C4⋊C4C2312yes—
207C4×C4⋊C4C2314yes—
208C4×C4⋊C4C2314yes—
209C4×C4⋊C4C2312yes—
210C4×C4⋊C4C2311yescharacteristic
211C4×C4⋊C4C2312yes—
212C4×C22⋊C4—24no—
213C4×C22⋊C4—22no—
214C4×C22⋊C4—22no—
215C4×C22⋊C4C2312yes—
216#376—22no—
217#376C2312yes—
218C2×#95—21no—
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×C4C2×D411yescharacteristic
251C23×C4C2×D411yescharacteristic
252C23×C4C2411yescharacteristic
253C2×SD16—28no—
254C2×SD16C22×C418yes—
255C2×D8—44no—
256C2×D8C22×C418yes—
257C2×M16—24no—
258C2×M16—24no—
259C2×M16—42no—
260C22×C8C2412yes—
261C22×C8C2412yes—
262C22×C8C2412yes—
263C22×C8C2412yes—
264C4⋊Q8—22no—
265C4⋊Q8C22×C414yes—
266#127—22no—
267#127C22×C414yes—
268#124—28no—
269#121—28no—
270#121—24no—
271#121C22×C418yes—
272C4×Q8—22no—
273C4×Q8—22no—
274C4×Q8—28no—
275C4×Q8—44no—
276C4×Q8C22×C414yes—
277C4×Q8C22×C414yes—
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—84no—
287C4×D4—88no—
288C4×D4C22×C414yes—
289C4×D4C22×C414yes—
290C4×D4C22×C418yes—
291#117—22no—
292#117—28no—
293#117—84no—
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—24no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—42no—
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—22no—
322C2×C22⋊C4—24no—
323C2×C22⋊C4—24no—
324C2×C22⋊C4—24no—
325C2×C22⋊C4—24no—
326C2×C22⋊C4C2×D414yes—
327C2×C42—21no—
328C2×C42—21no—
329C2×C42—21no—
330C2×C42—28no—
331C2×C42—22no—
332C2×C42—24no—
333C2×C42—24no—
334C2×C42—22no—
335C2×C42—24no—
336C2×C42—21no—
337C2×C42—21no—
338C2×C42—82no—
339C2×C42C22×C414yes—
340C2×C42C2×D411yescharacteristic
341C2×C42C2×D411yescharacteristic
342C2×C42#4114yes—
343C2×C42C2414yes—
344C2×C42C2411yescharacteristic
345C2×C42C2411yescharacteristic
346#107—28no—
347#107—44no—
348#107C22×C418yes—
349#106C22×C418yes—
350C4⋊C8C22×C414yes—
351C4⋊C8C22×C414yes—
352#103—24no—
353#103C22×C418yes—
354#103#4118yescomplemented
355#102—24no—
356#102—28no—
357#102—24no—
358#102—44no—
359#102C22×C418yes—
360#102C22×C418yes—
361#102C22×C418yes—
362C22⋊C8C22×C418yes—
363#97—28no—
364#97—22no—
365#97—22no—
366C4×C8—24no—
367C4×C8—44no—
368C4×C8C22×C418yes—
369#95—21no—
370#95—24no—
371#95—21no—
372#95—22no—
373#95—22no—
374#95—24no—
375#95—22no—
376#95—24no—
377#95—24no—
378#95C2×D412yes—
379#95C2×D412yes—
380#95C2×D414yes—
381#95C2×D412yes—
382#95C2×D412yes—
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—82no—
395C2×Q8C23×C412yes—
396C2×Q8C2×#4114yes—
397C2×D4—24no—
398C2×D4—28no—
399C2×D4—24no—
400C2×D4—24no—
401C2×D4—22no—
402C2×D4—24no—
403C2×D4—21no—
404C2×D4—22no—
405C2×D4—24no—
406C2×D4—22no—
407C2×D4—44no—
408C2×D4—48no—
409C2×D4—42no—
410C2×D4—42no—
411C2×D4—82no—
412C2×D4C23×C412yes—
413C2×D4C23×C414yes—
414C22×C4—21no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
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—44no—
431C22×C4—48no—
432C22×C4—44no—
433C22×C4—42no—
434C22×C4—42no—
435C22×C4—44no—
436C22×C4—82no—
437C22×C4—84no—
438C22×C4—84no—
439C22×C4—84no—
440C22×C4—82no—
441C22×C4C4×D414yes—
442C22×C4C4×D414yes—
443C22×C4C23×C412yes—
444C22×C4C22×D414yes—
445C22×C4C22×D414yes—
446C22×C4C22×D411yescharacteristic
447C22×C4C22×D411yescharacteristic
448C22×C4C22×D411yescharacteristic
449C22×C4C22×D411yescharacteristic
450C22×C4C22×D411yescharacteristic
451C22×C4C2×#4112yes—
452C22×C4C2511yescharacteristic
453SD16—216no—
454D8—48no—
455M16—48no—
456C2×C8—22no—
457C2×C8—24no—
458C2×C8—22no—
459C2×C8—24no—
460C2×C8—22no—
461C2×C8—22no—
462C2×C8—22no—
463C2×C8C23×C414yes—
464C2×C8C23×C414yes—
465C2×C8C2×#4114yes—
466C4⋊C4—24no—
467C4⋊C4—24no—
468C4⋊C4—24no—
469C4⋊C4—24no—
470C4⋊C4—24no—
471C4⋊C4—21no—
472C4⋊C4—21no—
473C4⋊C4—22no—
474C4⋊C4—22no—
475C4⋊C4—22no—
476C4⋊C4—24no—
477C4⋊C4—24no—
478C4⋊C4—22no—
479C4⋊C4—44no—
480C4⋊C4—48no—
481C4⋊C4—44no—
482C4⋊C4—82no—
483C4⋊C4—84no—
484C4⋊C4—84no—
485C4⋊C4C23×C412yes—
486C4⋊C4C23×C412yes—
487C4⋊C4C23×C414yes—
488C4⋊C4C2×#4114yes—
489C22⋊C4—24no—
490C22⋊C4—24no—
491C22⋊C4—24no—
492C22⋊C4—28no—
493C22⋊C4—28no—
494C22⋊C4—44no—
495C22⋊C4—48no—
496C22⋊C4—84no—
497C22⋊C4—84no—
498C22⋊C4—84no—
499C22⋊C4—84no—
500C22⋊C4—82no—
501C42—24no—
502C42—24no—
503C42—24no—
504C42—24no—
505C42—24no—
506C42—48no—
507C42—44no—
508C42—81no—
509C42—81no—
510C42—82no—
511C42—84no—
512C42—84no—
513C42C4×D414yes—
514C42C4×D414yes—
515C42C23×C412yes—
516C42C23×C412yes—
517C42C2×#4112yes—
518C42C2×#4112yes—
519C23—22no—
520C23—22no—
521C23—21no—
522C23—21no—
523C23—22no—
524C23—24no—
525C23—42no—
526C23—44no—
527C23—44no—
528C23—42no—
529C23—41no—
530C23—42no—
531C23—41no—
532C23—42no—
533C23C2×C4×D412yes—
534C23C23×D411yescharacteristic
535Q8—22no—
536Q8—24no—
537Q8—84no—
538Q8C4×#4114yes—
539D4—24no—
540D4—22no—
541D4—22no—
542D4—24no—
543D4—48no—
544D4—416no—
545D4—48no—
546D4—44no—
547D4—42no—
548D4—84no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—24no—
553C2×C4—22no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—28no—
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—24no—
570C2×C4—44no—
571C2×C4—48no—
572C2×C4—42no—
573C2×C4—44no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—48no—
579C2×C4—42no—
580C2×C4—44no—
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×C4C2×C4×D412yes—
590C2×C4C2×C4×D412yes—
591C2×C4C2×C4×D412yes—
592C2×C4C2×C4×D412yes—
593C2×C4C2×C4×D412yes—
594C2×C4#57612yes—
595C2×C4#57812yes—
596C2×C4C24×C411yescharacteristic
597C2×C4C22×#4112yes—
598C2×C4C22×#4111yescharacteristic
599C2×C4#58512yes—
600C8—24no—
601C8—24no—
602C8—24no—
603C8—42no—
604C22—22no—
605C22—22no—
606C22—22no—
607C22—42no—
608C22—42no—
609C22—42no—
610C22—44no—
611C22—44no—
612C22—42no—
613C22—42no—
614C22—44no—
615C22—84no—
616C22—82no—
617C22—82no—
618C22—84no—
619C22C22×C4×D411yescharacteristic
620C22D4×#4112yes—
621C22C2×#57611yescharacteristic
622C22C2×#57811yescharacteristic
623C22#358512yes—
624C4—22no—
625C4—22no—
626C4—22no—
627C4—22no—
628C4—24no—
629C4—42no—
630C4—42no—
631C4—44no—
632C4—44no—
633C4—44no—
634C4—44no—
635C4—84no—
636C4—82no—
637C4#295114yes—
638C4C2×C4×#4111yescharacteristic
639C4#342811yescharacteristic
640C4#347512yes—
641C2—42no—
642C2—82no—
643C2—84no—
644C2#3413611yescharacteristic
645C2C2×#295111yescharacteristic
646C2#6016811yescharacteristic
647C2#6155512yes—
648C2#6155812yes—
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. #34193C2semidirect
4. #23093C2semidirect
5. #23067C2semidirect
6. #23003C2semidirect
7. #22965C2semidirect
8. #22926C2semidirect
9. #22863C2semidirect
10. #21410C2semidirect
11. #21358C2semidirect
12. #21282C2semidirect
13. #21136C2semidirect
14. #21129C2semidirect
15. #20504C2semidirect
16. #20482C2semidirect
17. C4×#417C2semidirect
20. #3106C4semidirect
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
354. #103#41semidirect

Maximal subgroups

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

subgroupindex
C2×#31062
#342362
#341932
#230932
#230672
#230032
#229652
#229262
#228632
#214102
#213582
#212822
#211362
#211292
#205042
#204822
C4×#4172